Physics Reports 315 (1999) 1}284
Looking forward: frontiers in theoretical science. Memorial symposium in honor of Richard Slansky, Los Alamos NM, 20}21 May 1998 editors Fred Cooper, Geo!rey B. West Communicated by A. Schwimmer, R. Petronzio Contents Preface M. Jacob, Multiparticle production C.M. Bender, The complex pendulum N. Dombey, A. Calogeracos, Seventy years of the Klein paradox F. Cooper, Inclusive dilepton production at RHIC: a "eld theory approach based on a nonequilibrium chiral phase transition A.S. Goldhaber, Dual con"nement of grand uni"ed monopoles? G. Chapline, Is theoretical physics the same thing as mathematics? J.H. Schwarz, From superstrings to M theory A.B. Balantekin, Neutrino propagation in matter
Elsevier Science B.V.
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T. Pengpan, P. Ramond, M(ysterious) patterns in SO(9) R.W. Haymaker, Con"nement studies in lattice QCD C.-I. Tan, Di!ractive production at collider energies and factorization E. Timmermans, P. Tommasini, M. Hussein, A. Kerman, Feshbach resonances in atomic Bose}Einstein condensates J.N. Ginocchio, A relativistic symmetry in nuclei J. Patera, R. Twarock, Quasicrystal Lie algebras and their generalizations A.C. Hayes, Nuclear structure issues determining neutrino-nucleus cross sections J.E. Mandula, The gluon propagator
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Preface On May 20 and 21, 1998 a Symposium was held in Los Alamos, New Mexico to honor the memory of Richard Slansky, Director of the Theoretical Division at Los Alamos National Laboratory, who died unexpectedly from a brain aneurism. Dick was only 58 years old and in the prime of life. The shock and tragedy of his death has left a gaping hole in the lives of many who have lost a dear friend and colleague. We miss him terribly and mourn the loss. As a "tting tribute to Dick, the Symposium entitled: ¸ooking Forward: Frontiers in ¹heoretical Science brought together many who had known him at various stages of his career. It consisted of two closely connected parts. The "rst day was concerned more with Dick's contribution as a science administrator, particularly in fostering a climate of "rst-rate research at Los Alamos. The second day was primarily devoted to Dick's many interests in the area of High Energy and Elementary Particle Physics. Dick would probably not have approved of this separation since he did not like to think in us vs. them terms; throughout his years as part of management he always thought of himself as a scientist. Indeed, one of his most admirable qualities was his remarkable knowledge of the cutting edge of a wide range of exciting "elds. He was, however, a pragmatic man and, as such, will surely forgive us. It is this second day's activities that is recorded in this issue of Physics Reports. This is particularly "tting since Dick served as a long time Editor (1985}1998) of the journal and many of those assembled for the occasion had, at some time, been encouraged by Dick to contribute. Although Dick's major interest during his career was model building and Grand Uni"ed Theories, his interests were extremely broad. For example, when the issue of where to build the Superconducting Supercollider was in the forefront of attention, Dick single-handedly constructed a site atlas discussing the terrain of candidate sites. This included detailed calculations of problems of radiation and geology. In addition, he personally viewed several of the sites from a helicopter. In his early career he worked on various models for multiparticle production and was always interested in having experimental collaborators, such as Tom Ludlam, to con"rm his ideas. In many ways he was a frustrated experimentalist; this was re#ected throughout his career by an intense interest in the latest phenomenology and experimental results as well as in an active interest in the search for fractional charge and free quarks. Early on while at Yale, he developed a keen interest in group theory fostered by Feza Gursey and it is through his work in this area that he was best known. He wrote a famous Physics Reports classifying group representations relevant for model building that became a best-selling bible for model builders. In collaboration with Bob Moody and George Patera he extended this in a book which included the classi"cation of all possible Kac-Moody algebras. In addition to these important contributions, he has left his mark as one of the originators, together with Murray Gell-Mann and Pierre Ramond, of the much-cited see-saw mechanism for the generation of neutrino mass, which has become a major area of intense interest in the last few years. As an administrator, Dick was a champion for the preservation of excellence in the Theoretical Division at Los Alamos. Because of his broad interests and easy-going personality he was outstanding at being able to bridge the two cultures of fundamental basic research and applied programmatic research which coexisted in the Division. This was a particularly di$cult task in the post cold-war environment, especially when fundamental physics was under threat and funding was jeopardized. The talks on the "rst day of this symposium testi"ed to the success he had not
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Preface
only in keeping the Division strong but in attracting some of the best young researchers to Los Alamos during his tenure as Division Director. It is in this spirit that this Special Issue of Physics Reports, covering a broad spectrum of particle physics, is dedicated to the memory of our beloved colleague and friend, Dick Slansky, who is sorely missed by many of his colleagues not just at the Laboratory but world-wide. Fred Cooper Geo!rey B. West
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Multiparticle production M. Jacob CERN, DSU Division, CH-1211 Geneva 23, Switzerland
Abstract A collaboration with R. Slansky on the analysis of multiparticle production in the early 1970s is recalled, with a short review of the situation at that time, as seen by him in Physics Reports. Present approach to particle production is illustrated by a particular example namely the humpback inclusive distribution in jet fragmentation. 1999 Elsevier Science B.V. All rights reserved. PACS: 01.60.#q
1. Foreword Less than two months ago I retired from CERN, as one has to at the age of 65. This was the occasion for several events and I am much indebted for all the work which some of my friends invested into this celebration. However, there was a strong element of sadness in this overall joyful happening. Two close friends, who had been my junior collaborators in the past, at a 10-year interval, and who had both long expressed the wish to be associated with such an a!air when its time would come, had both met with an untimely death by the time it occurred. One of them is Claude Itzikson, with whom I worked in the early 1960s, and the other one is Dick Slansky, with whom I worked in the early 1970s. I very much enjoyed my collaborations with them and I was later very happy to see them both rising to research heights where I would have had di$culties following them. They both remained close friends over several decades and their passing away was a great pain for me. We are here to commemorate the memory of Dick Slansky and I would like to recall the spirit in which we worked together, already long ago. 2. A brief historical survey It started at Yale in the fall of 1971, when Dick was an assistant professor there and where I was visiting for a semester. Dick had come to Yale after being a postdoc at Caltech and he stayed there 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 8 - 9
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until he moved to Los Alamos in 1974, after having been promoted to associate professor. Dick was a westerner at heart and he was very happy to move to New Mexico. It was however at an extremely hard time for him since his "rst wife was then dying of cancer. Let us go back to Yale in the fall of 1971. At that time, Bubble chambers were producing a large amount of data and their phenomenological analysis was raising many challenges. With incident beams in the 15}30 GeV range one could study collisions in which several secondary particles were produced. They were shot forward in the laboratory system and one was already calling that jets. The mean charged particle multiplicity was of the order of 5, slowly rising with incident energy. The low multiplicity was partly due to the fact that a leading hadron was often keeping a considerable fraction of its incident energy. This was referred to as the leading particle e!ect. In the case of p}p collisions, a "nal state proton would thus keep on the average half of the incident energy. Yet a typical collision saw the formation of several secondaries, mainly n mesons. Feynman had already proposed his scaling property for inclusive spectra. Yang had advocated a limiting fragmentation approach. Multiperipheral models, on the one hand, and statistical models, on the other hand, were both claiming some elements of the truth, if not the ultimate truth, but many features of the actual data were already showing important departures from either one of these two extreme approaches, for a review see [1]. Talking with experimentalists at CERN, I had collected interest in the matter. Coming to Yale, I had some recent data and some questions with me. I remember vividly Dick coming to my o$ce to ask me what I was interested in. The next day we were collaborating. Within a month we had what we thought were some interesting results and we started to discuss their implications with experimental colleagues and, in particular with T. Ludlam and R. Adair, at Yale, and with R. Panvini, on the telephone, at Brookhaven. The prevailing attitude at that time was to start from an symptotic model, assumed to apply at very high energy, and to "nd out some of its predictions in the existing data. We took a complementary attitude, namely to see what was needed to reproduce the data in their already great variety. We developed a simple model, "tting reasonably well the available data. Most of them had to do with inclusive distributions, namely the longitudinal and transverse momentum distributions for a single secondary particle, summing over everything else. Fig. 1a shows such a distribution (n- in pp collision by R. Panvini et al.) and the way we could reproduce it, and actually predict it from other pieces of data [2]. Also shown (Fig. 1b) are the predictions which we had for Fermilab and CERN-ISR energies, with the rise in the central region which was later veri"ed. The value of the longitudinal momentum is, as usually done, normalized to its maximum accessible value. This de"nes the Feynman variable x, according to which Feynman had advocated that the data should scale, namely show eventually no variation with energy [1,3]. Fig. 1c shows other inclusive distributions and the calculated yields [2]. I cannot resist showing in Fig. 2 some ISR results [3] which provided in the early seventies a textbook illustration of the scaling property. The variable is here the pseudorapidity of the
This re#ects the situation as we were aware of it when we started to collaborate in 1971. The Feynman scaling variable is de"ned as x"2p /(s, where p is the longitudinal momentum and (s is the centre * * of mass energy. The rapidity is de"ned as ln(E#p )/(E!p ) and approximated as ln x(s/M for high energy E<M. * * Here M is the transverse mass M "(M#p. The pseudo rapidity is de"ned as n" ln(p#p )/(p!p ). It is very 2 * * close to the rapidity for energetic secondaries and simply relates to the production angle.
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Fig. 1. (a) Inclusive p-distribution in p}p collisions at 28.5 GeV together with the prediction of the fragmentation model. (b) Approach to scaling as predicted in the model; (c) Inclusive distributions in p}p collisions at 30 GeV as calculated in the model for di!erent types of secondary particles.
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Fig. 1. Continued.
charged particles produced in pp collisions with two incident beams of 15.4 and 26.7 GeV, respectively, and also in an asymmetric beam con"guration with a beam of 15.4 GeV and the other of 26.7 GeV. The data show clearly that the fragments of one proton have a pseudo-rapidity (in practice energy) distribution which is independent of the energy with which the fragmenting proton has been hit, provided that it has been hit hard enough. The transverse distributions were known to be sharply cut o! at large transverse momentum. This left for production only the so-called longitudinal phase space. Typical distributions available at that time, in 1971, are shown in Fig. 3a. It was only by the fall of 1972, with the advent of the ISR data, that inclusive distributions stopped showing an exponential fall-o! with transverse momentum, as production rates could be measured beyond 1 GeV and found surprisingly high. The observed yields were much larger than simple extrapolations would suggest, as shown in Fig. 3b. This was the "rst evidence for hard scattering among quarks and gluons (partons at that time) and
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Fig. 2. Feynman scaling as beautifully illustrated by the early ISR data.
it was quickly recognized that this was the case [4]. This behaviour led eventually to the beautiful and predicted jet results at the CERN p}pbar collider and now at the Fermilab Tevatron. Fig. 4 shows this two jet structure as clearly seen with CERN p}pbar data, in the early eighties [5]. This can be considered as a modern view of the Rutherford experiment whereby hard point like constituents within the colliding particles scatter at wide angle and materialize into jets of hadrons. Quantitative predictions could come when these processes could be analyzed in the framework of perturbative QCD [6]. I could present predictions for inclusive jet yields at the CERN p}pbar collider during a key note address at a symposium in Los Alamos in 1979, and they were veri"ed in 1983. This was my "rst visit to Dick here, which was to be followed by several others. Dick also came to CERN and in particular for a six month stay in 1977}1978. His main interest had then shifted to the applications of Group Theory to model building and to the study of higher symmetries. He wrote a great Physics Reports article along that line in 1981 [6]. This paper was so much in Dick's style. He had `taken the bull by the hornsa, fully mastered sophisticated techniques in Group Theory and written a very clear and comprehensive article in which the model builder could "nd all that was needed. But, back to particle production in 1971, the transverse momentum data were usually "tted to a Gaussian shape and one knew that all there was to be seen in practice was at very low transverse momentum. This was accordingly incorporated in any approach. The rare large transverse
Results from UA (1). For a review of collider physics, see [5].
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Fig. 3. (a) Transverse momentum distributions for di!erent secondary particle as measured at the ISR (data points) and at 24 GeV on a "xed target (dashed curves); (b) Large transverse momentum production at the ISR.
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Fig. 4. Jet production at the CERN p}pbar collider. Result from UA(1).
momentum events correspond to a largely independent hadronization with, in that case, a similar strong transverse momentum cut-o! with respect to the jet axis. Our model related the sharp fall o! of the transverse momentum distribution to the sharp fall o! also observed for the longitudinal momentum distribution of the secondary particles. We could reproduce inclusive yields (Fig. 1). We had the luck to come up with predictions which got veri"ed and it even continued when two-particle correlation data became available [7]. We were very excited even though we realized well that we had very little quantum dynamics in our model. We had mainly kinematics and some enforced speci"c features. We were actually turning the question around asking: are the data telling us something just interesting or something much deeper? How di$cult are they to reproduce? Two months later we were calculating many inclusive distributions, meeting with perhaps a too easy success with them at typically 15}30 GeV. We received the help of C.C. Wu, a graduate student of Dick, who eventually wrote a thesis along these lines. As I moved to Fermilab, Ed Berger, from Argonne, started collaborating with us. We had success with inclusive spectra and also with the Bubble chamber
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two particle correlation data which were just becoming available [7]. We had also some success with the "rst Fermilab results.
3. What were we doing? To put things in a nutshell, we were analyzing particle production in terms of that of clusters, with invariant mass of typically 1}2 GeV, which were excited from the incident particles and fragmenting isotropically into the observed "nal particles. Dick coined the word `Novaa for these clusters. It was a very good name translating the ephemeral #are of pions produced as the colliding particles were excited. But `clustera is the term which eventually remained, probably because `Novaa was too much associated with other features of the model which eventually had to be dropped, like their di!ractive production and isotropic decay up to high masses which the model kept. In any case, the key feature was that light particles such as pions, which are those dominantly produced, had to have, on average, the same velocity, or more speci"cally the same rapidity, as the rather massive clusters which they were originating from. As a result the inclusive spectra were falling rather steeply with momentum, in very good agreement with the data over several orders of magnitude, and one could "nd a rationale and easy "ts to the inclusive distribution of all the other secondaries. This is shown in Fig. 1. The leading particle e!ect was automatically inforced by the quasi-elastic production process which was assumed to dominate. On top of that, since a proton, as a member of a cluster, is produced with the same average rapidity as the pion fragments, it takes a centre of mass momentum typically 3 times larger than that of the pions. The invariant distribution of pions originating from a Nova of mass M is [2] f (x)"R e\V+ ) ,
(1)
where K is the mean momentum of the pions in the Nova rest frame, a quantity "xed by the sharply cut o! transverse momentum distribution, which was reproduced at the same time. R is here a normalization factor which was speci"ed in the model. With a typical mass of 2 GeV, one readily gets an e\V distribution, already in good agreement with the surprisingly fast (at that time) fall o! of the inclusive pion distributions in p}p collisions (Fig. 1). Scaling and the approach to scaling for secondaries produced at large x (typically 0.1(x(0.6), in the so-called fragmentation region, was associated with the dominantly di!ractive production of the lower mass Novas. This has stood the test of time. One could understand quantitatively how scaling was achieved at relatively low energies for pions and only at much higher energies for heavy particles. It had to be approached from below in the central region. In order to keep a compact and simple model, we bravely extended this di!ractive mechanism to higher and higher mass Novas still with globally isotropic decays. Since we advocated a step-bystep fragementation mechanism, isotropy was not speci"c to low masses. Considering di!ractive production we were led to assume dominance of single over double excitation. We advocated that one or two cluster(s) was all there was to be in practice in the 15}30 GeV range, even though production of more many clusters could eventually prevail at much higher energy [2]. Even though the di!ractive production of clusters was not a priori justi"ed at higher mass, it kept an a priori correct asymptotic behaviour for inclusive spectra when simply, and bravely, extended. It gave
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eventually a rapidity plateau approached from below. We thus kept it, up to the limit of phase space. It is actually natural and easy that way to obtain eventually a properly normalized rapidity plateau, and this was certainly good enough to reproduce the 15}30 GeV data, which hardly showed any sign of it. A cluster with isotropic decay distributes its products over typically two units of rapidity and, at such energies, there was little more available anyway for the larger multiplicity con"gurations. In large multiplicity con"gurations, secondaries were all in a central rapidity cluster with longitudinal and transverse momenta of the same order in the center of mass. However, with the advent of the Fermilab and then of the "rst CERN-ISR results, the rapidity interval was suddenly extended from 4 to 8 units in 1972}1973, exposing at long last the central rapidity plateau and allowing correlation measurements at central rapidity, or among particles emitted with relatively little energy in the center of mass [3]. Fig. 5 shows the inclusive rapidity distributions for several types of secondaries as they soon became available at the ISR [3], with the extending rapidity plateau long advocated by Feynman and which we had in our model. One could thus at long last separate a fragmentation region, covering about 2 units of rapidity, from a central region with its extending rapidity plateau slowly rising with energy. What we had been advocating and modeling was that there were only fragmentation regions at 15}30 GeV. Extending that without any change to ISR energy, the rapidity plateau was populated with the numerous fragments of massive Novas. This mechanism, which we had singled out in our model, and which was adequate at 15}30 GeV, turned out to be invalid for the central rapidity region appearing at ISR energies, despite the fact that we could still have correctly predicted the inclusive yield there. We started to be wrong with correlations. Fig. 6a shows the rapidity correlation among two charged secondaries (mainly pions) as measured at Fermilab and at the ISR [3]. Whereas at Fermilab (at that time) one still saw mainly a big clustering in the middle, at the ISR, one could see a developing ridge, showing that secondaries were indeed clustered in rapidity but in the same way over an extending rapidity range. One had to conclude that, now, di!erent lower mass clusters were being produced over the central rapidity range. Fig. 6b shows the central two-body correlation, extending over typically two units of rapidity, as expected from an isotropic cluster decay, but with a value independent of energy, as if the increase in collision energy was not extending the mass range (and the fragment multiplicity) of the clusters but rather resulted in the production of more and more of them, distributed rather uniformly over the central rapidity plateau. The naive extension of our model was totally failing on that whereas the data were far more in line with short range (in rapidity) order, as long advocated in the multiperipheral approach [8]. In that case one expects asymptotically a uniform spread of the secondaries over the full rapidity range. This was not holding for individual particles but it was holding for clusters. Electronic detectors were pouring out data at a much faster rate than the Bubble chambers. The properties of these central clusters were quickly analyzed. Their mean multiplicity is of the order of 3}4. Their mean invariant mass does not exceed 1.5 GeV. In present vernacular one is led to say that these clusters correspond to the hadronization of quark and gluon systems when one reaches the limit of validity of perturbative QCD. This is indeed the energy which it takes to pull a quark apart over a distance which corresponds to the penetration length of color into the vacuum. The typical cut-o! value for the invariant mass of a parton system which is used in jet fragmentation model is of the same order.
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Fig. 5. Inclusive rapidity distributions for di!erent types of secondaries as measured at the ISR, with the extending rapidity plateau, and compared with 24 GeV "xed target results.
Back at CERN, I got much involved with the analysis of all kinds of ISR data and corresponded regularly with Dick who had become very knowledgeable about all features of multi particle dynamics. In 1973}1974, he wrote in particular a review paper summarizing very well all that was then known [9]. At the same time, his beautiful work along that line caught the attention of P. Carruthers as he was forming a high-energy group of theorists at Los Alamos. P. Carruthers was then also much interested in multiparticle production [10]. This is this review paper which I now use to discuss particle production as seen by Dick at that time. He takes the reader through a very large amount of data extracting clearly all the important features. There is in particular one section which is called `how to build a successful modela in which he distinguishes very well what is most signi"cant.
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Fig. 6. (A) Two body rapidity correlations among charged pions at Fermilab and at the ISR, with evidence for the extending ridge. The di!erent plots correspond to (a) 102 GeV, (b) 400 GeV, (c) 11 against 11 GeV at ISR and (d) 31 against 31 GeV; (B) Two body correlations at central rapidity at the ISR. One secondary is taken at zero rapidity. Open dots are for 23 GeV and full dots are for 63 GeV in the centre of mass.
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4. Particle production circa 1973}1974 Dick "rst remarked that there are many models, from multiperipheral ones which tend to spread uniformly secondaries over the available rapidity range to those which emphasize the formation of heavy states, as the Nova model did. However, a successful model with inclusive data does not necessarily implies correct physics. He stressed that one should rather emphasize the complementarity between models and try to extract general features from the data. Instead of seeing a low invariant mass system always as a chain (with secondary equally spread in rapidity) one can see it as a cluster. On the other hand, a large rapidity gap in a multiperipheral chain, as frequently the case for lower multiplicities, leads to a di!ractive process. A `successfula model for particle production had to have a strong damping at large transverse momentum. We have good reasons to understand that for hadrons of "nite size (about 1 fm). Since what actually matters is the transverse mass, the lightest secondaries (pions) are automatically favored. The secondaries are in practice limited to longitudinal phase space (Fig. 2). Yet, they do not appear to take all the energy which they could have. The distribution are also sharply cut o! in x. The model has to have `weak inelasticitya, as associated with the leading particle e!ect. This is however not enough. The answer, as already said, is that the light pions have actually on the average the same rapidity as that of a much more massive object from which they originate. The rapidity of a fast particle (E
(2)
This is the same factor M which appears in (1) and gives the sharp fall o! of the inclusive distribution. This is how the shape is controlled in the fragmentation region. The large x distribution is dominated by the fragments of the lower mass clusters. Since their production (for global multiplicity less than 4) is mainly di!ractive, one could try di!raction dominance. One gets the correct sharp fall o! in x, the correct approach to scaling and eventually scaling. If one extends that to larger masses, the cluster production cross section as a function of mass should fall o! as M\ but, with a cluster multiplicity rising as M, this leads to a rapidity plateau [2,9]. Hence, even though the di!ractive picture is not correct at large multiplicity, many of its result will smoothly merge with those of a more realistic model. This is in particular the case for the mean multiplicity with eventually a logarithmic rise to a good approximation, but a sharper rise at "rst the heavier the type of secondary considered happens to be. This is shown in Fig. 7. The strong clustering picture leads to the sharp fall-of of the rapidity distribution for large multiplicity events, observed in the 15}30 GeV range and which is not a priori necessary from energy conservation alone. One can therefore extend the di!ractive picture to higher mass clusters a long way before meeting with obvious di$culties as they eventually occurred. Correlation data in the central #atter rapidity region exposed by the ISR results have to be properly reproduced and this is however not the case for a simple di!ractive model which obviously gives too strong correlations at central rapidity. As the contributions from large multicity con"gurations (n<4) become important one has also to acknowledge properly that their production is most of the times not di!ractive. The model has to include for them short-range order at high energy but however among clusters of particles of moderate invariant mass. At ISR energy, a moderate clustering e!ect is what remains of the approach "rst taylored to the 15}30 GeV data. One has to conclude that secondaries originate
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Fig. 7. Mean multiplicities for di!erent types of secondaries as a function of the center of mass energy squared.
from rather massive objects but that several of these, spread over the central rapidity range, are produced in the large multiplicity events. Fig. 8 shows the power and limitation of the di!ractive and short-range order approaches with data on topological cross sections, or corresponding to a "xed number of charged particle seen [9]. Whereas limiting fragmentation can hold as a reasonable approximation over a rather large energy range, extending all the way to the lowest energy Fermilab data, it becomes less and less tenable as the collision energy increases further. The dominant yields change as compared with a strictly (energy independant) di!ractive approach, but only slowly with energy. This is an example of what is referred to as `log s physicsa, a name which we coined and which caught. It translates the fact that it is the increase of the rapidity range, and not directly the energy, which sets the scale for changes among the dominant hadronic processes. In any case, the broadening of the multiplicity distribution remained an important feature of high-energy large multiplicity scattering and this implies strong clustering e!ects. All this is very well presented in Dick's review paper [9] where correlations are analyzed in great detail. Their implications for clustering could be studied with correlation data for each charged multiplicity. By 1975 the ISR results on particle production had been analyzed and the situation was reviewed by Foa [3]. Fig. 9 taken from this review shows the location and width of the rapidity distributions for di!erent numbers of prongs at top ISR energy. The two leading particles, associated with the protons on both sides, are removed in order to display the clustering of the dominant pion secondaries. For each event one determines the mean rapidity and the dispersion in rapidity of the clusters. This corresponds then to one point on a two-dimension plot. Piling up events one obtains
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Fig. 8. Topological cross sections as functions of the incident energy.
a pictural view of the clustering. For low multiplicities one still sees clearly typical di!ractive excitation bumps but they fade away in relative importance as the selected multiplicity rises. By the time one reaches the mean value of the multiplicity they have disappeared into a wide bulge centered at rapidity zero. The rise of parameter f , which would be 0 for a Poisson distribution stands as evidence for clustering. This, together with the rise of the mean charged multiplicity, is also shown in Fig. 9. Our Nova model gave naturally a strong rise of f with increasing energy as clustering became more and more pronounced [9]. This was a good feature with the lower energy Fermilab data and it gave us a good publicity. This was however far too strong for the higher energy ISR data, which quickly called for major revisions as to what was taking place at central rapidity. The ISR results provided however evidence for our good forward clusters still present at large rapidity. This is shown in Fig. 9 but can be studied more quantitatively with the rapidity correlations measured between a charged secondary and a more and more leading particle seen within a particular rapidity interval. These correlations show a strong forward clustering e!ect with a marked dip afterward before the correlation rises again because of the presence of numerous central rapidity secondaries in most of the events. For low multiplicities, the invariant mass associated with a clear leading cluster containing a proton is typically 2 GeV [3]. Our Nova model, emphasizing the role of the production of one or two clusters, had more or less the correct physics up to 100 GeV. The low mass clusters were di!ractively produced and
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Fig. 9. (a) Distribution of events versus the size and position of the observed clusters for low and medium multiplicities at the ISR; (b) The variations of the mean charged multiplicity and of the f parameter for negative secondaries with incident energy ("xed target or "xed target equivalent energy).
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controlled the high x distribution. The high-multiplicity con"gurations had to a good approximation only one big cluster at central rapidity, with secondaries piling up at low x. The model however failed at higher energies where one "nds instead more and more medium mass clusters, distributed more and more uniformly in rapidity. 5. Particle production now I insisted on this early 1970s period to revive the #avour of this work with Dick. We had no idea at that time of what QCD had in store but we could hit on this clustering e!ect which one "nds today at the end of the perturbative cascade, when one has eventually to stop describing quarks and gluons and shift in a still phenomenological way to the hadrons which are eventually observed. Despite many e!orts the observed clustering could not be related with prominent resonances alone. It appears to be a more general feature of particle production. This is what can now be seen as the hadronization of the energy which can be stored within an expanding bubble in the QCD vacuum, before it turns into particles. The attitude toward multiparticle production has dramatically changed since the advent of perturbative QCD, with its asymptotic freedom property. The interest naturally shifted to the hadronization of quark and gluon jets which one can follow a long way perturbatively. A quark can emit a gluon according to a speci"c strength and well de"ned kinematics with a double logarithmic behaviour at low energy and at small angle. The gluon can fragment into gluons or into a quark}antiquark pair. From either a quark or a gluon initially produced in a high momentum transfer reaction justifying a perturbative approach, one can follow the evolution of the jet. There can be branching into extra well identi"ed jets and, in any case, into a parton (quarks, antiquarks and gluons) population of the jets. One tries to follow the perturbative approach as far as possible, summing over all the leading logarithmic terms. In the state of the art approaches, subasymptotic corrections can also be included as power series in (a . This is referred to as the modi"ed leading Q logarithmic approach [11]. One ceases to wonder at the scaling property which comes as the lowest-order approximation to emphasize on the contrary scaling violations associated with the running coupling constant a and the jet evolution, which can be predicted. Yet it comes a stage in Q the jet evolution when one can no longer speak about partons and where one has to switch to hadrons, as the perturbative approach becomes no longer valid. One has to introduce some cut o! in the perturbative chain and reinterpret it in terms of hadrons. This is illustrated schematically by Fig. 10a. This is possible owing to the assumption of local parton}hadron duality according to which a partons density is turned into a hadron density for sets of particles with low relative momenta and therefore with relatively small invariant mass. Hadronization is taken as a soft process involving but little changes in transverse momentum and in rapidity. Even though it still cannot be described from "rst principle, hadronization can thus be properly parametrized within Monte Carlo models of increasing sophistication (Lund string model, cluster model, HERWIG, JETSET, ARIADNE). The most involved ones reproduce very well the jet topology, the jet parameters (such as thrust and energy #ow correlations, which are almost insensitive to the cut-o! energy) and several more speci"c features, even if they are more sensitive to it.
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Fig. 10. (a) The QCD cascade for jet fragmentation with eventual hadronization; (b) The humpback rapidity structure as predicted as a function of jet momentum; (c) The mean charged multiplicity as a function of jet momentum, as predicted and measured.
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In retrospect, one may say that the local parton}hadron duality, or the basically soft nature of hadronization processes, is the main legacy of all the work on multiparticle production which went on in the early 1970s. Beside providing a good picture of the many pieces of data on hadron production which were then available or which were later obtained, it gave us con"dence in our present soft way to switch from partons to hadrons and thus compare data with the result of a perturbative jet evolution. The most general feature of hadron production is its globally soft aspect. The "nal state hadrons can be strongly correlated but only at similar rapidity and with small relative transverse momenta, obvious and speci"c long-range kinematical correlations notwithstanding. The typical invariant mass at which one has to switch from parton distribution to a hadronic one is 1}2 GeV. I would like to illustrate present trends with a particularly striking and speci"c feature of jet evolution which is the predicted and now well-documented humpbacked rapidity plateau [11,12]. The perturbative evolution implies color coherence and interjet e!ects which are particularly strong among low-energy partons. In particular, it leads in practice to an angular ordering of the soft partons which strongly limits their overall multiplicity. As a result, it is the particle with intermediate energy (E say) which multiply most e!ectively in the QCD cascade. The e!ect is most clearly seen using the variable m"ln(1/x). As shown in Fig. 10b, the peak of the m distribution, which takes there to a good approximation a Gaussian shape, shifts linearly to higher and higher m values as the available rapidity range extends. The m bump which builds up the multiplicity also gets more and more pronounced. The mean charged multiplicity (taken as proportional to the parton density at the cut-o! value, of the order of 250 MeV) re#ects the humpback e!ect and is in very good agreement with the data, as shown in Fig. 10c. This is a particularly good illustration of the present approach to particle production following, as far as possible, the QCD evolution of the jets [11]. The predictions agree very well with PETRA and LEP results which span a very large energy range. One is indeed tempted to go as far as possible with partons, following the QCD cascade all the way to a rather low cut-o! value, of the order of the QCD scale. However, if, as seen in Fig. 10, one can do so with the inclusive spectra, one fails with two body correlations [11]. The simple parton}parton correlations tend to be too strong whereas two-body correlations agree with the data for hadron Monte Carlo model which now incorporate most features of the QCD cascade to end up with clusters of hadrons.
6. To conclude I very much enjoyed my collaboration with Dick when we were struggling with the particle production data in the early 1970s. He became a great expert in that phenomenology before he moved to the study of higher symmetries, grand uni"cation and possible departures from the Standard Model, with an expertise and passion which he had eventually to share with the increasing responsibilities which he took at Los Alamos. He always associated with his administrative duties an ever juvenal enthusiasm and wonderment for science. He spared no e!ort to cut across standard boundaries between disciplines. His beautiful contributions to symmetries and to possible departures from the Standard Model are reviewed separately at this symposium. I was very happy that he agreed to succeed me as one of the editors of Physics Reports, in 1986. Over 12 years spent as editor he brought to the journal much skill and enthusiasm and many good papers.
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References [1] D. Horn, Phys. Rep. 4 C (1972) 1. [2] M. Jacob, R. Slansky, Phys. Rev. D 5 (1972) 1847. M. Jacob, R. Slansky, C.C. Wu, Phys. Rev. D 6 (1973) 2494. [3] L. Foa, Phys. Rep. 12 C (1975) 1. [4] M. Jacob, P.V. Landsho!, Phys. Rep. 48 (1978) 61. [5] M. Jacob, Physics at collider energy, Carge`se Summer School Proceedings, 1981 and p}pbar collider Physics, Present and Prospects, Carge`se Summer School Proceedings, 1987. [6] R. Slansky, Phys. Rep. 79 (1981) 1. [7] E. Berger, M. Jacob, R. Slansky, Phys. Rev. D 6 (1972) 2580. W.D. Shephard et al. Phys. Rev. Lett. 28 (1972) 703. [8] A.H. Mueller, Phys. Rev. D 2 (1970) 2963 and Phys. Rev. D 4 (1971) 150. [9] R. Slansky, Phys. Rep. 11 C (1974) 99. [10] P. Carruthers, Minh Duong-Van, Phys. Lett. B 41 (1972) 597. [11] V.A. Khoze, W. Ochs, Mod. Phys. A 12 (1997) 2949. [12] Ya.I. Azimov, Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan, Z. Phys. C 27 (1985) 65 and Z. Phys. C 31 (1986) 213.
Physics Reports 315 (1999) 27}40
The complex pendulum Carl M. Bender Department of Physics, Washington University, St. Louis, MO 63130, USA
Abstract This talk proposes a generalization of conventional quantum mechanics. In conventional quantum mechanics one imposes the condition HR"H, where - represents complex conjugation and matrix transpose, to ensure that the Hamiltonian has a real spectrum. By replacing this mathematical condition with the weaker and more physical requirement HS"H, where ?"PT represents combined parity re#ection and time reversal, one obtains new in"nite classes of complex Hamiltonians whose spectra are also real and positive. These PT-symmetric theories may be viewed as analytic continuations of conventional theories from real to complex-phase space. This talk describes the unusual classical and quantum properties of PT-symmetric quantum-mechanical and quantum-"eld-theoretic models. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.!w; 03.65.Ge; 11.30.Er; 02.60.Lj Keywords: Non-Hermitian; PT-symmetry; Complex deformation
1. Introduction Several years ago, Bessis conjectured on the basis of numerical studies that the spectrum of the Hamiltonian H"p#x#ix is real and positive [1]. To date there is no rigorous proof of this conjecture. The reality of the spectrum of H is due to PT-symmetry. Note that H is invariant neither under parity P, whose e!ect is to make spatial re#ections, pP!p and xP!x, nor under time reversal T, which replaces pP!p, xPx, and iP!i. However, PT-symmetry is crucial. For
E-mail address:
[email protected] (C.M. Bender) This problem originated from discussions between Bessis and J. Zinn-Justin, who was studying Lee}Yang singularities using renormalization group methods. An i "eld theory arises if one translates the "eld in a theory by an imaginary term. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 4 - 1
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example, the Hamiltonian p#ix#ix has PT-symmetry, and our numerical studies indicate that its entire spectrum is positive de"nite; the Hamiltonian p#ix#x is not PT-symmetric, and the entire spectrum is complex. The connection between PT-symmetry and the reality of spectra can be understood as follows: We know that if two linear diagonalizable operators commute, then they can be simultaneously diagonalized. This theorem does not hold in general if the operators are not linear, and indeed PT is not a linear operator because it involves complex conjugation. However, if PT does commute with the Hamiltonian H and if we assume that we can simultaneously diagonalize these two operators, then we have E"EH, where E is an eigenvalue of H. Thus, E is real. The connection between PT-symmetry and the reality of spectra is nicely illustrated by some exactly solvable models. Consider the harmonic oscillator H"p#x, whose energy levels are E "2n#1. Adding ix to H does not break PT-symmetry, and the spectrum remains positive L de"nite: E "2n#. Adding !x also does not break PT-symmetry if we de"ne P as re#ection L about x", xP1!x, and again the spectrum remains positive de"nite: E "2n#. However, L adding ix!x does break PT-symmetry, and the spectrum is now complex: E "2n#1#i. L 2. A quantum-mechanical model The Hamiltonian studied by Bessis is just one example of a huge and remarkable class of complex Hamiltonians whose energy levels are real and positive. The purpose of this talk is to describe the properties of such Hamiltonians. We begin by examining the one-parameter class of quantum-mechanical Hamiltonians H"p!(ix), (N real) .
(1)
We "nd that, as a function of N, there are two phases with a transition point at N"2 at which the entirely real spectrum begins to develop complex eigenvalues. A full description of the behavior of the eigenvalues of H is given in [2,3]. We may conjecture that the underlying reason that the spectrum of a PT-symmetric theory is real is that the Hamiltonian is actually self-adjoint with respect to a new de"nition of adjoint: HS"H, where ?"PT. The Hilbert space consists of the set of vectors that can be represented as real linear combinations of the eigenfunctions (x) of H, which are also simultaneous eigenfuncL tions of PT. [Note that (x) are themselves complex functions because they solve a complex L di!erential equation.] The eigenfunctions of H are orthonormal: dx (x) (x)"d . We may K L KL de"ne the norm of a vector U(x) in the Hilbert space as dx [U(x)]; this norm is positive. The path of integration in the de"nition of the norm is a complex contour, as we will explain later. As the parameter N is varied, this path may cease to be continuous because of the presence of cuts in the complex-x plane. When this happens, we "nd that the eigenvalues of H become complex. We have studied the Hamiltonian (1) extensively using both numerical and analytical methods. As shown in Fig. 1, the spectrum of H exhibits three distinct behaviors as a function of N: When N52, the spectrum is in"nite, discrete, and entirely real and positive. (This region includes the case N"4 for which H"p!x; the spectrum of this Hamiltonian is positive and discrete and 1x2O0 in the ground state because H breaks parity symmetry!) At the lower bound N"2 of this region lies the harmonic oscillator. A phase transition occurs at N"2; when 1(N(2, there are
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Fig. 1. Energy levels of the Hamiltonian H"p!(ix), as a function of the parameter N. There are three regions: When N52, the spectrum is real and positive. The lower bound of this region, N"2, corresponds to the harmonic oscillator, whose energy levels are E "2n#1. When 1(N(2, there are a "nite number of positive real eigenvalues and an L in"nite number of complex conjugate pairs of eigenvalues. As N decreases from 2 to 1, the number of real eigenvalues decreases; when N41.42207, the only real eigenvalue is the ground-state energy. As N approaches 1>, the ground-state energy diverges. For N41 there are no real eigenvalues.
only a xnite number of positive real eigenvalues and an in"nite number of complex conjugate pairs of eigenvalues. In this region PT-symmetry is spontaneously broken [3]. As N decreases from 2 to 1, adjacent energy levels merge into complex conjugate pairs beginning at the high end of the spectrum; ultimately, the only remaining real eigenvalue is the ground-state energy, which diverges as NP1> [4]. When N41, there are no real eigenvalues. The SchroK dinger eigenvalue di!erential equation for the Hamiltonian (1) is !t (x)!(ix),t(x)"Et(x) .
(2)
Ordinarily, the boundary conditions that give quantized energy levels E are that t(x)P0 as "x"PR on the real axis; this condition su$ces when 1(N(4. However, for arbitrary real N we must continue the eigenvalue problem for (2) into the complex-x plane. Thus, we replace the real-x axis by a contour in the complex plane along which the di!erential equation holds and we impose the boundary conditions that lead to quantization at the endpoints of this contour. (Eigenvalue problems on complex contours are discussed in [5].) The regions in the cut complex-x plane in which t(x)P0 exponentially as "x"PR are wedges (see Fig. 2); these wedges are bounded by the Stokes lines of the di!erential equation [6]. The center of the wedge, where t(x) vanishes most rapidly, is called an anti-Stokes line. There are many wedges in which t(x)P0 as "x"PR. Thus, there are many eigenvalue problems associated with a given di!erential equation [5]. However, we choose to continue the eigenvalue
It is known that the spectrum of H"p!ix is null [4].
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Fig. 2. Wedges in the complex-x plane containing the contour on which the eigenvalue problem for the di!erential equation (2) for N"4.2 is posed. In these wedges t(x) vanishes exponentially as "x"PR. The wedges are bounded by Stokes lines of the di!erential equation. The center of the wedge, where t(x) vanishes most rapidly, is an anti-Stokes line.
equation (2) away from the harmonic oscillator problem at N"2. The wave function for N"2 vanishes in wedges of angular opening p centered about the negative-real and positive-real x-axis. For arbitrary N the anti-Stokes lines at the centers of the left and right wedges lie at the angles h
N!2 p N!2 p "!p# and h "! . N#2 2 N#2 2
(3)
The opening angle of these wedges is D"2p/(N#2). The di!erential equation (2) may be integrated on any path in the complex-x plane so long as the ends of the path approach complex in"nity inside the left wedge and the right wedge [7]. Note that these wedges contain the real-x axis when 1(N(4. As N increases from 2, the left and right wedges rotate downward into the complex-x plane and become thinner. At N"R the di!erential equation contour runs up and down the negative imaginary axis, and thus there is no eigenvalue problem at all. Indeed, Fig. 1 shows that the eigenvalues all diverge as NPR. As N decreases below 2, the wedges become wider and rotate into the upper-half x plane. At N"1 the angular opening of the wedges is p and the wedges are centered at p and p. Thus, the wedges become contiguous at the positive-imaginary x-axis, and the di!erential equation contour can be pushed o! to in"nity. Consequently, there is no eigenvalue problem when N"1 and, as we would expect, the ground-state energy diverges as NP1> (see Fig. 1). In the case of a Euclidean path integral representation for a quantum "eld theory, the (multiple) integration contour for the path integral follows the same anti-Stokes lines. See Ref. [12].
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To ensure the accuracy of our numerical computations of the eigenvalues in Fig. 1, we have solved the di!erential equation (2) using two independent procedures. The most accurate and direct method is to convert the complex di!erential equation to a system of coupled, real, second-order equations that we solve using the Runge}Kutta method; the convergence is most rapid when we integrate along anti-Stokes lines. We then patch the two solutions together at the origin. We have veri"ed those results by diagonalizing a truncated matrix representation of the Hamiltonian in Eq. (1) in harmonic-oscillator basis functions. 2.1. Semiclassical analysis WKB gives a good approximation to the eigenvalues in Fig. 1 when N52. The novelty of this WKB calculation is that it must be performed in the complex plane. The turning points x! are those roots of E#(ix),"0 that analytically continue o! the real axis as N moves away from N"2 (the harmonic oscillator): x "E ,e p \ ,, \
x "E ,e\ p \ , . >
(4)
These turning points lie in the lower (upper) x plane in Fig. 2 when N'2 (N(2). The leading-order WKB phase-integral quantization condition is (n#1/2)p" V>\dx (E#(ix),. It is crucial that this integral follow a path along which the integral is real. When V N'2 this path lies entirely in the lower-half x plane, and when N"2 the path lies on the real axis. When N(2 the path is in the upper-half x plane; it crosses the cut on the positive-imaginary axis and thus is not a continuous path joining the turning points. Hence, WKB fails when N(2. When N52, we deform the phase-integral contour so that it follows the rays from x to 0 and \ from 0 to x : (n#1/2)p"2 sin(p/N)E ,> ds (1!s,. We then solve for E : L >
E& L
C(3/2#1/N)(p (n#1/2) , ,> (nPR) . sin(p/N)C(1#1/N)
(5)
This result is quite accurate. The fourth exact eigenvalue (obtained using Runge}Kutta) for the case N"3 is 11.3143 while WKB gives 11.3042, and the fourth eigenvalue for the case N"4 is 18.4590 while WKB gives 18.4321. The spectrum of the "x", potential is like that of the !(ix), potential. The WKB quantization condition is like Eq. (5) except that sin(p/N) is absent. However, as NPR, the spectrum of "x", approaches that of the square-well potential [E "(n#1) p/4] while the energies of the !(ix), L potential diverge (see Fig. 1). 2.2. Asymptotic study of the ground-state energy near N"1 When N"1, the di!erential equation (2) can be solved exactly in terms of Airy functions. The anti-Stokes lines lie at 303 and at 1503. We "nd the solution that vanishes exponentially along each of these rays and then rotate back to the real-x axis: t
(x)"C Ai(Gx e! p #Ee! p ) .
(6)
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We must patch these solutions together at x"0 according to the patching condition (d/dx)"t(x)"" "0 to "nd the eigenvalues, but for real E the Wronskian identity for the Airy V function is (d/dx)"Ai(x e\ p #Ee\ p )"" "!1/2p (7) V instead of 0. Hence, there is no real eigenvalue. Next, we perform an asymptotic analysis for N"1#e, !t (x)!(ix)>Ct(x)"Et(x), and take t(x)"y (x)#ey (x)#O(e) as eP0#. We "nd that EPR as eP0 roughly like EJ(!ln e) . The explicit asymptotic formula is (2) 1& e e # E\ [(3 ln(2(E)#p!(1!c)(3]/8 .
(8)
To test the accuracy of this formula we have compared the exact ground-state energy E near N"1 with the asymptotic results in Eq. (8). For N"1.1, E "1.6837 while E "2.0955; for N"1.001, E "3.4947 while E "3.6723; for N"1.00001, E "4.7798 while E "4.8776; for N"1.0000001, E "5.8943 while E "5.9244. 2.3. Behavior near N"2 The most intriguing aspect of Fig. 1 is the transition that occurs at N"2. To describe quantitatively the merging of eigenvalues that begins when N(2, we let N"2!e and study the asymptotic behavior of the determinant of the matrix as eP0#. A conventional Hermitian perturbation of a Hamiltonian causes adjacent energy levels to repel, but in this case the complex perturbation of the harmonic oscillator (ix)\C&x!ex[ln("x"#ip sgn(x)] causes the levels to merge. (A complete description of this asymptotic study is given elsewhere [3].) The onset of eigenvalue merging is a phase transition. This transition occurs even at the classical level. We discuss the classical problem below.
3. Classical version of the theory The classical equation of motion (Newton's law) for the Hamiltonian (1) describes a particle of energy E subject to the complex forces. The trajectory x(t) of the particle obeys $dx[E#(ix),]\ "2 dt. While E and dt are real, x(t) is a path in the complex plane in Fig. 2. Consider the simple case N"2. Here, there are two turning points and there is one classical path that terminates at the classical turning points x in [4]. Other paths are nested ellipses with foci at ! the turning points (see Fig. 3). Cauchy's theorem implies that all these paths have the same period. When N"3, there is again a classical path that joins the left and right turning points and an in"nite class of paths enclosing the turning points (see Fig. 4). As these paths increase in size, they approach a cardioid shape. The indentation in the limiting cardioid occurs because paths may not cross, and thus all periodic paths must avoid the path that runs up the imaginary axis (see Fig. 5). When N is noninteger, we obtain classical paths that move o! onto diwerent sheets of the Riemann surface (see Fig. 6). In general, whenever N52, the trajectory joining x is a smile-shaped arc in the lower complex ! plane. The motion is periodic; thus, we have a complex pendulum whose (real) period ¹ is given by
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Fig. 3. Classical paths in the complex-x plane for the N"2 oscillator. The paths form a set of nested ellipses.
Fig. 4. Classical paths in the complex-x plane for the N"3 oscillator. In addition to the periodic orbits there is one path that runs o! to iR from the turning point on the imaginary axis.
the equation:
¹"2E\, , cos
(N!2)p C(1#1/N)(p . C(1/2#1/N) 2N
(9)
At N"2 there is a global change. Below N"2 a path starting at one turning point, say x , moves > toward but misses the turning point x . This path spirals outward, crossing from sheet to sheet on \ the Riemann surface, and eventually veers o! to in"nity asymptotic to the angle N/(2!N) p. Hence, the period abruptly becomes in"nite. The total angular rotation of the spiral is "nite for all NO2 and as NP2> but it becomes in"nite as NP2\ (see Fig. 7). The path passes many turning points as it spirals clockwise from x . [The nth turning point lies at the angle (3N!2!4n)/2Np \
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Fig. 5. Classical paths in the complex-x plane for the N"3 oscillator. As the paths get larger, they approach a limiting shape that resembles a cardioid. We have plotted the rescaled paths.
Fig. 6. Classical paths for the case N"2.5. These paths do not intersect; the graph shows the projection of the parts of the path that lie on di!erent sheets of the Riemann surface. As the size of the paths increases a limiting cardioid appears on the principal sheet of the Riemann surface. On the remaining sheets of the surface the path exhibits a knot-like topological structure.
(x corresponds to n"0).] As N approaches 2 from below, when the classical trajectory passes \ a new turning point, there is a corresponding additional merging of the quantum energy levels as shown in Fig. 1. This correspondence becomes exact in the limit NP2\ and is a manifestation of Ehrenfest's theorem.
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Fig. 7. Classical paths in the complex-x plane for N"1.8, 1.85 and 1.9. These paths are not periodic. The paths spiral outward to in"nity. As NP2 from below, the number of turns in the spiral increases. The lack of periodic orbits corresponds to a broken PT-symmetry.
4. Applications of complex Hamiltonians There appear to be many applications of complex, PT-invariant Hamiltonians in physics. Hamiltonians having an imaginary external "eld have been introduced recently to study delocalization transitions in condensed matter systems, such as vortex #ux-line depinning in type-II superconductors [7], or even to study population biology [8]. In these cases, initially real eigenvalues bifurcate into the complex plane due to the increasing external "eld, indicating the unbinding of vortices or the growth of populations. We believe that one can also induce dynamic delocalization by tuning a physical parameter (here N) in a self-interacting theory. The PT-symmetric Hamiltonian in Eq. (1) may be generalized to include a mass term mx. The massive case is more elaborate than the massless case; phase transitions appear at N"0 and at N"1 as well as at N"2 (see Fig. 8) [2]. Replacing the condition of Hermiticity by the signi"cantly weaker constraint of PT-symmetry allows one to construct new kinds of quasi-exactly solvable quantum theories [9]. An example of a quasi-exactly solvable Hamiltonian is H"p!x#2iax#(a!2b)x#2i(ab!J)x ,
(10)
where J is an integer and a and b are arbitrary parameters. A plot of the spectrum for various values of a and b is given in Fig. 9.
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Fig. 8. Energy levels of the Hamiltonian H"p#mx!(ix), as a function of the parameter N. There are now phase changes at N"0, N"1, and N"2.
Fig. 9. The spectrum for the QES Hamiltonian (10) plotted as a function of b for the case J"3 and a"0. For b' the QES eigenvalues are real and are the three lowest eigenvalues of the spectrum. When b goes below , two of the QES eigenvalues become complex and the third moves into the midst of the non-QES spectrum. We believe that the non-QES spectrum is entirely real throughout the (a, b) plane.
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Fig. 10. The spectrum for the Hamiltonian p#x(ix)C plotted as a function of e.
There are many classes of Hamiltonians for which one can construct complex deformations. For example, one can deform the potential x) by replacing it by x)(ix)C. For e positive the spectrum of the Hamiltonian is real. However, for e negative PT-symmetry is broken and the spectrum becomes complex. As e approaches !K, the eigenvalues gradually disappear into the complex plane until at e"!K there are no eigenvalues at all. In Figs. 10 and 11 we display the energy levels of the Hamiltonian H"p#x(ix)C (case K"2) as a function of the parameter e. This "gure is similar to Fig. 1, but now there are four regions: When e50, the spectrum is real and positive and it rises monotonically with increasing e. The lower bound e"0 of this PT-symmetric region corresponds to the pure quartic anharmonic oscillator, whose Hamiltonian is given by H"p#x. When !1(e(0, PT-symmetry is spontaneously broken. There are a "nite number of positive real eigenvalues and an in"nite number of complex conjugate pairs of eigenvalues; as a function of e the eigenvalues pinch o! in pairs and move o! into the complex plane. By the time e"!1 only eight real eigenvalues remain; these eigenvalues are continuous at e"1. Just as e approaches !1 the entire spectrum reemerges from the complex plane and becomes real. (Note that at e"!1 the entire spectrum agrees with the entire spectrum in Fig. 1 at e"1.) This reemergence is di$cult to see in Fig. 10 but is much clearer in Fig. 11 in which the vicinity of e"!1 is blown up. Just below e"!1, the eigenvalues once again begin to pinch o! and disappear in pairs into the complex plane. However, this pairing is di!erent from the pairing in the region !1(e(0. Above e"!1 the lower member of a pinching pair is even and the upper member is odd (that is, E and E combine, E and E combine, and so on); below e"!1 this pattern reverses (that is, E combines with E , E combines with E , and so on). As e decreases from !1 to !2, the number of real eigenvalues continues to decrease until the only real eigenvalue is the ground-state energy. Then, as e approaches !2>, the ground-state energy diverges logarithmically. For e4!2 there are no real eigenvalues. In Fig. 12 we display the energy levels of the Hamiltonian H"p#x(ix)C (case
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Fig. 11. A blow-up of Fig. 10 in the vicinity of the transition at e"!1. Just above e"!1 the entire spectrum reemerges from the complex plane, and just below e"!1 it continues to disappear into the complex plane. The spectrum is entirely real at e"!1.
Fig. 12. Energy levels of the Hamiltonian p#x(ix)C as a function of the parameter e. This "gure is similar to Fig. 10, but now there are "ve regions: When e50, the spectrum is real and positive and it rises monotonically with increasing e. The lower bound e"0 of this PT-symmetric region corresponds to the pure sextic anharmonic oscillator, whose Hamiltonian is given by H"p#x. The other four regions are !1(e(0, !2(e(!1, !3(e(!2, and e(!3. The PT-symmetry is spontaneously broken when e is negative, and the number of real eigenvalues decreases as e becomes more negative. However, at the boundaries e"!1 and e"!2 there is a complete positive real spectrum. When e"!1, the eigenspectrum is identical to the eigenspectrum in Fig. 10 at e"1. For e4!3 there are no real eigenvalues.
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K"3) as a function of the parameter e. This "gure is similar to Fig. 10, but now there are "ve regions in the parameter e. Quantum "eld theories analogous to the quantum-mechanical theory in Eq. (1) have astonishing properties. The Lagrangian L"(
)#m !g(i ), (N real) possesses PT invariance, the fundamental symmetry of local self-interacting scalar quantum "eld theory [10]. While the Hamiltonian for this theory is complex, the spectrum appears to be positive de"nite. Also, as L is explicitly not parity invariant, the expectation value of the "eld 1 2 is nonzero even when N"4 [11]. Thus, in principle, one can calculate directly (using the Schwinger}Dyson equations, for example [12]) the (positive real) Higgs mass in a renormalizable theory, such as !g or ig , in which symmetry breaking occurs naturally (without introducing a symmetry-breaking parameter). Replacing conventional g or g theories by !g or ig theories reverses signs in the beta functions. Thus, theories that are not asymptotically free become asymptotically free and theories lacking stable critical points develop such points. We believe that !g in four dimensions is nontrivial. Furthermore, PT-symmetric massless electrodynamics has a nontrivial stable critical value of the "ne-structure constant a [13]. We have examined supersymmetric PT-invariant Lagrangians [14] and "nd that the breaking of parity symmetry does not induce a breaking of the apparently robust global supersymmetry. We have investigated the strong-coupling limit of PT-symmetric quantum "eld theories [15]; the correlated limit in which the bare coupling constants g and !m both tend to in"nity with the renormalized mass M held "xed and "nite is dominated by solitons. (In parity-symmetric theories the corresponding limit, called the Ising limit, is dominated by instantons.)
Acknowledgements I thank my coauthors S. Boettcher, H. Jones, P. Meisinger, and K. Milton for their contributions to this research and D. Bessis, A. Wightman, and Y. Zarmi for useful conversations. This work was supported by the US Department of Energy.
References [1] [2] [3] [4] [5] [6]
D. Bessis, private discussion. C.M. Bender, S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. C.M. Bender, S. Boettcher, P.N. Meisinger, J. Math. Phys., to appear. I. Herbst, Commun. Math. Phys. 64 (1979) 279. C.M. Bender, A. Turbiner, Phys. Lett. A 173 (1993) 442. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. [7] N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77 (1996) 570, and Phys. Rev. B 56 (1997) 8651. [8] D.R. Nelson, N.M. Shnerb, condmat/9708071.
Ultimately we may de"ne ? as PCT, the fundamental symmetry of the world. There is no analog of the C operator in quantum mechanical systems having one degree of freedom and in scalar "eld theories.
40 [9] [10] [11] [12] [13] [14] [15]
C.M. Bender / Physics Reports 315 (1999) 27}40 C.M. Bender, R.F. Streater, C.M. Bender, C.M. Bender, C.M. Bender, C.M. Bender, C.M. Bender,
S. Boettcher, J. Phys. A: Math. Gen. 31 (1998) L273. A.S. Wightman, PCT, Spin & Statistics, and all that, Benjamin, New York, 1964. K.A. Milton, Phys. Rev. D 55 (1997) R3255. K.A. Milton, submitted for publication. K.A. Milton, J. Phys. A: Math. Gen. 32 (1999) L87. K.A. Milton, Phys. Rev. D 57 (1998) 3595. S. Boettcher, H.F. Jones, P.N. Meisinger, Phys. Rev. D, submitted for publication.
Physics Reports 315 (1999) 41}58
Seventy years of the Klein paradox N. Dombey * , A. Calogeracos Centre for Theoretical Physics, University of Sussex, Falmer, Brighton BN1 9QJ, UK Low Temperature Laboratory, Helsinki University of Technology, PO Box 2200, FIN-02015 HUT, Finland
Abstract The Klein paradox is examined. Its explanation in terms of electron}positron production is reassessed. It is shown that a potential well or barrier in the Dirac equation can produce positron or electron emission spontaneously if the potential is strong enough. The vacuum charge and lifetime of the well/barrier are calculated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling "rst calculated by Klein is time-independent. Furthermore, tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when it is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and not necessarily connected to particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of su$ciently large Z will bind positrons. Correspondingly, it is expected that as Z increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Pm; 11.10.Kk; 14.60.Cd Keywords: Klein paradox; Quantum tunnelling; Positron production
1. Introduction to the paradox(es) It is an honour to be back at Los Alamos to commemorate Dick Slansky. He had a special interest in quantum mechanics and so I think it appropriate to discuss here some work I have been doing with Calogeracos on one of the oldest problems associated with the Dirac equation: the
* Corresponding author. Talk given by N. Dombey. Permanent address: NCA Research Associates, PO Box 61147, Maroussi 151 22, Greece. E-mail address:
[email protected] (N. Dombey) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 3 - X
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Fig. 1. An electron of energy E scattering o! a potential step of height <. The electrons are shown with solid arrowheads; the hole state has a hollow arrowhead. The particle continuum (slant background) and the hole continuum (shaded background) overlap when m(E(
result that electrons should be able to tunnel through high potential steps. In 1928 Klein [1] submitted a paper for publication in which he calculated the re#ection and transmission of electrons of energy E, mass m and momentum k incident on the potential step <(x)"<, x'0,
<(x)"0, x(0
(1)
within the context of the new relativistic equation which had just been published by Dirac [2]. Klein found (see Section 2 below) that the re#ection and transmission coe$cients R , ¹ if <'2m 1 1 (Fig. 1) were given by R "((1!i)/(1#i)), ¹ "4i/(1#i) , 1 1 where i is the kinematic factor i"(p/k)(E#m)/(E#m!<)
(2)
(3)
and p is the momentum of the transmitted particle for x'0. It is easily seen from Eq. (3) that when E(
(5)
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With this choice of p, both R , ¹ are positive and satisfy R #¹ "1. So is the result of Eq. (2) 1 1 1 1 paradoxical? The general consensus is that it is for as the potential step
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2. The Dirac equation in one dimension In one dimension it is unnecessary to use four-component Dirac spinors. It is much easier to use two-component Pauli spinors instead. We adopt the convention c "p , c "ip . The above X V choice agrees with c c #c c "2g . The free Dirac Hamiltonian in one dimension is G H H G GH H "!p p#p m (6) W X and so the Dirac equation takes the form (p (R/Rx)!Ep #m)t"0 . V X
(7)
In what follows k stands for the wavevector, k for its magnitude and e""E""#(k#m. We try a plane wave of the form
A B
e IV\ #R
(8)
and substitute in (7). The equation is satis"ed by A"ik, B"E!m where E"$e. Note that in one dimension the negative energy solution is obtainable from the positive energy one simply by replacing e by !e unlike the case in three dimensions. The positive energy solutions have the form
ik N (e) e IV\ CR > e!m
(9)
and the negative energy solutions corresponding to hole states
ik e IV> CR , N (e) \ !e!m
(10)
where N (e) are appropriate normalisation factors. If we take the particle to be in a box of length ! 2¸ with periodic boundary conditions at x"!¸ and x"¸ , we normalise the two-dimensional spinor in (8) by requiring dx tRt"1. So we obtain 1 , N (e)" > (2¸(2e(e!m)
1 N (e)" . \ (2¸(2e(e#m)
(11)
Alternatively, we can go to the continuum limit ¸PR and use energy normalisation: then we require the wavefunctions to satisfy
dx tR(E, x)t(E , x)"d(E!E )
(12)
so that 1 , N (e)" > (2p(2e(e!m)
1 N (e)" . \ (2p(2e(e#m)
(13)
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2.1. Scattering ow the Klein Step The Hamiltonian is now H "!p p#<(x)#p m , (14) W X where < is the zeroth component of A . The electron charge is taken to be !1. The Dirac equation I reads (p (R/Rx)!(E!<(x))p #m)t"0 . V X Consider a wave incident from the left. The corresponding wavefunction is
ik
E!m
e IV#B
!ik
E!m
e\ IV
(15)
(16)
for x(0, and
F
!ip
e\ NV
(17)
for x'0 since that state is a hole state as we can see from Fig. 1: when
(18)
in terms of the quantity i de"ned by Eq. (3). This gives the expression for R of Eq. (2) above. 1 3. Pair production by a Klein step Consider the Klein step of Eq. (1) for <'2m. We will show that the expectation value of the current in the vacuum state in the presence of the step is non-zero. The derivation hinges on a careful de"nition of the vacuum state. The result implies that the Klein step produces electron}positron pairs out of the vacuum at a constant rate. This means (although it is not mentioned in the textbooks) that a source of energy must be available to maintain the potential. We use the derivation of CD2 [3]. 3.1. The normal modes in the presence of the Klein step An energy-normalised positive energy or particle solution to the Dirac equation can be written according to Eq. (13):
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e#m 2k
(19)
(20)
i e IV . k E#m
A negative energy or hole solution is
e!m 2k
i e IV . k E#m
Scattering is usually described by a solution describing a wave incident (say from the left) plus a re#ected wave (from the right) plus a transmitted wave (to the right). It is convenient here to use waves of di!erent form either describing a wave (subscript L) incident from the left with no re#ected wave or describing a wave (subscript R) incident from the right with no re#ected wave. Particle and hole wavefunctions will be denoted by u and v, respectively. It is clear that the non-trivial result we are seeking arises from the overlap of the hole continuum E(
(2i (2p u (E, x)" * i#1
E#m k
i e IVh(!x) k E#m
i!1
#
1!i E#m (2p u (E, x)" 0 2k 1#i
#
E#m 2k
(2i # i#1
i e N Vh(x) "p" E#m!<
i e\ N Vh(x) , !"p" E#m!<
(21)
i e IVh(!x) k E#m
i !k e\ IVh(!x) E#m
i e N Vh(x) . "p" E#m!<
(22)
We write "p" rather than p in these equations since the group velocity is negative for x'0 (cf. Eq. (17)). The factors (2p come from the energy normalisation factors in Eq. (13). We need to evaluate the currents corresponding to the solutions of Eqs. (21) and (22). According to our conventions a "c c "!p so V V W j ,!uR (E, x)p u (E, x)"(2i/p)/(i#1) , * * W *
(23)
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j ,!uR (E, x)p u (E, x)"!(2i/p)/(i#1) . 0 0 W 0
47
(24)
3.2. The dexnition of the vacuum and the vacuum expectation value of the current Now expand the wavefunction t in terms of creation and annihilation operators which refer to our left- and right-travelling solutions:
t(x, t)" dE +a (E)u (E, x)e\ #R#a (E)u (E, x)e\ #R#bR (E)v (E, x)e #R#bR (E)v (E, x)e #R, * * 0 0 * * 0 0 (25)
with tR given by the Hermitian conjugate expansion. The creation and annihilation operators in Eq. (25) satisfy the usual anticommutation relations +a (E), aR (E ),"id(E!E ), etc. Again * * the (2p factors in Eqs. (21) and (22) ensure that these anticommutation relations are consistent. We must now determine the appropriate vacuum state in the presence of the step. States described by wavefunctions u (E, x) and v (E, x) correspond to (positive energy) electrons and * * positrons, respectively, coming from the left. Hence with respect to an observer to the left (of the step) such states should be absent from the vacuum state, so a (E)"02"0, b (E)"02"0 . (26) * * Wavefunctions u (E, x) for E'm#< describe for an observer to the right, electrons incident from 0 the right. These are not present in the vacuum state hence a (E)"02"0, E'm#< . (27) 0 Wavefunctions v (E, x) describe, again with respect to an observer to the right, positrons incident 0 from the right; again b (E)"02"0 . (28) 0 The wavefunctions that play the crucial role in the Klein problem belong to the set u (E, x) for 0 m(E(
1 10" j"02"! dE dE +10"aR (E)a (E )"02uR (E, x)p u (E , x) * * * W * 2 #10"a (E)aR (E )"02uR (E , x)p u (E, x) * * * W *
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!10"aR (E)a (E )"02uR (E, x)p u (E , x) 0 0 0 W 0 #10"a (E)aR (E )"02uR (E , x)p u (E, x), . (31) 0 0 0 W 0 The "rst term in Eq. (31) vanishes due to (26). The second term becomes uR (E , x)p u (E, x)d(E!E ) * W * if we use the anticommutation relations and Eq. (26). The third term yields !uR (E, x)p u (E, x)d(E!E ) using Eq. (29) and the fourth term vanishes using the anticommuta0 W 0 tion relations (i.e. the exclusion principle; the state "02 already contains an electron in the state u hence we get zero when we operate on it with aR ). One energy integration is performed 0 0 immediately using the d function. The "nal result is
1 4i(E) 1 dE 10" j"02" dE (!j #j )"! * 0 2p (1#i(E)) 2 or
1 10" j"02"! dE ¹ (E) , 1 2p
(32)
where the energy integration is over the Klein range m(E(
4. Scattering by a square barrier We now turn our attention to a square barrier in place of the Klein step. Consider the square barrier <(x)"<, "x"(a; <(x)"0, "x"'a. For very wide barrier potentials with ma<1 we may expect to "nd similar results to those obtained by Klein for a potential step. We do, but the results are perhaps surprising. It is easy to show that the re#ection and transmission coe$cients are given for a barrier by [9] (1!i) sin(2pa) , R" 4i#(1!i) sin(2pa)
(33)
4i ¹" . 4i#(1!i) sin(2pa)
(34)
Note that Klein tunnelling is enhanced for a barrier if 2pa"Np
(35)
corresponding to E "
8i ¹ " . 8i#(1!i)
(36)
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It may seem unphysical that R and ¹ are not the same as the step re#ection and transmission coe$cients R and ¹ but it is not: it is well known in electromagnetic wave theory [11] that 1 1 re#ection o! a transparent barrier of large but "nite width (with two sides) is di!erent from re#ection o! a transparent step (with one side). R and ¹ thus involve Klein tunnelling but they arise from a more physical problem than the Klein step. The zero of potential is properly de"ned for a barrier whereas it is arbitrary for a step and the energy spectrum of a barrier or well is easily calculable. Spontaneous fermion emission from a barrier or well is described by supercriticality: the condition when the ground state energy of the system overlaps with the continuum (E"m for a barrier; E"!m for a well) and so any connection between particle emission and the timeindependent scattering coe$cients R and ¹ can be investigated.
5. Fermionic emission from a square well/barrier We discussed the "eld theoretic treatment of this topic in CDI [10]. We quickly review the argument. Spontaneous fermionic emission is a non-static process and in the case of a seemingly static potential, it is necessary to ask how the potential was switched on from zero. We follow CDI in turning on the potential adiabatically. The bound state spectrum for the well <(x)"!<, "x"(a; <(x)"0, "x"'a is easily obtained [5,10]: there are even and odd solutions given by the equations
tan pa"
(m!E)(E#<#m) , (m#E)(E#
tan pa"!
(m#E)(E#<#m) , (m!E)(E#
(37)
(38)
where now the well momentum is given by p"(E#<)!m. We have changed the sign of < so that it is now attractive to electrons rather than positrons in order to conform with other authors who have studied supercritical positron emission rather than electron emission. 5.1. Narrow well The simplest case to discuss is a very narrow deep well <(x)"!jd(x) which is the limit of a square well with j"2
E"!m cos j (o) .
(39)
When the potential is initially turned on and j is small there is one bound state just below E"m. As j increases, E decreases and at j"p/2, E reaches zero. For E'p/2, E becomes negative and if the well were originally vacant, we now have the absence of a negative energy state which we must interpret as the presence of a (bound) positron according to Dirac's hole theory.
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5.2. Digression on vacuum charge How is it possible to conserve charge and produce positrons out of the vacuum? The key to this question has been a fruitful ground for theorists in recent years. It is important to realise that the background "eld or potential that is being studied has its own in#uence on the charge properties of the system which must be taken into account: this introduces the concept of vacuum charge [12,13]. So at this point a single-particle interpretation of a potential in the Dirac equation must break down and "eld theory becomes necessary (as we have already seen in the discussion of pair production from the Klein step). But nevertheless it turns out that once the concept of vacuum charge is introduced, "rst quantisation is all that is necessary to determine its value. We brie#y review the treatment given in CDI. Let N , N be the number of positive and negative bound states, respectively. We expand t in > \ terms of the continuous spectrum wavefunctions and of the bound states: t(x, t)" +a (k, 0)u (k, x)e\ #R#a (k, 0)u (k, x)e\ #R > > I #cR(k, 0)u (k, x)e #R#cR(k, 0)u (k, x)e #R, \ \ > \ , , (40) # b (0)u (x)e\ #HR# dR(0)u (x)e\ #HR . H H H H H H Operators aR, a create and annihilate travelling electrons; cR, c are the corresponding ones for positrons. Operators b (bR) annihilate (create) bound electrons whereas d (dR) annihilate (create) H H H H bound positrons. The use of the (-) in Eq. (40) is dictated by the sign of the exponential and conforms to current literature. The Hermitian conjugate expansion is
tR(x, t)" +aR(k, 0)uR (k, x)e #R#aR(k, 0)uR (k, x)e #R > > I #c (k, 0)uR (k, x)e\ #R#c (k, 0)uR (k, x)e\ #R, \ \ ,> ,\ # bR(0)uR(x)e #HR# d (0)u (x)e #HR , (41) H H H H H H where we took into account the reality of u . The standard anticommutation relations are H obeyed
+a (k, t), aR (k , t,"d d +b (t), bR(t),"d , IIY G H GH +c (k, t), cR (k , t,"d d +d (t), dR(t),"d . IIY G H GH
(42)
We work in the Heisenberg picture throughout: The time dependence is carried by operators whereas state vectors are time-independent. However, basis ket vectors (and in particular the vacuum) are time-dependent. The vacuum "02 is de"ned by a (k)"02"c (k)"02"b "02"d "02"0 . G G
(43)
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The total charge is de"ned by (according to our conventions the electron charge is !1)
1 Q(t)" dx o(x, t)"! dx [tR(x, t), t(x, t)] . 2
(44)
Substituting (40), (41) and using (42) we get Q"Q #Q ,
(45)
where the normal-ordered or particle charge Q is given by Q " +!aR(k, t)a (k, t)#cR(k, t)c (k, t)!aR(k, t)a (k, t) I ,> ,\ # cR(k, t)c (k, t)! bRb # dRd H H H H H H and the vacuum charge Q by
(46)
1 Q " + N (E'0)! N (E(0), , (47) 2 I I where N (E'0) is the number of positive energy states and N (E(0) is the number of negative energy states. Given de"nition (43) of the vacuum we immediately get 10"Q"02"Q (48) so the vacuum charge turns out to be the spectral asymmetry of the Hamiltonian. It is important to note that we would not have obtained the connection between Q and the spectral asymmetry had we not identi"ed bound states with E(0 as positrons (Zeldovich and Popov [15] use a di!erent de"nition) or not used the commutator in Eq. (44). Note also that the total charge Q is always conserved. There are pitfalls when Eq. (47) for the vacuum charge Q is applied to actual systems and it is easy to obtain incorrect results. One way of doing things properly is to enclose the system in a box, impose periodic boundary conditions and take the ¸PR limit right at the end of the calculation (see [14] for details). Now back to the delta function potential. For j just larger than p/2, Q "#1 because of the presence of the positron and so the vacuum charge Q must now equal !1 to conserve charge. As the potential is increased further, j will reach p. Here E"!m which is the condition for supercriticality: the bound positron reaches the continuum and becomes free. This is the well-known scenario of spontaneous positron production "rst discussed [15,16] over 25 years ago. Note that at supercriticality j"p, the even bound state disappears and the "rst odd state appears. We can continue to increase j and count positrons: the total number of positrons produced for a given j is the number of times E has crossed E"0; that is Q "Int[(j/p)#(1/2)] ,
(49)
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where Int[x] denotes the integer part of x. The more interesting quantity for us is the number of supercritical positrons Q : the number of states which have crossed E"!m. This is given by 1 Q "Int[j/p] . (50) 1 Note that for any j there is at most one bound positron state. 5.3. Wide well We consider the general case of <'2m and then look in particular at the case ma<1 most closely corresponding to the Klein step. We follow the discussion given in CD2 [3]. We must "nd "rst the condition for supercriticality of the potential <, and then the number of bound and supercritical positrons produced for a given <. From Eq. (37) we see that the ground state becomes supercritical when pa"p/2 and therefore < "m#(m#p/4a. From Eq. (38) the "rst odd state becomes supercritical when pa"p and < "m#(m#p/a. Clearly, the supercritical potential corresponding to the Nth positron is (51) < "m#(m#Np/4a . , It follows from Eq. (51) that <"2m is an accumulation point of supercritical states as maPR. Furthermore, it is a threshold: a potential < is subcritical if <(2m. It is not di$cult to show for a given <'2m that the number of supercritical positrons is given by (52) Q "Int[(2a/p)(<!2m<] . 1 The corresponding value of the total positron charge Q can be shown using Eqs. (37) and (38) to satisfy Q !14Int[(2a/p)(<!m]4Q , so for large a we have the estimates
(53)
Q &(2a/p)(<!2m< . (54) Q &(2a/p)(<!m, 1 Now we can build up an overall picture of the wide square well ma<1. When < is turned on from zero in the vacuum state an enormous number of bound states is produced. As < crosses m a very large number Q of these states cross E"0 and become bound positrons. As < crosses 2m a large number Q of bound states become supercritical together. This therefore gives rise to a positively 1 charged current #owing from the well. But in this case, unlike that of the Klein step, the charge in the well is "nite and therefore the particle emission process has a "nite lifetime. Nevertheless, for ma large enough the transient positron current for a wide barrier is approximately constant in time for a considerable time as we shall see in the next section.
6. Emission dynamics We now restrict ourselves to the case <"2m#D with D;m. This is not necessary but it avoids having to calculate the dynamics of positron emission while the potential is still increasing beyond
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53
the critical value. We can assume all the positrons are produced almost instantaneously as the potential passes through <"2m. It also means that the kinematics are non-relativistic. Hence for a su$ciently wide well, Q &(2a/p)(2mD. The well momentum of the Nth supercritical positron 1 is still given by Eq. (35) p a"Np/2 which corresponds to an emitted positron energy , " E ""2m#D!(p #m'm. Note that the emitted energies have discrete values although for , , a large, they are closely spaced. The lifetime q of the supercritical well is given by the time for the slowest positron to get out of the well. The slowest positron is the deepest lying state with N"1 and momentum p "p/2a. Hence q+ma/p "2ma/p. So the lifetime is "nite but scales as a. But a large number of positrons will have escaped well before q. There are Q supercritical positrons initially and their average 1 (median) momentum p corresponds to N"Q /2; hence p "(mD/2 which is independent of a. 1 Thus a transient current of positrons is produced which is e!ectively constant in time for a long time of order q"ma/p "a(2m/D. We thus see that the square well (or barrier) for a su$ciently large behaves just like the Klein step: it emits a seemingly constant current with a seemingly continuous energy spectrum. But initially the current must build up from zero and eventually must return to zero. Hence, the well or barrier is a time-dependent physical entity with a "nite but long lifetime for emission of supercritical positrons or electrons with a discrete energy spectrum. In contrast a Klein step emits particles at a constant rate and with a continuous energy spectrum. In the appendix we show that if we assume that the transient current emitted from a well or barrier is constant in time (which it is not), then it is possible to obtain a relationship between this current and the transmission coe$cient just as is the case for the Klein step. Note again that the transmission resonances of the time-independent scattering problem coincide with the energies of particles emitted by the well or barrier. It is therefore tempting to use the Pauli principle to explain the connection. Following Hansen and Ravndal, we could say that R must be zero at the resonance energy because the electron state is already "lled by the emitted electron with that energy. But it is easy to show that the re#ection coe$cient is zero for bosons as well as fermions of that energy, and no Pauli principle can work in that case. Furthermore, emission ceases after time q whereas R"0 for times t'q. It follows that we must conclude that Klein tunnelling is a physical phenomenon in its own right, independent of any emission process. In our next section we consider further ways to investigate this tunnelling theoretically and experimentally.
7. Klein tunnelling and the Coulomb barrier It is clear from Eq. (33) that while the re#ection coe$cient R cannot be 0, neither is the transmission coe$cient ¹ exponentially small for energies E(<, even though the scattering is classically forbidden. The simplest way to understand this is to consider the negative energy states under the potential barrier as corresponding to physical particles which can carry energy in exactly the same way that positrons are described by negative energy states which can carry energy. It follows from Eq. (2) that R and ¹ correspond to re#ection and transmission coe$cients in 1 1 transparent media with di!ering refractive indices: thus i is nothing more than an e!ective fermionic refractive index corresponding to the di!ering velocities of propagation by particles in
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the presence and absence of the potential. On this basis, tuning the momentum p to obtain a transmission resonance for scattering o! a square barrier is nothing more than "nding the frequency for which a given slab of refractive material is tranparent. This is not a new idea. In Jensen's words `A potential hill of su$cient height acts as a Fabry}Perot etalon for electrons, being completely transparent for some wavelengths, partly or completely re#ecting for othersa [17]. We can now look in more detail at Klein tunnelling: both in terms of our model square well/barrier problem and at the analogous Coulomb problem. The interesting region is where the potential is strong but subcritical so that emission dynamics play no role and sensible time independent scattering parameters can be de"ned. For electrons scattering o! the square barrier <(x)"< we would thus require <(< "m#(m#p/4a together with <'2m so that positrons can propagate under the barrier. For the corresponding square well <(x)"!< there are negative energy bound states 0'E'!m provided that <'(m#p/4a [cf. Eq. (53)]. So when the potential well is deep enough, it will in fact bind positrons. Correspondingly, a high barrier will bind electrons. It is thus not surprising that electrons can tunnel through the barrier for strong subcritical potentials since they are attracted by those potential barriers. Another way of seeing this phenomenon is by using the concept of e!ective potential < (x) which is the potential which can be used in a Schrodinger equation to simulate the properties of a relativistic wave equation. For a potential <(x) introduced as the time component of a four-vector into a relativistic wave equation (Klein}Gordon or Dirac), it is easy to see that 2m< (x)"2E<(x)!<(x). Hence as the energy E changes sign, the e!ective potential can change from repulsive to attractive. For the Coulomb barrier, Anchishkin [18] has already suggested looking at scattering of p> o! heavy nuclei to see if there was any experimental evidence for tunnelling: the Klein}Gordon equation like the Dirac equation has negative energy solutions, so similar arguments apply. His calculations show a 20% enhancement in p>!U scattering compared with non-relativistic expectations. The analogous process for fermions would be positron-heavy nucleus scattering. For a positron of initial energy E incident on a heavy nucleus of charge Z the classical turning point r "Za/E. So it would be interesting to measure the wavefunction at the origin for positron scattering o! a heavy nucleus compared with the wavefunction at the origin for electron scattering o! the nucleus at the same energy in order to demonstrate tunnelling. Provided that Za(1, E'0 and normal Coulomb wavefunctions should be accurate enough for the calculation of the ratio o""t(0)" /"t(0)" of wavefunctions at the origin. The wavefunction at the origin for a Dirac particle is singular but the ratio is "nite. The exact continuum wavefunctions for a Dirac particle in a Coulomb potential are discussed by Rose [19]. We shall just write down his results: if the particles are non-relativistic then o"e\p8?# N ,
(55)
where p and E are the particle momenta and energies and this is of course exponentially small as pP0. But if the particles are relativistic o"fe\p8? ,
(56)
where f is a ratio of complex gamma-functions and is approximately unity for large Z. So o&e\p8?+ 10\ for Za&1 which is not specially small although it still decreases exponentially with Z.
N. Dombey, A. Calogeracos / Physics Reports 315 (1999) 41}58
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But while experiments for Za(1 should show tunnelling it will not yet amount to Klein tunnelling. For that we require Z large enough so that bound positron states may be present. This means that Z must be below its supercritical value Z of around 170 but large enough for the 1s state to have E(0. The calculations of Refs. [15,16] which depend on particular models of the nuclear charge distribution give this region as 150(Z(Z which unfortunately will be di$cult to demonstrate experimentally. Nevertheless, the theory seems to be clear: in this subcritical region positrons should no longer obey a tunnelling relation which decreases exponentially with Z such as that of Eq. (56). Instead the Coulomb barrier should become more transparent as Z increases, at least for some energies. The work of Jensen and his colleagues [9] shows that a transmission resonance (i.e. maximal transmission) for positron scattering on a Coulomb potential may well occur at Z"Z although the onset of supercriticality implies that time-independent scattering quantities will then no longer be well de"ned. We are now carrying out detailed calculations of positron scattering o! subcritical nuclei to clarify these matters.
8. Conclusions We hope that this discussion has demonstrated that the Klein step is pathological and therefore a misleading guide to the underlying physics. It represents a limit in which time-dependent emission processes become time-independent and therefore a relationship between the emitted current and the transmission coe$cient exists, as we show in the appendix. In general no such relationship exists. The underlying physics uncovered by Klein in his solution of the Dirac equation is not only particle emission but also tunnelling by means of the negative energy states which are characteristic of relativistic wave equations, similar to interband tunnelling in semiconductors [20]. It is time to "nally bring this 70 year old puzzle to a conclusion.
Acknowledgements We would like to thank A. Anselm, G. Barton, J.D. Bjorken, L.B. Okun and D. Waxman for their constant advice and help. One of us (ND) thanks NATO Scienti"c and Environmental A!airs Division for support which allowed him to attend this meeting. The other of us (AC) thanks G. Volovik and the Low Temperature Laboratory of Helsinki University of Technology for their hospitality, and EU Training and Mobility of Researchers Programme Contract No. ERBFMGECT980122 for its support.
Appendix A. The vacuum current in the presence of a wide potential barrier We now repeat the analysis of Section 3 for a square barrier for <'2m and ma<1. We shall obtain a relation between the vacuum current and the transmission coe$cient for the barrier just as we did for the Klein step in Eq. (32), namely
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N. Dombey, A. Calogeracos / Physics Reports 315 (1999) 41}58
1 8i 1 10" j"02"! dE "! dE ¹ (E) 2p (1!i)#8i 2p
(A.1)
provided that averages over phase angles are understood. This result can of course only be true in the approximation where the transient current is taken to be independent of time, since the right-hand side of Eq. (A.1) is time independent. We consider an observer OI to the left of the barrier. In the energy range that interests us where the continuum hole states under the barrier coincide in energy with the free particle continuum states, we have left incident (L) waves of the form
i i 1 E#m u (E, z)" e NV#A p e\ NV , x(!a , p * p 2(p E#m E#m
(A.2)
i 1 "E!<#m" u (E, z)" B e\ NV p * * p ! 2(p E!<#m #B 0
i e NV , !a(x(a , p E!<#m
i C E#m e NV, x'a. u (E, z)" p * p 2(p E#m
(A.3)
(A.4)
The above set of modes are energy-normalised. Note that despite the sign in the exponentials the "rst term in (A.3) describes a left-incident wave while the second term describes a right-incident wave: this is on account of Pauli's group velocity argument. Although we are interested in what observer OI sees, the current uR (e, x)a u (e, x) (with a "!p in one dimension) is constant, so it can * V * V W be conveniently calculated from Eq. (A.4). The answer is uR (e, x)a u (e, x)""C(E)"/2p , (A.5) * V * where "C(E)" is equal to the transmission coe$cient ¹ of Eq. (34). We can similarly write down the solution u (E, x) identical to the above with L and R inter0 changed. The corresponding current is uR (e, x)a u (e, x)"!"C(E)"/2p . (A.6) 0 V 0 If we were to treat the barrier problem as a genuine time-independent problem there would be no question of calculating any radiation: We would work in terms of parity eigenstates and the net current would clearly be zero. However, we are interested in a long barrier that becomes supercritical at t"0. We then have as we have seen in Section 5 a steady electron current moving out of the barrier for some large time q which is proportional to the barrier length a. We de"ne a state "02 as seen by observer OI to the left of the barrier. Then right-incident waves u represent 0 electrons escaping from the barrier and should thus be present in "02. On the other hand left-incident waves u should be absent. *
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The "eld can be expanded in the form
t(x, t)" dE+a (E)u (E, x)e\ #R#a (E)u (E, x)e\ #R,#rest * * 0 0
(A.7)
with a corresponding expression for tR. The terms rest above correspond to wavefunctions outside the energy range we are concerned with and are of no interest in what follows. The anticommutation relations read (the modes being energy normalised and i, j standing for L or R) +a (E), aR(E ),"d(E!E )d . G H GH According to the reasoning preceding (A.7) left-incident waves are absent, i.e. a (E)"02"0 . (A.8) * On the other hand, the outgoing (to the left) electron current is equivalent (as already mentioned) to aR (E)a (E )"02"d(E!E )"02 . (A.9) 0 0 Having speci"ed the vacuum state "02 the next and "nal step is the calculation of the expectation value of the current J"[tR(x), a t(x)] taken in the state "02: V (A.10) 10"J"02"(10"tRa t"02!10"t a tR"02) . V V Substituting (A.7) in (A.10) we end up with
1 10"J"02" dE dE +10"aR (E)a (E )"02uR (E)a u (E ) * * * V * 2 !10"a (E)aR (E )"02uR (E )a u (E)#10"aR (E)a (E )"02uR (E)a u (E ) * * * V * 0 0 0 V 0 !10"a (E)aR (E )"02uR (E )a u (E), . 0 0 0 V 0 The "rst term in brackets in (A.11) vanishes due to (A.8). The second term becomes
(A.11)
!uR (E , 0)a u (E, 0)d(E!E ) * V * if we use the anticommutation relations and (A.8). The third term yields !uR (E, 0)a u (E, 0)d(E!E ) 0 V 0 using (A.9) and the fourth term vanishes using the anticommutation relations (i.e. the exclusion principle; the state "02 already contains an electron in the state u hence we get zero when we 0 operate on it with aR ). One energy integration is performed immediately using the d function. The 0 "nal result is (using (A.5), (A.6))
"C(E)" "C(E)" 1 1 ! "! dE"C(E)" . J(0)" dE ! 2p 2p 2p 2
(A.12)
The sign of the current is consistent with a current of (previously bound) electrons to the left. In the limit we are interested in (a large) "C" should be averaged over a thus yielding
1 dE ¹ (E) . J(0)"! 2p
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Note that whereas for a Klein step we describe the particle emission in terms of electron}positron pair production while we say that a supercritical barrier will spontaneously emit electrons. But an observer just inside the barrier will see a corresponding current which he will interpret from its sign as a positron current. So from his point of view, there is pair production.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
O. Klein, Z. Phys. 53 (1929) 157. P.A.M. Dirac, Proc. Roy. Soc. 117 (1928) 612. A Calogeracos, N. Dombey, Klein Tunnelling and the Klein Paradox, Int. J. Mod. Phys. A 14 (1999) 631. A.A. Grib, S.G. Mamayev, V.M. Mostepanenko, Vacuum E!ects in Strong Fields, Friedmann, St. Petersburg, 1994. W. Greiner, B. Muller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, Berlin, 1985. A.I. Nikishov, Nucl. Phys. B 21 (1970) 346. A. Hansen, F. Ravndal, Phys. Scripta 23 (1981) 1033. J. Schwinger, Phys. Rev. 82 (1951) 664. H.G. Dosch, J.H.D. Jensen, V.F. Muller, Phys. Norv. 5 (1971) 151. A. Calogeracos, N. Dombey, K. Imagawa, Yad. Fiz. 159 (1996) 1331. (Phys. At. Nucl 159 (1996) 1275) J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, p. 512. See also D.S. Jones, The Theory of Electromagnetism, Pergamon, Oxford, 1964. M. Stone, Phys. Rev. B 31 (1985) 6112. R. Blankenbecler, D. Boyanovsky, Phys. Rev. D 31 (1985) 2089. A. Calogeracos, N. Dombey, The continuum limit and integral vacuum charge, JETP Lett. 68 (1998) 377. Ya.B. Zeldovich, S.S. Gershtein, Zh. Eksp. Teor. Fiz. 57 (1969) 654; Ya.B. Zeldovich, V.S. Popov, Usp. Fiz. Nauk 105 (1971) 403. W. Pieper, W. Greiner, Z. Phys. 218 (1969) 327. Quoted by F. Bakke, H. Wergeland, Phys. Scripta 25 (1982) 911. D. Anchishkin, J. Phys. A 30 (1997) 1303. M.E. Rose, Relativistic Electron Theory, Wiley, New York, 1961, p. 191. E.O. Kane, E.I. Blount, Interband tunneling, in: E. Burstein, S. Lundqvist (Eds.), Tunneling Phenomena in Solids, Plenum, New York, 1969, p. 79.
Physics Reports 315 (1999) 59}81
Inclusive dilepton production at RHIC: a "eld theory approach based on a non-equilibrium chiral phase transition Fred Cooper Theoretical Division, MS B285, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Dedicated to the memory of Richard Slansky, friend and colleague
Abstract Recently a real time picture of quantum "eld theory has been developed which allows one to look into the time evolution of a scattering process. We discuss two pictures for discussing non-equilibrium processes, namely the Schrodinger picture and the Heisenberg picture and show that a time-dependent variational method is equivalent to the leading order in large-N approximation in the Heisenberg picture. We then discuss the dynamics of a non-equilibrium chiral phase transition in mean "eld theory in the O(4) sigma model. We show how the pion spectrum can be enhanced at low momentum because of non-equilibrium e!ects. We then show how to use Schwinger's CTP formalism to calculate the inclusive dilepton spectrum coming from the pion plasma. We "nd that a noticeable enhancement occurs in this spectrum, but that there are large numerical uncertainties due to errors connected with the "nite times used to do our numerical simulations. 1999 Elsevier Science B.V. All rights reserved. PACS: 25.75.Gz; 11.30.Rd;12.38.Mh; 25.75.Dw Keywords: Dilepton production; Non-equilibrium processes; Closed time path (CTP) formalism
1. Introduction In the early 1970s, the study of inclusive Hadronic Interactions was at the forefront of theoretical research because of the opening of two major accelerators } FNAL in the United States and the ISR at CERN. As data poured in many of us tried to understand from various approaches such as Regge poles, Fireballs, and Hydrodynamics how to determine the single particle inclusive spectrum
E-mail address: [email protected]. (F. Cooper) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 3 - 7
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of particles. In that atmosphere, in 1974, Peter Carruthers founded the Particle Physics group at Los Alamos which initially included Dick Slansky, Geo! West, David Campbell, David Sharp, Mitch Feigenbaum and myself, all of whom were working on this problem when we "rst arrived. At that time QCD was not yet a well developed subject, computers were relatively archaic and the idea of a "rst principles approach to understand what happens after two particles collide was just a dream. Recently this dream has come closer to reality, and as a tribute to Dick and our early e!orts I would like to review progress we have made in this direction. I will begin by "rst reviewing two pictures for discussing non-equilibrium processes, namely the Schrodinger and Heisenberg pictures. In the Schrodinger picture one can reduce the number of degrees of freedom by using a time-dependent variational method with a trial density matrix which is Gaussian. In the Heisenberg picture one can use the large N expansion of the path integral in the closed time path formalism to give us a controllable expansion about the mean-"eld approximation. We will then review our model for a non-equilibrium phase transition, namely the O(4) linear sigma model, and set up the time evolution equations for the lowest order in large N approximation. In this model we will obtain natural cooling (quenching) of the plasma from the unbroken phase to the broken symmetry phase as a result of expansion into the vacuum with boost invariant kinematics imposed. We "nd that there is a growth of unstable modes when the order parameter (e!ective pion mass) goes negative during the evolution. This causes a distortion of the single particle distribution of pions. We also reconstruct typical classical con"gurations by sampling the quantum density matrix and see domains. We then obtain the dilepton spectrum from this time evolving plasma using Schwinger's closed time path formalism. 2. Strategies for studying time evolution problems in ju 5eld theory 2.1. Schrodinger picture In quantum mechanics, time evolution problems are usually discussed in the Schrodinger picture. The initial state in the x representation is the wave function W(x, t)"1x " W2 ,
(2.1)
which evolves in time according to the Schrodinger equation i RW(x, t)/Rt"HW(x, t) .
(2.2)
This equation can be obtained from Dirac's variational principle [1]. De"ne the action
C" dt1W "iR/Rt!H" W2 .
(2.3)
Minimimizing the action then yields the Schrodinger equation dC"0P+i R/Rt!H,"W 2"0 .
(2.4)
Here one thinks of H as an operator which in the x representation one has p"!i d/dx .
(2.5)
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Gaussian initial conditions for the wave function would be W(x, 0)"exp[!a(x!x )] . In "eld theory, the wave function is replaced by a wave functional. For example, a Gaussian wave functional is given by 1u"W2"t[u, t]
"exp !
[u(x)!u( (x)] V W
G\(x,y) !iR(x, y) [u(y)!u( (y)] 4
#ip( (x)[u(x)!u( (x)] .
(2.6)
The time evolution is now given by the functional Schrodinger equation
H" dx[!d/du#( u)#<(u)] .
i Rt/Rt"Ht,
(2.7)
Dirac's variational principle
C" dt1W"i R/Rt!H"W2 , (2.8)
dC"0P+i R/Rt!H,"W2"0
is the starting point for thinning the degrees of freedom. Instead of putting the functional Schrodinger equation on the computer, one assumes that the wave functional stays Gaussian, and uses the variational principle to determine the time evolution equations for the one and two point functions involved in the description of the Gaussian. This approximation is equivalent to mean "eld theory and at large-N becomes equivalent to the leading order large-N approximation. That is we assume a Gaussian trial wave functional: 1u"W 2"t [u, t] T T with t given by Eq. (2.6). The variational parameters have the following meaning: T u( (x, t)"1W "u"W 2; p( (x, t)"1W "!id/du"W 2 , T T T T G(x, y, t)"1W "u(x)u(y)"W 2!u( (x, t)u( (y, t) . T T The e!ective action for the variational parameters is
(2.9)
(2.10)
C(u( , p( , G, R)" dt1W "iR/Rt!H"W 2 T T
" dt dx[p(x, t)u (x, t)# dt dx dyR(x, y)GQ (x, y, t)! dt1H2 ,
(2.11)
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where
1H2" dx+p/2#2RGR#G\/8#1/2( u)!1/2 G#1/2< [u]G#1/8< [u]G, . (2.12)
Equations of motion that result from varying the action are p (x, t)" u!R1<2/Ru , u (x, t)"p ,
GQ (x, y, t)"2 dz[R(x, z)G(z, y)#G(x, z)R(z, y)] ,
RQ (x, y, t)"!2 dz[R(x, z)R(z, y)#G\/8#[ !R1<2/RG]d(x!y) . V
(2.13)
In theory if there is translational invariance, we can simplify these equations by Fourier transforming them in three-dimensional space to obtain 2G$ (k, t)G(k, t)!GQ (k, t)#4C(k, t)G(k, t)!1"0 ,
C(k, t)"k#m(t); m(t)"!k#j [dk]G(k, t) .
(2.14)
One can also linearize these equations by recognizing that the mean "eld approximation for the homogeneous problem is equivalent to a "eld theory with a time-dependent mass which is self-consistently determined. That is, if we assume a quantum "eld obeying (䊐#m(t)) (x, t)"0 ,
(2.15)
then one can satisfy the equation for G(x, y, t) by choosing G(x, y; t),1 (x, t) (y, t)2 .
(2.16)
2.2. Heisenberg picture } path integral approach } closed time-path approach of J. Schwinger In the Heisenberg picture, the operators are time-dependent, and the expectation value of the "elds in an initial state are the in"nite number of c-number variables. The in"nite heirarchy of coupled Green's functions need to be truncated by an approximation scheme. The large N approximation orders the connected Green's functions in powers of 1/N, with the connected four point function going as 1/N, 6 point function 1/N, etc. The formalism for preserving causality in initial value problems was invented by Schwinger [3] and was later cast in the form of path integrals [4]. The generating functional for initial value problem Green's functions is
Z[J>, J\, o]" dW i ¹H exp ! iJ u \ \ "Tr o¹H exp ! iJ u \ \
W
W
¹ exp iJ u > >
¹ exp iJ u i > > .
(2.17)
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This can be written as the product of an ordinary path integral times a complex conjugate one or as a matrix path integral
Z[J>, J\]" du Tr o exp i G > \
S[u ]# J u ! SH[u ]# J u > > > \ \ \
" du exp i(S[u ]#J u?)1u , i"o"u , i2,e 5 (? , (2.18) ? ? ? ? where 1u , i"o"u , i2 is the density matrix de"ning the initial state. We use the matrix notation u u?" > , a"1, 2 , (2.19) u \ with a corresponding two-component source vector
J > , a"1, 2 . J \ On this matrix space there is an inde"nite metric J?"
c "diag(#1,!1)"c?@ , ?@ so that, for example,
(2.20)
(2.21)
J?c u@"J u !J u . ?@ > > \ \ From the path integral we get the following matrix Green's function:
(2.22)
G?@(t, t )"d=/dJ (t)dJ (t )" , ? @ ( G(t, t ),G (t, t )"i Tr+ou(t)u (t ), , G(t, t ),G (t, t )"$i Tr+ou (t )u(t), , G(t, t )"i Tr+oT[u(t)u (t )], "H(t, t )G (t, t )#H(t , t)G (t, t ) , G(t, t )"i Tr+oTH[u(t)u (t )], "H(t , t)G (t, t )#H(t, t )G (t, t ) . We notice that G "G(t, t ) and G H"G(t, t ). $ $ We also will need the relationships
(2.23)
G (t, t )"iH(t!t )[U(t),UM (t )] "H(t!t )[G(t, t )!G(t, t )] , ! G (t, t )"!iH(t !t)[U(t),UM (t )] "H(t !t)[G(t, t )!G(t, t )] , ! as well as the relations between the Green's functions
(2.25)
G (t, t )"G (t, t )!G(t, t )"!G H(t, t )#G(t, t ) , $ $ G (t, t )"G (t, t )!G(t, t )"!G H(t, t )#G(t, t ) . $ $
(2.24)
(2.26)
(2.27)
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2.2.1. Large-N approximation The method for reducing the number of degrees of freedom in the Heisenberg picture is the large-N approximation [5]. If we have an N-component scalar "eld with Lagrangian ¸I [U]"(R U )(RIU )!(j/8N)(U U !(2Nk/j)) , G G G I G we can rewrite this as
(2.28)
s ¸I [U, s]"!U (䊐#s)U #(N/j)s #k , G G 2
(2.29)
where i"1,2, N and s"!k#(j/2N)U U . (2.30) G G If k'0, a spontaneous symmetry is breaking at the classical level. At this minimum the O(N) symmetry is spontaneously broken, s"0 and there are N!1 massless modes. Small oscillations in the remaining i"N (radial) direction describe a massive mode with bare mass equal to (2k"(jv . The generating functional for all graphs is given by [6]
Z[ j, K]" d ds exp+iS[ , s]#i [ j #Ks], .
(2.31)
Perform the Gaussian integral over the "eld :
Z[ j, K]" ds exp+iNS [s, j, K], ,
(2.32)
where
1 1 s i S " dx jG\[s] j#Ks# s #k # Tr ln G\[s] , 2 j 2 2 G\[s](x, y),+䊐#s,d(x!y) .
(2.33)
Because of the N in the exponent one is allowed to perform the integral over s by stationary phase. This leads to an expansion of Z in powers of 1/N the lowest term (stationary-phase point) is related to the previous Gaussian (Hartree) approximation. The e!ective action of the leading order is S [ , s]"S [ , s]#(i /2) Tr ln G\[s] . Varying the action leads to the mean "eld equations
(2.34)
+䊐#s, "0 , s"!k#(j/2N)( #(1/i)G(x, x; s)) .
(2.35)
We notice that this is the same equation found in the Gaussian approximation with m(t) being identi"ed with s.
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3. Dynamical evolution of a non-equilibrium chiral phase transition One important question for RHIC experiments is } can one produce disoriented chiral condensates (DCCs) in a relativistic heavy ion collision? Recently, Bjorken, Rajagopal and Wilczek and others proposed that a nonequilibrium chiral phase transition such as a quench might lead to regions of DCCs [7]. The model Rajagopal and Wilczek considered was the O(4) linear sigma model in a tree-level approximation, where a quench was assumed. Two de"ciencies of that model were its classical nature (it could not describe p}p scattering), and the quench was put in by hand. Our approach [8] instead was to look at the quantum theory in an approximation that captures the phase structure as well as the low-energy pion dynamics. We also used the natural expansion of an expanding plasma to cool the plasma and built into our approximation boost invariant kinematics which result from a hydrodynamic picture where the original plasma is highly Lorentz contracted. In the linear sigma model treated in leading order in the 1/N expansion the theory has a chiral phase transition at around 160 MeV and we choose the parameters of this theory to give a reasonable "t to the correct low-energy scattering data. We obtain natural quenching for certain initial conditions as a result of the expansion process. 3.1. Review of the linear p model The Lagrangian for the O(4) p model is ¸"RU ) RU!j(U ) U!v)#Hp .
(3.1)
The mesons form an O(4) vector U"(p, p ) . G As we discussed earlier in our discussion of the large-N approximation we introduce s"j(U ) U!v) and use the equivalent Lagrangian ¸ "! (䊐#s) #s/4j#sv#Hp . G G
(3.2)
The leading order in 1/N e!ective action which we obtain by integrating out the "eld and keeping the stationary phase contribution to the s integration is
i C[U, s]" dx ¸ (U, s, H)# N tr ln G\ , 2
(3.3)
G\(x, y)"i[䊐#s(x)]d(x!y) . This results in the equations of motion [䊐#s(x)]p "0, G
[䊐#s(x)]p"H ,
(3.4)
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and the constraint or gap equation s"!jv#j(p#p ) p)#jNG (x, x) . (3.5) We will introduce #uid proper time and rapidity variables to implement the kinematic constraint of boost invariance: 1 g, log(t!z/t#z) . 2
q,(t!z) ,
To implement boost invariance we assume that mean (expectation) values of the "elds U and s are functions of q only: q\R qR U (q)#s(q) U (q)"Hd , O O G G G s(q)"j(!v#U(q)#NG (x, x; q, q)) . (3.6) G To calculate the Green's function G (x, y; q, q ) we "rst determine the auxiliary quantum "eld (x, q) (q\R qR !q\ R!R #s(x)) (x, q)"0 . O O E , G (x, y; q,q ),1¹+ (x, q) (y, q ),2 . (3.7) We expand the quantum "eld in an orthonormal basis:
1
(g, x , q), [dk](exp(ikx) fk(q)ak#h.c.) , , q where kx,k g#k x , [dk],dk dk /(2p). The mode functions and s obey E , , E , f$k#((k/q)#k #s(q)#(1/4q)) fk"0 . E , 1 s(q)"j !v#U(q)# N [dk]" fk (q)"(1#2nk) . G q
(3.8) (3.9)
when s goes negative, the low-momentum modes with
((k#1/4)/q)#k ("s" E , grow exponentially. These growing modes then feed back into the s equation and get damped. Low momentum growing modes lead to the possibility of DCCs as well as a modi"cation of the low momentum distribution of particles. To "x the parameters of this mode we use the PCAC relation R AG (x),f mpG(x)"HpG(x) , I I p p and the de"nition of the broken symmetry vacuum, s p "mp "H , p p "f "92.5 MeV , p
m"!jv#jf #jN p p
K
[dk]
1 2(k#m p
.
(3.10)
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The mass renormalized gap equation is
1 j . s(q)!m"!jf #jU(q)# N [dk] " fk(q)" (1#2nk)! p p G q 2(k#m
(3.11)
j is chosen to "t low energy scattering data. We choose our initial data (at q "1) so that the system is in local thermal equilibrium in a comoving frame n "1/e@#I !1 , I
(3.12)
where b "1/¹ and E"((k/q)#k #s(q ). I E , The initial value of s is determined by the equilibrium gap equation for an initial temperature of 200 MeV and is 0.7 fm\ and the initial value of p is just H/s . The phase transition in this model occurs at a critical temperature of 160 MeV. To get into the unstable domain, we then introduce #uctuations in the time derivative of the classical "eld. For q "1 fm there is a narrow range of initial values that lead to the growth of instabilities 0.25("p"(1.3. The results of numerical simulations described in [8] for the order parameter s are shown in Fig. 1. Fig. 1 displays the results of the numerical simulation for the evolution of s ((3.8)}(3.9)). We display the auxiliary "eld s in units of fm\, the classical "elds U in units of fm\ and the proper time in units of fm (1 fm\"197 MeV) for two simulations, one with an instability (p" "!1) and O one without (p" "0). O We notice that for both initial conditions, the system eventually settles down to the broken symmetry vacuum result as a result of the expansion. We also considered a radial expansion and obtained similar results [9]. In the radial case, the outstate was reached earlier, but the number of oscillations where s became negative was similar. To determine the single particle inclusive pion spectrum we go to an adiabatic basis and introduce an interpolating number operator which
Fig. 1. Proper time evolution of the s "eld for two di!erent initial values of p.
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interpolates from the initial number operator to the out number operator. Introduce mode functions f which are "rst order in an adiabatic expansion of the mode equation I dy /dt"u , f "e\ WIO/(2u , I I I I 1
(g, x , q), [dk](exp(ikx)f (q)a (q)#h.c.) . , q
(3.13) (3.14)
In terms of the initial distribution of particles n (k) and b we have n (q),f (k , k , q)"1aR(q)a (q)2"n (k)#"b(k, q)"(1#2n (k)) , (3.15) I E , I I where b(k, q)"i( f (Rf /Rq)!(Rf /Rq)f ), n (q) is the interpolating number density. The distribution I I I I I of particles is f (k , k , q)"dN/p dx dk dg dk . E , , , E Changing variables from (g, k ) to (z, y) at a "xed q we have E dN dN " " p dz dx Jf (k , k , q) E , E , dk p dy dk ,
(3.16)
"A dk f (k , k , q)" f (k , k , q)kI dp . , E E , E , I
(3.17)
To compare our "eld theory calculation with some standard phenomenological approach, we considered a hydrodynamic calculation with boost invariant kinematics and determined the spectrum assuming that at hadronization the pions where at the breakup temperature ¹"m (as p well as ¹"1.4m ), with the distribution given by the Cooper}Frye}Schonberg formula [11] p dN dN " " g(x, k)kI dp . (3.18) E I dk p dk dy , Here g(x, k) is the single-particle relativistic phase-space distribution function. When there is local thermal equilibrium of pions at a comoving temperature ¹ (q) one has g(x, k)"g +exp[kIu /¹ ]!1,\ . (3.19) p I The comparison is shown in Figs. 2 and 3. We therefore "nd that a non-equilibrium phase transition taking place during a time evolving quark}gluon or hadronic plasma can lead to an enhancement of the low-momentum distribution of pions.
3.2. Determination of the ewective equation of state Equation of state is obtained in the frame where the energy momentum tensor is diagonal } we are already in that boost invariant frame ¹ "diag+e, p , p , . IJ E ,
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Fig. 2. Single-particle transverse momentum distribution for p"!1 initial conditions compared to a local equilibrium hydrodynamical calculation with boost invariance. Fig. 3. Single-particle transverse momentum distribution for p"0 initial conditions compared to a local equilibrium hydrodynamical calculation with boost invariance.
Fig. 4. Equation of state p/e as a function of q for the massless p model where we start from a quench.
When we have massless goldstone pions in the p model (H"0) then s goes to zero at large times. In the spatially homogenous case 1¹ 2"e, 1¹ 2"pd . GH GH The equation of state becomes p"e/3 at late times even though the "nal particle spectrum is far from thermal equilibrium as is seen in Fig. 4.
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3.3. Dephasing and looking for DCCs As we have shown in [10], dephasing justi"es the replacement of the exact Gaussian o by its diagonal elements. At large-N or in mean "eld theory the density matrix is a product of Gaussians in space: I 1u "o "u 2"(2pm)\ exp+!(p/8m)(u !u )!(1/8m)(u #u ), . (3.20) I I I I I I I I I I After a short while because of dephasing, the Gaussian distribution o! the diagonal u "u is I I strongly suppressed: m /p +( /2kn(k));m . I I I This is shown in Fig. 5. We "nd no support for `SchroK dinger cata states in which quantum interference e!ects between the two classically allowed macroscopic states at v and !v can be observed. An ensemble may be regarded as a classical probability distribution over classically distinct outcomes. The particle creation e!ects in the time-dependent mean "eld give rise to strong suppression of quantum interference e!ects and mediate the quantum to classical transition of the ensemble. If we project the density matrix onto an adiabatic number basis, we can reconstruct classical "eld con"gurations from the diagonal density by replacing the "eld operator a(k) by a(k)P[n(k)] e (I , with n(k) obtained by throwing dice on the density matrix and being randomly chosen between 0( (2p. Typical "eld con"gurations as a function of r (averaging over angles) are shown in Fig. 6.
Fig. 5. The Gaussian o for k"0.4 from Ref. [10] illustrating the strong suppression of o!-diagonal components due to dephasing. Fig. 6. Four typical "eld con"gurations drawn from the same classical distribution of probabilities.
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4. Inclusive dilepton production and Schwinger's closed time-path formalism Schwinger's CTP formalism is designed to allow one to calculate expectation values of operators in the initial density matrix. One quantity we are interested in for obtaining an e!ective hydrodynamics is the expectation value of the energy momentum tensor 1in"¹IJ(x)"in2,(e#p)uIuJ!pgIJ ,
(4.1)
where ¹IJ(x) is the "eld theory energy momentum tensor. Also by Fourier transforming this energy momentum tensor and looking in a comoving frame, we can ask how much energy is in the `freea part of various components and de"ne an equivalent number of quanta by dividing by u for each I species. If we consider the inclusive production of electron}positron pairs the probability amplitude is 1e\(k, s)e>(k , s )X " i2"1X"bd "i2 . I Q IY QY The inclusive distribution function for dileptons: (E /m)(E /m)(dN/[dk][dk ]),1i"d>b>bd "i2 . I I IY QY I Q I Q IY QY Using the relations between b, d to W and the free `outa "elds we obtain
1i " dx dx dx dx e IV\V+u> W(x ),+W>(x )u , I Q I Q ;e IYV\V+v> W(x ),+W>(x )v , " i2 . IY QY IY QY Now using the weak asymptotic condition [12] that W" "Z W , (4.2) R inside of matrix elements as well as the equation of motion of the spinors and the identity
R dF "F(t )!F(t ) . dt R We obtain
(4.3)
1e\(k, s)e>(k , s )X " P P 2
"iZ\ dx dx e IV>IYV u D 1X"T+W(x )WM (x ),"P P 2 DM v . I Q IY QY
(4.4)
Squaring this amplitude and summing over X we obtain
dN E E I I (k, k ; s, s )" dx dx dx dx e IV\Ve IYV\Vv D u D IY QY V I Q V m m [dk][dk ] 1P P "TH+W(x )WM (x ),T+W(x )WM (x ),"P P 2 DM u DM v . V I Q V IY QY
(4.5)
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The matrix element involved here, 1P P "TH+W(x )WM (x ),T+W(x )WM (x ),"P P 2 (4.6) is precisely the type of Green's function that is obtained from the generating functional of Schwinger's CTP formalism. The Lagrangian we will use to determine this four-point function is the O(4) linear p model # electrodynamics. This Lagrangian has three pieces: The mesons form an O(4) vector U"(p , p). This strongly G interacting Lagrangian is given by ¸ "! (䊐#s) #s/4j#sv#Hp , G G s"j(U ) U!v) .
(4.7)
To this we add the free lepton and photon Lagrangian ¸ "!F FIJ!(1/2a)(R ) A)#WM [ic RI!m]W . IJ I The interaction of the photons with the pion plasma and the leptons is given by e ¸ [ , A , W, WM ]" ( # )A AI#e ( R ! R )AI!eWM cIWA #LA . I I I I G I 2
(4.8)
(4.9)
If we treat the electromagnetic interactions perturbatively in e and the pions in the mean "eld approximation we obtain the graph shown in Fig. 7. The inverse propagators in the LSZ representation lop o! the external legs and put the leptons on mass shell. One is left with (E /m)(E /m)(dN/dkdk )"M (k, s; k s )=IJ(k, k ) , I IY IJ where
(4.10)
MIJ(k, s; k s )"v (k , s )cIu(k, s)u (k, s)cJv(k , s ) , = (k, k ),= #= #= IJ IJ IJ IJ
Fig. 7. Leading contribution from the plasma to dilepton production. The four fermion graph is to be evaluated using the matrix CTP Green's functions.
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"ie dy dy dz dz e I>IYX\W[D(y , y )PNH(y , z )D (z , z ) IN HJ #D(y , y )PNH (y , z )D (z , z )#D(y , y )PNH (y , z )D(z , z )] . IN HJ IN HJ If we want the invariant mass distribution function when M"q,
q"k#k ,
we obtain dN/dq"2 qdN/dM dq"R (q)=IJ(q) , IJ where
(4.11)
[dk] [dk ] d(q!k!k )¸M (k, k ) IJ E E I IY 1 2p 4m 2m " 1! 1# (qIqJ!gIJq) . (2p) 3 q q
R (q), IJ
(4.12)
If we were doing an ordinary perturbation theory calculation analytically we could use the translational invariance of the polarization tensor
P (y , z )" [dq]e\ OW\XP (q) IJ IJ
(4.13)
and the representation of the free photon propagator in Feynman gauge
g IJ D (z , z )" [dk]e\ IX\X $IJ k#ie
(4.14)
to obtain = (k, k )"!ie IJ
(2p)d(0) P (q) IJ q
"!ie<¹
dq d(q!k!k )P (q) . IJ q
(4.15)
The other terms have the photon on mass shell so they give no contribution to the particle production rates. In this homogeneous case we therefore obtain the usual result [13,14]
e ip PI(q) 4m 2m qdN I " 1! 1# . q q q dM dq<¹ (2p) 3
(4.16)
Let us "rst look at the case of a thermal plasma where we can calculate everything analytically [2]. The vacuum polarization graph can be found using the following expansion of the pion "eld to calculate the "nite temperature pion Green's functions
(x, t)"
[dk] [exp(ikx)a #exp(!ikx)bR] . I I 2u I
(4.17)
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The creation and annihilation operators obey the commutation relations [a , aR ]"[b , bR ]"(2p)d(k!k ) . I IY I IY And the phase space number densities n> and n\ are de"ned by I I 1aRa 2"(2p)n>d(k!k ) , I IY I 1bRb 2"(2p)n\d(k!k ) , I IY I so that the total number of positively charged particles is given by
N>" dk1aRa 2" dx dkn> . I I I
(4.18)
(4.19)
(4.20)
For the case of neutral plasma in thermal equilibrium at inverse temperature b we have n>"n\"1/(e@S!1) . I I Using the fact that for a free pion gas
dk 1 R(x) (y)2 " [n>e I V\W#(1#n\)e\ I V\W], 1 (x) R(y)2 I 2u (2p) I I dk [n\e I V\W#(1#n>)e\ I V\W] , " I 2u (2p) I I one "nds that the vacuum polarization tensor is given by
(4.21)
dk dk (2p)+(k !k ) (k !k ) !iP (q)" IJ I J 2u (2p) 2u (2p) ;[n\ n> d(q!k !k )#(1#n\ )(1#n> )d(q#k #k )] I I I I ;(k #k ) (k #k ) [n\ (1#n\ )#n> (1#n> )]d(q!k #k ), . I I I I J I :
(4.22)
The three terms in the above relation correspond to pion pair annihilation, creation and bremsstrahlung, respectively. The delta functions show that only the annihilation process survives for M above the dilepton threshold. We therefore obtain
4m !iPI(q)"q 1! p I q
S>du n>n\ , q I O\I S\
(4.23)
with u!"(q $q(1!4m/M)/2 and q "(M#q. Thus, the dilepton production rate from p a homogenous thermal plasma of pions is given by
1 dN aB M 4m l>l\ " 1! p <¹ dMdq 48p q M
S>du n>n\ . \ q I O\I S
(4.24)
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4.1. Finite time ewects For our numerical simulation of the interacting plasma we can only follow the time evolution of the plasma for a "xed time 2¹ which is typically about 100 Fermis. Therefore we need to make sure we have the causal formulation so that dileptons at time ¹ only get contributions from processes occurring at times t(¹. So we should not use Feynman propagators but instead use the CTP matrix propagators when we investigate the theory at "nite times. Also the interacting plasma is not time translationally invariant so we must use a noncovariant formalism. In the mean "eld approximation (as well as for the pion gas) there is factorization of four point functions so that the current}current correlation function takes the form 1JI(x)JRJ(y)2"1UR(x)U(y)21RIU(x)RJUR(y)2!1UR(x)RJU(y)21RIU(x)UR(y)2 !1RIUR(x)U(y)21U(x)RJUR(y)2#1RIUR(x)RJU(y)21U(x)UR(y)2 .
