Physics Reports vol.320

Physics Reports 320 (1999) 1}15

Long-range forces in the universe A.D. Dolgov* Teoretisk Astrofysik Center Juliane Maries Vej 30, DK-2100, Copenhagen, Denmark

Abstract Possible existence of new long range forces and their interaction strength are critically analyzed and reviewed.  1999 Elsevier Science B.V. All rights reserved. PACS: 04.90.#e

1. Introduction All forces in nature can be divided into two groups: the forces that are induced by an exchange of massless particles and fall as a power of distance, and the rest. The "rst group can be further sub-divided into Newton or Coulomb forces, which follow the celebrated inverse square decay law, F&1/r, and forces that fall faster or slower than that. The term `long-range forcesa usually relates to inverse square forces, though sometimes it is applied to all forces that fall slower than a power of distance. Among them are such extremes as the Van der Waals forces between electrically neutral atoms, which behave as 1/r, and quark con"nement forces, which presumably do not decrease with distance. The Van der Waals forces are not fundamental ones; they appear as a result of screening of the usual Coulomb interaction and we will not discuss them further. Historically the "rst long range forces that were quantitatively studied were gravitational forces, though gravitational coupling is the weakest in nature. Electric forces are stronger than gravitational ones by a factor of 10 but still the Coulomb law was not discovered until almost exactly a century after the Newton law. The reason for that is that gravity cannot be screened and large astronomical bodies possess huge `gravitational chargesa, proportional to their masses, and create strong (as we all feel) and easily measurable gravitational "elds. By contrast, macroscopic electric charges are usually screened and some e!orts are required to create a large Coulomb "eld. * Corresponding author: Tel.: #45-3532-5908; fax: #45-3532-5910. E-mail address: [email protected] (A.D. Dolgov)  Also at ITEP, Bol. Cheremushkinskaya 25, Moscow 113259, Russia. 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 7 0 - 8

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Macroscopically signi"cant long-range forces (behaving as 1/r) can be created through an exchange of massless bosons with di!erent spins, s"0, 1, 2. A massless bosonic "eld in the static case generically satis"es the equation *U "4po , (1) Q Q where o is the source density, i.e. charge density ( o ) for electromagnetic "eld and mass density ( o )   for gravitational "eld in non-relativistic limit. The solution to this equation is well known and can be expressed through the Green function:



o (r) P QQ , dr&&
U (r)" Q Q "r!r" r

(2)

where Q is the total electric (Q ), gravitational (Q "GM M ), or possibly a scalar (Q ) charge. Q  E    The potential U decreases as 1/r, so that the corresponding force falls as 1/r. One more essential Q point is that the potential is proportional to the total charge, Q"dro, so that macroscopically large "elds can be created. This is typical for massless particle exchange. It is worth noting that only in three-dimensional space an exchange of a massless boson creates force that decreases as 1/r. This makes possible stable Newtonian orbits in planetary systems and Coulomb orbits in atoms. Presumably life is only possible in three dimensions. For a massive "eld Eq. (2) is modi"ed as (*#m)U"4po

(3)

and the solution is



e\KP dro(r)eKrrYP . U" r

(4)

Thus the e!ective charge of the source is determined by the region with the thickness 1/m, while the transverse size may be much larger,&(d/m, where d is the smaller of the distance to the source and its size. Though the solution (2) looks the same for all spins, it is not quite so because of the di!erent properties of the sources. There is an essential di!erence between gravitational and electric "elds. There are particles with both positive and negative charges in nature. In particular, antiparticles have charges opposite to those of particles, the electric charge of an electron is opposite to the charge of a proton. Due to that electric charge is screened and matter is usually electrically neutral. A very large electric charge cannot exist even in vacuum. If the value of the charge exceeds a critical value the corresponding electric "eld creates electron}positron pairs from vacuum. As a result the charge is radiated away. As is well known, the critical value of the charge for a point-like particle is about e/a"137e, where e is the charge of the electron. For realistic ions of a "nite size the critical charge is somewhat larger. In contrast to electricity, gravitational "elds are almost always attractive, with a possible exception for unbounded sources, and cannot be screened. To see this more formally let us consider the Lagrangian of massless scalar , vector , and tensor "elds: I IJ (5) ¸"(* )(*I )!(* )(*I ?)#(* )(*I ?@)# j# jI# jIJ ,  I ?  I ?@ I IJ  I where j, jI, and jIJ are the sources of the corresponding "elds. Note that the sign of the kinetic term of the vector "eld is di!erent from those of scalar and tensor ones. The choice of signs is connected

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to the condition of the positive de"niteness of the energy of the "eld quanta. The sources jI and jIJ should be covariantly conserved, D jI"0 and D jIJ"0, otherwise the theory of massless "elds I I is not consistent. Scalar j and tensor jIJ sources are symmetric with respect to transition from particles to antiparticles while vector current jI is antisymmetric. This is well known from electrodynamics, where currents of particles and antiparticles are di!erent in signs. It can be proven [1,2] that there is only one conserved tensor source: the energy}momentum tensor ¹IJ. Its symmetry with respect to charge conjugation follows e.g. from the fact that ¹ is an energy density that should be positive both for particles and antiparticles. Static interaction is determined by the time components of the potentials and (in the   previous notations, "U and "U ). These components enter Lagrangian (5) with di!erent     signs. Writing Green functions for the corresponding "elds one can see that it gives rise to attraction of the same sign charges in gravity and to repulsion in electrostatics. The conservation of the energy}momentum tensor ¹IJ is linked to the freedom in choice of di!erent coordinate frames in the space-time. There is no other symmetry that gives rise to a conserved tensor source and so no massless spin two "elds except for the gravitational one exist in nature. As for conserved vector currents their existence is related to internal (gauge) symmetries and their number can be arbitrarily large. These symmetries ensure current conservation and the vanishing of the masses of the corresponding "elds not only on the classical but also on the quantum level, if anomalies are absent. There are several known charges in particle physics which may possibly be conserved. Among them are baryonic and di!erent leptonic (electronic, muonic, and tauonic) charges and in principle they might be coupled to massless vector "elds in the same way as the photon is coupled to electric charge. This possibility was "rst analyzed by Lee and Yang [3] and by Okun [4] who considered the couplings to baryonic and leptonic charges, respectively, and obtained very strong bounds on their magnitudes. We will discuss these bounds and some possible manifestations of new vectormediated long-range forces below. Due to chiral anomaly separate baryonic and total leptonic current conservations are broken but the combination (B!¸) remains non-anomalous and may be conserved. If this is the case, no massless vector "eld can be coupled to baryonic or leptonic current separately but the coupling to (B!¸) is permitted. The results obtained in the pioneering papers [3,4] could be applied to this case with little modi"cations. In principle, an exchange by massless fermions (e.g. by neutrinos, if they are massless) would also generate a power law force, but since one-fermion exchange changes quantum numbers of the source, the coherence would be lost and macroscopically interesting "elds could not be created. Some long-range forces can be generated by an exchange of a pair of massless fermions but such forces fall much faster than 1/r. For the case of ll-exchange the force behaves as 1/r [5]. In what follows, we will concentrate, however, on `reala long-range forces, that decrease as 1/r. 2. Equivalence principle and new long-range forces Let us forget for a while that baryonic charge, B, is not conserved and assume, following Ref. [2] that there exists a massless vector "eld coupled to B. Long-range forces generated by these baryo-photons would lead to `anti-gravitya between pieces of matter and to additional attraction

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between matter and antimatter. One would immediately conclude that this interaction must be weaker than the gravitational one, otherwise celestial bodies could not be formed, and thus the coupling constant should be quite small, a (10\+10\ a, where a"  is the electromag netic coupling constant and a is the baryonic one. This restriction can be very much improved if one invokes experiments testing equivalence principle [6}9]. The equivalence principle implies that the free-fall acceleration is the same for all bodies independently of their chemical content. This has been tested by EoK tvoK s et al. [6] for the acceleration towards the earth with the accuracy 10\ and with a somewhat better accuracy, a few;10\ by Renner [7]. Roughly a half century later this result, for the acceleration to the Sun, has been improved by Roll et al. [8] who have reached the accuracy about 10\ and by Braginsky and Panov [9] who have improved the accuracy down to 10\. Since the ratio of baryonic, B, (as well as leptonic charge, ¸) to mass is di!erent for di!erent samples, the baryonic long-range force would lead to a violation of the equivalence principle. It was shown in EoK tvoK s experiment that the ratio of the accelerations of copper and platinum are a /a "(4$2);10\. The ratios of ! . baryonic charge to mass for natural isotopes of Cu and Pt are, respectively, (B/m) "1.001 and ! (B/m) "0.999. Atomic weight here is given in terms of the weight of C. This permits to obtain . the following bound on the coupling constant of baryonic photons: a (10\

(6)

which is quite close to the original one found in Ref. [3]. This bound is based on the early experiment [6]. New more accurate tests [8,9] were made for aluminum and gold and aluminum and platinum, respectively. The variation of the ratio B/M for these elements is about 10\. It permits to improve the bound by 3}4 orders of magnitude, down to a (10\}10\

(7)

Similar arguments can be applied to leptonic long-range action [4]. The variation of the ratio of lepton number to the atomic weight for di!erent elements is about 10% and that makes the restriction two orders of magnitude stronger:



10\ from acceleration to the Earth , a ( * 10\}10\ from acceleration to the Sun .

(8)

These bounds are obtained under the assumption that neither baryonic nor leptonic charges of the Earth and the of Sun are screened by antibaryons or by antileptons. It is evidently true for baryonic charge because there are no antibaryons around to do the job. However, there are plenty of relic neutrinos and antineutrinos in the universe with the number density about 50/cm for any neutrino and antineutrino species, so that they may screen leptonic charges of celestial bodies. It was claimed in Refs. [10,11] that the screening of the Earth leptonic charge is almost complete and hence the tests of the equivalence principle could only give the bound a (10\}10\. This claim * was critically analyzed by Okun [12] and in more detail in Ref. [13]. It was shown that the screening of leptonic charge by cosmic neutrinos was negligible and the previously found limits survived. However it seems possible that the screening might be realized by a hypothetical light boson with a non-zero leptonic charge [12}14], if the latter existed. In short, the screening is impossible because the Bohr radius of neutrino bound state and/or the Debye screening length are both much larger than the normal atomic size. Therefore,

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macroscopically large pieces of matter with a non-compensated leptonic charge should exist. If a is * not too small, such pieces would be unstable, and for a small a the screening length would be * larger than the Earth radius and even the distance form the Earth to the Sun. This is in essence the arguments of Refs. [12,13]. In what follows somewhat di!erent arguments are presented, which also permit to reach the conclusion that lepton long-range forces, if exist, would not be screened. The coupling constant of the lepto}photonic interaction is bounded from above by the precise agreement with experiment of the QED predictions for anomalous magnetic moment of electron and for the Lamb shift. From these data follows, a (10\. The data on elastic l e-scattering give possibly a better bound by 2}3 *  orders of magnitude. However a much stronger restriction follows from astrophysics. If the coupling a is su$ciently large, lepto-photons would be abundantly produced in the Sun. Further, * depending on the value of the coupling, they would either propagate freely from the core to the surface and in this case they might carry away much more energy than photons, or if their mean free path is smaller than the core radius, R "10 cm, they would radiate away about the same  energy as photons. Both cases are not compatible with the existence of the Sun during 10 billion years. The arguments presented below are similar to that used by Bernstein et al. [15] and Okun [4] to restrict possible electric charges of di!erent types of neutrinos. Lepto-photons would be produced in the Sun through the reaction cePc e. Its cross-section * can be found from the Thomson cross-section with an evident substitution for the coupling constants: p ,p(cePc e)"8paa /3m , (9) * * *  where a"  is the electromagnetic "ne structure constant and a is an unknown hypothetical *  coupling constant of leptonic charge. The characteristic time of the production of lepto-photons is given by the expression (10) q ,(n */n )\"(p n n )\+3;10\ a\ s , *  A * * A A where the electron number density in the core, n "10 cm\ was substituted. If q is smaller than  * the c -escape time, q , thermal equilibrium with respect to c and the usual photons would be *  * established and the energy #ux of lepto-photons would be the same or even larger, because of a larger mean free path, than that of the usual photons. The mean free path of c in the core of the * Sun is given by Eq. (10) and is equal to l +10\a\ cm. It is smaller than the core radius if * * a '10\. In this case c are more or less in equilibrium with the usual photons. If a (10\, * * * lepto-photons very quickly escape the Sun and their energy #ux can be estimated as follows. From Eq. (10) one "nds that n *"3;10\ ) 0.24¹a\ where ¹ +1 keV is the core temperature. The  *  A total lepto-photon luminosity is 4pR 3;10\  0.72¹"3;10a ¸ (11) ¸ "  * > * a 3 * where we substituted 3¹ for the average energy of lepto-photons and ¸ "4;10 erg/s is the  > solar luminosity in normal photons. Demanding ¸ (¸ , we obtain * > a (3;10\ (12) * Possibly a more stringent bound can be established using the data from low background experiments. The #ux of lepto-photons from the Sun would excite atoms in the detectors and could be

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observed by the emitted light. However we do not need that here because using the limit (12) we will prove that lepton charge in the Earth and after that in the Sun are not screened and the limits (8) are valid. If the limit (12) is true, then the neutrino Bohr radius would be huge, r "1/a m '10/(m /eV) * * I J cm. There is a very large number of electrons inside this radius in the terrestrial material, in which the number density of electrons is n +10 cm\. Thus to "nd the radius of the bound  state one has to substitute the charge inside the orbit Z "4pRn /3. The corresponding poten*  tial behaves as ;(R)"a Z /R&R, i.e. it is a harmonic oscillator potential. The number of states * * with the principal quantum number i in this potential is N +i/2 (for a large i). The total G number of states with i(i is N (i)"i/6. The size of the orbit corresponding to i is  R &(iR , where  G R "(3/(8pa n m ))+0.07 cm a\m\ , (13)  *  J  J where a "10a and the neutrino mass is expressed in eV.  * Let us estimate now R , i.e. the radius of the piece of matter where compensation of leptonic  charge is incomplete, say, by 50%. In other words, N (i)"N /2. This condition means   N "4pn R /3"2i/6"R /3R (14)      From these line of equations one "nds R "2;10 cm/a m .   J The validity of normal gravitational interactions was veri"ed in laboratory experiments for bodies with a much smaller size than R . Assuming a very mild accuracy of 100%, we conclude  that the leptonic coupling must be weaker than the gravitational one, a (10\. That small * a implies that R is larger than the Earth radius, R "6;10 cm. Hence the leptonic charge of *  = the Earth created by electrons cannot be screened by neutrinos and the "rst of the limits (8) remains valid. Using this limit we can see that neutrinos could not screen the leptonic charge of the Sun as well, and the second of the limits (8) is also true. One may argue that the screening is achieved not by bound neutrinos but by neutrino gas inside the Earth. In this case the screening is given by the Debye screening length, l "m /en . One may " J * J check repeating the previous chain of arguments that even for n "n the screening is too weak to J  jeopardize the limits (8). Screening however may be quite essential [16] in another hypothetical case. If neutrinos are subject to a novel long-range force, their trajectories from the supernova 1987A would be bended and it would create a time dispersion dt/t&1/E [17}19], where E is neutrino energy. The energy spread of emitted neutrinos was about 10 MeV, the time of propagation t+5;10 s, and the duration of the signal dt was several seconds. The source of this hypothetical long-range force could be either electrons in the Galaxy, or protons, or dark matter particles, correspondingly with charges, q , q , or q . Depending on the nature of the source particles and their distribution in the C N "+ Galaxy, the limit



1 "q q "43;10\; J H m

for e and p ,

(15) /m for dark matter "+ N was derived [17}19]. The factor m /m appears for dark matter particles because in contrast to "+ N protons and electrons the mass density and not the number density of dark matter particles is known.

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The e!ect of screening on this bound (15) strongly depends upon the spatial distribution of the source particles in the Galaxy. If the latter are either electrons or protons, mostly concentrated in the central part of the Milky way, then the neutrino path practically lies outside the source charge distribution. In this case the Coulomb-like "eld of the source would be e!ectively screened by background neutrinos and no interesting bound on their charge can be obtained. Indeed, if neutrinos are very light, so that they remain relativistic today with temperature ¹+1.9 K, their Debye screening length is j &1/(q ¹ )+3;10\ kpc/q (16) " J J J This for q '10\ the screening radius is smaller than 3 kpc and neutrinos from SN 1987A would J not be e!ected by long-range forces from the galactic electrons or protons. If neutrinos are heavier, so that they are non-relativistic today, the screening radius would be even smaller. However if the source of the new long-range force are dark matter particles a restrictive limit on their charge can be obtained. The distribution of the neutrino charge Q (r) obeys the equation: J dQ /dr"(Q !Q )/j , (17) J 1 J " where Q (r) is the distribution of the source (dark matter) particles. For the localized charge of the 1 source the charge would be exponentially screened, Q "Q !Q &exp(!r/j ). For the dark  1 J " matter particles, which presumably have number density inversely proportional to r, the charge inside radius r behaves as Q (r)&r and the solution of Eq. (17) gives [16] "+ (18) Q (r)"Q (r)(j /r)(1!e\PH")  1 " Hence the charge is compensated only with the accuracy (j /R ), where R is the boundary radius "   where the charge of the source is cut-o!. Correspondingly the bound (15) with the account of the screening by cosmic neutrinos becomes q

(10\(m /m )(r/kpc)+10\(m /m ) (19) "+ "+ N "+ N The result does not depend on the charge of neutrino, q . J It was assumed above that the universe is neutral with respect to the charge to which massless vector bosons are coupled. Cosmological implications of possible violation of this assumptions are discussed in Section 4. The considerations presented above are valid only for electronic leptonic charge. The limits for muonic and tauonic charges are not so strong by an evident reason that neither muons nor tau-leptons are present in macroscopic quantities in any available samples of matter. Di!erent manifestations of possible long-range forces associated with muonic charge are discussed in Ref. [4]. If in addition to the usual photons, leptons are coupled to other massless vector bosons associated to di!erent leptonic charges, quantum corrections would generate mixings between these bosons. For example the usual photon would be transformed into muonic one through the muon loop. It would introduce a coupling of electrons to muonic photons or coupling of muonic neutrinos to the usual photons, or in other words, this diagram generates e!ective electric charge of muonic neutrinos. Due to gauge invariance, which is assumed to be unbroken for all the charges, the muon or other similar loops give contribution proportional to k, where k is the 4-momentum of the vector boson. So these loops do not generate photon mass. They create a mixing in the

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kinetic part of the Lagrangian: L "!F  !FJ #eF FJ , (20)    ?@  ?@ ?@ ?@ where l corresponds to a certain leptonic charge, electronic, muonic or tauonic, or their combination. Since all photons remain massless, this mixing do not create photon oscillations in vacuum. However, oscillations would appear in media because of di!erent refraction indices of the `photonsa, di!erent plasma frequencies, etc. Consideration of stellar evolution permits to put rather strong limits on the strength of possible couplings to leptonic charges, other than electronic charge [15,4,20]. This is closely analogous to the derivation of the limit (12). Due to induced electric charge of neutrinos the plasmons in the Sun or in other stars may decay into l l -pair. Demanding that these neutrinos do not radiate too much I I energy from the Sun, the bound a

(10\ (21)

 can be obtained. This bound is valid if l -mass is below solar plasma frequency, u &100 eV. I  A similar bound but valid for neutrinos with masses up to 10}20 keV can be deduced from white dwarfs. For a recent review on the subject see Ref. [21]. A more stringent bound follows from the consideration of photon oscillations inside the Sun or of the process cePc e through muonic loop, cPkkPc . In vacuum it may be absent because one I I could rede"ne photon wave function so that the state that interacts with electrons does not have mixing with muonic charge. It would not remain valid in plasma and the limit from solar luminosity obtained this way is approximately 10 times weaker (by 1/a) than (12). Still all the bounds for muonic and tauonic charges are by far weaker than those obtained for electronic charge from equivalence principle.

3. Nucleosynthesis bounds on new forces and light particles Consideration of big-bang nucleosynthesis (BBN) permits to put an upper bound on the number of particle species contributing to the energy density of the primeval plasma at temperatures ¹+1}0.1 MeV [22,23]. The limit is usually expressed in terms of e!ective massless neutrino species, k , which make the same contribution into the energy density as the particles in question. J Given the accuracy of the present day data, the safe upper bound is *k ,k !3(1. There are J J several recent papers [24}26] presenting di!erent limits, from *k (0.2 [24] to *k (2.3 [25]. J J The last result is obtained if the mass fraction of primordial He is quite high, > "0.250. We  assume in what follows that *k (1, but will keep in mind possibilities of higher and lower limits. J If there are new long-range forces, there are additional massless vector bosons which should be present at nucleosynthesis and, if these bosons have vector-like coupling, right-handed neutrino states would be also excited. If they all are in thermal equilibrium, then l would contribute *k " 0 J 1 for every neutrino #avor which possesses coupling to massless vector "elds, and *k " for the J  corresponding new photons. This issue was raised in Refs. [27,28]. The production of right-handed neutrinos and new photons, as considered in Ref. [27], proceeded through the reaction c #c  l #l ? ? ? ?

(22)

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where a"k, q. The coupling to electronic charge, as we have seen above, is so weak that this interaction is never in equilibrium. The cross-section of this reaction is [25] p "2pa/s(ln(s/k )!1) , (23) ?? ? " where s is the center-of-mass energy squared and k "4pa n ?/¹ is the Debye screening mo" ? J mentum with n ? being the number density of l . Thermal equilibrium would be established ? J if the rate of this reaction, C "p n ? would be bigger than the expansion rate, ?? ?? J H"(4pg /45)¹/m . With p inversely proportional to s&¹ equilibrium would be reached H . ?? at su$ciently small temperatures, ¹(¹ . Demanding that ¹ is below 1 MeV, so that primordial ? ? nucleosynthesis is not disturbed, the bound was obtained a (1.8;10\ [27]. It was shown in ? Ref. [28], however, that this limit may be invalid if either l are two-component massless particles ? or have Majorana mass. In the former case the theory may have infrared problems (see Section 5), while in the case of Majorana mass separate leptonic charges of, say, l and l would be I O non-conserved but it is possible to arrange conservation of Q !Q . A model of long-range forces J O coupled to the di!erence of muonic and tauonic charges was proposed by Okun [29] to avoid anomalous charge non-conservation. Thus if *k "1 is permitted then in the framework of the model [28,29] only one extra light J particle, c would be present at nucleosynthesis and the coupling a is not restricted by BBN. If I\O I\O however, *k (1 then the limit a (10\ should be true. We will show that a considerably J ? stronger limit for a or a can be obtained if one takes into account production of c through the I I\O ? reaction: c#kPc #k (24) ? This reaction goes in the "rst order in a and at ¹'m its probability is much larger than the ? I probability of the reaction (22) that goes in the second order in a . Making the same calculations as ? above but with the cross-section of reaction (24) p"8paa /3m we "nd that the new photon-like ? I particles c would not be in equilibrium at ¹"m if a (4;10\. However the number of ? I ? particle species at this temperature, including k! and three pions, is g (m )"17.25. It is 1.6 times H I larger than the usual number of particle species at BBN, g ,"10.75. If c dropped out of H ? equilibrium below ¹"m then their contribution into energy density at BBN is suppressed by the I factor [g ,/g (m )]"1.88, as follows from entropy conservation. Correspondingly c would H H I ? contribute at BBN as (8/7)/1.88"0.6 neutrino species. Thus a '4;10\ would be excluded if ? nucleosynthesis constraint is as strong as *k (0.6. With the present day accuracy, *k (1 the J J limit is approximately two orders of magnitude weaker. Indeed, the `photonsa c would be ? produced through the reaction (24) even below ¹"m , though the number of muons is Boltzmann I suppressed at lower ¹. At m /¹"4 the suppression factor is 0.1, so one may neglect the entropy I release by muon annihilation and decay. The contribution of c produced at this ¹ into cosmic ? energy density at BBN would be smaller than that of one neutrino species if a (2;10\ (25) ? Direct laboratory limits on a and possibility of observation of muonic photons in neutrino I experiments were considered in a recent paper [30]. The best bound follows from the measurement of the anomalous magnetic moment of muon, a (10\. As shown in Ref. [30] muonic photons I may be observed in high-energy neutrino experiments if their coupling is above 10\.

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4. Conserved and non-conserved charges, vanishing photon mass, and charge asymmetry of the universe If there is a strictly conserved charge coupled to a massless vector boson the universe must be neutral with respect to this charge, otherwise cosmology would be very much di!erent from the usual Friedman}Robertson}Walker one. In connection to baryonic photons this point was brie#y discussed in Ref. [31]. In what follows we will use the term `electric chargea, meaning any conserved charge coupled to massless vector bosons (photons). It is known that net electric charge of a closed universe must be zero as is enforced by the Gauss theorem. If the Universe is open it may contain a nonzero electric charge but a non-vanishing homogeneous charge density results in a cosmological disaster. Indeed in the Friedman}Robertson}Walker (FRW) background electric "eld E satis"es the equation * FG"!4pJ , (26) G where J is the electric charge density and the Maxwell "eld strength is related to electric "eld as E "F . It is easy to see that this equation has only solutions linearly rising with distance, G G E&Jl , and this would destroy the homogeneity of the Universe and create quadratically rising 3 energy density of electric "eld, (27) oE"E/8p+a(l n ) 3  where n ,b n is the number density of the excessive charged particles, (e.g. the di!erence   A between number densities of particles and antiparticles), b is the corresponding coe$cient of  charge asymmetry, and l is the present day horizon size, l &t +10 cm. However if the 3 3  coupling constant is extremely weak, this might be not so dangerous. The ratio of oE to cosmological energy density o &Hm &t\m is  . 3 . (28) rE"oE/o +an l /m   3 . In particular for baryonic charge n +3;10\ and this ratio is rE"10 a . It is negligible if  a (10\}10\, as restricted by the bounds (6}7). Charge particles in such long-range cosmic "eld would be accelerated up to energies E+an l +10 eVa(b /3;10\). For baryons this energy would be smaller than 10\ eV.  3  This kind of cosmological charge asymmetry would be noticeable for a much stronger coupling. It would create in particular an asymmetry in cosmic rays and an intrinsic dipole in cosmic microwave background radiation. However the universe may be charge asymmetric only locally but globally charge symmetric. If this is the case one should substitute instead of l a smaller size of 3 domains with a certain sign of charge asymmetry. It is possible that the considerations presented above are not cosmologically relevant because if a charge is strictly conserved, no charge asymmetry in the universe could be developed. One may make an unnatural assumption that a charge asymmetry existed ab inito but such hypothesis is incompatible with in#ationary cosmology with a su$ciently long period of in#ation [32,33]. If cosmological density of a conserved charge were non-vanishing, the energy density associated with this charge could not remain constant in the course of the universe evolution and in#ation would be impossible.

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The situation would be drastically di!erent if the photons have a small but non-vanishing mass. Cosmology of electrically charged universe both with zero and nonzero mass of the photon was considered in Refs. [33}38]. Though the photon mass is known to be very strongly bounded from above (for a review see [20,39,40]) cosmological e!ects associated with it in a charged universe might be essential. Analysis of the Jupiter magnetic "eld leads to m (6;10\ eV while magnetic A "elds in galaxies imply a much stronger limit m (3;10\ eV [41] which we will use in what A follows. With an addition of mass term the equation of motion for the vector "eld in FRW-background takes the form: D FI#mA "4pJ , (29) I J A J J where D is the covariant derivative in the external gravitational "eld. For a uniformly charged I universe this equation has a homogeneous and isotropic solution A "dR A (t) with A (t)" I I R R 4pJ /m. This solution is very much di!erent from the massless case, in particular, it does not make  A any electric "eld. There is no smooth limiting transition to the case of zero mass. The energy density of this solution is o "mA AI/2"abn/2m , (30) K A I A A where b is the corresponding charge asymmetry. The pressure density is equal to the energy density, p "o ; it is the sti!est equation of state [42}44]. The energy density decreases as 1/a, K K while the scale factor behaves as a&t. The energy density o would be smaller than the K cosmological energy density if electric charge asymmetry b is smaller than (31) b(m m /(at n +10\(m /3;10\ eV)(1/137a  A A A . For other possible charges with weaker couplings the limit on asymmetry is much less restrictive. One should keep in mind however that o scales as 1/a and thus it would dominate at an early K stage of the universe evolution. The condition that o did not dominate during primordial K nucleosynthesis permits to improve the bound (31) by 11}12 orders of magnitude. Another possibility to cure infrared problems of electrically charged universe, except of prescribing a small mass to the photon, is to introduce a non-minimal coupling of photons to gravitational "eld. The most general Lagrangian of this kind that contains only dimensionless coupling constants m has the form: G L "!F FIJ/4#mA AI/2#m RA AI/2#m RIJA A , (32)   IJ A I  I  I J where RIJ is the Ricci tensor and R is the curvature scalar. Three last terms in this expression should vanish if gauge invariance is unbroken. The upper bounds on the photon mass quoted above permit the constants m to be as large as 10. It can be shown that for the case of dominant G coupling to curvature scalar the universe expands according to the same law as in non-relativistic case, a(t)&t. Cosmology of charged universe with vector boson "eld described by the Lagrangian (32) may possess quite interesting and unusual features. In particular, it may possibly induce cosmic acceleration and in this sense mimic a non-zero cosmological constant. We will postpone a more detailed consideration of these problems to a future work.

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5. Problems of vanishing mass and charge conservation As we have already mentioned in passing, a massless vector boson must interact with a conserved current, otherwise the theory would be infra-red pathological. Even if the photon mass is non-zero but small in comparison with the characteristic energy scale of a reaction, it does not help to achieve infra-red regularization [45]. Emission of longitudinal photons would be catastrophic and perturbation theory breaks down. The probability of emission of k longitudinal photons is proportional to gaI(u/m )I, where g is the coupling constant of the interaction that breaks A current conservation. As was argued in Ref. [45], renormalization of the constant g due to virtual longitudinal photons is very strong and exponentially diminishes g almost to zero, g "g exp(!aK/4pm), where K is an ultraviolet cut-o! momentum. Thus even if electric charge  A were not conserved, the decay of electron would be very strongly suppressed. Similar problems may appear with baryonic long-range action. Baryonic charge is known to be non-conserved due to chiral anomaly. Thus baryonic photons, if exist, must be massive, otherwise the usual (massless) equation * FI"J? (33) I J J would be self-contradictory and one has to add the term mA , where a indicates the type of the ? J charge, e.g. it can be baryonic charge, a,B, or any other charge. If this is the case the upper bound on the coupling constant a would depend on the mass of baryo-photon, m . If m\ is smaller then the distance between the Sun and the Earth (this corresponds to m '10\ eV) the experiments testing validity of the equivalence principle by the acceleration to the Sun are not sensitive to the baryonic "eld. With rising m , when m\ becomes smaller then the radius of the Earth, R "6400 km, the bounds on a obtained from the EoK tvoK s-type experiments become less restrict= ive. An interesting possibility is that baryonic current is non-conserved, baryo-photons are massive but light, and the proton decay is suppressed because of renormalization of the baryon nonconserved coupling discussed above. Another infrared problem is related to vanishing masses of charged particles, e.g. neutrinos. It was analyzed in Refs. [46,47] in electrodynamics, where singularities in the limit of electron mass going to zero were considered. To avoid all these complications we assume that neutrinos have a small but non-vanishing mass and the theory would be well de"ned with at worst logarithmic singularities in the reaction amplitudes. Low mass neutrinos could be produced by leptonic "eld in the same way as e>e\-pairs are produced by electric "eld. The probability of production is proportional to the Schwinger exponent exp(!pm/"e E ") and with a very small m may be signi"cant and may lead to a screening of J * * J leptonic charge.

6. Other long-range forces As we have already mentioned, long-range forces can be associated with an exchange of bosons with spin 0, 1, or 2. Bosons with higher spins do not have interaction with conserved sources and the only massless boson with spin 2 is the graviton, that interacts with energy-momentum tensor

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[1,2]. In the case of massless vector bosons and graviton their vanishing masses are supported, respectively, by gauge invariance and general covariance. Long-range action due to a scalar particle exchange is less natural theoretically than that associated to vector or tensor bosons because no unbroken symmetry is known that would support zero mass of the corresponding scalar boson. Massless scalar "elds could arise as a result of continuous global symmetry breaking in accordance to the Goldstone theorem. However, a Goldstone "eld s has the pseudo-scalar coupling to matter, tM c ts, and so the corresponding potential  decreases as r\. Scalar type coupling, tM ts, may appear for pseudo-Goldstone "elds but simultaneously with this coupling the boson acquires a non-zero mass, m , and generates Yukawa type Q potential, ;&exp(!m r)/r. The only known theoretical principle which would ensure long-range Q action of a scalar "eld with Coulomb type potential, ;&r\, is conformal invariance and the corresponding scalar boson is called the dilaton. However, in realistic four-dimensional world conformal invariance is broken and massless dilaton does not exist. Even if a scalar "eld is massless at the classical level quantum corrections should generate its non-vanishing mass. By this reason the scalar companion of the graviton, , in the Brans}Dicke model, that interacts with the trace of the energy-momentum tensor of matter, would acquire mass and the scalar gravity would become short-range. The magnitude of the mass of generated by quantum e!ects cannot be unambiguously calculated. Even an order of magnitude estimate is very much uncertain, m "m/m . and (  . can give anything from m +m (m\+10\ cm) to m +10\ eV (m\+10 km) depending ( . ( ( ( on the characteristic mass scale of the theory m . The upper bound corresponds to m "m and   . the lower one corresponds to m "K +0.1 GeV. Because of that the Brans}Dicke "eld should  /!" not manifest itself on astronomical scales but might be observable in experiments testing gravity at small distances. Still if a massless scalar boson somehow exists and creates long-range forces these forces would not be screened and might be operative on astronomical scales. However a scalar force between a static source and relativistic fermion, e.g. neutrino, is suppressed by the Lorenz factor m /E. J There may exist free massless spin-2 bosons coupled only to gravity with a simple Lag-rangian, L"U UIJ_?. Since the coupling to matter is absent such bosons would not mediate long-range IJ_? forces but may possibly play a role in solution of the mystery of the cosmological term [48]. Very interesting and unusual long-range forces were proposed by Okun in the paper [49] where a model with macroscopic con"nement radius was considered. The theory is analogous to Quantum Chromodynamics but is based on a di!erent gauge group with a di!erent number of fundamental fermions. This permits to have the coupling constant a of the order of the electromagF netic one at microscopically small distances and simultaneously to have the con"nement radius 1/K between the size of atomic nuclei and the size of the universe, depending on the model F parameters. Such long-range forces do not decrease with distance and bearers of a non-zero h-charge would be connected by macroscopically long h-strings. Cosmological implications of this model were considered in Refs. [49,50].

7. Conclusion At the present day only two types of long-range forces are known: gravitational and electromagnetic. However it is not excluded and moreover is quite natural that the club of long-range forces is

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not con"ned only to those two. Though the strength of possible new long-range interactions should be quite weak, still due to possible macroscopically large charges their role in nature, in particular in cosmology and astrophysics may be signi"cant. They may be important for general cosmological evolution, large scale structure formation in the universe, and may be possibly imprinted on cosmic microwave background radiation. It is an interesting challenge to search and to "nd such new forces. It is my great pleasure to write a paper on the subject where my advisor and teacher Lev Borisovich Okun made such outstanding contributions and to dedicate this paper to his 70th anniversary.

Acknowledgements This work was supported by Danmarks Grundforskningsfond through its funding of the Theoretical Astrophysical Center. I am grateful for hospitality to the Weizmann Institute of Science, where this work was completed.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

S. Weinberg, Phys. Rev. 140 (1965) B515. S. Weinberg, Quantum Theory of Fields, Vol. I, Cambridge University Press, Cambridge, 1995, p. 537. T.D. Lee, C.N. Yang, Phys. Rev. 98 (1955) 1501. L.B. Okun, Yad. Fiz. 10 (1969) 358; English translation: Sov. J. Nucl. Phys. 10 (1969) 206. G. Feinberg, J. Sucher, Phys. Rev. 166 (1968) 1638. L. EoK tvoK s, D. Pekkar, F. Fekete, Ann. Phys. 68 (1922) 11. J. Renner, Mat. eH s termeH szettudomaH nyi ertsitoK 53 (1935) 542. P.G. Roll, R. Krotkov, R.H. Dicke, Ann. Phys. (NY) 26 (1964) 442. V.B. Braginsky, V.I. Panov, ZhETF 61 (1971) 873; English translation: Sov. Phys. JETP 34 (1972) 463. G.A. Zisman, Uchenye Zapiski LGPI No. 386 (1971) 80 (in Russian). V.M. Goldman, G.A. Zisman, R.Ya. Shaulov, Tematicheskii Sbornik LGPI, 1972 (in Russian). L.B. Okun, A remark on leptonic photons, June 1972 (unpublished, presented at ITEP seminar). S.I. Blinnikov, A.D. Dolgov, L.B. Okun, M.B. Voloshin, Nucl. Phys. B 458 (1996) 52. A.K. C7 iftc7 i, S. Sultansoi, S7 . TuK rkoK z, Phys. Lett. B 355 (1995) 494. J. Bernstein, M. Ruderman, G. Feinberg, Phys. Rev. 132 (1968) 1227. A.D. Dolgov, G.G. Ra!elt, Phys. Rev. D 52 (1995) 2581. J.A. Grifols, E. Masso, S. Peris, Phys. Lett. B 207 (1988) 493. J.A. Grifols, E. Masso, S. Peris, Astropart. Phys. 2 (1994) 161. G. Fiorentini, G. Mezzorani, Phys. Lett. B 221 (1989) 353. A.D. Dolgov, Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1. G.G. Ra!elt, hep-ph/9903472; Ann. Rev. Nucl. Part. Phys. 49 (1999), in preparation. V.F. Shvartsman, Pis'ma ZhETF 9 (1969) 315; English translation JETP Lett. 9 (1969) 184. G. Steigman, D.N. Schramm, J.E. Gunn, Phys. Lett. B 66 (1977) 202. S. Burles, K.M. Nollett, J.N. Truran, M.S. Turner, Phys. Rev. Lett. 82 (1999) 4176. K.A. Olive, D. Thomas, hep-ph/9811444. E. Lisi, S. Sarkar, F.L. Vilante, Phys. Rev. D 59 (1999) 123520. J.A. Grifols, E. Masso, Phys. Lett. B 396 (1997) 201. L.B. Okun, Mod. Phys. Lett. A 11 (1996) 3041.

A.D. Dolgov / Physics Reports 320 (1999) 1}15 [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]

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L.B. Okun, Phys. Lett. B 382 (1996) 389. V.A. Ilyin, L.B. Okun, A.N. Rozanov, Nucl. Phys. B 525 (1998) 51. R.H. Dicke, Phys. Rev. 126 (1962) 1580. A.D. Dolgov, Ya.B. Zeldovich, M.V. Sazhin, Kosmologiya Rannei Vselennoi (in Russian), MGU, 1988; Basics of Modern Cosmology, Edition Frontier, Dreux, 1990. A.D. Dolgov, Phys. Rep. 222 (1992) 309. R.A. Lyttleton, H. Bondi, Proc. Roy. Soc. (London) A 252 (1959) 313. A. Barnes, Astrophys. J. 227 (1979) 1. C. Ftaclas, J.M. Cohen, Astrophys. J. 227 (1979) 13. S. Orito, M. Yoshimura, Phys. Rev. Lett. 54 (1985) 2457. J.E. Kim, T. Lee, Mod. Phys. Lett. A 5 (1990) 2209. A.S. Goldhaber, M.M. Nieto, Rev. Mod. Phys. 43 (1971) 277. Particle Data Group, European Phys. J. C 3 (1998) 1. V. Chibisov, Sov. Phys. Uspekhi 19 (1976) 624. Ya.B. Zeldovich, ZhETF 41 (1961) 1609. Ya.B. Zeldovich, I.D. Novikov, Struktura i Evolyutsiya Vselennoi (in Russian), Nauka, Moscow, 1975; Structure and Evolution of the Universe, The University of Chicago Press, Chicago, 1983. I.Yu. Kobzarev, L.B. Okun, ZhETF 43 (1962) 1904. M.B. Voloshin, L.B. Okun, Pis'ma ZhETF 28 (1978) 156; English translation: JETP Lett. 28 (1978) 145. T.D. Lee, M. Nauenberg, Phys. Rev. 133 B (1964) 1549. A.D. Dolgov, L.B. Okun, V.I. Zakharov, Nucl. Phys. B 41 (1972) 197. A.D. Dolgov, Phys. Rev. D 55 (1997) 5881. L.B. Okun, JETP Lett. 31 (1980) 144. A.D. Dolgov, Yad. Fiz. 31 (1980) 1522.

Physics Reports 320 (1999) 17}25

Cooperation between scientists and the government in the US: bene"ts and problems夽 Sidney D. Drell Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305-5080, USA

Abstract This paper discusses the importance of maintaining close working relations between scientists and their government. Several examples of this cooperation in the U.S. are presented to illustrate the bene"ts, as well as problems, that result from such cooperation, or lack thereof. These examples include government support of scienti"c research, as well as contributions by science to help governments understand both the possibilities and the limitations of science as they formulate national policy.  1999 Elsevier Science B.V. All rights reserved. PACS: 01.78.#p Keywords: Science and government; Science and public policy; Science advice

For Lev Okun, a dear friend and admired colleague, on his 70th birthday. For his distinguished contributions to research and to teaching; for his holding to the highest principles during the most dizcult times in the Soviet Union; and for his continuing dedicated and selyess ewort to preserve the best science and scientists in Russia during the present dizculties, Lev commands the awection and highest respect of his colleagues worldwide.

In World War II, American scientists } from university and industrial research labs and including many refugees from persecution in Europe } were recruited to work in large projects which focused on developing the latest scienti"c advances in support of the military e!ort of the US

夽

This paper is adapted from a talk given at the Institute of Theoretical and Experimental Physics in Moscow on October 28, 1998 as co-recipient, with Professor A. Akhiezer of Kharkov, of the "rst Ya. Pomeranchuk Award. 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 5 5 - 1

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and its Allies. The Radiation Lab organized at MIT developed microwave radar into instruments that proved decisive in the aerial defense of England and in the ultimate defeat of German U-boat raiders against the life line of convoys crossing the Atlantic Ocean to England and up to Murmansk. Out of the latest developments in nuclear physics and the theory of "ssion, a successful nuclear chain reaction was achieved by Enrico Fermi and his collaborators at the University of Chicago's Metallurgical Labs on December 2, 1942. This led to the production of plutonium and eventually to the construction of the "rst atomic bombs at Los Alamos under J. Robert Oppenheimer's leadership. These are the two best known examples of the massive e!ort by US scientists to create new weapons of war, and they served as models for continuing close scienti"c}government cooperation; and not just in the US. Great national laboratories were created in all the major industrial powers, some devoted to basic and applied research, and some to secret military weapons projects. A major stimulus for creating these laboratories was the important role scientists had played in helping win WWII. The public stood in awe of the nuclear weapons, in particular, and was willing to spend money liberally to support strong scienti"c communities as a resource to rely on for national security, if for no other reason. Formal recognition of the broader importance of scientists for society came in the US with the publication of a very perceptive and pathbreaking report presented to President Truman in 1945 by Vannevar Bush who headed the O$ce of Scienti"c Research and Development for President Roosevelt during WWII. This report entitled `Science: The Endless Frontiera emphasized the need to build on wartime experience and laid out the requisites for a nation to develop and manage a major league scienti"c endeavor. Bush wrote as follows `Science, by itself, provides no panacea for individual, social, and economic ills. It can be e!ective in the national welfare only as a member of a team, whether the conditions be peace or war. But without scienti"c progress no amount of achievement in other directions can insure our health, prosperity, and security as a nation in the modern world.a And he went on to caution: `Scienti"c progress on a broad front results from the free play of free intellects, working on subjects of their own choice, in the manner dictated by their curiosity for exploration of the unknown. Freedom of inquiry must be preserved under any plan for Government support of science2a Furthermore, Vannevar Bush reminded Washington that research is a di$cult and often very slow voyage over uncharted seas, and therefore, for science to #ourish with governmental support, there must be funding stability over a period of years so that long-range programs may be undertaken and pursued e!ectively. This report, and the appreciation of what science had done, led to a remarkable period of support for science that lasted some 20 years after WWII in the US. I was of the generation entering graduate school after WWII that bene"ted greatly from all the opportunities opening for us at that time. Institutions such as the Air Force O$ce of Scienti"c Research and the O$ce of Naval Research supported our graduate studies and provided necessary facilities. In my "eld of highenergy physics, major national accelerator centers were created at Berkeley, Brookhaven, and Stanford, and accelerators were built at many universities including Cornell and MIT. The National Science Foundation was created. There was a veritable love a!air with science during a period of great support and cooperation. The cooperation intensi"ed in the late 1950s when the Soviet Union launched Sputnik, and the development of long-range ballistic missiles raised new threats to our security. This served to

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increase government reliance on scientists to address national security needs, and led to a growing appreciation of the need to forge even stronger bonds between government leaders and the scienti"c community. Cooperation and understanding extended beyond the realm of national security to the broader realms that, as V. Bush emphasized, contribute to the quality of life and to the protection we aspire to against diseases. As government reliance on scientists increased, so did the reliance of science on government. This is especially true in big science. The only way to make progress on some of the most fundamental frontiers, in particular the "elds of elementary particles and astrophysics, is in partnership with a government willing to pay some big bills. Governments are essential partners in the work, and it is in the self-interest of scientists to improve the understanding of government o$cials of what, why, and how we do the research that they are paying for. It is especially in scientists' self-interest to improve their understanding of the very nature of scienti"c progress, its reliance on basic research as the fuel and the engine of progress, and the need for stability and patience in their support of basic research. The importance of developing mechanisms to help maintain a strong bond and e!ective interactions between science and government in matters of US national security was already very clear in the 1950s. National security concerns were paramount, and rapid and, in some instances, revolutionary advances in military technology created a growing gap between science and government leaders. Our leaders were faced with di$cult new choices: In an era of H-bombs of mass indiscriminate destruction, what should we do to defend ourselves? What could we do? What about e!ects of worldwide fallout from continuing atmospheric nuclear weapon tests? What could we do in practice } taking realities into account } to reduce nuclear danger? President Eisenhower understood very well the importance of closing a growing gap } especially after Sputnik in 1957 and the emergence of the ICBM threat } between what scientists could foresee as the potential of these revolutionary technologies, and what our government leaders understood based on what they were familiar with. This led him to create the position of a full-time Science Advisor in the White House, and also to establish the President's Science Advisory Committee (PSAC). This mechanism was his resource for direct, in-depth analyses and advice as to what to expect from science and technology, both current and in prospect, in establishing realistic national policy goals. Members of PSAC, and consultants who served on its hardworking panels, were selected apolitically and solely on the grounds of demonstrated achievements in science and engineering, and of a commitment to work hard. They undertook studies and o!ered advice in response to White House interest over a broad range of national concerns that extended beyond issues of national security. Two things set PSAC apart from the then existing governmental line organizations and cabinet departments with operational responsibilities, and from non-governmental organizations engaged in policy research. First of all, members of PSAC had White House backing and, for the broad range of national security issues, access to the relevant, sensitive information for their studies. Secondly, the individual scientists were independent and presumably, therefore, immune from having their judgments a!ected by operational and institutional responsibilities. Therein lay their unique value. Beyond this, there had to be good `chemistrya between the President and his science advisor. The e!ectiveness of presidential science advisory mechanisms has waxed and waned in the US; and di!erent societies may choose very di!erent mechanisms for providing important technical

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advice to their leaders. I interpret it to be a healthy development for Russia that Yuri Osipov, President of your Academy of Science, has been included in Premier Primakov's inner cabinet so that the voice of science is being heard at the highest councils. We may hope he proves e!ective } and is being listened to. However, if but one scientist is being heard as a solitary advisor at the top, there is a danger that his own biases may color his advice. This danger was recognized in the US, and avoided by creating the PSAC to help balance out the inevitable parallaxes that can a!ect one's scienti"c visions. In the US the honeymoon between science and government began to come apart after 20 years, starting in the late 1960s. For one thing there were those in Washington who resented the fact that scientists who were feeding from the government trough for support of their research were speaking out more and more critically and publicly on policy issues of what to do with some of the new technical achievements resulting from science. There were particularly strong debates in the US on the dangers of atmospheric testing of nuclear weapons. The criticisms became very intense in the late 1960s as many scientists, and academics more broadly, became disillusioned with Washington and opposed strongly our involvement in Vietnam; and also became leading spokesmen in the debate about whether it was practical or desirable to employ ballistic missile defenses against intercontinental range missiles. No longer were scientists being perceived as major contributors to solving di$cult issues; more often than not, they were being viewed as part of a sustained opposition to government policies. Therein one sees a price for close cooperation with government. Some feel that science advisors on government committees must become captive to political policy decisions and quietly join in step with them. But such advisers are not members of the government, and most are academics and independent. Much of this work is pro bono and done out of a sense of duty to contribute to better policy decisions by the governments. Science advisers are not policy makers, and are listened to only on technical issues where they have some expertise to o!er. But their independence was not always welcome nor were they willing to be silent on other issues. There were also strains with many colleagues, especially during the Vietnam era, which were created by the barrier of secrecy between them and some of the advisers' activities. Nevertheless, whatever the di$culties, the need for governments to get the best counsel } not only in national security, but also in energy, environmental, and health care policy issues } is still of crucial importance today and will remain so for the future. The cost of making policy decisions, based on inadequate scienti"c understanding of what is possible, and what is pure fancy, can be catastrophic, whether we are considering the attack on disease or national security issues. This highlights the need to preserve an important emphasis on research and training } two key ingredients that are the foundations for progress in advancing our understanding } an area that requires close cooperation between science and government. It is tempting for government o$cials to "nd the strongest motivation for their support of science from the widely celebrated successes of speci"c scienti"c and technical projects that achieved the strategic goals for which they were created. It is much rarer to "nd an understanding in the political process of the long tortuous path of basic research that made it possible to accomplish those goals. Certainly, in the United States the legacy of the atom bomb project during World War II, and, more recently, of other similar major focused projects like the Apollo moon landing, have created a distorted picture of what science really is, and unrealistic expectations of what one can expect from it. This has led to frequent calls by society and governments

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to conquer diseases such as cancer and AIDS, and to overblown expectations that the victory will be timely and total. Anything less is viewed to be a failure. Often forgotten in evaluating the atomic bomb project are the decades of basic scienti"c achievements by the Chadwicks, Fermis, Hahns, Bohrs, Strassmans, Meitners, Frischs, Flerovs, Kharitons, Zel'dovichs, Semenovs, Kurchatovs that made it all possible. Political leaders must be reminded, and helped to appreciate, that a sound understanding of fundamental cellular processes underlies progress in combating disease. Several years ago the Nobel Laureate discoverer of the Hepatitis B virus, Baruch Blumberg, wrote in the Financial Times of London that the discovery of the Hepatitis B virus, which has led to life saving vaccines, would not have happened as fast, if at all, had he been assigned the task of "nding it, rather than `engaging in basic science without a speci"c application in minda. His voice echoes concerns of many biologists and medical personnel who keep reminding us that we face new dangers from the development of drug resistant organisms. Drug resistant bacterial pathogens like those which cause tuberculosis are increasing at a rapid rate, and antibiotics that have been used for the past 50 years are not working against them. Unless countered, that proliferation could result in nothing less than the end of the era of presumptive good health that we have all grown to take for granted with antibiotics at our disposal. It is true today that infectious diseases are still the leading cause of death in the world. We face a growing urgency to develop new strategies to combat widespread viral infection, like AIDS, and drug resistant bacteria, but this will not happen without a profound understanding of basic life processes with which to arm a successful counterattack. Close cooperation between science and government is needed to maintain a healthy balance of support for basic research and to avoid ill-advised political decisions to earmark funds toward speci"c goals that look popular, and are politically attractive, in the short term. Overemphasis on such projects will have the result of robbing support for the basic work that provides the seed corn for progress. There is universal agreement that basic research, that is, research not motivated by speci"c application, must largely be supported by the federal government. Private industry can hardly be expected to recapture the bene"ts of basic research on behalf of any one industry, or within a time interval which provides for a reasonable rate of return on the "nancial investment. The time interval between initial results in basic research and gainful applications has been recently in the 20}30 yr range, and of course many basic research results do not lead to applications at all. Generally, this time-span is so large that only the federal government can make the required investment. The late, lamented superconducting supercollider (SSC) is a clear example of the importance of government support and close cooperation with science, and of the loss when that relation breaks down. In 1993 the termination of the SSC by the United States Congress was a crippling blow to the United States program in this "eld. It raised major questions about the future of research in high-energy physics in our country. Ten years of intense e!ort by some of the best talent in the "eld, both the young and senior, devoted to the planning, design and beginning construction of that machine went down the drain. In the US we were faced with a disillusioned dispirited community. It had grown by perhaps 10% in numbers, in anticipation of the SSC, but with a decrease of 20% in real funding over a 5-year period for the rest of the program, and now had no coherent plan for the future. The "eld was in a widespread depression without a clear vision of where to go. If ever there was a need for scientists and the government to cooperate in setting priorities for maintaining a US high-energy physics program at the world frontier, consistent with

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a practical funding level from the government, that was it. We needed government support for an expanded commitment to international collaboration that would support US participation in actually building a big accelerator, the large hadron collider (LHC) at CERN. This required scientists educating and convincing Washington that such a commitment not only was essential for preserving a US leadership role in high-energy physics, but it would also be a real bene"t to the country's technical base. Above all, the scientists themselves had to unite behind a plan for the broader good, and be willing to look beyond their individual pet projects. Fortunately, in our country we have the High Energy Physics Advisory Panel (HEPAP) mechanism to address such problems. This is a standing panel, with a rotating membership of active, widely-respected physicists from the labs and the university user groups who are expected to take a broad view of the national program and set priorities within budget guidelines provided by the program managers whom they advise. The government asked these scientists through this mechanism to formulate recommendations for future research priorities and organizational structures for a national program. Their e!orts were successful. The community was able to unite in creating a practical and a!ordable program that was designed to build for a strong future. Its three elements are: E A #exible diverse and dynamic ongoing research e!ort to address scienti"cally compelling questions. E Vigorous studies to develop innovative technologies for future accelerators and detectors. (One example is a linear electron collider reaching above the 1 TeV energy region to be built by an international collaboration.) E Major participation at the highest energy frontier, the best current opportunity for which was identi"ed as joining the international collaboration building the large hadron collider at CERN. This proved to be a successful example of a crucially important cooperation between scientists and government, joined together e!ectively in the e!ort to restore a badly wounded US national program. For more than 30 years, the US high-energy physics community has relied on the so-called HEPAP process to develop program priorities with funding guidelines from Washington. This process created and sustained a community-wide consensus, or at least a working convenant, with a high batting average of success, up until the demise of the SSC. The SSC failure is a singular event. I am not sure that I understand all the factors leading to it, but aside from the SSC failure, our program's success was and is based on a process that has developed and sustained a good working cooperation between scientists and government. Let me turn back once again to national security issues of high technical content. Throughout the Cold War years, the issue of how best to discourage, deter, or defend ourselves against the use of nuclear weapons was on center stage. In particular, debates about the potential value, versus the dangerous illusions, of nationwide anti-ballistic missile defenses were ongoing. Though often driven by political considerations, these were serious debates in our country about strategic policy that touched a fundamental instinct of all human beings to protect our families and homes. But was it possible to meet the requirements of an e!ective ABM defense in an era with nuclear warheads of such enormous destructive potential?

S.D. Drell / Physics Reports 320 (1999) 17}25

23

Scientists in the United States, both in government and in universities and think tanks, were deeply involved in the questions which were at the root of almost all national security discussions. How easy was it, and how costly, to design and deploy o!ensive countermeasures to overpower any conceivable, a!ordable defense? Most experts saw an arms buildup between competing o!enses and defenses, and their countermeasures and counter-countermeasures, as harmful to strategic stability and future prospects of reducing the nuclear threat. The history and record of these debates in my country show how important it is to have an open channel of cooperation and dialogue between scientists and government. This was lacking in 1983 when President Reagan sought to escape the limitations of massive assured destruction by calling on emerging technologies of beam weapons and advanced sensors for a space-based defense. In the absence of careful independent analyses of practical technical realities, fanciful claims preceded more measured judgments. A largely political and highly acrimonious debate ensued that, for a while, was harmful to the e!orts to improve conditions for peaceful coexistence between our two countries. Looking back over this debate, one sees compelling evidence of the importance of high level cooperation between science and government on major national security issues with high technical content. This cooperation requires participation of scientists of top quality in detailed and comprehensive analysis of what to expect from science and technology, both current and future, in establishing realistic national policy goals. After all you cannot bend the laws of nature to satisfy your policy desires! Furthermore, these scientists must be independent and objective, as well as very good. Unfortunately, there was no process functioning to provide this during the ABM debate, which became more acrimonious than substantive in our country. In this connection I remember accompanying a bipartisan delegation of US Senate leaders in national security a!airs on a visit to Moscow in March, 1988 to meet your then President Gorbachev, Foreign Minister Shevardnadze and Marshal Akhromeyev. One evening I took them to Andrei Sakharov's apartment. When one of the senators, now a high o$cial in our government, questioned Sakharov on why should one not build an ABM, he received a strong, logically compelling lecture on the very same issues that I raised before about the harmful impact of an o!ense}defense race on stability, arms control, and the dangers of nuclear war. Andrei as usual was very eloquent and e!ective. A very important contribution by physicists to the ultimate resolution of that debate was the report prepared by the American Physical Society Study Group on `Science and Technology of Directed Energy Weaponsa co-chaired by N. Bloembergen of Harvard and C.K. Patel, then of Bell Labs. This study was an important example of the essential role of government cooperation with scientists. It would have been impossible without the government providing access to, and brie"ngs on, the essential technical details of the Star Wars program and ABM technology. It was published in the Reviews of Modern Physics, July 1987, four years after President Reagan's speech. This was a de"nitive analysis of the new and prospective technologies, along with the relevant operational issues. Laser and particle beams; beam control and delivery; atmospheric e!ects; beam}material interactions and lethality; sensor technology for target acquisition, discrimination, and tracking, systems integration including computing, power needs, and testing; survivability and system deployment were all analyzed carefully, as were some countermeasures. Aspects of boost-phase, mid-course, and terminal intercepts that were all parts of the Star Wars concept of a layered `defense in deptha were included in their comprehensive analysis.

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The sober "ndings of the APS Directed Energy Weapons Study are summarized in part as follows: `Although substantial progress has been made in many technologies of DEW [directed energy weapons] over the last two decades, the Study Group "nds signi"cant gaps in the scienti"c and engineering understanding of many issues associated with the development of these technologies. Successful resolution of these issues is critical for the extrapolation to performance levels that would be required in an e!ective ballistic missile defense system. At present, there is insu$cient information to decide whether the required extrapolations can or cannot be achieved. Most crucial elements required for a DEW system need improvements of several orders of magnitude. Because the elements are interrelated, the improvements must be achieved in a mutually consistent manner. We estimate that even in the best of circumstances, a decade or more of intensive research would be required to provide the technical knowledge needed for an informed decision about the potential e!ectiveness and survivability of directed energy weapon systems. In addition, the important issues of overall system integration and e!ectiveness depend critically upon information that, to our knowledge, does not yet exista. These questions are still central in 1998, although now, happy to say, it is no longer a question of US versus Russian missiles, but concern about proliferating missile technology as exhibited recently in North Korea and Iran; and the threat that is posed by small numbers of missiles, in contrast to the massive numbers that threatened a holocaust between the US and the former Soviet Union. Another major issue that we face today is the e!ort to end all underground nuclear explosions under a Comprehensive Test Ban Treaty (CTBT). Presently, 152 countries, including US and Russia, have signed a CTBT that bans, for all time, all nuclear explosions, anywhere, of any size. This treaty presents us with an historic opportunity. When President Clinton signed the CTBT at the United Nations on September 26, 1996, he said that it was `The longest sought, hardest fought prize in the history of arms control.a The e!ort to end all nuclear tests commenced four decades earlier. Upon leaving o$ce President Eisenhower commented that not achieving a nuclear test ban `would have to be classed as the greatest disappointment of any administration } of any decade } of any time and of any party2a. A decisive political and strategic reason for the powers with nuclear weapons to sign a ban on all nuclear testing was the importance of such a treaty for accomplishing broadly shared nonproliferation goals. This was made clear in the debate at the United Nations in May 1995 by 181 nations when they signed on to the inde"nite extension of the Non-Proliferation Treaty (NPT) at its "fth and "nal scheduled "ve-year review. A commitment by the nuclear-powers to cease testing and developing new nuclear weapons was a condition for many of the non-nuclear nations when they signed on to the Treaty. Not only will the CTBT help limit the spread of nuclear weapons through the non-proliferation regime, particularly if current negotiations succeed in strengthening the provisions for verifying that treaty and appropriate sanctions are applied for non-compliance, it will also dampen the competition among nations who already have nuclear warheads, but who now will be unable to develop and deploy with high con"dence more advanced ones at either the high or the low end of destructive power. The CTBT would also force rogue states seeking a nuclear capability to place con"dence in untested bombs. Notwithstanding a strong case for the CTBT, the United States and Russia } and the other nuclear powers } if they are to be signatories of this treaty, must be con"dent of a positive answer to

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25

the following question. Under a ban on all nuclear explosions, will it be possible to retain the currently high con"dence in the reliability of our nuclear arsenal over the long term, as the weapons age and the numbers are reduced through arms control negotiations? A group of independent scientists, working with government cooperation and support, addressed the scienti"c and technical challenge of answering this question. The "nding of this study was that the US could maintain con"dence in its enduring stockpile and meet our national security needs as currently perceived under a CTBT. To do so it was essential for the US to support a well-de"ned set of programs that are necessary to maintain the health of the enduring stockpile and to deepen our fundamental scienti"c understanding of the processes occurring during a nuclear explosion. They serve as the substitute for new nuclear test data. This conclusion helped form the technical base for President Clinton's decision for the United States to support and seek a true zero-yield CTBT announced in September, 1995. It is one more example of the importance of a close working relation between the scienti"c community and government. The CTBT must now be rati"ed by all 44 nations deemed to be nuclear capable, i.e. possessing nuclear warheads, nuclear power or research reactors, in order to enter into force. As we look to the future, we can see ominous new threats emerging which involve other weapons of indiscriminate destruction beyond the remaining 20 000 or so nuclear warheads still possessed today by at least 8 nations. Chemical and biological weapons in the hands of substate entities and terrorists are a growing concern. As the attack in the Tokyo subway system by the Aum Shinrikyo reminded us in 1995, these threats can no longer be ignored. Scientists will have to remain strongly involved in e!orts to build a safer 21st century, and this involvement means helping our government decision makers understand both the potential and the limits of science and technology.