(4.25)
If we insert the mode expansion of the charged pion "elds
U(x, t)" [dk][exp(ikx) f (t)a #exp(!ikx) f H(t)bR] I I I I and de"ne phase-space number densities N> and N\ by I I 1aRa 2"(2p)N>d(k!k ) , I IY I 1bRb 2"(2p)N\d(k!k ) . I IY I We obtain
dk dp 1 1J (x)J (y)2"e e\ k\p x\y G (k, p; t , t ) , I J V W (2p) (2p) i IJ
(4.26)
(4.27)
(4.28)
where 1 G (k, p; t , t )"A(k)K> ( p)#K\ (k)A( p)!N>J(k)M>I( p)!M\I(k)N\J( p) , V W IJ IJ i IJ
(4.29)
and
B(k),
$ik D(k), H K! (k), IJ !($)ik C(k), G k k A(k), G H D(k), M!(k), I !($)ik A(k), G C(k), N!(k), J !($)ik A(k), H
k"0 l"0 , k"0 l"j , k"i l"0 ,
(4.30)
k"i l"j , k"0 , k"i , l"0 , l"j ,
(4.31)
(4.32)
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and A(k; t , t )"(1#N )[ f H(t ) f (t )] H#N f H(t ) f (t ) , V W I I V I W I I V I W B(k; t , t )"(1#N )[fQ H(t ) fQ (t )] H#N fQ H(t ) fQ (t ) , V W I I V I W I I V I W C(k; t , t )"(1#N )[ f H(t ) fQ (t )] H#N f H(t ) fQ (t ) , V W I I V I W I I V I W D(k; t , t )"(1#N )[ fQ H(t ) f (t )] H#N fQ H(t ) f (t ) . V W I I V I W I I V I W Contracting G (k, p) with ¸I IJ"qIqJ!gIJq we obtain IJ ¸I IJG (kp; t , t )"[(q ) (p#k))#(q!q ) q)(p#k) ) (p#k)]A(p)A(k) IJ V W !iq (p#k) ) q[A(k)D(p)!D(k)A(p)!A(k)C(p)#C(k)A(p)] #q ) q[A(k)B(p)#B(k)A(p)!C(k)D(p)!D(k)C(p)] ,
(4.33)
(4.34)
where q"k!p . At the special case where q"0 we obtain ¸I IJG (kk; t , t )"4qk ) kA(k)A(k) . IJ V W Here k ) kP(q!4m) in the in"nite time limit.
(4.35)
4.2. Pion gas To see what the e!ects of "nite ¹ might be, let us look at the case where everything is analytically known, namely the pion gas we discussed earlier and which was also discussed in Ref. [2]. So we use the known values of A, B, C, D appropriate to the pion gas where f (t)"e\ SIR/(2u . I I First let us look at the e!ect of just putting a "nite time cuto! into the McLerran formula for =. Using the covariant form of the photon propagator and just cutting o! the internal integrations to run from !¹ to ¹ one obtains
2 2 e dy dz e\ OW\XPIJ(y , z ) =(k, k )"i dy dz IJ q \2 \2 e 2 2 "i< (4.36) dy dz e\ OW\X [dK]GIJ(K, K!q; y , z ) . q \2 \2 Again looking only at the place q"0 and keeping only the pion annihilation contribution
¸I IJG (kk; y , z )"(n/u)qk ) k e SIW\X . IJ I I We can perform the integration over ¹ and obtain the factor 4(sin[q !2u ]¹/q !2u ), I I
(4.37)
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which in the limit ¹PR becomes the factor 2¹;2pd(q !2u ) . I Thus, we see that in the in"nite time limit, the back to back dileptons get contributions only from pion pairs in the plasma with zero combined three momentum, each carrying energy "q /2. In what follows we want to see how doing a causal calculation changes the way in which we go on mass-shell from this simple replacement of the delta function by a representation of the delta function. We will "nd extra terms which only slowly go to zero with the time ¹. We now use the three-dimensional form for the propagators: D(x, y)"iH(x !y )g e\CV\W IJ IJ
[dq] q x y e \ [e\ O V\W!e O V\W] 2q
,iH(x !y )e\CV\Wg [dq]e q x\yD[q (x !y )] . IJ
(4.38)
In our numerical simulations we assumed spatial homogeneity so that the vacuum polarization has the form
PIJ(xy)"i1JI(x)JJ(y)2" [dK] [dP]e\ K\P x\yGIJ[K, P; x , y ] .
(4.39)
Inserting these into the expression for =IJ(k, k ) we then obtain
2
2 W X dy dz [dK]e\CW\We\CX\Xe OX\W dy dz \2 \2 \2 \2 ;D["q"(y !y )]GIJ(K, K!q; y , z )D["q"(z !z )] , (4.40)
=IJ(k, k )"ie<
where q"k#k . It is this contribution which persists when the time cuto! ¹PR. For the other two contributions we obtain
=IJ>(k, k )"!2e<
2
2 X X dz dy dy dz \2 \2 \2 \2
; [dK]Im[e OX\W[GIJ(K, K!q; y , z )!GIJ(!K, q!K; z , y )] e\ q W\W D["q"(z !y )] . ; 2"q"
(4.41)
Things simplify dramatically at the place where q"0. At that point the second and third contributions vanish and we have
1 dN 4q 2Bp " X kL> dk F[k, q"0, q ], <¹ dL>q (2p)L 3 L
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2 W X e 2 F[k, q"0, q ]"i dy dz dy dz ¹ \2 \2 \2 \2 ;e OX\WD (y !y )[A(k, y , z )]D (z !z ) , where
(4.42)
A(k, t, t )"(1#n )e\ SIR\RY/2u #n e SIR\RY/2u I I I I and D (t!t )"!i sin m (t!t )/m , X "2pL /C(n/2) , A A L where n is the number of spatial dimensions. In the limit m P0 we have A D (t!t )"!i(t!t )e\C R\RY The result for the annihilation part for massless photons is F
"!eN(k)/64qu(q !2u )¹[16u#q!8uq cos[2(2u !q )¹] , I I I I I I
(4.43)
with u"k#m . I We can rewrite this as
eN sin[(2u !q )¹] (q #2u ) I I # I "! . (4.44) 4qu (q !2u )¹ 16qu¹ I I I We would like to compare the representation of the delta function squared found here (renormalized to one at the delta function) to the result of just naively putting a cuto! into the covariant calculation which gave F
sin(+q !2u ,¹)/+q !2u ,¹. I I The CTP formalism which preserves causality instead gives
(4.45)
+sin[(2u !q )¹](q !2u )¹#(q #2u )/ 6qu¹, . I I I I The last term makes a very small di!erence at large ¹. In obtaining this result we assumed that e¹PR as eP0 . For the Brehmstrahlung contribution, we "nd en (1#n ) I I F "!4 (!2m #2m q !q !4m e!2q e A A A p¹(m #e)u(q !m) A I A !2e#2(m #e)(m !q #e) cos(2q ¹)#4q e(m #e) sin(2q ¹)) . (4.46) A A A so that the renormalized delta function squared for this case becomes !1/4q¹(m #e)(!2m #2m q !q !4m e!2q e A A A A !2e#2(m #e)(m !q #e) cos(2q ¹)#4q e(m #e) sin(2q ¹)) . A A A
(4.47)
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Here we have kept both a small photon mass (to regulate the infrared) as well as e. For the brehmstrahlung contribution one cannot set e to zero. At m P0 this becomes A (sin(q ¹))/q¹[1!q/e]#q/4e¹#1/e¹ .
(4.48)
In order to obtain our previous results we used e¹!'R. We now see to also have the unwanted terms going to zero we also need e¹!'R. The actual rate of production of dileptons gets from this expression a contribution which goes to zero as a constant divided by the total time. This constant is about the size of the entire dilepton production rate at a ¹"20 fermis. In the pion gas case when ¹"100 fermis, we need (in inverse units) e"1 for the brehmstrahlung contribution to be not that big a contamination, and for e to be small enough for the annihilation cross section to be reasonably accurate. To take the limit eP0 we choose for ¹'¹ "100: e!'(¹/¹ )\B, 0(d;1 , in order to smoothly go to the covariant cuto! result that is (sin(q ¹))/q¹ .
(4.49)
With this form for the e, the "nite q dependent contribution to the cross section goes to zero as 1/¹\B. Thus in doing numerical simulations, we "nd that in order to avoid contamination from brehmstrahlung processes we need to go to quite large hadronic time scales ¹'1000 to be in the asymptotic regime for the production of dileptons which is an electromagnetic process. For the brehmstrahlung process, the e!ective d function is independent of k so that one can do the integration over k for any q to obtain
dk (1#n )n I I. kL> C "X L L (2p)L u I
(4.50)
For b\"m we "nd C "0.127718, and C "0.0792387. p For the creation contribution we get a result similar to the annihilation but with q P!q , that is
eN sin[(2u #q )¹] (!q #2u ) I # I I F "! , 4qu (q #2u )¹ 16qu¹ I I I
(4.51)
so that the renormalized square of the delta function is +sin[(2u #q )¹]/(q #2u )¹#(q !2u )/16qu¹, I I I I vs. the cuto! McLerran formula result sin(+q #2u ,¹)/(q #2u )¹ . I I
(4.52)
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Fig. 8. Finite time e!ects for a pure pion gas. Here t "100f. Fig. 9. Comparison of quench (negative e!ective mass regime) vs. non-quench initial conditions for an interacting "eld theory described by the sigma model. The large rise for small e!ective dilepton mass is the artifact of the "nite time for the simulation. This e!ect slowly goes away as 1/t .
Using these formulae, we have evaluated the dilepton production at "xed time ¹ for both a pion gas and for the interacting "eld theory described by the p model described above. When the e!ective pion mass goes negative, there is signi"cant enhancement of the signal, however one can see the "nite time e!ects are still not controlled in our present simulations. In Fig. 8 we show the "nite time e!ects for a pure pion gas where one can determine the in"nited time limit analytically. The time here is 100 fermis. In Fig. 9 we see that quench conditions enhances signi"cantly the production of low mass dileptons over what one would "nd for a pure pion gas.
5. Conclusions We have shown how to use the CTP formalism to calculate the dilepton spectrum arising from a time evolving or a thermal plasma. For a plasma undergoing a chiral phase transition we expect a strong signal for existence of DCC-states in e>e\-channel M &2m , q (300 MeV. p , This would be visible by CERES if k"60 MeV . , In our calculations we have ignored the possible important e!ects of direct two body scattering in the plasma which arise only in next order in the 1/N approach. A similar enhancement seen by Boyanovsky et al. [15] in photon spectrum. Another problem for us is that our result is in#uenced by large "nite time corrections which are apparent for the free pion gas.
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Acknowledgements The work presented here was done in collaboration with Emil Mottola, Yuval Kluger, Volker Koch, Juan Pablo Paz Ben Svetitsky, Judah Eisenberg, Paul Anderson and John Dawson. This work was supported by the Department of Energy.
References [1] P.A.M. Dirac, Proc. Camb. Phil. Soc. 26 (1930) 376; A.K. Kerman, S.E. Koonin, Ann. Phys. 100 (1976) l332; R. Jackiw, A.K. Kerman, Phys. Lett. A 71 (1979) 158; F. Cooper, S.-Y. Pi, P. Stancio!, Phys. Rev. D 34 (1986) 3831. [2] Y. Kluger, V. Koch, J. Randrup, X.N. Wang, Phys. Rev. D 57 (1998) 280. [3] J. Schwinger, J. Math. Phys. 2 (1961) 407; K.T. Mahanthappa, Phys. Rev. 126 (1962) 329; P.M. Bakshi, K.T. Mahanthappa, J. Math. Phys. 4 (1963) 1; 4 (1963) 12; L.V. Keldysh, Zh. Eksp. Teo. Fiz. 47 (1964) 1515; [Sov. Phys. JETP 20 (1965) 1018]. [4] G. Zhou, Z. Su, B. Hao, L. Yu, Phys. Rep. 118 (1985) 1; R.D. Jordan, Phys. Rev. D 33 (1986) 44; E. Calzetta, B.L. Hu, Phys. Rev. D 35 (1987) 495. [5] K. Wilson, Phys. Rev. D 7 (1973) 2911; J. Cornwall, R. Jackiw, E. Tomboulis, Phys. Rev. D 10 (1974) 2424; S. Coleman, R. Jackiw, H.D. Politzer, Phys. Rev. D 10 (1974) 2491. [6] R. Root, Phys. Rev. D 11 (1975) 831; C.M. Bender, F. Cooper, G.S. Guralnik, Ann. Phys. 109 (1977) 165; C.M. Bender, F. Cooper, Ann. Phys. 160 (1985) 323. [7] J.D. Bjorken, Int. J. Mod. Phys. A 7 (1992) 4189; J.P. Blaizot, A. Krzywicki, Phys. Rev. D 46 (1992) 246; K. Rajagopal, F. Wilczek, Nucl. Phys. B 404 (1993) 577. [8] F. Cooper, Y. Kluger, E. Mottola, J.P. Paz, Phys. Rev. D 51 (1995) 2377; Y. Kluger, F. Cooper, E. Mottola, J.P. Paz, A. Kovner, Nucl. Phys. A 590 (1995) 581c; F. Cooper, Y. Kluger, E. Mottola, Phys. Rev. C 54 (1996) 3298. [9] M.A. Lampert, J.F. Dawson, F. Cooper, Phys. Rev. D 54 (1996) 2213. [10] S. Habib, Y. Kluger, E. Mottola, J.P. Paz, Phys. Rev. Lett. 76 (1996) 4660; F. Cooper, S. Habib, Y. Kluger, E. Mottola, Phys. Rev. D 55 (1997) 6471. [11] F. Cooper, G. Frye, E. Schonberg, Phys. Rev. D 11 (1975) 192. [12] H. Lehmann, K. Symanzik, W. Zimmerman, Nuovo Cimento 1 (1955) 205. [13] L.D. McLerran, T. Toimela, Phys. Rev. D 31 (1985) 545. [14] P.V. Ruuskanen, in: E.H.H. Gutbrod, J. Rafelski (Eds.), Particle Production in Highly Excited Matter, Plenum Press, New York, 1993, p. 593. [15] D. Boyanovsky, H.J. de Vega, R. Holman, S. Kumar, Phys. Rev. D 56 (1997) 5233.
Physics Reports 315 (1999) 83}94
Dual con"nement of grand uni"ed monopoles? Alfred Schar! Goldhaber Institute for Theoretical Physics, State University of New York, Stony Brook, NY 11794-3840, USA
Abstract A simple formal computation, and a variation on an old thought experiment, both indicate that QCD with light quarks may con"ne fundamental color magnetic charges, giving an explicit as well as elegant resolution to the `global colora paradox, strengthening Vachaspati's SU(5) electric}magnetic duality, opening new lines of inquiry for monopoles in cosmology, and suggesting a class of geometrically large QCD excitations } loops of Z(3) color magnetic #ux entwined with light-quark current. The proposal may be directly testable in lattice gauge theory or supersymmetric Yang}Mills theory. Recent results in deeply inelastic electron scattering, and future experiments both there and in high-energy collisions of nuclei, could give evidence on the existence of Z(3) loops. If con"rmed, they would represent a consistent realization of the bold concept underlying the Slansky}Goldman}Shaw `glowa model } phenomena besides standard meson}baryon physics manifest at long distance scales } but without that model's isolable fractional electric charges. 1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Av; 14.80.Hv; 11.27.#d Keywords: Quantum chromodynamics; Magnetic monopoles; Cosmic strings
1. Introduction A quarter century ago, the simultaneous and independent discoveries by Gross and Wilczek [1] and by Politzer [2] that quantum chromodynamics [QCD] is asymptotically free made this theory instantly what it still is } the unique candidate theory for describing the structure and interactions of baryons, as well as the mesons produced when baryons collide. For length scales below 0.1 fm and energy scales above 1 GeV, phenomena may be described accurately by perturbative techniques in terms of elementary quarks and gluons. At longer distances and lower energies, the most useful degrees of freedom become the baryons and mesons themselves, while the connection between these two regimes is less well determined because of the calculational di$culty associated with nonperturbative QCD. Nevertheless, a variety of experimental and theoretical approaches have produced so many successes that it would seem natural to assume there is little or no room for 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 4 - 9
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surprising new phenomena in QCD. That is especially plausible for the perturbative regime, so if there are surprises lurking they most likely will be found at long-distance scales. A rare if not unique proposal in this direction, which constitutes perhaps the boldest enterprise with Richard Slansky's name on it, is the paper of Slansky, Goldman, and Shaw [SGS] [3] suggesting explicit departures from naive QCD expectations at long-distance scales. Their work was stimulated by experimental indications of isolated electric charge with value or that of an electron charge. As Gordon Shaw explains [4], assuming that the manifest SU(3) gauge symmetry of QCD is reduced by a Higgs mechanism at very long-distance scales to SO(3), one may envision isolated SO(3) singlets made of two quarks, and therefore carrying fractional electric charge. While seeking such objects in the laboratory remains a worthwhile challenge, there are serious grounds to be hesitant about the explicit SGS proposal. The reason is that a Higgs mechanism is easy to formulate in a regime where the gauge coupling of the theory is weak, as in the standard model of electroweak interactions, but becomes very hard to interpret if the coupling is strong, which inevitably would be true for QCD on the scale they had in mind. There was little choice about this for SGS, because any shorter distance scale or higherenergy scale with such phenomena would have been prohibited by existing theoretical and experimental knowledge. Even now there is no experimental con"rmation of fractional electric charge, though searches continue [4], nor of any other long-range QCD e!ect. This paper presents a proposal which has signi"cant features in common with SGS, namely, new long-range phenomena beyond ordinary hadron dynamics (including con"nement of fundamental magnetic monopoles), but does not imply fractional electric charge. If the proposal turns out to have merit, then it could well be viewed as a vindication of the essence of SGS. Certainly the intellectual structure they developed was an important in#uence on my thinking. The presentation involves both `pusha and `pulla heuristic arguments, i.e., reasons to suspect the existence of new phenomena as well as appealing consequences which would follow if they occurred, but does not include a proof that they are inevitable. Because of the wide range of application, there likely will be a number of ways to test the proposal, including several outlined later. The focus begins with particles dual to quarks, namely, fundamental magnetic monopoles carrying both ordinary and color magnetic charge. Before plunging into the proposal, we should review some long-range e!ects which already are expected, resulting from hybridization between di!erent scales. If a region of space is su$ciently hot, then the temperature ¹ sets a scale which invokes asymptotic freedom, and thus allows one to describe the properties as if dealing with a gas of free quarks and gluons, commonly known as the `quark}gluon plasmaa. Clearly long-distance correlations in this regime should look quite di!erent from those at ¹"0. Thus, long-distance phenomena are altered in a predictable way, but in a high-energy rather than low-energy regime. Something similar should happen if, for example, compression of cold nuclear matter, as in the interior of a neutron star, were to produce baryon density much greater than that in normal nuclei. This time the high density gives a short-distance and therefore high-energy scale which implies asymptotic freedom and a change in long-distance correlations. Neither of these e!ects would be a surprise. Something a bit closer was the proposal of Ja!e [5] that a six-quark system of strangeness S"2 might be stable against decay to two K particles. This in turn raises a possibility discussed by Witten [6] that electrically neutral strange matter might be stable or metastable. However, even this e!ect if it occurred would be a consequence of relatively
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short-range interactions. Something still closer to what follows is the suggestion [7] that nuclei might have stable or metastable toroidal forms, where a quark-containing tube of color-magnetic #ux is bent into a closed curve, a structure which might exhibit rigidity and incompressibility as well as tension.
2. Heuristic arguments for monopole con5nement 2.1. Gauge invariance Though the existence of a magnetic monopole has yet to be con"rmed by experimental observation, even as a concept this object repeatedly has played the role of an intellectual aqua regia, exposing profound aspects of structure in physical systems. The most noted example is Dirac's realization [8] that the existence of isolable monopoles would require the quantization of electric charge q and magnetic charge g, through the quantum condition that the product of q with g is proportional to an integer. Dirac monopoles `inserteda into QED give a model for con"nement, because emanating from an elementary pole inside a Type II superconductor with its electron-pair condensate must be two strings of superconductor-quantized magnetic #ux, each terminating only on an antipole. If one pair of pole and antipole were slowly separated, clearly the two strings coming out of the pole would terminate on the antipole, implying a con"ning string tension holding the two together. By analogy, if the vacuum of QCD without light quarks comprised a monopole condensate [9,10], this then would con"ne heavy quarks. However, if there are elementary quarks light on the scale of K , then there is no con"nement of heavy quarks: Instead pair creation of light quarks allows /!" heavy-quark-containing mesons to be separated with no further energy cost. A general argument of 't Hooft [11] for QCD without light quarks shows that either heavy elementary quarks or heavy fundamental monopoles should be con"ned, but not both. If this argument still applied in the presence of light quarks, then monopole con"nement would be a triviality. In any case this makes it clear that there would be nothing obviously inconsistent about such con"nement. To understand why it might be expected, let us examine in the more familiar superconductor case the issue of screening, and what charges can or cannot be screened. In electrodynamics it is useful to distinguish two di!erent kinds of conserved charge, local or Gauss-law charge and Aharonov}Bohm [AB] or Lorentz-force charge [12]. Although the local charge of an electron-quasiparticle is completely screened inside a superconductor, the AB charge cannot be screened [13,14], because of the reciprocity requirement that an AB phase of p must occur whether a quasiparticle is di!racted around a #uxon (i.e., a superconductor quantum of #ux) or a #uxon is di!racted around a quasiparticle. If the e!ect on the #uxon is to be described by a local interaction, evidently the AB charge is not screened. Now let us look at the same issue for a fundamental monopole in QCD. The monopole may be characterized as the source of a Dirac string carrying color magnetic #ux which would produce an AB phase 2p/3 for a fundamental quark di!racted around it. Of course, in addition to this color #ux there must be an ordinary magnetic #ux in the string yielding a phase 4p/3 mod(2p). The fractional color #ux in the string implies that there must be a net nonzero color magnetic #ux coming out of the pole. A monopole whose Dirac string would carry full 2p color #ux has no such
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consequence, because that could be exactly compensated by an `adjointa monopole made from a classical con"guration of purely SU(3) gauge "elds. Thus, in the sense just de"ned, adjoint monopoles can be screened but fundamental monopoles cannot (a suggestive analogue to what happens with adjoint gluons and fundamental quarks). In the absence of dynamical quarks, this lack of screening may not matter, because vacuum #uctuations in the form of loops carrying #ux 2p/3 occur easily on arbitrarily large length scales, so that the magnetic charge is not de"nable as an eigenvalue. In this sense, it may be screened just like electric charge in a normal metal, i.e., with mean value zero but such large #uctuations that it is not de"ned as a sharp quantum observable. Thus the monopole charge, rather than being screened or compensated, may be hidden in much the same way a needle becomes invisible inside a haystack. Here is another perspective: In the theory with only adjoint "elds, such as those of the gluons, the gauge symmetry is SU(3)/Z(3), so that an arbitrarily thin tube of Z(3) magnetic #ux would be invisible even by the AB e!ect to all elementary excitations, hence could not excite the vacuum, and therefore need not carry an observable energy per unit length, as would have to be true for an observable string. On the other hand, once quarks are present, there could be a nonvanishing string tension for loops of Z(3) color magnetic #ux, so that geometrically large quantum #uctuations of these loops should be suppressed. A reason for suspecting this is that now the AB e!ect would make even the thinnest tube visible for those quark trajectories which surround the tube. Thus, the fractional color magnetic charge could become sharp, meaning that an observable color magnetic "eld, con"ned to a tube of "xed radius, emanates from the monopole out to in"nity. More formally, because now there are particles in the fundamental representation of SU(3), the full gauge symmetry applies, and so a nonzero Z(3) color magnetic #ux out of a fundamental monopole is at least potentially observable. 2.2. Ideal experiment Here is a thought experiment suggesting the same conclusion. Imagine a hadron such as a proton at rest near an SU(5) monopole, with its ordinary as well as color magnetic charge. If a deeply inelastic electron scattering sends a quark out of the proton with very high momentum parallel to that of the incident electron, then the quark's evolution in the beam direction can be described perturbatively for a time proportional to that momentum, which means the ordinary magnetic "eld of the monopole will de#ect it in such a way that only a fraction of a quantum of angular momentum will be transferred to the quark. This is inconsistent with conservation and quantization of angular momentum [15]. The analysis also can be carried out in the rest frame of the fastest "nal hadron. In this frame a perturbative computation is accurate for a "xed time of order 1 fm/c. However, the magnetic "eld of the monopole #ashes by the quark in a much shorter time because of the Lorentz contraction of the "eld con"guration, and therefore again there is a de"nite, but fractional transfer to the quark of angular momentum projected along the beam direction. One way to restore consistency is to assume that there is also a spherically symmetric color magnetic "eld, so that the combined "elds always transfer an integer number of angular momentum units to the quark. However, that assumption directly contradicts the most basic understanding of QCD, which requires a mass gap for color-carrying excitations, so that a longrange, `classicala, isotropic, color-magnetic "eld is impossible. How can these two requirements, of
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nonscreening and yet no isotropic long-range "eld, be reconciled? An obvious if not unique way to avoid the dilemma is by `escape into asymptotic freedoma: Color "elds make sense in the high-energy, short-distance, perturbative regime, so if the magnetic #ux comes out in a tube with radius of scale 4K\ it is consistent with knowledge about the low-energy behavior of the theory, /!" and at the same time satis"es the requirement of nonscreening. Evidently such a tube must have a "nite tension, so that the energy of a pole}antipole pair connected by the tube must rise linearly with separation, and this implies con"nement of fundamental monopoles. From the viewpoint of the deeply inelastic `thought experimenta, why shouldn't the monopole con"nement argument apply even if there are no dynamical quarks? In that case heavy `externala quark sources certainly are con"ned, and the failure of angular momentum quantization for a single pole}quark pair is acceptable, as there is never a single isolated quark moving in the "eld of the monopole. While the above arguments might be appealing, they surely do not constitute a proof of monopole con"nement. The reason is that even with light quarks it may be that large loops of magnetic #ux, at least of a certain cross sectional radius, still have arbitrarily low energy, in which case they would be part of the vacuum structure rather than physical excitations. Then the net #ux out of a monopole again would be hidden by vacuum #uctuations. However, because the quarks would be sensitive to arbitrarily thin tubes even with Z(3) #ux, there is now a much stronger constraint, from below as well as above, on the acceptable radii for #ux tubes with very low energy. Both because the thought experiment was the germinating element in my own thinking on this subject, and because more careful examination could tend either to strengthen or to weaken the argument, it seems worthwhile to focus more explicitly on the wave function evolution entailed by this process. In the presence of gauge "elds, the conventional (nongauge-invariant) momentum of an object whose charges couple to these "elds becomes unde"ned. Thus, in the directions transverse to the very high momentum of the struck quark, it makes no sense to think about the momentum of the quark by itself. However, the correlated wave function of the quark and the associated slower remnants might have a well-de"ned wave function in transverse momentum p , 2 a wave function which initially would be strongly peaked at p +0 and azimuthal angular 2 momentum about the beam direction also zero. Then the essential idea is that, absent any contribution from color magnetic "elds, the only e!ect feeding some change in this azimuthal angular momentum would be coming from the scattering of the fast quark on the ordinary magnetic "eld of the monopole. A fractional value for this angular momentum transfer gives the conclusion that something is inconsistent about this picture, and leads by elimination of alternatives to the inference that an observable string of color magnetic #ux emanates from the monopole. If we accept that inference, how do we "nd consistency restored? As an example, imagine that the observable string comes out of the monopole in the direction parallel to the fast quark momentum. We still may use gauge invariance to place the Dirac string of ordinary plus color #ux anywhere we like, and thus may choose it along the observable color string. In this case, for all except those trajectories which penetrate the observable, "nite-thickness string, the e!ective "eld is just that of a pole which is one end of a solenoid with ordinary magnetic #ux such that a quark going around it acquires a fractional phase 2p/3 mod 2p. Evidently, mesons or baryons generated by fragmentation of the fast quark will always have integer azimuthal angular momentum, but nevertheless the initial e!ect of fast passage of the quark by the monopole will be to generate a fractional change in the net
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angular momentum of the entire system interacting with the pole, something now allowed because an observable string with fractional magnetic #ux is present.
3. Consequences and applications Now let us look at how fundamental monopole con"nement would reorient perspectives on a variety of issues. 3.1. Paradox of `global colora A number of authors addressed the problem of generalizing a collective}coordinate quantization technique, accepted as describing the electric or `dyona charge of an SU(2) monopole, to the case of the SU(5) monopole [16}19]. They found that for the unbroken SU(3) of color the dyon charge of an isolated pole is not de"ned } an e!ect reminiscent of spontaneous symmetry breaking as in ferromagnetism. Evidently if monopoles with fundamental color charge are con"ned, this problem simply disappears. A more general and straightforward comment is that, with or without monopole con"nement, the paradox is ill-posed, because the collective}coordinate method has been used to quantize zero modes of the monopole placed in a perturbative QCD vacuum, which de"nitely is an incorrect description of the lowest-energy degrees of freedom on length scales large compared to K\ . Thus, while monopole con"nement eliminates the problem at the very beginning, the /!" signi"cance of that resolution perhaps is diminished because there might well not be such a problem if the right vacuum were understood well enough to be implemented for the analysis. 3.2. Electric}magnetic duality in a grand unixed model Recently Vachaspati [20,21] has described a remarkable duality of SU(5), clearly relevant for any grand uni"ed theory. The fundamental monopole is part of a family of tightly bound states, with magnetic charges 1, 2, 3, 4, and 6 times the fundamental charge. These "ve states can be identi"ed as dual partners of the "ve fundamental fermions in SU(5), three quarks, a lepton, and a neutrino. There is a possibly deep or possibly just technical issue, that the charge-2 state should be identi"ed with an antiquark. There are two other di$culties. First, the monopoles appear to be spinless, while the fermions of course have spin-. This problem arose already with the original Montonen} Olive proposal of duality between monopoles and gauge bosons [22,23], and eventually found two resolutions. One is to introduce supersymmetry, so that both monopoles and the dual elementary particles come in families with the same range of spins [24}28]. The other, acknowledging the possibility in principle of making a perfect correspondence through supersymmetry, is to be satis"ed with what might be called `virtual dualitya } a symmetry applied to all properties except spin. Whichever approach one prefers, with respect to this issue Vachaspati's system is in the same category as the older examples. The other di$culty [20,21] is that the e!ective long-range couplings of the monopoles and their dual partners are identical, except that quarks are con"ned, whereas previous discussions suggested that the colored monopoles are not. For this reason Vachaspati considered introducing the con"nement essentially by hand. The argument above that the monopoles which nominally carry
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nontrivial Z(3) color magnetic charge are automatically con"ned gives a way to perfect Vachaspati's duality, lending additional interest to pursuing it further. One side note: Con"nement could lead to loosely bound `baryonsa, but these then could collapse to the tightly bound `leptonsa already identi"ed in Vachaspati's scheme. Clearly, this is di!erent from the separate baryon and lepton conservation laws which apply at low energy scales, but as one expects those laws not to hold for particles on energy scales approaching the monopole masses this may well be a consistent result. Monopole evolution in cosmology Monopoles formed on a mass scale signi"cantly higher than the mass scale for in#ation would have disappeared during in#ation [29]; indeed, that is one of the attractive features of in#ationary models. However, lighter monopoles would need some other mechanism to explain why we do not see abundant evidence of their existence today. Many such mechanisms have been proposed, up to quite recent times. One possibility is that the dynamics at some intermediate era between monopole formation and the present would make the poles unstable, allowing them to disappear, even though any remnant which did survive would be stable now [30,31]. If monopoles were created at some early epoch and not swept away meanwhile, then the only way to explain their scarcity today would be by con"nement, exactly the phenomenon discussed here. How would that work? If monopoles were formed above the QCD phase transition expected at a temperature of order K , then con"nement below that transition would result in attachment /!" of Z(3) strings to each pole, either a single outgoing 2n/3 string, or two outgoing !2p/3 strings, with the opposite arrangement for antipoles. A pair connected by a single string likely would have disappeared by now, thanks to dissipative forces leading to gradual collapse and annihilation. On the other hand, a large loop with alternating negative and positive #ux connecting alternating pole and antipole could be much more durable. This kind of `cosmic necklacea, with the poles as `beadsa, was suggested by Berezinskii and Vilenkin [32] as a possible source of the highest-energy component of the cosmic-ray spectrum, through occasional annihilations of poles and antipoles, which might for example slowly drift together by sliding along the string. The evolution of networks of such strings is an interesting and nontrivial problem, which could be studied once the basic couplings associated with string crossings were determined. In particular, in principle a `fusiona of three strings converging together should be possible, which would allow three monopoles to be connected to each other in a dual version of a baryon. However, if this could happen easily then the problem of too many monopoles would be restored, so a crucial question is whether there is a substantial inhibition of such fusion.
4. New phenomena in QCD at accessible scales 4.1. Theoretical aspects Even though it is consideration of heavy, fundamental monopoles and their interactions which has led here to the suggestion that they would be held together by Z(3) color #ux strings, that statement clearly has a consequence for phenomena at much lower scales than the monopole mass. It means that even in the absence of such poles QCD must support excitations consisting of loops
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of color magnetic #ux, with the mass of a loop being proportional to its circumference. The loops would be unstable against shrinkage, but would give an interesting and nontrivial structure of QCD excitations on a length scale large compared to 1/K . This is reminiscent of the /!" Slansky}Goldman}Shaw proposal to explain experimental reports of fractional electric charge [3]. As mentioned earlier, they noted that if a Higgs mechanism at energy scales below, or length scales above, the scale associated with K could operate to reduce SU(3) of color to SO(3) of `glowa, /!" then diquarks could exist in isolation, and of course would carry fractional electric charge. Shortly after, Lazarides, Sha", and Trower [LST] [33] observed that such a Higgs e!ect automatically would imply con"nement of fundamental monopoles exactly like that argued above. As was also mentioned earlier, there is no natural starting point from which the phenomena of this particular Higgs mechanism could be deduced in a perturbative framework, the only recognized way to do it. This criticism applies equally to the deduction by LST. Of course, the fact that a conceivable route to a particular result turns out to be rocky and uncertain does not mean the result itself is necessarily wrong, only one still lacks evidence that it is right. Here the issue has been approached from the other end, and fundamental monopole con"nement derived. This does not necessarily imply the isolability of fractional electric charge or the screening of some QCD color-electric "elds, but it certainly does say there must be a new feature of QCD at large length scales, namely, loops of color-magnetic #ux, just as indicated by LST. Without light quarks, heavy quark con"nement implies loops of color-electric #ux, so familiar pictures would not be changed so enormously, just `dualizeda. This means that the change in structure of QCD as the mass of light quarks passes from above to below K would be quite subtle: Above there would be /!" at least metastable color-electric but substantial suppression of color-magnetic strings (more accurately, very low magnetic string tension), and below something more like the opposite would be true. The meaning of con"nement or non-con"nement needs a bit more attention. In terms of a four-dimensional euclidean path integral, con"nement is associated with exponential suppression, with the area of an appropriate loop appearing in the exponent, as opposed to e!ects associated with widely separated "nite-mass excitations, in which case only the length (perimeter) of the loop appears. For QCD with dynamical quarks, su$ciently large loops must exhibit a perimeter law, but the coe$cient of the perimeter term itself falls exponentially with quark mass because the tunneling leading to quark pair creation is exponentially suppressed. Thus a visible transition on a "nite lattice from area to perimeter law occurs at some "nite mass, presumably of order K , and should be rather smooth. For the proposed monopole con"nement, with mono/!" poles expected to be extraordinarily massive, the breaking of strings by monopole pair creation should be impossibly rare for observation on any "nite lattice. If the magnetic strings only exist for "nite quark mass, it becomes a delicate question exactly how the string tension depends on that mass. However, again one would expect a smooth transition, with the maximum tension approached for quark mass below K . This leads to the amusing conclusion that fundamental /!" dyons carrying both monopole and quark charges might exhibit an e!ective con"nement with very weak dependence on quark mass. If all this were con"rmed, it would be a vindication of the essential claim of SGS for nontrivial manifestations of fundamental QCD degrees of freedom at large length scales. These colormagnetic-#ux-loop excitations presumably should be an important class of what have been called `glueballsa, which likely would be drastically di!erent in character from what one would "nd in
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QCD without light quarks, and would NOT be pure glue, as the light quarks must be an essential part of their structure. As already stated, the fact that con"nement of fundamental monopoles would be equivalent to the existence of Z(3) magnetic #ux strings means that there is a way to test this proposal in familiar energy regimes of QCD. In particular, as lattice calculations grow steadily better at taking account of light quark degrees of freedom, it should become possible to study this issue on the lattice and obtain credible results. The best way to formulate the problem might be to insist a` la Wu and Yang [34] that along a straight line between monopole and antimonopole there is a gaugematching between vector potentials outside and inside the smallest plaquettes surrounding that line, involving one unit of Z(3) color #ux, and one Aharonov}Bohm unit of ordinary magnetic #ux. This means a phase of 2p associated with those plaquettes for u quarks encircling them, but 0 for d quarks. Of course, in all other respects there is a standard Dirac electromagnetic monopole vector potential for the pole}antipole system. All this implies, as stated earlier, that there must be a net Z(3) color magnetic #ux between monopole and antimonopole. If that #ux were observable and not hidden, then monopole con"nement would follow, and would be signaled by an area law for exponential suppression of monopole loops in the path integral, associated with the product of the separation between pole and antipole and the Euclidean time duration of that separation. There might be an analytic approach to determining whether or not con"nement occurs, a!orded by recent progress in studying supersymmetric nonabelian gauge theories [27,28,35,36]. In these theories it is often possible to make precise conjectures about the particle spectrum, and to verify the conjectures not by a direct proof but rather by subjecting the proposed forms to many di!erent consistency checks, and "nding that all are passed. To do this for our problem would require starting with at least an SU(5) theory (including a hypermultiplet containing quarks and leptons), and following an elaborate sequence of Higgs mechanisms to break the manifest symmetry down to SU(3) ;;(1) . This is surely much more complicated than anything which has been done so far with such systems, but might nevertheless be manageable. 4.2. Experimental aspects As physics is an experimental science, it surely is worth considering how the new kind of structure proposed here might be accessible to experimental observation. Up to this point in the paper, the main speculation has been the unproved proposal that Z(3) #ux loops may exist. To connect that with experiment entails more speculation. Conventional hadron collisions are not promising. First of all, any frequently occurring peculiar phenomena in such processes would have been noted already. Secondly, because Z(3) strings cannot break by creation of light-quark pairs, their coupling to conventional hadrons should be weak. This implies that they would not be generated easily in typical collisions. What couplings would be possible? Because u and d quark vacuum currents would circulate oppositely around the string, there should be a o -meson magnetic coupling `contact terma } i.e., only acting on sources which themselves overlap geometrically with the string. Thus a contracting string could release energy through emission of o mesons, but as these are rather massive there would be a poor match, between the likely small energy release in the contraction from one loop energy eigenstate to a lower one and the large mass of the emitted particle. All this implies a quite substantial lifetime for a large loop before collapse.
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Slow decay almost invariably goes together with low production rates, and helps to explain why even if they can exist Z(3) loops would not have leapt to our attention. Recently, experiments on deeply inelastic electron}proton scattering [37,38] have been interpreted as indicating a `hard-pomerona contribution to the reaction [39}41]. By familiar reasoning of Regge duality, such an e!ect should be associated with a new class of glueball excitations [39]. Could these new glueballs be the magnetic loops proposed here? If so, then it would not be strange if processes described by the hard pomeron also produced the loops. Perhaps detailed exclusive or semi-inclusive studies of such events would reveal structure related to the loops, formed as geometrically large and therefore high-energy excitations. It then becomes interesting to consider what kind of signal such an object would generate, but that is not easy to determine. All features of closed-string dynamics, many still obscure despite all the years of string studies, would appear to be relevant for the behavior of these Z(3) loops. Thus some caution is needed in guessing what should happen in these scattering processes. This time of course the coupling leading to production would be electromagnetic, but again would involve a contact interaction which at lowest order in momentum transfer would be to the anapole or toroidal moment of the #ux loop. This suggests that at the moment of appearance the loop would be quite small in size, but then could expand. If such primitive thinking covers the main features, then it becomes possible to suppose that in a suitable frame boosted along the beam direction there would be a fairly large isotropic ensemble of pions. Because of the decay energy mismatch mentioned earlier, the pions might be quite limited in their range of momenta. Such an e!ect could be quite striking, and very di!erent from typical results of deeply inelastic scattering. A di!erent picture, analogous to bremsstrahlung of photons, would be that with small probability virtual Z(3) loops exist in the neighborhood of the incident proton, and these are made real by the absorption of the highly virtual photon. To avoid enormous form-factor suppression, in a suitable Breit frame the initial and "nal momenta of the loop would both have to be large, implying Lorentz contraction which compensates for spatial oscillation of the phase factor in position space, an e!ect discussed some time ago for elastic scattering on deuterons [42]. If deeply inelastic electron}proton scattering gives an indirect hint of new long-distance dynamics in QCD, plus the potential to provide more direct evidence, then very high-energy nucleus}nucleus collisions at least have the possibility of generating such evidence in processes with large rate. By heating substantial volumes above the QCD phase-transition temperature, such collisions could permit formation of Z(3) loops, and if so their slow decay during the cooling process would give a characteristic signal, providing important evidence that quark}gluon plasma had formed. If the idea of slow decay is right, this would allow a loop to escape from the dense, highly excited formation region, and then to populate a small volume in pion momentum space with a large number of particles. While these thoughts about experimental signals are vague and sketchy, it may well be possible with further study to make more precise statements. What seems likely to be unchanged is the fact that a geometrically large object of high coherence, which decays slowly in small energy steps, will produce a signal di!erent from any more familiar system, including a large nucleus. [Of course, if the objects turned out to be at least stable against small perturbations, the signal would be even more striking.]
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5. Conclusions In the title the phenomenon proposed here is referred to as `dual con"nementa. Of course, this makes sense because familiar color-electric con"nement of heavy quarks is replaced by colormagnetic con"nement of heavy monopoles, but if the proposal is correct as stated then something deeper is at work. Usual discussions posit a duality between superconducting screening of one kind of charge and con"nement of the other. Here, however, the screening of color-electric charge is more powerful even than that by a superconductor, because for every heavy quark there is an attached light antiquark, exactly screening not only the local charge but also the Aharonov}Bohm charge. Total screening of the color electric charge carried by heavy quarks is the remnant of the heavy-quark con"nement which exists without light quarks. Thus the duality would be one between con"nement of fundamental color-magnetic monopoles and total screening of heavy quark color charge, not inconsistent with the familiar version but nevertheless clearly di!erent. If found, such a duality therefore would be something new. It is enticing to think that physics research is now at a stage where within a short time there might be direct evidence from a variety of directions on whether Z(3) strings occur in nature. Lattice gauge theory or supersymmetric gauge theory could give information, as could deeply inelastic electron scattering or high-energy nucleus}nucleus scattering. A positive answer would provide a "rm foundation for the theoretical and cosmological applications explored above. Perhaps even more satisfying if this happened would be the realization of a remarkable new consequence of QCD. This suggests a further challenge: Are there any other possible ways in which QCD could really give us a surprise? Not easy or obvious, but surely worth looking!
Acknowledgements This study was supported in part by the National Science Foundation. I have bene"ted over a period of time from conversations with Martin Bucher, Georgi Dvali, Edward Shuryak, Mikhail Stephanov, and Tanmay Vachaspati. Richard Slansky was a valued friend and colleague from student days on } about 40 years. Although we never collaborated on a paper, it was always a pleasure to discuss with him, and to experience his intelligent and discriminating enthusiasm for physics. His courage in facing physical challenges (in all senses!) was inspiring. Truly the word `glowa was as descriptive of his luminous personality as of the beautiful SGS idea. References [1] [2] [3] [4] [5] [6] [7] [8]
D.J. Gross, F. Wilczek, Phys. Rev. Lett. 30 (1983) 1343. H.D. Politzer, Phys. Rev. Lett. 30 (1983) 1346. R. Slansky, T. Goldman, G.L. Shaw, Phys. Rev. Lett. 47 (1981) 887. G.L. Shaw, unpublished (1998). R. Ja!e, Phys. Rev. Lett. 38 (1977) 195; 38 (1977) 617 (Erratum). E. Witten, Phys. Rev. D 30 (1984) 272. L. Castillejo, A.S. Goldhaber, A.D. Jackson, M.B. Johnson, Ann. Phys. 172 (1986) 371. P.A.M. Dirac, Proc. Roy. Soc. London, Ser A 133 (1931) 60.
94 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42]
A.S. Goldhaber / Physics Reports 315 (1999) 83}94 S. Mandelstam, Phys. Rep. C 23 (1976) 245. G. 't Hooft, Nucl. Phys. B 138 (1978) 1. G. 't Hooft, Nucl. Phys. B 138 (1978) 1. A.S. Goldhaber, S.A. Kivelson, Phys. Lett. B 255 (1991) 445. B. Reznik, Y. Aharonov, Phys. Rev. D 40 (1989) 4178. A.S. Goldhaber, R. MacKenzie, F. Wilczek, Mod. Phys. Lett. A 4 (1989) 21. A.S. Goldhaber, Phys. Rev. B 140 (1965) 1407. A. Abouelsaood, Nucl. Phys. B 226 (1983) 309. A.P. Balachandran, G. Marmo, N. Mukunda, J.S. Nilsson, E.C.G. Sudarshan, F. Zaccaria, Phys. Rev. Lett. 50 (1983) 1553. P. Nelson, A. Manohar, Phys. Rev. Lett. 50 (1983) 943. P. Nelson, S. Coleman, Nucl. Phys. B 237 (1984) 1. T. Vachaspati, Phys. Rev. Lett. 76 (1996) 188. H. Liu, G.D. Starkman, T. Vachaspati, Phys. Rev. Lett. 78 (1997) 1223. C. Montonen, D. Olive, Phys. Lett. B 72 (1977) 117. P. Goddard, J. Nuyts, D. Olive, Nucl. Phys. B 125 (1977) 1. E. Witten, D. Olive, Phys. Lett. B 78 (1978) 97. H. Osborn, Phys. Lett. B 83 (1979) 321. A. Sen, Int. J. Mod. Phys. A 9 (1994) 3707. N. Seiberg, E. Witten, Nucl. Phys. B 426 (1994) 19; B 430 (1994) 485 (Erratum). N. Seiberg, E. Witten, Nucl. Phys. B 431 (1994) 484. A.H. Guth, Phys. Rev. D 23 (1981) 347. P. Langacker, S.-Y. Pi, Phys. Rev. Lett. 45 (1980) 1. G. Dvali, H. Liu, T. Vachaspati, Phys. Rev. Lett. 80 (1998) 2281. V. Berezinskii, A. Vilenkin, Phys. Rev. Lett. 79 (1997) 5202. G. Lazarides, Q. Sha", W.P. Trower, Phys. Rev. Lett. 49 (1982) 1756. T.T. Wu, C.N. Yang, Phys. Rev. D 12 (1975) 3845. K. Intriligator, N. Seiberg, Nucl. Phys. Proc. (Suppl.) 45BC (1996) 1. M. Shifman, Prog. Part. Nucl. Phys. 39 (1997) 1. C. Adlo! et al., Nucl. Phys. B 497 (1997) 3. J. Breitweg et al., Phys. Lett. B 407 (1997) 432. A. Donnachie, P.V. Landsho!, Phys. Lett. B 437 (1998) 408. D. Haidt, Proceedings of the Workshop on DIS Chicago, April 1997, AIP, 1997. N.N. Nikolaev, B.G. Zakharov, V.R. Zoller, JETP Lett. 66 (1997) 138. H. Cheng, T.T. Wu, Phys. Rev. D 6 (1972) 2637.
Physics Reports 315 (1999) 95 } 105
Is theoretical physics the same thing as mathematics?夽 George Chapline Lawrence Livermore National Laboratory, Livermore, CA 94551, USA
Abstract The growing realization that the fundamental mathematical structure underlying superstring models is closely related to Langlands' program for the uni"cation of mathematics suggests that the relationship between theoretical physics and mathematics is more intimate than previously thought. We show that quantum mechanics can be interpreted as a canonical method for solving pattern recognition problems, which suggests that mathematics is really just a re#ection of the fundamental laws of physics. 1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.!w
1. Introduction One of the perennial mysteries of theoretical physics is why the laws of physics should so often have an elegant mathematical formulation } a circumstance often referred to as the `unreasonable e!ectiveness of mathematicsa [5]. In fact, natural phenomena very often exhibit regularities that from the mathematical point of view seem to involve especially unique and beautiful mathematical structures. A good example of this is provided by the apparently strong likelihood [1] that the parity violation observed in the weak interactions has its origins in the naturally chiral nature of a supersymmetric E gauge theory in 10 dimensions. Indeed the implied involvement in elementary particle physics of the exceptional Lie group E provides a connection between fundamental physics and various remarkable mathematical structures including octonians, selfdual lattices, perfect error correcting codes, Kummer surfaces, and non-standard Euclidean
夽
Based on a talk given at the Dick Slansky Memorial Symposium, May 21, 1998. E-mail address: [email protected] (G. Chapline) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 5 - 0
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spacetimes. Furthermore, the discovery of the anomaly structure of superstrings [2] implies [3] that the appearance of E in elementary particle physics is closely related to the theory of automorphic forms, which in turn suggests that the search for the fundamental laws in physics may be related to Langland's program for the uni"cation of mathematics. Apparent connections of the mathematical structure underlying superstring theories with the Monster sporadic group [4] and with number theory provide further evidence for a relationship between fundamental theoretical physics and Langlands' idea that there are fundamental structures related to automorphic forms which unify the various branches of mathematics. What is one to make of all this? My answer is that theoretical physics and mathematics are fundamentally the same thing. An obvious objection that one could easily imagine would immediately occur to anyone contemplating the idea that mathematics and theoretical physics are the same is that a great deal of mathematics was, or at least might have been, created by human beings to solve problems that have nothing to do with physics. For example, the di!usion equation might have been discovered not by a theoretical physicist seeking to describe the #ow of heat in a solid, but by an arbitrageur seeking to protect his employer's "nancial position in some asset. Indeed, it is often claimed now a days that mathematics is just a social}cultural}historical construction [6]. Arrayed against this view of mathematics as just a cultural phenomenon, though, is the powerful argument that the physical world as an embodiment of mathematical structures existed long before there were societies or even humanoids. There is little doubt, for example, that the planets were tracing out almost perfect ellipses long before there were any humans to observe the planets. Of course, with the exception of some isolated instances, such as the invention of the calculus, one might well question whether on the whole the mathematical regularities discovered in nature by theoretical physicists have any profound signi"cance for mathematics itself. In particular, a seemingly formidable obstacle to the idea that theoretical physics and mathematics are same is the fact that the core of theoretical physics } namely, quantum mechanics } has not played an important role in the development of pure mathematics. It is true that a number of "rst rate pure mathematicians have been fascinated with the formalism of quantum mechanics, and have made signi"cant contributions to the development of quantum mechanics. It is also noteworthy that the study of von Neumann algebras led to some surprising advances in knot theory [7] and supersymmetric quantum mechanics has been applied to topology [8]. Nonetheless on the whole the signi"cance of ordinary quantum mechanics for pure mathematics itself has remained obscure. Furthermore, and in my opinion, this is perhaps the most damaging criticism of the proposition that theoretical physics and mathematics are the same, ordinary quantum mechanics has not yet to this day found any practical applications unrelated to the microscopic properties of physical systems. On the other hand, there are some recent developments which promise to make it more di$cult if not impossible to take the position that quantum mechanics is just applied mathematics. First of all, it has been noticed that phase space quantization can be thought of as the geometric quantization of the =(R) group [9,10]. Secondly, the discovery that quantum computers could in principle be used to factor large numbers [11] and quickly search databases [12] o!er the hope that quantum mechanics will, in the not too distant future, be used to solve practical problems completely unrelated to microscopic physics. These developments suggest that, quite apart from its traditional role of providing a description for microscopic physical phenomena, quantum mechanics can also be given a purely mathematical interpretation. Our main objective in this paper will be
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to make a plausible case that indeed quantum mechanics might actually have been developed as a purely mathematical theory. As a bonus our argument will lead us to a provocative way of viewing the relationship between mathematics, theoretical physics, and neurobiology. Our basic argument is that quantum mechanics can be regarded as a fundamental theory of distributed parallel information processing and pattern recognition. It is worth noting in this connection that both Shor's [11] and Glover's [12] quantum mechanical algorithms depend on the unique ability of quantum mechanics to carry out parallel computations, and that Glover's database search algorithm can also be thought of as a pattern recognition algorithm. That quantum mechanics might be regarded as an underlying theory for conventional pattern recognition techniques, though, is to the author's knowledge a completely new idea, and will be the focus of this paper. However, some of the more sophisticated practitioners of the art of pattern recognition may have at least subliminally recognized that there might be such a connection. One of the fundamental problems of pattern recognition is feature vector quantization [13]; but despite an enormous e!ort the practical methods that have been developed to solve this problem remain largely ad hoc. Quantum mechanics, on the other hand, o!ers the possibility of a natural approach to vector quantization. Not unrelated to the problem of vector quantization is the problem of providing a physical de"nition of information [14]. Again quantum mechanics provides a natural solution to this problem; namely, the information that can be carried by a physical system is precisely determined by the dimension of Hilbert space [15]. Quantum mechanics also provides a natural theory of information #ow in a distributed system [16], and a rigorous bound on the rate of transmission of information [17]. It might be noted that in a sense these relatively recent developments concerning the physical properties of information were anticipated in the 1920s, when it was recognized that quantum mechanics could solve the long-standing problem of calculating the entropy constant of a gas. The rule of thumb that was eventually developed [18] was to divide phase space in `cellsa each of whose volume was equal to D, where f is the number of degrees of freedom. Later a rigorous way to quantize phase space, based on the Born}Jordan quantum mechanics, was developed by Wigner [19] and Moyal [20]. Because of its fundamental connection with information theory one might guess that the Wigner}Moyal formulation of quantum mechanics would be a good place to start when looking for a far-reaching interpretation of quantum mechanics as a theory of pattern recognition. In fact, we will show in the following that when the phase space to be quantized is a curved surface, Wigner}Moyal-like quantization not only leads us to a new way of formulating of quantum mechanics, but also a remarkable insight into why quantum computers should be so useful for pattern recognition. In the next section we review the procedure introduced in Ref. [10] for quantizing the parameterization of a curved surface. This generalization of the Wigner}Moyal phase space quantization formalism can also be given the `physicala interpretation as the Weyl quantization of a complete holographic representation of the surface. In particular, we note that in a paraxial ray-type approximation for the electromagnetic "eld the usual mode variables for the electromagnetic "eld can be replaced by "eld variables E(x) and E>(x) whose variation represents the transverse structure of a hologram. These "eld variables depend on both the position on the hologram and the orientation of the illuminating laser beam. The great advantage of using these variables is that quantization amounts to replacing the classical variables E(x) and E>(x) with
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ordinary creation and annihilation operators. This procedure is most interesting in the case of Riemann surfaces with non-trivial topology. We point out that representing a Riemann surface holographically amounts to a pedestrian version of a mathematically elegant characterization of a Riemann surface in terms of its Jacobian variety and associated theta functions. Applied to Riemann surfaces the end result of our holographic approach to quantization is a natural representation of the phase space for quantum systems with a "nite dimensional Hilbert space in terms of theta functions. While this representation is not in itself a new result } indeed it is equivalent to using the well known generalized coherent states for an SU(N) Lie algebra } our way of relating this representation to holograms of a Riemann surface points us in the direction of a new model for quantum mechanics. In Section 3 we make use of the connection between holographic representations of Riemann surfaces and "nite dimensional Hilbert spaces to formulate what is in e!ect an entirely new way of looking at quantum dynamics. Speci"cally we consider the problem of changing the shape of the Riemann surface so that if the relative phase of the illuminating laser beam at di!erent locations on the surface is varied, the holographic representation for the surface is unchanged. In the case of classical holograms this feedback control problem is equivalent to the classical principle of least action. However, when photon noise is introduced, we make use of results originally derived by Dyson [21] in the context of optimizing the performance of active optical systems to show that the problem becomes equivalent to the multi-channel quantum mechanical scattering theory of Newton and Jost. In Section 4 we discuss how this result may be related to a theory underlying superstring models, and suggest why it may lead to an explanation of the `unreasonable e!ectiveness of mathematicsa.
2. Quantum holography of Riemann surfaces As a generalization of the basic problem of quantizing a #at two-dimensional phase space parameterized by classical momentum and position variables p and q we now turn to the problem of quantizing the parameterization of a curved two-dimensional surface. In the case of a Riemann surface the curved surface can be represented by a collection of #at sheets glued together along branch cuts. Thus in this case the problem of quantizing parameterizations of the surface would appear to be very similar to the Moyal problem of quantizing #at p, q phase space, except that now the phase space quantizations on each sheet must be matched along the branch cuts. One might guess that such a system could be quantized by introducing a set of p, q variables with an index j which represented which sheet one was on. The usual Weyl operators a"p#iq and a>"p!iq are now replaced with sets +a , and +a>,. Thus the original Weyl}Heisenberg group will be H H replaced by a Lie group generated by operators of the form , , aHa, aHa and aa>, a a> . H H H H H H States playing much the same role as the coherent states that play such an important role in quantum optics [22] will be generated by the operators D(a)"exp(aa>!aHa)
(1)
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which are analogous to the displacement operators D(m, g). These operators obey the multiplication rule D(a)D(b)"e ' ?@M D(a#b) .
(2)
Replacing the Weyl}Heisenberg algebra with an arbitrary Lie algebra leads to the generalized coherent states of Perelomov and Gilmore [23,24]. For these generalized coherent states the space parameterized by a and aH is no longer the complex plane but a symmetric space G/H. These symmetric spaces have a natural sympletic structure [24] and provide a canonical phase space structure for any quantum system with a "nite-dimensional Hilbert space whose Hamiltonian acts simultaneously on at least two variables, as well as classical systems whose dynamics is described by Lax-pair-type equations [25]. Multi-phase solutions of the KdV equation can also be described using these states [26]. For these systems the symmetric space phase space structure is in fact similar to the p, q phase space of elementary mechanics. For example, in the SU(N) case in terms of the variables z"a(sinh aa)/(aa , the metric of the space parameterized by the a variables is proportional to dzdzH, and the Poisson bracket takes the simple form
Rf Rg Rg Rf ! . (3) + f, g,"!i Rz RzH RzH Rz H H H H Evidently, the variables z and zH play essentially the same role as the p and q variables in ordinary mechanics, and therefore we expect that the formalism introduced in Section 2 can be used to de"ne operators on the 2N-dimensional phase space parameterized by a and aH and introduce a Wignerlike distribution function on this space (in a way completely analogous to the way Wigner distributions are used in quantum optics, cf. [22]). It is interesting to note that when G/H is a complex torus then the coherent states generated by the D(a) can be identi"ed with the theta functions that play such an important role in the theory of the Jacobian varieties associated with Riemann surfaces [27]. Indeed the condition that the generalized coherent states be single valued on the torus means that the phase factor Im(abH) in Eq. (45) must be equal to 2p times an integer when a and b correspond to periods of the torus. Remarkably this is just the condition that a complex torus be an abelian variety; i.e. the Jacobian of a Riemann surface. Thus, it appears possible to regard the Jacobian of a Riemann surface as being very similar to the usual p, q phase space and to use theta functions to de"ne a quantization of this space in a way completely analogous to the way Wigner and Moyal used two-dimensional Fourier transforms to quantize p, q phase space. Since according to the classical Torelli theorem a Riemann surface can be reconstructed from its Jacobian and associated theta functions [27], one might assume that applying the Moyal formalism to the Jacobian of the Riemann surface e!ectively solves the problem of quantizing the geometry of a Riemann surface. We will now argue that what is involved here is essentially the assumption that quantizing the Riemann surface is equivalent to quantizing holographic representations of the surface. As a quick reminder all information concerning an arbitrarily curved surface in three-dimensions can be encoded onto a #at plane or the surface of a sphere by recording photographically or
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otherwise the interference of light scattered o! the surface with a reference beam. Of course, this is only useful if the illuminating and reference beams are su$ciently temporally and spatially coherent. In addition, if the surface is reentrant then the interference pattern must be recorded for various orientations of the illuminating beam in order to capture the shape of the entire surface. For the purpose of physically describing these holograms one may introduce a slowly varying `envelopea electric "eld whose rms magnitude corresponds to the intensity of the interference pattern. To this end we write the vector potential on the recording surface as a function of position h on the hologram in the form (for simplicity and because we later want to make contact with the theory of inverse scattering we will assume that our holograms are recorded on a large spherical surface parameterized by two angles that we collectively call h) A(h)"!(i/k)E(h, t) exp(ikz)#h.c. ,
(4)
where z is a coordinate for the direction perpendicular to the surface of the hologram and the factor exp(ikz) represents the rapidly varying phase of the reference and scattered beams. It is straightforward to quantize these "elds by substituting expression (47) into the standard radiation gauge Lagrangian for the electromagnetic "eld. One "nds the following commutation relations for the electric "eld operators on the recording surface [E (h), E>(h )]"2pud d(h!h ) . G H GH
(5)
The electric "eld E(h, t) receives contributions from various points on the surface. In the dipole approximation the contribution from each little patch of surface is determined by the cross product between the vector pointing from the patch of surface to the point x and the direction of the oscillating polarization induced in the surface patch by the illuminating beam. Since these two vectors are curl-free on the surface the sum of contributions over the surface has the character of an inner product of two harmonic di!erentials
(6) (u , u )" u u . 1 Now an inner product for harmonic di!erentials of the form (6) plays an important role in the theory of Riemann surfaces; speci"cally an inner product of the form (6) allows one to pick out from the space of all complex tori those tori that are abelian varieties; i.e. those complex tori that can be regarded as the Jacobians of a Riemann surface. Thus we see that the famous Torelli theorem has a `physicala interpretation in terms of holography. In one obviously important respect though the holographic representation of a Riemann surface using the electric "eld E(h, t) di!ers from the representation discussed at the beginning of this section that involved N-copies of the Weyl}Heisenberg group; namely, the holographic representation involves an in"nite number of annihilation and creation operators corresponding to the electric "eld intensity at any of the in"nite number of positions on the hologram, whereas the representation based on multiple copies of the Weyl}Heisenberg group involved only a "nite number of such operators. This discrepancy can be traced to the fact that the quantization problem most closely related to ordinary p, q phase space quantization assumes that the shape of the surface is "xed. Under such circumstances the electric "eld vectors at di!erent x points are not independent
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for a given orientation of the illuminating beam. Indeed upon re#ection it is clear that the only way to obtain algebraically independent annihilation and creation operators is to illuminate the di!erent handles of the Riemann surface in distinct ways. An aesthetic choice for these distinct illuminations would be to choose the di!erent illuminating beams in such a way that the polarizations induced on the surface by the di!erent beams can be identi"ed with a canonical basis for the "rst cohomology group. There are 2g 1-forms in such a canonical basis; one for each of the 2g 1-cycles associated with the g handles of the surface. Armed with such canonical illuminations we can construct a set of 2g holograms, from which the Riemann surface can be completely reconstructed. Corresponding to these 2g algebraically independent holograms are a set of 2g independent annihilation and creation operators, viz. E (x ) and E>(x ) for j"1,2, 2g, thus H H con"rming our previous guess concerning the structure of the quantized phase space that corresponds to a Riemann surface. The representation of this phase space in terms of generalized coherent states shows that we are dealing with an N-state system where N"2g. In the next section we will develop a model for the quantum dynamics of this system, based on our holographic representation for the phase space.
3. Adaptive optics model for quantum mechanics In Section 2 we have seen that the properties of theta functions allow us to de"ne a natural Wigner}Moyal-like quantization for Riemann surfaces, and that this natural quantization corresponds to the phase space of a quantum system whose Hilbert space is "nite dimensional; i.e. an `N-statea system. We now turn to the quantum dynamics of such a system, and show that viewing the quantization of the phase space from the point of view of holography of a Riemann surface leads to an interesting and novel interpretation for the quantum dynamics. This new interpretation for quantum dynamics can actually be summarized very succinctly: if the phase of the illuminating light beam is varied as a function of time and position, a small patch of the surface should move in such a way so as to exactly compensate for the changes in the e!ective path length of the beam illuminating that patch of surface. This formulation of quantum dynamics bears an obvious similarity to the engineering problem of adaptive optics, where one is interested in changing the shape of a mirror to compensate for the degradation of focal plane images by atmospheric turbulence. Indeed, we will make generous use of results from the theory of active optical systems in our formulation of quantum dynamics. It should be kept in mind, though, that there are close parallels with the theory of the KdV equation (cf. [26]), which could perhaps have been used as the basis for the following development. We begin by writing down equations which describe an interplay between deformations in the shape of a Riemann surface and changes in the intensity of light on a set of holograms that mimic the feedback control of an active optical system. The "rst equation states that we have a feedback control system that adjusts the absolute position X(p, t) of the surface with su$cient accuracy so that the intensity of light on the hologram at time t and position x is a linear function of the error e(p, t)"*X(p, t)#a(p, t); i.e.
I(h, t)"I (h)# dp B(x, p)e(p, t) , 1
(7)
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where I(h, t) is the recorded intensity of light on the hologram at time t and position h, and I (h) is the recorded intensity on the hologram when the Riemann surface is illuminated with a `standarda beam with no arti"cially imposed variations in phase with respect to position or time. In writing Eq. (7) we are neglecting the light propagation time between the Riemann surface and the hologram. Also we will be particularly interested in the following in the situation where I (x)"E>(x) ' E (x), where E (x) is the electric at position x on the hologram when the polarization H H H induced on the Riemann surface by the illuminating beam corresponds to one of the 2g harmonic 1-forms in a canonical basis for the "rst cohomology group of the surface. The second equation relates the deformation of the Riemann surface at point p, i.e. *X(p, t), produced by the feedback control system to the intensity of light on the hologram:
*X(p, t)" dX
R
\
dt A(p, h, t )I(h, t) .
(8)
When photon noise is neglected, i.e. when we regard the hologram as a classical object, then Eqs. (7) and (8) have the solution (in vector notation) e"[1!AB]\a .
(9)
Eq. (9) shows that when the negative feedback is strong the error e(p, t) is reduced to a small fraction of the perturbation in path length. That is, the position of the surface just tracks locally the change in optical path length. Upon re#ection one soon realizes that this just the classical principle of least action! What is perhaps most remarkable about this setup though is that the problem of changing the shape of the Riemann surface to compensate for changes in the optical path of the illuminating beam becomes equivalent to solving the SchroK dinger equation when the e!ects of photon noise on the hologram are taken into account. This situation is qualitatively di!erent from the classical case because the negative feedback in Eq. (9) will amplify noise, and so there is now a limit to how strong the negative feedback can be made. It is not hard to show that in the presence of photon noise the two point correlation function for the errors averaged over a time long compared to the characteristic time for photon number #uctuations has the form (again in operator notation) 1e e 2"[1!A B ]\[1!A B ]\+; #A A d I , ,
(10)
where ; is the average of a a over the same time and d "d(p !p )d(t !t ). In contrast with the classical case an optimal choice for the feedback matrix A is somewhat arbitrary. However if, following Dyson, we take as the criterion for optimizing the system that a quadratic function of the feedback errors should be minimized, then it can be shown [21] that the optimal feedback matrix A(p, h, t ) can be expressed in the form A"KB2I\ ,
(11)
where K(p , p , t !t ) is a causal matrix satisfying the non-linear operator equation K#K2#K(B2I\B)K2#;"0 .
(12)
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This equation is a matrix generalization of the Gelfand}Levitan equation, and is essentially equivalent to the multi-channel inverse scattering theory of Newton and Jost [28]. In our formulation of their theory time plays the role of their position coordinate r, and their discrete channel labels a"1,2, N are replaced by an index j"1,2, 2g running over the harmonic 1-forms u in a canonical basis for the cohomology of a Riemann surface. Therefore the matrices H A and K are "nite matrices describing surface deformations whose support coincides with the support of harmonic functions f such that u "df . H H H It is worth noting that in our formulation of quantum mechanics the dynamical behavior of the system is entirely determined by the pair correlation matrix U. This is a consequence of the linearity of the fundamental Eqs. (7) and (8) and exactly mirrors the fundamental theorem of quantum computation [29,30], which states that any quantum computation can be e!ected by quantum logic gate which consists of a unitary operator acting on an arbitrary choice of two input variables. Thus, the pattern recognition capabilities of quantum computers are intimately related to the pattern recognition capabilities of adaptive optics systems. It is amusing to note in this connection that our quantized version of Dyson's algorithm is formally similar to Hop"eld's scheme for `collective computationa using spins [31]. 4. Mathematics as theoretical physics? The meaning of mathematics and its relation to physics have been controversial subjects for millennia among mathematicians and philosophers. It has not been our purpose here to review these historical discussions. Instead our intention has been to add a new ingredient to these discussions that may in the course of time help to resolve some of the long-standing philosophical issues. This new ingredient has two di!erent but related aspects. The "rst aspect is that, prompted by the discovery of potentially useful applications for quantum computers, we have found a mathematical interpretation of quantum mechanics as a theory of pattern recognition. The second aspect is that our `adaptive opticsa model for quantum mechanics is an invitation to relate our formulation of quantum mechanics to a quantum theory of membranes, which is a putative candidate for the mathematical theory underlying superstring models [32]. Quantum #uctuations in a hologram due to photon noise give rise via the feedback control system introduced in Section 3 to quantum #uctuations in the shape of the surface, and this may perhaps be interpreted by saying that the classical geometry of the Riemann surface has been replaced with quantum geometry. Thus quantum holography might also be interpreted as a quantum theory of membranes. In addition as noted in Section 2 our holographic formulation of quantum kinematics is closely related to the classical theory of integrable dynamical systems. Now it happens that certain kinds of membranes correspond to the large N limit of integrable systems whose Lax-pair dynamics is similar to SU(N) quantum dynamics [33]. Furthermore, it has independently been suggested [34] that the same kind of Wigner}Moyal quantization as we have employed could be used to quantize these large N models for membranes. Therefore it appears that our holographic formulation of quantum mechanics may indeed be pointing us in the direction of a fundamental structure for theoretical physics. Although the exact nature of the mathematical structure underlying superstring models remains shrouded in mist, let us we accept for the moment that our formulation of quantum mechanics is
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actually closely related to the quantum theory of membranes being sought as the theory underlying superstring models. This allows us to immediately make a tentative yet intriguingly plausible conjecture as to the nature of the fundamental relationship between mathematics and theoretical physics. Namely, we are led to suggest that the fundamental connecting link between mathematics and theoretical physics is the pattern recognition capabilities of the human brain. On one hand, it seems quite reasonable to assume that the human brain's aptitude for mathematical reasoning evolved from the mammalian brain's more primitive function as a pattern recognition engine. On the other hand, the pattern recognition capabilities of the mammalian brain may be closely related to our formulation of quantum mechanics. Of course, given our present state of ignorance concerning how the mammalian brain actually functions this last assertion is highly speculative. However, the general idea that a quantum mechanical theory of information #ow (cf. [7]) can be looked upon as a model for the type of distributed information processing carried out in the brain has a lot to recommend it. For example, one of the fundamental heuristics of distributed information processing networks is that minimization of energy consumption requires the use of time division multiplexing for communication between the processors [35], and it would be natural to identify the local internal time in such networks as a quantum phase.
References [1] G. Chapline, R. Slansky, Dimensional reduction and #avor chirality, Nucl. Phys. B 209 (1982) 461. [2] M. Green, J.H. Schwarz, Anomaly cancellations in supersymmetric D"10 superstring theory, Phys. Lett. B 144 (1984) 117. [3] A. Schellekens, N. Warner, Anomolies and Modular Invariance in String Theory, Phys. Lett. B 177 (1985) 317. [4] G. Chapline, Uni"cation of elementary particle physics in 26 dimensions, Phys. Lett. B 158 (1985) 393; The Monster sporadic group and a theory underlying superstring models, in: Proceedings of the Strings '96 Conference, Institute for Theoretical Physics, 1996. [5] E.P. Wigner, The unreasonable e!ectiveness of mathematics in the natural sciences. Comm. Pure Appl. Math. 13 (February 1960). [6] R. Hersh, What is Mathematics, Really? Oxford University Press, Oxford, 1997. [7] L.H. Kau!man, On Knots, Princeton University Press, 1987. [8] E. Witten, Supersymmetry and Morse theory, J. Di!erential Geom. 17 (1982) 661; Topological Quantum Field Theory 117 (1988) 353. [9] T. Dereli, A. Vercin, =(R) covariance of the Weyl}Wigner}Groenewold}Moyal quantization, J. Math. Phys. 38 (1997) 5515. [10] G. Chapline, A. Granik, Moyal quantization, holography, and the quantum geometry of surfaces, Chaos, Solitons and Fractals, to be published. [11] P.W. Shor, Algorithms for quantum computation: discrete log and factoring, in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, IEEE Press, 1994. [12] L.K. Glover, Quantum mechanics helps in searching for a needle in a haystack, Phys. Rev. Lett. 79 (1997) 325. [13] T. Kohonen, Self-Organization and Associative Memory, Springer, Berlin, 1988. [14] B. Schumacher, Information from quantum measurements, in: W.H. Zurek (Ed.), Complexity, Entropy and the Physics of Information, Addison-Wesley, Reading, MA, 1990. [15] A.S. Kholevo, Bounds for the quantity of information transmitted by a quantum communication channel, Problems of Information Transfer 9 (1973) 3. [16] G. Chapline, Information #ow in quantum mechanics, in: T.D. Black et al. (Eds.), Santa Fe Workshop on Foundations of Quantum Mechanics, World Scienti"c, Singapore, 1988.
G. Chapline / Physics Reports 315 (1999) 95 } 105 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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C.M. Caves, P.D. Drummond, Quantum limits on bosonic communication rates, Rev. Modern Phys. 66 (1994) 481. E. Fermi, Notes on Thermodynamics and Statistics, University of Chicago Press, Chicago, 1966. E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749. J.E. Moyal, Quantum mechanics as a statistical theory, Proc. Camb. Phil. Soc. 45 (1949) 99. F.J. Dyson, Photon noise and atmospheric noise in active optical systems, J. Optical Soc. Am. 65 (1975) 551. D.F. Walls, G.J. Milburn, Quantum Optics, Springer, Berlin, 1994. A.M. Perelomov, Coherent states for an arbitrary Lie group, Commun. Math. Phys. 26 (1972) 222. W. Zang, D.H. Feng, R. Gilmore, Coherent states: theory and applications, Rev. Mod. Phys. 62 (1990) 867. M. Adler, P. van Moerbeke, Completely integrable systems, Lie algebras, and curves, Adv. Math. 38 (1980) 267. A.C. Newell, Solitons in Mathematics and Physics, SIAM, 185. H.M. Farkas, I. Kra, Riemann Surfaces, Springer, Berlin, 1992. R.G. Newton, R. Jost, The construction of potentials from the S-matrix for systems of di!erential equations, Nuovo Cimento 1 (1955) 590. D. Deutsch, A. Barenco, A. Ekert, Universality in quantum computation, Proc. R. Soc. London A 449 (1995) 669. S. Lloyd, Almost any quantum logic gate is universal, Phys. Rev. Lett. 75 (1995) 346. J.J. Hop"eld, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. B 209 (1984) 3088. P. Horava, E. Witten, Heterotic and Type I string dynamics from eleven dimensions, Nucl. Phys. B 460 (1996) 506. E.G. Floratos, G.K. Leontaris, Integrability of self-dual membranes in (4#1) dimensions and the Toda lattice, Phys. Lett. B 223 (1989) 153. C. Castro, A Moyal quantization of the continuous Toda "eld, Phys. Lett. B 413 (1997) 53. G. Chapline, Sentient networks, in: Proceedings of the First International Conference on Multisource}Multisensor Fusion, CREA Press Athens, Georgia, 1998.
Physics Reports 315 (1999) 107}121
From superstrings to M theory夽 John H. Schwarz California Institute of Technology, Pasadena, CA 91125, USA
Abstract In the strong coupling limit Type IIA superstring theory develops an 11th dimension that is not apparent in perturbation theory. This suggests the existence of a consistent 11D quantum theory, called M theory, which is approximated by 11D supergravity at low energies. In this review we describe some of the evidence for this picture and some of its implications. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.!w; 11.25.Sq; 04.50.#h Keywords: String theory; M theory; Supersymmetry; Superstrings
1. Introduction Superstring theory is currently undergoing a period of rapid development in which important advances in understanding are being achieved. The purpose of this review is to describe a portion of this story to physicists who are not already experts in this "eld. The focus will be on explaining why there can be an 11D vacuum, even though there are only 10 dimensions in perturbative superstring theory. The nonperturbative extension of superstring theory that allows for an 11th dimension has been named M theory. The letter M is intended to be #exible in its interpretation. It could stand for magic, mystery, or meta to re#ect our current state of incomplete understanding. Those who think that (2D) supermembranes (the M2-brane) are fundamental may regard M as standing for membrane. An approach called Matrix theory is another possibility. And, of course, some view M theory as the mother of all theories. Superstring theory "rst achieved widespread acceptance during the xrst superstring revolution in 1984}1985. There were three main developments at this time. The "rst was the discovery of an 夽
Work supported in part by the U.S. Dept. of Energy under Grant No. DE-FG03-92-ER40701. For a more detailed review see Ref. [1]. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 6 - 2
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anomaly cancellation mechanism [2], which showed that supersymmetric gauge theories can be consistent in 10 dimensions provided they are coupled to supergravity (as in type I superstring theory) and the gauge group is either SO(32) or E ;E . Any other group necessarily would give uncancelled gauge anomalies and hence inconsistency at the quantum level. The second development was the discovery of two new superstring theories } called heterotic string theories } with precisely these gauge groups [3]. The third development was the realization that the E ;E heterotic string theory admits solutions in which six of the space dimensions form a Calabi}Yau space, and that this results in a 4D e!ective theory at low energies with many qualitatively realistic features [4]. Unfortunately, there are very many Calabi}Yau spaces and a whole range of additional choices that can be made (orbifolds, Wilson loops, etc.). Thus there is an enormous variety of possibilities, none of which stands out as particularly special. In any case, after the "rst superstring revolution subsided, we had "ve distinct superstring theories with consistent weak coupling perturbation expansions, each in 10 dimensions. Three of them, type I theory and the two heterotic theories, have N"1 supersymmetry in the 10D sense. Since the minimal 10D spinor is simultaneously Majorana and Weyl, this corresponds to 16 conserved supercharges. The other two theories, called types IIA and IIB, have N"2 supersymmetry (32 supercharges) [5]. In the IIA case the two spinors have opposite handedness so that the spectrum is left}right symmetric (nonchiral). In the IIB case the two spinors have the same handedness and the spectrum is chiral. The understanding of these "ve superstring theories was developed in the ensuing years. In each case it became clear, and was largely proved, that there are consistent perturbation expansions of on-shell scattering amplitudes. In four of the "ve cases (heterotic and type II) the fundamental strings are oriented and unbreakable. As a result, these theories have particularly simple perturbation expansions. Speci"cally, there is a unique Feynman diagram at each order of the loop expansion. The Feynman diagrams depict string world sheets, and therefore they are 2D surfaces. For these four theories the unique ¸-loop diagram is a closed orientable genus-¸ Riemann surface, which can be visualized as a sphere with ¸ handles. External (incoming or outgoing) particles are represented by N points (or `puncturesa) on the Riemann surface. A given diagram represents a well-de"ned integral of dimension 6¸#2N!6. This integral has no ultraviolet divergences, even though the spectrum contains states of arbitrarily high spin (including a massless graviton). From the viewpoint of point-particle contributions, string and supersymmetry properties are responsible for incredible cancellations. Type I superstrings are unoriented and breakable. As a result, the perturbation expansion is more complicated for this theory, and the various worldsheet diagrams at a given order (determined by the Euler number) have to be combined properly to cancel divergences and anomalies [6]. An important discovery that was made between the two superstring revolutions is called T duality [7]. This is a property of string theories that can be understood within the context of perturbation theory. (The discoveries associated with the second superstring revolution are mostly nonperturbative.) T duality shows that spacetime geometry, as probed by strings, has some surprising properties (sometimes referred to as quantum geometry). The basic idea can be illustrated
A discussion with Richard Slansky helped to convince us that E ;E would work.
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by the simplest example. This entails considering one spatial dimension to form a circle (denoted S). Then the 10D geometry is R;S. T duality identi"es this string compacti"cation with one of a second string theory also on R;S. However, if the radii of the circles in the two cases are denoted R and R , then R R "a . (1) Here a "l is the universal Regge slope parameter, and l is the fundamental string length scale (for both string theories). The tension of a fundamental string is given by ¹"2pm"1/2pa , where we have introduced a fundamental string mass scale
(2)
m "(2pl )\ . (3) Note that T duality implies that shrinking the circle to zero in one theory corresponds to decompacti"cation of the dual theory. Compacti"cation on a circle of radius R implies that momenta in that direction are quantized, p"n/R. (These are called Kaluza}Klein excitations.) These momenta appear as masses for states that are massless from the higher-dimensional viewpoint. String theories also have a second class of excitations, called winding modes. Namely, a string wound m times around the circle has energy E"2pR ) m ) ¹"mR/a . Eq. (1) shows that the winding modes and Kaluza}Klein excitations are interchanged under T duality. What does T duality imply for our "ve superstring theories? The IIA and IIB theories are T dual [8]. So compactifying the nonchiral IIA theory on a circle of radius R and letting RP0 gives the chiral IIB theory in 10 dimensions! This means, in particular, that they should not be regarded as distinct theories. The radius R is actually a vev of a scalar "eld, which arises as an internal component of the 10D metric tensor. Thus types IIA and IIB theories in 10D are two limiting points in a continuous moduli space of quantum vacua. The two heterotic theories are also T dual, though there are technical details involving Wilson loops, which we will not explain here. T duality applied to type I theory gives a dual description, which is sometimes called I'. Names IA and IB have also been introduced by some authors. For the remainder of this paper, we will restrict attention to theories with maximal supersymmetry (32 conserved supercharges). This is su$cient to describe the basic ideas of M theory. Of course, it suppresses many fascinating and important issues and discoveries. In this way, we will keep the presentation from becoming too long or too technical. The main focus will be to ask what happens when we go beyond perturbation theory and allow the coupling strength to become large in type II theories. The answer in the IIA case, as we will see, is that another spatial dimension appears.
2. M theory In the 1970s and 1980s various supersymmetry and supergravity theories were constructed. (See [9], for example.) In particular, supersymmetry representation theory showed that 10 is the largest spacetime dimension in which there can be a matter theory (with spins 41) in which supersymmetry is realized linearly. A realization of this is 10D super Yang}Mills theory, which has 16
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supercharges [10]. This is a pretty (i.e., very symmetrical) classical "eld theory, but at the quantum level it is both nonrenormalizable and anomalous for any nonabelian gauge group. However, as we indicated earlier, both problems can be overcome for suitable gauge groups (SO(32) or E ;E ) when the Yang}Mills theory is embedded in a type I or heterotic string theory. The largest possible spacetime dimension for a supergravity theory (with spins 42), on the other hand, is 11. Eleven-dimensional supergravity, which has 32 conserved supercharges, was constructed 20 years ago [11]. It has three kinds of "elds } the graviton "eld (with 44 polarizations), the gravitino "eld (with 128 polarizations), and a three-index gauge "eld C (with 84 polarizaIJM tions). These massless particles are referred to collectively as the supergraviton. 11D supergravity is also a pretty classical "eld theory, which has attracted a lot of attention over the years. It is not chiral, and therefore not subject to anomaly problems. It is also nonrenormalizable, and thus it cannot be a fundamental theory. Though it is di$cult to demonstrate explicitly that it is not "nite as a result of `miraculousa cancellations, we now know that this is not the case. However, we now believe that it is a low-energy e!ective description of M theory, which is a well-de"ned quantum theory [13]. This means, in particular, that higher dimension terms in the e!ective action for the supergravity "elds have uniquely determined coe$cients within the M theory setting, even though they are formally in"nite (and hence undetermined) within the supergravity context. Intriguing connections between type IIA string theory and 11D supergravity have been known for a long time. If one carries out dimensional reduction of 11D supergravity to 10D, one gets type IIA supergravity [14]. Dimensional reduction can be viewed as a compacti"cation on circle in which one drops all the Kaluza}Klein excitations. It is easy to show that this does not break any of the supersymmetries. The "eld equations of 11D supergravity admit a solution that describes a supermembrane. In other words, this solution has the property that the energy density is concentrated on a 2D surface. A 3D world-volume description of the dynamics of this supermembrane, quite analogous to the 2D world volume actions of superstrings, has been constructed [15]. The authors suggested that a consistent 11D quantum theory might be de"ned in terms of this membrane, in analogy to string theories in 10 dimensions. Another striking result was the discovery of double-dimensional reduction [16]. This is a dimensional reduction in which one compacti"es on a circle, wraps one dimension of the membrane around the circle and drops all Kaluza}Klein excitations for both the spacetime theory and the world-volume theory. The remarkable fact is that this gives (previously known) type IIA superstring world-volume action [17]. For many years these facts remained unexplained curiosities until they were reconsidered by Townsend [18] and by Witten [13]. The conclusion is that type IIA superstring theory really does have a circular 11th dimension in addition to the previously known 10 spacetime dimensions. This fact was not recognized earlier because the appearance of the 11th dimension is a nonperturbative phenomenon, not visible in perturbation theory.
Unless the spacetime has boundaries. The anomaly associated to a 10D boundary can be cancelled by introducing E supersymmetric gauge theory on the boundary [12]. Most experts now believe that M theory cannot be de"ned as a supermembrane theory.
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To explain the relation between M theory and type IIA string theory, a good approach is to identify the parameters that characterize each of them and to explain how they are related. Eleven-dimensional supergravity (and hence M theory, too) has no dimensionless parameters. As we have seen, there are no massless scalar "elds, whose vevs could give parameters. The only parameter is the 11D Newton constant, which raised to a suitable power (!1/9), gives the 11D Planck mass m . When M theory is compacti"ed on a circle (so that the spacetime geometry is R;S) another parameter is the radius R of the circle. Now consider the parameters of type IIA superstring theory. They are the string mass scale m , introduced earlier, and the dimensionless string coupling constant g . An important fact about all "ve superstring theories is that the coupling constant is not an arbitrary parameter. Rather, it is a dynamically determined vev of a scalar "eld, the dilaton, which is a supersymmetry partner of the graviton. With the usual conventions, one has g "1e(2. We can identify compacti"ed M theory with type IIA superstring theory by making the following correspondences: m"2pRm , g "2pRm . Using these one can derive other equivalent relations, such as
(4) (5)
g "(2pRm ) , (6) m "g m . (7) The latter implies that the 11D Planck length is shorter than the string length scale at weak coupling by a factor of (g ) . Conventional string perturbation theory is an expansion in powers of g at "xed m . Eq. (5) shows that this is equivalent to an expansion about R"0. In particular, the strong coupling limit of type IIA superstring theory corresponds to decompacti"cation of the 11th dimension, so in a sense M theory is type IIA string theory at in"nite coupling. This explains why the 11th dimension was not discovered in studies of string perturbation theory. These relations encode some interesting facts. The fact relevant to Eq. (4) concerns the interpretation of the fundamental type IIA string. Earlier we discussed the old notion of double-dimensional reduction, which allowed one to derive the IIA superstring world-sheet action from the 11D supermembrane (or M2-brane) world-volume action. Now, we can make a stronger statement: The fundamental IIA string actually is an M2-brane of M theory with one of its dimensions wrapped around the circular spatial dimension. No truncation to zero modes is required. Denoting the string and membrane tensions (energy per unit volume) by ¹ and ¹ , one deduces that $ + ¹ "2pR¹ . (8) $ + However, ¹ "2pm and ¹ "2pm. Combining these relations gives Eq. (4). $ +
The E ;E heterotic string theory is also 11D at strong coupling [12].
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Type II superstring theories contain a variety of p-brane solutions that preserve half of the 32 supersymmetries. These are solutions in which the energy is concentrated on a p-dimensional spatial hypersurface. (The world volume has p#1 dimensions.) The corresponding solutions of supergravity theories were constructed by Horowitz and Strominger [19]. A large class of these p-brane excitations are called D-branes (or Dp-branes when we want to specify the dimension), whose tensions are given by [20] ¹ "2pmN>/g . (9) "N This dependence on the coupling constant is one of the characteristic features of a D-brane. It is to be contrasted with the more familiar g\ dependence of soliton masses (e.g., the 't Hooft}Polyakov monopole). Another characteristic feature of D-branes is that they carry a charge that couples to a gauge "eld in the RR sector of the theory. (Such "elds can be described as bispinors.) The particular RR gauge "elds that occur imply that even values of p occur in the IIA theory and odd values in the IIB theory. In particular, the D2-brane of type IIA theory corresponds to our friend the supermembrane of M theory, but now in a background geometry in which one of the transverse dimensions is a circle. The tensions check, because (using Eqs. (4) and (5)) ¹ "2pm/g "2pm"¹ . (10) " + The mass of the "rst Kaluza}Klein excitation of the 11D supergraviton is 1/R. Using Eq. (5), we see that this can be identi"ed with the D0-brane. More identi"cations of this type arise when we consider the magnetic dual of the M theory supermembrane. This turns out to be a "ve-brane, called the M5-brane. Its tension is ¹ "2pm. Wrapping one of its dimensions around the circle + gives the D4-brane, with tension ¹ "2pR¹ "2pm/g . (11) " + If, on the other hand, the M5-frame is not wrapped around the circle, one obtains the NS5-brane of the IIA theory with tension ¹ "¹ "2pm/g . (12) ,1 + This 5-brane, which is the magnetic dual of the fundamental IIA string, exhibits the conventional g\ solitonic dependence. To summarize, type IIA superstring theory is M theory compacti"ed on a circle of radius R"g l . M theory is believed to be a well-de"ned quantum theory in 11D, which is approximated at low energy by 11D supergravity. Its excitations are the massless supergraviton, the M2-brane, and the M5-brane. These account both for the (perturbative) fundamental string of the IIA theory and for many of its nonperturbative excitations. The identities that we have presented here are exact, because they are protected by supersymmetry.
In general, the magnetic dual of a p-brane in d dimensions is a (d!p!4)-brane.
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3. Type IIB superstring theory In the previous section we discussed type IIA superstring theory and its relationship to 11D M theory. In this section we consider type IIB superstring theory, which is the other maximally supersymmetric string theory with 32 conserved supercharges. It is also 10D, but unlike the IIA theory its two supercharges have the same handedness. Since the spectrum contains massless chiral "elds, one should check whether there are anomalies that break the gauge invariances } general coordinate invariance, local Lorentz invariance, and local supersymmetry. In fact, the UV "niteness of the string theory Feynman diagrams (and associated modular invariance) ensures that all anomalies must cancel. This was veri"ed also from a "eld theory viewpoint [21]. The low-energy e!ective theory that approximates type IIB superstring theory is type IIB supergravity [5,22], just as 11D supergravity approximates M theory. In each case the supergravity theory is only well de"ned as a classical "eld theory, but still it can teach us a lot. For example, it can be used to construct p-brane solutions and compute their tensions. Even though such solutions themselves are only approximate, supersymmetry considerations ensure that their tensions, which are related to the kinds of charges they carry, are exact. Another signi"cant fact about type IIB supergravity is that it possesses a global SL(2, R) symmetry. It is instructive to consider the bosonic spectrum and its SL(2, R) transformation properties. There are two scalar "elds } the dilation and an axion s, which are conveniently combined in a complex "eld o"s#ie\( .
(13)
The SL(2, R) symmetry transforms this "eld nonlinearly: oP(ao#b)/(co#d) ,
(14)
where a, b, c, d are real numbers satisfying ad!bc"1. However, in the quantum string theory this symmetry is broken to the discrete subgroup SL(2, Z) [23], which means that a, b, c, d are restricted to be integers. De"ning the vev of the o "eld to be 1o2"h/2p#i/g , (15) the SL(2, Z) symmetry transformation oPo#1 implies that h is an angular coordinate. More signi"cantly, in the special case h"0, the symmetry transformation oP!1/o takes g P1/g . This symmetry, called S duality, implies that the theory with coupling constant g is equivalent to coupling constant 1/g , so that the weak coupling expansion and the strong coupling expansion are identical! The bosonic spectrum also contains a pair of two-form potentials B and B, which transform IJ IJ as a doublet under SL(2, R) or SL(2, Z). In particular, the S duality transformation oP!1/o interchanges them. The remaining bosonic "elds are the graviton and a four-form potential C , IJMH with a self-dual "eld strength. They are invariant under SL(2). In the introductory section we indicated that types IIA and IIB superstring theories are T dual, meaning that if they are compacti"ed on circles of radii R and R one obtains equivalent theories for the identi"cation R R "l. Moreover, in Section 2 we saw that type IIA theory is actually M theory compacti"ed on a circle. The latter fact encodes nonperturbative information. It turns out to be very useful to combine these two facts and to consider the duality between M theory
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compacti"ed on a torus (R;¹) and type IIB superstring theory compacti"ed on a circle (R;S). Recall that a torus can be described as the complex plane modded out by the equivalence relations z&z#w and z&z#w . Up to conformal equivalence, the periods can be taken to be 1 and q, with Im q'0. However, in this characterization q and q "(aq#b)/(cq#d), where a, b, c, d are integers satisfying ad!bc"1, describe equivalent tori. Thus, a torus is characterized by a modular parameter q and an SL(2, Z) modular group. The natural, and correct, conjecture at this point is that one should identify the modular parameter q of the M theory torus with the parameter o that characterizes type IIB vacuum [24,25]! Then the duality gives a geometrical explanation of the nonperturbative S duality symmetry of the IIB theory: the transformation oP!1/o, which sends g P1/g in the IIB theory, corresponds to interchanging the two cycles of the torus in the M theory description. To complete the story, we should relate the area of the M theory torus (A ) + to the radius of the IIB theory circle (R ). This is a simple consequence of formulas given above mA "(2pR )\ . (16) + Thus the limit R P0, at "xed o, corresponds to decompacti"cation of the M theory torus, while preserving its shape. Conversely, the limit A P0 corresponds to decompacti"cation of the IIB + theory circle. The duality can be explored further by matching the various p-branes in 9 dimensions that can be obtained from either the M theory or the IIB theory viewpoints [26]. When this is done, one "nds that everything matches nicely and that one deduces various relations among tensions, such as ¹ "(1/2p)(¹ ) . (17) + + This relation was used earlier when we asserted that ¹ "2pm and ¹ "2pm. + + Even more interesting is the fact that the IIB theory contains an in"nite family of strings labelled by a pair of relatively prime integers (p, q) [24]. These integers correspond to string charges that are sources of the gauge "elds B and B. The (1, 0) string can be identi"ed as the fundamental IIB IJ IJ string, while the (0, 1) string is the D-string. From this viewpoint, a (p, q) string can be regarded as a bound state of p fundamental strings and q D-strings [27]. These strings have a very simple interpretation in the dual M theory description. They correspond to an M2-brane with one of its cycles wrapped around a (p, q) cycle of the torus. The minimal length of such a cycle is proportional to "p#qq", and thus (using q"o) one "nds that the tension of a (p, q) string is given by ¹ "2p"p#qo"m . (18) N O The normalization has been chosen to give ¹ "2pm. Then (for h"0) ¹ "2pm/g , as expected. Note that decay is kinematically forbidden by charge conservation when p and q are relatively prime. When they have a common division n, the tension is the same as that of an n-string system. Whether or not there are threshold bound states is a nontrivial dynamical question, which has di!erent answers in di!erent settings. In this case there are no such bound states, which is why p and q should be relatively prime. Imagine that you lived in the 9D world that is described equivalently as M theory compacti"ed on a torus or as type IIB superstring theory compacti"ed on a circle. Suppose, moreover, you had very high-energy accelerators with which you were going to determine the `truea dimension of
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spacetime. Would you conclude that 10 or 11 is the correct answer? If either A or R was very + large in Planck units there would be a natural choice, of course. But how could you decide otherwise? The answer is that either viewpoint is equally valid. What determines which choice you make is which of the massless "elds you regard as `internala components of the metric tensor and which ones you regards as matter "elds. Fields that are metric components in one description correspond to matter "elds in the dual one.
4. U dualities Maximal supergravity theories (ones with 32 conserved supercharges) typically have a noncompact global symmetry group G. For example, in the case of type IIB supergravity in 10 dimensions the group is SL(2, R). When one does dimensional reduction one "nds larger groups in lower dimensions. For example, N"8 supergravity in four dimensions has a noncompact E symmetry [28]. More generally, for D"11!d, 34d48, one "nds a maximally noncompact form of E , B denoted E . These are statements about classical "eld theory. The corresponding statement about B B superstring theory/M theory is that if we toroidally compactify M theory on R";¹B or type IIB superstring theory on R";¹B\, the resulting moduli space of theories is invariant under an in"nite discrete U duality group. The group, denoted E (Z), is a maximal discrete subgroup of the B noncompact E symmetry group of the corresponding supergravity theory [23]. An example that B B we will focus on below is E (Z)"SL(3, Z);SL(2, Z) . (19) The U duality groups are generated by the Weyl subgroup of E plus discrete shifts of axion-like B B "elds. The subgroup SL(d, Z)LE (Z) can be understood as the geometric duality (modular group) B of ¹B in the M theory picture. This generalizes the SL(2, Z) discussed in the preceding section. The subgroup SO(d!1, d!1; Z)LE (Z) is the T duality group of type IIB superstring theory B compacti"ed on ¹B\. These two subgroups intertwine nontrivially to generate the entire E (Z) B U duality group. Suppose we wish to focus on M theory and disregard type IIB superstring theory. Then we have a geometric understanding of the SL(d, Z) subgroup of E (Z) from considering M theory on B R\B;¹B. But what does the rest of E (Z) imply? To address this question it will su$ce to B consider the "rst nontrivial case to which it applies, which is d"3. In this case the U duality group is SL(3, Z);SL(2, Z). The "rst factor is geometric from the M theory viewpoint and nongeometric from the IIB viewpoint, whereas the second factor is geometric from the IIB viewpoint and nongeometric from the M theory viewpoint. So the question boils down to understanding the implication of the SL(2, Z) duality in the M theory construction. Speci"cally, we want to understand the nontrivial qP!1/q transformation. To keep the story as simple as possible, we will take the ¹ to be rectilinear with radii R , R , R (i.e., g &Rd ) and assume that C "0. Let us suppose that R corresponds to the `eleventha GH G GH dimension that takes us to the IIA theory. Then we have IIA theory on a torus with radii R and R . The nongeometric duality of M theory is T duality of IIA theory. T duality gives a mapping to
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an equivalent point in the moduli space for which R PR "l/R "l/R R , i"1, 2 (20) G G G G with l unchanged. Note that we have used Eq. (4), reexpressed as l"R l. Under a T duality the string coupling constant also transforms. The rule is that the coupling of the e!ective theory (8D in this case) is invariant: 1/g"4pR R /g"4pR R /(g ) .
(21)
Thus, g "g l/R R . What does this imply for the radius of the 11th dimension R ? Using Eq. (5), R "g l PR "g l . Thus,
(22)
R "g l/R R "l/R R . However, the 11D Planck length also transforms, because
(24)
l"g lP(l )"g l implies that
(25)
(23)
(l )"g l/R R "l/R R R . (26) The perturbative IIA description is only applicable for R ;R , R . However, even though T duality was originally discovered in perturbation theory, it is supposed to be an exact nonperturbative property. Therefore, this duality mapping should be valid as an exact symmetry of M theory without any restriction on the radii. Another duality is an interchange of circles, such as R R . This corresponds to the nonperturbative S duality of the IIB theory, as we discussed earlier. Combining these dualities we obtain the desired nongeometric duality of M theory on ¹ [29]. It is given by R Pl/R R and cyclic permutations, accompanied by
(27)
lPl/R R R . Eqs. (27) and (28) have a nice interpretation. Eq. (27) implies that
(28)
1/R P(2pR )(2pR )¹ . (29) + Thus it interchanges Kaluza}Klein excitations with wrapped supermembrane excitations. It follows that these six 0-branes belong to the (3, 2) representation of the U-duality group. Eq. (28) implies that ¹ P(2pR )(2pR )(2pR )¹ . + +
(30)
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Therefore, it interchanges an unwrapped M2-brane with an M5-brane wrapped on the ¹. Thus these two 2-branes belong to the (1, 2) representation of the U-duality group. This basic nongeometric duality of M theory, combined with the geometric ones, generates the entire U duality group in every dimension. It is a property of quantum M theory that goes beyond what can be understood from the e!ective 11D supergravity, which is geometrical. This analysis has been extended to allow C O0 [30]. In this case there are indications that the torus should be considered to be noncommutative [31].
5. The D3-brane and N:4 gauge theory D-branes have a number of special properties, which make them especially interesting. By de"nition, they are branes on which strings can end } D stands for Dirichlet boundary conditions. The end of a string carries a charge, and the D-brane world-volume theory contains a ;(1) gauge "eld that carries the associated #ux. When n Dp-branes are coincident, or parallel and nearly coincident, the associated (p#1)-dimensional world-volume theory is a ;(n) gauge theory. The n gauge bosons AGH and their supersymmetry partners arise as the ground states of oriented strings I running from the ith Dp-brane to the jth Dp-brane. The diagonal elements, belonging to the Cartan subalgebra, are massless. The "eld AGH with iOj has a mass proportional to the separation of the ith I and jth branes. This separation is described by the vev of a corresponding scalar "eld in the world-volume theory. The ;(n) gauge theory associated with a stack of n Dp-branes has maximal supersymmetry (16 supercharges). The low-energy e!ective theory, when the brane separations are small compared to the string scale, is supersymmetric Yang}Mills theory. These theories can be constructed by dimensional reduction of 10D supersymmetric ;(n) gauge theory to p#1 dimensions. In fact, that is how they originally were constructed [10]. For p43, the low-energy e!ective theory is renormalizable and de"nes a consistent quantum theory. For p"4, 5 there is good evidence for the existence nongravitational quantum theories that reduce to the gauge theory in the infrared. For p56, it appears that there is no decoupled nongravitational quantum theory [32]. A case of particular interest, which we shall now focus on, is p"3. A stack of n D3-branes in type IIB superstring theory has a decoupled N"4, d"4 ;(n) gauge theory associated to it. This gauge theory has a number of special features. For one thing, due to boson}fermion cancellations, there are no ;< divergences at any order of perturbation theory. The beta function b(g) is identically zero, which implies that the theory is scale invariant (aside from scales introduced by vevs of the scalar "elds). In fact, N"4, d"4 gauge theories are conformally invariant. The conformal invariance combines with the supersymmetry to give a superconformal symmetry, which contains 32 fermionic generators. Half are the ordinary linearly realized supersymmetrics, and half are nonlinearly realized ones associated to the conformal symmetry. The name of the superconformal group in this case is SU(4"4). Another important property of N"4, d"4 gauge theories is electric}magnetic duality [33]. This extends to an SL(2, Z) group of dualities. To understand these it is necessary to include a vacuum angle h and de"ne a complex coupling 7+ q"h /2p#i 4p/g . (31) 7+ 7+
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Under SL(2, Z) transformations this coupling transforms in the usual nonlinear fashion (qP(aq#b)/(cq#d)) and the electric and magnetic "elds transform as a doublet. Note that the conformal invariance ensures that q is a meaningful scale-independent constant. Now consider the N"4 ;(n) gauge theory associated to a stack of n D3-branes in type IIB superstring theory. There is an obvious identi"cation, that turns out to be correct. Namely, the SL(2, Z) duality of the gauge theory is induced from that of the ambient type IIB superstring theory. In particular, the q parameter of the gauge theory is the value of the complex scalar "eld o of the string theory. This makes sense because o is constant in the "eld con"guration associated to a stack of D3-branes. The D3-branes themselves are invariant under SL(2, Z) transformations. Only the parameter q"o changes, but it is transformed to an equivalent value. All other "elds, such as BG , IJ which are not invariant, vanish in this case. As we have said, a fundamental (1, 0) string can end on a D3-brane. But by applying a suitable SL(2, Z) transformation, this con"guration is transformed to one in which a (p, q) string } with p and q relatively prime } ends on the D3-brane. The charge on the end of this string describes a dyon with electric charge p and magnetic q, with respect to the appropriate gauge "eld. More generally, for a stack of n D3-branes, any pair can be connected by a (p, q) string. The mass is proportional to the length of the string times its tension, which we saw is proportional to "p#qo". In this way one sees that the electrically charged particles, described by fundamental "elds, belong to in"nite SL(2, Z) multiplets. The other states are nonperturbative excitations of the gauge theory. The "eld con"gurations that describe them preserve half of the supersymmetry. As a result their masses saturate a BPS bound and are given exactly by the considerations described above. An interesting question, whose answer was unknown until recently, is whether N"4 gauge theories in four dimensions also admit nonperturbative excitations that preserve 1/4 of the supersymmetry. To explain the answer, it is necessary to "rst make a digression to consider three-string junctions. As we have seen, type IIB superstring theory contains an in"nite multiplet of strings labelled by a pair of relatively prime integers (p, q). Three strings, with charges (p , q ), i"1, 2, 3, can meet at G G a point provided that charge is conserved [34,35]. This means that p " q "0 , (32) G G if the three strings are all oriented inwards. (This is like momentum conservation in an ordinary Feynman diagram.) Such a con"guration is stable, and preserves 1/4 of the ambient supersymmetry provided that the tensions balance. It is easy to see how this can be achieved. If one regards the plane of the junction as a complex plane and orients the direction of a (p, q) string by the phase of p#qq, then Eqs. (18) and (32) ensure a force balance. The three-string junction has an interesting dual M theory interpretation. If one of the directions perpendicular to the plane of the junction is taken to be a circle, then we have a string junction in nine dimensions. This must have a dual interpretation in terms of M theory compacti"ed on a torus. We have already seen that a (p, q) string corresponds to an M2-brane with one of its cycles wrapped on a (p, q) cycle of the torus. So now we join three such cylindrical membranes together. Altogether we have a single smooth M2-brane forming a >, like a junction of pipes. The three arms are wrapped on (p , q ) cycles of the torus. This is only possible topologically when Eq. (32) is G G satis"ed.
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We can now describe a pretty construction of 1/4 BPS states in N"4 gauge theory, due to Bergman [36]. Such a state is described by a 3-string junction, with the three prongs terminating on three di!erent D3-branes. This is only possible for n53, which is a necessary condition for 1/4 BPS states. The mass of such a state is given by summing the lengths of each string segment weighted by its tension. This gives a result in agreement with the BPS formula. Clearly, this is just the beginning of a long story, since the simple picture we have described can be generalized to arbitrarily complicated string webs. So long as the web is in a plane, charges are conserved at the junctions, and all string segments are oriented in the way we have described, the con"guration will be 1/4 BPS. Remarkably, arbitrarily high spins can occur. There are simple rules for determining them [37]. When the web is nonplanar, supersymmetry is completely broken, and reliable mass calculations become di$cult. However, one should still be able to achieve a reliable qualitative understanding of such excitations. In general, there are regions of moduli space in which such nonsupersymmetric states are stable.
6. Conclusion In this brief review we have described some of the interesting advances in understanding superstring theory that have taken place in the past few years. Many others, such as studies of black hole entropy, have not even been mentioned. The emphasis has been on the nonperturbative appearance of an 11th dimension in type IIA superstring theory, as well as its implications when combined with superstring T dualities. In particular, we argued that there should be a consistent quantum vacuum, whose low-energy e!ective description is given by 11D supergravity. The relevant quantum theory } called M theory } has important features, such as the nongeometric U duality described in Section 4, that go beyond what can be understood within ordinary (nonrenormalizable) 11D supergravity. What we have described makes a convincing self-consistent picture, but it does not constitute a complete formulation of M theory. In the past two years there have been some major advances in that direction, which we will brie#y mention here. The "rst, which goes by the name of matrix theory [38], bases a formulation of M theory in #at 11D spacetime in terms of the supersymmetric quantum mechanics of N D0-branes in the large N limit. This proposal has been generalized to include an interpretation for "nite N. In that case Susskind has proposed an identi"cation with discrete light-cone quantization of M theory, in which there are N units of momentum along a null compact direction [39]. Both versions of matrix theory have passed all tests that have been carried out, some of which are very nontrivial. At times there appeared to be discrepancies, but these were all the result of subtle errors that have now been tracked down. The construction has a nice generalization to describe compacti"cation of M theory on a torus ¹L [40]. However, it does not seem to be useful for n'5 [32], and other compacti"cation manifolds are (at best) awkward to handle. Another shortcoming of this approach is that it treats the eleventh dimension di!erently from the other ones. Another proposal relating superstring and M theory backgrounds to large N limits of certain "eld theories has been put forward recently by Maldacena [41] and made more precise by others [42]. In this approach, there is a conjectured duality (i.e., equivalence) between a conformally invariant "eld theory (CFT) in n dimensions and type IIB superstring theory or M theory on an
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Anti-de-Sitter space (AdS) in n#1 dimensions. The remaining 9!n or 10!n dimensions form a compact space, the simplest cases being spheres. The three examples with unbroken supersymmetry are AdS ;S, AdS ;S, and AdS ;S. This approach is sometimes referred to as AdS/CFT duality. This is an extremely active and very promising subject. It has already taught us a great deal about the large N behavior of various gauge theories. As usual, the easiest theories to study are ones with a lot of supersymmetry, but it appears that in this approach supersymmetry breaking is more accessible than in previous ones. For example, it might someday be possible to construct the QCD string in terms of a dual AdS gravity theory, and use it to carry out numerical calculations of the hadron spectrum. Indeed, there have already been some preliminary steps in this direction [43]. Despite all of the successes that have been achieved in advancing our understanding of superstring theory and M theory, there clearly is still a long way to go. In particular, despite much e!ort and several imaginative proposals, we still do not have a convincing mechanism for ensuring the vanishing (or extreme smallness) of the cosmological constant for nonsupersymmetric vacua. Superstring theory is a "eld with very ambitious goals. The remarkable fact is that they still seem to be realistic. However, it may take a few more revolutions before they are attained.
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Physics Reports 315 (1999) 123}135
Neutrino propagation in matter夽 A.B. Balantekin * Institute for Nuclear Theory, University of Washington, Box 351550 Seattle, WA 98195-1550, USA Department of Astronomy, University of Washington, Box 351580 Seattle, WA 98195-1580, USA
Abstract The enhancement of neutrino oscillations in matter is brie#y reviewed. Exact and approximate solutions of the equations describing neutrino oscillations in matter are discussed. The role of stochasticity of the media that the neutrinos propagate through is elucidated. 1999 Elsevier Science B.V. All rights reserved. PACS: 14.60.Pq; 26.30.#k; 26.65.#t; 96.40.Tv Keywords: Neutrino oscillations; The MSW e!ect; Solar neutrinos; Supernova neutrinos
1. Introduction Particle and nuclear physicists devoted an increasingly intensive e!ort during the last few decades to searching for evidence of neutrino mass. Recent announcements by the Superkamiokande collaboration of the possible oscillation of atmospheric neutrinos [1] and very high statistics measurements of the solar neutrinos [2] brought us one-step closer to understanding the nature of neutrino mass and mixings. Experiments imply that neutrino mass is small and the seesaw mechanism [3], to the development of which Dick Slansky contributed, is perhaps the simplest model which leads to a small neutrino mass. If the neutrinos are massive and di!erent #avors mix they will oscillate as they propagate in vacuum [4]. Dense matter can signi"cantly amplify neutrino oscillations due to coherent forward scattering. This behavior is known as the Mikheyev}Smirnov}Wolfenstein (MSW) e!ect [5]. Matter e!ects may play an important role in the solar neutrino problem [6}8]; in transmission of
夽
Expanded version of a talk at the Slansky Memorial Symposium, Los Alamos, May 1998. * Permanent address. Department of Physics, University of Wisconsin, Madison, WI 53706, USA. E-mail address: [email protected] (A.B. Balantekin) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 7 - 4
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solar [8] and atmospheric neutrinos [9,10] through the Earth's core; and shock re-heating [11] and r-process nucleosynthesis [12] in core-collapse supernovae. If the neutrinos have magnetic moments matter e!ects may also enhance spin-#avor precession of neutrinos [13]. The equations of motion for the neutrinos in the MSW problem can be solved by direct numerical integration, which must be repeated many times when a broad range of mixing parameters are considered. This often is not very convenient; consequently various approximations are widely used. Exact or approximate analytic results allow a greater understanding of the e!ects of parameter changes. The purpose of this article is to present a review of the solutions of the neutrino propagation equations in matter. Recent experimental developments and astrophysical implications of the neutrino mass and mixings are beyond the scope of this article. Very rapid developments make a medium such as the World Wide Web more suitable for the former and the latter was recently reviewed elsewhere [14]. Recent experimental developments can be accessed through the special home page at SPIRES [15] and theoretical results at the Institute for Advanced Study [16] and the University of Pennsylvania [17]. An assessment of the Superkamiokande solar neutrino data was recently given by Bahcall et al. [6]. A number of recent reviews cover implications of recent results for neutrino properties [18].
2. Outline of the MSW e4ect The evolution of #avor eigenstates in matter is governed by the equation [5,20]
u(x) R W(x) " i
Rx W (x) (K I where
(K
!u(x)
W (x) , W (x) I
(1)
u(x)"(1/4E)(2(2 G N (x)E!dm cos 2h ) $ for the mixing of two active neutrino #avors and
(2)
u(x)"(1/4E)(2(2 G [N (x)!N (x)/2]E!dm cos 2h ) $ for the active-sterile mixing. In these equations
(3)
(K"(dm/4E) sin 2h , (4) dm,m!m is the vacuum mass-squared splitting, h is the vacuum mixing angle, G is the $ Fermi constant, and N (x) and N (x) are the number density of electrons and neutrons, respective ly, in the medium. In a number of cases adiabatic basis greatly simpli"es the problem. By making the change of basis
W (x) cos h(x) " W (x) sin h(x)
!sin h(x)
cos h(x)
W (x) , W (x) I
(5)
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the #avor-basis Hamiltonian of Eq. (1) can be instantaneously diagonalized. The matter mixing angle in Eq. (5) is de"ned via sin 2h(x)"(K/((K#u(x))
(6)
cos 2h(x)"!u(x)/((K#u(x)) .