Physics Reports 320 (1999) 27}36

From Alexander of Aphrodisias to Young and Airy J.D. Jackson University of California, Berkeley, CA, USA and Lawrence Berkeley National Laboratory, University of Berkeley, Berkeley, CA, 94720, USA

Abstract A didactic discussion of the physics of rainbows is presented, with some emphasis on the history, especially the contributions of Thomas Young nearly 200 years ago. We begin with the simple geometrical optics of Descartes and Newton, including the reasons for Alexander's dark band between the main and secondary bows. We then show how dispersion produces the familiar colorful spectacle. Interference between waves emerging at the same angle, but traveling di!erent optical paths within the water drops, accounts for the existence of distinct supernumerary rainbows under the right conditions (small drops, uniform in size). Young's and Airy's contributions are given their due.  1999 Elsevier Science B.V. All rights reserved. PACS: 01.30.Rr; 42.15.Dp; 42.25.Fx; 42.68.Ge

*This pedagogical piece on rainbows is dedicated to Lev B. Okun, colleague and friend, on his 70th birthday. On an extended visit to Berkeley in 1990, Lev saw on my ozce wall a picture of a double rainbow with at least three supernumerary bows visible inside the main bow. As part of my `lecturea on the photograph, I showed Lev a copy of these 1987 handwritten notes prepared for a class. He said, `Are these published somewhere?a My answer was no, but now they are, in augmented form. Lev is an amazing man, a physicist-mensch } a brilliant researcher, mentor, and warm human being. I have a vivid memory of a wonderful trip to Yosemite National Park with an allegedly ailing Lev. In the early morning hours, we found Lev outside our tent in Curry Village perched on a sloping rock doing vigorous calisthenics! Lev, may you have Many Happy Returns!+ The rainbow has fascinated since ancient times. Aristotle o!ered an explanation (not correct), as did clerics and scholars through the ages. Newton and Descartes established the elementary theory, according to what e now know as geometrical optics. But long before Newton and Descartes, as early as the 13th century, the puzzling occasional phenomenon of supernumerary rainbows was noted. These `aberrationsa were inexplicable in terms of geometrical optics. It was not until the beginning of the 19th century that Thomas Young, promoting the wave theory of light against acolytes of Newton, o!ered the correct explanation of the supernumeraries as results of interference. Airy put the theory on a "rm mathematical footing in 1836. A scholarly treatment of the 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 8 8 - 5

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J.D. Jackson / Physics Reports 320 (1999) 27}36

history of the attempts to understand the rainbow by Boyer [1] contains much of interest, including striking paintings and photographs with the rainbow as subject. A semi-popular account of the theory of rainbows is presented by Nussenzveig [2]. The discussion that follows traces the theory of the rainbow from the simple Cartesian} Newtonian description to the interference-di!raction-caustic treatment of Airy.

1. Geometrical optics, no dispersion A light ray is incident on a water drop of radius a at impact parameter b, as shown in Fig. 1. The index of refraction of water at the wavelength of the sodium D lines (j"5890, 5896 As ) and at 203C is closely n". The ray has an angle of incidence i whose sine is sin i"b/a,x. The angle of  refraction r is given by Snell's law as r"sin\(x/n). The scattering angle h for the emerging light ray (de"ned here as the angle of emergence of the ray relative to the incident direction) can be computed by adding up the angular bends made by the ray: The entering bend is (i!r) Each internal re#ection bend is (p!2r) The exiting bend is (i!r) For m internal re#ections, the scattering angle is thus h ""2(i!r)#m(p!2r)" [Modulo 2p] . (1) K The primary rainbow has m"1, the secondary, m"2, and so on. Fig. 2 shows the scattering angle as a function of sin i"b/a for m"1 and m"2. At the extremes, the angle is either 0 or p,

Fig. 1. Geometrical optics of a primary rainbow.

J.D. Jackson / Physics Reports 320 (1999) 27}36

29

but for intermediate b/a values, the light is scattered at various angles. Note the gap between 1293 and 1383. This is a region of negligible scattering (from higher orders) and appears as a dark space between the primary and secondary rainbows (known as Alexander's dark space, after Alexander of Aphrodisias, a follower of Aristotle and head of the Lyceum in Athens around 200 AD). The feature that causes the rainbow is the extremum in angle as a function of impact parameter. For the primary rainbow (upper curve in Fig. 2), the minimum angle is h "1383 at x "0.86066   for n". Classically, the scattering cross section is  dp/dX""b db/(sinh dh)" (2) At the extremum, db/dh is in"nite, corresponding to a (classically) in"nite cross section. Wave aspects prevent the in"nity, of course, but it is indicative of a large cross section. The singular behavior is an example of a caustic. To examine the vicinity of the extremum and see its dependence on the index of refraction, we make a Taylor series expansion around the minimum. For the primary bow (m"1), we have h"p#2 sin\x!4 sin\(x/n) ,

(3)

where x"b/a. The "rst two derivatives are 2 4 dh " ! , dx (1!x (n!x

(4)

2x 4x dh " ! . dx (1!x) (n!x)

(5)

The extremum occurs for dh/dx"0, i.e., (n!x "2(1!x , or x "((4!n)/3 and    (1!x "((n!1)/3. 

Fig. 2. Rainbow scattering angles according to geometrical optics for index of refraction n"1.34. As indicated by the dotted lines, the dark band is somewhat wider in violet light, for which n"1.345. Typical m"1 and m"2 rays are shown in Fig. 4(a).

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The second derivative at x"x is of interest:  dh 9 (4!n h, " . (6) dx 2 (n!1) VV For n", h"9.780, and h "137.973.   For x near x , we have hKh #h(x!x )/2. In passing, we note that near h"h , the classical     scattering cross section is





2 x dp Ka ;  . (7) h(h!h ) sin h dX   As sketched in Fig. 4(b), scattering is concentrated at h"h , but also occurs for h'h . This   is what causes the white appearance `insidea the primary bow (and `outsidea the secondary bow).

2. Colors of the rainbow, dispersion The beautiful colors of the rainbow are a consequence of the variation of the index of refraction of water with wavelength of the light. This dispersion, as it is called, is shown quantitatively in Fig. 3. If we arbitrarily de"ne the visible range of wavelength to be from 400 nm (violet) to 700 nm (red), we "nd that the index of refraction di!ers by *n"1.3;10\ from one end of the range to the other. Now consider the e!ect of a change in n on h: R 4x dh "!4 [sin\(x/n)]" . Rn dn n(n!x

(8)

Fig. 3. Index of refraction of water as a function of wavelength. The visible light interval is between 400 and 700 nm.

J.D. Jackson / Physics Reports 320 (1999) 27}36

31

Fig. 4. Sketches to accompany the text.

At the rainbow angle,


 dh dn

2 " n 



4!n n!1

(9) V For n"4/3, dh/dn"  "2.536. With *n"1.3;10\, we "nd *h "3.3;10\ radians "1.893. V  The colors of the rainbow are spread over about 23 out of the 423 away from the anti-solar point (1803!1383). Since dn/dj(0, the red light emerges at a smaller angle than the violet.

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J.D. Jackson / Physics Reports 320 (1999) 27}36

The viewer thus sees the rainbow with the red at the outer side of the arc and the violet on the inner side, as indicated in Fig. 4(c). For the secondary bow, the order of the colors is opposite.

3. Consequences of the wave nature of light, supernumerary rainbows For rays incident at impact parameters close to b "x a, the scattering angle is equal to h ,    correct to "rst order inclusive. In fact, because of the quadratic dependence of h!h on  *x"x!x , two rays incident at impact parameters greater and less than b by an amount   "*x" will emerge with the same scattering angle. In the wave picture, as observed by Young [3] in 1803, these two waves emerging in the same direction can interfere. Whether the interference is constructive or destructive depends on the di!erent in optical path length of the two rays. This varies as a function of *x and so provides the potential for interference e!ects in addition to dispersion in rainbows. Referring to Fig. 1, we see as the solid line the critical ray, which emerges at h"h . On either  side are shown neighboring rays with small "*x" that emerge at angles di!ering from h only in  O(*x). The surfaces AA and BB are convenient ones for de"ning the optical path of a ray in the neighborhood of the critical ray. The optical path, or more appropriately, the phase accumulated along the ray, is given by

(x)"2ka(1!cos i#2n cos r) ,

(10)

where the 2(1!cos i) represents the sum of the distances from AA to the drop's surface and similarly for the exit leg, while 4n cos r is the length (times n) of the path interior to the drop. The free-space wave number is k"u/c"2p/j. In terms of x"b/a, the phase is

(x)"2ka[1!(1!x#2(n!x] .

(11)

We are interested in the behavior of (x) near x!x . Consider the derivative,  2x x d

"2ka ! . dx (1!x (n!x





(12)

Comparison with dh/dx in part(a) shows the relation, d /dx"kax dh/dx .

(13)

Writing x"x #m, we can put this equation in the form,  d /dm"ka[x dh/dm#m dh/dm] .  Integration on both sides from 0 to m yields

 





K dh m dm dm  K "ka x (h!h )#mh! h(m) dm .   

(m)! "ka x (h!h )#   





(14)

(15) (16)

J.D. Jackson / Physics Reports 320 (1999) 27}36

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Inserting h(m)"h #hm/2#O(m) in the integral, we "nd 

(m)" #ka[x (h!h )#hm/3#O(m)] .   

(17)

For two rays, a and b, as shown in Fig. 1, with equal and opposite small m values, the phase di!erence is d" (m)! (!m)"2kahm/3 .

(18)

If we equate this phase di!erence to 2pN and express m in terms of h!h , we "nd the angles of  constructive interference to be h !h K[(h)/2](3pN/ka) . , 

(19)

(Actually, a more correct procedure has N# replacing N. See Ref. [4, p. 243] and Section 3.21.)  The angles of constructive interference mark the positions of additional rainbows, called supernumerary rainbows. They lie at larger angles than h and so fall `insidea the main bow. Their  colors are in the same order as in the primary bow. They are rarely seen because conditions must be optimized for them to appear unobscured or not washed out. The angle (h !h ) depends on the ,  droplet size, varying as (ka)\. For large drops, the angle becomes very small and the supernumerary bows fall inside the various colors of the primary bow. Using N"5/4, h"9.780, and (h!h ) "3;10\ radians (corresponding to the spread caused by dispersion), we "nd   (ka) K2.5;10. With k appropriate to the sodium D lines, we obtain a K0.28 mm. Larger



 drops will cause the supernumeraries to be obscured by the e!ects of dispersion. Variation in drop size, even if the drops are small enough, also causes the maxima of the supernumerary bows to be washed out in angle. Thus, one needs small drops, uniform in size, in order to see clearly the supernumeraries. All this was understood by Young [3]. For very small drop size, a(50 lm, the whole pattern of primary peak (N") and supernumer ary peaks for a given wavelength is so spread in angle that dispersion e!ects are unimportant. All the colors have broad primary peaks lying almost on top of each other in angle. The result is a `white rainbowa or `fog bowa.

4. Huygens: construction for the rainbow, Airy integral The scalar di!raction theory of Huygens, Young, Fresnel and Kirchho! [5, Section 10.5] can be used to obtain an approximate description of the rainbow in wave theory, as was "rst done by George B. Airy (1836). Consider the line BB in Fig. 1, where we have evaluated the expression for the phase (m). A wave along this line will have the form tJexp[ik z#ik ) r #i (m)] , (20) , , , where we choose our axes so that z is in the direction of scattering at h and r is measured along  , BB, with value ax at the critical ray. If the wave is propagating in the direction h, then  k ) r "!ka(h!h )x , (21) , , 

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where the negative sign comes from the fact that k and r are antiparallel (h'h ). Since z is , ,  constant on BB, the relevant parts of the wave's overall phase are k ) r # (m)"ka[(h!h )(x !x)#(h/3)m]#

(22) , ,    "ka[!m(h!h )#(h/3)m]# . (23)   With the approximation, h!h "hm/2, we "nd  k ) r # (m)! "!(ka/6)hm#O(m) . (24) , ,  Along the line BB, the wave amplitude in the neighborhood of x"x or m"0 has the form  t(m)"exp(!ikahm/6) (25) assuming the slowly varying amplitude function is a constant. We can now use the simplest version of the Kirchho! integral for di!raction,



k e I0 t " t(x) da . 1  2pi R

(26)

With kRKkr!k ) x in the usual way, we "nd a scattering amplitude,



t J 1 



e I?F\F Ke\ I?FK dm .

(27) \ This can be put in the form of the Airy integral Ai(!g), as de"ned by Abramowitz and Stegun [6, p. 447]:



1  Ai(!g)" cos(t/3!gt) dt , (28) p  where g"(2ka/h)(h!h ). This function is shown in Fig. 4(d). For positive g, Ai(!g)  oscillates with an amplitude that decreases as g\. For negative g, Ai(!g) is exponential in character, falling rapidly to zero for !g'1. The maxima and minima occur successively at g"1.0188 (1.1155), 3.2482 (3.2616), 4.8201 (4.8263), 6.1633 (6.1671), 7.3722 (7.3748) . The numbers in parentheses are values of [(3p/2)(N#)] for N"0, 1, 2, 2, from our previous  discussion of the angles for constructive interference. For larger N values, the agreement is excellent. It is of interest to compare the angular positions of the supernumerary rainbows implicit in the tabulated g values with the examples quoted by Young [3]. Notorious for not giving details of his calculations, he only quotes answers. He states that for drops  inches in diameter, the reds of the  "rst and fourth supernumerary bows are approximately 23 and 43 inside the red of the primary (the "rst just clearing the violet of the primary). With h"9.912 and j"700 nm for red light, we "nd ka"1.50;10 and g"1.34 (h!h ), with the angles measured in degrees. With *g"2.23  and 6.35 for the "rst and fourth supernumeraries, we obtain *h"1.7 and 4.7 degrees, in rough agreement with Young. Incidentally, the fact that gO0 for N"0 (primary bow) explains the

J.D. Jackson / Physics Reports 320 (1999) 27}36

35

long-standing puzzle that the angular positions of some rainbows were observed to vary appreciably away from the Cartesian}Newtonian angle h (evidently dependent on drop size).  The peak intensities of the supernumerary bows, relative to the primary bow, are 0.612, 0.504, 0.446, 0.408, 2, falling o! only as g\ or (N#)\. Note that the g\ behavior is  just what our classical cross section gave. The wave aspect rounds the corners and gives interference. The intensity pattern for red and violet light is sketched in Fig. 4(e) for a"64 lm (ka+10 for violet light). The "rst supernumerary bow would be visible, but subsequent ones would not. Add some variation in drop size and everything except the primary bow will wash out. An approximate cross section for a given ka can be written in terms of Airy's integral by normalizing the average intensity at large g to the classical cross section. From Abramowitz and Stegun [6], one "nds that for large g the leading term in an asymptotic series is (29) Ai(!g)K(1/(p)(1/g)sin(g#p/4) .  The average value of its square is 1"Ai(!g)"2"1/(2p(g). With the expression for g in terms of h!h , this becomes  2 1 h  1  . (30) 1"Ai(!g)"2" ka h(h!h ) 2p 2  Comparison with the classical cross section, near h"h ,  dp x 2  Ka  , (31) dX sin h h(h!h )   leads to the cross section in the Airy approximation,



 






dp 2  x (ka)"Ai(!g)" . (32) K2pa  dX sin h h  (See Fig. 4(f ).) For n", h "137.973, sin h "0.66952, x "0.86066, and h"9.780. Ignoring     the loss of intensity from the refractions and re#ection, the cross section for a given component of the rainbow of "xed ka is thus dp K2.80(ka)"Ai(!g)"a . dX

(33)

At the peak of the rainbow, dp/dX"0.803(ka)a. For ka"10, this cross section is 32 times as great as an isotropic cross section, dp/dX"a/4.

5. Comment on polarizations and loss of intensity After Young, but before Airy, David Brewster showed in 1812 that the scattered rainbow light was almost completely polarized, con"rming earlier observations of Biot (of the Biot}Savart law in magnetism). The polarization comes about because the refracted and re#ected intensities at each of the interfaces are di!erent for di!erent polarizations. The formulas of pp. 305}306 of Jackson [5]

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J.D. Jackson / Physics Reports 320 (1999) 27}36

can be used to show that at h"h , the ratios of scattered amplitude to incident amplitude are  i 8/27 g E 1  "j  2!n E g 2 2n  k n#2 2#n







for E plane of incidence , ,



(34) for E plane of incidence . ,

For n", the intensity of perpendicular polarization is 8.78;10\ of the incident, while the  intensity of the parallel polarization relative to the perpendicular is 3.9;10\. The cross section quoted above must therefore be multiplied by approximately  ;1.039;8.78;10\ for un polarized light incident.

6. Note on notation Van de Hulst [4] de"nes his Airy integral to be

 

f (z)"



cos



p (zt!t) dt . 2

(35)

 His z and our g are related by z"(12/p)g. His function f (z) is f (z)"(2p/3)Ai(!g). Note that other notations are used for the Airy integral. For example, see [7, Section 59].

Acknowledgements Special thanks are owed to Gail Harper and Betty Armstrong, who TE X'ed the equation-rich handwritten manuscript, and to Don Groom for generating postscript "les from my hand-drawn "gures.

References [1] C.B. Boyer, The Rainbow, Princeton University Press, Princeton, 1959, 1987. [2] H.M. Nussenzveig, Scienti"c American, April 1977, p. 116. [3] T. Young, Experiments and Calculations Relative to Physical Optics, Bakerian Lecture, November 24, 1803, publ. Phil. Trans. 1804, in: G. Peacock (Ed.), Miscellaneous Works of Thomas Young, Vol. I, John Murray, London, 1855, p. 179, esp. pp. 185}187. [4] H.C. Van de Hulst, Light Scattering by Small Particles, Wiley, New York, 1957, Chapter 13; also Dover reprint. [5] J.D. Jackson, Classical Electrodynamics, 3rd Edition, Wiley, New York, 1998. [6] M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 1972. [7] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Pergamon, Oxford, 1962.

Physics Reports 320 (1999) 37}49

Particle creation by charged black holes I.B. Khriplovich 
Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia
Novosibirsk University, Russia Dedicated to Lev Borisovich Okun on his birthday

Abstract A simple derivation is given for the leading term (n"1) in the Schwinger formula for the pair creation by a constant electric "eld. The same approach is applied then to the charged particle production by a charged black hole. In this case, as distinct from that of a constant electric "eld, the probability of the charged particle production depends essentially on the particle energy. The production rate by black holes is found in the nonrelativistic and ultrarelativistic limits. The range of values for the mass and charge of a black hole is indicated where the discussed mechanism of radiation dominates the Hawking one.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.20.Ds; 04.70.Dy Keywords: External "elds; Black holes

The problem of particle production by the electric "eld of a black hole has been discussed repeatedly [1}6]. The probability of this process was estimated in these papers using in some way or another the result obtained previously [7}9] for the case of an electric "eld constant all over the space. This approximation might look quite natural with regard to su$ciently large black holes, for which the gravitational radius exceeds essentially the Compton wave length of the particle j"1/m. (We use the units with "1, c"1; the Newton gravitational constant k is written down explicitly.) However, in fact, as will be demonstrated below, the constant-"eld approximation, generally speaking, is inadequate to the present problem, and does not re#ect a number of its essential peculiarities. It is convenient to start the discussion just from the problem of particle creation by a constant electric "eld. In this paper we restrict ourselves to the consideration of the production of electrons E-mail address: [email protected] (I.B. Khriplovich) 0370-1573/99/$ - 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 7 8 - 2

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and positrons, primarily because the probability of emitting these lightest charged particles is the maximum one. Besides, the picture of the Dirac sea allows one in the case of fermions to manage without the second-quantization formalism, thus making the consideration most transparent. To calculate the main, exponential dependence of the e!ect, it is su$cient to restrict to a simple approach due to [7] (see also Refs. [10,11]). In the potential !eEz of a constant electric "eld E the usual Dirac gap (Fig. 1) tilts (see Fig. 2). As a result, a particle with a negative energy in the absence of the "eld, can now tunnel through the gap (see one of the horizontal dashed lines in Fig. 2) and go to in"nity as a usual particle. The hole created in this way is nothing but an antiparticle. The exponential factor in the probability of the particle creation depends obviously on the action only inside the barrier. This action does not change under a shift of the dashed line in Fig. 2 up or down, i.e., under a shift by *E of the energy E of the created particle. Being obviously an integral of motion, E is also the energy of the initial particle of the Dirac sea. If we put for instance E"!m,

Fig. 1.

Fig. 2.

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so that the particle enters the barrier at z"0, the squared four-dimensional momentum (E!e )!p"m becomes (!m#eEz)!p"m . For the time being we assume that the transverse part of the particle momentum p "(p ,p ), which , V W is also an integral of motion, is equal to zero. Inside the barrier the modulus of the momentum p(z)"p (z) is X "p (z)""(m!(m!eEz) . The action inside the barrier equals:



KC# pm dz" p(z)"" . 2eE  Finally, the exponential factor in the probability = is [7]: S"

(1)

=&exp(!2S)&exp(!pm/eE) .

(2)

One can easily take into account in the exponent (2) the transverse momentum p . This integral , of motion will, clearly, enter all the previous formulae in the only combination m#p . So, , expression (2) demands in this case the substitution mPm#p , , changing thus to





p(m#p ) , . =&exp ! eE

(3)

Let us calculate now the pre-exponential factor in the probability of particle creation. The obtained exponential (3) is the probability that a particle of the Dirac sea approaching the potential barrier from the left (see Fig. 2), will tunnel through it to the right, thus becoming a real electron. To obtain the total number of pairs created per unit volume per unit time, the exponential (3) should be multiplied by the current density of the particles of the Dirac sea j "ov . X X For the velocity we use the common relationship

(4)

v "RE/Rp (5) X (the subscript z of the longitudinal momentum p is again omitted here and below). The particle density is as usual o"2dp dp/(2p) , , the factor 2 being due to two possible orientations of the electron spin.

(6)

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I.B. Khriplovich / Physics Reports 320 (1999) 37}49

For a "xed coordinate z and "xed p the identity holds: , (RE/Rp) dp"dE .

(7)

On the other hand, it is obvious that the interval of energies dE of the tunneling particles is directly related to the interval dz of longitudinal coordinates of the points where the particles enter the barrier: dE"eE dz (up to an inessential sign). Being interested in the probability per unit volume in general, and per unit longitudinal distance in particular, we should delete thus the arising factor dz when calculating the e!ect. So, the total number of pairs created per unit volume per unit time is







dp p(m#p ) , exp ! , . = "2eE  (2p) eE

(8)

Now the trivial integration over the transverse momenta gives the "nal result = "(eE/4p)exp(!pm/eE) . 

(9)

The probability = in the above formulae is supplied with the subscript 1/2 to indicate that the result refers to particles of spin one half. Obviously, the notion of the Dirac sea, and hence the above derivation by itself, do not apply to boson pair creation. However, in the semiclassical approximation, the creation rate for particles of spin zero is almost the same. The only di!erence is that since these particles do not have two polarization states, the rate is two times smaller than (9): = "(eE/8p)exp(!pm/eE) . 

(10)

The corresponding exact results for a constant electric "eld are [9]






eE  1 pm = " exp !n , (11)  4p eE n L eE  (!1)L\ pm =" exp !n . (12)  n 8p eE L Obviously, the account for higher terms, with n52, in the sums (11), (12) makes sense only for very strong electric "elds, for eE9m. For smaller "elds, when eE;m, simple formulae (9) and (10) are correct quantitatively. The above straightforward derivation explains clearly some important properties of the phenomenon. First of all, the action inside the barrier does not change under a shift of the dashed line in Fig. 2 up or down. Owing to this property alone expressions (2) and (9) are independent of the energy of created particles. Then, for the external "eld to be considered as a constant one, it should change weakly along the path inside the barrier. Obviously, the length of this path l&m/eE di!ers essentially from the Compton wave length j"1/m of the particle. The ratio l/j is of the same order of magnitude as the action S inside the barrier, and therefore should be large for the semiclassical approximation to be applicable at all. Thus, one may expect that, generally speaking, the constant-"eld approximation is not applicable to the problem of a charged black hole radiation, and that the probability of particle






I.B. Khriplovich / Physics Reports 320 (1999) 37}49

41

production in this problem is strongly energy-dependent. The explicit form of this dependence will be found below. We restrict ourselves in the present work to the case of a non-rotating black hole. We start the solution of the problem by calculating the action inside the barrier. The metric of a charged black hole is well-known: ds"f dt!f\ dr!r(dh#sinh d ) ,

(13)

where f"1!2kM/r#kQ/r ,

(14)

M and Q being the mass and charge of the black hole, respectively. The equation for a particle 4-momentum in these coordinates is f \(e!eQ/r)!fp!l/r"m .

(15)

Here e and p are the energy and radial momentum respectively, of the particle. We assume that the particle charge e is of the same sign as the charge of the hole Q, ascribing the charge !e to the antiparticle. Clearly, the action inside the barrier is minimum for the vanishing orbital angular momentum l. It is rather evident therefore (and will be demonstrated below explicitly) that after the summation over l just the s-state de"nes the exponential in the total probability of the process. So, we restrict for the moment to the case of a purely radial motion. The equation for the Dirac gap for l"0 is e (r)"eQ/r$m(f , !

(16)

which is presented in Fig. 3. It is known [12] that at the horizon of a black hole, for r"r "kM#(kM!kQ, the gap vanishes. Then, with the increase of r the lower boundary > e (r) of the gap decreases monotonically, tending asymptotically to !m. The upper branch e (r) at \ > "rst, in general, increases, and then decreases, tending asymptotically to m. It is clear from Fig. 3 that those particles of the Dirac sea whose coordinate r exceeds the gravitational radius r and whose energy e belongs to the interval e (r)'e'm, tunnel through > \ the gap to in"nity. In other words, a black hole loses its charge due to the discussed e!ect, by emitting particles with the same sign of the charge e, as the sign of Q. Clearly, the phenomenon takes place only under the condition eQ/r 'm . >

(17)

For an extreme black hole, with Q"kM, the Dirac gap looks somewhat di!erent (see Fig. 4): when Q tends to kM, the location of the maximum of the curve e (r) tends to r , and the value of > > the maximum tends to eQ/r . It is obvious however that the situation does not change qualitatively > due to it. Thus, though an extreme black hole has zero Hawking temperature and, correspondingly, gives no thermal radiation, it still creates charged particles due to the discussed e!ect.

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Fig. 3.

In the general case Q4kM the doubled action inside the barrier entering the exponential for the radiation probability is

 

2S"2

P

dr"p(r,e)"

P P dr r (!pr#2(eeQ!kmM)r!(e!km)Q . "2  r!2kMr#kQ P 

(18)

Here p "(e!m is the momentum of the emitted particle at in"nity, and the turning points  r are as usual the roots of the quadratic polynomial under the radical; we are interested in the  energy interval m4e4eQ/r . Of course, the integral can be found explicitly, though it demands > somewhat tedious calculations. However, the result is su$ciently simple: 2S"2p[m/(e#p )p ][eQ!(e!p ) k M] . (19)    Certainly, this expression, as distinct from the exponent in formula (2), depends on the energy quite essentially. Let us note that the action inside the barrier does not vanish even for the limiting value of the energy e "eQ/r . For a nonextreme black hole it is clear already from Fig. 3. For an extreme K > black hole this fact is not as obvious. However, due to the singularity of "p(r,e)", the action inside the

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Fig. 4.

barrier is "nite for e"e "eQ/r for an extreme black hole as well. In this case the exponential K > factor in the probability is exp(!p((km/e)kmM) .

(20)

Due to the extreme smallness of the ratio (km/e&10\ ,

(21)

the exponent here is large only for a very heavy black hole, with a mass M exceeding that of the Sun by more than "ve orders of magnitude. And since the total probability, integrated over energy, is dominated by the energy region e&e , the semiclassical approach is applicable in the case of K extreme black holes only for these very heavy objects. Let us note also that for the particles emitted by an extreme black hole, the typical values of the ratio e/m are very large: e/m&e /m"eQ/kmM"e/(km&10 . K In other words, an extreme black hole in any case radiates highly ultrarelativistic particles mainly. Let us come back to nonextreme holes. In the nonrelativistic limit, when eQ/r Pm and, > correspondingly, the particle velocity vP0, the exponential is of course very small: exp(!2pkmM/v) .

(22)

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Therefore, we will consider mainly the opposite, ultrarelativistic limit where the exponential is exp(!p(m/e) eQ) .

(23)

Of course, here also the energies e&e &eQ/kM are essential, so that the ultrarelativistic limit K corresponds to the condition eQ
(24)

But then the semiclassical result (23) is applicable (i.e., the action inside the barrier is large) only under the condition kmM<1 .

(25)

Let us note that this last condition means that the gravitational radius of the black hole (r &kM) > is much larger than the Compton wave length of the electron 1/m. In other words, the result (23) refers to macroscopic black holes. Combining (24) with (25), we arrive at one more condition for the applicability of formula (23): eQ<1 .

(26)

We shall return to this relationship later. Let us note that in [4] the action inside the barrier was being calculated under the same assumptions as formula (23). However, the answer presented in [4], 2S"pmr /eQ, is totally > independent of energy (and corresponds to formula (2) which refers to the case of a constant electric "eld). I do not understand how such an answer could be obtained for the discussed integral in the general case eOe . K The obtained exponential is the probability that a particle approaching the turning point r (see  Figs. 3 and 4) from the left, will tunnel through the potential barrier. One should recall that in the general case the position of the turning point depends not only on the particle energy e, but on its orbital angular momentum l as well. The total number of particles with given e and l, approaching a spherical surface of the radius r in unit time, is equal to the product of the area of this surface:  S"4p r(e,l ) (27)  times the current density of the particles jP(e,l )"o/(g (dr/dt)  (see, e.g., [13], A 90). The particle velocity is vP"dr/dt"Re/Rp

(28)

(29)

(the subscript r of the radial momentum p is again omitted). To obtain an explicit expression for the particle density o, we will use the semiclassical approximation (the conditions of its applicability for the region r 4r4r will be discussed later). Let us note that the volume element of the phase >  space 2

dp dp dp dx dy dz V W X (2p)

(30)

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is a scalar. On the other hand, the number of particles in the elementary cell dx dy dz equals (see [13], A 90) o(cdx dy dz ,

(31)

where c is the determinant of the space metric tensor. Since all the states of the Dirac sea are occupied, we obtain by comparing formulae (30) and (31) that the following expression o

2 dp dp dp 2 dp dp dp " V W X" V W X (2p) (2p) (g c (!g (g   should be plugged in formula (28) for the current density (the summation here and below is performed with "xed e and l, see (28)). In our case the determinant g of the four-dimensional metric tensor does not di!er from the #at one, so that the radial current density of the particles of the Dirac sea is dp Re jP(e,l )"2 . (2p) Rp

(32)

The summation on the right-hand-side reduces in fact to the multiplication by the number 2l#1 of possible projections of the orbital angular momentum l onto the z-axis and to the integration over the azimuth angle of the vector l, which gives 2p. Using identity (7), we obtain in the result 2p(2l#1) jP(e,l )"2 . (33) (2p)r(e,l )  Finally, the pre-exponential factor in the probability, di!erential in energy and orbital angular momentum, is 2(2l#1) . p

(34)

Correspondingly, the number of particles emitted per unit time is



dN 2 " de (2l#1)exp[!2S(e,l)] . (35) dt p J In the most interesting, ultrarelativistic case dN/dt can be calculated explicitly. Let us consider the expression for the momentum in the region inside the barrier for lO0:




"p(e,l,r)""f \




l m# r



eQ  f! e! . r

(36)

The main contribution to the integral over energies in formula (35) is given by the region ePe . In K this region the functions f (r) and e!eQ/r, entering expression (36), are small and change rapidly. As to the quantity k(r,l )"m#(l/r) ,

(37)

one can substitute in it for r its average value, which lies between the turning points r and r .   Obviously, in the discussed limit ePe the near turning point coincides with the horizon radius, K

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r "r . And the expression for the distant turning point is in this limit  >





2k (kM!kQ . (38) r "r 1#  > e !k r K > Assuming that for estimates one can put in formula (37) r&r , one can easily show that the > correction to 1 in the square bracket is bounded by the ratio l/(eQ). Assuming that this ratio is small (we will see below that this assumption is self-consistent), we arrive at the conclusion that r +r , and hence k can be considered independent of r: k(r,l)+m#l/r . As a result, we  > > obtain 2S(e,l)"peQ(m/e#l/r e) . > Now we "nd easily

(39)

dN/dt"m(eQ/pmr )exp(!pmr /eQ) . (40) > > Let us note that the range of orbital angular momenta, contributing to the total probability (40), is e!ectively bounded by the condition l:eQ. Since eQ<1, this condition allows one to change from the summation over l in formula (35) to the integration. On the other hand, this condition justi"es the used approximation k(r,l)+m#l/r . > However, up to now we have not considered one more condition necessary for the derivation of formula (40). We mean the applicability of the semiclassical approximation to the left of the barrier, for r 4r4r . This condition has the usual form >  (d/dr)(1/p(r))(1 . (41) In other words, the minimum size of the initial wave packet should not exceed the distance from the horizon to the turning point. Using the estimate r (eQ!er ) > p(r)& > (r!r )(r!r ) > \ for the momentum in the most essential region, one can check that for an extreme black hole the condition (41) is valid due to the bound eQ<1. In a non-extreme case, for r !r &r , the > \ > situation is di!erent: the condition (41) reduces to e((eQ!1)/r &eQ/r . (42) > > Thus, for a non-extreme black hole in the most essential region ePe the condition of the K semiclassical approximation is not valid. Nevertheless, the semiclassical result (40) remains true qualitatively, up to a numerical factor in the pre-exponential. In concluding this section few words on the radiation of light charged black holes, for which kmM(1, i.e., for which the gravitational radius is less than the Compton wave length of the electron. In this case the "rst part, e((eQ!1)/r , >

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of inequality (42), which guarantees the localization of the initial wave packet in the region of a strong "eld, means in particular that eQ"Za'1

(43)

(we have introduced here Z"Q/e). It is well-known (see, e.g. [14,15]) that the vacuum for a point-like charge with Za'1 is unstable, so that such an object loses its charge by emitting charged particles. It is quite natural that for a black hole whose gravitational radius is smaller than the Compton wave length of the electron, the condition of emitting a charge is the same as in the pure quantum electrodynamics. (Let us note that the unity in all these conditions should not be taken too literally: even in quantum electrodynamics, where the instability condition for the vacuum of particles of spin 1/2 is for a point-like nucleus just Za'1, for a "nite-size nucleus it changes [14,15] to Za'1.24. On the other hand, for the vacuum of scalar particles in the "eld of a point-like nucleus the instability condition is [16,17]: Za'1/2.) As has been mentioned already, for a light black hole, with kmM(1, the discussed condition eQ'1 leads to a small action inside the barrier and to the inapplicability of the semiclassical approximation used in the present article. The problem of the radiation of a charged black hole with kmM(1 was investigated numerically in [18]. The exponential exp(!pmr /eQ) > in our formula (40) coincides with the expression arising from formula (2), which refers to a constant electric "eld E, if one plugs in for this "eld its value Q/r at the black hole horizon. As > has been mentioned already, an approach based on formulae for a constant electric "eld was used previously in Refs. [1}6]. Thus, our result for the main, exponential dependence of the probability integrated over energies, coincides with the corresponding result of these papers. Moreover, our "nal formula (40) agrees with the corresponding result of Ref. [6] up to an overall factor 1/2. (This di!erence is of no interest by itself: as has been noted above, for a non-extreme black hole the semiclassical approximation cannot guarantee an exact value of the overall numerical factor.) Nevertheless, we believe that the analysis of the phenomenon performed in the present work, which demonstrates its essential distinctions from the particle production by a constant external "eld, is useful. First of all, it follows from this analysis that the probability of the particle production by a charged black hole has absolutely nontrivial energy spectrum. Then, in no way are real particles produced by a charged black hole all over the whole space: for a given energy e they are radiated by a spherical surface of the radius r (e), this surface being close to the horizon for the  maximum energy. (It follows from this, for instance, that the derivation of the mentioned result of Ref. [6] for dN/dt has no physical grounds: this derivation reduces to plugging E"Q/r to the Schwinger formula (9), obtained for a constant "eld, with subsequent integrating all over the space outside the horizon.) Let us compare now the radiation intensity I due to the e!ect discussed, with the intensity I of & the Hawking thermal radiation. Introducing additional weight e in the integrand of formula (35), we obtain I"pm(eQ/pmr ) exp(!pmr /eQ) . > >

(44)

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I.B. Khriplovich / Physics Reports 320 (1999) 37}49

As to the Hawking intensity, the simplest way to estimate it, is to use dimensional arguments, just to divide the Hawking temperature ¹ "(r !r )/4pr & > \ > by a typical classical time of the problem r (in our units c"1). Thus, > I &1/4pr . (45) & > More accurate answer for I di!ers from this estimate by a small numerical factor &2;10\, but & for qualitative estimates one can neglect this distinction. The intensities (44) and (45) get equal for p (mr ) p (kmM) > & eQ& . (46) 6 ln(mr ) 6 ln(kmM) > (One cannot agree with the condition eQ&1/(4p) for the equality of these intensities, derived in Ref. [6] from the comparison of e "eQ/r with ¹ "(r !r )/(4pr ).) K > & > \ > Let us consider in conclusion the change of the horizon surface of a black hole, and hence of its entropy, due to the discussed non-thermal radiation. To this end, it is convenient to introduce, following Refs. [19,20], the so-called irreducible mass M of a black hole:  (47) 2M "M#(M!Q ;  here and below we put k"1. This relationship can be conveniently rewritten also as M"M #Q/4M . (48)   Obviously, r "2M , so that the horizon surface and the black hole entropy are proportional to >  M.  When a charged particle is emitted, the charge of a black hole changes by *Q"!e, and its mass by *M"!eQ/r #m, where m is the deviation of the particle energy from the maximum one. > Using the relationship (48), one can easily see that as a result of the radiation, the irreducible mass M , and hence the horizon surface and entropy of a non-extreme black hole do not change if the  particle energy is the maximum, eQ/r . In other words, such a process, which is the most probable > one, is adiabatic. For m'0, the irreducible mass, horizon surface, and entropy increase. As usual, an extreme black hole, with M"Q"2M , is a special case. Here for the maximum  energy of an emitted particle e "e, we have *M"*Q"!e, so that the black hole remains K extreme after the radiation. In this case *M "!e/2, i.e., the irreducible mass and the horizon  surface decrease. In a more general case, *M"!e#m, the irreducible mass changes as follows:




e!m # *M "!  2



e m M ! # m.  2 4

(49)

Clearly, in the case of an extreme black hole of a large mass, already for a small deviation m of the emitted energy from the maximum one, the square root is dominating in this expression, so that the horizon surface increases. I am grateful to I.V. Kolokolov, A.I. Milstein, V.V. Sokolov, and O.V. Zhirov for their interest in this work and useful comments. The work was supported by the Russian Foundation for Basic

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Research through Grant No. 98-02-17797, by Grant No. 96-15-96317 Leading Science Schools, and by the Federal Program Integration-1998 through Project No. 274.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

M.A. Markov, V.P. Frolov, Teor. Mat. Fiz. 3 (1970) 3. W.T. Zaumen, Nature 247 (1974) 531. B. Carter, Phys. Rev. Lett. 33 (1974) 558. G.W. Gibbons, Comm. Math. Phys. 44 (1975) 245. T. Damour, R. Ru$ni, Phys. Rev. Lett. 35 (1975) 463. I.D. Novikov, A.A. Starobinsky, Zh. Eksp. Teor. Fiz. 78 (1980) 3 [Sov. Phys. JETP 51 (1980) 3]. F. Sauter, Z. Phys. 69 (1931) 742. W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714. J. Schwinger, Phys. Rev. 82 (1951) 664. V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum Electrodynamics, Pergamon Press, Oxford, 1989. W. Greiner, J. Reinhardt, Quantum Electrodynamics, Springer, Berlin, 1994. N. Deruelle, R. Ru$ni, Phys. Lett. 52 B (1974) 437. L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Butterworth-Heinemann, London, 1995. Ya.B. Zel'dovich, V.S. Popov, Uspekhi Fiz. Nauk 105 (1971) 403 [Sov. Phys. Uspekhi 14 (1972) 673]. A.B. Migdal, Uspekhi Fiz. Nauk 123 (1977) 369 [Sov. Phys. Uspekhi 20 (1972) 879]. A. Sommerfeld, Wave Mechanics, Dutton, New York, 1930. H. Bethe, Intermediate Quantum Mechanics, Benjamin, New York, 1964. D.N. Page, Phys. Rev. D 16 (1977) 2402. D. Christodoulou, Phys. Rev. Lett. 25 (1970) 1596. D. Christodoulou, R. Ru$ni, Phys. Rev. D 4 (1971) 3552.

Physics Reports 320 (1999) 51}58

Decoherence}#uctuation relation and measurement noise L. Stodolsky Max-Planck-Institut fu( r Physik (Werner-Heisenberg-Institut), Fo( hringer Ring 6, 80805 Mu( nchen, Germany

Abstract We discuss #uctuations in the measurement process and how these #uctuations are related to the dissipational parameter characterizing quantum damping or decoherence. On the example of the measuring current of the variable-barrier or QPC problem we discuss the extra noise or #uctuation connected with the di!erent possible outcomes of a measurement. This noise has an enhanced short time component which could be interpreted as due to `telegraph noisea or `wavefunction collapsesa. Furthermore, the parameter giving the strength of this component is related to the parameter giving the rate of damping or decoherence.  1999 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz Keywords: Decoherence; Mesoscopic; Measurement

1. Introduction I have always shared Lev Okun's interest in the fundamentals of quantum mechanics, and it is a pleasure to share the following thoughts with him for his 70th birthday. Our topic has to do with `quantum dampinga or `decoherencea, the description of how a quantum system loses its coherence when in contact with an external system or environment. This is an interesting and amusing subject with many aspects. In studying the loss of coherence one may say we are seeing how a quantum system `gets classicala. On the other hand, the environment in question may be a `measuring apparatusa and so we get involved with the `measurement problema. Finally, in a more pedestrian vein, many of the problems and equations are those of ordinary kinetic theory. We have studied these issues over the years and applied the results in many contexts, ranging from optically active molecules [1}3] to neutrinos [4] to gravity [5] and E-mail address: [email protected] (L. Stodolsky) 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 6 5 - 4

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L. Stodolsky / Physics Reports 320 (1999) 51}58

quantum dots [6]. In the particularly simple case of the two-state system, such as the two states of a handed molecule, two mixing neutrino #avors, or an electron tunneling between two quantum dots, it is possible to give a fairly complete phenomenological treatment of the problem. Here we would like to discuss a further idea, that there are certain #uctuations in the measuring system and these are connected with the parameter characterizing the damping or decoherence of the observed system.

2. Damping}decoherence parameter In the description of the loss of coherence a certain parameter D arises, which can be thought of as the quantum damping or decoherence rate. We "rst describe how this parameter arises. Our description of the two-state system is in terms of the density matrix, which is characterized by a three-component `polarization vectora P, via (1) o"(I#P ) p) ,  where the p are the pauli matrices. P gives the probability for "nding the system in one of the two X states (l or l , electron on the left or right dot and so on) via P "Prob(¸)!Prob(R). (We shall C I X refer to our two states as L and R). Hence P gives the amount of the `qualitya in question. The X other components, P , contain information on the nature of the coherence. "P""1 means the  system is in a pure state, while "P""0 means the system is completely randomized or `decohereda. P will both rotate in time due to the real energies in the problem and shrink in length due to the damping or decoherence. The time development of P is given by a `Bloch-likea equation [1}3] PQ "V;P!DP . (2)  The three real energies V give the evolution of the system in the absence of damping or decoherence, representing for example the neutrino mass matrix or the energies for level splitting and tunneling on the quantum dots. The second term of Eq. (2), our main interest here, describes the damping or decoherence. D gives the rate at which correlations are being created between the `systema (the neutrinos, the external electron on the dots) and the environment or detector, and this is the rate of damping or decoherence. The label `tra means `transversea to the z axis. The damping only a!ects P because  the `dampinga or `observinga process does not induce jumps from one state to another: the neutrino interaction with the medium conserves neutrino #avor, the read-out process for the quantum dots leaves the electron being observed on the same dot; the observing process conserves P "1p 2. X X A formula for D can be given [1}3] in terms of the S matrices for the interaction of the environment or measuring apparatus with the two states of our system. There is a certain complex quantity K whose imaginary part gives the damping D and whose real part gives an energy shift to the system being measured. K is given by K"i( -ux)1i"1!S SR "i2 . (3) * 0 The factor -ux is the #ux or probing rate, where in the QPC application to be discussed one can use the Landauer formula -ux"e< /p , with < the voltage in the detector circuit [7]). The label  

L. Stodolsky / Physics Reports 320 (1999) 51}58

53

i refers to the initial or incoming state of the probe and the S's are the S matrices corresponding to the di!erent states of the observed system. In our problem with two barriers (¸ and R) we may express the barrier penetration problem in S matrix form, calculate K according to the above formula and take the imaginary part to "nally obtain D [6]. In doing so, we "nd that D, as might have been expected, involves the di!erence in transmission by the two barriers. However, because of the various phases which are in general present in S, there are some further and more subtle phase-dependent e!ects which can contribute to D. These e!ects are quite interesting, (and not uncontroversial) but we shall not discuss them here and simply con"ne ourselves to the most straightforward situation where the only contributions to D are due to the di!erence in transmission by the two barriers. Thus the `measurementa consists solely in the fact that each barrier passes a di!erent current. With the S matrix parameterized such that the transmission coe$cients for the two barriers are called cos h ,cos h , * 0 corresponding to transmission probabilities p "cos h , p "cos h , we "nd [6] with this * * 0 0 neglect of phases D"( -ux)+1!cos *h, ,

(4)

where *h"h !h . Hence D is maximal for very di!erent transmission probabilities or large *h, * 0 while for the case of *h small: (*h) . D+( -ux) 2

(5)

D is a phenomenological parameter representing a kind of dissipation, resembling in some ways, say, the electrical resistance. Now for resistance and similar dissipative quantities there is the famous #uctuation}dissipation theorem [8] which relates the resistance or similar parameter to #uctuations in the system, as in the relation between resistance and Johnson noise. Should there be such a relation here? Of course here it is not energy that is being dissipated. Rather it is `coherencea that is being lost or perhaps entropy that is being produced. Nevertheless we should expect such a relation. For an interesting treatment of the #uctuation}dissipation theorem not based on energetic considerations see Ref. [9]. Indeed, looking at the Bloch equation Eq. (2) in its original context as the description of the polarization in nuclear magnetic resonance, it is quite natural to see the decay of the polarization as due to #uctuating magnetic "elds in the sample. Here, however, we wish to consider not "elds in a sample but something being observed by a `measuring apparatusa. We will nevertheless reach a similar conclusion, in that a `measuring apparatusa is something that reacts di!erently according to the state of the thing being observed. If it does not react di!erently, it does not `measurea, obviously. Hence we expect a measuring apparatus to show #uctuations related not only to the state of the system being observed, but also to how strongly it reacts to di!erences in that system. Furthermore since D, according to Eq. (4), is determined by these same di!erences, we expect some relation between the damping or decoherence rate and the #uctuations. In the following we would like to show how such expectations are realized in the variable barrier or QPC (quantum point contact) measuring process, where the `measuring apparatusa is a current determined by a variable tunneling barrier.

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3. The current in the variable-barrier problem Brie#y, the measurement process using a quantum point contact (QPC) detector [10] can be described as the modi"cation of a barrier whose transmission varies [11] according to whether an external electron is nearby or farther away. When the external electron is close by there is a certain higher barrier, and when it is farther away, there is a reduced barrier. Given an incident or probing #ux on the barrier (in practice also electrons), the modi"cation of the resulting current through the barrier, thus `measuresa where the external electron is located. Thus we have a quantum system, the external electron, which is being `observeda by the detector current. The state of this external electron, in the case where it represented as a two-state system is given by the density matrix, evolving as in Eq. (2). The density matrix elements o and o , give the probabilities of the system ** 00 being observed being found in the state ¸ or R (o #o "1 and o !o "P ). ** 00 ** 00 X Experiments of this type give a fundamental insight into the nature of measurement, and in an elegant experiment Buks et al. [10] * stimulated by the work of Gurvitz [12] * saw the expected loss of fringe contrast in an electron interference arrangement when one of the interferometer paths was `under observationa by a QPC. Here, however, we wish to focus not so much on the object being measured but rather on the behavior of the `measuring apparatusa * the current through the variable barrier or QPC. We wish to examine certain `extraa #uctuations in this current due to the measurement process. We thus turn to the current through the two-barrier system. Consider the probability for a given sequence of transmissions and re#ections through the barriers. Let 1 represent a transmission and 0 a re#ection for the probing electrons. Also let p be the probability of transmission and q that for a re#ection, (q#p"1), where each of these quantities has a label ¸ or R. We write the probability that in N probings the "rst probing electron was transmitted, the second re#ected, 2 the (N!1)th transmitted and the Nth transmitted, as Prob[11201]. Now it is relatively easy to write down a formula for this probability in the situation where the time in which the N probings take place is small compared to the time in which the observed system changes its state. Taking there to many probings in this time, we "nd [6] Prob[11201]"o (p p 2q p )#o (p p 2q p ), N;N , ** * * * * 00 0 0 0 0



(6)

where N is the number of probings in the time it takes the observed system to change states. The

 main point about this formula is that each sequence contains factors of only the ¸ or R type. We need the restriction N(N because for longer times the observed system can change states and

 the string 2q p will get `contaminateda with factors with R labels. For short times, where this is * * not a problem, the formula applies and may be arrived at either by thinking about the amplitude for the whole multi-electron process or by repeated wavefunction `collapsesa. Using it, one can understand the continuum from `almost no measurementa to `practically reduction of the wavefunctiona by varying the parameters p and p . The former occurs for approximate equality of * 0 p and p and the latter in the opposite limit where, say, p is one and p is zero. One can also * 0 * 0 understand the origin of `telegraphica signals resembling a `collapse of the wavefunctiona in this latter case: for p and p close to zero and one, respectively, we will have predominantly sequences * 0 of either transmissions or re#ections with high probability, while mixed sequences are improbable [6]. Here we would like to use Eq. (6) to examine the #uctuations in the current.

L. Stodolsky / Physics Reports 320 (1999) 51}58

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4. Measurement noise Eq. (6) says that for short times N;N we have a simple combination of two processes, each

 one consisting of a sequence characterized by statistical independence. Naturally, a distribution consisting of the (normalized) sum of two such distinct distributions will have a variance greater than the average of the variances from individual distributions. For example, in the classical limit we would have two distinct peaks for the transmitted current. Hence we expect greater #uctuations than we would have with just a single distribution. To quantify this we calculate the variance <"Q!QM  in the number of particles transmitted in N probings of the barrier. To do this we need the probability Prob(Q, N) for Q transmissions in N probings. Since the two terms of Eq. (6) are essentially those leading to the binomial distribution in statistics, the combinatorics are that of the binomial distribution, and we have Prob(Q, N)"o P (Q)#o P (Q), N;N , (7) ** * 00 0

 where P (Q) is the binomial expression for the probability of Q transmissions in N trials given the * single-trial probability p . (For N large and p small this is approximated by the poisson distribu* tion P (Q)+((QM )//Q!) e\/M *, with QM "p N.) * * * * Now calculating < for this distribution, that is the variance in the number of particles transmitted in N trials, we obtain <(N)"o < #o < #o (1!o )(QM !QM ), N;N , (8) ** * 00 0 ** ** * 0

 where < , < are the variances for the distributions P , P , and we have used o "(1!o ). * 0 * 0 00 ** The "rst two terms give just the average of the variances of the component distributions, as might have been expected. The last term, however, represents some extra noise or #uctuation due to the fact that a measurement is taking place, that the current responds di!erently to the two di!erent states of the `observeda. A justi"cation for calling the last term an `extraa noise is also provided when we recognize that the "rst terms in Eq. (8) are simple shot noise -like contributions (in Ref. [13] called `partition noisea) that would disappear in the current if the electron charge were not quantized, i.e. in the limit of "nite current but with eP0 (see below). Note that, as is reasonable, this extra noise is absent if the observed system is de"nitely in one state or the other, if either o or (1!o ) are zero. If only one state is present there is only one 00 ** thing to measure and so no extra #uctuations. Henceforth, to simplify the writing we concentrate on the relaxed state of the two-state system and put o "o "(1!o )". ** 00 **  These considerations were all for N;N . What happens for N


 compared to any transition or relaxation time? Well, then we have N/N independent, uncor  related intervals, each one with a < given by Eq. (8). The variance from j independent objects, each one with variance < is <"< j, so for long intervals we have from Eq. (8) with N"N , and so H H

 < "p N * *  p #p (p !p ) (p !p ) 0# * 0 N 0 N N" * N, N




 * 0 2 4 4




going as N, or the time, as it should be.



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We draw two main conclusions, that there is an extra #uctuation beyond the simple partition or shot noise-like contributions < , < and that this contribution involves the measuring strength or * 0 analyzing power squared (p !p ). * 0 5. Relation between D and < + Both these extra #uctuations and the damping D are related to the di!erence in transmission coe$cients. Calling this extra `measurement noisea < ,
 and for D we have Eq. (4). The relation between < and D is implicit but we can "nd a simple + explicit relation in the approximation of working to lowest order in *h"h !h : * 0 (QM !QM )+4p(1!p)N(*h) , (11) * 0 where p is the average transmission probability (p #p )/2. The factor p(1!p) re#ects the fact that * 0 with all transmissions or all re#ections there are no #uctuations. Now since both < and D are + proportional to (*h) we can write a `decoherence}#uctuation relationa relating D to the extra #uctuation < . K < +p(1!p)N(D/-ux), N;N (12) +

 for short times, while for long times < +p(1!p)N N(D/-ux), N


 Hence D can be seen either through the loss of coherence in the `observed systema or through the #uctuations of the `measuring systema, here the current. 6. Conclusions and experimental implications With quantum dots and QPCs an experimental realization of these ideas could be one in which the two-state system controlling the variable barrier is an external electron resident on and tunneling between two dots [6]. Other variants, both for the two-state system and the probing current or beam, can undoubtedly be contemplated, for example a single particle or atom in a trap continually probed by some beam. Note that we are now discussing the totally relaxed condition of the two-state system, so unlike the situation where we study oscillations [1,6] no particular starting time or injection time need be experimentally de"ned. We just have certain extra #uctuations even when measuring in the incoherent or relaxed state. The extra #uctuations may be said to be due to "nding sometimes one state and sometimes the other. For comparison with experiment, we reexpress the above results in the more usual language of a current and its variance. It should be noted, however, that we have not accounted for those #uctuations which may already be present in the incoming #ux, as from the reservoirs providing the current [13].

L. Stodolsky / Physics Reports 320 (1999) 51}58

57

The parameter N, which played an important role in the above, was the number of sequential probings under consideration. N is essentially a measure of the time and may be considered as giving a time interval *t"N/-ux to be used in de"ning the current. That is, a certain averaging or integration time *t will be involved in converting the number of transmissions Q into a current j"eQ/*t. We thus introduce a label *t on the current to indicate the averaging time or number of probings used in its de"nition. This averaging time may be purely theoretical or represent, for example, the response time of the instrumentation. With e the electron charge we have j ,eQ/*t"eQ( -ux/N) . R

(14)

Above we considered two regimes, N(N for times short compared to the relaxation time for

 the two-level system controlling the current, and N'N for long times compared to this

 relaxation time. We can now rewrite Eq. (8) as a relation for the variance of the current for short averaging times






p #p * 0#(p !p ) (e -ux), N;N * 0

 N

( j )!( j )" R R

(15)

while for long averaging times Eq. (9) becomes




( j)!( j )" R



(p !p ) p #p 0 N * 0# * (e -ux), N


 N N

(16)

The #uctuations of the current are seen to depend on the time scale used in its de"nition. While the common "rst term ((p #p )/N)(e -ux)"eN(p #p )/(*t) is the usual statistical behavior * 0 * 0 expected from the (N law for (<, the `measurement induceda (p !p ) term shows quite * 0 di!erent behavior in the two regimes. For long averaging times, Eq. (16), it has the same N behavior as the "rst term and so gives an additional constant level to the partition or shot noise-like contribution. For short averaging times, Eq. (15), the `measurementa term does not decrease with N at all and for short times but large N can completely dominate the #uctuations. This increased noise at high frequencies, with no `ultraviolet cuto! a is one of the most interesting points here. These strong short-time #uctuations are connected with the `telegraph noisea alluded to above. If one wishes, this can be thought of as coming from `wavefunction collapsesa. Since these `collapsesa have no time scale, i.e. are instantaneous, there is nothing at short times to smooth or `softena the noise. We stress, however, that this `collapsea language is not necessary, everything simply comes from Eq. (6), which can be obtained in an amplitude approach [4], without `collapsesa. In both Eqs. (15) and (16) one can see, as mentioned earlier, that only the `measurementa terms survive in the limit where the charge eP0 and NPR such that the current remains "nite. This shows how the noise represented by the "rst terms has its origin in the discreetness of the charge, while that of the second terms has to do with something else, `measurementa. Finally, we reexpress the results of Section 5 in terms of the current. Using again the subscript `Ma for the extra, `measurement induceda #uctuation we have for Eq. (12) [( j )!( j )] +8p(1!p)D(e -ux), N;N R +

 R

(17)

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L. Stodolsky / Physics Reports 320 (1999) 51}58

for short integrating times, with the `harda behavior that it does not decrease with *t, while Eq. (13) becomes N [( j )!( j )] +8p(1!p)  D(e -ux), N
 N

(18)

for long integrating times, with a conventional 1/*t fall o! . These express, in the two regimes, the `decoherence}#uctuationa relation. References [1] R.A. Harris, L. Stodolsky, J. Chem. Phys. 74 (4) (1981) 2145. [2] R.A. Harris, L. Stodolsky, Phys. Lett. B 116 (1982) 464. [3] L. Stodolsky, Quantum damping and its paradoxes, in: J.S. Anandan (Ed.), Quantum Coherence, World Scienti"c, Singapore, 1990. [4] G. Ra!elt, G. Sigl, L. Stodolsky, Phys. Rev. Lett. 70 (1993) 2363. [5] L. Stodolsky, Acta Phys. Polon. B 27 (1996) 1915. [6] L. Stodolsky, Phys. Lett. B 459 (1999) 193. [7] D.J. Thouless, Topological Quantum Numbers and Nonrelativistic Physics, World Scienti"c, Singapore 1998, p. 79. [8] H.B. Callen, T.A. Welton, Phys. Rev. 83 (1951) 34. [9] D. Chandler (Ed.), Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1987 (Chapter 8). [10] E. Buks, R. Schuster, M. Heiblum, D. Mahalum, V. Umanksy, Nature 391 (1998) 871. [11] M. Field, C.G. Smith, M. Pepper, D.A. Ritchie, J.E.F. Frost, G.A.C. Jones, D.G. Hasko, Phys. Rev. Lett. 70 (1993) 1311. [12] S. Gurvitz, quant-ph 9607029, Phys. Rev. B 56 (1997) 15215, and quant-ph 9808058. [13] M. Reznikov, R. de Picciotto, M. Heiblum, D.C. Glattli, A. Kumar, L. Saminadayar, Quantum Shot Noise.