(7)
and
In the adiabatic basis the evolution equation takes the form
!(K#u(x) R W(x) i
" Rx W (x) i h (x)
!i h (x)
(K#u(x)
W (x) , W (x)
(8)
where prime denotes derivative with respect to x. Since the 2;2 `Hamiltoniana in Eq. (8) is an element of the SU(2) algebra, the resulting time-evolution operator is an element of the SU(2) group. Hence it can be written in the form [19]
;"
W (x) W (x)
!WH(x) , WH(x)
(9)
where W (x) and W (x) are solutions of Eq. (8) with the initial conditions W (x )"1 and W (x )"0. If the matter mixing angle, h(x), is changing very slowly (i.e., adiabatically) its derivatives in Eq. (8) can be set to zero. In this approximation the `Hamiltoniana in the adiabatic basis is diagonal and the system remains in one of the matter eigenstates. To calculate the electron neutrino survival probability Eq. (1) needs to be solved with the initial conditions W "1 and W "0. Using Eq. (9) the general solution satisfying these initial conditions I can be written as W (x)"cos h(x)[cos h W (x)!sin h WH(x)]#sin h(x)[cos h W (x)#sin h WH(x)] ,
(10)
where h is the initial matter angle. Once the neutrinos leave the dense matter (e.g. the Sun), the solutions of Eq. (8) are particularly simple. Inserting these into Eq. (10) we obtain the electron neutrino amplitude at a distance ¸ from the solar surface to be W (¸)"cos h [cos h W !sin h WH ]exp(i(dm/4E)¸) 1 1 # sin h [cos h W !sin h WH ]exp(!i(dm/4E)¸) , 1 1
(11)
where W and W are the values of W (x) and W (x) on the solar surface. The electron neutrino 1 1 survival probability averaged over the detector position, ¸, is then given by P(l Pl )"1"W (¸)"2 "# cos 2h cos 2h (1!2"W ") * 1 ! cos 2h sin 2h (W W #WH WH ) . 1 1 1 1
(12)
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If the initial density is rather large, then cos 2h &!1 and sin 2h &0 and the last term in Eq. (12) is very small. Di!erent neutrinos arriving the detector carry di!erent phases if they are produced over an extended source. Even if the initial matter density is not very large, averaging over the source position makes the last term very small as these phases average to zero. The completely averaged result for the electron neutrino survival probability is then given by [21] P(l Pl )"# cos 2h 1cos 2h 2 (1!2P ) ,
(13)
where the hopping probability is P ""W " , 1
(14)
obtained by solving Eq. (8) with the initial conditions W (x )"1 and W (x )"0. Note that, since in the adiabatic limit W remains to be zero P "0. 1 3. Exact solutions Exact solutions for the neutrino propagation equations in matter exist for a limited class of density pro"les that satisfy an integrability condition called shape invariance [22]. To illustrate this integrability condition we introduce the operators AK "i R/Rx!u(x), \
AK "i R/Rx#u(x) . >
(15)
Using Eq. (15), Eq. (1) takes the form AK W (x)"(KW (x), I \
AK W (x)"(KW (x) . > I
(16)
The shape invariance condition can be expressed in terms of the operators de"ned in Eq. (15) [23] AK (a )AK (a )"AK (a )AK (a )#R(a ) . \ > > \
(17)
We also introduce a similarity transformation which formally replaces a by a : ¹K (a )O(a )¹K \(a )"O(a ) .
(18)
The MSW equations take a particularly simple form using the operators [24] BK "AK (a )¹K (a ), > >
BK "¹K \(a )AK (a ) , \ \
(19)
which satisfy the commutation relation [BK , BK ]"R(a ) , \ >
(20)
where a is de"ned using the identity R(a )"¹K (a )R(a )¹K \(a ) L L\
(21)
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127
with n"1. Two additional commutation relations [BK BK , BK L ]"(R(a )#R(a )#2#R(a ))BK L , > \ > L >
(22)
[BK BK , BK \L]"(R(a )#R(a )#2#R(a ))BK \L > \ \ L \
(23)
and
can easily be proven by induction. Using the operators introduced in Eq. (19), Eq. (1) can be rewritten as BK BK W (x)"KW (x) . > \
(24)
Eqs. (22) and (23) suggest that BK and BK can be used as ladder operators to solve Eq. (24). > \ Introducing
W&exp !i u(x; a ) dx , \
(25)
one observes that AK (a )W"0"BK W . \ \ \ \
(26)
If the function L f (n)" R(a ) I I
(27)
can be analytically continued so that the condition f (k)"K
(28)
is satis"ed for a particular (in general, complex) value of k, then Eq. (22) implies that one solution of Eq. (24) is BK I W. Similarly, the wave function > \
W& exp #i u(x; a ) dx , >
(29)
satis"es the equation BK W"0 . > >
(30)
Then a second solution of Eq. (24) is given by BK \I\W. Hence for shape invariant electron \ > densities the exact electron neutrino amplitude can be written as [24]
W (x)"bBK I exp !i u(x; a ) dx #cBK \I\ exp #i u(x; a ) dx , \ >
(31)
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where b and c are to be determined using the initial conditions W (x )"1 and W (x )"0. For the linear density pro"le N (x)"N !N (x!x ) , 0 where N is the resonant density: 2(2G N E"dm cos 2h , $ using the technique described above we can easily write down the hopping probability
(32)
P ""W (x )""exp(!pX) , where
(34)
dm sin 2h N . X" 4E cos 2h N This is the standard Landau}Zener result [21,25]. For the exponential density pro"le N (x)"N e\?V\V0 , where N is the resonant density given in Eq. (33), the hopping probability is [26] P "(e\pd\ F!e\pd)/(1!e\pd) , where we de"ned dm d" . 2Ea
(33)
(35)
(36)
(37)
(38)
4. Supersymmetric uniform approximation The coupled "rst-order equations for the #avor-basis wave functions can be decoupled to yield a second-order equation for only the electron neutrino propagation ! RW (x)/Rx![K#u(x)#i u (x)]W (x)"0 . (39) The large body of literature on the second-order di!erential equations of mathematical physics motivates using a semiclassical approximation for the solutions of Eq. (39). The standard semiclassical approximation gives the adiabatic evolution [27]. For a monotonically changing density pro"le supersymmetric uniform approximation yields [28]
i dm PH dr[f(r)!2f(r) cos 2h #1] , (40) P "exp(!pX), X" p 2E P where rH and r are the turning points (zeros) of the integrand. In this expression we introduced the scaled density f(r)"2(2G N (r)/dm/E , $
(41)
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129
where N is the number density of electrons in the medium. By analytic continuation, this complex integral is primarily sensitive to densities near the resonance point. The validity of this approximate expression is illustrated in Fig. 1. As this "gure illustrates the approximation breaks down in the extreme non-adiabatic limit (i.e., as dmP0). Hence it is referred to as the quasi-adiabatic approximation. The near-exponential form of the density pro"le in [29] motivates an expansion of the electron number density scale height, r , in powers of density: N (r) (42) !r , " b NL , L N (r) L where prime denotes derivative with respect to r. In this expression a minus sign is introduced because we assumed that density pro"le decreases as r increases. (For an exponential density pro"le, N &e\?V, only the n"0 term is present). To help assess the appropriateness of such an expansion the density scale height for the Sun calculated using the Standard Solar Model density pro"le is plotted in Fig. 2. One observes that there is a signi"cant deviation from a simple exponential pro"le over the entire Sun. However the expansion of Eq. (42) needs to hold only in the
Fig. 1. The electron neutrino survival probability for the Sun [28]. The solid line is calculated using Eq. (40). The dashed line is the exact (numerical) result. The dotted line is the linear Landau}Zener result. In the top "gure, the lines are indistinguishable. An exponential density with parameters chosen to approximate the Sun was used [29]. Fig. 2. Electron number density scale height (cf. Eq. (42)) as a function of the radius for the Sun [29]. The dashed line is the exponential "t over the whole Sun. The shaded are indicates where the small angle MSW resonance takes place for neutrinos with energies 5(E(15 MeV.
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MSW resonance region, indicated by the shaded area in the "gure. Real-time counting detectors such as Superkamiokande and Sudbury Neutrino Observatory, which can get information about energy spectra, are sensitive to neutrinos with energies greater than about 5 MeV. For the small angle solution (sin 2h&0.01 and dm"5;10\ eV), the resonance for a 5 MeV neutrino occurs at about 0.35R and for a 15 MeV neutrino at about 0.45R (the shaded area in the "gure). In that > > region the density pro"le is approximately exponential and one expects that it should be su$cient to keep only a few terms in the expansion in Eq. (42) to represent the density pro"le of the Standard Solar Model. Inserting the expansion of Eq. (42) into Eq. (40), and using an integral representation of the Legendre functions, one obtains [30]
dm L b dm L [P (cos 2h )!P (cos 2h )] , b (1!cos 2h )# X"! L> 2n#1 L\ 2E 2(2G E L $
(43)
where P is the Legendre polynomial of order n. The n"0 term in Eq. (43) represents the L contribution of the exponential density pro"le alone. Eq. (43) directly connects an expansion of the logarithm of the hopping probability in powers of 1/E to an expansion of the density scale height. That is, it provides a direct connection between N (r) and P (E ). Eq. (43) provides a quick and J J accurate alternative to numerical integration of the MSW equation for any monotonically changing density pro"le for a wide range of mixing parameters. The accuracy of the expansion of Eq. (43) is illustrated in Fig. 3 where the spectrum distortion for the small angle MSW solution is plotted. In this calculation we used the method of Ref. [31] and neglected backgrounds. The neutrino-deuterium charged-current cross-sections were calculated using the code of Bahcall and Lisi [32]. One observes that for the Sun, where the density pro"le is nearly exponential in the MSW resonance region, the "rst two terms in the expansion provide an excellent approximation to the neutrino survival probability.
5. Neutrino propagation in stochastic media In implementing the MSW solution to the solar neutrino problem one typically assumes that the electron density of the Sun is a monotonically decreasing function of the distance from the core and ignores potentially de-cohering e!ects [33]. To understand such e!ects one possibility is to study parametric changes in the density or the role of matter currents [34]. In this regard, Loreti and Balantekin [35] considered neutrino propagation in stochastic media. They studied the situation where the electron density in the medium has two components, one average component given by the Standard Solar Model or Supernova Model, etc, and one #uctuating component. Then the Hamiltonian in Eq. (1) takes the form HK "((!dm/4E) cos 2h#(1/(2)G (N (r)#NP (r)))p #((dm/4E) sin 2h)p , $ X V where one imposes for consistency 1NP (r)2"0
(44)
(45)
A.B. Balantekin / Physics Reports 315 (1999) 123}135
131
Fig. 3. (a) Spectrum distortion at SNO for the small-angle MSW solution (dm&5;10\ eV and sin 2h&0.01). The solid line is the exact numerical solution. The dashed, dot}dashed, and dotted lines result from values of n up to 0, 1, and 2 in Eq. (43). The error bars on the exact numerical result correspond to two and "ve years of data collection. The dot}dot}dot}dashed line is the spectrum without MSW oscillations, normalized to the same total rate as with MSW oscillations. Note that on the scale of this "gure the n"1 and 2 lines are not distinguishable from the exact answer. (b) The relative error arising from the use of Eq. (43).
and a two-body correlation function 1NP (r)NP (r )2"bN (r)N (r ) exp(!"r!r "/q ) .
(46)
In the calculations of the Wisconsin group the #uctuations are typically taken to be subject to colored noise, i.e. higher-order correlations f 2"1NP (r )NP (r )22
(47)
are taken to be f "f f #f f #f f
(48)
and so on. Mean survival probability for the electron neutrino in the Sun is shown in Fig. 4 [36] where #uctuations are imposed on the average solar electron density given by the Bahcall}Pinsonneault model. One notes that for very large #uctuations complete #avor de-polarization should be achieved, i.e. the neutrino survival probability is 0.5, the same as the vacuum oscillation probability for long
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Fig. 4. Mean electron neutrino survival probability in the sun with #uctuations. The average electron density is given by the Standard Solar Model of Bahcall and Pinsonneault [37] and sin 2h"0.01.
distances. To illustrate this behavior the results from the physically unrealistic case of 50% #uctuations are shown. Also the e!ect of the #uctuations is largest when the neutrino propagation in their absence is adiabatic. This scenario was applied to the neutrino convection in a corecollapse supernova where the adiabaticity condition is satis"ed [38]. Similar results were also obtained by other authors [39}42]. It may be possible to test solar matter density #uctuations at the BOREXINO detector currently under construction [43]. Propagation of a neutrino with a magnetic moment in a random magnetic moment has also been investigated [35,44]. Also if the magnetic "eld in a polarized medium has a domain structure with di!erent strength and direction in di!erent domains, the modi"cation of the potential felt by the neutrinos due polarized electrons will have a random character [45]. Using the formalism sketched above, it is possible to calculate not only the mean survival probability, but also the variance, p, of the #uctuations to get a feeling for the distribution of the survival probabilities [36] as illustrated in Fig. 5. In these calculations the correlation length q is taken to be very small, of the order of 10 km, to be consistent with the helioseismic observations of the sound speed [46]. In the opposite limit of very large correlation lengths are very interesting result is obtained [38], namely the averaged density matrix is given as an integral
1 dx exp[!x/(2b)]o( (r, x) , (49) lim 1o( (r)2" (2nb \ O reminiscent of the channel-coupling problem in nuclear physics [47]. Even though this limit is not appropriate to the solar #uctuations it may be applicable to a number of other astrophysical situations.
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Fig. 5. Mean electron neutrino survival probability plus minus p in the sun with #uctuations. The average electron density is given by the Standard Solar Model of Bahcall and Pinsonneault and sin 2h"0.01. Panels (a), (b), (c), and (d) correspond to an average #uctuation of 1%, 2%, 4%, and 8%, respectively.
Acknowledgements This work was supported in part by the U.S. National Science Foundation Grant No. PHY9605140 at the University of Wisconsin, and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. I thank Institute for Nuclear Theory and Department of Astronomy at the University of Washington for their hospitality and Department of Energy for partial support during the completion of this work.
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[40] C.P. Burgess, D. Michaud, Ann. Phys. 256 (1997) 1; P. Bamert, C.P. Burgess, D. Michaud, Nucl. Phys. B 513 (1998) 319. [41] E. Torrente-Lujan, hep-ph/9602398; hep-ph/9807361; hep-ph/9807371; Phys. Lett. B 441 (1998) 305. [42] K. Hirota, Phys. Rev. D 57 (1998) 3140. [43] H. Nunokawa, A. Rossi, V. Semikoz, J.W.F. Valle, Nucl. Phys. B Proc. Suppl. 70 (1999) 345. [44] S. Pastor, V.B. Semikoz, J.W.F. Valle, Phys. Lett. B 369 (1996) 301; S. Sahu, Phys. Rev. D 56 (1997) 4378; S. Sahu, V.M. Bannur, hep-ph/9806427. [45] H. Nunokawa, V.B. Semikoz, A.Yu. Smirnov, J.W.F. Valle, Nucl. Phys. B 501 (1997) 17. [46] J.N. Bahcall, M.H. Pinsonneault, S. Basu, J. Christensen-Dalsgaard, Phys. Rev. Lett. 78 (1977) 17. [47] A.B. Balantekin, N. Takigawa, Rev. Mod. Phys. 70 (1998) 441.
Physics Reports 315 (1999) 137}152
M(ysterious) patterns in SO(9) Teparksorn Pengpan*, Pierre Ramond Institute for Fundamental Theory, Department of Physics, University of Florida, Gainesville FL 32611, USA
Abstract The light-cone little group, SO(9), classi"es the massless degrees of freedom of 11-dimensional supergravity, with a triplet of representations. We observe that this triplet generalizes to four-fold in"nite families with the quantum numbers of massless higher spin states. Their mathematical structure stems from the three equivalent ways of embedding SO(9) into the exceptional group F . 1999 Elsevier Science B.V. All rights reserved. PACS: 0.2.20.Sv; 04.65.#e; 11.30.Pb; 12.60.Jv
1. N ⴝ 1 supergravity in eleven dimensions It has been recently pointed out that 11-dimensional supergravity is the local limit of a much bigger theory, called M-theory [1], that also contains in di!erent limits all known string theories in 10 dimensions. At present, it is still elusive, and only a partial formulation [2] exists in the literature. Since M-theory lives in 11 dimensions, its massive degrees of freedom must be expressible as multiplets of SO(10), the Lorentz little group of eleven dimensions and its massless degrees of freedom must form in representations of SO(9). Among those are the "elds of the local supergravity theory which reveal themselves in the local limit. While it is likely that some of the physical objects in M-theory are not local, one still expects that they would be expressible in terms of in"nite towers of representations of these little groups. There is a pervading lore against interacting theories that contain massless states of higher spin. It is based on several no-go theorems, formulated in terms of local "eld theory [3]. They state that relativistically invariant theories with a "nite number of local massless "elds of spin higher than
* Corresponding author. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 8 - 6
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two, and with a "nite number of derivatives in their interactions, do not exist. It follows that any such theory with an in"nite tower of "elds, and arbitrarily high derivative couplings escapes the no-go theorems and could conceivably exist. Even with supersymmetry, building such a theory seems like a hopeless task, and the many published attempts have met with partial success. A four-dimensional formulation [4] uses an in"nite-dimensional superalgebra, with the interesting feature that it necessarily contains a cosmological constant. Hence it seems that the lore against massless high-spin interacting theories is mainly based on the di$culties associated with their construction rather than on their impossibility. Since M-theory is most likely non-local, it may evade the no-go theorems, and could contain an in"nite number of "elds. It is therefore interesting to examine the SO(9) properties of 11-dimensional supergravity, whose massless states are local limit of M-theory. In the following, we would like to draw attention to a remarkable mathematical fact, which shows that the supergravity triplet of SO(9) representations is actually the tip of a mathematical iceberg. We will start by presenting group-theoretical evidence that the supergravity representations are the "rst of an in"nite family of massless states of higher spin. Then we will o!er a mathematical resolution in terms of embeddings of SO(9) into the exceptional group F , as well as some generalizations. Since there are no coincidences in the study of these highly constrained theories, it is tempting to muse that these extra higher-spin massless states represent the degrees of freedom of M-theory, even though we have not been able to obtain any dynamical evidence for this conjecture.
2. Group phenomenology of SO(9) The classical Lie group SO(9) plays an important dual role in the study of theories in ten and eleven dimensions, as the light-cone little group of Lorentz-invariant theories in 10 space and one time dimensions, and as the little group of massive representations of theories in 9 space and one time dimensions. The representations of SO(9) are best described in Dynkin's language, which Dick Slansky used to great e!ectiveness in particle physics [5]. As a rank 4 Lie algebra, it takes four positive integers to label its irreducible representations, in the form [a a a a ]. Its four basic representations are: E Vector, [1000], with nine components, < , G E Adjoint, [0100], with 36 components, B , GH
E Three-form, [0010], with 84 components, B , GHI
E Spinor, [0001], with 16 components, t . ? All representations with odd a are spinorial. The irreps of SO(9) are characterized by "ve generalized Dynkin indices I , wN, p"0, 2, 4, 6, 8 , (1) N where w are the weights in the representation. Thus I is the dimension of the irrep, and I is related to the quadratic Casimir invariant by C "36I /I .
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139
Table 1 Irrep
[1001]
[2000]
[0010]
I I I I I
128 256 640 1792 5248
44 88 232 712 2440
84 168 408 1080 3000
N"1 supergravity in 11 dimension is a local "eld theory that contains three di!erent massless "elds, two bosonic that describe gravity and a three-form, and one Rarita}Schwinger spinor. Its physical degrees of freedom are classi"ed in terms of the light-cone little group, SO(9), E Graviton as a symmetric second-rank tensor, [2000], G , GH E Third-rank antisymmetric tensor, [0010], B , GHI
E Rarita}Schwinger spinor-vector, [1001], W . ?G Their group-theoretical properties are summarized in Table 1 [6]. We note that these indices, except for I , match between the fermion and the two bosons. As is well known, equality of the bosonic and fermionic dimensions is an indication of supersymmetry. On the light-cone, the supersymmetry algebra reduces to +Q , Q ,"d , (2) ? @ ?@ where the supersymmetric generators transform as the 16 spinor of SO(9). They split into creation and annihilation operators under the decomposition SO(9)MSO(6);SO(3), 16"(4, 2)#(4, 2)
(3)
and we obtain a Cli!ord algebra +QI , QI R,"1 ,
(4)
where QI transforms as (4, 2), and QI R as (4, 2). The states of the Sugra multiplet are then obtained by successive applications of the QI R on the vacuum state, to yield 128 bosons and 128 fermions +1, QI R,(QI R),2, (QI R), (QI R),"02 .
(5)
The equality between the number of bosons and fermions is manifest. All three irreps have the same quadratic Casimir invariant, since they have the same I /I ratio. Surprisingly, we have found that some higher spin representations of SO(9) also occur in triples with the same quadratic Casimir invariant, and show remarkable group-theoretical kinships with the supergravity triplet. The higher-spin triplets appear in four di!erent types, S}¹}¹, S}S}S, ¹}¹}S, and ¹}S}¹, where S describes fermionic (odd a ), and ¹ bosonic (even a ) degrees of freedom. The largest representation is listed "rst, and its dimension is equal to the sum of dimensions of the other two irreps of the triplet. Thus only triplet of the S}¹}¹ type display supersymmetry-like properties.
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Table 2 Irrep
[2100]
[0110]
[1101]
I I I I I
910 3640 19 864 130 840 977 944
1650 6600 34 920 217 320 1 498 344
2560 10 240 54 784 348 160 2 466 304
2.1. S}T}T triples In Dynkinese, these triples are of the form [1#p#2r, n, p, 1#2q#2r] [2#p#2q#2r, n, p, 2r] [p, n, 1#p#2r, 2q] ,
(6)
labelled by four integers, n, p, q, r"1, 2,2; the sum of the Dynkin invariants I , and I , I , I , over the bosons match those of the fermion representation. All three have the same quadratic Casimir invariant. The simplest of this class is the supergravity multiplet which we have already discussed, and only the lowest of these triples has manifest supersymmetry. The number of fermions and bosons of each triplets are equal, and a multiple of 128, but their construction does not follow that of the supergravity multiplet as polynomials of QI R acting on some state. They appear to be supersymmetric without supersymmetry, the simplest being: E The supertriple, with n"1, p"q"r"0, contains [2100] #[0110] #[1101] , with group-theoretic numbers given by Table 2 and described by "elds of the form
(7)
h #A #W . (8) GHIJ GHIJK ?GHI Their index structure indicates the appearance of higher spin "elds. It is not possible to generate this triple by repeated use of the light-cone supersymmetry algebra acting on some "eld "j2, with dimension equal to 20 (1, QI R, (QI R),2, (QI R), (QI R))"j2 .
(9)
This would imply that "j2 appears twice in the triple, but the triple contains no duplicate representations of SO(6);SO(3) that add up to dimension equal to 20. These "elds appear in the Kronecker product of the supergravity triplet with the two-form [0100], [2000] [0100]"[2100] [2000] +[1010] [0100], ,
(10)
[0010] [0100]"[0110] [0010] +[1002] [1000] [0110] [1100] [0002], ,
(11)
[1001] [0100]"[1101] [1001] +[2001] [1101] [1001] [0101] [0011] [0001], .
(12)
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Table 3 Irrep
[3010]
[1020]
[2011]
I I I I I
7700 46 200 384 360 3 938 760 46 646 664
12 012 72 072 585 624 5 748 792 64 127 736
19 712 118 272 969 984 9 687 552 110 529 792
Irrep
[4000]
[0012]
[1003]
I I I I I
450 2200 15 160 130 360 1 325 944
4158 20 328 131 784 1 016 520 8 839 560
4608 22 528 146 944 1 146 880 10 103 296
Table 4
A suitable product can be found that automatically traces out the extra representations (in the curly brackets), subtracts all traces, and all totally antisymmetric tensors. The states of this triple could be understood as some sort of bound state between the supergravity states and something having the Lorentz properties of a 2- or 7-form in the light cone little group. Whether this union can be consumated through actual dynamics remains to be seen. Although this feature usually associated with supersymmetry remain, it is not a supermultiplet. E The second tower of triples is obtained by multiplying all its representations by [1010], and performing suitable subtractions. This representation appears in the antisymmetric product of two second-rank antisymmetric tensor "elds. It could therefore be generated by applying two two-forms on the supergravity multiplet, resulting in a bound state between supergravity and two branes. The simplest with p"1, n"q"r"0, contains [3010] #[1020] #[2011] , (13) with group theory table (Table 3). E The third tower is more complicated, multiplying the fermion and the second boson by [0002], and the "rst boson by [2000]. The simplest in this series, with q"1 n"p"r"0, and contains (see Table 4) [4000] #[0012] #[1003] . (14) E The fourth in"nite tower is also twisted. The simplest of this series has r"1, with content [4002] #[0030] #[3003] and group-theory mugshot (Table 5).
(15)
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Table 5 Irrep
[4002]
[0030]
[3003]
I I I I I
32 725 261 800 2 938 280 41 127 080 673 801 256
23 595 188 760 2 055 768 27 239 256 414 212 568
56 320 450 560 4 994 048 68 366 336 1 084 279 808
Table 6 Irrep
[2003]
[4001]
[0021]
I I I I I
18 480 117 040 1 010 992 10 640 944 128 166 448
5280 33 440 297 632 3 303 584 43 030 688
13 200 83 600 713 360 7 337 360 85 922 192
2.2. S}S}S triples Here all three representations are spinors [2#p#2r, n, p, 3#2q#2r] [4#p#2q#2r, n, p, 1#2r] [p, n, 2#p#2r, 1#2q] (16) and the dimension of the "rst is the sum of the other two. The simplest example is [2003] [4001] [0021] .
(17)
Its group-theory mugshot is given in Table 6. 2.3. T}T}S triples In this class, the dimension of the largest boson (listed "rst) is equal to that of the spinor and the second boson, [1#p#2r, n, p, 2#2q#2r] [3#p#2q#2r, n, p, 2r] [p, n, 1#p#2r, 1#2q] .
(18)
The lowest member of this class is [1002] [3000] [0011] with mugshot (Table 7).
(19)
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Table 7
Irrep
[1002]
[3000]
[0011]
I I I I I
924 3080 13 400 68 216 3 82 328
156 520 2392 13 432 87 544
768 2560 11 008 54 784 2 99 776
Irrep
[2002]
[3001]
[0020]
I I I I I
3900 18 200 1 14 920 8 75 720 7 549 064
1920 8960 57 728 4 55 936 4 148 096
1980 9240 57 192 4 19 784 34 53 384
Table 8
2.4. T}S}T triples The last class contains the representations [2#p#2r, n, p, 2#2q#2r] [3#p#2q#2r, n, p, 1#2r] [p, n, 2#p#2r, 2q] .
(20)
Its lowest-lying member is [2002] [3001] [0020]
(21)
with mugshot (Table 8). There are several triples which only match dimensions and quadratic Casimir invariants; we found one made entirely of spinors [1033] [7001] [0305],
[7122] [6008] [4018]
(22)
with the dimension of the "rst equal to the sum of the other two, and all with the same quadratic Casimir, but their I do not match. 2.5. Basic operations It is possible to understand these di!erent triples in terms of four basic operations, which starting from the supergravity multiplet, generate all triples: E D : Increase the Dynkin labels all three irreps within a triple by [0100]. E D : Increase the Dynkin labels of all three irreps within a triple by [1010].
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E D : Increase the Dynkin labels of the "rst and third irreps by [0001], the second by [1000]. E D : Increase the Dynkin labels of the "rst and second irreps by [1001], the third by [0010]. The D operations may be simplest to understand as they can be generated by applying representations that appear either as the light-cone 2-form [0100], or in its twice-antisymmetrized product, since ([0100] [0100]) "[0100] [1010] . (23) A light-cone 2-form may indicate a brane state, and these triples could then be understood as bound states of the supergravity "elds with these branes. The third and fourth operations are more complicated as they treat the di!erent members di!erently. However, starting from the supergravity multiplet, they generate all other triples, as shown in the diagram below, where the upward arrow denotes D , and the downward arrow denotes D :
It is clear that the supergravity multiplet sits at the beginning of a very intricate and beautiful complex of irreps of SO(9). Limited by the two dimensions of the paper, we have not shown the e!ect of the D operations which act uniformly on any of the triples in the picture. The whole pattern is summarized by the general form of the triples [1#a #a , a , a , 1#a #a ] [2#a #a #a , a , a , a ] [a , a , 1#a #a , a ] , where a are non-negative integers. G
(24)
3. Mathematical origin of the triples So far we have only o!ered numerical evidence for the remarkable structure of the SO(9) representations. A recent paper by Gross et al. [7] and one of us (P.R.), unveils its mathematical
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145
origin. The following is a watered-down version of its contents. It points to a construction of a more general character, but does not (yet) seem to shed light on its physical interpretation. The triples stem from the triality of SO(8), which is explicitly realized in F , and the three equivalent ways to embed SO(9) into F . That very triality is already familiar to particle physicists: the three equivalent ways to embed SU(2);;(1) in SU(3), called I-spin, ;-spin, and <-spin [9]. The general idea behind the mathematical construction goes as follows. Let F and B be two Lie algebras of equal rank such that FMB. The Weyl group of F, W(F), is bigger than that of B, W(B), with r-time as many operations. The fundamental Weyl chamber of F, is the sliver of weight space that contains all weights with positive or zero Dynkin labels; it is r times smaller than the fundamental Weyl chamber of the subgroup B. Let j be the highest weight of an irrep of F; it lies either inside or at the boundary of the Weyl chamber. We can choose r operations of W(F) not in W(B) which map the fundamental Weyl chamber of F into that of B. When applied to this highest weight, they produce r copies inside the chamber of B (unless the weight is at the chamber boundary). In order to make sure it is inside the chamber, we add to it the weight o"[1, 1, 1,2,1], the Weyl vector (or half sum of positive roots). Then we are sure the Weyl group will act on this weight non-trivially. We now construct the r weights j ,w (j#o )!o , i"1, 2,2, r , G G $
(25)
where w are the operations W(F) not in W(B). They all lie on the fundamental Weyl chamber of B, G and on its boundary, and therefore describe an r-plet of irreps of B. Apply this reasoning to the case F MSO(9). The Weyl group of F has dimension 1152, that of SO(9), 384 so that r"3. Starting from any representation of F this construction generates a triplet of representations of SO(9). There remains to identify those three elements of the Weyl group, the reason for the relations among their invariants and the emergence of supersymmetry in the construction. To understand the origin of the triality in F , the octonion language is convenient since the adjoint of F is generated by antihermitian traceless 3;3 matrices over the octonions, supple mented by their automorphism group, G . Triality is then related to the three inequivalent ways of picking out one of the matrix's o!-diagonal elements, and this construction generalizes to the Lie algebras of the magic square [8]. Octonions, together with real numbers, R, complex numbers, C, quaternions, Q, are the four Hurwitz (division) algebras. 3;3 matrices with elements belonging to these algebras generate interesting mathematical structures. E For real numbers, these matrices generate the Lie algebra SO(3). Its maximal subgroup is SO(2). E For complex numbers, they generate the Lie algebra A &SU(3), and singling out one of the three o!-diagonal elements picks out the subgroup SU(2);;(1)&SO(3);SO(2). E For quaternions, together with their automorphism group A &SU(2), they generate C &Sp(6). Two o!-diagonal elements are treated equally by the subgroup Sp(4);Sp(2)& SO(5);SO(3). E With octonions, and their automorphism group G , they generate the exceptional group F . Its subgroup B &SO(9) naturally picks out one of the three o!-diagonal elements.
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Under F MSO(9), its adjoint breaks up as 52"36#16, where 36 is the adjoint of SO(9) and 16 its spinor representation. Another way to look at this embedding is to say that it generates a 16dimensional coset space acted on by the orthogonal group SO(16). It yields the anomalous embedding SO(16)MSO(9) according to which the spinor of SO(9) "ts in the vector of SO(16).
4. The magic square Starting from the four Hurwitz algebras, it is possible to construct the so-called composition algebras, which include all the exceptional groups, except G , the automorphism group of the octonions. That construction [8] relies on the triality of both SO(8) and on the structure of 3;3 matrices. Start from 3;3 antihermitian traceless matrices with elements over the product of any two of the four Hurwitz algebras, with three o!-diagonal elements, and two diagonal elements which are pure imaginary. They are acted on by the automorphism groups of the algebras, and each of their parameters generate a Lie algebra transformation, to produce one of the 10 Lie algebras in the magic square (Table 9). In particular, the exceptional group F is generated by 3;3 traceless antihermitian matrices over the octonions, together with G , the automorphism group of the octonion multiplication table. This produces the 3;8#2;7#14"52 parameters of the algebra. Each of the algebras appearing in the magic square have subalgebras of equal rank, and we can apply our mathematical construction to each. The results are summarized in Table 10 below, "rst for the exceptional groups, and then for the non-exceptional groups in the square (Table 11). The connection with supersymmetry occurs through the generation of representations of the subgroup in terms of polynomials in Cli!ord charges.
Table 9
R C Q O
R
C
Q
O
SO(3) SU(3) Sp(6) F
SU(3) SU(3);SU(3) SU(6) E
Sp(6) SU(6) SO(12) E
F E E E
Table 10 Group
Subgroup
Coset dimension
r
E E E F
SO(16) SO(12);SO(3) SO(10);SO(2) SO(9)
128 64 32 16
135 63 27 3
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Table 11 Group
Subgroup
Coset dimension
r
SO(12) SU(6) Sp(6) SU(3);SU(3) SU(3)
SO(8);SO(4) SO(6);SO(3);SO(2) Sp(4);Sp(2) SO(3);SO(3);SO(2);SO(2) SO(3);SO(2)
32 16 8 16 4
30 15 3 3 3
4.1. SU(3) triples By studying the simplest non-trivial example, of the embedding of SU(2);;(1) into SU(3), the emergence of supersymmetry in our general construction should become clear. We begin with some well-known facts about SU(3), the algebra generated by 3;3 antihermitian traceless matrices over the complex numbers. Its maximal subgroup is SU(3)MSU(2);;(1)&SO(3);SO(2) .
(26)
There are three equivalent ways to imbed this subalgebra in SU(3), corresponding, in particle physics language, to I-spin, ;-spin and <-spin. These three embeddings can be understood in terms of the Weyl group. The Weyl group of SU(3) is S , the permutation group on three objects. It is three times as big as the Weyl group of its maximal subgroup SO(3);SO(2). All the states spanned by this algebra live in a two-dimensional lattice. In the Dynkin basis, the weights are labelled by two integers [a , a ]. The action of the Weyl group is simplest acting on an orthonormal basis +e , in which the roots have components which are half-integers between !2 G and 2. The simple roots of SU(3) are given in terms of the three vectors e which span a three? dimensional the Euclidean space a "e !e ,
a "e !e .
(27)
A weight is labelled in the orthonormal basis by three numbers, (b , b , b ), such that b #b #b "0. It can also be expressed in the Dynkin basis as w"a u #a u ,
(28)
where the fundamental vectors u are determined through 2(a , u )/(a , a )"d . G H G G GH
(29)
It follows that the same weight has the orthonormal basis components (30) b "(2a #a ), b "(a !a ), b "!(a #2a ) . The action of the Weyl group on the orthonormal components is just the permutation group on the three b 's. Furthermore, the fundamental Weyl chamber is described by the inequalities G
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b 'b 'b . We will use the Weyl vector, de"ned as half the sum of the positive roots: o "(1, 0,!1)&[1, 1] , (31) 13 where the square brackets indicate the Dynkin basis and the curved brackets the orthonormal basis. Start from an irrep of SU(3), labelled by its maximum Dynkin weight [a , a ], or orthonormal components (b , b , b ), and add the Weyl vector, so that the resulting weight is inside the chamber. Since the Weyl group of SU(3) is three times as large as that of SU(2), the fundamental Weyl chamber of SU(2), is three times as large as that of SU(3). Identify the three Weyl transformations (permutations) that produce a weight in the fundamental Weyl chamber of SU(2), de"ned by b 5b . These yield three weights (b #1, b , b !1), (b #1, b !1, b ), (b , b !1, b #1) . (32) Next, we subtract the SU(2) Weyl vector, o "(,!)&[1], and revert to the Dynkin basis. 13 This construction yields the SU(3)-generated triples, with the ;(1)&SO(2) charge indicated as a subscript [a #a #1] [a ] . (33) SU(3): [a ] ?\? ?>?> \ ?>?> The three Weyl operations are best identi"ed by their action on the fundamental irrep of SU(3), u, d, s: w : (u, d, s)P(u, d, s) , (34) w : (u, d, s)P(u, s, d) , (35) w : (u, d, s)P(s, u, d) . (36) Starting from the singlet of SU(3), with a "a "0, we get the simplest triple [0] [1] [0] . (37) \ This triplet of representations forms a supersymmetric multiplet, generated by a charge that is a doublet under the SU(2) and has ;(1) value of !. To understand its origin, note that the group acting on the four-dimensional coset SU(3)/SU(2);;(1) is SO(4),SO(3);SO(3), de"ning an embedding of SO(4)MSU(2);;(1). The states of this lowest triple correspond to the decomposition of the SO(4) spinor. An obvious interpretation is to view SO(3);SO(2) as the compact subgroup as the light cone little group of either SO(4, 1);SO(2), or SO(3);SO(3, 1), which lead to theories in d"5 and 4 dimensions, respectively. Two fundamental operations generate the higher triples, in one to one correspondance with the rank of the mother algebra SU(3). The operation that increases a by two, produces on [0] the in"nite chain [0] [2] [4] 2 . \ \ \ The SO(2) charge is the energy and the SO(3) representation determines the spin.
(38)
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149
4.2. Sp(6) triples We now apply the same construction to the next entry in the magic square. The Lie algebra Sp(6),C is generated by the 3;3 traceless antihermitian matrices over quaternions, augmented by SU(2), the automorphism group of the quaternions. By picking out one of its o!-diagonal element, we obtain the embedding Sp(6)MSp(4);Sp(2)&SO(5);SO(3)
(39)
with an eight-dimensional coset space acted on by SO(8). Sp(6) has three simple roots, given by a "e !e , a "e !e , a "2e . (40) The same weight, written in Dynkin and orthonormal bases, has components [a , a , a ] and (b , b , b ), respectively, with the relations b "a #a #a , b "a #a , b "a , (41) derived from Eq. (29), where now i, j"1, 2, 3, after carefully accounting for the long root. Its Weyl vector is given by (42) o "(3, 2, 1)&[1, 1, 1] . ! The Weyl group of C contains S , which permutes the three b 's. As in the previous case, three of its G operations relate the three equivalent ways to embed SO(5);SO(3). Add to the highest weight of an irrep of C , [a , a , a ] the Weyl vector, to put it inside the fundamental chamber of Sp(6). There are three elements of the Weyl group which map this weight into the fundamental chamber of the subgroup SO(5);SO(3), de"ned by b 5b 50, and b 50: the identity element, the parity P , which interchanges b and b , and the cyclic element C which e!ects (b , b , b )P(b , b , b ). After subtracting the Weyl vector of the subgroup o"(2, 1; 1), where the last entry is for the SO(3) subgroup, we obtain the three weights (b #1, b #1; b ), (b #1, b ; b #1), (b , b ; b #2) . (43) To rewrite these in the Dynkin basis, we use the expression for the simple roots of SO(5) in the orthonormal basis a "e !e , a "e , which yield the Sp(6)-generated triple
(44)
[a , a #a #1; a ] [a #a #1, a ; a #a #1] [a , a ; a #a #a #2] . The scalar irrep of Sp(6) yields the simplest triplet
(45)
[0, 1; 0] [1, 0; 1] [0, 0; 2] ,
(46)
or in terms of the dimensions of the representations of SO(5);SO(3), (5, 1) (4, 2) (1, 3) ,
(47)
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the representation content of super-Yang}Mills in 10 dimensions. The orthogonal group SO(8) acts on the eight-dimensional coset space, de"ning an embedding of the spinor of SO(5);SO(3) into the vector irrep of SO(8): 8 "(4, 2). The triple is just the decomposition of the spinors of SO(8): 4 8 "(4, 2), 8 "(5, 1)#(1, 3). The subgroup can be interpreted as the light-cone little group of 1 1 either SO(6, 1);SO(3), or SO(5);SO(4, 1), which imply Lorentz-invariant theories in either d"7 or 5 space-time dimensions. 4.3. F triples The F simple roots are given by a "e !e , a "e !e , (48) (49) a "e , a "(e !e !e !e ) . Through the use of Eq. (29), we "nd the relation between the components of any weight in the Dynkin basis [a , a , a , a ] and the orthonormal basis (b , b , b , b ) b "a #2a #a #a , b "a #a #a , (50) b "a #a , b "a . (51) The Weyl vector is given in the orthonormal and Dynkin bases by o "(11, 5, 3, 1)&[1, 1, 1, 1] . (52) $ To generate the triples, we start with an irrep of F , [a , a , a , a ], and add to it the Weyl vector to put it inside the fundamental chamber. Express it in the orthonormal basis. The three elements of the F Weyl group which map weights inside the fundamental chamber of SO(9), are the identity, a parity and an anticyclic permutation. These are most easily expressed by their action on the fundamental of F , which is a 3;3 hermitian traceless octonionic matrix. The parity interchanges two o!-diagonal octonion elements, and the cyclic permutes the three o!-diagonal octonion elements. The "rst is the original weight, the second and third are obtained in terms of permutations on the simple roots of D , for which we have a "e !e , a "e !e , (53) a "e !e , a "e #e . (54) In this numbering, a is at the center of the Dynkin diagram, and does not move under the permutations. The two permutations that produce the required weights are S : a Ca ,a Ca ,a Ca , (55) C : a Ca Ca Ca . (56) Weights in the chamber of B are determined by the inequalities b 5b 5b 5b 50. Next, to get in the closure of the Weyl chamber, we subtract the Weyl vector of SO(9), o "(7, 5, 3, 1)&[1, 1, 1, 1]
(57)
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and convert in the Dynkin basis of SO(9), using the B relations a "e !e , a "e !e , a "e !e , a "e , from which
151
(58) (59)
a "b !b , a "b !b , (60) a "b !b , a "2b . (61) This leads back to the F generated triples in the Dynkin basis [1#a #a , a , a , 1#a #a ] [2#a #a #a , a , a , a ] [a , a , 1#a #a , a ] . (62) This formula corresponds to that previously derived empirically. For the simplest case, the triple is the supergravity multiplet [1, 0, 0, 1] [2, 0, 0, 0] [0, 0, 1, 0] .
(63)
The group SO(16) acts on the 16-dimensional coset, providing the embedding of the spinor of SO(9) into the vector of SO(16). Then the lowest order triple is the decomposition of the two spinors of SO(16) into SO(9), which explains the supersymmetry. 4.4. E
-multiples
As indicated by the decompositions of the E-like exceptional groups in the magic square, we can generate higher-order Cli!ord algebras. E The embedding E MSO(10);SO(2) produces a 32-dimensional coset space. The lowest mul tiplet contains 27 terms which are decomposition of the two spinor irreps of SO(32) in terms of SO(10);SO(2). These are generated by a Cli!ord algebra with 2 elements, half-fermions, half-bosons. The SO(32) Cli!ord is also generated by the magic square embedding SO(12)MSO(8);SO(4). E With E MSO(12);SO(3), a 64-dimensional coset space is generated. The two spinor irreps of SO(64) break into 63 irreps of the subgroup, with 2 states generated by a Cli!ord algebra, again half of them fermions. E Finally, E MSO(16) produces 2 states describing the two spinors of SO(128) in terms of 135 irreps of SO(16)! In all these cases, the lowest multiplet that is generated contains states of spin higher than two, which means that any theory based on these structures probably have no local limit, except perhaps by compacti"cation. Finally we note that the magic square embedding SU(6)MSO(6);SO(3);SO(2) yields in the lowest multiplet the states generated by the SO(16)-Cli!ord, corresponding to the dimensional reduction of the sugra multiplet, and it could be viewed as coming from d"eight-, "ve-,
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four-dimensional theories or even on AdS ;S . For the reader interested in proofs concerning the properties of these multiplets, we refer to [7], where a novel formula between representations of Lie algebras of the same rank is derived. In conclusion, we have described a very rich mathematical structure, which contains as special cases, the supermultiplets of 11-dimensional supergravity and 10-dimensional super-Yang}Mills. While there is at present no clear interpretation of these structures, it is tempting to believe that they may shed some light on the eventual structure of M-theory.
Acknowledgements One of us (P.R.) acknowledges the hospitality of the Aspen Center for Physics and illuminating conversations with Profs L. Brink, R. Moody and S. Sternberg. This work was supported in part by the United States Department of Energy under grant DE-FG02-97ER41029.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
A. Sen, hep/th-9802051. T. Banks, W. Fishler, S. Shenker, L. Susskind, Phys. Rev. D 55 (1997) 5112. L. Durand III, Phys. Rev. 128 (1962) 434; S. Weinberg, E. Witten, Phys. Lett. 96B (1980) 59. M.A. Vasiliev, Ann. Phys. 190 (1989) 59; Phys. Lett. B 257 (1991) 111; Int. J. Mod. Phys. D 5 (1996) 763; E. Sezgin, P. Sundell, CTP TAMU-19/98, hep-th/9805125. R. Slansky, Phys. Rep. 79 (1981) 1. See, W.G. McKay, J. Patera, Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras, Marcel Dekker, New York, 1981. B. Gross, B. Kostant, P. Ramond, S. Sternberg, Proceedings of the National Academy of Sciences 95 (1998), pp. 8441. B. Gross, B. Kostant, S. Sternberg, private communication; P. Ramond, Introduction to Exceptional Groups and Algebras, CALT-68-577, December, 1976, unpublished. S. Meshkov, C. Levinson, H. Lipkin, Phys. Rev. Lett. 10 (1963) 361.
Physics Reports 315 (1999) 153}173
Con"nement studies in lattice QCD Richard W. Haymaker Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA
Abstract We describe the current search for con"nement mechanisms in lattice QCD. We report on a recent derivation of a lattice Ehrenfest}Maxwell relation for the Abelian projection of SU(2) lattice gauge theory. This gives a precise lattice de"nition of "eld strength and electric current due to static sources, charged dynamical "elds, gauge "xing and ghosts. In the maximal Abelian gauge the electric charge is anti-screened analogously to the non-Abelian charge. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.15.Ha; 11.30.Ly Keywords: Lattice QCD; Con"nement; Abelian projection
1. Introduction Quenched lattice QCD calculations of the static quark anti-quark potential have "rmly established a linearly rising behavior over all distances obtainable in state-of-the-art simulations [1]. However, this situation satis"es no one, least of all a Dick Slansky. A &black box' calculation at limited quark separation gives supporting evidence that QCD con"nes quarks, but it o!ers no explanation nor reveals a principle governing the phenomenon. Lattice QCD is more than just an algorithm to calculate quantities at strong coupling. The lattice is a regulator for QCD, parametrized by the lattice spacing a. Gauge symmetry is preserved at all costs. This regulation scheme is perhaps the only one that gives a completely self-consistent cuto! model in its own right. The dynamical variables are elements of the symmetry group rather than the Lie algebra. As a consequence, many of the topological features that are &likely suspects' in the physics of con"nement have natural de"nitions on the lattice. These include conserved ;(1), Z(N), SU(N)/Z(N) monopole loops, Dirac sheets, Z(N) and SU(N) vortex sheets and other features. Lattice work [2}6] is coming into its own in sorting through some of the seminal ideas on con"nement [7}12] proposed in the 1970s. I venture to say that Dick Slansky could easily have 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 1 9 - 8
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been drawn into the fray because it has all the elements that typically captured his imagination. The central questions are fundamental. They involve interesting issues in group theory, topology, and duality. Candidate mechanisms are proliferting and fundamental questions remain unsettled. Con"nement is a consequence of a disordered state characterized by the expectation value of a Wilson loop suppressed to an area law rather than a perimeter law for an ordered system. Contributions from topological objects, e.g. monopoles and vortices can accomplish this. On the lattice these objects are not singular. They occur abundantly in SU(N) lattice theories. It is only in the continuum limit of zero lattice spacing where these approach singular structures. In lattice simulations, one can tamper with these objects and see if suppression is accompanied by the loss of an area law. In this way one has a laboratory to study candidate disordering mechanisms. It is not realistic to try to give a proper review of the many competing views in this brief article. Further, the studies are possibly diverging more rapidly than at any point in the past considering the variety of papers on the subject over the past year. However, I would like to touch on them and further to argue that all the studies are describing genuine properties of QCD, seen through the eyes of subsets of the full dynamical variables. The controversies have to do with which scenario will lead to the most compelling explanation of con"nement. Although these lattice studies reveal relevant con"nement physics, the goal is not completely clear. Most would agree, however, that if we knew the dual form of QCD it would give a de"nitive description of the physics of con"nement. Consider the example of superconductivity. By identifying the carriers of the persistent current one can discover an instability in the normal vacuum. The Ginzburg}Landau e!ective theory describes the consequence of the instability which of course elucidates the fundamental principle underlying the phenomenon which is the spontaneous breaking of the ;(1) gauge symmetry [13]. We might imagine that an understanding of the topological structures might in the same way lead to a discovery of a spontaneous gauge symmetry breaking. The problem is that a de"nitive scenario still seems quite distant.
2. Lattice con5nement studies } a brief survey In this section I wish to call attention to some of the approaches in lattice studies of con"nement that have been active over the past 12 months. This is principally to give a #avor of the work and a list of recent references, and rely in the references therein for a more complete bibliography. Our most recent work, which we describe in Section 5 is based on the Abelian projection. We will restrict our attention to SU(2) theories throughout this article, believing that the essential issues will be revealed in this simplest case.
At Lattice '98, the XVI International Symposium on Lattice Field Theory in Boulder, there were 52 contributed papers on color con"nement, up from a dozen or so in the early 1990s.
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2.1. Abelian projection, Abelian dominance In the Georgi}Glashow model with a gauge "eld coupled to the adjoint Higgs one can de"ne a gauge invariant Abelian "eld strength [14]. This is the construction used to identify the magnetic "eld of a 'tHooft}Polyakov monopole. Further, the Higgs "eld can be used to de"ne an Abelian projected theory. This is accomplished by gauge transforming the Higgs "eld into the 3 direction. The resulting partially gauge "xed theory is still invariant under ;(1) gauge transformations } rotations about the 3 axis. Pure gauge SU(2) has no adjoint Higgs "eld and so there is no straightforward way to de"ne Abelian variables. It is still possible to de"ne an Abelian Projection [15,16,18]. One can "nd a &collective' Higgs "eld transforming under the adjoint representation which we denote as an &adjoint "eld'. Consider an arbitrary Wilson line starting and ending at a particular site. This can be parametrized by =(x)"cos s#i ) r sin s where is a normalized adjoint "eld. The Wilson line could be (i) an open (no trace) Poliakov loop, (ii) an open plaquette in, e.g. the (1, 2) plane, or (iii) an arbitrary sum of such lines then normalized to construct an SU(2) element. (iv) One can generalize to de"ne the adjoint "eld self-consistently by requiring that the adjoint "eld at one site be equal to the normalized sum of the eight neighboring adjoint "elds after parallel transport to that site. In these four examples, one can further "x the gauge by rotating the adjoint "eld into the third direction. This then "xes (i) to the Polyakov gauge, (ii) to a plaquette gauge (iv), e.g. a clover gauge, and (iv) in the maximal Abelian gauge. The &collective' adjoint "eld itself is gauge covariant, it does not necessitate a gauge "xing. Di!erent adjoint "elds correspond to di!erent Abelian projections. The maximal Abelian gauge appears to be the most promising choice. The condition, equivalent to the one given just above [17], maximizes (;#;!;!;), where ;"; #ir ) U, ;#U"1. The maximization brings the links as close as possible to the values ; "; "0 and ;#;"1, leaving a ;(1) invariance. A new and interesting alternative has been proposed recently by Van der Sijs [19]. This gauge "xing is based on the lowest eigenstate of the covariant Laplacian operator: i.e. sum the adjoint "elds at the eight neighboring sites } parallel transported to a given site } and subtract 8; the "eld at the site. This de"nes a gauge with no lattice Gribov copies. And it identi"es the positions of monopoles as singularities (zeros) of the adjoint "eld. &Abelian dominance' asserts further that operators built out of the ;(1) links de"ned by the &Abelian projection' will dominate the calculation of string tension and other observables. There are a large number of numerical tests which support Abelian dominance. However, the successful tests are for the most part restricted to (a) certain speci"c quantities, e.g. string tension in the fundamental representation, and (b) the gauge choice of the maximal Abelian gauge. In other gauges, e.g. plaquette gauge, and/or other quantities such as string tension in other representations, Abelian dominance is not well supported and in some cases is strongly violated. On the other hand, in a series of papers Di Giacomo et al. [20}24] have studied the "nite temperature con"ning phase transition and shown that the disorder parameter is very insensitive to the choice of Abelian projection. This is the foundation of the dual superconductor scenario. This is not in con#ict with the above results because it does not require the Abelian dominance ansatz. Chernodub et al. [25] have formulated this problem in terms of the e!ective potential and found similar results.
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Using the Abelian projection and Abelian dominance, the problem is reduced to a ;(1) gauge theory albeit with a complicated action coming from the e!ects of gauge "xing. As in a pure ;(1) gauge theory, Dirac monopoles can be identi"ed as the charge carriers of the persistent currents in the dual superconductor scenario [18,26}29]. Bakker et al. [30] have shown that Abelian monopole currents, de"ned in the maximal Abelian gauge are physical objects: there is a strong local correlation between monopoles and enhancements in the SU(2) gauge invariant action. Other recent results of Polikarpov et al. [31}33] include Abelian monopoles in the maximal Abelian gauge are dyons. There are strong correlations between magnetic charge, electric charge and topological charge density. These same connections have been reported independently by Ilgenfritz et al. [34]. In order to understand the disordering e!ect of monopoles on large Wilson loops, Hart and Teper [35] have studied the clustering properties of monopole loops and found two classes of clusters: A single cluster permeates the whole lattice volume, the remaining are small localized clusters. In a series of papers, Ichie and Suganuma [36}39] have sought a deeper understanding of Abelian dominance. They look at the residual degrees of freedom after Abelian projection, i.e. the coset "elds and argue that in a random variable approximation that Abelian dominance is exact. Ogilvie [40] has argued in the context of the Abelian projection that gauge "xing is in principle unnecessary, that results are the same whether the gauge is "xed or not. This is at odds with many simulations, leaving much to be sorted out. Grady [41,42] has given evidence that casts doubt on whether the Abelian monopole con"nement mechanism carries over to the continuum limit. A number of authors, including our group have reported that a well-de"ned electric vortex forms between static sources [43,17,44]. The vortex is very well described by a dual Ginzburg}Landau e!ective theory. In e!ect we are able to show that there is a local relation between the electric #ux and the curl of the monopole current, de"ned by Abelian projection of full SU(2) "eld con"gurations that gives a damping of "elds as they penetrate the dual superconductor. This directly accounts for con"nement. In Section 5 we derive a precise de"nition of the electric "eld which tightens up the de"nition of vortices [45]. 2.2. Vortices in Z(N);SU(N)/Z(N) formulation In the early 1980s there was an e!ort by Mack and Petkova [46], Ya!e [47], Tomboulis [48], Yoneya [49], Cornwall [50], and Halliday and Schwimmer [51], to understand con"nement in terms of SU(N) vortices. These are singular gauge con"gurations characterized by their topological properties. They are multiple valued in SU(N) but single valued in SU(N)/Z(N). There are more recent developments by Tomboulis and Kovacs [52}54]. Their study is based on a decomposition of the SU(2) partition function into variables de"ned over the center Z(2) and variables de"ned over the quotient group SU(2)/Z(2) (" SO(3)) [48]. Whereas the SU(2) manifold has a trivial topology, SO(3) does not: p (SO(3))"Z(2). A thorough review of the topological issues in this approach can be found in Ref. [52].
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To accomplish this reformulation, one appends new Z(2) valued variables p , which live on plaquettes, to the usual set of SU(2) links. This allows one to write the standard SU(2) Wilson action partition function in terms of these Z(2) variables, p , and link variables, +;K ,, de"ned over the manifold of SO(3) rather than SU(2). The latter distinction is manifest in the reformulated partition function in that the SU(2) links +; , occur only in Z(2) invariant combinations, i.e. invariant under the transformation ; P$; . Consider a single (1, 2) plaquette with p "!1, all others "#1. A Z(2) monopole occurs in each of the two (1, 2, 3) cubes adjacent to the face of the negative plaquette and similarly in the two (1, 2, 4) cubes. On the dual lattice, each cube becomes a dual link orthogonal to the cube and this forms a conserved Z(2) monopole current loop (the smallest such loop) on the dual lattice. The negative plaquettes become dual plaquettes on the dual lattice. They form a surface bounded by the monopole loop. This surface is called a &thin vortex' sheet. One can build larger monopole current loops and vortices by laying out stacks of negative p plaquettes. There are also the usual plaquettes built from the trace of 4 SU(2) link variables. Again consider a con"guration in which all such plaquettes are positive except for a single (1, 2) plaquette which is negative. An SO(3) monopole occurs in each of the two (1, 2, 3) cubes adjacent to the negative plaquette and similarly in the two (1, 2, 4) cubes. This is an SO(3) object because every link involved in its construction occurs quadratically and hence is invariant under a Z(2) transformation of each link. Analogously, there is an SO(3) monopole current loop on the dual lattice. The surface bounded by the loop is a Dirac sheet which di!ers from the &thin vortex' sheet in that it can be moved arbitrarily by multiplying the SU(2) links by elements of the center $ without costing any action. The equivalence of the two forms of the partition function requires that every Z(2) monopole be coincident with an SO(3) monopole and vice versa. Hence we can visualize the vortex structure as two currents on coincident loops, bounded by two surfaces. This is called a &hybrid' vortex. We can also shrink the monopole loop to zero, leaving either a pure &thin vortex' sheet, or in the other case a pure Dirac sheet. The SO(3) monopoles also have associated non-Abelian (non-integer valued) #ux which costs action. These con"gurations are denoted &thick vortices'. The Dirac sheet is the return #ux required by the topological group p (SO(3))"Z(2). In this picture, it has been shown that thin vortices are exponentially suppressed at weak coupling and cannot account for con"nement. However SO(3) thick vortices can occur at weak coupling for con"gurations in which all terms in the plaquette action are close to one. In recent papers Kovacs and Tomboulis [52,53] have shown that the static quark potential can be reproduced by contributions from SO(3) thick vortices linking the Wilson loop in SU(2); and in SU(3); and that exclusion of vortices results in a perimeter law. See Gavai and Mathur [55] for a study of Z(2) monopoles and the decon"nement phase transition. Also see Grady [56] for variation on the SO(3)}Z(2) monopole construction. 2.3. Center projection vortices There are other closely related approaches that also focus on the center of the group as the key to understanding con"nement. In a number of recent papers, Greensite et al. [57] have introduced the
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&Center projection' as a way of identifying Z(2) vortices for SU(2) which they denote as &Projection vortices'. They di!er from the thick and thin vortices of Tomboulis. They are de"ned in the maximal center gauge. In this gauge, one maximizes ;. Hence ; will be as close as possible to $1. The Z(2) links are then de"ned Z"sign(; ). This gives a Z(2) gauge theory that has &P vortex' excitations which are somewhat analogous to the Tomboulis &thin' vortices. They also de"ne ¢er vortices' which can be much larger than one lattice spacing and are analogous to the Tomboulis &thick vortices'. Greensite et al. [58}60] have reported a growing list of encouraging results. Among the results are: P vortices are well correlated with SU(N) vortices; no P vortices no con"nement; P vortices account for the full string tension; P vortex density scales; Center vortices thicken as the lattice cools; P-vortex locations are correlated among Gribov Copies; preliminary successful generalizations to SU(3); and center vortices are compatible with Casimir scaling. See also Langfeld et al. [61,62] and Stephenson [63] for further results on center vortices. A variation in the center projection procedure is to "rst "x to the maximal Abelian gauge, and then maximize cos s where s is the ;(1) link angle. This is denoted the indirect maximal center gauge, as opposed to the above procedure denoted the direct maximal center gauge. In this gauge, a sheet consisting of monopole loops alternating with anti-monopole loops coincides with the P vortex sheet [58,60] indicating a possible overlap between this and other scenarios. 2.4. Dual variables One expects that the disorder regime in QCD would be described by an ordered regime in dual QCD. A de"nitive form of dual QCD would probably take us a long way toward an understanding of con"nement. We would like to call attention to recent lattice work on dual variables by Cheluvaraja [64], and continuum work by Majumdar and Sharatchandra [65], Sharatchandra et al. [66], Mathur [67], Faddeev and Niemi [68], and Chan and Tsun [69]. The work by Majumdar and Sharatchandra supports the dual QCD ansatz by Baker et al. [70]. Seiberg and Witten [71] exploited duality to establish con"nement for supersymmetric QCD. 2.5. Instantons An important aspect of con"nement studies is to identify what objects cannot account for con"nement. Instantons were likely suspects at one time. Smoothing techniques are very important in the identi"cation of instantons. Unlike monopoles, small instantons can fall through the holes in the lattice and further can be swamped by short distance #uctuations. New smoothing techniques which overcome these di$culties, denoted &renormalization group mapping', have been applied to con"gurations by DeGrand et al. [72] to elucidate the role of instantons in the QCD vacuum. They come to the strong conclusion that instantons alone do not con"ne. See also other recent works by Narayan and Neuberger [73]; Narayan and Vranas [74]; Ph. de Forcrand et al. [75] and B. Alles et al. [76].
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2.6. Interconnections Even though the possible explanations for con"nement seem to be diverging at this point, I would like to re-emphasize that di!erent descriptions can in some cases be describing the same underlying physics: Abelian monopoles in the maximal Abelian gauge correlate with E E E E
SU(2) enhancements in the gauge invariant action [30], gauge invariant topological charge [31}34], P vortices in the maximal center gauge [59], Instantons [77,78]. Therefore, the monopoles cannot be thought of as merely gauge artifacts.
3. ;(1) gauge theories and superconductivity The onset of superconductivity is governed by the spontaneous breaking of the ;(1) gauge symmetry via a non-zero vacuum expectation value of a charged "eld [13]. An immediate consequence of this is the generation of a photon mass and, for type II superconductors, the formation of magnetic vortices which con"ne magnetic #ux to narrow tubes [79] as revealed by the Ginzburg}Landau e!ective theory. Lattice studies of dual superconductivity in SU(N) gauge theories seek to exploit this connection in establishing the underlying principle governing color con"nement. In ;(1) lattice pure gauge theory (no Higgs "eld), this same connection is seen to be present, not in the de"ning variables, but rather in the dual variables. More speci"cally: 1. A "eld with non-zero magnetic monopole charge, U, has been constructed [80]. It is a composite 4-form living on hypercubes constructed from gauge "elds. There are also monopole current 3-forms. On the dual lattice this monopole operator is a 0-form living on dual sites. The monopole currents are 1-forms living on dual links. These currents either form closed loops or end at monopole operators. The monopole operator has a non-zero vacuum expectation value in the dual superconducting phase, 1U2O0, thereby signaling the spontaneous breaking of the ;(1) gauge symmetry. 2. Dual Abrikosov vortices have been seen in simulations [27,81]. They are identi"ed by the signature relationship between the electric "eld and the curl of the monopole current in the transverse pro"le of the vortex. The dual coherence length, m measures the characteristic distance from a dual-normal-superconducting boundary over which the dual-superconductivity turns on. The dual London penetration length, j measures the attenuation length of an external "eld penetrating the dual-superconductor. The dual photon mass &1/j and the dual Higgs mass &1/m . A signal 1U2O0 without the consequent signal of a dual photon mass does not imply con"nement. An observation of a dual photon mass, i.e. vortex formation, without 1U2O0 does not reveal the underlying principle governing the phenomenon.
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3.1. Higgs ewective theory The Higgs theory, treated as an e!ective theory, i.e. limited to classical solutions, and considered in the dual sense, provides a model for interpreting simulations of the pure gauge theory that can reveal these important connections. Recalling the Higgs' current J "!(ie/2)( H(R !ieA ) ! (R #ieA ) H) , I I I I I and spontaneous gauge symmetry breaking through a constraint Higgs potential h"ve CSV, v"constant ,
(1)
(2)
leads to the London theory of a type II superconductor. Using Eqs. (1) and (2) we obtain J "!ev(A !R u) , I I I (R J !R J )#m(R J !R J )"0 , I J J I A I J J I 1 e;J# B"0 , j
(3)
where ev"m"1/j . A Using Ampere's law e;B"J, we obtain
(4)
eB"B/j , identifying j as the London penetration depth. If the manifold is multiply connected, then the gauge term in Eq. (3) can contribute, as long as eu(x) is periodic, with period 2p on paths that surround a hole.
(B#je;J ) ) n da" (A#jJ) ) dl , 1 ! " eu ) dl , ! 2p "N "Ne "U . K K e
where N quanta of magnetic #ux penetrates the hole in the manifold. In real superconductors, the hole is a consequence of the large magnetic "eld at the center which drives the material normal. A cylindrically symmetric vortex solution is given by B#je;J"U d(r )n , K , X (1!je )B (r )"U d(r ) , , X , K , U B (r )" K K (r /j) . X , 2pj ,
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The delta function core of this vortex is normal, i.e. no spontaneous symmetry breaking, and the exponential tail of the vortex is a penetration depth e!ect at the superconducting-normal boundary. The key point is that the modulus of the Higgs "eld must be independent of position to get these idealized vortices. For a &Mexican hat' Higgs potential, there is a coherence length setting the length scale from a normal-superconducting boundary over which the vacuum expectation value of the Higgs "eld changes from zero to its asymptotic value. On the lattice, the same phenomenon occurs. We can generate vortices from "nite con"gurations. In the continuum these objects are singular. Since the lattice formulation is based on group elements, rather than the Lie algebra the periodic behavior of the compact manifold is manifest. This gives the 2pN ambiguity in the group angle leading to N units of quantized #ux. To see how this works, consider the lattice Higgs action (" (x)") , S"b (1!cos h (x))!i ( H(x)e FIV (x#k)#H.c.)# < & IJ V V IJ V I where h (x) is the curl of the gauge "eld, IJ h (x)"D>h (x)!D>h (x) , IJ I J J I and where (x) is the Higgs "eld and (x#k) refers to the Higgs "eld at the neighboring site in the k direction and D> is the forward di!erence operator. The electric current is given by J a JC (x)"Im( H(x)e FIV (x#k)) , ei I where a is the lattice spacing. Let us choose a Higgs potential that constrains the Higgs "eld " (x)""1. Then if sin[h#2Np]+h , we "nd a relation between the "eld strength tensor and the curl of the current 1 a (D>JC(x)!D>JC (x)) 2pN 1 I J J I " "Ne , F ! Ka IJ ei e a a
(5)
where eaF "sin[h (x)#h (x#k)!h (x#l)!h (x)] . IJ I J I J If N"0 then this is a London relation which implies a Meissner e!ect. For NO0 there are N units of quantized #ux penetrating that plaquette, indicating the presence of an Abrikosov vortex. 3.2. Pure U(1) gauge theory In a pure ;(1) lattice gauge simulation (without Higgs "eld), lattice averages over many con"gurations exhibit superconductivity in the dual variables. The superconducting current
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carriers are monopoles. They can be de"ned in a natural way on the lattice using the DeGrand Toussaint [26] construction. They arise in non-singular con"gurations again because of the 2pN ambiguity in group elements. These are the magnetic charge carriers for dual superconductivity. For a review of the monopole construction and vortex operators in ;(1) gauge theory see, e.g. the 1995 Varenna Proceedings. (For a review and further references to superconductivity on the lattice see e.g. [79].) As a brief summary, consider the unit 3-volume on the lattice at "xed x . The link angle is compact, !p(h 4p. The plaquette angle is also compact, !4p(h 44p and de"ned I IJ eaF (x)"h (x)"D>h (x)!D>h (x) , IJ IJ I J J I where a is the lattice spacing. This measures the electromagnetic #ux through the face. Consider a con"guration in which the absolute value of the link angles, "h ", making up the cube are all small I compared to p/4. Gauss' theorem applied to this cube then clearly gives zero total #ux. Because of the 2p periodicity of the action we decompose the plaquette angle into two parts h (x)"hM (x)#2pn (x) . (6) IJ IJ IJ where !p(hM 4p. If the four angles making up one of the six plaquette are adjusted (so that, IJ e.g. h 'p) then there is a discontinuous change in hM by !2p and a compensating change in n . IJ IJ IJ We can clearly choose the con"guration that leaves the plaquette angles on all the other faces safely away from a discontinuity. We then de"ne a Dirac string n passing through this face (or better IJ a Dirac sheet since the lattice is 4D). hM measures the electromagnetic #ux through the face. IJ This construction gives the following de"nition of the magnetic monopole current: a JK(x)"e D>hM (x) . (7) IJNO J NO e I K This lives on dual links on the dual lattice. Although Eq. (6) is not gauge invariant, Eq. (7) is. Further it is a conserved current, satisfying the conservation law D>JK(x)"0. I I In simulations of a pure ;(1) gauge theory we "nd that lattice averages give a relation similar to Eq. (5), but in the dual variables 1HF 2!j IJ
1D\JK(x)!D\JK(x)2 1 I J J I "Ne , a a
where HF is dual of F . This is the signal for the detection of dual vortices [79]. IJ IJ 4. Non-Abelian theory The link of these considerations to con"nement in non-Abelian gauge theory is through the Abelian projection [15,16]. One "rst "xes to a gauge while preserving ;(1) gauge invariance. The non-Abelian gauge "elds can be parametrized in terms of a ;(1) gauge "eld and charged coset "elds. The working hypothesis is that operators constructed from the ;(1) gauge "eld alone, i.e. Abelian plaquettes, Abelian Wilson loops, Abelian Polyakov lines and monopole currents, will exhibit the correct large distance correlations relevant for con"nement.
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In the continuum limit the maximal Abelian gauge condition is (R $gA(x))A!(x)"0 . I I I This is achieved on the lattice by a gauge con"guration that maximizes R, where 1 R[;], Tr(p ; (n)p ;R(n)) , I I 2 L I and where ; (n) is the link starting a site n and extending in the k direction. I cos( (n))e FIL sin( (n))e QIL I I . ; (n)" I cos( (n))e\ FIL !sin( (n))e\ QIL I I After gauge "xing, the SU(2) link matrices may be decomposed in a &left coset' form:
; (n)" I
cos( (n)) I
sin( (n))e AIL I
e FIL
0
,
(8)
cos( (n)) !sin( (n))e\ AIL 0 e\ FIL I I Under a ;(1) gauge transformation, +g(n)"exp[ia(n)p ],, h (n)Ph (n)#a(n)!a(n#k( ), c (n)Pc (n)#2a(n) . (9) I I I I In other words, the left coset "eld derived from the link ; (n) is a doubly charged matter "eld living I on the site n and is invariant under ;(1) gauge transformations at neighbouring sites. The c ,cos( ) are real-valued "elds which near the continuum &1#O(a) where a is the I I lattice spacing. The o!-diagonal w ,sin( )e AI become the charged coset "elds ga= (x), and I I I h the photon "eld gaA(x). [The SU(2) coupling b"4/g in 3#1 dimensions.] I I The static potential constructed from Abelian links gives as de"nitive a signal of con"nement as the gauge invariant static potential as found by Suzuki et al. [18,28], Stack et al. [29] and Bali et al. [82]. Bali et al. "nd the Abelian string tension calculation gives 0.92$0.04 times the full string tension for b"2.5115. Whether this approaches 1.0 in the continuum limit remains to be seen. Eq. (4) gives the connection between the non-zero vacuum expectation value of an order parameter and the photon mass or equivalently the London penetration depth. This connection between order parameter and penetration depth is the key to connecting spontaneous symmetry breaking to vortex formation and hence con"nement. Dual superconductivity studies seek to establish a connection, of course calculated from the original variables. Di Giacomo et al. [20}24] have reported extensive studies of the behavior of the order parameter for the dual theory (denoted disorder parameter) at the con"ning transition. See also Polikarpov et al. [83]. Our group and others have reported dual vortex formulation between sources allowing the determination of the London penetration depth [43,17,44]. We can demonstrate a qualitative correspondence between these two indicators of dual superconductiviy in that they are both observed and both show the correct behavior on the two sides of the transition. However, technical di$culties have eluded a direct comparison check of the dual form of Eq. (4).
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Fig. 1. (From Ref. [20]) o vs. b for SU(2) gauge theory. The peak signals decon"ning phase transition. Here monopoles are de"ned by the Abelian projection on Polyakov line.
Fig. 2. (From Ref. [17]) Transverse pro"le of the electric "eld and curl of the monopole current in the midplane between a static qq pair in the maximal Abelian gauge at "nite temperature for a con"ning (left) and uncon"ning (right) phase.
Fig. 1 shows a plot of o"2d ln 1k2/db as a function of b near the transition. The spike indicates step behavior in 1k2 at the transition. Electric dual vortices between sources are well established [43,17,44] in this gauge. The typical behavior is shown in Fig. 2. The London relation is seen in the con"ning case for transverse distances larger than about two lattice spacing. The dual coherence length m +2, i.e. the onset of
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Fig. 3. (From [44]) Check of the dual Ampere law in the dual vortex pro"le. E is the electric "eld, k is the monopole current. Fig. 4. (From [44]) Fit of the electric "eld vortex pro"le E and the tangential component of the magnetic monopole current k to the Ginzburg Landau theory. F
the violation of the London relation. The uncon"ned case is also shown where curl J is almost zero, and the electric "eld falls more gradually than in the con"ning case. Bali et al. [44] have done a large-scale simulation of these vortices. Again "tting the electric "eld and the monopole current to the Ginzburg}Landau theory their results are shown in Figs. 3 and 4. Their data is very well described by the two G}L parameters j"1.84$0.24 and m"3.10$0.40. The ratio G}L parameter i"j/m"0.59$0.14. They "nd the total #ux in the vortex, U"1.10$0.02 in units of the quantized #ux. They concluded tentatively that type I dual superconductivity is indicated.
5. De5nition of electric 5eld strength Central to "nding the e!ective theory is the de"nition of the "eld strength operator in the Abelian projected theory, entering not only in the vortex pro"les but also in the formula for the monopole operator. All de"nitions should be equivalent in the continuum limit, but use of the appropriate lattice expression should lead to a minimization of discretization errors. In a recent paper [45] we exploited lattice symmetries to derive such an operator that satis"es Ehrenfest relations; Maxwell's equations for ensemble averages irrespective of lattice artifacts. This gives a precise lattice de"nition of current and charge density independent of lattice size, and independent of the continuum limit. In the Abelian projection SU(2) link variables are parametrized by ;(1) links and charged coset "elds. The latter are normally discarded in Abelian projection, as are the ghost "elds arising from the gauge "xing procedure. Since the remainder of the SU(2) infrared physics must arise from these, an understanding of their role is central to completing the picture of full SU(2) con"nement. In the maximal Abelian gauge the supposedly unit charged Abelian Wilson loop has an upward renormalization of charge due to this current. A localized cloud of like polarity charge is induced in
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the vacuum in the vicinity of a source, producing an e!ect reminiscent of the antiscreening of charge in QCD. In other gauges studied, the analogous current is weaker, and acts to screen the source. We show that this current can be quantitatively written as a sum of terms from the coset and ghost "elds. The contribution of the ghost "elds in the maximally Abelian gauge in this context is found to be small. The e!ect of the Gribov ambiguity on these currents is argued to be slight. 5.1. U(1) We "rst introduce and review the method due to Zach et al. [84], in pure ;(1) theories. Consider a shift in the ;(1) link angles in the partition function containing a Wilson loop source term
Z (+hQ ,)" [d(h #hQ )] sin h e@ FIJ . I I 5 5 I Since the Haar measure is invariant under this shift, Z is constant in these variables. Absorbing 5 the shift into the integration variable and taking the derivative R Z "0 , RhQ (x ) 5 J we get the relation
0" [dh] (cos h !sin h bD sinh ) e@ FIJ . 5 5 I IJ This can be cast into the form 1sin h D (1/e) sin h 2 5 I IJ "ed 1D F 2 , "J . I IJ 5 V V5 J 1cos h 2 5 This is a well-known technique to generate exact relations between Green's functions that is useful in generating Ward identities, or Schwinger}Dyson equations, or in this case we denote as Maxwell}Ehrenfest relations. We use the term Ehrenfest because it is the expectation value of what is normally a classical extremum of the path integral } an Euler Lagrange equation. 5.2. SU(2) no gauge xxing Before addressing the full problem we "rst generalize from ;(1) to SU(2) without the complication of gauge "xing:
Z (+;Q,)" [d(;;Q)] = (;)e@13 . 5 The size of the source is irrelevant so we choose it to be the simplest case, i.e. a plaquette: = , Tr(;R;R;;ip ) .
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We choose the shift to be in the 3 direction
i ;Q (x )" 1! e (x )p . I 2I The invariance d/de (x ) Z "0 I 5 gives the Ehrenfest relation b
1= S 2 I "d . V V5 1=2
For b"2.5, b1= S 2"0.0815$0.0002, and 1=2"0.0818$0.0001, and the di!erence I "0.0003$0.0003. The notation S denotes an e derivative [45]. The denominator is just the ordinary plaquette: I =, Tr(;R;R;;) . To cast this into the form of Maxwell's equation we decompose the link into diagonal D and I o!-diagonal parts O I ; (x)"D (x)#O (x) . I I I Further we simplify notation 122 ,1= 22/1=2 . 5 We then group terms involving the diagonal part on the left and take all terms having at least one factor of the o!-diagonal link to the right. 1 [b1S 2 ] " 1D F 2 . I 5 3" e I IJ 5 Finally, note d
"(1/e)J , V V5 J giving the "nal form of the Ehrenfest relation 1D F 2 "1J 2 #J . I IJ 5 J 5 J This then tells us how to choose a lattice de"nition of "eld strength that satis"es an Ehrenfest relation: 11 Tr(DRDRDDip ) . F " IJ IJ e 2 5.3. Gauge xxed SU(2), U(1) preserved The e!ect of gauge "xing gives
Z (+;Q,)" [d(;;Q)]= (;)D d[F]e@13 , $. 5
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where we have introduced
dg (y) d[F (;+EHW,; x)] , H G H W G V and integrated out the g variables in the standard way. In this case Z is not invariant. The shift is inconsistent with the gauge condition. However, it is 5 invariant under an in"nitesimal shift together with an in"nitesimal &corrective' gauge transformation that restores the gauge "xing 1"D
$.
i G(x)" 1! g(x) ) r . 2 Use of the invariance of the measure under combination of a shift and a &corrective' gauge transformation we obtain
Rg (z) R R # I Z "0 . Re (z ) Rg (z) 5 Re (z ) I I I X I In shorthand notation, the Ehrenfest relation reads
(D ) (D ) $. I # $. I #bS I D D $. $. Gauge "xing has introduced three new terms: (= ) " #(= ) " #= I I
"0 .
(10)
(= ) " comes from the corrective gauge transformation acting on the source which is ;(1) I invariant but not SU(2) invariant .
(D ) $. I is due to the shift of the Faddeev}Popov determinant . D $. (D ) $. I is due to the corrective gauge transformation of the Faddeev}Popov determinant . D $. The latter two derivatives are subtle. The key is to "rst consider the constraint up to "rst order in the shift and the corrective gauge transformations RF (x) RF (x) F (x)# G de(z )# G dg (z)"0 , G Rg (z) I Re(z ) I X I and then de"ne the Faddeev}Popov matrix as a derivative of the corrected constraint: M
RF (x) R RF (x) G dg (z) . #dM " F (x)# G de(z )# GV_ HW GV_ HW Rg (y) G Rg (z) I Re(z ) H I X I
See Ref. [45].
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Table 1 Terms in the Ehrenfest relation, Eq. (terms) Source Ehrenfest term
=
= (;PD)
1(= ) " 2 I 1(= ) " 2 I D 1= D$.$.I" 2 D 1= D$.$.I" 2 1b(S) " 2 I Zero
0.65468(10) 0.06095(7)
0.63069(20) 0.04463(4)
0.00127(21)
0.00132(50)
0.00529(3) !0.72246(68) !0.00026(77)
0.00564(3) !0.68275(50) !0.00045(64)
Note: The column labeled &= ' corresponds to the source described in the text. In the second column the links are replaced by the diagonal parts of the links in order to test a second example. The theorem gives zero for the sum. b"2.5, 4 lattice.
Finally we evaluate the derivative using D /D"Tr[M\M ] . I I A check of the Ehrenfest theorem is given in Table 1. Some of the individual terms on the right-hand side require a 2N;2N matrix inversion, where N is the lattice volume. Hence to test the result numerically, we chose as small a lattice as possible. The result does not involve the size of the lattice which is 4 in Table 1. The last column employs a di!erent source. The links making up the plaquette are replaced by the diagonal parts only. By separating the links into diagonal and o!-diagonal parts we get the "nal form of the Ehrenfest}Maxwell relation. 1D F 2"1J 2#J " #J " #1J$." 2#1J$." 2 . I IJ J J J J J The "rst term in the current comes from the excitation of the charged coset "elds, the static term has an extra non-local contribution coming from the corrective gauge transformation, and the last two contributions are from the ghost "elds. These terms give a non-vanishing charge density cloud around a static source. The left-hand side can be used as a lattice operator to measure the total charge density. 5.4. Abelian point charge has cloud of like charge As a simple application we use this de"nition of #ux to calculate div E on a source and the total #ux away from the source. In Table 2, we see that the integrated #ux on a plane between the charges plus the integrated #ux on a back plane of the torus is larger than the div E on the source. The interpretation is that the bare charge is dressed with same polarity charge by the interactions and the neighborhood has a cloud of like charge. Hence there is anti-screening. This charge density has contributions from all terms in the Ehrenfest relation.
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Table 2 div E normalized to 1/b for a classical point charge b
div E (cl.pt.charge)"1/b
div E (on source)
Total #ux
10.0 (almost classical)
0.1
0.1042(1)
0.0910(8) (mid) 0.0148(8) (back) 0.1092(8) (total)
0.4166
0.5385(19)
0.7455(70) (mid) 0.0359(72) (back) 0.7815(95) (total)
2.4
Note: Source is a 3 Wilson loop. div E measured on a source. Electric #ux measured on the midplane centered on the Wilson loop. Also included is the #ux through a plane on the far side of the torus, and the sum being the total #ux. 8 lattice, 3;3 Wilson loop.
5.5. Summary We have exploited symmetries of the lattice partition function to derive a set of exact, nonAbelian identities which de"ne the Abelian "eld strength operator and a conserved electric current arising from the coset "elds traditionally discarded in Abelian projection. The current has contributions from the action, the gauge "xing condition and the Faddeev}Popov operator. Numerical studies on small lattices veri"ed the identity to within errors of a few per cent. We have found the Faddeev}Popov current in particular to be unusually sensitive to systematic e!ects such as low numerical precision and poor random number generators, but the origin of any remaining, subtle biases, if they exist, is not clear; we have already considered all terms in the partition function. In a pure ;(1) theory the static quark potential may be measured using Wilson loops that correspond to unit charges moving in closed loops, as demonstrated by "1D\F 2""d . In J JI 5 Abelian projected SU(2) the same measurements in the maximally Abelian gauge yield an asymptotic area law decay and a string tension that is only slightly less than the full non-Abelian value. In other gauges it is not clear that an area law exists } certainly it is more troublesome to identify. We have seen that in the context of the full theory the Abelian Wilson loop must be reinterpreted. The coset "elds renormalize the charge of the loop as measured by "1D\F 2" and charge is also J JI induced in the surrounding vacuum. Full SU(2) has anti-screening/asymptotic freedom of color charge, and in the maximally Abelian gauge alone have we seen analogous behavior, in that the source charge is increased and induces charge of like polarity in the neighboring vacuum. Whether this renormalization of charge can account for the reduction of the string tension upon Abelian projection in this gauge is not clear. In other gauges, where Abelian dominance of the string tension is not seen, the coset "elds appear to have a qualitatively di!erent behavior, acting to suppress and screen the source charge. In conclusion, the improved "eld strength expression de"ned by the Ehrenfest identity does not coincide with the lattice version of [17] of 't Hooft's proposed "eld strength operator [14]. The Abelian and monopole dominance of the string tension invites a dual superconductor hypothesis for con"nement. If this is to be demonstrated quantitatively such as by veri"cation of a (dual)
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London equation then a careful understanding of the "eld strength operator is required. The Ehrenfest identities may provide this [85].
Acknowledgements I wish to thank G. Di Cecio, A. Di Giacomo, J. Greensite, G. Gubarev, A. Hart, M.I. Polikarpov Y. Sasai, E.T. Tomboulis and J. Wosiek for discussions. This work was supported in part by United States Department of Energy grant DE-FG05-91 ER 40617.
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Physics Reports 315 (1999) 175}198
Di!ractive production at collider energies and factorization Chung-I Tan Department of Physics, Brown University, Providence, RI 02912, USA
Abstract One of the more interesting developments from recent collider experiments is the "nding that hadronic total cross sections as well as elastic cross sections can be described by the exchange of a `soft Pomerona pole in the near-forward limit. The most important consequence of Pomeron being a pole is the factorization property. However, due to Pomeron intercept being greater than 1, the extrapolated single-di!raction dissociation cross section based on a classical triple-Pomeron formula is too large leading to a potential unitarity violation at Tevatron energies, which has been referred to as `Dino's paradoxa. We review the resolution which involves a proper implementation of "nal-state screening correction, with `#avoringa for Pomeron as the primary dynamical mechanism for setting the relevant energy scale. In this approach, factorization remains intact, and unambiguous predictions for double-Pomeron exchange, doubly di!raction dissociation, etc., both at Tevatron and at LHC energies, can be made. 1999 Elsevier Science B.V. All rights reserved. PACS: 13.85.!t
1. Introduction One of the more interesting developments from recent collider experiments is the "nding that hadronic total cross sections as well as elastic cross sections in the near-forward limit can be described by the exchange of a `soft Pomerona pole, i.e., the absorptive part of the elastic
E-mail address: [email protected] (C.-I. Tan) This review is written in honor of Richard Slansky, who helped me greatly in appreciating the importance of understanding inelastic production as the origin of di!ractive processes. In this article, which replaces an earlier draft (hep-ph/9706276), we concentrate on soft di!ractive production. In a future article, we plan to discuss the relation between soft and hard Pomeron [2] with particular emphasis on the `#avoringa mechanism leading to the picture of a `Heterotic Pomerona (hep-ph/9302308). 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 0 - 4
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amplitudes can be approximated by Im ¹ (s, t)Kb (t)s?PRb (t). The Pomeron trajectory has ? @ ? @ two important features. First, its zero-energy intercept is greater than one, aP(0),1#e, eK0.08}0.12, leading to rising p(s). Second, its Regge slope is approximately a PK 0.25}0.3 GeV\, leading to the observed shrinkage e!ect for elastic peaks. The most important consequence of Pomeron being a pole is factorization. For a singly di!ractive dissociation process, factorization leads to a `classical triple-Pomerona formula, (1) dp/dt dmPdp /dt dm,F P (m, t)p P (M, t) , ? @ where M is the missing mass variable and m,M/s. The "rst term on the right-hand side of Eq. (1) is the so-called `Pomeron #uxa, and the second term is the `Pomeron-particlea total cross section. Eq. (1) is, in principle, valid only when m\ and M are both large, with t small and held "xed. However, with e&0.1, it has been observed [6] that the extrapolated pp single-di!raction dissociation cross section, p, based on the standard triple-Pomeron formula is too large at Tevatron energies by as much as a factor of 5}10 and it could become larger than the total cross section. Let us denote the singly di!ractive cross section as a product of a `renormalizationa factor and the classical formula, dp/dt dm"Z(m, t; s)dp /dt dm .
(2)
It was argued by Goulianos [6] that agreement with data could be achieved by having an energy-dependent suppression factor, Z(m, t; s)PZ (s),N(s)\41 [6,7]. However, the modi"ed % triple-Pomeron formula no longer has a factorized form. An alternative suggestion has been made recently by Schlein [8,9]. It was argued that phenomenologically, after incorporating lower triple-Regge contributions, the renomalization factor for the triple-Pomeron contribution could be described by an m-dependent suppression factor, ZPZ (m). 1 In view of the factorization property for total and elastic cross sections, the `#ux renormalizationa procedure appears paradoxical and could undermine the theoretical foundation of a soft Pomeron as a Regge pole from a non-perturbative QCD perspective. We shall refer to this as `Dino's paradoxa. Finding a resolution that is consistent with Pomeron pole dominance for elastic and total cross sections at Tevatron energies will be the main focus of this study. In particular, we
For "ts for soft Pomeron parameters, see Ref. [1]. We use s "1 GeV as the basic energy scale throughout this paper. These `classicala expressions are: P P F P (m, t)"(1/16p)b (t)(m\)? R\, and p P (M, t)"g(t)(M)? \b (0). The triple-Pomeron coupling g(t) can be in ? ? @ @ principle determined by data below (s430 GeV where cross sections are relatively insensitive to the choice of the Pomeron intercept value used. As far as I am aware of, the "rst attempt in using a `triple-Reggea analysis to interpret real experimental data was made in [3]. A formal postulation was "rst made in [4]. For other contemporaneous works, see [5]. This `renormalizationa factor is chosen so that the new `Pomeron #uxa, F (s, m, t),N(s)\F P (m, t), is normalized to , N unity for s5s , (s K22 GeV. With m (s)"1.5/s, m "0.1, eK0.08, N(s), dtK F P (m, t) dmK0.25sC'1, for
N
\ K (s522 GeV, and N(s),1 for (s422 GeV. For a di!erence in opinion, see Ref. [11]. See also Ref. [10].
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want to maintain the following factorization property: dp P F (m, t)p (M) (3) I I dt dm I when m\ and M become large. A natural expectation for the resolution to this paradox lies in implementing a large screening correction to the classical triple-Pomeron formula. However, this appears too simplistic. In the absence of a new energy scale, a screening factor of the order 5}10, if obtained, would apply both at Tevatron energies and at ISR energies. This indeed is the case for the eikonalization analysis by Gotsman et al. [12] as pointed out by Goulianos. Since a successful triple-Pomeron phenomenology exists up to ISR energies, a subtler explanation is required. We shall assume that any screening e!ect can supply at the most 10}20% suppression and it cannot serve as the primary mechanism for explaining the paradox. Triple-Regge phenomenology has had a long history. It has enjoyed many successes since early 1970s, and it should emerge as a feature of any realistic representation of non-perturbative QCD for high-energy scattering. In particular, it should be recognized that, up to ISR energies, triple-Pomeron phenomenology has provided a successful description for the phenomenon of di!ractive dissociation. A distinguishing feature of the successful low-energy triple-Pomeron analyses is the value of the Pomeron intercept. It has traditionally been taken to be near 1, which would lead to total cross sections having constant `asymptotic valuesa. In contrast, the current paradox centers around the Pomeron having an intercept greater than 1, e.g., eK0.1. Instead of trying to ask `how can one obtain a large suppression factor at Tevatron energiesa, an alternative approach can be adopted. We could "rst determine the `triple-Pomerona coupling by matching the di!ractive cross section at the highest Tevatron energy. A naive extrapolation to lower energies via a standard triple-Pomeron formula would of course lead to too small a cross section at ISR energies. We next ask the question: E Are there physics which might have been overlooked by others in moving down in energies? E In particular, how can a high-energy "t be smoothly interpolated with the successful low-energy triple-Pomeron analysis using a Pomeron with intercept at 1, i.e., eK0. A key observation which will help in understanding our proposed resolution concerns the fact that, even at Tevatron energies, various `subenergiesa, e.g., the missing mass squared, M, and the di!ractive `gapa, m\, can remain relatively small, comparable to the situation of ISR energies for the total cross sections. (See Fig. 1.) Our analysis has identi"ed the `yavoringa of Pomeron [15] as the primary dynamical mechanism for resolving the paradox. A proper implementation of
See Ref. [13]. We include both NNM and cc production as well as other e!ects. The e!ective degrees of freedom involved are `diquarksa and charm quarks, respectively. For color counting, a baryon is considered as a bound state of a quark and diquark. In a more modern approach, baryons are to be considered as skyrmions in a large N approach. Again, they should be treated as independent degrees of freedom from mesons. See also Ref. [14]. We do not include `semi-harda production in the current treatment for soft Pomeron. Flavoring will indeed be the primary mechanism in our construction of a `Heterotic Pomerona.
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Fig. 1. Phase space for single-di!raction dissociation from ISR to LHC in terms of rapidity variables y,log m\ and y ,log M. The dashed lines are for `#avoringa scale, y , chosen to be 9 for illustration. Dotted lines are y K3 lines
for both y and y . The dashed}dotted lines are constant center of mass energy lines for E , i"1,2,6, equals to
G 15, 30, 60, 630, 1800, 14,000 GeV, respectively.
"nal-state screening correction (or xnal-state unitarization), assures a unitarized `gap distributiona. However, the onset of this suppression cannot take place at low energies; we "nd that #avoring sets this crucial energy scale. We also "nd that initial-state screening remains unimportant, consistent with the pole dominance picture for elastic and total cross-section hypothesis at Tevatron energies. In the usual usage of classical triple-Regge formulae, the basic energy scale is always in terms of s K1 GeV. We demonstrate that there are at least two other energy scales, s ,eW and s ,eW, y K3}5 and y K8}10, which must be incorporated properly. The "rst is associated with the physics of light quarks and the scale of chiral symmetry breaking. The second is the `#avoringa scale and is associated with `heavy #avora production. In a non-perturbative QCD setting, both play an important role in our understanding of a bare Pomeron with an intercept greater than unity [15]. In our treatment, initial-state screening remains unimportant, consistent with the pole dominance picture for elastic and total cross-section hypothesis at Tevatron energies. The factor F P(m, t) from the Pomeron contribution will be referred to as a `unitarized Pomeron #ux factor,a ? and we shall occasionally refer to our procedure as `unitarization of Pomeron yuxa. We shall demonstrate that in our unitarization scheme the total renormalization factor has a factorized form Z(m, t; s)"Z (m, t)Z (M)"[S (m, t)R(m\)]R(M) , (4) B K where S is due to "nal-state screening (Eq. (32)). R is a #avoring factor, given by Eq. (11), and there is one #avoring factor for each Pomeron propagator. With the pole dominance hypothesis, we demonstrate that the unitarized #ux factor must satisfy a normalization condition.
dt dm F P(m, t)gPPP(t)mC,b (b (0) , ? ? ? \
(5)
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where F P(m, t),Z (m, t)F P(m, t). The required damping to overcome the divergent behavior of ? B ? F P(m, 0) at small m comes from both the screening factor S (m, t) and the factor mC from the ? `Pomeron-particlea total cross section. In Section 2, we "rst review the dynamics of soft Pomeron at low energies. We next discuss in Section 3 the origin of #avoring scale and indicate its relevance for di!ractive dissociation cross sections. In Section 4, we explain why, given the Pomeron pole dominance hypothesis, the initial-state screening cannot be large at Tevatron energies and emphasize the importance of "nal-state absorption. We study in Section 5 the e!ect of "nal-state screening via an eikonal mechanism, consistent with the Pomeron pole dominance at collider energies. Putting these together, we provide the "nal resolution to Dino's paradox in Section 6. We present a phenomenological analysis which yields an estimate for the `high-energya triplePomeron coupling: gPPP(0)K0.14}0.20 mb .
(6)
This value is consistent with our #avoring expectation (Eq. (34)). Surprisingly, the amount of screening required at Tevatron energies seems to be very small. Predictions for other related cross sections are discussed in Section 7. Comparison of our approach to that of Refs. [6,8] as well as other comments are given in Section 8.
2. Soft Pomeron at low energies In order to be able to answer the questions we posed in the Introduction, it is necessary to "rst provide a dynamical picture for a soft Pomeron and to brie#y review the notion of `Harari}Freunda duality. 2.1. Harari}Freund duality Although Regge phenomenology pre-dated QCD, it is important to recognize that it can be understood as a phenomenological realization of non-perturbative QCD in a `topological expansiona, e.g., the large-N expansion. In particular, an important feature of a large-N expectation is A A emergence of the Harari}Freund two-component picture [16]. For P 420 GeV/c, it was recognized that the imaginary part of any hadronic two-body amplitude can be expressed approximately as the sum of two terms Im A(s, t)"R(s, t)#P(s, t). From the s-channel point of view, R(s, t) represents the contribution of s-channel resonance while P(s, t) represents the non-resonance background. From the t-channel point of view, R(s, t) represents the contribution of `ordinarya t-channel Regge exchanges and P(s, t) represents the di!ractive part of the amplitude given by the Pomeron exchange. Three immediate consequences of this picture are: 1. Imaginary parts of amplitudes which show no resonances should be dominated by Pomeron exchange, (RK0, and PKconstant). 2. Imaginary parts of A(s, t) which have no Pomeron term should be dominated by s-channel resonances,
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3. Imaginary parts of amplitudes which do not allow Pomeron exchange and show no resonances should vanish, Point (2) can best be illustrated by partial-wave projections of pNPpN scattering amplitudes from well-de"ned t-channel isospin exchanges. Point (3) is best illustrated by examining the K>pPKp, where, by optical theorem, Im A(K>pPKn)Jp (K>p)!p (Kn). The near-equality of these two cross sections, from the t-channel exchange view point, re#ects the interesting feature of exchange degeneracy for secondary Reggeons. Finally, let us come to point (1). From the behavior of p ! , p ! , p and p , one "nds that the near-constancy for the P-contribution corresponds to NN p N ) N NN having an e!ective `low-energya Pomeron intercept at 1, i.e., a P (0)K1 . 2.2. Shadow picture and inelastic production A complementary treatment of Pomeron at low energies is through the analysis of inelastic production, which is responsible for the non-resonance background mentioned earlier. Di!raction scattering as the shadow of inelastic production has been a well established mechanism for the occurrence of a forward peak. Analyses of data up to ISR energies have revealed that the essential feature of non-di!ractive particle production can be understood in terms of a multiperipheral cluster-production mechanism. In such a picture, the forward amplitude at high energies is predominantly absorptive and is dominated by the exchange of a `bare Pomerona. In a `shadowa scattering picture, the `minimum biaseda events are predominantly `short-range ordereda in rapidity and the production amplitudes can be described by a multiperipheral cluster model. Under a such an approximation to production amplitudes for the right-hand side of an elastic unitary equation, Im ¹(s, 0)" "¹ ", one "nds that the resulting elastic amplitude is L L dominated by the exchange of a Regge pole, which we shall provisionally refer to as the `bare Pomerona. Next consider singly di!ractive events. We assume that the `missing massa component corresponds to no-gap events, thus the distribution is again represented by a `bare Pomerona. However, for the gap distribution, one would insert the `bare Pomerona just generated into a production amplitude, thus leading to the classical triple-Pomeron formula. Extension of this procedure leads to a `perturbativea expansion for the total cross section in the number of bare Pomeron exchanges along a multiperipheral chain. Such a framework was proposed long time ago [17], with the understanding that the picture can make sense at moderate energies, provided that the intercept of the Pomeron is near one, a(0)K1, or less. However, with the acceptance of a Pomeron having an intercept greater than unity, this expansion must be embellished or modi"ed. It is quite likely that the resolution for Dino's paradox lies in understanding how such an e!ect can be accommodated within this framework, consistent with the Pomeron pole dominance hypothesis. 2.3. Bare Pomeron in non-perturbative QCD In a non-perturbative QCD setting, the Pomeron intercept is directly related to the strength of the short-range-order component of inelastic production and this can best be understood in
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a large-N expansion [18]. In such a scheme, particle production mostly involves emitting `low-mass pionsa, and the basic energy scale of interactions is that of ordinary vector mesons, of the order of 1 GeV. In a one-dimensional multiperipheral realization for the `planar componenta of the large-N QCD expansion, the high-energy behavior of a n-particle total cross section is primarily controlled by its longitudinal phase space, p K(gN/(n!2)!)(gN log s)L\s(\. Since L there are only Reggeons at the planar diagram level, one has J "2a !1 and, after summing 0 over n, one arrives at Regge behavior for the planar component of p where a "(2a !1)#gN . (7) 0 0 At next level of cylinder topology, the contribution to partial cross section increases due to its topological twists, p K(g/(n!2)!)2L\(gN log s)L\s(\, and, upon summing over n, one L arrives at a total cross-section governed by a Pomeron exchange, p(>)"ge?P7, where the Pomeron intercept is (8) aP"(2a !1)#2gN . 0 Combining Eqs. (7) and (8), we arrive at an amazing `bootstrapa result, aPK1. Having a Pomeron intercept near 1 therefore depends crucially on the topological structure of large-N non-Abelian gauge theories [18]. In this picture, one has a K0.5}0.7 and gNK0.3}0.5. 0 With a K1 GeV\, one can also directly relate a to the average mass of typical vector mesons. 0 Since vector meson masses are controlled by constituent mass for light quarks, and since constituent quark mass is a consequence of chiral symmetry breaking, the Pomeron and the Reggeon intercepts are directly related to fundamental issues in non-perturbative QCD. This picture is in accord with the Harari}Freund picture for low-energy Regge phenomenology. Finally we note that, in a Regge expansion, the relative importance of secondary trajectories to the Pomeron is controlled by the ratio e?0W/e?PW"e\?P\?RW. It follows that there exists a natural scale in rapidity, y (aP!a )\(y K 3}5. The importance of this scale y is, of course, well 0 known: When using a Regge expansion for total and two-body cross sections, secondary trajectory contributions become important and must be included whenever rapidity separations are below 3}5 units. This scale of course is also important for the triple-Regge region: There are two relevant rapidity regions: one associated with the `rapidity gapa, y,log m\, and the other for the missing mass, y ,log M.
2.4. Conyict with Donnachie}Landshow picture It has become increasingly popular to use the Donnachie}Landsho! picture [1] where Pomeron intercept above one, i.e., e&0.1. Indeed, it is impressive that various cross sections can be "tted via Pomeron pole contribution over the entire currently available energy range. However, it should be pointed out that Donnachie}Landsho! picture is not consistent with the Harari}Freund picture at low energies. In particular, it underestimates the Pomeron contribution at low energies, and it leads to a strong exchange degeneracy breaking.
A phenomenological realization of QCD emphasizing the topological structure of large-N gauge theories is dual parton model (DPM). For a recent review, see Ref. [19].
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It can be argued that the di!erence between these two approaches should not be important at high energies. This is certainly correct for total cross sections. However, we would like to stress that this is not the true for di!ractive dissociation, even at Tevatron energies. This can best be understood in terms of rapidity variables, y and y . Since y#y K>, >,log s, it follows that,
even at Tevatron energies, the rapidity range for either y or y is more like that for a total cross
section at or below the ISR energies. (See Fig. 1.) Therefore, details of di!ractive dissociation cross section at Tevatron would depend on how a Pomeron is treated at relatively low subenergies.
3. Soft Pomeron and 6avoring Consider for the moment the following scenario where one has two di!erent "ts to hadronic total cross sections: 1. High-energy xt: p (y)Kb b eCW for y
(9)
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where e (y) has the properties E e KeK0.1 for yCW>?YR. This representation can now be extended down to the region y&y . ?@ We shall adopt a particularly convenient parameterization for e (y) in the next section when we discuss phenomenological concerns. To complete the story, we need also to account for the scale dependence of Pomeron residues. What we need is an `interpolatinga formula between the high-energy and low-energy sets. Once a choice for e (y) has been made, it is easy to verify that a natural choice is simply b(y)"b e C\CW W. It follows that the total contribution from a `#avoreda Pomeron corresponds ? ? to the following low-energy modi"cation ¹? @(y, t)PR(y)¹ (y, t) , (10) ? @ ? @ where ¹ (y, t),b b e>C>? PRW is the amplitude according to a `high-energya description with ? @ ? @ a "xed Pomeron intercept, and R(y),e\ C\CW W\W
(11)
is a `#avoringa factor. The e!ect of this modi"cation can best be illustrated via Fig. 2. This #avoring factor should be consistently applied as part of each `Pomeron propagatora. With the normalization R(R)"1, we can therefore leave the residues alone, once they have been determined by a `high-energya analysis. For our single-particle gap cross section, since there are three Pomeron propagators, the renormalization factor is given by the following product: Z"R(y)R(y ). It is instructive to plot in Fig. 3 this combination as a function of either m or M for
various "xed values of total rapidity, >. 3.2. Flavoring of bare Pomeron We have proposed sometime ago that `baryon paira, together with other `heavy #avora production, provides an additional energy scale, s "eW, for soft Pomeron dynamics, and this e!ect can be responsible for the perturbative increase of the Pomeron intercept to be greater than unity, aP(0)&1#e, e'0. One must bear this additional energy scale in mind in working with a soft Pomeron [15]. That is, to fully justify using a Pomeron with an intercept aP(0)'1, one must restrict oneself to energies s's where heavy #avor production is no longer suppressed. Conversely, to extrapolate Pomeron exchange to low energies below s , a lowered `e!ective trajectorya must be used. This feature of course is unimportant for total and elastic cross sections at Tevatron energies. However, it is important for di!ractive production since both m\ and M will sweep right through this energy scale at Tevatron energies. Flavoring becomes important whenever there is a further inclusion of e!ective degrees of freedom than that associated with light quarks. This can again be illustrated by a simple one-dimensional multiperipheral model. In addition to what is already contained in the Lee}Veneziano model, suppose that new particles can also be produced in a multiperipheral chain.
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Fig. 2. E!ect of #avoring factor R(s) when applied to a standard rising cross section: p "bsC, e"0.1 and b"16 mb, given by the solid curve. With R(y) given by Eq. (31), the dashed}dotted curve has e "0, j "1, and #avoring scale y "9, and the dotted curve corresponds to e "!0.04. Fig. 3. Renormalization factor due to #avoring alone, Z (m; s),R(m\)R(M), as a function of rapidity y"log m\ for various "xed center of mass energies. These curves correspond to parameters used for the solid line in Fig. 4.
Concentrating on the cylinder level, the partial cross sections will be labeled by two indices (12) p K(g/p!q!)2N>O(gN log s)N(gN log s)Os(\ , N O where q denotes the number of clusters of new particles produced. Upon summing over p and q, we obtain a `renormalizeda Pomeron trajectory a P "a P #e ,
(13)
where a P K1 and eK2gN. That is, in a non-perturbative QCD setting, the e!ective intercept of Pomeron is a dynamical quantity, re#ecting the e!ective degrees of freedom involved in nearforward particle production [15]. If the new degree of freedom involves particle production with high mass, the longitudinal phase space factor, instead of (log s)O, must be modi"ed. Consider the situation of producing one NNM bound state together with pions, i.e., p arbitrary and q"1 in Eq. (12). Instead of (log s)N>, each factor should be replaced by (log(s/m ))N>, where m is an e!ective mass for the NNM cluster. In terms of rapidity, the longitudinal phase space factor becomes (>!d)N>, where d can be thought of as a one-dimensional `excluded volumea e!ect. For heavy-particle production, there will be an energy range over which keeping up to q"1 remains a valid approximation. Upon summing over p, one "nds that the additional contribution to the total cross section due to the production of one heavy-particle cluster is [13] p &p (>!d)(2gN)log(>!d)h(>!d), where a P K1. Note O the e!ective longitudinal phase space `threshold factora, h(>!d), and, initially, this term represents a small perturbation to the total cross section obtained previously (corresponding to q"0 in Eq. (12)), p . Over a rapidity range [d, d#d ], where d is the average rapidity required for producing another heavy-mass cluster, this is the only term needed for incorporating this new degree of freedom. As one moves to higher energies, `longitudinal phase space suppressiona
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becomes less important and more and more heavy-particle clusters will be produced. Upon summing over q, we would obtain a new total cross section, described by a renormalized Pomeron, with a new intercept given by Eq. (13). We assume that, at Tevatron, the energy is high enough so that this kind of `thresholda e!ects is no longer important. How low an energy do we have to go before one encounter these e!ects? Let us try to answer this question by starting out from low energies. As we have stated earlier, for >'3}5, secondary trajectories become unimportant and using a Pomeron with aK1 becomes a useful approximation. However, as new #avor production becomes e!ective, the Pomeron trajectory will have to be renormalized. We can estimate for the relevant rapidity range when this becomes important as follows: y '2d #1q2 d . The "rst factor d is associated with leading
particle e!ect, i.e., for proton, this is primarily due to pion exchange. d is the minimum gap associated with one heavy-mass cluster production, e.g., nucleon}antinucleon pair production. We estimate d K2 and d K2}3, so that, with 1q2 K2, we expect the relevant #avoring rapidity
scale to be y K8}10. 4. Pomeron dominance hypothesis at Tevatron energies We shall "rst explore consequences of the observation that both total cross sections and elastic cross sections can be well described by a Pomeron pole exchange at Tevatron energies. Absorption correction, if required, seems to remain small. Since the singly di!ractive cross section, p, is a sizable part of the total, it must also grow as sC. This qualitative understanding can be quanti"ed in terms of a sum rule for `rapidity gapa cross sections. This in turn imposes a convergence condition on our unitarized Pomeron #ux, F P(m, t). ? To simplify the discussion, we shall "rst ignore transverse momentum distribution by treating the longitudinal phase space only. For instance, for singly di!raction dissociation, the longitudinal phase space can be speci"ed by two rapidities, y,log(m\) and y ,log M. The "rst variable
speci"es the rapidity gap associated with the detected leading proton (or antiproton), and the second variable speci"es the rapidity `spana of the missing mass distribution. At "xed s, they are constrained by y#y K>,log s (see Fig. 1), and we can speak of di!erential di!ractive cross
section dp/dy. We shall in what follows use +m\, M, and +y,log m\, y ,log M, inter changeably. Dependence on transverse degrees of freedom can be reintroduced without much e!ort after completing the main discussion. Consider the process a#bPc#X, where the number of particles in X is unspeci"ed. However, unlike the usual single-particle inclusive process, the superscript for X indicates that all particles in X must have rapidity less than that of the particle c, i.e., the detected particle c is the one in the "nal state with the largest rapidity value. Kinematically, a single-gap cross section is identical to the singly di!raction dissociation cross section discussed earlier. Under the assumption where all transverse motions are unimportant, one has y Ky ,> and the di!erential gap cross section, A ?
dp /dy, can also be considered as a function of y and y , with y#y K>.
Because the detected particle c has been singled out to be di!erent from all other particles, this is no longer an inclusive cross section, and it does not satisfy the usual inclusive sum rules. Upon integrating over the rapidity gap y and summing over particle type c, no multiplicity enhancement factor is introduced and one obtains simply the total cross section, i.e., a gap cross section satis"es
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the following exact sum rule:
dp (14) dy ?@ A, p " p "p . ?@ A L ?@ dy A L A Interestingly, this allows an identi"cation of the total cross section as a sum over speci"c gap cross sections, p , each is `deriveda from a speci"c `leading particlea gap distribution. Since no ?@ A restriction has been imposed on the nature of the `gap distributiona, e.g., particle c can have di!erent quantum numbers from a, the notion of a gap cross section is more general than a di!raction cross section. For di!raction dissociation, c must have same quantum numbers as the incident particle a. For de"niteness, we discuss leading particle gap distribution relative to the incoming particle a with a positive, large rapidity, and assume c"a. For pp collision, actual di!ractive cross section, p, is arrived at by taking into account contributions involving di!raction at both p and p vertices. It follows that the singly di!ractive dissociation cross section, p, is a part of p . ?@ ?@ ? Consider next our factorized ansatz, Eq. (3). For y and y large, it leads to a gap distribution,
dp /dy"e\WF (y)p (y ). If p (y )"gb eCW , it follows that contribution from each gap distribu?@ A ? A @ @ @ tion is Regge behaved, p KbA eC7b , where the total Pomeron residue is a sum of `partial residuesa ?@ A ? @ b " bA " dy FA (y)ge\>CW . (15) ? ? ? A A For above integral to converge, each #ux factor must grow slower than eCW. That is, FA (y)e\CWP0 as ? yPR. In a traditional Regge approach, the large rapidity gap behavior for each #ux factor is controlled by an appropriate Regge propagator, e?G>?H\W. Clearly, a standard triple-Pomeron behavior with aP'1 is inconsistent with the pole dominance hypothesis. Unitarity correction must supply enough damping to provide convergence. There is yet another way of expressing the consequence of the pole dominance hypothesis. Dividing each gap di!erential cross section by the total cross section, factorization of Pomeron leads to a `limiting distributiona: o (y,>)Po (y), oA (y). That is, the limit is independent of the A ? ?@ ? total rapidity, >, and the gap density is normalizable, dy o (y)" dy oA (y)" (bA /b )"1. A A ? ? ? ? This normalization condition for the gap distribution is precisely Eq. (15). Let us now restore the transverse distribution and concentrate on the di!ractive dissociation contribution, which can be identi"ed with the high M and high m\ limit of dp (t, m; M). In ?@ ? terms of m, M, and t, the di!erential cross section at large M under our factorizable ansatz takes on the following form, dp/dt dmKF P(m, t)p P (M), where p P (M)"gPPP(t)(M)Cb (0). It fol@ @ @ ?@ ? lows from Eq. (15) that F P(m, t) must satisfy the following bound: ? K (16) dm F PgPPP(t)(m, t)mC4 dt dt dm F P(m, t)gPPP(t)mC,b (b (0) . ? ? ? ? \ K Q \ The hypothesis of a Pomeron pole dominance for the total and elastic cross sections is of course only approximate. However, to the extend that absorptive corrections remain small at Tevatron energies, one "nds that a modi"ed Pomeron #ux factor must di!er from the `classicala Pomeron #ux at small m in such a way so that the upper bound in Eq. (16) is satis"ed. We shall refer to F P(m, t) as the `unitarized Pomeron #uxa. How this can be accomplished via "nal-state screen? ing will be discussed next. Note both the similarity and the diwerence between Eq. (16) and the
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`#ux normalizationa condition mentioned in the Introduction. Here, this convergent integral yields asymptotically a "nite number, b , and the ratio b /b (0) can be interpreted as the probability of ? ? ? having a di!ractive gap at high energies. 5. Final-state screening The best-known example for implementing the idea of `screeninga in high-energy hadronic collisions has been the `expanding diska picture for rising total cross sections. Di!raction scattering as the shadow of inelastic production has been a well established mechanism for the occurrence of a forward peak. Analyses of data up to collider energies have revealed that the essential feature of non-di!ractive particle production can be understood in terms of a multiperipheral clusterproduction mechanism. In such a picture, the forward amplitude is predominantly absorptive and is dominated by the exchange of a `bare Pomerona. If the Pomeron intercept is greater than one, it forces further unitarity corrections as one moves to higher energies. For instance, saturation of the Froissart bound can be next understood through an eikonal mechanism, with the absorptive eikonal s(s, b) given by the bare Pomeron amplitude in the impact-parameter space. The main problem we are facing here is not so much on how to obtain a `most accuratea #ux factor F P(m, t) at very small m. We are concerned with a more di$cult conceptual problem of how ? to reconcile having a potentially large screening e!ect for di!raction dissociation processes and yet being able to maintain approximate pole dominance for elastic and total cross sections up to Tevatron energies. We shall show using an expanding disk picture that absorption works in such a way that inelastic scattering can only take place on the `edgea of disk. Therefore, once applied using "nal-state screening, the e!ect of initial-state absorption will be small, hence allowing us to maintain Pomeron pole factorization for elastic and total cross sections. 5.1. Expanding disk picture Let us brie#y review this picture which also serves to establish notations. At high energies, a near-forward amplitude can be expressed in an impact-parameter representation via a twodimensional Fourier transform,
¹(s, t),2is dbe q b fI (s, b),
fI (s, b)"(4ins)\ dqe\ b q¹(s, t) ,
(17)
where tK!q. Assume that the near-forward elastic amplitude at moderate energies can be described by a Born term, e.g., that given by a single Pomeron exchange where we shall approximate it to be purely absorptive. Let us denote the contribution from the Pomeron exchange to fI (s, b) as s(s, b). With aP(t)"1#e#a Pt, and approximating b(t) by an exponential, we "nd s(s, b)K X(s)e\@ Q, X(s),s(s, 0)Kp (s)/4B(s) ,
(18)
This mechanism has been recognized before. It was used in Ref. [20] to explain why a `maximal odderona cannot be allowed in any hadronic scheme which admits an expanding-disk interpretation at high energies.
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where B(s)"b #a P log s and p (s)"p sC. With e'0, the Born approximation would eventually violate s-channel unitarity at small b as s increases. A systematic procedure which in principle should restore unitarity is the Reggeon calculus. However, our current understanding of dispersion-unitarity relation is too qualitative to provide a de"nitive calculational scheme. The key ingredient of `screeninga correction is the recognition that the next-order correction to the Born term must have a negative sign. (The sign of double-Pomeron cut contribution.) In an impact-representation, Reggeon calculus assures us that the correction can be represented as fI (s, b)K!(1/2!)k(s)s(s, b) , (19) where k is positive. To go beyond this, one needs a model. A physically well-motivated model which should be meaningful at moderate energies and allows easy analytic treatment is the eikonal model. Writing fI (s, b)"fI (s, b)#fI (s, b)#fI (s, b)#2, the expansion alternates in sign, and with simple weights such that fI (s, b)"[1!e\IQQ @]/k, and
¹(s, t)"
2is k
dbe q b+1!e\IQQ @, .
(20)
Conventional eikonal model has k"1. We keep k41 here so as to allow the possibility that screening is `imperfecta. Observe that the eikonal derived from the Pomeron exchange, s(s, b), is a monotonically decreasing function of b, taking on its maximum value X(s) at b"0, which increases with s due to e'0. The eikonal drops to zero at large b and is of the order 1 at a radius, b (s)K A (B(s)log kX(s)&log s. Within this radius, fI (s, b)"O(1) and it vanishes beyond. This is the `expansion diska picture of high-energy scattering, leading to an asymptotic total cross section O(b (s)). A 5.2. Inelastic screening In order to discuss inelastic "nal-state screening, we follow the `shadowa scattering picture in which the `minimum biaseda events are predominantly `short-range ordereda in rapidity and the production amplitudes can be described by a multiperipheral cluster model. Substituting these into the right-hand side of an elastic unitary equation, Im ¹(s, 0)" "¹ ", one "nds that the resulting L L elastic amplitude is dominated by the exchange of a Regge pole, which we shall provisionally refer to as the `bare Pomerona. Next consider singly di!ractive events. We assume that the `missing massa component corresponds to no gap events, thus the distribution is again represented by a `bare Pomerona. However, for the gap distribution, one would insert the `bare Pomerona just generated into a production amplitude, thus leading to the classical triple-Pomeron formula. Extension of this scheme leads to a `perturbativea treatment for the total cross section in the number of bare Pomeron exchanges along a multiperipheral chain. Such a scheme was proposed long time ago [17], with the understanding that the picture could make sense at moderate energies, provided that the intercept of the Pomeron is one, a(0)K1, or less. However, with the acceptance of a Pomeron having an intercept greater than unity, this expansion must be embellished. Although it is still meaningful to have a gap expansion, one must improve the descriptions for parts of a production amplitude involving large rapidity gaps by taking into account absorptions for the
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gap distribution. We propose that this `partial unitarizationa be done for each gap separately, thus maintaining the factorization property along each short-range ordered sector. This involves "nal-state screening, and, for singly di!raction dissociation, it corresponds to the inclusion of `enhanced Pomeron diagramsa in the triple-Regge region. To simplify the notation, we shall use the energy variable, m\, and the rapidity gap variable, y"log(m\), interchangeably. Let us express the total unitarized contribution to a gap cross section in terms of a `unitarized #uxa factor, F(y, t),(eW/16p)"g (y, t)",(1/16pm)" f (m, t)", so that it B B reduces to the classical triple-Pomeron formula as its Born term. That is, the corresponding Born amplitude for f (m, t) is the `square-roota of the triple-Pomeron contribution to the classical B formula, fP (m, t)"bP(t)(m\)?PR\. Screening becomes important if large gap becomes favored, i.e., when e'0. Let us work in an impact representation, g(y , t),2idqe b qg (y , b). Consider an expansion B B B g (y, b)"s (y, b)#g (y, b)#g (y, b)#2 where, under the usual exponential approximation for B the t-dependence, the Fourier transform for the Born term is s (y, b)"(pB(y)/4B (y)) e\@ BW, with B B B (y)"b #a Py and pB(y)"p eCW. The key physics of absorption is B B B g (y, b)K!k s(y, b)s (y, b) . (21) B B The proportionality constant k can be di!erent from the constant k introduced earlier for the B elastic screening, either for kinematic or dynamic reasons, or both. For a generalized eikonal approximation, one has g (y, b)"s +1!k s# (k s)!2,"s (y,b)e\IBQW @. If we de"ne the B B r B B `"nal-state screeninga factor as the ratio between the unitarized #ux factor and the classical triple-Pomeron formula, F P(y, t)"S(y, t; X)F P(y, t), we then have ? ? S(y, t; X)"" f (y, t)/f (y, t)" . (22) B B We shall use this expression as a model for probing the physics of inelastic screening in an expanding disk picture. Let us examine this eikonal screening factor in the forward limit, t"0, where dbs (y, b)e\IBQW @ B . (23) S(y, 0; X) " dbs (y , b) B B Unlike the elastic situation, the integrand of the numerator is strongly suppressed both in the region of large b and in the region `insidea the expanding disk. The only signixcant contribution comes from a `ringa region near the edge of the expanding disk [20]. Since the value of the integrand is of O(1) there, one "nds that the numerator varies with energy only weakly. On the other hand, the denominator is simply pB(y), which increases as eCW. Therefore, this leads to an exponential cuto! in y, S(y, 0; X)&e\CW. This damping factor precisely cancels the m\C behavior from the classical triple-Pomeron formula at small m, leading to a unitarized Pomeron #ux factor.
It is possible also to apply an eikonal model to study initial-state screen for singly di!ractive dissociation cross section. This indeed has been performed before, however, without taking "nal-state screening into account [12]. The fact that inelastic absorption takes place at small impact parameter, with surviving scattering allowed only at the edge of the expanding disk has also been noted there. Since our "nal-state absorption would remove all scattering within the disk, applying an eikonal initial-state absorption procedure becomes unnecessary.
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To be more precise, let us work in a simpler representation for S(y, 0; X) by changing the variable bPz,e\@ W. With our Gaussian approximation, one "nds that
S(y, 0; X)" r
dz z P\e\VX
, (24) VIB6W PPW , where where r(y),B(y)/B (y). This expression can be expressed as +rx\PC(x, r)," B B VI 6W PPW C(x, r)"V dz zP\e\X is the incomplete Gamma function. In this representation, one easily veri"es that the screening factor has the desired properties: As k P0, screening is minimal and one B has S(y, 0; X)P1. On the other hand, for y large, X(y) increases so that S(y, 0; X)P[k X(y)]\, B as anticipated. Similarly, we "nd that the logarithmic width for the unitarized #ux D(y, t) at t"0 has increased . As k P0, from 2B to 2B where B "B +!r log dz zP\e\VX, B B VI 6W PPW B B B B P B PB . For y very large, B PB log k X(y)&b (y)Jy. This corresponds to a faster B B B B A shrinkage than that of ordinary Regge behavior. Averaging over t, one "nds at large di!ractive rapidity y, the "nal-state screening provides an average damping 1S 2Pe\CW"mC . (25) This leads to a unitarized Pomeron #ux, F P(m, t), which automatically satis"es the upper bound, ? Eq. (16), derived from the Pomeron pole dominance hypothesis [12].
6. Final recipe Having explained earlier the notion of #avoring and its e!ects both on Pomeron intercept and on its residues, we must build in this feature for the "nal-state screening. As we have shown in the last section, inelastic screening is primarily driven by the `unitarity saturationa of the elastic eikonal (Eq. (18)). However, because of #avoring, screening sets in only when the Pomeron #avoring scale is reached. This picture is consistent with the fact that, while low-mass di!raction seems to be highly suppressed, high-mass di!raction remains strong at Tevatron energies. Since a Pomeron exchange enters as a Born term, i.e., the eikonal for either the elastic or the inelastic di!ractive production, #avoring can easily be incorporated if we multiply both s(y, b) and s (y, b) by a #avoring factor R(y). That is, if we adopt a generalized eikonal model for "nal-state B screening, the desired screening factor becomes S (m, t)"S(y, t; R(y)X(y), k ) (26) B where S is given by Eq. (22). We have also explicitly exhibited the dependence on the maximal value of the #avored elastic eikonal, RX, and on the e!ectiveness parameter k . B Let us now put all the necessary ingredients together and spell out the details for our proposed resolution to Dino's paradox. Our xnal recipe for the Pomeron contribution to single-di!raction dissociation cross section is (27) dp/dt dm"F P(m, t)pP (M) , @ ? where the unitarized #ux, F P, and the Pomeron-particle cross section, pP , are given in @ ? terms of their respective classical expressions by F P(m, t),Z (m, t)F P(m, t) and pP (M), @ ? B ?
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Z (M)p P (M, t). It follows that the total suppression factor is @ K Z(m, t; M)"Z (m, t)Z (M)"[S (m, t)R(m\)]R(M) (28) B K with the screening factor given by Eq. (26) and the #avoring factor given by Eq. (11). Finally, we point out that the integral constraint for the unitarized #ux (Eq. (5)), when written in terms of these suppression factors, becomes
dt dm S (m, t)R(m\)F P(m, t)g(t)mC"b (b (0) , ? ? ? \ where F P(m, t),(1/16p)b (t)(m\)?PR\. ? ?
(29)
6.1. Phenomenological parameterizations Both the screening function and the #avoring function depend on the e!ective Pomeron intercept, and we shall adopt the following simple parameterization. The transition from a (0)"1#e to a(0)"1#e will occur over a rapidity range, (y, y). Let y ,(y#y) and j\,(y!y). Similarly, we also de"ne e ,(e#e ) and *,(e!e ). A convenient parameterization for e we shall adopt is e (y)"[e#* tanhj (y!y )] . (30) The combination [e!e (y)] can be written as (2e )[1#(s/s )H]\ where s "eW. Combining this with Eq. (11), we arrive at a simple parameterization for our #avoring function (31) R(s),(s /s)C >Q Q H \. With a P K1, we have e K0, e K*Ke/2, and we expect that j K1}2 and y K8}10 are reasonable range for these parameters. To complete the speci"cation, we need to provide a more phenomenological description for the "nal-state screening factor. First, we shall approximate the screening factor by an exponential in t: (32) S (y, t)KS (y, 0)e BWR , where S (y, 0)"+rx\PC(x, r),, with x"k R(y)X(y) and r"r(y). The width, *B (y), can be ob B B tained by a corresponding substitution. Note that S (y, 0) depends on B(y), B (y), X(y), and k . B B Phenomenological studies allow us to approximate B(y)Kb #0.25y and B (y)Kb #0.25y, B B b Kb /2}2.3 GeV\. B The only quantity left to be speci"ed is the e!ectiveness parameter k . Since the physics of B "nal-state screening is that driven by a Pomeron with intercept greater than unity, the relevant rapidity scale is again y . Let us "x k "rst by requiring that screening is small for y(y , i.e., B S (y, 0)&1 as one moves down in rapidity from y to y . Similarly, we expect screening to approach its full strength as one moves past the #avoring threshold y . We thus "nd it economical to
By choosing e (0, it is possible to provide a global `averagea description mimicking `secondary trajectorya contributions for various low energy regions. In Ref. [13], acceptable estimates are e K!0.11 to !0.5.
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parameterize k (y)K(k /2)+1#tanh[j (y!y )], (33) B B and we expect y &y and j &j . This completes the speci"cation of our unitarization procedure. B 6.2. High-energy diwractive dissociation The most important new parameter we have introduced for understanding high-energy di!ractive production is the #avoring scale, s "eW. We have motivated by way of a simple model to show that a reasonable range for this scale is y K8}10. Quite independent of our estimate, it is possible to treat our proposed resolution phenomenologically and determine this #avoring scale from experimental data. It should be clear that one is not attempting to carry out a full-blown phenomenological analysis here. To do that, one must properly incorporate other triple-Regge contributions, e.g., the PPR-term for the low-y region, the nnP-term and/or the RRP-term for the low-y region, etc.,
particularly for (s4(s &100 GeV. What we hope to achieve is to provide a `caricaturea of the interesting physics involved in di!ractive production at collider energies through our introduction of the screening and the #avoring factors [13]. Let us begin by "rst examining what we should expect. Concentrate on the triple-Pomeron vertex g(0) measured at high energies. Let us for the moment assume that it has also been measured reliably at low energies, and let us denote it as g (0). Our #avoring analysis indicates that these two couplings are related by g(0)Ke\CW g (0) .
(34)
With eK0.08}0.1 and y K8}10, using the value g (0)"0.364$0.025 mb [21], we expect a value of 0.12}0.18 mb . Denoting the overall multiplicative constant for our renormalized triple-Pomeron formula by K, (35) K,b(0)gPPP(0)b (0)/16p . @ ? With bK16 mb, we therefore expect K to lie between the range 0.15}0.25 mb. N We begin testing our renormalized triple-Pomeron formula by "rst turning o! the "nal-state screening, i.e., setting S "1. We determine the overall multiplicative constant K by normalizing the integrated p to the measured CDF (s"1800 GeV value. With e"0.1, j "1, this is done for a series of values for y "7, 8, 9, 10. We obtain respective values for K"0.24, 0.21, 0.18, 0.15, consistent with our #avoring expectation. As a further check on the sensibility of these values for
The published CDF p values at (s"546 and 1800 GeV are 7.89$0.33 and 9.46$0.44 mb, respectively. These values correspond to m "0.15. We shall restrict m(0.05 and t to be inside the extreme forward peak. For m "0.05,
we reduce these values by&8% while maintaining their relative ratio of 0.834. For t to be within the extreme forward di!raction peak we scale down the ISR di!ractive cross sections also by approximately 8%. This is appropriate for our determination of the triple-Pomeron coupling.
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Fig. 4. Fits to representative single-di!raction dissociation cross sections from ISR to Tevatron [6]. The solid line corresponds to e"0.08, e "!0.07, j "1, y "9, with a small amount of "nal-state screening, k "0.1. The dotted M and the dashed}dotted curves correspond to k "0.2 and k "0, i.e., no screening, respectively.
the #avoring scales, we "nd for the ratio o,p(546)/p(1800) the values 0.63, 0.65, 0.68, 0.72, respectively. This should be compared with the CDF result of 0.834. Next we consider screening. Note that screening would increase our values for K, which would lead to large values for g. Since we have already obtained values for triple-Pomeron coupling which are of the correct order of magnitude, the only conclusion we can draw is that, at Tevatron, screening cannot be too large. With our parameterization, we "nd that screening is rather small at Tevatron energies, with k K0.0}0.2. This comes as somewhat as a surprise! Clearly, screening will become important eventually at higher energies. After #avoring, the amount of screening required at Tevatron is apparently greatly reduced. Having shown that our renormalized triple-Pomeron formula does lead to sensible predictions for p at Tevatron, we can improve the "t by enhancing the PPR-term as well as RRP-terms which can become important. Instead of introducing a more involved phenomenological analysis, we simulate the desired low-energy e!ect by having e K!0.06 to !0.08. A remarkably good "t M results with e"0.08}0.09, y "9 and k K0}0.2 [13]. This is shown in Fig. 4. The ratio o ranges from 0.78 to 0.90, which is quite reasonable. The prediction for p at LHC is 12.6}14.8 md. Our "t leads to a triple-Pomeron coupling in the range of gPPP(0)K0.12}0.18 mb ,
(36)
exactly as expected. Interestingly, the triple-Pomeron coupling quoted in Ref. [6] (g(0)" 0.69 mb ) is actually a factor of 2 larger than the corresponding low-energy value [21]. Note that this di!erence of a factor of 5 correlates almost precisely with the #ux renormalization factor N(s)K5 at Tevatron energies. We believe, with care, the physics of #avoring and "nal-state screening can be tested independent of the speci"c parameterizations we have proposed here. In particular, because our unitarized Pomeron #ux approach retains factorization along the `missing massa link, unambiguous predictions can be made for other processes involving rapidity gaps.
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7. Predictions for other gap cross sections For both double Pomeron exchange (DPE) and doubly di!ractive (DD) processes, one is dealing with three rapidity variables which can become large. We will treat these two cases "rst before turning to more general situations. 7.1. Prediction for DPE cross sections For double Pomeron exchange (DPE), we are dealing with events with two large rapidity gaps. The "nal-state con"guration can be speci"ed by "ve variables, t , t , m , m , and M. For t and t small, one again has a constraint, m\Mm\Ks. Alternatively, we can work with rapidity variables, y ,log (m\), y ,log (m\), and y ,log M, with y #y #y K>"log s. The
appropriate DPE di!erential cross section can be written down, with no new free parameter. Let us introduce a renormalization factor dp dp "Z (y , t , y , y , t ) . ".#
dy dt dy dt dy dt dy dt One immediately "nds that, using Pomeron factorization for the missing mass variable,
(37)
Z "[S (y , t )R(y )]R(y )[S (y , t )R(y )] ".#
"Z (m , t )Z (M)Z (m , t ) . (38) B B Alternatively, we can express this cross section in terms of singly di!ractive dissociation cross sections as
dp dp dp ?@ +R(y )p (y ),\ ?@ , ?@ " (39)
@? dy dt dy dt dy dt dy dt where p (y )"b b eCW . This clean prediction involves no new parameter, with the understanding @? @ ? that, when y is low, secondary terms must be added.
7.2. Prediction for DD cross sections For double-di!raction dissociation (DD), there are two large missing mass variables, M, M, separated by one large rapidity gap, y, and its associate momentum transfer variable t. Again, for t small, we have the constraint y #y #yK>. K K (y, t)p P(y ), where the classical The classical di!erential DD cross section is p P(y )FM P @ K ? K `gap distributiona function is FM P (t, y)"(1/16p)eC>? PRW. After taking care of both #avoring and "nal-state screening, one obtains for the renormalization factor Z (y , y, t, y )"R(M)[SM (y, t)R(y)]R(M),Z (M)ZM (m, t)Z (M) . (40) "" K K
B
A new screening factor, SM (y, t), has to be introduced because of the di!erence in the t-distribution associated with two factors of triple-Pomeron coupling. It can be obtained from S (y, t) by replacing B (y) by BM (y)"bM #a Pt, where, by factorization, bM "2b !b (bM is the t-slope associated with the B B triple-Pomeron coupling).
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Alternatively, this cross section can again be expressed as a product of two single di!ractive cross sections
dp dp dp ?@ +ZM (y, t)p (y, t),\ ?@ , ?@ " (41) ?@ dy dt dy dt dy dt dy K K K K where p (y, t),(1/16p)"b (t)b (t)eC>? PRW". Other than the modi"cation from S to SM , this predic?@ ? @ tion is again given uniquely in terms of the single-di!raction dissociation cross sections. 7.3. Other gap cross sections We are now in the position to write down the general Pomeron contribution to the di!erential cross section with an arbitrary number of large rapidity gaps. For instance, generalizing the DPE process to an n-Pomeron exchange process, there will now be n large rapidity gaps, with n!1 short-range ordered missing mass distributions alternating between two gaps. The corresponding renormalization factor is ZL "Z (m , t )Z (M)ZM (m , t )Z (M)2Z (M )Z (m , t ) . (42) .# B B
L\ B L L Other generalizations are all straightforward. However, since these will unlikely be meaningful phenomenologically in the near future, we shall not discuss them here. It is, nevertheless, interesting to point out that, if any cross section does become meaningful experimentally, #avoring would dictate that it is most likely the classical triple-Regge formulas with aP(0)K1 that would be at work "rst.
8. Comments Let us brie#y recapitulate what we have accomplished. Given Pomeron as a pole, the total Pomeron contribution to a singly di!ractive dissociation cross section can in principle be expressed as (43) dp/dt dm"[S (s, t)][F P(m, t)][pP (M)] , @ ? F P(m, t)"S (m, t)FP (m, t) . (44) ? ? E The "rst term, S , represents initial-state screening correction. We have demonstrated that, with a Pomeron intercept greater than unity and with a pole approximation for total and elastic cross sections remaining valid, initial-state absorption cannot be large. We therefore can justify setting S K1 at Tevatron energies. E The "rst crucial step in our alternative resolution to the Dino's paradox lies in properly treating the "nal-state screening, S (m, t). We have explained in an expanding disk setting why a "nal-state screening can set in relatively early when compared with that for elastic and total cross sections. E We have stressed that the dynamics of a soft Pomeron in a non-perturbative QCD scheme requires taking into account the e!ect of `#avoringa, the notion that the e!ective degrees of freedom for Pomeron is suppressed at low energies. As a consequence, we "nd that FP (m, t)"R(m\)F P (m, t) and pP (M)"R(M)p P (M) where R is a `#avoringa factor. ? ? @ @
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It is perhaps worth contrasting what we have achieved with the #ux renormalization scheme of Goulianos [6]. By construction, the normalization factor N(s) is of the form which one would have obtained from an initial-state screening consideration. Although this breaks factorization, one might hope perhaps the scheme could be phenomenologically meaningful at Tevatron energies. Note that, for (s'22 GeV, the renormalization factor N(s) has an approximately factorizable form: N(s)&0.25sC"0.25(m\)C(M)C. It follows that the di!ractive di!erential cross section remains in a factorized form (45) m dp/dt dm&0.25[mC>F P (m, t)][(M)\Cp P (M, t)] . N N It can be shown that Eq. (45) leads to a di!ractive cross section p which, up to log s, is asymptotically constant. That is, the di!ractive dissociation contribution no longer corresponds to the part of total cross sections represented by the Pomeron exchange. This is not in accord with the basic hypothesis of Pomeron dominance for total and elastic cross sections at Tevatron energies. Our "nal resolution shares certain common features with that proposed by Schlein [8]. At a "xed m, Z (M)K1 as sPR so that it is possible to identify our renormalization factor
Z (m, t)"S (m, t)R(m\) with the #ux damping factor Z (m) of Schlein. In Ref. [8], it was emphaB 1 sized that the behavior of Z (m) can be separated into three regions. (i) (m , m ) where Z K1, (ii) 1 1 (m , m ) where Z drops from 1 to 0.4 smoothly, and (iii) (0, m ) where Z (m)P0 rapidly as mP0. The 1 1 boundaries of these regions are m &0.015 and m &10\. The "rst boundary m can be identi"ed with our energy scale, s &m\&eW. If we identify the boundary between region-(ii) and region-(iii) with our #avoring scale y by s\"e\W"m , one has y K9, which is consistent with our estimate. Since S (m, t)K1 for m's\ and R(m\) drops from R(1)KsC to 1 at s , their Z (m) behaves 1 qualitatively like our renormalization factor. If one indeed makes this connection, what had originally been a mystery for the origin of the scale, m , can now be related to the non-perturbative dynamics of Pomeron #avoring. It should be stressed that our discussion depends crucially on the notion of soft Pomeron being a factorizable Regge pole. This notion has always been controversial. Introduced more than 30 years ago, Pomeron was identi"ed as the leading Regge trajectory with quantum numbers of the vacuum with a(0)K1 in order to account for the near constancy of the low-energy hadronic total cross sections. However, as a Regge trajectory, it was unlike others which can be identi"ed by the
Ultimately, these two schemes can be di!erentiated by confronting experimental data. For our scheme, because of Pomeron pole dominance, it leads to a normalizable limiting gap distribution, o (y; >), i.e., o(y, t; >)Po (y, t). For ? ? y(y ;>, it is cut-o! in y at least as fast as o (y, t)Je\CW. In contrast, the #ux renormalization scheme, Eq. (45), leads to ? a gap distribution of the form o (y; >)Je\C7eCW, for 1;y;>. Test of these two alternatives for either the normaliz? ation and the y-distribution can, in principle, be carried out by comparing data at two Tevatron energies by focusing on the region of "xed small t and 0.025m50.002 (y (y(y ). Interestingly, both behaviors seem to provide acceptable
"ts based on data presented in Ref. [11]. There are also several di!erences between our result and that of Schlein. First, our renormalization factor Z(m, t; s) is t-dependent whereas Schlein's is not. At very small m, our suppression factor does not vanish as fast as that of Schlein: (Z (m)&m, whereas ours behaves as mC). Furthermore, since we have found that there is very little screening needed at 1 Tevatron energies, our slow cuto! might not set in until much higher energies so that it could indeed be possible to observe the m\C behavior for dp/dy dt at Tevatron energies.
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particles they interpolate. With the advent of QCD, the situation has improved, at least conceptually. Through large-N analyses and through other non-perturbative studies, it is natural to A expect Regge trajectories in QCD as manifestations of `string-likea excitations for bound states and resonances of quarks and gluons due to their long-range con"ning forces. Whereas ordinary meson trajectories can be thought of as `open stringsa interpolating qq bound states, Pomeron corresponds to a `closed stringa con"guration associated with glueballs. However, the di$culty of identi"cation, presumably due to strong mixing with multi-quark states, has not helped the situation in practice. In a simpli"ed one-dimensional multiperipheral realization of large-N QCD, the non-Abelian gauge nature, nevertheless, managed to re-emerge through its topological structure [18]. The observation of `pole dominancea at collider energies has hastened the need to examine more closely various assumptions made for Regge hypothesis from a more fundamental viewpoint. It is our hope that by examining Dino's paradox carefully and by "nding an alternative resolution to the problem without deviating drastically from accepted guiding principles for hadron dynamics, Pomeron can continue to be understood as a Regge pole in a non-perturbative QCD setting. The resolution for this paradox could therefore lead to a re-examination of other interesting questions from a "rmer theoretical basis. For instance, to be able to relate quantities such as the Pomeron intercept to non-perturbative physics of color con"nement represents a theoretical challenge of great importance.
Acknowledgements I would like to thank K. Goulianos for "rst getting me interested in this problem during the Aspen Workshop on Non-perturbative QCD, June 1996. Intensive discussions with K. Goulianos, A. Capella, and A. Kaidalov at Rencontres de Moriond, March, 1997, have been extremely helpful. I am also grateful to P. Schlein for explaining to me details of their work and for his advice. I want to thank both K. Goulianos and P. Schlein for helping me to understand what I should or should not believe in various facets of di!ractive data! Lastly, I really appreciate the help from K. Orginos for both numerical analysis and the preparation for the "gures. This work is supported in part by the D.O.E. Grant CDE-FG02-91ER400688, Task A.
References [1] A. Donnachie, P.V. Landsho!, Phys. Lett. B 296 (1992) 227; J.R. Cuddel, K. Kang, S.K. Kim, Phys. Lett. B 395 (1997) 311; R.J.M. Covolan, J. Montanha, K. Goulianos, Phys. Lett. B 389 (1996) 176; M. Block, to be presented at VIIth Blois Workshop on Elastic and Di!ractive Scattering, Seoul, June 10}14, 1997. [2] G. Ingelman, P. Schlein, Phys. Lett. B 296 (1992) 227. [3] D. Silverman, C.-I. Tan, Relation between the multi-Regge model and the missing-mass spectrum, Phys. Rev. D 2 (1970) 233. [4] C. DeTar et al., Phys. Rev. Lett. 26 (1971) 675. [5] D. Horn, F. Zachariasen, Hadron Physics at Very High Energies, Benjamin, New York, 1973. [6] K. Goulianos, Phys. Lett. B 358 (1995) 379.
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[7] K. Goulianos, Proceedings of the Third Workshop on Small-x and Di!ractive Physics, Argonne National Laboratory, USA, September 1996; K. Goulianos, Proceedings of the Fifth International Workshop on Deep Inelastic Scattering and QCD (DIS-97), Chicago, USA, April 1997. [8] P. Schlein, Proceedings of the Third Workshop on Small-x and Di!ractive Physics, Argonne National Laboratory, USA, September 1996; P. Schlein, Proceedings of the Fifth International Workshop on Deep Inelastic Scattering and QCD (DIS-97), Chicago, USA, April 1997. [9] S. Erhan, P. Schlein, Saturation of the Pomeron #ux factor in the proton by damping small Pomeron momenta, Phys. Lett., submitted for publication. [10] A. Brandt et al., Measurements of single di!raction at (s"630 GeV; Implication for the Pomeron #ux factor, Nucl. Phys., submitted for publication. [11] K. Goulianos, Comments on the Erhan-Schlein model of damping the Pomeron #ux at small-x, hep-ph/9704454. [12] E. Gotsman, E.M. Levin, U. Maor, Phys. Rev. D 49 (1994) 4321. [13] T.K. Gaisser, C.-I. Tan, Phys. Rev. D 8 (1973) 3881; C.-I. Tan, Proceedings IX Rencontres de Moriond, Meribel, France, 1974. [14] J.W. Dash, S.T. Jones, Phys. Lett. B 157 (1985) 229. [15] C.-I. Tan, in: K. Goulianos (Ed.), Proceedings of Second International Conference on Elastic and Di!ractive Scattering, Editions Frantieres, Dreux, 1987, p. 347; C.-I. Tan, in: D. Schi!, J.T.V. Tran (Eds.), Proceedings of Nineteenth International Symposium on Multiparticle Dynamics, Arles, Editions Frontieres, Dreux, 1988, p. 361. [16] H. Harari, Phys. Rev. Lett. 20 (1968) 1395; P.G.O. Freund, Phys. Rev. Lett. 20 (1968) 235. [17] W. Frazer, D.R. Snider, C.-I. Tan, Phys. Rev. D 8 (1973) 3180. [18] H. Lee, Phys. Rev. Lett. 30 (1973) 719; G. Veneziano, Phys. Lett. B 43 (1973) 314; F. Low, Phys. Rev. D 12 (1975) 163. [19] A. Capella, U. Sukhatme, C.-I. Tan, J.T.V. Tran, Phys. Rep. 236 (1994) 225. [20] G. Finkelstein, H.M. Fried, K. Kang, C.-I. Tan, Phys. Lett. B 232 (1989) 257. [21] R.L. Cool, K. Goulianos, S.L. Segler, H. Sticker, S.N. White, Phys. Rev. Lett. 47 (1981) 701.
Physics Reports 315 (1999) 199}230
Feshbach resonances in atomic Bose}Einstein condensates Eddy Timmermans *, Paolo Tommasini, Mahir Hussein , Arthur Kerman T-4, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Institute for Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics 60 Garden Street, Cambridge, MA 02138, USA Instituto de Fn& sica, Universidade de SaJ o Paulo, C.P. 66318, CEP 05315-970 SaJ o Paulo, Brazil Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Abstract The low-energy Feshbach resonances recently observed in the inter-particle interactions of trapped ultra-cold atoms involve an intermediate quasi-bound molecule with a spin arrangement that di!ers from the trapped atom spins. Variations of the strength of an external magnetic "eld then alter the di!erence of the initial and intermediate state energies (i.e. the &detuning'). The e!ective scattering length that describes the low-energy binary collisions, similarly varies with the near-resonant magnetic "eld. Since the properties of the dilute atomic Bose}Einstein condensates (BECs) are extremely sensitive to the value of the scattering length, a &tunable' scattering length suggests highly interesting many-body studies. In this paper, we review the theory of the binary collision Feshbach resonances, and we discuss their e!ects on the many-body physics of the condensate. We point out that the Feshbach resonance physics in a condensate can be considerably richer than that of an altered scattering length: the Feshbach resonant atom}molecule coupling can create a second condensate component of molecules that coexists with the atomic condensate. Far o!-resonance, a stationary condensate does behave as a single condensate with e!ective binary collision scattering length. However, even in the o!-resonant limit, the dynamical response of the condensate mixture to a sudden change in the external magnetic "eld carries the signature of the molecular condensate's presence: experimentally observable oscillations of the number of atoms and molecules. We also discuss the stationary states of the near-resonant condensate system. We point out that the physics of a condensate that is adiabatically tuned through resonance depends on its history, i.e. whether the condensate starts out above or below resonance. Furthermore, we show that the density dependence of the many-body ground-state energy suggests the possibility of creating a dilute condensate system with the liquid-like property of a selfdetermined density. 1999 Elsevier Science B.V. All rights reserved.
* Corresponding author. E-mail address: [email protected] (E. Timmermans) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 5 - 3
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PACS: 03.75.Fi; 05.30.Jp; 32.80.Pj; 67.90.#z Keywords: Bose}Einstein condensates; Feshbach resonance; Coherent matter wave dynamics
1. Introduction The recent observation of Feshbach resonances in the inter-particle interactions of a dilute Bose}Einstein condensate of Na-atoms by Ketterle's group at MIT [1] was an eagerly anticipated event. The signi"cance of this experimental breakthrough appears of singular importance as its consequences are far-reaching in two sub"elds of physics: (i) Atomic and Molecular Physics. Although predicted theoretically [46], technical di$culties had previously prevented the observation of the low-energy Feshbach resonances [2]. This situation abruptly changed when the experimental e!orts recently culminated in the observation of resonances in ultra-cold Na at MIT, in Rb by Heinzen's group at U.T. Austin [3] and by the Wieman-Cornell collaboration at JILA [4], as well as in Cesium by Chu's group at Stanford [5]. This cascade of results indicates that the "eld of atomic trapping and cooling has achieved the necessary amount of control and precision to carry out systematic studies of the resonances in a variety of atomic systems. (ii) BEC-physics. An important distinguishing feature of the MIT-experiment [1], is that the Feshbach resonances were observed in an atomic BEC-system [6}8]. The resonances were observed by varying an external magnetic "eld, thereby altering the &detuning' (de"ned as the di!erence between the initial and intermediate state energies). Similarly, the e!ective scattering length that describes the low-energy atom}atom interaction, varies with magnetic "eld and is consequently &tunable'. As all quantities of interest in the atomic BECs crucially depend on the scattering length, a tunable interaction suggests very interesting studies of the many-body behavior of condensate systems. Previous searches for the low-energy Feshbach resonances had been unsuccessful [2]. The di$culties that had to be overcome were multiple: the magnetic traps can only trap low "eld seeking states, the resonant magnetic "eld strengths are rather high and the margin of error on the predicted values for the resonant magnetic "elds were considerable due to the uncertainties in the interatomic potential curves. Conversely, the measured values of the resonant "elds and the &widths' will be of great help in re"ning the potential curves that characterize the inter-atomic interaction. This, in turn, has important applications in spectroscopy, high-frequency resolution measurements and atomic clocks. The Feshbach resonance has been listed as one of the prospective schemes to alter the e!ective inter-particle interactions of the cold-atom systems. A variable interaction strength is a highly unusual degree of freedom in experimental studies of many-body systems. Especially for the atomic condensate system, the &tunability' of this parameter suggests very interesting applications as virtually all observable quantities, as well as the stability of the system, sensitively depend on its value. This aspect which motivated much of the theoretical work [28}33] was emphasized in [1], as well as the Nature article that accompanied the original report [9]. Other schemes have been proposed to alter the inter-particle interactions, e.g. by means of external electrical "elds [10].
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Among the many applications suggested by this novel degree of freedom, we name but a few: (i) Study of negative scattering length condensates. A dilute gas condensate of bosons that experience an e!ective mutual attraction is unstable in the absence of an external potential. A trapping potential can &stabilize' a negative scattering length condensate of limited number of bosons [11}15]. Once the population of the condensate exceeds this critical number, the condensate collapses. Although most theories agree on the collapse and the critical number of particles, the details and the mechanism of the collapse are still being debated in the current literature (for a discussion see, for example, Ref. [15]). Measurements taken by Hulet's group at Rice university with Li provide valuable data on this interesting system, but a study of the collapse at variable values of the interaction strength will give de"nitive tests of the theoretical predictions and yield much needed insight in the dynamics of the collapse. (ii) Study of the condensate phase separation. Overlapping condensates are unstable if the strength of the unlike boson interactions exceeds the geometric mean of the like-boson interaction strengths. Mutually repelling condensates then separate spatially and act as immiscible #uids. As the phase separation criterion depends solely on the interaction strengths, one could by varying one of the strengths [16}18], render the condensates miscible or immiscible. Whether the BEC-systems will have practical applications remains to be seen, but a two-#uid system that can be made miscible or immiscible at will, does suggest applications in areas such as data-storage, or perhaps even quantum computation. (iii) Study of Josephson oscillations. A mixture of same species condensates in di!erent internal states can exchange bosons coherently, for example by interacting with coherent near-resonant laser light [19}21]. Such coherent inter-condensate particle exchange processes are often referred to as inter-condensate &tunneling' processes because of the strong analogy with Josephson tunneling [22,23]. Interestingly, unlike the condensed matter Josephson junctions, the dilute condensate mixtures can actually probe the non-linear regime of the Josephson oscillations [19}22]. Like the DC Josephson junction, the number of bosons in each condensate oscillates when the values of the chemical potentials of the respective condensates di!er. For a single dilute condensate, the chemical potential is equal to the product of the interaction strength and the density. Thus, a sudden change of the interaction strength e!ects precisely such a chemical potential di!erence. (iv) Condensate Dynamics. While the near-equilibrium dynamics of dilute single condensate systems are well-understood, at least in the low-temperature limit, the far-from equilibrium condensate dynamics poses a problem that has not been satisfactorily resolved. A detailed understanding of the condensate formation, in particular of the formation time [24}27], will have important implications in a variety of "elds, such as "eld theory and early universe theories. Experimental studies with a variable interaction strength will give a "rm understanding of the role of the inter-particle interactions. Clearly, each of the above applications represents an exciting prospect. It is in this context, the creation of a tunable interaction strength, that the motivation for much of the previous research on Feshbach resonances has been situated [9,28}33]. However, we would urge caution in interpreting the e!ects of the Feshbach resonance on the condensate solely as altering the inter-boson interaction. We believe that the e!ects of the Feshbach resonance are considerably more profound. In particular, in this paper, we review some of our previous research results ([34,35], see also Ref. [36] for a brief review of the tunneling aspect) that show that in a near-resonant BEC, the Feshbach resonant atom-molecule coupling creates a second condensate component of quasibound molecules. The many-body dynamics predicts that the expectation value of the molecular
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"eld does not vanish in a near-resonant magnetic "eld, 1tK 2O0: the hallmark of Bose condensa tion. Whether the presence of the molecular condensate predicts a behavior that di!ers from that of a single condensate with altered scattering length depends on the values of several relevant parameters: the detuning, the rate of change of the detuning (in a dynamical experiment with changing magnetic "eld), the lifetime of the quasi-bound molecules, limited } most importantly } by collisions with other atoms or molecules that change the molecular state. O!-resonance, the stationary condensate system does behave as a single condensate with a scattering length that has the value predicted by the binary collision description. However, even in the o!-resonant limit, the dynamical response of the condensate system to a sudden change of the external magnetic "eld, can di!er signi"cantly from that of a single condensate with e!ective scattering length. We "nd that, subsequent to a sudden change of the external magnetic "eld, the number of atoms and molecules that occupy the respective condensates oscillate. These oscillations are damped out on the time scale of the single molecule lifetime. The o!-resonant condensate lives much longer than the individual molecules, since the atomic condensate acts as a reservoir of atoms, continually replenishing the molecular condensate. The oscillations can then be observed by illuminating the condensate with light that is near resonant with a transition of the quasi-bound molecule. The oscillations in the molecular condensate population then reveal themselves as an oscillating intensity of the image. The oscillations, like the oscillating current observed in Josephson junctions, are caused by the coherent inter-condensate exchange of particles. Unlike the Josephson tunneling, the exchange involves boson pairs. This, as we shall show, has a profound e!ect on the stationary state properties of the BEC system. For instance, we "nd that, for a near-resonant detuning, the homogeneous ground state system is always unstable in the limit of ultra low atomic particle density. Unlike the negative scattering length condensate, this instability does not necessarily lead to collapse. At higher densities the near-resonant condensate system can be stabilized by the inter-particle interactions of the atoms and molecules. In that case, the many-body energy, as a function of the atomic particle density, goes through a minimum. The many-body system can "nd this minimum by decreasing its volume and, if given enough time, can relax to the state of minimal energy and self-determined density, a typical liquid-like property. This self-determined density would still be of the order of 10 cm\, so that these considerations suggest the possibility of creating the world's "rst rari"ed liquid! The paper is organized as follows. In Section 2, we review the binary collision theory of low-energy Feshbach resonances. In Section 3, we specialize to the magnetically controlled Feshbach resonances recently observed in atomic traps. In that section, we set up the Hamiltonian for the many-body problem and we argue that a second molecular condensate component is formed. A compelling argument follows from the equations that describe the many-body dynamics. We also discuss the e!ects of the most important destructive in#uence that the molecular condensate undergoes: collisions of the molecules with other atoms or molecules that change the vibrational state of the quasi-bound molecule. In Section 4, we investigate the stationary states of the near-resonant condensate system, neglecting, for the time being, the e!ects of particle loss. We point out that the state the near-resonant condensate system "nds itself in, depends on its history. If the system was brought near resonance by adiabatically increasing the detuning, its state di!ers from that of the system created by lowering the detuning. Furthermore, we show that, in contrast to the e!ective scattering length description, the condensate system does not have to collapse, as it is tuned adiabatically through resonance. Finally, we conclude in Section 6.
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2. Low-energy Feshbach resonances 2.1. Introduction In this paper, we focus on the implications of a Feshbach resonance on the many-body behavior of Bose}Einstein condensed systems. To this purpose, we develop the necessary theoretical framework to describe the relevant binary collision physics. The following discussion does not aim at being comprehensive with regard to Feshbach resonance physics. For such treatment, we refer the reader to more speci"c articles [37}39]. By de"nition, Feshbach resonances involve intermediate states that are quasi-bound, so that they are sometimes referred to as closed-channel collisions. These intermediate states are not bound in the true sense of the word, as they acquire a "nite lifetime due to the interaction with continuum states of other channels (such as the channel of the incident projectile/target system). For example, in electron}atom and electron}ion scattering, the intermediate states generally decay by ejecting the electron captured in the intermediate state. These states are known as auto-ionization states. In the atom}atom scattering Feshbach resonances of interest here, the intermediate states are molecules with electronic and nuclear spins that have been rearranged from the spins of the colliding atoms by virtue of the hyper"ne interaction. The intermediate molecular states interact with the continuum states of the incident channel that are the scattering states of the single-channel atom}atom scattering problem. In the next section, we discuss these single-channel scattering states. 2.2. Low-energy potential scattering As the interactions of interest involve bosonic atoms at ultra-low translational energies, the collision physics reduces to the description of s-wave scattering. Speci"cally, the angular momentum potential barrier of &height' &( /M¸), where ¸ is the range of the interatomic interaction and M the mass of a single atom, prevents colliding atoms with relative motion of lower kinetic energy and non-vanishing angular momentum from entering the inter-atomic interaction region. The magnitude of the angular momentum potential barrier height is of order ( /M¸)"( /m a);(a /¸);(m /M)&10}100 mK, where m represents the electron mass, (m /M)&10\}10\, and where a denotes the usual Bohr radius (¸/a )&10. Thus, in cold atom samples of temperature below 1 mK, bosonic atoms undergo pure s-wave scattering. The atomic Bose}Einstein condensates have temperatures of the order of 1 lK. Before we proceed with the treatment of low-energy Feshbach resonances, we describe the solution to the single-channel scattering problem for the collision of two of such indistinguishable atoms. In the center-of-mass frame the problem reduces to an integration of the radial s-wave Schrodinger equation with the corresponding molecular potential. The resulting regular solution, u(r), where r denotes the internuclear distance and ®ular' means that lim u(r) is "nite, has to be P normalized. We normalize u(r) to u (r) by requiring its asymptotic behavior to be similar to that of , the zeroth-order spherical Bessel function, which is the regular solution to the free atom Schrodinger equation u (r)&sin(kr#d )/(kr), rPR , , where d is the s-wave phase shift.
(1)
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For future reference, we relate u (r) to the s-wave components of scattering states that have been , normalized using alternative schemes. One useful normalization consists of separating the scattered wave into an incident plane wave of wave vector k and an outgoing spherical wave: uk(r)& exp(ik ) r)#f exp(ikr)/r, rPR ,
(2)
where f is the scattering amplitude. By comparing its asymptotic behavior to that of u , Eq. (1), we , "nd the usual expression for the s-wave scattering amplitude: f"[ exp(2id )!1]/2ik , (3) where k denotes the wave number, which is the magnitude of the k-vector. Furthermore, the s-wave component of uk is equal to [uk(r)] " exp(id )u (r) . (4) , An alternative normalization that we shall consider, requires the regular solution to the Schrodinger equation to be a superposition of incident and outgoing spherical waves: u>(r)& exp(!ikr)/r!S exp(ikr)/r, rPR ,
(5)
where the coe$cient of the outgoing wave, S, is the scattering (or S)-matrix for the single-channel s-wave scattering problem, S" exp(2id ). By comparing the asymptotic behaviors of u and u>, , we "nd that u>(r)"!2ik exp(id ) u (r) . (6) , The alkali atoms in the atomic-trap condensates interact through molecular potentials that support bound states. In accordance with Levinson's theorem, u (r) has then nodes in the , inter-atomic interaction region. The number of nodes is equal to the number of bound states of the corresponding potential. Furthermore, u (r) for the ultra-cold collision energies is essentially , independent of the energy. To see that, we start by noting that for the relevant collision energies the de Broglie wavelength, (2p/k), vastly exceeds the range ¸ of the inter-atomic potential. Outside the range of the inter-atomic interaction, r'¸, but well within the de Broglie wavelength, r(k\, ru(r) is approximately linear: ru(r)Jr!a, where a is the scattering length. By scaling u(r) to the normalized function u , which in this region of the internuclear distance, ¸(r(k\, takes , on the form u (r)+1#(d /kr), we see that d "!ka and , u (r)+1!(a/r) where ¸(r(k\ , (7) , independent of the collision energy. 2.3. Low-energy Feshbach resonances In describing the Feshbach resonant collision, we distinguish the channels of the continuum incident projectile/target state and the closed (molecular) channels to which it is coupled. To this purpose, we introduce the projection operators P and M that denote, respectively, the projections onto the Hilbert subspace of the incident channel and the subspace of the closed (molecular)
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channels. These projection operators satisfy the usual projection operator properties 0"MP"PM , P"P and M"M .
(8)
The time-independent SchroK dinger equation, satis"ed by the total scattering state "W2 of the binary atom system, (E!H)"W2"0 ,
(9)
takes on the form of coupled equations: (E!H )P"W2"H M"W2 , .. .+ (E!H
)M"W2"H P"W2 , ++ +.
(10) (11)
where we use the notation PHP"H , etc. .. We may obtain the projection of the scattering state onto the Hilbert space of the quasi-bound molecules, M"W2, by formally inverting Eq. (11): M"W2"(E!H
)\H P"W2 . ++ +.
(12)
The substitution of Eq. (12) into the projection of the SchroK dinger equation onto the continuum channel (10) then yields an e!ective SchroK dinger equation for the continuum scattering state, (E!H )P"W2"0, with an e!ective Hamiltonian, 1 H , H "H #H +. .. .+ E!H ++
(13)
that exhibits a strong dependence on the energy, E, of the colliding particles. For the purpose of treating the scattering problem, it is, in fact, more instructive to start by inverting Eq. (10) for P"W2, by means of the propagator for outgoing waves, g>(E)"(E!H #ig)\ , . ..
(14)
where g is an in"nitesimally small positive number. This inversion leads to P"W2""u>2#g>(E)H M"W2 . N N .+
(15)
The u>-state in the above expression is a scattering state of the single-channel (P) scattering N problem, (E!H )"u 2"0. In addition, we choose the asymptotic boundary condition of the .. N scattering state so that u> is the superposition of incident and outgoing spherical waves shown in N Eq. (5) for the P-channel, lim u>(r)"exp(!ikr)/r!S exp(ikr)/r. Upon insertion of the P N expression for P"W2 from Eq. (15) into Eq. (11), we "nd the e!ective SchroK dinger equation satis"ed by M"W2: (E!H
)M"W2"H "u>2#H g>(E)H M"W2 . ++ +. N +. N .+
(16)
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Finally, substitution of the formal solution for M"W2 from Eq. (16), 1 H "u>2 , +. N !H g>(E)H ++ +. N .+ into the initial expression for P"W2 of Eq. (15) yields the following result: M"W2" E!H
(17)
1 H "u>2 . (18) +. N !H g>(E)H ++ +. N .+ The asymptotic dependence of P"W2 on the radial inter-nuclear distance r supplies the S-matrix that characterizes the low-energy collision with Feshbach resonance. At the end of this section, we determine S in this manner. P"W2""u>2#g>(E)H N N .+ E!H
2.4. Width The experimentally observed low-energy Feshbach resonances are narrow } each individual resonance is well-separated in frequency space from the other resonances. Near a particular resonance m, with a single intermediate molecular state "u 2 of speci"c ro-vibrational quantum K number, we may further simplify the expression (18) by keeping only a single diagonal matrix element in evaluating the energy denominator of Eq. (18). Speci"cally, we replace 1 1 P"u 2 1u " , K E!E #iC /2 K E!H !H g>(E)H K K ++ +. N .+ where the energy E and width C of the resonance are equal to K K E "Re1u "H #H g>(E)H "u 2 , K K ++ +. N .+ K C K"!Im1u "H g>(E)H "u 2 . K +. N .+ K 2
(19)
(20)
Furthermore, the coupling between the continuum and molecular states for these resonances is weak enough that we may evaluate "u 2 in the spirit of perturbation theory as the eigenstate of the K molecular potential, and approximate E by its eigenvalue. K In particular, the low-energy dependence of the width is of importance to the observed resonances and we will evaluate C (E) in detail. The expansion of g>(E) in continuum states "k2 K N gives with Eq. (14) the following expression:
"1u "H "k2" C K +. K"!Im "p "1u "H "k2"d(E!Ek) . K +. E!Ek#ig 2 k k
(21)
In this equation, the continuum states, "k2, are not plane waves, but properly normalized scattering states uk (introduced in Eq. (2)). We shall work in box normalization so that in coordinate space, [exp(ik ) r)#f exp(ikr)/r] 1r"k2"uk(r)/(X& , rPR , (X
(22)
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where X represents the macroscopic volume to which the binary atom system is con"ned. At the ultra-low collision energies of interest, the matrix elements 1u "H "k2, are dominated by K .+ the s-wave component of the "k2-wave. Within the molecular interaction range, the amplitudes of the higher partial waves have all but vanished at these energies as the colliding particles lack the energy to overcome the angular momentum potential barrier. Furthermore, the low-energy s-wave, u (r) is essentially energy independent, so that 1u "H "k2 is not only independent of the direction , K .+ of the k-vector, but also of its magnitude:
exp(id ) exp(id ) a, dr u (r)HK u (r)" 1u "H "k2+ K +. , K +. (X (X
(23)
where the a-parameter denotes the integral over the relative internuclear position, a"dr u (r)HK u (r). Consequently, the width is proportional to the square of a and the K +. , remaining &phase space factor' k d(E!E )/X: I
1 C (E) K "pa d(E!Ek) Xk 2 or
M
C (E)"a k , K 2p #
(24)
where M is the mass of a single atom and k the wave number corresponding to the relative velocity # of a pair of atoms with total kinetic energy E in the center-of-mass frame. Note that the width depends on the energy of the colliding atoms through the phase space factor, which is a measure of the phase space volume available to the binary atom system after the collision. Evidently, this remark is of importance to the Feshbach resonances in the ultra-cold atoms systems, where the relative velocity of the interacting atoms rigorously vanishes. It is customary to introduce a &reduced width', c, which makes the k-dependence of the width explicit:
M , C (E)"2ck where c"a K 4p
(25)
where it is understood that k is the wave number corresponding to E. Under condensate conditions, kP0, and C (E)P0, although the value of the coupling constant, a, remains constant. K The corresponding phase shift also vanishes linearly with k, but the e!ective scattering length tends to a well-de"ned "nite value, as we shall see below. 2.5. Scattering matrix We evaluate the continuum scattering wave P"W2, 1 1u "H "u>2 . P"W2""u>2#g>(E)H "u 2 N N .+ K E!E #iC /2 K +. N K K
(26)
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As discussed above, "u>2 is normalized by requiring its asymptotic r-dependence to take on the N form u>(r)"exp(!ikr)/r!exp(2id )exp(ikr)/r, where d is the s-wave phase shift. In the lowN energy regime, d "!ka, where a is the scattering length. Furthermore, we remarked in Eq. (6) that u>(r)"!2ik exp(id )u (r), where u is the regular solution to the SchroK dinger equation with N , , u (r)+1!r/a outside the range of the molecular potential. , In evaluating the state (26) in the asymptotic region of coordinate space, the following asymptotic expansion of the s-wave component of the g>-propagator is useful: N M exp(ikr) exp(id ) u (r ) where rPR, (27) [g>(E; r, r )] P! , N 4p r
and where we replaced the mass in the usual expression for the propagator by the reduced mass, M/2. The asymptotic behavior of g>(E)H "u 2 of Eq. (26) is then given by N .+ K M exp(ikr) lim 1r"g>(E)H "u 2"! exp(id ) dr u (r)HK u (r) , .+ K N .+ K 4p r P M exp(ikr) "! exp(id )a . (28) 4p r
Finally, with 1u "H "u>2"!2ik exp(id )a, we obtain the desired asymptotic behavior: K +. N M exp(ikr) ak 1r"g>(E)H "u 21u "H "u>2"i exp(2id ) N .+ K K +. N 2p r
"i exp(2id )C (E) K
exp(ikr) . r
(29)
Consequently, we obtain the following expression for the asymptotic r-dependence of the scattering state of Eq. (26):
iC (E) exp(ikr) exp(!ikr) K ! 1! exp(2id ) . (30) 1r"P"W2& E!E #iC (E)/2 r r K K By identifying the factor in square brackets in Eq. (29), with the scattering matrix S, we obtain
C (E) K , (31) E!E #i C (E)/2 + K where we used that d "!ka. Note that the S-matrix is unitary, "S""1, as be"ts scattering without loss-channels. As a consequence, we can describe the scattering by means of an e!ective scattering length, S"exp(!2ia k), where a "a#a , with C K exp(!2ika )"1!i E!E #iC /2 K K E!E !iC /2 K K . " (32) E!E #iC /2 K K S"exp(!2ika) 1!i
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Thus, we "nd that the e!ective scattering length for collision energy E is equal to
1 C ;[E!E ] K K a (E)"a# tan\ . (33) 2k (E!E )#C /4 K K In the ultra-low-energy limit appropriate for condensate systems, E and C (E) vanish and K E!E P!e, where we will refer to e, the energy of the molecular state relative to the continuum K level of the P-channel, as the detuning of the Feshbach resonance. In this limit, we need to expand Eq. (33) to lowest order in k. With C "2ck, E" k/M and tan\(x)+x if x;1, we "nd K c lim a (E)"a! . (34) e # Furthermore, in describing the weakly interacting many-body system, it is notationally more convenient to work with the interaction strength j than with the scattering length a. If the inter-particle scattering can be described in the Born approximation, then the j-parameter represents the zero-momentum Fourier component of the inter-particle interaction potential. The zero momentum Fourier component is related to the scattering length a, calculated in the same approximation as j"(4p /M)a. However, the Born approximation cannot be used in describing the low-energy binary atom collisions. In that case, the interaction strength is still proportional to the scattering length, although the latter has to be determined more accurately from the full potential scattering problem. In the same spirit, we may introduce an &e!ective' strength, j "(4p /M)a , to describe the binary atom interaction, related to j as j "j!(a/e) . (35) In Eq. (35) we made use of the expression for the reduced width, c"a(M/4p ). 3. Feshbach resonances in atomic condensate systems 3.1. Magnetically controlled, hyperxne-induced Feshbach resonance In each of the experiments, the low-energy Feshbach resonances were observed by studying the behavior of the ultra-cold-atom systems under variations of an external magnetic "eld. The resonance in the binary-atom interactions is caused by the hyper"ne interaction which #ips the electronic and nuclear spins of one of the colliding atoms, bringing the collision system from the continuum (P)-channel into a closed channel, M, of di!erent spin arrangement. The M-channel is closed by the external magnetic "eld which has raised the continuum level of the binary spin #ipped atom system. While in the M-channel, the colliding atoms reside in a quasi-bound molecular state m. A second hyper"ne-induced spin #ip breaks up the molecule, returning the system to the initial P-channel. If the energy of the intermediate quasi-bound molecule is equal to the continuum level of the P-channel, the above-described collision process is &on resonance'. Variations of the external
Strictly speaking, the usual scattering length is only de"ned in the limit EP0, but Eq. (33) with S"exp(!2ia k) gives the correct low energy scattering matrix.
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magnetic "eld shift the relative energies of the M molecule and the P-continuum level and tune, or detune, the interacting atoms close to, or far from resonance. We now describe the physics of this collision in more detail. We "nd it instructive to discuss the collision "rst in the limit of high external magnetic "eld, although this is not the regime in which the experiments were conducted. A high magnetic "eld, B"Bz( , aligns the electronic (s) and nuclear (i) spins of the atoms. To be de"nite, we consider a system of ultra-cold Na atoms (i") with spin projections m "!, and m "!. The G Q interaction of two such atoms is described by the triplet potential, where &triplet' refers to the total electronic spin of the interacting atoms. In the triplet state, the valence electrons of the atoms behave as indistinguishable fermions and &avoid each other', thereby reducing the Coulomb repulsion of the electrons. In contrast, if the spins of the colliding atoms are arranged in a singlet state, the valence electrons do not avoid each other and the Coulomb repulsion generally reduces the depth of the inter-atomic potential, as compared to the triplet potential. Thus, the inter-atomic interaction depends on the magnitude of the total electronic spin, S, where S"s #s . In this case, the spins of the initial binary atom system are in a state "S 2""m ", m "!; Q G m "!, m "!2, with electronic spins arranged in a pure triplet state, S"1. Consequently, Q G the atoms interact through the molecular triplet potential. However, the binary atom hyper"ne interaction, < "(a / );[s ) i #s ) i ], does not commute with S and can #ip the electronic spins of a triplet state to a singlet con"guration. At large internuclear separation, this singlet channel corresponds to the binary atom system with a single spin-#ipped atom. Consequently, the continuum of the singlet channel lies an energy D above the continuum of the incident triplet channel, where D"B[2k #k ], and where k and k denote the electronic and nuclear magnetic , , moments. Under near-resonant conditions, the singlet potential supports a quasi-bound molecular state u (r)"S 2, of energy E near the continuum of the P-channel and total (electronic and nuclear) K HY K spin state "S 2. HY In this context, the atom}molecule coupling, H #H , is provided by the binary atom .+ +. hyper"ne interaction, < . The corresponding a-parameter that indicates the strength of this inter-channel coupling is the product of the spin matrix element and the overlap of the regular triplet wave function with the molecular singlet wave function,
a"1S "< "S 2; dr u (r)u (r) , K , HY
(36)
which characterizes the resonance. The observed resonances were created at intermediate values of the magnetic "eld for which the actual spin state of the individual atoms are not states of good m and m -quantum number. Instead G Q the single atom spin degrees of freedom occupy a state that diagonalize the spin-dependent part of the single-atom Hamiltonian:
a [2k s !k i ] , . H " s ) i #B ) 1
(37)
The "rst term on the right-hand side of Eq. (37) is the single-atom hyper"ne interaction, characterized by a , an energy that depends on the isotope (e.g. a "42.5 mK for Na, see, e.g. Ref. [40]). At zero magnetic "eld, B"0, the diagonalization yields the hyper"ne states of good &f ' quantum
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number, where f represents the &total' single atom spin, f"s#i. At high magnetic "eld strengths B G Q ! 1 The system of two alkali atoms, each atom in this particular spin state, is then generally in a linear combination of a singlet and triplet state. We represent the interatomic potential interaction
[2k ) S!k I ] a a , # < PK # S ) I H " S ) I#B ) 1 1 1
2 2 1 a "H # < PK # S ) I . (40) 1 1 1 2 1 The expression of Eq. (40) suggests that a convenient partitioning into collision channels divides the Hilbert space of the binary atom/spin system into subspaces j of states of good I and S-quantum numbers with the proper superposition of "I, S; M , M 2 states that diagonalizes the ' 1 H -operator. The corresponding eigenvalues determine the continuum levels of the j-chan1 nels. By construction, the atom}atom potential interaction,
a " 1S "S ) I "S ; j2 dr u (r)u (r) . H HY_ K HY K , H 2
(41)
Since the strength of the magnetic "eld is the experimental knob that controls the resonance detuning, it is helpful to make the magnetic "eld dependence explicit. The detuning, e, is the energy
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Fig. 1. Schematic representation of the molecular potentials of the incident and intermediate state channels. The energy di!erence of the continuum levels, D, is the sum of the binding energy, denoted here by E , of the quasi-bound state and the &detuning' e.
di!erence of the quasi-bound molecular state, E , and the continuum of the incident atoms, K pictured in Fig. 1. When B takes on its resonant value, B , the energy of the quasi-bound molecule K lines up with the continuum of the incident atoms so that D equals the binding energy of the bound state, D"E . Near resonance, D+E #(RD/RB);[B!B ], and thus e"D!E +(RD/RB); K K K K [B!B ]. K Consequently, the near-resonant e!ective scattering length of Eq. (33) depends on the external magnetic "eld strength as
*B a "a 1! B!B
c , where *B" . a;RD/RB
(42) K Thus, the dependence on the magnetic "eld strength is &dispersive'. Likewise the e!ective interaction strength of Eq. (34) takes the form
*B j "j 1! B!B
.
(43)
K
3.2. Feshbach resonant interactions in many-body systems We treat the many-body physics of the atoms in second quantization. In describing the &homogeneous' many-body system, it is convenient to work with the creation and annihilation operators c( k and c( kR of the single-particle plane wave states. In a more general description, we work with the "eld operators tK (r, t), which, for the homogeneous system, take the form tK (r, t)" k exp(ik ) r)c( (t)/(X, where we use Box normalization, and where X represents the I macroscopic &box' volume. The dynamics of the single-species low-temperature dilute gas can be taken to be governed by a Hamiltonian operator HK "dr H K (r), where we denote the Hamiltonian
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density operator by H K ,
j H K (r)"tK R(r) ! #< (r) tK (r)# tK R(r)tK R(r)tK (r)tK (r) , 2m 2
(44)
and where < represents the external potential experienced by the atoms. In the ultra-dilute atomic-trap systems of density, n&10}10 cm\, giving na&10\}10\, particles interact essentially as binary atom systems. Speci"cally, the atom}atom interactions can be described as in binary collisions, partly because the collision complex is so short lived that its interaction with other particles may be neglected. This assumption is a crucial ingredient of the formal many-body treatment which reintroduces the binary atom scattering length as a result of the &ladder approximation' in treating the atom}atom interactions. The "nal result of this treatment is the result quoted above Eq. (35): j"(4p /M)a. The low density of the gas might suggest the prescription jPj , where j is the e!ective interaction strength of Eq. (43) that describes binary atom collisions, 1 a , (45) jPj "j! RD/RB B!B K to describe the e!ects of the Feshbach resonance. However, this recipe, which we shall refer to as the j -description, becomes problematic near resonance, eP0. One reason, of course, is that the system does not remain dilute in this description, na PR. Alternatively, we may consider the time that two atoms, approaching each other at relative velocity v, spend in each others presence during the collision. If we estimate this time by the delay time, q " (Rd /RE), we can naturally " break up q into a contribution that describes the time that the atoms spend in each others " potential well, "q ""a/v, and the time that the atoms spend in the intermediate molecular state, " N "q """c/(ev)"(kP0). Note that lim "q /q ""R, which means that the atoms spend an " K C " K " N in"nitely longer time in the intermediate molecular state near resonance. The validity of approximating the interactions as &one-on-one' collisions, is then not self-evident. Furthermore, as we discuss in Section 4, the j -description leads to unphysical predictions for the many-body behavior of a stationary on-resonant condensate. In an approach that avoids making any &a priori' assumptions, we reformulate the many-body problem while treating the particles as compound particles. We account for the Feshbach resonances by including the intermediate state molecules explicitly, as well as the spin-#ip interactions that access the molecular states. We include the compound character of the particles by adding a subscript to the "eld operators or annihilation/creation operators, tK and c( k for the atoms and ? ? tK and c( k for the m-state molecules. The hyper"ne-induced spin #ips that create the molecules K K formed in the intermediate state of the Feshbach resonance, are then described by the following term in the many-body Hamiltonian of a homogeneous system:
1 1K, m"< "k, k 2c( R Kc( kc( k , (46) HK " K ? ? Y +. K k k (2 Y where < denotes the two-particle hyper"ne interaction, and where the interaction matrix element is an integral over the unsymmetrized two-particle wave functions of the m-molecule with center-of mass momentum K, "K, m2, and the binary atom continuum state of a-atoms colliding with
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momenta k and k . The factor of 1/(2 has to be included to insure that the matrix elements of H calculated in "rst and second quantization are equal, as follows from the example of an +. HK -matrix element with single-molecule bra and a binary atom ket-state. Likewise, the hyper+. "ne-induced breakup of the molecules is described in the Hamiltonian by HK , the hermitian .+ conjugate of HK . +. We now evaluate the matrix element explicitly in coordinate space. The wave functions then depend on the positions r and r of both nuclei. The dependence of the molecular state on the relative position, r !r , is given by the molecular wave function, u (r !r ), and the dependence K on the center-of-mass position by the box-normalized plane wave state exp(iK ) [r #r ]/2)/(X. The binary atom state depends on the relative coordinates through a scattering state with s-wave component u ("r !r "). Its dependence on the center-of-mass coordinate is expressed by the plane , wave exp(i[k#k ] ) [r #r ]/2)/X. Using the notation introduced in the previous section, the resulting expression for the matrix element of HK is equal to +. 1 a 1S "S ) I "S ; j2 1K, m"< "k, k 2" HY X 2
K ;exp(id ) dr dr uH(r !r ) exp !i ) [r #r ] K 2
;u ("r !r ") exp(i[k#k ] ) [r #r ]/2) . (47) , The substitution to &sum' [r #r ]/2"R and &di!erence' coordinates, r !r "r, then factorizes the integral into the product of an R-integral, dR exp(!i[K!k!k ] ) R)"XdK k k , and an >Y r-integral, proportional to the Feshbach resonance a-parameter: (48) 1K, m"< "k, k 2"(1/(X)dK k k a , >Y where we have used that exp(id )+1. Consequently, the Feshbach-resonant interactions, in second quantization, are described by
a a 1 [c( R dr tK R (r)tK (r)tK (r) . c( c( ]" HK " K ? ? +. (X k k (2 K k>kY ? k ? kY (2 Y
(49)
Similar to the &elastic' inter-atomic interactions, the low-energy conditions imply a coupling strength to the molecular channel that is independent of the momentum transfer. Consequently, the Feshbach-resonant interactions are also characterized by a single parameter: the atom}molecule coupling strength a. In accordance with these results, we generalize the Hamiltonian density for the many-body system, Eq. (44), to describe the many-atom/molecule system:
j
j # ? tK RtK tK #tK R ! #e# K tK R tK tK (r) H K "tK R ! ? ? ? K ? 2M 2 4M 2 K K K a #j tK RtK tK R tK # [tK R tK tK #tK tK RtK R] , ? ? K K (2 K ? ? K ? ?
(50)
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where the position dependence of the operators is understood, and where j , j and j represent the ? K strengths of the atom}atom, molecule}molecule and atom}molecule interactions, respectively. In Eq. (50), we have assumed that the atoms and molecules do not experience external potentials. We note that the same Hamiltonian correctly reproduces the Feshbach-resonant behavior of a binary atom system. 3.3. Dynamics of the condensate mixture A new phenomenon occurs in the many-body physics: the appearance of a molecular condensate component, which follows from the many-body dynamics. A convenient starting point to describe this dynamics are the Heisenberg equations of motion for the atomic and molecular "eld operators: i tKQ (x, t)"[HK , tK (x, t)] , ? ? i tKQ (x, t)"[HK , tK (x, t)] . K K
(51)
With the following expressions for the relevant commutators, [tK R(r, t), tK (x, t)]"d d(r!x), and G H G H [tK (r, t), tK (r, t)]"0, i, j"a or m, we obtain the following coupled operator equations: H G
i tQK "! tK #j tK RtK tK #jtK R tK tK #(2atK tK R , ? ? ? ? ? K K ? K ? 2M ?
a i tQK "! tK #etK #j tK R tK tK #jtK RtK tK # tK RtK R , K K K K K K ? ? K (2 ? ? 4M K
(52)
where it is understood that all "eld operators depend on the same position. The operator equations (52), which provide an &exact' description of the many problem, are generally very di$cult to solve. However, for dilute condensates, we obtain a closed set of equations for the condensate "elds,
(r)"1tK (r)2 and (r)"1tK (r)2, by taking the expectation value of Eqs. (52). Furthermore, we K ? K ? assume that for the dilute gas systems considered here, we may take the condensed "elds to be totally coherent in the sense that the expectation value of the products is equal to the product of the expectation values, e.g. 1tK tK 2+ . This corresponds to a particular Gaussian trial wave ? ? ? functional in the Dirac time-dependent variational scheme [41]. We "nd
i
Q " ! #j " "#j" " #(2a H , ? ? ? K ? K ? 2M
a i
Q " ! #e#j " "#j" " #
. K K K ? K (2 ? 4M
(53)
These coupled non-linear equations replace the usual Gross}Pitaevskii equation that describes the time evolution of the dilute single condensate system [42,43]. Note that the -"eld has a source K term J so that the expectation value of the molecular "eld operator is forced to take on a ? "nite value when O0: the atom}molecule coupling creates a molecular condensate component ? in the presence of an atomic condensate.
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The Gaussian trial wave function, which leads to an e!ective classical Hamiltonian density,
j
j H " H ! # ?" " # H ! #e# K" " # j" "" " ? ? K K ? K 2M 2 ? 4M 2 K #
a
[ H # H] , K ? (2 K ?
(54)
gives equations of motion identical to Eq. (53). 3.4. Ewects of particle loss The experimental lifetime of the m-molecules is not only determined by the hyper"ne-induced spin #ips, but also by collisions with other atoms, or even molecules. The importance of such three-body collisions is particularly pronounced as the recent experiments &resonate' on a molecular state m of high vibrational quantum number (e.g. l"14 for the MIT experiment). The created quasi-bound molecules are consequently fragile and a collision with a third particle, atom or molecule, likely causes the molecule to decay into a state of lower vibrational quantum number. Such collisions that &quench' the internal molecular state are a potential problem for molecular Bose condensation. They are also the most likely culprit for the particle loss that served as a signal to detect the Feshbach resonances in the MIT experiment. Particle loss in atomic traps is usually described to su$cient accuracy, by very simple rate equations n "n [!c n !c n ] , ? ? ?? ? ?K K (55) n "n [!c n !c n ] . K K K? ? KK K In the above equation, n and n represent the particle densities of atoms and molecules, n "" ", ? K G G where i"a or m, if all particle are Bose condensed. The c -coe$cients represent the rate G H coe$cients for collisions between particles i and j that change the internal state of the i-particle. Typical values for the alkali atoms are c &10\}10\ cm s\. The fragility of the loosely ?? bound alkali dimers is expressed by atom}molecule and molecule}molecule state changing collision rates that exceed the atom}atom rates by several orders of magnitude: 10\}10\ cm s\, where these numbers are estimates based on calculations with hydrogen molecules [44]. A &pure' molecular condensate of density 10 cm\ of such dimers is then not expected to survive longer than 10\ s. Nevertheless, this time scale might actually su$ce to study interesting molecular condensate physics. Furthermore, o!-resonance, the molecular condensate is signi"cantly smaller than the atomic condensate, which keeps &replenishing' the small molecular condensate with bosons. The lifetime of the condensate mixture is then equal to the molecular lifetime divided by the fraction of molecules, e.g. a condensate system with a 1 percent molecular condensate can survive 100 times longer than a single-molecule embedded in an atomic condensate of the same density. Of equal importance is the question whether, and how, the atomic/molecular condensate system reaches its equilibrium as the detuning e is altered by varying the magnetic "eld. From the two-"eld coupling in the equations of motion, Eq. (53), we expect, as we discuss below, that a sudden change
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of the detuning is followed by oscillations in the atomic and molecular populations. From Eqs. (53), it would appear as if these oscillations persist inde"nitely, or, at least until the condensates have disappeared due to state changing collisions or other three-body recombination processes. However, we point out that a correct treatment of the e!ects of state-changing collisions on the condensate predicts that the number oscillations damp out on a time scale that is the lifetime of a single-molecule. As we noted above, the molecular condensate can survive for a much longer time and the condensate system can reach a &quasi-equilibrium' state on a time scale small compared to its lifetime. To build in the e!ects of state-changing collisions, we treat the loss-processes by considering the channels of all chemical reactions that remove particles from the atomic and molecular condensates. The elimination of the two-body collision channels in perturbation theory modi"es the equations of motion (53) in a predictable way: the interaction strengths become absorptive with an imaginary part that determines the loss-rates. In the -equation, for instance, the interaction K strengths are replaced by j Pj !i c /2 and jPj!i c /2. We de"ne the o!-resonant regime K K KK K? to correspond to values of the detuning that exceed, in absolute value, the molecular kinetic energy, as well as any of the single-particle interaction energies e<j n , jn , j n , jn and n ;n . In this K K K ? ? ? K ? regime, we can neglect the variations of an initially constant atomic density and the main e!ect of the atomic condensate is that of a coherent (i.e. one that preserves the phase information) reservoir of atoms for the molecular condensate. In that limit, the molecular "eld equation of motion is linear: (56) i
Q "[e(t)#jn !i c /2] #(a/(2) (t) , ? K ? K K where c corresponds to the molecule loss-rate: c / "c n . In the same o!-resonant limit, the K K K? ? atomic condensate "eld, to lowest order, propagates without feeling the e!ect of the molecular condensate: (t)+(n exp(!ij n t/ ). The solution to Eq. (56) for a detuning that is suddenly ? ? ? ? shifted from its initial value to e then gives the time dependent "eld (t): D K it it c t
(t)" exp ! (2j n ) #[ ! ]; exp ! (e #jn ) exp ! K , (57) K ?
? ?
D 2
where represents the initial value of the molecular "eld, " (t"0), and is the value that K the molecular "eld tends to at large times, t<( /c ), "!an /[(2(e #jn !2j n !ic /2)]+ D ? ? ? K K ? !an /((2e ), which is the quasi-equilibrium value of the molecular condensate "eld. Note ? D that the molecular condensate density has an oscillating contribution &2"( ! ) H " cos([e #jn !2j n ]t/ ) exp(!c t/2 ). This oscillation in the molecular population is caused by D ? ? ? K the interference of the propagation of the initial molecular "eld amplitude with the amplitude stemming from the coherent inter-condensate exchange of atom pairs. Thus, the oscillations are pure quantum e!ects and, if observed, provide strong evidence for the presence of a molecular condensate. Similarly, the atomic condensate density will oscillate with twice the amplitude of the molecular density oscillations, since each molecule that appears takes out two atoms from the condensate. These oscillations of the population imbalance are similar to the oscillating current observed at Josephson junctions. Here, due to the additional concern of boson decay, the oscillations damp out on a time scale of ( /c ). Thus, in addition to leading to particle loss, the state K changing collisions serve as relaxation processes, allowing the molecular condensate to approach its &static' value +!an /((2e). For a more general variation of the detuning with time, the ? oscillations appear if the rate of change of the detuning "e " exceeds "ec / ", or alternatively, if the rate K
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of change of the magnetic "eld exceeds "BQ /[B!B ]"<(c / ). In the opposite limit, "e /e"(c / , the K K K system adiabatically follows its quasi-equilibrium state, (t)+!an (t)/[(2e(t)]. K ? With the expression for the static value of the molecular "eld, we can also obtain the loss rate of the condensate system in the o!-resonant regime. We de"ne the loss of the condensate system as the loss of atomic particles that occupy the atomic/molecular condensate. The total density of atomic condensate particles, n, is then equal to n"n #2n , where we count each molecule as two ? K atomic particles. The corresponding loss rate of the atomic condensate particles is given by n "n #2n "!n [c n !2c n ] , (58) ? K ? ?? ? K? K where, in the o!-resonant regime, we only account for atom}atom and molecule}atom collisions, n +!c n and n +!c n n . In the case of a slowly varying magnetic "eld, n +[an/2e], ? ?? ? K K? K ? K ? so that, with n +n, we "nd ? n "!c n!(c a/e)n . (59) ?? K? As a three-body collision process, the molecular-state changing collisions contribute a term bn, proportional to n. From Eq. (59) we "nd that the three-body collision rate is equal to b"c a/e. K? We note that the above expressions are only valid in the far o!-resonant limit. Closer to resonance, the non-linearities of the equations become important and have to be included. However, in a qualitative sense, the response of a condensate to a sudden variation of the magnetic "eld agrees with the o!-resonant behavior: out-of-phase oscillations of the atomic and molecular densities are damped out, after which the total condensate density decays exponentially, as shown in Fig. 2.
Fig. 2. Plot of the particle densities: the total condensate density, n"n #2n , in full line, the atomic density n in ? K ? dashed line and the molecular density n in dash-dotted line. The calculation is for a homogeneous BEC that was initially K in equilibrium at density n"10 cm\ when the detuning experienced a sudden shift from e"50jn to e"2jn. The order of magnitude of the interaction parameters, jn"j n"j n"a(2n"10 Hz, and of the decay parameters, K ? c "c "5;10\ cm s\ (while neglecting the atomic decay) are realistic. K? KK
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3.5. Conclusions and remarks regarding molecular condensate formation The population imbalance oscillations, discussed above in the far-o! resonance regime, illustrate that the presence of a molecular condensate leads to observable e!ects. Thus, the appearance of a molecular condensate component is not a semantic issue but one that implies fundamental and observable di!erences in the many-body physics of the condensate system. Before proceeding, we point out that the formation of a molecular condensate component does not defy common sense. For instance, in the limit that the molecules are destroyed extremely rapidly, the oscillations that occur in response to a sudden change of the magnetic "eld are &overdamped', and we will not "nd a signal of the molecular condensate. It is furthermore interesting and important to note that the lifetime of the molecular condensate can greatly exceed the lifetime of a single molecule. We also point out that the appearance of a condensate of molecules does not violate conservation of energy. It is true that in the binary collision the energy and momentum the molecule receives from the incident atoms are generally &o!-shell', so that the creation of the molecule can only be understood in the sense of a &virtual' particle. However, this consideration does not preclude the appearance of a molecular condensate. In a condensate system, it is not possible to assign a de"nite energy to a single-particle component of the full wave function. Nevertheless, we may assign a chemical potential to each condensate: k represents the chemical potential of the atomic and ? k the chemical potential of the molecular condensate. These chemical potentials include the K non-linear self-interaction terms (e.g. j " " is one contribution to k ) and k includes the single ? ? ? K molecule energy e. The energy required to make a single molecule is then equal to k !2k . K ? Interestingly, in the equilibrium condensate mixture, k "2k , as we discuss in the next section. K ? Thus, it costs no energy to make a molecule in the equilibrium condensate mixture, even though e may be di!erent from zero. In that sense the energy and momentum of the molecules are &on shell' and the molecules may be treated as long-lived &real' particles without violating any conservation laws.
4. Statics Under the present experimental conditions, particle loss is an important e!ect to be reckoned with. Nevertheless, in the near future, it might be possible to &resonate' on less fragile molecular states (i.e. with lower values for the c -collision rates). Even with the present systems, it might be K? possible to let a near-resonant condensate relax to its quasi-equilibrium state. In this section, we discuss relevant aspects of this equilibrium. We "nd, for example, that some of the unphysical predictions of the j description on resonance are avoided in the condensate mixture description. Furthermore, we show that the stationary state of the system depends on its history. For instance, the state of the system at a near-resonant value for the detuning may be di!erent if it was obtained by an adiabatic lowering or by an adiabatic increase of the detuning. Near resonance, the homogeneous BEC ground state is unstable in the limit of very low atomic particle density. At higher densities, on the other hand, the condensate can be stable with respect to density #uctuations, in contrast to the predictions of the j -description. At higher atomic particle densities, n, the inter-particle interaction dominates over the atomic-molecular condensate coupling and can
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stabilize the system. As a function of the density n, the energy goes through a minimum at a density where the inter-condensate coupling interaction is of the order of the inter-particle interaction energy per particle. The appearance of a minimum suggests the possibility of creating condensate systems with the liquid-like property of a self-determined density. A low-density condensate would spontaneously adjust its volume to settle in a state with density equal to the value at which the energy is minimized. From previously calculated numbers, it appears that the corresponding density is only of the order of 10}10 cm\, and the resulting system would be a rari"ed liquid! In the j -description of the on-resonant BEC, the dispersive dependence of the e!ective scattering length on the magnetic "eld gives a diverging inter-boson interaction. For the purpose of visualizing the inter-particle interactions, one can picture the bosons as hard spheres with radius equal to the scattering length (assuming a'0). The fraction of space occupied by the spheres diverges as e approaches zero, eP0, and the system is not a weakly interacting single condensate. Not surprisingly, a naive extrapolation of the dilute condensate results leads to unphysical predictions for macroscopic measurable quantities in this limit. For instance, the chemical potential, k, and the pressure, P, of dilute condensates of interaction strength j and density n are given by k"jn,
P"jn .
(60)
Hence, a simple replacement jPj gives diverging values that change discontinuously from #R to !R at the pole, e"0. Of course, one can argue that near resonance, the scattering length description of the collision breaks down, or, if nothing else, that expressions (60), valid only for weakly interacting condensates, do not apply. However, in the hybrid atomic/molecular condensate picture, we "nd that the unphysical divergencies do not occur. Near and on-resonance, the equilibrium system simply arranges the atomic and molecular densities so as to insure that k "2k , where k and k respectively denote the chemical potentials of the atoms and the K ? K ? molecules. 4.1. Static equations In this section, we will assume that the condensate system has relaxed to a stationary state and that the e!ects of particle loss are negligeable on the time scale that the system is observed. We determine the stationary state variationally. A "rst remark of relevance to the determination of the ground state of the system is that the inter-condensate coupling interaction (HK #HK ) necessar.+ +. ily lowers the ground state many-body energy. Indeed, the coupling contribution, &a H #h.c., K ? sensitive to the relative phase of the atomic and molecular condensate "elds, can and will be negative in the lowest energy state. As usual in describing the static system, we may choose the "elds to be real, but the relative sign of the ground state "elds is determined by minimizing a . ? K Without loss of generality, we will assume that a'0, and we will choose the phase of to be zero, ?
'0, so that the ground-state value of is negative, (0. ? K K If we neglect particle loss, the total number of atomic particles, N" dr n(r)" dr; [" (r)"#2" (r)"], is a conserved quantity. To account for this, we may vary the Hamiltonian with ? K a Lagrange multiplier k, d(H !kN)/d H(r)"0, and d(H !kN)/d H(r)"0. These equations ? K
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take on the form of coupled time-independent Gross}Pitaevskii equations, k "[! /2M#j #j ] #(2a , ? ? ? K ? K ? (61) 2k "[! /4M#e#j #j ] #(a/(2) , ? K K K ? K of the form of Eq. (53) with the time derivatives replaced by the corresponding chemical potential, i
Q Pk . Note that k in the equation plays the role of the atomic chemical potential, k "k, G G G ? ? and 2k in the equation plays the role of the molecular chemical potential, k "2k. Thus, the K K condition of chemical equilibrium is satis"ed: k "2k . K ? In the limit of large detuning, by which we mean here that "e" greatly exceeds the kinetic energy as well as any of the molecule interaction energies, "e"<j n , jn , we "nd that the corresponding K K ? approximations in the Gross}Pitaevskii equation of the molecular condensate lead to a
+!
, "e"<0 , K (2e ?
(62)
in agreement with the value of the quasi-equilibrium molecular "eld found in Eq. (57). The insertion of this expression into the -equation of (61) yields an e!ective single condensate Gross}Pitaevskii ? equation for : ?
a # j! n . (63) k " ! ? ? 2M e ? ?
Thus, in the o!-resonant limit, the stationary atomic condensate behaves e!ectively as a single atomic condensate with an inter-atomic interaction described by the e!ective interaction strength, Eq. (35), of the binary atom system. However, the e!ective single condensate Gross}Pitaevskii equation (63) does not describe the appearance of the small molecular condensate, and consequently, cannot describe the interesting dynamical e!ects discussed in the previous section. Note that the molecular condensate in the o!-resonant limit of negative detuning takes on the same sign as the atomic condensate. This does not contradict the remark that the H interaction .+ is minimized by "elds of opposite sign. The variational procedure with Lagrange multiplier k "nds an extremum, but does not guarantee that the extremum is a minimum. In fact, the actual ground state in the o!-resonant regime of negative detuning is the &trivial' all-molecule solution to Eqs. (61),
"0. Interestingly, the state +!an /(2e, e;0, represents a local maximum of the energy. ? K ? To see that, we reformulate the variational problem. Instead of introducing the chemical potential, we insure particle conservation by substituting n "n!2n . If the system is large ? K enough to neglect surface e!ects, we may omit the kinetic energy and work with an energy density, (64) u"j n#j n #jn n #en #(2an . K ? K ? K ? ? K K We parametrize the &population imbalance' or the fraction of atoms that are converted to molecules by a scaled molecular "eld x, "x(n/2, which can take on values between !1 and K #1. In terms of the molecular "eld parameter, the atomic condensate density is n "n(1!x), ? and the energy per particle, e"u/n is a simple fourth order polynomial in x: e"j n#a(nx#+[!j #j/2]n#e/2,x!a(nx#[j /2!j /4!j/2]nx . ? ? K ? Minimization of the energy e, yields the value of the ground state expectation value of x.
(65)
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In the interest of simplicity, we start by considering Feshbach resonances in ideal gas condensates, i.e. we consider the limit j"j "j "0. In that case the energy per atomic particle is equal ? K to
x e "a(n e #x(1!x) , 2
(66)
where we introduced the scaled detuning, e "e/[a(n]. We determine the expectation value of the molecular condensate "eld, "x(n/2, for a system of atomic particle density n, n"n #2n , ? K K variationally by "nding the extrema of e (x, n) for "xed value of n, Re /Rx"0. Within the limitation that "x"41, we "nd two solution for any given value of e "e/[a(n]:
[e #(e #12]/6 x (e )" > #1 [e !(e #12]/6 x (e )" \ !1
if e 42 , if e '2 , if e 5!2 , if e (!2 .
(67)
The x "#1 and x "!1 solution correspond to the above-mentioned trivial all-molecule > \ solution, "0. Although the all-molecule solution represents a third solution in the detuning ? interval !2(e (#2, the analysis of the next paragraph shows that this solution is unstable in this detuning region. The previously obtained o!-resonant limit (62), +!an/(2e or K x+!1/e , corresponds to x if e<0 and x if e;0. The molecular "eld solution of positive sign, \ > x , corresponds to a maximum of the energy, as we can see from Fig. 3. Nevertheless, as we discuss > below, the molecular condensate "eld value x cannot simply &roll down' on the energy curve. Such changes of the relative population of the atomic and molecular condensates also involve a change of the relative phase. From the dynamical treatment of the population changes, it follows that both solutions, x and x , are stable with respect to small perturbations of the condensate populations. > \ This is a signi"cant conclusion, since it suggests that the near-resonant BEC-system can reside in two distinct stationary states. 4.2. Near-equilibrium population dynamics How can a maximum in the energy correspond to a dynamically stable state? To gain insight into this issue, we construct the classical Hamiltonian that describes the dynamics of position independent atomic and molecular "elds. Again, we limit the discussion to the &ideal gas' condensate mixture, j"j "j "0, although the treatment can easily be generalized to include the ? K non-linear interactions. In terms of the density and phase variables, "(n exp(ih ), ? ? ?
"(n exp(ih ), we obtain the following expression for the Lagrangian density of the Dirac K K K variational principle: i
L" [ H Q ! Q H # H Q ! Q H ]!H , K K ? ? K K 2 ? ? "! hQ n ! hQ n !en !a(2n (n cos (h !2h ) . K K K ? ? ? K K ?
(68)
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Fig. 3. The energy per atom particle of a homogeneous mixture of Feshbach resonant coupled atomic and molecular condensates, as a function of the scaled molecular "eld x ( "x(n/2), at a "xed detuning, chosen so that K e "e/[a(n]"0.5. The energy is shown in units of the tunneling energy, a(n, and the curve was calculated in the assumption of non-interacting, or &ideal', atomic and molecular condensates (j"j "j "0). ? K
We now simplify the expressions by making use of particle conservation, n "n!2n . We also ? K introduce the relative phase variable, h"2h !h . In terms of the relative phase and the total ? K number of molecules, N , the Lagrangian-per-particle, ¸" dr L/N, reads K ¸"! hQ # hQ (N /N)!e(N /N)!a(n (N /N)[1!2(N /N)]cos (h) , K K K K
(69)
where N represents the total number of atoms. The fraction of molecules (N /N), which K can take values from 0 to , is the (dimensionless) conjugate momentum of the relative phase variable, R¸/RhQ " (N /N)" p . K
(70)
The hQ -dependence of the Lagrangian does not contribute to the equations of motion, ? expressing the fact that one of the condensate phases can be chosen arbitrarily and that h is the only dynamical phase variable. From Eqs. (69) and (70) we "nd the classical Hamiltonian H that governs the time evolution of the two-"eld system by identifying H from ¸"hQ p!H: H"ep#a(n (2p (1!2p) cos h .
(71)
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The corresponding equations of motion,
1!6p RH
hQ " "e#[a(n] cos h Rp (2p RH
p "! " sin h , Rh
(72)
lead to the stationary states x , x (1, by requiring RH/Rp"RH/Rh"0 at the equilibrium > \ solution (h , p ). The p "0 equation gives sin (h )"0, corresponding to atomic and molecular "elds of the same, h "0, or opposite, h "p, signs. The second requirement, hQ "0, results in a quadratic equation for (2p""x" (where x denotes the scaled molecular "eld of the previous paragraph). The solution to this equation gives the stationary states: (2p "[(e #12#cos h e ]/6,
(73)
identical to the solutions of Eq. (67) in the regime "x "(1. Since at (h , p ), RH/RpRh"0, the small ! amplitude #uctuations, hI "h!h and p "p!p , evolve in time according to the following linearized equations of motion:
1#6p RH p "!cos h a(n p ,
hQI + ((2p) Rp RH hI "!cos h (2p[1!2p]a(n hI .
p +! Rh
(74)
Consequently, h$I "!uhI , where u is a real-valued frequency, regardless of the sign of cos h "$1: (1#6p ) (1!2p ) RH RH . "[a(n]
u" 2p Rp Rh
(75)
Thus, the #uctuation variables, hI and p describe a harmonic motion in phase space, and the position-independent "eld #uctuations experience a restoring force, both at h "0 (corre sponding to x ) and at h "p (corresponding to x ). The reason for the dynamical stability > \ of the #uctuations around the h "0, in spite of the local maximum in the energy, follows from the analysis. Although the &inverse mass' is negative, m\"RH/Rp(0, so is the &spring constant', K"RH/Rh(0. The resulting frequency, u"(K/m is consequently realvalued. Likewise, we may study the stability of the &all-molecule' solution, "0. In that case, the ? relative phase h is not a meaningful quantity, and it is more convenient to start from the "eld equations of motion: i
Q "(2a H, K ? ?
i
Q "e #(a/(2) . K K ?
(76)
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These equations have, in fact, been solved analytically in the "eld of non-linear optics [45]. Near the all-molecule solution, "0, the molecular "eld is approximately equal to ?
+(n/2 exp(!iet/ ). In the same limit, the #uctuations of the -"eld are approximately ? K decoupled from the -#uctuations, and evolve according to K (77) i
Q "a(n exp(!iet/ ) H . ? ? Substitution of the following linear combination of positive and negative frequency "elds of small amplitude,
"exp(!iet/2 )[a exp(iut)#a exp(!iut)] , (78) ? > \ into Eq. (77) yields a set of linear equations in the a and a -coe$cients. The frequency at which > \ the system near the all-molecule state oscillates after it has been perturbed by breaking up a few molecules into atoms, follows from the requirement of a non-trivial solution to the linear set of equations in a and a : > \
u"(e/2)!(a(n) . (79) Thus, u is imaginary if "e"(2a(n, and the all-molecule system is unstable in the regime where both "x "(1 and "x "(1. > \
Fig. 4. Plot of the molecule fraction (N /N), as a function of the detuning, scaled by the &tunneling energy', a(n, for both K stationary state solutions, called x and x in the text to denote a molecular "eld of, respectively, positive and negative > \ sign. The atomic and molecular condensates were assumed to be &ideal' in this calculation, i.e. j"j "j "0. The ? K all-molecule branch (N /N)"1/2, of the x -state is shown in dotted line to indicate that the homogeneous x -branch is K > > unstable in this regime, as explained in the text.
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Thus, at any value of the detuning, we "nd two (and not three) di!erent stationary states the system can occupy. Which state the system will "nd itself in, depends on its history. For instance, if the condensate was brought near-resonance by adiabatically increasing the detuning from &below resonance', the system is expected to reside in the x -state. In Fig. 4, we plot the molecule fraction > of both stationary states as a function of the detuning. Interestingly, the analysis of the next paragraph shows that the all-molecule branch of the x -solution is unstable with respect to > density #uctuations. In the "gure, we indicate the instability by plotting the unstable branch in a dotted line. 4.3. Considerations regarding the mechanical stability The previous analysis does not give insight into the stability of the homogeneous system with respect to position dependent #uctuations of the densities. That this is a major concern in describing the dynamics of these systems will become clear from the discussion below. A "rst example that illustrates the importance of position dependent "eld #uctuations, is the instability of the homogeneous all-molecule solution if e'2a(n. To see the instability, we allow for position dependent #uctuations and solve for plane wave -#uctuations around the all? molecule solution,
"exp(!iet/2 )[a exp(!i[k ) r!ut])#a exp(i[k ) r!ut])] . (80) ? > \ Upon insertion of this expression in the equations of motion (74), after restoring the kinetic energy term, we "nd that the resulting frequency,
k e !(a(n) , !
u" I 2M 2
(81)
does become imaginary for a "nite range of wave numbers if e'2a(n. Speci"cally, the modes with k vectors of magnitude (M(e!2a(n )( k((M(e#2a(n ) grow exponentially: the homogeneous x "#1-branch is unstable and the system spontaneously generates a "nite atomic > condensate. Consequently, we show the corresponding branch of the x -state in Fig. 4 in dotted > line, to indicate that the system will not remain in that state. Whether the other branches are stable, is a question that can be answered from a more general RPA-study, and will be the subject of further research. However, even without the knowledge of the excitation modes, we can make statements about the mechanical stability of the system by investigating the pressure, for example. Whereas we could omit the inter-particle interactions to get an accurate qualitative picture of the population dynamics, their e!ects in the mechanical stability of the homogeneous system are all-important and they have to be considered. Indeed, the ground state of the homogeneous near-resonant ideal gas-condensate is always unstable. This follows from the pressure, P, which we determine from the energy per atomic particle, e, P"n(de/dn): P"n "n
Re Re dx #n Rn Rx dn Re , Rn
(82)
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where x represents the actual value of the scaled molecular "eld, which follows from Re/Rx"0. In the ideal gas condensate, e"e "(e/2)x#a(n(1!x)x, we "nd P "n(a(n/2)x (1!x) , (83) so that its pressure has the same sign as the molecular "eld. In the ground state system, for which x "x (0, P is negative and the homogeneous system is unstable. Furthermore, we expect the \ stationary state of positive molecular "eld, x , to be equally unstable, as its energy can be lowered > by spatially separating the atomic and molecular condensates. The inter-particle interactions can stabilize the homogeneous condensate system, but only at higher values for the density or in the o!-resonant regime. The pressure that follows from Eq. (82) can be written as j a(n j nx(1!x) . P" ? n# K n #jn n # K ? 2 2 K 2 ?
(84)
In this expression, we recognize the &elastic' inter-particle interaction contributions in addition to the atom-molecule Feshbach resonant coupling term that was the sole contribution to P . The inter-particle interaction contributions are proportional to n, j n/2, where j is an interaction P P strength of a magnitude that is representative of j , j and j. The a-contribution to the ground ? K state pressure, on the other hand, is of order !an if the system is near-resonance, e+0, and of order !an ;(a(n/[2e]) in the o!-resonant regime. Near-resonance, the inter-particle interaction and atom-molecule coupling contributions are of the same order of magnitude if n+n "(a/j ). In the limit of vanishing density, n;n "(a/j ), the density-dependent contribuP P P P tion to e is dominated by the Feshbach-resonant interaction and the system behaves as the near-resonant ideal gas system. In contrast, at &high' densities, n
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the order of typical atomic-trap condensate densities, n &10}10 cm\. Moreover, the P atomic-trap set-up lends itself very well to the observation of this unusual property. After switching the trap-potential o!, the condensate of self-determined density does not #y apart, but stays together as a &blob' of approximately uniform density. The above considerations suggest that the equilibrium Feshbach-resonant BEC can be an ultra-dilute gas with the liquid-like property of a self-determined density. Whether the liquid-like system is really stable, will have to follow from an RPA-study of its excitations. Nevertheless, it is certainly a fascinating thought that the dilute atomic condensates with Feshbach resonances might yield the "rst example of a rari"ed liquid!
5. Conclusions In this paper, we discussed the theoretical description of the hyper"ne-induced low-energy Feshbach resonances in the neutral atom interactions of condensates. Such resonances were recently observed in atomic-traps, of the type used to achieve Bose-Einstein condensation. In fact, one of the observations of the resonance was carried out in a BEC [1]. We pointed out that the Feshbach resonances a!ect the condensate physics in a more profound manner than the alteration of the e!ective inter-particle interaction. The atom}molecule coupling that gives rise to the resonance in binary-atom collisions, creates a second molecular condensate component in the many-body BEC-system. Even if the molecular condensate density remains small in the o!-resonant detuning regime (an/e& a few percent or so), the presence of the molecular condensate can still be detected by suddenly varying the magnetic "eld and observing oscillations in the atomic and molecular populations. If the molecular condensate survives long enough to reach its equilibrium, the resulting double condensate displays fascinating properties: (i) The state of the near-resonant condensate depends on its history. For instance, the behavior of the BEC that is taken through the resonance by adiabatically varying the detuning, will be di!erent if the system starts out above or below the resonance. (ii) Although, in the limit of vanishing density, the ground state of the near-resonant BEC can experience negative pressure, in agreement with the j description, the homogeneous higher density system can be stabilized by the inter-particle interactions of the atoms and molecules. In that case, the ground state energy goes through a minimum as the atomic particle density is increased. This suggests the remarkable, liquid-like property of self-determined density. Thus, by tuning an external magnetic "eld near resonance, an atomic condensate could become the "rst rari"ed liquid to be observed in nature.
Acknowledgements This paper is dedicated to the memory of R. Slansky, who, we think, would have been amused by this application of "eld theory to the many-body condensate problem. One of us, E.T., thanks P. Milonni and A. Dalgarno for interesting discussions. The work of M.H. was supported in part by the Brazilian agencies CNPq and Fapesp. and of A. K. in part by the US Department of Energy (D.O.E.) under cooperative research agreement DEFC0294ER418.
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References [1] S. Inouye, M.R. Andrews, J. Stenger, H.-J. Miesner, D.M. Stamper-Kurn, W. Ketterle, Nature 392 (1998) 151}154. [2] N.R. Newbury, C.J. Myatt, C.E. Wieman, Phys. Rev. A 51 (1995) R2680}2683. [3] Ph. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, B.J. Verhaar, Phys. Rev. Lett. 81 (1998) 69. [4] J.L. Roberts, N.R. Claussen, J.P. Burke, Jr., C. Greene, E.A. Cornell, C.E. Wieman, preprint, 1998. [5] V. Vuletic, A.J. Kerman, C. Chin, S. Chu, preprint, 1998. [6] M.H. Anderson, J.R. Ensher, M.R. Mathews, C.E. Wieman, E.A. Cornell, Science 269 (1995) 198}201. [7] C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet, Phys. Rev. Lett. 75 (1995) 1687}1690; C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys. Rev. Lett. 78 (1997) 985. [8] K.B. Davis et al., Phys. Rev. Lett. 75 (1995) 3969}3973. [9] K. Burnett, Nature 392 (1998) 125. [10] P.W. Milonni, A. Smith, Phys. Rev. A 53 (1996) 3484; P.O. Fedichev, Yu. Kagan, G.V. Shlyapnikov, J.T.M. Walraven, Phys. Rev. Lett. 77 (1996) 2913. [11] C.A. Sackett, H.T. Stoof, R.G. Hulet, Phys. Rev. Lett. 80 (1998) 2031. [12] M. Ueda, A.J. Leggett, Phys. Rev. Lett. 80 (1998) 1576. [13] Yu. Kagan, G. Shlyapnikov, J. Walraven, Phys. Rev. Lett. 76 (1996) 2670. [14] E. Shuryak, Phys. Rev. A 54 (1996) 3151. [15] M. Ueda, K. Huang, preprint, 1998. [16] J. Stenger, S. Inouye, D.M. Stamper-Kurn, H.-J. Miesner, A.P. Chikkatur, W. Ketterle, Nature 396 (1999) 345. [17] C.J. Myatt et al., Phys. Rev. Lett. 78 (1997) 586. [18] E. Timmermans, Phys. Rev. Lett. 81 (1998) 5718. [19] D.S. Hall, M.R. Matthews, C.E. Wieman, E.A. Cornell, preprint, 1998. [20] J. Williams, R. Walser, J. Cooper, E. Cornell, M. Holland, preprint, cond-mat/9806337, 1998. [21] A. Smerzi, S. Fantoni, S. Giovanazzi, S. Shenoy, Phys. Rev. Lett. 79 (1997) 4950. [22] R.E. Packard, Rev. Mod. Phys. 70 (1998) 641. [23] A. Barone, G. Paterno, Physics and Applications of the Josephson E!ect, Wiley, New York, 1982. [24] C.W. Gardiner, P. Zoller, R.J. Ballagh, J.M. Davis, Phys. Rev. Lett. 79 (1997) 1793. [25] C. Josserand, S. Rica, Phys. Rev. Lett. 78 (1996) 1215. [26] H.T.C. Stoof, Phys. Rev. Lett. 66 (1991) 3148; Phys. Rev. A 45 (1992) 8398. [27] Yu. Kagan, B.V. Svistunov, Phys. Rev. Lett. 79 (1997) 3331. [28] E. Tiesinga, A.J. Moerdijk, B.J. Verhaar, H.T.C. Stoof, Phys. Rev. A 46 (1992) R1167}R1170. [29] E. Tiesinga, B.J. Verhaar, H.T.C. Stoof, Phys. Rev. A 47 (1993) 4114}4122. [30] A.J. Moerdijk, B.J. Verhaar, A. Axelsson, Phys. Rev. A 51 (1995) 4852}4861. [31] J.M. Vogels et al., Phys. Rev. A (1997) R1067}1070. [32] H.M.J.M. Boesten, J.M. Vogels, J.G.C. Tempelaars, B.J. Verhaar, Phys. Rev. A 54 (1996) R3726}R3729. [33] J.P. Burke, J.L. Bohn, preprint, 1998. [34] E. Timmermans, P. Tommasini, R. Co( te, M. Hussein , A. Kerman, preprint, cond-mat/9805323, 1998. [35] P. Tommasini, E. Timmermans, M. Hussein, A. Kerman, preprint, cond-mat/9804015, 1998. [36] R. Co( teH , E. Timmermans, P. Tommasini, Super#uidity and Feshbach resonances in BEC, in: C.T. Whelan, R.M. Dreizler, J.H. Macek, H.R.J. Walters (Eds.), Proceedings of E.U. meeting on &New Directions in Atomic Physics', held in Cambridge, U.K., Plenum Press, New York, 1999, to be published. [37] H. Feshbach, Theoretical Nuclear Physics, Wiley, New York, 1992. [38] V.I. Lengyel, V.T. Navrotsky, E.P. Sabad, Resonance Phenomena in Electron-Atom Collisions, Springer, Berlin, 1992. [39] G. Shulz, Rev. Mod. Phys. 45 (1973) 378. [40] A.A. Radzig, B.M. Smirnov, Reference Data on Atoms, Molecules and Ions, Springer, Berlin, 1985. [41] A. Kerman, P. Tommasini, Ann. Phys. (NY) 260 (1997) 250. [42] E.P. Gross, Nuovo Cimento 20 (1961) 454. [43] L.P. Pitaevskii, Sov. Phys.-JETP 13 (1961) 451.
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[44] These estimates follow from calculations on hydrogen and helium, recently reported in N. Balakrishnan, R.C. Forrey, A. Dalgarno, Phys. Rev. Lett. 80 (1998) 3224. [45] J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan, Phys. Rev. 127 (1962) 1918}1939. [46] W.C. Stwalley, Phys. Rev. Lett. 37 (1976) 1628.
Physics Reports 315 (1999) 231}240
A relativistic symmetry in nuclei Joseph N. Ginocchio Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Abstract We review the status of a quasi-degenerate doublets in nuclei, called pseudospin doublets, which were discovered about 30 years ago and the origins of which have remained a mystery, until recently. We show that pseudospin doublets originate from an SU(2) symmetry of the Dirac Hamiltonian which occurs when the scalar and vector potentials are opposite in sign but equal in magnitude. Furthermore, we survey the evidence that pseudospin symmetry is approximately conserved for a Dirac Hamiltonian with realistic scalar and vector potentials. We brie#y discuss the relationship of pseudospin symmetry with chiral symmetry and the implications of pseudospin symmetry for the antinucleon spectrum in nuclei. 1999 Elsevier Science B.V. All rights reserved. PACS: 21.60.Cs; 21.10.!k; 21.30.Fe; 24.10.Jv; 03.65.Pm
1. Introduction Dick Slansky and I met almost 30 years ago as assistant professors at Yale University. Although, for the most part, Dick's published research at that time was in hadron collisions [1], he became interested in group theory and symmetry, particularly applied to elementary particles physics, during his stay at Yale. I had similar interest in group theory and symmetry, but primarily applied to the many-body problem, particularly in nuclear physics. In the mid 1970s Dick and I each accepted positions at Los Alamos National Laboratory and we continued to discuss symmetry in physics. At Los Alamos he published a number of papers in group theory including his very popular review on `Group Theory for Uni"ed Model Buildinga [2], and also published a book on a$ne Lie algebras [3]. In 1989 he became Theoretical Division Leader at the Los Alamos National Laboratory, and, technically, my `bossa. At the time we were at Yale I had heard about an observation that certain single-nucleon levels in spherical nuclei were found to clump into quasi-degenerate doublets [4,5]. The term pseudospin E-mail address: [email protected] (J.N. Ginocchio) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 1 - 6
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doublets was applied to these states, although the quasi-degeneracies we assumed to be accidental. About two years ago I reexamined these pseudospin doublets, the origin of which remained a mystery, and I discovered that they are a consequence of a relativistic symmetry [6,7]. I discussed this revelation with Dick and he suggested that this symmetry may have some connection with chiral symmetry. In this paper I would like to discuss the progress that has been made in understanding pseudospin symmetry, including an enticing connection with chiral symmetry via QCD sum rules.
2. Pseudospin symmetry The spherical shell model orbitals that were observed to be quasi-degenerate have nonrelativistic quantum numbers (n , l, j"l#) and (n !1, l#2, j"l#) where n , l, and j are P the single-nucleon radial, orbital, and total angular momentum quantum numbers, respectively [4,5]. This doublet structure is expressed in terms of a `pseudoa orbital angular momentum lI "l#1, the average of the orbital angular momentum of the two states in doublet and `pseudoa spin, s ". For example, (n s , (n !1)d) will have lI "1, (n p, (n !1) f) will have lI "2, etc. These doublets are almost degenerate with respect to pseudospin, since j"lI $s for the two states in the doublet; examples are shown in Fig. 1. Pseudospin `symmetrya was shown to exist in deformed nuclei as well [8,9] and has been used to explain features of deformed nuclei, including superdeformation [10] and identical bands [11,12]. However, the origin of pseudospin symmetry remained a mystery and `no deeper understanding of the origin of these (approximate) degeneraciesa existed [13]. A few years ago it was shown that relativistic mean "eld theories gave approximately the correct spin}orbit splitting to produce the pseudospin doublets [14]. In this paper we shall review
Fig. 1. Examples of pseudospin doublets in the Pb region. n is the radial quantum number of the state, l is the orbital angular momentum, j the total angular momentum.
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more recent developments which show that pseudospin symmetry is a relativistic symmetry [6,15,7].
3. Symmetries of the Dirac Hamiltonian The success of the shell model implies that nucleons move in a mean "eld produced by the interactions between the nucleons. Normally, it su$ces to use the Schrodinger equation to describe the motion of the nucleons in this mean "eld. However, in order to understand the origin of pseudospin symmetry, we need to take into account the motion of the nucleons in a relativistic mean "eld and thus use the Dirac equation. The Dirac Hamiltonian, H, with an external scalar, < , 1 and vector, < , potentials is given by 4 H"a ) p#b(m#< )#< , 1 4
(1)
where we have set "c"1, a, b are the usual Dirac matrices [16], m is the nucleon mass, and p is the three momentum. The Dirac Hamiltonian is invariant under an SU(2) algebra for two limits: < !< " constant and < #< "constant. For "nite nuclei the constant will be zero since each 1 4 1 4 potential will go to zero at large distances [17]. (For in"nite nuclear matter the constant could be non-zero.) The generators for the SU(2) algebra, SK , which commute with the Dirac Hamiltonian, G [H, SK ]"0, for the case when < "< are given by [17] G 1 4 (1#b) a ) ps( a ) p (1!b) G #s( , SK " G 2 G 2 p
(2)
where s( "p /2 are the usual spin generators and p the Pauli matrices. This reduces to G G G
s( SK " G G 0
0 , s( G
(3)
where 2s ) p s( "; s( ; " p !s( . G NG N G p G
(4)
In Eq. (4) ; "(r ) p/p) is the momentum}helicity unitary operator introduced in [14] that N accomplishes the transformation from the normal shell model space to the pseudo-shell model space while preserving rotational, parity, time-reversal, and translational invariance. This symmetry limit leads to spin doublets, but we know that the spin}orbit splitting in nuclei is very large so this limit is not applicable to nuclei. However, we shall see that it is relevant for the antinucleon spectrum. We shall show that, in the limit of < "!< , the conserved symmetry is pseudospin symmetry. 1 4 The generators for the SU(2) algebra, SIK , which commute with the Dirac Hamiltonian, [H, SIK ]"0, G G
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for the case when < "!< are given by [17] 1 4 a ) ps( a ) p (1#b) (1!b) G #s( . SKI " G G p 2 2
(5)
This reduces to
s( SIK " G G 0
0
lIK ¸IK " G G 0
0
s(
.
(6)
G For < "!< , the eigenfunctions of the Dirac Hamiltonian, HW "E W are doublets 1 4 O I O O I (SI "1/2, k"$1/2) with respect to the SU(2) generators SIK of Eq. (6) G SKI W "kW , X O I O I SIK W "((1/2Gk)(3/2$k) W , (< "!< ) , (7) O I! ! O I 1 4 where SIK "SIK $iSIK . The eigenvalue q refer to the other necessary quantum numbers. ! V W The fact that the pseudo-spin generators (6) have only the spin operator s( operating on the lower G compoent of the Dirac wave function has the consequence that the spatial wave functions for the two states in the pseudospin doublet are identical in the limit of < "!< to within an overall 1 4 phase. This symmetry for < "!< is general and applies to deformed nuclei as well as spherical 1 4 nuclei. In the case for which the potentials are spherically symmetric, the Dirac Hamiltonian has an additional invariant SU(2) algebra; namely, the pseudo-orbital angular momentum,
, (8) lK G where lIK "; lK ; , lK "r;p. In this limit, the Dirac wave functions are eigenfunctions of G N G N G the Casimir operator of this algebra, ¸KI ) ¸KI "W lI 2"lI (lI #1)"W lI 2, where we have O H KH O H KH used a coupled basis, and j is the eigenvalue of the total angular momentum operator 2"j ( j#1)"W lI 2, and m is the eigenvalue of JK . Thus pseudoJK "¸IK #SIK , JK ) JK "W lI G O H KH G O H KH H X G orbital angular momentum as well as pseudospin are conserved in the spherical limit and < "!< . From Eq. (8), we see that the lower component wave function will have spherical 1 4 harmonic of rank lI coupled to spin to give total angular momentum j. Since r ) p conserves the total angular momentum but p changes the orbital angular momentum by one unit because of parity conservation, the upper component also has total angular momentum j, but orbital angular momentum l"lI $1. If j"lI #, then it follows that l"lI #1, whereas if j"lI !1/2, then l"lI !1. This agrees with the pseudospin doublets originally observed [4,5] and discussed at the beginning of this paper. For axially symmetric deformed nuclei, there is a U(1) generator corresponding to the pseudoorbital angular momentum projection along the symmetry axis which is conserved in addition to the pseudospin for < "!< , 1 4 KKI 0 , (9) jKI " 0 KK
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where KKI "; KK ; . In this case the Dirac wave functions are eigenfunctions of jKI , N N jIK "W KI X2"KI "W KI X2, where X is total angular momentum projection, X"KI #k, which has the O O same value for the upper and lower components since r ) p conserves the total angular momentum projection. Thus X"KI $, corresponding exactly to the quantum numbers of the pseudospin doublets for axially deformed nuclei discussed in [8,9]. However, the exact symmetry limit cannot be realized in nuclei, because, if < "!< , there are 1 4 no Dirac bound valence states and hence nuclei cannot exist. However, we now show that pseudospin symmetry is approximately realized for < +!< . 1 4 4. Free nucleons For a free Dirac nucleon, < "< "0, and hence both symmetries are valid. The free nucleon 1 4 wave function is
u I e NXX\#NR , W "N N I INXSI K>#N
(10)
where u are the Dirac spinors u "(), u "(), E "(p#m, and N is the normalization. I \ N This wave function has de"nite helicity and hence is an eigenfunction of the helicity operator, ; W "2kW . N N I N I
(11)
The wave functions with k"$ are doublets for SK and, since SKI "; SK ; , the wave functions G G N G N WI "; W "2kW are doublets for SIK , the pseudospin. N N I N I G N I 5. Realistic mean 5elds A near equality in the magnitude of mean "elds, < +!< , is a universal feature of the 1 4 relativistic mean "eld approximation (RMA) of relativistic "eld theories with interacting nucleons and mesons [18] and relativistic theories with nucleons interacting via zero range interactions [19], as well as a consequence of QCD sum rules [20]. We shall discuss QCD sum rules in the next section. Recently, realistic relativistic mean "elds were shown to exhibit approximate pseudospin symmetry in both the energy spectra and wave functions [15,21,22]. In Fig. 2 we show the energy splittings between pseudospin doublets normalised by 2lI #1 as a function of the average binding #elI )/2. We see that the energy splitting for the same pseudo-orbital energy 1e2"(elI > \ angular momentum decreases as the radial quantum increases; that is, as the binding energy decreases. Also for the same binding energy, the energy splitting increases as the pseudo-orbital angular momentum increases. These features follow from the square well potential [6]. In Table 1 we tabulate some of the calculated energy splittings compared to the measured splittings. What is interesting is that the measured splittings are smaller than those calculated by relativistic mean
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Fig. 2. Pseudospin doublet energy splittings normalised by 2lI #1 as a function of the average binding energy 1e2 [15]. n is the radial quantum number of the state with the lower orbital quantum number.
Table 1 Pb pseudospin doublet energy splittings for RMA [15] compared to the experimental values lI
4 2 3 1
ps doublets Neutrons 0h !1f 1f !2p Protons 0g !1d 1d !2s
elI !elI (RMA) > \
elI !elI (EXP) > \
2.575 0.697
1.073 !0.328
4.333 1.247
1.791 0.351
"eld theory indicating that pseudospin symmetry breaking is overestimated by the relativistic mean "eld approximation. Pseudospin doublets will manifest themselves for deformed potentials as well when < +!< . 1 4 In Fig. 3 the single particle (s.p.) energies of the doublets are plotted versus the deformation. We see
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Fig. 3. Single-particle (s.p.) energies for neutron pseudospin partners as a function of deformation b [21].
that for each value of pseudo asymptotic quantum [NI n KI ] numbers there is a quasi-degenerate pseudospin doublet with X"KI $. As mentioned in the last section the relativistic SU(2) pseudo-spin symmetry implies that the spatial wave function for the lower component of the Dirac wave functions will be equal in shape and magnitude for the two states in the doublet. For spherical nuclei the Dirac wave function for "(g lI [>lI s]HlI > , if lI [> s]HlI > ), the two states in the doublet are W O > > K O > lI K O HlI > K lI lI "(g lI [> s]H \ , if lI [> s]H \ ) where g, f are the radial wave W lI O \ lI \ K O \ lI K H \ K functions, >l are the spherical harmonics, s is a two-component Pauli spinor, and [2]H means coupled to angular momentum j. For a square well potential, the overall phase between the two amplitudes will be a minus sign [6] so we expect that, in the symmetry limit for realistic (r)"!flI (r). In Fig. 4 we see that, for realistic zero range potentials, potentials, flI > \ (r)+!flI (r) [15]. flI > \ These results are also valid for the relativistic mean "eld approximation to a nuclear "eld theory with meson exchanges [21]. In Fig. 5a the pseudospin doublets in the vicinity of the Fermi surface for neutrons and protons are shown. The upper (g) and lower components (f ) of the pseudospin doublets are also shown in Fig. 5b}d. While the upper components are very di!erent with di!erent nodal structure, the lower components are almost identical. We also note that the lower components are small with respect to the upper components, which is consistent with the non-relativistic shell model.