Physics Reports 320 (1999) 59}78

Anatomy of a con"ning string V.I. Zakharov Max-Planck Institut f u( r Physik, Fo( hringer Ring 6, 80805 Mu( nchen, Germany

Abstract We review recent papers on the anatomy of the con"ning string in the Abelian Higgs model with condensed Higgs "eld. The basic observation is that apart from the well known Abrikosov}Nielsen}Olesen strings of "nite transverse size there exist in"nitely thin topological strings. These mathematically thin strings are responsible, in particular, for a stringy correction to the potential between con"ned charges at short distances. Possible implications for QCD are brie#y discussed.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.27.#d; 12.38.Aw; 12.38.Qk Keywords: Quantum Chromodynamics; Strings; Classical solutions

1. Introduction Lev Borisovich Okun has always been keen to introduce new hypothetical objects and phenomena into the well-ordered world of particle physics. I remember one of the "rst talks which I heard as a student was by him where he introduced at once a multitude of novel particles like baryons with integer spin. I may not say that I was always enthusiastic about these bizarre creatures. However, at least once I was indeed delighted to hear Okun's speculations on hypothetical new interactions. I mean the possibility of new non-abelian interactions which are similar to the interactions of the standard quarks and gluons in all respect except the range of interaction [1]. Which could be as large as one can imagine. And then Lev Borisovich described in an artistic way how a pair of new (non-relativistic) particles, so called h-particles are produced in Moscow, reach freely, say, Far East and then learn that they are actually con"ned and cannot separate themselves

E-mail address: [email protected] (V.I. Zakharov) 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 6 1 - 7

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V.I. Zakharov / Physics Reports 320 (1999) 59}78

from each other. Then they develop a con"ning string which forces them back to Moscow and they reiterate their enchanted motion again and again until annihilate into the corresponding glueballs, or thetonia. Thus, the con"ning strings could be thought of as classical objects with macroscopic sizes. In fact, this picture could have provoked disturbing thoughts. What actually happens to the strings when the h-quarks come back close to each other? Are they retaining their length and become just sloppy or do they disappear and then resurrect themselves from ashes? Moreover, it seems unnatural that the "rst motion of this slow stringy pendulum is di!erent from the subsequent ones. Does it mean that the strings were at the very beginning? It took me many years to realize the questions and, simultaneously, the fact that we do not need the new h-particles to pose all these questions about the con"ning strings. The QCD itself is the same well suited for this purpose. Indeed, the change of all the scales by 106 where X is a large number is only of psychological help at best. Moreover, this kind of questions occupied Gribov for a long time [2]. The question addressed by Gribov was how could it be reconciled with causality, that quarks are produced as free particles at short distances, #y away and then still have time to communicate and build up a kind of con"ning color "eld con"guration in between them. The answers which Gribov suggested were highly non-trivial. Like supercritical properties of the QCD vacuum and the related existence of relatively large-size states. The question we would like to address here [3,4] is what is left from an Abrikosov} Nielsen}Olesen (ANO) string [5] when the end-point monopoles are brought together to a distance much smaller than the size of the ANO string? The answer promoted here is that what is left of the string is a string again. But this time it is an in"nitely thin string which can be thought of as a Dirac string [6] or a topological string [7] (which is the same). A physical manifestation of this string is a stringy, or linear piece of potential between the con"ned charges which persists even at in"nitely small distances. These answers are obtained within the Abelian Higgs model, while the relation to QCD is provided by the dual superconductor con"nement mechanism [8]. This note is a mini-review based primarily on the original papers written in collaboration with Gubarev and Polikarpov [3,4] from the Institute of Theoretical and Experimental Physics in Moscow. It is of special pleasure for me that our collaboration is so closely related to the same ITEP where in old days I enjoyed working with L.B. Okun on many common papers.

2. Field theoretical solenoid A key element in the conjectured mechanism of the quark con"nement [8] is the Abrikosov} Nielsen}Olesen string [5]. Although eventually it will not be this string that captures our attention, it is useful to begin with a brief review of the ANO string. The strings are classical solutions to the Abelian Higgs model (AHM). The AHM describes a gauge "eld A interacting with a charged scalar "eld U as well as self-interactions of the scalar I "eld. The corresponding action is

 

S" dx



1 1 1 F # "(R!iA)U"# j("U"!g) 4e IJ 2 4

(1)

V.I. Zakharov / Physics Reports 320 (1999) 59}78

61

where e is the electric charge, j, g are constants and F is the electromagnetic "eld-strength tensor, IJ F ,R A !R A . The scalar "eld condenses in the vacuum, 1U2"g, and the physical vector and IJ I J J I scalar particles are massive, m "eg, m "2jg. In the perturbative regime, interaction of the 4 & massive scalar and vector particles can be readily calculated order by order. There exists, however, a topologically non-trivial stringy solution to the classical equations of motion which possesses a cylindrical symmetry and is characterized by a "nite energy per unit of its length. In a way, it realizes an analog of a solenoid in "eld theory. The key element is to look for a solution with a non-vanishing electric current j "e(UH(R U)!(R UH)U) . I I I Moreover, if one chooses the scalar "eld of the form U(r)"e\ L(f (o) ,

(2)

(3)

then the electric current is circular and reminds, therefore, the solenoid current. The value of n in Eq. (3) is integer for the "eld U to be a unique function of the coordinates. Since the current has only a non-vanishing j component, the functional form of the vector potential is ( A(r)"!e( A(o)/o . (4) ( The central question is whether it is possible to ensure "niteness of the energy (per unit length) with the ansatz (3) and (4). The behavior of the "elds at oP0 and oPR is crucial at this point and it is easy to check that the conditions f (0)"A(0)"0,

f (R)"g,

A(R)"!n/e ,

(5)

su$ce to make the energy "nite. From these conditions alone, one can derive the magnetic "eld #ux carried by the ANO string. Indeed,





2p H dS" A dx " n , I I e

(6)

where to evaluate the latter integral we used the asymptotic value of A , see Eq. (5). Thus, the #ux is ( an integer of a minimal magnetic monopole charge. It is obvious then that the strings can end up with monopoles. Moreover, if the strings are long enough, the solution with cylindrical symmetry is approximately valid. Explicit form of the functions f (o), A(o) introduced above can be found by solving the classical equations of motion: ie

A " (UHR U!UR UH)!eA , I I I I 2 (7)

U#2ieA RIU!eA AI"jgU!jU"U" . I I In particular, one concludes from these equations that the functions f (o), A(o) approach their asymptotic values (5) exponentially. In case of the vector "eld, it is exp(!m o) and in case of the 4 scalar "eld it is exp(!m o). The full functions f (o), A(o) can be obtained by numerical methods. 1

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V.I. Zakharov / Physics Reports 320 (1999) 59}78

Details of the solution depend on the ratio of the masses, m and m . Two limiting cases are of 4 1 special interest. In the so called Bogomolny limit, m "m , (8) 4 1 the equations simplify greatly and can be solved in fact analytically, by expanding in series in mo. Another useful case is the so called London limit, m<m . (9) 1 4 In this limit, the scalar "eld is frozen as U"g and one can neglect in most cases the #uctuations of the "eld U around its vacuum value. However, the string tension, or energy per unit length becomes logarithmically divergent in this limit: p+(pm /2e) ln m/m . (10) 4 1 4 Indeed, without the Higgs "eld one cannot construct a solution with "nite string tension. From the physical point of view, the most important manifestation of the ANO string is the linearly growing potential energy for an external monopole}antimonopole pair: lim <(r)"p ) r , (11) P where the distance r is much larger than the inverse masses, m\. Thus, the magnetic charges are 14 con"ned. This property of the Abelian Higgs model is the basis for the dual superconductor mechanism of the quark con"nement [8]. Namely, one assumes that the QCD vacuum is similar to the vacuum of the AHM in the sense that a scalar "eld with non-vanishing magnetic charge condenses in the QCD vacuum. Then the (color) electric charge is con"ned. The usage of the language of the magnetic and electric charges is most transparent in the so called ;(1) projection of QCD [7] which treats diagonal gluon "elds as photon-like gauge "elds while non-diagonal gluons play the role of extra charged "elds.

3. Potential at short distances Property (11) is most famous and best known about the AHM and the con"nement mechanism. The linear growth of the potential is derived at large distances, where the scale is set by the inverse masses. The question central for the present note is whether there are short strings whose sizes are much smaller than the inverse masses. But "rst let us de"ne what we understand by a short string. We will accept a physical de"nition of the short string, namely, as existence of a stringy or linear potential at small distances [3,4]. In view of this de"nition, a few general remarks on the potential at short distances are now in order. We begin with a trivial Yukawa attractive potential and expand it in powers of (mr) at small r: C Crm Crm C # #2 , < (r)"! e\KP"! #Cm! 7 r 2 6 r

(12)

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63

where C is a positive constant. This simple equation carries an important message, namely, that the physics behind the odd and even powers of the mass m in the potential is di!erent. Indeed, naively we should have had only powers of m since m is the only parameter entering the Lagrangian of a massive boson. Thus, our idea could be that we start with a Coulomb like potential at small r and develop a perturbative expansion in mr. Eq. (12) implies that such an approach would fail because of infrared divergences. This can be readily checked explicitly of course. But it su$ces to notice that the odd powers in (12) could arise only from the infrared divergences which are cut o! at distances of order 1/m. Appearance of the terms non-analytical in m allows [9] to make important conclusions about the heavy quark potential in QCD. Namely, if a physical observable is sensitive to large distances at the level of a certain power correction then it cannot be evaluated to such accuracy because the e!ective coupling is large. Rather, one should reserve for the strength of the correction as a phenomenological parameter. In our case, we can conclude from Eq. (12) that the quark}antiquark potential in QCD can be parameterized as C < M (r)"! \#C K #C rK #2  /!"  /!" // r

(13)

where C is calculable perturbatively as an expansion in a (r) while C account phenomenologi\ Q  cally for the contribution of large distances. Moreover, C can be included into de"nition of  the (heavy) quark masses and the message is that the power corrections to the potential at short distances start with r terms [10]. A salient feature of Eq. (13) is the absence of a linear in r term at short distances. The proof is in two steps. First, we learn from (12) that the linear correction to the potential is not contaminated by large-distance e!ects and, then, conclude that these corrections vanish since the gluon mass in QCD is vanishing. Thus, presence of the linear term at short distances in the quark potential would signal new non-perturbative physics at short distances [11]. According to the nomenclature we are introducing now, the validity of the Eq. (13) would imply that there are no short strings in QCD because there is no linear correction to the potential. In case of the AHM model the situation is di!erent at "rst sight since there is a non-vanishing mass of the gauge boson and the linear term is obviously present, see Eq. (12). Note, however, the sign of the linear term. It is negative while for a string it should be positive. Apparently, we should include this condition of the positiveness of the slope of the potential at short distances into our de"nition of a short string. With this completion in mind, we would say that if the potential of a monopole}antimonopole pair at short distances in the AHM is described by a massive photon exchange, then there are no short strings. The reader might "nd this discussion excessive. Is not it much simpler just to evaluate directly "rst the photon propagator and, then, the corresponding potential? The point is that this calculation is much more tricky than one might assume. The actual calculation of the potential will be presented in the two subsequent sections. Here we outline a straightforward calculation of the propagator which goes back more than 20 years [12]. The main technical problem is that we have now both magnetic and electric charges interacting with the same photon and most of the conventional formalisms do not apply under the circumstances. We will follow, therefore, Zwanziger [13] to introduce formally two vector "elds, A and I

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B interacting with electric and magnetic currents, respectively. The action reads: I



S (A, B)" dx+(n ) [RA])#(n ) [RB])   8 # (n ) [RA])(n ) [[RB]]B)! (n ) [RB])(n ) [[RA]]B), ,   where n is a unit space-like vector, n"1 and I [AB] "A B !A B , (n ) [AB]) "n (AB) , IJ I J J I I J JI [(G)]B "e G . IJ  IJHM HM The classical equations of motion are:

(14)

(nR)A !n (nR)(RA)!R (nR)(nA)#n R(nA)!i(nR)e n R A "!iejC , I I I I IJHM H M J I (15) (nR)B !n (nR)(RB)!R (nR)(nB)#n R(nB)#i(nR)e n R B "!igjK . I I I I IJHM H M J I Note that both jK and jC are conserved as a consequence of the equations of motion, RjKC"0. Although the theory contains two gauge "elds A and B , it still describes one physical photon I I with two physical degrees of freedom. This follows from a careful treatment of the Hamiltonian dynamics of the system [13]. In simpli"ed terms, the reason is that there is a constraint that the "eld strength tensor constructed on the potential A is dual to the "eld strength tensor constructed I on B . I To work out propagators one needs to "x the gauge. A convenient choice of the gauge-"xing term is:



S "1/2 dx(M(nA)#M(nB)) .  

(16)

The propagators then take the following form in the momentum space:






k#M k k 1 1 I J! (k n #k n ) 1A A 2"1B B 2" d # J I I J I J M (kn) (kn) I J k IJ 1 1A B 2" ([nk]B) . I J IJ k(kn)

(17)

Notice an apparent dependence on an arbitrary vector n. The meaning of this vector is that there are Dirac strings attached to the magnetic charges and the Dirac strings are directed along the vector n . Having in mind the presence of the Dirac strings, the form of the propagators (17) can I actually be readily understood on physical grounds. The condition that the dependence on the n is I only super"cial yields the Dirac quantization condition eg"2pm. Moreover, there are general proofs that upon imposing this condition the physical results are in fact independent on the choice of the vector n [13]. I When the Zwanziger formalism is combined with the spontaneous breaking of the ;(1) symmetry apparent inconsistencies arise. Indeed, let us assume that a charged scalar "eld acquires a non-vanishing vacuum expectation value or, even simpler, a mass term (m /2)A is added to the 4 I Lagrangian. Then a straightforward diagonalization of the bilinear terms in the Lagrangian results

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65

in the following propagator of the B-"eld (see, e.g., [12]): k n #k n k k m X 4 (d !n n ) . J I# I J IJ , X "d ! I J (18) 1B B 2, IJ IJ I J I J (k ) n) (k ) n) (kn) IJ k#m 4 Expressions (18) can be readily obtained by inserting the mass term and combining 1A, B2 propagators worked out above for a massless photon (see Eq. (17)). Next it would be natural to evaluate, as usual, the potential energy of a monopole pair due to the (massive) photon exchange in terms of the propagator (18):




(19)

where D is the same as 1B B 2 above. However, the n -dependence does not drop o! from
4. London limit Now we will venture to uncover a structure of the con"ning strings in the AHM. To this end we will consider distances small compared to the inverse masses. Concentrate "rst [3] on the London limit, that is distances r satisfying m\
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Basically, we are dealing in this section with two problems. First, we found above that the photon propagator depends on the choice of an unphysical vector n and we cannot, therefore, to I explore the standard relation between the propagator and the potential energy. Second, the treatment of an ANO string whose length is much smaller than its transverse size seems a complicated technical problem because of the edge e!ects. The solution to the latter problem will be the introduction of a dual representation of the same Abelian Higgs model in terms of strings rather than "elds. To circumvent the former problem, we will evaluate the potential energy of a monopole}antimonopole pair through the Wilson loop which provides an explicitly gauge invariant expression for the potential <(r). As for the string formulation of the AHM, we will use the results of the papers in Ref. [14]. In more detail, the partition function of AHM in the London limit (jPR) can be exactly rewritten in terms of the world-sheet coordinates XI (p) of closed ANO strings [14]:



DR e\1R , lim Z " &+ R B  H where the action for the ANO strings reads:

(21)

S(R)"(p/e)m (R, KK R) 4 p " m dpdp e?@R XI I(p)R XI J(p)K(XI (p)!XI (p))e?Y@YR XI I(p)R XI J(p) ? @ ?Y @Y e 4



(22)

and the kernel K(x) satis"es the equation (!R#m )K(x)"d(x). 4 Coming back to the static potential energy of a monopole}antimonopole pair in the vacuum of the AHM, we represent it "rst in terms of expectation value of the Wilson loop:



1  +  , K U U 1H( jK)2" DADBDUDUM e\18 > V .\ C  >H  \E  > E H   , Z

(23)

where j is the world trajectory of the monopole charge. Integrating over the "eld B gives: K I 1 DADUDUM e\1&+UUM H( jK) , 1H( jK)2" Z &+ (24) 1 2p 1   H( jK)"e  V ! (RA# [R ]B)# (RA) 4 e ! 4





where [R ]B"(nR)\[[njK]]B, dR "jK, and we used Dirac quantization condition eg"2p. ! ! Note that the surface [R ]B is spanned on the monopole loop j and parallel to the vector n. ! K Moreover, in the London limit one can integrate over the (non-singular) phase of the Higgs "eld and the gauge "eld A to get



DRe\1R>RA (25) lim 1H( jK)2"e\pCHK)K HK R B  H where ( jK, KK jK)" C C Cdx Cdx K(x!x) and the string action S(R) is the same as in (22).  Y I I

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67

Eq. (25) is convenient to demonstrate that by using the string representation we solved also the problem of the apparent dependence of the potential on the vector n discussed in the preceding I section. The reason is that 1H( jK)2 does not depend in fact on the shape of surface spanned on the monopole loop jK. Indeed, by change of variables, RPR!R the integral over closed surfaces ! (dR"0) is reduced to an integral over surfaces bounded by the loop C which corresponds to the current jK:



DR e\1R . (26) lim 1H( jK)2"e\pCHK)K HK K R B H H Thus the dependence on the direction n of the Dirac string disappears and 1H( jK)2 depends only I on the loop C, or the monopole trajectory. For a static monopole}antimonopole pair located at distance r an explicit form of jK is jK(x)"d [d(x!r/2)!d(x#r/2)] I I and the monopole}antimonopole potential is calculated as 1 <(R)"! ln 1H( jK)2 . ¹

(27)

(28)

As for the "rst term (2p/e)( jK, KK jK) in (26), it gives in case of the contour (27):



2p p e\K4P ( jK, KK jK)"(self energy)! dt e e r

(29)

which results in the Yukawa-type contribution to the potential <(R). As for the integration over the surfaces R, it is more involved. At the classical level one has to "nd the surface of minimal area bounded by the contour C. For the loop de"ned by (27) the minimal surface is the #at surface parameterized as follows: XI "t t3(!R;#R) , XI G"rGp p3(!1;#1) .  The calculation of the action S(R  ) is then straightforward. Since for R  : ! ! RG e?@R XI I(p)R XI J(p)"(dIdJG!dJdIG) ? @ 2

(30)

(31)

one has

   

1 pRm  dk 1 4 S(R  )" dt dt dp dp e I6I N\6I NY ! 2 (2p) k#m e 4 dk sin(kr/2) 1 1 4pRm  4 dt . " (2p) k#m (kr) e 2 4 Collecting all the above we have for the static monopole}antimonopole potential:





p e\K4P 1 4prm  dk sin(kr/2) 1 4 <(r)"! # . e (2p) k#m (kr) e r 2 4

(32)

(33)

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Choosing the Higgs mass m as the UV cut-o! we "nally obtain: 1  p e\K4P pm m 2 e\P(V>K4 <(r)"! # 4 R ln 1 ! # dx e r [x#m ] 2e m m  4 4 4 pm e\K4P " 4 !2 #m r ln[m/m ]!2[1!e\K4P]#2m r Ei[m r] (34) 4 1 4 4 4 2e m r 4  (!x)I e\R dt"C#ln[x]# Ei[x]"! kk! t V I which completes the evaluation of the potential energy of the static monopole pair in the approximation considered. To summarize, we have found the potential <(r) in the London limit. It does contain a linear correction to the Coulomb-like potential at short distances with a positive slope. Thus, there exist short strings in this limit. The resulting potential does not depend on the choice of the vector n in I the Zwanziger formalism, as is expected. The use of a dual representation of the AHM is crucial to see the e!ect of short strings. Indeed, in the language of `solenoidsa we had a short solenoid and expected large edge e!ects. In the dual description in terms of strings, we deal with a short string.

 









5. Topological strings We proceed now to distances much smaller than m\. In this case the London limit does not help 1 and we simply assume that m and m are of same order. The main result [4] is that a stringy 4 1 correction to the Coulomb-like potential at short distances is still there due to what can be called topological strings. The notion of topological string goes back in fact to the paper in Ref. [7]. Here, the strings come to the consideration through analysis of gauge "xing in AHM. Because of the gauge invariance, the phase of the charged "eld U can be rotated to an arbitrary value. In particular, the unitary gauge corresponds to this phase equal zero, Im U"0. This condition "xes the gauge unless the real part of the "eld U also vanishes: Im U"0, Re U"0 .

(35)

Viewed as two constraints in the four dimensional world, the conditions (35) de"ne a world sheet swept by a moving string. It is worth emphasizing that these are in"nitely thin, or mathematical strings. The observation crucial for our purposes [7] is that these strings are either closed or end up with magnetic charges. The language of the Dirac string [6] can be useful to substantiate the point. Indeed, the in"nitely thin line discussed above is nothing else but the Dirac string connecting the monopoles. The possibility of its dynamical manifestations arises from the fact that the Dirac string cannot coexist with UO0 and U vanishes along the string. Indeed, self-energy of the Dirac string is normalized to be zero in the perturbative vacuum. To justify this one can invoke duality and ask for equality of self-energies of electric and magnetic charges. Since the electric charge has no string attached, the requirement would imply vanishing energy for the Dirac string. However, if the Dirac

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69

string would be embedded into a vacuum with 1U2O0 then its energy would jump to in"nity since there is the term 1/2"U"A in the action and APR for a Dirac string. Hence, U"0 along the I I string and it is just the condition mentioned above. In other words, Dirac strings always rest on the perturbative vacuum which is de"ned as the vacuum state obeying the duality principle. Therefore, even in the limit rP0 there is a deep well in the pro"le of the Higgs "eld U. This might cost energy which is linear with r even at small r. Here we consider the classical approximation, hence the problem of "nding the potential <(r) is equivalent to solving classical equations of motion with boundary condition U"0 along the straight line connecting the monopoles. Let us note that numerical results for the potential <(r) can be found in a number of papers [15]. However, prior to [4] there were no measurements dedicated speci"cally to small corrections to the Coulombic potential at rP0. We will consider the unitary gauge, Im U"0. Then the most general ansatz for the "elds consistent with the symmetries of the problem is: U"g f (o, z), Im f"0 , A "e x( A(o, z), A "A "0 , ? ?@ @   o"[x x ], z"x , x( "x /o, ? ?  ? ?

(36) a"1, 2 .

Let us introduce also a new variable i"(2j/e"m /m and measure all dimensional quantities & 4 in terms of m "eg. Then the energy functional is: 4

 

p > > E(r)" dz o do+[(1/o)R (oA)#R]#[R A] M X e \  # [R f ]#[R f ]#f A#i( f !1), M X 






(37)



 1 r dm d z!m . R" d(o) ) o 2 \

(38)

In the limit rP0 the Coulombic contribution becomes singular. The easiest way to separate the singular piece is to change the variables A"A #a, where A is the solution in the absence of the B B Higgs "eld: A "(1/2o)[z /r !z /r ], z "z$r/2, r "[o#z ] . B \ \ > > ! ! !

(39)

Upon this change of variables the energy functional and the classical equation of motion take the form E(r)"E

!p/r#EI (r) ,  

 



> p > EI (r)" dz o do e \ 



1  R (oa) #[R a] X M o



#[R f ]#[R f ]#f (a#A )#i( f !1) , M X B 

(40)

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R





1 R (oa) #Ra"f (a#A ) , M o M X B

1 1 R [oR f ]#R f"f (a#A )# if ( f !1) . X B 2 o M M

(41)

(42)

The energy functional has been minimized numerically. The numerical results [4] for various i values clearly demonstrate that there is a linear piece in the potential even in the limit r;1. The slope p at rP0 was de"ned by the "tting the numerical data to: 




EI (r)"C   



1!e\P !1 #(p #C )r"p r#O(r) ,     r

(43)

The resulting slope p depends smoothly on the value of i, For the purpose of orientation let us  note that for i"1 the slope of the potential at rP0 is the same as at rPR. That is, within error bars: p +p  

(44)

where p determines the value of the potential at large r.  Thus, existence of short strings is proven in the classical approximation to the Abelian Higgs model [4]. The linear piece in the potential at small distances re#ects the boundary condition that U"0 along the straight line connecting the monopoles.

6. Lessons for QCD Similarity between QCD and the Abelian Higgs model becomes transparent in the ;(1) projection of QCD which is a certain way to "x gauge in a non-Abelian theory [7]. The interest in the ;(1) projection has been related mostly to the con"nement mechanism. There exist detailed numerical simulations on the lattice which con"rm the dual-superconductor picture of the con"nement (for review and references see [16]). Moreover, condensation of a scalar "eld U with

 magnetic charge is con"rmed within the ;(1) projection as well. We will concern ourselves here with implications of short strings discussed above for QCD. To begin with, existence of a ;(1) gauge invariant operator "U " of dimension d"2 at "rst sight,

 causes problems. Indeed, because of the condensation of U in the vacuum, this operator has

 a non-vanishing vacuum-to-vacuum matrix element. On the other hand, in QCD the lowest dimension of operator which may have a non-vanishing matrix element over the vacuum is d"4 [17]. It is (G? ) where G? is the gluon "eld strength tensor. The dimensions of local gauge IJ IJ invariant operators play a crucial role in applications of the operator product expansion (OPE) and we come to a disturbing conclusion that applying the OPE in the Abelian projection would be inconsistent with QCD. The paradox is resolved through the observation [4] that the OPE breaks in the Abelian Higgs model, as is explained below. Since the ;(1) projection of QCD is similar to the AHM, we expect, therefore, that the OPE breaks down in this projection as well. Moreover, the

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OPE results in certain predictions for observables and the breaking of these predictions may not depend on the use of the ;(1) projection. The breaking of the OPE in the AHM at the level of 1/Q corrections re#ects the presence of the topological strings discussed above. Indeed, validity of the OPE is equivalent to the assumption that at short distances the e!ects of particle exchanges are calculable explicitly, while the e!ects of large distances are included into matrix elements of various operators constructed from the "elds propagating at short distances. It is clear, on the other hand, that short strings cannot be reconstructed from particle exchanges and should be added as new objects. The simplest objects to apply the OPE are propagators (for review and references see, e.g., [19]). The standard logic can be illustrated by an example of the photon propagator connecting two electric currents which can be fully reconstructed within the AHM from the OPE:






1 1 1 1 1 1 d IJ . ! e1U2 # e1U2 e1U2 !2 " (45) D (Q)"d IJ IJ Q Q Q Q Q Q Q#m 4 Thus, one uses "rst the general OPE assuming "Q"<eU, then substitutes the vacuum expectation of the Higgs "eld U and upon summation of the whole series of the power corrections reproduces the propagator of a massive particle. The latter can also be obtained by solving directly the classical equations of motion. This approach fails, however, if there are both magnetic and electric charges present. Indeed, as is shown in Section 3 the propagator constructed along these lines contains a singular piece m 4 (d !n n ) , I J (Q#m )(Qn) IJ 4 see Eq. (18). The presence of such a term immediately implies that the standard OPE does not work at the level of Q\ corrections. Indeed, choosing Q large does not guarantee now that the m correction is small since the factor (Qn) in the denominator may become zero. 4 The di$culties with the propagator stem from the fact that one cannot avoid overlap of the Dirac strings and trajectories of charged particles in case of vacuum condensation of a charged "eld U. In other words, the singularities in the (Qn) variable signal presence of the Dirac string. The issue cannot be settled by formal manipulations with these stringy singularities. To the contrary, the Dirac string acquires a physical meaning in terms of energy and controls the 1/Q corrections to the potential, see the preceding section. As a result, instead of a 1/Q correction associated with "U" and large distances which is expected on the basis of the OPE, there is a 1/Q correction associated with small distances. We expect similar transmutation to happen within QCD. Then emergence of the operator "U " signals rather 1/Q corrections associated with small distances than violation

 of the OPE with respect to the infrared corrections.

 Appearance of the 1/Q corrections which go beyond the standard OPE is not in con#ict with QCD because of the ultraviolet renormalon. The ultraviolet renormalon signals that the perturbative QCD is inconsistent at the level of 1/Q terms and short strings may be thought of a non-perturbative counterpart which is needed to make the theory unique. There were many attempts to build up a phenomenology based on the ultraviolet renormalon (see, e.g., [18]). Phenomenological consequences of the short strings appear to be di!erent from the schemes considered so far.

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There is a common question in QCD, how the con"nement a!ects the structure of singularities of the gluon propagator (see, e.g., [2]). The prevailing viewpoint is that the con"nement is manifested through a 1/Q singularity in the infrared (see, e.g., [21]). There are also arguments that because of the Gribov's copies the propagator, to the contrary, vanishes in the infrared limit [22]. The AHM allows for a fresh look at the problem. Namely, one argues [20] that it is the same stringy singularities in the (Qn) variable that signal con"nement. Indeed, on one hand the AHM is a con"ning theory. On the other hand, in the London limit the propagator of the photon connecting two (con"ned) magnetic charges can be found explicitly [20]. The result is that, apart from the term (18), one should account only the e!ect of the scattering of the "eld A o! closed strings world sheets: I D  (Q)"[1/(Q#M )]X (Q)#D(Q) , IJ 4 IJ IJ

(46)



1 dp 1 nn m e\ ? @ 1R (Q)R (!p)2R . D(Q)"! 4 J@ IJ Q#M (Q ) n) (2p) p#M (p ) n) I? 4 4

(47)

Note that the expression (46) for the gluon propagator is exact in the London limit (there are no loop contributions to Eq. (46))! Moreover, the string}string correlation function in Eq. (46) is de"ned as follows: 1 1O2R" Z



DR e\1RO, .  R



Z " 

DR e\1(R) . 

(48)

R

and the string action S(R) is the same as in Eq. (22). The string}string correlator is parametrized in terms of a single function DR(Q): 1R (Q)R (!p)2R"(2pe)d(Q!p)e e Q Q DR(Q) . I? J@ I?KM J@DM K D

(49)

The behavior of the function DR(Q) in the infrared region, QP0, can be estimated as follows: C , DR(Q)" Q#m 

(50)

where m is the mass of the lightest glueball with quantum numbers of the photon, J."1\.  Collecting all the factors we get for the propagator in the axial gauge:






Q n #Q n QQ 1 I J! I J F(Q)! (d !n n )G(Q) D  " d # I J IJ I J IJ (Q ) n) (Q ) n) (Q ) n) IJ

(51)

where






1 Cm m 4  F(Q)" 1# , Q#m (Q#m )(Q#m ) 4 4 




(52)



m Cm m 4 4  G(Q)"! 1! . Q#m (Q#m )(Q#m ) 4 4 

(53)

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Thus, we do not reproduce any special behavior of the propagator at QP0. In particular, there is neither Q\ singularity nor vanishing of the propagator as QP0. The only unusual feature is the singularities in (Q ) n). To summarize, the Abelian Higgs model demonstrates breaking of the OPE at large Q at the level of 1/Q corrections due to the topological strings [4]. Also, one can derive an expression for the photon propagator which is exact in the London limit [22]. The expression demonstrates that the con"nement is manifested in the propagator rather through stringy singularities than through a particular behavior at small Q.

7. Phenomenological applications The existence of short strings in the AHM goes back to the analysis, how unique is the gauge "xing in this model [7]. Similarly, it was argued [7] that the ;(1) projection can be introduced uniquely everywhere except for the world trajectories of magnetic charges. One can also trace strings connected to the monopoles. These observations are crucial to conjecture that short strings exist in the ;(1) projection of QCD as well [4]. However, there is no direct way to evaluate the e!ective tension for the Dirac string embedded into the physical vacuum of QCD. The reason is that the topology of the gauge "xing is described now not directly in terms of the "elds appearing in the Lagrangian but rather in terms of foreign objects like eigenvalues of some operators. There exists convincing evidence that the dual Abelian Higgs model does describe the lattice gluodynamics in the infrared limit [23]. Moreover, the structure of the observed string which determines the QM Q potential at large distances is well described by the classical Landau}Ginzburg equations [24]. In particular, results of the measurements of the QQM potential on the lattice has been compared with the predictions of the (dual) AHM since a long time (see, e.g., [25] and references therein). What we can add to this analysis is the prediction for the behavior of the QQM potential at small distances [4]. Indeed, the de"nitions of the monopoles and strings relevant to the ;(1) projection apply in fact at small distances. Thus, we predict a non-vanishing slope p of the potential at rP0.  Numerically, the prediction for p depends on the ratio of m and m . It is known [23] that the  4 1 realistic case of quantum gluodynamics is close to the Bogomolny limit, m "m . In this case the 4 1 prediction for the potential at short distances is especially simple (see Eq. (44)): p "p ,   i.e. the slope does not depend on the distance r. It is amusing therefore that the lattice simulation [26] does not indicate indeed any change in the slope of the QM Q potential at all the distances available, r50.1 fm. Also, the data on the bound states of heavy quarks are much better described if one assumes that the linear piece in the potential persists at short distances as well [27]. The short strings is the "rst theoretical explanation of such a behavior of the potential at rP0 since according to the standard picture the correction to the Coulombic potential is of order r at small distances [10]. The linear piece in the potential is an example of a 1/Q correction where Q is a generic large mass parameter. In this case it is a correction to the leading, Coulomb-like piece in the potential and Q+1/r. It would be of course very important to see, whether there are other detectable

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corrections due to the short strings in QCD. Let us emphasize, however, that not all the 1/Q corrections in QCD are associated with short distances. For example, in case of DIS there are 1/Q corrections coming from the infrared region and this is perfectly consistent with the OPE. Thus, the class of theoretical objects for which an observation of the 1/Q corrections would signify going beyond the OPE is limited. One example was the potential <(r) discussed above. Other examples are the correlator functions (54)



P (Q)"i exp(iqx)10"¹+j(x), j(0),"02 (q,!Q) , H

(54)

where j(x) are various currents constructed from the quark and gluon "elds and we suppress for the moment the Lorentz indices. In case of the correlators (54) the infrared sensitive power corrections start with Q\ terms and are proportional to the gluon condensate, 10"a (G? )"02 [17]. The Q IJ di!erence from the DIS case where the 1/Q corrections come from large distances is that the matrix element of the T-product is taken over the vacuum in (54) while in case of DIS matrix elements over the nucleon are involved. We will proceed now to summarize phenomenological consequences from a heuristic assumption that short strings are imitated by a short-distance tachyonic gluon mass [28]. After our discussions of the heavy quark potential, the introduction of a tachyonic gluon mass is straightforward. Indeed, the linear term in <(r) at small r can obviously be imitated [29] by the Yukawa potential with a tachyonic gluon mass j: (2a /3)j&!p . (55) Q  As far as the potential is concerned the substitution (55) is an identity since the potential is given by a Born graph. However, the hypothesis (55) enhances the predictive power of the theory greatly since the tachyonic gluon mass can be consistently used in one-loop approximation as well. Of course, Eq. (55) may serve only for a rough estimate. First of all, the value of the slope p at short  distances is not known well because the measurements [26] were not dedicated to the short distances. Moreover, Eq. (55) assumes that the short-distance potential is due to a vector-like exchange while at large distances the pr term corresponds to a scalar exchange and there is no evidence for a change [26]. However, these reservations should not veil the fact that the gluon mass is large according to the estimate (55). And the real question is [28] whether a kind of large tachyonic gluon mass is acceptable in view of the known properties of various P (Q). H One of the basic quantities to be determined from the theory is the scale at which the parton model for the correlators (54) gets violated considerably via the power corrections. Technically, one studies usually P(M) rather than the original P (Q), where [17]: H P (M),[QL/(n!1)!](!d/dQ)LP (Q) (56) H H in the limit where both Q and n tend to in"nity so that their ratio M,Q/n remains "nite. Moreover, within the standard OPE the correlators P(M) at large M is represented as P (M)+(parton model) . H a c H 1# # H #O((ln M)\, M\) ln M/K M /!"






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where the constants a , c depend on the channel, i.e. on quantum numbers of the current j. Terms H H of order 1/ln M and M\ are associated with the "rst perturbative correction and the gluon condensate, respectively. To characterize the scale of the power corrections, one may introduce [30] the notion of M which is de"ned as the value of M at which the power corrections become,   say, 10% from the unity. The meaning of M is that at lower M the power corrections blow up.   In the o-channel, M (o-channel)&0.6 GeV (59)   which is determined by the value of the gluon condensate, 1a (G? )2 and agrees well with Q IJ independent evaluations of M from the experimental data on the e>e\ annihilation [17].   If one proceeds to other channels, in particular to the p-channel and to the 0!-gluonium channels, nothing special happens to M associated with the gluon condensate. However, it was   determined independently that the actual values of M do vary considerably in these channels   [30]: M (n-channel)51.8 GeV , (60)   M (0!-gluonium channel)515 GeV . (61)   These lower bounds on M are obtained from the coupling of the pion to the q c q current (which    in turn is a function of f and quark masses) in the pion channel, and from a low-energy theorem in L the gluonic channel. Such values of M cannot be reconciled with the the standard sum rules (59).   Now, a new term proportional to j is added to the theoretical side of P (M) which becomes H a b c (62) P(M)+(parton model) 1# H # H # H ln M M M






where c is calculable in terms of j [28]: H c +4c "(4a /3p)c "(4a /p)j . (63) L M Q    Q Phenomenologically, in the o-channel there are severe constraints on the new term c /M (see H second paper in Ref. [18]): c +!(0.03!0.07) GeV . (64) M Remarkably enough, the sign of c does correspond to a tachyonic gluon mass (if we interpret M c this way). Moreover, when interpreted in terms of j the constraint (64) does allow for a large j, M say, j"!0.5 GeV. As for for the n-channel one "nds now a new value of M associated with jO0:   M (n-channel)+4M (o-channel) (65)     which "ts nicely the Eqs. (60) and (61) above. Moreover, the sign of the correction in the n-channel is what is needed for phenomenology [30]. Fixing the value of c to bring the theoretical P (M) L L into agreement with the phenomenological input one gets j+!0.5 GeV .

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Finally, we can determine the new value of M in the scalar-gluonium channel and it turns to be   what is needed for the phenomenology, see Eq. (61). Thus, qualitatively the phenomenology with a tachyonic gluon mass which is quite large numerically stands well to a few highly non-trivial tests. Further crucial tests of the model with the tachyonic gluon mass could be furnished with measuremants of various correlators P (M) on the H lattice. The list of the predictions can be found in [28].

8. Conclusions Most applications of QCD refer to small distances since the e!ective coupling is small and one can rely on the perturbation theory. Namely, one deals with point-like quarks which in the zero approximation move freely and whose interaction is described by exchange of gluons weakly coupled to the quarks. At large distances, there are no analytical methods available since the e!ective coupling is large. One relies either on qualitative models or numerical simulations. The resulting picture is also simple. However, it is formulated in terms of di!erent objects, namely, strings stretched between quark and antiquarks. The strings carry "nite energy per unit length. As a result the potential energy between quarks grows linearly with distance and the quarks cannot be separated and observed as free particles. In this way the con"nement of quarks gets a natural explanation [8]. One of the long-standing problems in QCD is to understand better the relation between the description in terms of particle (gluon) exchange at short distances and the strings at large distances (for review and references see [31]). The hope now is to develop a dual description in terms of particles and fundamental strings [31]. These fundamental strings, unlike the ANO strings described above, have no thickness and cannot be constructed from "eld theory. In case of the four dimensional world relevant to QCD the theory of fundamental strings nowadays is rather a goal than reality. It is remarkable that an elementary string was introduced "rst more than 50 years ago by Dirac in conjunction with the notion of magnetic charges. The string may be thought of as an in"nitely thin solenoid which transports the magnetic #ux to the monopole. To avoid the possibility to observe the string via the Bohm}Aharonov e!ect one introduces the quantization condition for the magnetic charges. The Dirac string by construction does not have any energy, however, and in this respect is di!erent from the fundamental strings. In this review, we emphasized the point [3,4] that the energy of the Dirac string can be normalized to zero only once, in the perturbative vacuum. Moreover, the "eld inside the Dirac string is so strong that it pushes away all other "elds and any condensate vanishes along the Dirac string. But since formation of the condensate in vacuum is favored energetically, the destruction of the condensates along the string shifts the energy upward. The Dirac strings may end in monopoles. Therefore, evaluating the energy of a pair of magnetic charges one should take account that the Dirac string destroys the condensate along a line connecting the charges. The vanishing of the condensates is imposed now as a boundary condition on the classical equations of motion. Solving the equations one can prove that the appearance of a mathematically thin line in the boundary condition implies a term in the potential energy which grows linearly with the distances between the magnetic charges. Since the Dirac string is mathematically thin, this linear piece survives at any

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small distance and the Dirac string, when embedded into a non-trivial vacuum, does have a kind of induced "nite energy per unit length. In this sense the Dirac string behaves itself like the conjectured fundamental string. These results were established directly within the Abelian Higgs model. From the numerical studies it is known from many examples that this model in classical approximation does imitate full gluodynamics (see, e.g., [23]). Therefore, we expect that also in case of the potential energy of a quark}antiquark pair there is a stringy, or linear piece in the potential which survives at short distances [3,4]. On the other hand, if only physical, or thick ANO-type strings are relevant to QCD (as has been always assumed tacitly) then at distances much smaller than the thickness the linear piece should disappear from the potential [10]. It is remarkable therefore, that the numerical simulations of QCD do indicate linear piece in the potential at all distances studied (down to 0.1 fm) [26]. Moreover, within the same Abelian Higgs model which describes well gluodynamics at large distances one can "nd the numerical value of the linear piece at short distances as well. The prediction agrees with the measurements (which are not dedicated to short distances, however). Introduction of tachyonic gluon mass at short distances, as a manifestation of short strings in QCD, allows to extend greatly the range of phenomenological applications of the short strings [28]. So far the novel phenomenology was successful to explain old puzzles. There is a possible link of the short strings to ultraviolet renormalons (see, e.g., [18]) which demonstrate breaking of perturbative expansions in QCD at the level of the 1/Q corrections. Short strings might be a non-perturbative counterpart of the ultraviolet renormalon. Although "rst phenomenological applications look favorable for short strings, further checks are desirable and possible, mostly through measurements of various current correlators on the lattice (see [28]). It goes without saying that there are many open problems on the theoretical side with description of the topological strings at the quantum level.

References [1] L.B. Okun, Nucl. Phys. B 173 (1980) 1. [2] V.N. Gribov, Orsay lectures on con"nement. 2 (hep-ph/9404332); The theory of quark con"nement (hepph/9902279). [3] F.V. Gubarev, M.I. Polikarpov, V.I. Zakharov, Phys. Lett. B 438 (1998) 147. [4] F.V. Gubarev, M.I. Polikarpov, V.I. Zakharov, Short strings in the Abelian Higgs model (hep-th/9812030). [5] A.A. Abrikosov, ZhETF 32 (1957) 1442; H.B. Nielsen, P. Olesen, Nucl. Phys. B 61 (1973) 45. [6] P.A.M. Dirac, Proc. Roy. Soc. A 133 (1931) 60. [7] G. 't Hooft, Nucl. Phys. B 190 (1981) 455. [8] Y. Nambu, Phys. Rev. D 10 (1974) 4662; G. 't Hooft, in: A. Zichichi (Ed.), High energy physics, Proc. EPS Intern. Conf., Editrici Compositori, 1976; S. Mandelstam, Phys. Rep. 23 (1976) 245; A.M. Polyakov, Phys. Lett. B 59 (1975) 82. [9] M. Beneke, V.M. Braun, V.I. Zakharov, Phys. Rev. Lett. 73 (1994) 3058. [10] Ya.Ya. Balitskii, Nucl. Phys. B 254 (1985) 166; H.G. Dosch, Yu.A. Simonov, Phys. Lett. B 205 (1988) 339. [11] R. Akhoury, V.I. Zakharov, Phys. Lett. B 438 (1998) 165. [12] A.P. Balachandran, H. Rupertsberger, J. Schechter, Phys. Rev. D 11 (1975) 2260; T. Suzuki, Prog. Theor. Phys. 81 (1989) 752. [13] D. Zwanziger, Phys. Rev. D 3 (1971) 343; R.A. Brandt, F. Neri, D. Zwanziger, Phys. Rev. D 19 (1979) 1153. [14] P. Orland, Nucl. Phys., B 428 (1994) 221; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.I. Polikarpov, U.-J. Wiese, M.A. Zubkov, Phys. Lett., 309 B (1993) 133.

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[15] A. Jevicki, P. SenjanovicH , Phys. Rev. D 11 (1975) 860; J.W. Alcock, M.J. Bur"tt, W.N. Cottingham, Nucl. Phys. B 226 (1983) 299; J.S. Ball, A. Caticha, Phys. Rev. D 37 (1988) 524; S. Kamizawa, Y. Matsubara, H. Shiba, T. Suzuki, Nucl. Phys. B 389 (1993) 563; M. Baker, N. Brambilla, H.G. Dosch, A. Vairo, Phys. Rev. D 58 (1998) 034010. [16] M.N. Chernodub, M.I. Polikarpov, Abelian Projection and Monopoles, Lectures given at NATO advanced study institute on Con"nement, Duality and Nonperturbative Aspects of QCD, Cambridge, UK, 23 June}4 July 1997 (hep-th/9710205); A. Di Gaicomo, Evidence for dual superconductivity of QCD ground state, Workshop on quantum chromodynamics, Paris, France, 1}6 June 1998 (hep-th/9809047); G.S. Bali, The mechanism of quark con"nement, Talk given at 3rd International Conference in Quark Con"nement and Hadron Spectrum (Con"nement III), Newport News, VA, 7}12 June 1998 (hep-ph/9809351). [17] M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 1979 385, 448. [18] V.I. Zakharov, Nucl. Phys. B 385 (1992) 452; S. Narison, Phys. Lett. B 300 (1993) 293; A.I. Vainshtein, V.I. Zakharov, Phys. Rev. Lett. 73 (1994) 1207. G. Altarelli, P. Nason, G. Ridol", Z. Phys. C 68 (1995) 257; M. Neubert, Nucl. Phys. B 463 (1996) 511; S. Peris, E. de Rafael, Nucl. Phys. B 500 (1997) 325; M. Beneke, V.M. Braun, N. Kivel, Phys. Lett. B 404 (1997) 315. [19] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Fortsch. Phys. 32 (1985) 585; Nucl. Phys. B 249 (1985) 445. [20] M.N. Chernodub, M.I. Polikarpov, V.I. Zakharov, Phys. Lett. B 457 (1999) 147. [21] G.B. West, Phys. Lett. 115B (1982) 468; M. Baker, J.S. Ball, F. Zachariasen, Phys. Rev. D 37 (1988) 1036; Erratum-ibid. D 37 (1988) 3785. [22] V. N. Gribov, Nucl. Phys. B 139 (1978) 1; D. Zwanziger, Nucl. Phys. B 364 (1991) 127. [23] S. Kato, S. Kitahara, N. Nakamura, T. Suzuki, Nucl. Phys. B 520 (1998) 323; M.N. Chernodub, S. Kato, N. Nakamura, M.I. Polikarpov, T. Suzuki, Various representations of infrared e!ective lattice SU(2) Glyodynamics (hep-lat/9902013). [24] K. Schilling, G.S. Bali, C. Schlichter, Nucl. Phys. Proc. Suppl. 73 (1999) 638. [25] M. Baker, J.S. Ball, F. Zachariasen, Phys. Rev. D 56 (1997) 4400. Yu.A. Simonov, Phys. Usp. 39 (1996) 313. [26] G.S. Bali, K. Schilling, A. Wachter, Quark}anti-quark forces from SU(2) gauge theories on large lattices (heplat/9506017); Phys. Rev. D 55 (1997) 5309. [27] A.M. Badalian, V.L. Morgunov, Determination of alpha(s) (1-GeV) from the charmonium "ne structure. (hepph/9901430). [28] K.G. Chetyrkin, S. Narison, V.I. Zakharov, Short distance tachyonic gluon mass and 1/Q corrections (hepph/9811275). [29] V.I. Zakharov, Prog. Theor. Phys. (Suppl.) 131 (1998), 107; hep-ph/9802416. [30] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 191 (1981) 301. [31] A.M. Polyakov, Int. J. Mod. Phys. A 14 (1999) 645; I.R. Klebanov, From three-branes to large N gauge theories (hep-th/9901018).

Physics Reports 320 (1999) 79}106

Precision #avour physics and supersymmetry Ahmed Ali *, David London
Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
Laboratoire Rene& J.-A. Le& vesque, Universite& de Montre& al, C.P. 6128, succ. centre-ville, Montre& al, QC, Canada H3C 3J7

Abstract We review the salient features of a comparative study of the pro"le of the CKM unitarity triangle, and the resulting CP-violating phases a, b and c in B decays, in the standard model and in several variants of the minimal supersymmetric standard model (MSSM), reported recently by us. These theories are characterized by a single phase in the quark #avour mixing matrix and give rise to well-de"ned contributions in the #avour-changing-neutral-current transitions in K and B decays. We analyse the supersymmetric contributions to the mass di!erences in the B}B and B}B systems, DM and DM , respectively, and to the B B Q Q B Q CP-violating quantity "e" in K decays. Our analysis shows that the predicted ranges of b in the standard model and in MSSM models are very similar. However, precise measurements at B-factories and hadron machines may be able to distinguish these theories in terms of the other two CP-violating phases a and c.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.15.Ff; 12.15.Hh; 12.60.Jv Keywords: Quark #avour mixing; CP violation; CKM matrix; Unitarity triangle; Supersymmetry

1. Introduction In this article, written to honour the scienti"c achievements of Lev Okun, we discuss some selected topics in quark #avour physics. In particular, we review the present status of quark #avour mixing in the Standard Model (SM) and in some variants of the Minimal Supersymmetric Standard Model (MSSM). The idea is to present contrasting pro"les of quark #avour physics in these theoretical frameworks which can be tested in the next generation of experiments in #avour physics. The emphasis in this paper is on CP violating asymmetries and particle}antiparticle

* Corresponding author. E-mail address: [email protected] (A. Ali) 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 7 5 - 7

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mixings induced by weak interactions. These topics are close to Lev Okun's own scienti"c research. In fact, the possibility of observing violation of CP-invariance in heavy particle decays was proposed in an important paper by Okun, Pontecorvo and Zakharov in 1975, just after the discovery of the charmed hadrons [1]. To be speci"c, these authors studied the consequences of DD pair production, subsequent D}D mixing, and CP violation for the "nal states involving same-sign l!l! and opposite-sign l>l\ dileptons. In particular, as a measure of D}D mixing they proposed the measurement of the ratio of the same sign to the inclusive dilepton events, N>>#N\ \ 1 (C !C )#4(DM ) 1 * 1* , R , " (1) " N>\#N\>#N>>#N\ \ 2 (C #C )#4(DM ) 1 * 1* where C and C are the widths of the (short-lived) D and (long-lived) D mesons, respectively, and 1 * 1 * DM is their mass di!erence. They also suggested the measurement of the charge asymmetry 1* d ,(N>>!N\ \)/(N>>#N\ \)K4Re e , (2) " " as a measure of CP violation. Here, e is the CP-violating parameter in the wave functions of " D and D mesons, analogous to the corresponding parameter e in the K-system [2] 1 * ) D &D #e D , D &D #e D , (3) 1  "  *  "  where D and D are the pure CP states. So far, neither R nor d have been measured [2]. In fact,   " " in the Cabibbo}Kobayashi}Maskawa (CKM) theory of quark #avour mixing [3], which is now an integral part of the SM, no measurable e!ects are foreseen for either of the ratios R and d , due to " " the experimentally established hierarchies in the quark mass spectrum and the CKM matrix elements. Typically, one has in the SM [4], DM /(C #C )KO(10\), (C !C )/(C #C );1 , (4) 1* 1 * 1 * 1 * with d completely negligible. By virtue of this, the quantities R and d have come to be " " " recognized as useful tools to search for physics beyond the SM [5,6]. The OPZ formulae also apply to the time-integrated e!ects of mixing and CP violation in the B}B and B}B systems, and they were used in the analysis [7] of the UA1 data on inclusive Q Q B B dilepton production [8]. Calling the corresponding mixing measures R B and R Q, respectively, present experiments yield R BK0.17 and R QK1/2 [2]. These measurements are consistent with the more precise time-dependent measurements, yielding DM B"0.471$0.016 (ps)\ [9] and the 95% CL upper limit DM Q'12.4 (ps)\ [10]. However, the corresponding CP-violating charge asymmetries d B and d Q in the two neutral B-meson systems have not been measured, with the present best experimental limit being d B"0.002$0.007$0.003 from the OPAL collaboration [2] and no useful limit for the quantity d Q. These charge asymmetries are expected to be very small in the SM, re#ecting essentially that the width and mass di!erences DC and DM in the B}B and B B B}B complexes are relatively real. Typical estimates in the SM are in the range d B"O(10\) Q Q and d Q"O(10\). Hence, like d , they are of interest in the context of physics beyond the SM " [11,12]. With the advent of B factories and HERA-B, one expects that a large number of CP asymmetries in partial decay rates of B hadrons and rare B decays will become accessible to experimental and

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theoretical studies. Of particular interest in this context are the #avour-changing neutral-current (FCNC) processes which at the quark level can be thought of as taking place through induced bPd and bPs transitions. In terms of actual laboratory measurements, these FCNC processes will lead to DB"1, DQ"0 decays such as BP(X ,X )l>l\ and BP(X ,X )c, where X (X ) Q B Q B Q B represents an inclusive hadronic state with an overall quantum number S"$1(0), as well as their exclusive decay counterparts, such as BP(K, KH, p, o,2)l>l\ and BP(KH, o, u,2)c. Of these, the decays BPX c and BPKHc have already been measured [2]. The DB"2, Q DQ"0 transitions lead to B}B and B}B mixings, brie#y discussed above. Likewise, non-trivial Q Q B B bounds have been put on the CP-violating phase sin 2b from the time-dependent rate asymmetry in the decays B/BPJ/tK [13]. In K decays, the long sought after e!ect involving direct CP Q violation has been "nally established through the measurement of the ratio e/e [14,15]. This and the measurement of the CP-violating quantity "e" in K Ppp decays [2] represent the sPd FCNC * transitions. Likewise, there exists great interest in the studies of FCNC rare K decays such as K>Pp>ll and K Ppll [16], of which a single event has been measured in the former decay * mode [17]. The FCNC processes and CP asymmetries in K and B decays provide stringent tests of the SM. The short-distance contributions to these transitions are dominated by the top quark, and hence these decays and asymmetries provide information on the weak mixing angles and phases in the matrix elements < , < and < of the CKM matrix. Some information on the last of these matrix RB RQ R@ elements is also available from the direct production and decay of the top quarks at the Fermilab Tevatron [18]. The measurement of < will become quite precise at the LHC and linear colliders. R@ Moreover, with advances in determining the (quark) #avour of a hadronic jet, one also anticipates being able to measure the matrix element < (and possibly also < ). RQ RB We shall concentrate here on the analysis of the data at hand and in forthcoming experiments which will enable us to test precisely the unitarity of the CKM matrix. These tests will be carried out in the context of the Unitarity Triangles (UT). The sides of UTs will be measured in K and B decays and the angles of these UTs will be measured by CP asymmetries. Consistency of a theory, such as the SM, requires that the two sets of independent measurements yield the same values of the CKM parameters, or, equivalently the CP-violating phases a, b and c. We are tacitly assuming that there is only one CP-violating phase in weak interactions. This is the case in the SM but also in a number of variants of Supersymmetric Models, which, however, do have additional contributions to the FCNC amplitudes. In fact, it is the possible e!ect of these additional contributions which will be tested. In this case, quantitative predictions can be made which, in principle, allow experiments to discriminate among these theories [19]. As we shall see, the case for distinguishing the SM and the MSSM rests on the experimental and theoretical precision that can achieved in various input quantities. Of course, there are many other theoretical scenarios in which deviations from the pattern of #avour violation in the SM are not minimal. For example, in the context of supersymmetric models, one may have non-diagonal quark}squark}gluino couplings, which also contain additional phases. These can contribute signi"cantly to the magnitude and phase of bPd, bPs and sPd transitions, which would then violate the SM #avour-violation pattern rather drastically. In this case it is easier to proclaim large deviations from the SM but harder to make quantitative predictions. This paper is organized as follows. In Section 2, we discuss the pro"le of the unitarity triangle within the SM. We describe the input data used in the "ts and present the allowed region in o}g

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space, as well as the presently allowed ranges for the CP angles a, b and c. We also discuss the "ts in the superweak scenario, which di!ers from the SM "ts in that we no longer use the constraint from the CP-violating quantity "e". The superweak "ts are not favoured by the data and we quantify this in terms of the 95% CL exclusion contours. We turn to supersymmetric models in Section 3. We review several variants of the MSSM, in which the new phases are essentially zero. Restricting ourselves to #avour violation in charged-current transitions, we include the e!ects of charged Higgses H!, a light scalar top quark (assumed here right handed as suggested by the precision electroweak "ts) and chargino s!. In this scenario, which covers an important part of the SUSY parameter space, the SUSY contributions to K}K, B}B and B}B mixing are of the same form B B Q Q and can be characterized by a single parameter f. Including the NLO corrections in such models, we compare the pro"le of the unitarity triangle in SUSY models, for various values of f, with that of the SM. We conclude in Section 4.

2. Unitarity triangle: SM pro5le Within the SM, CP violation is due to the presence of a nonzero complex phase in the quark mixing matrix < [3]. A particularly useful parametrization of the CKM matrix, due to Wolfenstein [20], follows from the observation that the elements of this matrix exhibit a hierarchy in terms of j, the Cabibbo angle. In this parametrization the CKM matrix can be written approximately as






1!j j Aj(o!ig)  .
(5)

The allowed region in o}g space can be elegantly displayed using the so-called unitarity triangle (UT). The unitarity of the CKM matrix leads to the following relation: <
(6)

Using the form of the CKM matrix in Eq. (5), this can be recast as
(7)

which is a triangle relation in the complex plane (i.e. o}g space), illustrated in Fig. 1. Thus, allowed values of o and g translate into allowed shapes of the unitarity triangle. Constraints on o and g come from a variety of sources. Of the quantities shown in Fig. 1, "< " and A@ "< " can be extracted from semileptonic B decays, while "< " is probed in B}B mixing. The S@ RB B B interior CP-violating angles a, b and c can be measured through CP asymmetries in B decays [21]. Additional constraints come from CP violation in the kaon system ("e"), as well as B}B mixing. As Q Q the constraints that are expected to come from the rare B and K decays mentioned earlier are not of interest for the CKM phenomenology at present, we shall not include them in our "ts.

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Fig. 1. The unitarity triangle. The angles a, b and c can be measured via CP violation in the B system.

2.1. Input data The CKM matrix as parametrized in Eq. (5) depends on four parameters: j, A, o and g. We summarize below the experimental and theoretical data which constrain these CKM parameters. 2.1.1. "< ", "< " and "< /< " SQ A@ S@ A@ We recall that "< " has been extracted with good accuracy from KPpel and hyperon decays SQ [2]: "< ""j"0.2196$0.0023. The determination of "< " is based on the combined analysis of the SQ A@ inclusive and exclusive B decays: "< ""0.0395$0.0017 [2], yielding A"0.819$0.035. The A@ knowledge of the CKM matrix element ratio "< /< " is based on the analysis of the end-point S@ A@ lepton energy spectrum in semileptonic decays BPX lll and the measurement of the exclusive S semileptonic decays BP(p,o)lll. Present measurements in both the inclusive and exclusive modes are compatible with "< /< ""0.093$0.014 [10]. This gives (o#g"0.423$0.064. S@ A@ 2.1.2. "e", BK ) The experimental value of "e" is [2] "e""(2.280$0.013);10\ .

(8)

In the standard model, "e" is essentially proportional to the imaginary part of the box diagram for K}K mixing and is given by [22] G f  M M (9) "e"" $ ) ) 5BK (Ajg)(y +g( f (y ,y )!g( ,#g( y f (y )Aj(1!o)) , A AR  A R AA RR R  R 6(2pDM ) ) where y ,m/M , and the functions f and f are the Inami}Lim functions [23]. Here, the g( are G G 5   G QCD correction factors, calculated at next-to-leading order: (g( ) [24], (g( ) [25] and (g( ) [26]. The AA RR AR theoretical uncertainty in the expression for "e" is in the renormalization-scale independent parameter BK , which represents our ignorance of the matrix element 1K"(dM cI(1!c )s)"K2. Recent )  calculations of BK using lattice QCD methods are summarized at the 1998 summer conferences by ) Draper [27] and Sharpe [28], yielding BK "0.94$0.15 . )

(10)

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2.1.3. DM , fBBK B B The present world average for DM is [9] B DM "0.471$0.016 (ps)\ . B

(11)

The mass di!erence DM is calculated from the B}B box diagram, dominated by t-quark B B B exchange: G DM " $ M M ( f BBK B)g( y f (y )"
(12)

where, using Eq. (5), "
(13)

2.1.4. DM , f QBK Q Q The B}B box diagram is again dominated by t-quark exchange, and the mass di!erence Q Q between the mass eigenstates DM is given by a formula analogous to that of Eq. (12): Q (14) DM "(G/6p)M M Q( f QBK Q)g( Qy f (y )"
 

Re(e/e)"(28.0$3.0 (stat)$2.6 (syst)$1.0 (MCstat));10\ ,

(17)

in agreement with the earlier measurement by the CERN experiment NA31 [15], which reported a value of (23$6.5);10\ for the same quantity. The present world average is Re(e/e)" (21.8$3.0);10\. This combined result excludes the superweak model [30] by more than 7p.