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Fig. 4. Pb lower component wave functions f lI (r) (dash line), !f lI (r) (dot}dash line) for the (2s ,1d ) O \ O > pseudospin doublet (lI "1) as a function of the radius r [15].
Fig. 5. Pb: (a) energy spectrum; (b}d) upper (g) and lower (f ) components of the Dirac wave function for pseudospin doublets as a function of the radius r [21].
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6. QCD sum rules Applying QCD sum rules in nuclear matter, the scalar and vector self-energies were determined to be [20] R "!4pp o /mm , Q , ,
(12) R "32po /m , T , where o is the nuclear density, and m the quark mass. p is the sigma term which arises from the , , breaking of chiral symmetry [23]. The ratio then becomes R /R "!p /8m . (13) Q T , For reasonable values of p and quark masses, this ratio is close to !1. The implication of these , results is that chiral symmetry breaking is responsible for the scalar "eld being approximately equal in magnitude to the vector "eld, thereby producing pseudospin symmetry.
7. Antinucleon spectrum The antinucleon states are obtained by charge conjugation, C, applied to the negative energy eigenstates of the Dirac Hamiltonian [16]. This leads to a spectrum which has quasi-degenerate spin doublets, not pseudospin doublets. This follows from the fact that, under charge conjugation,
s( CRSIK C" G G 0
0 "SK . (14) G s( G Thus in Eq. (14) the spin operator s( operates on the upper component and hence the spatial wave G functions for the upper components of the states in spin doublet will be very similar. Likewise for spherical nuclei, the pseudo-orbital angular momentum goes into the orbital angular momentum, and for axially deformed nuclei, pseudo-orbital projection goes into orbital projection along the body "xed z-axis. This symmetry in the antinucleon spectrum also follows from the fact that the antinucleon potentials are <M "CR< C"< , and <M "CR< C"!< . Thus <M +<M and the symmetry 1 1 1 4 4 4 1 4 of the Dirac Hamiltonian generated by Eq. (3) applies and spin doublets are produced in the antinucleon spectrum [17].
8. Summary We have shown that pseudospin symmetry is a broken SU(2) symmetry of the Dirac Hamiltonian which describes the motion of nucleons in realistic scalar and vector mean "eld potentials, < +!< . This symmetry predicts that the spatial wave functions of the lower components for 1 4 states in the doublet will be very similar in shape and size and this has been substantiated by
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relativistic mean "eld approximations of relativistic nuclear "eld theories and relativistic nuclear Lagrangians with zero range interactions. This symmetry has been linked via QCD sum rules to chiral symmetry breaking in nuclei. Finally, the antinucleon spectrum is shown to have a spin symmetry rather than a pseudospin symmetry. Future applications of pseudospin symmetry will involve the testing of the wave function through the relationships between transitions within pseudospin doublets which follow from pseudospin symmetry [24].
Acknowledgements This work was supported by the United States Department of Energy.
References [1] R. Slansky, Phys. Rep. 11 (1974) 99. [2] R. Slansky, Phys. Rep. 79 (1981) 1. [3] S. Kass, R.V. Moody, J. Patera, R. Slansky, A$ne Lie Algebras, Weight Multiplicities, and Branching Rules, University of California Press, Berkeley, 1990. [4] K.T. Hecht, A. Adler, Nucl. Phys. A 137 (1969) 129. [5] A. Arima, M. Harvey, K. Shimizu, Phys. Lett. 30B (1969) 517. [6] J.N. Ginocchio, Phys. Rev. Lett. 78 (1997) 436. [7] J.N. Ginocchio, A. Leviatan, Phys. Lett. B 425 (1998) 1. [8] A. Bohr, I. Hamamoto, B.R. Mottelson, Phys. Scripta 26 (1982) 267. [9] T. Beuschel, A.L. Blokhin, J.P. Draayer, Nucl. Phys. A 619 (1997) 119. [10] J. Dudek, W. Nazarewicz, Z. Szymanski, G.A. Leander, Phys. Rev. Lett. 59 (1987) 1405. [11] W. Nazarewicz, P.J. Twin, P. Fallon, J.D. Garrett, Phys. Rev. Lett. 64 (1990) 1654. [12] F.S. Stephens et al., Phys. Rev. C 57 (1998) R1565. [13] B. Mottelson, Nucl. Phys. A 522 (1991) 1. [14] A.L. Blokhin, C. Bahri, J.P. Draayer, Phys. Rev. Lett. 74 (1995) 4149. [15] J.N. Ginocchio, D.G. Madland, Phys. Rev. C 57 (1998) 1167. [16] W. Greiner, B. MuK ller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, New York, 1985. [17] J.S. Bell, H. Ruegg, Nucl. Phys. B. 98 (1975) 151. [18] B.D. Serot, J.D. Walecka, The relativistic nuclear many-body problem, in: J.W. Negele, E. Vogt (Eds.), Advances in Nuclear Physics, vol. 16, Plenum Press, New York, 1986. [19] B.A. Nikolaus, T. Hoch, D.G. Madland, Phys. Rev. C 46 (1992) 1757. [20] T.D. Cohen, R.J. Furnstahl, K. Griegel, X. Jin, Prog. Part. Nucl. Phys. 35 (1995) 221. [21] G.A. Lalazissis, Y.K. Gambhir, J.P. Maharana, C.S. Warke, P. Ring, Phys. Rev. C 58 (1998) R45; LANL archives nucl-th/9806009. [22] J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring, A. Arima, Phys. Rev. C 58 (1998) R628. [23] T.P. Cheng, L.F. Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, New York, 1984. [24] J.N. Ginocchio, Phys. Rev. C 59 (1999), in press; LANL archives nucl-th/9812025.
Physics Reports 315 (1999) 241}256
Quasicrystal Lie algebras and their generalizations Jir\ mH Patera , Reidun Twarock * Centre de Recherches Mathe& matiques, Universite& de Montre& al, Montre& al, Que& bec, Canada Arnold Sommerfeld Institut, TU-Clausthal, 38678 Clausthal, Germany
Abstract We review and extend the results about quasicrystal Lie algebras of Patera et al. [Phys. Lett. A 246 (1998) 209], which is a new family of in"nite dimensional Lie algebras over the real and complex number "elds, whose generators are in a one-to-one correspondence with the points of a one-dimensional quasicrystal. Some new properties of quasicrystal Lie algebras and further details on their representation theory are pointed out and the concept of generalized quasicrystal Lie algebras is presented. The latter allows to associate to the generators of the Lie algebra quasicrystal points of one-dimensional quasicrystals with acceptance windows symmetric around 0, which was not possible in the framework of Patera et al. 1999 Elsevier Science B.V. All rights reserved. PACS: 02.20.Sv; 61.44.Br Keywords: Lie algebras; Quasicrystals; Witt (Virasoro) algebra
1. Introduction Recent development in the theory of quasicrystals has led to the study of the so-called CUT AND also called model sets, or quasicrystals. The latter we will use here for simplicity. Such quasicrystals can be viewed as rather idealized models of physical quasicrystals studied in laboratories, or as a generalization of lattices. PROJECT POINT SETS,
* Corresponding author. E-mail addresses: [email protected] (J. Patera), [email protected] (R. Twarock) Work supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR of Quebec. Work supported by the Ministry for Science and Culture of Lower Saxony in the framework of the DorotheaErxleben Program. 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 2 - 8
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Quasicrystals share some important properties with lattices. Both are deterministic point sets so that the removal or the addition of a single point in either of them creates a defect in the structure. Both are also uniformly dense throughout the entire space and uniformly discrete, a property technically referred to as the Delone property. The crucial distinction between lattices and quasicrystals is the periodicity of the former in contrast to the complete lack of such periodicity in the latter. A quasicrystal, in spite of its determinism, does not contain a periodic subset. There is a further general class of common properties between lattices and quasicrystals which only recently were fully discovered [1,2]. These are the scaling symmetries. In lattices, such symmetries are rarely exploited because most of the pertinent results can be obtained using the periodicity, but in quasicrystals there is no choice but to use scaling symmetries, which represent an important tool in the description, generation, and theoretical handling of quasicrystals. There are di!erent types of quasicrystals. They can be split into families distinguished by the irrationality which is built into the coordinates of their points. The best studied among them are the quasicrystals involving quadratic irrationalities, given by the so-called Pisot numbers which are de"ned via solutions of certain algebraic equations. Among Pisot numbers related to quadratic equations, the ones by far most studied in connection with quasicrystals are the ones related to the equation x"x#1 and thus to its solutions q"(1#(5),
q "(1!(5) .
(1)
The presence of two solutions (1) allows for a new symmetry-like transformation in the theory, namely the interchange q q . It turns out that this transformation is of fundamental importance for the theory of quasicrystals. In physics, quasicrystals related to this type of irrationality were "rst observed in 1984 [3]. The aperiodic structures were recognized by the presence of localities with non-crystallographic 5-fold re#ection symmetry in the X-ray di!raction pattern. Note that irrationality q is the golden ratio known for a number of reasons since Antiquity and plays a fundamental role in nature, e.g. in the spiral formed by a shell and the curve of a fern [4}9]. The observation of the re#ection symmetry generating 5-fold rotation brought up a further relation between quasicrystals and lattices through the "nite Coxeter groups, which split into two types, the crystallographic and non-crystallographic ones. The former are "nite symmetry groups of lattices and it is known that in three-dimensional lattices one "nds 2-, 3-, 4-, and 6-fold symmetries. Correspondingly, in the general n-dimensional case, lattice symmetries are given by the appropriate Coxeter group. Furthermore, crystallographic Coxeter groups are in a 1}1correspondence with the semisimple "nite dimensional Lie algebras (over the complex number "eld) and play a fundamental role in physics. The non-crystallographic Coxeter groups exists (as irreducible groups) only in dimensions 2, 3, and 4. There are in"nitely many such groups in dimension 2, which are the dihedral groups well known in physics; in dimensions 3 and 4 there is precisely one non-crystallographic group in each of them, namely the groups H and H , both containing the 5-fold symmetry dihedral group H . The analogous properties of lattices and quasicrystals, particularly the link between crystallographic and non-crystallographic Coxeter groups, suggests the possibility of a link at the level of Lie theory. Are there Lie algebras related to non-crystallographic Coxeter groups of any kind? The
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answer for now is de"nitely negative, or at least there is no standard link between such groups and Lie algebras. Thus at best one may hope for some kind of non-traditional relation between them. In this article we show two constructions of Lie algebras built around one-dimensional quasicrystals. The Lie algebras we construct here resemble more to the Witt algebra than to the semisimple ones. In fact, our algebras may be perfect but are never semisimple.
2. Mathematical preliminaries In this section we want to introduce quasicrystals and recall their pertinent properties. Furthermore, we recall some information from Lie theory for later use. The quasicrystal points in our case are represented by numbers on the real axis, more precisely numbers of the form a#bq where a and b are integers. Thus the stage for our construction is the algebraic number "eld Q[q], although in most cases it su$ces to use its ring of integers Z[q]"Z#Zq. The de"nition of a quasicrystal below is based on the (Galois) automorphism, denoted by prime, which exists in Q[q] and hence also in Z[q] and interchanges q with q . In the general theory of quasicrystals in higher dimensions the automorphism is combined with the corresponding mapping of the bases of these spaces by what is called the star map. In one-dimension, we also call this automorphism a star map, even if it is here just the Galois automorphism in Q[q]. A point x becomes x according to x"a#b(5 x "a!b(5, a, b3Z . We stress here a remarkable property of the star map, which underlies the de"nition of a quasicrystal and is therefore crucial in our context: the star map is everywhere discontinuous. Two arbitrarily close points on the real line can have very distant images under the star map, as can be seen for example from the pairs of numbers 1 and 1#q\, with 1 and 1#q \"1!q after the mapping. It is known that Z[q] is a Euclidean domain, in particular Z[q] is a unique factorization domain. For a set FLZ[q], one has gcd+F ,"(gcd+F,) . The group of units of Z[q] consists of +$qI " k3Z,. It is also known that each prime p of Z of the form p,$2 (mod 5) remains prime in Z[q]. Each prime p of Z of the form p,$1 (mod 5) splits as a conjugated pair p"qq with qOq . We now introduce the de"nition for a one-dimensional quasicrystal [2,10}12]. Such an object is speci"ed by a bounded interval (r, t) (acceptance window) in R. A point x belongs to the quasicrystal provided its star map image x is in (r, t). De5nition 1. Let (r, t) be a bounded interval. The quasicrystal R((r, t)) is the point set R((r, t)) " : +x3Z[q] " x 3(r, t), .
(2)
The interval (r, t) is said to be the acceptance window of R((r, t)). Note that in the de"nition one could take the interval to be open or closed, or even semi-open. The corresponding quasicrystals di!er in these cases by two or only one point. Therefore, for simplicity, we take the interval open whenever possible in order to avoid the exceptional points in
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the quasicrystal. We further note that the density of points in a quasicrystal is not in#uenced by the position of the acceptance window in R. The density is determined by the length "r!t" of the acceptance window. It is conventional to consider distances between adjacent points of a quasicrystal as tiles. Quasicrystals with the acceptance window satisfying "r!t""qI for some integer k are two-tile quasicrystals, otherwise there are three tiles in the quasicrystal [13]. A useful example is the quasicrystal R([0, 1]). The acceptance condition is given as 04x "a#q b"a!q\b41 which leads to a formula for the points of R([0, 1]) given by R([0, 1])"+[1#b/q]#qb " b3N,6+0, ,
(3)
where [A] denotes the integer part of A. The points of R([0, 1]) nearest to the origin are the following: 2,!q, 0, 1, 1#q, 2#2q, 2#3q, 3#4q, 4#5q, 4#6q, 5#7q, 5#8q,2
(4)
It is obvious from (3) that there are only three possible tiles in R([0, 1]), one tile occurring precisely once between 0 and 1. The other tiles are q or 1#q"q. We remark that rescaling of the boundaries by qI with k an integer, produces a rescaled version of the quasicrystal, i.e. the tiles are rescaled accordingly, but the tiling sequence remains the same. Furthermore, the tiling sequence is constrained by the fact that the ratio between adjacent tiles is 1 : q or 1 : 1 [13]. A further important feature of quasicrystals R([a, b]) is their re#ection symmetry around the point (a #b )/2. Thus, in the case of R([0, 1]) there is a re#ection symmetry around , and therefore its negative points are readily obtained from those greater than 0. Indeed, for every x3R([a, b]), one has y"(b #a )!x3R([a, b]) because x 3[a, b]0y "b#a!x 3[a, b]. Correspondingly, for R([!1, 0]) we obtain R([!1, 0])"+[n/q]#qn " n3N, which can be seen to be given by the points of R([0, 1]) re#ected through the origin. From the de"nition of quasicrystals given in (2) some properties typical of quasicrystals can be readily inferred: E Selfsimilarity: If the acceptance interval (r, t) is open, any "nite pattern P3R((r, t)) is repeated in"nitely often. Since (r, t) is assumed to be open, x #P L(r, t) if x is chosen such that x is small enough. E Absence of any periodic subset: A translational symmetry would mean that we have x#R((r, t))LR((r, t)), i.e. x #(r, t)L(r, t) and also for the closure of the interval x #[r, t]L[r, t]. However, the latter implies x "0 and thus x"0. E Central q-in#ation symmetry and quasiaddition: Instead of translational symmetries, we have for intervals (!r, r) symmetric around the origin qLx3R((!r, r)) with n3N whenever x3R((!r, r)). Furthermore, R((!r, r)) is invariant under quasiaddition xq!yq for x, y3R((!r, r)). The "rst can be seen from the fact that (q )Lx 3(!r, r). The second follows from the fact that (q) and !q are both positive and add up to 1.
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We remark that the q-in#ation mentioned above is only one representative of an in"nite family of scaling symmetries [2].
3. Quasicrystal Lie algebras In this section we review the construction of a particular type of in"nite dimensional Lie algebras over the real and/or complex number "elds, called quasicrystal Lie algebras, which was "rst introduced in [1]. Its generators are in a one-to-one correspondence with points from a onedimensional quasicrystal. Like the quasicrystals introduced in (2) they depend crucially on the choice of the acceptance window, and they are therefore denoted by Q(X), where the acceptance window X is a bounded interval. There is a certain resemblance between the quasicrystal Lie algebras and the Witt algebra [14}16], particularly some of the rank-2 Witt algebras of Patera and Zassenhaus [17], because the latter have served as a building block for the quasicrystal Lie algebras. But despite this super"cial resemblance, there are substantial di!erences between the two types of Lie algebras, which stem from the quasicrystal set underlying the construction of quasicrystal Lie algebras. To anticipate only a few of the di!erences at this stage, we remark that quasicrystal Lie algebras do not allow for a central extension, so that we do not have, strictly speaking, a quasicrystal analog to the Virasoro algebra. However, quasicrystal Lie algebras contain an abundance of "nite dimensional subalgebras, which might open interesting possibilities for applications to physics. In order to de"ne quasicrystal Lie algebras Q(X) corresponding to a one-dimensional quasicrystal R(X) with bounded X according to Patera et al. [1], the acceptance interval has to be restricted to a positive or negative interval, i.e. 0 is not allowed to be an inner point. This restriction is necessary to establish the Lie algebra structure of the quasicrystal Lie algebras. Then we have the following. De5nition 2. Let F be any number "eld such that FMQ[q] and let aOb be real numbers such that 04ab(R. Let X be one of the intervals [a, b], (a, b], [a, b), or (a, b). The quasicrystal Lie algebra Q(X) over F is the F-span of its basis B(Q(X))"+¸ " n3R(X), , L with the commutation relations of the basis elements given by
(m!n)¸ if n#m3R(X) , L>K [¸ , ¸ ]" L K 0 if n#m , R(X) .
(5)
Whereas the antisymmetry of the commutators is obvious, the Jacobi identity has to be ensured by the condition that either XLRY or XLRX, i.e. it holds provided ab50 and requires the restriction of the acceptance intervals as mentioned before. We note that it is always possible to enlarge the algebra Q([a, b]) with ab'0 to Q([a, b]6+0,). This will be important for the representation theory. The commutation relations (5) resemble the standard Witt algebra [14}17], but in contrast to the latter they contain many commutators equal to 0, because the operation `additiona is often not
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compatible with the quasiperiodic structure of the quasicrystal, i.e. it does not follow from m, n3R(X) that one must have m#n3R(X). As an example consider some of the commutators in Q([0, 1]): [¸ ,¸ ]"!q¸ , [¸ ,¸ ]"(1#q)¸ , >O >O >O >O >O >O [¸ ,¸ ]"0, [¸ ,¸ ]"(2#3q)¸ , >O >O >O >O >O [¸ ,¸ ]"!(3#6q)¸ , [¸ ,¸ ]"!(4#7q)¸ . >O \\O >O \\O \O The behaviour of the corresponding quasicrystal points under addition is illustrated in Fig. 1. It is often more convenient to rewrite the commutation relations (5) using the characteristic function sX of the interval X: [¸ , ¸ ]"(m!n)sX(n #m )¸ . (6) L K L>K In real life quasicrystals are always "nite size objects, fragments of the ideal in"nite quasicrystals we have been considering throughout the paper so far. A "nite quasicrystal implies that also the window contains only a "nite number of points, forming another quasicrystal of "nite size. The star map is then a duality transformation between the two pictures of one (?) quasicrystal: one in the quasicrystal space, the other in the space of the acceptance window. We note here that it is also possible to discretize the acceptance window X of the quasicrystal R(X) by viewing it as another quasicrystal,now with an acceptance window X lying in the original quasicrystal space. The discretization of X to X leads to a restriction of R(X) to R(X ) which is B B a "nite subset of R(X), i.e. a "nite quasiperiodic set. The algebra corresponding to this set, i.e. Q(X ), B is "nite dimensional and is in one-to-one correspondence with points m3X which under the star map fall into the acceptance interval X. It takes the form (7) [¸ , ¸ ]"(m!n)sX(n #m )sX (n#m)¸ Y L>K L K with n, m3X and n , m 3X and it is easily veri"ed that antisymmetry and the Jacobi identity hold, so that the object is again a Lie algebra. As an example, consider Fig. 2. The points in X correspond to generators of a "nite dimensional Lie algebra with "ve non-vanishing commutators given by [¸ , ¸ ]"(2!q)¸ , [¸ , ¸ ]"¸ , \>O O \>O \>O O \>O [¸ , ¸ ]"(1#q)¸ , [¸ , ¸ ]"(1!q)¸ , \>O O \>O \>O O \>O [¸ , ¸ ]"¸ . \>O O \>O Further examples may be obtained by changing the size of X and X .
Fig. 1. Addition of quasicrystal points corresponding to the Lie algebra in the example above.
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Fig. 2. Illustration of projection method for "nite quasicrystal Lie algebras.
4. Properties of quasicrystal Lie algebras Here we illustrate some properties of quasicrystal Lie algebras following from De"nition 2.1. Some of them can be found also in [1], others are stated here for the "rst time. E The algebras Q(X) and Q(!X) are isomorphic under ¸ ¸ for any x3R(X). V \V The reason for this is that x 3X is equivalent to !x 3!X. E From now on we assume that the boundary points of X satisfy 04a(b. E Any Q([a, b]) with 2a5b is an Abelian Lie algebra. Any Q([a, b]) with 2a(b is non-Abelian. The commutator between ¸ and ¸ is zero if n#m is not in R([a, b]), i.e. if n #m are L K not in [a, b]; also, the commutator of an element with itself is zero. Hence all commutators vanish for 2a5b. Otherwise, there can always be found two points which lead to a non-trivial commutator. E From now on we discuss only the non-Abelian algebras unless otherwise stated. E The Lie algebra Q([a, b]) has a non-trivial center precisely if a'0. Its center is the Lie algebra Q((b!a, b]). The center contains elements ¸ with k#n'b for all n3[a, b]. Hence, k3(b!a, b]. I : +¸ " m3R LR(X), of Q(X) is E Let r " : inf R1 m . Then the centralizer Q(X) of a subset S " 1 K 1 KZ given as Q(X) "Q((b!r, b]), if the in"mum is attained, i.e. r " : min R1 m , and as 1 KZ Q(X) "Q([b!r, b]) otherwise. 1 With r as above, the centralizer contains ¸ with k#r'b (resp. k#r5b), so that k3(b!r, b] I (resp. k3[b!r, b]).
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E For a subalgebra A " : +¸ " m3R([r, b]), a4r(b, of Q([a, b]) the normalizer A is given by K , A "Q([a, b]). For a subalgebra AI " : +¸ " m3R([r, l]), r5a, l(b, 2r'b, of Q([a, b]) the , K normalizer AI is given by AI "+¸ " m3R((b!r, b]),6+¸ , if a"0 and , , K AI "+¸ " m3R((b!r, b]), otherwise. , K For the case of A this follows from the fact that m #n 5r if m3[r, b] and n3[a, b]. In the case , of AI notice "rst of all that the condition 2r'b is necessary to ensure that it is indeed a subalgebra. Then the claim follows from the condition: r4n #m 4l or n #m 5b for m 3[r, l] and n 3[a, b]. E In the case of the toy-model R([0, 1]), it can be shown that any n points of this quasicrystal never add up to 0. As a consequence, the lower or upper central series of the quasicrystal Lie algebra Q([0, 1]) never produces the generator ¸ . This can be seen by implementing (3). E The non-Abelian part of Q([a, b]) is indecomposable. The commutation of two Lie algebra generators corresponds on the quasicrystal level to the addition of primed indices in X. Because XLR, every point of it can be written as a sum of (in"nitely) many pairs of other points of X and hence Q([a, b]) is indecomposable. E The Lie algebras Q([a, b]) and Q([qa, qb]) are isomorphic, ¸ ¸ . More generally, non-Abelian V OV Lie algebras Q([a, b]) and Q([c, d]) are isomorphic precisely if a"qIc and b"qId for k3Z. It is a consequence of the fact that qZ[q]"Z[q]. E The algebra Q([c, b]) is an ideal of Q([a, b]) provided 04a(c(b. Consequently, there are no semi-simple quasicrystal Lie algebras. It follows from the fact that n #m 5c if n 3[c,b] and m 3[a, b]. E The derived algebra of Q([a, b]) is Q((2a, b]). Only the algebras Q((0, b]) and Q((0, b)) are perfect. The algebras Q([a, b]) are solvable and nilpotent if a'0. The "rst claim is again a consequence of the fact that commutation of generators in the Lie algebra means addition of primed coe$cients in the acceptance window of the quasicrystal. Solvability and nilpotency hold because the upper and lower central series break after the primed indices leave the acceptance window under addition. This happens after a "nite number of steps for a'0. E The algebras Q(X) admit only a trivial central extension [15], i.e. the direct sum Q(X) Fc. It follows from the standard cocycle condition, if one uses the fact that in the quasicrystals admissible for the construction of quasicrystal Lie algebras, points are never centrally symmetric around 0, i.e. if n3R(X) one does not have !n3R(X).
5. Finite dimensional subalgebras of quasicrystal Lie algebras In this section we want to illustrate an important property of quasicrystal Lie algebras, which is its abundance of "nite dimensional subalgebras mentioned earlier. This structure could be crucial for applications of quasicrystal Lie algebras in physics, because quasicrystal Lie algebras substantially di!er in this respect from the standard Witt algebras as well as the rank-2 Witt algebras. The generators of the latter are indexed by the elements of Z[q] (see [17]), but the corresponding algebras have no indecomposable subalgebras of "nite dimension greater than 3. The phenomenon is described by the following.
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Lemma 3. The closure of any xnite set of generators from B(Q(X)) under the commutations (5) is a xnite dimensional subalgebra of Q(X). Proof. Let ¸ ,2, ¸ I3B(Q(X)) be a "nite set of basis elements. This implies x ,2, x 3X. Since V I V X is bounded, there is only a "nite number of linear combinations n x #n x #2#n x in I I X with n ,2, n 3ZY. Hence only "nitely many multiple commutators of the elements of the set I are non-zero. 䊐 As an example consider the subalgebra of Q([0, 1]) generated by the basis elements ¸ and >O ¸ . The closure of the elements ¸ and ¸ under the commutation relations is \\O >O \\O a 12-dimensional subalgebra, which we denote by D. Setting a"2#3q and b"!1!2q, we get the rest of the basis elements by repeated communications of ¸ "¸ , ¸ "¸ : ? >O @ \\O [¸ , ¸ ]"(b!a)¸ "(!3!5q)¸ , ? @ ?>@ >O [¸ , [¸ , ¸ ]]"b(b!a)¸ "(13#21q)¸ , ? ? @ ?>@ >O [¸ , [¸ , ¸ ]]"a(b!a)¸ "(!21!34q)¸ , @ ? @ ?>@ \O [¸ , [¸ , [¸ , ¸ ]]]"b(b!a)¸ "(34#55q)¸ , ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , ¸ ]]]"2ab(b!a)¸ "(178#288q)¸ , ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , ¸ ]]]"a(b!a)¸ "(!55!89q)¸ , @ @ ? @ ?>@ \\O [¸ , [¸ , [¸ , [¸ , ¸ ]]]]"b(b!a)(2a#b)¸ "(322#521q)¸ , ? ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , [¸ , ¸ ]]]]"3ab(b!a)¸ "(699#1131q)¸ , @ ? ? ? @ ?>@ >O [¸ , [¸ , [¸ , [¸ , ¸ ]]]]"3ab(b!a)¸ "(699#1131q)¸ , ? @ @ ? @ ?>@ [¸ , [¸ , [¸ , [¸ , [¸ , ¸ ]]]]]"b(b!a)(2a#b)(3a#b)¸ "(5257#8506q)¸ . ? ? ? ? ? @ ?>@ >O D is solvable: Its upper central series consists of the derived algebra D and D " : [D , D ] generated by the following basis elements: D "+¸ , ¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,, >O >O >O >O >O >O >O \O \\O D "+¸ , ¸ ,. >O Furthermore, D is nilpotent: Its lower central series consists of D , E " : [D, D ],2, E " : [D, E ], given by E "+¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,, \\ \O >O >O >O >O >O >O E "+¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,¸ ,, \\ >O >O >O >O >O E "+¸ , ¸ ,¸ ,¸ ,, >O >O >O E "+¸ ,. >O
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6. Representations of the quasicrystal Lie algebra Q(X) Here we provide information on the representation theory of quasicrystal Lie algebras Q(X). For simplicity, we "rst consider the Lie algebra Q([0, a]) and its representation in a space
(8)
form a basis of Q K P Q
(9)
Proof. We need to verify the commutation rule (6): (¸ ¸ ) " (¸ ) (¸ ) L P I K I Q L K P Q IZR " d s (n #k )(j#k)d s (m #s )(j#s) P L>I X I K>Q X IZR . "sX(n #m #s )sX(m #s )(j#s)(j#m#s)d P L>K>Q With this, one obtains (¸ ¸ ) !(¸ ¸ ) "(sX(m #s )(j#m#s)!sX(n #s )(j#n#s)) L K P Q K L P Q ;sX(n #m #s )(j#s)d
P L>K>Q
.
s (n #m #s )(j#s) However, this is equal to (m!n)sX(n #m )d P L>K>Q X sX(n #m #s )O0, one has sX(n #m )"sX(m #s )"sX(n #s )"1.
because,
for
Note that representation (9) holds also for domains X"[a, b] with a'0, because the corresponding quasicrystal Lie algebra is a subalgebra of Q([0, b]). As in (9), it can be shown that the commutation relations are also satis"ed by , (¸ ) "sX(m #s #j )(j#s)d P K>Q>H K P Q>H
(10)
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which resembles more closely to the corresponding results for the usual Witt algebra. To see this, consider the realization ¸ " : zK(zR ) acting on "s#j2 " : zQ>H, i.e. K X ¸ "s#j2"(s#j)"s#m#j2 K and its matrix form (¸ ) "(j#s)d (11) K P Q>H P K>Q>H and notice that (11) di!ers from (10) only by the factor sX(m #s #j ). Correspondingly, a realization of quasicrystal Lie algebras in terms of di!erential operators can only be LOCAL in the sense that the operator realization depends on the vector of the representation space it is acting on. This dependence is then controlled by the function sX(m #s #j ). In particular, we have ¸ "s#j2"zK(zR )sX(m #s #j )zQ>H"(s#j)sX(m #s #j )"s#m#j2 . K X Up to now it has been assumed that m and s (resp. s#j) are from the same quasicrystal, i.e. restricted by the same acceptance window. This assumption can be weakened by the introduction of another acceptance window X , such that ¸ zQ"zK(zR )sX (m #s )zQ"ssX (m #s )zK>Q Y K X Y again ful"lls the commutation relations of the quasicrystal Lie algebra. It can be shown that X is restricted by the following: Lemma 5. For given X"[a, b] the acceptance window X "[c, d] is constrained by the condition c5d!b, i.e. the largest X compatible with X is given by X "[d!b, d]. Proof. The left-hand side of (6) can be calculated } under the assumption that both X and X are simultaneously positive or negative } to give [¸ , ¸ ]zI"k(m!n)sX (m #n #k )zL>K>I . Y L K On the other hand,
(12)
[¸ , ¸ ]zI"(m!n)sX(m #n )¸ zI"k(m!n)sX(m #n )sX (m #n #k )zL>K>I . L K L>K Y To guarantee equality, we need sX (m #n #k )"1NsX(m #n )"1, which is violated if we have Y n #m 'b and c#n #m 4d, i.e. for c#b(d. 䊐
7. Generalized quasicrystal Lie algebras A crucial feature of quasicrystal Lie algebras is the one-to-one correspondence of their generators with the points of a suitably chosen one-dimensional quasicrystal. For technical reasons, the possible quasicrystals had to be restricted to those with an acceptance window of the form [a, b], (a, b], [a, b) or (a, b) with 04ab(R in order to guarantee that the quasicrystal Lie algebras are
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indeed Lie algebras. The idea of this section is to generalize the concept of quasicrystal Lie algebras so as to admit quasicrystals with acceptance windows containing 0 as an inner point. In particular, the generators of generalized quasicrystal Lie algebras are in one-to-one correspondence with one-dimensional quasicrystals with acceptance windows of the form X"[!b,!a]6+0,6[a, b]. The advantage of this generalization is that it opens the possibility to use quasicrystal Lie algebras for modeling situations in physics which require such a type of central symmetry, e.g. the interaction of particles with opposite electric charge or particles and their antiparticles, by assigning to one of the particles the quasicrystal points associated with the positive acceptance interval and to the other one the quasicrystal points with negative acceptance interval. Several (related) generalizations of the concept of quasicrystal Lie algebras in the above sense are possible. All of them have in common that their generators are in a one-to-one correspondence with one-dimensional quasicrystals associated to an acceptance window X"X 6+0,6X with X "!X and that the commutation relations of the basis elements are de"ned separately for the four cases (n, k)3X ;X with i, j3+1, 2,. G H The prototype of a generalized quasicrystal Lie algebra is of the following form (Referred to as VERSION Ia): For n3X and k3X : [¸ , ¸ ]"!(k#n)+¸ sX(n !k )!¸ sX(k !n ), . I\L L I L\I
(13)
For n3X and k3X : [¸ , ¸ ]"(k#n)+!¸ sX(n !k )#¸ sX(k !n ), . I\L L I L\I
(14)
For n, k3X and n, k3X : s (!k !n ), . [¸ , ¸ ]"(k!n)+¸ sX(n #k )!¸ \I\L X L I L>I
(15)
Similarly, in VERSION Ib, the sign on the right-hand side of the de"ning relations (13) and (14) can be simultaneously reversed without changing the Lie algebra structure of the object. VERSION Ia and Ib have the interesting property that a restriction of X to X or X , i.e. if sX is replaced by sXG for i"1, 2, respectively, leads to the fact that the last identity reproduces the commutation relations which de"ne the quasicrystal Lie algebras (6). Alternatively, the following VERSION IIa, which is obtained from VERSION Ia via sign changes in the right-hand side of the de"ning relations, can be used, because it also de"nes a Lie algebra structure: For n3X and k3X : [¸ , ¸ ]"!(k#n)+¸ sX(n !k )#¸ sX(k !n ), . I\L L I L\I
(16)
For n3X and k3X : [¸ , ¸ ]"(k#n)+#¸ sX(n !k )#¸ sX(k !n ), . I\L L I L\I
(17)
For n, k3X or n, k3X : s (!k !n ), . [¸ , ¸ ]"(k!n)+¸ sX(n #k )#¸ \I\L X L I L>I
(18)
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As before, a simultaneous sign #ip on the right-hand side of the "rst two identities (the resulting identities will be referred to as VERSION IIb) is consistent with the Lie algebra structure. A restriction of X to X or X in the last identity reproduces for X the commutation relations which de"ne the quasicrystal Lie algebras in [1], but for X only reproduces them up to a sign. The four generalized quasicrystal Lie algebras are of two principally di!erent types: E
VERSION
Ia and
VERSION
IIb are of a similar structure. For these algebras we have
[[¸ , ¸ ], ¸ ]"0 L I J for all possible choices of the coe$cients n, k and l. The upper central series thus brakes up already in the second step and the Lie algebras are not perfect. Hence, a universal central extension cannot be expected. E In the case of VERSION Ib and VERSION IIa, we "nd perfect Lie algebras with a structure similar to the quasicrystal Lie algebras. Here as well, there is no non-trivial central extension. In order to decide which of the four versions of the generalized quasicrystal Lie algebras is most suitable for applications, it is convenient to view them in another basis. With the notation R " : ¸ sX(n )!¸ sX(!n ) , \L L L S " : i(¸ sX(n )#¸ sX(!n )) , K L \L one obtains the following identities: E For VERSION Ia: If n, m3X , i"1, 2 G [R , R ]"0 , L K [R , S ]"0 , L K [S , S ]"4(n!m)R . L K L>K If n3X and m3X with iOj, i, j3+1, 2, G H [R , R ]"0 , L K [R , S ]"0 , L K [S , S ]"4(n#m)R . L K L\K E For VERSION IIa: If n, m3X [R , R ]"0 , L K [R , S ]"0 , L K [S , S ]"4i(m!n)S L K L>K
(19) (20)
(21) (22) (23)
(24) (25) (26)
(27) (28) (29)
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and with reversed sign in the last line if n, m3X . If n3X and m3X [R , R ]"0 , L K
(30)
[R , S ]"0 , L K
(31)
[S , S ]"!4i(n#m)S , L K L\K
(32)
and with reversed sign in the last line if n3X and m3X . E For VERSION Ib: If n, m3X , i"1, 2 G [R , R ]"4(n!m)R , L K L>K
(33)
[R , S ]"0 , L K
(34)
[S , S ]"0 . L K
(35)
If n3X and m3X with iOj, i, j3+1, 2, G H [R , R ]"4(n#m)R , L K L\K
(36)
[R , S ]"0 , L K
(37)
[S , S ]"0 . L K
(38)
E For VERSION IIb: If n, m3X [R , R ]"!4i(m!n)S , L K L>K
(39)
[R , S ]"0 , L K
(40)
[S , S ]"0 , L K
(41)
and with reversed sign in the "rst line if n, m3X . If n3X and m3X [R , R ]"!4i(n#m)S , L K L\K [R , S ]"0 , L K [S , S ]"0 , L K and with reversed sign in the "rst line if n3X and m3X .
(42) (43) (44)
Finally, we indicate matrix representations for the generalized quasicrystal Lie algebras. Since the representation theory for VERSION Ia and VERSION IIb is similar, we only present it for the case of VERSION Ia.
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255
In this case, R and S are de"ned separately on the sets X and X as follows: L L
0 zL R " : , n3X , L 0 0
(45)
0 !z\L , n3X , R " : L 0 0
(46)
zL zLzR X , S " : (2i) L 0 zL
(47)
S " : (2i) L
n3X ,
z\L z\LzR X , n3X . 0 z\L
(48)
Notice, that the de"nition is such that S "S , because n!m3X implies m!n3X . L\K K\L Similarly, as a representative of the cases VERSION IIa and Ib, we show explicitly only matrix representations of VERSION IIa:
* 0 R " : , n3X , i3+1, 2, L G 0 0
(49)
is admissible for any freely chosen entry *. Furthermore, one has
0 0 S " : (4i) L 0 zLzR
, n3X ,
(50)
(51)
X
0 0 S " : (4i) , n3X . L 0 z\LzR X
In particular, one may choose * in (49) in such a way that the corresponding anticommutator of the generators R ([R , R ] " : R R #R R ) looks similar to the commutator-relation for the S . L L K > L K K L L For instance, choosing * to be zL for n3X and z\L for n3X leads to the following relations: [R , R ] "2R , n, m3X , i3+1, 2, , L K > L>K G
(52)
[R , R ] "2R , n, 3X , m3X , iOj3+1, 2, . L K > L\K G H
(53)
8. Conclusion and outlook Quasicrystal Lie algebras and their various generalizations discussed here might be a useful tool in physics in all those areas where the Witt and the Virasoro algebra play a crucial role. In particular, they might be useful due to their abundance of "nite dimensional subalgebras, which are not so extensively found in the case of the Witt or the Virasoro algebra.
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One possible example for an application of quasicrystal Lie algebras might be the construction of a discrete quantum mechanics on a discretization of S along the lines of Dobrev et al. [18]. Due to the one-to-one-correspondence of quasicrystal Lie algebras with points of one-dimensional quasicrystals, they might be viewed as `quasicrystal analogsa to the Witt algebra. Because of the lack of a central extension, we do not have, strictly speaking, a `quasicrystal Virasoro algebraa. The generalized quasicrystal Lie algebras, which allow for acceptance windows symmetric about the origin, could be used in relation to physical models which have a built-in symmetry similar to a `charge conjugationa type of symmetry, e.g. models related to interactions of particles with opposite electric charges or particle anti-particle con"gurations.
Acknowledgements We would like to thank Dr. H. de Guise for a careful reading of the manuscript. R.T. is grateful for the hospitality extended to her during this work at the Centre de Recherches MatheH matiques, UniversiteH de MontreH al.
References [1] J. Patera, E. Pelantova, R. Twarock, Quasicrystal Lie algebras, Phys. Lett. A 246 (1998) 209}213. [2] Z. MasaH kovaH , J. Patera, E. PelantovaH , In#ation centers of the cut and project quasicrystals, J. Phys. A: Math. Gen. 31 (1998) 1443}1453. [3] D. Shechtman, I. Blech, D. Gratias, J. Cahn, Metallic phase with long-range order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951}1953. [4] H. Huntley, The Divine Proportion: A Study In Mathematical Beauty, Dover Publ., New York, 1970. [5] M. Ghyka, The Geometry of Art and Life, Dover Publ., New York, 1977. [6] D.R. Hofstader, GoK del, Escher, Bach: an Eternal Golden Braid, Basic Books, New York, 1979. [7] B. Grunbaum, G. Shephard, Tilings and Patterns, W.H. Freeman & Co., New York, 1987. [8] R. Lawlor, Sacred Geometry } Philosophy and Practice, Thames & Hudson, New York, 1989. [9] J. Kappra!, Connections: The Geometric Bridge Between Art and Science, McGraw-Hill, New York, 1991. [10] R.V. Moody, J. Patera, Quasicrystals and icosians, J. Phys. A: Math. Gen. 26 (1993) 2829}2853. [11] L. Chen, R.V. Moody, J. Patera, Noncrystallographic root systems, in: J. Patera (Ed.), Quasicrystals and Discrete Geometry, Fields Institute Monograph Series, vol. 10, Amer. Math. Soc., Providence, RI, 1998. [12] J. Patera, Noncrystallographic root systems and quasicrystals, in: R.V. Moody (Ed.), Mathematics of Long Range Aperiodic Order, Kluwer, Dordrecht, 1997. [13] Z. MasaH kovaH , J. Patera, E. PelantovaH , Minimal distances in quasicrystals, J. Phys. A: Math. Gen. 31 (1998) 1539}1552. [14] R. Moody, A. Pianzola, Lie Algebras With Triangular Decompositions, Wiley-Interscience, New York, 1995. [15] V. Kac, A. Raina, Highest Weight Representations of In"nite Dimensional Lie Algebras, World Scienti"c, Singapore, 1987. [16] S. Kass, R. Moody, J. Patera, R. Slansky, A$ne Lie Algebras, Weight Multiplicities and Branching Rules, Univ. of Calif. Press, Los Angeles, 1990. [17] J. Patera, H. Zassenhaus, Higher rank Virasoro algebras, Comm. Math. Phys. 136 (1991) 1}14. [18] V.K. Dobrev, H.-D. Doebner, R. Twarock, A discrete, nonlinear q-SchroK dinger equation via Borel quantization and q-deformation of the Witt algebra, J. Phys. A: Math. Gen. 38 (1997) 1161}1182.
Physics Reports 315 (1999) 257}271
Nuclear structure issues determining neutrino-nucleus cross sections A.C. Hayes Los Alamos National Laboratory, Los Alamos, NM 87544, USA
Abstract We investigate the nuclear structure issues that determine the neutrino-nucleus cross sections of interest in nuclear, particle, and astrophysics. We discuss the uncertainties involved in calculating these cross sections, and the expected accuracy of model predictions. 1999 Elsevier Science B.V. All rights reserved. PACS: 25.30.Pt Keywords: Neutrino interactions; Nuclear structure; Shell model
1. Introduction Several of the searches for neutrino oscillations involve measuring the interaction of neutrinos with the nucleus. A primary example is the recently announced evidence [1] for neutrino oscillations at Super-Kamiokande, which comes from comparing the ratio of muons to electrons created by the scattering of atmospheric neutrinos from oxygen in the Cherenkov water detector. Another detector based on neutrino-nucleus scattering is the Sudbury Neutrino Observatory (SNO) which will measure the interaction of B solar neutrinos with deuterium in heavy water. Both the Liquid Scintillator Neutrino Detector (LSND) and KARMEN experiments involve neutrino scattering from carbon in mineral oil. The search for neutrino oscillations (l Pl and l Pl ) at the BooNE I I experiment at Fermi Lab will be a search for l CPe\N and l CPe>B quasi-elastic scattering. In addition to acting as a signal for neutrino oscillations, neutrino-nucleus reactions appear as background cross sections in these experiments and provide important checks on neutrino #ux or
E-mail address: [email protected] (A.C. Hayes) 0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 6 - 5
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detector e$ciency. Crucial to all these experiments is our understanding of neutrino-nucleus scattering and our ability to predict the cross sections to su$ciently high accuracy. Another "eld in which neutrino-nucleus scattering plays a signi"cant role is that of nucleosynthesis. As the core of a massive star collapses to form a neutron star, the #ux of neutrinos is so large that signi"cant nuclear spallation occurs, despite the small cross-sections. It has been pointed out by Woosley et al. [2] that neutrinos of all #avors excite nuclei to particle unbound states through the neutral current, and the A(l,l X)A reaction, or the l-process, may be an important process for nucleosynthesis of a number of elements. The temperature of the neutrinos is #avor-dependent and is ¹ I K O O&8 MeV and ¹ &4}5 MeV. This translates to transferring an average of 25 MeV J J J J S J J of energy to the nucleus, with a Fermi}Dirac tail up to &80 MeV. Thus, an understanding of both the neutral-current excitation and the subsequent multi-particle breakup of the nucleus are needed. Neutrino-nucleus scattering also plays an important role in r-process nucleosynthesis, which is responsible for the formation of half of the elements with A'70. In a stellar environment where a neutron gas exists alongside nuclei, neutron capture becomes the dominant mode for synthesizing medium and heavy mass nuclei. Under stellar conditions where the neutron capture rate is fast compared to b-decay, the conditions for the rapid or r-process, the nucleosynthesis rate becomes proportional to the the b-decay rate. The r-process is thought to take place in the expanding &hot bubble' of a type II supernova, which would mean that the #ux of neutrinos is su$ciently intense to cause signi"cant neutrino-nucleus reactions. The competition between neutron capture and neutrino scattering can be used to determine the distance of the r-process site from the neutron star and to estimate the time scale for the process. Extracting information on the issues requires knowledge of the neutrino capture cross sections by the very neutron-rich nuclei lying along the r-process path. The physics determining the various neutrino-nucleus cross sections of interest to particle and astrophysics varies with neutrino energy and with the structure of the nuclei involved. In this article we discuss the main nuclear physics issues involved and examine the dependence of the predicted cross-sections on models of nuclear structure.
2. Formalism Neutrino absorption on the nucleus occurs through the charged-current of the weak interaction l!#(Z, A)P(ZG1, A)#l! ,
(1)
where the incoming #ux is a beam of either neutrinos or anti-neutrinos, and the leptons are either electrons or muons or their anti-particles. The expression for neutrino absorption on the nucleus in terms of nuclear structure matrix elements has been derived by O'Connell [3] and by Walecka [4].
> G d(cos(h)d(E !E #E !E )p E "M" , p(E )" J J 2p \ D
(2)
where G is the weak interaction coupling constant, G/( c)"1.16639;10\ GeV\, E !E is the mass di!erence between initial and "nal nuclear states, p and E are the neutrino momentum and J J
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energy in the laboratory, p and E are the lepton momentum and energy, and cos(h)"(p ) p )/"p ""p " . (3) J J The matrix element "M" involves combinations of the "ve lepton traces listed in [3] with nuclear matrix elements of the seven multipole operators detailed in Donnelly and Haxton [5]. The seven operators, which are functions of the position coordinate and of the magnitude of the momentum transfer, q""q"""p !p " are J M (qx)"j (qx)> (X ) , (4) ( ( ( V D (qx)"M (qx) ) (1/q) , ( (( J J#1 1 M # M D (qx)" ! ) , ( (> ( (\ q ( 2J#1 2j#1
R (qx)"M (qx) ) r , ( (( J J#1 M # M )r, R (qx)" ! ( (> ( 2J#1 2J#1 ( (\
R (qx)" (
J#1 J )r, M # M ( (> ( (\ 2J#1 2J#1
1 X (qx)"j (qx)> (X )r ) , ( ( ( V q where M (qx) ) e"j (qx)[> (X ) e](. (* * * V The single particle matrix elements of the above operators, 1 j ""O("" j 2, have been tabulated by ? @ Donnelly and Haxton. Matrix elements between many-body nuclear states are given by 1J ""O(""J 2"R ? @1J ""[aR?, a @]( ""J 21 j ""O("" j 2 . (5) D H H D H H ? @ Here the matrix elements 1J ""[aR?, a @](""J 2 are the one-body density matrix elements which D H H describe the probability of removing a particle from an orbit j in the initial state J and creating @ a particle in the orbit j to form the "nal state J . These one-body density matrix elements contain @ D all the nuclear structure information determining the neutrino cross sections. If the neutrino #ux is known, the model dependence involved in determining the one-body densities matrix elements represents the uncertainty of the predicted neutrino-nucleus cross sections.
3. Atmospheric neutrinos The most recent evidence (1) for neutrino oscillations comes from the anomaly in the number of k- and e-type neutrinos reaching the Super-Kamiokande detector after being produced in the atmosphere by cosmic rays. The observed ratio of muons to electrons is about a factor of two less than expected. The addition of a signi"cant zenith angle dependence in the ratio of muon to electron events, where muon neutrino coming from larger distances (zenith angle 903) evidence
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larger depletion, while muon neutrino coming from overhead show no loss, is strongly suggestive of neutrino oscillations. The evidence of neutrino disappearance is reported as the observed ratio of muon to electron events divided by the ratio of events calculated in a Monte Carlo simulation. The atmospheric neutrino #ux involves neutrinos in the 0.1}3.0 GeV range. These neutrinos interact in the water detector through a charge exchange on oxygen O(l, l). The neutrino}oxygen cross sections are calculated [6] using the relativistic Fermi-gas model (RFG) [7], which incorporates binding energy e!ects through an average separation energy. At the higher neutrino energies (E '1 GeV) the J RFG is expected to work reasonably well, although the accuracy of its predictions maybe unclear. Given the signi"cance of the nuclear cross sections there is a clear need to examine these and determine whether there is any room for model dependence. This has been done by Engel et al. [8] who investigated carefully the neutrino-nucleus cross sections for Kamiokande. They examined several phenomena beyond the scope of Fermi-gas models. These included the role of bound states and resonances in oxygen, the Coulomb interactions of the outgoing leptons and nucleons with the residual A"15 nucleus, and the two-body interaction between nucleons in O. None of these e!ects are accurately represented in the Monte Carlo simulations used to predict event rates at Kamiokande and IMB. Nonetheless, Engel et al. concluded that the neglected physics could not account for the anomalous k to e ratio, nor were these e!ects likely to change the absolute event rates by more than 10}15%. They therefore conclude that the k/e ratio is a robust measure of the anomaly. The agreement between the more detailed nuclear structure model for O and the Fermi gas model re#ects the neutrino energies involved. For the most part, the atmospheric neutrinos excite the nucleus to a region well above the giant resonances, minimizing the importance of details of the structure of the nucleus. The Fermi gas model provides a reasonable description of quasi-free scattering, and this excitation energy range dominates the cross section at these neutrino energies.
4. Solar neutrino detection Four of the "ve solar neutrino experiments involve detecting neutrino-nucleus reactions, namely, Homestake, SAGE, GALLEX, and SNO. At Kamiokande solar neutrinos are detected via neutrino}electron scattering. The neutrino-nucleus cross sections measured in these detectors have been examined in detail in [9]. For completeness we summarize the key issues involved here. The primary source of neutrinos from the sun is the proton}proton burning chain, with an additional weaker source of neutrinos from the CNO cycle. Table 1 (taken from Bachall [9]) lists the nuclear reactions involved in the solar neutrino #ux, with the corresponding neutrino energies and the calculated #uxes. 4.1. Cl(l , e\)Ar The experiment at the Homestake Mine in South Dakota detects solar neutrinos by the reaction Cl(l , e\)Ar using a 0.6 kton perchloroethylene (C Cl ) detector. The experiment counts the number of individual Ar atoms produced by the l interactions. With accurate knowledge of the expected cross section one can check the #ux of solar neutrinos reaching the detector.
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Table 1 Neutrinos from the pp chain in the sun Reaction
Label
l energy (MeV)
Predicted #ux (10 cm\ s\)
p#pPH#e>#l p#e\#pPH#l Be#e\PLi#l
pp pep Be
6.0 0.014 0.47
BPBe#e>l He#pPHe#e>#l
B hep
40.42 1.442 (90%) 0.861 (1010%) 0.383 (15 418.77
5.8;10\ 8;10\
The threshold for the charge-exchange reaction on Cl is 0.814 MeV, which is above the maximum energy of the pp neutrinos (Table 1). The detector is sensitive primarily to Be, and B neutrinos, and also to the weaker #uxes of CNO(N, N, F), hep and pep neutrinos from the sun. Of these, only the B neutrinos have enough energy to excite states of Ar lying higher than the ground state, and approximately 77% of the event rate is expected to come from B neutrinos. Most of the uncertainty in the expected cross sections on Cl comes from the excited states contribution. The cross section to the ground state is known because it is determined by the measured half-life of Ar, which decays exclusively by electron capture to the Cl . The B b-decay spectrum involves neutrinos up to energies of about 15 MeV, so that many excited states of Ar can be populated. However, only those states below particle threshold (up to about 8.4 MeV of excitation) contribute to the production of Ag . Of these, the Fermi transition to the isobaric analog of Cl dominates. In the absence of small radiative corrections or isospin mixing, the Fermi transition rate is independent of nuclear structure and is dependent only by the matrix element of the isospin raising operator. Thus, this contribution to the event rate is known to high accuracy. The uncertainty in the total neutrino-nucleus cross section at Homestake is dominated by the cross section to the remaining excited states of Ar. For a pure Gamow}Teller transition, ¸"0, J"1, in the limit of small momentum transfer qP0, the matrix element entering the expression for the neutrino cross section can be expressed in terms of the B(GT) value from beta decay. (6) "M"P(1!cos(h))B(GT) , where B(GT) is the strength of the equivalent beta decay transition and is related to the beta-decay ft value by ft"(6146/B(GT)) s .
(7)
cos(h) is as de"ned in Eq. (3). For solar neutrinos only Fermi and Gamow}Teller transitions contribute signi"cantly to the cross section. All other transitions to excited states, which would involve ¸O0, are so-called forbidden and are expected to contribute little for an average neutrino energy of 7 MeV and as much as 10% for the end-point 15 MeV neutrinos. Thus, for the B neutrino spectrum only
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Gamow}Teller contributions to the excited state cross section need to be considered. These have been determined [10] to within 3% accuracy from a measurement of the b-delayed protons from the b>-decay of Ca. Ca is the isobaric analog of Cl. If isospin is a good symmetry, the Cl(l , e\)Ar excited state cross sections are determined by the ft values of the Ca b> decays to the K isobaric analogues of the Ar levels. 4.2. Ga(l , e\)Ge At both SAGE and GALLEX solar neutrinos are detected through the neutrino absorption reaction Ga(l , e\)Ge in large (tens of tons) gallium detectors. The threshold for the charge exchange reaction on Ga is low, E "0.233 MeV. Thus, all low-energy neutrinos from both the pp and CNO chains can be detected. This unique feature of the gallium detectors make them the sole detectors sensitive to the pp neutrinos (E 40.42 MeV). Because of the neutrino energies NN and #uxes involved most of the capture cross section is to the ground state of Ga, though the excited state contributions represent about 88% of the total B part of the cross section. The ground state of Ge decays 100% to Ga by electron capture with a known half-life, so that the ground state to ground state neutrino capture cross section can be determined accurately. The "rst excited state of Ge lies at 0.175 MeV and a number of excited states can be populated by the higher energy neutrino #uxes. Of the excited states the largest nuclear matrix element is the Fermi transition to the isobaric analog 3/2\(8.89 MeV) state (IAS). However, Champagne et al. [11] measured the decay properties of this state and found it to decay primarily by neutron emission with a negligible gamma-decay branch. They, therefore, concluded that there is very little contribution to the detection sensitivity of the Ga detectors from the neutrino population of the IAS. The Gamow}Teller transitions to other excited states of Ga are di$cult to calculate accurately. The transition matrix elements can be deduced from (p, n) measurements but the uncertainties remain large. The resulting uncertainty into total capture rate is &10%, and it is unlikely to be improved upon without a signi"cantly more accurate extraction of the GT transition matrix elements from (p, n) measurements. 4.3. d(l , e\) and d(l, l ) The Sudbury Neutrino Observatory (SNO) will measure interactions of B and hep neutrinos with deuterium using a heavy water (D O) detector. SNO will measure both neutrino absorption by deuterons through the charged current l #dPp#p#e\ and neutrino-disintegration of the deuteron through the neutral current l#dPl #p#n .
(8)
(9)
The restriction to B and hep neutrinos is because the thresholds for these two reactions is 1.442 and 2.225 MeV, respectively. SNO will also be sensitive to neutrino}electron elastic scattering. These scattered electron are peaked strongly in the forward direction and can be distinguished for those electrons produced in the charge exchange reaction on deuterium.
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A measurement of the shape of the recoil electron spectrum in the charged-current reaction will provide a direct measure of the spectrum for the incident neutrinos. The neutral current reaction is #avor blind and therefore gives a measure of the total neutrino #ux and a test of the solar models. The ratio of the charged-current to neutral-current events will allow a determination of whether electron neutrinos have `disappeareda and oscillated to other neutrino #avors. For both the charged-current test of the total #ux and the ratio of charged-current to neutralcurrent test of the electron neutrino #ux, the measured quantity has to be referenced to a theoretical expectation. The experiment will search for a departure form theory, which would signal either (or both) physics beyond the Standard Model of particle physics or the Standard Solar Model. Clearly, the expected theoretical cross sections have to be known to high accuracy. Bachall et al. [12] have calculated the expected "rst and second moments of the recoil electron's kinetic energy and the ratio of the number of charged to neutral-current events expected at SNO. They determined shifts in the SNO observables for various neutrino-oscillation scenarios. All Standard Model corrections to these estimates need to be examined. The most signi"cant of these include the role of meson-exchange currents and of radiative corrections in l#d scattering. Towner [13] has investigated the impact that radiative corrections in neutrino}deuterium scattering for the charged-current and neutral-current channels have on the observables to be measured at SNO. The calculations showed that the corrections are generally small and can be neglected. However, in the case where internal Bremsstrahlung photons emitted in the reaction l #dPp#p#e\#c are detected by the Cherenkov detectors, the ratio of the number of charged-current to neutral-current events seen at SNO was found to be shifted by about one standard deviation. Calculations for the meson-exchange corrections for neutrino}deuterium scattering have yet to be calculated.
5. Neutrino}Carbon scattering at LSND and KARMEN At both LSND and KARMEN the signal for neutrino oscillations involves neutrino interactions in mineral oil (CH ), and understanding neutrino scattering from carbon is important for these experiments. The neutrino source at both these experiments comes from the decay of pions produced in the beam stop. The vast majority of the pions decay at rest producing one muon neutrino, muon anti-neutrino and electron neutrino. Of these only the electron neutrino has enough energy to cause a nuclear charge-exchange reaction on carbon. However, at LSND 3.4% of the pions decay while in #ight and produce muon neutrinos of su$cient energy for such reactions. The oscillation of decay-in-#ight (DIF) muon neutrinos, (l Pl ), is detected at LSND by the I appearance of high-energy electrons from the l CPNe\ reaction. Extracting oscillation para meters from this search requires knowledge of the expected cross section. In addition, the measured l CPNk\ cross-section acts as a test of the l DIF #ux and of the detector e$ciency. The I I KARMEN experiment also has the e$ciency to measure the lCPBk> reaction. At both LSND and KARMEN the inclusive l CPXe\ and the exclusive l CPN cross-sections are mea sured for the electron neutrinos from the decay of the pion at rest. The Michel spectrum for these decay-at-rest (DAR) electron neutrinos is known, and the l C cross sections provide a nice constraint on the nuclear structure models used for carbon.
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The DAR neutrino spectrum involves neutrino energies 0}52 MeV, with an average neutrino energy E & 32 MeV. The Q-value for the charge-exchange reaction (l , e\) on C is about J 17 MeV. The DAR inclusive cross section is then dominated by low multipoles (1>, 1\, 2\) and by excitation of the giant resonances. In the case of the l C cross section, the DIF muon I neutrino #ux involves an average neutrino energy of about E &150 MeV, but the #ux is J "nite up to E &250 MeV. Because of the mass of the muon and the mass di!erence between J C and N the Q-value for the (l , k\) reaction is close to 123 MeV. Calculations for this I cross section need to include both a good description of the giant resonance region (E &15}40 MeV of excitation in C) and of higher excitation energy regions (up to 80}100 MeV). V Furthermore, all multipoles j&0}5 make signi"cant contributions to the inclusive cross section. The di!erence in the momentum and energy transfers between the DAR (q&0.2 fm\) and the DIF (q&1.0 fm\) cross sections results in the latter being considerably more di$cult to calculate accurately. In calculating very low-energy or very high-energy neutrino-nucleus cross sections the choice of model space to describe the structure of the nucleus is usually straightforward. In the case of low-energy processes it is most important to provide a very detailed description of the nuclear wave functions for the initial and "nal states involved. Thus, in a shell model sense, one attempts to include all con"gurations, and full con"guration mixing, within a small number of shells. In contrast, for high neutrino energies the details of con"guration mixing are unlikely to be important, whereas the inclusion of highly excited particle}hole con"gurations crucial. As discussed in the case of atmospheric neutrinos, high-energy neutrino reactions are reasonably well described by Fermi-gas model calculations. However, the DIF l C cross section at LSND I poses an especially di$cult problem in that the neutrino energies involved are intermediate between these two situations. As discussed below, both details of con"guration mixing and the inclusion of particle}hole excitations across many shells play a signi"cant role in determining the cross section. The inclusive C(l , k\)X cross section measured at LSND was "rst calculated by the Caltech I group [14] using a sophisticated continuum RPA model. The calculated cross sections overestimated experiment by almost a factor of two, and suggested that the measured cross sections may be inconsistent with other observables for C. The key observables that need to be considered as checks on model calculations are k-capture, the DAR C(l , e\)X, (e, e ), photoabsorption, and b-decay. These di!erent probes involve di!erent energy and momentum transfers and, thus, constrain di!erent aspect of the calculations. The disagreement between theory and experiment for the DIF l cross section caused a #urry of theoretical activity [15}17] in an e!ort to uncover I possible shortcomings of the RPA calculations. Studies of the e!ect of di!erent nuclear structure assumptions on the predicted neutrino-nucleus cross sections were quite revealing, and we discuss these in some detail below. In the simplest model, C consists of four neutrons and protons in the p-shell outside a closed He core. Excited states reached by the (l , k\) reactions would be simple particle}hole states built I on the ground state. However, the structure of both the C and the continuum states N involve considerably more sophisticated con"gurations. Since there is a limit on the size of the model space that can be included in any calculation, one is forced to make approximations in the hope that these incorporate all the essential physics for the problem under consideration. In the case of the neutrino cross sections on carbon the assumptions or approximations that are
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most likely to a!ect the predicted cross sections are: 1. 2. 3. 4. 5.
the C ground state p-shell structure; the treatment of ground state correlations beyond the p-shell; the model space truncation, especially for the "nal states; con"guration mixing in the "nal states; and the size parameter entering the calculations. In the next few subsections we discuss the e!ect of each of these on the predicted cross section.
5.1. Ground state p-shell structure There are two simple, but almost opposite, starting points for describing the C . The "rst of these is to assume that C consists of the p-shell equivalent of three a-particles. In this model the g.s. is in an ¸"0 S"0 state, and has good SU(4) symmetry [44 4]. A description of C in terms of three tightly bound a-particles is a reasonable starting point. Indeed, the time-honoured Cohen}Kurath p-shell interaction predicts that this state makes up 78% of the C ground state wave function. The second starting point is to assume that the four protons and neutrons "ll the lowest available p-shell orbit, i.e., that C corresponds to a closed p -shell. This starting point is quite distinct from the assumption of three a's or from the Cohen}Kurath wave function. It involves a ground state that is 84% SO0. Table 2 summarizes the spin structure of 12C under di!erent model assumptions. The most serious problem that arises in assuming a closed p -shell is the overestimation of transitions within the p-shell. This is clearly demonstrated by considering the exclusive chargeexchange cross section to the 1> N , which is dominated by the pPp Gamow}Teller (GT) transition. The GT operator, pq , conserves SU(4) symmetry. However, there is no way to make > a ¹"1 1> state with [44 4] symmetry. Thus, in the 3-a model of C the GT transition to the N is forbidden. On the other hand, starting with a closed p -shell leads to a strong p Pp transition, which is a factor of six larger than that experiment. When one includes [14] RPA correlations the factor of six is reduced to a factor of four. The Cohen}Kurath interaction, which was "tted to p-shell nuclei with particular attention paid to GT transitions, provides a good description of this transition. The transition to the lowest 2> state of N is also overestimated by a factor of several when C is restricted to a closed p -shell. Table 2 Spin structure of the C ground state Model
S"0 (%)
S"1 (%)
S"2 (%)
% [44 4]
Three a-particles Cohen}Kurath interaction (p )
100 81 16
0 18 59
0 (2 25
100 78 6
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In the case of particle}hole excitations out of the p-shell, the p-shell structure of the ground state does not a!ect the total sum-rule for a given operator. However, the very di!erent g.s. spin structures for the models discussed above means that the strength for a given operator will be distributed di!erently over the "nal state multipoles. The net result is that the predicted #ux averaged cross section will be di!erent for di!erent C p-shell wave functions. Table 3 shows the distribution of B(*J)"1J ""[r*, p] (""J 2 strength for the isovector 1 u D G dipole and octupole multipoles in C. We note that for the spin-independent multipoles the strength is independent of the p-shell structure of the ground state, while the spin-multipole strengths are model-dependent. Table 4 summarizes the 0 uP0 u and 0 uP1 u contributions to k-capture, (l , e\) DAR and (l , k\) DIF for the open-shell Cohen}Kurath and the closed p ground state wave functions. I The largest model dependence is seen in the GT transition to the N . However, even when one excludes the ground state cross section, there remains a signi"cant di!erence in the predicted inclusive k-capture rate and the (l , k\) DIF cross section. This is particularly true in the latter case, I where the two calculations di!er by 25%. Table 3 Distribution of isovector dipole and octupole 1 u strength for di!erent C ground states C model
Dipole 1\
SU(4) 3a-particles Closed p
11.6 11.6
SU(4)3a-particles Closed p
Spin dipole 0\ 3.9 5.6
Spin dipole 1\ 11.6 14.1
Spin dipole 2\
Total
19.3 15.0
46.2 46.2
Oct. 3\
Spin Oct. 2\
Spin Oct. 3\
Spin Oct. 4\
Total
67.7 67.7
48.4 16.1
67.7 56.4
87.0 130.5
270.8 270.8
Table 4 Model dependence in (0#1) u calculations for neutrino reactions k-capture rate (10 s\) Closed p
Open p
(l , e\) DAR 10\ cm Closed p
Open p
(l , k\) DIF I 10\ cm Closed p
Open p
pPp 1> 2> 3> pP(sd) 0\ 1\ 2\ 3\ 4\
37.46 1.10 *
6.41 0.30 0.08
86.25 0.24 *
9.92 0.04 0.01
4.35 2.42 *
0.87 0.92 0.24
4.15 21.35 12.15 0.07 0.06
3.05 18.0 13.10 0.06 0.05
0.01 3.11 3.79 0.0 0.0
0.03 3.09 4.04 0.0 0.0
0.09 6.79 3.75 1.27 1.43
0.07 4.85 4.13 1.33 1.14
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5.2. Ground state correlations The restriction of the C ground state to a p-shell wave function, be it a closed p or the Cohen}Kurath wave function, leads to an overestimate of the di!erent neutrino cross sections. The one-body operators that determine the neutrino cross sections (Eqs. (4)) connect the ground state to 1p}1h excited states. When 2p}2h excitations are included in the ground state they lead to destructive interference, the so-called `backward-going diagramsa of RPA. The inclusion of these 2p}2h correlations, either in an RPA or a shell model calculation, reduces the neutrino cross sections by about a factor of two. In the shell model all 2p}2h con"guration allowed up to a given
u of excitation are normally treated on an equal footing. In contrast, in RPA calculations the 2p}2h correlations are restricted to the type "(h\, p )L SJ,¹; (h\, p )L S1J, ¹ : 002 ,
(10)
where the 2p}2h states are made up only from the coupling of two 1p}1h states of the spin, isospin J, ¹ that comprise the basis states of N. Here we are labeling the 1p}1h states by their excitation, n u, in an oscillator model. In Table 5 we compare the predictions of two calculations (17), which start with a closed p wave function for C, both which di!er in that the calculation labeled TDA does not include 2p}2h ground state correlations. Both calculations treat the excited states of N as 1p}1h states "N2""(h\, p)L SJ,¹2, n"0, 1, 2, 3, 4 .
(11)
It is clear that for all observables considered, (l , k\)DIF, (l , e\), k-capture and photoabsorption, I the inclusion of 2p}2h correlations reduces the predictions considerably (& a factor of two), and brings the calculated values in closer agreement with experiment.
Table 5 Inclusive cross sections involving a continuum of N states, including the ground state (l , k\) DIF I p;10\ cm
(l , e\) DAR p;10\ cm
k-capture K ;10 s\
Closed-p -shell TDA 0 u (0#1) u (0#1#2) u (0#1#2#3) u (0#1#2#3#4) u
5.62 18.21 25.61 30.36 31.85
52.29 59.53 57.52 57.59 57.62
35.56 72.98 75.41 75.23 75.21
Closed-p -shell#correlations 0 u (0#1) u (0#1#2) u (0#1#2#3) u (0#1#2#3#4) u
2.56 9.60 13.78 16.10 16.77
19.50 23.60 27.37 27.34 28.46
13.13 34.01 38.17 37.61 38.38
Photoabsorption p ;10\ cm
27.28 27.31
14.84 14.50
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5.3. Model space truncation In calculating the inclusive neutrino cross sections one has to include all "nal states that contribute signi"cantly. The size of the required "nal state model space clearly depends on the neutrino energy. For the DAR neutrino spectrum (E (52 MeV) the calculated cross section J converges at 1 u of excitation, i.e., the inclusive cross section is almost entirely accounted for in term of pPp and pPsd transitions. In addition, multipoles higher than j"2 do not contribute signi"cantly to the cross section. For the (l , k\) DIF neutrino #ux (E (250 MeV) only 57% of I J the cross section is accounted for in a (0#1) u calculation. Approximately, 85% of the cross section is accounted for in a (0#1#2) u model space and about 97% in a (0#1#2#3) u calculation. All multipoles up to j"5 contribute signi"cantly to the DIF cross section. The need to include excitations up to 4 u of excitation while paying careful attention to the ground state structure of C makes the (l , k\) DIF cross section particularly di$cult to calculate accurately. I The size of the model space increases considerably as the number of shells included increases. At some stage it is no longer possible to diagonalize the Hamiltonian. 5.4. Final state conxguration mixing Up till now, we have only considered 1p}1h "nal states in N. However, in reality these 1p}1h con"gurations are strongly mixed with 2p}2h as well as more complicated con"gurations. The e!ect of including these con"gurations is to spread the 1p}1h strength in energy thus changing the #ux-averaged cross section. Detailed calculations [17] "nd that the DIF (l ,k\) cross section is I decreased by a few percent, while the (l , e\) DAR cross section, k-capture and photoabsorption cross section all increase by about 6}25%. The di!erent e!ect the inclusion of 2p}2h "nal state con"gurations has on the di!erent probes results from the fact that each sample has di!erent linear combinations of multipoles. 5.5. Nuclear size parameter A parameter that has to enter all the nuclear calculations for the neutrino calculations in the size of the nucleus. In shell model calculations this enters through the oscillator parameter. Changing the oscillator parameter changes the shape of the axial-vector and vector form factors entering the neutrino cross sections. The oscillator parameter is normally chosen to reproduce key observables, for example, the ground state charge radius. For the ground state of C this suggests an oscillator parameter of b"1.64 fm. This value of b has been used in all the neutrino calculations listed above for C. However, the shape of the measured (e,e ) form factor to the 1\ and 2\¹"1 high-lying states of C suggests the need for a larger oscillator parameter, b"1.82 fm. The M2 contribution is signi"cant for all the neutrino}carbon reactions under discussions. Fig. 1 shows the M2 vector form factor for b"1.64 and 1.82 fm. For momentum transfers below the "rst maximum for the form factor the neutrino cross section increases as b increases. In contrast, momentum transfers beyond the "rst peak in the form factor result in the predicted cross section being decreased as b increases. For higher multipoles the "rst maximum in the form factor occurs at higher q, so that even at q&1 fm\ an increase in the oscillator parameter means an increase in the neutrino cross section.
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Fig. 1. M2 vector form factor for the 1 u transition pP(sd). The shape of the form factor is shown for two choices of the oscillator parameter, b"1.64 and 1.82 fm. These two choices are suggested by the measured ground state charge form factor and the excited state (e,e ) form factors, respectively. As discussed in the text and shown in Table 6, their e!ect on the neutrino reactions depends on the momentum transfer involved.
We have examined the dependence on the calculated (l , e\) DAR, (l , k\) DIF cross sections I and the k-capture rates on the assumed oscillator parameter by comparing the predictions for b"1.64 and 1.82 fm. Relative to the b"1.64 fm calculation the inclusive (l , e\) DAR cross section increased by 17%, the (l , k\) DIF cross section decreased by 9%, while the k-capture rate I increased by 14%. 5.6. Shell-model predictions In Table 6 we present the results of the most complete to date shell model results for the various neutrino reactions on carbon of interest. The results are presented as a function of the size of the model space, increasing from 0 u to 4 u. The 4 u calculation, which includes 2p}2h correlations in both the ground state of C and the "nal states of N predicts a value for the (l , k\) cross I section of 14.5;10\ cm for b"1.64 fm and 13.4;10\ cm for b"1.82 fm. This is to be compared with the experimental value of 12.3;10\ cm [18] and the RPA prediction of (18!20);10\ cm. The smaller prediction for this cross section in the shell model arises from a combination of the e!ects discussed above, namely, the ability to include in a shell model formalism an open p-shell and 2p}2h correlations in the "nal states. Both the (l , e\) cross section
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Table 6 Inclusive cross sections involving a continuum of N states, but excluding the ground state (l , k\) DIF I p;10\ cm
(l , e\) DAR p;10\ cm
k-capture K ;10 s\
Photoabsorption p ;10\ cm
Shell-model b"1.64 0 u (0#1) u (0#1#2) u (0#1#2#3) u (0#1#2#3#4) u
1.16 8.09 11.95 13.74 14.16
0.27 4.85 4.75
0.69 22.06 24.55
14.75
Shell-model b"1.82 (0#1#2) u (0#1#2#3#4) u
8.06 12.89
5.56
28.0
18.17
Expt.
11.7
5.7(8) LSND Collab. 6.1(13) [19]
31.0
and the k-capture rate show agreement with the experiment for the larger value of b, but theory is somewhat low compared to experiment when the smaller oscillator parameter is used. A direct comparison between these latter reactions and the (l , k\) cross section can be misleading because I there is a very di!erent distribution of the strength over the di!erent multipoles in each case. 5.7. Theoretical uncertainties Of the cross sections discussed in this article, the theoretical DIF (l , k\) cross section on carbon I probably carries the largest uncertainty. For the solar neutrino cross sections, the neutrino energies are low enough to allow only Fermi and Gamow}Teller transitions. For the nuclei of interest, these can usually be determined from experimental constraints, e.g., beta-decay GT strengths. In the case of atmospheric neutrinos, the neutrino energies are high enough that the cross section is dominated by quasi-free scattering, thus minimizing the importance of details of nuclear structure. In contrast, the predictions for (l, l\) cross sections on carbon can be quite model dependent. Calculations restricted to a Tamm}Danco! approximation built on a closed p ground state overestimate all neutrino reactions by about a factor of 3. As one builds in more correlations into both the ground state and the "nal states the predictions of the calculations come closer to experiment. When these correlations are mostly included, the remaining di!erences between the predictions of the model calculations and the dependence of the predictions on nuclear size suggest that the theoretical uncertainties are of the order of 25}35%.
References [1] Super-Kamiokande Collaboration, Phys. Rev. Lett. 81 (1998) 1562. [2] S.E. Woosley, D.H. Hartmann, R.D. Ho!man, W.C. Haxton, Astrophys. J. 356 (1990) 272.
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[3] J.S. O'Connell, T.W. Donnelly, J.D. Walecka, Phys. Rev. C 6 (1972) 719. [4] J.D. Walecka, in: V.W. Hughes, C.S. Wu (Eds.), Muon Physics, Academic Press, New York. [5] T.W. Donnelly, W.C. Haxton, Atomic Data Nucl. Data Tables 23 (1979) 103; T.W. Donnelly, W.C. Haxton, Atomic Data Nucl. Data Tables 25 (1980) 1. [6] M. Nakahata et al., J. Phys. Soc. Jpn 55 (1986) 3786. [7] R.A. Smith, E.J. Moniz, Nucl. Phys. B 43 (1972) 605. [8] J. Engel, E. Kolbe, K. Langanke, P. Vogel, Phys. Rev D 48 (1993) 3048. [9] J.N. Bachall, em Neutrino Astrophysics, Cambridge University Press, Cambridge, 1998. [10] A. Garcia, E.G. Adelberger, P.V. Magnus, H.E. Swanson, O. Tengblad, ISOLDE Collaboration, D.M. Moltz, Phys. Rev. Lett. 67 (1991) 3654. [11] A.E. Champange, G.E. Dodge, R.T. Kouzes, M.M. Lowry, A.B. MacDonald, M.W. Roberson, Phys. Rev. C 38 (1987) 900. [12] J. Bachall, P.I. Krastev, E. Lisi, Phys. Rev. C 55 (1997) 494. [13] I.S. Towner, Phys. Rev. C 58 (1998) 1288. [14] K. Kolbe, K. Langanke, F.K. Thielemann, P. Vogel, Phys. Rev. C 52 (1995) 3437; K. Kolbe, K. Langanke, S. Krewald, Phys. Rev. C 49 (1994) 1122. [15] N. Auerbach, N. Van Giai, O.K. Vorov, Phys. Rev. C 56 (1997) R2368. [16] S.K. Singh, N.C. Mukhopadhyay, E. Oset, Phys. Rev. C 57 (1998) 2687. [17] I.S. Towner, A.C. Hayes, in preparation. [18] LSND collaboration using most recent evaluation of DIF #ux, private commun. [19] B.E. Bodmann et al., Phys. Lett. B 332 (1994) 251.
Physics Reports 315 (1999) 273}284
The gluon propagator Je!rey E. Mandula* Division of High Energy Physics, US Department of Energy, Washington, DC 20585, USA
Abstract We discuss the current state of what is known non-perturbatively about the gluon propagator in QCD, with emphasis on the information coming from lattice simulations. We review speci"cation of the lattice Landau gauge and the procedure for calculating the gluon propagator on the lattice. We also discuss some of the di$culties in non-perturbative calculations } especially Gribov copy issues. We trace the evolution of lattice simulations over the past dozen years, emphasizing how the improvement in computations has led not only to more precise determinations of the propagator, but has allowed more detailed information about it to be extracted. 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 11.15.!q; 11.15.Ha; 14.70.Dk Keywords: Gluon; Propagator; Gauge; Lattice
1. Introduction In quantum "eld theory, the Green's functions carry all the information about the theory's physical and mathematical structure. Aside from the vacuum expectation values of "elds, the moduli which parametrize the phase structure of a "eld theory, the two-point functions are its most basic quantities. From this point of view, the gluon propagator may be thought of as the most basic quantity of QCD. Even without quarks, in a pure Yang-Mills theory, the gluon propagator is well de"ned. At short distances or equivalently large momentum transfers, because of asymptotic freedom we expect that perturbation theory should be su$cient to describe any Green's function. By contrast, at large distances or small momenta, there is no available analytic method to pin down the
* Corresponding author. E-mail address: [email protected] (J.E. Mandula) 0370-1573/99/$ - see front matter 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 2 7 - 7
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behavior of the Green's functions. Furthermore, the fact that there are no asymptotic gluon states raises the possibility that the gluon propagator may be quite di!erent than those associated with stable particles. As a result of its centrality from the "eld theortic perspective, the infrared limit of the gluon propagator has been subject of calculation and speculation since QCD was accepted as the correct theory of the strong interactions in the early 1970s. Ideas about the structure of the gluon propagator have been informed by many attractive hypotheses and conceptual puzzles. Among the recurring themes are: E The relation of the gluon propagator, speci"cally its infrared behavior, to con"nement; E The behavior of Green's functions in a theory in which one expects that none of the quanta of the fundamental "elds are physical particles in that theory; E How the Green's functions express such general properties of "eld theory as spectral positivity; E How the absence of any physical gluon states can be compatible with any non-zero gluon propagator. Note that despite the absence of physical asymptotic gluon states, gluons are real } every bit as real as quarks. They are also observed in the same way } through the jets of hadrons that result when they are produced in high-energy collisions. Their presence was indirectly inferred from the original deep inelastic electron}proton scattering experiments performed at SLAC in the late 1960s and early 1970s. There they were needed to account for the fraction of the momentum of the proton that was not attributable to quarks, the proton's electrically charged constituents. This fraction, according to the so-called momentum sum rule, was close to 50%. Gluon jets were directly observed experimentally in experiments at DESY in the 1980s. The problem of having fundamental constituents that only occur con"ned inside hadrons is not a physical nor a conceptual one. It is purely a problem of reconciling this physical structure with our notions of how Green's functions behave in "eld theory. The plan of this paper is the following: In Section 2, we brie#y review the situation in the ultraviolet, where perturbation theory holds, and use this discussion to "x notation. In Section 3, we discuss some of the ideas that have been advanced regarding the behavior of the gluon propagator in the infrared. In Section 4, we discuss the procedure for calculating the gluon propagator on the lattice. We introduce the lattice Landau gauge, and discuss its implementation. We also discuss some of the di$culties in non-perturbative calculations } especially Gribov copy issues. In Section 5 we trace the evolution of lattice simulations over the past dozen years, emphasizing how the improvement in computations has led not only to more precise determinations of the propagator, but has allowed more detailed information about it to be extracted. In Section 6 we summarize what we now know, from non-perturbative lattice studies, about the gluon propagator, and also what is still obscure about it.
2. The gluon propagator in the ultraviolet Since in this discussion we shall be exclusively concerned with the gluon propagator, we restrict our considerations to pure Yang}Mills theory, with no quarks. The gluon propagator is the Fourier transform of the time-ordered matrix element of two gluon "elds A? (x), where for gauge I
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group S;(N) the index a runs from 1 to N!1.
D?@ (q)"!i dxe O V 10"¹(A? (x)A@(0)"02. IJ I J
(1)
In covariant gauges the propagator has the kinematic form D?@ (q)"!id?@[(g !(q q /q))D(q)#a(q q /q)D*(q)] . IJ IJ I J I J The parameter a speci"es the gauge, and the Landau gauge is a"0. To zeroth order in perturbation theory, the propagator is the same as in QED,
(2)
D (q)"1/q . (3) To any "nite order in perturbation theory, this power dependence remains valid. It is possible that when all orders are summed, there could result an anomalous dimension D(q)&1/(q>A) .
(4)
What the value of c may be, or even if it is non-zero, is not known.
3. The gluon propagator in the infrared In the infrared, the situation regarding the gluon propagator is much murkier. It is instructive to summarize brie#y some of the more thoroughly studied ideas and speculations about the infrared behavior of the gluon propagator: 3.1. The gluon propagator explicitly displays conxnement If the con"nement potential is linear, it can be expressed in terms of the exchange of quanta whose momentum space propagator behaves like D(q)&1/q, (qP0) .
(5)
This behavior was hypothesized by Mandlestam [1] in the late 1970s, and it was made the characteristic of the phenomenological model studied for many years by Baker et al. [2]. More recently, by studying certain truncations of the Schwinger}Dyson equations, this behavior has been advocated by Brown and Pennington [3]. From the point of view that con"nement is a result of a sort of &&Dual Meissner E!ect'', the idea that the gluon propagator should express con"nement directly is in some sense rather heretical. 3.2. The propagator has a non-zero anomalous dimension Marenzoni et al. [4] carried out lattice simulations of the gluon propagator and interpreted their results, speci"cally the observation that the propagator fell o! quite di!erently than in perturbation
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theory, in terms of an anomalous dimension. Their "ts to lattice data were consistent with the anomalous dimension being consistent with c+1. Their preferred functional form was 1
q ? Z m#q K
.
(6)
3.3. The propagator acquires an ewective mass The earliest lattice simulations of the gluon propagator in the Landau gauge, by Mandula and Ogilvie and by Gupta et al. [6] were interpreted in terms of a massive particle propagator. D(q)&1/(q#m) .
(7)
3.4. The propagator vanishes at vanishing momentum This behavior, which is often described as the propagator having a pair of poles with conjugate complex masses m"be! p was hypothesized by Gribov [7], in connection with his study of gauge copies. It has been advocated on a number of di!erent grounds, by Stingl [8], Cudell and Ross [9], Smekal et al. [10], Zwanziger [11], Namislowski [12], and Bernard et al. [13]. The gluon propagator in this scenario may take the explicit form in the infrared D(q)&q/(q#m) .
(8)
This form is an explicit realization of a theorem due to Zwanziger [11], namely that on the lattice, for any "nite spacing but in the in"nite volume limit, the gluon propagator must vanish at q"0. lim D(q"0)"0 , , where N is the number of sites on each side of the lattice.
(9)
3.5. The propagator takes its perturbative form The form for the gluon propagator incorporated into all QCD models used in simulations to design experimental detectors and interpret the results of collider experiments is simply the perturbative form D(q)"1/q .
(10)
4. The gluon propagator on the lattice Lattice simulations of the gluon propagator have been carried out since the late 1980s, and are still being pursued. The goal of these studies has remained to try to arrive at a de"nitive understanding of the gluon propagator's infrared behavior. More speci"cally, one wishes to have as
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accurate a numerical determination of D(q) as one can, over the largest range of momenta, and for the numerical result to converge to an analytic picture of the gluon propagator. One hopes to articulate the character of D(q) such that it expresses physics of a theory without physical gluon asymptotic states. Each of the possibilities listed above is sensibly motivated, in that each incorporates some known or expected property of QCD. As with all lattice calculations, the quality of the results, measured by the statistical precision, the size of the lattice spacing, or the total volume of space}time simulated, have steadily improved over time. In order to appreciate what has and has not been learned, we must "rst review the proper de"nitions of operators and gauge conditions, and the sources of errors on the lattice. 4.1. The kinematics of discretization On a "nite lattice, momentum is a periodic discrete variable. Denoting the lattice spacing by a and the number of sites per side by N, each component of the momentum takes the values q "0, $2n/aN, $2(2n/aN), 2, $n/a . I The kinematic range of the dimensionless momentum,
(11)
aq,( aq aq I I I
(12)
aq3[0, 2p] .
(13)
is
The free propagator on the lattice is a periodic function of the lattice momentum and we can use it to de"ne a lattice corrected momentum. 1 1 D(q)" , . (14) (4/a) sin(aq /2)#m q( #m I The momentum so de"ned absorbs much of the lattice artifact errors in propagators. It has the kinematic range aq( 3[0, 4] .
(15)
For small momentum it approaches the ordinary dimensionless momentum. 4.2. The lattice Landau gauge On the lattice, where the basic variables are the unitary matrices ; (x) that express parallel K transport, the gauge potential may be de"ned as [5,14] ; (x)!;R(x) ; (x)!;R(x) I . I !Tr I A (x)" I I 6iag 2iag
(16)
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The lattice Landau gauge condition could be de"ned in terms of it by D A (x), A (x)!A (x!k( )"0 . (17) I I I I I However, it is truer to the continuum situation to formulate it as a maximization condition Max ReTr;E (x) , I EV V I ;E (x),g(x); (x)g(x#k( )R . (18) I I The maximization condition implies the "nite di!erence one, and has the virtue that it excludes very unsmooth gauge con"gurations which would maximize the trace on some sites and minimize it on others, in the extreme case on alternating sites. The maximization condition, in this sense, carries the smoothness character of the continuum Landau gauge. Any gauge condition expressible as f (;)"0 can be implemented by following the Fade'evPopov procedure. This consists of writing the path integral of any quantity O(;) with a gauge invariant measure, multiplying by 1 in the form of a delta function of the gauge condition times the reciprocal of its Jacobian, the Fade'ev-Popov determinant, judiciously interchanging the order of the path integrals, and "nally reexpressing the result in terms of an un"xed measure again.
D;e\QO(;)" " D; Dg e\QO(;)D (;)d( f (;E)) D3 $. " Dg D;e\QO(;)D (;)d( f (;E)) $. " D;e\QO(;M ) .
(19)
Here, for "xed g, ;M is the gauge transform of ; to the f (;)"0 gauge: ;M [;]";E, f (;M )"0 .
(20)
In contrast to perturbation theory, where the evaluation of D gives rise to the introduction of $. ghost "elds, in simulations there is no need to compute the Fade'ev-Popov determinant. The correct adjustment to the measure is built into the simulation recipe: Perform the simulation without speci"cation of the gauge, but for each lattice con"guration, transform the link variables to the f (;)"0 gauge before evaluating and averaging the path integrand. 4.3. Gribov copies Gribov copies [7] are a serious conceptual problem in using lattice simulation to understand the behavior of Green's functions in a gauge theory. Their existence on the lattice was investigated starting in the early 1990s [15}17]. Expressing the gauge condition as a maximization condition at each site avoids some trivial lattice artifact copies, but all the analogues of the continuum Gribov copies are still present. From the "rst it was clear that the treatment of Gribov copies in
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a simulation could have a substantial impact on its results for gauge propagators [15,18]. Because of the uncertainty they lend to simulations of gauge dependent quantities, Gribov copies have been studied by several groups, and studies of their impact on lattice simulations continue [19}21]. Variant gauge conditions and algorithms continue to be investigated as well [22}24]. A conceptual procedure for selecting one out of several copies is to take the maximization condition as an absolute maximum. The space of con"gurations which are absolute maxima is called the fundamental modular domain, and the rule that the path integral should be restricted to the fundamental modular domain removes copies in principal, except on the boundary of the domain [25}28]. Unfortunately, there is no practical procedure known for actually "nding the fundamental modular domain. All methods of gauge "xing are liable to end on copies. Experience in simulations has shown that there are typically many copies, and that quite di!erent con"gurations can have nearly the same value of the maximization functional. An inconvenience, though a serious one, is that gauge "xing is a notoriously slow process. Many procedures have been advocated for accelerating the process, and all work fairly well if the size of the lattice is not too large. However, for very large lattices gauge "xing seems to become much slower, and all acceleration methods seem to loose much of their e!ectiveness. The situation is in some sense as bad as it can be: the role of Gribov copies is not fully understood, there is no perfect recipe for dealing with them, yet they do seem to matter in that there is ample evidence from simulations that the manner in which they are treated sometimes has a signi"cant impact on the "nal result of a simulation.
5. Results from the lattice In this section we will describe the progress that has been made over the dozen or so years that the simulations of the gluon propagator have been carried out. 5.1. Earliest simulations The "rst lattice simulations of the gluon propagator were carried out by Mandula and Ogilvie [5] and by Gupta et al. [6] in 1987. With the computers available at that time, the statistical quality of their signals deteriorated very quickly with lattice time. Therefore, they expressed their results in terms of Euclidean lattice time at zero spacial momentum. The graphs of their results are shown in Figs. 1 and 2. Mandula and Ogilvie's results come from using a 4;10 lattice at b"5.8. The open circles are the time}time component of the D (qo "0o , t) propagator, which is #at in lattice Landau gauge, IJ while the "lled circles are the space}space components, which carry the dynamical information. The work of Gupta et al. [6] used the largest lattices that had been employed for lattice gauge simulations to that time, 18;42. They used b"6.2 as the lattice coupling, and employed the most powerful supercomputer that was available for lattice simulations, a CRAY at Los Alamos National Lab. The major conclusion from these analyses follow from the fact that the dynamical components of the propagator seem to fall linearly over an extended range, which, since the plots are on a semi-log scale, is the behavior expected for a massive particle. At the largest distances the lattice periodicity leads to an enhancement, clearly observable from Fig. 2.
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Fig. 1. The zero spatial momentum gluon propagator vs. lattice time, from Mandula and Ogilvie [5].
Fig. 2. The zero spatial momentum gluon propagator vs. lattice time, from Gupta et al. [6].
Another salient feature evident from the "gures is that the data clearly follow a curve which is concave downwards for small lattice time. This is quite strange behavior, because it implies that propagator's spectral function is not positive de"nite. This is easy to see. The curvature of an arbitrary positive linear combination of straight exponential ("xed mass) decays is
d ln c e\KGR " G dt G
c e\KGR G G
c m e\KGR ! c m e\KGR G G G G G G c e\KGR G G
.
(21)
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Fig. 3. The e!ective gluon mass vs. lattice time, from Bernard et al. [13].
For all positive c , this expression is positive as a consequence of the Schwartz inequality. G 5.2. The ewective gluon mass The time slice to time slice fallo! of the gluon propagator provides a natural de"nition of an e!ective gluon mass. D(q"0, t#1) m (t),!ln . D(q"0, t)
(22)
For such a de"nition to be useful requires more computational resources than were available in 1987, when the best that could be done was a global "t to determine the best average mass over the full extent in lattice time. With considerably greater computational resources, in 1993 Bernard et al. [13] and Marenzoni et al. [4] carried out simulations of the gluon propagator with su$cient precision to infer an e!ective mass value as a function of the lattice time. The results from Ref. [13] are shown in Fig. 3. Bernard et al. used 16;40 lattices at b"6.0. Evidently, the e!ective mass grows with (Euclidean) time, at least for small times where the lattice data are best. There is no conclusion to be drawn from this analysis about whether or not it levels out to a "xed asymptotic value. This is another demonstration that the propagator is not described by a positive spectral function. For any such, the e!ective mass would be a monotonically falling function of Euclidean time. 5.3. The most recent lattice results The state of gluon simulations continues to improve, as much more powerful computers have become available for such studies. Recently Leinweber et al. [29] have carried out a high statistics study simulation on a very large lattice, 32;64 sites, at b"6.0. They expressed their results in momentum space, Fig. 4.
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Fig. 4. The momentum space gluon propagator scaled by q, from Leinweber et al. [29].
The "gure gives aqD(q), the momentum-space propagator scaled by aq, for all values of the lattice 4-momentum. The factor aq is the lattice corrected momentum. The dispersion in the values for moderate q, which is well outside their statistical errors, indicates that lattice artifacts are still present. This graph displays in yet a di!erent way the fact that the gluon propagator is not describable in terms of a positive spectral function. For a free, massless particle, the graph would be #at, and for a free massive particle, or a general propagator with a positive spectral density, the graph would be monotonically increasing and concave downwards everywhere. The graph is also incompatible with the Gribov form, D(q)"q/(q#m) .
(23)
This would also give a monotonically increasing curve. Finally, if there is an anomalous dimension, it is certainly very small.
6. Conclusions: What have we learned? The essential conceptual problem about the gluon propagator is that there are no asymptotic states associated with it. In a Lehmann}KaK llen representation, all the intermediate states that contribute are non-physical, that is, they lie in gauge-variant sectors of the full Hilbert space of the quantum "eld theory. It is just the unfamiliarity of this situation that has given rise to suggestions about the analytic structure of the gluon propagator ranging from its being an entire function to its having an essential singularity at the origin. The most striking observation about the gluon propagator, one that was seen from the earliest simulations, has held up in subsequent analyses. It is that the spectral function describing the gluon propagator is not positive de"nite. Further simulations have not yet given a de"nitive picture of the propagator's structure, but they have ruled out some of the plausible suggestions, including some inspired by the earlier simulations.
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Simulations of the gluon propagator are no longer compatible with a &&complex mass'' form a% la Gribov. The point is not that one is certain that the gluon propagator is "nite at q"0, but rather that it seems that it cannot vanish as rapidly as q as qP0. The question of an anomalous dimension is still open, although a substantial one cannot be squared with the latest simulations, at least for large q. One should note though that in those simulations, the propagator falls much more steeply than 1/q for intermediate values of the momentum. A "nal puzzle is Zwanziger's theorem, speci"cally that even on the largest lattices there is no sign of the vanishing of the propagator at q"0. Here the problem may be analytic rather than computational. The statement of the theorem is that at q"0, all gluon Green's functions vanish as the lattice volume goes to in"nity. Even for an in"nite lattice, the theorem and its proof give no indication of what the rate of approach to 0 might be. It might be very weak, and might also strongly depend of the total lattice volume. If the goal of getting a de"nitive picture of the gluon propagator has not been fully realized as yet, the progress in the past 12 years has been impressively substantial.
Acknowledgements The occasion for this Symposium was the untimely death of Dick Slansky. The author wishes to express his appreciation to Fred Cooper and Geo! West for organizing this meeting to honor Dick's memory. He also wishes to thank them for encouraging this contribution to the Symposium } a review of a subject that Dick supported from its beginnings.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
S. Mandelstam, Phys. Rev. D 20 (1979) 3223. M. Baker, J.S. Ball, F. Zachariasen, Nucl. Phys. B, 186 (1981) 531, 560. N. Brown, M.R. Pennington, Phys. Lett. B 202 (1988) 257; Phys. Rev. D 38 (1988) 2266, 39 (1989) 2723. P. Marenzoni, G. Martinelli, N. Stella, M. Testa, Phys. Lett. B 318 (1993) 511. J.E. Mandula, M. Ogilvie, Phys. Lett. B 185 (1987) 127. R. Gupta, G. Guralnik, G. Kilcup, A. Patel, S. Sharpe, T. Warnock, Phys. Rev. D 36 (1987) 2813. V.N. Gribov, Nucl. Phys. B 139 (1978) 1. M. Stingl, Phys. Rev. D 34 (1986) 3863. J.R. Cudell, D.A. Ross, Nucl. Phys. B 359 (1991) 247. L. v Smekal, A. Hauck, R. Alkofer, Phys. Rev. Lett. 79 (1997) 3591; Ann. Phys. 267 (1998) 1. D. Zwanziger, Nucl. Phys. B 364 (1991) 127; Phys Lett. B 257 (1991) 168. J.M. Namislowski, 1993, unpublished. C. Bernard, C. Parrinello, A. Soni, Nucl. Phys. B (Proc. Suppl.) 30 (1993) 535. L. Giusti, M.L. Paciello, S. Petrarca, B. Taglienti, M. Testa, Phys. Lett. B 432 (1998) 196. A. Nakamura, M. Plewnia, Phys. Lett. B 255 (1991) 274. Ph. de Forcrand, J.E. Hetrick, A. Nakamura, M. Plewnia, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 194. A. Hulsebos, M.L. Laursen, J. Smit, A.J. van der Sijs, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 199. P. Coddington, A. Hey, J. Mandula, M. Ogilvie, Phys. Lett. B 197 (1987) 191. S. Petrarca, Nucl. Phys. B (Proc. Suppl.) 26 (1992) 435. H. Suman, K. Schilling, Phys. Lett. B 373 (1996) 314; Nucl. Phys. B (Proc. Suppl.) 53 (1997) 850.
284 [21] [22] [23] [24] [25] [26] [27] [28] [29]
J.E. Mandula / Physics Reports 315 (1999) 273}284 A. Cucchieri, T. Mendes, Nucl. Phys. B (Proc. Suppl.) 63 (1998) 841; A. Cucchieri, Nucl. Phys. B 508 (1997) 353. D. Zwanziger, Nucl. Phys. B 412 (1994) 657. M. Mizutani, A. Nakamura, Nucl. Phys. B (Proc. Suppl.) 34 (1994) 253. P. van Baal, Nucl. Phys. B 367 (1992) 259. P. van Baal, hep-th/9711070, 1997. G. Dell'Antonio, D. Zwanziger, Commun. Math. Phys. 138 (1991) 291. P. van Baal, Nucl. Phys. B 417 (1994) 215; hep-th/9511119 (1995). A. Cucchieri, Nucl. Phys. B 521 (1998) 365. Leinweber, Skullerud, Williams, and Parrinello, Phys. Rev D (Rapid Commun.) 58 (1998) 031501-1; heplat/9811027, 1998.
D. Bailin, A. Love/Physics Reports 315 (1999) 285}408
ORBIFOLD COMPACTIFICATIONS OF STRING THEORY
D. BAILIN , A. LOVE Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, UK Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20-0EX, UK
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Orbifold compacti"cations of string theory D. Bailin , A. Love Centre for Theoretical Physics, University of Sussex, Brighton BN1 9QJ, UK Department of Physics, Royal Holloway and Bedford New College, University of London, Egham, Surrey TW20 0EX, UK Received October 1998; editor: J.A. Bagger
Contents 1. Orbifold constructions 1.1. Introduction 1.2. Toroidal compacti"cations 1.3. Point groups and space groups 1.4. Orbifold compacti"cations 1.5. Matter content of orbifold models 1.6. Lattices 1.7. Asymmetric orbifolds 2. Orbifold model building 2.1. Introduction 2.2. Wilson lines 2.3. Modular invariance for toroidal compacti"cation 2.4. orbifold modular invariance 2.5. GSO projections 2.6. Modular invariant Z orbifold compacti"cations 2.7. Untwisted sector massless states 2.8. Twisted sector massless states 2.9. Anomalous ;(1) factors 2.10. Continuous Wilson lines 3. Yukawa couplings 3.1. Introduction 3.2. Vertex operators for orbifold compacti"cations 3.3. Space group selection rules 3.4. H-momentum conservation 3.5. Other selection rules 3.6. 3-point functions from conformal "eld theory
288 288 289 293 296 302 303 309 314 314 315 316 318 320 322 324 325 327 327 328 328 329 331 332 334 335
3.7. 3-point function for Z orbifold 3.8. B "eld backgrounds 3.9. Classical part of 4-point function from conformal "eld theory 3.10. Quantum part of the 4-point function 3.11. Factorisation of the 4-point function to 3-point functions 3.12. Yukawa couplings involving excited twisted sector states 3.13. Quark and lepton masses and mixing angles 4. KaK hler potentials and string loop threshold corrections to gauge coupling constants 4.1. Introduction 4.2. KaK hler potentials for moduli 4.3. KaK hler potentials for untwisted matter "elds 4.4. KaK hler potentials for twisted sector matter "elds 4.5. String loop threshold corrections to gauge coupling constants 4.6. Evaluation of string loop threshold corrections 4.7. Modular anomaly cancellation and threshold corrections to gauge coupling constants 4.8. Threshold corrections with reduced modular symmetry 4.9. Uni"cation of gauge coupling constants
0370-1573/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 2 6 - 4
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353 353 355 358 363 365 367
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D. Bailin, A. Love / Physics Reports 315 (1999) 285}408 5. The e!ective potential and supersymmetry breaking 5.1. Introduction 5.2. Non-perturbative superpotential due to gaugino condensate(s) 5.3. E!ective potential
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5.4. Supersymmetry breaking 5.5. Cosmological constant 5.6. A-terms and B-terms 5.7. Further considerations 6. Conclusions and outlook References
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Abstract The compacti"cation of the heterotic string theory on a six-dimensional orbifold is attractive theoretically, since it permits the full determination of the emergent four-dimensional e!ective supergravity theory, including the gauge group and matter content, the superpotential and KaK hler potential, as well as the gauge kinetic function. This review attempts to survey all of these calculations, covering the construction of orbifolds which yield (four-dimensional space}time) supersymmetry; orbifold model building, including Wilson lines, and the modular symmetries associated with orbifold compacti"cations; the calculation of the Yukawa couplings, and their connection with quark and lepton masses and mixing; the calculation of the KaK hler potential and its string loop threshold corrections; and the determination of the non-perturbative e!ective potential for the moduli arising from hidden sector gaugino condensation, and its connection with supersymmetry breaking. We conclude with a brief discussion of the relevance of weakly coupled string theory in the light of recent developments on the strongly coupled theory. 1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.-w; 12.10.-g; 12.60.Jv
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1. Orbifold constructions 1.1. Introduction It is well known that the construction of a consistent quantum string theory is possible only for speci"c dimensionalities of the (target) space}time. For the bosonic string the required dimension is D"26, while for the superstring dimension D"10 is required. Thus from the outset we are forced to consider the `compacti"cationa of the (spatial) dimensions which are surplus to the d"4 dimensions of the world that we inhabit, if we are to have any chance of connecting the string theory with experimental (particle) physics. The string theory which is best placed to generate such a connection is the heterotic string [117], a theory of closed strings, in which the right-moving degrees of freedom of the superstring are adjoined to the twenty-six left-moving degrees of freedom of the bosonic string. To endow such a construction with a geometrical interpretation sixteen of the left-movers are compacti"ed by associating them with a 16-dimensional torus, with radii of order the Planck length (l &10\ m). Just as the compacti"cation of one dimension onto a circle in the (original) . "ve-dimensional Kaluza}Klein theory [135,141] generates a gauge boson, so here the compacti"cation generates gauge "elds, including some of a stringy origin which derive from the possibility of the string winding around the torus. In this way, the 16 left-movers generate an `internala gauge symmetry with the (rank 16) gauge group E ;E being consistent with the cancellation of gauge and gravitational anomalies which is essential for a satisfactory quantum theory [113]. Although this scenario explains in a satisfying way how a gauge symmetry can emerge from string theory, there are serious problems which remain. Firstly, there is the fact that the symmetry group E ;E is far larger than the (rank 4) SU(3);SU(2);;(1) gauge symmetry which we observe. Secondly, there remains a ten-dimensional space}time, six of whose dimensions must be compacti"ed before we even contemplate questions like gauge symmetries and matter generations. The orbifolds [79,80], which are the subject of this review are one method of compactifying the unobserved six dimensions. An orbifold is obtained when a six-dimensional torus (¹) is quotiented by a discrete (`pointa) group (P), as we shall see shortly. The identi"cation of points on ¹ under the action of the point group generates a "nite number of "xed points where the orbifold is singular. At all other points the orbifold is (Riemann) #at. It is for this reason that we are able to calculate rather easily all of the parameters and functions of the emergent supergravity theory: the gauge group and matter content; the Yukawa couplings and KaK hler potential, which determine the quark and lepton masses and mixing angles; the gauge kinetic function, including string loop threshold corrections, which in turn determine the uni"cation scale of the gauge coupling constants. We shall see also how modular invariance constrains the e!ective potential, and hence determines the actual value of the coupling constants at uni"cation, as well as the nature of the supersymmetry breaking mechanism. There are, of course, other methods of string compacti"cation including Calabi}Yau manifolds [43,115,116], free fermion models [139,3], and N"2 superconformal "eld theories [107,108,140], and (some) orbifold models are connected to some of these models [138,98,13,14,24]. However, none of the alternatives has so far been as fully worked out as the orbifold theories, and it is for this reason that we have focused upon them. If for no other reason, they illustrate the sort of predictive power which we should eventually like string theory to have (even if it should transpire that nature does not in fact utilize orbifolds!)
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1.2. Toroidal compactixcations The construction of the ten-dimensional heterotic string has been fully described elsewhere (see, for example, [114,132,35]) and we need not review it here. As already noted, to have any chance of a realistic theory it is obviously essential that six of the (nine) spatial dimensions have to be compacti"ed to a su$ciently small scale as to be unobservable at current accelerators. The simplest way to do this is to compactify on a torus. This ensures that the simple linear string (wave) equations of motion are una!ected, since the torus is #at. We work in the light-cone gauge. Then there are eight transverse bosonic degrees of freedom denoted by XG(q,p) where i"1,2 labels the two transverse four-dimensional space}time coordinates, and XI(q,p) where k"3,2,8 labels the remaining six spatial degrees of freedom. (q, p with 0)p)p are the world sheet parameters.) XG and XI are split into left and right moving components in the standard manner XGI(q,p)"XGI(q!p)#XGI(q#p) . (1.1) 0 * In addition there are eight right-moving transverse fermionic degrees of freedom WG (q!p), 0 WI (q!p), and the 16 (internal) left-moving bosonic degrees of freedom X' (q#p) (I"1,2,16) 0 * which generate the E ;E gauge group of the ten-dimensional heterotic string. The (toroidal) compacti"cation of the six spatial coordinates XI(q,p) (k"3,2,8) does not a!ect the mode expansions of XG(q,p), WG (q!p), WI (q!p) or X' (q#p), so 0 0 * 1 1 i aG e\ LO\N# aG e\LO>N , (1.2) XG(q,p)"xG#pGq# n L n L 2 L$
WGI(q!p)" dGIe\ LO\N (R) 0 L P
(1.3)
or " bGIe\ PO\N (NS) (1.4) P PZ8> depending on whether the world-sheet fermion "eld obeys periodic (Ramond, R) or anti-periodic (Neveu}Schwarz, NS) boundary conditions t (q!p!p)"#t (q!p) (R) , 0 0 t (q!p!p)"!t (q!p) (NS) . 0 0 The mode expansion of the gauge degrees of freedom is i a' X' (q#p)"x' #p' (q#p)# L e\ LO>N * * * 2 n L$ with the momenta p' lying on the E ;E root lattice. * In an orthonormal basis, vectors on the E root lattice the form (n ,n ,2,n ) or (n #,n #,2,n #)
(1.5)
(1.6)
(1.7)
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with n integers and G n "0 mod 2 . (1.8) G G There is an alternative formulation of these internal degrees of freedom which replaces the 16 bosonic left movers (X') compacti"ed on the E ;E lattice by 32 real fermionic left-movers (j ,jM (A"1,2,16)) where j ,jM may separately have either periodic (R) or antiperiodic (NS) boundary conditions. Then j " j e\ LO>N (R) L L (1.9) " j e\ PO>N (NS) , P PZ8> and similarly for the second set jM . (j ,jM ) transform as the (16,1)#(1,16) representation of the maximal subgroup O(16);0(16) L E ;E . The compacti"cation of XI entails the identi"cation of the corresponding centre-of-mass coordinates xI with points which are separated by a lattice vector of the torus. Thus xI,xI#2p¸I ,
(1.10)
where the factor 2p is for convenience and the vector L with coordinates ¸I belongs to a sixdimensional lattice K
(1.11) K, r e " r 3Z , R R R R where e (t"3,2,8) are the basis vectors of the lattice. Then the closed string boundary conditions R for the coordinates XI may also be satis"ed when XI(q,p)"XI(q,0)#2p¸I
(1.12)
corresponding to the string winding around the torus. The compacti"cation also requires the quantization of the eigenvalues of the corresponding momentum operators pI. The eigenfunctions exp(i pIxI) are single-valued only if I pI¸I3Z . (1.13) I Thus, the momenta are quantized on the lattice KH which is dual to K
KH" m eH " m 3Z , R R R R where the basis vectors eH of KH satisfy eHIeI,eH ) e "d . R I R S R S RS
(1.14)
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Then the generalized mode expansions are 1 i XI (q!p)"xI #p)(q!p)# aI e\ LO\N , 0 0 0 n L 2 L$ 1 i XI (q#p)"xI #pI (q#p)# aIe\ LO>N * * * n L 2 L$
(1.15) (1.16)
with pI ,(pI!2¸I) , (1.17) 0 pI ,(pI#2¸I) , (1.18) * xI"xI #xI , 0 * where p3KH and L3K. The mass formula for the right movers in ten-dimensional heterotic string theory, which derives from the constraint equations, yields the four-dimensional mass formula m"N(b)#pI pI !a(b) , (1.19) 0 0 0 where b"R, NS labels the boundary conditions of the fermionic right-movers, and the number operators N(b) is given by N(b)"N #N (b) , D
(1.20)
with N " (aG aG #aI aI) , (1.21) \L L \L L L N (R)" (ndG dG #ndI dI) , (1.22) $ \L L \L L L N (NS)" (rbG bG #rbI bI) . (1.23) $ \P P \P P P a(b) arises from the normal ordering of the operator ¸ in the Virasoro algebra and has the values a(R)"0 , (1.24) a(NS)" . (1.25) (Sums over i"1,2 and k"3,2,8 are implied by the repeated su$xes.) Similarly, the fourdimensional mass formula for the left movers is m"NI #pI pI #p' p' !1 , * * * * * where a sum over I"1,2,16 is also implied and NI " (aG aG #aI aI#a' a') . \L L \L L \L L L
(1.26)
(1.27)
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If the fermionic formulation of the (left-moving) internal degrees of freedom is used the mass formula becomes (1.28) m(b,c)"NI (b,c)#pI pI !a (b,c) * * * when b,c"R,NS labels the (independent) boundary conditions for the two sets of real fermions j ,jM , and NI (b,c)"NI #NI (b)#NI (c) , $ $ where NI " (aG aG #aI aI) , \L L \L L L NI (R)" n(j j #jM jM ) , $ \L L \L L L NI (NS)" r(j j #jM jM ) . $ \P P \P P P Similarly the normal ordering constant a (b,c)"a #a (b)#a (c) , $ $ where
(1.29)
(1.30) (1.31) (1.32)
(1.33)
a ", a (R)"!, a (NS)" . (1.34) $ $ The mass formulae (1.19), (1.26) and (1.28) all include contributions from momenta pI ,pI in the 0 * compacti"ed manifold, which, as we have shown in Eqs. (1.17) and (1.18), are quantized. As we shall see, the lattice K and hence its dual KH generically have some arbitrary scale factors R , the lengths R of the basis vectors e , and angles between basis vectors. So, except for certain isolated values of R these parameters, massless states, in particular, only arise when momenta and winding numbers on the compact manifold are zero pI "0"pI . (1.35) 0 * In fact, the particles we observe in nature must all derive from massless string states, since otherwise their masses would be of the order of the string scale (10 GeV). We may now see why the simple toroidal compacti"cation under consideration is unacceptable for phenomenological reasons. Let us consider a massless state, so m"0"m . (1.36) * 0 Suppose we "x the (massless) left-mover state; for example, we may use one of the a operators on \ the left-movers' ground state "02 , or use momentum p' on the E ;E lattice with p' p' "2. To * * * * each such left-moving state we may attach a massless right-moving state bG "02 (i"1,2) utilizing \ 0 the NS fermionic oscillators. Since i"1, 2 corresponds to the two transverse space}time dimensions, the overall string state transforms as a space}time vector or a space}time tensor, the latter case arising only if the left-moving state is aH "02 ( j"1,2). Alternatively, we may attach the \ *
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(massless) Ramond groundstate "02 to the "xed left-moving state. This transforms as an eight0 component SO(8) chiral spinor, the opposite chirality spinor having been deleted by the GSO projection used in the superstring construction. This eight-component SO(8) chiral spinor may be decomposed into representations of SO(2);SO(6)LSO(8), the SO(2) corresponding to the two transverse space}time coordinates, and the SO(6) to the six compacti"ed coordinates. Then 8 "(#12);4#(!12)4 (1.37) * and it is clear that there are four space}time spinor particles of each chirality. Thus, if the (bosonic) string state constructed "rst was a vector particle, the fermionic state we have just constructed is four gauginos whereas if the bosonic state "rst constructed was a space}time tensor, the graviton, the fermionic state is four gravitinos. Evidently the toroidal compacti"cation under consideration leads inevitably to N"4 space}time supersymmetry, and hence to a non-chiral gauge symmetry. The observed cancellation of the gauge chiral anomaly within each generation of fermions strongly suggests (but does not conclusively prove) that the gauge symmetry is chiral, and hence that there can be at most N"1 space}time supersymmetry; N*2 supersymmetries automatically cancel chiral anomalies within each supermultiplet. 1.3. Point groups and space groups In the previous section we considered the compacti"cation of the ten-dimensional heterotic string in which the six left-movers and six right-movers XI ,XI (k"3,2,8) are compacti"ed onto 0 * the (same) torus ¹ generated by the lattice K, with the 16 left-movers X' compacti"ed on the * (self-dual) E ;E torus ¹#"#. This latter torus is generated by the root lattice of the group E ;E . A torus is de"ned by identifying points of the underlying space which di!er by a lattice vector l3C"2pK x,x#l .