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A great deal of theoretical e!ort has gone into calculating this quantity at next-to-leading order accuracy in the SM [31}33]. The result of this calculation can be summarized in the following form due to Buras and Silvestrini [34]: Re(e/e)"Im j [!1.35#R (1.1"r"B#(1.0!0.67"r")B)] . (18) R Q 8  8  Here Im j "Im <





2g(1!o) . sin(2b)" (1!o)#g

(21)

Thus, a measurement of sin 2b would put a strong contraint on the parameters o and g. However, the CDF measurement gives [13] sin 2b"0.79>  , (22) \  or sin 2b'0 at 93% CL. This constraint is quite weak * the indirect measurements already constrain 0.524sin 2b40.94 at the 95% CL in the SM [19]. (The CKM "ts reported recently in the literature [10,37,38] yield similar ranges.) In light of this, this measurement is not included in the "ts. The data used in the CKM "ts are summarized in Table 1.

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Table 1 Data and theoretical input used in the CKM "ts Parameter

Value

j "< " A@ "< /< " S@ A@ "e" DM B DM Q m (m (pole)) R R m (m (pole)) g( A A g( AA g( AR g( RR BK ) f B(BK B

0.2196 0.0395$0.0017 0.093$0.014 (2.280$0.013);10\ (0.471$0.016) (ps)\ '12.4 (ps)\ (165$5) GeV 1.25$0.05 GeV 0.55 1.38$0.53 0.47$0.04 0.57 0.94$0.15 215$40 MeV

m Q

1.14$0.06

2.2. SM xts In the "t presented here [19], ten parameters are allowed to vary: o, g, A, m , m , g , g , f B(BK B, R A AA AR BK , and m . The DM constraint is included using the amplitude method [39]. The rest of the ) Q Q parameters are "xed to their central values. The allowed (95% CL) o}g region is shown in Fig. 2. The best-"t values of the CKM parameters are: j"0.2196 ("xed), A"0.817, o"0.196, g"0.37 .

(23)

The `best-"ta values of the CKM matrix elements are as follows (Note that we have rounded all elements except < and < to the nearest 0.005): S@ RB




0.975




0.220

0.002!0.003i

0.975

0.040

0.007!0.003i !0.040 1

.

(24)

Now, turning to the ratios of CKM matrix elements, which one comes across in the CKM phenomenology, the `best-"ta values are: "< /< ""0.19, "< /< ""2.12, RB RQ RB S@ The 95% CL ranges are:

"< /< ""0.091 . S@ A@

0.154"< /< "40.24, 1.304"< /< "43.64, 0.064"< /< "40.125 . RB RQ RB S@ S@ A@

(25)

(26)

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Fig. 2. Allowed region in o}g space in the SM, from a "t to the ten parameters discussed in the text and given in Table 1. The limit on DM is included using the amplitude method [39]. The theoretical errors on f B(BK B, BK and m are treated Q ) Q as Gaussian. The solid line represents the region with s"s #6 corresponding to the 95% CL region. The triangle

 shows the best "t. (From Ref. [19].)

With the above "ts, the `best-"ta value of DM is DM "16.6 (ps)\. The corresponding 95% CL Q Q allowed range is 12.4 (ps)\4DM 427.9 (ps)\ . (27) Q The CP angles a, b and c can be measured in CP-violating rate asymmetries in B decays. These angles can be expressed in terms of o and g. Thus, di!erent shapes of the unitarity triangle are equivalent to di!erent values of the CP angles. Referring to Fig. 2, we note that the preferred (central) values of these angles are (a,b,c)"(933,253,623). The allowed ranges at 95% CL are 6534a41233, 1634b4353, 3634c4973 .

(28)

These ranges are similar to the ones obtained in [37,38], but not identical as the input parameters di!er. Of course, the values of a, b and c are correlated, i.e. they are not all allowed simultaneously. We illustrate these correlations in Figs. 3 and 4. Fig. 3 shows the allowed region in sin 2a}sin 2b space allowed by the data. And Fig. 4 shows the allowed (correlated) values of the CP angles a and c. This correlation is roughly linear, due to the relatively small allowed range of b (Eq. (28)). 2.3. CKM xts in superweak theories As we mentioned earlier, superweak theories of CP violation are now ruled out by the measurements of the ratio e/e [14,15]. We show in this section that a non-trivial constraint on the CKM phase g also results from the present data leaving out the information on "e" (we have not included the measurement of e/e in the analysis either, as discussed in the context of our SM-based

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Fig. 3. Allowed 95% CL region of the CP-violating quantities sin 2a and sin 2b in the SM, from a "t to the data given in Table 1. (From Ref. [19].)

Fig. 4. Allowed 95% CL region of the CP-violating quantities a and c in the SM, from a "t to the data given in Table 1. (From Ref. [19].)

"ts presented earlier). The input parameters for this "t are given in Table 1, except that now we leave out "e" and BK from the analysis. Thus, we have one data input less compared to the SM-"t ) and only nine parameters to "t (compared to ten in the SM-case). The most sensitive theoretical parameter in the "ts is now f B(BK B. To show the dependence of the allowed CKM-parameter space on this quantity, we "x its value in performing the "ts and vary it in the range 170 MeV4f B(BK B4280 MeV. The results for the allowed 95% CL contour are shown in Fig. 5 for the six values f B(BK B"190,210,220,240,260 and 280 MeV. The resulting unitarity triangle for the choice f B(BK B"170 MeV is very similar to one shown for f B(BK B"190 MeV and hence we do not display it. The triangle drawn in each case is the best-"t solution. From these "gures we see that the case g"0 (superweak model) is ruled out for all values

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89

of f B(BK B in the Lattice-QCD range f B(BK B"215$40 MeV. Only for very high values of f B(BK B, illustrated here by the case f B(BK B"280 MeV, is the superweak theory still compatible with data. Restricting to the range 190 MeV4f B(BK B4240 MeV, given by the upper four plots in Fig. 5, we see that at 95% CL the CKM-phase g is determined to lie in the range 0.204g40.55.

Fig. 5. Allowed regions in o}g space in the Superweak theories obtained by leaving out the constraint from "e", and performing a "t to the remaining parameters given in Table 1. The limit on DM is included using the amplitude method Q [39]. The input values for f B(BK B are shown on top of the individual "gures. The solid lines represent the region with s"s #6 corresponding to the 95% CL region. The triangles show the best "ts.



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Fig. 6. Allowed regions in o}g space in the Superweak theories obtained by leaving out the constraint from "e", and performing a "t to the remaining parameters given in Table 1, assuming that theoretical errors are Gaussian-distributed. The limit on DM is included using the amplitude method [39]. The input values for f B(BK B used in the "ts are shown on Q top of the individual "gures. The solid lines represent the region with s"s #6 corresponding to the 95% CL region.

 The triangles show the best "ts.

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91

This can also be seen in Fig. 6 (uppermost of the three curves), where we show the resulting 95% CL allowed contour using f B(BK B"215$25 MeV, but assuming that the errors are Gaussian distributed. A comparison of the allowed (o}g)-contours in Figs. 5 and 6 also shows that the speci"c distribution assumed for the theoretical error is not crucial. Thus, with the input f B(BK B"215$25 MeV, present data predict a value of g well within a factor 3. However, the assumption on the error of f B(BK B does play a signi"cant role in determining the allowed range of g. For example, using f B(BK B"215$40 MeV as input, the 95% CL allowed contour comes close to the o-axis, making the superweak value g"0 just barely incompatible with data. This is shown in Fig. 6 (second of the three curves shown here). The superweak theory becomes compatible with the available data if the theoretical error on f B(BK B is further increased to $60 MeV. The resulting contour for f B(BK B"215$60 MeV is displayed in Fig. 6 (lowest of the three curves). We conclude that with the present theoretical knowledge f B(BK B"215$40 MeV, the superweak case is ruled out at 95% CL from the CKM "ts, though the value of g is not determined precisely.

3. Unitarity triangle: A SUSY pro5le In this section we examine the pro"le of the unitarity triangle in supersymmetric (SUSY) theories. The most general models contain a number of unconstrained phases and so are not su$ciently predictive to perform such an analysis. However, there is a class of SUSY models in which these phases are constrained to be approximately zero, which greatly increases the predictivity. In the following subsections, we discuss aspects of more general SUSY theories, as well as the details of that class of theories whose e!ects on the unitarity triangle can be directly analyzed. 3.1. Flavour violation in SUSY models * overview We begin with a brief review of #avour violation in the minimal supersymmetric standard model (MSSM). The low energy e!ective theory in the MSSM can be speci"ed in terms of the chiral super"elds for the three generations of quarks (Q , ;A, and DA) and leptons (¸ and EA), chiral super"elds for G G G G G two Higgs doublets (H and H ), and vector super"elds for the gauge group   S;(3) ;S;(2) ;;(1) [40]. The superpotential is given by ! ' 7 = "f GHQ D H #f GHQ ; H #f GHE ¸ H #kH H . +11+ " G H  3 G H  * G H   

(29)

The indices i, j"1,2,3 are generation indices and f GH, f GH, f GH are Yukawa coupling matrices in the " 3 * generation space. A general form of the soft SUSY-breaking term is given by !¸

"(m )G q q RH#(m )G dI dI RH#(m )G u u RH#(m)G e e RH  /H G "H G 3H G #H G #(m)G lI lI RH#DhR h #DhR h !(Bkh h #h.c.) *H G        

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#(AGHq dI h #AGHq u h #AGHe lI h #h.c.) " G H  3 G H  * G H  M M M BI BI # = # I = I # GI GI #h.c. ,




2

2

2



(30)

where q , u , dI , lI , e , h and h are scalar components of the super"elds Q , ; , D , ¸ , E , H and G G G G G   G G G G G  H , respectively, and BI , = I and GI are the ;(1), S;(2) and S;(3) gauge fermions, respectively. The  SUSY-breaking parameters (m )G , with m "m ,m ,m ,m and the trilinear scalar couplings AGH,AGH $H $ " 3 * # " 3 and AGH are 3;3 matrices in the #avour space. It is obvious that supersymmetric theories have an * incredibly complicated #avour structure, resulting in a large number of a priori unknown mixing angles, which cannot be determined theoretically. Present measurements and limits on the FCNC processes do provide some constraints on these mixing angles [41]. We shall not follow this route here. Alternatively, one could put restrictions on the soft SUSY-breaking (SSB) terms. The ones most discussed in the literature are those which "nd their rationale in supergravity (SUGRA) models, in which it is assumed that the SSB terms have universal structures at the Planck scale, following from the assumption that the hidden sector of N"1 SUGRA theory is #avour blind. This results in the universal scalar mass, m , with (m )G "(m)G "2"mdG ; D"D"D, universal A-terms,  /H #H  H    AGH"f GHAm , etc., and universal gaugino masses M , de"ned as M "M "M "M . These " "  E    E universal structures are required in order to suppress FCNC processes. The scenario with the additional constraint m "D is called the minimal SUGRA model. In other theoretical scenarios,   it is not necessary to invoke universal SSB terms. In order to make testable predictions it is su$cient to restrict all #avour violations in the charged-current sector, which are determined by the known CKM angles [42]. We shall be mostly dealing here with this scenario, known as minimal yavour violation, as well as SUGRA-type scenarios. Even in these restricted scenarios, one is confronted with the complex phases residing in the = and ¸ part of the supersymmetric lagrangian. In general, MSSM models have three +11+  physical phases, apart from the QCD vacuum parameter hM which we shall take to be zero. The /!" three phases are: (i) the CKM phase represented here by the Wolfenstein parameter g, (ii) the phase h "arg(A), and (iii) the phase h "arg(k) [43]. The last two phases are peculiar to SUSY models  I and their e!ects must be taken into account in a general supersymmetric framework. In particular, the CP-violating asymmetries which result from the interference between mixing and decay amplitudes can produce non-standard e!ects. Concentrating here on the DB"2 amplitudes, two new phases h and h arise, which can be parametrized as follows [44]: B Q 1B "H1317"BM 2 1 BQ  BQ , (31) h " arg BQ 2 1B "H1+"BM 2 BQ  BQ where H1317 is the e!ective Hamiltonian including both the SM degrees of freedom and the SUSY contributions. Thus, CP-violating asymmetries in B decays would involve not only the phases a, b and c, de"ned previously, but additionally h or h . In other words, the SUSY contributions to the B Q real parts of M (B ) and M (B ) are no longer proportional to the CKM matrix elements <





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However, the experimental upper limits on the electric dipole moments (EDM) of the neutron and electron [2] do provide constraints on the phases h and h [45]. In SUGRA models with I  a priori complex parameters A and k, the phase h is strongly bounded with h (0.01p [46]. The I I phase h can be of O(1) in the small h region, as far as the EDMs are concerned. In both the  I DS"2 and DB"2 transitions, and for low-to-moderate values of tan t, it has been shown that h does not change the phase of either the matrix element M (K) [43] or of M (B) [46]. Hence, in    SUGRA models, arg M (B)" "arg M (B)" "arg(m), where m "
 In supersymmetric jargon, the quantity tan b is used to de"ne the ratio of the two vacuum expectation values (vevs) tan b,v /v , where v (v ) is the vev of the Higgs "eld which couples exclusively to down-type (up-type) quarks and leptons. S B B S (See, for example, the review by Haber in Ref. [2].) However, in discussing #avour physics, the symbol b is traditionally reserved for one of the angles of the unitarity triangle. To avoid confusion, we will call the ratio of the vevs tan t.

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These features lead to an enormous simpli"cation in the #avour structure of the SUSY contributions to #avour-changing processes. In particular, SUSY contributions to the transitions bPs, bPd and sPd are proportional to the CKM factors, <
(35)

Im A (K)"Ajg(y +g( f (y ,y )!g( ,#g( y f (y )Aj(1!o)) . (36) 1+ A AR  A R AA RR R  R The expressions for Im A !(K), Im A !(B) and Im A  (B) can be found in Refs. [47,52,56]. & Q E For the analysis reported here, we follow the scenario called minimal yavour violation in Ref. [42]. In this class of supersymmetric theories, apart from the SM degrees of freedom, only charged Higgses, charginos and a light stop (assumed to be right-handed) contribute, with all other supersymmetric particles integrated out. This scenario is e!ectively implemented in a class of SUGRA models (both minimal and non-minimal) and gauge-mediated models [57], in which the "rst two squark generations are heavy and the contribution from the intermediate gluino-squark states is small [46,52}55].

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For these models, the next-to-leading-order (NLO) corrections for DM , DM and "e" can be B Q found in Ref. [58]. Moreover, the branching ratio B(BPX c) has been calculated in Ref. [42]. We Q make use of this information and quantitatively examine the unitarity triangle, CP-violating asymmetries and their correlations for this class of supersymmetric theories. The phenomenological pro"les of the unitarity triangle and CP phases for the SM and this class of supersymmetric models can thus be meaningfully compared. Given the high precision on the phases a, b and c expected from experiments at B-factories and hadron colliders, a quantitative comparison of this kind could provide a means of discriminating between the SM and this class of MSSM's. The NLO QCD-corrected e!ective Hamiltonian for DB"2 transitions in the minimal #avour violation SUSY framework can be expressed as follows [58]: G H " $ (<
(37)

where the NLO QCD correction factor g(

(B) is given by [58]: 1 a (m ) D g( (B)"a (m )c/(2b ) 1# Q 5 #Z D , 1 Q 5 LD L 4p S








(38)

in which n is the number of active quark #avours (here n "5), the quantity Z D is de"ned below, D D L are the lowest order perturbative QCD b-function and the anomalous dimension, and c and b LD respectively. The operator O "O is the one which is present in the SM, previously de"ned in the **  discussion following Eq. (32). The explicit expression for the function S can be obtained from Ref. [56] and for D it is given in Ref. [58], where it is derived in the NDR (naive dimensional regularization) scheme using MS-renormalization. The Hamiltonian given above for B}B mixing leads to the mass di!erence B B G (39) DM " $ (<








    a (M ) A @  a (m ) A@ Q @ Q 5   g( (K)"a (m )A@  Q A a (m ) a (m ) Q A Q @ a (m ) a (m ) a (M ) D #Z ; 1# Q A (Z !Z )# Q @ (Z !Z )# Q 5      S 4p 4p 4p








(40)

.

(41)

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Here c c Z D" LD ! , (42)  b L LD 2b 2b LD LD and the quantities entering in Eqs. (38) and (41) are the coe$cients of the well-known beta function and anomalous dimensions in QCD: The ratio






g( (B)(NLO) a (M ) D 1 "1# Q 5 #Z , (43)  g( (B)(LO) S 4p 1 is worked out numerically in Ref. [58] as a function of the supersymmetric parameters (chargino mass m , mass of the lighter of the two stops m I 0, and the mixing angle in the stop sector). This R Q ratio is remarkably stable against variations in the mentioned parameters and is found numerically to be about 0.89. Since in the LO approximation the QCD correction factor g( (B)(LO) is the same 1 in the SM and SUSY, the QCD correction factor g( (B)(NLO) entering in the expressions for DM 1 B and DM in the MSSM is found to be g( (B)(NLO)"0.51 in the MS-scheme. This is to be Q 1 compared with the corresponding quantity g( "0.55 in the SM. Thus, NLO corrections in DM B (and DM ) are similar in the SM and MSSM, but not identical. Q The expression for g( (K)(NLO)/g( (K)(LO) can be expressed in terms of the ratio 1 1 g( (B)(NLO)/g( (B)(LO) given above and the #avour-dependent matching factors Z D: 1 1 L g( (K)(NLO) g( (B)(NLO) a (m ) a (m ) 1 " 1 # Q A (Z !Z )# Q @ (Z !Z )     g( (K)(LO) g( (B)(LO) 4p 4p 1 1 K0.884 , (44) where we have used the numerical value g( (B)(NLO)/g( (B)(LO)"0.89 calculated by Krauss and 1 1 So! [58], along with a (m )"0.34 and a (m )"0.22. Using the expression for the quantity Q A Q @ g( (K)(LO), which is given by the prefactor multiplying the square bracket in Eq. (41), one gets 1 g( (K)(NLO)"0.53 in the MS-scheme. This is to be compared with the corresponding QCD 1 correction factor in the SM, g( "0.57, given in Table 1. Thus the two NLO factors are again very RR similar but not identical. Following the above discussion, the SUSY contributions to DM , DM and "e" in supersymmetric B Q theories are incorporated in our analysis in a simple form: DM "DM (SM)[1#f (m ! ,m I 0,m !,tan t, )] , B B B Q R & DM "DM (SM)[1#f (m ! ,m I 0,m !,tan t, )] , Q Q Q Q R & Gf M M (45) "e"" $ ) ) 5 BK (Ajg)(y +g( f (y ,y )!g( , ) A AR  A R AA 6(2pDM ) #g( y f (y )[1#f (m ! ,m I ,m !,tan t, )]Aj(1!o)) . RR R  R C Q R & Here, is the ¸R-mixing angle in the stop sector. The quantities f , f and f can be expressed as B Q C g( (B) g( (K) f "f " 1 RDB(S), f " 1 RDB(S) , (46) B Q C g( g( RR

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where RDB(S) is de"ned as DM (SUSY) S B RDB(S), (LO)" . (47) DM (SM) y f (y ) B R R The functions f , i"d,s,e are all positive de"nite, i.e. the supersymmetric contributions add G constructively to the SM contributions in the entire allowed supersymmetric parameter space. We "nd that the two QCD correction factors appearing in Eq. (46) are numerically very close to one another, with g( (B)/g( Kg( (K)/g( "0.93. Thus, to an excellent approximation, one has 1 1 RR f "f "f ,f. B Q C How big can f be? This quantity is a function of the masses of the top squark, chargino and the charged Higgs, m I 0, m  ! and m !, respectively, as well as of tan t. The maximum allowed value of Q & R f depends on the model (minimal SUGRA, non-minimal SUGRA, MSSM with constraints from EDMs, etc.). We have numerically calculated the quantity f by varying the SUSY parameters

, m I 0, m , m ! and tan t. Using, for the sake of illustration, m ! "m I 0"m !"100 GeV, Q & Q R & R m ! "400 GeV and tan t"2, and all other supersymmetric masses much heavier, of O(1) TeV, we Q "nd that the quantity f varies in the range: 0.44f40.8

for " "4p/4 ,

(48)

with the maximum value of f being at "0. This is shown in Fig. 7, where we have plotted the function f against (upper "gure), and against m ! , m I 0 and m ! (lower "gure), varying one R & Q parameter at a time and holding the others "xed to their stated values given above. These parametric values are allowed by the constraints from the NLO analysis of the decay BPX #c Q reported in Ref. [42], as well as from direct searches of the supersymmetric particles [2]. The allowed value of f decreases as m I 0, m  and m ! increase, though the dependence of f on m ! is Q & & R rather mild due to the compensating e!ect of the H! and chargino contributions in the MSSM, as observed in Ref. [42]. Likewise, the allowed range of f is reduced as tan t increases, as shown in Fig. 8 for tan t"4, in which case one has 0.154f40.42 for " "4p/4. This sets the size of f allowed by the present constraints in the minimal #avour violation version of the MSSM. If additional constraints on the supersymmetry breaking parameters are imposed, as is the case in the minimal and non-minimal versions of the SUGRA models, then the allowed values of f will be further restricted. A complete NLO analysis of f would require a monte-carlo approach implementing all the experimental and theoretical constraints (such as the SUGRA-type mass relations). In particular, the NLO correlation between B(BPX c) and f has to be studied in an Q analogous fashion, as has been done, for example, in Refs. [54,55] with the leading order SUSY e!ects. In this paper we adopt an approximate method to constrain f in SUGRA-type models. We take the maximum allowed values of the quantity RDB(S), de"ned earlier, from the existing LO analysis of the same and obtain f by using Eq. (46). For the sake of de"niteness, we use the updated work of Goto et al. [54,55]. From the published results we conclude that typically f can be as large as 0.45 in non-minimal SUGRA models for low tan t (typically tan t"2) [54], and approximately half of this value in minimal SUGRA models [46,53,54]. Relaxing the SUGRA mass constraints, admitting complex values of A and k but incorporating the EDM constraints, and imposing the constraints mentioned

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Fig. 7. Dependence of the supersymmetric function f on the LR-mixing angle in the stop sector, (upper "gure), and on m  ! , m ! and m I 0 (lower "gure), for tan t"2; values of the other supersymmetric parameters are stated in the text. Q & R

above, Baek and Ko [50] "nd that f could be as large as f"0.75. In all cases, the value of f decreases with increasing tan t or increasing m  ! and m I 0, as noted above. Q R 3.3. SUSY xts For the SUSY "ts, we use the same program as for the SM "ts, except that the theoretical expressions for DM , DM and "e" are modi"ed as in Eq. (45). We compare the "ts for four B Q representative values of the SUSY function f (0,0.2,0.4 and 0.75) which are typical of the SM, minimal SUGRA models, non-minimal SUGRA models, and non-SUGRA models with EDM constraints, respectively. The allowed 95% CL regions for these four values of f are all plotted in Fig. 11. As is clear from this "gure, there is still a considerable overlap between the f"0 (SM) and f"0.75 regions. However, there are also regions allowed for one value of f which are excluded for another value. Thus a su$ciently precise determination of the unitarity triangle might be able to exclude certain values of f (including the SM, f"0). From Fig. 11 it is clear that a measurement of the CP angle b will not distinguish among the various values of f: even with the naked eye it is evident that the allowed range for b is roughly the same for all models. Rather, it is the measurement of c or a which has the potential to rule out certain values of f. As f increases, the allowed region moves slightly down and towards the right in

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Fig. 8. Dependence of the supersymmetric function f on the ¸R-mixing angle in the stop sector, (upper "gure), and on m  ! , m ! and m I 0 (lower "gure), for tan t"4; values of the other supersymmetric parameters are stated in the text. Q & R

the o}g plane, corresponding to smaller values of c (or equivalently, larger values of a). We illustrate this in Table 2, where we present the allowed ranges of a, b and c, as well as their central values (corresponding to the preferred values of o and g), for each of the four values of f. From this table, we see that the allowed range of b is largely insensitive to the model. Conversely, the allowed values of a and c do depend somewhat strongly on the chosen value of f. Note, however, that one is not guaranteed to be able to distinguish among the various models: as mentioned above, there is still signi"cant overlap among all four models. Thus, depending on what values of a and c are obtained, we may or may not be able to rule out certain values of f. One point which is worth emphasizing is the correlation of c with f. This study clearly shows that large values of f require smaller values of c. The reason that this is important is as follows. The allowed range of c for a particular value of f is obtained from a "t to all CKM data, even those measurements which are una!ected by the presence of supersymmetry. Now, the size of c indirectly a!ects the branching ratio for BPX c: a larger value of c corresponds to a smaller value of "< " Q RQ through CKM unitarity. And this branching ratio is among the experimental data used to bound SUSY parameters and calculate the allowed range of f. Therefore, the above c}f correlation indirectly a!ects the allowed values of f in a particular SUSY model, and thus must be taken into account in studies which examine the range of f. For example, it is often the case that larger values

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Table 2 Allowed 95% CL ranges for the CP phases a, b and c, as well as their central values, from the CKM "ts in the SM ( f"0) and supersymmetric theories, characterized by the parameter f de"ned in the text f

a

b

c

(a,b,c) 

f"0 (SM) f"0.2 f"0.4 f"0.75

65}1233 70}1293 75}1343 86}1413

16}353 16}343 15}343 13}333

36}973 32}903 28}853 23}733

(933,253,623) (1023,243,543) (1103,233,473) (1193,223,393)

Table 3 Allowed 95% CL ranges for the CP asymmetries sin 2a, sin 2b and sin c, from the CKM "ts in the SM ( f"0) and supersymmetric theories, characterized by the parameter f de"ned in the text f

sin 2a

sin 2b

sin c

f"0 (SM) f"0.2 f"0.4 f"0.75

!0.91}0.77 !0.98}0.65 !1.00}0.50 !1.00}0.14

0.53}0.94 0.52}0.93 0.49}0.93 0.45}0.91

0.35}1.00 0.28}1.00 0.22}0.99 0.16}0.91

of f are allowed for large values of c. However, as we have seen above, the CKM "ts disfavour such values of c. For completeness, in Table 3 we present the corresponding allowed ranges for the CP asymmetries sin 2a, sin 2b and sin c. Again, we see that the allowed range of sin 2b is largely independent of the value of f. On the other hand, as f increases, the allowed values of sin 2a become increasingly negative, while those of sin c become smaller. The allowed (correlated) values of the CP angles for various values of f can be clearly seen in Figs. 9 and 10. As f increases from 0 (SM) to 0.75, the change in the allowed sin 2a}sin 2b (Fig. 9) and a}c (Fig. 10) regions is quite signi"cant. In Section 2.1, we noted that "< /< ",BK and f B(BK B are very important in de"ning the allowed S@ A@ ) region in the o}g plane. At present, these three quantities have large errors, which are mostly theoretical in nature. Let us suppose that our theoretical understanding of these quantities improves, so that the errors are reduced by a factor of two, i.e. "< /< ""0.093$0.007 , S@ A@ . BK "0.94$0.07 , ) f B(BK BB"215$20 MeV

(49)

How would such an improvement a!ect the SUSY "ts? We present the allowed 95% CL regions ( f"0,0.2,0.4,0.75) for this hypothetical situation in Fig. 12. Not surprisingly, the regions are quite

A. Ali, D. London / Physics Reports 320 (1999) 79}106

101

Fig. 9. Allowed 95% CL region of the CP-violating quantities sin 2a and sin 2b, from a "t to the data given in Table 1. The upper left plot (f"0) corresponds to the SM, while the other plots (f"0.2,0.4,0.75) correspond to various SUSY models. (From Ref. [19].)

a bit smaller than in Fig. 11. More importantly for our purposes, the regions for the di!erent values of f have become more separated from one another. That is, although there is still a region where all four f values are allowed, precise measurements of the CP angles have a better chance of ruling out certain values of f. In Table 4 we present the allowed ranges of a, b and c, as well as their central values, for this scenario. Table 5 contains the corresponding allowed ranges for the CP asymmetries sin 2a, sin 2b and sin c. The allowed sin 2a}sin 2b and a}c correlations can be seen in [19]. As is consistent with the smaller regions of Fig. 11, the allowed (correlated) regions are considerably reduced compared to Figs. 9 and 10. As before, although the measurement of b will not distinguish among the various values of f, the measurement of a or c may. Indeed, the assumed reduction of errors in Eq. (49) increases the likelihood of this happening. For example, consider again Table 2, which uses the original data set of Table 1. Here we see that 6534a41233 for f"0 and 8634a41413 for f"0.75. Thus, if experiment "nds a in the range 86}1233, one cannot distinguish the SM ( f"0) from the SUSY model with f"0.75. However, consider now Table 4, obtained using data with reduced errors. Here, 6734a41163 for f"0 and 9734a41373 for f"0.75. Now, it is only if experiment "nds a in the range 97}1163 that one cannot distinguish f"0 from f"0.75. But this range is quite a bit smaller than that obtained using

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Fig. 10. Allowed 95% CL region of the CP-violating quantities a and c, from a "t to the data given in Table 1. The upper left plot (f"0) corresponds to the SM, while the other plots (f"0.2,0.4,0.75) correspond to various SUSY models. (From Ref. [19].)

Fig. 11. Allowed 95% CL region in o}g space in the SM and in SUSY models, from a "t to the data given in Table 1, with the (hypothetical) modi"cations given in Eq. (49). From left to right, the allowed regions correspond to f"0 (SM, solid line), f"0.2 (long dashed line), f"0.4 (short dashed line), f"0.75 (dotted line). (From Ref. [19].)

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Fig. 12. Allowed 95% CL region in o}g space in the SM and in SUSY models, from a "t to the data given in Table 1, with the (hypothetical) modi"cations given in Eq. (49). From left to right, the allowed regions correspond to f"0 (SM, solid line), f"0.2 (long dashed line), f"0.4 (short dashed line), f"0.75 (dotted line). (From Ref. [19].)

Table 4 Allowed 95% CL ranges for the CP phases a, b and c, as well as their central values, from the CKM "ts in the SM ( f"0) and supersymmetric theories, characterized by the parameter f de"ned in the text f

a

b

c

(a,b,c) 

f"0 (SM) f"0.2 f"0.4 f"0.75

67}1163 74}1243 83}1303 97}1373

20}303 19}293 18}293 16}283

42}903 36}823 31}733 26}593

(933,243,633) (1023,243,543) (1103,233,473) (1193,223,393)

Note: We use the data given in Table 1, with the (hypothetical) modi"cations given in Eq. (49). Table 5 Allowed 95% CL ranges for the CP asymmetries sin 2a, sin 2b and sin c, from the CKM "ts in the SM ( f"0) and supersymmetric theories, characterized by the parameter f de"ned in the text f

sin 2a

f"0 (SM) f"0.2 f"0.4 f"0.75

!0.80 !0.93 !0.99 !1.00

sin 2b } } } }

0.71 0.53 0.23 !0.23

0.64 0.61 0.57 0.52

} } } }

sinc 0.86 0.85 0.85 0.83

0.44 0.34 0.27 0.19

} } } }

1.00 0.98 0.91 0.73

Note: We use the data given in Table 1, with the (hypothetical) modi"cations given in Eq. (49).

the original data. This shows how an improvement in the precision of the data can help not only in establishing the presence of new physics, but also in distinguishing among various models of new physics.

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4. Conclusions In the very near future, CP-violating asymmetries in B decays will be measured at B-factories, HERA-B and hadron colliders. Such measurements will give us crucial information about the interior angles a, b and c of the unitarity triangle. If we are lucky, there will be an inconsistency in the independent measurements of the sides and angles of this triangle, thereby revealing the presence of new physics. If present, this new physics will a!ect B decays principally through new contributions to B}B mixing. If these contributions come with new phases (relative to the SM), then the CP asymmetries can be enormously shifted from their SM values. In this case there can be huge discrepancies between measurements of the angles and the sides, so that the new physics will be easy to "nd. A more interesting possibility, from the point of view of making predictions, are models which contribute to B}B mixings and "e", but without new phases. One type of new physics which does just this is supersymmetry (SUSY). There are some SUSY models which do contain new phases, but they su!er from the problem described above: lack of predictivity. However, there is also a large class of SUSY models with no new phases. In this paper we have concentrated on these models. In these models, there are new, supersymmetric contributions to K}K, B}B and B}B B B Q Q mixing. The key ingredient in our analysis is the fact that these contributions, which add constructively to the SM, depend on the SUSY parameters in essentially the same way. That is, so far as an analysis of the unitarity triangle is concerned, there is a single parameter, f, which characterizes the various SUSY models within this class of models ( f"0 corresponds to the SM). For example, the values f"0.2, 0.4 and 0.75 are found in minimal SUGRA models, non-minimal SUGRA models, and non-SUGRA models with EDM constraints, respectively. We have therefore updated the pro"le of the unitarity triangle in both the SM and some variants of the MSSM. We have used the latest experimental data on "< ", "< /< ", DM and DM , as well as A@ S@ A@ B Q the latest theoretical estimates (including errors) of BK , f B(BK B and m ,f B(BK B/f Q(BK Q. In Q ) addition to f"0 (SM), we considered the three SUSY values of f: 0.2, 0.4 and 0.75. We "rst considered the pro"le of the unitarity triangle in the SM, shown in Fig. 2. At present, the allowed ranges for the CP angles at 95% CL are 6534a41233, 1634b4353, 3634c4973 ,

(50)

or equivalently, !0.914sin 2a40.77, 0.524sin 2b40.94, 0.354sin c41.00 .

(51)

We have also performed CKM "ts for the superweak model. This is done by leaving out the constraint from "e". The resulting allowed unitarity triangle now depends on the value of f B(BK B. With the present estimate f B(BK B"215$40 MeV, the superweak case, i.e. g"0, is ruled out at 95% CL. However, unless the theoretical error on this quantity is reduced, the resulting value of g has a large uncertainty. We then compared the SM with the di!erent SUSY models. The result can be seen in Fig. 11. As f increases, the allowed region moves slightly down and to the right in the o}g plane. The main conclusion from this analysis is that the measurement of the CP angle b will not distinguish among

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the SM and the various SUSY models * the allowed region of b is virtually the same in all these models. On the other hand, the allowed ranges of a and c do depend on the choice of f. For example, larger values of f tend to favour smaller values of c. Thus, with measurements of c or a, we may be able to rule out certain values of f (including the SM, f"0). However, we also note that there is no guarantee of this happening * at present there is still a signi"cant region of overlap among all four models. Finally, we also considered a hypothetical future data set in which the errors on "< /< ", BK and S@ A@ ) f B(BK B, which are mainly theoretical, are reduced by a factor of two. For two of these quantities ("< /< " and f B(BK B), this has the e!ect of reducing the uncertainty on the sides of the unitarity S@ A@ triangle by the same factor. The comparison of the SM and SUSY models is shown in Fig. 12. As expected, the allowed regions for all models are quite a bit smaller than before. Furthermore, the regions for di!erent values of f have become more separated, so that precise measurements of the CP angles have a better chance of ruling out certain values of f.

Acknowledgements We thank Laksana Tri Handoko for his help in producing Figs. 7 and 8.

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Physics Reports 320 (1999) 107}118

Tilting the brane, or some cosmological consequences of the brane Universe G. Dvali *, M. Shifman
Physics Department, New York University, 4 Washington Place, New York, NY 10003, USA
Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA

Abstract We discuss theories in which the standard-model particles are localized on a brane embedded in space}time with large compact extra dimensions, whereas gravity propagates in the bulk. In addition to the ground state corresponding to a straight in"nite brane, such theories admit a (one parameter) family of stable con"gurations corresponding to branes wrapping with certain periodicity around the extra dimension(s) when one moves along a noncompact coordinate (tilted walls). In the e!ective four-dimensional "eld-theory picture, such walls are interpreted as one of the (stable) solutions with the constant gradient energy, discussed earlier [1,2]. In the cosmological context their energy `redshiftsa by the Hubble expansion and dissipates slower than the one in matter or radiation. The tilted wall eventually starts to dominate the Universe. The upper bound on the energy density coincides with the present critical energy density. Thus, this mechanism can become signi"cant any time in the future. The solutions we discuss are characterized by a tiny spontaneous breaking of both the Lorentz and rotational invariances. Small calculable Lorentz noninvariant terms in the standard model Lagrangian are induced. Thus, the tilted walls provide a framework for the spontaneous breaking of the Lorentz invariance.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.25.Mj; 12.10.!g; 12.60.!i; 12.60.Jv Keywords: Brane Universe; Compacti"cation; Non-BPS walls; Tilted walls; Critical energy density

1. Introduction Nonobservation of low-energy supersymmetry may be due to the fact that we live on a non-BPS topological defect, or brane, embedded in higher-dimensional space}time [1]. While the external

* Corresponding author. E-mail address: [email protected] (G. Dvali) 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 6 6 - 6

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space may or may not be supersymmetric, the e!ective low-energy theory on the non-BPS brane is not supersymmetric. The idea of the brane Universe is especially motivated by the solution of the hierarchy problem through lowering of the fundamental scale of quantum gravity down to TeV [3,4] (see also [5,6] Lowering the GUT scale was suggested in [7,8]). In the model suggested in Ref. [3] the standardmodel particles are localized on a topological defect, or 3-brane, embedded in space with large extra compact dimensions of size R, in which gravity can propagate freely. In other words, the original (4#N)-dimensional space}time is assumed to be split into M ;M , where M is the  ,  four-dimensional Minkowskian space while M is a compact manifold. We will refer to it as to the , external space. Within the scenario [3] one can lower the fundamental scale of gravity M  down to . TeV, or so. The observed weakness of gravity at large distances is then due to a large volume of the extra space R<M\ . The relation between the fundamental (M ) and four-dimensional (M ) . . . Planck scales can be derived [3] by virtue of the Gauss law, < , (1) M"M,> , . . where < is the volume of the external space with radius R. Although this relation is valid for any , values of M  and R, the case most interesting phenomenologically is M & TeV. (For R& 1 mm, . . this implies that the extra space is two dimensional.) The most important `technicala question to be addressed is dynamical localization of the standard-model particles on the brane. In the "eld-theoretic context the fermions can be localized due to an index theorem, as was suggested in [9], whereas the localization of the gauge "elds requires the outside medium to be con"ning [10,11]. In particular, this implies that free charges cannot exist in the bulk. On the other hand, in the string-theoretic context, the most natural framework for the braneworld picture is through the D-brane construction (for a review, see e.g. [12]). In this context the standard-model particles can be identi"ed with the open string modes stuck on the brane, whereas gravity comes from the closed string sector propagating in the bulk [4,13}16]. For the purposes of the present paper the precise nature of localization will be unimportant, since we will exploit the low-energy e!ective "eld-theory approach for which the high-energy nature of the brane is beyond `resolutiona. In this paper we will present new theoretical observations regarding the branes on the manifolds M ;M where M is compact. Then we discuss some possible cosmological consequences of the  , , brane Universe with the low-scale quantum gravity. We point out that these theories admit stable solutions which could manifest themselves through a tiny spontaneous breaking of the Lorentz and rotational invariances. In four-dimensional "eld theory the solutions above can be interpreted as a recently discovered new class of stable vacuum con"gurations in supersymmetric theories, with the constant gradient energy, which may or may not break supersymmetry [1,2]. In the latter case these solutions generalize the notion of the BPS saturation to in"nite values of the central charge [2]. Below we will show how these solutions can naturally emerge in the brane Universe picture.  By the in"nite central charges we do not mean trivial in"nities associated with the area of the wall or the length of the string.

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Before proceeding, let us note that other possible cosmological implications of branes can be due to `brane in#ationa driven by a displaced set of branes [17] or due to nonconservation of global quantum numbers in the brane Universe [18]. In a very di!erent context a class of time-dependent cosmological solutions was discussed [19,20] within the Hor\ ava}Witten approach [21]. Some aspects of thermal cosmology were considered in [22]. These issues are not directly related, however, to the present work. For de"niteness assume (4#N)-dimensional space}time to be M ;T , where M is four ,  dimensional Minkowskian space while T is an N-torus with the radius R. The standard-model , particles are the modes living on a 3-brane embedded in this space}time. Let the brane tension (the energy per unit three-volume) be ¹. Throughout the paper we will assume that ¹&M (the only . fundamental mass scale in the theory). Obviously, ¹'0 if the brane is to be regarded as a dynamical object (the wall), since otherwise it will be unstable. Then, in the absence of matter and radiation, the e!ective four-dimensional energy density o has two contributions: a positive one  from the brane tension, and a (must-be) negative one from the bulk: o "¹#K < 
  ,

(2)

as seen in experiments at distances
r
(3)

Fig. 1. Domain walls (branes) in M ;T . A * the straight untilted brane, B * tilted brane, C * curved brane. The  , length of the cylinder in the x direction is ¸&H\.

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where r is a typical curvature radius of the brane, not to be confused with the Friedmannian curvature radius, which we take in"nite. (In the case of the tilted brane r is its longitudinal dimension, r&¸. In this case Eq. (3) can be obtained as follows: (4) o &M (a) ,
  . provided that the tilt angle a&R/¸;1, see below.) This energy provides an e!ective force resisting to bending; the force tends to straighten out the brane. What would be the cosmological signi"cance of this excess energy? To estimate its impact we have to know r. It is natural to assume that in the scale smaller than the Hubble size the brane is straightened out (no excessive crumpling). Whatever mechanism solves the horizon and isotropy problems, it would also help to this straightening. So, it seems reasonable to assume that r9H\9o\ M , (5)  . where o is the critical density of the Universe today. This estimate obviously refers also to the  longitudinal dimension of the tilted brane, ¸9H\. Now, substituting this in Eq. (3) and using Eq. (1) with N52 we get o :o . (6)
   The upper bound which can only, but not necessarily, appear for N"2 is intriguing since it suggests that the brane can serve as some sort of dark matter in Universe. We will study the nature of its energy below. The potential importance of the domain walls in the cosmological considerations was recognized long ago [23]. In Ref. [23] it was shown that were the domain walls within our universe, serious (potentially terminal) cosmological problems might arise in the theory. Situation dramatically changes if our Universe itself is a wall.

2. What is the tilted wall? To illustrate the idea we will consider, for simplicity, the four-dimensional space M compacti "ed in M ;T. The situation is general and does not depend on particular details.  Let us start from M and assume that the dynamical theory under consideration has multiple  discrete degenerate vacua. The simplest example is the theory of the real scalar "eld with the potential of the double-well type. The simplest supersymmetric example is the minimal Wess}Zumino model with the cubic superpotential. The "eld con"guration depending only on one coordinate (call it z) that interpolates between one vacuum at zP!R and another at zPR is the domain wall. The width of the wall d in the z direction is of order d&M\ where M is the mass scale of the "eld(s) of which the wall is made. It is assumed that d is much smaller than any other relevant scale of dimension of length. On M it is meaningless to speak of the tilted wall * the direction z can be chosen arbitrarily.  Now, we compactify z and consider the theory on the cylinder M ;T. The underlying dynamical  theory must be modi"ed accordingly, in order to allow for the existence of the walls. In the case at hand it is su$cient to assume that the wall-forming "eld U lives on a circle, i.e. one can consider the

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model of the sine}Gordon type or its supergeneralizations. Both, the superpotential and the KaK hler potential must be periodic in U, with commensurate periods. For simplicity we assume these periods to be 2n. Following an old tradition, we rename the compact coordinate, zPX. The noncompact coordinates (including time) will be denoted by x . We look for the topologically I nontrivial solutions of the soliton type on the cylinder, depending on one coordinate only. The solution of the type U (X), which is independent of x , is the straight wall, see A in Fig. 1. It satis"es  I the condition U (X#2pR)"U (X)#2n. This wall is aligned `parallela to the cylinder. The tilted   wall (B in Fig. 1) is a solution of the type U (X cos a#x sin a) where a is the tilt angle. Note that the ? function U does not coincide with U , generally speaking. In the limit of small a the di!erence ?  between U and U is O(a). The condition U ((X#2pR)cos a#x sin a)"U (X cos a#  ? ? ? x sin a)#2p must be satis"ed. It is not di$cult to see that the tilted wall solution exists, if so does the straight wall solution. The tilted wall is stable provided the solution is `naileda at the points 1, 2 at the boundaries (see Fig. 1). Alternatively, one can glue the boundaries of the cylinder converting it into a two torus. The tilted wall solution is automatically stable then if the wall winds around the two-torus. It is not di$cult to prove that the domain walls on the cylinder cannot be BPS-saturated, strictly speaking. However, if d/R;1 the straight walls may be very close to the BPS saturation, achieving the BPS saturation in the limit d/RP0, when the wall becomes the `genuine branea. The tilted walls are never BPS-saturated, their tension being larger than that of the straight wall. This e!ect * the increase of the internal tension ¹ of the tilted wall compared to that of the straight wall * is proportional both to a and to (d/R). It can be made arbitrarily small in the limit d/RP0. We will neglect it in what follows, using one and the same tension ¹ for the tilted and straight walls. The tilting does produce an impact on o since the area of the tilted wall per unit length of the cylinder in the x direction is larger. This e!ect is most conveniently described by the e!ective low-energy theory of the zero modes on the wall. The corresponding discussion is presented in the next section. 3. Calculating q brane for nonvanishing tilt angles To deal with the long wave length (j


* *

g ( ) , L"M dx  . *x *xI I

(7)

 More exactly, the requirement of periodicity applies to the derivative of the superpotential d= and to the KaK hler metric.  This assertion is subject to reservations. Under certain special conditions the BPS walls can exist on the cylinder. This issue will be treated in detail elsewhere.

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where g is the external metrics depending on the structure of the manifold on which the "elds

live. For instance, in the case of the domain wall in Minkowski space g "d . All dimensional constants in the low-energy theory of the zero modes are related to the order parameter, which is the brane tension ¹&M. The dynamics of the Goldstone bosons on the brane is described [1] by . (3#1)-dimensional "eld theory which may or may not be supersymmetric (in the latter case the brane must be non-BPS saturated). If we deal with the supersymmetric theory g is obtained from the KaK hler potential. Let us assume for the time being that we have only one Goldstone boson and g "1. Then, the solution "ax goes through the equations of motion of theory (7). This is the constant energy density vacuum, discussed in Refs. [1,2]. To make contact with the discussion above, we note that the vacuum "ax represents the tilted wall described by the solution U in the full theory. The ? additional contribution to o compared to the straight brane is obviously *o "¹a/2 .
 

(8)

This result has a very transparent interpretation in the full theory. It exactly reproduces the increase of the brane surface per unit length of the cylinder for a nonvanishing tilt angle a, see Fig. 2. Given this interpretation, one might ask why one needs to consider the e!ective low-energy theory at all. The point is that at the next step we want to switch on gravity in the bulk. Having the low-energy theory of the zero modes, describing matter on the brane, helps analyze the impact of gravity. In the language of the e!ective "eld theory (7), the bending of the brane is equivalent to

 acquiring some x dependence in the vacuum. The generic (x) con"guration is unstable and I will decay producing waves (the sound waves on the brane). This is the mechanism of eliminating foldings on the brane. Since these waves travel with the speed of light, the brane will iron itself out in the horizon scale. Eventually it will evolve to a tilted brane.

Fig. 2 . The map of the cylinder of Fig. 1. The ratio of the surfaces of the branes B to A per unit length in the x direction is 1#a/2.

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In the state with an arbitrarily bent brane we will distinguish two components. One can be viewed as a collection of all possible Goldstone waves traveling with the speed of light, which `redshifta away like ordinary matter. Another component is the vacuum solution (more precisely, a family of solutions) that would be stable if it were not for the expansion of the Universe. This con"guration `redshiftsa away slower than matter and can be called the tilted brane con"guration (or wrapped, if there are several windings on the length of the cylinder ¸). It can only `redshifta through the `stretchinga triggered by the Universe expansion. In the case of the generic massless "elds with the #at g , the solution "ax is stable under  any localized deformations. If 's are the Goldstone bosons arising due to the spontaneous  breaking of some compact symmetry, 's are periodic (in fact, they are phases de"ned modulo  shifts). This is exactly what happens in our case since the extra dimensions are assumed to be compact. Then the solution "ax must be modi"ed appropriately, see Section 2. Here we will add a comment regarding the issue of stability. To explain the point we will use a simple example of the Goldstone "eld produced as a result of breaking of some global ;(1) symmetry (a similar model was treated in Section 6 of Dvali and Shifman [1]). Start from the Lagrangian L""* U"!j/2("U"!v) . I The equations of motion have a solution U"c e IV,

c"(v!k/j

(9)

(10)

which corresponds to winding of the phase with the period 2p/k as one moves along x. This solution has both the gradient and potential energies (cf. the solutions discussed in [2]). The potential term scales as &k/j. It is seen that at jPR the solution (10) becomes pure gradient energy (v must scale accordingly, of course, i.e. v&j\). This is because in this limit, the x-dependence cannot a!ect the order parameter and, thus, the potential energy. In the case of the tilted brane this would mean that the brane with a constant tilt carries purely gradient energy in the limit when bending cannot a!ect its tension. This is true for any brane in the limit when one can ignore its thickness. The corresponding con"guration cannot decay into the waves. Its energy is reduced only through the Friedmann expansion of the Universe and, thus, scales as&a\, the scale factor in the Friedmann Universe. This means, in turn, that the state at hand will sooner or later dominate over both, the radiation and the matter densities. We will discuss the observational implications of this fact in Section 5. Before, however, let us discuss the e!ect of higher dimensional bulk gravity. 4. Switching on gravity in the bulk The e!ective Lagrangian for the zero modes localized on the brane becomes L

"M gIJ(* * )g #fermions , I  J
  .

(11)

 The analogy is much more precise than one might naively think, since the translation in the external space can be regarded as an internal U(1) rotation. In the presence of gravity, this is a gauge rotation gauged by the external components of the graviton (graviphoton(s)). We can neglect the coupling to gravity for the time being, due to the large sizes of extra dimension(s).

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where g and g are the external and induced metrics, respectively. The fermion terms are relevant  in the supersymmetric case. So far, we have discussed the brane dynamics on the cylinder neglecting gravity. Now we want to take it into account. In the scenario under consideration gravity is not con"ned to the wall, but, rather, propagates in the bulk. The components of the graviton belonging to M and T, require  separate treatment. The e!ect which is most important for us is due to the higher-dimensional components of the graviton (the so-called graviphotons). For simplicity we will consider only a single extra dimension compacti"ed on a circle parametrized by the coordinate X. The zero-mode component of g is the graviphoton A (x). Viewed as a four-dimensional gauge "eld, A (x) gauges I I I the translation in X, i.e. XPX#f (x ), which, from the four-dimensional standpoint, is an internal I U(1) gauge symmetry. Since the brane spontaneously breaks the translational invariance, the corresponding gauge symmetry is realized nonlinearly. The Goldstone mode is eaten up by the gauge "eld A (which gets a mass of order (1 mm)\ [24]). As a result, the tilted wall solution I

"ax we have considered previously is pure gauge. It can be compensated by the gauge "eld A "a and presents no physically observable e!ect. Thus, if we have only one zero mode on the V wall, the tilted wall is indistinguishable from the straight one in the presence of gravity. To make the idea work we must have two or more zero boson modes on the wall. Then the tilted wall (the constant gradient energy solution) will lead to a physically observable excess in o. Let us explain this in more detail. Our solution can be made physical by `projecting outa the graviphoton, provided the compact manifold breaks translational invariance in the extra space. The simplest dynamical realization is as follows. Consider topologically stable winding con"gurations [1]. (Such con"gurations may anyway be needed for supersymmetry breaking [1] or for stabilizing radii at large distances [25,26]). Consider a "ve-dimensional scalar "eld m(x , X) I transforming under an internal U(1) symmetry as mPe ?m. Assume that a potential forces the ' condensate 1m2O0 to develop. The simplest choice can be <"("m"!v)/m, or any other similar function. Then the vacuum manifold is a circle, and there are topologically stable winding con"gurations m"w e L60,

w"(v!nm/R

(12)

with integer n. They correspond to giving a vacuum expectation value to di!erent Kaluza}Klein modes and, therefore, are topologically stable due to the mapping of the vacuum circle on the external compact space. Thus we are free to choose any of these states as the ground state. Con"gurations with nonzero n break spontaneously the X translations and, thus, give mass to the graviphoton A . As a result, the graviphoton "eld can be integrated out in the low-energy e!ective I theory. Then the brane Goldstone remains a physical "eld. More precisely, the picture is as follows. The theory at hand has two U(1) symmetries from the very beginning: the internal U(1) and `externala U(1) gauge symmetry under translations in ' # the extra dimension. However, from the point of view of a four-dimensional observer living on the brane, both of them are internal symmetries, one global and another gauged by A . The I

 Strictly speaking, we have to restrict ourselves to n(vR/m since larger winding numbers force the "eld to vanish, and the solution will `unwinda. For our purposes, however, it is su$cient to consider small enough n.

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condensate 1m2O0, with nO0, breaks both U(1)'s down to a global U(1) describing the change of a relative position of the brane in the extra space. The brane breaks the latter down to nothing, but since there is no gauge degree of freedom left, the corresponding Goldstone is physical. A related mechanism of `projecting outa the graviphoton while still keeping at least one Goldstone mode may be provided by multicomponent walls suggested in Refs. [27,28]. It was noted that in supersymmetric theories with multiple degenerate vacua, under certain conditions there exists a variety of the BPS-saturated walls. Some of them may coexist together; then, the tension of such `multilayera con"guration is exactly equal to the sum of the tensions of the individual walls, independently of the distance between the layers (individual walls). A similar e!ect takes place for the D-branes which are the BPS-states in the limit of unbroken supersymmetry, and the net force between two straight parallel branes vanishes (see [12] for a review). This means that, apart from the zero mode corresponding to the overall translation, there are extra zero modes corresponding to shifting the layers with respect to each other, without changing the position of the center of mass. As was mentioned, on the cylinder the exact BPS saturation can be achieved only in the limit d/RP0. Hence, only the "rst zero mode * the one related to the overall translations * is exactly zero, others become quasi-zero. The zero mode is eaten up by the graviphoton, eliminating it from the game. The quasi-zero modes remain physical, their dynamics is described by a lowenergy Lagrangian. The supersymmetry breaking in general will give mass to this pseudoGoldstones since the net force between branes (or walls) is nonzero. If there are no other massless states in the bulk, at large distances the only force between the branes is due to gravity which gives a small r,-suppressed mass to the Goldstones (r is the interbrane separation). This mass can be made arbitrarily small if the separation is large and, practically, it can be neglected. All dynamical consequences are the same as discussed in Section 3.

5. Implications for cosmology Now let us discuss cosmology in more detail. We will treat the problem from the standpoint of the e!ective four-dimensional "eld theory at scales <
(13)

has the form (14) s"(¹cx#s e NV  (modulo the Lorentz transformations), where c is a constant and the second term is some collection of massless plane waves. The energy stored in the second term will just `redshifta away, like massless matter. However, the "rst term produces a rather di!erent contribution. In the present

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context this term describes the tilted brane with the tilt angle given by c&R/r. In other words, when we move along x, the brane is wrapping around the extra dimension with the period &r. In the absence of gravity, in the limit of an in"nitely thin brane, this is a stable con"guration for any c. In the presence of gravity, however, its energy `redshiftsa away as &a\, due to the Friedmann expansion. Therefore, eventually this energy will dominate over matter. When this happens actually depends on the initial condition for r(c) and on the subsequent evolution of the scale factor. Assume that initially c&RH , where H is the Hubble parameter at that time. In other words, we   assume that, when the brane was `formeda, it wrapped around the extra dimension once (on average) per the causally connected region &H\. Now, at present time, this region must have  had evolved into a volume comparable to the present Hubble size (or larger). This is required by whatever mechanism solving the horizon problem. This means that a (15) r &   9H\ .     H a   Thus, the energy density of a tilted brane would be :o . In particular, it is su$cient to have A a period of in#ation with the number of e-foldings H M  . , (16) ¹ ¹   0 and the subsequent reheating temperature ¹ , where ¹ &3 K3. For instance, a brief period of 0   the `brane in#ationa [17] can do the job. In reality, however, we expect in#ation to have more e-foldings and, thus, o :o is rather
  A natural. NKln

6. The Lorentz symmetry breaking The important fact is that the solution at hand spontaneously breaks the Lorentz and rotational invariances in four dimensions. This would result in a global anisotropy in the expansion if o were to dominate the Universe. Thus, we must require o to be subdominant today, but it
 
  can become dominant at any time in the future. In this respect any observational evidence of a global anisotropy would be extremely important for constraining o .
  The tilted brane would induce rotational (or Lorentz) noninvariant terms in the e!ective four-dimensional standard-model Lagrangian. The important fact is that the brane Goldstones (or pseudo-Goldstones, which are similar in this respect) necessarily couple to all particles living on the brane through an induced metric on the brane. Thus, the operators of the form * *

* *

tc *xJt# F FJ?#2 *x *xJ I? *x *xJ I I I

(17)

 The wrapped brane can provide an additional (time-dependent) force stabilizing R. This may have implications for the early cosmology [29}32].

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will appear (accompanied by the appropriate powers of ¹). Here, t are the matter fermions and F stands for the matter gauge "elds. In the background of a titled brane "ax, these terms will I? induce e!ective Lorentz-violating interactions among the standard-model "elds. Can one have tilted branes and still avoid the anisotropy and violations of the rotational invariance? In principle, the answer is positive. To this end one must deal with more complicated manifolds, such that the topology of the manifold M is the same as that of our three-dimensional , space M (of course, in any case, locally M must be Minkowskian). In other words, M must   , include unshrinkable surfaces C that are isomorphic to M . In this case the brane that corres  ponds to a point on C can move around when one moves along the brane in M . The   topologically nontrivial con"guration of interest emerges when this motion maps C onto M . For   such a con"guration, one can "nd a preferred reference frame for which rotations on M will not be  broken, but the Lorentz invariance will be broken in the arbitrary reference frame. To be more speci"c, let us give an example. Imagine that we have an external space, with three extra dimensions, which forms a manifold K of radius R (say, one can think of K as of a three-sphere). Assume that `oura three-dimensional space is also a very large manifold K, its radius is much larger than H\. Let us call these two manifolds `smalla and `biga, respectively. Clearly, at human scales the `largea manifold (plus time) is identical to M , but not globally. Now, our brane corresponds to  a point on the `smalla K. Imagine a con"guration such that when one moves in `largea K, the point in `smalla K also moves in the same way. Thus, we get a mapping KPK, which is certainly topologically stable. Roughly speaking, the brane wraps isotropically in all directions. Such a wrapping is isotropic and stable. What is the low-energy picture corresponding to this construction? Since K has three dimensions, the brane breaks three translational invariances. Thus, there are three massless Goldstone modes in the e!ective low-energy theory, U, A"1, 2, 3. Let x be three coordinates on our large S which locally look as the Cartesian coordinates in our Minkowski space. Then, the solution is  U"ax. This solution is isotropic because of the spherical symmetry of the problem. Its energy density will still scale J1/a because this is essentially the same gradient-energy solution we have discussed previously.

7. Conclusions The idea of con"ning our Universe to a wall which ensures an appropriate supersymmetry breaking [1] seems to be promising. At the very least, it deserves further investigation. Being combined with the idea of compacti"cation of the extra dimensions and allowing gravity to propagate in the bulk [3] it leads to potentially realistic and reach phenomenology. In this paper we have shown that the walls on M ;M where M is compact generate peculiar theoretical  , , e!ects due to tilting. The situation becomes especially interesting when gravity is switched on. The appropriate theoretical framework for its analysis is provided by the e!ective low-energy theories of the Goldstone modes on the brane. After gravity is switched on one of these modes is eaten up by the graviphoton (making it massive and eliminating it from the massless particle spectrum). We presented models where there are residual physical Goldstone (or pseudo-Goldstone) modes. These solutions produce a framework for the spontaneous breaking of the Lorentz and rotational invariance and may have observable consequences.

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Acknowledgements We would like to thank A. Dolgov, I. Kogan, K. Olive, T. Piran, and A. Vilenkin for useful discussions and comments. This work was supported in part by DOE under the grant number DE-FG02-94ER40823.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

G. Dvali, M. Shifman, Nucl. Phys. B 504 (1997) 127. G. Dvali, M. Shifman, Phys. Lett. B 454 (1999) 277. N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 516 (1998) 70. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 436 (1998) 257. E. Witten, Nucl. Phys. B 471 (1996) 135. J. Lykken, Phys. Rev. D 54 (1996) 3693. K.R. Dienes, E. Dudas, T. Gherghetta, Phys. Lett. B 436 (1998) 55. K.R. Dienes, E. Dudas, T. Gherghetta, Nucl. Phys. B 537 (1999) 47. V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 125 (1983) 136. G. Dvali, M. Shifman, Phys. Lett. B 396 (1997) 64. G. Dvali, M. Shifman, Erratum, Phys. Lett. B 407 (1997) 452. J. Polchinski, String Theory, Cambridge University Press, Cambridge, 1998. G. Shiu, S.-H.H. Tye, Phys. Rev. D 58 (1998) 106007. G. Shiu, S.-H.H. Tye, Nucl. Phys. B 542 (1999) 45. Z. Kakushadze, S.-H.H. Tye, Nucl. Phys. B 548 (1999) 180. I. Antoniadis, B. Pioline, Nucl. Phys. B 550 (1999) 41. G. Dvali, H. Tye, Phys. Lett. B 450 (1999) 72. G. Dvali, G. Gabadadze, hep-ph/9904221. A. Lukas, B.A. Ovrut, D. Waldram, hep-th/9806022. A. Lukas, B.A. Ovrut, D. Waldram, hep-th/9902071. P. Hor\ ava, E. Witten, Nucl. Phys. B 475 (1996) 94. M. Maggiore, A. Riotto, Nucl. Phys. B 548 (1999) 427. Ya.B. Zeldovich, I.Yu. Kobzarev, L.B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [Sov. Phys. JETP 40 (1974) 1]. N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. B 429 (1998) 263.



[25] R. Sundrum, Phys. Rev. D 59 (1999) 08 5010. [26] [27] [28] [29] [30] [31] [32]

N. Arkani-Hamed, S. Dimopoulos, J. March-Russell, hep-th/9809124. M. Shifman, Phys. Rev. D 57 (1998) 1258. M. Shifman, M. Voloshin, Phys. Rev. D 57 (1998) 2590. N. Kaloper, A. Linde, Phys. Rev. D 59 (1999) 101303. A. Mazumdar, hep-ph/9902381. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, J. March-Russell, hep-ph/9903239. N. Arkani-Hamed, S. Dimopoulos, N. Kaloper, J. March-Russell, hep-ph/9903224.

Physics Reports 320 (1999) 119}126

Enhanced electroweak radiative corrections in SUSY and precision data I.V. Gaidaenko , A.V. Novikov , V.A. Novikov , A.N. Rozanov
*, M.I. Vysotsky ITEP, Moscow, Russia
CPPM, IN2P3, CNRS, France and ITEP, Moscow, Russia ITEP, Moscow, Russia and INFN, Sezione di Ferrara, Italy

Abstract Enhanced radiative corrections generated in SUSY extensions of the Standard Model spoil the "t of the precision data (Z-boson decay parameters and W-boson mass). This negative e!ect is washed out for heavy enough squarks, because of the decoupling property of SUSY models. We "nd that even for light squarks the enhanced radiative corrections can be small. In this case substantial t t mixing is necessary.  1999 *0 Elsevier Science B.V. All rights reserved. PACS: 12.15.Lk; 13.38.Dg Keywords: SUSY; Radiative corrections; Stop mixing

A number of predictions of the Standard Model are now tested with an accuracy of the order of 0.1%. There are two (more or less) free parameters in the Standard Model: the value of the strong coupling constant a( (m ) and the mass of the Higgs boson m . Making a "t of the latest set of  8 & precision data reported in [1] (Z-boson decay parameters, W-boson and top-quark masses), we obtain m "(71>) GeV , & \ a( (m )"0.119$0.003 ,  8 s/n "15.0/14 .    * Corresponding author. CNRS-CPPM-INP3-Univ. MediterraneH , Marseille Cedex 9, France. E-mail address: [email protected] (A.N. Rozanov) 0370-1573/99/$ - 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 5 6 - 3

(1) (2) (3)

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The quality of this "t is very good, which imposes strict constraints on the possible extensions of the Standard Model. In the SUSY extensions a lot of new particles are introduced. Their contributions to electroweak observables come through loops and for m 'm are of the order of a (m /m ), where 1317 5 5 5 1317 m characterises the mass scale of the new particles. These contributions were calculated and 1317 analysed in number of papers [2}5]. A large violation of SU(2) symmetry in the third family of squarks by a large value of m +175 GeV leads to an enhancement of the corresponding oblique corrections by the factor  (m /m )+16 to be compared with the numerous terms of the order of a (m /m ). (The  5 5 5 1317 presence of terms &m in SUSY models was recognised long ago [6].) Inspired by this fact,  calculations of the enhanced corrections to the functions < , < and < were made [7]. The  0

functions < describe electroweak radiative corrections to Z-boson couplings to leptons g and G  g /g , and to W-boson mass [8]. To calculate these enhanced terms we expanded the polarisation 4  operators of the vector bosons R (k) at k"0. The terms enhanced as m/m m come from 4  5 1317 R (0), while those enhanced as m/m come from R (0). The higher-order derivatives of 5  1317 58 self-energies are suppressed as (m /m ), and are therefore not taken into account. 58 1317 In the present paper the in#uence of these new terms on the precision data "t will be analysed. To begin with, we should expand the analysis of [7] and take into account the main SUSY corrections to hadronic Z-decays as well. Vertices with gluino exchange generate (potentially) large corrections of the order of a( (m /m ) in the limit m 'm . For hadronic Z decays we use the following  8 1317 1317 8 expression for the width [9]: "12[g R #g R ]C , C "C   4 4   8

(4)

where C "(1/24(2p)G m.  I 8 Corrections induced by gluino exchanges lead to the following SUSY shifts of the factors R  [10]: G dR "dR "1#[a( (m )/p]D ,   8  4

 



(5)



4  \X xyz z   D "! , dz ln 1! dz    3 x#(z #z )(y!x)    

(6)

where x"(m /mq ), y"(m /mg ). We take these strong SUSY corrections into account in our 8 8 analysis. The weak SUSY corrections to Z decays into hadrons are taken into account by the corrections to the functions < and < calculated in [7].  0 The stop exchange contributes to the vertex corrections to the ZPbbM decay amplitude; but since there are no terms enhanced as (m /m ) [11], we will not take the corresponding corrections into  5 account in the present paper. Let us start the discussion of the SUSY corrections to the functions < from the description of the G stop sector of the theory. The t t mass matrix has the following form: *0




mt *



m A   , m A mt 0  

(7)

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121

where t t mixing is proportional to a large value of the top-quark mass and is not small. *0 Diagonalizing matrix (7) we obtain the following eigenstates: t "c t #s t , 0  * t "!s t #c t ,  * 0 where c ,cos h , s ,sin h , h being the t t mixing angle, and *0  *0  *0 *0 tan h "(m!mt *)/(mt *!m), m5mt *5m , *0     where m and m are the mass eigenvalues:   4mA mt #mt 0 "mt *!mt 0"   $ 1# . m " *  (mt *!mt 0) 2 2

(8)

(9)



(10)

The following relation between mt * and mb * takes place: (11) mt *"mb *#m#m cos(2b)c ,  8 5 where c ,cos h "0.77 (h is the electroweak mixing angle) and tan b is equal to the ratio of 5 5 5 the vacuum averages of two Higgs neutrals, introduced in SUSY models. Relation (11) is central for the present paper; it demonstrates a large breaking of SU(2) symmetry in the third generation of squarks. The only hypothesis that is behind this relation is that the main origin of the large breaking of this SU(2) is in the quark Higgs interaction. The enhanced electroweak radiative corrections induced by squarks of the third generation depend on three parameters: m , m and mb *. The dependence on the angle b is very moderate and,   in numerical "ts, we will use the rather popular value of tan b"2. In what follows we will write mb instead of mb *, bearing in mind that the b squark does not contribute to the corrections under 0 investigation (let us note that b b mixing is proportional to mb and can be neglected). * 0 Let us present the formulas from [7], which describe the enhanced SUSY corrections to the functions < : G (12) d*0 < "(1/m)[cg(m , mb )#sg(m , mb )!csg(m , m )] ,       1317  8   (13) d*0 < "d*0 < #> [c ln(m/mb )#s ln(m/mb )]!csh(m , m ) ,         1317 0 1317   *  d*0 < "d*0 < #> s[c ln(m/mb )#s ln(m/mb )] 1317 1317   *     (14) #[(c!s)/3][ch(m , mb )#sh(m , mb )]!(cs/3)h(m , m ) ,         where g(m , m )"m#m!2[mm/(m!m)] ln(m/m) , (15)           m 5 4mm (m#m)(m!4mm#m)   #       ln  h(m , m )"! # (16)   m 3 (m!m) (m!m)      and > is the left-doublet hypercharge, > "Q #Q ". * * 
 Now that we have all the necessary formulas at our disposal, let us start with analysing the data in the simplest case of the absence of t t mixing, sin h "0. In this case the corrections d < *0 *0 1317 G




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Table 1 Fit of the precision data with SUSY corrections taken into account in the case of the absence of t t mixing, sin h "0 *0 *0 and m taken as a free parameter. For mb '300 GeV, SUSY corrections become negligible and the SM "t of the data is & reproduced mb (GeV)

m (GeV) &

a( 

s/n   

100 150 200 300 400 1000

850> \ 484> \ 280> \ 152> \ 113> \ 77> \

0.113$0.003 0.116$0.003 0.117$0.003 0.118$0.003 0.119$0.003 0.119$0.003

20.3/14 18.1/14 17.3/14 16.3/14 15.8/14 15.2/14

Table 2 The same as Table 1, but with a value of the lightest Higgs-boson mass m "120 GeV that is about the maximum allowed  value in the simplest SUSY models mb (GeV)

a( 

s/n   

100 150 200 300 400 1000

0.110$0.003 0.115$0.003 0.116$0.003 0.118$0.003 0.119$0.003 0.119$0.003

30.2/15 21.9/15 18.6/15 16.4/15 15.8/15 15.5/15

depend on parameter mb only, and this dependence is shown in Figs. 1}3 of paper [7]. For mb :300 GeV the theoretical values of all three functions < become larger than the experimental G values. Let us begin our "t by taking the value of the Higgs boson mass as a free parameter. In this "t, the mass of the Higgs boson becomes larger than its Standard Model "t value (1). The results of the "t are shown in Table 1. (To reduce the number of parameters we take mg "mb in this paper. Let us stress that light quarkinos with masses of the order of 100}200 GeV are usually allowed only if gluinos are heavy, mg 5500 GeV [12]. In the case of the heavy gluino correction D (Eq. (6))  becomes power-suppressed and we return to the Standard Model "t value of a( "0.119.) We see  how in the SUSY extension of the Standard Model with light superpartners the "t gets worse. However, in SUSY the mass of the Higgs boson is no longer a free parameter. Of the three neutral Higgs bosons, the lightest should have a mass less than approximately 120 GeV. If other Higgs bosons are considerably heavier, this lightest boson has the same couplings to gauge bosons as in the Standard Model, so formulas for the Standard Model radiative corrections can be used since deviations are suppressed as (m /m ) (m being the mass of the heaviest Higgs). For the maximal    allowed value m "120 GeV, the results of the "t are shown in Table 2. In what follows, we will  always take m "120 GeV since, for 90 GeV(m (120 GeV, the results of the "t are practically   the same. This table demonstrates that superpartners should be heavy if we want to have a good-quality "t of the data. The next step is to take into account t t mixing. *0

I.V. Gaidaenko et al. / Physics Reports 320 (1999) 119}126

123

Fig. 1. Values of the d< , d< and d< at mb "150 GeV.  0

In Fig. 1 we show the dependence of SUSY corrections d < on m and m for mb "150 GeV; 1317 G   the same in Fig. 2 for mb "200 GeV. One clearly sees from these "gures that there exists, even for such low values of mb , domain of low m values where the enhanced radiative corrections are  damped. In Fig. 2 there is a valley where d < reach the minimum values, which are consider1317 G ably smaller than 1. This valley starts at m +mb , m +1000 GeV and goes to m +100 GeV,    m +400 GeV. The smallness of the radiative corrections at the point m +mb , m +1000 GeV    can be easily understood: here h +p/2, so in d < only the term proportional to g(m , mb ) *0 1317   remains. However, for m "mb this term equals zero. At this end-point of the valley,  t +t , t +t , so mt 0<mt *, which prevents the relation between mt 0 and mt * occurring in a wide  *  0 class of models. In these models (e.g. in the MSSM) the left and the right squark masses are equal at the high-energy scale and, renormalizing them to low energies we have mt *'mt 0. Almost along the whole valley we have tan h '1, which means that mt 0'mt *. This possibility to suppress *0 radiative corrections was discussed in [13]. However, in the vicinity of the end-point m "300 GeV, m "70 GeV the value of tan h becomes smaller than 1 and mt 0(mt *.   *0 In Table 3 we present the results of the "t, assuming mb "150 GeV and m "120 GeV, along the  line of minimum s, which is formed at m +150 GeV. We see that for heavy m the quality of the  

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Fig. 2. Values of the d< , d< and d< at mb "200 GeV.  0

Table 3 For light mb "150 GeV, light m "mb and m "120 GeV, the quality of the "t can be the same as in the Standard Model,   if m is heavy enough  m (GeV) 

a( 

s/n   

482 743 1289

0.116(3) 0.117(3) 0.117(3)

16.3/15 15.8/15 15.6/15

"t is not worse than in the Standard Model. In Table 4 we show values of s along its valley of minimum, which is formed for mb "200 GeV. Once more we observe that a good quality of the "t is possible for light superpartners if t t mixing is taken into account. This e!ect is clearly seen as *0 a valley of low s on the plots of the allowed regions in the plane m }m both for mb "150 GeV in   Fig. 3a and for mb "200 GeV in Fig. 3b.

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125

Fig. 3. m !m exclusion plot for SUSY model, assuming m "120 GeV, m g "mb "mb *, tan(b)"2 and Eq. (11).    Three-parameter "t (m , a( , a) is done to EWWG-99 data. Figure (a) corresponds to mb "150 GeV, while (b) corresponds   to mb "200 GeV.

Table 4 For "xed values of mb "200 GeV and m "120 GeV, results of the "t along the valley of minimum s  m (GeV) 

m (GeV) 

a( 

s/n   

1296 888 387 296

193 167 131 72

0.118$0.003 0.118$0.003 0.118$0.003 0.117$0.003

15.6/15 15.8/15 16.1/15 16.7/15

The formulas for the enhanced radiative corrections in SUSY extensions of the Standard Model obtained in [7] were used in the present paper to "t the data of the precision measurements of Z-boson decay parameters at LEP and the SLC, the value of m and m at the Tevatron. The "t 5  with SUSY corrections, assuming a small value of mb , the absence of t t mixing, and *0 m "120 GeV, leads to the growth of the s value. Thanks to the decoupling property of the SUSY  extension in the case of heavy squarks, the results of the Standard Model "t are reproduced. However, even for comparatively light sbottom and small mass of one of the two stops, values of t t mixing can be found that have d < small and s almost the same as in the Standard Model. *0 1317 G We are grateful to L.B. Okun for stimulating discussions. A.V. Novikov is grateful to G. Fiorentini for his hospitality in Ferrara, where part of this work was done. A.N. Rozanov is grateful to CPPM-IN2P3-CNRS for supporting this work. The research of I.G., A.N., V.N. and

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M.V. is supported by RFBR grants 96-02-18010, 96-15-96578, 98-02-17372 and 98-02-17453; that of A.N., V.N. and M.V. by INTAS-RFBR grant 95-05678 as well. We thank Susy Vascotto for the kind help in the preparation of the manuscript.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

LEP EWWG and SLD HFEWG, CERN-EP/99-15, 1999. P. Chankowski, A. Dabelstein, W. Hollik, W. MoK sle, S. Pokorsky, J. Rosiek, Nucl. Phys. B 417 (1994) 101. D. Garcia, R.J. Jimenez, J. Sola, Phys. Lett. B 347 (1995) 309, 321. W. de Boer, A. Dabelstein, W. Hollik, W. MoK sle, U. Schwickerath, Z. Phys. C 75 (1997) 627. J. Erler, D.M. Pierce, Nucl. Phys. B 526 (1998) 53. L. Alvarez-GaumeH , J. Polchinski, M. Wise, Nucl. Phys. B 211 (1983) 495; R. Barbieri, L. Maiani, Nucl. Phys. B 224 (1983) 32. I.V. Gaidaenko, A.V. Novikov, V.A. Novikov, A.N. Rozanov, M.I. Vysotsky, JETP Lett. 67 (1998) 761. V.A. Novikov, L.B. Okun, M.I. Vysotsky, Nucl. Phys. B 397 (1993) 35; M.I. Vysotsky, V.A. Novikov, L.B. Okun, A.N. Rozanov, Usp. Fiz. Nauk 166 (1998) 539. V. Novikov, L. Okun, A. Rozanov, M. Vysotsky, LEPTOP, preprint ITEP 19-95/CPPM-95-1. K. Hagiwara, H. Murayama, Phys. Lett. B 246 (1990) 533. M. Boulware, D. Finnell, Phys. Rev. D 44 (1991) 2054. S. Lammel, FERMILAB-CONF-98-055-E (1998). P. Chankowski, Proceedings of the Quantum e!ects in the MSSM, Barcelona 1997, p. 87, preprint IFT/97-18, hep-ph/9711470.

Physics Reports 320 (1999) 127}129

On non-supersymmetric CFT in four dimensions Nikita Nekrasov 
*, Samson L. Shatashvili Institute of Theoretical and Experimental Physics, 117259, Moscow, Russia
Jewerson Laboratory of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, Yale University, New Haven, CT 06520, USA

Abstract We show that the N"0 theories on the self-dual D3-branes of Type 0 string theory are in the class of the previously considered tadpole-free orbifolds of N"4 theory (although they have SO(6) global symmetry) and hence have vanishing beta function in the planar limit to all orders in 't Hooft coupling. Also, all planar amplitudes in this theory are equal to those of N"4 theory, up to a rescaling of the coupling.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.10.Kk; 11.25.Sq Keywords: Type 0 theories; Non-susy theories; CFT in four dimensions

1. Vanishing of the planar beta function Recently, the non-supersymmetric gauge theories were studied in the framework of string theory/gauge theory correspondence [1}3]. The case [4] of N"4 super-Yang}Mills theory can be modi"ed in several ways leading to the theories with lower supersymmetry. One possibility is to restrict the set of "elds of N"4 theory according to the following principle: take a discrete subgroup C of the R-symmetry group SO(6) and let it act on the color indices in some representation R : g C c 3;(dim R). Let U? denote a "eld of the N"4 theory. The index a belongs to the E representation of the R-symmetry group, and both color and Lorentz indices are omitted. Impose the invariance condition: U?"g? [c\U@c ] @ E E * Corresponding author. E-mail address: [email protected] (N. Nekrasov)  On leave of absence from St. Petersburg Steklov Mathematical Institute. 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 5 9 - 9

(1)

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N. Nekrasov, S.L. Shatashvili / Physics Reports 320 (1999) 127}129

for any g3C. It is known [5,6] that if the representation R is such that for any group element gO1 TrRc "0 (2) E then the planar graphs in the resulting theory are equal to those of the parent N"4 theory up to a rescaling of coupling. All these theories occur in the studies of the Type IIB theory compacti"ed on 1;1/C [7] and they were suggested as examples of large N conformal theories dual to certain string theory backgrounds [8]. They are known to be one-loop "nite for all C [9] while for higher loops they are known to have vanishing gauge beta functions in the planar limit (see [9] for two loops in the N"1 case [5,6] for general case). Klebanov and Tseytlin [2,3] suggested to study the D-branes in the Type 0B theory [10,11]. This theory has in the low-energy spectrum no fermions, the same NS bosons as Type IIB theory but the doubled set of RR bosons. The doubled set of RR "elds leads to the doubled set of D-branes. In particular, there are two types of D3-branes: electric and magnetic ones. It turns out that the low-energy theory on the stack of N D3-branes is a truncated version of the N"4 theory. In the case of pure electric or pure magnetic D3-branes one has a theory of ;(N) gluons coupled to six adjoint scalars while in the case of (1, 1) branes one has ;(N);;(N) gauge group, six adjoint scalars for each gauge factor and two sets of bi-fundamental fermions (N, NM ) and (NM , N). Those fermions correspond to the open strings connecting electric and magnetic branes [2,12]. In the paper [3] the two-loop "niteness of this self-dual theory was proven in the leading-N approximation. Also very interesting work has been done in [13]. The purpose of our paper is to point out that this theory has in fact vanishing planar beta function to all orders in 't Hooft coupling, just like any other orbifold theory studied in [5,6,8,9]. The point is that both the electric and self-dual theories are also orbifolds of N"4 theory. The slightly subtle point is that the R-symmetry group of N"4 theory is the spin cover of SO(6), i.e. S;(4). This group has a center Z+9 . The group C"9 LZ can also be used for orbifolding just   like any other subgroup of S;(4). By inserting projectors on the C-invariant "elds (1) into the 't Hooft diagrams one immediately sees that if the representation of C in the gauge group obeys (2) then the planar gauge coupling beta function vanishes. For the self-dual theory one starts with ;(2N) N"4 theory and represents C"+1, u, in the ;(2N) as follows:






1 0 , c " S 0 !1 where the blocks are N;N. Clearly this representation obeys (2). For the purely electric theory one starts with ;(N) N"4 gauge theory and takes the trivial representation of C which does not obey (2). Hence the theory has non-trivial beta function even in the large N limit. 2. Another curious theory Among the theories N"0 which were not explicitly considered in [9] there are also theories with C"9 . We would start with ;(4N) N"4 gauge theory and represent C"+1, u, u, u, 

N. Nekrasov, S.L. Shatashvili / Physics Reports 320 (1999) 127}129

129

as c "diag(1, g, g, g)1 . In this case, the orbifold group acts on the scalars by changing the E ,", sign of all six of them. The "eld content of the resulting theory is gauge group ;(N), two sextets of bi-fundamental scalars (N , NM ), four sets of bi-fundamental fermions: (N , NM ), i"0, 1, 2, 3, G G> G G> 4,0. This theory has a string theory realization. It is the theory of threebranes of Type IIB at the singularity of the form 1/9 where 9 re#ects all the coordinates. It can be also viewed as   9 orbifold of the Type 0 theory. The space}time aspects as well as its AdS ;1/ dual realization   will be discussed elsewhere [14]. Note added. After completion of the manuscript, a paper [15] appeared which studies further aspects of self-dual threebranes in type 0 string theory.

Acknowledgements We are grateful to I. Klebanov for discussions. The research of N. N. is supported by Harvard Society of Fellows, partly by NSF under grant PHY-98-02-709, partly by P''N under grant 98-01-00327 and partly by grant 96-15-96455 for scienti"c schools. The research of S. S. is supported by DOE grant DE-FG02-92ER40704, by NSF CAREER award, by OJI award from DOE and by Alfred P. Sloan foundation.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

A.M. Polyakov, Nucl. Phys. Proc. Suppl. 68 (1998) 1; Int. J. Mod. Phys. A 14 (1999) 645. I. Klebanov, A. Tseytlin, Nucl. Phys. B 546 (1999) 155; B 547 (1999) 143. I. Klebanov, A. Tseytlin, JHEP 9903 (1999) 015. J. Maldacena, Mod. Phys. Lett. A 13 (1998) 219. M. Bershadsky, Z. Kakushadze, C. Vafa, Nucl. Phys. B 523 (1998) 59. M. Bershadsky, A. Johansen, Nucl. Phys. B 536 (1998) 141. M.R. Douglas, G. Moore, hep-th/9603167. S. Kachru, E. Silverstein, Phys. Rev. Lett. 80 (1998) 4855. A. Lawrence, N. Nekrasov, C. Vafa, Nucl. Phys. B 533 (1998) 199. L. Dixon, J. Harvey, Nucl. Phys. B 274 (1986) 93. N. Seiberg, E. Witten, Nucl. Phys. B 276 (1986) 272. O. Bergman, M. Gaberdiel, Nucl. Phys. B 499 (1997) 183. J. Minahan, hep-th/9811156, JHEP 9904 (1999) 007. I. Klebanov, N. Nekrasov, S. Shatashvili, in preparation. A. Tseytlin, K. Zarembo, Phys. Lett. B 457 (1999) 77.

Physics Reports 320 (1999) 131}138

Nonperturbative QCD vacuum and colour superconductivity N.O. Agasian*, B.O. Kerbikov, V.I. Shevchenko Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia

Abstract We discuss the possibility of existence of colour superconducting state in real QCD vacuum with nonzero 1a GG2. We argue, that nonperturbative gluonic "elds might play a crucial role in colour superconductivity  scenario.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Lg; 12.38.!t Keywords: Colour superconductivity; Gluon condensate

We wish to start with a few introductory words. We formulate the problem in the paper which we cannot solve, at least now, without resorting to simple, sometimes naive estimates. Not everybody accepts such an approach and believes in it. Probably, the success or failure depends here on the insight and physical intuition. Inspiring example of solving complicated problems by surprisingly simple tools can be found in many works of Lev Okun. But sure the fact that we are lucky to work near Lev Borisovich at ITEP does not mean that his favourite tool would work in our hands. It is the reader of the paper who will judge.

1. Introduction The behaviour of QCD at high density has become recently a compelling subject due to (re)discovery of colour superconductivity [1,2]. The essence of the phenomenon is the formation

* Corresponding author. E-mail addresses: [email protected] (N.O. Agasian), [email protected] (B.O. Kerbikov), shevchen@ heron.itep.ru (V.I. Shevchenko) 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 8 0 - 0

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of the BCS-type diquark condensate at densities exceeding the normal nuclear density [3] by a factor (2}3). Colour superconductivity has been studied within di!erent versions of the Nambu}Jona}Lasinio-type model [1,4,3] or the instanton model [2,5]. To our knowledge, however, no attention has been paid to the fact, that as compared to real QCD, both approaches miss important nonperturbative gluonic content of the theory. The NJL model contains the gluon degrees of freedom in a very implicit way: it is argued, that high-frequency mode (one-gluon exchange) after being integrated out gives rise to the e!ective four quark interaction (see, e.g. [6,7]). The instanton model deals only with speci"c "eld con"gurations * instantons and antiinstantons. The NJL model with gluon condensate included has been considered in [8] while con"ning background superimposed on instantons has been treated in [9]. The role played by the gluon degrees of freedom in the problem under consideration is essentially twofold. First, it is assumed, that they are responsible for producing the quark}quark attractive interaction leading at small enough temperatures to the Cooper pairing (thus gluons play the role of phonons over ion lattice speaking in condensed matter terms). On the other hand, the vacuum gluon "eld #uctuations should be a!ected by colour superconducting state itself in a way analogous to the Meissner e!ect in ordinary superconductor. The crucial point is which force will win, i.e. whether superconductivity will survive or will be destroyed by gluonic "elds as it happens in the standard BCS theory in the presence of strong enough external magnetic "eld. The studies of idealized QCD performed by several authors have shown, that depending on the chemical potential k and the temperature ¹ the system displays three possible phases: (1) chiral symmetry breaking without diquark condensation; (2) mixed phase with nonzero values of both chiral and diquark condensates; (3) diquark condensation without symmetry breaking. The system may be described by the thermodynamic potential X( ,D;k,¹), where and D are order parameters, related to the chiral and diquark condensates, respectively (explicit de"nitions will be given below). The potential X is expressed in terms of quantum e!ective action C as X"C¹/< .  Suppose, that the potential X has been calculated within the framework of some NJL-type model, i.e. with the gluon sector excluded (except for contribution of high-frequency modes, giving the necessary attraction). Consider for simplicity the phase (3) of the system with "0, DO0 and ¹"0. Let X( "0,D ;k,0) be the stationary value of the thermodynamic potential, where D is   the solution of the gap equation *X/*D"0 (see below). Now we superimpose the nonperturative vacuum gluon "elds on the above picture. The detailed knowledge of the nonabelian Meissner e!ect is unfortunately absent. Anyway, it is obvious that the corresponding microscopic picture is far from being trivial. The nonlinear character of the equations of motion for gluon "elds is of prime importance here, while usual Meissner e!ect for abelian "elds is essentially linear phenomenon. On the contrary, general symmetry arguments tell us, that part of gluon degrees of freedom becomes massive if the colour gauge invariance is spontaneously broken. It means e!ective screening of low-frequency modes and therefore it is reasonable to assume, that the formation of the colour gauge invariance * breaking diquark condensate should lead to the decrease of the gluon condensate by some factor, which we assume to be about a few units (but, presumably, not to exactly zero value, as it happens in abelian case). Then it will be energetically favourable for the system to remain in the colour superconducting state with DO0 only if the quantity e(i)"!(1!1/i)[b(a )/16a ]1G G 2   IJ IJ

(1)

N.O. Agasian et al. / Physics Reports 320 (1999) 131}138

133

is less than X( "D"0;k,0)!X( "0,D ;k,0). The factor i in (1) represents the unknown rate of  decrease of the gluon condensate due to superconducting diquark state formation. Note, that e(iPR)"!e . In what follows we will show, that these two energy gaps have the same order  of magnitude unless the i is close to 1. It should be noted, that the analogy with the BCS superconductor in external "eld is somewhat loose here for at least two reasons. First, strong nonperturbative gluonic "elds are inherent for the QCD vacuum. The consistent way of analysis should include a set of gap equations for the free energy of the system depending on gluon and di!erent types of quark condensates, determining energetically best values for all of them simultaneously. The second point is the relation between scales, characterizing colour superconductivity and nonperturbative gluon #uctuations. In particular, only modes with the wavelengths larger than the Cooper pair radius are responsible for the supercurrent while the rest do not admit simple interpretation in terms of Ginzburg}Landau theory. We leave the analysis of these complicated problems for the future. Another important remark is in order. It might be naively assumed, that if the system under study is in the decon"nement region, the vacuum gluonic content may be taken as purely perturbative. There are several reasons, however, why it is not the case. The most important one is the following. Finite density breaks Euclidean O(4) rotational invariance and hence chromoelectric and chromomagnetic components of the correlator 1a G2 enter on the di!erent footing. In  I$ particular, decon"nement, i.e. zero string tension is associated with the vanishing of the electric components, while it is energetically favourable for the magnetic ones to stay nonzero (the same phenomenon takes place for the temperature phase transition [10]). At the same time it is just strong magnetic "eld, which is able to destroy the superconductivity.

2. General formalism We start with the QCD Euclidean partition function



Z" DA DtM Dt exp(!S) ,

(2)

where





1 S" F F dx# tM (!ic D !im#ikc )t dx . I I  4g IJ IJ

(3)

We supress colour and #avour indicies and also introduce chemical potential k (only the case N "2; N "3 is considered in this paper). Performing integration over the gauge "elds one gets   e!ective fermion action in terms of cluster expansion








Z" DtM Dt exp ! dx ¸ !S  

(4)

with ¸ "tM (!ic * !im#ikc )t and e!ective action S "  11hL22/n! where h"  I I   L  dx tM (x)c A (x)t t(x) and double brackets denote irreducible cumulants. I I

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To proceed, one is to make considerable simpli"cation of Eq. (4). First, only the lowest, four-quark interaction term is usually kept in S . Second, it is instructive to consider instead of the  original nonlocal kernel some idealized local one, respecting the given set of symmetries. In the problem under study it is common to choose either instanton-induced four-fermion vertex (the choice, adopted in [2,5]) or di!erent versions of the NJL model [1,4,3] including the one, motivated by one gluon exchange [4]. In the latter case, one gets



S " dx dy(tM (x)c t t(x))(tM (y)c t t(y))D(x!y) I I 

(5)

with D(x!y)"Gd(x!y) and where the coupling constant G with the dimension [m]\ is introduced. We assume, as it has already been mentioned, that this localized form of the kernel does not mimic all gluonic content of the original theory, in other words only some part of gluon degrees of freedom participates in the condensate formation. One way to analyse the role, played by other ones, would be to consider more realistic nonlocal functions D(x!y) (which, in principle, encode all necessary information if we keep only four-quark interaction). This will be done elsewhere, while in the present paper we work with the local form of the action. Performing colour, #avour and Lorentz Fierz [7] transformations and keeping only scalar terms in both tM t and tt channels, we arrive at ¸ "G[(tM (x)Kt(x))(tM (y)Kt(y))!(tM (x)U t!(y))(tM !(y)U t(x))] ,  ? ?

(6)

where K"(i/(6)1) q ,  $

U "(1/(12) e c q , ? ?@A  $

and t!"CtM 2"c c tM 2. We note, that with only scalar terms kept, Lagrangian (6) is no more   chiral invariant. The attraction in scalar colour antitriplet channel (which also exists if one starts from the instanton-induced interaction) could lead to the formation of the condensate, breaking colour S;(3). In close anology with [3] we replace the common coupling constant G by two independent constants G and G corresponding to the two terms in (6). Next step is to write   down the partition function and to perform its bosonization. We adopt the standard Hubbard}Stratonovich trick and get






Z" DtM Dt exp ! dx(¸ #¸ )  








" D DD DDR exp




# Tr Ln

dx+![ /4G#DDR/G ]  

i*K #i(m# )!ic k  i*K 2!i(m# )#ic k 2DRC\UR  2UCD



.

(7)

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135

For the system of the massless quarks at the phase (3) and ¹"0 the thermodynamic potential reads (we remind, that Z&exp(!S)&exp(#C))






2UCD i*K !ic k "D" 1 C  ! Tr Ln . X" " g < < i* K 2#ic k 2DRC\UR     The value of the diquark condensate is determined by the gap equation

(8)



*X "0 . (9) *D D D  By going in (8) to wave number-frequency space and making use of the gap Eq. (9), we arrive at the following expression for the thermodynamic potential at its minimum X( "D"0; k, 0)!X( "0, D ; k, 0)KD/[G ln(M/D)] , (10)     where M is the NJL cuto! which is typically about 0.8 GeV [1,3]. Alternatively, the cuto! may enter via the formfactor or the instanton zero-mode. In the NJL-type calculations the values of the cuto! M and the coupling constant G are "tted simultaneously, but no unique `standarda "t exists  so far [11}13]. To estimate the r.h.s. of (10) we have taken for cuto! M"0.8 GeV, for coupling G"12G"(15}40) GeV\ and the value of the gap in the diquark scalar sector  D "(0.1}0.15) GeV. With these parameters, one gets  X( "D"0; k, 0)!X( "0, D ; k, 0)&(1}5) ) 10\ GeV . (11)  To be on a robust quantitative footing and to consider "nite temperatures one can replace estimate (11) by the result of direct numerical calculations of the thermodynamic potential performed in [3]. Our result (11) is larger than the corresponding value, presented in [3]. The discrepancy may be due to di!erent value of the coupling constant g adopted in [3]. Needless to say, that the larger is the estimate of (10) the larger is the critical "eld extinguishing superconductivity. Now, let us estimate expression (1). We use di!erent sets of data from [14}16] and take for the gluon condensate





a G " G G "(0.014}0.026) GeV.  p IJ IJ Then for two #avours one gets in one loop









1 1 4 e(i)" 1! ) 11! ) 32 i 3

a G G p IJ IJ

1 & 1! ) (4}8);10\ GeV . i

 (12)

It is seen, that (10) and (12) have the same order of magnitude for i52. It should be noted, that estimate (11) given above is rather optimistic in the following sense. If one naively assume the colour superconductor to be the BCS one and take its typical parameters, for example, from [2], then one has at ¹"0 for the value of the critical magnetic "eld H "0.64(D )(m p ) ,     $

(13)

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where p"k!m, m is the constituent quark mass. We take k"(0.4}0.5) GeV and corre$   sponding D "(0.04}0.10) GeV from the paper [2] which are smaller, than the typical values we  have analysed before. The maximum of (13) with respect to p at the "xed k is reached for $ k"(2p and it gives $ H "(0.8}8);10\ GeV .  

(14)

Strictly speaking, we are not allowed to apply formulas like (13) to the colour "elds and interpret them in terms of Meissner e!ect, but we note, that (14) even without any numerical factors is about the order of magnitude smaller than (12).

3. Finite density e4ects The comparison made in the previous section was intuitively based on the analogy with the Meissner e!ect in ordinary superconductors. The actual value of the gluon condensate in (12) was taken to be the vacuum one. This is not quite correct, however. Even without the formation of any diquark condensate, the gluon condensate in the hadronic matter is di!erent from that in the vacuum. In order to get an idea about such dependence, let us consider e!ective dilaton Lagrangian [17,18] ¸(p)"(* p)!<(p), <(p)"(j/4)p(ln p/p !) ,  I  

(15)

where dilaton "eld is de"ned according to (m/64"e ")p(x)"!h (x)"![b(a )/4a ]G (x)G (x) ,   II   IJ IJ and j"m/16"e ", p"16"e "/m ,      where e "1h 2 is the nonperturbative vacuum energy density and m the dilaton (i.e. glueball   II  0>> in our case) mass. Low-energy dilaton physics can be used for description of the gluon condensate behaviour at "nite density and temperature [19,20]. In the chiral limit masses of nucleons in QCD are determined by the nucleon}dilaton vertex ¸ "mHq q with the e!ective mass  , mH"m ) (p/p ). In isotopically symmetric system the energy density takes the form: , , 



e(p ,n)"<(p )!<(p )#4 L L 

N$ dp (p#mH , , (2p) 

(16)

where n"n #n "2p/3p is the baryon density and the chemical potential k"p#mH. L N $ $ , Being interested in the densities close to the nuclear density n"0.1 GeV we minimize the total  energy density with respect to the dilaton "eld



*e *p

"0 ,

NN

L

N.O. Agasian et al. / Physics Reports 320 (1999) 131}138

and "nd in the region nm ;16"e " ,  a 1(a /p)G G 2 "  G G  IJ IJ L p IJ IJ





) (1!cn/n ) , 

137

(17)

 where the factor c"m n /4"e "K0.05. It means, that the decrease of the condensate at the densities ,   of interest is about (10}25)%. However, we should point out, that the matter may have already the structure of cold quark}gluon plasma instead of being hadronic at the densities of interest. Presumably, it would require another type of estimates for the value of the gluon condensate. The reader is referred to the paper [21] where it is argued, that the long-wave part of colour magnetic "eld which can lead to the Meissner e!ect is suppressed compared to the critical "eld by the perturbative coupling constant (which is assumed to be small due to asymptotic freedom). It is important to distinguish the quantum "elds considered in [21] which propagate in the quark}gluon plasma from the nonperturbative "elds discussed by us here, hence the magnitude of the latter is an `external parametera with respect to the colour superconductivity problem itself. This question needs further investigation.

4. Discussion We have con"ned ourselves in this letter to a very modest aim * to compare the energy gap, typical for the colour superconductor and the contribution to the vacuum energy density, coming from the gluon condensate. We argued, that if there is a competition between colour antitriplet scalar diquark and gluon condensates, which is natural to assume from abelian analogy, then it happens on the same energy scale and therefore to take this e!ect into account is important for the self-consistent picture. It is clear, that further, more quantitative analysis is needed. The idealized case of two #avours was investigated in the paper. For chemical potentials exceeding the strange quark mass the N "N "3 scenario is more physical. It seems, that   inclusion of the third #avour does not change our analysis crucially. Moreover, due to the phenomenon of colour}#avour `lockinga [4], eight of the nine ((N!1) #1 ) gauge degrees  13 3 of freedom acquire mass due to the Higgs mechanism in N "3 case and therefore colour  superconductivity is `completea. According to the line of reasoning adopted in this paper, it means, that i(N "3) should be larger than i(N "2) (for the scalar channel).   To conclude, we have discussed the role played by gluon degrees of freedom in the colour superconductivity and, in particular, we have argued, that nonperturbative gluon #uctuations might be strong enough to destroy the phenomenon. More quantitative analysis of this problem is in progress now.

Acknowledgements The work was supported in part by RFFI-DFG grant 96-02-00088G. The work of B.K. was also supported by RFFI grant 97-02-16406. V.Sh. acknowledges the ICFPM-INTAS 96-0457 grant.

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The authors are grateful to I. Dobroskok, D. Ebert, J. Hos\ ek, H.-J. Pirner, K. Rajagopal and Yu.A. Simonov for useful discussions and comments. V.Sh. would like to thank the theory group of R[ ez\ Nuclear Physics Institute and especially J. Hos\ ek for their kind hospitality.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

M. Alford, K. Rajagopal, F. Wilczek, Phys. Lett. B 422 (1998) 247. R. Rapp, T. Schaefer, E. Shuryak, M. Velkovsky, Phys. Rev. Lett. 81 (1998) 53. J. Berges, K. Rajagopal, Nucl. Phys. B 538 (1999) 215. M. Alford, K. Rajagopal, F. Wilczek, Nucl. Phys. B 537 (1999) 443. G. Carter, D. Diakonov, Nucl. Phys. A 642 (1998) 78. J. Bijnens, C. Bruno, E. de Rafael, Nucl. Phys. B 390 (1993) 501. D. Ebert, H. Reinhardt, M. Volkov, Progr. Part. Nucl. Phys. 33 (1994) 1. D. Ebert, M. Volkov, Phys. Lett. B 272 (1991) 86. Yu. Simonov, Yad. Fiz. 57 (1994) 1491. E. Gubankova, Yu. Simonov, Phys. Lett. B 360 (1995) 93. S. Klevansky, Rev. Mod. Phys. 64 (1992) 649. U. Vogl, W. Weise, Progr. Part. Nucl. Phys. 27 (1992) 195. G. Ripka, Quarks Bound by Chiral Fields, Clarendon Press, Oxford, 1997. H.G. Dosch, Proceedings of the International School of Physics `Enrico Fermia, Selected Topics in Nonperturbative QCD, IOS Press, Amsterdam, 1996. M. D'Elia, A. DiGiacomo, E. Meggiolaro, Phys. Lett. B 408 (1997) 315. S. Narison, Phys. Lett. B 387 (1996) 162. A.A. Migdal, M.A. Shifman, Phys. Lett. B 114 (1982) 445. J. Ellis, J. Lanik, Phys. Lett. B 150 (1985) 289. N.O. Agasyan, JETP Lett. 57 (1993) 208. N.O. Agasyan, D. Ebert, E. Ilgenfritz, Nucl. Phys. A 637 (1998) 135. D.T. Son, Phys. Rev. D 59 (1999) 094019.

Physics Reports 320 (1999) 139}146

Quantum corrections from nonresonant == scattering Michael S. Chanowitz Theoretical Physics Group, Ernest Orlando Lawrence Berkeley National Laboratory, University of California, Berkeley, CA 94720, USA

Abstract An estimate is presented of the leading radiative corrections to low energy electroweak precision measurements from strong nonresonant == scattering at the TeV energy scale. The estimate is based on a novel representation of nonresonant scattering in terms of the exchange of an e!ective scalar propagator with simple poles in the complex energy plane. The resulting corrections have the form of the corrections from the standard model Higgs boson with the mass set to the unitarity scale for strong == scattering.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.15.Lk; 11.15.Ex; 12.60.Fr Keywords: Radiative corrections; Strong == scattering

I "rst met Lev Okun at the 1976 `Rochestera conference held in the USSR, in Tbilisi, Georgia. Sakharov was under strong attack by the government for his human rights activities and was originally not invited but was permitted to attend after he protested the lack of an invitation to the Soviet Academy. Understandably even those Soviet physicists who were sympathetic to Sakharov and his ideas were cautious about associating with him during the meeting. While there may well have been others I did not observe, to me it was remarkable to see one Soviet physicist who did not hesitate to stroll openly with Sakharov on the streets of Tiblisi. This was of course Okun. His behavior then demonstrated the same simple idealism and courage that is re#ected now in the decision he has taken since the dissolution of the USSR to remain in Moscow, to preserve the unique physics environment at ITEP, when he could easily have accepted more comfortable positions outside of Russia.

E-mail address: [email protected] (M.S. Chanowitz) 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 7 7 - 0

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Not unrelated to his moral character is the clarity, depth and humanity with which Okun practices physics. This gives me a sel"sh reason for submitting the work presented here: I would like to have his view of it. It has a plausible conclusion reached by a strange method and raises questions I do not understand. It is based on a representation of an exactly unitary model of nonresonant == scattering in terms of an e!ective scalar propagator with simple poles in the complex energy plane. The method was applied and veri"ed for tree approximation amplitudes and is used here to estimate quantum corrections.

1. Introduction The electroweak symmetry may be broken by weakly coupled Higgs bosons below 1 TeV or by a new sector of quanta at the TeV scale that interact strongly with one another and with longitudinally polarized = and Z bosons. Precision electroweak data favors the "rst scenario [1,2], but the conclusion is not de"nitive, because the relevant quantum corrections are open to contributions from many forms of new physics. Occam's (an archaic spelling of Okun's?) razor favors the simplest interpretation, which assumes that the only new physics contributing signi"cantly are the quanta that directly form the symmetry breaking condensate. In that case the data do favor weak symmetry breaking by Higgs scalars. But nature may have dealt us a more complicated hand, with other, probably related, new physics also contributing to the radiative corrections. Then the precision data tells us nothing about the symmetry breaking sector * unless we can `unscramblea the di!erent contributions, which in general we do not know how to do * and implementation of the Higgs mechanism by strong, dynamical symmetry breaking remains a possibility. The nature of the symmetry breaking sector can only be established de"nitively by its direct discovery and detailed study in experiments at high energy colliders. Strong == scattering is a generic feature of strong, dynamical electroweak symmetry breaking [3]. The longitudinal polarization modes = scatter strongly above 1 TeV because the enforce* ment of unitarity is deferred to the mass scale of the heavy quanta that form the symmetry-breaking condensate. To the extent that QCD might be a guide to dynamical symmetry breaking we expect the a partial wave to smoothly saturate unitarity between 1 and 2 TeV. Like the SM (standard  model) Higgs boson, nonresonant strong == scattering would also contribute to the low-energy radiative corrections probed in precision electroweak measurements. This note presents an estimate of those corrections, based on a novel representation of nonresonant strong == scattering as an e!ective-Higgs boson exchange amplitude. Strong == scattering models are customarily formulated in R-gauges. The e!ective-Higgs representation allows them to be reexpressed gauge invariantly and, in particular, in unitary gauge [4,5]. It applies to the leading s-wave amplitudes with I"0,2. The e!ective-Higgs representation has a signi"cant practical advantage: it predicts the experimentally important transverse momentum distributions of the "nal state quark jets and the == diboson in the collider process qqPqq==, which cannot be obtained from the conventional method based on the e!ective = approximation [1,6}8]. The method has been veri"ed numerically for tree amplitudes [4] and gauge (i.e., BRST) invariance has been demonstrated [5]. The K-matrix model is a useful model of strong == scattering which smoothly extrapolates the == low-energy theorems [3,9,10] in a way that exactly satis"es elastic unitarity. The

M.S. Chanowitz / Physics Reports 320 (1999) 139}146

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e!ective-Higgs representation of the K-matrix model has a surprisingly simple form: the singularities of the propagator are simple poles in the complex s plane, like an elementary scalar. It is then easy to compute the contribution to the = and Z vacuum polarization tensors from which the `obliquea corrections [11,12] are obtained. The "nal result for the oblique parameters S and ¹ is like the SM Higgs contribution with m replaced by a combination of the unitarity scales for strong scattering in the I"0,2 channels, & determined in turn by the low-energy theorems as noted in [3]. S and ¹ are given by



 



16pv 32pv 1 1 ln # ln S" k k 2 18p

,

16pv 32pv 1 !1 ln # ln ¹" k k 2 8p cos h 5

(1) ,

(2)

where v"((2G )\, h is the weak interaction mixing angle and k is the reference scale. For $ 5 k"1 TeV the corrections are SK0.036 and ¹K!0.11. Similar results follow from the cut-o! nonlinear sigma model when the unitarity scales are used for the cuto!s [13,14]. In the following sections I review the K-matrix model, derive the e!ective scalar propagator, deduce the oblique corrections, raise some theoretical issues, and "nally discuss the physical interpretation of the result.

2. K-matrix model for WWPZZ In the SM the Higgs sector contribution to ==PZZ is given by just the s-channel Higgs pole. Therefore, we use the K-matrix model for ==PZZ to abstract the e!ective-Higgs propagator. The model is summarized in this section. As is conventional we use the ET [3,15}18] (see the review cited in [3] for additional references) (equivalence theorem) to de"ne the model in terms of the unphysical Goldstone bosons, w! and z. Partial wave unitarity is conveniently formulated as Im 1/a "!1 . (3) '( The K-matrix model is constructed to satisfy the low-energy theorems and partial wave unitarity. It is de"ned by 1/a) "1/R !i , '( '( where R are the real threshold amplitudes that follow from the low-energy theorems, '( R "s/16pv ,  R "!s/32pv .  The corresponding s-wave T-matrix amplitudes are M)(s)"16pa) ' '

(4)

(5a) (5b)

(6)

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for I"0,2. Finally, the wwPzz amplitude is M)(w>w\Pzz)"(M)!M))   

(7)

3. E4ective-Higgs propagator To obtain the e!ective-Higgs propagator we `transcribea the K-matrix model from R-gauge to U-gauge [4,5]. The heart of the matter is to "nd the contribution of the symmetry-breaking sector in U-gauge, which encodes the dynamics speci"ed in the original R-gauge formulation of the model. This is accomplished using the ET as follows. Suppose that the longitudinal gauge boson modes scatter strongly. At leading order in the weak gauge coupling g we write the amplitude =>=\PZZ as a sum of gauge-sector and Higgs-sector * * terms, M "M #M , (8) 2 %  1 where SB denotes the symmetry-breaking (i.e., Higgs) sector. Gauge invariance ensures that the contributions to M that grow like E cancel, leaving a sum that grows like E, given by %  M "g(E/om )#O(E, g) (9) %  5 where o"m /(cos h m). The neglected terms of order E and of higher order in g include the 5 5 8 electroweak corrections to the leading strong amplitude. The order E term in Eq. (9) is the residual `bad high-energy behaviora that is cancelled by the Higgs mechanism. It is also precisely the low-energy theorem amplitude, M "s/ov"M #O(s, g) (10) *#2 %  using m "gv/2 and s"4E. Eqs. (8) and (9) may be used to derive the low-energy theorem 5 without invoking the ET. Now consider an arbitrary strong scattering model, designated as model `Xa, formulated in the usual way in an R-gauge in terms of the unphysical Goldstone bosons, M6 (wwPzz). The %  total gauge boson amplitude is gauge invariant and the ET tells us that for E<m it is 5 approximately equal to the Goldstone boson amplitude, i.e., M6 (= = )KM6 (ww) (11) 2 * * %  in the same approximation as Eq. (9). Eq. (8) holds in any gauge. Specifying U-gauge we combine it with Eqs. (9)}(11) to obtain the U-gauge Higgs sector contribution for model X, M6 (= = )"M6 (ww)!M . 1 * * %  *#2

(12)

 If the symmetry-breaking force is strong, the quanta of the symmetry-breaking sector are heavy, m <m , and 1 5 decouple in gauge boson scattering at low energy, M ;M . Then the quadratic term in M dominates 1 %  %  M for m ;E;m , which establishes the low-energy theorem without using the ET [9,10]. 2 5 1

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The preceding result applies to any strong scattering amplitude. Now we specialize to s-wave ==PZZ scattering and use Eq. (12) to obtain an e!ective-Higgs propagator with standard `Higgsa-gauge boson couplings. Neglecting m ;s and higher orders in g as always, the e!ective 5 scalar propagator is P (s)"!(v/s)M6 (= = ) 6 1 * *

(13)

Eqs. (10) and (12) with o"1 then imply P (s)"!(v/s)M6(ww)#1/s 6 0

(14)

The term 1/s, corresponding to a massless scalar, comes from M in Eq. (12). It ensures good *#2 high-energy behavior while the other term in Eq. (14) expresses the model-dependent strong dynamics. Finally, we substitute the K-matrix amplitude, Eq. (7), into Eq. (14) to obtain the e!ective propagator for the K-matrix model as the sum of two simple poles






1 1 1 2 # , P " ) 3 s!m 2s!m  

(15)

where m and m are   m"!16piv 

(16)

m"#32piv . 

(17)

and

It is surprising to "nd such a simple expression involving only simple poles. It is not surprising that the poles are far from the real axis since they describe nonresonant scattering. Interpreted heuristically as Breit}Wigner poles they correspond to resonances with widths twice as big as their masses.

4. Oblique corrections The oblique corrections are evaluated from the vacuum polarization diagrams that in the SM include the Higgs boson [11]. In place of the SM propagator, P "1/(s!m ), we substitute 1+ & P from Eq. (15). Where the SM contribution depends on the log of the Higgs boson mass, ) ¸ "ln(m /k), we now "nd instead the combination ¸ , 1+ & )










2 1 m m m ¸ "ln & P¸ " ln  # ln  , ) 3 1+ 3 k k k where m are complex masses de"ned in Eqs. (16)}(17). 

(18)

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The results quoted in Eqs. (1) and (2) follow from the usual expressions for S,¹ where we use the real part of ¸ in place of ¸ , ) 1+ S"Re(¸ )/12p (19) ) and ¹"!3 Re(¸ )/16p cos h . (20) ) 5 The imaginary part of ¸ is an artifact which we discard; it results from the fact that our ) approximation neglects the = mass, as in any application of the ET. At q"0, where the oblique corrections are computed, there is no contributution to the imaginary part of the vacuum polarization from the relevant diagrams. Combining the I"0 and 2 terms in Eq. (18) we have Re(¸ )"ln(216pv/k) . (21) ) Evaluating Eq. (21) we "nd that the oblique correction from the K-matrix model is like that of a Higgs boson with mass 2.0 TeV.

5. Questions The I"2 component of the e!ective propagator has peculiar properties, perhaps due to the fact that for the I"2 channel we are representing t- and u-channel dynamics by an e!ective s-channel exchange. The minus sign in the I"2 low-energy theorem, Eq. (5b), which may be thought of as arising from the identity t#u"!s, leads to interesting di!erences between the I"0 and I"2 components of the e!ective propagator P . ) First, the I"2 component of the e!ective scalar propagator has a negative pole residue, which would correspond to a unitarity violating ghost if it described an asymptotic state (which it does not). In fact the sign is required to ensure unitarity, since it is needed to cancel the bad high-energy behavior of the gauge sector amplitude which has a negative sign in the I"2 channel. In Eq. (15) for P the I"2 pole appears with a positive sign because of a second minus sign from the isospin ) decomposition, Eq. (7). Neither pole of the e!ective propagator has a negative (ghostly) residue. In any case the amplitude is exactly unitary by construction. The sign di!erence between the pole positions, m and m in Eqs. (16) and (17), may also be   traced to the phases of the low-energy theorems in Eq. (5). The position of m on the negative  imaginary axis of the complex s plane corresponds to poles in the fourth and second quadrants of the complex energy plane, consistent with causal propagation as in the conventional m!ie prescription. But the position of m on the positive imaginary axis corresponds to poles in the "rst  and third quadrants of the complex energy plane. This would imply acausal propagation if the poles are on the "rst sheet but not if they are on the second sheet. Working in the limit of massless external particles as we are it is not apparent on which sheet they occur.

 I thank Henry Stapp for a discussion of this point.

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I conclude that the sign of the pole residue arising from the I"2 amplitude is not problematic but that the implications of the pole position requires better understanding.

6. Physical interpretation We have used a convenient representation of the K-matrix model to estimate the low-energy radiative corrections from strong == scattering. The result that the corrections are like those of a Higgs boson with mass at the unitarity scale is plausible and agrees with an earlier estimate using the cut-o! nonlinear sigma model [13,14]. The estimate establishes a &default' radiative correction from the strongly coupled longitudinal gauge bosons in theories of dynamical symmetry breaking. In general there will be additional contributions from other quanta in the symmetry breaking sector. Those contributions are model dependent as to magnitude and sign. In computing their e!ect it is important to avoid double-counting contributions that are dual to the contribution considered here. Current SM "ts to the electroweak data prefer a light Higgs boson mass of order 100 GeV with a 95% CL upper limit that I will conservatively characterize as :300 GeV [2]. Since the corrections computed here are equivalent to those of a Higgs boson with a mass of 2 TeV, they are excluded at 4.5 standard deviations. Therefore, there must be additional, cancelling contributions to the radiative corrections from other quanta in the theory if strong == scattering occurs in nature. This would not require "ne-tuning although it would require a measure of serendipity. There are good reasons for the widespread view that a light Higgs boson is likely and for the popular designation of SUSY (supersymmetry) as The People's Choice. But SUSY also begins to require a measure of serendipity [19}21] to meet the increasing lower limits on sparticle and light Higgs boson masses. While the community of theorists has all but elected SUSY, the question is not one that can be decided by democratic processes. At the end of the day only experiments at high-energy colliders can tell us what the symmetry breaking sector contains. Collider experiments, particularily those at the LHC, should be prepared for the full range of possibilities, including the capability to measure == scattering in the TeV region.

Acknowledgements I wish to thank David Jackson, and Henry Stapp for helpful discussions. This work was supported by the Director, O$ce of Energy Research, O$ce of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contracts DE-AC0376SF00098.

References [1] M.I. Vysotsky, V.A. Novikov, L.B. Okun, A.N. Rozanov, Phys. Atom. Nucl. 61 (1998) 808}811; Yad. Fiz. 61 (1998) 894}897.

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[2] The LEP collaborations, LEP EWWG, and SLD Heavy Flavour and Electroweak Groups, CERN-EP/99-15, 1999. [3] M.S. Chanowitz, M.K. Gaillard, Nucl. Phys. B 261 (1985) 379. For a recent review including a discussion of experimental signals see M. Chanowitz, in: D. Graudenz, Proceedings of the Summer School on Hidden Symmetries and Higgs Phenomena, p. 81, (Zuoz, Switzerland, August 1998) PSI-Proceedings 98-02 and eprint hepph/9812215. [4] M.S. Chanowitz, Phys. Lett. B 373 (1996) 141, hep-ph/9512358. [5] M.S. Chanowitz, Phys. Lett. B 388 (1996) 161, hep-ph/9608324. [6] M.S. Chanowitz, M.K. Gaillard, Phys. Lett. 142 B (1984) 85. [7] G. Kane, W. Repko, B. Rolnick, Phys. Lett. B 148 (1984) 367. [8] S. Dawson, Nucl. Phys. B 29 (1985) 42. [9] M.S. Chanowitz, M. Golden, H.M. Georgi, Phys. Rev. D 36 (1987) 1490. [10] M.S. Chanowitz, M. Golden, H.M. Georgi, Phys. Rev. Lett. 57 (1986) 2344. [11] M.E. Peskin, T. Takeuchi, Phys. Rev. D 46 (1991) 381. [12] G. Altarelli, R. Barbieri, Phys. Lett. B 253 (1991) 161. [13] M.K. Gaillard, Phys. Lett. B 293 (1992) 410. [14] O. Cheyette, Nucl. Phys. B 361 (1988) 183. [15] J.M. Cornwall, D.N. Levin, G. Tiktopoulos, Phys. Rev. D 10 (1974) 1145. [16] C.E. Vayonakis, Lett. Nuovo Cimento. 17 (1976) 383. [17] B.W. Lee, C. Quigg, H. Thacker, Phys. Rev. D 16 (1977) 1519. [18] H-J. He, W. Kilgore, Phys. Rev. D 55 (1997) 1515, hep-ph/9609326. [19] P.H. Chankowski, J. Ellis, S. Pokorski, Phys. Lett. B 423 (1998) 327}336, e-Print Archive: hep-ph/9712234. [20] R. Barbieri, A. Strumia, Phys. Lett. B 433 (1998) 63-66, e-Print Archive: hep-ph/9801353. [21] P.H. Chankowski, J. Ellis, M. Olechowski, S. Pokorski, Nucl. Phys. B 544 (1999) 39}63, e-Print Archive: hep-ph/9808275.

Physics Reports 320 (1999) 147}158

Non-renormalization of induced charges and constraints on strongly coupled theories S.L. Dubovsky*, D.S. Gorbunov, M.V. Libanov, V.A. Rubakov Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, 117312 Moscow, Russia

Abstract It is shown that global fermionic charges induced in vacuum by slowly varying, topologically non-trivial background scalar "elds are not renormalized provided that expansion in momenta of background "elds is valid. This suggests that strongly coupled theories obey induced charge matching conditions which are analogous, but generally not equivalent, to 't Hooft anomaly matching conditions. We give a few examples of induced charge matching. In particular, the corresponding constraints in softly broken supersymmetric QCD suggest non-trivial low-energy mass pattern, in full accord with the results of direct analyses.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.15.Me

1. Introduction Four-dimensional gauge theories strongly coupled at low energies often exhibit interesting content of composite massless fermions. This property is potentially important for constructing composite models of quarks and leptons, which is long being considered as a possible `major step in our way into the nature of mattera [1]. Powerful constraints on the low-energy spectrum are provided by 't Hooft anomaly matching conditions [2]. These are extensively used, in particular, in establishing duality properties of supersymmetric gauge theories (see, e.g., Refs. [3,4] and references therein).