(1.38)
This identi"cation is called `moddinga and in the six-dimensional toroidal case we write ¹"R/C .
(1.39)
We may generalize this process by identifying points on the torus which are related by the action of an isometry h. To be well-de"ned on the torus h must be an automorphism of the lattice, i.e. hl32pK if l32pK and preserve the scalar products he ) he "e ) e . R S R S The isometry group is called the point group (P) and an orbifold X is de"ned as X"¹/P;¹#"#/G ,
(1.40)
(1.41)
where G is the embedding of P in the gauge group E ;E . P and therefore G are discrete groups. Evidently the six-dimensional orbifold ¹/P may be obtained by identifying points of the underlying space (R) which are related by the action of the point group, up to a lattice vector l x,hx#l .
(1.42)
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We may regard the right-hand side as the action of the pair (h,l) upon the point x, and the set of all such pairs S,+(h,l) " h3P,l32pK, ,
(1.43)
de"nes a group S, the space group, with the product de"ned in the obvious way by [(h ,l )(h ,l )]x,(h ,l )[(h ,l )x] . Thus we may also write ¹/P"R/S .
(1.44)
(1.45)
The solution of the string equations propagating on an orbifold are almost as straight forward as for a toroidal compacti"cation, since the orbifold is #at almost everywhere. The exceptions are the points of the torus which are left "xed by the point group. Modding out the point group identi"es di!erent lines on the torus passing through the "xed points, so that a conical singularity occurs and the orbifold is not locally isomorphic to R at such points. It follows from Eq. (1.42) that the "xed points satisfy x "hx #l (1.46) D D so if 1!h is singular there are "xed lines or tori, rather than isolated "xed points. The full de"nition of an orbifold compacti"cation requires the speci"cation of ¹ or equivalently the lattice C, the discrete point group P, and its embedding G in the gauge degrees of freedom. The elements h 3 P act upon the bonsonic coordinates XI(q,p) (k"3,2,8) of the string as SO(6) rotations. Possible choices of P are further restricted by the phenomenological requirement to obtain an N"1 space}time supersymmetric spectrum; no supersymmetry (N"0) might also be acceptable, but the conventional wisdom is that N"1 supersymmetry is preferred because of the solution to the technical hierarchy problem which it a!ords. To get N"1 supersymmetry the point group P must be a subgroup of SU(3) [43] PLSU(3) .
(1.47)
This may be seen by recalling that SO(6) is isomorphic to SU(4), so if P satis"es Eq. (1.47) there is a covariantly constant spinor on the six-dimensional orbifold, and it is this extra symmetry which generates the required supersymmetry. For the present we restrict our attention to the cases when the point group P is abelian. Then it must belong to the Cartan subalgebra of SO(6) associated with XI (k"3,2,8). We denote the generators of this subalgebra by M ,M ,M . Then in the complex basis de"ned by (1.48) Z,(1/(2)(X#iX) , Z,(1/(2)(X#iX) ,
(1.49)
Z,(1/(2)(X#iX)
(1.50)
the point group element h acts diagonally and may be written h"exp[2pi(v M#v M#v M)]
(1.51)
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with 0)"v "(1 (i"1,2,3). The condition that Eq. (1.47) is satis"ed then gives G $v $v $v "0 (1.52) for some choice of signs; this may be seen by noting that the eigenvalues of h acting on a spinor are e p!T!T!T. The requirement that h acts crystallographically on the lattice C plus the condition (1.52) then leads to the conclusion [79,80] that P must either be Z with N"3,4,6,7,8,12 or Z ;Z with , + , N a multiple of M and N"2,3,4,6. Some of the point groups have two (inequivalent) embeddings in SO(6), i.e. they are realized by the inequivalent sets of v ,v ,v . The complete list is given in Tables 1 and 2. These results are the six-dimensional analogue of the famous result that crystals in three dimensions have only N"2,3,4,6-fold rotational symmetries, (augmented by the N"1 space}time supersymmetry requirement (1.52)). In all cases it is possible to "nd a lattice upon which P acts crystallographically, and in many cases there are several lattices for a given P. Often the massless spectrum and gauge group of the orbifold are independent of the choice of lattice, and are determined solely by P. However, we shall see in Section 2 that when the full space group, not just Table 1 Point group generators for Z L SU(3) orbifolds h"exp 2pi(v M#v M#v M) , Point group
(v , v , v )
Z Z Z !I Z !II Z Z !I Z !II Z !I Z !II
(1,1,!2) (1,1,!2) (1,1,!2) (1,2,!3) (1,2,!3) (1,2,!3) (1,3,!4) (1,4,!5) (1,5,!6)
Table 2 Point group generators for Z ;Z L SU(3) orbifolds h"exp 2pi(v M#v M#v M); u"exp 2pi(w M# + , w M#w M) Point group
(v ,v ,v )
(w ,w ,w )
Z ;Z Z ;Z Z ;Z Z ;Z Z ;Z !I Z ;Z !II Z ;Z Z ;Z
(1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1) (1,0,!1)
(0,1,!1) (0,1,!1) (0,1,!1) (0,1,!1) (0,1,!1) (1,1,!2) (0,1,!1) (0,1,!1)
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the point group, is embedded in the E ;E group then the orbifold properties, not surprisingly, do depend upon the lattice K. 1.4. Orbifold compactixcations The existence of the point group P means that there are additional ways, over and above the toroidal conditions (1.12), in which the closed string boundary conditions may be satis"ed. Let the Z point group be generated by an element h, so that the general element is hL (0)n)N!1). , (The generalization to Z ;Z generated by h,u is trivial.) Then the identi"cation (1.42) means that + , the closed string boundary conditions for the coordinates XI (k"3,2,8) may also be satis"ed when X(q,p)"(hL,l)X(q,0)"hLX(q,0)#l .
(1.53)
Evidently the `untwisteda sector (n"0) corresponds to the toroidal compacti"cation discussed in the previous section. However, there are additional `twisteda sectors, satisfying Eq. (1.53), with nO0 , and these generate new string states which were not present in the toroidal compacti"cation. Before considering these new states, however, an immediate question arises: what feature of the orbifold removes the unwanted gaugino and gravitino states which we showed are a generic feature of toroidal compacti"cations, and which are present in the untwisted sector of the orbifold compacti"cation? We have explained that the de"nition of an orbifold requires the speci"cation of a discrete group G comprising the space group S and its embedding in the gauge degrees of freedom. Thus to each element of g3G there corresponds an operator g which implements the action of g on the Hilbert space. Because the orbifold is de"ned by modding out the action of G, it follows that physical states must be invariant under G. That is to say, they are eigenvectors of g with eigenvalue unity. Now consider the four gravitino states in the untwisted sector "02 aH "02 ( j"1,2) . (1.54) 0 \ * Since j"1, 2 corresponds to the transverse space}time coordinates which are una!ected by the point group transformations, it is clear that g acts trivially on the left-moving piece of the state. The right moving piece is the Ramond sector ground state, which is an SO(8) chiral spinor. The decomposition (1.37) is given explicitly by 8 "(,,,),(,,!,!)#(!,!,!,!),(!,!,,) , (1.55) 0 where the underlining indicates that all (three) permutations are included, and the individual entries are the eigenvalues of M, M, M, M respectively. The point group generator h is given by Eq. (1.51), and we see that acting on the "rst four states its eigenvalues are hM "exp[ip(v #v #v )],exp ip(v !v !v ),exp[ip(v !v !v )],exp [ip(v !v !v )] (1.56) with the second four states having complex conjugate eigenvalues. Condition (1.52) ensures that at least one of these states have hM "1. Suppose, for example, that v #v #v "0 .
(1.57)
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(Similar arguments are easily constructed for the other possibilities.) Then the eigenvalues of the above four states are hM "1,exp(2piv ),exp(2piv ),exp(2piv ) . (1.58) So provided that v , v and v are all non-zero, the last three states all have hM O1. It follows that they are not invariant under the action of the point group, and are therefore not space-group invariant either. Thus three of the four gravitinos are deleted, as required if we are to obtain an N"1 space}time supersymmetric theory. On the other hand, if one of v is zero only two of the four gravitinos are deleted and we have at least N"2 supersymmetries surviving. It is for, this reason that Table 1 lists only point the nine group elements with v all non-zero. Similarly in Table 2 we list point group elements of the Z ;Z orbifolds which for a"1,2,3 have v and w not both zero. + , ? ? The twisted sectors of the orbifold string theory are de"ned by Eq. (1.53) with nO0. Let us consider the case of a Z orbifold and the n"1 twisted sector. The extension to n'1 and Z ;Z , , + is easily done. The "rst thing to note is that the modi"ed boundary conditions lead to a di!erent form of the various mode expansions. In this complex basis de"ned in (1.48)}(1.50), the mode expansion of the string world sheet is
1 1 i b? e\ L>T?O\N# bI ? e\ L\T?O>N (1.59) Z?"z? # D 2 n#v L>T? n!v L\T? ? ? L$ where a"1,2,3 labels the three complex planes. The fractional modings are needed to supply the phase factors exp(2piv ) acquired by Z? under the action of the point group. z? is a complex "xed ? D point, constructed from the real "xed points (1.46) analogously to (1.48)}(1.50). Evidently the full speci"cation of a twisted sector requires not only the point group element (h in this case) but also the particular "xed point (or torus) which appears in the zero mode part of the world sheet. Note too that the boundary conditions require that the momentum is zero, since h acts non-trivially in all planes; this is not necessarily the case in all twisted sectors of non-prime orbifolds. For example it is clear from Table 1 that in the h-sectors of the Z -orbifold the mode expansion of Z will have non-zero, but quantized, momentum. The complex conjugate mode expansion is 1 M 1 i bM ? ?e\ L\T?O\N# bI ? e\ L>T?O>N (1.60) ZM "z ? # L\T ? D 2 n#v L>T? n!v ? ? L$ which appear in Z? and ZM ? obey the commutation relations and operators b? ?, bI ? ?, bM ? ?, bIM L>T L\T L\T L>T? [b? ?,bM A A]"d?A(n#v )d , L>T K\T ? K>L [bI ? ?,bIM A A]"d?A(n!v )d . L\T K>T ? K>L Thus the b with n#v'0 are (proportional to) annihilation operators and the bM the L>T \L\T associated creation operators. Likewise the b with n#v(0 are creation operators and the L>T bM the associated annihilation operators. Similarly for bI and bIM . \L\T L\T L>T The point group also acts upon the right-mover fermionic degrees of freedom, so that in the h-twisted sector the boundary conditions are modi"ed: t? (q!p!p)"ep T?t? (q!p) (R) , 0 0 t? (q!p!p)"!ep T?t? (q!p) (NS) , 0 0
(1.61)
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where the complex t? (a"1,2,3) are constructed from the tI just as the Z? are de"ned in terms of 0 0 the XI (k"3,4,2,8) in (1.48)}(1.50). Thus the modi"ed mode expansions are t? (q!p)" e? ?e\ L>T?O\N (R) 0 L>T L " c? ?e\ P>T?O\N (NS) P>T P
(1.62)
and tM ? (q!p)" e ? ?e\ L\T?O\N (R) 0 L\T P " c ? ?e\ P\T?O\N (NS) , P\T P
(1.63)
where , +e? ?,e @ @,"d?@d K>L L>T K\T +c? ?,c @ ,"d?@d . (1.64) P>T Q\T@ P>Q The space group may also be embedded in the gauge degrees of freedom, and in general, it must be, as we shall see. The element (h,l) of the space group is generally mapped on to (H,V) where H is an automorphism of the E ;E lattice and V is a shift on the lattice. In this section we only address the (compulsory) embedding of the point group elements (h,0) in the gauge group. The (optional) embedding of the lattice elements (1,l), Wilson lines, is discussed in Section 2.2. It is easiest to consider "rst the embedding using the fermionic formulation of the gauge degrees of freedom. The 16 real fermions j transform as the vector representation of O(16)LE . The simplest non-trivial embedding is achieved by picking an O(6) subgroup of O(16), in which the vector representation decomposes into a (six-dimensional) vector representation of SO(6) plus (ten) SO(6) singlets. We next form 3 complex fermions from the 6 real fermions, precisely as we did for the right-moving fermions tI (k"3,2,8), and then take the action of the point group on these 0 3 complex fermions to be precisely what it is on the three complex right-moving fermions t? ; the 0 other ten-fermions are untransformed. This is called the standard embedding [80]. Evidently the mode expansions of these three complex gauge fermions will be modi"ed precisely as are those of the complex fermionic right-movers. The second set of fermions (jM ) are left completely untransformed. This embedding amounts to a shift on the E ;E lattice when we use the bosonic formulation. To see why we need the relationship t'(q#p) ": exp(2iX' ): (1.65) * between the bosonic toroidal coordinates X' and the complex fermions. Then multiplying t by 0 a phase factor exp(2pi<) amounts to adding p<' to the bosonic coordinates X' . Thus the * embedding of (h,0) on the E ;E lattice is realized as (1,p<'), and the h-twisted sector boundary conditions for the X' become * X' (q#p#p)"X' (q#p)#p<' (1.66) * *
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up to (p times an E ;E root lattice vector, and the mode expansion satisfying this is i 1 X' "x' #(p' #<')(q#p)# a'e\ LO>N . * * * 2 n L
299
(1.67)
Evidently the net e!ect of the twist h is to shift the momentum p' by <'. In the standard embedding, * which we have so far discussed, <'"(v v ,v ,0)(0) , (1.68) where v (a"1,2,3) are the twists of the 3 complex compacti"ed coordinates. ? However, we may also entertain the possibility of more general (non-standard) embeddings. Then (so far) the only constraint on the shift <' is that for a Z orbifold NV' is on the E ;E root , lattice (so that in the h,"1 sector the momenta p' #NV' are on the same lattice as the p' are): * * NV'3K . (1.69) # "# The requirement (1.52) on the v (a"1,2,3) ensures that the above constraint is always satis"ed ? by the standard embedding. In the absence of Wilson lines, the embedding of (h,0) can always be realized as a shift (1,p<') on the E ;E lattice, and sometimes this shift is also realizable on an automorphism H of the lattice. The changes in the mode expansions which we have described feed through into the calculations of the generators ¸ , ¸I of the Virasoro algebra, and in particular to changes in the expressions for K L ¸ ,¸I which lead to the mass formulae.. These now involve fractional number operators associated with the fractional-modings. The fractional modings also a!ect the calculations of the normal ordering constants. The general results are that a complex bosonic coordinate with moding shifted by v ("v"(1) contributes (1.70) a (v)" !"v"(1!"v") to the subtraction constant, while a complex Ramond fermion with moding shifted by v contributes a (v)"! #"v"(1!"v") . (1.71) $ The standard Neveu}Schwarz fermion may for these purposes be regarded as a Ramond fermion with shift v". Then a complex Neveu}Schwarz fermion with moding shifted by v contributes (1.72) a (v)"a ("v#"), !1(v( ,1 $ for "a ("v!"), (v(1 (1.73) $ The upshot of these changes is that the mass formula for the right movers in the h-twisted sector has the general structure (1.74) M"N #N (b)!a !a (b) , $ $ 0 where, as in Eq. (1.20), b"R, NS labels the (shifted) boundary conditions satis"ed by the fermionic right movers, N " aG aG # b? # b? bM ? , bM ? \L L \L>T? L\T? \L\T? L>T? L ? LL>T? ? LL\T?
(1.75)
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N (R)" ndG dG # e? # e ? , (n#v )e ? (n!v )e? $ \L L ? \L\T? L>T? ? \L>T? L\T? L L ?L>T? ? LL\T?
(1.76)
c? # c ? (1.77) N (NS)" rbG bG # (r#v )c ? (r!v )c? $ \P P ? \P\T? P>T? ? \P>T? P\T? P ?P\T? PZ8> P P ?P T? and 1 1 (1.78) a " ! "v "(1!"v ") , ? ? 3 2 ? 1 1 a (R)"! # "v "(1!"v ") , ? ? $ 3 2 ? !5 1 1 1 a (NS)" # v # 1! v # . (1.79) $ ? ? 24 2 2 2 ? (The form of a (NS) assumes that !1(v ( for all a, with the obvious change (1.73) to be made $ ? for any v satisfying (v (1.) Note that there is no momentum contribution to m, since, as ? 0 ? already observed, p is zero in a twisted sector (when all v O0). 0 ? The mass formula for the left movers in the h-twisted sector is
m"NI #(p' #<')!a , * * where NI has the same form as N in Eq. (1.75) but with all operators replaced by their left-moving analogues. The subtraction constant a 1 (1.80) a "1! "v "(1!"v ") . ? ? 2 ? (The extra compared with a derives from the 16 internal bosonic left-movers.) There is a corresponding formula for m when the fermionic formulation of the gauge degrees of freedom is * used. However we shall not quote it. We may now see why the embedding of the point group in the gauge group is compulsory. First note that the mass formula (1.74) shows that the Ramond sector ground state "02 is a (twisted 0 sector) massless right-moving state, since a #a (R)"0 (1.81) $ and, by de"nition, no oscillators are utilized. Level matching then requires that there is a massless left-moving state. Now, since NI involves fractionally moded operators, it is easy to see that its eigenvalues are also fractional. For a Z orbifold , NNI 3Z (1.82) so to obtain a massless left moving state, it follows from the mass formula (1.4) that N(<!v)32Z
(1.83)
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using Eq. (1.52) and the fact that p' and N<' are on the E ;E root lattice. This constraint is * trivially satis"ed by the standard embedding (1.68), but is not in general satis"ed by the trivial embedding (<"0). In fact, it follows from Tables 1 and 2 that of the Z orbifolds only the Z and , Z orbifolds allow the trivial embedding; none of the Z ;Z orbifolds do. It is in this sense that + , we say that the embedding of the twist in the gauge degrees of freedom is generally compulsory. Condition (1.83) is su$cient to ensure level matching in the Neveu}Schwarz sector, and at general higher levels [183,105]. In fact, it is necessary and su$cient to ensure the modular invariance of the theory, as we shall see in Section 2.3.4; modular invariance means that the one-loop toroidal amplitude does not depend on the choice of the (two) basis vectors which generate the lattice de"ning the torus. We have mentioned already that the (essential) primary virtue of orbifold models over toroidal compacti"cations is that the unwanted gravitinos (in the untwisted sector) are removed by the requirement of point group invariance. This point group invariance also reduces the gauge symmetry when the point group is embedded in the gauge degrees of freedom, as it has to be, in general. Precisely what gauge symmetry survives depends upon the details of the particular orbifold. However, we can make a general statement when the standard embedding is adopted. Then the constraint (1.47) ensures that the point group is embedded is an SU(3) subgroup of one of the E groups. Since E ME ;SU(3) (1.84) it is clear that the surviving gauge symmetry will always include E ;E . Further, the rank of the gauge group is una!ected by the embedding since the gauge bosons associated with the Cartan sub-algebra are all invariant under the action of the point group: They are given by bG "02 a' "02 . (1.85) \ 0 \ * We have already observed that the right-moving state is invariant under the action of P, and its embedding as a shift < on the E;E lattice means that the oscillators a' are also untransformed. L Thus the standard embedding gives a gauge group of at least E ;;(1);E . The `chargeda gauge bosons of E ;E are given by bG "02 "p' 2 (1.86) \ 0 * with (p' )"2, and we shall show in Section 2.7 that, when the point group is embedded as a shift * <' on the lattice, the surviving gauge bosons satisfy p' <'"0 mod 1 . (1.87) * Then, with the standard embedding, only the Z and Z orbifolds have more gauge symmetry. Z has E ;SU(3);E and Z has E ;SU(2);;(1);E . Non-standard embeddings, which embed non-trivially in both E factors, may also be con sidered. They are constrained by Eqs. (1.69) and (1.83). Then, besides the trivial embeddings (<"0) for the Z and Z orbifolds, the number of independent new embeddings ranges from three, for the Z -orbifolds, to 602 for the Z -II orbifold. Full details may be found in [124,125,102,104, 106,49,50,137]. As we have seen, the standard embedding breaks one of the E symmetries to a smaller group with the same rank, while leaving the other E unbroken. This a!ords the prospect of achieving further symmetry breaking, by Wilson lines, for example, leaving a realistic gauge
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symmetry. For this reason the broken E is called the `observablea gauge group, and the unbroken E the `hiddena gauge group. 1.5. Matter content of orbifold models We have seen that the gauge symmetry in orbifold models is determined entirely by the point group P and its embedding in the gauge degrees of freedom. In particular the six-dimensional lattice ¹ on which the orbifold is compacti"ed does not a!ect these results, so long as we do not embed the ¹ lattice vectors in the E ;E lattice. The same is true of the matter content of orbifold models: one just constructs massless, space group invariant, N"1 chiral supermultiplets in all sectors using the fractionally moded creation operators and shifted momenta appropriate to the point-group twist. It might be thought that the lattice enters via the "xed points, which we have emphasized label the di!erent twisted sectors. However, the number of "xed points (n ) under an SO(6) automorphism (h) depends only upon the automorphism, and not in the speci"c lattice. In fact, n may be calculated using the Lefschetz "xed point theorem which gives n "s(h)"det(1!h) , (1.88) where s(h) is the Euler character and h is given in the vector representation of SO(6). The matter with which we shall be primarily concerned consists of chiral supermultiplets transforming non-trivially with respect to the observable gauge group. We have seen that the standard embedding breaks the E symmetry to at least E , so the matter transforms as some representations of this group. It is easy to see the only representations which occur are the 27 and 27. First note that we can construct (scalar) E matter analogously to the gauge bosons: bI "02 "p' 2 (k"3,2,8) (1.89) \ 0 * using the compacti"ed untwisted oscillators bI , rather than the transverse space}time oscil\ lators bG . However, since the right movers transform non-trivially under the action of the point \ group, the left-movers must too. Under the decomposition E ME ;SU(3) the adjoint 248-dimensional representation of E decomposes as 248"(78,1)#(1,8)#(27,3)#(27,3) .
(1.90)
(1.91)
Thus the only matter which transforms non-trivially with respect to E and with respect to PLSU(3) is the (27,3) and (27,3). Each 27 can accommodate one generation of fermions, together with some extra matter. This can be seen using the decomposition E MSO(10) in which 27"16#10#1 .
(1.92)
(1.93)
Then the 16 accommodates the observed 15 chiral states together with an SU(3);SU(2);U(1) singlet, presumably the right-chiral neutrino state.
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For the standard embedding the net number of chiral generations is given by the formula [43,79,187] n ,n(27)!n(27)"s % 1 s(h,g) , (1.94) " 2"P" F E where "P" is the order of the point group P, and s(h,g) is the number of "xed points common to the elements g,h3P. As we have seen, this last quantity does not depend on the lattice, and may easily be calculated using Eq. (1.88). This calculation is especially easy for the prime order orbifolds Z , Z , since the "xed points of the generator h are "xed points of all hL (1)n)N!1), and then s"(1/N)(N!1) det(1!h) . (1.95) Remarkably in all abelian orbifolds n is a multiple of 12. % The orbifolds of even order all have "xed tori in some sectors. For example the Z orbifold of Table 1 has a "xed torus (the third complex plane) in the h sector. In such sectors we e!ectively have N"2 supersymmetries and there are two invariant space-time spinors with opposite helicity. Equivalently such sectors contribute 27#27 pairs to the matter content. The full determination of the matter content of Z orbifolds may be found in [137] for Z orbifolds and in [98,143] for , , Z ;Z orbifolds. It is clear that as they stand none of them has a realistic gauge group and/or + , matter content, and it is for this reason that in Section 2 we are led to study the embedding of the full space group S, not just P, in the gauge degrees of freedom. 1.6. Lattices The complete speci"cation of an orbifold requires the choice of a lattice ¹ upon which the point group P acts as an automorphism. In general there are several lattices for any given point group, but, as we saw in Section 1.4, many properties of the orbifold-compacti"ed string theory do not depend on the choice of the lattice. However, when we embed the lattice in the gauge degrees of freedom non-trivially, as we do in Section 2, then the resulting theory manifestly depends upon ¹. We consider the lattices of semi-simple Lie groups of rank 6. Inner automorphisms of such lattices are provided by the Weyl group of the algebra. It is generated by elements s whose action ? upon a vector x is to re#ect it in the simple root e : ? s (x)"x!2(x ) e )e /(e ) e ) . (1.96) ? ? ? ? ? Such re#ections are not SU(3) transformations, so the Weyl group is not contained in SU(3) and therefore cannot be the point group of any of our orbifolds. However it has some subgroups which are contained in SU(3). In particular, there is the cyclic subgroup generated by the Coxeter element [161,137,143] C,s s s s s s which satis"es C,"1 ,
(1.97)
(1.98)
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where the order N of the cyclic group is the Coxeter number. For a simple Lie algebra the Coxeter number is given by number of non-zero roots N" . rank of Lie algebra
(1.99)
It is these `Coxetera orbifolds which we shall describe. We include in this class also the cyclic subgroups of SU(3) generated by the generalized Coxeter element(s), in which one (or more) of the Weyl re#ections is replaced by an outer automorphism of the Dynkin diagram. Let us consider the rank 4 Lie algebra SO(8). The Coxeter element is C "s s s s with N"6 . (1.100) 1- The Dynkin diagram has two automorphisms: (i) s , in which two of the (outer) roots, say e e are interchanged, and (ii) s , in which the outer roots are cyclically permuted e Pe Pe Pe . (e is the central root.) Then s (x)"x/(e ) e ) , s (x)"x![(x ) e )(e !e )#(x ) e )(e !e )#(x ) e )(e !e )]/(e ) e ) (1.101) s is of order 2, and s of order 3. Thus there are two generalized Coxeter elements associated with the SO(8) algebra: "s s s s with N"8 , C 1-
(1.102) with N"12 , C "s s s 1- where the numbers in square brackets give the order of the outer automorphism used in the generalized Coxeter element. By considering products of such lattices, with Lie algebra having rank less than or equal to six we can "nd all Coxeter orbifolds. The results for the Z orbifolds are given , in Table 3. Even though we have speci"ed the lattices upon which the various point groups act, it is important to recognize that there remain a number of `deformation parametersa which are not "xed. Generically there remain some undetermined scale factors, characterizing the size of the orbifold, as well as some undetermined angles between basis vectors, the complex structure of the lattice. Under the action of the point group h a lattice vector e is transformed as e Pe "h e , h 3Z . R R SR S SR Since h is an isometry we require
(1.103)
(he ) he )"(e ) e ) R S R S so that
(1.104)
G"h2Gh
(1.105)
where G ,e ) e RS R S is the metric on the lattice.
(1.106)
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Table 3 Z Coxeter orbifolds , Point group
Lattice
Z Z Z Z Z !I Z !I Z !II Z !II Z !II Z !II Z !II Z !II Z Z !I Z !I Z !II Z !II Z !II Z !I Z !I Z !I Z !II Z !II
(SU(3)) (SU(4)) SO(5);SU(4);SU(2) (SO(5));(SU(2)) (G );SU(3) (SU(3) );SU(3) SU(6);SU(2) SO(8);SU(3) SO(7);SU(3);SU(2) G ;SU(3);(SU(2)) SU(3) ;SU(3);(SU(2)) SU(4) ;SU(3);SU(2) SU(7) SO(9);SO(5) SO(8) ;SO(5) SO(8) ;(SU(2)) SO(10);SU(2) SO(9);(SU(2)) E F ;SU(3) SO(8) ;SU(3) SO(4);F SO(8) ;(SU(2))
We have seen that the speci"cation of an orbifold includes the identi"cation of the (sixdimensional) metric of the compacti"ed space. We have also seen that besides the (symmetric) graviton and dilaton states the 10-dimensional spectrum also includes anti-symmetric tensor particles. Thus we may consider a more general situation than that which we have considered hitherto, in which there is an antisymmetric background "eld (B) besides the symmetric background metric "elds (G). The possibility of doing this may also be seen by considering a generalization of the original Polyakov action
¹ S "! dp(!h) [h?@G R XIR XJ#e?@B R XIR XJ#2] , IJ ? @ IJ ? @ 2
(1.107)
where p? (a"1,2) are the world sheet coordinates q and p, h is the world sheet metric, e the ?@ ?@ anti-symmetric two-dimensional, tensor, and G , B are the (constant) target space metric and IJ IJ antisymmetric tensor "eld. The unexhibited terms include Wilson line contributions (A' ) linking I the (ten-dimensional) string world sheet to the (16-dimensional) left-moving gauge degrees of freedom. These will be discussed in Section 2. The background "eld B is taken to be non-zero only IJ in compacti"ed dimensions. Then the new term is easily seen to be a total divergence, so the "eld
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equations and mode expansions are unaltered. Nevertheless, its presence a!ects the compacti"cation because the "eld conjugate to XI becomes P "!¹(G XQ J#B X J) , I IJ IJ where
(1.108)
XQ J,R XJ, X J,R XJ . (1.109) Using the standard mode expansion for XJ yields the momentum operator conjugate to xI as p "p #2B l¸l I I I and it is p , rather than p, which has eigenvalues on the lattice KH dual to K. The upshot is that the left and right mover mode expansions still have the form (1.15), (1.16), but now p ,p are given by 0 * pI "p I!¸I!BIl¸l , (1.110) 0 (1.111) pI "p I#¸I!BIl¸l . * with p 3KH and L3K. The full six-dimensional compacti"ed space is evidently associated with 36 quantities, 21 associated with the (symmetric) metric parameters and 15 with the antisymmetric background "eld. In most applications far fewer parameters are non-zero, since the lattice is de"ned in terms of lower dimensional constructions. Many of these use two-dimensional lattices, which are speci"ed by just four quantities G ,G ,G ,B . It is customary to combine these into two complex quantities ¹, ; de"ned as follows. The metric quantities G , are de"ned by two basis vectors whose relative size GH and orientation may be characterized by the complex number ; which speci"es the end point of the vector e , in the complex plane when e is normalized to the unit vectors lying along the real axis of the Argand diagram. Then ; is given by (1.112) i;"(1/G )(G #i(det G) and is called the `complex structurea. As it involves only ratios of terms in the metric it carries no information about the overall size of the (two-dimensional) torus. This information is supplied by the complex numbers i¹,2(B #i(det G) so that
(1.113)
det(G$B)""¹"/4 and the (square of the) imaginary part of ¹ gives the area of the fundamental torus. 1.6.1. Example: Z orbifold with standard embedding [79,80] We illustrate the foregoing generalities by applying them to the Z -orbifold, the simplest of the (symmetric) abelian orbifolds. The point group generator h satis"es h"1
(1.114)
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and its action on the compacti"ed dimensions is given by Eq. (1.51) with (1.115) (v ,v ,v )"(1,1,!2) . We have already noted that the gauge bosons arise in the untwisted sector, and are given by the states bG "02 a' "02 \ 0 \ * corresponding to the Cartan sub-algebra, and the states
(1.116)
bG "02 "p' 2 \ 0 * with (p' )"2 and p' <'3Z corresponding to the charged state of SU(3);E ;E . <' is the * * standard embedding of the point group in the gauge group and is given by Eq. (1.68). Similarly the chiral gauge non-singlet matter is given by the states c ? "02 "p' 2 , (1.117) \ 0 * where the c ? (a"1,2,3) are the untwisted fermionic oscillators in the complex basis (1.48)}(1.50). \ The right-movers are eigenstates of the operator hM , which implements the action of h on the Hilbert space, with eigenvalue hM "e\p .
(1.118)
Then the corresponding left-mover momentum states "p' 2 are those with * (1.119) (p' )"2, p' <'"mod 1 * * and it is easy to see that such states transform as the (27,3) representation of E ;SU(3). (The anti-particles have p' <'" mod 1.) Thus the untwisted sector generates a total of nine chiral * matter generations. The Z point group is realized on the lattice K which comprises three copies of the root SU(3) lattice. The SU(3) lattice has two basis vectors e ,e satis"es e ) e "e ) e "!2e ) e (1.120) Its Coxeter element is C"s s where s ,s are de"ned in Eq. (1.96). Then Ce "e , Ce "!e !e and C"1
(1.121)
(1.122)
(1.123)
as required. In this basis the matrix representing C is
C"
0
!1
1
!1
(1.124)
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so from Eq. (1.88) the number of "xed points in each plane is det(1!C)"3 .
(1.125)
It is easy to see that, up to a lattice vector, these "xed points of C are given by (1.126) x "n (e #2e ) (n "0,1,2) . D The (six-dimensional) point group generator h is de"ned as the product of the Coxeter elements associated with each of the (three) SU(3) lattices, so h has a total of 27 "xed points x "n (e #2e )#n (e #2e )#n (e #2e ) D with n "0,1,2 for each i"1,2,3. G For the twists (1.115) a and a (NS), given in Eqs. (1.78) and (1.79), both vanish $ a "0"a (NS) $ so the right movers' twisted ground state "02 has 0 m"0 . 0 Similarly, from Eq. (1.80), we "nd
(1.127)
a " so far a massless left-moving twisted state we require
(1.130)
M"NI #(p' #<')!"0 . * * The only solutions with NI "0 have
(1.131)
p' #<'" *
"
(1.128)
(1.129)
1 $10 3
(1.132)
1 1 ! ($ ) 6 2
(1.133)
2 0 , ! 3
(1.134)
where the underlining signi"es that all ("ve) permutations are to be taken, and in Eq. (1.133) an odd number of # entries is required. Evidently, the above solutions constitute 10, 16 and 1 repres entations of SO(10), and are all singlet representations of SU(3). Thus the twisted matter states with
"02 "p' #<'2 0 *
(1.135)
(1.136) (p' #<')" * transform as the (27, 1) representation of the E ;SU(3) gauge group, and in fact there is one such representation associated with each of the 27 "xed points. (The antiparticles, which transform as (27 ,1) representations, occur in the h-twisted sector associated with the same "xed point. In this respect the Z orbifold is atypical, since in general chiral matter in 27 representations may arise in any sector.)
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Including the 9 chiral generations from the untwisted sector, we get a total matter content of 36 generations in the Z orbifold. (This of course agrees with the general formula (1.95).) It is clear from the de"nition (1.120) that the SU(3) lattice has a "xed ; modulus (1.137) i;"!#i(3/2"ep while the ¹ modulus specifying the overall size of the orbifold is arbitrary. Thus in the Z orbifold all three ; moduli have the common "xed value given above, and all three ¹ moduli are unconstrained. As we have already said, these moduli are derived from the background "eld values associated with the (10-dimensional) graviton, dilaton and antisymmetric "eld. In the untwisted sector these (gauge singlet) particles are given by c ? "02 bI A "02 (a,c"1,2,3) , (1.138) \ 0 \ * where bI A (c"1,2,3) are the untwisted left-moving oscillators in the complex basis (1.48)}(1.50). Evidently the ¹ ,; (a"1,2,3) moduli "elds are associated with the diagonal (a"c) gauge singlet ? ? particles. There are also (massless) gauge singlet states in the twisted sector. They are "02 bI A "02 , 0 \ * "02 bIM A bIM B "02 (c,d"1,2,3) , 0 \ \ * and are associated with the so called `blowing up modesa (BUMs). When the background "elds associated with the BUMs are taken to in"nity, the conical orbifold singularities are `blown upa, repaired, and we are left with a Calabi}Yau manifold [80]. 1.7. Asymmetric orbifolds [162,163] The treatment of orbifolds which we have presented so far rests on the geometrical notion of compactifying six spatial coordinates on a torus and then modding out an automorphism of the associated lattice. The mode expansions for the compacti"ed left and right movers then follow from this geometrical construction. The action of the point group on the (left-moving) gauge degrees of freedom is then speci"ed, consistent with modular invariance. This symmetric treatment of the six compacti"ed spatial coordinates contrasts with the asymmetric construction of the original heterotic string. In this we "rst consider the torus (C#C), one C for each E group, turn on an appropriate anti-symmetric B-"eld, and then the left and right momenta are given by (P ,P ), where P and P each belong to the E ;E root lattice. The * 0 * 0 standard heterotic string is then obtained by restricting to momenta of the form (P ,0) and using * only left-moving oscillators to construct the states of the Hilbert space. It is natural to wonder whether this asymmetry has to be restricted to the gauge degrees of freedom or whether it can be continued further into the ten-dimensional space-time coordinates. The work of Narain [164] and collaborators [165] has shown that the combined left-right momentum gives rise to an even self-dual lattice with a Lorentzian metric. For a six-dimensional toroidal compacti"cation the signature is [(#1)>,(!1)] .
(1.139)
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The combined momenta have the form P"(P ,P ) , * 0
(1.140)
where P is a 22-dimensional vector, P is six-dimensional, and P belongs to a lattice C . We * 0 construct an orbifold by considering automorphisms of this lattice which do not necessarily treat the left- and right-moving components symmetrically. In doing this it is essential that the right and left-moving Hilbert spaces are not mixed. Then a general element g of the space group may be de"ned to act on the momentum degrees of freedom as follows: g"P ;P 2"exp[2pi(P ) a !P ) a )]"h P ;h P 2 , (1.141) * 0 * * 0 0 * * 0 0 where h and h are 22-dimensional and six-dimensional rotations, and a and a are 22* 0 * 0 dimensional and six-dimensional shifts. The action of g on the bosonic oscillators is then simply their rotation by the matrices h or h . Similarly this action on the (right-moving) NSR world sheet * 0 fermions is also given by the h rotation. Note that the action on the gauge degrees of freedom is 0 already speci"ed, as these are a part of the C lattice. The principal di$culty in constructing asymmetric orbifolds arises from the twisted sectors, i.e. string states which close only up to the action of the space group. Since the action of g is de"ned on momentum states, it does not give a sensible action on the con"guration space (x) coordinates. In particular, the "xed points of the symmetric orbifold, de"ned in Eq. (1.46), have no immediate generalization to the asymmetric case because the action of the space-group may have a di!erent number of "xed points for left- and right-moving degrees of freedom. However, we may use the requirement of modular invariance (see Section 2) to obtain information about the twisted sector before constructing it. Then it can be shown that the generalization of the Lefschetz "xed point result (1.88) is
n
"
det(1!h )det(1!h ) det(1!h) * 0" , "IH/I" "IH/I"
(1.142)
where the determinant is over eigenvalues of h"(h ,h ) which are not equal to unity; I is the * 0 subspace of lattice vectors in C which are invariant under the action of h, and IH is its dual. "IH/I" denotes the index of I in IH. It is far from obvious, but nevertheless true, that the formula (1.142) ensures that n is an integer. The number of "xed points is of course the degeneracy of the twisted sector ground state, and the formula suggests that we should "rst consider a symmetric orbifold, and somehow take the square root of the number of "xed points. To do this we "rst consider the lattice C but with a euclidean signature [(#1),(#1)], and denote it by CI to avoid confusion. Now we consider a symmetric orbifold withwindings allowed on CI and momenta on its dual. Although C is self-dual with the Lorentzian signature, CI is not self-dual because of its Euclidean signature. However, it is easy to see that if (p ,p )3CI ,
(1.143)
(p ,!p )3CI H .
(1.144)
then
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Thus we consider a lattice C_ with momenta (P ;P ) having the general form (1.110),(1.111): * 0 (1.145) P "PI !¸!B¸ , 0 (1.146) P "PI #¸!B¸ , * where the windings ¸ are on CI (1.147) ¸"(p ,p ) and the momenta PI are on its dual PI "2(p !p ) . The antisymmetric "eld B
(1.148)
(p ,p !p ;p !p ,!p ) which is generated by vectors of the form
(1.150)
maybe chosen so that if the vectors e generate the lattice CI IJ G e ) Be "e .Ge mod 2 , (1.149) G H G H where G has the Lorentzian signature. Then the momentum vectors (P ;P ) on the C lattice * 0 have the form
with
(k 0;0,!k ),(0,!k ;k ,0)
(1.151)
(k ,k )3CI . (1.152) Then, analogous to the E ;E compacti"cation, we obtain the untwisted sector of the asymmetric orbifold by restricting to momenta of the form (k 0;0,!k ) (1.153) and using only the "rst 22 left-moving oscillators and the last 6 right-moving oscillators. Now consider the twisted sector of the symmetric orbifold. As in Eq. (1.46), the "xed points x satisfy D (1!h)x "l (1.154) D so each "xed point is associated with a lattice vector l3CI . Of course, since we identify points which di!er by a lattice vector x ,x #l if l 3 CI D D x is also associated with l#(1!h)l . D Let us denote by I the subspace of CI which is left invariance by h I"+w3CI " (1!h)w"0,
(1.155)
(1.156)
Evidently the lattice vector l associated with x is orthogonal to every vector in I. Thus l is in the D subspace N of CI which is orthogonal to I. Clearly, (1!h)CI is a subspace of N, and the number of inequivalent "xed points is given by the index of (1!h)CI in N n
""N/(1!h)CI "
(1.157)
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and it can be shown [162] that this is precisely the square of n
given in Eq. (1.142): n
"(n
) . (1.158) We can associate with each lattice vector l"(l ,l )3N an untwisted state of the asymmetric orbifold having momentum P"(l ,0;0,!l ) (1.159) as in Eq. (1.153). Then the vertex operators for the emission of such states include matrices ¹. which act upon the ground states of the twisted sector. The number of inequivalent l3CI is given by Eq. (1.157), and the matrices ¹. constitute a representation of a group G with dimension n
"(n
) . (1.160) We could, of course, equally well have associated l3N with the untwisted state of a (dual) asymmetric orbifold having momentum PI "(0,!l ;l ,0) . (1.161) Then the matrices ¹.I generate a group GI isomorphic to G. In fact the "xed point set constitutes an (n
,n
) representation of G;GI , and for the symmetric orbifold (where we keep both P, PI ) we have a single irreducible representation. For the asymmetric orbifold generated by P we evidently have n
copies of the n
-dimensional representation of G. Each of these copies gives rise to identical physics, and we retain only the n
states in any single representation. This is what is meant by taking the `square root of the "xed point seta. We illustrate the foregoing ideas by constructing an asymmetric Z -orbifold which for the left-movers looks like a toroidal compacti"cation and for the right-movers looks like a Z -orbifold. We take the even self-dual lattice C to comprise C "C#C#3C ,
(1.162)
where C is the root lattice of E , and C is de"ned by C "+(p ,p ) " p ,p 3=, p !p 3R, (1.163) * 0 * 0 * 0 where R is the (two-dimensional) root lattice of SU(3) and = is its (dual) weight-lattice. Then the 22-dimensional left momentum has the form P "(p' ,p ',p ,p p ) (1.164) * * * * * * with p',p ' (I"1,2,8) the E ,E momenta and p (a"1,2,3) the left momenta on the C latti *? ces. Similarly the six-dimensional right moving momentum is P "(p ,p ,p ) (1.165) 0 0 0 0 Under the asymmetric Z -action the state "P ,P 2 transforms as * 0 "P ,P 2Pep ? 4"P ,hP 2 , (1.166) * 0 * 0 where, as in the symmetric Z -orbifold, h denotes a simultaneous rotation by 2p/3 in all three tori, and < is the standard embedding (1.68) (with v given in Eq. (1.115)) of the twist in the gauge degrees ? of freedom by means of a shift.
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The physical states in the untwisted sector are simply those in the toroidal compacti"cation which are invariant under the action of the (asymmetric) Z point group. As before, the graviton, antisymmetric tensor, and dilaton states and their N"1 space}time supersymmetric partners, are easily seen to survive, and again, as in Section 1.7, the gauge boson states bG "02 a' "02 , (1.167) \ 0 \ * bG "02 "p' 2 (1.168) \ 0 * * with (p' )"2 and p' <'3Z corresponding to the gauge group SU(3);E ;E also survive. * * However, because the action of the point group on the left-movers is now toroidal, additional vectors states survive bG "02 a? "02 , \ 0 \ * bG "02 "p ,p ,p 2 , \ 0 * * *
(1.169) (1.170)
with p #p #p "2 (1.171) * * * and these generate a further SU(3) gauge symmetry. The untwisted states (5.96) also survive and are of course singlets with respect to the (new) SU(3) gauge symmetry. More interesting things happen in the twisted sector. First we construct the Euclidean lattice CI "C #C #3CI . There the invariant lattice I is given by
(1.172)
I"C #C #3(R,0) , (1.173) where as before R is the root lattice of SU(3), and in the same notation the normal lattice is N"3(0,R) .
(1.174)
Since the action of the point group on the left-movers is toroidal (1!h)CI "N
(1.175)
it follows from Eqs. (1.157) and (1.158) that n
"1 . (1.176) (For the symmetric Z orbifold it will be recalled that there are 27 "xed points.) In fact [162], there is a single matter "eld in the E ;SU(3) representation (27,3,1,1,1)#(27,1,3,1,1)#(27,1,3,1,1)#[(1,3,3,3,1)#(1,3,3,3,1) # (1,3,3,3,1)#(1,3,3,3,1)#perms] ,
(1.177)
where `permsa indicates the representations needed to make the last bracket symmetric with respect to the last three SU(3)s. As for the symmetric Z orbifold, the h twisted sector gives the antiparticles of the h twisted sector. Other examples of asymmetric orbifold compacti"cation may be found in Refs. [110,180,85,146,147].
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2. Orbifold model building 2.1. Introduction As we have seen in Section 1, the observable gauge group of an orbifold compacti"ed string theory is quite large e.g. E ;SU(3) for the Z orbifold with the standard embedding of the point group. It is therefore necessary to "nd mechanisms to break the gauge group to that of the standard model. The usual mechanism in an SU(5),SO(10) or E grand uni"ed theory is to employ Higgs bosons to spontaneously break the grand uni"ed group. However, this requires the presence in the theory of massless scalar states in the adjoint representation or some larger representation of the gauge group. In a supersymmetric grand uni"ed theory not derived from string theory, we can introduce any representations of the gauge group we require at will. On the other hand, in a grand uni"ed theory derived from string theory, the spectrum of massless states is prescribed by the string theory for any speci"c compacti"cation. Although by sifting through consistent orbifold compacti"cations we can "nd a range of possibilities for the massless spectrum, this range is not in general wide enough to permit the presence of adjoint or larger representations, as we now discuss. The largest representations of the gauge group that can occur in a string theory are controlled by the level of the Kac}Moody algebra [111] (or current algebra) for the left movers, which is de"ned as follows. The vertex operator N,
z "e\O\ N
(2.2)
with q and p the Wick rotated world sheet variables. In general, J satis"es the operator product ? expansion J (z)J (w)&kM d (z!w)\#if J (z!w)\#2 (2.3) ? @ ?@ ?@A A with f the structure constants of the gauge group. The level k of the Kac}Moody algebra (or ?@A current algebra of the currents J ) is a non-negative integer de"ned by ? k"2kM /t , (2.4) where t is the highest root of the Lie algebra. In particular, for simply laced groups with normalisation t"1, (2.5) kM "k . The states of the string theory not only fall into representations of the Lie algebra of the gauge group but also into representations of the Kac}Moody algebra [111]. In practice, we are interested in unitary representations of the Lie algebra with a mass spectrum that is bounded below. For these representations, there is a bound on the highest weights of the representations of the Lie algebra that can occur, namely, % n m )k , G G G
(2.6)
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where n are the Dynkin labels of the highest weight of the representation, and m are positive G G integers that are "xed for a given Lie algebra G, and can be found tabulated in various places. For level 1 (k"1) the representations of the Lie algebra that can occur in string theory are very limited. In particular, for SO(10) or SU(5), the adjoint or larger representations do not occur [84]. This means that the usual spontaneous symmetry breaking mechanisms for breaking the symmetries in SO(10) or SU(5) grand uni"ed theory can not be used in string theory with level 1 Kac}Moody algebras. It is possible [152] to use theories with Kac}Moody algebras with level greater than 1, but then a plethora of large exotic representations of the grand uni"ed group occurs [99] for which it is di$cult to generate large masses to remove them from a low energy theory. It is therefore attractive to stick with level 1 Kac}Moody algebras and to look instead for another mechanism to achieve some preliminary breaking of the gauge group, before spontaneous symmetry breaking, using the available smaller representations of the gauge group is applied. Such a mechanism exists in the form of Wilson lines, which we shall discuss in the next section. 2.2. Wilson lines In Section 1, the point group was embedded in the gauge group in order to achieve some breaking of the gauge group [79], and, in the case of Z orbifolds other than Z and Z to ensure , a modular invariant theory. Further breaking of the gauge group can be achieved (in a modular invariant way) by embedding the complete space group in the gauge group [80,123,15]. This means that not only should the point group element be embedded as a shift on the E ;E bosonic degrees of freedom, but also the various basis vectors of the torus lattice underlying the orbifold should be embedded as such shifts. As we shall see, not only does this produce gauge symmetry breaking but it also modi"es the matter "eld content, so that 3 generation models can be obtained [123,16,17]. Consider a twisted sector with boundary conditions twisted by the space group element (h,l), where h is a point group element with l as a lattice vector, l" r e , (2.7) M M where r are some integral coe$cients and e are basis vectors of the 6 torus. To embed the space M M group in the gauge group, the point group element h will be embedded as the shift p<', as before, and the lattice basis vector e as the shift pa' . To ensure that we have an embedding we must check M M that we obtain a homomorphism. Thus, we must correctly image the product of two space group elements (h ,l ) and (h ,l ), (h ,l )(h ,l )"(h h ,l #h l ) . (2.8) For a Z point group generated by h, , (h,l),"(I,o) . (2.9) Consequently, we must require that N(<'#r a' )3K M M # "# which implies that N<'3K # "#
(2.10) (2.11)
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and N a' 3K , M # "# so that
(2.12)
N <'"0 mod 2 'Z#
N <'"0 mod 2, 'Z#
(2.13)
and N a' "0 mod 2, N a' "0 mod 2 . (2.14) M M 'Z# 'Z# In addition, the embedding of the space group must be chosen in such a way that the fundamental modular invariance property of the theory is preserved. The way to ensure a modular invariant theory is the subject of the next two sections. 2.3. Modular invariance for toroidal compactixcation In the "rst instance, the evaluation of a string loop amplitude, such as Fig. 1, involves a path integral over world sheet metrics as well as over the bosonic and fermionic string degrees of freedom. The essential subtlety of the one loop string amplitudes for present purposes is contained in the toroidal world sheet of the vacuum to vacuum amplitude of Fig. 2. In"nities may arise in evaluating this amplitude (and other 1 loop amplitudes) unless we are careful to avoid including the contribution of equivalent world sheet tori in"nitely many times. Tori may be characterised by the modular parameter q, which is de"ned as follows. First construct the complex variable z"p#iq
(2.15)
from the world sheet coordinates p and q. Then a world sheet torus may be de"ned by making the identi"cations z,z#p(n j #n j ) ,
(2.16)
Fig. 1. The one-loop string amplitude. Fig. 2. The vacuum-to-vacuum string amplitude.
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where j and j are two "xed complex numbers, and n and n are arbitrary integers. Points on the torus may be written as (2.17) z"p j #p , 0)p , p (p . H Because conformal invariance may be applied to rescale j to 1 if we wish, it is only the ratio q"j /j (2.18) that is relevant for characterising tori. Not all values of q specify inequivalent tori. If we consider the modular transformations
(2.19)
ad!bc"1 ,
(2.20)
j a b j " , j c d j where a,b,c and d are integers satisfying
then n j #n j "n j #n j , (2.21) where n and n are also arbitrary integers. Thus, j and j de"ne the same torus as j and j when the identi"cation (2.16) is made. The corresponding transformations on q q "(aq#b)/(cq#d)
(2.22)
constitute the (world sheet) modular group SL(2,Z), and tori whose modular parameters are related by Eq. (2.22) are equivalent. In"nities in the vacuum-to-vacuum amplitude (and other one-loop string amplitudes) may now be avoided by restricting the path integral over world sheet metrics to the range !)Re q(, Im q*0, "q"*1 (2.23) which ensures that inequivalent tori are counted only once. For this to work, it is necessary that the q dependent path integral over the bosonic and fermionic string degrees of freedom for the vacuum-to-vacuum amplitude should be invariant under the modular transformations (2.22). This path integral is referred to as the partition function Z and after converting the Euclidean path integral to a determinant it is given by Z"Tr(q&*q &0)
(2.24)
where the Hamiltonian has been written in terms of left and right mover contributions H and * H as 0 H"H #H (2.25) * 0 and q"e pO ,
q "e\ pO .
(2.26)
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2.4. Orbifold modular invariance It will be convenient for the moment to use the fermionic formulation of the heterotic string to study the modular invariance of the orbifold partition function [80,183]. For the space group element (h,l), let the twists on the boundary conditions of the 3 complex right moving fermionic degrees of freedom associated with the compact manifold be ep TG, i"1,2,3, and let the twists on the boundary conditions of the 16 complex left moving fermionic degrees of freedom associated with the E ;E gauge group be ep T ', I"1,2,16. These latter twists include the e!ect of embedding the lattice vectors e as well as the point group element h, i.e. they include the Wilson M lines. From Section 2.2, after switching from the bosonic to the fermionic formulation of the heterotic string we must have v '"<'#r a' (2.27) M M for a Z orbifold. , The orbifold partition function will be a sum over terms corresponding to the various choices of twisted boundary conditions in the p and p directions on the torus. For example, for a left moving complex fermionic degree of freedom with boundary conditions twisted by h"ep U and g"ep S in the p and p directions, respectively, the generalisation of Eq. (2.24) to an orbifold is (2.28) ZU"Tr(q&*Uep S\ ,$U) , S where H (w) is the left-mover Hamiltonian for boundary conditions twisted by ep U and N (w) is * $ the fermion number (see, for example, Ref. [35, Section 11.2]). Evaluation of the trace gives
ZU"e\p S\Uh S
u
w
,q ,
(2.29)
where
,q "q\U\U> ep S\U (1!qL\Uep S)(1!qL>U\e\p S) . (2.30) w L For the purpose of studying the way in which partition function terms transform under modular transformations it is useful to note that the Jacobi h function of Eq. (2.29) has the modular property h
u
h a
u
,q "e h ? w
u
w
,a\q ,
(2.31)
where a:qP(aq#b)/(cq#d) , a
u
ab
u
cd
w
"
w
,
(2.32) (2.33)
and e is a 12th root of unity independent of u and w. Also useful are the shift properties ? u ,q (2.34) h((S>),q)"!e\p Uh U w
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and
h((S ),q)"!ep Sh U>
u
w
,q .
(2.35)
A partition function term is a product of fermionic and bosonic factors for both right and left movers, but the phases arising from the modular transformation of the boundary conditions of the bosonic factors cancel between right and left movers. Consequently, only the fermionic factors need be considered for present purposes. A modular transformations (2.31) has the e!ect on the boundary conditions, (h,g)P(h ,g ) where (h ,g )"(hBgA,h@g?) .
(2.36)
If we consider a modular transformation that leaves the boundary conditions unaltered then, in order that the partition function can be uniquely de"ned, we must require that partition function terms, for given boundary conditions, transform into themselves without any modi"cation, whereas, potentially, a phase factor could arise. In particular, if we consider the boundary conditions (h,g)"(h,I)
(2.37)
and the modular transformation qPq#N ,
(2.38)
where h is of order N, then (h ,g ) is the same as (h,g). The corresponding partition function factor for a left moving complex fermionic degree of freedom with boundary conditions twisted by h"ep U in the p direction undergoes the modular transformation ZUPep ,U\UZU . (2.39) Similarly, for a right mover partition function factor ZM U the corresponding transformation under the same modular transformation is ZM U( Pe\p ,U( \U( ZM U . For a partition function term
(2.40)
(2.41) Z" ZM TG ZT ' , G\ ' where vG are the twists on the right moving complex fermionic degrees of freedom, and v ' are the twists on the left-moving E ;E complex fermionic degrees of freedom, the transformation induced by the modular transformation (2.38) is
ZPexp !piN vG(1!vG)! (1!v ') Z . (2.42) G ' Thus, to ensure that this partition function term transforms identically to itself, without any phase factor, we must require that
N vG(1!vG)! v '(1!v ') "0 mod 2 . G '
(2.43)
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The homomorphism condition (2.10), together with the requirement N vG"0 mod 2 G for the action of the point group to be of order N acting on the spinor representation of SO(8), allow Eq. (2.43) to be simpli"ed to
N (vG)! (v ') "0 mod 2 , (2.44) G ' with v ' given by Eq. (2.27) for the h twisted sector of the orbifold with Wilson lines. For the hL twisted sector,
(2.45) N n (vG)! (n<'#r a' ) "0 mod 2 M M G ' with n"0,2,N!1 and r "0,2,N!1. In particular, embeddings of the point group in the M gauge group consistent with modular invariance are required to satisfy
(<')! (vG) ,N(<!v)"0 mod 2 , ' G and Wilson lines consistent with modular invariance are required to satisfy N
N (a' ),Na"0 mod 2 , M M ' N a' a' ,Na ) a "0 mod 1, oOp M N M N '
(2.46)
(2.47) (2.48)
and N <'a' ,N< ) a "0 mod 1 . M M '\ These results may be extended to Z ;Z orbifolds. + ,
(2.49)
2.5. GSO projections As well as modular invariance imposing restrictions on the choice of point group embeddings and Wilson lines, it also imposes (generalised) GSO projections on the states [18,124,171]. For a Z orbifold with point group generated by h, the complete partition function has the form , 1 (2.50) Z" g(m,n)Z K L , F F N K L where Z K L is the partition function for twists hK and hL in the p and p directions, respectively, F F and g(m,n) are phase factors "xed by modular invariance of the complete partition function Z, and determined by considering modular transformations that map one term in the sum (2.50) into another.
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In the absence of Wilson lines, the contribution to (2.50) for boundary conditions twisted by hK in the p direction is 1 ,\ (2.51) Z" s(hK,hL)Tr(DLq&*FKq &0FK)#2 , N L where D"ep B
(2.52)
d"(h#mv( ) ) v( !(P#m<) ) <#(m/2)(<!v( )#e .
(2.53)
with
In Eq. (2.53), h is the so called H momentum for the bosonised right moving NSR fermionic degrees of freedom, P is the momentum on the E ;E lattice, v( ,(0,v,v,v) (2.54) describes the action of the point group on the compact manifold, and ep C is the action of h on the left mover oscillators involved in the construction of the state. (We shall only be interested in massless states, in which case these are no right mover oscillators.) The factor s(hK,hL) as de"ned to be 1 for the untwisted sector hK"I and, otherwise, it is the number of simultaneous "xed points of hK and hL on the subspace rotated by hK. (This last remark is necessary to take account of the possibility of hK possessing "xed tori.) The GSO projection deriving from Eq. (2.51) is particularly simple in the case of the h twisted sector (m"1) and in the absence of "xed tori in the h twisted sector. Then, s(h,hL)"s(h) for all n ,
(2.55)
where s(h) is the number of "xed points of h, all of which must be "xed by hL. Thus, Eq. (2.49) simpli"es to
,\ 1 (2.56) Z" s(h)Tr DLq&*Fq &0F N L and states with D"1 survive the GSO projection. It turns out that all massless states in the h sector have D"1, so that all massless states in this sector survive. More generally, [124,142,100] the "xed points of hK and hL di!er, and s(hK,hL) does not have the same value for all n. This prevents us pulling out the s(hK,hL) factor from the summation to leave a simple GSO projection. Instead, it is necessary to evaluate the degeneracy factor in the partition function. 1 ,\ (2.57) D(hK)" s(hK,hL)DL N L and states for which D(hK) is zero are projected out. In the presence of Wilson lines, Eq. (2.51) still applies if DL is replaced by DI (n,m) where DI (n,m)"ep BI L K
(2.58)
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with dI (n,m)"(h#mv( ) ) nv( !(P#m<#r "(mn/2)v( #ne .
a ) ) (n<#r a )#(m<#r a ) ) (n<#r a ) M K M M L M M K M M L M (2.59)
In Eq. (2.59), the space group elements associated with "xed points in the hK and hL twisted sectors have been written as (hK,r e #(1!hK)K) and (hL,r e #(1!hL)K). The degeneracy factor is M K M M L M then 1 ,\ (2.60) D(hK)" s(hK,hL,r )DI (n,m) . M L N M L L P For a given "xed point in the hK twisted sector (a choice of r ) the degeneracy factor now has M K separate terms for each "xed point in the hL twisted sector (each choice of r ). The factor M L s(hK,hL,r ) now counts the number of simultaneous "xed points of hK and hL associated with M L a particular choice of r . For example, for the h twisted sector of the Z orbifold, with one Wilson M L line a , the 27 "xed points split into 3 sets of 9 associated with <#r a ,r "0, $1. With 2 Wilson lines a and a , the 27 "xed points split into 9 sets of 3 associated with <#r a #r a , r ,r "0,$1. 2.6. Modular invariant Z orbifold compactixcations The simplest case [80,123] in which to illustrate the way in which modular invariance restricts the consistent choices of point group embeddings and Wilson lines is the Z orbifold. In that case, the inequivalent choices of the point group embedding <' may be determined as follows. First write, <"(< ,< ) , (2.61) where < and < are the components of < shifting the E and E lattices, respectively. Two shifts < and < that di!er by an E lattice vector are equivalent, as are two shifts that di!er by a Weyl re#ection of the E lattice, and similarly for < . The homomorphism conditions are 3 <( "0 mod 2 (
(2.62)
and 3 <)"0 mod 2 . ) For the Z orbifold, v"(,,) so that the modular invariance condition (2.46) is 3(<#<)"0 mod 2 .
(2.63)
(2.64)
(2.65)
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Combining Eqs. (2.62)}(2.64), requires <"q , <"q , (2.66) where q and q are integers. Any shift < , i"1,2, is within a distance 1 from some lattice point on an E lattice. Thus, by G subtracting o! an appropriate lattice vector we can always arrange that <)1, <)1 . Then, up to interchanging < and < the only inequivalent possibilities are <"<"0 , <"0 , <", <", <" , <"<" and
(2.67)
(2.68) (2.69) (2.70) (2.71)
<", <" . (2.72) There is a large range of choices for the Wilson lines a . As for < we have the homomorphism M conditions 3 a("0 mod 2 M (Z#
(2.73)
and 3 a)"0 mod 2 . M )Z# Also, by subtracting o! appropriate lattice vectors we can arrange that
(2.74)
(a())1, (a)))1 . (2.75) M M (Z# )Z# On the other hand, we can no longer use Weyl re#ections to reduce the possibilities further because equivalent theories are connected by Weyl re#ections on < and a simultaneously. However, not all M Wilson lines satisfying (2.73)}(2.75) and the modular invariance conditions 3a"0 mod 2 , M
(2.76)
and 3< ) a "0 mod 2 (2.77) M are independent. If the action of the point group element h on the basis vectors for the compact manifold lattice is he "M e , M MN N
(2.78)
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then a "M a #K , (2.79) M MN N where K is an E ;E lattice vector, re#ecting the fact that these are inequivalent paths on the torus that are equivalent on the orbifold. One approach [48] to writing down all possible models is to list all possible choices of a(,J3E , M and then to use the modular invariance conditions on a to limit the possible choices of a), K3E , M M consistent with the choice of a(. There are various other transformations on the Wilson lines, and M the <' and Wilson lines together, that give equivalent models. Phenomenologically promising models can then be selected by imposing requirements such as standard model gauge group, 3 generations and absence of extra colour triplets which may mediate rapid proton decay [125,49,50,102,103,100]. 2.7. Untwisted sector massless states Only initially massless states rather than states with masses on the string scale are directly relevant to the low energy world. It will be convenient to bosonise the NSR right mover fermionic degrees of freedom. Then, the 8 real fermions or 4 complex fermions become 4 real bosons with momentum on an SO(8) lattice. Denote the momentum components on the SO(8) lattice (the so-called H momentum) by hG, i"0,1,2,3. Then, the formulae for massless states of Section 1 become M"M"0 , 0 * where 1 1 1 M"N# (hG)! 2 2 4 0 G
(2.80)
(2.81)
and 1 1 (2.82) M"NI !1# (P') , * 2 4 ' where P' is the E ;E lattice momentum. Now N contains only the contribution of transverse bosonic oscillators, and hG"1 mod 2 (2.83) G because of the GSO projection. As discussed in Section 1, the untwisted sector massless states include the gauge "elds with NS sector right movers bG "02 , i"1,2, created from the vacuum by space}time fermionic oscil\ 0 lators. In the case of the untwisted sector, the generalised GSO projections are equivalent to straightforward space group invariance without any phase factors. In the bosonic formulation, the space group element (h,l) with l the linear combination of lattice basis vectors e M l"r e (2.84) M M
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induces the translation on the E ;E lattice p(<'#r a' ), so that the action on a state with M M momentum P' is exp(2ip(<#r a ) ) P). Since bG does not transform under the space group, M M \ space-group invariance requires P ) <"0 mod 1
(2.85)
and P ) a "0 mod 1 for all o (2.86) M for the gauge "elds. These conditions result in breaking of the original E ;E gauge group. For the Z orbifold, a complete classi"cation has been given with 1, 2 or 3 Wilson lines (the maximum independent number.) When 3 Wilson lines are deployed, examples of models with SU(3);SU(2); ;(1)N gauge group can be obtained. (The breaking of the extra ;(1) factors not required by the standard model will be discussed later.) Left chiral right movers for matter "elds transform as 3 of SU(3) contained in 4 "3 #1 when the SO(8) spinor Ramond sector ground state is decomposed as 4#4 of the compact manifold SO(6). Thus, the left chiral right movers transform with a phase factor e\p under h. Consequently, the condition (2.85) for space group invariance is modi"ed for (2.87) P ) <" mod 1 , for left chiral matter "elds, and the condition (2.86) is unaltered. The surviving matter "eld content can be adjusted by adjusting the choice of Wilson lines. 2.8. Twisted sector massless states In general, additional massless matter occurs in the twisted sectors of the orbifold. For the h twisted sector of a Z orbifold, let the twists on the boundary conditions of the NSR fermions , associated with the compact manifold be ep TG, i"1,2,3. Then, in the hL twisted sector, the shift on the boundary conditions of the bosonised NSR fermions is nv( , with v( as in Eq. (2.54), so that the H momentum is replaced by h#nv( . Also, the shift on the E ;E degrees of freedom, including possible Wilson lines, is p(n<'#r a' ) so that the E ;E lattice momentum P' is replaced by M M P'#n<'#r a' . M M With the bosonic formulation of the heterotic string and the bosonised version of the NSR fermions, the only twisted boundary conditions are for the right and left-moving compact manifold bosonic degrees of freedom. Thus, the modi"cation to the normal ordering constant is vG(1!vG), for both right and left movers. Then, the formulae for massless states become G M"M"0 , (2.88) 0 * where
and
1 1 M"N# (hG#nv G)!a 0 4 2 G
(2.89)
1 1 M"NI # (P'#n<'#r a' )!a M M 4 * 2 '
(2.90)
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with 1 1 a" ! vG(1!vG) 2 2 G
(2.91)
and 1 a "1! vG(1!vG) . (2.92) 2 G The oscillator terms N and NI now contain contributions from fractionally moded bosonic oscillators associated with the compact manifold. For a Z ;Z orbifold, generated by point group elements h and u, for the hIul twisted sector, + , nv G must be replaced by kv G#luG where v G and uG are the twists for h and u, respectively. Also n<'#r a' must be replaced by k<'#l='#r a' , where <' and =' are the embeddings in the M M M M gauge group of h and u. Naive space group invariance can not be applied in the twisted sectors, and the surviving states are those allowed by the generalised GSO projections [18,124,171]. In the case of the prime order orbifolds (Z and Z ) all twisted sector massless states survive these GSO projections, for arbitrary embeddings of the point group and arbitrary Wilson lines [18,124]. By a careful choice of Wilson lines, Z orbifold models with 3 generations of quarks and leptons and SU(3);SU(2);;N(1) observable gauge group can be obtained [125,49,50,102,103,100]. This outcome depends crucially on the fact that Wilson lines di!erentiate the various "xed points [123,16,17], so that some "xed points have associated quark and lepton generations and others do not. Generically, additional massless matter with exotic gauge quantum numbers occurs [11]. This unwelcome matter will have to be removed from the observable low energy world, possibly by con"nement due to non-trivial quantum numbers under a large non-abelian factor in the hidden sector gauge group [86]. Large non-abelian factors in the hidden sector gauge group have the additional virtue of providing the gaugino condensates necessary for non-perturbative supergravity, as will be discussed in Section 5. A simple Z orbifold example with a single Wilson line may be obtained by taking 112 0 (0) (2.93) <'" 333 and 112 a' "a' "a' " 0 (0) . (2.94) 333
In this case, the observable sector gauge group is [SU(3)] where the "rst three SU(3) factors may be interpreted as SU (3);SU (3);SU (3). The untwisted sector massless matter "elds are 9 copies ! * 0 of (1,3,3 ,1) of [SU(3)]. The Wilson line di!erentiates the twisted sector "xed points so that the representations of [SU(3)] arising are 9 copies of (1,3 ,3,1)#(3,3,1,1)#(3 ,1,3 ,1) from the twisted sectors with r "0 mod 3, 9 copies of (3,3 ,1,1)#(1,3,1,3 )#(3 ,1,1,3)#3(1,1,3,1) from the twisted MM sectors with r "1 mod 3, and 9 copies of (3 ,1,3,1)#(1,1,3 ,3 )#(3,1,1,3)#3(1,3 ,1,1) from the MM twisted sectors with r "2 mod 3. MM With the de"nition of the electric charge Q "¹*#¹0#> #> * 0
(2.95)
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the twisted sectors with r "0 mod 3 contain 9 generations of quarks and leptons, together with MM associated states to make up 9 copies of the 27 of E . However, the other twisted sectors contain only exotic massless matter which can form fractionally charged colour singlet states. In this example, not all exotic matter can be con"ned by the extra SU(3) factor in the gauge group. A complete classi"cation of models in the absence of Wilson lines, their gauge groups and massless matter content, has been carried out for all Z orbifolds [137]. Potentially realistic models , with Wilson lines producing standard model gauge group and 3 generations of quarks and leptons have been obtained in the cases of Z as just discussed and Z orbifolds [51] though a complete classi"cation has not been carried out. It is worth noting that there is never any need to adjust the theory to be free of gauge (and gravitational) anomalies due to chiral fermions. Freedom from such anomalies comes as an automatic consequence of the modular invariance of the string theory [172]. 2.9. Anomalous ;(1) factors In the "rst instance, model building leads to theories with SU(3);SU(2);;N(1) gauge group with p'1. To reach the standard model, it is necessary for all but one of the ;(1) factors in the (observable) gauge group to be broken at a large scale. Frequently, one of the ;(1) factors is anomalous [76,12,75] with an anomaly arising from diagrams with 3 non-abelian gauge bosons, or one ;(1) gauge boson and two gravitons, as external legs. Then, at string one loop order a Fayet-Iliopoulos D-term is generated for this ;(1) factor, ; (1), and the corresponding D-term, D , in the Lagrangian takes the form g qG # qG " " , (2.96) D " G 192p G G whereas, for a non-anomalous ;(1), say ; (1), D " qG " " , (2.97) G G where qG and qG are the corresponding ;(1) charges of the scalar "elds . Since, in general, these G
carry not only the anomalous ;(1) charge but also other ;(1) charges, many ;(1) factors may be G broken in this way [52,104]. As a consequence of selection rules on the Yukawa couplings and non-renormalizable couplings in an orbifold theory (which we shall discuss later) the e!ective potential often possesses F #at directions. Then, spontaneous symmetry breaking may occur along such a direction, with ;(1) factors in the gauge group being broken at a very large scale. 2.10. Continuous Wilson lines The discussion of Wilson lines so far has assumed that the point group is embedded in the gauge group as a shift <' on the bosonic E ;E degrees of freedom. An alternative is to embed the point group in the gauge group as a discrete rotation [126,124] of the E ;E lattice, still in the bosonic formulation of the heterotic string. Then, the space group element (h,l), with l a lattice vector as in
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Eq. (2.7), is embedded as (h ,a) where h is a rotation and a'"p r a' (2.98) M M M is a shift, on the E ;E degrees of freedom. For those components of a' that are rotated by h , no M restriction is imposed on them by the homomorphism condition because, for these components (h ,a' ),"(I,0) , (2.99) M when h is of order N. The components of a' that are not rotated by h are restricted by Eq. (2.14), as M usual. A priori, modular invariance might put conditions on the rotated components of a' . However, M this does not happen, and would not be expected to happen, because these components of the Wilson lines do not a!ect the mass operator for the twisted sector, and so do not a!ect level matching between left and right movers. Thus, the components of a' that are rotated by h are continuously M variable parameters (additional moduli) which are referred to as continuous Wilson lines. Unlike the usual discrete Wilson lines, continuous Wilson lines are able to reduce the rank of the gauge group. The gauge "elds associated with the Cartan subalgebra are of the form bG "02 a' "02 , I"1,2,16, where i is a four dimensional space-time index. While h acts trivally \ 0 \ * on bG ,h has a non-trivial action on some of the left-mover bosonic oscillators a' . As \ \ a consequence of point group invariance, some of the gauge "elds of the Cartan subalgebra are projected out of the theory, leaving only that part of the Cartan subalgebra for while a' is \ unrotated by h . This is not the whole story because it is possible for there to be h invariant combinations of E ;E momentum states "P'2, of the form "P'2#"h P'2#2"(h ),\P'2, in the Z case, which play the part of Cartan subalgebra states. However, the GSO projections due to , Wilson lines generically project out some of the states, so that the rank of the gauge group is indeed reduced. When the point group embedding in E ;E is a rotation h rather than a shift, twisted sector states consistent with the boundary conditions will have to have centre-of-mass coordinates at a "xed point (or torus) of h , as well as at a "xed point (or torus) of h. If E' are basis vectors for the E ;E lattice, then the "xed points X' will be of the form D X' "[(I!h )\(a#t E )]' , (2.100) D where t are integers, and the form of a depends on the "xed point on the compact manifold according to Eqs. (2.98) and (2.7). Because X' has only left-moving components, we are dealing here with an asymmetric orbifold. Consequently, the vacuum degeneracy for the twisted sector due to the E ;E degrees of freedom is the square root of the number of "xed points determined in this way.
3. Yukawa couplings 3.1. Introduction In this section, we shall discuss superpotential terms, focussing on the trilinear terms that give rise to Yukawa couplings. Before a Yukawa coupling is fully determined it is necessary to normalise
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correctly the "elds involved. This requires a knowledge of the KaK hler potential which will be deferred to Section 4. Quite a lot can be learned about the Yukawa couplings in an orbifold compacti"ed theory using the various selection rules which will be presented in the next few sections. In later sections, the detailed construction of Yukawa couplings will be discussed. The most important aspect of this is the leading exponential dependence of couplings amongst twisted sector states on the deformation parameters or moduli of the orbifold. This gives a possible starting point for understanding the hierarchy of quark and lepton masses, and the chapter closes with a discussion of progress to date in "tting the quark and lepton masses using orbifold compacti"cations. 3.2. Vertex operators for orbifold compactixcations The vertex operators for untwisted sector states will be described "rst before the modi"cations necessary for twisted sector vertex operators are given. So far as the right movers are concerned, the vertex operator <0 in the !1 picture for emission of a scalar boson with four-dimensional space \ time momentum p is of the form (3.1) <0 "e\(e N 60tG . 0 \ This vertex operator is for a boson with right mover bG "02 where i"1,2,3 are complex basis \ 0 indices for the three complex planes of the compact manifold, and is the phase of the bosonised superconformal ghost "eld. The corresponding 0 picture vertex operator <0, which is required for superpotential terms with more than three chiral "elds, is of the form
RXG p <0"e N 60 2iz 0# t tG Rz 2 0 0
(3.2)
with z as in Eq. (2.2). The vertex operator <0 in the !1/2 picture for the emission of a fermionic \ state is given by (3.3) <0 "e\( e N 60S , \ where S is the spin "eld. It will often be convenient in what follows to bosonise the complex NSR world sheet fermionic degrees of freedom in the form e &K, m"1,2,5. In terms of the world sheet bosonic degrees of freedom H the vertex operator (3.3) takes the form K 0 (3.4) <0 "e\( e N 6 e F & , \ where (3.5) h"a "($,2,$) with an even number of plus signs to satisfy the constraint on the ten-dimensional chirality of the state due to the GSO projection. The vertex operators for boson emission can also be recast in terms of H-momentum h by bosonising the NSR fermions so that <0 "e\(e N 60e F & , \ where, in this case, for a scalar boson h"a "(0,0$1,0,0) , T
(3.6) (3.7)
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where underlining denotes permutations, and
RXG p <0"e N 60 2iz 0# t e ?T & . Rz 2 0
(3.8)
So far as the left movers are concerned, states with left movers of the type aH "02 , where j is \ * a compact manifold index in the complex basis, have vertex operators R < "2iz XH e N 6* * Rz *
(3.9)
with z as in Eq. (2.2). This includes the moduli discussed in Section 1.6 Gauge "elds in the Cartan subalgebra have left mover vertex operators of the same type, but with compact manifold index j replaced by an E ;E index I. Gauge "elds not in the Cartan subalgebra, and also massless matter "elds with non-trivial E ;E quantum numbers, have vertex operators associated with momentum P' on the E ;E lattice (satisfying (P')"2 for the untwisted sector) ' ' ' (3.10) < "e N 6*e . 6* . * For twisted sector vertex operators some modi"cations are required. The vertex operator <"< < (3.11) 0 * must now contain a product of twist "elds that construct the twisted sector vacuum from the untwisted vacuum. The product of twist "elds for the right moving NSR fermions is analogous to the spin "eld and is given by e LJ( & for the hL twisted sector, where v( describes the action of the point group on the compact manifold, as in (2.54). Then the h momentum in Eqs. (3.4) and (3.6) is replaced by h"a #nv( Q
(3.12)
or h"a #nv( (3.13) J respectively. The bosonic E ;E degrees of freedom are untwisted (except in models with continuous Wilson lines) but the momentum P' is shifted by the embedding of the point group and the Wilson lines so that P' is replaced in Eq. (3.10) by P'#n<'#r a' as in Section 2.8. M M It is di$cult to give useful explicit expressions for the twist "elds pG for the bosonic degrees of freedom, to be discussed later, but in practice what is usually su$cient is a knowledge of the operator product expansions involving these twist "elds which will be given in Section 3.6. The ! and !1 picture vertex operators then contain a factor ppp for the twist "elds associated with the three complex planes of the compact manifold. The 0 picture vertex operator is more complicated and contains excited twist "elds. Tree level correlation functions (involving untwisted or twisted sector states) have to be constructed to cancel the ghost charge 2 of the vacuum (where eO( has ghost charge q.) A U term in the superpotential may be extracted from a Yukawa coupling of the form t t , for which we need a 3-point function of the type 1< (z ,z )< (z ,z )< (z ,z )2, and a UL> superpotential \ \ \
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term may be extracted from a non-renormalisable coupling of the form t t L> for which we need an n#3 point function of the type 1< (z ,z )< (z ,z )< (z ,z )< (z z )2< (z ,z )2 . \ \ \ L> L> 3.3. Space group selection rules For a non-zero correlation function, the product of space group elements associated with the twisted sector states involved should contain the identity element of the space group [78,120]. In particular, consider a 3-point function with the three states associated with the space group elements (a,l ),(b,l ) and (c,l ) where a,b and c are point group elements and l "(I!a) f #(I!a)K , ? ? l "(I!b) f #(I!b)K , (3.14) @ @ l "(I!c) f #(I!c)K , A A where f , f and f are "xed points in the a, b, and c twisted sectors, respectively, and K , K and ? @ A ? @ K are arbitrary lattice vectors. Then, A (a,l )(b,l )(c,l )"(abc, l #al #abl ) . (3.15) For the identity element of the space group to be included, there is the requirement that abc"I
(3.16)
which is the point group selection rule, and the additional requirement (which we shall sometimes refer to as the space group selection rule) that l #al #abl "0 (3.17) for some choice of K , K and K . This can be simpli"ed with the aid of the point group selection ? @ A rule to l #l #l "0 (3.18) for some choice of K ,K and K . In other words, ? @ A (I!a) f #(I!b) f #(I!c) f "0 (3.19) ? @ A up to the addition of (I!a)K , (I!b)K or (I!c)K . This restricts the "xed points which can ? @ A couple. A simple example is provided by the Z orbifold. As we saw earlier, the "xed points f for the h twisted sector are given by 1 (2e #e ) f" p H\ H\ H 3 H for integers p ,p and p , with associated space group elements (h,l) where l"(I!h) f#(I!h)K"p e #p e #p e #(I!h)K .
(3.20)
(3.21)
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If we consider a coupling of three states, each in the h twisted sector, associated with "xed points f , f and f characterised by integers p, p and p, o"1,3,5, respectively, then the point group M M M selection rule is trivially satis"ed and the space group selection rule gives p("0 mod 3, o"1,3,5 . (3.22) M ( For non-prime order orbifolds [124,142,100,53], the discussion is a little more complicated. In this case, as we saw earlier, the "xed points of hK are not necessarily the same as those of hL, for mOn, and when constructing physical states we have to take linear combinations of "xed points to get an eigenstate of h. If f is a "xed point of hI and n is the smallest integer such that hKf &f (up to I I I a lattice vector) then we have to make the linear combinations
with
K\ "p2" e\ AP "hPf 2 , I P
(3.23)
2pp c" , p"0,1,2,m!1 , m
(3.24)
which have eigenvalues e A of h. A subset of these survive the GSO projection. Then, a 3-point function will couple three states of the form (3.23). It can then be seen that, if the space group selection rule is satis"ed by the states " f 2, " f 2 and " f 2, it is satis"ed by these linear combinations. 3.4. H-momentum conservation When the NSR right-moving fermionic degrees of freedom are bosonised, as discussed in Section 3.2, there is a conserved H-momentum associated with these bosonic degrees of freedom [78,120,65,104,142]. In the untwisted sector, spin 0 bosonic states in the NS sector have Hmomentum h given by Eq. (3.7), and for super"eld of a particular chirality we may "x attention on T (3.25) h "(0 0 100) . T The 10-dimensional space}time N"1 supercharge carries H-momentum h!"($1,G1,1,1,1)/2 , / so that the superpartners of these bosonic states have H-momentum
(3.26)
h!"h !h!"(G,$, ,!,!) , (3.27) Q T / where again the underlining denotes permutations. For a 3-point coupling of two fermions and one boson, conservation of H-momentum means that or
h#h!#h8"0 T Q Q
h("(1,1,1) , T ( where we are now displaying only the (non-zero) compact manifold components.
(3.28)
(3.29)
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Untwisted sector bosons in chiral supermultiplets have right movers bG "02 where i refers to \ 0 the complex basis and corresponding H-momenta (100), (010) and (001) as in (3.25). If we denote the bosons with i associated with the three complex planes by ; , ; and ; , then the only coupling allowed by H-momentum conservation is ; ; ; . For a Z orbifold, spin 0 bosons in the hL twisted sector have H-momentum of the form , (3.30) h "(0,0 1,0,0)#nv( T for super"elds of a particular chirality, with v( as in Eq. (2.54). (For the hI ul twisted sector of a Z ;Z oribifold this becomes kl( #lu( .) The fermionic superpartners have H-momentum + , h!"h !h! (3.31) Q T / and the H-momentum conservation law for 3-point couplings remains (3.29) but with the modi"ed form of h(. T For n#3 point couplings all but three of the vertex operators are in the zero picture. The picture changing operation is implemented by the superconformal current for the compact manifold degrees of freedom ¹ "2iz(R XG tM G #R tM G tG ) (3.32) $ X 0 0 X 0 0 which adds H-momentum (!100) when picture changing <0 with compact manifold index \ i"1,2 or 3 in the complex basis. If we write a "(1,0,0), a "(0,1,0), a "(001) , (3.33) then the H-momentum conservation law for a vertex with two fermions and n#1 bosons as in Section 3.2 is L> L> h(! a("(1,1,1) . T G ( (
(3.34)
Table 4 H-momenta for massless spin 0 bosons for the twisted sectors of the Z ;Z orbifold Twisted sector
Notation
h for massless states T
h u hu h u hu hu Untwisted Untwisted Untwisted
A B D AM BM C CM ; ; ;
(1,0,2) (0,1,2) (1,1,1) (2,0,1) (0,2,1) (1,2,0) (2,1,0) (1,0,0) (0,1,0) (0,0,1)
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A simple illustration of the application of H momentum conservation to couplings involving twisted sector states is provided by the Z ;Z orbifold. In this case, the point group elements h and u are represented by v( "(1,0,!1)
(3.35)
and u( "(0,1,!1) , respectively, and the H-momentum for spin 0 bosons in the hIul twisted sector as
(3.36)
h "(100)#kv( #lu( . (3.37) T The H-momenta for the massless states in the various twisted sectors are given in Table 4, where we continue to suppress the zero entries of the H-momentum. The Yukawa couplings consistent with the point group selection rule that are also allowed by H-momentum conservation are ; ; ; ,AAM ; ,BBM ; ,CCM ; DDD,AM BC,ABM CM ,ACD,BCM D,AM BM D .
(3.38)
3.5. Other selection rules For a Z orbifold, the element h generating the point group can be written in the complex space , basis (corresponding to (1/(2) (X#iX), etc.) as (ep T,ep T,ep T). These three-phase rotations of the complex coordinates for the compact manifold are automorphisms of the lattice and are thus symmetries of the 6-torus that are left unbroken by the construction of the orbifold [78,120,65,104]. We shall refer to this symmetry as point group invariance to distinguish it from the topological point group selection rule discussed in Section 3.3. If the action of the phase rotation on the ith complex plane is of order M the correlation functions involving (R XM G)K(R XM G)L(R XG)N(R XM G)O X U S T are allowed by point group invariance only if m#p!n!q"0 mod M .
(3.39)
For a 3-point function, where the bosonic vertex operators are all in the !1 picture, the fact that there are no bosonic oscillators involved in the construction of massless right movers means that Eq. (3.39) simpli"es to m!n"0 mod M .
(3.40)
This then restricts the allowed Yukawa couplings of massless twisted sector states for which bosonic oscillators act on the left mover ground state (excited twisted sector states.) For a 4 or more point function, even with no bosonic oscillators involved in the construction of the right mover state, derivatives of bosonic degrees of freedom can arise from the construction of zero picture vertex operators and the full expression (3.39) is required. We shall discuss in Section 3 one further selection rule where derivatives of bosonic degrees of freedom at the same "xed point are involved.
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3.6. 3-point functions from conformal xeld theory The dependence of superpotential terms for Yukawa couplings upon the moduli or deformation parameters can be calculated by studying a 3-point function for vertex operators using conformal "eld theory methods [78,120]. However, the determination of the overall normalisation of the superpotential term requires the factorisation of a 4-point function into 3-point functions and this will be discussed in later sections. Of course, there is also the question of the correct normalisation of the "elds involved using their kinetic terms. This requires a knowledge of the KaK hler potential and discussion of this will be delayed to Section 4. Schematically, we are interested in 1< <$<$2 where < denotes a bosonic vertex operator and <$ denotes a fermionic vertex operator. The factors involving e N 60, e N 6*, e . 6* and e F & can be evaluated straightforwardly. The di$cult part is the expectation value of the product of twist "elds which, for twisted sector ground states, is a product of factors of the type (z z )2 , (3.41) ZG,1pG G ?(z ,z )pGlG @(z ,z )pG G lG I , D , D \I > , DA where i"1,2,3 labels the three complex planes of the 6 torus for the compact manifold and pG is a twist "eld referring to that complex plane. (The case of twisted sector excited states will be discussed in Section 3.12.) The twists k /N, l /N and !(k #l )/N are for that complex plane and G G G G f , f and f are the "xed points involved. The twist "elds are de"ned to construct the twisted sector ? @ A ground state, denoted by "p 2 from the untwisted ground state "02, so that I , "p 2"p (0,0)"02 . (3.42) I , I , The twisted sector boundary conditions for the ith complex plane are of the form XG(q,p#p)"e\p IG ,XG(q,p)
(3.43)
and similarly for the other twists, or equivalently, after continuation to Euclidean space, XG(ep z,e p z )"ep IG ,XG(z,z ) .
(3.44)
We shall mostly suppress display of the "xed points in what follows and shall often suppress the index i labelling the "xed plane. The expectation value ZG factors enter a quantum piece ZG and a classical piece ZG , so that ZG"ZG ZG , (3.45) where (3.46) ZG " exp(!SG ) , G 6 where XG are the solutions for the classical "eld and the action SG continued for Euclidean space is 1 RXG RXM G RXG RXM G SG" dz # (3.47) p Rz Rz Rz Rz
with z and z as in Eq. (2.2). The string equations of motion RXG/RzRz "0
(3.48)
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require that RXG/Rz and RXG/Rz are functions of z and z alone, respectively, which must be chosen to respect the boundary conditions implicit in the operator product expansions (OPEs) with the twist "elds. These OPEs may be deduced from the mode expansions of the string degrees of freedom. If we write i cR i b L\I , z\L\I ,! L>I , zL>I L , X "x # 0 0 2 (n!k/N) (n#k/N) 2 L L where b and c are oscillator modes, then for zP0, RX RX i " 0+! cR zI ,\ Rz 2 I , Rz
(3.49)
(3.50)
dropping the annihilation operator term, and so RX RX i "p 2" p (0,0)"02&! zI ,\cR "p 2 . I , I , Rz I , Rz I , 2
(3.51)
Thus, R Xp (0,0)&z\\I ,q (0,0) , (3.52) X I , I , where q (0,0) acting on the untwisted ground state creates an excited state of the twisted sector. I , Restoring the z and z dependence of the conformal "elds, we conclude that the relevant OPEs are RX p (w,w )&(z!w)\\I ,q (w,w ) , I , Rz I , RXM p (w,w )&(z!w)\I ,q (w,w ) , I , Rz I , RX p (w,w )&(z !w )\I ,q (w,w ) , I , Rz I , RXM p (w,w )&(z !w )\\I ,q (w,w ) , I , Rz I ,
(3.53)
where q,q ,q and q are four varieties of excited twist "elds. For p , k/N is replaced by 1!k/N in \I , these expressions. The classical solutions of the string equations of motion (3.48) with the correct boundary conditions in the presence of the twist "elds as zPz , z and z are of the form l l RX /Rz"a(z!z )\\I ,(z!z )\\ ,(z!z )\I ,> , , l RXM /Rz "a (z !z )\\I ,(z !z )\\ ,(z !z )\I ,>l , , l RX /Rz "b(z !z )\I ,(z !z )\ ,(z !z )\\I ,\l , , l (3.54) RXM /Rz"bM (z!z )\I ,(z!z )\ ,(z!z )\\I ,\l , ,
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where a and b are constants to be determined, and we are continuing to suppress the index i labelling the complex plane. The constants a and b in Eq. (3.54) are determined by certain integrations round closed contours referred to as global monodromy conditions. In practice, only one of RX /Rz and RX /Rz , say RX /Rz, is an acceptable classical solution, because the other gives a divergent classical action. Then, we may set b to zero. Integrating round a closed contour C such that X is shifted by an amount v but not rotated we have
dz(RX /Rz)"v . (3.55) ! For example, if we choose C to go l times round z counterclockwise and k times round z clockwise, then X is unrotated. To "nd the shift v we have to multiply together the (powers of ) space group elements associated with the "xed points. If we write a for the point group element with action e\p I , in this complex plane and b for the point group element with action e\p l ,, then what we need is (a,(I!a) f #(I!a)K)l(b\,(I!b\) f #(I!b\)K)I @ ? l " (I,(I!a )( f !f #K)) ? @ " (I,v) ,
(3.56)
where, in each case, K is an arbitrary lattice vector. Strictly, if the complex plane in question is, for example, the 1#i2 plane, then we need the component of (1!al)( f !f #K), on the 1#i2 ? @ direction to give v for X> . The integral (3.55) now determines the constant a as follows. For convenience, take z "0, z "1, z "z "R , using SL(2,C) symmetry. Then, using the integral
! leads to
dz z\\I ,W(z!1)\\l ,W"!2i sin(klpy)
C((k#l)/N)v i(!z )I>l , a" . 2 sin(klp/N)C(k/N)C(l/N)
(3.57) C(ky)C(ly) C((k#l)y)
(3.58)
(3.59)
Consequently, after performing the dz, using Appendix A of [118], Z (with the index i sup pressed) is given by Eq. (3.46) with "v" "sin(kp/N)""sin(lp/N)" S " 4p sin(klp/N) "sin((k#l)p/N)"
(3.60)
with v as in Eq. (3.56) and the sum over X reducing to a sum over the choices of v parameterised by the arbitrary lattice vector K. When there are two independent twists k and l involved [42] there are two distinct global monodromy conditions [177,88] which can be obtained by encircling the pairs of points z and z and z and z in turn. This leads to two di!erent expressions for S . Consistency between these
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two expansions has to be achieved by restricting the sum over the initially arbitrary lattice vectors K occurring in the expression for v. When the two twists k and l are identical, there is only a single independent global monodromy condition and this subtlety does not arise. Another subtlety is that if one of the twisted sectors involved has a "xed plane then the factor ZG in the 3-point function corresponding to this complex plane reduces to a two twist "eld correction function and can be normalised to 1. 3.7. 3-point function for Z orbifold The ideas of the previous section can be illustrated and the result made explicit so far as the moduli and "xed point dependence is concerned by considering the coupling of three states each in the h twisted sector of the Z orbifold. This coupling is allowed by the point group selection rule and also by H-momentum conservation because it is analogous to the DDD coupling for the Z ;Z orbifold discussed earlier. The space group selection rule (3.22) speci"es the "xed point with which the third h twisted sector state must be associated given the "xed points for the "rst 2 states. The twists for the three complex planes are l 2 k G" G" , i"1,2,3 . 3 3 3
(3.61)
It then follows from Eq. (3.60) that "v " SG " G , 2p(3
i"1,2,3
(3.62)
and thus
!1 "v " , (3.63) Z & exp G 2p(3 G T where v is the component of v in the ith complex plane (e.g. v is the component of v in the 1#i2 G direction.) Also, from Eq. (3.56) v"(I!h)( f !f #K) . For the Z orbifold, with f of the form given in Eq. (3.20), v takes the form v" d (e #e )#(I!h)K , H\ H\ H H where d
"p !p H\ H\ H\ and the integers p( , J"1,2, take the values H\ p( "0,$1 . H\
(3.64)
(3.65)
(3.66)
(3.67)
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In addition, K is an arbitrary linear combination with integral coe$cients of the basis vectors e , o"1,2,6, and using the action of the point group on the basic vectors we can write M v" [(d #2k !k )e #(d #k #k )e ] (3.68) H\ H\ H H\ H\ H\ H H H with k and k arbitrary integers. H\ H The orbifold possesses deformation parameters or moduli which are continuously variable quantities corresponding to radii and angles de"ning the underlying torus. These parameters can be absorbed into the de"nition of the basis vectors e . The most general lattice basis compatible M with the point group is obtained by requiring that all scalar products e ) e are preserved by the M N point group action. Here, we restrict attention to the moduli R ""e """e " . (3.69) H\ H\ H The angles h de"ned in terms of the scalar products e ) e are "xed at 2p/3 for H\ H H\ H compatibility with the point group. The other six angles are also moduli and the dependence of the Yukawa couplings on these can be found elsewhere [53,144]. The orthonormal space basis g , r"1,2,6, in which the point group element h has the action P
h(g #ig )"ep (g #ig ) (3.70) H\ H H\ H for j"1,2,3, is related to the lattice basis e , o"1,2,6 by M #i sin pg ) . (3.71) e "R g , e "R (cos pg H H\ H\ H\ H H\ H\ Consequently, the component of v in the direction g #ig corresponding to the jth complex H\ H plane is v "(d #2k !k )R #(d #k #k )R cos p H H\ H\ H H\ H\ H\ H H\ # i(d #k #k )R sin p . H\ H\ H H\ Thus, in Eq. (3.63)
(3.72)
"v ""[(d #2k !k )#(d #k #k )(2k !k )]R . (3.73) G G\ G\ G G\ G\ G G G\ G\ To "nd the leading term in the Yukawa for large values of R we need to minimise the G\ coe$cient of R in Eq. (3.73) with k and k arbitrary integers, and, as a consequence of Eq. G\ G G\ (3.67) d
"0,$1,$2 . G\ The result is
(3.74)
"v " "0 for d "0 G +', G\ (3.75) "R for d "$1,$2 . G\ G\ In this approximation, the Yukawa coupling between three h twisted sectors takes the form
1 "v " Z &exp ! 2p(3 G G +',
(3.76)
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with the moduli and "xed-point-independent constant of proportionality to be "xed by Z . It will be noticed that the size of the Yukawa coupling is controlled by the `distancesa d between the G\ "xed points on the 3 complex planes. Such calculations have been carried out for all Z and , Z ;Z orbifolds [78,120,42,53,177,88,144,22,19,178,20,21]. + , 3.8. B xeld backgrounds If the components on the compact manifold of the anti-symmetric tensor "eld always present in heterotic string theory develop vacuum expectation values then this B "eld background a!ects the Yukawa couplings [89]. The B "eld background is included through the term in the action
1 p (3.77) dp dq e?@B R XPR XQ , S "! P1 ? @ 2p where the indices r and s refer to the real space basis. In the complex basis and after Wick rotation,
iB RXG RXM G RXG RXM G RXG RXM G RXG RXM G 1 # ! G\ G dz ! , S" dz Rz Rz p Rz Rz Rz Rz Rz Rz p
(3.78)
where we have retained only the background B "elds with both indices in a single complex plane. Assuming, as before, that RXG /Rz gives a divergent classical action and should be dropped, we obtain
!(1!iB )"v ""sin(k p/N)""sin(l p/N)" G\ G G G G (3.79) Z " exp(!S )" exp 4psin((k l p/N)) "sin((k l )p/N)" G G G G> G TG 6 as the generalisation of Eq. (3.60). In the case of the Z orbifold, R is replaced in Eq. (3.73) by (1!iB ) R . This can be G\ G\ G G\ written in terms of the moduli ¹ as follows. In general, for the ith complex plane, G (3.80) i¹ "2(b #i((det g) )(2p)\ , G G\ G G where, in terms of the basis vectors e for the lattice of the 6 torus, ? g "e ) e , (3.81) ?@ ? @ b "eP B eQ (3.82) ?@ ? PQ @ and the determinant refers to the 2;2 matrix for the ith complex plane. For the Z orbifold, the (deformed) lattice basis vectors are e "(1,0)R , e "(cos p,sin p)R and similarly for the other complex planes with R and R replacing R . Thus, , ((det g) "((3/2)R G\ G b "((3/2)R B , G\ G G\ G\ G (1!iB )R "(¹ /(3)(2p) . G\ G G\ G
(3.83)
(3.84) (3.85) (3.86)
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The e!ect of the B "eld background on Z , and so on the Yukawa coupling, is therefore to replace R by ¹ (2p)/(3 so that the Yukawa couplings are functions of the ¹ moduli. G\ G 3.9. Classical part of 4-point function from conformal xeld theory The determination of the moduli independent normalisation of a Yukawa coupling requires the factorisation of a 4-point function into 3-point functions. The di$cult part is the expectation value of a product of 4 twist "elds and (for twisted sector ground states) we are interested in factors ZG for the ith complex plane of the form ZG "1pG G (z z )pG G (z ,z )pG lG (z ,z );pGlG (z ,z )2 . , D \I , D G I , D \ , D This can be written as a product of a classical and a quantum part as
(3.87)
ZG "(ZG ) (ZG ) ,
(3.88)
(ZG ) " exp(!SG ) , 6G
(3.89)
with
RXG RXM G RXG RXM G 1 # SG " dz p Rz Rz Rz Rz
(3.90)
and the classical solutions now in the presence of four twist "elds. The solutions of the string equations of motion with correct boundary conditions as zPz , z , z and z in the presence of the twist "elds are of the form RX/Rz"au l (z) , I , , (z ) , RXM /Rz "a u I , l , (z ) , RX/Rz "bu \I , \l , (z) , RXM /Rz"bM u \I , \l ,
(3.91)
where (3.92) u l (z)"(z!z )\I ,(z!z )I ,\(z!z )\l ,(z!z )l ,\ , I , , where we have suppressed the index i. For the case k"l there are two independent contours for global monodromy conditions [177,88] to "x a and b which can be taken to be C and C of Fig. 3. For kOl, there are three independent contours and, much as for the 3-point function, this results in a restriction on the sum over initially arbitrary lattice vectors in the "nal expressions. We shall focus on k"l. If v is the H shift in X in going round the contour C then H RX RX dz# dz , (3.93) v" H Rz Rz !H !H where
v "(I!hI)( f !f #K)
(3.94)
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Fig. 3. Independent contours for global monodromy conditions for the 4-point function. In general, the points z , z and z may be encircled more than once.
and v "(I!hI)( f !f #K) (3.95) with projection of the shifts onto the appropriate complex planes. Carrying out the contour integrals and solving for a and b leads to 1 +"v "#"q""v "#iq (ep I ,v v !e\p I ,v v ), , pS " 4q sin(kp/N) where SL(2,C) symmetry has been used to set z "0, z "x, z "1, z "z "R , the quantity q is de"ned by iF(1!x) q"q #iq " F(x)
(3.96)
(3.97)
(3.98)
and F(x) is the hypergeometric function F(x),F
k k ;1! ;1;x . N N
(3.99)
The real and imaginary parts of q are thus given by i (FM (x )F(1!x)!F(x)FM (1!x )) q " 2 "F"
(3.100)
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and q "I(x,x )/2"F(x)" where
(3.101)
I(x,x ),F(x)FM (1!x )#F(x)F(1!x) .
(3.102)
3.10. Quantum part of the 4-point function The quantum part of Z is determined with the aid of a di!erential equation for its dependence on the variable z derived using the stress tensor method [78]. (The variables z , z and z can be "xed to constant values using SL(2,C) symmetry as in (3.97).) This method relies on the operator product expansion (OPE) of the stress tensor ¹(z) with the twist "eld p , namely, I , h p (w,w ) R p (w,w ) # U I , #2 , (3.103) ¹(z)p (w,w )& I , I , I , (z!w) (z!w) where h
1k k " 1! I , 2N N
(3.104)
is the conformal dimension of the twist "eld. The stress tensor is the normal ordered product ¹(z)"!:R XR XM : (3.105) X X and h can be identi"ed by considering the expectation value of ¹(z) between twisted sector I , ground states "p 2 and "p 2. The method also relies on the OPE I , \I , 1 1 ! R X(z)R XM (w)&¹(z)# #2 . (3.106) U 2 X (z!w) As usual, the index i labelling the particular complex plane of the 6 torus is being omitted. The OPE (3.103) implies that
h (z!z ) 1¹(z)p (z )p (z )2! I , R ln(Z ) " lim (z )p (z )p . X \I , I , \l , l , (z!z ) Z X X Moreover, the OPE (3.106) implies that
1 1¹(z)p (z )p (z )p l (z )pl (z )2/Z "lim g(z,w;z )! , \I , I , \ , , H (z!w) X U where g(z,w;z )"1!R X(z)R XM (w)p (z )p (z )2/Z . (z )p (z )p H X U \I , I , \l , l ,
(3.107)
(3.108)
(3.109)
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Thus, R ln(Z ) can be derived if we can calculate g(z,w;z ). Because of the OPEs of R X and R XM , X H X U with the twist "elds and the OPE of R X with R XM , g has the behaviour for z,wPz ,2,z and X U zPw, g(z,w;z ) &(z!w)\#finite, zPw G &(z!z )\I ,, zPz l zPz &(z!z )\ ,, &(z!z )\\I ,, zPz zPz &(z!z )\\l ,, &(w!z )\\I ,, wPz wPz &(w!z )\\l ,, &(w!z )\I ,, wPz wPz . &(w!z )\l ,, In terms of the holomorphic function de"ned in Eq. (3.92), g is "xed to be of the form
P(z,w) #A(z ,z ) , (w) g(z,w;z )"u l (z)u H H \I , \l , H I , , (z!w)
(3.110)
(3.111)
where P(z,w) is a polynomial quadratic in z and w separately, P(z,w)" a wGzH . (3.112) GH G H The coe$cients a are determined by requiring that there is no simple pole for zPw and that the GH numerator of the double pole is 1. This "xes all coe$cients a except for a , a and a , for which GH there are only two equations. This freedom corresponds to the freedom to absorb the constant part of P(z,w)/(z!w) into A. It is convenient to "x all coe$cients a before calculating A, without loss GH of generality. Specialising to the case k"l, a convenient choice is
k k a " z z # 1! z z . N N
(3.113)
Then, k P(z,w)" (z!z )(z!z )(w!z )(w!z ) N
k # 1! (z!z )(z!z )(w!z )(w!z ) . N
(3.114)
Using SL(2,C) symmetry to set z "0, z "x, z "1, z "z "R ,
(3.115)
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the expression (3.107) for the derivative of ln(Z ) now reduces to k k 1 1 AI (x,x ) R ln(Z ) "! 1! ! ! , V N N x (1!x) x(1!x)
345
(3.116)
where AI (x,x )" lim (!z )\A(0,x,1,z ) . (3.117) X Before (Z ) can be calculated it remains to determine A. The global monodromy conditions for the quantum part of X using the same two contours C G (Fig. 3) as were used for the classical part of X give
D GX " !
! and consequently
RX dz# Rz G
RX dz "0 Rz G !
(3.118)
dz g(z,w)# dz h(z ,w)"0 , (3.119) !G !G where an auxiliary correlation function h(z ,w;z ) is de"ned by G h(z ,w;z )"1!R X(z,z )R XM (w,w )p (z )p (z )p l (z )pl (z )2/Z . (3.120) G X U \I , I , \ , , The most general form of h consistent with the OPEs of R X and R XM with the twist "elds and the X U non-singular OPE of R X with R XM is U X (z )u (w)B(z ,z ) . (3.121) h(z ,w;z )"u ,\I , ,\l , G G G ,\I , ,\l , Specialising to k"l, dividing through by u (w), choosing z ,2,z as in Eq. (3.115), and ,\I , ,\I , taking the limit wPR, gives a pair of equations for A and B which can be solved to give AI (x,x )"x(1!x)R ln I(x,x ) , V where I(x,x ) was de"ned in Eq. (3.102). Now that AI (x,x ) is known, Eq. (3.116) can be integrated to give
(3.122)
(Z ) "c "x(1!x)"\I ,\I ,[I(x,x )]\ , (3.123) where c is a constant. Multiplying together Eqs. (3.89) and (3.123) to obtain Z gives c"x(1!x)"\I ,\I , e\1 T T , (3.124) Z " q "F(x)" T T where we have used Eq. (3.101), the constant c is c /2, and S (v ,v ) is given by Eq. (3.96). 3.11. Factorisation of the 4-point function to 3-point functions To derive the Yukawa couplings in which we are interested we now have to factorise the 4-point function by writing it as a sum of terms which are products of 3-point functions [78,177,88]. We
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Fig. 4. u channel factorisation of 4-point function. Fig. 5. s channel factorisation of 4-point function.
"rst factorise in the u channel (Fig. 4) to derive the required Yukawa coupling up to a moduli and "xed point independent normalisation factor and then factorise in the s channel (Fig. 5) to establish the normalisation. To study the u channel factorisation it is necessary to take the limit xPR. Using the asymptotic form for F(x), we then obtain $1 S + ("v "#"v ") for xPR , (3.125) 4p sin(2kp/N) where v "v !v , v "ep I ,v #v (3.126) and the plus sign and minus sign correspond to k/N(1!k/N and k/N'1!k/N, respectively. The factorisation of the 4-point function into a sum of terms that are products of 3 point functions depends on the OPE of two twist "elds. In general, for conformal "elds A and A of conformal G H weights h ,hM and h ,hM the OPE can be written in the form G G H H C A (w,w ) GHI I A (z,z )A (w,w )& (3.127) G H (z!w)FG>FH\FI(z !w )FM G>FM H\FM I I for zPw, where C are some coe$cients. For twist "elds, we can write GHI p (x,x )p (z ,z )& >I L p (3.128) (z z )"x!z "\FI ,\FI , I , D I , D D D D I , D D for xPz , where the sum is over "xed points, the coe$cients > can be interpreted as Yukawa couplings, as will be seen shortly, and the conformal weights are given by h
1k k "hM " 1! I , I , 2N N
(3.129)
as in Eq. (3.104). The 4-point function can then be written in the form valid for xPz , Z "1p (0)p (x)p (1)p (z )2 \I , D I , D \I , D I , D (0)p (1)p (z )2;"x!z "\FI ,\FI , . & >I L 1p D D D \I , D \I , D I , D D
(3.130)
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Moreover, for conformal "elds A ,A and A , G H I 1A (z ,z )A (z ,z )A (z ,z )2"C (z !z )\FGH(z !z )\FM GH G H G H I GHI G H GH
(3.131)
with h "h #h !h GH G H I and similarly for hM . In the case of twist "elds, GH "z "\FI , 1p (0)p (1)p (z )2">HI , \I , D \I , D I , D DDD Consequently, Z takes the form for xPR,
(3.132)
(3.133)
>HI , . (3.134) Z +"x"\FI ,\FI ,"z "\FI , >I , DDD DDD D To complete the factorisation, we have to use the requirement that the u channel "xed points f summed over must be consistent with the space group selection rule, which takes the form (1!hI)( f#K)"hI(1!hI)( f #K)#(1!hI)( f #K) , where the action of the point group element in this complex plane is h"ep , .
(3.135)
(3.136)
The relation (3.135) is also correct if we interchange f and f , and there are similar relations with f and f replacing f and f . Aided by the space group selection rule we can show that v "h\I(1!hI)( f!f #K) (3.137) and v "!(1!hI)( f!f #K) . This allows Z to be written in the factorised form for xPR, and k/N(1!k/N, "x"I ,I ,\(C(1!k/N)) e\1I T e\1I T Z +c cos(kp/N)(C(1!2k/N)) D T T with "v " . SI (v )" 4p sin(2kp/N)
(3.138)
(3.139)
(3.140)
It remains to "x the normalisation constant in Eq. (3.139). This can be done by considering the s channel factorisation (Fig. 5) which gives the coupling for the annihilation of two twisted states into an untwisted state. To study these s channel couplings we need to Poisson resum e\1 so T T as to write Z in terms of momenta on the dual KH of the lattice K corresponding to the momenta of untwisted S channel states. Because the sum over v is over the coset (1!hI)( f !f #K) rather than K, it is necessary "rst to arrange for a sum over K by writing
pk ( f !f #q) v "!2ie pI ,sin N
(3.141)
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where q3K. Writing S in terms of q and using the Poisson resummation identity 1 exp(!p)(q# )2A\(q# )!2pid2(q# )
(3.142)
"x(1!x)"\I ,\I , Z "2c exp(2pip ) ( f !f ))=N>T = M N\T , (3.143)
(3.145)
To carry out the s channel factorisation it is now necessary to consider the limit xP0. The relevant OPE of twist "elds is < (x,x ) , (3.146) (0,0)p (x,x )& (!x)F\FI ,(!x )FM \FI ,CI , p I , D D D N U N U \I , D N U where < is the twist invariant vertex operator for the emission of an untwisted sector state with N U p3KH and winding number w3(1!h)( f !f #K), and h and hM are the conformal dimensions of the corresponding untwisted state. Then, for xP0, Z+ CI , (1)p (z )2 . (!x)F\FI ,(!x )FM \FI ,1< (x)p D D F U N U \I , D I , D N U Moreover, using Eq. (3.131), for xP0,
(3.147)
(1)p (z )2+CI , "z "\FI ,(!1)F>FM 1< (x)p I , D N U D D N U \I , D which results in
(3.148)
CI , xF\FI , x FM \FI ,"z "\FI , . (3.149) Z + CI , D D N U N U D D N U The 4-point function can now be normalised by considering the contribution to the sum over untwisted states from I, which is the untwisted state with p"w"0 and h"hM "0. Taking f "f and f "f , in Eq. (3.148), ""z "FI ,1Ip (1)p (z )2"1 , (3.150) C \I , D I , D D D where the 2-point function for two twist "elds is normalised (consistently with Eq. (3.127)) by 1p (1)p (z )2""z "\FI , . \I , D I , D
(3.151)
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Thus, the contribution from I on Eq. (3.149) can be written as Z +"x"\FI ,"z "\FI , . (3.152) Comparing with the term with p"v"0 in Eq. (3.143) the corresponding contribution is Z +c"x"\FI ,/
kp "z "\FI , . N
(3.153)
(3.154)
Returning to Eq. (3.139) with constant of proportionality now "xed, and comparing with Eq. (3.134), gives the result for the 3-point function
kp C(1!k/N) >I , e\1I T , "
(3.155)
(3.156)
and v is given by Eq. (3.138). If there are any complex planes that are unrotated by one of the three twists involved, the normalisation factor should be restricted to the rotated complex planes, because, for the unrotated plane, the 3-point function reduces to a 2-point function that can be normalised to 1. 3.12. Yukawa couplings involving excited twisted sector states The vertex operators for excited states, i.e. states with oscillators acting on the ground state, involve derivatives of string degrees of freedom as well as twist "elds. For correlation functions involving excited states there is then the selection rule [78,120,65,102}104] that a correlation function for which the product of vertex operators contains the factor (R XG)N(R XM G)O must have X X p!q"0 mod N (3.157) if the action of a point group element in the ith complex plane is of order N. To calculate the moduli dependence [78,21,22] consider for simplicity the situation where there are two excited twisted sector states involved each of which is created from the vacuum by a single bosonic left mover oscillator. The description of excited twisted sector states requires the excited twisted "elds q and q that occur in the OPEs (3.53). Thus, the non-trivial part of the 3-point I , I , function we wish to consider if of the form (z ,z )2 , (3.158) (Z ) "1q (z ,z )ql (z ,z )p I , , \I>l , where the index i referring to the complex plane and the "xed point dependence have been suppressed. Consideration of the 2-point function 1q (z ,z )q (z ,z )2 and the twisted sector I , \I , mode expansions shows that the excited twist "elds that create normalised states are (2k/N)\ q I ,
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and (2(1!k/N))\ q . For an acceptable solution with convergent classical action, (Z ) is \I , found to have the same moduli dependence as the 3-point function with unexcited twist "elds. However, the overall normalisation of the 3-point function changes. This normalisation, which depends on the twisted sectors involved, can be "xed by considering the 4-point function (Z ) "1q (z ,z )p (z ,z )q l (z ,z )pl (z ,z )2 . \I , I , \ , , With the aid of the OPEs (3.53) this can be written as
(3.159)
(Z ) " lim (w !z )I ,(z !z )\l , U X X X (z ,z )p (z ,z )2 ;1R XR XM p (z ,z )p (z ,z )p X U \I , I , \l , l , " e\1Ql1R X R XM 2 # e\1QlR X lR XM l(Z ) . (3.160) X U X Q U Q 6 6 In Eq. (3.160), X has been separated into a classical and a quantum part, 1R X R X 2 is the X U expectation value in the presence of the four twist "elds, and (Z ) is given by
(z ,z )p (z ,z ) . (Z ) " DX e\1p (z ,z )p (z ,z )p \I , I , \l , l ,
(3.161)
Using operator product expansion methods, setting z "0, z "x, z "1 and z "z (3.162) using SL(2,C) invariance, and taking the limit xPz to achieve u channel factorisation, leads to 1q (0)q l (1)p l (z )2 \I , \ , I> , "2 l (!1)I ,\l , for I (1! l , , , , 1p (0)p l (1)p l (z )2 \I , \ , I> , 1q (0)q l (1)p l (z )2 I> , "2(1!I )(!1)I ,\l , for I '1! l . \I , \ , (3.163) , , , 1p (0)p l (1)p l (z )2 \I , \ , I> , Taking account of the normalisation of the excited twist "elds discussed above and powers of !1 from the conformal "eld theory of the 3-point function, the Yukawa coupling ># involving excited states should be de"ned by 1 l " ># \ (1!I )\ (!1)l ,\I ,1q (0)q l (1)p l (z )2 I> , \I , \l , I>l , 2(N) , \I , \ ,
(3.164)
and consequently ># \I , \l , I>l , "( l , for I (1! l , 1p (0)p l (1)p l (z )2 , \I , , \I , \ , I> , ># \I , \l , I>l , "(\I , for I '1! l . (3.165) l 1p (0)p l (1)p l (z )2 , , , \I , \ , I> , There are thus twist-dependent suppression factors arising in the excited twisted sector Yukawa couplings relative to the Yukawa couplings amongst twisted sector ground states [21,22].