*Corresponding author. E-mail address: [email protected] (S.L. Dubovsky) 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 7 3 - 3

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The basis for anomaly matching is provided by the Adler}Bardeen non-renormalization theorem [5]. In non-Abelian theories, the absence of radiative corrections to anomalies is intimately connected to topology: if one introduces background gauge "elds corresponding to the #avor symmetry group, the anomalies in global currents are proportional to the topological charge densities of these background "elds. Integer-valuedness of global fermionic charges, on the one hand, and integer-valuedness of topological charges of background gauge "elds, on the other, imply that anomaly equations should not get renormalized. Gauge "eld backgrounds are not the only ones that may have topological properties. Topology is inherent also in scalar background "elds of Skyrmion type. Indeed, one-loop calculations [6] show that slowly varying in space, static scalar "elds induce, in vacuum, fermionic global charges which are proportional to the topological charges of the background. By analogy to triangle anomalies, this suggests that induced charges do not receive radiative corrections, and hence may serve as constraints for low-energy spectrum of strongly coupled theories [7]. Unlike triangle anomalies, however, the one-loop expressions for the induced charges are promoted to full quantum theory only if the expansion in momenta of the background "elds is valid in the full theory. The latter property can often be established to all orders of perturbation theory (exceptions are easy to understand); in some models the validity of the expansion in momenta can be also shown non-perturbatively. We will see that induced charge matching conditions emerging in this way have a certain relation to anomaly matching. However, in some cases the two sets of matching conditions are inequivalent, so the induced charges give additional information on the properties of the low-energy theory. This information is particularly interesting in softly broken supersymmetric gauge theories. Fermionic charges in vacuum are induced due to Yukawa interactions of fermions with background scalar "elds. These interactions introduce masses to some of the fermions in the fundamental theory and hence explicitly break a subgroup of the #avor group. As a consequence, some of the fermions of the low-energy e!ective theory acquire masses. Induced charge matching conditions constrain the resulting mass pattern of the e!ective theory. We will see that these conditions are satis"ed automatically (provided the triangle anomalies match) if all composite fermions charged under explicitly broken #avor subgroup become massive. The latter situation is very appealing intuitively; however, we are not aware of any argument implying that it should be generic. This paper is organized as follows. In Section 2 we show that global charges induced in vacuum by slowly varying background scalar "elds do not get renormalized provided that the derivative expansion is valid. In Section 3 we discuss exceptional situations by presenting a model where the derivative expansion fails at the one-loop level itself. In Section 4 we give several examples of induced charge matching (ordinary QCD, supersymmetric N"1 QCD exhibiting the Seiberg duality [8], SQCD with softly broken supersymmetry). We conclude in Section 5 by discussing the relation between induced charges and triangle anomalies.

2. Non-renormalization of induced charges To be speci"c, let us consider QCD with N colors and N massless fermion #avors. Let t? and   tI  , a, a "1,2, N , denote left-handed quarks and anti-quarks, respectively. To probe this theory,  ?

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we introduce background scalar "elds m? (x) of the following form, @   m?(x)"m ;?(x) , @  @ where m is a constant and ; is an S;(N ) matrix at each point x. Let these "elds interact with   quarks and anti-quarks, (1) L "tI  m? t@#h.c.  ? @ Besides the global S;(N ) ;S;(N ) symmetry, the theory exhibits non-anomalous baryon * 0 symmetry, tPe ?t, tI Pe\ ?tI , mPm. The baryonic current is conserved and obtains nonvanishing vacuum expectation value in the presence of the background scalar "elds. To the leading order in momenta, the one-loop expression for this induced current is [6] 1jI 2"(N /24p)eIJHMTr(;R ;R;R ;R;R ;R) . (2)  J H M This expression can be obtained by considering a con"guration which in the vicinity of a given point x has the following form, ;(x)"1#e(x) ,

(3)

where e(x) is a small and slowly varying anti-Hermitean background "eld. To the leading order in momenta, one-loop-induced baryonic current is, then, as shown in Fig. 1 with fermions of mass m running in the loop. The complete expression (2) is reconstructed by making use of  S;(N ) ;S;(N ) global symmetry. * 0 A remarkable property of Eq. (2) is that the baryonic charge induced in vacuum by slowly varying, time-independent background "eld ;(x) with ;(x)P1 as "x"PR is proportional to the topological number of the background, 1B2"N N[;] ,  where

(4)



1 dxeGHITr(;R ;R;R ;R;R ;R) . N[;]" G H I 24p The higher derivative terms omitted in Eq. (2) do not contribute to 1B2. Let us see that Eq. (4) does not get renormalized in the full quantum theory provided the expansion in momenta of the background "eld works. Let us consider the same theory with the gauge coupling a depending on coordinates x. The induced current is now a functional of a(y) and ;(y), 1 jI (x)2"jI[x;a(y);;(y)] . At slowly varying a(y) we expand 1 jI (x)2 in derivatives of a at the point x, 1 jI (x)2"JI[x;a(x);;(y)]#BIJ[x;a(x);;(y)]R a(x)#O[(Ra)] , J

(5)

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Fig. 1. Leading order contribution to the induced baryonic current.

where the coe$cients on the right-hand side are now functions, rather than functionals, of a(x) (but still functionals of ;(y)). The structure analogous to the right-hand side of Eq. (2) appears in the derivative expansion of the "rst term on the right-hand side of Eq. (5), JI"( f(a)/24p)eIJHMTr(;R ;R;R ;R;R ;R)#2 J H M Our purpose is to show that f (a) is independent of a, and hence f (a)"N .  Let us make use of the conservation of 1 jI (x)2, R 1 jI (x)2"0 . I If f were a non-trivial function of a, the divergence of JI would contain the term

(6)

(1/24p)eIJHMTr(;R ;R;R ;R;R ;R)(Rf/Ra)R a . J H M I The only possible source of cancellation of this term in Eq. (6) is the second term on the right-hand side of Eq. (5). The cancellation would occur i! BIJ contained the term of the following structure bIJ[;]Rf/Ra with R bIJ[;]"!(1/24p)eJNHMTr(;R ;R;R ;R;R ;R) . (7) I N H M However, the right-hand side of Eq. (7) is not a complete divergence of any tensor that is invariant under the #avor group (recall that the right-hand side of Eq. (7) is a topological current). Hence, the conservation of the baryonic current requires that Rf/Ra"0. This argument is straightforward to be generalized to the other conserved currents and to the theories other than QCD. As discussed in Section 1, it implies that induced charges should match in fundamental and low-energy theories. Examples of such a matching are given in Section 4.

3. Failure of derivative expansion: an example An important ingredient in the above argument is the derivative expansion. While it works in QCD and many other models, at least to all orders of perturbation theory, one can design models

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Fig. 2. Dangerous diagram in the model of Section 3. Heavy and light lines correspond to massless and massive fermions, respectively.

where the derivative expansion fails, and the induced charges cannot be reliably calculated even within perturbation theory. As an example, let us consider a model of free left-handed fermions t?, tI  and s? , a, a "1,2, N , with the mass term  ? L "k tI  s? #h.c. (8) I  ? The model is invariant under the global S;(N ) ;S;(N ) symmetry under which t, tI , and * 0 s transform as (N ,1), (1,NM ), and (1,N ), respectively. The `baryon numbersa of fermions t,tI and    s are #1, !1 and #1, respectively. Let us introduce background "elds m? (x) and their @ interaction with fermions t and tI in the same way as in Eq. (1). To see that the derivative expansion is not reliable in this model, let us again consider the background "eld of the form (3). At e"0, fermions tI and m"const ) (k s#m t) form massive   Dirac "eld, while g"const ) (m s!k t) remains massless Weyl "eld. Both types of fermions   interact with the background "eld e(x). In an attempt to calculate the induced baryonic current, one faces diagrams with massless internal fermion lines like the one shown in Fig. 2. It is straightforward to see that the derivative expansion of these diagrams is singular. The fact that the derivative expansion fails in this model manifests itself in di!erent values of induced charges in various limits. Namely, at k <m one can ignore the background "elds, and   1B2"0. On the other hand, at k ;m , the mass term (8) becomes irrelevant, so 1B2"N[;]. As   outlined above, this phenomenon is due to the fact that not all fermions charged under S;(N ) ;S;(N ) obtain masses upon introducing the background "elds m(x). * 0 This example shows that the validity of the derivative expansion requires that the background scalar "elds provide masses to all relevant fermions. This will be the case in all examples presented in the next section. 4. Examples of induced charge matching 4.1. QCD We again consider conventional S;(N ) QCD with N massless #avors. Let us generalize slightly   the discussion of Section 2 by introducing background "elds (9) mN (x)"m ;N (x) O  O

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which give x-dependent masses to N quark #avors only, L "mN (x)tI  tO#h.c. The "elds m(x) O N   are N ;N matrices; hereafter the indices p,q,r;p ,q ,r run from 1 to N with N (N . (N !N )        #avors remain massless. In all the examples of this section we consider background "elds that are constant at spatial in"nity; by a global S;(N ) rotation we set ;(x)P1 as "x"PR. Besides the * baryon number symmetry ;(1) , we will be interested in a vector subgroup ;(1)D of the original  S;(N ) ;S;(N ) #avor group, whose (unnormalized) generator is * 0 ¹D"diag(1,2,1,!N /(N !N ),2,!N /(N !N )) .        The background "elds are charged under neither ;(1) nor ;(1)D.  As all fundamental fermions that interact with the background scalar "elds acquire masses due to this interaction, the derivative expansion is justi"ed, at least order by order in perturbation theory. Hence, for slowly varying m(x) one has 1B2"N N[;] , (10)  1¹D2"N N[;] . (11)   Let us see that the low-energy e!ective theory of QCD * the non-linear sigma-model * indeed reproduces Eqs. (10) and (11). In the absence of the background "elds, the non-linear sigma-model action contains only derivative terms for the S;(N ) matrix valued dynamical sigma-model "eld  <(x), including the usual kinetic term and the Wess}Zumino term. The background "eld m(x) introduces a potential term into the low-energy e!ective Lagrangian, *L "Tr(mR<#



<(x)"



;(x)

0

0

1

.

(12)

Hence, the induced baryonic charge appears at the classical level [9]; as the baryonic charge of <(x) is equal to its topological number N[<] times N , the induced baryonic charge is indeed given by  Eq. (10). Likewise, it follows from the structure of the Wess}Zumino term that the ¹D current of the  con"guration of the form (12) is (cf. Ref. [10]) jD "N /24pe Tr(;R ;R;R ;R;R ;R) , I  IJHM J H M so the ¹D charge of the con"guration (12) is given by Eq. (11).  We see that the induced charges in QCD and its low-energy e!ective theory match rather trivially. The way the induced charges match becomes more interesting when low-energy theories (in the absence of background scalar "elds) contain massless fermions. 4.2. Supersymmetric QCD Let us now consider supersymmetric QCD with N colors and N #avors. To be speci"c, we   discuss the case 3N 'N 'N #3. This theory exhibits the Seiberg duality [8]: the fundamental    theory contains the super"elds of quarks QG and anti-quarks QI I , while its e!ective low-energy H

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counterpart at the origin of moduli space is an S;(N !N ) magnetic gauge theory with magnetic   quarks q , magnetic anti-quarks q HI and mesons MGI with the superpotential qMq . H G Let us probe this theory by adding the scalar background "elds mO (x) with the same properties as N above, i.e., by introducing the term (13) mO (x)QI  QN N O into the superpotential of the fundamental theory. Let us take for de"niteness 24N  (N !N !1. The calculation of the induced baryon and ¹D charges in the fundamental theory    proceeds as above, and we again obtain Eqs. (10) and (11). We now turn to the e!ective low-energy theory. For slowly varying m(x), the term (13) translates into Tr(mM), so the total superpotential of the magnetic theory is qMq #k Tr(mM) , (14)  where k is the dimensionful parameter inherent in the magnetic theory. The ground state near the  origin of the moduli space has the following non-vanishing x-dependent expectation values of the magnetic quarks and anti-quarks, 1qN2"kN, p"1,2,N , q"1,2,N , O O   (here the upper and lower indices refer to magnetic color and #avor, respectively)

(15)

1q O 2"kO , p"1,2, N , q "1,2, N N N   (here the lower index refers to magnetic color). The expectation values obey

(16)

kP (x)kN(x)"!k mP (x) . N O  O They also satisfy the D-#atness condition at each point in space, kROkP "kO kRP . With our choice of N O N O background "elds, Eq. (9), one has I (x) , k"$(k m =(x), k"G(k m =     where = and = I are N ;N unitary matrices obeying   (= I =)(x)";(x) .

(17)

Indeed, at m"m ) 1, the matrices k and k are proportional to N ;N unit matrix, up to magnetic    color rotation. At m"m ;(x) one has k"$(k m ; ;, k"G(k m ;R where ; (x) is         a slowly varying matrix belonging to S;(N ) subgroup of the magnetic color group. The explicit  form of ; (x) is to be found from the minimization of the gradient energy, and it is not important  for our purposes. Since the gradient energy has to vanish at spatial in"nity, =(x) and = I (x) are constant at "x"PR, so they can be characterized by their winding numbers N[=] and N[= I ]. Because of Eq. (17) one has N[=]#N[= I ]"N[;] .  Hereafter we use the same notations for super"elds and their scalar components.

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In this ground state, the magnetic color is broken down to S;(N !N !N ). At small m , the     ground state (15), (16) is close to the origin of the moduli space, so the magnetic description is reliable. Both the baryon number and ¹D are broken in this vacuum. However, there exist combinations  of these generators and magnetic color generators that remain unbroken. Recalling [8] that the baryon number of magnetic quarks equals N /(N !N ) and that the magnetic quarks and    anti-quarks transform as (NM ,1) and (1,N ), respectively, under the global S;(N ) ;S;(N ) group,   * 0 the unbroken generators are B"B!(N /(N !N ))¹  ,    

(18)

¹ "¹D#¹  ,   

(19)

where ¹  is the following generator of the magnetic color group, 






N N   ¹ "diag 1,2,1,! , ,! .  N !N !N 2 N !N !N       As the fundamental quarks and gluons are singlets under magnetic color, the induced charges 1B2 and 1¹ 2 calculated in the magnetic theory should match Eqs. (10) and (11). Let us check that this  is indeed the case. The induced charges appear in the magnetic theory through x-dependent mass terms of fermions. These are generated by the expectation values (15), (16). The mass terms coming from the superpotential (14) are kN (x)WG tO#kN(x)tI HI WOI , O N G N H O

(20)

where W, t and tI are fermionic components of mesons, magnetic quarks and magnetic antiquarks, respectively. The gauge interactions give rise to other mass terms, kRN(x)t? jO!kRN (x)j? tI O , O N ? O N ?

(21)

where j@ is the gluino "eld, a,b"1,2,(N !N ) are the magnetic color indices. ?   To calculate the induced baryon number 1B2 we observe that the only fermions carrying non-zero B are magnetic quarks t? with i"1,2,N , G  a"(N !N !N #1),2,(N !N ), magnetic anti-quarks tI HI and gluinos j? , jN. Their ? N ?      B-charges are






N N N N  !    t?: ! " , G N !N N !N N !N !N N !N !N           N  j? : , N N !N !N    N  . tI HI ,jN: ! ? ? N !N !N   

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Hence, the induced B is due to the x-dependent mass term (21) and is equal to N  1B2"! ) (N !N !N )(N[=R]#N[= I R]) .    N !N !N    It follows from Eq. (17) that 1B2 indeed coincides with N N[;], the induced baryon number  calculated in the fundamental theory. The induced charge 1¹ 2 is calculated in a similar way. The relevant ¹ charges of magnetic   quarks are N !N   t? :! , N N !N !N    N  , u"(N #1),2,N , tS :   N N !N   and similarly for magnetic anti-quarks, gluinos and mesons. We "nd that both the x-dependent mass terms, (20) and(21), contribute to 1¹ 2, and obtain  N  ) (N !N )(N[=]#N[= 1¹ 2" I ])    N !N   N !N   ) (N !N !N )(N[=R]#N[= I R]) . #    N !N !N    This is equal to N N[;], so the induced ¹ charges also match in the fundamental and low-energy   theories. 4.3. Softly broken SQCD As our last example, let us consider supersymmetric QCD with small soft masses of scalar quarks, m , that explicitly break supersymmetry [11]. We again probe this theory by introducing / the term (13) into the superpotential. The restrictions on N , N and N are the same as in the    previous example. The induced charges, as calculated in the fundamental theory, are still given by Eqs. (10) and (11). The low-energy theory near the origin is still the magnetic theory, but now with soft mass terms of scalar mesons and scalar magnetic quarks [11]. The scalar potential of the magnetic theory near the origin at small m is determined both by the superpotential (14) and by the soft terms, / <(M,q,q )""q q#k m"#"qM"#"Mq "#m MRM#m(qRq#q Rq )#D-terms , (22)  + O where m and m are proportional to m . Were the soft terms in Eq. (22) positive, the ground state + O / of this theory at m'k m would be at the origin, 1q2"1q 2"1M2"0. The masses of fermions O   in the magnetic theory would vanish, the induced charges 1B2 and 1¹D2 would be zero, so the  induced charge matching would not occur. Hence, the induced charge matching requires that either m and/or m are negative, so that the ground state even at m "0 is far away from the origin, or O +  m"0, m 50 with the ground state being the same as in the previous example. This is in O + accordance with the explicit calculations. Indeed, it has been found in Ref. [12] (see also Ref. [13])

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that m(0 at N #1(N (3N /2, i.e., when the magnetic theory is weakly coupled. As regards O    the conformal window 3N /24N (3N , the complete analysis of the infrared soft masses is still    lacking. However, the existing results [14] (see also Ref. [12]) suggest that either the origin q"q "M"0 is again unstable, or m"m "0. We conclude that the induced charge matching O + provides qualitative understanding of the behavior of soft masses in low energy description of softly broken SQCD. One can show that the same phenomenon occurs in softly broken supersymmetric theories with SO(N ) and Sp(2k) gauge groups and fundamental quarks at N , N and k such that the Seiberg    duality holds. In particular, in theories with weakly coupled magnetic description, the soft masses of magnetic squarks are negative. This is again in full accordance with induced charge matching conditions. It is worth noting that there exists an example [12] where soft masses of fundamental scalar quarks single out the vacuum at the origin of the moduli space (in the absence of the background "elds m(x)). This is the theory with Sp(2k) gauge group and 2k#4"2N quarks Q , i"1,2,2N ,  G  in the fundamental representation. The low-energy e!ective theory [15] contains antisymmetric mesons M and has superpotential Pf M. One can probe this theory by adding x-dependent mass GH terms mNO (x)Q  Q where p"1,2,N , q "(N #1),2,2N . In the theory without soft supersymO N    metry breaking, the induced charges match in a similar way as in the previous example: scalar mesons obtain the expectation values 1M  (x)2JmR (x) which give x-dependent masses to ferON ON mionic mesons. After the soft scalar quark masses are introduced, the scalar potential of the low-energy theory contains soft meson masses, m MRM where m '0 at k'1 [12]. At the "rst + + sight, this ruins the induced charge matching at small m , as the ground state appears to be at  M"0 and no x-dependent masses of fermionic mesons seem to be generated. However, the symmetries of the theory allow for a linear supersymmetry breaking term in the scalar potential, m f(m ,m )mM, which shifts the ground state to 1M2JmR and in this way restores induced charge / /  matching. Hence, we argue that this linear term is indeed generated in the low-energy theory.

5. Discussion Let us discuss the relation between induced charges and triangle anomalies; we consider induced baryon number in QCD as an example. We use the notations of Section 4.1. The x-dependence of the background "eld m(x) can be removed at the expense of modi"cation of the gradient term in the quark Lagrangian. Namely, after the S;(N ) rotation of the left-handed quark "elds, * t(x)P;\(x)t(x), tI (x)PtI (x), "rst N quarks and anti-quarks have x-independent masses m ,   and the gradient term of these quarks becomes tM ic ) (D#(1!c/2)A*)t where A*"0, A*";R ;\, the covariant derivative D contains dynamical gluon "elds, and we  G G I switched to four-component notations. The addition to the gradient term may be viewed as the interaction of massive quarks with the background pure gauge vector "elds corresponding to S;(N ) subgroup of the #avor group; these background "elds are small, time-independent, and * slowly varying in space. Now, consider an adiabatic process (either in Minkowskian or in Euclidean space}time) in which the background vector "elds A*(x) (in the gauge A*"0) change in time from A*"0 to  G A*";R ;\, always varying slowly in space and vanishing at spatial in"nity (an example of such G G

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a process is an instanton of large size). Suppose that this process begins with the system in the ground state which has zero-induced charges because of the triviality of the background. As the background vector "elds interact with massive degrees of freedom only, the system remains in its ground state in the entire process, at least order by order in perturbation theory. The induced baryon number in the "nal state * the quantity we are interested in * is equal to 1B2"dxR jI I which in turn is determined by the anomaly in the gauge-invariant baryonic current jI . Hence, we recover Eq. (10):



N 1B2"!  dx TrF* FI * "N N[;] . IJ IJ  16p

(23)

This observation relates induced charges and anomalies. Let us note in passing that the phenomenon discussed in Section 3 may be understood in this language as follows. After the background "eld m(x) is introduced, and its phase ;(x) is rotated away, there remain massless fermions interacting with the background "eld A*(x). In this situation the adiabatic process does not necessarily end up in the ground state, because some energy levels of massless fermions may cross zero. In that case the baryon number induced in the ground state by the background "eld A*(x)";R ;\(x) would be di!erent from Eq. (23), as the anomaly deterG G mines the total change in the baryon number. This is the reason for the dependence of the induced charges on the parameters of the theory (m /k in the example of Section 3).   One may wonder whether similar phenomenon (fermion level crossing) might occur even if all relevant fundamental fermions obtain masses upon introducing the background scalar "elds, i.e., whether the "nal state might actually contain excitations carrying non-zero net baryon number. To argue that this does not happen, let us consider QCD again. The appearance, in the "nal state, of excitations with non-zero net baryon number would show up as a non-vanishing index of the four-dimensional Euclidean Dirac operator D[A*]"c ) (D#(1!c/2)A*(x))#m , so that the  vacuum-to-vacuum amplitude would vanish while matrix elements of baryon number violating operators between the initial and "nal vacua would not. However, for arbitrary gluon "elds, the eigenvalues u of the operator D[A*"0]"c ) D#m obey "u"'m (the Euclidean operator c ) D   is anti-Hermitean) so the operator D[A*] has no zero modes when the background "elds A*(x) are small (A*(x);m at all x) and slowly vary in space}time. This argument implies that Eq. (23) is  valid in full quantum theory even at m (K . Although the situation in theories with funda /!" mental scalars is more complicated, it is likely that analogous arguments are designed in these theories as well. Finally, the same adiabatic process may be considered within the low-energy e!ective theory. The induced baryon number is now related to the anomaly in the e!ective theory, provided all low energy degrees of freedom interacting with S;(N ) gauge "elds become massive upon introducing * the mass m to N #avors of fundamental quarks. As the ;(1) ;S;(N ) ;S;(N ) anomalies are   * 0 the same in the fundamental and low-energy theories, the induced baryon numbers match automatically in this case. This observation has an obvious generalization: a su$cient condition for induced charge matching is that no low energy degrees of freedom transforming non-trivially under a subgroup of the #avor group remain massless when this subgroup is explicitly broken by masses of some fermions of the fundamental theory. This property is certainly valid in supersymmetric theories where no phase transition is expected to occur as the masses of some of the #avors #ow

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from small to large values, i.e., where massive #avors smoothly decouple. On the other hand, this property does not seem to be guaranteed in non-supersymmetric models, though it is intuitively appealing and may well be quite generic.

Acknowledgements The authors are indebted to E. Akhmedov, D. Amati, G. Dvali, Yu. Kubyshin, A. Kuznetsov, V. Kuzmin, A. Penin, K. Selivanov, A. Smirnov, P. Tinyakov and S. Troitsky for numerous helpful discussions. This research was supported in part by Russian Foundation for Basic Research under grant 99-02-18410. The work of S.D. and D.G. is supported in part by INTAS under grant 96-0457 within the research program of the International Center for Fundamental Physics in Moscow and by ISSEP fellowships. The work of S.D., D.G., and M.L was supported in part by the Russian Academy of Sciences, JRP grant 37. V.R. would like to thank Professor Miguel Virasoro for the hopsitality at the Abdus Salam International Center for Theoretical Physics, where part of this work was carried out.

References [1] L.B. Okun, Particle Physics Prospects: August '81, Closing talk given at 10th International Symposium on Lepton and Photon Interactions at High Energy, Bonn, West Germany, August 24}28, 1981. [2] G. 't Hooft, in: G. 't Hooft et al. (Eds.), Recent Developments in Gauge Theories, Plenum Press, New York, 1980, p.135. [3] K. Intriligator, N. Seiberg, Lectures on supersymmetric gauge theories and electric-magnetic duality, Nucl. Phys. Proc. Suppl. 45 BC, (1996) (1) hep-th/9509066. [4] E. Poppitz, S.P. Trivedi, Dynamical supersymmetry breaking, hep-th/9803107. [5] S.L. Adler, W.A. Bardeen, Phys. Rev. 192 (1969) 1517. [6] J. Goldstone, F. Wilczek, Phys. Rev. Lett. 47 (1981) 986. [7] V.A. Rubakov, Induced charges as probes of low-energy e!ective theories, hep-th/9812128. [8] N. Seiberg, Nucl. Phys. B 435 (1995) 129. [9] A.P. Polychronakos, Phys. Rev. D 35 (1987) 1417. [10] E. D'Hoker, E. Farhi, Nucl. Phys. B 248 (1984) 59. [11] O. Aharony, J. Sonnenschein, M.E. Peskin, S. Yankielowicz, Phys. Rev. D 52 (1995) 6157. [12] N. Arkani-Hamed, R. Rattazzi, Phys. Lett. B 454 (1999) 290. [13] H.-C. Cheng, Y. Shadmi, Nucl. Phys. B 531 (1998) 125. [14] A. Karch, T. Kobayashi, J. Kubo, G. Zoupanos, Phys. Lett. B 441 (1998) 235, hep-th/9808178. [15] K. Intriligator, P. Pouliot, Phys. Lett. 353 B (1995) 471.

Physics Reports 320 (1999) 159}173

TeV-scale leptoquarks from GUTs/string/M-theory uni"cation S.S. Gershtein *, A.A. Likhoded , A.I. Onishchenko
State Research Center of Russia **Institute for High Energy Physics++, Protvino 142284, Moscow region, Russia
Institute for Theoretical and Experimental Physics, Moscow 117218, Russia

Abstract The question of a possible existence of the `baroleptonsa (leptoquarks) at the TeV energy scale is considered. Leptoquark mass and coupling bounds coming from rare meson decays are brie#y reviewed.  1999 Elsevier Science B.V. All rights reserved. PACS: 14.80.!j Keywords: Leptoquark; GUT; Extra dimensions; Strings; M-theory

1. Introduction In 1964 in the paper `On possible types of the elementary particlesa [1] Okun has discussed the question on the particles not observed by that time, which supposed to have unusual properties and whose existence does not contradict to fundamental laws of the nature. Let us remind what types of particles and related questions were discussed in [1]: 1. 2. 3. 4. 5. 6.

Are there baryons which are not subjected to strong interactions, i.e. which are not hadrons? Are there strongly interacting leptons? Are there baryons and leptons with the integer spin? Are there mesons with the half-integer spin? Are there baroleptons? Are there weakly interacting leptons, which are heavier than the muon?

* Corresponding author. E-mail addresses: [email protected] (S.S. Gershtein), [email protected] (A.A. Likhoded)  Let us note that `hadrona is the term later introduced by L.B. Okun. 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 6 3 - 0

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For many theorists such a statement seemed to be too arti"cial, as the existence of such particle was proved neither theoretically nor experimentally (besides the fact that those objects are not forbidden by the general conservation laws and quantum mechanics principles). The fruitfulness of these questions, however, has revealed itself later, at the moment of the q-lepton discovery. In paper [1] the possibility for leptons, heavier than the muon, to exist was argued in the following way: as for masses of such leptons, which are heavier than the K-meson, only the electromagnetic interactions could be the production source for these particles and their lifetimes should be shorter than 1 ps. Such leptons could easily escape from the experiments conducted at that time. It should be noted that it was Sakharov who discussed in 1948 for the "rst time the question of the possibility of the existence of a lepton, heavier than the muon, and its search in the electromagnetic interaction, just after the muon discovery. However, due to the fact that in his paper `Passive mesonsa [2] Sakharov has suggested the idea of the muon catalysis, this paper was security-restricted and remained unknown to a wide audience for a long time. Further the question of the existence of heavier leptons was discussed in a number of papers [3,4], however, the bulk of e!orts was aimed to the search of anomalous interactions of the muon or muon neutrino to explain the electron and muon mass di!erence [4]. The situation has drastically changed after the universality of the weak interactions of muon and electron (and their neutrinos) had obtained the experimental support. It became evident that the k!e mass di!erence could not be explained in terms of the muon anomalous interactions. This led to more serious treatment of the problem of heavier lepton existence. In this connection the modern classi"cation of heavier leptons has been given from CVC [5,6] and their decays modes have been considered in details. In particular, it was pointed out that the qPl oPl #nn channel [5], which is 1.5 O O times larger than that one for qPkl l or qPel l , could be the main decay mode for such I O C O lepton. The important role of the qPl A Pl 3n has been revealed from PCAC [7]. New stream O  O into the search of heavier leptons has been brought by e>e\-collider experiments [8]. Basing on the assumption of the short lifetime of such leptons (that could be expected from the universality of the weak interaction) the experimental search criteria have been developed, the main of which is to search for uncorrelated isolated k!e pairs in e>e\-collisions. This method has been used in experiments at the ADONE collider to search for heavy leptons [9]. The ADONE energy turned out to be insu$cient for q-lepton discovery. However, namely the uncorrelated k!e-pair observation became the "rst signal of q-lepton discovered by Perl [10]. The example with the q-lepton discovery demonstrates that the statement of the question of exotic particle existence and their properties is extremely important for their discovery, as it mainly determines the experimental strategy. The principal approach by Okun [1] (everything is allowed that is not prohibited) in connection with questions (3), (4), (5) turned out to be absolutely correct as well. The answer on these questions has been given after the origin of the idea of the supersymmetry and construction of the GUT models. It is evident, that baryons and leptons with the integer spin are the superpartners of ordinary particles. Thus, questions (3) and (4), in fact, raised the problem of the possibility of the broken supersymmetry existence. If the scale of the supersymmetry breaking were not so large, as it results from the experimental data, then the existence of mesons with half-integer spin (which consist of quarks q and their supersymmetric partner q ) could be quite possible. However, with the modern lower limits 

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on supersymmetric quark masses the lifetimes of such mesons turn out to be much smaller than their hadronization times, so as the bound state with ordinary quarks does not have time to be produced. From the point of view of experimental bounds the only possibility remains to be open * the existence of longliving light gluino, m  (5 GeV, although it is beyond the MSSM. E The realization of this possibility could lead to the presence of the objects discussed in [1]: mesons (hybrids) with half-integer spin, udM g , etc., and baryons with half-integer spin, uudg , etc. As for baroleptons discussed in [1], they appear in a natural way in the GUT models in the form of leptoquarks. If leptoquark masses are comparable with the GUT scale, these leptoquarks are hardly to be discovered in current experiments. However, following the logics of [1], the question of possibility for `lighta leptoquarks to exist and possible bounds on their masses can be considered. Namely this question is a theme of the paper presented.

2. Models Here we present a short review of some models, in which one could have light leptoquarks with the masses in the TeV range. Among the possibilities are: E Grand Uni"ed Theories (GUTs) [12] and their supersymmetric analogues [11], where leptons and quarks usually appear in the same multiplet. E Heterotic superstring models, with and without intermediate grand uni"cation, in particular their free fermionic realizations. E TeV-scale GUTs and TeV-scale superstrings, where low scale of grand uni"cation is achieved through the appearance of extra space}time dimensions. E Extended technicolour theories [13], where quarks and leptons individually appear in multiplets of the dynamically broken extended technicolour group. The other particles in each multiplet are new fermions, that would appear at low energies in fermion}anti-fermion bound states, some of which are leptoquarks. E Substructure or compositeness models [14], where the `preonsa in a quark and lepton could combine to form a scalar or vector leptoquark. Any discussion of the above leptoquark (LQ) models has been historically based on a set of assumptions due to BuchmuK ller, RuK ckl and Wyler (BRW) under which consistent LQ models can be constructed [15]. These authors had also classi"ed the possible leptoquark states, according to their possible spins and fermion number thus leading to the 10 states displayed in Table 1. These assumptions may be stated as follows: (a) LQ couplings must be invariant with respect to the SM gauge interactions. (b) LQ interactions must be renormalizable. (c) LQs couple to only a single generation of SM fermions. (d) LQ couplings to fermions are chiral. (e) LQ couplings separately conserve baryon and lepton numbers. (f ) LQs only couple to the SM fermions and gauge bosons.

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2.1. Non-SUSY and SUSY GUTs Historically, leptoquarks "rst appeared in Pati and Salam's SU(4) model [16], where the idea of lepton number as a fourth colour was implemented. The minimal group for models of this type is S;(4) ;S;(2) ;;(1) 0 and matter of one generation is sitting in the following representations:  * 2 (4,2,0), (4,1,#), (4,1,!). The leptoquarks, arising in these models induce #avour-changing neu  tral currents and lepton family and baryon number violation, but their contribution to proton decay is severely suppressed [17]. Constraints on Pati}Salam-type leptoquarks have been discussed in [16] and [18,19]. One would expect these vector leptoquarks to have full-strength couplings to a lepton and a quark of the same generation, giving, for instance, a large (&j/m , j&1) contribution to KPek  (s#kPd#e). In general, in grand uni"ed models the intermediate scale of symmetry breaking is large of the order of 10}10 GeV, and so the vector leptoquarks, appearing in these models, are of no interest to us. In paper [20] the model with Pati}Salam SU(4)SU(2) SU(2) group, broken at * 0 the lowest energy scale that phenomenology [21] allows (1000 TeV) was discussed. There, in order to generate large mass for right-handed q-neutrino the `3;3 see-saw mechanisma was used [22]. For this one introduces a singlet fermion S &(1,1,1) and the simple Higgs multiplet s&(4,1,2). * The additional Yukawa term then has the following form: "nSM Tr[sRf ]#h.c. (1) 7 * 0 when combined with the standard electroweak Yukawa terms this yield a neutrino mass matrix of the form L




0 m  0

m  0

n1s2

n1s2

0

0



(2)

in the [l , (l ), S ] basis (m is the top-quark mass). This will give us one massless eigenstate which * 0 *  we identify with the standard neutrino, and a massive Dirac neutrino. For n1s2<m , the massless  state has approximately standard electroweak interactions. Because the light eigenstate is massless for all values of the nonzero entries of the mass matrix, 1s2 can be reduced to about 1000 TeV. The implementation of this mechanism of symmetry breaking in SUSY grand uni"ed theories with the SO(10) gauge group and symmetry breaking chain SO(10)PS;(2) ;S;(2) ;S;(4) PMSSM * 0 .1 are also possible, because, as it is shown in the analysis of [23], the reason for high intermediate scale of mass breaking is the use of (1,3,10) Higgs "eld to generate right-handed neutrino masses, which is not needed in the mechanism, described above. In papers [24,25] in the framework of left}right supersymmetric models the phenomenology of light doubly charged particles was analysed. In the Pati}Salam case they "nd that entire leptoquark (3,1,10) multiplets can remain light. The discussion of prospects for including additional light colour triplets and anti-triplets in the spectrum of SUSY grand uni"ed theories was done in [26]. In this paper the authors proposed

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particular string-inspired Pati}Salam model, which has light colour triplet of charge ! plus  antitriplet of charge  with masses of order 200 GeV in its low-energy spectrum.  It was shown in [27] that the minimal S;(5) model plus a light (m &100 GeV) S;(2) doublet  leptoquark would be compatible with present experimental results on proton decay and sin h . 5 The authors of [28] have built a low-energy compatible S;(4)-type model for vector leptoquarks with mass &1 TeV. And "nally, we would like to mention S;(15) grand uni"ed theory as a candidate theory with light leptoquarks, where all 14 types of leptoquarks, listed in Table 1 can be present and the GUT scale may be about 10 GeV. The main motivation for the models with this gauge group is the gauge uni"cation without proton decay. In this model gauge bosons have de"nite B and ¸ numbers, because of the only de"ning representation 15 for the fermions. This situation is contrary to that with the conventional GUTs based on S;(5), SO(10) and E(6), in which proton decay occurs at leading order via gauge bosons with de"nite B!¸ but inde"nite separate B and ¸. The attractive feature of these models is that they predict a large set of weak-scale scalar leptoquarks (for more details see [29]). 2.2. Heterotic superstring models The "eld theory limit of the heterotic superstring is some GUT-like gauge "eld theory, which is broken at or below the string scale to the Standard Model gauge group and possibly some extra ;(1)s:S;(3);S;(2);;(1)L>. Thus, this low-energy limit of heterotic string theory may contain leptoquarks for the same reason as grand uni"ed theories themselves. For example, certain Calabi}Yau or orbifold compacti"cations have G"E /H as the four dimensional gauge group below the compacti"cation scale, where H is some discrete symmetry group that can be chosen such that G is the Standard Model (;;(1)L as we already noted). The low-energy "elds are then generally sit in the 27 of E . For each generation of quarks and leptons  one then would expect a right-handed neutrino, an extra SO(10) singlet, the two Higgs doublets of the MSSM, and a pair of S;(2) singlets, which may have their couplings of either diquarks or leptoquarks, but not both simultaneously [58], as this would lead to the proton decay [50]. Detailed investigation of the uni"cation of gauge couplings within the framework of a wide class of realistic free-fermionic string models, including the #ipped S;(5), SO(6);SO(4), and various S;(3);S;(2);;(1) models has been done in [30]. It was shown, that if the matter spectrum below the string scale is that of the MSSM, then string uni"cation is in disagreement with experiment. The one-loop string threshold corrections in free-fermionic string models, the e!ect of non-standard hypercharge normalization, light SUSY thresholds and intermediate-scale gauge structure cannot resolve the disagreement with low-energy data, and, only the inclusion of extra colour triplets and electroweak doublets beyond the MSSM at the appropriate thresholds lead to the gauge couplings uni"cation. The constraints on leptoquarks in such superstring-derived models were calculated in [58], and are extensively reviewed in [59]. 2.3. Extra dimensions Recently, in [31] extra large space dimensions was used as a radical new proposal for avoiding the gauge hierarchy problem by lowering the Planck scale to the TeV scale. The extra dimensions

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Table 1 Quantum numbers and fermionic coupling of the leptoquark states. No distinction is made between the representation and its conjugate. Bl is the branching fraction of the LQ into the ej "nal state, Q is its electric charge and F * fermion number (F"3B#¸) Leptoquark

SU(5) Rep

Q

Coupling

Bl

S * S 0 SI 0 S *

5 5 45

     

j (e>u ), j (ldM ) * * j (e>u ) 0 j (e>dM ) 0 !(2j (e>dM ) * !j (e>u ), !j (ldM ) * * (2j (lu ) * j (e>u) * j (lu) * j (e>u) 0 !j (e>d) 0 j (e>d) * j (ld) *

1/2 1 1

Scalars F"!2

F"0

45

S *

45

S 0

45

SI *

10/15



    !            ! 

  

1 1/2 0 1 0 1 1 1 0

Vectors F"!2

F"0

< *

24

< 0

24


10/15

< * < 0
10 10 75 40



            ! 



      



    ! 

j (e>dM ) * j (ldM ) * j (e>dM ) 0 j (e>u ) 0

1

j (e>u ) * j (lu ) * j (e>d), j (lu) * * j (e>dM ) 0 j (e>u) 0 (2j (e>u) * !j (e>d), j (lu) * * (2j (ld) *

1

0 1 1

0 1/2 1 1 1 1/2 0

required to achieve this are in the (sub-)millimeter range, and thus imply a profound change in Newton's gravitational force law at these distances. After that in [33}37], it was shown how extra large dimensions could also be used to lower the fundamental string scale to the TeV scale. The idea of taking the string scale in the TeV range was "rst considered in [38]. In [32,35]

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a complementary proposal of lowering the fundamental GUT scale to the TeV range was presented. As it was shown in these papers extra space}time dimensions modify the running of the Standard Model gauge couplings in such a way that not only is the uni"cation preserved, but in fact it occurs more rapidly. So, with the help of extra dimensions, it becomes possible to have grand uni"cation as low as the TeV scale. In these models Standard Model gauge "elds (together with corresponding charged matter) may reside inside of p49 spatial dimensions p * branes (or a set of overlapping branes), while gravity lives in a larger (10 or 11) dimensional bulk of space}time. So, in this scenario we might have light TeV * scale vector leptoquarks. It could turn out that coupling of these leptoquarks to the Standard Models quarks and leptons are severely suppressed due to some anomaly free gauge discrete symmetries, as it is the case for B and ¸ non-preserving leptoquarks, mediating proton decay. Here its worth to mention the TeV * scale supersymmetric Standard Model (TSSM) [39]. However, there are some Pati}Salam like Type I string models [34], so that the leptoquarks, which arise here can give sizeable contributions to rare processes. 2.4. Extended technicolour Another possibility to have vector leptoquarks in the TeV range is the models with the dynamical electroweak and #avour symmetry breaking. One such class of models is extended technicolour (ETC) [40,41]. In these models it is assumed that there is a new gauge interaction, besides Standard Model interactions, called `technicoloura, with gauge group G , and gauge 2! coupling a that becomes strong in the vicinity of a few hundred GeV. The dynamical symmetry 2! breaking is than realized via the condensation of technifermions at this scale, in the same way as chiral symmetry breaking in QCD via light quark condensation. The breaking of quark, lepton and technifermion #avour symmetries is achieved by embedding technicolour, colour and part of electromagnetic ;(1) into the gauge group G , which breaks at a high scale K
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Goldstone bosons (axions) and light charged pseudo-Goldstone bosons; thus, uni"cation of the known forces near the Planck scale cannot be achieved in the models with extended technicolour. The next simplest possibility, G ;S;(4) ;S;(2) ;;(1) 0, cannot lead to uni"cation of the 2! .1 * 2 Pati}Salam and weak gauge groups near the Planck scale. However, superstring theory provides relations between couplings at the Planck scale without the need for an underlying grand-uni"ed gauge group, which allows uni"cation of the S;(4) and S;(2) couplings at di!erent Kac}Moody .1 * levels with the M between 10 and 10. .1 2.5. Compositeness Light leptoquarks frequently arise in composite or substructure models. The motivation for considering quarks and leptons as composite is the absence of explanation in Standard Model why quarks and leptons come in three generations, which except for their masses (#avours), are identical replications. Why the quarks and leptons are similar in the weak sector? Next, the Higgs sector experiences the gauge hierarchy problem, when Standard Model results from spontaneous breakdown of higher symmetry at some high-energy scale as in GUTs. Present lower limits on the characteristic scale of the particle substructure K in various processes exceed one to several TeV. It is here where one of the most di$cult problems associated with composite fermions occurs * the lightness of the quarks and leptons compared to the scale of compositeness. The almost masslessness of the observed fermions, on this scale, could be attributed to a nearly exact chiral symmetry, realized in the Wigner}Weyl sense. As the consequence, these theories must satisfy a criterion proposed by't Hooft [47]: one can test theories for this chiral symmetry by comparing gauge anomalies as realized by composite fermions and by their constituents. The results should be the same. However, the construction of fully realistic models satisfying this anomaly-matching condition has proved extraordinarily di$cult (for review see [48]). It may be that supersymmetry provides the necessary ingredient to ensure light composite fermions, which are now supersymmetric partners of Goldstone bosons in a theory with a spontaneously broken approximate global symmetry. Their masses then remain small even after the breaking of supersymmetry and the global symmetry. In dealing with composite gauge bosons we face a even more di$cult problem. The Case}Gasiorowicz}Weinberg}Witten theorem forbids the presence of interacting massless composite vector particles, as a consequence all composite vector bosons should have masses of the order of the compositeness scale [49]. TeV-scale leptoquarks may naturally appear in any substructure models. If a constituent particle of Standard Model fermion carries quark or lepton number, as it is the case in the `F-Sa model (in this model quarks and leptons contain a single subunit fermion ; or D and scalar SM , the index i"1,2,3, l denotes three colours and lepton G number, thus realizing Pati and Salam suggestion for lepton number as a fourth colour), then the composite quark could turn into a lepton in the presence of a composite lepton by exchanging the appropriate constituents. For example, in the model based on superstrong S;(4) and discussed in [53], the spinless (SSM )12 states form two scalar leptoquarks, a colour triplet with charges , a colour   The FSM picture of quarks and leptons was proposed explicitly by Greenberg and Sucher [51] in 1981, and is implicit in an earlier model [52], in which S itself is a composite of two fermions, one carrying a horizontal (family) symmetry and the other carrying colour or lepton number.

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antitriplet with charges !. They have to be quite heavy (5100 TeV) in order to suppress decays  like K Pke. This is also true of vector leptoquarks, the gauge bosons which become massive when * S;(4) breaks down to S;(3) ;;(1) , if S;(4) is a gauged symmetry. If a condensate ! ! \* 2 , 1(S SM #S SM #S SM !3S SM )/(122O0 is responsible for breaking S;(4) to S;(3) ;;(1) 2 ! \*         the leptoquark SSM bosons could be absorbed into longitudinal components of S;(4) / ! S;(3) ;;(1) gauge bosons and thus be removed from the low-lying spectrum. ! \* 3. Mass and coupling bounds Presenting the experimental constraints on the couplings and masses of the B- and ¸-preserving leptoquarks we will closely follow [50]. Discussing the phenomenology of the leptoquarks arising in the context of the extended models one can "nd that there are seven renormalizable quark}lepton}boson couplings, which conserve B and ¸ and are consistent with the S;(3);S;(2);;(1) symmetries of the Standard Model for both scalar and vector leptoquarks. The parts of the Lagrangian for scalar and vector interactions are [15] L "+(j q  iq l #j u  e )SR#j I dM  e SI R#(j u l 01 0 0  01 0 0  *1 0 * 1 *1 *  * # j q iq e )SR #j I dM l SI R #j q  iq qo l ) So R ,#h.c. *1 0 *  *1 *  *  01 *  0 

(3)

and L "+(j Mq c l #j MdM c e )
(5)

(eGHeKL/m )(q HPqL)(lM KP lG) 

(6)

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(see Tables 3 and 4) where P and P are chiral projection operators "¸ or R, m is the leptoquark  mass, and a coupling e"j (e"j/2) for vectors (scalars), and i, j, m, n are the generation indices. A number of constraints coming from meson decay have been considered in the literature [5,18,19,54}58,60], and in this paper we will summarize some of them. 3.1. n-decays In the Standard Model, charged pions decay only via kl channel as the decay to el is suppressed by angular momentum conservation. The experimental ratio [61] C(n>Pe l) R , "1.231$0.006;10\  C(n>Pkl)

(7)

agrees well with the Standard Model prediction [62] m (m!m) C (1#D)"1.235$0.004;10\ , (8) R " C L  m (m!m) I I L where D is a radiative correction. Con"ning the contribution to this ratio from the interference of the Standard Model axial vector amplitude and the leptoquark-induced e!ective pseudoscalar operators one gets (S$P contribution) (eLe)(5;10\(m /100 GeV), (eLe)(10\(m /100 GeV) , (9) * 0  * 0  where n is a light neutrino index. Under the assumption of nonequality of the leptoquark couplings to muons and electrons (eOe) one gets (<$A contribution) * * m   . (10) eeL, eeL(2;10\ * * * * 100 GeV






It should be noted that if the leptoquark coupling is independent of the lepton generation, i.e. e"e, then leptoquark contributions will not change the R ratio. However, in this case e is still constrained by quark}lepton universality. One can con"ne e!ective leptoquark couplings from the upper bound BR(n>Pkl )(8;10\ C (e/m )(0.25G , (e e /m )(5;10\ G m /m . (11) *  $ * 0  $ I L 3.2. K-decays Charged kaons decay into kl "nal states with BR"63.5% and a lot of other "nal states mostly I involving pions. As in the pion case the decay to el channel is suppressed by angular momentum C  We will con"ne ourselves to the constraints, coming from rare meson decays.

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conservation, so requiring the leptoquark contribution to be less than the experimental error, one can estimate pseudoscalar contribution and get bounds on the leptoquark couplings from the KPkl and KPel decays. For the kaon decay via a leptoquark it is natural to have kl in the "nal state as kl . The upper C I bound on the ratio [63] BR(KPkl ) C (6.3;10\ BR(KPkl ) I

(12)

leads to the following bounds: e * , 8m G"< "  $ 

(e e )m * 0 ) (6.3;10\ . 8m G"< "f  m  $  ) I

(13)

There are also bounds from the absence of the #avour-changing decays KPnl lM , where   l lM "ee , kk, ke or ek. In this case one has   e BR(KPn>llM ) * (4G"< " . $  m BR(KPnMlM l) 

(14)

Another source of bounds on the leptoquark couplings are the leptonic decays of neutral kaons. Most scalar and vector leptoquarks would allow the decays K Pkk, ee , ke , which are suppressed * in the Standard Model by the absence of #avour-changing neutral currents. Both axial vector and pseudoscalar quark matrix elements can contribute to these decays. For the decay width one can derive C(K Pk8e!)K[(m !m)/64nm ](e/m )f m(1.2;10\ GeV , * ) I ) *  ) I

(15)

which implies






m   . e(6;10\ * 100 GeV

(16)

3.3. D-decays Like in the pion case, the decay D>Pk>l is suppressed by angular momentum conservation (besides usual Cabbibo suppression), as the mass of the D> meson is much greater than that of the muon. So this decay may severely constrain leptoquark couplings and masses (as the leptoquarkmediated subprocesses do not have such suppression). However, the present upper bound on the rate for D>Pk>l is not very tight, but it can be used to con"ne e!ective pseudoscalar quark vertices involving c, d, k and any neutrino: (m !m) (e e ) " I f  m * 0 (4.5;10\ GeV , " " m 64n 

(17)

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which implies






m   . e e (6;10\ * 0 100 GeV

(18)

Two other sources of bounds on the leptoquark couplings are connected with: (a) The CKM matrix element < , which is measured in *C"1 neutrino interactions  l#dPk#c. Requiring that leptoquarks contribute less than the total observed rate gives e((4G /(2)"< "m .   * $

(19)

(b) The CKM matrix element < measurement from the decay DMPK\e>l, which gives the  following bounds (e/m )(4G /(2 , *  $

(20)

where e/m is the e!ective coupling for an axial vector vertex involving c,s,e, and any neutrino *  species. The e!ective coupling can be also constrained from the upper bound on the branching ratio BR(D>Pn>llM )



q  BR(D>Pn>llM ) . e/m (2/(4G "< ")( " $  *  q >BR(DPn\le ) "

(21)

As in the case of kaons, leptoquarks can induce #avour-changing D decays to lepton pairs (kk, e e, ke). From the limit BR(DPke )(10\ [63] one has e e (4;10\(m /100 GeV) . * 0 

(22)

From BR(DPkk)(1.1;10\ [63] e e(10\(m /100 GeV) , * 0 

(23)

and from BR(DPee )(1.3;10\ [63] e e(4;10\(m /100 GeV) . * 0 

(24)

3.4. B-decays The leptoquark couplings to b quarks can be constrained from the upper bounds on the #avour-changing decays BPllM X (llM "kk, ee ), BPl lM K (l lM "kk,ke ,ee ), and BMPkk,e e,ke.     It was observed that the decay B\PelX (B\PklX) has a branching ratio of 12% (11%). This is the rate expected in the Standard Model with "< ""0.05, so we can require that the leptoquarks 
mediating this interaction satisfy (e/m )44G /(2"< " . 
*  $

(25)

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The branching ratio for BPqlX has been measured to be 4.2#0.72!0.68$0.46% by Aleph [64]. So one gets the bounds BR(B>PllM X) (e/m )(8G"< " *  $ 
BR(B>PllM X)

(26)

on four fermion vertices involving elbc or elbu. The width of the decay BMPe k via an e!ective axial vector quark operator will be as follows: C(BPe!k8)"[(m!m)/64nm](e/m ) f m . I *  I

(27)

The CLEO bound BR(BPe k)(6;10\ implies C(BPe!k8)(3;10\ GeV ,

(28)

that leads to the bounds on the pseudoscalar vertex (e e /m )42.6;10\1/GeV . * 0 

(29)

4. Conclusions The statement of a question about the possible existence of exotic particles and interactions [1] proceeding from the principle `everything is allowed that is not prohibiteda turned out to be quite fruitful. Theoretical investigation of such objects stimulates corresponding experimental searches. It took place with q-lepton and continues with searches for supersymmetrical particles. The existing symmetry between leptons and quarks naturally infers the possibility of existence of leptoquarks (or according to the terminology of [1] * baroleptons). In this context the models with relatively `lighta leptoquarks, connected with the TeV-scale uni"cation models or possible leptoquarks existence at the scale lower than the GUT scale, are of interest from the point of view of direct experimental discovery or observation of the leptoquark traces in weak decays.

References [1] L.B. Okun, Zh. Eksp. Teor. Fiz. 47 (1964) 1777. [2] A.D. Sakharov, Collected Scienti"c Publications of FIAN, 1948, Scienti"c Publications, 1995. [3] Y. Fujii, Prog. Theoret. Phys. 21 (1959) 232; J.J. Sakutai, Ann. Phys. 11 (1960) 1; Ya.B. Zel'dovich, UFN 78 (1962) 549 (Sov. Phys. Usp. 5 (1963) 931); E.M. Lipmanov, Zh. Eksp. Teor. Fiz. 43 (1962) 893; Zh. Eksp. Teor. Fiz. 46 (1964) 1917 (Sov. Phys. JETP 16 (1963) 634). [4] I.Yu. Kobsarev, L.B. Okun, JETP 41 (1961) 1205. [5] S.S. Gershtein, V.N. Folomeshkin, Yad. Fiz. 8 (1968) 768. [6] J.D. Bjorken, C.H. Liewllyn Smith, Phys. Rev. D (1973) 887.

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[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] [43] [44] [45] [46] [47]

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[51] O.W. Greenberg, J. Sucher, Phys. Lett. B 99 (1981) 339. For fermion-scalar models of composite fermions see also O.W. Greenberg, Phys. Rev. Lett. 35 (1975) 1120; M. Veltman, in: T.B.W. Kirk, H.D.I. Abarbanel (Eds.), Proceedings of the 1979 International Symposium on Lepton and Photon Interactions at High Energies, Fermilab, August 23}29, 1979, (Fermi National Accelerator Laboratory, Batavia, IL, 1979), p. 529; H. Fritzsch, G. Mandelbaum, Phys. Lett. B 102 (1981) 319; R. Casalbuoni, R. Gatto, Phys. Lett. B 103 (1981) 113; O.W. Greenberg, R.N. Mohapatra, S. Nussinov, Phys. Lett. B 148 (1984) 465; M. Suzuki, Phys. Rev. D 45 (1992) 1744. [52] H. Terazawa, Y. Chikashige, K. Akama, Phys. Rev. D 15 (1977) 480; H. Terazawa, Phys. Rev. D 22 (1980) 184. [53] J. Rosner, hep-ph/9812537. [54] M. Leurer, Phys. Rev. D 46 (1992) 3757. [55] R. Mohapatra, G. SegreH , L. Wolfenstein, Phys. Lett B 145 (1984) 433. [56] I. Bigi, G. KoK pp, P.M. Zerwas, Phys. Lett. B 166 (1986) 238. [57] W. Buchmuller, D. Wyler, Phys. Lett. B 177 (1986) 377. [58] B.A. Campbell, J. Ellis, K. Enqvist, M.K. Gaillard, D.V. Nanopoulos, Int. J. Mod. Phys. A 2 (1987) 831. [59] J. Hewett, T. Rizzo, Phys. Rep. 183 (1989) 193. [60] A.J. Davies, X. He, Phys. Rev. D 43 (1991) 225. [61] D.I. Britton et al., Phys. Rev. Lett. 68 (1992) 3000; G. Czapek et al., Phys. Rev. Lett. 70 (1993). [62] S. Berman, Phys. Rev. Lett. 1 (1958) 468; T. Kinoshita, Phys. Rev. Lett. 2 (1957) 477; W.J. Marciano, A. Sirlin, Phys. Rev. Lett. 36 (1976) 1425 (1.233$ 0.004; 10\); T. Goldman, W. Wilson, Phys. Rev. D 15 (1977) 709 (1.239$ 0.001; 10\); W. Marciano, see [61], (1.2345$ 0.0010; 10\). [63] Review of Particle Physics, Eur. Phys. J. C 3 (1}4) (1998). [64] ALEPH Collaboration, contributed paper at the XXVI International Conference on High Energy Physics, Dallas, Texas, 1992.

Physics Reports 320 (1999) 175}186

How unitarity imposes a steep small-x rise of spin structure function g "g #g and breaking of DIS sum rules *2   I.P. Ivanov 
, N.N. Nikolaev *, A.V. Pronyaev, W. SchaK fer

IKP(Theorie), KFA Ju( lich, D-52428 Ju( lich, Germany
Novosibirsk University, Novosibirsk, Russia L.D. Landau Institute for Theoretical Physics, GSP-1, 117940, ul. Kosygina 2, Moscow 117334, Russia Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0435, USA

Abstract We derive a unitarity relationship between the spin structure function g (x,Q)"g (x,Q)#g (x,Q), the *2   LT interference di!ractive structure function and the spin-#ip coupling of the pomeron to nucleons. Our di!ractive mechanism gives rise to a dramatic small-x rise g (x,Q)&g(x,QM )&(1/x)>B, where d is an *2  exponent of small-x rise of the unpolarized gluon density in the proton g(x,QM ) at a moderate hard scale QM  for light #avour contribution and large hard scale QM &m for heavy #avour contribution. It invalidates D the Burkhardt}Cottingham sum rule. The found small-x rise of di!raction driven g (x,Q) is steeper than *2 given by the Wandzura}Wilczek relation under conventional assumptions on small-x behaviour of g (x,Q).  1999 Elsevier Science B.V. All rights reserved.  PACS: 11.55.Hx; 13.60.Hb; 13.88.#e Keywords: Polarized deep inelastic scattering; Unitarity; Burkhardt}Cottingham sum rule; Wandziera}Wilczek relation; Transverse spin asymmetry

1. Introduction and motivation The combination g (x,Q)"g (x,Q)#g (x,Q) of familiar spin structure functions g and g of *2     deep inelastic scattering (DIS) is related to the absorptive part of amplitude A (D"0) of forward IMJH (¹) transverse to (¸) longitudinal photon scattering accompanied by the target nucleon spin-#ip, p "[1/(Q#=)]Im A (D"0)"4pa /Q ) 4m /(Q ) xg (x,Q) , *2 *2 \\*> CK N * Corresponding author. Tel. #49-2461-616472; fax: #49-2461-613930. 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 7 2 - 1

(1)

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where D is the momentum transfer and k,l"$1,¸ and o,j"$ are helicities of particles in  cHp PcHp scattering, Q,= and x"Q/(Q#=) are standard DIS variable. The motto of J H I M high-energy QCD * the quark helicity conservation, the common wisdom that high-energy scattering is spin-independent, some model considerations including [1] the vanishing onepomeron exchange contribution to A (D"0), all suggest that the corresponding spin \\*> asymmetry A "p /p vanishes in small-x limit of DIS.  *2 2 In this communication we demonstrate that this is not the case. We "nd about x-independent spin asymmetry A and scaling and steeply rising g (x,Q) at small x,  *2 g (x,Q)&(G(x,QM ))/x , *2

(2)

where G(x,Q)"xg(x,Q)&(1/x)B is the conventional unpolarized gluon structure function of the target nucleon and QM  is #avour-dependent scale to be speci"ed below. The case of the helicity amplitude A (D) is quite tricky. On the one hand, \\*> QCD-motivated considerations strongly suggest a nonvanishing pomeron spin-#ip in di!ractive nucleon}nucleon scattering [2]. On the other hand, recent studies have shown that the s-channel helicity nonconserving (SCHNC) LT interference cross section p" of di!ractive *2 DIS [3] and related SHCNC spin-#ip amplitudes of di!ractive vector meson production do not vanish [4,5] at small x. As Zakharov emphasized [2] such spin-#ip does not con#ict the quark helicity conservation because in scattering of composite objects helicity of composite states is not equal to the sum of helicities of quarks, which arguably holds way beyond the perturbative QCD (pQCD) domain. The recent work on SCHNC vector meson production illustrates this point nicely [3}5]. Consequently, pomeron exchange well contributes to this helicity amplitude but the Procrustean bed of Regge factorization enforces the forward zero, A (D)JD, and vanishing p in \\*> *2 one-pomeron exchange approximation. The principal point behind our result (2) is Gribov's observation [6] that such kinematical zeros can be lifted by two-pomeron exchange (two-pomeron cut) which can contribute to helicity amplitudes vanishing in one-pomeron exchange approximation. A good example is a recent derivation [7] of a rising tensor structure function b (x,Q) for DIS  o! spin-1 deuterons. In de"ance of common wisdom, it gives rise to dependence of total cross section on the deuteron tensor polarization which persists at small x. Such a rise of b (x,Q)  invalidates the Close}Kumano sum rule [8]. Incidentally, it derives for the most part from di!ractive mechanism which we pursue in this paper. Another example due to Karnakov [9] is a di!erence of cc total cross sections for parallel and perpendicular linear polarizations of colliding photons * the quantity which vanishes in one-pomeron exchange approximation. The keyword behind these new e!ects is unitarity [10], two-pomeron cut is simply a "rst approximation to imposition of unitarity constraints.

2. Regge theory expectations and sum rules We recall that our expectations for small-x behaviour of di!erent structure functions, &(1/x)B, have been habitually driven by the Regge picture of soft interactions, in which the exponent (intercept) d"a!1 is controlled by quantum numbers of the relevant t-channel exchange (a good

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summary is found in textbook [11]). For instance, the Regge theory suggests a/&1 for helicitydiagonal pomeron (vacuum) exchange dominated F (x,Q) and F (x,Q) and a & for the  * 0  secondary reggeon (o,A )-exchange quantities like F (x,Q)!F (x,Q) and u-exchange F (x,Q).  N L  The dominant A and f reggeon exchange in the axial vector channel suggests a &0 for xg (x,Q).     These Regge theory intercepts are not stable against QCD evolution, but extensive studies of small-x asymptotics of generalized two-gluon and quark}antiquark ladder diagrams have revealed only marginal modi"cations of the above hierarchy of intercepts (for the BFKL pomeron exchange see [12], for reggeon exchange and/or non-singlet structure function see [13], for di!erent spin structure functions see: g (x,Q) in [14], g (x,Q) in [15], F (x,Q) in [16]). The corollary of these   A studies is that g (x,Q) and g (x,Q) of two-parton ladder approximation have the x-dependence   typical of the reggeon exchange and their contributions to spin asymmetries A and A do indeed   vanish in the small-x limit. We recall that works [14}16] focused on exactly forward, D"0, Compton scattering amplitudes. Burkhardt and Cottingham [1] argued that because neither pomeron nor high-lying reggeon exchanges contribute to A (D"0), then unsubtracted (superconvergent) dispersion \\*> relation holds for this Compton scattering amplitudes. Precisely superconvergence has been the principal assumption behind the much discussed BC sum rule [1]







dl Im A (Q,l,D"0)"0 (3)  / for thorough reviews see [11,17}19]. The tricky point is that the BC amplitude A (Q,l,D) (which  di!ers from our A (D) only insigni"cantly) receives a contribution from pomeron \\*> exchange, and the integral



dxg (x,Q)J 



dl Im A (Q,l,D)  / would diverge at any "nite DO0, which makes the BC sum rule quite a singular one. As we emphasized above, A (D)JD and vanishes at D"0 only because of rigours of Regge \\*> factorization, Gribov's two-pomeron exchange breaks Regge factorization and gives A (D"0)O0, see also [20]. The specter of resulting dramatic small-x rise of g (x,Q) \\*>  and of divergence of the BC integral permeates the modern literature on spin structure functions (see textbook [11] and recent reviews [17}19]). The aforementioned breaking of the Close}Kumano sum rule is of the same origin; if cast in the Regge language, the scaling and rising tensor structure function found in [7] falls into the pomeron-cut category. Numerical estimates show that tensor asymmetry is quite large, the related evaluations of g (x,Q) are as yet lacking. *2 (x,Q) 3. Di4ractive DIS and unitarity-driven gU LT In this communication we "ll this gap and report the "rst ever evaluation of unitarity or di!ractive driven contribution to g (x,Q) in terms of the two other experimentally accessible spin *2 observables: the SCHNC LT interference di!ractive DIS structure function [3] and the pomeron spin-#ip amplitude in nucleon}nucleon and pion}nucleon scattering [2]. By unitarity relation, the

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Fig. 1. The unitarity diagram with (a) &elastic' pX intermediate state and (b) &inelastic' pHX intermediate state with excitation of the proton to resonances or low-mass continuum pH.

opening of di!ractive DIS channel cHpPpX a!ects the elastic scattering amplitude ([10] and references therein). The best-known unitarity e!ect is Gribov's absorption or shadowing correction [21] to one-pomeron exchange. Besides simple shadowing, for spinning particles unitarity corrections can give rise to new spin amplitudes absent in one-pomeron exchange, which was precisely the case with tensor structure function for DIS o! spin-1 deuteron [7]. In the related evaluation of unitarity-driven p3 we start with the eikonal unitarity diagram in *2 Fig. 1. Here the eikonal refers to the &elastic' pX intermediate state, the e!ect of so-called &inelastic' intermediate states pHX when the proton excites into resonances or low-mass continuum states will be commented on below. Hereafter all unitarity corrections will be supplied by a superscript (U). As an input we need amplitudes A" of di!ractive DIS cHp PX p , where k stands for spin states of IMJH J H I M the di!ractive state X. Applying the optical theorem to this unitarity contribution to forward scattering amplitude, we "nd [10]



A" , (4) p3"Re 1/(16p(=#Q)) dD dM A" \\IM IM*> *2 IM where M is the invariant mass of the intermediate state. In order for this unitarity diagram to contribute to p , the r.h.s. of (4) must have a structure which in the convenient polarization *2 vector-spinor representation has the form p J1 f "(r[neR(!)])"in2 , *2 where r is the nucleon spin operator, n is the unit vector along the cHp collision axis, and

(5)

e(l)"!1/(2(l,i) is the photon polarization vector for helicity l. In the polarization-vector representation the factorized one-pomeron amplitude for di!ractive DIS reads [4,5]






p A"" i# d/ +¹




r (6) ; 1#i  (r[nD]) , m N where ¹ is the imaginary part of di!ractive amplitude for an unpolarized target, V stands for the IJ transverse polarization vector of di!ractive state X and r "r (0)exp(!B D) is the ratio of the   

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spin-#ip to non-#ip pomeron}nucleon couplings. The signature factor g"i#(p/2)d/

(7)

is de"ned through the Q- and x-dependent e!ective intercept d/"(d log Im ¹(x,Q))/(d log 1/x) ,

(8)

which is the same as d taken at a relevant average hard scale. The slight di!erence of this scale and  of d/ thereof for di!erent helicity amplitudes can be neglected for the purposes of our discussion and to a good approximation r can be considered a real valued quantity.  The Regge factorization (6) is equally applicable to one-pomeron exchange elastic scattering, in which case it dictates (hereafter we suppress the helicity (!)) A (D)J¹ (eRD)(r[nD]) , (9) \\*> \* which, as we mentioned in the Introduction, vanishes in the forward case D"0. We need the ¸¹ transition in either of the di!ractive cHX vertices and spin-#ip transition in either of the pomeron}nucleon vertices in unitarity diagram of Fig. 1, the other two vertices are spin non-#ip ones. The both spin-#ip transitions are o!-forward with "nite momentum transfer D to the intermediate state and the integrand of Eq. (4) will be J(eRD)(r[nD]). Summing over the phase space of the intermediate state X includes an integration over azimuthal angle of D of the form



d

1 (eRD)(r[nD])" D(r[neR]) , 2p 2

(10)

which has precisely the desired spin structure (5). Now we notice that the LT interference term in di!erential cross section of di!ractive DIS on unpolarized nucleons cHp PX p equals J H I M dp" /dMdD"1/(16p(=#Q)) A"H A" (11) *2 \HIM IM*H IMH and di!ers from the r.h.s. of (4) only by complex conjugation of one of di!ractive amplitudes. In principle, p" can be measured experimentally. The scaling properties of dp" have been estab*2 *2 lished in [3]. The conventionally de"ned LT interference di!ractive structure function F is *2 twist-3 [3], for the purposes of the present discussion it is convenient to factor out the kinematical factor D/Q and de"ne the scaling and dimensionless LT di!ractive structure function g" (x/,b,Q) *2 such that (M#Q)p" dMdD"4pa /Q ) (De)/Q ) (1#"r "D/m) *2 CK  N (12) ) g" (x/,b,Q)B exp(!B D) , *2 *2 *2 where b"Q/(Q#M) and x/"x/b are di!ractive DIS variables. In what follows we shall neglect corrections J"r ", because nucleon spin-#ip e!ects are numerically very small within the  di!raction cone. Then, making use of (11), (12) and of the factorization property of one-pomeron amplitude (6), we obtain g3 (x,Q)"1/x ) r (0)sin(pa/) ) *2 



db B *2 ) ) g" (x/,b,Q) . b 4m(B #B ) *2 N *2  V

(13)

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4. The model evaluations of gLT

Eq. (13) is our central result and up to now we have been completely model independent. In principle, the g" can be measured experimentally, in the lack of such direct data in our numerical *2 estimates of p we resort to QCD model for di!ractive DIS developed in [3]. We refer to this *2 paper for details, here we only recall the salient results. The driving term of di!ractive DIS is excitation of qq Fock states of the photon (Fig. 2). We notice that only qq pairs with the sum of helicities zero contribute to p . Consider "rst the *2 contribution from intermediate heavy #avour excitation, in which case the mass m of a heavy D quark provides the large pQCD hard scale [22,23] QM +m/1!b . (14) D The lower blobs in the diagram of Fig. 2a are related to skewed unintegrated gluon structure functions of the proton which can be approximated by the conventional diagonal unintegrated gluon structure functions taken at x "x/. To a log QM  accuracy, gross features of g" (x/,b,Q) *2   are described by [3] (15) g" +e/3B m ) b(1!b)(2!3b)a(QM )G(x/,QM ) , *2 D *2 D 1  where e is the quark charge in units of the electron charge, a is the strong coupling, and we D 1 assumed Q<4m. Notice that the hard scale (14) rises as bP1. The QCD scaling violations in the D gluon structure function are strong and at moderate values of QM  the crude approximation is (16) G(x/,QM )JQM A(1/x/)BJbB/(1!b)A(1/x)B ,  where c&1 and 2d & 0.4}0.5 for moderate Q and x:10, see below. Both the scaling violations  and, to a lesser extent the small-x rise of gluon densities, enhance the contribution from b&1. Then the contribution from intermediate open heavy #avour to the l.h.s. of (13) can be evaluated as r (0)e 1  D ) a(QM )G(x,QM +m) . g3(x,Q)+! (17) D *2  30x(B #B )2mm 1 *2  N D The numerical factor in the r.h.s. of (17) is only a crude estimate for c+1, it depends strongly on the pattern of scaling violations which for heavy #avours is under the control of pQCD. The detailed discussion of pQCD hard scale for light #avour contribution to g" is found in [3]. *2 We only emphasize that because the dominant contribution to p comes from b&1 where the *2

Fig. 2. (a) The QCD model for unitarity diagram of Fig. 1. The lower blobs are related to unintegrated gluon structure function of the proton and contain the pomeron spin-#ip amplitude. (b) The pQCD Feynman diagram content of the shaded upper blobs.

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hard scale (14) is enhanced by the factor J1/1!b, even for light #avours p receives a dominant *2 contribution from hard to semihard gluons. The "nal result for g" is similar to (15) with the *2 substitution of m by the semihard scale QM +(0.5}1) GeV. Still it is not under the full control of D pQCD because of substantial contribution of soft momenta in the quark loop integration. Also, because g" changes the sign, the numerical results for the moment (13) depend on the pattern of *2 scaling violations at the semihard scale QM +(0.5}1) GeV which can be a!ected by soft dynamics. Furthermore, the overall result for unitarity driven g is proportional to 1D2, which is the soft *2 quantity controlled by the size of the target proton and large transverse size of di!ractive qq states [24]. All this sensitivity to soft input notwithstanding, our unitarity driven g3 has precisely the *2 same QCD status as standard di!ractive structure functions: it exists, it is a scaling phenomenon, its QCD evolution is reasonably well understood [25], but its numerical magnitude is not calculable from "rst principles of pQCD, although QCD-motivated models do correctly reproduce all features of di!ractive DIS [23,26]. To this end we recall that no one has ever requested pQCD to provide the input for standard DGLAP evolution of the proton structure function. The small-x dependence of xg is the same [3] as of the unpolarized di!ractive structure *2 function [22,23] and, in principle, could be borrowed from experiment. The e!ective exponent d depends on x , at x "10\ for light #avour contribution we "nd 2d +0.4, for the numerically     smaller charm contribution 2d +0.6.  (U)

5. Numerical estimates: unitarity-driven gLT vs. the Wandzura}Wilczek relationship The nucleon spin-#ip de"nes a brand new skewed gluon distribution [27,28], without going into details we only state that anomalous dimensions which control the small-x dependence of this skewed structure function are identical to those for unpolarized gluon distribution and the spin-#ip parameter r would depend on neither x nor Q. Zakharov's sound arguments [2] in favour of  non-vanishing r do not require pQCD and the existence of our unitarity-driven g (x,Q) is  *2 beyond doubts. However, as a soft parameter r is quite sensitive to models of soft wave function of  the nucleon [2]. Incidentally, it is of great interest for the polarimetry of stored proton beams and the whole spin physics program at RHIC [29]. Di!erent model estimates of r and the experi mental situation are summarized in recent review [29]. The experimental data on pion}nucleon scattering give "r ""0.2$0.3, the experimental data on proton}proton scattering leave a room for  quite a strong spin-#ip, r "!0.6 with about 100% uncertainty. The theoretical models give  "r ": 0.1}0.2, the sign of r remains open.   Even if r were known, there will be corrections to our eikonal estimate (13) from proton  excitations pH (resonances and continuum) in the intermediate state (Fig. 1b). Although SCHC di!raction excitation amplitudes are smaller than elastic ones (for suppression of di!raction excitation by the node e!ect see [30]), the inelastic (p,pH) spin-#ip transitions can be enhanced compared to elastic (p,p) ones [5]. Consequently, one cannot exclude that the contribution of pH excitations would enhance the e!ective r by a large factor. Here for the sake of de"niteness we  evaluate xg" (x,Q) assuming the conservative value r "!0.1. We take B "10 GeV for light *2  *2 #avours and B "5 GeV\ for charm as evaluated in [3], the slope B remains unknown and we *2  put B "0. This conservative estimate is shown in Fig. 3, at the moment we cannot exclude even  one order in magnitude larger e!ect. At Q:4m the charm contribution to di!raction is small, the 

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Fig. 3. The conservative estimate assuming r (0)"!0.1 for the unitarity driven xg3(x,Q) vs. the expectation from  *2 WW relation (the dashed curve) for GS [36] parameterization for g (x,Q). The di!erence between the curves for Q"5  GeV (diamonds) and Q"100 GeV (triangles) is due to the charm contribution at large Q.

di!erence between small- and large-Q curves in Fig. 3 illustrates the signi"cance of charm contribution to g . A crude parameterization of our numerical results for x:10\ and Q"5 *2 GeV is xg3(x,Q)+r (0);10\(0.001/x)  . *2  It corresponds to spin symmetry

(18)

(19) A +6;10\ ) r (0)m /(Q ,   N which is approximately #at in the region of x"(10\}10\), cf. the x-dependence of tensor asymmetry in [7]. The steep rise (18) of g3 cannot go forever as it would con#ict the unitarity bounds. The same is *2 true of the experimentally observed rise of unpolarized structure functions. The scale of unitarity e!ects is set by the ratio of di!ractive to non-di!ractive DIS [25,31]. One could evaluate higher-order unitarity e!ects consistently by the technique developed in [32], here we only note that higher-order unitarity corrections are known to play marginal role in related nuclear shadowing in DIS on even heaviest nuclei [33]. Above we focused on the two-pomeron cut contribution which dominates at very small x. The related contribution from secondary reggeon}pomeron cut will be of the form xg0/J(1/x)B>?0\ . *2 Because of a large spin-#ip coupling of secondary reggeons (for the review see [29]), this subleading term can well dominate the unitarity correction at moderately small x. It will de"nitely dominate the small-x behaviour of proton}neutron di!erence g !g . It could eventually be evaluated *2 *2 with the further progress in QCD modelling of reggeon e!ects in di!ractive DIS [34]. With the

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reference to reggeon studies in [34], here we only emphasize that g0/(x,Q) is a scaling function of *2 Q. As a reference value for the comparison with our small-x result, we show in Fig. 3 the so-called Wandzura}Wilczek (WW) result [35]



dy g (y,Q) . (20) y  V The standard parameterizations and QCD ladder estimate [14] give g (x,Q)&(1/x)B with the  exponent d &0 } 0.5. Then xg55(x,Q) would vanish at small x as x\B. To the extent that they  *2 are "tted to the same experimental data, all available parameterizations of g give approximately  the same WW integral, our WW curve shown in Fig. 3 is for the parameterization [36]. Our di!ractive mechanism takes over at x:10\. The uncertain status of the WW relation has been much discussed in the literature [17}19]. The WW relation has never been supposed to, and evidently does not, hold in presence of such a di!ractive component of g . However, in the spirit of duality sum rules one may still hope that *2 full g minus our di!ractive component minus the R/ cut contribution has the required *2 superconvergence properties and one can hypothesize that the WW relation is applicable to g !g3. In other words, it is tempting to identify our unitarity e!ect g3 with the long sought *2 *2 *2 deviation from the WW relation and to hypothesize that WW relation would hold approximately at large and moderate x where the di!ractive component is numerically small. This seems to be the case in the experimentally studied range x90.01 [37]. As well known, g (x,Q) does not admit any obvious parton model interpretation. The OPE *2 content of unitarity corrections to structure functions deserves a dedicated study. We only notice that in close similarity to leading twist unpolarized di!ractive structure function [25], the upper blob in diagrams of Fig. 2 receives a substantial contribution from large and moderate transverse distances. So to say, in the quark loop we are way along the light cone, but "nite, not 1/Q, transverse distance from the light cone. xg55(x,Q)"x *2

(U)

6. Unitarity-driven gLT from vector meson production? We wish to point out that di!ractive vector mesons o!er a direct experimental window at e!ective r (0) which sums up the elastic and inelastic rescattering contributions. Evidently,  unitarity diagrams of Figs. 1 and 2 do contribute to cHp P< p too. Our di!ractive mechanism J H I M would give rise to "nite A4 (D"0), whereas in two-parton ladder (" one pomeron \>> exchange) A4 (D)JD. The marginal di!erence from the above evaluation of di!ractive \>> g is emergence of qq < vertex instead of one of the pointlike QED qq c vertices and that the two *2 gluon distributions in vector meson production are skewed di!erently than in the calculation of g3, but the ratio of di!raction driven SCHNC LT and SCHC conserving dominant one-pomeron *2 amplitudes can be worked out. What counts is that the lower blob of the diagram is identical to that in the case of g3. For the experimental determination of A (D"0) one needs to *2 \>> isolate the polarization dependence of production of longitudinally polarized vector mesons by circular polarized photons on transverse polarized targets, which can be done experimentally because decays of vector mesons are self-analysing. Note, that unitarity considerations in

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Section 3 were quite general and did not require an applicability of pQCD, "nite Q was only needed to have longitudinal photons. In the case of vector mesons, non-vanishing A4 (D"0) is possible not only with virtual photons in polarized DIS, but also for \>> circular polarized real photons. Although real photoproduction of o and mesons will be utterly nonperturbative process, the cross sections are large, high luminosity external real photon beams can readily be produced either by laser backscattering or coherent bremsstrahlung in crystals, and vector meson production seems to be an ideal testing ground for existence of di!raction-driven g3. *2 Because much of the dependence on the model of the vector meson wave function would cancel in the ratio of SCHNC spin-#ip and SCHC non-#ip amplitudes, even nonperturbative real photoproduction of vector mesons would provide useful constraints on r (0).  7. Can the Burkhardt}Cottingham sum rule be salvaged? Our di!raction driven contribution to g (x,Q) rises at small x faster than g (x,Q). Incidentally, *2  the di!ractive mechanism does not contribute to the di!erence helicity of amplitudes which gives rise to the spin asymmetry A . Consequently, at small x the di!raction driven g3 is dominated by  *2 g . The resulting small-x rise of g invalidates the superconvergence assumption behind the   derivation of the BC sum rule [1]. There were suggestions reviewed in [17,19] that the BC sum rule might be salvaged because the residues of pomeron cuts might vanish at large Q. This is not the case with our di!ractive g3(x,Q) which is a manifestly scaling function of Q. To this end we recall *2 that unpolarized di!ractive DIS is a well established scaling phenomenon ([22,25,26], for the corresponding phenomenology and review of the HERA data on di!ractive DIS see [23,38]).

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8. Impact of di4raction-driven gLT to extraction of g1 from longitudinal asymmetry As well known there is a J1/Q correction from g in the extraction of the small-x spin  structure function g from asymmetry A :   A "(1/F (x,Q))+g (x,Q)!(4mx)/Qg (x,Q),    N  "(1/F (x,Q))+g (x,Q(1#(4mx)/Q)!(4m/Q)xg (x,Q), . (21)   N N *2 The small-x growth of xg3(x,Q) invalidates the common lore assumption that this correction *2 can be neglected at low x. Quite to the contrary, it is not dissimilar to, or even somewhat faster than, the usually discussed small-x rise of g (x,Q) and this term cannot be neglected o! hand. Our  conservative estimate (18) suggests that it is small, though.

9. Summary and conclusions The unitarity considerations are hardly ever mentioned in lecture courses and seldom applied in practice in the parton era. We described here one spectacular application of the unitarity relation which modi"es dramatically our ideas on high-energy behaviour of spin e!ects. We have shown

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how s-channel helicity nonconserving LT interference in di!ractive DIS in conjunction with the pomeron spin-#ip in di!ractive nucleon}nucleon scattering gives rise to a steep rise Eq. (1) of the spin structure function g (x,Q) at small x. The transverse spin asymmetry considered in this *2 paper and tensor spin asymmetry discussed earlier in [7] fall into a broad family of unitarity (di!raction) driven spin e!ects which, in the opposite to the common wisdom, persist in high energy and/or small-x limit (the work on straightforward extension of the above to related spin structure functions in DIS o! photons is in progress). The rate of rise of di!raction driven g3(x,Q) is related to that of the other experimental *2 observable * the unpolarized gluon structure function of the target proton or, still better, to the experimentally measurable x/ dependence of unpolarized di!ractive structure function. Whether the di!raction driven xg3(x,Q) is numerically large or small is not an issue, the crucial point is *2 that the found rate of the small-x rise of g (x,Q) invalidates the superconvergence assumptions *2 behind the Burkhardt}Cottingham sum rule and behind the Wandzura}Wilczek relation. There exists an interesting possibility of testing the existence of di!ractive mechanism for g in vector *2 meson production by circular polarized real photons on transverse polarized proton targets, which deserves dedicated study.

Acknowledgements One of the authors (NNN) had the privilege of being a graduate (1968}1970) and postgraduate (1970}1973) student of Lev Borisovich Okun at the Institute for Theoretical and Experimental Physics. This contribution is a humble tribute to what I learned from Lev Borisovich those unforgettable years and way beyond the studentship: Thank you, Lev Borisovich! Our warm thanks go to B.G. Zakharov for inspiration and much insight at early stages of this work. We are grateful to B. Ermolaev and I. Ginzburg for useful discussions. IPI thanks Prof. J. Speth for the hospitality at the Institut f. Kernphysik of Forschungszentrum JuK lich. The work of IPI has been partly supported by RFBR, the work of A.V.P. was supported partly by the US DOE grant DE-FG02-96ER40994.

References [1] H. Burkhardt, W.N. Cottingham, Ann. Phys. (USA) 56 (1970) 453. [2] B.G. Zakharov, Yad. Fiz. 49 (1989) 1386; Sov. J. Nucl. Phys. 49 (1989) 860; B.Z. Kopeliovich, B.G. Zakharov, Phys. Lett. B 226 (1989) 156. [3] N.N. Nikolaev, A.V. Pronyaev, B.G. Zakharov, Phys. Rev. D 59 (1999) 091501. [4] E.V. Kuraev, N.N. Nikolaev, B.G. Zakharov, JETP Lett. 68 (1997) 667. [5] I.P. Ivanov, N.N. Nikolaev, JETP Lett. 69 (1999) 268. [6] V.N. Gribov, Sov. J. Nucl. Phys. 5 (1967) 138; Yad. Fiz. 5 (1967) 197. [7] N.N. Nikolaev, W. Schafer, Phys. Lett. B 398 (1997) 245; Erratum-ibid. B 407 (1997) 453. [8] F.E. Close, S. Kumano, Phys. Rev. D 42 (1990) 2377. [9] V.M. Karnakov, Yad. Fiz. 37 (1983) 1258; Sov. J. Nucl. Phys. 37 (1983) 748. [10] V.A. Abramovskii, V.N. Gribov, O.V. Kancheli, Yad. Fiz. 18 (1973) 595; Sov. J. Nucl. Phys. 18, (1974) 308. [11] B.L. Io!e, V.A. Khoze, L.N. Lipatov, Hard Processes, North-Holland, Amsterdam, Oxford, New York, Tokyo, 1984.

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[12] E.A. Kuraev, L.N. Lipatov, S.V. Fadin, Sov. Phys. JETP 44 (1976) 443; Sov. Phys. JETP 45 (1977) 199; L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904; L.N. Lipatov, Phys. Rept. 286 (1997) 131; V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429 (1998) 127. [13] B.I. Ermolaev, S.I. Manaenkov, M.G. Ryskin, Z. Phys. C 69 (1996) 259; R. Kirschner, L.N. Lipatov, Nucl. Phys. B 213 (1983) 122. [14] J. Bartels, B.I. Ermolaev, M.G. Ryskin, Z. Phys. C 70 (1996) 273; Z. Phys. C 72 (1996) 627. [15] B.I. Ermolaev, S.I. Troian, e-Print Archive: hep-ph/9703384. [16] B. Ermolaev, R. Kirschner, L. Szymanowski, Eur. Phys. J. C 7 (1999) 65. [17] R.L. Ja!e, Comm. Nucl. Part. Phys. 19 (1990) 239. [18] M. Anselmino, A. Efremov, E. Leader, Phys. Rep. 261 (1995) 1; Erratum-ibid. 281 (1997) 399}400. [19] B. Lampe, E. Reya, MPI-PHT-98-23; e-Print Archive: hep-ph/9810270. [20] R.L. Heimann, Nucl. Phys. B 64 (1973) 429. [21] V.N. Gribov, Sov. Phys. JETP 29 (1969) 483. [22] M. Genovese, N. Nikolaev, B. Zakharov, Phys. Lett. B 378 (1996) 347; B 380 (1996) 213. [23] M. Bertini, M. Genovese, N.N. Nikolaev, A.V. Pronyaev, B.G. Zakharov, Phys. Lett. B 422 (1998) 238. [24] N.N. Nikolaev, A.V. Pronyaev, B.G. Zakharov, JETP Lett. 68 (1998) 634. [25] N.N. Nikolaev, B.G. Zakharov, Z. Phys. C 53 (1992) 331; JETP 78 (1994) 598; Z. Phys. C 64 (1994) 631. [26] M. Genovese, N. Nikolaev, B. Zakharov, JETP 81 (1995) 625; JETP 81 (1995) 633. [27] A.V. Radyushkin, Phys. Lett. B 385 (1996) 333. [28] X. Ji, Phys. Rev. D 55 (1997) 7114. [29] N.H. Buttimore et al., Phys. Rev. D 59 (1999) 114010. [30] N.N. Nikolaev, Surveys High Energ. Phys. 7 (1994) 1; Int. J. Mod. Phys. E 3 (1994) 1; Erratum-ibid. E 3 (1994) 995. [31] V. Barone, M. Genovese, N.N. Nikolaev, E. Predazzi, B.G. Zakharov, Phys. Lett. B 326 (1994) 161. [32] B.G. Zakharov, Phys. Atom. Nucl. 61 (1998) 838; Yad. Fiz. 61 (1998) 924. [33] V. Barone et al., Z. Phys. C 58 (1993) 541. [34] W. SchaK fer, Deep inelastic scattering and QCD-DIS'98. in: Gh. Corenmans, R. Roosen (Eds.), Proceedings of 6th International Workshop, Brussels, Belgium, 4}8 April 1998, World Scienti"c, Singapore, pp. 404}407. [35] S. Wandzura, F. Wilczek, Phys. Lett. B 72 (1977) 195. [36] T. Gehrmann, W.J. Stirling, Phys. Rev. D 53 (1996) 6100. [37] P.L. Anthony et al., SLAC-PUB-7983, 1999; E-print Archive: hep-ex/9901006. [38] J. Bartels, J. Ellis, H. Kowalski, M. Wustho!, Eur. Phys. J. C 7 (1999) 443; J. Bartels, C. Royon, hep-ph/ 9809344.

Physics Reports 320 (1999) 187}198

The Higgs boson: shall we see it soon or is it still far away? D.I. Kazakov Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Physics, 141980 Dubna, Moscow Region, Russia

Abstract The status of the Higgs boson mass in the Standard Model and its supersymmetric extensions is reviewed and the perspectives of Higgs searches are discussed. The parameter space of the Minimal Supersymmetric Standard Model (MSSM) is analysed with the emphasis on the lightest Higgs mass. The infrared behaviour of renormalization group equations for the parameters of MSSM is examined and infrared quasi-"xed points are used for the Higgs mass predictions. They strongly suggest the Higgs mass to be lighter than 100 or 130 GeV for low and high tan b scenarios, respectively. Extended models, however, allow one to increase these limits for low tan b up to 50%.  1999 Elsevier Science B.V. All rights reserved. PACS: 14.80.Bn; 14.80.Cp

`¹he search for the Higgs boson is the task C 1 of high energy physicsa ¸.B. Okun', ¹alk at ¸.D. ¸andau Memorial Seminar, Moscow, January 1998 [1].

1. Introduction The last unobserved particle from the Standard Model is the Higgs boson. Its discovery would allow one to complete the SM paradigm and con"rm the mechanism of spontaneous symmetry breaking. On the contrary, the absence of the Higgs boson would awake doubts about the whole picture and would require new concepts. The Higgs mechanism is the simplest and most minimal mechanism which allows one to provide masses to all the particles of the SM, preserving the renormalizability of a theory. It introduces a single new particle * the Higgs boson, which is considered to be a point-like particle, or a bound

E-mail address: [email protected] (D.I. Kazakov) 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 6 8 - X

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state in some approaches, and is supposed to be neutral and massive with the mass of an order of the electroweak breaking scale, i.e. 10 GeV. Experimental limits on the Higgs boson mass come from a direct search at LEP II and Tevatron and from indirect "ts of electroweak precision data, "rst of all from the radiative corrections to the W and top quark masses. A combined "t of modern experimental data gives [2] m "78> GeV (1)  \ which at the 95% con"dence level leads to the upper bound of 260 GeV. At the same time, recent direct searches at LEP II for the c.m. energy of 189 GeV give the lower limit of almost 95 GeV [2]. Within the Standard Model the value of the Higgs mass m is not predicted. The e!ective  potential of the Higgs "eld at the tree level is < "!m"H"#(j/2)("H") . (2)  The minimum of < is achieved for non-vanishing v.e.v. of the Higgs "eld 1H2"v equal to  v"m/(j, which gives the mass m "(2jv as a function of the vacuum expectation value of the  Higgs "eld, v"174.1 GeV, and the quartic coupling j which is a free parameter. However, one can get the bounds on the Higgs mass. They follow from the behaviour of the quartic coupling which obeys the following renormalization group equation describing the change of j with a scale: dj/dt"(1/16p)(6j#6jh!6h#gauge terms) (3) R R with t"ln(Q/k). Here h is the top-quark Yukawa coupling. Since the quartic coupling grows R with rising energy inde"nitely, an upper bound on m follows from the requirement that the theory  be valid up to the scale M or up to a given cut-o! scale K below M [3]. The scale K could .  .  be identi"ed with the scale at which a Landau pole develops. The upper bound on m depends  mildly on the top-quark mass through the impact of the top-quark Yukawa coupling on the running of the quartic coupling j. On the other hand, the requirement of vacuum stability in the SM (positivity of j) imposes a lower bound on the Higgs boson mass, which crucially depends on the top-quark mass as well as on the cut-o! K [3,4]. Again, the dependence of this lower bound on m is due to the e!ect of the R top-quark Yukawa coupling on the quartic coupling in Eq. (3), which drives j to negative values at large scales, thus destabilizing the standard electroweak vacuum. From the point of view of LEP and Tevatron physics, the upper bound on the SM Higgs boson mass does not pose any relevant restriction. The lower bound on m , instead, is particularly  important in view of the search for the Higgs boson at LEPII and Tevatron. For m &174 GeV and R a (M )"0.118 the results at K"10 GeV or at K"1 TeV can be given by the approximate Q 8 formulae [4]

 

m '72#0.9[m !174]!1.0  R

 

a (M )!0.118 Q 8 , K"10 GeV , 0.006

m '135#2.1[m !174]!4.5  R

a (M )!0.118 Q 8 , K"1 TeV , 0.006

where the masses are in units of GeV.

(4) (5)

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Fig. 1. Strong interaction and stability bounds on the SM Higgs boson mass. K denotes the energy scale up to which the SM is valid.

Fig. 1 [5] shows the perturbativity and stability bounds on the Higgs boson mass of the SM for di!erent values of the cut-o! K at which new physics is expected. We see from Fig. 1 and Eqs. (4) and (5) that indeed for m &174 GeV the discovery of a Higgs particle at LEPII would imply that R the Standard Model breaks down at a scale K well below M or M , smaller for lighter %32 .  Higgs. Actually, if the SM is valid up to K&M or M , for m &174 GeV only a small range %32 .  R of values is allowed: 134(m :200 GeV. For m "174 GeV and m (100 GeV [i.e. in the LEPII  R  range] new physics should appear below the scale K& a few to 100 TeV. The dependence on the top-quark mass however is noticeable. A lower value, m K170 GeV, would relax the previous R requirement to K&10 TeV, while a heavier value m K180 GeV would demand new physics at an R energy scale as low as 10 TeV. The previous bounds on the scale at which new physics should appear can be relaxed if the possibility of a metastable vacuum is taken into account [6]. In this case, the lower bounds on m follow from requiring that no transition at any "nite temperature occurs, so that all space  remains in the metastable electroweak vacuum. In practice, if the metastability arguments are taken into account, the lower bounds on m become gradually weaker, though the calculations  become less reliable. On the other hand, this low limit is only valid in the SM with one Higgs doublet: it is enough to add a second doublet with the mass lighter than K to avoid it. A particularly important example of a theory where the bound is avoided is the Minimal Supersymmetric Standard Model.

2. The Higgs boson mass in minimal supersymmetry Supersymmetric extensions of the Standard Model are believed to be the most promising theories at high energies. An attractive feature of SUSY theories is a possibility of unifying various forces of Nature. The best-known supersymmetric extension of the SM is the Minimal

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Supersymmetric Standard Model (MSSM) [7]. The parameter freedom of the MSSM comes mainly from the so-called soft SUSY breaking terms, which are the sources of uncertainty in the MSSM predictions. The most common way of reducing this uncertainty is to assume universality of soft terms, which means an equality of some parameters at a high-energy scale. Adopting the universality, one reduces the parameter space to a "ve-dimensional one [7]: m , m , A, k, and B.   The last two parameters are convenient to trade for the electroweak scale v"v#v, and   tan b"v /v , where v and v are the Higgs "eld vacuum expectation values.     Contrary to the SM, in the MSSM there are at least two Higgs doublets. At the tree level the Higgs potential containing the neutral components and therefore responsible for the masses of physical scalars has the form g#g ("H "!"H ") . <"m"H "#m"H "!m(H H #h.c.)#          8

(6)

There are "ve physical eigenstates: CP-even Higgses h and H, CP-odd Higgs A and a pair of charged Higgses H!, which at tree level have the following masses: m "m#m ,    m " [m #MG((m #M8)!4m Mcos 2b ] , &   8   8 (7) m !"m #M .  5 & For m <M the mass of the lightest Higgs is less than the Z boson mass  8 m KM "cos 2b" (8)  8 independently of any other parameters. However, the inequality m (M is violated by radiative  8 corrections, and the lightest Higgs mass can exceed a hundred GeV, but not very much [8,9]. The detailed analysis of the MSSM parameter space can be performed by minimization of a s function. This analysis implies also that a number of constraints on the parameters like the gauge coupling uni"cation, b}q uni"cation, radiative electroweak symmetry breaking, dark matter density, bPsc decay rate, etc. are imposed. Details of this analysis can be found in Refs. [10,11]. For low tan b the present Higgs limit severely constrains the parameter space, as can be seen from Fig. 2, which shows the excluded regions in the (m , m ) plane for di!erent signs of k. The   experimental Higgs limit of 90 GeV is also valid for the low tan b scenario (tan b(4) of the MSSM. As is apparent from Fig. 2 this limit clearly rules out the k(0 solution, in agreement with other studies [12]. However, this "gure assumes m "175 GeV. The top dependence on the Higgs mass is R slightly steeper than linear in this range and may move the contours within $5 GeV. Adding about one p to the top mass, i.e. m "180 GeV, implies that for the contours in Fig. 2 one R should add 6 GeV to the numbers shown. Even in this case the k(0 solution is excluded for a large region of the parameter space. Only the small allowed region with m '700 GeV is still available  for m "180 GeV. Note that in this region the squarks are well above 1 TeV, so in this case the R cancellation of the quadratic divergencies in the Higgs masses, which is only perfect if sparticles and particles have the same masses, starts to become worrying again. For k'0 practically the whole plane is allowed, except for the left bottom corner shown on the top right-hand side of Fig. 2, although the latest experiment has almost covered this region [2].

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Fig. 2. Contours of the Higgs mass (solid lines) in the m , m plane (above) and the Higgs masses (below) for both signs   of k for the low tan b solution tan b"1.65 for m "175 GeV. R

The upper limit for the mass of the lightest Higgs is reached for heavy squarks, but it saturates quickly, as is apparent from the bottom row in Fig. 2. For m "1000, m "1000, which   corresponds to squarks masses of about 2 TeV, one "nds for the upper limit on the Higgs mass in the low tan b scenario [13]: m "97$6 GeV ,  where the error is dominated by the uncertainty from the top mass. If one requires the squarks to be below 1 TeV, these upper limits are reduced by 4 GeV. For high tan b the upper limit on the Higgs mass in the Constrained MSSM is [13] m "120$2 GeV .  The error from the top mass is small since the high tan b "ts anyway prefer top masses around 190 GeV.

3. Infrared quasi-5xed point scenario One of the possible ways to reduce the parameter freedom of the MSSM is to use the fact that some low-energy parameters are insensitive to their initial high-energy values. This allows one to

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"nd them without detailed knowledge of physics at high energies. To do this one has to examine the infrared behaviour of renormalization group equations (RGEs) for these parameters and use possible infrared "xed points to further restrict them. Notice, however, that the true IR "xed points, discussed e.g. in Ref. [14] are reached only in the asymptotic regime. More interesting is another possibility connected with the so-called infrared quasi-"xed points (IRQFPs) "rst discussed in Ref. [15] and then widely studied by other authors [16}29]. These "xed points usually give the upper (or lower) bounds for the relevant solutions. The well-known example of such infrared behaviour is the top-quark Yukawa coupling > "h /(4p) for low tan b in the framework of the MSSM. In this case the corresponding one-loop R R RGE has exact solution > (t)"> E(t)/(1#6> F(t)) , (9) R R R where E(t) and F(t) are some known functions. It exhibits the IRQFP behaviour in the limit > "> (0)PR [15,18}20,22,30] where the solution becomes independent of the initial R R conditions: > (t)N>$."E(t)/6F(t) . (10) R R A similar conclusion is valid for the other couplings [20,22}24,29,30]. It has been pointed out that the IRQFPs exist for the trilinear SUSY breaking parameter A [22], for the squark R masses [20,23] and for the other soft supersymmetry breaking parameters in the Higgs and squark sector [29]. In the case of large tan b the system of the RGEs has no analytical solution and one can use either numerical or approximate ones. It has been shown [31] that almost all SUSY breaking parameters exhibit IRQFP behaviour. For the IRQFP solutions the dependence on initial conditions > , A and m disappears at low G   energies. This allows one to reduce the number of unknown parameters and make predictions for the MSSM particle masses as functions the only free parameter, namely m , or the gaugino mass,  while the other parameters are strongly restricted. The strategy is the following [32,31]. As input parameters one takes the known values of the top-quark, bottom-quark and q-lepton masses (m , m , m ), the experimental values of the gauge R @ O couplings [2] a "0.118, a "0.034, a "0.017, the sum of Higgs vev's squared v"v#     v"(174.1 GeV) and the "xed-point values for the Yukawa couplings and SUSY breaking  parameters. To determine tan b the relations between the running quark masses and the Higgs v.e.v.s in the MSSM are used: m "h v sin b , (11) R R m "h v cos b , (12) @ @ m "h v cos b . (13) O O The Higgs mixing parameter k is de"ned from the minimization conditions for the Higgs potential. Then, one is left with a single free parameter, namely m , which is directly related to the gluino  mass M . Varying this parameter within the experimentally allowed range, one gets all the masses  as functions of this parameter.

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For low tan b the value of sin b is determined from Eq. (11), while for high tan b it is more convenient to use the relation tan b"(m /m )/(h /h ), since the ratio h /h is almost a constant in the R @ @ R R @ range of possible values of h and h . R @ For the evaluation of tan b one "rst needs to determine the running top- and bottom-quark masses. One can "nd them using the well-known relations to the pole masses (see e.g. [33,34,30]), including both QCD and SUSY corrections. For the top-quark one has m  R , (14) m (m )" R R 1#(*m /m ) #(*m /m ) R R /!" R R 1317 where m "(174.1$5.4) GeV [35]. Then, the following procedure is used to evaluate the R running top mass. First, only the QCD correction is taken into account and m (m ) is found in the R R "rst approximation. This allows one to determine both the stop masses and the stop mixing angle. Next, having at hand the stop and gluino masses, one takes into account the stop/gluino corrections. For the bottom quark the situation is more complicated because the mass of the bottom quark m is essentially smaller than the scale M and so one has to take into account the running of this @ 8 mass from the scale m to the scale M . The procedure is the following [34,36,37]: one starts with @ 8 the bottom-quark pole mass, m "4.94$0.15 [38] and "nds the SM bottom-quark mass at the @ scale m using the two-loop QCD corrections @ m (m )1+"m /(1#(*m /m ) ) . (15) @ @ @ @ @ /!" Then, evolving this mass to the scale M and using a numerical solution of the two-loop SM RGEs 8 [34,37] with a (M )"0.12 one obtains m (M ) "2.91 GeV. Using this value one can calculate  8 @ 8 1+ the sbottom masses and then return back to take into account the SUSY corrections from massive SUSY particles m (M )"m (M )1+/1#(*m /m ) . (16) @ 8 @ 8 @ @ 1317 When calculating the stop and sbottom masses one needs to know the Higgs mixing parameter k. For the determination of this parameter one uses the relation between the Z-boson mass and the low-energy values of m  and m  which comes from the minimization of the Higgs potential: & & M m #R !(m #R )tan b 8#k" &  &  , (17) 2 tan b!1 where R and R are the one-loop corrections [11]. Large contributions to these functions come   from stops and sbottoms. This equation allows one to obtain the absolute value of k, the sign of k remains a free parameter. Whence the quark running masses and the k parameter are found, one can determine the corresponding values of tan b with the help of Eqs. (11) and (12). This gives in low and high tan b cases, respectively, tan b"1.47$0.15$0.05 for k'0 , tan b"1.56$0.15$0.05 for k(0 , tan b"76.3$0.6$0.3

for k'0 ,

tan b"45.7$0.9$0.4

for k(0 .

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The deviations from the central value are connected with the experimental uncertainties of the top-quark mass, a (M ) and uncertainty due to the "xed-point values of h (M ) and h (M ).  8 R 8 @ 8 Having all the relevant parameters at hand it is possible to estimate the masses of the Higgs bosons. With the "xed-point behaviour one has the only dependence left, namely on m or the  gluino mass M . It is restricted only experimentally: M '144 GeV [2] for arbitrary values of the   squarks masses. Let us start with low tan b case. The masses of CP-odd, charged and CP-even heavy Higgses increase almost linearly with M . The main restriction comes from the experimental limit on the  lightest Higgs boson mass. It excludes the k(0 case and for k'0 requires the heavy gluino mass M 5750 GeV. Subsequently one obtains  m '844 GeV, 

m !'846 GeV, &

m '848 GeV for k'0 , &

i.e. these particles are too heavy to be detected in the nearest experiments. For high tan b already the requirement of positivity of m excludes the region with small M . In   the most promising region M '1 TeV (m '300 GeV) for the both cases k'0 and k(0 the   masses of CP-odd, charged and CP-even heavy Higgses are also too heavy to be detected in the near future: m '1100 GeV for k'0,  m !'1105 GeV for k'0, & m '1100 GeV for k'0, &

m '570 GeV for k(0 ,  m !'575 GeV for k(0 . & m '570 GeV for k(0 . &

The situation is di!erent for the lightest Higgs boson h, which is much lighter. As has been already mentioned, for low tan b the negative values of k are excluded by the experimental limits on the Higgs mass. Further on we consider only the positive values of k. Fig. 3 shows the value of m for k'0 as a function of the geometrical mean of stop masses * this parameter is often  identi"ed with a supersymmetry breaking scale M . One can see that the value of m quickly 1317  saturates close to &100 GeV. For M of the order of 1 TeV the value of the lightest Higgs mass 1317 is [32] m "(94.3#1.6#0.6$5$0.4) GeV for M "1 TeV ,  1317

(18)

where the "rst uncertainty comes from the deviations from the IRQFPs for the mass parameters, the second one is related to that of the top-quark Yukawa coupling, the third re#ects the uncertainty of the top-quark mass of 5.4 GeV, and the last one comes from that of the strong coupling. One can see that the main source of uncertainty is the experimental error in the top-quark mass. As for the uncertainties connected with the "xed points, they give much smaller errors, of the order of 1 GeV. Note that the obtained result (18) is very close to the upper boundary, m "97 GeV, obtained in  Refs. [30,13] (see the previous section). For the high tan b case the lightest Higgs is slightly heavier, but the di!erence is crucial for LEP II. The mass of the lightest Higgs boson as a function of M is shown in Fig. 3. One has the 1317

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Fig. 3. (A) The dependence of the mass of the lightest Higgs boson h on M "(m m ) (shaded area) for k'0, low 1317 R R tan b. The dashed line corresponds to the minimum value of m "90 GeV allowed by experiment. (B), (C) The mass of the  lightest Higgs boson h as function of M for di!erent signs of k, large tan b. The curves (a, b) correspond to the upper 1317 limit of the Yukawa couplings and to m/m "0 (a) or to m/m "2 (b). The curves (c, d) correspond to the lower limit     of the Yukawa couplings and to m/m "0 (c) or to m/m "2 (d). Possible values of the mass of the lightest Higgs     boson are inside the areas marked by these lines.

following values of m at a typical scale M "1 TeV (M +1.3 TeV) [31]:  1317  m "128.2!0.4!7.1$5 GeV for k'0 ,  m "120.6!0.1!3.8$5 GeV for k(0 .  The "rst uncertainty is connected with the deviations from the IRQFPs for mass parameters, the second one with the Yukawa coupling IRQFPs, and the third one is due to the experimental

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uncertainty in the top-quark mass. One can immediately see that the deviations from the IRQFPs for mass parameters are negligible and only in#uence the steep fall of the function on the left, which is related to the restriction on the CP-odd Higgs boson mass m . In contrast with the low tan b  case, where the dependence on the deviations from Yukawa "xed points was about 1 GeV, in the present case it is much stronger. The experimental uncertainty in the strong coupling constant a is Q not included because it is negligible compared to those of the top-quark mass and the Yukawa couplings and is not essential here contrary to the low tan b case. One can see that for large tan b the masses of the lightest Higgs boson are typically around 120 GeV that is too heavy for observation at LEP II.

4. Summary and conclusion Thus, one can see that in the IRQFP approach all the Higgs bosons except for the lightest one are found to be too heavy to be accessible in the nearest experiments. This conclusion essentially coincides with the results of more sophisticated analyses. The lightest neutral Higgs boson, is on the contrary always light. In the case of low tan b its mass is small enough to be detected or excluded in the next two years when the c.m.energy of LEP II reaches 200 GeV. On the other hand, for the high tan b scenario the values of the lightest Higgs boson mass are typically around 120 GeV, which is too heavy for the observation at LEP II leaving hopes for the Tevatron and LHC. However, these SUSY limits on the Higgs mass may not be so restricting if non-minimal SUSY models are considered. In a SUSY model extended by a singlet, the so-called Next-to-Minimal model, Eq. (8) is modi"ed and at the tree level the bound looks like [39] mKM cos 2b#jv sin 2b ,  8

(19)

Fig. 4. Dependence of the upper bound on the lightest Higgs boson mass on tan b in MSSM (lower curve), NMSSM (middle curve) and extended SSM (upper curve) [39].

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where j is an additional singlet Yukawa coupling. This coupling being unknown brings us back to the SM situation, though its in#uence is reduced by sin 2b. As a result, for low tan b the upper bound on the Higgs mass is slightly modi"ed (see Fig. 4). Even more dramatic changes are possible in models containing non-standard "elds at intermediate scales. These "elds appear in scenarios with gauge mediated supersymmetry breaking. In this case, the upper bound on the Higgs mass may anyway increase up to 155 GeV [39] (the upper curve in Fig. 4), though it is not necessarily saturated. One should notice, however, that these more sophisticated models do not change the generic feature of SUSY theories, the presence of the light Higgs boson.

Acknowledgements The author is grateful to A.V. Gladyshev and M. Jurc\ is\ in for useful discussions and help in preparing the manuscript. Financial support from RFBR grant C 98-02-17453 is acknowledged.

References [1] L.B. Okun', Uspekhi Fiz. Nauk. 168 (1998) 625. [2] LEP Electroweak Working Group, CERN-EP/99-15, 1999, http://www.cern.ch/LEPEWWG/lepewpage.html. [3] N. Cabibbo, L. Maiani, G. Parisi, R. Petronzio, Nucl. Phys. B 158 (1979) 295; M. Lindner, Z. Phys. C 31 (1986) 295; M. Sher, Phys. Rev. D 179 (1989) 273; M. Lindner, M. Sher, H.W. Zaglauer, Phys. Lett. B 228 (1989) 139. [4] M. Sher, Phys. Lett. B 317 (1993) 159; C. Ford, D.R.T. Jones, P.W. Stephenson, M.B. Einhorn, Nucl. Phys. B 395 (1993) 17; G. Altarelli, I. Isidori, Phys. Lett. B 337 (1994) 141; J.A. Casas, J.R. Espinosa, M. Quiros, Phys. Lett. 342 (1995) 171. [5] T. Hambye, K. Reisselmann, Phys. Rev. D 55 (1997) 7255; H. Dreiner, hep-ph/9902347. [6] G. Anderson, Phys. Lett B 243 (1990) 265; P. Arnold, S. Vokos, Phys. Rev. D 44 (1991) 3260; J.R. Espinosa, M. Quiros, Phys. Lett. 353 (1995) 257. [7] H.P. Nilles, Phys. Rep. 110 (1984) 1; H.E. Haber, G.L. Kane, Phys. Rep. 117 (1985) 75; A.B. Lahanas, D.V. Nanopoulos, Phys. Rep. 145 (1987) 1; R. Barbieri, Riv. Nuovo Cimento 11 (1988) 1; W. de Boer, Progr. Nucl. Part. Phys. 33 (1994) 201; D.I. Kazakov, Surv. High Energy Phys. 10 (1997) 153. [8] A. Brignole, J. Ellis, G. Ridol", F. Zwirner, Phys. Lett. B 271 (1991) 123. [9] M. Carena, M. Quiros, C.E.M. Wagner, Nucl. Phys. B 461 (1996) 407. [10] W. de Boer, R. Ehret, D. Kazakov, Z. Phys. C 67 (1995) 647; W. de Boer et al., Z. Phys. C 71 (1996) 415. [11] A.V. Gladyshev, D.I. Kazakov, W. de Boer, G. Burkart, R. Ehret, Nucl. Phys. B 498 (1997) 3. [12] S.A. Abel, B.C. Allanach, Phys. Lett. B 431 (1998) 339. [13] W. de Boer, H.-J. Grimm, A.V. Gladyshev, D.I. Kazakov, Phys. Lett. B 438 (1998) 281. [14] B. Pendleton, G. Ross, Phys. Lett. B 98 (1981) 291. [15] C.T. Hill, Phys. Rev. D 24 (1981) 691; C.T. Hill, C.N. Leung, S. Rao, Nucl. Phys. B 262 (1985) 517. [16] E. Paschos, Z. Phys. C 26 (1984) 235; J. Halley, E. Paschos, H. Usler, Phys. Lett. B 155 (1985) 107. [17] J. Bagger, S. Dimopoulos, E. Masso, Nucl. Phys. B 253 (1985) 397; J. Bagger, S. Dimopoulos, E. Masso, Phys. Lett. B 156 (1985) 357; S. Dimopoulos, S. Theodorakis, Phys. Lett B 154 (1985) 153. [18] W. Barger, M. Berger, P. Ohman, Phys. Lett. B 314 (1993) 351. [19] P. Langacker, N. Polonsky, Phys. Rev. D 49 (1994) 454. [20] M. Carena et al., Nucl. Phys. B 419 (1994) 213. [21] W. Bardeen et al., Phys. Lett. B 320 (1994) 110. [22] P. Nath et al., Phys. Rev. D 52 (1995) 4169.

198 [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

D.I. Kazakov / Physics Reports 320 (1999) 187}198 M. Carena, C.E.M. Wagner, Nucl. Phys. B 452 (1995) 45. M. Lanzagorta, G. Ross, Phys. Lett. B 364 (1995) 163. J. Feng, N. Polonsky, S. Thomas, Phys. Lett. B 370 (1996) 95; N. Polonsky, Phys. Rev. D 54 (1996) 4537. B. Brahmachari, Mod. Phys. Lett. A 12 (1997) 1969. P. Chankowski, S. Pokorski, hep-ph/9702431, in: G.L. Kane (Ed.), Perspectives on Higgs Physics II, (World Scienti"c, Singapore, 1998). I. Jack, D.R.T. Jones, K.L. Roberts Nucl. Phys. B 455 (1995) 83; P.M. Ferreira, I. Jack, D.R.T. Jones Phys. Lett. B 357 (1995) 359. S.A. Abel, B.C. Allanach, Phys. Lett. B 415 (1997) 371. J. Casas, J. Espinosa, H. Haber, Nucl. Phys. B 526 (1998) 3. M. Jurc\ is\ in, D.I. Kazakov, Mod. Phys. Lett A 14 (1999) 671; hep-ph/9902290. G.K. Yeghiyan, M. Jurc\ is\ in, D.I. Kazakov, Mod. Phys. Lett A 14 (1999) 601; hep-ph/9807411. B. Schrempp, M. Wimmer, Prog. Part. Nucl. Phys. 37 (1996) 1. D.M. Pierce, J.A. Bagger, K. Matchev, R. Zhang, Nucl. Phys. B 491 (1997) 3; J.A. Bagger, K. Matchev, D.M. Pierce, Phys. Lett. B 348 (1995) 443. M. Jones, for the CDF and D0 Coll., talk at the XXXIIIrd Recontres de Moriond, (Electroweak Interactions and Uni"ed Theories), Les Arcs, France, March 1998. H. Arason, D. Castano, B. Keszthelyi, S. Mikaelian, E. Piard, P. Ramond, B. Wright, Phys. Rev. D 46 (1992) 3945. N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C 48 (1990) 673. C.T.H. Davies et al., Phys. Rev. D 50 (1994) 6963. M. Masip, R. Munos, A. Pomarol, Phys. Rev. D 57 (1998) 5340.

Physics Reports 320 (1999) 199}221

Ultra-high-energy cosmic rays and in#ation relics Vadim A. Kuzmin *, Igor I. Tkachev 
Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary Prosp. 7a, Moscow 117312, Russia
TH Division, CERN, CH-1211 Geneva 23, Switzerland Dedicated to Lev Okun, for his continuous inspiration

Abstract There are two processes of matter creation after in#ation that may be relevant to the resolution of the puzzle of cosmic rays observed with energies beyond GZK cut-o!: 1) gravitational creation of superheavy (quasi)stable particles, and 2) non-thermal phase transitions leading to the formation of topological defects. We review both possibilities.  1999 Elsevier Science B.V. All rights reserved. PACS: 96.40.!z; 95.35.#d; 98.80.Cq Keywords: Cosmic rays; Dark matter; Early universe

1. Introduction Cosmological and astrophysical considerations are able to provide the strongest restrictions on parameters of particle physics models and even rule out some classes of models entirely. This is especially valuable when the model is unrestricted by laboratory experiments (which is often the case). Among famous results which made a strong impact on model building is the cosmological domain wall problem which appears in models with spontaneous breaking of discrete symmetries [1] and the problem of magnetic monopoles in Grand Uni"ed Theories (GUT) [2]. In return, studies of cosmological phase transitions [3] and of the dynamics of bubbles of a metastable vacuum [4] lead to the change of basic concepts of the cosmology of the early Universe, and in#ationary cosmology [5}8] was born (for reviews see [9,10]). In#ation gives a possible solution to horizon, #atness and homogeneity problems of `classicala cosmology [6]. In#ation was

* 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 6 4 - 2

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designed to solve the problem of unwanted relics, like magnetic monopoles. It was promptly realized [11] that in#ation can generate small amplitude large-scale density #uctuations which are the necessary seeds for the galaxy and the large-scale structure formation in the Universe. This elevates in#ation from the rank of a `broad brush problem solvera to the rank of a testable hypotheses. And testable in "ne details, as rapidly accumulating data on cosmic microwave background #uctuations (CMB) (starting from COBE detection [12] through numerous balloon and ground-based CMBR experiments and with culmination at MAP [13] and PLANK [14] anticipated detailed maps of anisotropy of the microwave sky) and huge galaxy catalogs like the already collecting data SLOAN digital sky surview [15] will provide a wealth of cosmological information. In#ation is generally assumed to be driven by the special scalar "eld known as the inyaton. During in#ation, the in#aton "eld slowly rolls down towards the minimum of its potential. In#ation ends when the potential energy associated with the in#aton "eld becomes smaller than the kinetic energy, which happens when magnitude of the in#aton "eld decreases below the Plank scale, :M and `colda coherent oscillations of the in#aton "eld commence. These oscillations . contained all the energy of the Universe at that time. All matter in the Universe was created by reheating, which is nothing but decay of the zero momentum mode of in#ation oscillations. The process is obviously of such vital importance that here too one may hope to "nd some observable consequences, speci"c to the process itself and for particular models of particle physics, despite the fact that scales relevant for the reheating are very small. And, indeed, we now believe that there may be some clues left. Among those are: topological defects production in non-thermal phase transitions [16], GUT scale baryogenesis [17], generation of primordial background of stochastic gravitational waves at high frequencies [18], just to mention a few. However, the most interesting could be a possible relation to a mounting puzzle of the ultra-high-energy cosmic rays (UHECR) [19]. When a proton (or neutron) propagates in CMB, it gradually looses energy colliding with photons and creating pions. There is a threshold energy for the process, so it is e!ective for very energetic nucleons only, which leads to the Greisen}Zatsepin}Kuzmin (GZK) cuto! [20] of the high-energy tail of the spectrum of cosmic rays. All this means that the detection of, say, 3;10 eV proton would require its source to be within &50 Mpc. However, several well established events above the cut-o! were observed by Yakutsk [21], Haverah Park [22], Fly's Eye [23] and AGASA [24] collaborations (for the recent reviews see Refs. [25,26]). Results from the AGASA experiment [27] are shown in Fig. 1. The dashed curve represents the expected spectrum if conventional extragalactic sources of UHECR were distributed uniformly in the Universe. This curve exhibits the theoretical GZK cut-o!, but one observes events which are way above it. (Numbers attached to the data points show the number of events observed in each energy bin.) Note that no candidate astrophysical source, like powerful active galaxy nuclei, were found in the directions of all six events with E'10 eV [27] (at these energies cosmic rays experience little de#ection by galactic magnetic "elds). Is some unexpected astrophysics at work here or is this at last an indication of the long awaited new physics? There are two logical possibilities to produce UHE cosmic rays: either charged particles have to be accelerated to energies E'10 eV, or UHECR originate in decays of heavy X-particles,

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Fig. 1. AGASA data set [27], February 1990}October 1997.

m '10 GeV. The maximum energy which can be achieved in an accelerating site of the size 6 R which has the magnetic "eld strength B is [28] E(10Z(B/kG) (R/Mpc) eV .

(1)

A magnetic "eld is required either to keep the particle con"ned within the accelerating region or to produce an accelerating electric "eld. For protons (Z"1) a few sources satisfy this condition: pulsars, active galactic nuclei (AGNs) and radio-galaxies. However, energy losses (pair production and meson photoproduction) restrict the maximum energy to E(10 eV in pulsars and AGNs [25,29], while radio-galaxies that lie along the arrival directions of UHECR are situated at large cosmological distances, 9100 Mpc [30], i.e. beyond the GZK radius. Similar conclusions seem to be true with respect to cosmological gamma ray bursts as a possible source of UHECRs [31]. New astrophysics which may work is a possibility to generate UHECR within GZK sphere in remnants of dead quasars [32] (these are dormant galaxies which harbour a supermassive spinning black hole). New physics suggested as an explanation of UHE cosmic rays, ranges all the way to the violation of the Lorentz invariance [33]. Among less radical extensions of the standard model are: E The existence of a particle which is immune to CMB in comparison with nucleons. In this scenario the primary particle is produced in remote astrophysical accelerators (e.g. radiogalaxies) and is able to travel larger cosmological distances while having energies above the GZK cut-o!. There are variations on this scheme. Supersymmetric partner of gluon, the gluino, can form bound states with quarks and gluons. If gluino is light and quasistable (see e.g. [34,35]), the lightest gluino containing baryons will have su$ciently large GZK threshold to be such a messenger [34] and as a hadron it will be able to produce normal air showers in the Earth's atmosphere. However, there are strong arguments due to Voloshin and Okun [36], against light quasistable gluino based on constraints on the abundance of anomalous heavy isotopes which also will be formed as bound states with gluino.