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3.13. Quark and lepton masses and mixing angles The exponential suppressions [78,120,112] due to the moduli dependence of twisted sector Yukawa couplings can lead to a hierarchial quark and lepton mass matrix [127,54,55]. By utilising all the possible embeddings of the point group and all possible choices of Wilson lines a huge number of models can be obtained for each Z or Z ;Z orbifold. The strategy that has been , + , adopted [55] in exploring the possibilities for the quark and lepton masses (and weak mixing angles) has been to allow the quarks and leptons and Higgses to be assigned to arbitrary twisted sectors and arbitrary "xed points. In general, the Lagrangian terms ¸ for the quark masses take the form O d u (3.166) ¸ "(dM ,s ,bM ) M s #(u ,c ,tM ) M c #h.c. , * S O * B b t 0 0 where M and M are matrices deriving from couplings to Higgses H and H . In Eq. (3.166), B S (u ) ,(d ) ,(c ) ,(s ) ,(t ) and (b ) are the [SU(2)] doublet quark "elds, in terms of which the weak * * * * * * * current JI coupled to the = boson takes the form > d . (3.167) JI "(u ,c ,tM ) cII s > * b * On the other hand, in terms of the states u,d,c,s,t and b that diagonalise the quark mass matrix the weak current JI takes the form > d
JI "(u ,c ,tM ) cI< s > * b
,
(3.168)
* where the matrix < is the usual Kobayashi}Maskawa matrix
C <" !C S S S where
C S C C C !S S e B !C C S !C S e B
S S C C S #C S e B , !C S S #C C e B
C "cos h , S "sin h . G G G G For massless neutrinos, the Lagragian terms ¸l for the lepton mass take the form
(3.169)
(3.170)
e
¸l"(e ,k,q) M k * C q
0 and diagonalisation of the lepton mass matrix should not be required.
(3.171)
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To reproduce the Kobayashi}Maskawa matrix it is necessary for the quark mass matrices M and M to have o!-diagonal entries. Whether this is possible depends on the space group B S selection rules. For the prime order orbifolds Z and Z , all the Yukawa couplings are diagonal in the sense that any 2 quark or lepton or Higgs states can only couple to a unique third state. This derives from 2 twisted sectors coupling to a unique third twisted sector because of the point group selection rule, from two "xed points coupling to a unique third "xed point because of the space group selection rule, and from there being only one state of given gauge quantum numbers associated with a particular "xed point. Then the (renormalisable) Yukawa couplings can not reproduce the Kobayashi}Maskawa matrix. Moreover the (diagonal) elements of the mass matrix do not have any observable phases because they can be observed into a rede"nition of the right handed quark states. However, non-renormalisable superpotential terms occur in general and can give rise to e!ective Yukawa couplings amongst quarks, leptons and Higgses when some gauge singlet scalars in the non-renormalisable coupling acquire expectation values. This gives the scope to obtain o!diagonal entries and phase factors in the quark mass matrices. In general, we can obtain quark and lepton mass matrices M , M and M of the form B S C e a b
M" a
A
c ,
bI
c
B
(3.172)
where a,b,c,a ,bI ;e,A,B because they are induced by non-renormalisable terms in the superpotential. It is also natural to assume that e;A,B because of the smallness of the "rst generation quark and lepton masses, and so to assume that e also derives from a non-renormalisable term. Things are more complicated for non-prime-order orbifolds. However, it is still unlikely that a suitable Kobayashi}Maskawa matrix can arise from non-renormalisable terms, and it therefore still appropriate to look for matrices M of the same form. The strategy that has been adopted [55] has been to try to "t the second and third generation quarks and lepton masses with A and B in M given in terms of all the moduli (deformation parameters) of the orbifold, under the assumption that the smaller "rst generation masses are induced by non-renormalisable terms. Then the relevant Yukawa couplings ¸ are 7 ¸ "h Q cAH #h Q SAH #h Q tAH #h Q bAH #h ¸ kAH #h ¸ qAH , (3.173) 7 A A 1 A R R @ R I I O O where Q and ¸ denote quark and lepton doublets. At this time, the expectation values of H and H are additional parameters constrained by m . (3.174) 1H e#1H e"2 5 g The masses obtained at the string scale have to be run to 1 GeV using renormalisation group equations to make contact with the point at which quark and lepton masses are usually given. A subtlety is that the Yukawa couplings have to be run from the string scale of about 0.5;10 GeV whereas, because we know that gauge coupling constants unify at about 10 GeV (perhaps because of string loop threshold corrections), the gauge coupling constants should only be run from about 2;10 GeV.
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Of the Z orbifolds, only Z , Z , Z !I and Z are able [55] to "t the quark and lepton masses. L However, a number of possible e!ects have been neglected in these calculations. The e!ect of the tree level moduli dependent Kahler potential in normalising the matter states has not been included, nor have the twist dependent suppression factors if the Yukawa couplings are between excited twisted sector states, nor have the string loop threshold corrections to the Yukawa couplings from the one-loop Kahler potential. In the absence of a de"nite model for the entries of the mass matrix deriving from the non-renormalisable superpotential terms, the Kobayashi}Maskawa mixing angles and phases cannot be determined. However, a simple model for M and M with vanishing (11), (13) and (31) B S entries and opposite phases for the Eqs. (23) and (32) entries can give mixing angles consistent with experiment together with an approximately maximal weak CP violating angle d+953.
4. KaK hler potentials and string loop threshold corrections to gauge coupling constants 4.1. Introduction A supergravity theory is speci"ed by the superpotential, the KaK hler potential and the gauge kinetic function. The light shed by orbifold compacti"cations of superstring theory on the form of the superpotential (especially the renormalisable terms) was the subject of the previous section. The KaK hler potential and the gauge kinetic function, which yields the gauge coupling constants of the theory, will be studied in this section. A knowledge of the KaK hler potential allows the normalisation of the states of the theory to be carried out and is also necessary for the construction of the e!ective potential. On the other hand, a knowledge of the gauge kinetic function is necessary to determine the values of the string loop corrected gauge coupling constants at the string scale, which, with the aid of the renormalisation group equations, can be compared with the measured low energy values. Like the Yukawa coupling, the KaK hler potential and the gauge kinetic function both depend on the moduli of the orbifold discussed in Sections 3.7 and 3.8 and the values of the moduli are required before conclusions can be drawn. The determination of the moduli from the e!ective potential will be one of the topics discussed in the next section. 4.1.1. Modular properties of the KaK hler potential Associated with the ith complex plane of the underlying 6-torus of the orbifold, all abelian orbifolds have a modulus ¹ de"ned in Eq. (3.80) by G i¹ "2(b #i(+(det g) ,), i"1,2,3 , G G\ G G
(4.1)
where the matrices g and b are the metric and anit-symmetric tensor in the lattice basis, as in (3.81)}(3.82), and the determinant refers to the 2;2 matrix for the ith complex plane. When the point group acts as Z in the ith complex plane there is also a ; modulus, ; , de"ned by G 1 i; " (g #i((det g) ) . G g G G G G\ G\
(4.2)
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On the other hand, when the point group acts as Z with NO2 in the ith complex plane the , modulus ; is forced to take a "xed value and only ¹ survives as a continuous modulus. Speci"c G G orbifolds possess additional ¹ moduli but in what follows we shall focus on the moduli de"ned above. The ¹ moduli may be thought of as continuously variable quantities corresponding to deformations of the underlying torus. Moduli may also be thought of as expectation values of scalar "elds in the corresponding supergravity theory for which the e!ective potential is #at to all orders. Looked at this way, the existence of ¹ and ; moduli is equivalent to the existence of "02 aH "02 . In untwisted sector states of the string theory of the type bG "02 aHM "02 or bG \ 0 \ * \ 0 \ * general, depending on the point group, the "rst type of state can exist for i"j and for some choices of iOj. The second type of state is only permitted by point group invariance for i"j and then only if the ith complex plane is a Z plane (a plane in which the point group acts as Z ). The states in Eqs. (4.1) and (4.2) are the states with i"j. Orbifold compactifactions of string theory are known to possess certain modular symmetries to all orders in string perturbation theory. Generically, these symmetries are transformations of the form ¹ P(a ¹ !ib )/(ic ¹ #d ) G G G G G G G
(4.3)
and ; P(a ; !ib )/(ic ; #d ) G G G G G G G where a , b , c , d , a , b , c and d are integers, G G G G G G G G a d !b c "1 G G G G and
(4.4)
(4.5)
a d !b c "1 . (4.6) G G G G These symmetries are thus PSL(2,Z) modular groups, referred to as target space modular symmetries if these is a need to distinguish them for the world sheet modular symmetries discussed in Section 2. In some cases, string loop corrections can restrict the symmetries to subgroups of PSL(2,Z), or, equivalently can restrict the allowed range of values of these integers, as we shall see later. Further subtleties are that beyond string tree level the dilaton "eld S can participate in the modular transformations, and that, if Wilson line moduli are present, these may also enter the modular symmetries. We shall see in subsequent sections that at string tree level the KaK hler potential takes the form K"KK # " " (¹ #¹M )LG?#2 , ? G G ? G where
(4.7)
KK "!ln(S#SM )! ln(¹ #¹M ) . (4.8) G G G Here, only the diagonal ¹ moduli, ¹ , i"1,2,3, have been retained, the ; moduli have not been G displayed, and K has been taken to quadratic order in the matter "elds . The powers nG are ? ? referred to as the modular weights of the matter "elds.
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In the absence of matter "elds, the transformation of the KaK hler potential under a modular transformation on ¹ is G KPK#ln"ic ¹ #d " , (4.9) G G G which is a speci"c KaK hler transformation. For G"K#ln"=" ,
(4.10)
where = is the superpotential, to be invariant under modular transformations, = must transform with modular weight !1, by which is meant =P=(ic ¹ #d )\ . (4.11) G G G Because of non-renormalisation theorems this must be true to arbitrary orders in perturbation theory. If the matter "elds are now introduced, then, to retain the modular invariance of G, the matter "elds must transform with modular weights nG , by which is meant ? (4.12)
P (ic ¹ #d )LG? . ? ? G G G The modular properties of a Yukawa coupling ="h (¹ )
(4.13) ?@A G ? @ A in the superpotential may then be deduced. For = to have modular weight !1 we must have h
(¹ )Ph (¹ )(ic ¹ #d )\>LG?>LG@>LGA . ?@A G ?@A G G G G
(4.14)
4.2. KaK hler potentials for moduli There are several di!erent approaches to deriving KaK hler potentials from orbifold compacti"cations of string theory, including truncation of the corresponding 10-dimensional supergravity theory to four dimensions [188,92,23,93,94] identi"cation of accidental symmetries of the string action which can then be applied to the supergravity action [66,67,58,59,69], and comparison of amplitudes calculated in the string theory and in the supergravity theory with the aid of the N"2 superconformal algebra [81,25,26]. In this section, we shall present the second of these methods, and, very brie#y, in a discussion of the dilaton KaK hler potential, the "rst of these methods. In the next section, the last of these three methods will be used in a discussion of the matter "eld contribution to the KaK hler potential. Any of these methods may be used to discuss the moduli and matter "eld KaK hler potentials but it is useful here to present a di!erent method in each section to illuminate di!erent aspects of the origin of the form of KaK hler potentials. Employing the accidental symmetry approach [66,67], let consider "rst a complex plane of the underlying 6-torus for which the action of the point group is Z with NO2. Then, there is , associated with this complex plane only a ¹ modulus ((1,1) modulus) and no ; modulus ((1,2) modulus.) The background "eld term in the string action for this ¹ modulus may be written as
1 S" dz(¹R XR XM #h.c.) , X X p
(4.15)
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where the index i referring to the complex plane has been suppressed. This action possesses the `accidentala symmetries XPKX#C ,
(4.16)
¹P¹K\KM \ ,
(4.17)
where K and C are arbitrary complex numbers, and ¹P¹#iD ,
(4.18)
where D is an arbitrary real number. These symmetries of the world sheet action must appear as symmetries of the low-energy e!ective action for the moduli. The most general Lagrangian compatible with these symmetries is L"k(¹#¹M )\R ¹RI¹M , (4.19) I where k is a constant. The constant may be "xed by comparing the four moduli amplitude calculated at tree level in the supergravity theory and the string theory with the result that L"(¹#¹M )\R ¹RI¹M I which derives from the KaK hler potential K"!ln(¹#¹M ) .
(4.20)
(4.21)
If instead we consider a complex plane for which the action of the point group is Z , then there is both an associated ¹ modulus and an associated ; modulus. It is then convenient to introduce the metric g and anti-symmetric tensor b background "elds on the (real) lattice basis. The MN MN corresponding background "eld term in the string action is
1 S" dz F R XK MR XK N , MN X X p
(4.22)
where o,p"1,2, XK M and the string degrees of freedom in the lattice basis, de"ned by XK M"eMXP , P where r refers to the real space basis,
(4.23)
eM,eHP P M are basis vectors of the dual lattice, and
(4.24)
F "g #b . MN MN MN The ¹ and ; moduli for this complex plane are then de"ned by
(4.25)
¹"¹ #i¹ "2((det g!ib )
(4.26)
1 ;"; #i; " ((det g!ig ) , g
(4.27)
and
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or, consequently, the matrices g and b are given by
1 1 Re ¹ g" 2 Re ; !Im ;
!Im ; ";"
(4.28)
and
0 1 b" 2 Im ¹
!Im ¹ 0
.
(4.29)
The string (world sheet) action has the `accidentala symmetries XK MPM XK N#CM , MN F PF (M\) (M\) , MN HO HM ON
(4.30) (4.31)
and F PF #D , (4.32) MN MN MN where M is a real non-singular matrix, CM are real constants and D is a real anti-symmetric matrix. Applying these symmetries to the low energy supergravity e!ective action for the moduli, the most general consistent form of Lagrangian is L"!Tr((F#F2)\R F2(F#F2)\RIF) I "!Tr(g\R gg\RIg!g\R bg\RIb) , (4.33) I I where the overall multiplication constant has been "xed by comparing the ggbb amplitude in the low-energy supergravity theory and the string theory using the vertex operators coupled to the background "elds g and b. Substituting for g and b in terms of the ¹ and ; moduli gives L"(¹#¹M )\R ¹RI¹M #(;#;M )\R ;RI;M I I which derives from the KaK hler potential K"!ln(¹#¹M )!ln(;#;M ) .
(4.34)
(4.35)
Another modulus, in the sense of a "eld with #at e!ective potential to all orders in the corresponding supergravity theory, is the dilation S. A simple way of deriving the KaK hler potential for the dilaton "eld is by truncation to 4 dimensions of the 10-dimensional supergravity that is the e!ective "eld theory below the string scale. The supergravity multiplet of supergravity in 10dimensions contains bosonic states which are the symmetric metric tensor g , the antisymmetric tensor b and the 10-dimensional dilaton scalar , where A and B range over the 10-dimensional space. The dilaton S for the 4-dimensional reduction of the 10-dimensional supergravity is constructed from the degrees of freedom of the 10-dimensional theory as S"(De(#3(2iD
(4.36)
where D"det g GH
(4.37)
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with i and j referring to the compact six-dimensional manifold and D being the dual of the b "eld, IJ where k and l refer to four-dimensional space}time. The "eld D is given in terms of the "eld strength h for b as IJM IJ e(Dh "e RND . (4.38) IJM IJMN The kinetic term for S in the dimensionally reduced Lagrangian derives from the KaK hler potential term K"!ln(S#SM ) .
(4.39)
This is present not just in toroidal compacti"cations but also in the untwisted sector of any orbifold compacti"cation, constructed in this approach by truncating the dimensionally reduced theory. This is done by retaining only singlets under the action of some "nite subgroup of the rotation group SO(6) on the compact manifold designed to leave only an N"1 supergravity in four dimensions [188,23,92}94]. 4.3. KaK hler potentials for untwisted matter xelds The method described in this section, which can be found in greater detail in the original literature, [81] relies on the fact that the N"2 super Virasoro algebra for the left movers for a string theory with N"1 space}time supersymmetry, relates the left mover vertex operators W! for 27 and 27 matter "elds in the 10 of the SO(10) subgroup of E (apart from E ;E factors in the vertex operator) to other left mover vertex operators U! in the same N"2 chiral multiplets of this algebra. In general, the left mover vertex operators for moduli "elds M associated with matter "elds can be written as linear combinations of the vertex operators U!. Thus, the vertex operators for the (1,1) moduli denoted by M? with associated 27's denoted by can be identi"ed ? by M? ;?U> (4.40) ? ? for some coe$cients ;?, and the vertex operators for the (1,2) moduli denoted by MK with ? associated 27's denoted by can be identi"ed by I MK ;I U\ (4.41) K I for some coe$cients ;I . Let us de"ne the KaK hler metrics g M and G M for (1,1) moduli and 27 matter ?@ K ?@ "elds by M @"RK/RM?RM M @ g M ,RG/RM?RM ?@
(4.42)
and (4.43) G M ,RG/R R M "RK/R R M ? @ ? @ ?@ and similarly for the (1,2) moduli and 27 matter "elds. The two point functions for vertex operators W! can be related to 2-point functions for vertex operators U! with the result that the KaK hler metrics are related by g M ";?G M ;M @ ? ?@ @ ?@
(4.44)
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and similarly for (1,2) moduli. Thus, if the matrices ;? can be calculated relations can be found ? between moduli and matter "eld KaK hler metrics. A rather messy, but easily solved, di!erential equation involving the moduli metric matrix g and the matrix of coe$cients ; can be derived by "rst using the N"2 super-Virasoro algebra to relate pure moduli amplitudes of the type MMPM M M M to pure matter "eld amplitudes of the type
P M M and also to relate amplitudes of the type M PM M M to amplitudes of the type
P M M . In each case, because it is the vertex operators U! and W! that belong to N"2 supermultiplets, the matrices ; occur. In the second stage of the derivation, the various amplitudes are calculated from the corresponding supergravity theory as follows [81]. In the case of MMPM M M M and M PM M M amplitudes, there are contributions from sigma model interactions due to the nonminimal KaK hler potential and from graviton exchange. In the case of
P M M amplitudes, at leading order in the momenta, there are contributions from 4 scalar F terms, from gauge boson exchange and from corresponding D terms. The reason that the calculation will be able to determine the matrix ; and so to determine the combinations of gauge singlet scalars that are moduli "elds, is that the (de"ning) #atness of the e!ective potential with respect to moduli to all orders has been used to drop all moduli}moduli interactions other than sigma model interactions. The details of the calculation depend on the gauge group assumed. We shall assume for the moment that the gauge group is simply E ;E . When the gauge group is instead E ;E ;;N(1) or E ;E ;SU(3);;N(1), there are extra gauge boson exchanges and corresponding D term contributions as well as F terms modi"ed by modi"ed Yukawa couplings that a!ect the
P M M amplitudes. Finally, the expressions at leading order in the momenta for the amplitudes derived from the supergravity theory are inserted in the string relations between amplitudes. In this way, after elimination of the terms containing the matter "eld Yukawa coupling coe$cients between equations, a matrix equation involving the matrices g and ; is arrived at, namely, i R (;Rg\;R M (;\g(;R)\)) "(;RR (g\R M g)(;R)\) # R R M (K !K )d ?A ? B ?A ?A ? B 3 ? B
(4.45)
M ?, and the and a similar equation with a,d replaced by m,n, where R and R denote R/RM? and R/RM ? ? pure moduli term KK in the KaK hler potential has been decomposed in the form (proved possible in Ref. [81]) KK "K #K (4.46) with K depending only on M? and M M ? and K depending only on MK and M M K. Eq. (4.45) has the solution ;?"
(4.47)
and ;I "
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The matter "elds may be chosen in such a way as to replace the < matrices by identity matrices so that ;?"d? exp i(K !K ) ? ?
(4.49)
and (4.50) ;I "dI exp i(K !K ) . K K Knowing ;, the connection between matter "eld and moduli KaK hler metrics following from Eq. (4.44) is G M "g M exp i(K !K ) ?@ ?@
(4.51)
and (4.52) G "g exp i(K !K ) . KL KL Returning to the equations derived from the string relations between amplitudes before elimination of the matter "eld Yukawa coupling coe$cients between equations and utilising Eqs. (4.49)}(4.52) yields equations that relate the KaK hler metric for the moduli to the matter "eld Yukawa couplings. Once these Yukawa couplings have been speci"ed, the KaK hler metric can be solved for in speci"c cases [81]. A more realistic case is obtained [81,25] if the gauge group is E ;;N(1);E or E ;SU(3);;N(1);E . If, for example, [25] we take the gauge group to be E ;SU(3);E then this corresponds to the Z orbifold with standard embedding of the point group in the gauge degrees of freedom. In that case, the matter "elds are in (27,3) of E ;SU(3) and singlet under E and we denote the vertex operators for matter "elds in the 10 of the SO(10) subgroup of ?G E by W , where a is a global index labelling the various copies of (27,3) and i"1,2,3 is an SU(3) ?G index labelling the basis states of 3 of SU(3). (The free fermion factor in the vertex operator carrying the SO(10) quantum numbers is not displayed.) Associated with these matter "elds are the E ;SU(3) singlet scalars which are members of the same N"2 chiral multiplets and whose vertex operators we denote by U . In this case, there are only (1,1) moduli "elds, denoted by M , where ?G ' A and I are both global indices. It is convenient to decompose the global index on the modulus "eld in this way to mirror the decomposition of the index ai on the corresponding matter "eld into a global index a and an SU(3) index i. The vertex operators for the (1,1) moduli are in general linear combinations which can be identi"ed by M ;?G cU , (4.53) ' ' ?G where the ;?G are functions of the moduli and their conjugates. There is some arbitrariness in the ' de"nition of ; because we can make a rede"nition of the matter "elds by taking a linear combination of the various (27,3)'s or by making a change of basis in the SU(3) space. Thus, new matter "eld vertex operators W may be de"ned by ?YGY (4.54) W "R?Y(M)SGY(M)W ?G ? G ?YGY where R and S are functions of the moduli but not their conjugates, in order to preserve the KaK hler geometry, and S is unitary. Consequently, there is the freedom to replace ;?G by ;I ?YGY where ' ' ;I ?YGY";?G R?Y(M)SGY(M) . (4.55) ' ' ? G
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The N"2 superconformal algebra now relates the moduli metric g M M to the matter "eld metric ' ( G M M through ?G @H (4.56) g M M ";?G G ;M @MM GM M , ' ?@ ( ' ( where the unbroken SU(3) gauge symmetry has been used to block diagonalise G in the form G M M "G M d M . (4.57) ?G @H ?@ GH Equations involving g,G,; and the Yukawa coupling coe$cients for matter "elds may again be derived [25] by studying amplitudes for moduli and matter "eld with the following slight di!erences. Yukawa couplings have to be modi"ed to take account of the SU(3) indices so that the corresponding superpotential terms are (4.58) ="= (M)e #2 . GHI ?G @H AI ?@A The four scalar vertex contribution to the
P M M amplitude is then modi"ed by the modi"cation of the F terms in the e!ective potential. In addition, the
P M M amplitude is a!ected by SU(3) gauge boson exchanges and corresponding D terms. After elimination of the matter "eld Yukawa coupling coe$cients between equations a matrix equation involving g and ; is arrived at, which now takes the form i R [;Rg\;R M M (;\g(;R)\)] "[;RR (g\R M M g)(;R)\] ! g M M d d ?G AI ' "* ?G AI ' "* 3 ' "* ?A GI !
i (;M j ;)BM M C M (;\)#+ g (;M \)$M M l,M d (j ) , Cl #+ $M ,M B ?A M IG M "* ' 6 (4.59)
where the j are the Gell}Mann matrices for SU(3), M (;M j ;)BM M C M ,;M BM M lM M (j )lM M ;CG , (4.60) M "* ' "* M G ' M . To obtain the solution, we also require one of the and R and R M M denote R/RM and R/RM ' ' ' ' 2 original equations derived from the matter "eld and moduli amplitudes using the N"2 superconformal algebra for left movers, which may be taken to be i\R M M "g M M g M M #g M M g M M ! exp(iKK )g$M ,M #+(=;;;)GHE ' !) ( "M *M ' "* ( !) ' !) ( "* ' ( #+ ;(= M ;M ;M ;M ) M M M M M M e e l , !) "* $, GHF I F where
(4.61)
(=;;;)GHE ,= ;?G ;@H ;CE ' ( #+ ?@C ' ( #+ and the Riemann tensor of the KaK hler geometry is given by
(4.62)
R M M "R R M M g M M !R g M M g#M +M $,R M M g M M . ' !) ( "M *M ' "* ( !) ' ( #+ "* $, !) Eqs. (4.59) and (4.63) have the solution
(4.63)
KK "!i\ ln det B ,
(4.64)
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where KK is the pure moduli term in the KaK hler potential, B "M #M M ' ' ' or, as a matrix,
(4.65)
B"M#MR
(4.66)
;?G "X > ' ? 'G
(4.67)
>"B\ < ,
(4.68)
and
with
where < is an arbitrary matrix, = "we ?@A ?@A
(4.69)
1 det (XXR)" 2"w"
(4.70)
and
with X a function of M but not of MM . The degree of arbitrariness occuring in the solution is consistent with Eq. (4.54). If we make the choice (2X"<"I ,
(4.71)
then the solution for ; simpli"es to ;?G "(1/(2)d (B\ ) . ? 'G ' It follows that the moduli and matter "eld metrics are g
' M (M
"iB\ B\ '(M M
(4.72)
(4.73)
and (4.74) G M M "2iB\ M d M . '( ' ( If we retain only the diagonal moduli of Section 4.3, `switch o!a the other moduli and write ¹ ,M , i"1,2,3 , G GG then the moduli and corresponding matter "eld metrics simplify to g M "2i(¹ #¹M )\ G G GG
(4.75)
(4.76)
and G M "2i(¹ #¹M )\ . G G GG In terms of rede"ned matter "elds
K ,(2
GG G
(4.77)
(4.78)
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the KaK hler potential K to quadratic order in the matter "elds is K"!i ln(¹ #¹M )#i (¹ #¹M )\" K "#2 . (4.79) G G G G G G G For an orbifold possessing a complex plane where the point group acts as Z , so that there is both a ¹ modulus and a ; modulus associated with this complex plane, the situation is slightly more complicated. If the Z complex plane is the jth complex plane then the corresponding contribution to the KaK hler potential takes the form K"!ln[(¹ #¹M )(; #;M )!(B #CM )(BM #C )] , (4.80) H H H H H H H H where B and C are two complex matter "elds. H H All of the above discussion assumes that there are no Wilson lines breaking the gauge symmetry whereas in practice this will be necessary if the gauge group is to be reduced to a subgroup of E ;SU(3) as a suitable starting point for spontaneous symmetry breaking to the standard model. However, the KaK hler potential of the moduli and certain of the matter "elds in the theory with Wilson lines can be calculated from the corresponding terms in the KaK hler potential in the underlying theory without Wilson lines as a consequence of two observations [81]. First, the amplitudes are the same for the states which survive the GSO projections as in the original theory, and, second, the relationships amongst vertices that follow from the N"2 superconformal algebra are also unmodi"ed. This means that the KaK hler potential in the theory with Wilson lines may be derived by calculating in the theory without Wilson lines the KaK hler potential of the moduli and matter "elds associated with moduli that survive the GSO projections due to the Wilson lines. 4.4. KaK hler potentials for twisted sector matter xelds In the previous section, the KaK hler potential was derived for the moduli and the untwisted sector matter "elds related to the moduli by the superconformal algebra. However, such methods can not be employed when, as occurs for matter "elds in twisted sectors with Wilson lines, the matter "elds are not related to moduli. Other methods are then required [129,27,28]. One approach [27,28] is to make a direct comparison of amplitudes in the string theory with amplitudes in the supergravity theory without the bene"t of the superconformal algebra (in the spirit of the earliest papers [119,150] on the derivation of low energy supergravity from string theory.) First notice that holomorphic rede"nitions of the "elds allow the moduli and matter "eld KaK hler potentials and metrics to be written in a variety of forms [81]. For example, the KaK hler potential K"!ln(¹#¹M !" ") ,
(4.81)
where is a matter "eld, may be written in the form K"!ln(1!"¹I "!" I ")
(4.82)
by the rede"nition 1!¹I , ¹" 1#¹I
(2 I
" . 1#¹I
(4.83)
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These are equivalent KaK hler potentials because they di!er only by h#hM with
h"ln
1#¹M (2
.
(4.84)
In this new form, the KaK hler potential may be expanded in powers of the moduli as well as in powers of the matter "eld. It is this form of KaK hler potential that arises naturally in calculations of string amplitudes. In consequence, the KaK hler potential at quadratic order in the matter "elds (4.7) and (4.8) arises in the form K"! ln(1!"¹I ")# " I " (1!"¹I ")LG? G ? G G\J ? G so that the matter "eld KaK hler metric is
(4.85)
G M (¹I ,¹MI )"d M (1!"¹I ")LG? . (4.86) ?@ G G ?@ G G Information about the matter "eld metric may be derived from the two moduli } two matter "eld amplitude of Fig. 6. With zero moduli expectation values and to quadratic order in the momenta the supergravity amplitude is given by (4.87) A(¹I , I , MI ,¹MI )" (SQd M d M #sG M M (0,0)), R GH ?@ ?@ GH G ? @ H where the indices i and jM on G M M denote derivatives with respect to ¹I and ¹MI , and s, t and u are the H ?@ GH G usual Mandelstam variables. s"!(k #k ), t"!(k #k ), u"!(k #k ) . (4.88) A string theory calculation of this amplitude determines the matter "eld metric to quadratic order in the moduli (expectation values). G M (¹I ,¹IM )"d M #G M M (0,0)¹I ¹IM j#2 . (4.89) ?@ ?@ G G ?@ GH G Once this is known to quadratic order the values of the modular weights nG are obtained by ? comparison with Eq. (4.86). Explicit expressions for the modular weights in terms of the powers of oscillators involved in the construction of the twisted sector matter states may be found in Refs. [129,28].
Fig. 6. Two moduli } two twisted matter "eld scattering amplitude.
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These expressions allow all possible values of matter "eld modular weights for a speci"c orbifold (with arbitrary choices of point group embedding and Wilson lines) to be determined. In general, for a massless left mover the oscillator number N is given by * N "a !h , (4.90) * * )+ where a is the normal ordering constant for the particular orbifold twisted sector and h is the * )+ contribution to the conformal weight of the state from the E ;E algebra. For level 1 gauge group factors it is given by ¹(R ) dim G ? ? , (4.91) " )+ dim R (C(G )#1) ? ? ? where C(G ) is the quadratic Casimir for the adjoint representation of G and ¹(R ) is the quadratic ? ? ? Casimir for the representation R of G to which the state belongs ? ? ¹(R )"Tr Q , (4.92) ? ? where Q is any generator of G in the representation R . For a speci"c gauge group e.g. ? ? ? SU(3);SU(2);;(1) of the standard model, #ipped SU(5);;(1), [SU(3)] or SO(6);SO(4) and chosen representations for the matter "elds, we should use Eq. (4.91) to set a lower bound on h for each matter "eld to allow for the possibility of additional contributions to h from any )+ )+ extra ;(1) factors in the gauge group which are spontaneously broken along #at directions at a large energy scale, as frequently happens in orbifold theories. In this way, it is possible to derive the allowed range of modular weights [129,28] for the various twisted sectors of the Z and , Z ;Z orbifolds for a speci"c gauge group and matter "eld representations. This knowledge is + , useful in studying string loop threshold corrections to gauge coupling constants, as we see later. h
4.5. String loop threshold corrections to gauge coupling constants It is possible to derive e!ective low energy theories by integrating out the "elds with masses above a chosen scale to leave a theory containing only "elds with masses below this scale which can be employed in low energy calculations. [185]. So far as gauge coupling constants are concerned this means that renormalisation group equations may be run from the chosen scale to any lower energy with the coe$cients in the renormalisation group equations calculated using only the light states provided a threshold correction is made to the gauge coupling constants at the chosen scale which contains the contributions from the heavy states. In the case of heterotic string theory, the gauge coupling constant g (k) at energy scale k is related to the string scale coupling constant ? g by 120',% M 120',% #D , (4.93) 16pg\(k)"16pk g\ #b ln ? ? ? 120',% ? k
where k is the level of the gauge group factor G , ? ? M +0.53g ;10 GeV 120',% 120',% and g
+0.7 120',%
(4.94)
(4.95)
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is the common value of the gauge coupling constants [109,136] at the string tree level uni"cation scale M . We shall usually assume that all gauge group factors have level 1 (with the ;(1) 120',% factors suitably normalised.) The threshold correction D has been derived in terms of the complete ? spectrum of states for any four-dimensional heterotic string theory that is tachyon free [136]. It is given by
dq (B (q,q)!b ) , ? ? C q where, for convenience, we are denoting the modular parameter q of Section 2 by q, D" ?
(4.96)
q"q #iq and C as the fundamental domain,
(4.97)
C:!4q 4, q 50, "q"51. In Eq. (4.96),
(4.98)
(!1)Q>QdZ (s s ,q) R Tr (Q(!1)Q,$q&*q &0) , B (q,q)""g(!iq)"\ Q ? ? 2pi dq Q Q $ where q and q are as in Eq. (2.26), g(q) is the Dedekind g function,
(4.99)
g(!iq)"q (1!qL) (4.100) L and Z is the light cone gauge partition function for a single complex free fermion with s and R s taking the values 0 and 1 for NS or R boundary conditions for the two directions on the world sheet torus; the trace is over the internal degrees of freedom i.e. all degrees of freedom other than those of four-dimensional space}time. The charge Q is any generator of the factor G of the gauge ? ? group, and N is the `fermion numbera. Explicitly, $ Z (s ,s ,q)"q \ (1#q L\), (s ,s )"(0,0) R L "q \ (1!q L\), (s ,s )"(0,1) L "2q (1#q L), (s ,s )"(1,0) L "0, (s s )"(1,1). (4.101) Specialising to the case of an abelian orbifold, with point group G [82], the trace can be written as a sum over twisted sectors (h,g). Then, the trace factor in Eq. (4.99) is 1 Tr (Q(!1)Q,$q&*q &0)" Tr( )(Qg(!1)Q,$Fq&*Fq &0F) 1 ? F Q ? "G" F EC% which is just the orbifold partition function with the insertion of Q. ?
(4.102)
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For an orbifold theory with N"1 space}time supersymmetry the point group must be a "nite subgroup of the SU(3) group which is a subgroup of the SO(6) acting on the compact manifold degrees of freedom [80]. Any element of such a group either rotates all but one of the three complex planes for the compact manifold, rotates all 3 complex planes or rotates none of the complex planes. The corresponding twisted sectors are then referred to as N"2, N"1 or N"4 sectors, respectively. For an N"1 sector the boundary conditions do not allow any momentum or winding number associated with the compact manifold. As we shall see later, the moduli enter the Hamiltonian through the left and right mover momenta (or, equivalently, through the momenta and winding numbers) for the compact manifold, and so there can be no dependence of the threshold correction on the moduli for an N"1 sector. When there is at least one complex plane unrotated by h, in general there is a moduli dependent threshold correction for the h twisted sected. We must then ask what is the e!ect of g on the pair of boundary conditions (h,g) for the world sheet torus. The answer is that g must leave the same complex plane unrotated as h does if there is to be moduli dependence because the trace projects out states with non-trivial winding numbers or momenta if g rotates the complex plane. In the special case when h is the identity (the N"4 sector), if g is also the identity then there is no contribution to the threshold correction. This is because the (h"I, g"I) sector is a self-contained N"4 supersymmetric theory and both the renormalisation group coe$cients and the 1 loop threshold corrections vanish in such a theory. Thus, the moduli dependent threshold corrections come from (h,g) twisted sectors where h leaves a single complex plane unrotated (the N"2 sectors) and in addition g leaves the same complex plane unrotated [82]. Moduli dependence in threshold corrections is important because, as we shall see later, it provides a possible mechanism to move the uni"cation scale for gauge coupling constants down from the tree level string scale to the lower scale `observeda empirically [2,90]. 4.6. Evaluation of string loop threshold corrections The "rst step in evaluating the moduli dependent part of the threshold correction D is the ? observation that the contribution to the threshold correction from a twisted sector with a "xed plane (an N"2 sector) can be factorised in the form [82] B (q,q)"Z (q,q)C (q) , (4.103) ? 2-031 ? where Z is the zero-mode partition function for the 2-dimensional torroidal compacti"cation 2-031 corresponding to the "xed plane and the holomorphic function C (q) is the contribution from all ? other string degrees of freedom. Because q Z is modular invariant and also q ((B (q,q)/k ) 2-031 ? ? !B (q,q)/k ) for two di!erent factors G and G of the gauge group is also known to be modular @ @ ? @ invariant, it may be concluded that C (q)/k !(C (q)/k ) is also modular invariant. The theory of ? ? @ @ modular forms then requires this function to be a constant which must equal b /k !(b /k ) by ? ? @ @ taking the limit of Z and B , B for qPiR, and noting that 2-031 ? @ lim Z "1 2-031 O G
(4.104)
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and, as shown in Ref. [51], that lim B (q,q)"b . ? ? O G Thus, we are able to conclude that
b b B (q,q) B (q,q) ?! @ ? ! @ "Z 2-031 k k k k @ @ ? ? so that
(4.105)
(4.106)
dq (Z (q,q)!1) (4.107) 2-031 C q with the understanding that the formula is only to be applied to the di!erence D /k !D /k . ? ? ? @ The problem of evaluating the contribution to D from a particular N"2 twisted sector thus ? reduces to the evaluation of Z in the "xed plane for this sector [82]. To calculate this quantity 2-031 it is necessary to express the left and right mover Hamiltonians H and H in terms of windings and * 0 momenta in this "xed plane, to which the two-dimensional toroidal compacti"cation corresponds. The windings and momenta enter the right and left mover mode expansions through D "b ? ?
XP (t!p)"xP #pP (t!p)#oscillator terms , 0 0 0
(4.108)
XP (t#p)"xP #pP (t#p)#oscillator terms , * * * where
(4.109)
and
pP "(pP#2¸P) (4.110) pP "(pP!2¸P), * 0 with pP and ¸P the momenta and winding numbers, respectively, and r a real index for the space basis. (The world sheet variables are being denoted by (t,p) rather than (q,p) to avoid confusion with the modular parameter q.) In the conventions being used here XP"XP (t!p)#XP (t#p) . (4.111) 0 * In terms of the basis vectors eP , o"1,2,6, of the lattice for the 6 torus and with the `radiia M absorbed into the de"nition of the basis vectors, ¸P" mMeP (4.112) M M where mM are integers. For convenience, we are using basis vectors here that are smaller by than those used in Section 3 by a factor of 2p. Symmetric and anti-symmetric background "elds G and B are introduced in the world sheet PQ PQ action S through the term
1 p dp dt[G R XPR XQ#e?@B R XPR XQ] . S"! PQ ? ? PQ ? @ 2p
(4.113)
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In the presence of the background "elds, the conjugate momentum which is quantized on the dual lattice with basis vectors eHP,eM M P
(4.114)
is p "G pQ#2B ¸Q" n eM (4.115) P PQ PQ M P M where n are integers. In terms of p and the windings ¸P, the momentum pP is given by M P pP"GPQp !2GPQB ¸R , (4.116) Q QR where GPQ is the inverse of G . PQ It will be convenient to write all quantities in the lattice basis in which the string degrees of freedom are XK M,eMXP . P Then, we de"ne b ,eP B eQ , MN M PQ N g ,eP G eQ , MN M PQ N p ,g pN M0 MN 0
(4.117)
(4.118) (4.119) (4.120)
and p ,g pN , (4.121) M* MN * where pM and pM are the coe$cient of t!p and t#p in XK M and XK M , respectively. In terms of the 0 * 0 * background "elds p "n !g mN!b mN MN MN M0 M
(4.122)
and p "n #g mN!b mN MN MN M* M which may be written succinctly as p "n!(g#b)m 0
(4.123)
(4.124)
and p "n"(g!b)m . * The Hamiltonian is
(4.125)
H"H #H 0 *
(4.126)
H "p gMNp ,p2g\p N0 0 0 0 M0
(4.127)
with
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and H "p gMNp ,p2g\p * M* N* * * and the world sheet momentum is
(4.128)
P"H !H . * 0 The zero-mode partition function Z for the 6 torus
with
(4.129)
Z" q&*q &0 N0 N*
(4.130)
q "e\ pO
q"e pO,
(4.131)
may now be written in the form
with
Z" ep OL2K;exp(!pq [n2g\n!2n2g\bm#2m2(g!bg\b)m!2n2m]) L K
(4.132)
q"q #iq . (4.133) What we require is the partition function Z for the 2-dimensional toroidol compacti"cation 2-031 corresponding to the "xed plane of an N"2 twisted sector. Choosing the labelling of the complex planes such that it is the "rst complex plane that is the "xed plane, we should then take m and n of the form
m m
m"
0 0
,
n"
n n 0 0
0
0
0
0
.
(4.134)
The zero-mode partition function Z may then be cast in terms of m,m,n ,n ,b ,g ,g and 2-031 g as !pq "!¹;m#i¹m!i;n #n " , Z " ep OKL>KLexp (4.135) 2-031 ¹ ; L K where the moduli ¹ and ; associated with the N"2 complex plane are de"ned as in Eqs. (4.26) and (4.27). Returning to Eq. (4.107) and performing the q integration, as described in detail in Ref. [82], gives the contribution to the threshold correction from this N"2 twisted sector
D "!b [ln((¹#¹M ) " g(¹)")#ln((;#;M ) " g(;)")]#(moduli independent constant) , ? ? (4.136)
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where the Dedekind eta function is g(¹)"e\p2 (1!e\pL2) . (4.137) L The string loop threshold correction (4.136) can be seen to be invariant under the (target space) modular transformation (4.138)
¹P(a¹!ib)/(ic¹#d) corresponding to Eq. (4.3) by observing that under this transformation ¹#¹M P(¹#¹M )/"ic¹#d"
(4.139)
g(¹)P(ic¹#d) g(¹)
(4.140)
and and similarly for the ; dependent term. The complete threshold correction to the gauge coupling constant may be obtained as follows [82]. If the ith complex plane is left unrotated by a subgroup G of the point group G, then ¹/G is G G an orbifold with N"2 space}time supersymmetry. The threshold correction D for the original ? orbifold ¹/G may be written as
(b,)G"G " ln((¹G#¹M G)"g(¹G)") G #(moduli independent terms) , (4.141) D "! ? ? "G" #ln((; #;M )"g(; )") G G G G where the sum over i is over the N"2 complex planes i.e. the complex planes left unrotated in at least one twisted sector of the original orbifold and the moduli independent part of the threshold corrections contains the contribution of the N"1 complex planes. Here, (b,)G is the renor? malisation group coe$cient for the N"2 orbifold ¹/G . If the ith complex plane is a Z plane G + with MO2 then ; is not a (continuously variable) modulus but takes a "xed value, so that the G ; dependent term in Eq. (4.141) is just an additional constant term. To arrive at Eq. (4.141) it G should be noticed that the complete set of N"2 twisted sectors of the original orbifold ¹/G for which the ith complex plane is unrotated constitutes the twisted sectors of the N"2 orbifold ¹/G G and that the N"4 untwisted sector does not contribute to the threshold correction nor to b . Thus, ? combining the contributions of all these N"2 sectors of the original orbifold yields a coe$cient which is the renomalisation group coe$cient (b,)G. ? Although the derivation of the string loop threshold correction presented in this section is a one string loop order calculation it has been shown in an alternative approach using integrability conditions that there are no additional contributions from higher orders in string-perturbation theory [4,5]. 4.7. Modular anomaly cancellation and threshold corrections to gauge coupling constants The form of the moduli-dependent threshold corrections to gauge coupling constants can be partly understood in terms of cancellation of (target space) modular anomalies [71]. This approach also gives an alternative form for the numerical coe$cient in the threshold correction which involves the modular weights of the light states and is often more useful in practice.
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In the following discussion, we shall focus attention on the three moduli ¹ with modular G transformation as in Eq. (4.3). The transformation induced on the KaK hler potential is a particular KaK hler transformation as in Eq. (4.9) which we may write as KPK#h (¹ )#hM (¹M ) G G G G
(4.142)
with h (¹ )"ln(ic ¹ #d ) G G G G G and rewriting Eq. (4.12) the transformation on the scalar matter "elds is
(4.143)
P eLG?FG , (4.144) ? ? where nG is the modular weight of , with a corresponding transformation on the fermionic ? ? partners t of the scalar matter "elds [44,71] and on the gauginos j chosen to maintain modular ? ? invariance of the supergravity Lagrangian at classical level. However, this classical symmetry acting on chiral fermions is potentially broken at quantum level by anomalies due to triangle diagrams [44,71] with two gauge bosons plus a number of moduli as external legs and massless fermionic matter "elds and gauginos as internal lines. The one-loop anomaly for the gauge group factor G is a variation of the Lagrangian of the form ? (4.145) dL"(CM ) (h !hM )F@ FI IJ , G IJ @ ?G G where FI IJ is the dual "eld strength and the real constants (CI ) will be given shortly. This derives @ ?G from the variation of a supersymmetric Lagrangian term of the form
dL " dh((CI ) h =?= #h.c.) , ,-+ *-31 ? G G @ @?
(4.146)
where =? is the "eld strength (spinor) super"eld. @ The coe$cients (CI ) are calculated from the interaction terms in the low energy supergravity ?? theory that contribute to the anomaly triangle diagrams and these interaction terms are controlled by the KaK hler potential. For the gauge group factor G , the resulting coe$cient is ? (CI )G"(b )G/8p (4.147) ? ? with (b )G"!C(G )# ¹(R?)(1#2nG ) , (4.148) ? ? ? ? ? where C(G ) is the quadratic Casimir for the adjoint representation of G and ? ? ¹(R?)"Tr Q , (4.149) ? ? where Q is any generator of G in the representation R? to which the matter "eld belongs. ? ? ? ? In general, there can be two di!erent contributions to the cancellation of the modular anomaly to restore modular invariance at the quantum level. The "rst of these contributions is generated by a Green}Schwarz-type mechanism which involves allowing the dilaton "eld S, which
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does not transform under modular transformations at tree level, to undergo a transformation of the form dG SPS! %1 h 8p G G at one string loop level for some real coe$cients dG . The tree-level gauge kinetic term %1
(4.150)
" dh( f =?= #h.c.) @A @ A? %)
(4.151)
f "Sd @A @A then transforms into L #dL with %) %) dG dL "! %1 dh(h =?= #h.c.) %) G @ @? 8p
(4.152)
L with
(4.153)
and this cancels a part of the anomaly that is the same for each factor of the gauge group. To maintain modular invariance of the KaK hler potential, KK of Eq. (4.8) must be modi"ed at one string loop level to KK "!ln >! ln(¹ #¹M ) G G G
(4.154)
with dG (4.155) >"S#SM ! %1 ln (¹ #¹M ) . G G 8p G The remainder of the anomaly, which is not universal for all factors of the gauge group, will have to be cancelled by massive string mode contributions. Thus, the massive string mode contribution will have to cancel the variation
((b )G!dG ) %1 dh(h =?= #h.c.) . dL " ? G @ @? + 11*#11 +-"#1 8p
(4.156)
At this point, the knowledge gained in the previous section (in particular Eqs. (4.141) and (4.140)) suggests that the appropriate Lagrangian terms whose variation cancels the remainder of the anomaly is L
((b )G!dG ) %1 dh(ln(g(¹ ))=?= #h.c.) . "! ? G @ @? + 11'4# +-"#1 8p
(4.157)
This is a holomorphic term as would be expected for a term obtained from integrating out massive modes. In general, the (non-holomorphic) anomalous massless mode contribution dL is ,-+ *-31 generated by a non-local Lagrangian term. However, if we focus on the F@ FIJ term, with a view to IJ @ obtaining the string loop correction to the gauge coupling constant, then (for covariantly constant
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moduli) this part of ¸
is the local Lagrangian term ,-+ *-31 (b )G L " ? ln(¹ #¹M )F@ FIJ#2 . G G IJ @ ,-+ *-31 64p
(4.158)
Noticing that, for any function ,
( # M ) ( ! M ) dh( =?= #h.c)"! F@ FIJ# F@ (FI IJ)#2 . @ @? IJ @ IJ @ 8 8
(4.159)
and combining the F@ FIJ terms from Eqs. (4.151),(4.157) and (4.158), yields the string loop IJ @ corrected gauge coupling constant g given by [71] ? ((b )G!dG ) > %1 ln((¹ #¹M )"g(¹ )") , (4.160) g\" ! ? G G G ? 16p 2 G where the running of the gauge coupling constants has been ignored. Including the "eld theoretic one loop running of the gauge coupling constants g (k) at energy scale k, ? M 120',% #D (4.161) 16pg\(k)"16pg\ #b ln ? ? 120',% ? k
for level 1 gauge group factors, where (4.162) g\ "> 120',% gives the (rede"ned) gauge coupling constant at the string scale excluding the threshold correction D , and ? D "! ((b )G!dG )ln((¹ #¹M )"g(¹ )") . (4.163) ? ? %1 G G G G The Green}Schwarz coe$cients dG may be determined by comparing the threshold correction %1 (4.163) in the approach of this section with the threshold correction (4.141) in the approach of the previous section. We then see that dG "(b )G!(b,)G"G "/"G" . %1 ? ? G In general, the N"2 renormalisation group coe$cient is given by
(4.164)
(b,)G"!2C(G )#2 ¹(RG ) , (4.165) ? ? ? G where the sum over i is a sum over matter N"2 hypermultipletes in representations RG for the ? N"2 orbifold ¹/G . A special case is when there is a pure gauge hidden sector. Then, G (4.166) (b )G"!C(G )"b ? ? ? and dG "b (1!2"G "/"G") . G %1 ?
(4.167)
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Table 5 Non-¹#¹ Coexter Z orbifolds. For the point group generator h we display (m ,m ,m ) such that the action of h in the , complex orthogonal space basis is (ep K,ep K,ep K) Orbifold
Point group generator h
Lattice
Z !a Z !b Z !II!a Z !II!b Z !II!c Z !II!a Z !I!a
(1,1,!2)/4 (1,1,!2)/4 (2,1,!3)/6 (2,1,!3)/6 (2,1,!3)/6 (1,3,!4)/8 (1,!5,4)/12
SU(4);SU(4) SU(4);SO(5);SU(2) SU(6);SU(2) SU(3);SO(8) SU(3);SO(7);SU(2) SU(2);SO(10) E
In particular, if the ith complex plane is a Z plane, i.e. a plane where the point group acts as Z , then dG is zero. %1 4.8. Threshold corrections with reduced modular symmetry In Section 4.6, the assumption was made in the derivation of the threshold corrections to the gauge coupling constants that, whenver a twisted sector has a "xed plane, a decomposition of the 6-torus ¹"¹#¹ can be made with the "xed plane lying in ¹. When this assumption is not correct, which we shall refer to as the case of non ¹#¹ orbifolds, the discussion can be generalised as follows [158,29,30]. We shall see that the resulting threshold corrections have modular symmetries that are subgroups of PSL(2,Z). Non ¹#¹ Coxeter Z orbifolds are , tabulated in Table 5. Analogously to Eq. (4.107) we start from
dq dq Z2-031(q,q)!b, , (4.168) D " bF E F E ? ? ? C q C q F E where only the twisted sectors (h,g) for which there is a complex plane of the 6 torus ¹ "xed by both h and g contribute i.e. sectors which are twisted sectors of an N"2 space}time supersymmetric theory. In D , Z2-031 is the moduli-dependent part of the zero mode partition function for the ? F E 2 dimensional toroidal compacti"cation corresponding to the "xed plane of the (h,g) twisted sector, bF E is the contribution of the massless states in the (h,g) sector to the one-loop renormalisation ? group equation coe$cient and b, is the contribution of all N"2 twisted sectors. Unlike the ? ¹#¹ case, b, no longer factors out from the "rst term in Eq. (4.168) because Z2-031 now ? F E depends on the particular twisted sector. It is convenient to write D in terms of a subset (h ,g ) of ? N"2 twisted sectors (referred to as the fundamental elements) with the integration over an enlarged region CI depending on (h ,g ). Then,
D " bF E ? ? F E
dq dq (q,q )!b, Z2-031 . ? F E CI q CI q
(4.169)
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Here, the single twisted sector (h ,g ) replaces a set of twisted sectors which can be obtained from it by applying those PSL(2,Z) transformations that generate the fundamental regionI C of the world from the fundamental region of PSL(2,Z). In general, sheet modular symmetry group of Z2-031 F E is invariant under a congruence subgroup of PSL(2,Z) obtained by restricting the paraZ2-031 F E meters a,b,c,d in the PS¸(2,Z) transformation qP(aq#b)/(cq#d) . If we denote such groups by C (n) de"ned by c"0 (mod,n)
(4.170)
(4.171)
and C(n) de"ned by b"0 (mod,n) then, for example, for C (3), CI "+I,S,S¹,S¹,C ,
(4.172)
(4.173)
where S and ¹ are the PSL(2,Z) transformations S:qP1/q
(4.174)
¹:qPq#1
(4.175)
and
To calculate Z2-031 , for an orbifold with point group generated by h we "rst write the action of F E h on the basis vectors eP of the lattice of the 6-torus as M h:eP PeP Q . (4.176) M N NM Then the action of h on m and n of Eqs. (4.112) and (4.115) is h:mPm "Qm
(4.177)
h:nPn "(Q2)\n .
(4.178)
and
For the hI twisted sector, the "xed plane assocated with g "hI is determined by QIm"m, ((Q2)\)In"n
(4.179)
and m and n in the "xed plane are then parameterised by two integers. Using this form for m and n in Eq. (4.132), and introducing a metric g and an anti-symmetric tensor b for the two, , dimensional sublattice of the "xed plane with the moduli ¹ and ; de"ned in terms of g and b , the , , q integrations may be performed to obtain an expression for Z . For all non-¹#¹ orbifolds, ' E it is found that all fundamental sectors can be generated from fundamental sectors of the form (I,g )
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by applying world sheet modular transformations. The "nal result for the threshold correction is always of the form [158,29,30]
C ¹ D "! ((b )G!dG ) ln(¹ #¹M )# GK ln g G G G ? ? %1 2 l GK K G CI ; ! ((d )G!dI G ) ln(; #;M )# GKln g G , (4.180) ? %1 G G 2 lI GK G K where the sum over i is restricted to complex planes which are unrotated in at least one twisted sector (N"2 complex planes), and for the ; moduli is further restricted to complex planes for which the point group acts as Z . The coe$cients (d )G are de"ned analogously to Eq. (4.148) with ? the modular weights with respect to ; moduli replacing modular weights with respect to ¹ moduli, and the dI G are the Green Schwarz parameters for the ; modular transformations. The values of %1 G C , l , CI and lI are given in Table 6 for the various non-¹#¹ Coxeter Z orbifolds. In the GK GK GK GK , case of Z !II!b, the modulus ; is understood to be replaced by ; !2i. The range over which m runs depends on the value of i but always C " CI "2 . (4.181) GK GK K K In Eq. (4.180), the coe$cients (b,)G have been identi"ed using Eq. (4.164). ? The threshold correction D now has (target space) modular symmetries that are subgroups of ? PSL(2,Z), e.g. for the Z !II!a orbifold, the part of the threshold correction involving ¹ and ; has the form D "!((b )!d )(ln(¹ #¹M )"g(¹ )"(; #;M )"g(; )") ? ? %1 ¹ (4.182) #ln (¹ #¹M ) g (; #;M )"g(3; )" , 3
Table 6 Values of C , l , CI and lI for non-¹#¹ Coxeter Z orbifolds GK GK GK GK , Orbifold
C ,CI GK GK
l , lI GK GK
Z !a
C "2 CI "2 C "C "1 CI "CI "1 C "2, C "C "1 CI "CI "1 C "2, C "C "1 CI "CI "1 C "2, C "C "1 CI "CI "1 C "C "1 CI "CI "1 C "2
l "2 lI "1 l "1, l "2 lI "1, lI " l "2, l "1, l "3 lI "1, lI " l " l "1, l "3 lI "3, lI "1 l " l "1, l "3 lI "1, lI " l "1, l "2 lI "1, lI " l "2
Z !b Z !IIa Z !IIb Z !IIc Z !IIa Z !Ia
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which for modular transformations on ¹ is invariant under C(3) and for modular transforma tions on ; is invariant under C (3) with C(3) and C (3)de"ned by imposing the conditions (4.171) or (4.172) in Eq. (4.3) or (4.4). The modular symmetries of the threshold corrections for the non-¹#¹ case may also be determined without explicit calculation of the threshold corrections [31] by using a method [151,174,175] which explores the modular group that leaves invariant the spectrum of the twisted sectors. In the presence of discrete Wilson lines, knowledge of the explicit threshold corrections [160] is limited to the case without moduli but the modular symmetries may be determined by a generalisation of the above approach [87,174,31,153] both for the ¹#¹ case and for the non-¹#¹ case. On the other hand, explicit calculations of the e!ect of Wilson line moduli on the threshold corrections are available [6,45,159]. 4.9. Unixcation of gauge coupling constants From Eq. (4.161), the running of the gauge coupling constants (assume level 1 gauge group factors) is given by 16pg\(k)"16pg\ #b ln ? 120',% ?
M 120',% #D ? k
(4.183)
with D given by Eq. (4.163) for ¹#¹ orbifolds and by Eq. (4.180) to include non-¹#¹ ? orbifolds, and with g
+0.7 120',%
(4.184)
and M
+0.53 g ;10 GeV , 120',% 120',%
(4.185)
where g is the common value of the gauge coupling constants at the string tree level 120',% uni"cation scale M . 120',% If there are no additional states, over and above the (minimal supersymmetric) standard model states, with masses interemediate between the electroweak scale and the string scale, [7,32] then it may be necessary to explain the di!erence between the `observeda uni"cation of gauge coupling constants at M +2;10 GeV 6
(4.186)
and tree-level un"ciation scale M by the occurrence [130,129,33] of suitable moduli depen120',% dent threshold corrections D . (The moduli-independent part of the threshold correction is small ? [136,8,133].) If g and g are gauge coupling constants for two factors of the SU(3);SU(2);;(1) ? @ standard model gauge group then M 120',%" a@ ?G\@ @G @?\@@ aB ?H\B @H @?\@@ , G H M 6 H G
(4.187)
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where the product over i is over N"2 complex planes and the product over j is over N"2 complex planes for which the point group acts as Z , GK ¹ ! a "(¹ #¹M ) g G (4.188) G G G l GK K and
; !I HK . (4.189) a "(; #;M ) g H H H H lI HK K The coe$cients (b )G and (d )H may be written in terms of the modular weights of 3 generations of ? ? quarks and lepton and the electroweak Higgses h and hM in the supersymmetric standard model as (b )G"3# (2nG #nG #nG ) , /E SE BE E (b )G"5#nG #nGM # (3nG #nG ) F F /E *E E
(4.190) (4.191)
and 33 3 1 (b )G" # (nG #nGM )# (nG #8nG #2nG #3nG #6nG ) (4.192) /E SE BE *E CE F 5 5 F 5 E with similar expressions for (d )H with modular weights with respect to ; replacing modular ? H weights with respect to ¹ , where g labels the generations and ¸(g) and Q(g) are lepton and quark G SU (2) doublets. * It is possible to generate all possible modular weights of the massless matter with quark, lepton and Higgs quantum numbers in the twisted sectors of an arbitrary Z or Z ;Z orbifold with , + , SU(3);SU(2);;N(1) gauge group (allowing for extra ;(1) factors to be spontaneously broken along #at directions at a high-energy scale), as discussed in Section 4.5. Then, under the simplifying assumption that a single ¹ modulus is dominating the threshold corrections (either one of the ¹ or G ¹"¹ "¹ "¹ ) the Z and Z ;Z orbifolds that permit a uni"cation solution with , + , M (M can be identi"ed [129]. For any particular choice of modular weights that permits 6 120',% such a solution the value of the dominant modulus to achieve uni"cations at 2;10 GeV can then be calculated. In general, this results in values of the dominant modulus ¹ that are unnaturally B large in Planck scale units. (Re ¹ &20 is typical). Smaller values can be obtained in a variety of B ways e.g. by including Wilson line moduli as well as ¹ and ; moduli in the threshold correction [169] or by assuming that uni"cation at M occurs to a gauge group larger than the standard 6 model [34,36] with the massless states of the supersummetric standard model below that scale. In this latter case, the renormalisation group coe$cients b are those of the supersymmetric standard ? model but the threshold corrections, which are determined by the massless states at the string scale, are modi"ed. For a review of other options see Dienes [72]. A more radical possibility is that the di$culty lies not in the running of the gauge coupling constants but in the gravitational constant which is modi"ed by the appearance of a "fth dimension above a certain energy in M-theory [190].
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5. The e4ective potential and supersymmetry breaking 5.1. Introduction We have seen in earlier sections that the moduli dependence of the Yukawa couplings, the gauge kinetic functions, and the KaK hler potential can, in principle, account for many otherwise puzzling features of low-energy phenomenology, such as the hierarchy of fermion masses and the `precociousa uni"cation of the observed gauge couplings at an energy scale a factor of 20 or so below the string scale. Our foregoing discussion, however, does not address the question of why and whether the moduli have the particular values needed to solve these problems. Nor does it explain why the N"1 space-time supersymmetry is broken at a hierarchically low energy scale compared with the string scale; this is required phenomenologically, both in order to protect the TeV scale of electroweak symmetry breaking from string scale corrections, and in order to achieve the `observeda uni"cation of the gauge coupling strengths. We shall see in this section how these two shortcomings are related, and remedied. The obvious approach to the "rst problem, the stabilisation of the moduli, is to calculate the e!ective potential for the relevant "elds and determine which values of the moduli minimise it. However, the moduli potential is #at to all orders in string perturbation theory, when space}time supersymmetry is unbroken [74,189]. This follows from a non-renormalisation theorem in string theory directly analogous to the familiar non-renormalisation theorems in supersymmetric "eld theories. The point is that each (scalar) moduli "eld is a component of a chiral supermultiplet which necessarily contains a pseudoscalar partner of the scalar mode. In the case of the T-modulus de"ned in (Section 4.1), the real part, associated with the overall size of the torus, is (the expectation value of ) a scalar "eld while the imaginary part is (the expectation value of ) a pseudoscalar "eld. The vertex operator for the pseudoscalar "eld is
< J dz BR X(R XM #k ) W ) WM )e\ I 6 , X X
(5.1)
where k is the four-momentum, and X is the complex world sheet made from the two compacti"ed dimensions under consideration. At zero momentum only the "rst (bosonic) term survivies, and this vanishes because it is a total derivative. Thus the zero momentum mode decouples, and the theory is invariant under the Peccei-Quinn axionic symmetry (see Eq. (4.18)). BPB#const ,
(5.2)
as noted in Eq. (4.18). As a result, the superpotential is independent of the pseudoscalar "eld B. However, because of supersymmetry, B can only appear in the combination ¹ given in Eq. (4.1). So the superpotential is independent of ¹ and the moduli e!ective potential is therefore #at to all orders in string perburbation theory. It follows that the moduli, and in particular the size of the compacti"ed dimensions, have their values "xed by non-perturbative e!ects and/or supersymmetry breaking. A similar argument applies to the dilaton moduli "eld S. The non-perturbative mechanism which has attracted most attention, and upon which we shall concentrate in this section, is hidden sector gaugino condensation [91,70,73]. Because of asymptotic freedom gauge coupling strengths increase as the energy
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scale M is reduced from the string scale (m ). The quantitative relationship is given by the renormalisation group equation which to one loop order gives Mep @E+"m ep @EK , where b"!3c(G)# ¹(R?) ? determines the leading term of the beta function b b g#2 g# b(g)" (16p) 16p
(5.3)
(5.4)
c(G) is the quadratic Casimir for the adjoint representation of the (simple) gauge group G and ¹(R?) the usual Casimir for chiral supermultiplets: ¹(R?)"Tr(Q?)
(5.5)
where Q? is the matrix representing any generator of G in the representation R? to which the chiral matter belong. The gauge coupling becomes large at a scale K where exp(8p/bg(K)) is of order unity, and is given by (5.6) KKm ep @EK which is exponentially suppressed relative to the string scale. When this occurs we entertain the possibility of gaugino condensation in which the quantity j j , bilinear in the gaugino "elds, ? @ acquires a non-zero vacuum expectation value (VEV) with "1j j 2 "&K . (5.7) ? @ In this regime the use of a "eld theoretic description in terms of gauge and gaugino "elds alone is inadequate. In a globally supersymmetric theory the supersymmetry can only be broken by the F-term of a chiral supermultiplet acquiring a non-zero VEV. The gaugino bilinear j j is (proportional to) ? @ the lowest component of the (composite) chiral super"eld =?= , where =? is the usual "eld ? @? ? strength chiral super"eld, and is therefore not an F-term. Thus gaugino condensation does not break global supersymmetry, and this is con"rmed by explicit claculations [184]; this also agrees with conclusions following from Witten's index theorem [186,187]. However, in a locally supersymmetric theory, a supergravity theory, things are di!erent [70,73,179]. Under a local supersymmetry transformation of the spinor component t of a chiral supermultiplet, this gaugino bilinear j j does appear in dt. So 10"dt"02O0 if a gaugino ? @ condensate occurs, and supersymmetry is broken; for this to happen the gauge kinetic function must be non-minimal. This breaking of local supersymmetry in the hidden sector provides a seed for supersymmetry breaking in the observable sector, which is coupled to the hidden sector only by gravitational interactions.
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5.2. Non-perturbative superpotential due to gaugino condensate(s) In the strongly interacting regime we need more than just the usual gauge kinetic piece of a globally supersymmetric Lagrangian: L
" dhf (U)=?= #h.c. , @A @ A? %)
(5.8)
where f (U) is the gauge kinetic function, dependent on the gauge singlet chiral super"elds U, @A including the moduli super"elds; = is the standard (spinor-valued, chiral) gauge "eld strength A? super"eld whose lowest dimension component is the gaugino "eld j . In addition, we need an A? e!ective Lagrangian to describe the interactions of the (bound states and) possible gaugino condensate, which arise as consequences of the strong gauge interactions. We therefore construct a composite supermultiplet ; to describe the lightest of the non-perturbative states. This is assumed to be a gauge singlet chiral super"eld ;,4=?= (5.9) @ @? which has the (singlet) gaugino, bilinear combination j j as its lowest dimension [M] compon@ @ ent. Then ; develops a vacuum expectation value if the gaugino condensate forms. To determine if it does we need an e!ective Lagrangian for the composite "eld ;. Because of its non-canonical dimensions the kinetic term for ; is [184]
9 L " dh dh (;;) , ) c
(5.10)
where c is a dimensionless constant. The inclusion of this term in the e!ective theory means that the KaK hler potential K, discussed in the previous section is modi"ed by these non-perturbative e!ects. If we now denote by KI the KaK hler potential in the absence of a condensate then the complete KaK hler potential is given by K"KI #K ,
(5.11)
where
9 K"!3 ln 1# e)I (;;M ) (S SM )\ c
(5.12)
with S a `chiral compensatora super"eld with scaling dimension unity [63,148]. The choice of S determines the normalization of the gravitational action L J e\) S SM " M R (5.13) FF so (5.14) e\) S SM " M "1/16pG Jm , FF and we see that the condensate contribution to the KaK hler potential is suppressed by the square of the Planck mass m.
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The Lagrangian must also be augmented by a term which reproduces the anomalies of the underlying theory [148,131]. The anomalies in question are the chiral anomaly, the scaling (energymomentum trace) anomaly, and the supersymmetry current c-trace anomaly, and all are proportional to (di!erent) components of the composite super"eld ;. The chiral anomaly, for example, is given by RIJ"!(b(g)/2g)F FI IJ (5.15) I ?IT ? where b(g) is given in Eq. (5.4); F is the usual non-Abelian "eld strength, and FI is its dual. In ?IJ ?IT the same notation the anomaly of the energy-momentum tensor is hI"(b(g)/2g)F FIT I ?IT ? and the supercurrent trace anomaly is
(5.16)
cIS "(b(g)/g)F pIJj . (5.17) I ?IT ? In order to ensure that the anomalous Ward identities are satis"ed to tree order we add a term [182] L
b(g) dh ; ln(c;/S)#h.c. "! 6g
to the Lagrangian (c is an unknown constant). L ;(x,h,hM )P e ?;(x,h e\ ? ,hM e ? )
(5.18)
is chosen so that under chiral transformations (5.19)
and under scale transformations ;(x,h,hM )PeA;(x eA,h eA ,hM eA )
(5.20)
the variation of the action dx L gives precisely the required chiral, scaling and superconfor mal anomalies. It is easy to see how this works. The F term of ; ln ; clearly includes, among others, the term F ln u, where u is the scalar component of ; and F is the F-part. Under the transformations (5.19) 3 3 and (5.20) above, F transforms covariantly whereas 3 ln uPln u#3ia (5.21) and ln uPln u#3c ,
(5.22)
respectively. Then L generates the anomalous terms ia(b(g)/2g)(F !FR ) and c(b(g)/2g); 3 3 (F #FR ) which are just the required anomalies. 3 3 Taking the (hidden sector) gauge kinetic function in Eq. (5.8) to be f (U)"f (U)d @A % @A we see that we may combine L
(5.23)
and L to yield the non-perturbative superpotential %) = I " f (U);!(b(g)/6g); ln(c;/S) . (5.24) %
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Although we have taken proper account of the scaling and chiral anomalies, we must also ensure that the e!ective theory is invariant under the target space modular transformations [95,101,41,166] de"ned in Eq. (4.3): a ¹ !ib G (i"1,2,3) (5.25) ¹P G G G ic ¹ #d G G G with a ,b , c , d integers satisfying G G G G a d !b c "1 . (5.26) G G G G Since the KaK hler potential KI in the absence of the condensate(s) already satis"es Eq. (4.9) KI PKI #ln"ic ¹ #d " , (5.27) G G G we require that the additional piece K arising from the condensate is modular invariant. Then from Eq. (5.12) we infer that ;/S has modular weight !1 (;/S)P(;/S)(ic ¹ #d )\ . (5.28) G G G The modular invariance of G,K#ln "=/S" (5.29) requires that =/S has modular weight !1, as noted in Eq. (4.11). In general, for this to be satis"ed by the non-perturbative contribution = given in Eq. (5.24), we have to include some further ¹ -dependence in =. It follows from Eqs. (4.161), (4.162) and (4.163) that (the holomorG phic part of ) the gauge kinetic function is 1 (5.30) f (U)"S! (b G !dG )ln g(¹ ) , % %1 G % 8p G where the second term derives from string loop threshold corrections to the (hidden sector) gauge group coupling constant, and the dG are to cancel anomalies under the target space duality %1 transformations. We may write 1 f (U)"R! bG ln g(¹ ) , % % G 8p G where
(5.31)
1 R,S# dG ln g(¹ ) . (5.32) %1 G 8p G Then under the duality transformations, it follows from Eqs. (4.150) and (4.140) that R is invariant and 1 f (U)Pf (U)! b G ln(ic ¹ #d ) . % % G G G 8p %
(5.33)
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To ensure that = has the required modular weight we replace = I in Eq. (5.24) by b(g) =" f (U);! ; ln[;v(¹ )/S] , % G 6g where v(¹ ) has modular weight n . Then =/S has weight !1 provided G G 3g bGY "1!3bGP/b n "1! % % G 16pb(g) %
(5.34)
keeping only the "rst term of b(g) given in Eq. (5.4). Now v(¹ )J g(¹ )LG (5.35) G G G has weight n , and Ferrara et al. [95}97] have argued that this is the unique ¹ dependence which G G does not lead to unphysical zeros or poles in the upper-half of the i¹ complex plane. Thus "nally G we obtain the superpotential
b(g) 1 ; ln c; g(¹ )LG/S =" f (U);! G 6g 4 % G 1 b " ;R! % ; ln c; g(¹ )/S . (5.36) G 4 96p G The above treatment is easily generalised to the formation of several gaugino condensates, associated with hidden sector (non-abelian simple) gauge groups G (n"1,2,p). There are then L p composite chiral super"elds ; , and the non perturbative superpotential is L N 1 b =" ; R! L ln c ; g(¹ )/S (5.37) L L G 4 L 96p L G with b determining the leading term of the beta function b (g ) of G , and the c unknown L L L L L constants. To determine whether gaugino condensation, and hence supersymmetry breaking, actually occurs we need to calculate the e!ective potential deriving from the supergravity theory we have obtained, and to see whether the scalar component(s) u of ; have non-zero values at the L L minimum. This is the calculation to which we now turn.
5.3. Ewective potential The e!ective potential in any supergravity theory is given by < "e%[G (G\) G !3] , where G,K#ln "="
(5.38)
(5.39)
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with K the KaK hler potential and = the superpotential, and we are keeping only the scalar M are components u and uH of the chiral super"elds U and U? in terms of which K, = and = de"ned. The derivatives of G are written as and
G ,RG/Ru ,
G ,RG/RuH
(5.40)
G ,RG/Ru RuH . (5.41) Then (G\) is the inverse of the matrix G . In the case under consideration the chiral super"elds involved are those whose scalar components are the dilation "eld S, de"ned in Eq. (4.36); the orbifold moduli "elds ¹ , ; , de"ned in Eqs. G G (4.1) and (4.2), some of which are "xed by the point group; the condensates u ; and other matter L "elds u , including Higgs "elds H and H . Evidently the calculation and minimization of < in ? full generality is a formidable calculation when several moduli and gauge condensates are active. The calculation of < in the case of a single (overall) modulus ¹, and when the dilation "eld S is modular invariant (dG "0), but with several gaugino condensates, has been done by Taylor [181]. %1 He notes the existence of a zero-energy local, but not global, minimum, which corresponds to the weak coupling (i.e. Re SPR) limit. In this limit < "="0, corresponding to a supersymmetric vacuum. This supersymmetry is not surprising. The weak coupling limit corresponds to in"nite Planck mass, since as we have seen in Section 4 the KaK hler potential has a leading term K&!ln(Re S)
as Re SPR
(5.42)
and then from Eq (5.14) we see that m PR as Re SPR . (5.43) In this limit only global supersymmetry survives, and we have already noted that a gaugino condensate cannot break global supersymmetry. In this weak coupling limit the potential is minimised when R=/R; "0 , (5.44) L i.e. the global F-terms vanish. Using Eq. (5.37) we "nd that the condensate is then given by k u (S,¹ )" epR @L g(¹ )\ . (5.45) L G G ce L G Substituting Eq. (5.45) into = eliminates the dependence upon the condensate and we obtain the `truncateda superpotential b = " L u (R,¹ ) (5.46) G 96p L L entirely in terms of the moduli "elds. As we have said, this is a good approximation provided that Re S is stabilised at a `largea value at the minimum of the e!ective potential. Phenomenologically we require 1 g(m ) & (5.47) (4pRe S)\" 24 4p
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for coupling constant uni"cation, so 1Re S2&2 which is not particularly large. Further, we shall see that for a single condensate at least, the e!ective potential does not have a local minimum at a "nite value of S [73,101,95]. So we shall assume that some other mechanism is responsible for stabilising the dilation <E<. In view of the complexity of minimizing the full e!ective potential, it is desirable to "nd a more economical procedure, and the one which has received considerable attention consists of using the truncated superpotential (5.46), in which the condensates are assumed to have the form (5.45), rather than the full non-perturbative superpotential (5.37). The justi"cation for doing this is "rst to note that the form (5.46)
= "X(R) g(¹ ) , G G where
(5.48)
X(R)" d epR @L L L
(5.49)
with d "b k/96pc e"constant , (5.50) L L L is essentially required by the fact that = must have modular weight !1. It has further been noted [46,156,56] in the case of a single overall ¹ modulus and dG "0, that for small values of %1 "u "/"k", the form (5.45) for the condensate can be deduced from the extremum conditions on the L L full e!ective potential with the assumption of modular covariance. Then for Re S'!b /24p , (5.51) L "u ";"k", and it follows that the full e!ective potential is well approximated by the truncated L L e!ective potential obtained using = and the original (condensate-independent) KaK hler poten tial KI . The above condition is satis"ed for a wide range of values of Re S including the realistic case where Re S&2. 5.3.1. Pure gauge hidden sector For the remainder of this section we shall therefore use the truncated superpotential (5.48) and the e!ective potential which derives from it using the KaK hler potential K. The simplest case is when the hidden sector is a pure gauge Yang}Mills theory, i.e. there is no hidden sector matter. Then the KaK hler potential is given in (4.154) KI "!ln >! ln(¹ #¹M ) G G G and >, given in Eq. (4.155), can be written 1 dG ln(¹ #¹M )"g(¹ )" . >"R#R! %1 G G G 8p G
(5.52)
(5.53)
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The e!ective potential is calculated using Eqs. (5.38) and (5.39):
< ">\ (¹ #¹M )\"g(¹ )"\ "X!>XR"!3"X" G G G G > 1 # X! dG XR (¹ #¹M )"GK G" , %1 G G >!(1/8p)dG 8p %1 G where
dg GK G,(¹ #¹M )\#2g(¹ )\ G G G d¹
(5.54)
(5.55) G
and XR,dX/dR .
(5.56)
The hope is that this potential has a minimum at "nite values for the moduli ¹ and R, and that G the consequent value of > corresponds to a realistic values 2g\ . Unfortunately, this does not happen generically. For a reasonable value of R (and hence >) the potential does develop a minimum at "nite values of ¹ . If R is "xed, then for reasonable values and the case of a single G overall modulus ¹,¹ "¹ "¹ , there is always a minimum [101,68] with ¹&1.23. However, as mentioned previously, the potential does not obviously have a minimum at a "nite value of R: in fact for a single condensate the only stationary point of < at "nite R is a maximum [47]. The condition for a stationary point is R gives
> (¹ #¹M )"GK G"(XM !d XM RM ) !>XRR(XM !>XM RM ) (X!>XR) 2XM ! G G (>!d ) G G G d G (¹ #¹M )"GK G"(XM !d XM RM ) , ">XRR G G >!di G G where d ,dG /8p , G %1 XR,RX/RR, etc . In the case that dG "0, so >"2Re R"2Re S, the above condition reduces to %1 (2!GI )(X!>XR)XM ">XRR(XM !>XM RM ) ,
(5.57)
(5.58) (5.59)
(5.60)
where GI , (¹ #¹M )"GK G" , G G G which may be satis"ed trivially, when X!>XR"0 ,
(5.61)
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or non-trivially. De Carlos et al. [47] have shown that if the trivial solution gives a reasonable value of >, then it will always correspond to a minimum of < , whereas the non-trivial solution is never a minimum. Further, in the trivial case, we see by inspection that the minimum of < occurs at any zero of the modi"ed Eisenstein function GK G, and in particular at the "xed points ¹"1 and e p of the modular group. These statements are easily veri"ed for the case of a single condensate X(R)"de\?R
(5.62)
a"!24p/b'0 .
(5.63)
with
Eq. (5.61) gives (5.64)
>"!1/a(0 an unphysical value. The non-trivial solution with >'0 is >"(2!GI /a
(5.65)
which is clearly a maximum of e\?7 [(1#a>)!3#GI ] . < J >
(5.66)
The situation is not much better when we have two or more condensates. For realistic values of > 24p Re R/"b "<1 L as already noted. Then the trivial (minimum) condition (5.61) reduces to XR"0
(5.67)
(5.68)
or c\ epR @L"0 . L L So for two condensates we get
(5.69)
1 1 1 1 \ c ln , >"Re R" ! (5.70) 2 c 24p b b and for the unknown constants c of order unity this is typically small, and therefore unrealistic. L Similar conclusions are reached for three or more condensates. The foregoing conclusion is largely una!ected by consideration of the more realistic case with dG O0, although the complexity of Eq. (5.57) necessitates a numerical treatment. In essence, the %1 parameter d , in which the dG appears in Eq. (5.57), is generically small, so the e!ects may be G %1 calculated perturbatively in d . In any case, it is important to note that dG O0 severely constrains G %1 the formation of multiple pure gauge condensates. The reason is that any complex plane i which is
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not an N"2 plane does not contribute to the threshold corrections to the gauge coupling constants, and consequently not to the gauge kinetic function either. So for this particular plane bGY"dG (5.71) L %1 for each gauge group G . However, for a pure gauge condensate we see from Eq. (4.166) that L (5.72) bGY"b , L L so b "b ,b (5.73) L K for any two of the hidden sector gauge group G and G , since the right-hand side of Eq. (5.71) is L K independent of n. Thus each of the condensates has the same exponential exp(24pR/b) and the system e!ectively has just one condensate. This eliminates all Z orbifolds from consideration, , since each of them has at least one non N"2 complex plane, as is apparent from Table 1. The Z ;Z models are, however, una!ected. + , 5.3.2. Hidden sector with matter In view of the di$culty in stabilizing the dilaton "eld > at an acceptable value with a pure gauge hidden sector, the natural recourse is to study the e!ects of hidden matter [156,182,56,46,157,9,134] Then, besides the "eld strength supermultiplets =??, with a labelling the generators of the gauge group G, we have chiral matter multiplets QG , with m"1,2,M labelling the multiplets, and K i labelling the components of the representation of G to which Q belong. We assume that for each K multiplet QG there is a chiral supermultiplet QM belonging to the complex conjugate representation K KG of G to which QG belongs. Then, in the strong coupling regime discussed in Section 5.1, besides the K formation of a gaugino condensate, we entertain the possible formation of chiral matter condensates 1q qG 2 O0 and bound states, just as in QCD we get mesons from quark anti-quark G KG K bound states; in a supersymmetric theory we have also the possibility of bound squark}antisquark states. We assume too that the charged matter "elds QG and QM are coupled to gauge singlet K KG super"elds A by trilinear terms in the perturbative superpotential K =" h (¹ )A QG QM K G K K KG K G such that the `quarksa develop non-zero masses
(5.74)
m "h (¹ )1A 2 (5.75) K K G K when the gauge singlet "elds develop non-zero VEVs. The trilinear terms give a contribution
L " dh h (¹ )A QG QM K G K K KG K G to the Lagrangian. To describe the bound states we de"ne the M gauge singlet composite chiral super"elds < , QG QM (m"1,2,M) (5.76) K K KG G which contain the squark}antisquark bilinear q q G as the lowest dimension [M] components. G KG K
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In the absence of the mass terms the global symmetry of the (supersymmetric hidden sector gauge) theory is SU(M) ;SU(M) ;;(1) ;;(1) ;;(1) (5.77) * 0 4 0 if the M `quarka super"elds all belong to the same representation R of G. In any case there is an extra ;(1) symmetry compared with the non-supersymmetric case which relates to the gaugino "eld. The chiral ;(1) acts on the gaugino composite super"eld as in Eq. (5.19) and on the matter composite super"elds < as K < (x,h,h)Pe ?< (x,he\ ? ,hM e ? ) , (5.78) K K while the ;(1) symmetry acts only on the matter super"elds so 0 ;(x,h,hM )P;(x,h,hM ) , (5.79) < (x,h,hM )Pe @< (x,h,hM ) . K K Both of the above ;(1) symmetries are broken at the quantum level by the Adler}Bell}Jackiw anomaly. Under the chiral ;(1) we get dL "!a(b/32p)FFI (5.80) with b de"ned in Eq. (5.3), so b"!3c(g)#2M¹(R)
(5.81)
in the case that the M `quarka super"elds are all in the representation R of G. Under the ;(1) 0 transformation dL "b(2c/32p)FFI 0 where
(5.82)
c"2 ¹(R )"2M¹(R) . (5.83) K K As before, we need an e!ective Lagrangian expressed in terms of the composite super"elds, which reproduces these anomalies, and which yields an e!ective non-perturbative superpotential with the correct modular weight (!1); the modular weight of the gaugino composite "eld is !1, as before, and it is easy to see that the matter composite "elds < /S have modular weight !2/3. Then K proceeding as before we obtain the full non-perturbative superpotential to be
b 1 ; ln c;>A @ <\20 @ g(¹ )/S ! h (¹ )A < . (5.84) =" ;R! K G K G K K 96p 4 K G K Also as before, we shall instead use the `truncateda superpotential which is obtained by eliminating the composit super"elds ;,< using K R=/R;"0"R=/R< . (5.85) K
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(It is unclear whether this enjoys the same numerical justi"cation as in the pure gauge case.) This gives ; 2¹(R) <" , K 32p h (¹ )A K G K 2¹(R) 32pe @>A A\@ \@ @\A ;"k epR @\A c g(¹ ) [h (¹ )A ]20 @ G K G K 32p 2¹(R) G K
(5.86)
and b!c = " ;. 96p
(5.87)
The form of the trilinear coupling (5.74) may be generalised to the form =" h (¹ )A QG QM , (5.88) ?KL G ? K LG K L ? where there are arbitrary number of gauge singlet super"elds A with more general couplings. The ? e!ect is the replacement in = h (¹ )u Pdet M , K G K K where
(5.89)
M , h (¹ )A (5.90) KL ?KL G ? ? is the `quarka mass matrix. The dependence of the Yukawa couplings h on the moduli ¹ is ?KL G well-understood, as we saw in Section 3. Non-trivial dependence arises only when all three of the coupled "elds are (point group) twisted sector states. It is also easy enough to generalize to the case when the `quarka composite "elds < belong to K di!erent representations R of G. However, the multi-gaugino condensate is typically di$cult to K handle. The reason is that in general the quark "eld Q belongs to non-trivial representations K R of several gauge groups G , just as the quark "elds in the standard model belong to non-trvial KL L representations of SU(3) and SU(2). Thus the di!erent gaugino condensates are coupled to each other unless, for each m, R is non-trivial for precisely one n. In that case the quark condensate is KL proportional to a single gaugino condensate, just as in the single condensate case already discussed. To procede further we need the KaK hler potential K for the matter "elds A . At tree graph level K ? we have seen in Eq. (4.79) that for untwisted matter, and for orbifolds whose point group does not act as Z in any complex plane (so the ; moduli are "xed) the matter contribution to the KaK hler potential is K " (¹ #¹M )\"u " K G G G G
(5.91)
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so the "elds u have modular weight !1. For Z planes the situation is slightly more complicated, G and for twisted matter we have seen in Section 4.5 that (5.92) K " (¹ #¹M )L?G"u ", K?"u " , ? K ? K G G ? ? G where the modular weights n are model-dependent and calculable. ?G The e!ect of the matter condensate has been studied [156] in the simpli"ed case that there is a single overall ¹ modulus ¹ "¹ "¹ "¹ and a single untwisted gauge singlet "eld A having modular weight !1, and dG "0. Then %1 K"!ln(S#SM )!3ln(¹#¹M !"A") (5.93) and the e!ective potential is given by
1 < "(S#SM )\(¹#¹M !"A")\ "(S#SM )= !="# (¹#¹M !"A")"= #AM = " 1 2 3
1 = # (¹#¹M !"A") = !3 !3"=" , 2 3 ¹#¹M !"A"
(5.94)
where (5.95)
= ,R=/RS,etc. 1 In this case the truncated superpotential reduces to
epR @\A = J g(¹)@AA
(5.96)
and then the e!ective potential is
1 3c < " "="(S#SM )\(¹#¹M )\(1!"AI ")\ 3(1!"AI ")\" f "# "AI "\ 3 1 b!c
#
3b ["(¹#¹M )GK "!1] , b!c
(5.97)
where "AI ","A"/(¹#¹M )
(5.98)
is duality unvariant, = f ,1!(S#SM ) 1 1 =
(5.99)
and GK is de"ned in Eq. (5.55). As before, for the single gaugino condensate under consideration < has no stable minimum for the dilaton at a "nite value, so LuK st and Taylor [156] take S and f as free parameters whose value 1 is "xed by some other mechanism. The modular invariance of < means that the self-dual points ¹"1 and ¹"e p are stationary points, but they may be maxima, minima or saddle points
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depending on the parameters b,c and f . In the case b#2c(0, < always has a non-trivial 1 minimum with AI O0, so non-vanishing `quarka masses are dynamically generated, and local supersymmetry is spontaneously broken. Further, the parameter f can be "ne-tuned so that < is 1 zero at this minimum; in other words the cosmological constant vanishes. Simultaneously the compacti"cation scale is determined to be of order the Planck mass. In the case b#2c'0, however, there is always a zero energy minimum of < at AI "0, the condensates are zero, and supersymmetry is unbroken; so there is no dynamical mass generation and the compacti"cation radius is undetermined. The continuing di$culty of stablizing the dilaton has led de Carlos, Casas and Muno z [46] to study multiple gaugino condensate, with the (tacit) assumption that the matter "elds transform non-trivially with respect to only one of the gauge groups. Again they take a single overall modulus ¹, a single gauge singlet "eld A, and dG "0. Their numerical analysis indicates that < does not %1 have a true minima even for two condensates. This is understood by noting that the VEV of AI , de"ned in Eq. (5.98), is expected to be small, since it vanishes perturbatively. Then, since the superpotential has a power dependence of AI , see Eq. (5.96), the dominant contribution to < in Eq. (5.94) comes from the term proportional to "= ", except for a small region where = "0. Thus (5.100) < &(S#SM )\(¹#¹M )\"= " which has an absolute minimum at = "0 . (5.101) However, it is clear that this cannot be satis"ed for a single condensate of the form (5.96). The authors note that this de"ciency can be remedied if the superpotential is augmented by a perturbative contribution ="A
(5.102)
which models the generic cubic self-interaction of the gauge singlet "elds A : ? =" hK (¹ )A A A . ?@A G ? @ A ? @ A Then Eq. (5.101) gives c c =" ;. A" 96p b!c
(5.103)
(5.104)
If we now substitute back into = we get an e!ective superpotential as a function of S and ¹ alone. Not surprisingly it has the form (5.48) previously derived from the requirements of modular invariance and the consideration of anomalies =JepR @g(¹)\
(5.105)
although the value of b is now includes contributions from the matter as well as the pure gauge contributions. Again, of course, the dilaton cannot be stablised with a single condensate gauge group.
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However it is now rather easy to do so with two gauge groups [56,181,145] with the unknown constants taking values of order unity. As before, the minimum occurs at a value ¹&1.23 in all cases [101,47], but the value of > depends upon the exact gauge groups and the representations occupied by the hidden matter. There is no di$culty in obtaining physically reasonable values of > [47]. 5.4. Supersymmetry breaking Spontaneous breakdown of local supersymmetry occurs when the Goldstone fermion is `eatena by the gravitino, thereby giving it the extra degrees of freedom needed for a massive spin 3/2 particle. The supergravity Lagrangian contains a four fermion term (5.106) L "f t pIJj t c j #h.c. , @* T* I A0 $ @A * where j are the gaugino "elds of the (hidden) gauge group G, t is the gravitino "eld, t are the @ J fermionic components of the chiral super"elds U , f (U) is the (non-minimal) gauge kinetic ?@ function and f ,Rf /Ru . (5.107) @A @A Evidently, if there is a gaugino condensate, the above term mixes the Goldstone fermion "eld g"f 1j j 2t (5.108) @A @* A0 with the gravitino "eld. Thus, provided that the gaugino condensate and f are non-zero at the @A minimum of the e!ective potential we have been examining, the local supersymmetry is broken, and the gravitino acquires a non-zero mass m "e% m , where G is the value of G"K#ln "="
(5.109)
(5.110)
at the minimum. This conclusion is in accord with the general result that for spontaneous supersymmetry breaking to occur the variation of at least one of the "elds in the theory must have a non-zero VEV. The variation dt of t under a local supersymmetry transformation contains the terms dt "!(2e% (G\) G m!f (G\) j j #2 , @A @ A where
(5.111)
f ,Rf /Ru H (5.112) @A @A so again a non-zero condensate and non-minimal gauge kinetic function with f is non-zero, @A indicate a breakdown of local superstring. This breaking of supersymmetry by the hidden sector gaugino condensate leads to soft supersymmetry breaking in the observable sector. In particular, it is easy to see that (all) gauginos
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acquire non-zero masses, while the corresponding gauge "elds remain massless. The mass terms derive from the following two-fermion term in the supergravity Lagrangian (5.113) L "e% fK (G\) G jK jK , @A @ A $ where we fK is an observable sector gauge kinetic function and jK are observable sector gaugino @A @ "elds. Thus, with a diagonal gauge kinetic function, the mass of the canonically normalised gaugino jK "(Re fK ) jK is (5.114) M"m (Re fK )\fK (G\) G , where the su$x &0' indicates that the quantity is evaluated at the minimum of the e!ective potential. Formula (5.114) gives the gaugino mass at the string scale where (Re fK )\"g( (m )"4pa( (m ). We may use the renormalization group equation
(5.115)
M(k)/a(k)"M(m )/a(m ) to determine the gaugino mass M K (k) at the scale k. If we also use the form
(5.116)
="X(R) g(¹ ) G G for the e!ective non-perturbative potential, as discussed in the previous sections, then
(5.117)
1 1 MK (k)"2pa( (k)m !>f ! (bK G !dG ) [(1!f )d !>](¹ #¹M )"GK G" , (5.118) 1 8p % %1 >!d 1 G G G G G where f ,1!>XR/X , 1 d ,dG /8p. (5.119) G %1 GK G is the (modular covariant) Eisenstein function, de"ned in Eq. (5.55), and bG and de"ned in Eq. % (4.148). In deriving Eq. (5.118) it is necessary to augment the form (5.114) in order to obtain a modular invariant expression for the gaugino mass; in particular the term 2b G g(¹ )\dg/d¹ % G G which arises from fK G is replaced by 2b G GK G. % 2 The supersymmetry breaking also generates non-zero masses for the matter scalar "elds u . With ? the form (5.92) for the KaK hler potential, valid for small values of u , we may expand the e!ective ? potential to quadratic order in u and read o! the scalar masses. This gives ? "d (1!f )!>" 1 (¹ #¹M )"GK G"nG , (5.120) m?"< #m 1# G G G ? P (>!d ) G G where < is the ground state energy, the cosmological constant, given by
< "m " f "!3# >\(>!d )\"d (1!f )!>"(¹ #¹M )"GK G" . 1 G G 1 G G G
(5.121)
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We have already noted that < is minimized with values of ¹ close to the self-dual points at which GK G is zero, and because of this the last term of both < and m? is generally small. Further, f is zero 1 P at the minimum in the case that dG is zero, and in general f too is small. It follows that %1 1 < &!3m (5.122) and that the scalar masses squared are generally negative (5.123) m?&!2m , P completely unacceptable predictions, which cast doubt on either the validity or the relevance of the whole gaugino condensate mechanism for supersymmetry breaking. The scalar mass problem would be solved if the cosmological constant were small, and indeed the observed #atness of the universe on large scales supports the view that < is zero, or very small. It is worth noting, however, that in principle the cosmological constant is not necessarily the same as the particle physics vacuum energy. The observed #atness on large scales may be an average value of highly curved values at very small scales [40]. Nevertheless, we shall take the economical view that < is zero, and that we must therefore seek mechanisms to achieve this. In particular we regard the philosophy of setting < "0 in contradiction to the prediction (5.122) as being unacceptable. 5.5. Cosmological constant The vanishing of the cosmological constant < evidently requires the existence of additional matter whose contribution cancels those discussed hitherto, although we shall not attempt to explain why this should be so when supersymmetry is broken. We assume that this extra matter arises only in an additional term K of the KaK hler potential. Then the new KaK hler potential is K"KI #K (X,XM ) (5.124) with KI as in Eq. (4.154) and G"K#ln "=" .
(5.125)
The consequence is that the e!ective potential becomes < "e%v , where 1 ">!d (1!fR)"(¹ #¹M )"GK G"#K\6"K " v"" fR"!3# G G G 6 6 >(>!d ) G G and, as before
(5.126)
(5.127)
fR"1!>XR/X ,
(5.128)
K "RK /RX etc . 6
(5.129)
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Evidently, we can tune the X-dependent terms to ensure that the cosmological constant vanishes by arranging that v "0 , (5.130) where v is the value of v at the minimum of < . With this requirement the minimum of < is obtained by minimising v. So for the case d "0 we "nd that the ¹ are near the "xed point value at G G which GK G is zero and fR"0. The additional contribution to G means that the gravitino mass is now given by m , (5.131) m "e% but otherwise has no e!ect on the formula (5.118) for the observable sector gaugino masses. Similarly the scalar masses are still given by (5.120) but with the cosmological constant now tuned to zero. So
m?&m P which is quite acceptable, in principle.
(5.132)
5.6. A-terms and B-terms The generic cubic term (4.14) in the perturbative superpotential = "h (¹ )U U U , (5.133) ?@A G ? @ A where U are chiral super"elds, generates Yukawa couplings and quartic scalar couplings in the ? @ A supersymmetric "eld theory. In the presence of supersymmetry breaking e!ects, such as we are considering, it also generates (soft), trilinear couplings of the scalar "elds u of the form ? @ A L "A hK u u u (5.134) ?@A ?@A ? @ A and it is straightforward to calculate these; including the contribution = to the superpotential we merely expand < to third order in the scalar "elds. Then >!d (1!fM R) G A m\"!fM R# (¹ #¹M )GKM G ?@A G G >!d G G Rlnh ?@A , ; !(1#nG #nG #nG )#(¹ #¹M ) (5.135) ? @ A G G R¹ G where
"e) (KM)\ h (5.136) ?@A K ?@A M? @ A with KM the KaK hler potential for the matter "eld u , and nG its modular weight, see Eq. (5.92). It is K M M well-known that to avoid axions, and to break the observable sector electroweak symmetry successfully, it is necessary to include a `k-terma hK
= "k H H 5
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bilinear in the two Higgs "elds H into the perturbative superpotential. Then, as for the trilinear terms, there are corresponding soft terms, bilinear in the scalar "elds, induced by the supersymmetry-breaking hidden sector L "!B k( h h , 5 where
(5.137)
B m\"1!fM R(1!>k R) 5 >!d (1!fM R) R ln k R ln k G 5#d 5!(1#nG #nG ) (¹ #¹M )GK G (¹ #¹M ) # G G G G R¹ G RR >!d G G G (5.138)
and
k( "e) (KM)\ k K 5 M k can be calculated [6], and for the Z !IIb orbifold we have 5 R ln g(¹ )g(¹ /3)R ln g(; )g(; /3) k J= 5 R¹ R; and
(5.139)
(5.140)
n"n"!1, nG "nG "0, iO3 . There is also a term [6] in the KaK hler potential
(5.141)
K "ZH H #h.c. , 8 where
(5.142)
Z"(¹ #¹M )\(; #;M )\ which generates a soft scalar bilinear term
(5.143)
L "!B kh h #h.c. , 8 8 where
(5.144)
k""="Z[1!(¹ #¹M )GK (¹ ,¹M )!(; #;M )GK (; ,;M )] 8 is the coe$cient of the higgsino bilinear term in the Lagrangian and
(5.145)
!m\B k"=Z !1#(¹ #¹M )(GK (¹ ,¹M )#h.c.)# (; #;M )(GK (; ,;M )#h.c.) 8 8 # (¹ #¹M )(; #;M )GK (¹ ,¹M )GK (; ,;M )#" fR" (¹ #¹M ) G ">!d (1!fR)""GK (¹ ,¹M )" . ! G (5.146) G G G >(>!d ) G G The calculations of de Carlos et al. [47] show that a gravitino mass m in the range 10 GeV(m (10 GeV is easily obtained in models with hidden sector matter. There is
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therefore every reason to suppose that the incorporation of these supersymmetry breaking terms into the renormalization group equations will yield a sparticle spectrum on the same scale. 5.7. Further considerations The stabilization of the dilaton was achieved by using two or more gaugino condensates with suitably chosen hidden sector matter content [56,181,145]. An alternative, which requires only a single condensate has recently been proposed [57,39,173]. This utilises the observation that there are good reasons to believe that there are sizeable stringy non-perturbative corrections to the KaK hler potential. The e!ect is to replace the ln > term in Eq. (4.154) by a so far unknown function P(>). Then Casas [57] has shown in several examples how P(>) can be chosen so that the dilaton is stabilized with just a single condensate. However, it has not so far been possible to do this while simultaneously achieving a zero cosmological constant. It is straightforward to generalize the foregoing calculations of the supersymmetry breaking to this case [37,38]. We saw in Section 5.2 how the requirement that the non-perturbative physics preserves the modular invariance severely constrains the form of the non-perturbative superpotential. It was observed [68] that superpotentials involving the modular invariant function j(¹) may in principle arise in orbifold theories with gauge non-singlet states which become massless at special values of the moduli, although examples are lacking. j(¹) must appear in a function H(¹)"( j!1728)K jL P( j)
(5.147)
multiplying = (m,n are integers and P is a polynomial) =P=H( j)
(5.148)
in order to avoid singularities in the fundamental domain F"+¹: "¹"51, 04Im ¹41, .
(5.149)
This observation has been given added force recently [83,64] by the discovery that F-theory constructions of = are indeed modular forms, in fact E theta functions. Although the appear ance of H( j) does not a!ect the stabilization of the dilaton when there is a single condensate, it clearly does a!ect the values of the ¹ moduli at the minimum of < . One interesting feature is that G mimima arise in the interior of the fundamental domain [37,38] F, whereas previously they were on the boundary [68]. It is natural to wonder whether the minimization of < at complex values of the moduli might induce CP-violation via the moduli dependence of the soft supersymmetry breaking terms [128,40,1] calculated in the previous sections, although it has been argued [77,60] that there is no explicit CP-violation in string theory, perturbative or non-perturbative. Indeed the CP-violating phases of the soft supersymmetry breaking A and B terms are constrained to be less than O(10\) by the current limit on the electric dipole moment of the neutron [40]. Thus if CP-violation does arise in this way the challenge to string theory is to explain why these phases are so small. It is found [37,38] that the phases are either zero or well below the experimental bounds, unless both a non-minimal KaK hler potential, as discussed above, and the modular invariant function j(¹) is present via the appearance of H(j) multiplying =. In those circumstances CP-violation comparable to the current upper bounds does occur.
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6. Conclusions and outlook The `observeda uni"cation [2] of the SU(3);SU(2);U(1) gauge coupling strengths of the minimal supersymmetric extension of the standard model (MSSM) is to date the best evidence that the low energy world really is supersymmetric. Compacti"ed string theory naturally generates an e!ective four-dimensional supergravity } Yang Mills theory and, as we have seen in Eq. (4.155), it requires coupling constant uni"cation at a value a
,g /4p"(4p Re S)\
(6.1)
determined by the dilaton S, ignoring contributions D from the string loop threshold corrections ? and the Green}Schwarz anomaly cancelling coe$cients dG for the present. If/when we understand %1 the non-perturbative physics which stabilizes the dilaton "eld at a value with 1Re S2&2
(6.2)
the observed uni"cation with a&1/25 would also be evidence for an underlying string theory. However, to date we have no a priori convincing theory which leads to this result. In addition (and unlike a grand uni"ed theory, which also requires uni"cation), string theory predicts the energy scale at which uni"cation is achieved to be m K4;10 GeV
(6.3)
as follows from Eq. (4.185) using the `observeda value of g , which is a factor of 20 or so higher than the `observeda uni"cation scale (4.186). In Section 4.9 we discussed the feasibility of bridging this gap using calculations of the string loop threshold corrections D calculated in various orbifold ? compacti"cations. Our conclusion is that it is possible that these can remove the discrepancy, but that large values of the ¹ modulus 1Re ¹2&20
(6.4)
are required to do so. However, we saw in Section 5.3 that when the ¹ modulus is stabilized by hidden sector gaugino condensation, its value is generically of order unity, so again we have no a priori convincing theory as to how such a large value might arise. Of course, as we noted, the assumption that the only the matter content is that of the MSSM might be wrong, but here too we have no a priori convincing reason for including the extra matter needed to remove the discrepancy. Thus, although not conclusive, at face value the `observeda uni"cation is also the best evidence to date that (perturbative) string theory is wrong. We can see this another way. With six dimensions compacti"ed on a space of volume <, the 10-dimensional e!ective supergravity theory arising from heterotic string theory relates the four-dimensional gravitational coupling G and the uni"ed gauge coupling strength a to the , string tension a and the dilaton "eld u as follows: G "(a )eP/64p< , ,
(6.5)
a
(6.6)
"(a )eP/16p< .
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Since eP and < enter in the same combination in both expressions we can eliminate both and relate the string tension, and hence the string energy scale, to the coupling strength and the Planck mass m ,G\ . , m ,(a )\ "a m . (6.7) &m . if we use the `observeda value of the uni"ed gauge coupling. De"ning m "a<\ , with %32 14a42p expected, it follows that
m p %32 "ae\P a m 4 . a & 79
G m " , %32
(6.8)
if we require that eP(1, so that a perturbative treatment is justi"ed. In contrast the `observeda uni"cation scale is m K3;10 GeV (6.9) %32 and m "1.22;10 GeV, so the observed ratio is far smaller than the perturbatively predicted . lower bound. In fact to get the observed value requires eP&10
(6.10)
way beyond any perturbative validity. One possibility, therefore, is that in the real world string theory is strongly coupled, and that the perturbative treatment underlying this review is irrelevant to particle phenomenology. Developments in the past few years have shown that what were formerly regarded as di!erent string vacua may all be related using a web of duality transformations. (Two theories A, compacti"ed on a space X, and B, compacti"ed on >, are `duala to each other if the physics in the common uncompacti"ed space M is identical [10].) In particular, it has been established that the (10-dimensional) strongly coupled E ;E heterotic string theory, compacti"ed on a Calabi Yao threefold X, is dual to a new 11-dimensional M-theory [190,121,122], compacti"ed on X;S/Z . In the "eld theory limit M-theory reduces to an 11-dimensional supergravity theory with two E super-Yang Mills theories on each of the (two) 10-dimensional hyperplanes corresponding to the "xed points of the S/Z orbifold. It is beyond the scope of this review to give much detail of this. Su$ce it to say that in this case, when the theory is compacti"ed on a Calabi Yao space of volume <, the theory relates the four-dimensional gravitational coupling G and the uni"ed gauge coupling strength a to , the 11-dimensional gravitational coupling i and the length R "po of the orbifold interval as follows: G "i/8pR < , , a "(4pi) /< .
(6.11) (6.12)
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Thus de"ning m "a<\ , as before, with 14a42p expected %32 m ,i\ "(4pa\ ) m /a %32 2.266 K m %32 a
403
(6.13) (6.14)
using the observed value of a, and
R\"32pa\
m %32 a\m %32 m .
0.238 m K a %32
(6.15) (6.16)
if we use the observed uni"cation scale. So the length scale associated with the GUT is of the same order as, or a bit larger than, the fundamental scale m\of the 11-dimensional theory at which uni"cation of the GUT and gravitational forces presumably occurs, and the orbifold length scale R is R &9.5am\ (6.17) an order of magnitude larger than the fundamental scale. In this picture, at low energies the world is four-dimensional with gauge couplings evolving logarithmically and power law evolution of the gravitational coupling. Around R\ a "fth dimension opens up, and the power law evolution of the gravitational coupling changes; the logarithmic evolution of the gauge couplings is una!ected since the gauge "elds are con"ned to the walls at the "xed points of the extra dimension. Finally, at m the gauge couplings unify and six further dimensions open up; the theory is now 11%32 dimensional and has (sixth) power evolution of the couplings. Although weakly coupled at this scale, the gauge and gravitational couplings unify at m with a value a&1. Thus, unlike the weakly coupled heterotic string theory, analysed above, M-theory allows a consistent incorporation of the parameters associated with `observeda uni"cation. However, there are several points which should be borne in mind. One is that M-theory does not explain the parameters, any more than perturbative string theory did. As in the weakly coupled heterotic string, the e!ective supergravity theory emerging from the compaci"ed M-theory has two model independent moduli with 1Re S2,(1/4p)(4pi)\ < , 1Re ¹2,6 (4pi)\ R < . Using the previous formulae (6.13) and (6.15), we "nd [62] 1Re S2"1/g &2
(6.18) (6.19)
(6.20)
and
6 a am . &39a 1Re ¹2" 32p m %32
(6.21)
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and, as before, we have no a priori convincing theory of why the moduli should have these values. Further, it is amusing to observe that the large value required for 1Re ¹2 would su$ce to bridge the previously noted uni"cation gap of the weakly coupled theory, thereby dispensing with the need for a strongly coupled theory! Physically, the most important feature distinguishing between M-theory and the weakly coupled string theory is that the gravitational "elds propagate in the bulk (compacti"ed) 11-dimensional world, while the gauge and matter "elds are con"ned to the (compacti"ed) 10-dimensional hyperplanes. One e!ect of this is that because of the variation with the extra coordinate, the e!ective (four-dimensional) supergravities di!er at the two ends. In particular, the gauge kinetic function of the (observable sector) E gauge "elds is f "S#a¹ (6.22) with a an integer determined by the Hodge numbers of the Calabi}Yao threefold X upon which the theory is compacti"ed. (In the `standarda embedding the gauge connection of one of the E theories is set equal to the spin connection of X, and this breaks the gauge symmetry (in the observable sector) to E .) The (hidden sector) E has gauge kinetic function f "S!a¹ . (6.23) These expressions have a striking similarity to those f "S$e¹ (6.24) which occur when the weakly coupled heterotic string is compacti"ed, with the e¹ terms arising from the string loop threshold corrections (in the large ¹ limit) and e determined by the anomaly. The other quantities needed to specify the e!ective supergravity theory have also been calculated [61,167,168,154], and these may be applied straightforwardly to determine the soft supersymmetry breaking terms. So the second point to note is that since, as we have previously observed, it is not yet unambiguously determined that we are in the strongly coupled regime, it is important to have calculations for both the weakly coupled case and the strongly coupled case in order to decide the matter phenomenologically. In any case, it is already known that some features of the weakly coupled regime (e.g. the KaK hler potential) carry across to the strongly coupled case with little or no modi"cation, so some results from the former have a wider validity than their parentage might indicate. At the time of writing, the form of M-theory compacti"ed on a Calabi Yao threefold with standard [121,122,190], and non-standard [149,155] embeddings, is known, and there are some results for orbifold compacti"cations [170,176], with the concomitant modular symmetry groups. For this, and all of the previously given reasons, we hope that this review of weakly coupled orbifold compacti"cations of heterotic string theory will also be of relevance to the exciting new developments that are now occurring.
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W. Nakel, C.T. Whelan/Physics Reports 315 (1999) 409}471
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RELATIVISTIC (e, 2e) PROCESSES
Werner NAKEL, Colm T. WHELAN Physikalisches Institut der UniversitaK t TuK bingen, Auf der Morgenstelle 14, D-72076 TuK bingen, Germany Department of Applied Mathematics & Theoretical Physics, University of Cambridge, CB3 9EW, UK
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 315 (1999) 409}471
Relativistic (e, 2e) processes Werner Nakel , Colm T. Whelan Physikalisches Institut der Universita( t Tu( bingen, Auf der Morgenstelle 14, D-72076 Tu( bingen, Germany Department of Applied Mathematics & Theoretical Physics, University of Cambridge, CB3 9EW, UK Received July 1998; editor: J. Eichler Contents 1. Introduction 1.1. Notations and de"nitions 2. Experimental techniques 2.1. Electron beams 2.2. Targets 2.3. Electron spectrum analyzers 2.4. Scattering arrangements 2.5. The determination of absolute cross sections 3. An outline of theory 3.1. Low-energy (e,)2e) processes 3.2. Simple relativistic approximations
411 412 412 414 415 416 417 419 419 419 420
3.3. Relativistic distorted-wave Born approximation 4. Comparison between theory and experiment 4.1. Experiments with unpolarized primary beams 4.2. Experiments with transversely polarized beams 5. Summary and conclusions Acknowledgements Appendix A. Appendix B. References
422 426 427 454 463 465 465 467 468
Abstract In this report we review the experimental and theoretical development in the study of relativistic (e, 2e) processes. In these (e, 2e) experiments one observes electron impact ionization processes where both the ejected and scattered electrons are detected in coincidence after angular and energy analysis. This allows a complete determination of the collision kinematics and hence provides a sensitive test of theoretical models. The goal of the investigations, reviewed here, is the basic understanding of the inner-shell ionization process by relativistic electrons up to the highest atomic numbers, probing the quantum mechanical Coulomb problem in the regime of high energies (up to 500 keV) and strong "elds. (e, 2e) experiments with polarized electron beams represent an important step towards the ideal of a quantum mechanical complete analysis of the ionization process. In the calculation of the pertinent triply di!erential cross section we will show that only a fully relativistic theory where the strong distorting e!ects of the heavy atom are properly included in both the incident and "nal channels can approach a realistic description of the problem and that the simplest viable approximation is the relativistic distorted-wave Born approximation. 1999 Elsevier Science B.V. All rights reserved. PACS: 34.80.-i Keywords: Relativistic electron-impact ionization; Spin asymmetries; Relativistic distorted-wave Born approximation
0370-1573/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 1 2 9 - X
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1. Introduction The ionization of atoms by electron impact is one of the basic processes of atomic physics. An (e, 2e) experiment allows one to measure such a process where the two outgoing electrons are detected in coincidence with their energies and angles resolved. It is, thus, a measurement which is kinematically complete and provides a demanding test of all theoretical models. Since the "rst (e, 2e) measurements of Ehrhardt et al. [1] and Amaldi et al. [2], in the late 1960s, experimental and theoretical activity has been intense (for recent reviews see [3}5]). Much e!ort has been devoted to the study of the dynamics of the scattering process itself and in using the technique as an analytic tool to probe the nature of target wave functions. For such work the electron impact and binding energies lie in an energy regime typically between 10 eV and 10 keV, and the primary focus has been on hydrogen, helium and outer shell electrons. Work in the relativistic region began in 1982 with the absolute (e, 2e) experiments of SchuK le and Nakel [6] at an incident energy of 500 keV on the K shell of silver. These kinematically complete experiments on the inner shell states of high-Z atoms probe the fundamental ionization mechanism in the regime of relativistic energies and strong "elds up to the K shell of uranium. The last few years have seen great strides in the area and recently experiments have been performed with transversely polarized electron beams [7] which represent an important step towards the ideal of a complete quantum mechanical scattering experiment. The description of these processes has necessitated the development of new theoretical and computational methods. This is hardly surprising since the problem is fully relativistic, both the incoming and exiting electrons are fast, the target electron feels the full e!ect of the "eld of the highly charged nucleus, it is a many-body problem with long-range Coulomb forces. The calculation of the pertinent triply di!erential cross section (TDCS) opens up a whole new area of theoretical study and o!ers a direct insight into the subtilities of spin-dependent and other purely relativistic e!ects in atomic physics. It is our intention in this article to review the experimental and theoretical developments in the study of relativistic (e, 2e) processes. We will begin in Section 2 with a survey of the experimental techniques used. Section 3 is devoted to theory. It will be informative to "rst look at the low-energy situation; here the distorted-wave Born approximation (DWBA) has proved remarkably useful both in the description of multiple scattering e!ects and inner-shell ionization. Then we will give a sketch of simple relativistic approximations. In order to overcome the limitations of these approximations, Walters et al. [8] proposed to implement a fully relativistic distorted-wave Born approximation (rDWBA). We will argue that it is the simplest possible approximation which we could hope to employ if we are to gain an understanding of relativistic (e, 2e) processes. We will stress the vital importance of formulating the problem in a fully relativistic way and show that semirelativistic wave functions quickly lead one into error. In Section 4 we will present a detailed comparison between theory and measurement. We will look at the key geometries used also at low energies and will show that each has something to o!er in helping us to understand relativistic processes. Spin and relativity are inexorably mixed and the study of spin-dependent (e, 2e) processes o!ers the possibility of a much deeper understanding of relativistic atomic physics. With this in mind, Section 4.2 will be devoted to the study of spin-dependent processes with transversely polarized primary beams. Some short reports overviewing the area have already appeared [9}11] but this is, to our knowledge, the "rst complete review.
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1.1. Notations and dexnitions In the following, atomic units ( "m "e"1) are used so the numerical value of the vacuum velocity of light is c"137.0359895; the metric tensor is diag(g )"(1,!1,!1,!1) , (1.1) IJ contravariant four vectors are written xI"(t, x) and the summation convention is understood. We assume that the incident electrons have energy E and momentum k and that two electrons having energies E , E , momenta k , k are emitted into solid angles X , X . We denote the TDCS as dp/(dX dX dE). All the experiments we will consider are coplanar and we follow the convention that the positions of the outgoing electrons will be de"ned by angles h , h left and right of the beam direction (Fig. 1). Without loss of generality we assume that E 5E . Table 1 gives some notations and abbreviations used in the paper.
2. Experimental techniques To date all relativistic (e, 2e) experiments have been performed in the energy range between E "300 and 500 keV, whereas the (e, 2e) experiments at lower energies were done typically between 10 eV and 10 keV [5]. The higher energies e!ect not only the design of the facilities for the primary electron beams (Section 2.1) but also mean that the experimental techniques have to be di!erent from the low energy analogs, as for instance the use of magnetic electron spectrum analyzers (Section 2.3) instead of electrostatic ones or the use of foil targets (Section 2.2) instead of atomic beams. (In recent (e, 2e) studies of conduction bands in solids [12] typical impact energies are of the order of 20 keV.) The relativistic (e, 2e) measurements have all been performed in one of two experimental setups at the University of TuK bingen. Schematic diagrams of the setups are shown in Fig. 2a and b and in Fig. 3. It will be convenient to use the abbreviations setup A (indexed 1 or 2 according to the di!erent electron spectrum analyzer used) and setup B. Setup A: Setup A (Fig. 2) operates with an unpolarized primary electron beam with energies up to 500 keV produced by a van de Graa! generator (Section 2.1.1). The scattering arrangement is described in Section 2.4.1 and the targets in Section 2.2. Setup A may be run with two types of
Fig. 1. Schematic diagram of a coplanar (e, 2e) experiment showing the energies, momenta and angles of the incoming and the two outgoing electrons, respectively, indexed as 0, 1 and 2.
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Table 1 E E ,E k k ,k K, q h ,h h k P Z Abbreviations TDCS DDCS DWBA rDWBA CBA rCBA srCBA FBA rFBA BEBR
Kinetic energy of the incident electrons Kinetic energy of the fast and slow outgoing electrons, respectively Momentum of the incident electrons Momenta of the fast and slow outgoing electrons Momentum transferred to the target by the incident electron (K,q"k !k ) Angle between the incident and the outgoing electrons Angle between the two outgoing electrons Recoil momentum of the ion Degree of polarization of the incident electrons Atomic number of the target atoms Triply di!erential cross section Doubly di!erential cross section Distorted-wave Born approximation Relativistic distorted-wave Born approximation Coulomb}Born approximation Relativistic Coulomb}Born approximation Semirelativistic Coulomb}Born approximation First Born approximation Relativistic "rst Born approximation Bound-electron Bethe ridge
Fig. 2. Sketch of the coplanar (e, 2e) experiment operating with an unpolarized electron beam with energies up to 500 keV produced by a van de Graa! generator. (a): In setup A both electron spectrum analyzers each consist of a magnetic spectrometer with a scintillation detector. (b): In setup A both of the electron spectrum analyzers consist of a nondispersive, i.e. triply focusing magnet in front of a surface barrier detector. The scattering arrangements are described in Section 2.4.1.
electron spectrum analyzers. Setup A (Fig. 2a): Both electron spectrum analyzers consist of a dispersive magnet with a scintillation detector (Section 2.3.2). Setup A (Fig. 2b): Both electron spectrum analyzers consist of a nondispersive, i.e., triply focusing magnet in front of a surface barrier detector (Section 2.3.1).
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Fig. 3. Sketch of the coplanar (e, 2e) experiment operating with a transversely polarized electron beam of 300 keV produced by a GaAs electron source and a cascade generator. The spin direction is perpendicular to the scattering plane. The scattering arrangement is described in Section 2.4.2.
Setup B: This setup, Fig. 3, generally, operates with a transversely polarized primary electron beam up to 300 keV produced by a cascade generator (Section 2.1.2). Setup B can only be run with electron spectrum analyzers each consisting of a dispersive magnet with a scintillation detector, (Section 2.3.2). The scattering arrangement is described in Section 2.4.2, whereas details on the targets are given in Section 2.2. 2.1. Electron beams Electrostatic accelerators are used to accelerate the electron beams up to 500 keV. The electron sources for the spin-polarized electron beams (300 keV) used the photoemission of electrons from GaAs crystals irradiated by circularly polarized light (Section 2.1.2), while unpolarized electrons (Section 2.1.1) were generated by a directly heated tantalum cathode (500 keV). 2.1.1. Unpolarized electron beam Electron beams with energies up to 500 keV were produced by a single-stage van de Graa! accelerator, horizontally mounted (model AS-700, High Voltage Engineering Europa B.V.). The charging belt, in this system, is provided with staples of steel and charged by a capacitive plate. To obtain a well collimated beam with a low background suitable for the (e, 2e) experiments, the original cathode system had to be modi"ed. The electron source } a directly heated tantalum cathode } originally consisted of a wire of 1 mm diameter. It was replaced by a pointed cathode. The divergence of the beam is de"ned by a diaphragm located in the "rst stage of the acceleration tube. A spot size on target smaller than 1 mm in diameter and a beam divergence of about $0.2% could be obtained. This was achieved without the use of any slits, thereby avoiding the production of additional background by scattering at the edges (the slits for the voltage stabilization were also not in use). The beam transport system consists of a 903-bending magnet, the focusing coils and the de#ector coils are of conventional design. The unscattered beam is collected by a Faraday cup, which is retractable to allow measurements at small electron scattering angles.
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2.1.2. Polarized electron beam Two di!erent photoemission sources have been used for the production of the polarized electrons. In both cases circularly polarized light is used to irradiate gallium arsenide crystals, but the sources di!er in the character of the crystals employed, i.e. strained GaAs or GaAsP. The sources were designed to operate at the high voltage terminal of an electrostatic accelerator (300 kV). (i) Polarized source with a GaAsP crystal: In our "rst experiments a gallium arsenid phosphide (GaAsP) crystal was used, requiring a photon energy of 1.9 eV which is matched closely by the wavelength of a HeNe laser. A polarization of 35}40% could be achieved. The source was originally developed for bremsstrahlung experiments [13] and is described elsewhere [14]. This source was used for measurements of the asymmetry in relativistic (e, 2e) processes due to the spin}orbit interaction of the continuum electrons (Section 4.2.1). To get higher degrees of polarization the source is now replaced by a setup using a strained GaAs crystal, described in (ii). (ii) Polarized source with a strained GaAs crystal: Very recently a source operating with a strained GaAs crystal has been built up delivering a polarization degree up to 70%. The pertinent experimental setup is based on that of the GaAsP source mentioned above and described in detail by Mergl et al. [14]. Therefore we will only summarize the basic features of the version currently in use. The source operates in the high-voltage terminal of a 300 kV accelerator. In order to keep the electron optics simple no di!erential pumping stage is used. Rather the source chamber was separated from the accelerator by a commercially available platinum diaphragm with a diameter of 150 lm, thus maintaining a pressure di!erence of four orders of magnitude between the ultrahigh vacuum of the source chamber and the high vacuum of the accelerator. In order to generate spin-oriented photoelectrons the strained GaAs crystal has to be irradiated by circularly polarized light of a wavelength in the range of 820}850 nm. The laser diode used produces 100 mW linearly polarized light. Its intensity, and hence the beam current, can be varied with an electron-optical modulator. The circular polarization is achieved by a second light modulator used in the quarter-wave retardation mode. Switching the polarity of the voltage across the modulator reverses the helicity of the circular polarization. In this way the sign of the spin polarization can easily be #ipped, thus avoiding any problems associated with rotating optical devices. Strained GaAs crystals were supplied by the Spire Corporation, USA. The photoelectrons emitted from the cathode are accelerated to 300 eV and focused by the extraction lens into an electrostatic 903 de#ector. In this way the initially longitudinal polarization could be transformed into a transverse polarization. A second lens accelerates the beam to 600 eV and focuses the electrons onto the 150 lm diaphragm where a maximum transmission of about 15% is achieved. After this the electron beam is aligned with the axis of the 300 keV accelerator tube. The degree of spin polarization can be measured by a Mott analyzer placed in the beam line in front of the entrance to the scattering chamber. In the Mott analyzer the electrons scattered through 1203 by a gold foil are detected by a pair of ion-implanted silicon detectors. The polarized source with the strained GaAs crystal has been used for measurements of the spin asymmetry in relativistic (e, 2e) processes due to the "ne-structure e!ect (Section 4.2.2). 2.2. Targets In the relativistic (e, 2e) experiments, to date, only solid targets have been used with foil thicknesses between 5 and 60 lg/cm and atomic numbers 29 (Cu), 47 (Ag), 73 (Ta), 79 (Au) and
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92(U). One cannot use foils at low energies because it is highly improbable that the electrons could leave the solid without signi"cant further interactions (see Section 2.5). These plural scattering processes are very much reduced at relativistic energies and one has the great advantage that one can use thin foil targets and determine with comparative ease the absolute cross section. The problem of obtaining absolute cross sections at the lower energies used in gas-phase experiments has produced an extensive literature (see [15,16]) and continues to present a very substantial challenge. The foil targets are on a metal frame and have a diameter of 8 mm. The uranium, gold and tantalum foils are on carbon backing, the silver and copper foils are, depending on the thickness, free or on carbon backing. The target thickness was determined from measurements of the energy loss of a particles passing through the foil. The conversion factor was taken from the tables of Barkas and Berger [17]. The product of target thickness and beam current was checked by measuring the elastically scattered electrons with a surface barrier detector. For tantalum and gold the target thickness was determined also from measurements of the characteristic K X-rays with a Ge(Hp) detector at 1203. Using the total cross section for K-shell ionization, the #uorescence yield and the beam current, the respective target thickness was calculated. 2.3. Electron spectrum analyzers The electrostatic spectrum analyzers which have been used in nonrelativistic (e, 2e) measurements (see e.g. [5]), are not suitable for energies of the order of 100 keV. Therefore two types of analyzers have been developed speci"cally for relativistic (e, 2e) experiments: E electron spectrum analyzers consisting of a nondispersive, i.e., triply focusing magnet in front of an energy dispersive detector (used in setup A , see Section 2.4.1); E electron spectrum analyzers consisting of a dispersive magnet followed by the detector (used in setups A and B, see Sections 2.4.1 and 2.4.2, resp.). These analyzers are discussed in the next two sections. 2.3.1. Electron spectrum analyzers with a nondispersive magnet In an (e, 2e) experiment in which both electron detectors are able to record a wide energy range simultaneously energy-sharing measurements become possible using a two-parameter coincidence technique. In this approach, however, considerable problems can arise by intense but undesired parts of the spectra. For example, it is essential to avoid detecting elastically scattered electrons, since the count rate for these is orders of magnitude higher than that for inelastic processes. To eliminate these elastically scattered electrons before they reach the detector, while at the same time allowing the transmission of inelastically scattered electrons with a broad energy band, a nondispersive and doubly focusing (i.e. triply focusing) magnet was inserted between the de"ning aperture and the detector. The magnet and the energy-dispersive (surface barrier) detector together formed the electron spectrometer. In constructing such triply focusing magnets SchuK le and Nakel [6] were fortunate to be able to follow previous developments by Komma [18], and Komma and Ruo! [19]. Basically, each magnet was a dipole with a homogeneous "eld ("eld index n"0). The energy focus was achieved by an appropriate bending of the pole pieces. The magnet focused, in an energy independent way, electrons having a spatial divergence in the symmetry plane of the magnet as well as in the perpendicular direction. Inside the magnet, electrons with di!erent momenta were
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spatially separated thus allowing the elastically scattered electrons to be eliminated. A detailed description of the electron spectrum analyzer with a nondispersive magnet is given by Ruo! et al. [20]. The "rst relativistic (e, 2e) experiments [6,21] were done using this type of analyzer (see Section 4.1.1.1.(b)). Unfortunately, surface barrier detectors have some unfavorable properties which make them unsuitable for certain applications. For example, for target atoms with lower atomic number Z, the energy resolution may not be su$cient to separate the K and L shells, or, in the case of higher Z, to separate the L and M shells. Moreover, the low-energy tails in the pulse-height response of the numerous electrons coming from outer shells may mask the electrons from the inner shells one is primarily interested in. (The problem is discussed in detail in [20].) Additionally, the timing properties of the surface barrier detectors were in some cases not su$cient for the requirements of the coincidence measurements. Therefore, a di!erent type of spectrum analyzers was designed and constructed; it will be described in the next section. 2.3.2. Electron spectrum analyzers with a dispersive magnet Bonfert et al. [22] constructed electron spectrum analyzers which consisted of a magnet for the energy analysis and a plastic scintillation detector for good time resolution. These analyzers are described in detail in [20] and only a brief description will be given here. The magnet consisted of a doubly focusing homogeneous sector "eld shaped by an iron core with a de#ection angle of 1413 and a momentum resolution up to 0.4%. Its medium plane was identical with the scattering plane. The fringing "elds were responsible for vertical focusing. Entrance and exit angles of 573 determined that the vertical focus occurred at the same point as the horizontal, their use also resulted in relatively large object and image distances and a large focal length. Thus, target and scintillator with photomultiplier were outside the fringing "eld of the magnet. The large focal length allowed one to be able to position several slits in front of the energy de"ning slit and thus discriminate against the numerous elastically scattered electrons. There were also several traps inside the gap of the magnet which served to eliminate electrons that did not have the desired momenta. The magnets were tested by observing the image of the target spot produced by elastically scattered electrons on a #uorescent screen placed at the energy slit. Using a surface barrier detector behind the energy slit, electron spectra for di!erent energies selected by the magnet were recorded. In this way the apparatus transmission and the suppression of undesired electrons could be checked. Using a scintillation detector the peak of the elastically scattered electrons was measured by increasing the magnetic "eld in small increments, thus determining the experimental value of the momentum resolution. All measurements using a polarized beam and most of the measurements using an unpolarized beam were performed with this type of electron spectrum analyzer. 2.4. Scattering arrangements The van de Graa! accelerator with the unpolarized beam, 500 keV, as well as the cascade generator with the polarized beam, 300 keV, both have their own scattering chambers, electron spectrum analyzers and electronics. Thus two complete and independent (e, 2e) apparatii have been used to perform the (e, 2e) experiments at the University of TuK bingen. As above we shall refer to these as setups A and B. Conventional signal processing and coincidence circuitry are used. It will be convenient through the text to refer back to the experimental arrangements as outlined in this
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Section. We therefore give in the Appendix A a brief summary of all the relevant experiments that have been performed and the speci"c setups used. 2.4.1. (e, 2e) apparatus with an unpolarized beam up to 500 keV, setups A and A The electron beam coming from the van de Graa! accelerator (see Section 2.1.1) was transported by a 903-bending magnet, focusing coils and de#ector coils into the scattering chamber. The cylindrical vacuum chamber had a diameter of 820 mm and height of 410 mm. It was designed to work with electron detection systems which could be situated inside, setup A , or outside, setup A , the scattering chamber. Setup A : For measurements of angular distributions it was advantageous to have the possibility of changing the detector position continuously and without breaking the vacuum. Therefore, a pair of electron spectrum analyzers were designed to operate inside the scattering chamber and these could be rotated around the target, located at the center of the chamber, by a microprocessorcontrolled stepping motor. Scattering angles can be selected in steps of 0.13. These spectrum analyzers consisted of a dispersive magnet and a scintillation detector as described in Section 2.3.2. Since the position of the energy-de"ning slit was "xed, the magnetic "eld had to be measured to an accuracy which was much better than the momentum resolution of the magnet, (0.4%). This is done by use of a rotating coil driven by a synchronous motor to an accuracy of about 0.05%. The induced ac signal from the coil was fed into a special electronic network and then processed by a microcomputer. This device allowed one to exactly select desired momentum values and control properly the coil currents. For further details see [20]. Setup A : The electron detector systems consisting of nondispersive magnets and surface barrier detectors (see Section 2.3.1) were designed for working outside the chamber. To this end the chamber was furnished with 26 welded necks that allow coupling the housings of the detector systems. In the scattering plane, the chamber was extended by an 803 segment (height 45 mm) closed by a thin plastic foil (thickness 50 lm). The foil was chosen to be transparent to characteristic X-rays and bremsstrahlung photons. A special feature of the chamber was the fact that it was divided into two cylinders below the scattering plane. The lower cylinder was "xed to the #oor and accommodated the pump, the target slide at the center of the chamber, the aperture-de"ning diaphragms for the outer magnets, and the cog wheel for the inner magnets. After breaking the vacuum the upper cylinder could be rotated to bring the welded necks and the segment into suitable positions. The operating pressure inside the chamber is about 10\ mbar. 2.4.2. (e, 2e) apparatus with polarized beam of 300 keV, Setup B The beam was produced by the polarized electron source described in Section 2.1.2 and then accelerated to 300 keV using a high-voltage cascade generator. It was then transported into a scattering chamber after "rst passing through a Mott analyzer to measure the degree of polarization. The cylindrical stainless-steel vacuum chamber had a diameter of 500 mm and a height of 400 mm. It contained only one of the two electron spectrum analyzers each consisting of a dispersive magnet and a scintillation detector as described in Section 2.3.2. The system was mounted on the lid of the chamber and could be rotated around the target. When the vacuum was broken the lid could be rotated to move the electron detector to a di!erent scattering angle. The second electron spectrum analyzer was mounted outside the scattering chamber in a separate housing which could be coupled to the chamber by welded necks. To measure and control the
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magnetic "elds of the spectrum analyzers high-sensitivity Hall probes were used (Digital Teslameter DTM-141, Group 3 and Gaussmeter 9000, F.W. Bell). 2.5. The determination of absolute cross sections The absolute triply di!erential cross sections were determined from dp N " , (2.1) dX dX dE N n *X *X *E where N is the number of true coincidences, N the number of primary electrons, n the number of target atoms per unit area, *X , *X are the solid angles of the electron detectors and *E is the energy width for registration of coincidence events (FWHM). The solid angles were determined from the geometry and checked by the measurement of elastic scattering. The energy width was determined from the product function of the two single transmission curves of the spectrometers. Since magnetic spectrometers were used (*p/p " constant), *E is dependent on the value of the energy sharing. The momentum resolution of one of the spectrometers was, for instance, chosen to be 0.9% instead of 0.4%, (the best value possible) to ensure a total transmission of 100%. The great advantage of being able to use thin-foil rather than gaseous targets is the (relative) ease with which absolute cross sections can be obtained. However, in spite of the high electron energies used, there remains a "nite probability that the electrons will be scattered before and/or after the ionizing e!ect (e.g. by Mott or M+ller scattering). The low-energy outgoing electron is especially a!ected. The in#uence of plural scattering can be checked by using di!erent thicknesses of target foil. A possible source of further true coincidences is the process of electron}electron bremsstrahlung, i.e. the emission of a photon in the collision of two electrons [13]. Therefore on the outer shells electron}electron coincidences can be produced which "t the kinematical conditions of the (e, 2e) process on an inner shell. The only available calculation of the elementary process of electron}electron bremsstrahlung on an initially bound electron was done by Haug and Keppler [23]. No experimental evidence has been found for the process in our (e, 2e) measurements [24].
3. An outline of theory 3.1. Low-energy (e, 2e) processes Over the past several years much e!ort has been applied to the understanding of the physics of (e, 2e) and related processes and much progress has been made in the study of atomic outer shells at low energies [3}5]. The theoretical description of K and L shell ionization of heavier targets opens up a whole new range of problems. Relativistic e!ects are globally important, the target electron is in a deep inner shell, the incident and "nal electrons have velocities that are a signi"cant proportion of the speed of light, retardation, magnetic and spin-dependent interactions need to be considered. It will be useful to draw analogies to the low-energy regime and to contrast the structures observed
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and the geometries employed, and we will do so below. Here we will only present a few brief remarks. Let us emphasize once again that by carefully choosing the geometrical arrangement and the energies of the incoming and exiting electrons one can predicate the physics which will dominate the shape and magnitude of the triply di!erential cross section which is observed. For the simplest atoms, H and He, an impressive body of literature has grown up and it has become clear that in many geometries delicate few body e!ects have to be included both in the incident and "nal channels [86,25,26]. While there have been some successes the theoretical situation for all setups and kinematics is far from resolved. One approach which has proved useful is the distorted-wave Born approximation. In its most usual form [27] it includes the elastic scattering of the incident electron to all orders in the static "eld of the atom, and the same for the outgoing electrons in the "eld of the ion. The ionizing collision occurs only once, i.e. the interaction between the incident and target electron is treated only to "rst order and no account of electron}electron repulsion in the "nal state or polarization in the incident is included. Capture cannot take place. Exchange e!ects manifest themselves both in the indistinguishability of the particles in the "nal channel, and in the exchange potentials in the elastic channels. In other words it allow for the possibility that the incident electron could exchange with one of the atomic electrons `beforea the ionizing collision and that either of the outgoing electrons could exchange with the remaining `spectatora electron in the "nal channel. What it does contain is the possibility of including multiple scattering e!ects in the elastic channels. Whelan and Walters [29] argued that these e!ects were the source of the large angle peak in coplanar symmetric geometry, see below. Zhang et al. [28] also considered the role of multiple scattering e!ects in inner-shell ionization for medium Z atoms at nonrelativistic energies. Earlier calculations, based on the "rst Born approximation, had been unable to represent the shape of the observed TDCS [64]. Zhang et al. [28] argued that one could not neglect the in#uence of the atomic potential on any of the electrons until much higher energies. This they claimed was especially true for inner shell ionization since one would naively expect that the ionization process to occur relatively close to the nucleus i.e. in a region where the static potential is at its strongest. It was argued that the simplest viable approximation was the DWBA and excellent agreement with experiment was obtained. 3.2. Simple relativistic approximations The "rst theoretical analysis of excitation and ionization in relativistic situations was given as far back as 1932 by M+ller [32], who developed a "rst order time dependent perturbation theory. All of the approaches discussed in this report can be related to this perturbative formalism (for a di!erent approach see [34,35,41]). Basically all the approximations, considered here, construct the S matrix with the electron}electron interaction, i.e. the ionizing interaction, given by the QED photon propagator, and the pre and post-collisional interactions between the electron and the atom/ion being represented by purely electrostatic and radial symmetric external "elds, (this is discussed in detail below when we develop the relativistic form of the DWBA). In simple terms the plane wave approximation has zero "eld on all unbound electrons, the "rst Born approximation uses Dirac plane waves, (no "eld), for the incoming and fast scattered electrons, and the Coulomb Born treats all free particles as if they saw the full nuclear charge. The rDWBA uses a representation of the potential of the atom in all free channels, i.e. the many body problem is replaced by
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e!ective one body problems. It is, however, quite a numerical task to solve the Dirac equation for elastic scattering by a Coulomb potential and many of the calculations have resorted to further simplifying approximations, the e!ect of these extra approximations will be discussed below. (a) Plane wave models: The most straightforward approach to the description of (e, 2e) processes involves using solutions of the free Dirac equation, i.e. relativistic plane waves states, to represent all unbound electrons. This model was "rst put forward by Fuss et al. [36] in an attempt to provide a theoretical basis for extending the technique of electron momentum spectroscopy (see [37] and references therein) to the relativistic domain. This paper therefore used additional approximations such that the TDCS factorized into a purely kinematic and an initial state structure depended part. Along the same lines, Bell [38] proposed to combine the cross section for free electron}electron scattering (involving a suitable o!-shell initial state) with the Compton pro"le of the initial bound state to arrive at the relativistic plane wave impulse approximation (PWIA). An exhaustive study of di!erent PWIA-type models has been given in [39]. The "rst evaluation of the plane wave model without any additional approximations [40] was carried out to provide benchmark results for the implementation of the rDWBA approach (see below). As a by-product, this work showed the additional approximations alluded to above led to nontrivial changes in the results, indicating that the idea of factorizing the cross section function is problematic in the relativistic context. More importantly, all these studies left no doubt about the fact that the plane wave model is unsuited for the description of the existing experimental data, typically mispredicting the absolute cross sections by an order of magnitude. A large portion of the deviations could be attributed to the lack of orthogonality between bound and plane wave states [40]. (b) First Born approximation: The "rst Born approximation (FBA) has been extensively used in asymmetric geometries at nonrelativistic energies [42,43]. The incident and fast scattered electrons are treated as plane waves while the ejected electron is treated as a Coulomb wave. Normally, the ejected electron wave function and the target wave function are chosen to be orthogonal, this not only avoids spurious autoionizing terms but gives the correct high-energy limit for the cross section, i.e., the plane wave approximation goes like 1/q, where q is the momentum transfer while the "rst Born approximation has the correct 1/q, behaviour. Generally speaking it works well on atomic outer shells at high impact energies. It is however very much less successful on inner shells [28]. Considerable e!ort has been devoted to extending this method to the relativistic domain. The "rst (e, 2e) studies in this spirit were carried out by Das and collaborators [44,45] who employed a semirelativistic Sommerfeld}Maue function for one of the outgoing electrons. Subsequently, Jakuba{a-Amundsen evaluated the "rst-order S-matrix element using semirelativistic Coulomb waves, i.e., neglecting the relativistic contraction of the bound state and approximating the continuum Coulomb state by a nonrelativistic Coulomb wave times a free spinor. This model did well in predicting integrated cross sections [30]; however it yielded a value for the absolute TDCS which was very much too large. Walters et al. [8] used this simple model to establish for the "rst time the importance of spin-#ip terms in coplanar symmetric geometries and Jakuba{a-Amundsen used it to consider the contraction of the bound orbital [47]. (c) Coulomb}Born approximations: Jakuba{a-Amundsen [46] argued that one could not neglect the Coulomb potential in the treatment of inner shell ionization of high-Z atoms. She put the full nuclear charge on all free particles. This was justi"ed by the assumption that the inner region close to the nucleus would have a very strong in#uence on the resulting TDCS. However, because of the
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great numerical and computational di$culties involved she was forced to use semirelativistic Coulomb waves which had the advantage of being available in closed analytic form. This theory was a de"nite improvement over earlier calculations, and agreement with experiment was encouraging especially for intermediate values of Z. The merits and shortcomings of this theory have been analyzed in [50]. Very recently, a fully relativistic version has been produced [51] which shows that the original physical insight was essentially correct and that most of its shortcomings lay in the use of semirelativistic rather than fully relativistic wave functions. However, from the discussion of this and the preceeding section it is quite obvious that what really is required for even starting to understand the relativistic (e, 2e) process is a relativistic extension of DWBA model, as was "rst noted by Walters et al. [8]. First encouraging results of such calculations were reported in [52,53]. In the mean time, rDWBA has established itself as the standard method for the interpretation of relativistic (e, 2e) experiments. Therefore, in the following section, we will, following mainly [53], discuss this model in some more detail. 3.3. Relativistic distorted-wave Born approximation As we spelled out in Section 3.1 the key elements of DWBA are the all order description of elastic scattering in the incident and "nal electron states within the "rst-order ionization matrix elements. At the relativistic energies, some simpli"cations arise: exchange in the elastic channels is likely to be negligible, "nal state electron repulsion will certainly play no major role [49]. However, as mentioned earlier, we are now dealing with a fully relativistic problem. Speci"cally, this implies that exact relativistic elastic scattering wave functions (Dirac spinors) need to be used, and that the full QED photon propagator that mediates the interaction in the "rst-order M+ller matrix element needs to be included. The TDCS for the relativistic (e, 2e) process, where the spins are not resolved, can be written quite generally (2p) k k N dp E E E G "1k e , k e "SK "k e , ie 2" , " (3.1) 2N c k dX dX dE
CCCC where SK is the S-matrix operator; 0, 1, 2 and b refer to the incoming, the two outgoing and the initially bound electron, respectively, E , E , E and k , k , k are the on shell total energies and momenta of the unbound particles where E"kc#c , and we are using i to denote the quantum numbers of the atomic bound states, e are the spin projections with respect to the quantization axis, which we take in the beam direction. In the form Eq. (3.1) the TDCS is insensitive to spin polarization, we have averaged over the initial spins, e , e , and summed over the "nal e , e . (Hence the factor N /N : N is the occupation number of state G G i and N the number of degenerate states with this set of quantum numbers.) We will discuss the
question of explicitly spin-dependent measurements in a little while but for the time being we will assume that all experiments are in no way spin resolved. The starting point of the derivation of the rDWBA model is the M+ller "rst-order approximation, in which the S matrix in Eq. (3.1) takes the form 1k e , k e "SK "k e , ie 2"5 "S !S
(3.2)
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with
S "!i dx tM (x)c AI(x)t (x) , I
(3.3)
where the four potential AI(x) is given by
AI(x)" dy D (x!y)JJ(y) . IJ
(3.4)
D (x!y) is our free photon propagator which we can write as IJ
dk e IV\W !4p ig D (x!y)" IJ IJ (2p) k#ie c
(3.5)
and JJ(y) is the fermion current JJ"tM (y)cJt (y) (we are using the notation of [33]). In other words
(x)D (x!y)tR (y)ccJt (y) , S "i dx dy tR (x)ccIt I C I C IJ I C G C
(3.6)
where c are the usual Dirac matrices and t is an exact stationary solution of the "rst quantized I C Dirac equation. It has been assumed that the electromagnetic "eld of the nucleus and the atomic electrons could be incorporated in the form of an e!ective classical "eld. In all calculations, to date, this "eld was assumed to be purely electrostatic and radially symmetric in the reference frame of the calculations (which was taken to be the rest frame of the nucleus). This meant that the Dirac equation, that had to be solved in the elastic channels, could be separated in spherical coordinates. In all the calculations reported a self-consistent relativistic Kohn}Sham local density approximation potential was used [54] to represent the atomic potential. For this choice of potential the Dirac equation is separable and solutions can be written
t(x)"e #R A
g (r )N (XK ) G V G I V , i f (r )N (XK ) G V \G I V
(3.7)
where now the quantum number i"!(2j#1)( j!l) represents good total angular momentum, j and parity (!1)J. The photon propagator can be represented in polar coordinates and then expanded in multipoles. Keller et al. [53], used the program of Salvat and Mayol [55], to numerically evaluate the radial functions, t , and the Clebsch}Gordan coe$cients were extracted from Burgess and Whelan [56]. I C They approximated the K-shell electron by a relativistic hydrogenic 1s state. The full partial wave/multipole analysis is exceedingly tedious and we will not reproduce it as it is given in all its glory in [53].
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The most serious challenge faced in developing the rDWBA code was that one ended up needing to evaluate to a high accuracy a large number, typically of the order of 60 000, integrals of the form
R"
dr r V V
dr r j (or )[!y (or )#i j (or )] (r ) (r ) (r ) (r ) , W W L L L V V W @ W
(3.8)
where j, y are regular and irregular spherical Bessel functions, o"(E !E )/c, r "max(r , r ), V W r "min(r , r ) and each represents a large or small component of the radial Dirac solution. V W There were three main di$culties: 1. The wave-function components are only known analytically in the asymptotic region. It is therefore impossible to tailor the integration algorithm explicity to the analytic structure of the integrand in the vicinity of the nucleus where one would expect the dominant contributions. 2. Only the term is bounded, the 's corresponding to the free particles are highly oscillatory, as are the spherical Bessel functions arising from the multipole expansion of the potential. These oscillations are intensi"ed as the linear momenta and the energy transfer are increased. 3. These integrals are needed to high accuracy. Due to the large number of radial matrix elements and the relative signs entering through the vector coupling coe$cients, cancellation e!ects are severe. The major problem faced was being sure that the integrals were su$ciently accurately evaluated especially for the larger values of the angular momentum. The method developed depended on applying the Bethe approximation, see Whelan [57], which allowed the reduction to two threedimensional integrals one of which contained a bound state. The numerical problems were thus concentrated in the evaluation of integrals over three highly oscillatory terms. Consider the term
dr r j (or ) j (k r ) j (k r ) . V V L V J V J V
(3.9)
This integral is characteristic of the entire problem. In Fig. 4 we plot its integrand for a typical set of parameters and the reader will appreciate just oscillatory these functions are. Whelan [57] succeeded in "nding an analytic solution to Eq. (3.9). This allowed a crucial bench mark for the numerical methods. The reader is referred to [53] for full details of the computational approach and the extensive testing of the program that was undertaken. It should be noted that in [53] and subsequent papers it was assumed that all distorted waves could be generated in the "eld of a neutral atom. The complication of adding Coulomb boundary conditions for one or both electrons has been the subject of a recent publication, [51] and we will discuss it in Section 4 below where we compare theory and experiment. In summary, the success of the DWBA at nonrelativistic energies and the failure of the simple relativistic models discussed in Section 3.2 provoked an attempt to develop a relativistic equivalent. The rDWBA is however a fully relativistic approximation where the same philosophy was used as in the nonrelativistic case, but no attempt was made to retain the mathematical formalism which was suitable at low energies. We note that there are some other key di!erences, in the
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Fig. 4. Plot of the integrand of term Eq. (3.9) for a typical set of parameters.
low-energy DWBA work the incoming electron and exiting electrons were scattered in the static exchange potential of the atom/ion while at high energies we used a pure static atomic potential in all channels. It is important to be clear, in both the low- and high-energy work we allow for the indistinguishability of the two "nal state electrons but in the low-energy case we also allow for
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exchange e!ects in the elastic scattering channels. For example, at low energies the approximation allows for the slow exiting electron to exchange with the remaining ionic electrons on its way to the detector, the rDWBA does not. This simpli"cation is justi"ed when one considers the speed of the ejected electron in the relativistic case, however this di!erence between the two approximations has to be borne in mind when one interprets low- and high-energy spin asymmetry experiments. We will return to this point below. In principle, any theoretical model of relativistic (e, 2e) processes provides an approximate representation of the S matrix as function of all quantum numbers, continuous and discrete, so that the outcome of any possible experiment can be predicted by taking suitable averages over unobserved degrees of freedom. However, in reality, experiments cannot be performed with pure quantum mechanical states. This problem is particularly evident in the case of (e, 2e) processes with spin-polarized primary electrons, that cannot be produced with 100% polarization. In order to take these e!ects into account in the theoretical description of such experiments, the density matrix formalism (see, e.g. [58,59]) has to be employed. The application of this technique to relativistic (e, 2e) processes is straightforward, though some care has to be taken because the spin projection of relativistic continuum states is only de"ned for free electrons in their own rest frame. Therefore, the corresponding observables are not de"ned by the Pauli matrices, but by more complex operators that account for the Lorentz transformation to the observer frame required [60,61]. However, it can be shown [62] that, once the proper de"nition of the projection quantum numbers e of the asymptotic bound and scattering states required in (e, 2e) theory has been implemented, the density matrices themselves take the same form as in standard nonrelativistic spinor theory. The "nal result of these formal manipulations is a relation expressing the measured asymmetry as function of the various S-matrix elements corresponding to di!erent combinations of the projection quantum numbers [62]. It should be mentioned that, for the speci"c asymmetry parameter to be discussed below, an equivalent expression results from directly manipulating pure states [63], though this approach would be di$cult to extend to other observables.
4. Comparison between theory and experiment The great advantage of the (e, 2e) method is that by carefully choosing the geometrical arrangement and the energies of the incoming and outgoing electrons one can predicate the physics which will dominate the shape and magnitude of the triply di!erential cross section which is observed. The geometry of the experiment and the physics it reveals are inexorably mixed. In Section 4.1 we will discuss di!erent geometrical arrangements using unpolarized primary beams. One of the key features of the relativistic experiments was that many of them are absolute. This meant that unlike the low-energy inner shell experiments [64] the theory had both to predict the size as well as the shape of the TDCS. We will look carefully at this absolute data and the theoretical attempts to reproduce it. In the light of our observations we will argue that the rDWBA is the simplest possible means at our disposal to describe deep inner shell electron impact ionization. The measurement of the angular and energy distributions of the TDCS gives a very re"ned picture of the ionization process and by carefully exploiting the geometrical and kinematical setup of these experiments one can focus on "ne details of these interactions which can be very
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revealing. For example, in Section 4.1.1, we will report on a process of the interference of contributions to the S matrix describing the exchange of longitudinal and transverse photons. This process is responsible not only for a large fraction of the shift of the binary maximum away from the direction of momentum transfer, but also for the appearance of a pronounced secondary maximum in the forward quadrant. Spin is a characteristic of relativistic phenomena and the possibility of doing experiments which are spin resolved presents a substantial challenge to theory. Very recently there have been measurements with transversely polarized beams on both K and L shells which has allowed a direct insight into the interactions of the continuum electrons in the strong nuclear "eld and of target coupling e!ects. Again the rDWBA does well. As the rDWBA does not include exchange e!ects in the elastic scattering channels (in contrast to the treatment of the nonrelativistic case), we can conclude that the measurement must be due to other mechanisms. We will consider all these in Section 4.2. 4.1. Experiments with unpolarized primary beams Most of the measurements using an unpolarized beam have been done with setup A (see Section 2.4.1), in some cases setup B has been run in the unpolarized mode. 4.1.1. Coplanar asymmetric geometry } Ehrhardt geometry In the coplanar asymmetric geometry } often called Ehrhardt geometry } fast outgoing electrons are detected which have a small scattering angle with respect to the primary beam and the angular distribution of the slow outgoing electrons is registered. In this arrangement, the two electrons have greatly di!erent energies and so the exchange amplitude is small and post-collisional e!ects weak. Intuitively, at least for ionization from the outer shell, one would except it to be a geometry that is particularly favourable to a perturbative approach. For nonrelativistic energies the characteristic parameters and the di!erent dynamical situations are discussed in detail in a paper by Ehrhardt et al. [65] from which we have taken the instructive Fig. 5 with the polar diagrams of the TDCS. The incoming electron originates from the bottom of the graph. The most important kinematic parameter is the momentum transfer q which is the di!erence of the momenta of the incoming and the fast outgoing electron. (In this article we will use for the momentum transfer the symbols K or q.) The direction of the momentum transfer vector is shown as an arrow and the symmetry axis of the TDCS is the dotted line. In the case of the "rst Born approximation the TDCS is symmetric about the direction of momentum transfer q, i.e. one gets a peak in the forward direction with a maximum at q, the direction an electron would move in a pure binary collision, while there is a second smaller maximum in the direction of !q, in the direction of maximum recoil of the ion. These peaks are usually called the `binarya and `recoila peaks, respectively. For lower impact energies, one sees signi"cant deviations away from the binary and recoil directions [65] as higher-order terms in the Born series switched. This simple picture has to be treated with caution when one looks at inner shell process [28], where it is found that the binary peak is strongly in#uenced by distortion e!ects due to the static potential of the atom. We will follow the custom of referring to these peaks as `binarya and `recoila even though in the relativistic case is partially directed into the forward quadrant and even the use of the name has been recently questioned [66].
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Fig. 5. TDCS in the nonrelativistic regime for the ejection of s or p electrons for small, medium and large momentum transfer (from Ehrhardt et al. [65]).
For K shells for which most of the relativistic (e, 2e) measurements have been done so far, we will discuss the subject in terms of the binary and the recoil region rather than reviewing the di!erent measurements one after the other. 4.1.1.1. Measurements on K shells. (a) ¹he binary region: angular distributions (i) Absolute cross sections: In Fig. 6 the absolute TDCS for gold measured by Bonfert et al. [22] is shown (E "500 keV, E "100 keV, h "!153). The absolute scale of the cross section measurements has an error of approximately $15%. The measurement is compared with several calculations, namely the results of the fully relativistic "rst-order Born theory [48], two approximations to this theory [8,44], the Coulomb}Born theory of Jakuba{a-Amundsen [46,47] which employs nonrelativistic Coulomb scattering waves times a free electron spinor to represent all unbound electrons, and the rDWBA. It is obvious that the distortion of all unbound wave functions by the atomic potential in the rDWBA leads to a signi"cant decrease of the TDCS and to a shift of the binary peak, resulting in a good agreement with the absolute experimental data. There exist drastic discrepancies in absolute magnitude between the relativistic and semirelativistic "rst-order Born results. Keller et al. [48] have shown that the TDCSs for the ionization of s states of high-Z atoms by relativistic electrons is rather sensitive to errors introduced by the use of approximate semirelativistic scattering wave functions. Their result resolves a long-standing problem in this "eld in explaining why (and how) previous applications of the "rst-order Born approximation failed so seriously in the description of the experimental results. In Figs. 7 and 8 [31,48] we show the absolute TDCS for silver (E "500 keV, E "100 keV, h "!73, and h "!153). As one would expect, the agreement of the approximate "rst-order theories with the experiment in the region of the binary peak is somewhat improved because here the atomic "eld is substantially weaker compared to gold. The srCBA is within the error bars. We see that in general the rDWBA gives consistently good results although there is an indication that agreement is best for the heavier atom at the higher energies.
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Fig. 6. Absolute TDCS for K-shell ionization as a function of h in coplanar asymmetric geometries: E "500 keV, E "100 keV; Z"79, h "!153, Symbols: experimental data [22], full curve: fully relativistic "rst order Born theory [48]; dashed curve: results of approximate "rst-order Born theory [8]; dash-dotted curve: results of theory of [44]; short-dashed curve: results of Coulomb}Born theory of [46,47]; dotted curve: results of rDWBA [52,53].
Fig. 7. Absolute TDCS for (e, 2e) on silver, E "500 keV, E "100 keV, h "!153. Full squares experimental results [22]; short-dashed curve, srFBA [47]; long-dashed curve, rFBA [62]; full curve, rDWBA [53]; dash-dotted curve, srCBA [50] (from [31]).
(ii) ¹he position of the binary peak: In the nonrelativistic "rst Born approximation, as mentioned above, the TDCS is symmetric about the momentum transfer K"k !k . For a K shell this results in the appearance of two peaks, one at K, the binary, and one about !K the recoil. The corresponding experiments carried out at nonrelativistic energies revealed that this symmetry is
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Fig. 8. Absolute TDCS for K-shell ionization as a function of h in coplanar asymmetric geometries: E "500 keV, E "100 keV, Z"47, h "!73. Symbols: experimental data [22]; full curve: fully relativistic "rst-order Born theory [48]; long-dashed curve: results of approximate "rst-order Born theory [8]; dash-dotted curve: results of theory of [44]; short-dashed curve: results of Coulomb}Born theory of [46,47]; dotted curve: results of rDWBA [52,53] (from [48]).
broken and that the binary (and recoil) peaks are shifted towards larger detection angles. (see, for example, Fig. 5). It is now generally accepted that the observed shift is due to post-collisional interactions between the slow ejected and the fast scattered electrons [5]. This interaction is implicitly contained in the second-order Born term and entirely absent from the standard DWBA calculations. At relativistic energies Bonfert et al. [22] also observed a shift in the binary peak in coplanar asymmetric geometry when they studied the K-shell ionization of gold with 500 keV electrons (Fig. 9). In this case the shift has another origin. Notwithstanding the fact that postcollisional interactions are found to be negligible at these energies, there is already a clear indication of the shift in both semirelativistic and fully relativistic "rst-order Born calculations [46,48]. In order to analyze the shift Ast et al. [67], consider the fully relativistic "rst Born approximation described in Section 3.2(b). Crucially, this model contains the full photon propagator Eq. (3.5). Now in relativistic physics the electron}electron interaction is mediated by a photon exchange while the nonrelativistic interaction can be viewed as an instantaneous interaction between charge densities. In order to understand the role of magnetic and retardation e!ects Ast et al. [67] replaced the propagator de"ned by Eq. (3.5) by that corresponding to the Coulomb potential, i.e.
dk e IV\W (4.1) D! (x!y)"!4pd(x !y )lim k#g E and calculated the TDCS for the K shell of gold (500 keV) in asymmetric geometry, in the "rst-order Born approximation. We reproduce these results here (Fig. 9). The di!erence DI (x!y)"D !D! (x!y) IJ IJ
(4.2)
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Fig. 9. Absolute TDCS for the electron-impact K-shell ionization of gold, E "500 keV, E "100 keV, h "!153. TDCS plotted as a function of the emission angle of the slow outgoing electron. Experiment (T) Bonfert et al. [22], theory: "rst Born approximation (- - -) (normalized to experimental data), Walters et al. [8]; model calculation (*) (normalized to experimental data); rDWBA (.....) (from [67]).
represents the relativistic contributions. The Coulomb potential can be regarded as the nonrelativistic limit of the photon propagator. The term DI (x!y) can thus be looked upon as the correction IJ due to magnetic and retardation e!ects. The spatial components of this propagator DI (x!y) HI describe the magnetic interaction between the currents of the two moving charges, the component DI (x!y) incorporates retardation e!ects. One notes that the position of the peak in the model calculation agrees with the direction of momentum transfer (h "35.23). Clearly, the shift obtained in the "rst Born (h "423) is caused by magnetic and retardation e!ects. The position of the binary maximum is insensitive to the use of a semirelativistic wave function [46,8], or a fully relativistic wave function [48] to describe the ejected electron, we remark that only by doing a full rDWBA calculation one can achieve the full shift to the experimentally observed value h "503. Indeed only then as we observed earlier can we achieve agreement with absolute size of the cross section. A similar analysis of the shift can be carried out for (e, 2e) measurements in coplanar symmetric geometry (see Section 4.1.2). In summary, the symmetry about the direction of momentum transfer is in the relativistic case already broken at the "rst Born level due to the e!ect of retardation and magnetic contributions to the electron}electron interaction as well as distortion e!ects in the incoming and outgoing channels. The shift of the binary peak has also been observed for atoms of lower atomic number as silver [22] and copper [68]. For the case of copper we refer to the analysis by Keller and Dreizler [66]. (iii) Measurements under Bethe ridge conditions: SchroK ter et al. [69,70] reported on absolute triply di!erential cross section measurements using kinematical conditions to meet the so-called bound-electron Bethe ridge [71] and its neighbourhood. At the bound-electron Bethe rigde (BEBR) the full momentum K"k !k transfered by the incident electron to the target is
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absorbed by the atomic electron (K"k ), i.e. the recoil momentum k is zero. The concept of the BEBR stems from the nonrelativistic (e, 2e) regime. There measurements on the BEBR yield good agreement between theory and experiment [71]. The primary energies used by SchroK ter et al. [69,70] amounted to 300 and 500 keV. The measurements were performed on the K shell of silver (E "25.5 keV) with setup A (see Section 2.4.1). Fig. 10a}Fig. 10f [72] show the absolute TDCS of the K shell of silver on the BEBR at 500 and 300 keV, respectively, as a function of the scattering angle h . To meet the condition k "0 along the curve the angular positions of the detectors as well as the energy sharing had to be changed from one measuring point to the other. The rDWBA overestimates the experimental data whereas the semirelativistic Coulomb}Born approximation (srCBA) is in good agreement. The good agreement is astonishing if one considers that the fully relativistic Coulomb}Born approximation (rCBA) overestimates the data. Therefore, Ast [72] suggests that the good agreement of the srCBA might be an artefact of the simultaneous semirelativistic approximations for bound and continuum states. (b) ¹he binary region: energy distributions In most of the relativistic (e, 2e) experiments angular distributions have been measured where the energy sharing of the excess energy between the outgoing electrons was constant. The advantage of using electron-spectrum analyzers consisting of nondispersive magnets and surface barrier detectors (see Section 2.3.1) is that one can make use of a two-parameter coincidence technique in which both electron detectors simultaneously record a very wide energy range of the outgoing electrons. In this way, energy-sharing experiments had been performed in which correlations between a large number of ionization processes were measured simultaneously. The energy conservation in the collision process demands that E "E #E #E where the excess energy E !E can be distributed between the kinetic energies E and E of the two outgoing electrons. The ion recoil energy is negligible. The expected kinematical curves are straight lines. The position of the lines of the particular shells are given by the respective binding energies. The ability to separate the lines experimentally depends on the energy resolution of the detectors. Using coplanar asymmetric geometry and "xed angles of the detectors, SchuK le and Nakel [6] have measured energy distributions on the K shell of silver (Z"47) and Ruo! and Nakel [21] on the K shell of tantalum (Z"73). These measurements had been performed at a primary energy of 500 keV using setup A (see Section 2.4.1). The experimental data represent the "rst results from a relativistic (e, 2e) experiment. Examples of two-parameter plots of the true coincidences as a function of the energies of the outgoing electrons are shown in a paper of Ruo! et al. [20]. In Fig. 11 (from [72]) we compare the measurements with the results of di!erent calculations. Plotted is the absolute TDCS against the kinetic energy E of the outgoing electrons detected at h . Again, as in the case of gold just discussed, the distortion of all unbound wave functions by the atomic potential in the rDWBA leads to a signi"cant decrease of the TDCS resulting in a good agreement with the absolute experimental data (error in the absolute scale $15%). The nearly perfect agreement for silver is surprising since in the other cases (Figs. 7 and 10) the rDWBA overestimates the cross section for silver. (c) ¹he recoil/secondary peak region: angular distributions (i) ¹he K shell of copper (Z"29): Very recently, Besch et al. [68] reported on an (e, 2e) measurement at an energy of 300 keV on the K shell of copper (E "9 keV). Using coplanar asymmetric kinematics (h "!93, E "220 keV) the relative TDCS was measured as a function
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Fig. 10. Absolute TDCS of K-shell ionization under BEBR conditions. (a) E "300 keV, E "74.5 keV, h "!26.073, (b) E "500 keV, E "74.5 keV, h "!18.463, (c) E "300 keV, E "137.25 keV, h "!38.873, (d) E "500 keV, E "237.25 keV, h "!37.093, (e) measurement on the Bethe ridge, E "300 keV, (f) measurement on the Bethe ridge, E "500 keV. Experiment: (䢇) SchroK ter et al. [69], () SchroK ter [24]. Theory: (.....) srCBA (Jakuba{a-Amundsen [46]), (*) rDWBA with atomic potential in the outgoing channel, (- - -) rDWBA with ionic potential in the outgoing channel, (-.-.-) rCBA (from [72]).
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Fig. 11. Absolute TDCS as a function of the outgoing electron energy, E "500 keV. Experiment (䢇) SchuK le and Nakel [6], (o) Ruo! and Nakel [21]. Theory semirel. 1. Born. (-.-.-) Das and Konar [44], (.....) Walters et al. [8]; srCBA (- - -) Jakuba{a-Amundsen [46]; rDWBA (*). (a) silver, h "!203, h "403, (b) silver, h "!253, h "403, (c) tantalum, h "!203, h "403, (d) tantalum, h "!303, h "303 (from [72]).
of the scattering angle of the outgoing slow electrons (Fig. 12, symbols, the error bars represent one standard deviation). Apparatus B (see Section 2.4.2) with an unpolarized beam was used. The experimental result is compared with a rDWBA calculation of Dreizler et al. [73]. Both experimental and theoretical data are normalized to unity at the binary maximum. The most remarkable feature is a distinct double-peak structure in the region commonly called the recoil region. A broad maximum appears for emission of both electrons into the same (forward) quadrant and an additional peak structure for observation angles !1203(h (!853 of the slow electron. Fig. 12 shows that the rDWBA calculation of [73] reproduces all these features and the position of the binary peak quantitatively. Hence } as Keller and Dreizler argue [66] } it should be possible to
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Fig. 12. Relative TDCS for K-shell ionization of copper versus scattering angle h of the slow outgoing electron E "71 keV. Primary energy E "300 keV, fast outgoing electron (E "220 keV) detected at h "!93. The error bars represent one standard deviation. The full curve is a calculation according to the rDWBA of Keller et al. [53,73]. Both experimental and theoretical data are normalized to unity at the binary maximum. (from Besch et al. [68]).
relate all experimental "ndings to the peculiarities of the exchange of a virtual photon between the two active electrons, and their elastic scattering o! the spectator ion. The purpose of the study [66] was to provide a detailed interpretation of the experiment along these lines, with particular emphasis on understanding the mechanism leading to the secondary maximum. As a result of the investigation the authors concluded that the structures may be interpreted in terms of the interplay of two distinct mechanisms. The destructive interference of contributions to the S matrix describing the exchange of longitudinal and transverse photons, respectively, is responsible for the appearance of the pronounced secondary maximum in the forward quadrant, whereas a two-step mechanism of elastic electron}ion scattering in the incident channel, followed by an ionizing binary electron}electron collision leads to the additional peak structure in the cross section at the (critical) angle of h "!953. In addition, the authors [66] have observed that the magnetic interaction between projectile and spectator ion also in#uences the magnitude of the secondary maximum, whereas elastic backscattering of the ionized electrons o! the residual ion (the classical recoil mechanism) plays no signi"cant role. In view of these "ndings, these authors suggested that a term like `(magnetic) interference peaka might be more adequate to address the secondary maximum of the TDCS than the standard expression `recoil peaka. The observation of a two-step mechanism of elastic electron}ion scattering in the incident channel, followed by an ionizing binary electron}electron collision is quite similar to a result obtained in the theory of the well-known large angle peak in coplanar symmetric geometry (see, e.g. [74] and references therein and Section 4.1.2). (ii) ¹he K shell of silver (Z"47): A close inspection of the rDWBA calculations shows that there are hints for a double-peak structure in the recoil region as for instance in the calculations of Ast et al. [52] for silver and gold. However, in this cases the structures are too weak to give a statistically signi"cant signal in the experiment. Moreover, the measurement of Bonfert et al. [22] on silver (Fig. 8) was not extended far enough into the range of negative h to observe it. Fig. 8
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shows that the simpler theories (see Section 3.2) are not able to describe the order of magnitude of the recoil peak relative to the binary peak. Again, the rDWBA is quite successful in this respect. The discrepancy between theory and experiment at h "!503 has to be reconsidered experimentally. Recent measurements of the relative TDCS at 300 keV by Sauter et al. [76] show good agreement with rDWBA calculations in the recoil regime. (iii) ¹he K shells of gold (Z"79) and uranium (Z"92): The fact that only the rDWBA is able to predict the recoil peak indicates that this structure should be understood as an e!ect of the electron}atom interaction and hence should depend strongly on the nuclear charge. Fig. 13 (taken from [53]) shows TDCSs calculated according to the rDWBA for the kinematical con"guration of Fig. 8 and for di!erent atoms, where in each case the maximum of the binary peak has been normalized to unity. The calculations indicate a drastic build up of the relative magnitude of the secondary maximum. Recent measurements of Sauter et al. [76] for gold and silver con"rm this feature. 4.1.1.2. Measurements on L shells. With (e, 2e) experiments using relativistic primary electron beams it is mainly K-shell ionization processes that have been investigated so far. With the rDWBA a qualitative and in many cases quantitative description of these data has become possible. This allows the conclusion that the K-shell ionization process can be described in terms of a one-step process, provided that the strong electromagnetic "eld of the spectator ion as well as relativistic e!ects are properly taken into account. We now consider the question whether the ionization of the L shell can adequately be described in terms of the rDWBA. (i) Measurements on the 2p subshell of gold: 1s states are characterized by a density maximum at the location of the atomic nucleus so that the collision dynamics is essentially
Fig. 13. Relativistic DWBA calculations for TDCS at E "500 keV, E "100 keV, h "!73 with di!erent targets. Cross sections normalized to unity at the maximum of the binary peak. Full curve, uranium target; long dashed curve, gold target; short dashed curve, silver target. (from Keller et al. [53]).
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determined at extremely small radial distances. It is therefore of considerable interest to check relativistic collision theory in the complementary situation of a deeply bound inner-shell state with vanishing density at the origin so that the contributions from very small radial distances do not dominate the ionization cross section. This can only be achieved by considering the "ne structure split p substates. Kull et al. [77] reported recently the "rst experiment of this type. Apart from being of considerable interest in its own right the study was an important preparation for a measurement of the relativistic "ne-structure e!ect (see Section 4.2.2). The authors have measured the angular distribution of the absolute triply di!erential cross section for 300 keV electronimpact ionization of the 2p subshell of gold as a function of the scattering angle of the outgoing slow electrons (Fig. 14). The binding energies of the three subshells 2s , 2p and 2p are 14.4, 13.7 and 11.9 keV, respectively. The authors succeeded in separating the 2p from the 2s and 2p shells which are not resolved. Setup A (see Section 2.4.1) has been used which had, however, to be improved in several points for the present measurement. The authors reduced the ripple of the high voltage of the van de Graa! generator by a factor of 6 down to 70 V. A further important step was the improvement of the stability of the magnetic "eld measuring device (better than 5;10\). One electron analyzer had an energy acceptance of 0.48 keV (FWHM) at a detection energy of 78.1 keV the other 1.50 keV at 210 keV. The angular acceptance was $1.73 in the scattering plane and $2.73 out of plane. The in#uence of plural scattering was checked by using targets of di!erent thicknesses. The measurement was performed using a gold target of 5 lg/cm on a carbon backing of 6 lg/cm. An estimation of the in#uence of plural scattering at even smaller target thicknesses gave no indication of a further change of the result. The measurement was performed in a coplanar asymmetric geometry at the bound-electron Bethe ridge (see Section 4.1.1.1) and in its neighbourhood. To match the Bethe ridge point in the
Fig. 14. Absolute TDCS for the 2p -shell ionization of gold (E "300 keV, E "11.9 keV) in coplanar asymmetric geometry for E "210 keV, E "78.1 keV and h "!27.33. The full curve is a calculation according to the rDWBA by Keller et al. [62], the dashed line the CBA by Jakuba{a-Amundsen [63]. The error bars represent one standard deviation of the statistical error. The vertical line indicates the position of the bound-electron Bethe ridge (K"k , k "0). (from Kull et al. [77]).
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angular distribution measurement for "xed kinetic energies (E "210 keV, E "78.1 keV) the scat tering angle of the outgoing fast electrons is determined to be h "!27.33. To give an overview of the parameters chosen for the measurement, Fig. 15 displays the TDCS for the 2p -shell ionization calculated according to the relativistic plane wave impulse approximation (see Section 3.2) of Bell [38]. The full circles indicate the combinations of scattering angles chosen for the measurement. The sixth point from left complies with the condition of the bound-electron Bethe ridge K"k , i.e. k "0. The PWIA essentially maps out the momentum pro"le of the bound state wave function. Since the p-state momentum wave function has a node at the origin, the PWIA predicts a vanishing cross section at the Bethe ridge point. The full circles of Fig. 14 represent the absolute TDCS for the ionization of the 2p shell of gold as a function of the scattering angle of the outgoing slow electrons. The error bars represent one standard deviation of the statistical error only. The systematical error in the absolute scale is estimated to be $20% mainly due to the uncertainties of the target thickness and the e$ciency and energy width of the electron analyzers. The curves represent the results of calculations carried out in rDWBA (full curve, Keller et al. [62]) and semirelativistic Coulomb}Born approximation (CBA) (dashed curve, [63]). The agreement with the rDWBA is good while the discrepancies to the CBA are somewhat larger. In contrast to the result of the PWIA the rDWBA and CBA predict the local minimum not at the bound electron Bethe ridge (k "0) but at larger angles. This is caused by the relativistic contributions to the electron}electron interaction [67]. The relative magnitude of both peaks and the "lling up of the minimum are presumably due to distortion e!ects of the ejected electron-wave function [77]. A study of the dependence of the binary-peak structure on the system parameters according to the CBA is given by Jakuba{a-Amundsen [63]. To clarify the remaining discrepancies between rDWBA and experiment in the peak at larger scattering angles should be a matter of further investigations. (ii) Measurement on the ¸ shell of gold } nonresolved ,ne structure: Prinz and Keller [78] have measured the relative TDCS of the L-shell ionization of gold (Fig. 16). The measurement was done
Fig. 15. Calculated TDCS for the 2p -shell ionization of gold for E "300 keV and a "xed energy sharing E "210 keV and E "78.1 keV versus the angles h and h of the outgoing electrons in coplanar asymmetric geometry according to the PWIA of Bell [38]. The combination of scattering angles for the measured data points are indicated by the full circles (from Kull et al. [77]).
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Fig. 16. Relative TDCS for electron impact ionization of the L shell of gold, as a function of the observation angle h of the slow outgoing electron. Impact energy 300 keV, fast outgoing electron 200 keV, h "!103. Symbols: experiment (error bars indicate statistical error), full curve: relativistic DWBA calculation, normalized to give best visual "t to experimental data (from Prinz and Keller [78]).
with apparatus B (unpolarized beam) described in Section 2.4.2 using a primary beam energy of 300 keV. Fast outgoing electrons of 200$5.7 keV energy were detected at h "!103, while the second detector sampled electrons of 86.9$2.9 keV. The energy width of the detectors chosen represent a compromise between the necessities of separating the L shell from higher shells, and covering the ionization events from all magnetic substates of the L shell. The corresponding theoretical calculations were carried out using the rDWBA code described in Keller et al. [53,62]. TDCSs were calculated for ionization of the individual magnetic sublevels and for the same energy of the fast outgoing electron, i.e. here the di!erent level binding energies were balanced by di!erent secondary electron energies. The results were added incoherently to give the subshell averaged cross sections sought. Tests using simpler theoretical models showed that the explicit convolution of theoretical data with the experimental energy and angle acceptance functions leads to results in complete agreement with those obtained from the above procedure. In Fig. 16 the experimental and theoretical relative TDCS are compared. All major features of the experimental data, notably the asymmetric shape of the binary maximum (which is due to the superposition of the binary maxima of the subshell ionization cross sections that take their maxima at di!erent angles), the double-peak structure around h "!603 and the ratios of the peaks, are reasonably well reproduced by the calculation, demonstrating once more that rDWBA provides an adequate framework for the analyzes of relativistic (e, 2e) data. 4.1.2. Coplanar symmetric geometry } Pochat geometry In this geometry the two outgoing electrons have equal energies and their momenta k , k lie in the same plane as the incident momentum k and make same angles h with the beam direction. This can be described as a very `harda collision, with the incident electron losing over half its initial
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momentum. This geometry was "rst studied by Amaldi et al. [2] and is used in the nonrelativistic regime as a method for directly studying target wave functions in momentum space (see e.g. [79]). In the relativistic regime wave function measurements play only a minor role; the deep inner shell states of heavy atoms are well approximated by hydrogen-like wave functions. (e, 2e) experiments in symmetric geometry are, however, interesting for the investigation of the ionization process itself both in the relativistic as well as in the nonrelativistic regime. A systematic experimental study of the impact energy and charge dependence of the relativistic TDCS of equal energy sharing collisions in coplanar symmetric geometry has been carried out by Bonfert et al. [22] and Walters et al. [8] the results of which are given in Fig. 17a}Fig. 17f. For all measurements setup A (Section 2.4.1) has been used. The K shells of gold, silver and copper at 300 and 500 keV have been investigated. The error bars of the experimental values represent the standard deviation only. The systematic error of the absolute scale was estimated to be $15%. (a) Absolute cross sections: For gold, (Fig. 17a and Fig. 17b), we "nd good agreement between the experimental results and the rDWBA calculations which for the 500 keV case (b) is almost perfect; it is the only calculation available that is able to reproduce the decrease in the experimental data for detection angles smaller than 303, even though no post-collisional interaction (PCI) is contained in the theory. It can be estimated [47] that the onset of the PCI e!ects at these high energies is at much smaller angles. The observed increase in the theoretical data for detection angles smaller than 153 is not in con#ict with the experimental observations. We believe that this increase is of relativistic origin and not a consequence of the lack of PCI. For gold all the other calculations overestimate the experimental data considerably. In a theoretical analysis Keller et al. [48] showed that the TDCS for the ionization of s states of high-Z atoms by relativistic electrons is rather sensitive to errors introduced by the use of approximate semirelativistic scattering wave functions. The authors pointed out that for the discussion of the "rst-order Born approximation the symmetric geometry is particular instructive because, in this case, for a given emission angle the momentum transfer K is the same in all contributions to the S matrix, allowing a more detailed analysis of the theory. However, it is obvious from the outset that the "rst Born approximation is not well suited for the description of this experiment because it does not treat the two outgoing electrons on an equal footing. Fig. 17 illustrates that indeed, in the "rst-order Born theories, large discrepancies persist between theory and experiment, in particular for the heavier targets. As one would expect, the Coulomb}Born and rDWBA theories that use the same type of wave function for both outgoing electrons do much more better here. It should be noted that the semirelativistic "rst-order Born results actually use an additional approximation, namely a Darwin-type bound state wave function. As was shown in [47], the e!ect of this approximation on the TDCS is negligible for silver and copper targets, while for gold it increases the cross section by about 30% in the binary regime. Therefore, while the discrepancies between the semirelativistic and relativistic "rst-order Born results shown for gold are somewhat exaggerated by this e!ect, they cannot be satisfactorily explained by the use of di!erent bound state wave functions. In the case of silver and copper the rDWBA overestimates the experimental data by 30}50%. The reason for this is not clear. The semirelativistic Coulomb}Born approximation of Jakuba{aAmundsen [47] while in poor accord with gold does better for silver and copper. (b) Peak position: In Section 4.1.1.1(a) we discussed in detail the angular position of the binary peak for coplanar asymmetric geometry. Ast et al. [67] performed a similar analysis for equal
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Fig. 17. Absolute TDCS for K-shell ionization of gold, silver and copper vs. scattering angle h in the coplanar symmetric geometry. (a) Z"79, E "300 keV; (b) Z"79, E "500 keV; (c) Z"47, E "300 keV; (d) Z"47, E "500 keV; (e) Z"29, E "300 keV; (f ) Z"29, E "500 keV. Symbols: experimental data [22,8], full curve: results of rDWBA [52,53], dotted curve: results of approximate "rst-order Born theory [8], dash-dotted curve: results of relativistic "rst-order Born theory [48], dashed curve: results of semirelativistic Coulomb}Born theory [47] (from [72]).
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energy sharing collisions in coplanar symmetric geometry and compared the results with experimental data [22,8]. In Fig. 18, the TDCS for the K shell of gold and E "500 keV is plotted as a function of the ion recoil momentum k "0 which is always collinear with the beam. The binary maximum in this geometry, in the nonrelativistic "rst Born approximation, corresponds to k "0. In the relativistic case there is a signi"cant shift away from zero recoil. Again from a comparison of the results of the full and the model calculation in relativistic "rst Born approximation it is evident that the peak shift is a purely relativistic e!ect. We would like to mention that in contrast to the asymmetric case there is a di!erence between the peak position in the semirelativistic "rst Born approximation [46,8] and the fully relativistic "rst Born approximation [48]. This e!ect can be traced to the structure of the exact and semirelativistic "rst Born matrix elements ([48] where the corresponding plot could be found). For lower atomic numbers as silver and copper the peak shift gets smaller [8]. (c) ¸arge angle peak: In Section 4.1.1.1(c) we discussed a large angle peak which has been observed in asymmetric geometry. To show the situation in symmetric geometry we display in Fig. 19 the calculations for gold already given (in Fig. 17) but now extended to larger scattering angles and on a log scale. The TDCSs calculated in the plane wave model and "rst Born approximation decreases steadily whereas the rDWBA predicts a large angle peak. Measurement with relativistic energies have not yet been performed in this angular regime. However, such a peak is well known from the low-energy (e, 2e) physics. The "rst indication of a large angle structure was found by Pochat et al. [75] in an investigation of the ionization of He at 100 and 200 eV. This prompted Whelan and Walters [29] to speculate that a multiple scattering mechanism might be at work [80]. They argued that the large angle structure could be interpreted in terms of the incident electron "rst elastically back scattering from the atom, essentially the nucleus, and then colliding
Fig. 18. Absolute TDCS for the K-shell ionization of gold, E "500 keV, E "E "209.65 keV, symmetric geometry. TDCS plotted as a function of the recoil momentum k . Experiment: (T) Bonfert et al. [22] and Walters et al. [8]; theory: "rst Born approximation (- - -) (normalized to experimental data) Keller et al. [62]; model calculations (*) (normalized to experimental data); rDWBA (.....) Whelan et al. [74] (from Ast et al. [67]).
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Fig. 19. Absolute TDCS for (e, 2e) on gold at 500 keV impact energy in coplanar symmetric geometry plotted against the detection angle of one outgoing electron. Experiment: (䊉) Bonfert et al. [22], Walters et al. [8], Theory: (-.-.-) PWBA, Keller and Whelan [40], (.....) "rst Born, Walters et al. [8]; (###) Coulomb Born, Jakuba{a-Amundsen [47], (*) rDWBA (from [74]).
with the target electron. If this were a collision between free particles, the two electrons would be detected at 903 to each other, i.e. at h"1353. Motivated by this simple intuitive model, a DWBA calculation was performed [81]. In Fig. 20 we show a comparison between this calculation and a measurement at an impact energy of 200 eV [82]. It is worth noting the very sharp dip that theory predicts and experiment "nds between the two peaks. The DWBA is the simplest viable approximation which allows for the mechanism suggested by Whelan and Walters [29] and the good agreement obtained with experiment appears to validate their description. It is further supported by the fact that the large angle peak is observed to be enhanced with respect to the binary as the atom size is increased [83,84]; an e!ect which was also predicted by Whelan and Walters [29] from their multiple scattering interpretation. It is worth noting that the results shown in Fig. 20 are plotted on a log scale and remarking that the large angle peak is very much smaller than the binary. We will comment on both these features in the following Section 4.1.3 where we come to look at a special feature of (e, 2e) processes in symmetric geometry, the relative importance of spin-#ip and non spin-#ip contributions to the TDCS. 4.1.3. Spin-dependent e+ects using unpolarized beams on unpolarized targets In a relativistic collision the spin is not conserved and in such a process a #ip of the spin of one or both the particles involved is possible. Let us "rst have a look onto the importance of spin-#ip processes in the cases so far investigated. In the relativistic regime experiments have been performed under either highly asymmetric or fully symmetric conditions. In Fig. 21 from [85] an asymmetric collision is considered. The TDCS
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Fig. 20. E "200 eV, helium target, Theory DWBA, Experiment [82].
obtained in rDWBA are plotted against the detection angle of the slow outgoing electron for a K-shell ionization of gold at an impact energy of 500 keV. The detection angle of the fast outgoing electron (E "319.3 keV) is "xed at h "!103, the slow electron carries an energy of 100 keV. The solid curve represents the complete TDCS for all channels and the dashed one the partial TDCS for the non-spin-#ip contributions. Here, at these impact energies, spin-#ip processes contribute only a little to the TDCS. However, the picture changes completely for a fully symmetric collision. If we consider Eq. (3.1) there are 16 possible spin channels, only six of which would exist in an nonrelativistic theory, i.e. s "s "s "s , @ s "s "!s "!s , @ s "!s "!s "s , @ where s is either # or !. For channel (i) above, both the direct and exchange S matrices are non-spin-#ip while for (ii) S and (iii) S are spin-#ip channels. In coplanar symmetric geometry it is clear that channel (i) is zero, i.e. the non-spin-#ip channels with equal spins for both electrons are blocked. Therefore, the relative importance of the spin-#ip channels is enlarged compared to the (i) (ii) (iii)
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Fig. 21. Absolute TDCS for the K-shell ionization of gold by electron impact plotted against the angle of the slow outgoing electron. Impact energy 500 keV, energy of the slow outgoing electron 100 keV, detection angle of the fast outgoing electron h "!153, 䉫 Bonfert et al. [22], (*) complete rDWBA, (- - -) rDWBA but non-spin-#ip channels only (from [85]).
asymmetric collision. This was "rst noted by Walters et al. [8], while investigating the semirelativistic "rst Born approximation. In Fig. 22 the TDCS for gold at an impact energy of 500 keV is plotted against the detection angle h of one of the outgoing electrons who share energy (E "E "209.65 keV). The solid line represents the TDCS obtained in rDWBA including all spin-#ip contributions and the dashed curve represents the partial TDCS for the non spin-#ip channels only. The spin-#ip channels contribute up to 40% to the TDCS, in the binary region and several orders of magnitude at the minimum point. It is instructive to compare these results with those of Fig. 20. We note that just as in the nonrelativistic case the non-spin-#ip calculation produces a very distinct minimum between the two peaks but the addition of the spin-#ip terms act to signi"cantly "ll up the minimum. Both calculated cross sections are compared with the absolute experimental results [22]. Whereas the calculated data of the TDCS (solid line) agrees very well with the experimental results, the partial non-spin-#ip TDCS underestimates the experimental results signi"cantly. To investigate the physics of the distortion potentials in the incident and "nal channels, Whelan et al. [74] have looked closer into the collision with gold at an impact energy of 300 keV and performed two additional calculations. In the "rst, the distorted wave in the incident channel was replaced by a plane wave and the distorted waves for the outgoing electrons retained (dashed line in Fig. 23) and in the second the distorted wave in the incident channel was retained but two plane waves were used in the outgoing channel, each orthogonalized to the bound state (dotted line, Fig. 23). These model calculations were used as a means of exploring the in#uence of the elastic scattering of the electrons in the atomic "eld separately for the incoming and outgoing electrons
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Fig. 22. Absolute TDCS for (e, 2e) on gold at 500 keV impact energy in coplanar symmetric geometry plotted against the detection angle of one outgoing electron. (䉫) Bonfert et al. [22], (*) rDWBA (all channels), (- - -) rDWBA but non-spin-#ip channels only (from [85]).
Fig. 23. TDCS for (e, 2e) on gold at 500 keV impact energy in coplanar symmetric geometry plotted against the detection angle of one outgoing electron. (*) complete rDWBA with distortion in the incident and "nal channels, (- - -) rDWBA with a plane wave in the incident channel and distorted waves in the "nal channels; (.....) rDWBA with a distorted wave for the incoming electron and plane waves for the outgoing electrons. (from [74]).
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and the interference between these e!ects. The large angle peak was only found if there was a distortion in the incoming channel, i.e. if we have no elastic scattering in the incident channel then there is no large angle peak exactly as is observed in the nonrelativistic case above. In the calculations of [74], however, evidence for strong interference e!ects was observed: the position of the large angle peak was shifted signi"cantly when distortion was included in all the unbound channels compared to the model situation with only a distortion in the incident channel. In the latter the large angle peak is maximal at about 1303. This is more or less the position predicted by the intuitive model mentioned above. Having distorted waves in all channels shifted the maximum of the peak to 1103 (703 seen with respect to the backward direction). This additional shift goes beyond the simple model. Interference e!ects and the in#uence of the heavy target atom are clearly signi"cant. It would be very useful, if the existence of the large angle peak could be veri"ed experimentally. However Whelan et al. [74] noted that the magnitude of the large angle peak is less than 0.05 mb/sr keV and speculated that this might prove di$cult for experiment. They further argued that as the large angle peak is clearly driven by the interaction of the atom with the incoming and outgoing electrons, a setup with maximal distortion could lead to a larger TDCS for the large angle region. With this in mind they performed calculations on uranium which is the heaviest element suitable for an (e, 2e) experiment with the present experimental setups and timescales. With its binding energy of 115 keV, the in#uence of the atom on the in- and outgoing electrons should be most signi"cant. In Fig. 24, the TDCS for the K-shell ionization of uranium calculated with the rDWBA is plotted for 500 keV (dashed curve), 300 keV (full curve) and 200 keV (dotted curve) impact energy against the detection angle [74]. Here the large angle peak is indeed more
Fig. 24. TDCS for (e, 2e) on uranium in coplanar symmetric geometry plotted against the detection angle of one outgoing electron: (- - -), 500 keV impact energy; (*), 300 keV; and (.....), 220 keV, all calculations in complete rDWBA (from [74]).
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pronounced and gains in height. A cross section of more than 0.3 mb/sr keV, which is six times bigger than in the collision with gold at 300 keV. Such experiments should be well within present experimental capabilities. Another interesting observation could be made here. Compared with gold the large angle peak is shifted to even smaller scattering angles. This is another strong indication of the importance of the in#uence of the strong atomic potential on the process. Clearly, it would also be of interest to make measurements at the minimum point between the binary and large angle peaks. As we noted above, the theoretical calculations predict that spin-#ip channels will give a maximal contribution at this point. It is clearly of interest to look for geometrical arrangements where distorting e!ects will be particularly strong. With this in mind it is valuable to consider coplanar experiments where the angle between the exiting electrons is held "xed and both are rotated around the beam direction. This geometry was earlier considered by Whelan et al. [86,87] while studying Coulomb three body e!ects at low energy. Its principal signi"cance at these energies is that one may, as a "rst approximation, treat the "nal state electron}electron interaction as separable and thus one may concentrate on initial channel e!ects. In the relativistic case neither electron}electron repulsion, post-collisionally, nor polarization in the incident channel are expected to be of any signi"cance. The reasons for the use of this geometry in a relativistic situation are completely di!erent. The principal advantage of the new geometry is that it allows one to study the role of strong distortion in the incident and "nal channels and the e!ect of their interference. We remark that for existing experiments the TDCS exhibits only a small amount of structure and the di!erence between the various theoretical approximations manifests itself mainly through di!erences in absolute size. In contrast, for the new geometry, much structure is observed and this is crucially dependent on the degree of distortion in the individual channels. Following Whelan et al. [88], we show in Fig. 25,
Fig. 25. TDCS plotted against the detection angle of electron 2 for gold at 300 keV, the outgoing electrons share energy, h "603, continuous line, rDWBA: dotted line, plane waves in all channels; long broken line is a distorted wave in incident channel and plane waves in "nal channels; short broken line, plane waves in incident channel and distorted wave in "nal channel; dash-dotted line, relativistic Born approximation (from [88]).
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the TDCS for the K-shell ionization of gold obtained in the rDWBA, plotted against the detection angle of the second electron. Also shown are various model calculations where some or all the distorted waves are replaced by plane waves. The relative angle h between the two outgoing electrons is "xed to 603, the impact energy is 300 keV and the two outgoing electrons share energy (E "E "109.65 keV). One recognizes a double peaked primary structure and a secondary maximum left and right of the primary. Each point accessed in the coplanar constant h geometry has one point in common with a speci"c asymmetric Ehrhardt geometry. By comparison with the TDCS obtained in Ehrhardt geometry one recognizes that the primary structure samples over binary and the secondary maxima over recoil peaks in di!erent Ehrhardt geometries. We note further that the saddle point in the primary structure has the coordinates !h "h "h and is therefore also the point in common with the symmetric Pochat geometry. Because of the symmetry of the problem the point !h "h "h must be either an extremum or the TDCS must be a constant in the neighbourhood of this point. It is very instructive to see why a minimum is realized here. In calculation of the TDCS, one sums over asymptotic spin projection channels for the outgoing electrons and averages over the initial ones. Now as before the six channels labeled (i), (ii), (iii), s "$, are dominant. The remaining channels which contain only spin #ip amplitudes (which would not exist in a nonrelativistic theory) make a much smaller contribution. For channel (i) where all spins are equal it is clear that at the symmetric point !h "h "h the S matrix is exactly zero. However for the two other channels the S-matrix elements are "nite at this point. Indeed for (ii) "S""<"S#" and for (iii) "S"";"S#" over the entire angular range. Spin #ip amplitudes are smaller compared to non-spin-#ip amplitudes at the impact energies under consideration here. In Fig. 26 we plot the partial TDCS in rDWBA corresponding to the spin
Fig. 26. Partial TDCS obtained in rDWBA plotted against the detection angle of electron 2 for gold at 300 keV impact energy, the outgoing electrons share energy, the relative angle h between the detectors is "xed at 603 (*) partial cross section channel (i), (} } }) partial cs channel (ii), (* * *) partial cs channel (iii) (from [85]).
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channels (i), (ii) and (iii) with s " and h "603 for gold at 300 keV impact energy. We see that the dip observed in the TDCS arises because of the relative importance of the individual channel contributions as we move away from the point of symmetry. It is now instructive to vary the relative angle between the two outgoing electrons. In Fig. 27 the TDCS is plotted as a function of the detection angle h for gold at 300 keV impact energy for (a) h "103, (b) 303, (c) 903, (d) 1153, (e) 1403 and (f) 1803. Signi"cant changes in the shape of the curve are observed if the relative angle h is enlarged or decreased with respect to our earlier example h "603. If h is enlarged the primary structure decreases with respect to the secondary maxima which remains more or less unchanged and for h "1403 the primary structure it has completely vanished. The case (f ) h "1803 is in somewhat special because for an energy-sharing collision the direction of the momentum transfer q with respect to one electron coincides with minus the direction of momentum transfer !q with respect to the other electron. In these circumstances, the terms binary and recoil become essentially meaningless. However, the physically more interesting case is the decrease of the "xed relative angle h . Here the primary structure remains strong but the dip vanishes and the point of symmetry converts from a local minimum to a maximum. To investigate this in greater detail consider once more (see Fig. 28) the contribution of the di!erent spin channels to the TDCS, but now for the case h "103. From Fig. 28 we learn that the contribution of spin channel (i) is dramatically decreased in absolute size and with respect to the channels (ii) and (iii), not only in the neighbourhood of the point of symmetry !h "h "h but over the entire angular range. This is a direct result of the anti-symmetry of the two fermion wavefunctions. Despite the fact that we have neglected electrostatic repulsion the possibility of "nding two continuum electrons in the same spin state decreases as h P0. This e!ect is an example of Pauli blocking familiar from atomic structure physics. Experiments in this geometry are in progress. For further discussion see [88,85]. 4.1.4. Inyuence of Coulomb boundary conditions [89,51] A feature of all the rDWBA calculations we have discussed so far is that we have assumed that all the exiting electrons asymptotically moved in the "eld of a neutral atom. It was never expected that this would impose a serious limitation on the calculations as the two "nal state electrons are so fast that a residual ionic charge of at most one was not expected to alter either the shape or the absolute size of the cross section. Nevertheless, it was felt that having the facility to impose Coulomb asymptotic conditions within the code was valuable not only to con"rm the validity of the earlier calculations but also to allow the construction of a fully relativistic Coulomb}Born approximation which would lead to a better understanding of the e!ect of using semirelativistic wave functions. The necessary changes to include Coulomb asymptotic conditions are described in detail in [51], and we will not go into them here. We only remark that the method used for generating the asymptotic Coulomb waves is a relatively straightforward generalization of the phase amplitude method of Burgess [90]. In comparing the results of various calculations it will be convenient to follow the notation of [51] and label each of the three unbounded electrons, in the order incoming, fast and slow, with a label and subscript. The letter D or C will indicate whether the electron is a distorted wave or a pure Coulomb wave, while the subscript labels the asymptotic charge Z seen by the electrons i.e. the usual rDWBA will be denoted by D D D , the Coulomb}Born by C C C and the 8 8 8 variant of the rDWBA where the two exiting electrons see an asymptotic charge of 1 by D D D . In
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Fig. 27. TDCS obtained in rDWBA plotted against the detection angle of electron 2 for gold at 300 keV impact energy, the outgoing electrons share energy, the relative angle h is "xed at (a) 153, (b) 303, (c) 903, (d) 1153, (e) 1403, (f) 1803 (from [85]).
Fig. 29 we consider the ionization of gold in symmetric geometry with an impact energy of 300 keV, as expected the D D D and D D D results are essentially identical. The semirelativistic Coulomb}Born approximation of [46] is in poor accord while the fully relativistic version, C C C is in much better accord. We should conclude from this that using semirelativistic
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Fig. 28. Same as Fig. 26 but for h "103.
Fig. 29. TDCS (in mb/sr keV) for the ionization of the K shell of gold (Z"79) as a function of h "!h in coplanar symmetric geometry. E "300 keV, E "E "109.65 keV. Full curve, rDWBA with physical boundary conditions (D D D ); dotted curve, rDWBA without physical boundary conditions (D D D ); broken curve, rCBA (C C C ); chain curve, srCBA [47]. The experimental points [22] are given with their error bars representing the standard deviation } systematic error $15% (from [51]).
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wavefunctions is not likely to be successful for any of these calculations. In Fig. 30 we show results for asymmetric geometry for 500 keV incident energy and E "100 keV, h "!153 the pattern repeats itself but here the C C C calculations underestimates, slightly the data. In Fig. 31, we show the TDCS for 500 keV electrons incident on silver, with E "100 keV, h "!73. The
Fig. 30. TDCS for the ionization of the K shell of gold (Z"79) in coplanar asymmetric geometry as a function of h for E "500 keV, E "100 keV, and h "xed at !153. Same legend as in Fig. 29 (from [51]).
Fig. 31. TDCS for the ionization of the K shell of silver (Z"47) in coplanar asymmetric geometry as a function of h for E "500 keV, E "100 keV, and h "xed at !73. The key is the same as in Fig. 29, with the label C C C replacing C C C . The semirelativistic Coulomb}Born data (srCBA) are from Jakuba{a-Amundsen [50] (from [51]).
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Fig. 32. Sketch of the arrangement of an (e, 2e) experiment with transversely polarized electrons.
D D D and D D D approximations are seen to be the best in predicting the maximum positions and di!er by about 5% at the peak. The fully relativistic Coulomb}Born, C C C underpredicts the maximum and shifts its position to slightly too large an angle. 4.2. Experiments with transversely polarized beams (e, 2e) experiments with polarized electrons represent an important step towards the ideal of a quantum mechanically complete analysis of the elementary process of electron-impact ionization. Recently, two types of relativistic (e, 2e) experiments using transversely polarized electron beams and unpolarized targets have been carried out to measure spin asymmetries. In the "rst type spin-up spin-down asymmetries have been measured which are caused by spin-dependent forces, i.e. by Mott scattering (essentially the spin}orbit interaction of the continuum electrons moving with relativistic energies in the Coulomb "eld of the atomic nucleus) (Section 4.2.1). In experiments of the second type the spin asymmetries are interpreted in terms of the so-called "ne-structure e!ect for the case of electron-impact ionization of states with nonzero orbital angular momentum (Section 4.2.2). Spin projections in relativistic quantum mechanics are not a good quantum number outside of the rest frame of a particle and care is needed in formulating the theoretical framework in which one can properly describe spin and angular momentum sensitive (e, 2e) experiments. The only really appropriate way of giving a description in this case is in terms of the density matrix, and Keller et al. [62] have taken great care to set the rDWBA calculations within the correct framework (see Section 3.3). In this section when we talk about the rDWBA or indeed the fully relativistic "rst-order Born approximation then we mean these approximations in the density matrix formalism as de"ned in [62]. A sketch of the experimental arrangement used for the measurements is shown in Fig. 32. 4.2.1. Spin asymmetries due to spin}orbit interaction of the continuum electrons Currently, measurements of the spin asymmetries due to spin}orbit interaction of the continuum electrons are available for the K shell of silver [7] and for the L shell of gold [78]. Measurements of spin asymmetries due to spin}orbit interaction of continuum electrons have been performed very recently for the K-shell of copper, silver and gold by Sauter et al. [104].
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4.2.1.1. Measurements on the K shell of silver. The "rst relativistic (e, 2e) experiment using a transversely polarized electron beam was designed by Prinz et al. [7] to look for a spin asymmetry caused by the spin}orbit interaction of the continuum electrons in the Coulomb "eld of the atomic nucleus, i.e. by spin-dependent forces. The spin}orbit interaction arises from the interaction of the magnetic moment of the electrons with the magnetic "eld felt in the rest frame of the electrons because of their motion in the Coulomb "eld of the target nucleus (Mott scattering). As a result a spin-up-down asymmetry is to be expected in the TDCS of electron-impact ionization. Measurements on the K shell of silver were performed using electrons spin polarized perpendicular to the scattering plane with the outgoing electrons detected in asymmetric geometry with h "!103, E "300 keV and E "200 keV. The quantity measured directly is the counting rate of the true coincidences alternately for spin-up and spin-down electrons of the primary beam. The spin asymmetry is de"ned as the relative cross section di!erence A"(dp(#)!dp(!))/(dp(#)#dp(!)) ,
(4.3)
where dp(#) and dp(!) are the TDCSs for impinging electrons with spin-up and spin-down. The asymmetry A is obtained as the ratio A"N/P where P is the polarization of the beam and for spin up and spin down coincidence counting rates N and N , respectively > \ N"(N !N )/(N #N ). Setup B was used for the measurements (see Section 2.4.2). The result > \ > \ of the measurement of the spin asymmetry A as a function of the scattering angle h of the outgoing slow electrons is shown in Fig. 33. To visualize the angular position of the binary peak and the secondary maximum, Fig. 34 shows the measured relative TDCS (averaged over the spin directions of the primary beam). We compare the measurements with the results of the rDWBA [62] and Coulomb}Born approximations [91,63]. In the region of the secondary peak only the rDWBA is
Fig. 33. Spin up-down asymmetry A of the TDCS for electron-impact ionization of the K shell of silver as a function of the scattering angle h of the outgoing slow electrons of energy E "74.5 keV. E "300 keV, E "200 keV, h "!103. The error bars represent the standard deviations only; the systematic error of the asymmetry scale was estimated to be $2%. Symbols, experimental data of Prinz et al. [7]; dotted curve, CBA calculation of [91]; dashed curve, CBA calculation of [63]; full curve, rDWBA (from [62]).
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Fig. 34. Relative TDCSs for the asymmetry experiment [7] (data normalized at binary maximum). E "300 keV, E "200 keV, h "!103. Symbols, experimental data [7]; dotted curve, CBA calculation of [91]; dashed curve, CBA calculation of [63]; full curve, rDWBA (from [62]).
capable of reproducing the trend and order of magnitude of the TDCS observed. For the discussion of asymmetries, it is important to note that the de"nition of A (Eq. (4.3)) favors large asymmetries in regions where the spin-inclusive TDCS is small even if the spin di!erential cross sections dp(#) and dp(!) are of similar magnitude. This e!ect is well known from elastic electron atom scattering, where the correlation of large asymmetries with the minima of the di!erential cross section is interpreted as an indication of weak spin}orbit coupling (for a detailed example, see ([92] p. 54). Due to the fact that in the present case the asymmetry is de"ned by the correlated observation of two particles, its interpretation is not so clear. However, it is safe to interpret large values of the asymmetry function in terms of large spin related e!ects if the corresponding spin inclusive TDCS is also large. We will therefore concentrate on the discussion of the asymmetry in the region of the maxima of the TDCS. The experimental data (Fig. 33) show a small asymmetry in the region of the binary peak and a pronounced maximum of the asymmetry associated with the secondary maximum of the TDCS. All calculations correctly reproduce these gross features. However, both CBA calculations mispredict the shape of the asymmetry parameter curve observed. Only the rDWBA correctly describes the position of the experimental maximum of the asymmetry function. In the work of Prinz et al. [7], the observed asymmetries were interpreted in terms of spin}orbit interaction of the continuum electrons in the strong Coulomb "eld of the nucleus. It was argued the the maximum close to the direction of momentum transfer is mainly due to electron}electron collisions with the nucleus merely acting as a spectator, so that spin}orbit coupling resulting from electron nucleus scattering should be of little importance. The secondary maximum can only be present if the nucleus takes up considerable recoil momentum so that in this region the asymmetry is expected to be much larger. The experimental data are clearly compatible with this intuitive picture. It is instructive to consider the Z dependence of the asymmetry calculated according to the rDWBA. Following [62], we exhibit in Fig. 35 a series of rDWBA calculations for copper (Z"29),
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silver (Z"47), gold (Z"79) and uranium (Z"92). The impact energy was "xed by taking the following parameters for gold, E "300 keV, E "200 keV, h "!103, keeping the ejection angle "xed and the ratios E /E , E /E constant for all the other atoms. The ratio of recoil to binary peak is signi"cantly enhanced as Z is increased, this is a clear indication of nuclear in#uence [29,83]. Signi"cantly, the corresponding spin asymmetry (Fig. 36) is also enhanced. Recall that the spin interaction is, to lowest order in 1/c, the spin}orbit interaction energy and is proportional to r\d<(r)/dr where <(r) is the radial symmetric scattering potential. We would, therefore, intuitively expect an enhancement in spin asymmetry due to spin}orbit interactions of the continuum
Fig. 35. rDWBA results for TDCSs for ionization of the K shell of di!erent target species, normalized to unity at the binary maximum. Impact energy for gold 300 keV, constant ratios of E /E "3.72 and E /E "3 in all systems, and h "!103. Full curve, uranium (Z"92); long-dashed curve, gold (Z"79); short-dashed curve, silver (Z"47); dotted curve, copper (Z"29) (from [62]).
Fig. 36. Asymmetry corresponding to Fig. 35 (from [62]).
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electrons. It needs to be emphasized once again, however, that in the rDWBA calculations one treats everything fully relativistically. Thus, while one automatically includes `spin}orbita interactions one also includes all higher-order e!ects in the elastic scattering. In other words, the rDWBA gives a correct description of the elastic (Mott) scattering of all the unbounded electrons in the strong "eld of the atom, these processes give rise to the asymmetries that Prinz et al. [7] have measured (see &Note added in proof'). That we get a spin asymmetry at all for the K-shell ionization is a clear indication of relativistic e!ects. In the nonrelativistic limit spin}orbit coupling can be neglected so that spin and orbital angular momentum projections are separately conserved and it is straightforward to see that [62], the asymmetry A is zero. If, however, the target is polarized parallel or antiparallel to the incoming beam polarization then one can get a nonzero as asymmetry at low energies. This is the arrangement used in the experiments of Baum et al. [93] where a beam of low-energy spin-polarized electrons merged with a beam of lithium atoms with a polarized valence shell. These results were interpreted as a measurement of the ratio of singlet- and triplet-ionization triply di!erential cross section and are in reasonable agreement with the results of DWBA calculations of Zhang et al. [94]. Recall that the rDWBA approximation makes no allowance for exchange in the elastic scattering channels, while the standard DWBA of Section 3 and [94] is entirely nonrelativistic, hence we see that the spin asymmetries observed by Prinz [7] are relativistic in origin while those observed by Baum et al. [93] are related to exchange e!ects in the elastic channels. 4.2.1.2. Measurements on the L shell of gold.Prinz and Keller [78] have studied the importance of continuum spin}orbit coupling in the ionization of the L shell of gold. Apart from being of considerable interest in its own right such a study was an important preparatory work for the investigation of the relativistic "ne structure e!ect discussed in the following Section 4.2.2. The apparatus used was the same as that for the measurement on silver described above but now the transversely polarized beam was directed onto a 50 lg/cm gold-foil target. Fast outgoing electrons of 200$5.7 keV kinetic energy were detected at h "!103, while the second detector sampled electrons of 86.9$2.9 keV. The energy width of the detectors chosen represent a compromise between the necessities of separating the L shell from higher shells, and covering the ionization events from all magnetic substates of the L shell (the relevant binding energies are 14.3, 13.7, and 11.9 keV, respectively). The results of the measurement of the spin asymmetry, A (Eq. (4.3)), as a function of the scattering angle h of the outgoing slow electrons are depicted in Fig. 37a. To visualize the angular position of the recoil and the binary peak, Fig. 37b shows the measured relative triply di!erential cross section (averaged over the spin directions of the primary beam). Corresponding theoretical calculations were carried out using the rDWBA code described in Keller et al. [53,62]. Comparison of spin asymmetry with cross section shows that both measured and calculated asymmetries are very small in the binary region, while larger spin e!ects are associated with the recoil region. These results resemble closely with those found for K-shell ionization of silver discussed above, and the same simple physical picture suggests itself for the present case. To con"rm this interpretation, Prinz and Keller [78] compare the calculated total asymmetry with the corresponding results for individual magnetic sublevels (Fig. 38). The numerical data give strong evidence that the subshell averaged asymmetry is dominated by the contribution of the 2s state, while the p-state asymmetries appear to cancel out almost completely. In fact,
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Fig. 37. (a) Relative TDCS for electron impact ionization of the L shell of gold, as a function of the observation angle h of the slow outgoing electron. Impact energy 300 keV, fast outgoing electron energy 200 keV, h "!103. Symbols: experiment (error bars indicate statistical error), full curve: rDWBA calculation, normalized to give best visual "t to the experimental data (from Prinz and Keller [78]). (b) Spin-up}down asymmetry (as given by Eq. (4.3)) of the TDCS for the kinematics of (a). Symbols: experiment (error bars indicate statistical error); full curve: results of rDWBA calculation, averaged over subshells (from Prinz and Keller [78]).
Fig. 38. Results of rDWBA calculations for spin up-down asymmetries for the kinematics of Fig. 37. Full curve: asymmetry, averaged over subshells; chain curve: asymmetry for 2p state ionization; broken curve: asymmetry for 2p state ionization; dotted curve: asymmetry for 2s state ionization (from Prinz and Keller [78]).
the large 2p and 2p state asymmetries are quite similar in shape except for an overall (negative) factor. As was observed earlier by Keller et al. [62] in the relativistic regime this pattern is characteristic for the presence of the "ne structure e!ect, discussed below in Section 4.2.2, which indeed vanishes if the two levels are not discriminated in the experiment [95].
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4.2.2. Spin asymmetries due to the relativistic xne-structure e+ect Hanne [98] has predicted that spin-up-down asymmetries might be observed in (e, 2e) experiments even if spin-dependent forces on the continuum electron are negligible. This observation has prompted some interesting experimental [96,97] and theoretical work [99] at low energies. The essential point is the following: one will observe a spin asymmetry provided only that a "nestructure multiplet may be experimentally resolved in an ionization process (i.e. if the total angular momentum J is known). Due to conservation of total angular momentum this e!ect may also be interpreted in terms of an orientation of the spectator ion in the "nal state. In Appendix B we give a brief summary of the low-energy derivation as given in [95]; see also [62,100]. It is clear from this derivation that this "ne-structure e!ect will arise even if there is negligible Mott scattering and no account is taken of exchange in the elastic channels. However, for the low-energy experiments mentioned above exchange e!ects has been found to be extremely signi"cant, in particular exchange between the slow ejected electron and the residual ion in almost all cases gives the dominant contribution to the asymmetry [96]. At our high energies these exchange e!ects are negligible [101], they are therefore not included in the rDWBA program. In the following we will argue that it is possible to identify a spin asymmetry which has the `pure "ne-structure charactera in relativistic (e, 2e) collisions. Of course, for inner shell ionization of high-Z atoms the question arises how to separate the "ne-structure e!ect from asymmetries caused by spin-dependent forces of the continuum electrons as discussed in Section 4.2.1. In a very recent paper of Besch et al. [102] a relativistic (e, 2e) experiment is reported using a transversely polarized electron beam of 300 keV and the 2p state of the L shell of uranium. Since for the present the authors were interested in measuring the "ne-structure e!ect exclusively they chose kinematical conditions in a way that the asymmetry coming from the continuum electrons were expected to be as small as possible. To achieve this the binary region was chosen. Indeed, as discussed in Section 4.2.1, Prinz et al. [7] could show in a former (e, 2e) experiment on the K shell of silver that the asymmetry in the binary peak is close to zero whereas in the recoil region a distinct asymmetry was found. (Asymmetries due to the "ne-structure e!ect vanish for K-shell ionization because of the absence of orbital angular momentum in the s state.) To con"rm this feature of a small in#uence of the spin}orbit coupling of the continuum electrons the authors have repeated the former (e, 2e) experiment in the binary peak of the K shell of silver [7] with similar kinematical parameters as in the present measurement of uranium. Due to the smaller radius of the K-shell orbital of silver compared to the 2p orbital of uranium the spin}orbit interaction of the continuum electrons should be about a factor of three stronger compared to the 2p orbital of uranium. Nevertheless, the authors got only asymmetries between 0% and 3% indicating that any larger asymmetry observed under comparable kinematical conditions cannot be explained by continuum spin}orbit coupling. Apparatus B was used for the experiment (see Section 2.4.2). The source for the polarized electron beam (see Section 2.1.2) used the photoemission of electrons from a strained GaAs crystal irradiated by circularly polarized light of a laser diode and delivered a continuous beam of 300 keV with a polarization degree in the range of 60}65%. As the target an uranium foil with a thickness of 60 lg/cm was used. The measurement was performed on the 2p shell of uranium (E "17.2 keV) as the energy distance to the 2p shell is 3.8 keV and it is therefore relatively easy to experimentally resolve. In coplanar asymmetric geometry the outgoing fast electrons of E "210 keV are observed at a "xed scattering angle of h "!24.83 with respect to the primary beam direction. The detector for the coincident
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slow electrons was adjusted to an energy of 72.8 keV in order to select (e, 2e) processes from the 2p shell. The result of the measurement of the spin asymmetry A as a function of the scattering angle of the outgoing slow electrons is shown in Fig. 39 (full circles). The error bars are the standard deviations. The solid line is a theoretical prediction of the rDWBA of Keller et al. [62]. To visualize the angular distribution of the triply di!erential cross section in the binary peak we show in Fig. 39 the corresponding result of the rDWBA calculation, averaged over spin degrees of freedom. The experiment shows large asymmetries (up to 18%) which, according to the argument given above, cannot be explained in terms of spin}orbit coupling of the continuum electrons. In view of the fact that in this energy domain there are no known mechanisms other than spin}orbit coupling and possibly the "ne-structure e!ect that could lead to spin asymmetries, the authors conclude that the experiment clearly evidences the existence of the "ne-structure e!ect in relativistic (e, 2e) collisions.
Fig. 39. (a) Spin up-down asymmetry A of the TDCS for electron-impact ionization of the 2p state of the L shell of uranium as a function of the scattering angle h of the outgoing slow electrons of energy E "72.8 keV. The primary electron energy amounted to E "300 keV. The outgoing fast electrons of E "210 keV were observed at an angle of h "!24.83 Experiment: (䊏) Besch et al. [102]. The error bars represent the standard deviations only, the systematic error of asymmetry scale was estimated to be $2%. The solid line is a theoretical prediction of the rDWBA of Keller et al. [62] (from [102]). (b) Angular distribution of the TDCS according to the rDWBA for an unpolarized beam.
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In particular, the characteristic change of sign of the asymmetry near the so-called Bethe ridge (ion recoil momentum k "0) could be con"rmed. It appears for relativistic as well as for nonrelativis tic primary energies. The interpretation is con"rmed by the reasonable agreement between the experimental data and the calculated rDWBA results, which include just the two mechanisms mentioned above but ignores elastic channel exchange e!ects completely. It is interesting to note that a plane wave calculation leads to a vanishing asymmetry [62]. For the present high energies the active electrons are well decoupled from the spectator target electrons. Therefore, no exchange between the continuum electrons and the target atom is included for the incoming and outgoing elastic channels in the rDWBA calculations of the "ne-structure e!ect [62]. Compared to this a calculation of the asymmetry for lower incident energies done by Dorn et al. [103] includes the exchange between each outgoing electron and the remaining target electrons. There it turned out that this exchange mechanism can be the major factor under particular kinematical conditions; see also [99]. In view of this fact the asymmetry measured in the experiment of Besch et al. [102] can be regarded to be due to a more basic scattering mechanism since there is only the exchange between the two active electrons involved. An interesting task would be to link up with the work of Prinz and Keller [78]. Particularly, in the binary region, they observed a small asymmetry in an inclusive measurement on the L shell of gold in contrast to comparatively large asymmetry values predicted by their theoretical investigations for the 2p subshells. This phenomenon is supposed to be due to the averaging over the di!erent "ne-structure levels [78]. It will be interesting to con"rm this feature experimentally by carrying out inclusive and exclusive measurements under the same kinematical conditions. To conclude this section, we follow Keller et al. [62] in comparing calculations of the relativistic "ne-structure e!ect with the "ne-structure e!ect in the nonrelativistic limit. In the nonrelativistic theory [95], a characteristic signal for this e!ect is a ratio of the asymmetries Ap "!2Ap (see Appendix B). By contrast, in the absence of spin}orbit coupling the asymmetry vanishes for s-state ionization. This is illustrated in Fig. 40 for the case of an argon target at an nonrelativistic input energy. Fig. 41 shows the asymmetries for the same ratios of impact and binding energies and outgoing electrons as in Fig. 40 but for an uranium target. Here all active electrons are relativistic, so that in addition to target coupling e!ects there is substantial spin}orbit coupling. Furthermore, due to the choice of Bethe ridge conditions, the nodes in the momentum-space wave functions lead to minima in the TDCSs, which in#uence the asymmetry function. In contrast to the argon results the uranium asymmetries exhibit a rich structure, Ap is nonzero and while there is something of a similarity between the Ap and Ap the simple relation Ap "!2Ap no longer holds. However, there is an indication here that something of the character of the target coupling e!ects has been imposed on the asymmetry, in particular in the region of the binary maximum. This impression is further strengthened by the comparison between the full relativistic "rst-order Born approximation, the rDWBA and the semirelativistic Coulomb Born approximation given in Fig. 42. These approximations di!er substantially in their treatment of the Mott scattering of the continuum electrons and indeed predict signi"cantly di!erent asymmetries away from binary maximum but tend to give very similar asymmetries in the binary region. This suggests that measurements around the binary maximum would show a signi"cant sensitivity to target coupling e!ects.
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Fig. 40. rDWBA results for asymmetries for ionization of the n"2 magnetic sublevels. Z"18, E "3.52 keV, and E "2.38 keV, and h "!303 (Bethe ridge conditions). Full curve, 2p state; dashed curve, 2p state; dotted curve, 2s state (from [62]).
Fig. 41. rDWBA results for asymmetries for ionization of the n"2 magnetic sublevels. Z"92, E "300 keV, E "200 keV, and h "!273 (Bethe ridge conditions). Full curve, 2p state; dashed curve, 2p state; dotted curve, 2s state (from [62]).
5. Summary and conclusions Since the "rst relativistic (e, 2e) experiments carried out more than 15 years ago, this "eld of research has grown into a mature branch of atomic physics. This progress has only been possible due to the development and permanent re"nement of experimental setups that have allowed to
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Fig. 42. Theoretical results for asymmetries for ionization of the 2p shell; Z"79, E "300 keV, and E "210 keV, and h "!27.33 (Bethe ridge conditions). Full curve, rDWBA; dashed curve, CBA [63]; dotted curve, relativistic "rst-order Born approximation [48] (from [62]).
establish a broad and "rm data basis of relative and absolute triply di!erential cross sections and spin asymmetries. These experimental results have in turn served to provide critical tests for the theoretical modeling of the inner-shell ionization process, that we are now able to understand in some detail. We have learned that an adequate theoretical description cannot be achieved unless E E E E
one one one one
uses fully relativistic wave functions to describe the active electrons; includes all spin #ip channels; includes the in#uence of the large atom in all channels; uses the full photon propagator, including magnetic and retardation e!ects.
The relativistic distorted-wave Born approximation is the simplest possible approximation which meets these criteria, in particular it includes the in#uence of the high central charge and the (supposedly) passive electrons at least in the form of an e!ective potential in all channels. It has been used to good e!ect to describe experiments with both polarized and unpolarized electron beams. We are able to observe a number of manifestations of the relativistic nature of the process, as for instance the interference process responsible not only for a large fraction of the shift of the binary maximum away from the direction of momentum transfer, but also for the appearance of a pronounced secondary maximum in the forward quadrant, or the appearance of substantial continuum spin}orbit coupling e!ects in the recoil regime and the importance of spin-#ip e!ects. Fine-structure e!ects are clearly apparent in the binary regime for 2p substate ionization. In general, agreement between the rDWBA, the absolute and relative experimental TDCSs for K and L shells as well as spin asymmetries is highly encouraging. However, we have seen that for some of the lighter atoms and the lower energies experimental data tend to be somewhat smaller, even though theory continues to accurately predict "ne details of the shape of the TDCS. It is not clear
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what is the source of this discrepancy and more work needs to be done to understand it. There is also an urgent need for more experiments to test the ever increasing number of theoretical predictions. Predictions which suggest that the joint experimental and theoretical study of relativistic (e, 2e) processes will continue to make signi"cant contributions to our understanding of subtilities of relativistic few-body interactions.
Acknowledgements We are deeply indebted to our collaborators who have worked with us on this enterprise over many years. We thank our friends in Frankfurt am Main (HansjoK rg Ast and Stefan Keller), TuK bingen (Karl-Heinz Besch, JuK rgen Bonfert, HansjoK rg Graf, Thomas Kull, Hans-Thomas Prinz, Herbert Ruo!, Michael Sauter, Claus Dieter SchroK ter and Edmund SchuK le) and Cambridge (Ugo Ancarani and Jens Rasch) for their e!orts and o!er a special word of gratitude to Reiner Dreizler and James Walters from both of whom we have learnt enormously. We would like to thank HansjoK rg Ast and Stefan Keller for the permission to use unpublished material from their Habilitation theses. One of us (WN) will never cease to be grateful to Reiner Dreizler who initiated the Belfast-Cambridge-Frankfurt collaboration who developed the relativisticdistorted-wave Born approximation. WN should also like to thank Doris Jakuba{a-Amundsen for working out the Coulomb}Born approximation and for helpful discussions. The experimental work would not have been possible without the generous support of the Deutsche Forschungsgemeinschaft over many years (Na 102/6-1 to 11-3). We have been fortunate to live at a time where collaboration between European Laboratories is actively encouraged and have bene"ted from "nancial support from the British Council and Deutscher Akademischer Austauschdienst (DAAD) through the ARC programs, Nato (CRG 950407) and the European Union. The numerical calculations shown in this work were performed in a number of di!erent sites, in particular using the workstation clusters of the Gesellschaft fuK r Schwerionenforschung (GSI), Darmstadt, the Hochschulrechenzentrum UniversitaK t Frankfurt, the Hitachi at Cambridge and several di!erent parallel processors at Daresbury Laboratory. One of us, (CTW), is grateful to the hospitality of the sta! of the Institute of Theoretical Atomic and Molecular Physics at the Harvard Smithonian Center for Astrophysics for where a major part of his contribution to this report was written.
Appendix A. Summary of the experiments 1. Impact energy: 500 keV. 䡩 Target : Ag. Coplanar asymmetric geometry at "xed angles h "!203,!253; h "403, energy of ejected electron: 210 keV4E 4350 keV Setup: A , Ref. [6].
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2. As in (1) for Ta with the speci"cations h "!203, h "403, 100 keV4E 4360 keV, h "!303, h "303, 110 keV4E 4320 keV. Setup: A , Ref. [21]. 3. Impact energy: 500 keV. coplanar asymmetric geometry at "xed energy. 䡩 Target: Ag. E "375 keV, E "100 keV, h "!153, 2534h 4753, h "!73, !8034h 4!403, h "!73, 2534h 4753. 䡩 Target: Au. E "310 keV, E "100 keV, h "!153, 2034h 4753. Setup: A , Ref. [22]. 4. Impact energy: 300, 500 keV. energy-sharing coplanar symmetric geometry (h "!h "h) 䡩 Target: Au, 300 keV: E "E "110 keV, 1034h4603, 500 keV: E "E "210 keV, 1034h4603. 䡩 Target: Ag, 300 keV: E "E "137 keV, 2034h4553, 300 keV: E "E "237 keV, 1034h4503. 䡩 Target: Cu, 300 keV: E "E "145.5 keV, 3034h4503, 500 keV: E "E "245.5 keV, 2434h4563. Setup: A , Refs. [8,22]. 5. Impact energy: 300 keV 䡩 Target: Cu. coplanar asymmetric geometry E "220 keV, E "71 keV, h "!93, !15034h 4!303, 1534h 41353 Setup: B, Ref. [68].
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6. Impact energy: 300 keV 䡩 Target: Ag. measurements of spin asymmetry, coplanar asymmetric geometry at "xed energy E "200 keV, E "74.5 keV, h "!103, !15534h 4!303, 2534h 41503. Setup: B, Ref. [7]. 7. Impact energy: 300 keV 䡩 Target: Ag. Triply di!erential cross sections near or on Bethe ridge conditions (k "0), coplanar asymmetric geometry at E "200 keV, E "74.5 keV, h "!26.13, 3034h4753, E "E "137.25 keV, h "!37.93, 1534h4603. Setup: A , Refs. [69,70]. 8. Impact energy: 300 keV, spin asymmetry, relative TDCS. Unresolved L-shell data for Au E "200$5.7 keV, E "86.9$2.9 keV, h "!103,
!12035h 4703 .
Setup: B, Ref. [78]. 9. Impact energy: 300 keV, 2p subshell for Au. Absolute TDCS for the "ne-structure selected state, E "200 keV, E "78.1 keV, h "!27.33, 3035h 4753 Setup: A , Ref. [77]. 10. Impact energy: 300 keV. 2p spin asymmetry for uranium, E "210 keV, E "72.8 keV, h "!24.83, 2535h 4753 . Setup: B, Ref. [102].
Appendix B. Nonrelativistic limit for L-shell ionization In the nonrelativistic limit, Keller et al. [62] showed that for p-state ionization their density matrix formalism reduces to the same form as in the previous low-energy work of Jones et al. [95]. It is instructive to consider the derivation given in the latter paper. Explicitly, one assumes the "nal ion contains a p-hole and that the "nal j state may be experimentally resolved, one assumes that ¸S coupling is valid and that one is in the natural frame (i.e. the beam is polarized perpendicular to the scattering plane). Given these assumptions the m projection is the sum of the m and m projections. H J Q
468
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Consider the case of a spin up electron ionizing an atom leaving the ion in the state j", m ". H Clearly, this means that we must have m "1 and s"! for the ion; consequently the ionized J electron must have spin #. This "nal state for the ion may be reached either by direct scattering f J or exchange scattering g J . Since the direct and exchange processes cannot be distinguished, K K i.e. both "nal state electrons have the same spin, the amplitudes add coherently and the cross section is given by (B.1) "N" f !g " , p KH where the comes from the vector coupling coe$cients and N is a constant depending on how we normalize our continuum waves. Next consider the case of a spin up electron leaving the ion in the state j", m "!, in this H case we must have the ionized electron with spin down. Again the "nal state may be reached by the direct scattering process, f J or the exchange process. In this case it should be possible to K experimentally distinguish between the two cases since the two exiting electrons have di!erent energies and spins. Consequently, the amplitudes add incoherently and the cross section is given by (B.2) "N(" f "#"g ") . p \ \ KH Now since the experiments we are concerned with cannot distinguish the "nal m states of the ion H these states must be summed over i.e. p (#)"N(" f !g "#" f "#"g ") \ \ and in a similar manner
(B.3)
p (!)"N(" f !g "#" f "#"g ") \ \
(B.4)
and p (#)"N(1/3" f !g "#" f "#"g "#1/3" f "#1/3"g "#" f !g ") , (B.5) \ \ \ \ p (!)"N(1/3" f !g "#" f "#"g "#1/3" f "#1/3"g "#" f !g ") . (B.6) \ \ \ \ Now f J, g J will be di!erent for j" and j" since di!erent wave functions and di!erent energies K K should be used in the evaluation of the amplitudes, however if one assumes the states to be degenerate and that one can use the same approximate wave functions for both states then it is clear that A "!2A .
(B.7)
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473 CONTENTS VOLUME 315
F. Cooper, G.B. West (editors). Looking forward: frontiers in theoretical science. Symposium to honor the memory of Richard Slansky, Los Alamos NM, 20}21 May 1998 D. Bailin, A. Love. Orbifold compacti"cation of string theory W. Nakel, C.T. Whelan. Relativistic (e, 2e) processes
Nos. 1}3, p. 1 Nos. 4 & 5, p. 285 No. 6, p. 409
CONTENTS VOLUMES 311}314 I.V. Ostrovskii, O.A. Korotchcenko, T. Goto, H.G. Grimmeiss. Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces 311, No. 1, p. 1 R. Singh, B.M. Deb. Developments in excited-state density functional theory 311, No. 2, p. 47 D. Prialnik, O. Regev (editors). Processes in astrophysical #uids. Conference held at Technion } Israel Institute of Technology, Haifa, January 1998, on the occasion of the 60th birthday of Giora Shaviv 311, Nos. 3}5, p. 95 S.J. Sanders, A. Szanto de Toledo, C. Beck. Binary decay of light nuclear systems 311, No. 6, p. 487 B. Wolle. Tokamak plasma diagnostics based on measured neutron signals 312, Nos. 1/2, p. 1 F. Gel'mukhanov, H. Agren. Resonant X-ray Raman scattering 312, Nos. 3}6, p. 87 J. Fineberg, M. Marder. Instability in dynamic fracture 313, Nos. 1/2, p. 1 Y. Hatano. Interactions of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states 313, No. 3, p. 109 J.J. Ladik. Polymers as solids: a quantum mechanical treatment 313, No. 4, p. 171 D. Sornette. Earthquakes: from chemical alteration to mechanical rupture 313, No. 5, p. 237 S. Schael. B. physics at the Z-resonance 313, No. 6, p. 293 D.H. Lyth, A. Riotto. Particle physics models of in#ation and the cosmological density perturbation 314, Nos. 1/2, p. 1 R. Lai, A.J. Sievers. Nonlinear nanoscale localization of magnetic excitations in atomic lattices 314, No. 3, p. 147 A.J. Majda, P.R. Kramer. Simpli"ed models for turbulent di!usion: theory, numerical modelling, and physical phenomena 314, Nos. 4/5, p. 237 T. Piran. Gamma-ray bursts and the "reball model 314, No. 6, p. 575 E.H. Lieb, J. Yngvason. Erratum. The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1}96) 314, No. 6, p. 669 G. Zwicknagel, C. Toep!er, P.-G. Reinhard. Erratum. Stopping of heavy ions at strong coupling (Physics Reports 309 (1999) 117}208) 314, No. 6, p. 671