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High-energy (anti)neutrinos produced in distant astrophysical sources will annihilate via Z resonance on the relic neutrinos and produce energetic gammas or nucleons [37]. The relic neutrino masses in the eV range are consistent with this scenario [38], as well as with the Super-Kamiokande results. The required high density of the relic neutrinos is achieved if gravitational clumping takes place [37] or if the Universe has a signi"cant lepton asymmetry in background neutrinos [38]. Even then the total luminosity of the neutrino sources in the Universe must be as high as 10\}10 of its photon luminosity, and, therefore, neutrino-only sources are called for by the upper bound from the #ux of the cosmic rays [39]. An independent constraint on the density of the relic neutrinos comes from CMBR and already the present data start to be challenging for models with large neutrino asymmetry [40]. E Another class of suggestions is related to topological defects. UHECR are produced when topological defects decompose to constituent "elds (X-particles) which in turn decay [41]. Maximum energy is not a problem here, but in models which involve string [42] or superconducting string [41] networks, the typical separation between defects is of order of the Hubble distance and thus these models are subject to a GZK cut-o!. Models in which defects can decay `locallya include networks of monopoles connected by strings (necklaces) [43], vortons (charge and current carrying loops of superconducting strings stabilized by angular momentum) [44], and monopolonium (bound monopole}antimonopole pairs) [45]. Finally, magnetic monopoles accelerated by intergalactic magnetic "elds have also been considered as primary UHECR particles [46]. E Conceptually, the simplest possibility is that UHECR are produced cosmologically locally in decays of some new particle [47,48]. GZK cut-o! is automatically avoided but the candidate X-particle must obviously obey constraints on mass, number density and lifetime.

2. UHECR from decaying particles In order to produce cosmic rays in the energy range E'10 GeV, the decaying primary particle has to be heavy, with a mass well above GZK cut-o!, m '10 GeV. The lifetime, q , cannot be 6 6 much smaller than the age of the Universe, t +10 yr. Given this shortest possible lifetime, the 3 observed #ux of UHE cosmic rays will be generated with the rather low number density of X-particles, X &10\, where X ,m n /o , n is the number density of X-particles and o is 6 6 6 6   6   the critical density. On the other hand, X-particles must not overclose the Universe, X (1. With 6 X &1, the X-particles may play the role of cold dark matter and the observed #ux of UHE cosmic 6 rays can be matched if q &10 yr. 6 Spectra of UHE cosmic rays arising in decays of relic X-particles were successfully "tted to the data for m in the range 10(m /GeV(10 [49,50]. For example, the "t of Berezinsky et al. 6 6 [49] to observed #uxes of UHECR assuming m +10 GeV is shown in Fig. 2. Beside the mass of 6 the X-particle there is another parameter which controls the #ux of the cosmic rays from decaying particles: namely, the ratio of X-particles number density and their lifetime. For the "t in Fig. 2 one used (X /X )(t /q )"5;10\. 6 !"+ 3 6 The problem of the particle physics mechanism responsible for a long but "nite lifetime of very heavy particles can be solved in several ways. For example, some otherwise conserved quantum number carried by X-particles may be broken very weakly due to instanton transitions [47],

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Fig. 2. Predicted #uxes from decaying X-particles, as calculated in Ref. [49] and the data. Latest AGASA results, Fig. 1, are not shown.

or quantum gravity (wormhole) e!ects [48]. If instantons are responsible for X-particle decays, the lifetime is estimated as q &m\ ) exp(4p/a ), where a is the coupling constant of the relevant 6 6 6 6 gauge interaction. The lifetime will "t the allowed window if the coupling constant (at the scale m ) 6 is a +0.1 [47]. 6 A class of natural candidates for superheavy long-lived particles which arise in string and M theory was re-evaluated recently in Refs. [51] and particles with desired masses and long life-times were identi"ed. Other interesting candidates were found among adjoint messengers in gauge mediated supergravity models [52] and in models of superheavy dark matter with discrete gauge symmetries [53]. Superheavy dark matter candidates in superstrings and supergravity models were considered also in Ref. [54]. Below we address the issue of X-particle abundance.

3. Superheavy particle genesis in the early Universe Superheavy particles can be created in the early Universe by several mechanisms. Among those are: E Non-equilibrium `thermala production in scattering or decay processes in primordial plasma [47,48]. E Production during decay of in#aton oscillations (`preheatinga) [55}58]. E Direct gravitational production from vacuum #uctuations during in#ation [59,19,60]. In any case the "nal ratio of the density in X-particles to the entropy density is normalized by the reheating temperature. The reheating temperature is limited to the value below 10}10 GeV in supergravity models with decaying heavy gravitino [61]. This restricts model parameters when `thermala mechanism of heavy particle production is operative (but does not rule it out [47,48,62]).

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The last two mechanisms are closely related to each other and both can be described on equal footing within frameworks of a single uni"ed approach: particle creation in an external time varying background. However, while the outcome of the second mechanism is highly dependent upon the strength of the coupling of the X-"eld to the in#aton, no coupling (e.g. to the in#aton or plasma) is needed in the third mechanism when the temporal change of the metric is the single cause of particle production. Even absolutely sterile particles are produced by the third mechanism which may be relevant for very long-lived superheavy particles. The resulting abundance is quite independent of the detailed nature of the particle which makes the superheavy (quasi)stable X-particle a very interesting dark matter candidate. We concentrate here on the second and third mechanisms and from the start we introduce coupling of the X-"eld to the in#aton for uniformity of discussion. The limit of zero coupling will correspond to purely gravitational production. In the case of a heavy scalar "eld-X we consider the model (2) ¸"(* )!<( )#(* X)!(M !mR)X!(g/2) X .  I  6  I Here <( ) is the in#aton potential. In simple `chaotica [8] models of in#ation the in#aton is either `massivea with the scalar potential <( )"m /2, or `masslessa, <( )"j /4. Normalization to ( the large-scale structure requires m/M +10\ in the former model and j+10\ in the latter ( . model. The constant m describes direct coupling to the space}time curvature R, with m"0 corresponding to the minimal coupling and m"1/6 to conformal coupling. A fermion "eld (spin ) is conformally coupled to gravity. In addition to standard kinetic and  mass terms it also may have coupling to the in#aton, < "g XM X.  It is convenient to work in conformal metric ds"a(g)(dg!dx) with rescaled "elds, u, a(g) and s,Xa(g)Q, where s"1 and s" for scalar and fermion "elds, respectively. In what  follows we measure time and space intervals in units of in#aton mass, q,mg. 3.1. Quantum xelds in classical backgrounds Here we summarize the basic formalism of particle creation in external classical background (e.g. space}time metric of an expanding universe or oscillating in#aton "eld). For more details see e.g. Refs. [63}66]. (i) Spin 0 bosons. A real scalar "eld is Fourier expanded in a comoving box s(q, x)" [s (q)ak#sH(q)aR k]e kx . I I \ k

(3)

Annihilation and creation operators commute except for [ak, aRk ]"1. The mode functions, s ,s (q) of a scalar Bose "eld are solutions of the oscillator equation I I s#u(q)s "0 (4) I I I with the time-dependent frequency u(q)"k#(a/a)(6m!1)#ma#4qu , I Q

(5)

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where ,d/dq and m,M /m, q,g (0)/4m . (6) Q 6 Here (0) is the value of the in#aton "eld when it starts to oscillate (which corresponds to normalization u(0)"1). Let u at some time interval satisfy the adiabatic condition "u "/u;1. We can choose solutions I I I of Eq. (4) which enter decomposition Eq. (3) to be positive-frequency modes s "u\e\ SIO and I I de"ne vacuum state ak"02"0. The number of particles in a non-vacuum state will be constant during evolution through this adiabatic interval. Let the adiabatic condition be violated for some time and then the system enters another adiabatic interval. In that interval we can de"ne another set of positive-frequency modes and corresponding vacuum. Initial positive-frequency modes evolved through non-adiabatic region will not coincide with `outa state modes, but one set of modes can be expressed in terms of the other. This decomposition is called Bogolyubov transformation [67]. Since one and the same "eld is expanded with the use of two di!erent sets of mode functions, the Fourier coe$cients are also related to each other a k "akak#bH k aR k .

(7)

It follows immediately that the initial vacuum state at late times contains particles (8) 10"aRk outa k "02""b " . I Technically, it is easier to "nd Bogolyubov's coe$cients by diagonalizing Hamiltonian of the "eld X. For any time q this procedure gives "b ""("s "#u"s "!2u)/4u , I I I where mode functions are solutions of Eq. (4) with initial (vacuum) conditions

(9)

s (0)"u\, s (0)"!ius . (10) I I I (ii) Spin  fermions.  The relevant mode functions of the Fermi "eld satisfy the oscillator equation with the complex frequency s#(u!im )s "0 , I I  I

(11)

where the real part of the frequency is given by u"k#m and m "m a#(qu. We choose I   Q s (0)"(1!m /u, s (0)"!ius , I I I  as the initial conditions. In this case we "nd per spin state

(12)

"b ""(u!m !Im(s sHY))/2u . I  I I Finally, the number density of X-particles created by time-varying background is

(13)



n "1/2pa "b "k dk , 6 I Q

(14)

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where is the sum over spin states. The expression (14) gives the number density of particles only, Q with an equal amount of antiparticles being created in the case of charged "elds. 3.2. Gravitational creation of particles It has been noticed [59,19] that superheavy particles are produced gravitationally in the early Universe from vacuum #uctuations and that their abundance can be correct naturally, if the standard Friedmann epoch in the Universe evolution was preceded by the in#ationary stage. This is a fundamental process of particle creation unavoidable in the time varying background and it requires no interactions. Temporal change of the metric is the single cause of particle production. Basically, it is the same process which during in#ation generated primordial large-scale density perturbations. No coupling (e.g. to the in#aton or plasma) is needed. All one needs are stable (very long-living) X-particles with mass of order of the in#aton mass, m +10 GeV. 6 In#ationary stage is not required to produce superheavy particles from the vacuum. Rather, the in#ation provides a cut o! in excessive gravitational production of heavy particles which would happen in the Friedmann Universe if it would start from the initial singularity [19]. Resulting abundance is quite independent of the detailed nature of the particle which makes the superheavy (quasi)stable X-particle a very interesting dark matter candidate. So, we start our consideration with gravitational creation of particles, i.e. we put g"0 (or equivalently q"0) in the formulas above. 3.2.1. Friedmann cosmology For particles with conformal coupling to gravity (fermions or scalars with m"), it is the particle  mass which couples the system to the background expansion and serves as the source of particle creation. Therefore, just on dimensional grounds, we expect n Jm a\ (15) 6 6 at late times when particle creation diminishes. In Friedmann cosmology, aJ(mt)?J(m/H)? (a"  and a" for radiation and matter dominated expansion respectively). We conclude that the  anticipated formulae for the X-particles abundance can be parameterised as n "C m (H/m )? . (16) 6 ? 6 6 On the other hand, it is expansion of the Universe which is responsible for particle creation. Therefore, this equation which describes simple dilution of already created particles, is valid when expansion becomes negligible, H;m . This means also that particles with m
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Fig. 3. The coe$cient C , de"ned in Eq. (16), is shown as a function of a for the background cosmology with a power law ? scale factor aJt?, Ref. [60].

There is no room for superheavy particles in our Universe if it started from the initial Friedmann singularity [19], since the value of the Hubble constant is limited from above only by the Planck constant in this case. 3.2.2. Inyationary cosmology However, this restriction will not be valid if in#ation separates initial conditions, whatever they were, from the observable Universe. In an in#ationary cosmology the Hubble constant (in e!ect) did not exceeded the in#aton mass, H(m . The mass of the in#aton "eld has to be m &10 GeV ( ( as constrained by the amplitude of primordial density #uctuations relevant for the large scale structure formation. Therefore, direct gravitational production of particles with m 'H&10 6 GeV has to be suppressed in in#ationary cosmology. Particle creation from vacuum #uctuations during in#ation (or in the de Sitter space) was extensively studied [68,69], usually in the case of small m and in application to generation of 6 density #uctuations necessary for the large scale structure formation. The characteristic quantity which is usually cited in these applications, the variance of the "eld, 1X2, is de"ned by an expression similar to Eq. (14), in the typical case a +!b the integrand is being multiplied I I by the factor 2 sin(u q)/u . For example, for the scalar Bose "eld with the minimal coupling I I to the curvature, 1X2"3H/8pm if m ;H [68,69]. For massless self-interacting "eld G 6 6 G 1X2+0.132H/(j [70]. Particle creation for the speci"c case of the Hubble-dependent e!ective G mass, m (t)JH(t), was considered in Ref. [71]. 6 Results [60] of direct numerical integration of gravitational creation of superheavy particles in chaotic in#ation model with the potential <( )"m /2 shown in Fig. 4. ( This "gure was calculated assuming ¹ "10 GeV for the reheating temperature. (At reheating 0 the entropy of the Universe was created in addition to X-particles. In general, multiply this "gure by the ratio ¹ /10 GeV and devide it by the fractional entropy increase per comoving volume if it 0 was signi"cant at some late epoch.) The reheating temperature is constrained, ¹ (10 GeV, in 0 supergravity theory [61]. We "nd that X h(1 if m +(few);10 GeV. This value of mass is in 6 6 the range suitable for the explanation of UHECR events [19]. Gravitationally created superheavy

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X-particles can even be the dominating form of matter in the Universe today if X-particles are in this mass range [59,19]. 3.2.3. Isocurvature yuctuations in superheavy particle matter In numerical calculations, Ref. [60], it was found that the variance 1X2 of the "eld X measured at the end of in#ation is independent upon m if the mass of X is small and coupling to curvature is 6 minimal. At some later epoch when H+m (which will be long after the end of in#ation if X is 6 a light "eld) the "eld X starts to oscillate on all scales, including k"0. Only at this time, which we denote by t , all "eld #uctuations are transformed into the non-zero particle density and we can use 6 o "m n +m 1X2. The variance of X #uctuations was unchanged on large scales, starting 6 6 6 6 from the end of in#ation down to the time t . So, when the "eld starts to oscillate o Jm . 6 6 6 However, the energy density of the in#aton "eld, o"3H/8pG, decreased during this time interval in proportion to H(t )/H(0)+m /H(0). That is why the ratio of the energy density in X-particles 6 6 to the total energy density does not depend on m when measured at t't , see Fig. 4. 6 6 Variance of the "eld X is di!erent from the usually calculated one for the "xed de Sitter in#ationary background because we consider the actual evolution of the scale factor and the value of the Hubble parameter is not constant during in#ation, being larger at earlier times. Correspondingly, the number of created particles per decade of k grows logarithmically towards small k if m is small. (Power spectrum behaves similarly.) Examples of the particle 6 number, kn (k), for several values of m are shown in Fig. 5 at the moment corresponding 6 6 to 10 completed in#aton oscillations. The particle momentum is measured in units of the in#aton mass. In contrast to this, one has in the "xed de Sitter background 4p1X2+Hd ln k (k/H)\J with (3!2l)+2m /3H at small m , and consequently 1X2J1/m . Note that the power 6 6 6 spectrum in the "xed de Sitter background grows towards large values of k, which is opposite to the behavior of Fig. 5. Therefore, calculations which would be based on the customary procedure of matching a "xed de Sitter background to a subsequent Friedmann stage would give wrong results, with X PR at 6 m P0. 6

Fig. 4. Ratio of the energy density in X-particles, gravitationally generated in in#ationary cosmology, to the critical energy density is shown as a function of X-particle mass, Ref. [60].

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Fig. 5. Spectrum of created particles, kn(k), in a model with massive in#aton is shown for several choices of the mass of scalar X-particle with the minimal coupling (solid lines) and the conformal coupling (dotted line), Ref. [60]. Masses and momenta, k, are given in units of the in#aton mass.

Matching is also dangerous in the case of large m . When the change is too abrupt, it generates 6 arti"cial particles. This may easily happen for m 'm , see e.g. [72] where excessive production 6 ( was found. At m 9m the number of created particles decreases exponentially with m . 6 ( 6 As Fig. 5 shows, the power spectrum of #uctuations in X-particles is almost scale independent at small k if m /m +1. Therefore, if such particles constitute a considerable fraction of dark matter, 6 ( these #uctuations will be transformed into isocurvature density perturbations at late times and can a!ect large-scale structure formation. Isocurvature #uctuations produce 6 times larger angular temperature #uctuations in cosmic microwave background radiation (CMBR) for the same amplitude of long-wavelength density perturbations compared to the adiabatic case [73]. Therefore, to "t observations with a single spectrum, the mass #uctuation spectrum in isocurvature cold dark matter cosmology must therefore be tilted (with respect to scale invariant spectrum) to favor smaller scales. Let the power spectrum of the "eld #uctuations be kP (k)Jk@. A "t to the second moments of 6 the large-scale mass and cosmic microwave background distributions requires b90.25. Models with b ranging from 0.3 to 0.6 were considered in Refs. [74,75]. It is interesting that the spectrum is indeed correctly tilted for m which is somewhat larger than the in#aton mass, see Fig. 5. On the 6 other hand, we see that lighter particles, m (m , with minimal coupling to gravity and X +1 are 6 ( 6 excluded. 3.3. Preheating Considering gravitational creation of particles only, we would be left today with an oscillating in#aton "eld dominating the Universe. To reheat the Universe we must couple the in#aton to some other "elds. In oscillating background and in #at space}time the number density of particles

 Exclusions from this rule exist if in#aton potential does not have a minimum and energy density in the in#aton "eld after in#ation decreases faster than in conventional matter [76].

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grows exponentially: parametric resonance is always e!ective, see e.g. Ref. [66]. This would lead to explosive decay of oscillations. However, in an expanding universe the resonance is blocked by the redshift, which removes created particles out of the resonance bands. Thus, coherent oscillations of e.g. axion [77], or moduli "elds [78], do not decay via parametric resonance in the expanding Universe. It is interesting that the "eld trapped in a (self-) gravitational well can in principle burst in radiation [79,80], but limitations on the relation between density and size of the clump make it hard to achieve critical conditions when elaborated dark matter models, like the axion [79,81] are considered (axion miniclusters are promising objects in this respect though [82]). Narrow width of the resonance band is a prime reason of stability and such "elds will decay only when the Hubble constant H becomes smaller than the particle width C (i.e. the lifetime of individual particle becomes smaller than the age of the Universe). This results in reheating temperature ¹ &(CM . For a long time it was believed [83] that this is the end of the story for 0& . the in#aton "eld, until it was realized [55] that the resonance parameter q, Eq. (6) can naturally be extremely large in the in#aton case. Indeed, the coupling constant is multiplied by the ratio of initial amplitude of the "eld to its mass, squared, which for the in#ation is &10. The resonance is broad and it is impossible to red-shift the system out of the resonance. The resulting explosive decay is not actually a parametric resonance, since resonance parameter can change by orders of magnitude just during one period of oscillation [56,85], but decay occurs anyway because each period of oscillations the adiabaticity conditions are violated during some short time intervals [56,85]. It is possible that the in#aton oscillations decay after only a dozen of oscillations. Let us now consider this process and introduce a non-zero coupling of the in#aton and the X "elds. For the su$ciently large value of q in Eq. (6) the gravitational production of particles is negligible. If the amount of particles produced is relatively small, the process of their creation may still be considered as creation by external time varying background (oscillating zero mode of the in#aton). However, production can be very e$cient at large q and back reaction of produced particles on the motion of the in#aton "eld has to be included. This restores conservation laws and, as a matter of fact, we shall now consider the decay of the in#aton into X. Here we restrict ourselves to the case of scalar X-particles. The equation of motion for the in#aton "eld is u! u!(aK /a)u#au#4qsu"0 .

(17)

Now the background (i.e. the in#aton "eld) is not homogeneous anymore and the complete quantum problem is not easy. However, the e$ciency of particle creation saves the day: the system rapidly becomes classical [86] and the problem can be turned over to a computer [56}58]. Instead of the number density of created particles it is more convenient (and rigorous) to measure the related quantity: the variance of the #uctuation "eld



1s2"1/(2p) dk"s o " . I

(18)

Initially #uctuations are small and the problem can be followed in the Hartree approximation, viz., the "eld s in Eq. (17) is replaced by its average, 1s2, and spatial gradients of the in#aton "eld are

 For a discussion of the parametric resonance induced by an in#aton "eld see also Refs. [84].

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neglected. The formalism of Section 3.1 then applies. In the situation of e$cient particle creation occupation numbers grow and at some point quantum averages can be approximated by classical averages computed with the help of a certain distribution function [86,87]. At late times the problem becomes classical and it has to be supplemented with appropriate initial conditions which re#ect early quantum evolution. Those are speci"ed in the Fourier space as follows [56,86]. Amplitudes of mode functions are distributed with the probability density (19) P[s o ]Jexp[!(2 (0)/m)u (0)"s o (0)"] I I I and mode functions have random phases. The initial `velocitiesa are locked to their corresponding `coordinatesa s o (0)"!iu (0)s o . (20) I I I The parameter m/ (0) sets the scale of 1s2 which separates regions of quantum and classical #uctuations in this model. The semiclassical description is reliable as long as g/4p" qm/4p (0);1. The full non-linear problem can now be solved numerically on the lattice (namely, Eq. (17) and corresponding equation for the "eld s(q, x) are solved directly in the con"guration space). This classical problem accounts for all the e!ects of particle creation, their rescattering, inverse decays, etc. Initial stage of intensive growth of #uctuations attained (similar to parametric resonance, but not equivalent to it since the resonance parameter q changes rapidly in the expanding Universe) is followed by the stage which is similar to the Kolmogorov turbulence when smooth power spectra of #uctuations are established which slowly approach equilibrium [57,58,86,88]. An important quantity is the maximum strength of #uctuations achieved during the course of evolution (#uctuations are diluted by the expansion of the Universe after decay slows down). A compilation of results of Refs. [56,57] for the realistic case m "10\M is shown in Fig. 6 as ( . a function of model parameters. The stars are results of the lattice calculation which takes all back reaction e!ects into account, solid lines correspond to the computationally less expensive Hartree approximation.

Fig. 6. The maximum value of the variance 1X 2 as a function of q. Solid lines correspond to the Hartree

 approximation with di!erent values of m ,m /m , Ref. [56]. The stars are results of fully non-linear lattice calculations Q 6 ( with m ;m , Ref. [57]. 6 (

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To compare this with the gravitational creation note that in the Hartree approximation o /o&2 6 when #uctuations reach their maximum. Even the lowest level of #uctuations shown in Fig. 6 exceeds the amount of gravitationally created particles. However, while stable particles will be generally overproduced, this mechanism still can be relevant for dark matter and UHECR phenomenon, when e.g. for a given value of m the value of q is low enough to prevent e!ective 6 particle creation, but not negligible. Fig. 6 shows that particles 10 times heavier than in#aton are produced and even this is not a limit in the case of very large q [89]. In the opposite situation when q is su$ciently large and #uctuations develop to the full strength, an interesting and important phenomenon of non-thermal phase transitions [16] can occur which is the subject of discussion in the following section. 4. Topological defects and in6ation Decaying topological defect can naturally produce very energetic particles, and this may be related to UHECR [41}46], for recent reviews see [26]. However, among motivations for in#ation there was the necessity to get rid of unwanted topological defects. And in#ation is doing this job excellently. Since temperature after reheating is constrained, especially severely in supergravity models, it might be that the Universe was never reheated up to the point of GUT phase transitions. Topological defects with a su$ciently high scale of symmetry breaking cannot be created. Then, how could such topological defects populate the Universe? The answer may be provided by non-thermal phase transitions [16] which can occur in preheating after in#ation. Explosive particle production caused by stimulated decay of in#aton oscillations leads to anomalously high "eld variances which restore symmetries of the theory even if the actual reheating temperature is small. The defects are formed when variances are reduced by the continuing expansion of the Universe and a phase transition occurs. 4.1. Non-thermal phase transitions Let us "rst describe ideas qualitatively. Let the in#aton oscillations which have amplitude

(0)&M decay rapidly into some "eld X. In the instant process of decay the energy is conserved . and we have mM &k1X2, where on the left-hand side of this equality we write the initial ( . in#aton energy and on the right-hand side we write the "nal energy stored in the X-"eld. The in#aton decays much faster than the system thermalizes, therefore typical momentum of X particles is of order of the in#aton mass, k&m . We "nd 1X2&M . In thermal equilibrium with the ( . temperature ¹ we would have 1X2"¹/12. In this respect the strength of #uctuations is the same as it would be in equilibrium with the Plankian temperature despite the fact that the real reheating temperature is much lower. Of course, this is an extreme estimate, mainly because the expansion of the Universe was neglected. The realistic 1X2 is shown in Fig. 6, but it can still exceed equilibrium temperature by many orders of magnitude. This e!ect is important for the behavior of spontaneously broken symmetries at preheating. Let us consider the model in which in the vacuum state the symmetry is broken by an order parameter U. At the tree level this can be described by the potential <(U)"!kU/2#jUU/4 .

(21)

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The parameter k is related to the symmetry breaking scale via k"jUU. If the "eld U and the  product of in#aton decay, X, are coupled (with corresponding interaction term in the Lagrangian aXU/2), at non-zero density of X-particles the e!ective mass of U "eld changes to m "!k#a1X2. The symmetry is restored if the e!ective mass became positive,  1X2'jUU/a.  The real problem is complicated and model dependent. While some features can be anticipated and some quantities roughly estimated, the issue requires numerical studies. In recent papers [90}92] the defect formation and even the possibility of the "rst-order phase transitions during preheating was demonstrated explicitly. 4.2. Topological defects in simple models Let us consider for simplicity the system when one and the same "eld serves as the in#aton and the symmetry breaking parameter, U, . We derive the set of models from the prototype potential <( )"j/4( !v)!(g/2) X .

(22)

The in#aton scalar "eld has M components, " + , and interacts with an N-component G G scalar "eld X, X" , X. For simplicity, the "eld X is taken massless and without selfG G interaction. The "elds have minimal coupling to gravity in a FRW universe with a scale factor a(t). It is assumed that the in#aton oscillations start along direction. In the e!ective mass of there  are contributions from g1X2 as well as Jj1(d )2. We review several models, in the order of increasing complexity. Initial conditions in all cases correspond to the vacuum for #uctuations. System then evolved through particle creation, their rescattering, phase transitions and "nally defects creation. 4.2.1. Domain walls [93] Consider one component in#aton "eld, M"1, and no X-"elds. Relevant topological structures in the double-well potential are domain walls. Even in the model without "elds X, #uctuations of the "eld grow dramatically (because of the self-interaction Jj, the zero mode can decay in the process 2 P4 ; in addition to that the spinodal decomposition is e!ective in the present model). This leads to the formation of a domain structure at certain values of j. The domain structure which emerged in such situation is shown in Fig. 7. 4.2.2. Strings [91] Again, consider the model without X-"elds, where the in#aton has two components, M"2. Relevant topological structures in the Mexican hat potential are strings. Fluctuations of the "elds and during preheating restore symmetry along direction, but    along the direction symmetry is broken. For some period of time the universe is divided into  domains "lled with the "eld +$v. Gradually the amplitude of #uctuations of the "elds 

decreases, and symmetry P! also breaks down. At this moment the seed domain G   structure is transformed into a string network. Fig. 8 shows the string distribution in a simulation with the symmetry breaking scale v"3;10 GeV, when a pair of `in"nitea strings and one big loop were formed. Size of the box is comparable to the Hubble length at this time.

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Fig. 7. Domain wall structure generated in simulation of preheating in the model Eq. (22) with M"1, g"0, Ref. [93].

Fig. 8. String network generated in simulation of preheating in the model Eq. (22) with M"2, g"0 (see Ref. [91]) is shown at two successive moments of time.

The "nal result is the string formation, but the sequence of symmetry-breaking patterns would be unusual for thermal phase transitions. For example, the O(2) symmetry was only partially restored. Another peculiarity of non-thermal phase transitions is the possible presence of an oscillating zero mode during "nal stages of the phase transition. This makes probability of formation of long strings (as well as domains in the previous model) to be a non-monotonic function of v, at least for large v and in simplest models considered so far. In more complicated models in#aton oscillations can decay completely before the phase transition, and the phase transition can be even of the "rst order. Defect formation in such models will be more robust and resembling more closely phase transitions in thermal background.

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4.2.3. First-order phase transition [90] Here we consider a one component in#aton "eld, M"1 and several X-"elds. In the case g/j<1 the phase transition can be of the "rst order. This can be expected by analogy with the usual thermal case [94]. The necessary conditions for this transition to occur and to be of the "rst order are as follows. (i) At the moment of phase transition a local minimum of the e!ective potential should be at

"0, which gives g1X2'jv. (ii) At the same time, the typical momentum p of X particles should be smaller than gv. This is H the condition of the existence of a potential barrier. Particles with momenta p(gv cannot penetrate the state with " "+v, so they cannot change the shape of the e!ective potential at " "+v. Therefore, if both conditions (i) and (ii) are satis"ed, the e!ective potential has a local minimum at

"0 and two degenerate minima at +$v. (iii) Before the minima at +$v become deeper than the minimum at "0, the in#aton's zero mode should decay signi"cantly, so that it performs small oscillations near "0. Then, after the minimum at " "+v becomes deeper than the minimum at "0, #uctuations of drive the system over the potential barrier, creating an expanding bubble. All these conditions can be met more easily at large g/j<1 and with several X-"elds. Results discussed below were obtained [90] on a 128 lattice in the model with parameters g/j"200 and v"0.7;10\M +0.8;10 GeV, for a two-component X, with the expansion of the universe . assumed to be radiation dominated (similar behavior was observed in the model with single X "eld, i.e. N"1, as well). Time dependence of the zero mode is shown in Fig. 9. Initially oscillates with a large   amplitude M
Fig. 9. Time dependence of the zero-momentum mode of in units of its vacuum value for two runs with di!erent realizations of random initial conditions for #uctuations in simulation of preheating in the model Eq. (22) with M"1, gO0, Ref. [90].

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Fig. 10. Spontaneously nucleated bubble of the new phase at the initial stage of its expansion, Ref. [90].

the bare potential. This can occur only because the e!ective potential acquires a minimum at "0 due to interaction of the "eld with 1X2. The amplitude of oscillations is decreasing much faster than it would decrease due to the simple expansion of the Universe. This happens because of continuing decay of the zero mode into X-particles. At still later times, q9720 in Fig. 9, the zero mode of decays completely. This should be interpreted as restoration of the symmetry P! by non-thermal #uctuations. Finally, at q9860 (when X-#uctuations were diluted su$ciently by the expansion) a phase transition occurs and the symmetry breaks down. In runs with di!erent input for initial (`vacuuma) #uctuations the system ends up either in #v or in !v vacuum and the transition happens at di!erent times. It was shown that transition is triggered in this model by a spontaneous nucleation of a bubble of the new phase and the bubble's subsequent expansion until it occupies the whole integration volume. The "eld con"guration at the beginning of the phase transition is shown in Fig. 10, where the surface of the constant "eld "!0.7v is plotted at the beginning of the phase transition. Inside the surface

(!0.7v. (Minimum of the e!ective potential is shifted somewhat from the vacuum value at this time, see Fig. 9, because the contribution of #uctuations is still signi"cant.) Models that exhibit behavior shown in Fig. 9 will lead to the domain structure surviving until present. This would be a cosmological disaster and such a class of models is ruled out [1]. However, a di!erent behavior of the zero mode is however observed at smaller values of g/j or N (but still g/j<1). There, the phase transition can occur when the zero mode still oscillates near

"0 with a relatively large amplitude. This is a new, speci"cally non-thermal type of a phase transition. In such cases, bubbles of #v and }v phases will be nucleated in turn (most often at the maximum of amplitude of the zero mode when it is in the closest proximity to the top of the potential barrier), but their abundances need not be equal, and for certain values of the parameters one of the phases may happen not to form in"nite domains. Such models are not ruled out and in fact may have interesting observable consequences, e.g. an enhanced (by bubble wall collisions) background of relic gravitational waves produced by the mechanism proposed in Ref. [18].

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There are no reasons to doubt that in more complicated models non-thermal phase transitions may lead to creation of magnetic monopoles, or magnetic monopoles connected by strings (necklaces [43]), the latter being the most favorable topological defect candidate for explanation of UHECR [49].

5. Conclusions Next generation cosmic-ray experiments, like the Pierre Auger project [95], the high-resolution #y's eye [96], the Japanese telescope array project [97], and the orbiting wide-angle light-collector (OWL) [98], will tell us which model for UHECR may be correct and which has to be ruled out. Very weakly interacting superheavy X-particles with m "(a few);10 GeV may naturally 6 constitute a considerable fraction of cold dark matter. These particles are produced in the early Universe from vacuum #uctuations during or after in#ation. Decays of X-particles may explain UHE cosmic-rays phenomenon. Related density #uctuations may have left an imprint in #uctuations of cosmic microwave background radiation if scalar X-particles with minimal coupling to gravity are approximately twice heavier than the in#aton and X &1. 6 If UHE cosmic rays are indeed due to the decay of these superheavy particles, there has to be a new sharp cut-o! in the cosmic-ray spectrum at energies somewhat smaller than m . 6 Since the number density n depends exponentially upon m /m , the position of this cut-o! 6 6 ( is "xed and can be predicted to be near m +10 GeV, the very shape of the cosmic ray spectrum ( beyond the GZK cut-o! being of quite a generic form that follows from the QCD quark/gluon fragmentation. A very discriminating signature is related to the anisotropy of cosmic rays. If particles immune to CMBR are present, the UHECR events should point towards distant (i.e. beyond GZK sphere), extraordinary astrophysical sources [99]. If superheavy relic particles are in the game, the Galaxy halo will be re#ected in anisotropy of the UHECR #ux [100}102]. If neither will be true but UHECR will point instead to `locala galaxies with evidence for a central supermassive black hole, that would imply that the existence of a black hole dynamo is not a su$cient condition for the presence of pronounced jets and UHECRs are created by the remnants of dead quasars [32]. If none of the above will be true, arrival directions are almost isotropic on large angular scales but UHECRs will cluster on small scales, then perhaps extragalactic magnetic "elds are much stronger than previously thought [103], or perhaps cosmic necklaces do exist. It is remarkable that we might be able to learn about the earliest stages of the Universe's evolution by studying UHECRs. Discovery of heavy relic X-particles will mean that the model of in#ation is likely to be correct, or that at least standard early Friedmann evolution from the singularity is ruled out, since otherwise X-particles would have been inevitably overproduced [19,104]. We conclude that the observations of ultra-high-energy cosmic rays can probe the spectrum of elementary particles in the superheavy range and can give an unique opportunity for investigation of the earliest epoch of evolution of the Universe, starting with the ampli"cation of vacuum #uctuations during in#ation through "ne details of gravitational interaction and down to the physics of reheating.

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Acknowledgements The work of V.K. was supported partially by the Russian Fund for Fundamental Research grant 98-02-17493-a. References [1] Ya.B. Zeldovich, I.Yu. Kobzarev, L.B. Okun, Zh. Eksp. Teor. Fiz. 67 (1974) 3 [Sov. Phys. JETP 40 (1974) 1]. [2] Ya.B. Zeldovich, M.Yu. Khlopov, Phys. Lett. 79B (1978) 239. [3] D.A. Kirzhnits, JETP Lett. 15 (1972) 529; D.A. Kirzhnits, A.D. Linde, Phys. Lett. 42B (1972) 471; Sov. Phys. JETP 40 (1974) 628; Ann. Phys. 101 (1976) 195; S. Weinberg, Phys. Rev. D 9 (1974) 3320; L. Dolan, R. Jackiw, Phys. Rev. D 9 (1974) 3357. [4] I.Yu. Kobzarev, L.B. Okun, M.B. Voloshin, Yad. Fiz. 20 (1974) 1229 [Sov. J. Nucl. Phys. 20 (1975) 644]; S. Coleman, Phys. Rev. D 15 (1977) 2922; S. Coleman, F. DeLuccia, Phys. Rev. D 21 (1980) 3305; V.A. Berezin, V.A. Kuzmin, I.I. Tkachev, Phys. Lett. 120B (1983) 91; J. Ipser, P. Sikivie, Phys. Rev. D 30 (1984) 712; V.A. Berezin, V.A. Kuzmin, I.I. Tkachev, Phys. Rev. D 36 (1987) 2919; S.K. Blau, E.I. Guendelman, A.H. Guth, Phys. Rev. D 35 (1987) 1747. [5] A.A. Starobinskii, Phys. Lett. 91B (1980) 99. [6] A.H. Guth, Phys. Rev. D 23 (1981) 347}356. [7] A.D. Linde, Phys. Lett. 108B (1982) 389; A. Albrecht, P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [8] A.D. Linde, Phys. Lett. 129B (1983) 177. [9] A.D. Linde, Particle Physics and In#ationary Cosmology, Harwood, Chur, Switzerland, 1990. [10] E.W. Kolb, M.S. Turner, The Early Universe, Addison-Wesley, Reading, MA, 1990. [11] V.F. Mukhanov, G.V. Chibisov, JETP Lett. 33 (1981) 532; S.W. Hawking, Phys. Lett. 115B (1982) 295; A.A. Starobinsky, Phys. Lett. 117B (1982) 175; A.H. Guth, S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. [12] G.F. Smoot et al., Astrophys J. 396 (1992) L1; http://www.gsfc.nasa.gov/astro/cobe/cobe}home.html. [13] http://map.gsfc.nasa.gov/. [14] http://astro.estec.esa.nl/Planck/. [15] http://www.sdss.org/. [16] L. Kofman, A. Linde, A.A. Starobinsky, Phys. Rev. Lett. 76 (1996) 1011; I.I. Tkachev, Phys. Lett. B 376 (1996) 35. [17] E.W. Kolb, A.D. Linde, A. Riotto, Phys. Rev. Lett. 77 (1996) 4290; E.W. Kolb, A. Riotto, I.I. Tkachev, Phys. Lett. B 423 (1998) 348. [18] S.Y. Khlebnikov, I.I. Tkachev, Phys. Rev. D 56 (1997) 653. [19] V.A. Kuzmin, I.I. Tkachev, JETP Lett. 68 (1998) 271. [20] K. Greisen, Phys. Rev. Lett. 16 (1966) 748; G.T. Zatsepin, V.A. Kuzmin, Pisma Zh. Eksp. Teor. Fiz. 4 (1966) 114. [21] N.N. E"mov et al., in: M. Nagano, F. Takahara (Eds.), Proceedings of the International Symposium on Astrophysical aspects of the most energetic cosmic rays, World Scienti"c, Singapore, 1991, p. 20; T.A. Egorov et al., in: M. Nagano (Ed.), Proceedings of the Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic Rays, ICRR, U. of Tokyo, 1993; B.N. Afanasiev, in: M. Nagano (Ed.), Proceedings of the International Symposium on Extremely High Energy Cosmic Rays: Astrophysics and future observations, Institute for Cosmic Rays Research, Tokyo, 1996. [22] M.A. Lawrence, R.J.O. Reid, A.A. Watson, J. Phys. G 17 (1991) 733; http://ast.leeds.ac.uk/haverah/hav-home.html. [23] D.J. Bird et al., Phys. Rev. Lett. 71 (1993) 3401; Astrophys J. 424 (1994) 491; 441 (1995) 144. [24] N. Hayashida et al., Phys. Rev. Lett. 73 (1994) 3491; http://icrsun.icrr.u-tokyo.ac.jp/as/project/agasa.html. [25] V.S. Berezinsky, S.V. Bulanov, V.A. Dogiel, V.L. Ginzburg, V.S. Ptuskin, Astrophysics of Cosmic Rays, Elsevier, Amsterdam, 1990. [26] P. Bhattacharjee, G. Sigl, astro-ph/9811011; V. Berezinsky, astro-ph/9811268. [27] M. Takeda et al., Phys. Rev. Lett. 81 (1998) 1163. [28] M.A. Ruderman, P. Sutherland, Astrophys J. 196 (1975) 51; R.D. Blandford, R. Znajek, MNRAS 179 (1977) 433; A.M. Hillas, Ann. Rev. Astron. Astrophys 22 (1984) 425.

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Physics Reports 320 (1999) 223}248

Search for exotic baryons with hidden strangeness in di!ractive production processes L.G. Landsberg State Research Center Institute for High Energy Physics, Protvino, Moscow region 142284, Russia

Abstract Experimental results of the SPHINX Collaboration on studying proton di!ractive production processes are presented. Evidences for new baryon states with masses 92 GeV were obtained in hyperon-kaon e!ective mass spectra in several reactions. Unusual properties of these massive states (small enough decay widths, large branching ratios for their decays with strange particles in "nal states) make them serious candidates for cryptoexotic pentaquark baryons with hidden strangeness "qqqss 2. New results for these states are presented here, as well as recent data on large violation of the OZI rule in proton di!ractive dissociation.  1999 Elsevier Science B.V. All rights reserved. PACS: 14.40.Gk; 13.85.Fb Keywords: Exotic baryon; Di!ractive production; OZI rule

1. Introduction The last two decades have been marked by a signi"cant increase in studies in the "eld of spectroscopy of hadrons, i.e., the particles participating in strong interactions. It has been "rmly established that hadrons are not truly elementary, but composite particles. Similar to atomic nuclei, which consist of nucleons, hadrons are bound systems composed of fundamental particles known as quarks. Quarks are those structural elements that de"ne the variety of the hadronic matter. In addition to a fractional electric charge and baryon number, di!erent #avors (isospin, strangeness, charm, etc.) quarks have a `strong-interaction chargesa called colors. The interactions between quarks go through the exchange of color virtual particles, gluons, in a similar way as the interactions between electric charges are e!ected by exchange of virtual photons. However, in contrast to neutral photons, which bear no charge and do not interact directly with each other, gluons are characterized by color `chargesa and are capable of interacting not only with color quarks but between themselves, thus forming even bound gluon states 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 6 0 - 5

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} glueballs. The interaction between color quarks and gluons is described by quantum chromodynamics. Apparently, color quarks and gluons cannot exist in a free state. Such a hypothesis is experimentally based on the long-standing unsuccessful e!orts to "nd free quarks. Therefore, the concept of con"nment (`the quark imprisonmenta) arose, according to which only the particles without color charge can freely exist, the so-called `colorlessa or `whitea hadrons. All these hadrons are divided by their quantum numbers and their quark content into two large groups: baryons and mesons. Baryons are characterized by their baryon number B (B"1 for baryons and B"!1 for antibaryons) and are produced in pairs to conserve the baryon number of the whole system. Mesons have B"0. Rapid development of hadron spectroscopy has led to a signi"cant advance in the systematics of ordinary hadrons with simplest valence quark structure: qq for mesons and qqq for baryons. Here, we consider only the so-called valence quarks, which determine the hadron nature and its main characteristics (quantum numbers). According to the current theoretical perceptions, well con"rmed by numerous experiments, the valence quarks in a hadron are surrounded by a `clouda of virtual quark}antiquark pairs and gluons, which are constantly emitted and absorbed by the valence quarks. This `clouda or, as one says now, the `seaa of quark}antiquark pairs and gluons, is physical reality determining many properties of hadron (for example, space distribution of its electric charge and magnetic moment, intrinsic distribution of quark and gluon constituents over momenta, etc.). A very important question is whether there exist `colourlessa hadrons with a more complex inner valence structure, such as multiquark mesons (M"qqq q ) and baryons (B"qqqq q), dibaryons (D"qqqqqq), hybrid states (M"qq g, B"qqqg), and glueballs, which are mesons composed only of gluons (M"gg, ggg). Of course, in such new forms of hadronic matter, known as exotic hadrons, e!ects of the quark}gluon `seaa must also appear in addition to the valence quark and gluon structure. The discovery of the exotic hadrons would have a far-reaching consequences for quantum chromodynamics, for the concept of con"nement and for speci"c models of hadron structure (lattice, string, and bag models). Detailed discussions of exotic hadron physics can be found in recent reviews [1}7]. Exotic hadrons can have anomalous quantum numbers not accessible to three-quark baryons or quark}antiquark mesons (open exotic states), or even usual quantum numbers (cryptoexotic states). Cryptoexotic hadrons can be identi"ed only by their unexpected dynamical properties (anomalously narrow decay widths, anomalous decay branching ratios and so on). As is clear from review papers [1}7], in the last decade searches for exotic mesons have led to a considerable advance in this "eld. Several new states have been observed whose properties cannot be explained in the framework of naive quark model of ordinary mesons with qq valence quark structure. These states are serious candidates for exotic mesons (most of them are of cryptoexotic type). At the same time the situation for exotic baryons is far from being clear. There are also some examples of possible unusual baryon resonances [8}11], but these data are not su$cently precise and are not supported by some new experimental results [2,12}16]. Extensive studies of the di!ractive baryon production and search for cryptoexotic pentaquark baryons with hidden strangeness (B ""qqqss 2, here q"u, d quarks) are being carried out by the (

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SPHINX Collaboration at the IHEP accelerator with 70 GeV proton beam. This program was described in detail in reviews [2]. The recent data of the SPHINX experiment [16}25] gave new important evidence of possible existence of cryptoexotic baryons with hidden strangeness X(2050)>PR(1385)K> and X(2000)>PRK>. We shall summarize these data in Section 3, after a general description of the nature and expected properties of cryptoexotic baryons, as well as some promising ways for their production and observation. We also present here the SPHINX results in favour of strong violation of the OZI rule in proton di!ractive dissociation reactions [26}28] which may be connected with direct strangeness in the nucleon quark structure.

2. Exotic baryons and possible mechanisms of their production There arise three main questions tightly connected with the exotic searches in the SPHINX experiments: 1. How to identify cryptoexotic B ""qqqss 2 baryons without open exotic values of their ( quantum numbers and how to distinguish them from several dozens of well-known NH and D isobars? 2. How to produce the exotic baryons in the most e!ective way? 3. How to reduce background processes and to make easier the exotic baryon observation? We will try to "nd some qualitative answers to these questions because of the lack of theoretical models for the description of exotic hadrons. 2.1. Properties of exotic baryons with hidden strangeness As has been stated before, cryptoexotic baryons do not have external exotic quantum numbers, and their complicated internal valence structure can be established only indirectly by examining their unusual dynamical properties that are quite di!erent from those of ordinary baryons "qqq2. In this connection, we consider the properties of multiquark baryons with hidden strangeness "qqqss 2. If such cryptoexotic baryon structure consists of two color parts spatially separated by a centrifugal barrier, its decays into the color-singlet "nal states may be suppressed because of a complicated quark rearrangement in decay processes. The properties of multiquark exotic baryons with the internal color structure "qqqqq 2 ""(qqq) ;(qq ) 2    (color octet bonds) or

(1)

(2) "qqqqq 2 ""(qqq ) ;(qq) 2    (color sextet}antisextet bonds) are discussed in [29}32]. Here, subscripts 1c, 8c, and so on specify representations of the color S;(3) group. If the mass of a nonstrange baryon with hidden  strangeness is above the threshold for decay modes involving strange particles in "nal states, the main decay channels must be of the type B ""qqqss 2P>K#kn (

(3)

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(k"0, 1,2). Another possibility is associated with the decays B P"qqqss 2P N(gN; gN)#kn (4) ( which involve the emission of particles with a signi"cant ss component in their valence-quark structure. It should also be emphasized that g and, particularly, g mesons are strongly coupled to gluon "elds and, hence, to the states with an enriched gluon component. Therefore, baryon decays of the type BPNg, Ng may be speci"c decay modes for hybrid baryons (see, for example, [2]). The nonstrange decays of baryons with hidden strangeness, B PN#kn, must be suppressed by ( the OZI rule. Thus, the e!ective phase-space factors for the decays of the massive baryons B would ( be signi"cantly reduced because of this OZI suppression (owing to a high mass threshold for the allowed decays B P>K with respect to the suppressed decay B PNn, Dn). The mechanism of ( ( quark rearrangement of color clusters in the decays of particles with complicated inner structure of type (1) or (2) can further reduce the decay width of cryptoexotic baryons and make them anomalously narrow (their widths may become as small as several tens of MeV). Here, theoretical predictions are rather uncertain. For this reason, only experiments can answer conclusively the question of whether such narrow baryon resonances with hidden strangeness really exist. Thus, it is desirable to perform systematic searches for the cryptoexotic baryons B with ( anomalous dynamical features listed below. (i) The dominant OZI-allowed decay modes of the baryons B are those with strange particles in ( the "nal states: R("qqqss 2"BR["qqqss 2P>K]/BR["qqqss 2Ppnn; Dn]91.

(5)

For ordinary "qqq2 isobars, R(D; NH)&(several %) [33]. (ii) The cryptoexotic baryons B can simultaneously possess a large mass (M'1.8}2.0 GeV) and ( a narrow decay width (C450}100 MeV). This is due to a complicated internal color structure of these baryons, which leads to a signi"cant quark rearrangement of color clusters in decay processes, and due to a limited phase space for the OZI-allowed decays B P>K. At the same time, ( typical decay widths of the well-established "qqq2 isobars with similar masses are not less than 300 MeV. 2.2. Diwractive production mechanism and search for exotics It was emphasized in a number of studies [1,2,8,10,30,32,34] that di!ractive production processes featuring Pomeron exchange o!er new possibilities in searches for exotic hadrons. Originally, interest was focused on the model of Pomeron with a small cryptoexotic (qqq q ) component [30,32]. According to modern concepts, the Pomeron is a multigluon system owing to which exotic hadrons can be produced in gluon-rich di!ractive processes (see Fig. 1). It is apparent that only the states with the same charge and #avor as those of the primary hadrons can be produced in di!ractive processes. Moreover, there are some additional restrictions on the spin and parity of the formed hadrons which are stipulated by the Gribov}Morrison selection rule. According to this rule, the change in parity *P occurring as a result of the transition from the primary hadron to the di!ractively produced hadronic system, is connected with the corresponding change in the spin *J through the relation *P"(!1) (. For example, because of this rule, in the proton di!ractive dissociation (for proton J."1/2>), only baryonic states with

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Fig. 1. Diagrams for exotic baryon production in the di!raction processes with the Pomeron exchange. The Pomeron P is a multigluon system.

natural sets of quantum numbers 1/2>, 3/2\, 5/2>, 7/2\, etc., can be excited. The Gribov}Morrison selection rule is not a rigorous law and has an approximate character. The Pomeron exchange mechanism in di!ractive production reactions can induce the coherent processes on the target nucleus. In such processes the nucleus acts as descreet unit. Coherent processes can be easily identi"ed by the transverse momentum spectrum of the "nal-state particle system. They manifest themselves as di!ractive peaks with large values of the slope parameters determined by the size of the nucleus: dN/dP Kexp(!bP ), where bK10A GeV \. Owing to 2 2 the di!erence in the absorption of single-particle and multiparticle objects in nuclei, coherent processes could serve as an e!ective tool for separation of resonance production against nonresonant multiparticle background (see e.g. [35]). The conditions for coherent reactions are destroyed by absorption processes in nuclei. Thus, the coherent suppression of nonresonant background takes place: p (res) p (res)   ' . (6) (nonres. BG) p (nonres. BG) p   The separation of coherent reactions can be achieved by studying dN/dP distributions for 2 processes under investigation. As is seen from dN/dP spectra in the SPHINX experiments, there are strong narrow forward 2 cones in these distributions with the slope b950 GeV\, which correspond to a coherent di!ractive production on carbon nuclei (see, for example, Fig. 2). For identi"cation of the coherently produced events we used `softa transverse momentum cut P (0.075}0.1 GeV. With this cut noncoherent 2 background in the event sample can be as large as 30}40%. It is possible to reduce this background with more stringent P cut at the cost of some decrease of the signal statistics. 2

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Fig. 2. dN/dP distribution for typical di!ractive production reaction (p#NP[RK>]#N). The P -regions for 2 2 coherent reaction on carbon nuclei (P 10.075}0.1 GeV) and for nonperipheral processes (P '0.3 GeV) are shown. 2 2

2.3. Processes with large transverse momenta As was discussed above, coherent di!ractive production reactions with small transverse momenta seem quite promising for the search for exotic hadrons, but, of course, they do not exhaust the existing opportunities. Certainly, these searches can be also performed for all di!ractive-type processes (e.g. without coherent cuts for small P ). And of special interest is the study of 2 nonperipheral production reactions which can be the most e!ective way to seek for certain exotic states, especially those that are formed at short ranges and are characterized by broad enough transverse momentum distributions. In this case, the most favorable conditions for identifying exotic hadrons are achieved at higher transverse momenta (P '0.3}0.5 GeV), where the back2 ground from peripheral processes is strongly suppressed. For instance, the unusually narrow meson states X(1740)Pgg [36] and X(1910)Pgg [37] were observed in studying the chargeexchange reactions n\#pP[gg]#D and n\#pP[gg]#n after the selection of events with

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Fig. 3. Diagram for di!ractive-type reaction with multiple Pomeron rescattering (these processes might be signi"cant at P 90.3 GeV). 2

P 50.3 GeV. These anomalous states are good candidates for cryptoexotic mesons. The rescat2 tering mechanism involving multipomeron exchange in the "nal state (a gluon-rich process) may explain X(1740) and X(1910) nonperipheral production [38]. In the very high primary energy region which, for example, corresponds to the search for exotic states with heavy quarks, di!ractive production reactions with rescattering can be used, instead of the charge-exchange processes with rescattering (see the diagram in Fig. 3). The cross sections of these di!ractive processes also do not die out with energy rise. The nonperipheral P region 2 for these processes is shown in Fig. 2. 2.4. Electromagnetic mechanism The search for exotic hadrons with hidden strangeness can be also carried out in another type of hadron production processes caused by electromagnetic interactions. The example of such process is the formation reaction with s-channel resonance photoproduction of strange particles c#NP"qqqss 2P>K

(7)

(see diagram in Fig. 4a). It is possible in principle to study the s-channel resonance production by detailed energetic scanning of the cross sections and angular distributions for Eq. (7) and by performing the subsequent partial-wave analysis. As is seen from Fig. 4a and from VDM (with its signi"cant coupling of photon with ss pair through -meson), reaction (7) can provide a natural way to embed the ss quark pair into intermediate resonance state and to produce the exotic baryon with hidden strangeness. The existing data on reactions (7) are rather poor and insu$cient for such systematical studies. But one can hope that good data would be produced in the near future in the experiments on strong current electron accelerators CEBAF and ELSA (see, for example, [39]). Electromagnetic production of exotic hadrons can be searched for not only in the resonance photoproduction reactions but in the collisions of the primary hadrons with virtual photons of the Coulomb "eld of the target nuclei [40}44], e.g. in the Primako! production reactions h#ZPa#Z

(8)

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Fig. 4. Electromagnetic mechanisms for production of exotic baryons with hidden strangeness: (a) formation reaction with s-channel resonance photoproduction; (b) Coulomb production reaction h#ZPa#Z.

(see diagram in Fig. 4b). The Coulomb production mechanism plays the leading role in the region of very small transfer momenta, where it dominates over the strong interaction process [40}42]. The total cross section of the Coulomb reaction (8) is 2J #1 ? C (aPh#c) . p[h#ZPa#Z]" Kp ! 
 2J #1 F

(9)

The value p is obtained in the QED calculations. In the narrow width approximation for the  resonance as in Eq. (9) p has the form 





M  O [q!q ] ?

 "F (q)" dq . p "8naZ X  M!m q O  ? F

(10)

Here Z is the charge of nucleus; a"1/137 is the narrow structure constant; C(aPh#c) is the radiative width of a; J , J and M , m are the spins and masses of a and h particles; F (q) is ? F ? F X the electromagnetic formfactor of nucleus; q "[[M!m]/2E ] is the minimum square

 ? F F momentum transfer q; E is the primary hadron energy. In the high-energy region, q is very F

 small and q"P #q KP . 2

 2 It must be borne in mind that in the Coulomb production reactions with primary protons one studies the same processes as in ordinary photoproduction reactions. But the Coulomb production in the experiments with unstable primary particles (pions, kaons, hyperons) opens the unique possibility to study the photoproduction reactions with these unstable `targetsa.

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3. The experiments with the SPHINX setup The search for exotic baryons with hidden strangeness was performed in the experiments on the proton beam of the 70 GeV IHEP accelerator with the SPHINX spectrometer. The SPHINX setup includes the following basic components: 1. A wide aperture magnetic spectrometer with proportional wire chambers, drift chambers, drift tubes, scintillator hodoscopes. 2. Multichannel c-spectrometer with lead}glass Cherenkov total absorption counters. 3. A system of gas Cherenkov detectors for identi"cation of secondary charged particles (including a RICH detector with photomatrix equipped with 736 small phototubes; this is the "rst counter of this type used in the experiments). 4. Trigger electronics, data acquisition system and on-line control system. The SPHINX spectrometer works in the proton beam with energy E "70 GeV and intensity N I+(2}4);10p/cycle. The measurements were performed with a polyethylene target to optimize the acceptance, sensitivity and secondary photon losses. The "rst version of the SPHINX setup was described in [12]. The next version of this setup after partial modi"cation (with a new c-spectrometer and with better conditions for K and R identi"cation) was discussed in [21]. To separate di!erent exclusive reactions, a complete kinematical reconstruction of events was performed by taking into account information from the tracking detectors, from the magnetic spectrometer, from the c-spectrometer, and from all Cherenkov counters of the SPHINX setup. At the "nal stage of this reconstruction procedure, the reactions under study were identi"ed by examining the e!ective-mass spectra for subsystems of secondary particles. Several photon-induced di!ractive production processes were studied in the experiments of the SPHINX Collaboration [12}25,27,28]: p#NP[pK>K\]#N ,

(11)

P[p ]#N vK>K\ P[K(1520)K>]#N , vK\p

(12)

(13)

P[RH(1385)K>]#N vKn P[RH(1385)K>]#N#(neutrals) , vKn P[RK>]#N , vKc P[R> K]#N vpnvn>n\

(14)

(15) (16) (17)

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P[pg]#N, vn>n\n

(18)

P[pg]#N, vn>n\gPn>n\2c

(19)

P[pu]#N, vn>n\n P[pn>n\]#N, P[D>>p\]#N , vpp>

(20) (21) (22)

and several other processes. Here N is nucleon or C nucleus for the coherent processes. The separation of coherent di!ractive processes is obtained by studying their dN/dP distributions, as 2 is shown in Fig. 2.

4. Previous data on the coherent di4ractive reactions pⴙCP[R 0Kⴙ ]ⴙC and pⴙCP[R*(1385)0Kⴙ ]ⴙC One of the major results obtained with the SPHINX setup was the study of RK> system produced in di!ractive process (16). These data were obtained in two di!erent runs on the SPHINX facility: (a) the "rst run with the old version of this setup (`old runa, [12,17}20]); (b) the second run with partially upgraded SPHINX apparatus (`new runa [21}23]). As a result of this upgrading the detection e$ciency and purity of K and R events were signi"cantly increased. The main results of these measurements can be summarized as follows: (1) Old [16}20] and new [21}23] data from coherent di!ractive reaction (16) were obtained under di!erent experimental conditions, with a signi"cantly modi"ed apparatus, with di!erent background and systematics. Nevertheless, the RK> invariant mass spectra from both runs are in a good agreement which makes them more reliable. (2) The combined mass spectrum M(RK>) for coherent reaction (16) from the old and new data (with P (0.1 GeV) is presented in Fig. 5. This spectrum is dominated by the X(2000) peak with 2 parameters in Table 1. (3) There is also some near threshold structure in this M(RK>) spectrum in the region of &1800 MeV (see Fig. 5 and Table 1). Such a shape of the RK> mass spectrum (with an additional structure near the threshold) proves that the X(2000) peak cannot be explained by a non-resonant Deck-type di!ractive singularity. Therefore, most likely this peak has a resonant nature. A strong in#uence of P cut for the production of this X(1810) state was established: this 2 structure is produced only at very small P (:0.01}0.02 GeV) } see below. 2

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Fig. 5. Combined mass spectrum M(RK>) for coherent di!ractive reaction (16) in old and new runs on the SPHINX setup (P (0.1 GeV). The parameters of X(2000) peak in this spectrum are: M"1997$7 MeV; C"91$17 MeV. 2

We have also some data in studying the RH(1385)K> system in reaction (14), which were obtained only in the old run (data on this reaction from the new run are now in process of analysis). Coherent events of (14) were singled out in the analysis of dN/dP distribution as a strong forward 2 peak with the slope b950 GeV\. In order to reduce the noncoherent background and to obtain the R(1385)K> mass spectrum for the `purea coherent production reaction on carbon nuclei a `tighta requirement P (0.02 GeV was imposed and the mass spectra of R(1385)K> for the 2 coherent events of (14) were obtained (see, for example, Fig. 6). In these spectra some very narrow structure X(2050) was observed. The "ts of the spectra with Breit}Wigner peaks and polynomial smooth background were carried out, and the average values for the main parameters of X(2050) structure are presented in Table 1. Certainly, one needs further con"rmation of the existence of X(2050) in the new data with increased statistics. Up to now it is impossible also to exclude completely the feasibility for X(2000) and X(2050) to be in fact two di!erent decay modes of the same state. In studying coherent reactions (21) p#CP[pp>p\]#C and (22) p#CP[D(1232)>>p\] #C under the same kinematical conditions as of processes (14) and (16) a search for other decays channels of the X(2000) and X(2050) baryons was performed [18,19]. No peaks in 2 GeV mass range were observed in the mass spectra of pp>p\ and D(1232)>>p\ systems produced in reactions (21) and (22), respectively. Lower limits on the corresponding decay branching ratios R (see (5))

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Table 1 The main results of the previous SPHINX data for coherent di!ractive reactions p#CP[RK>]#C and p#CP[RH(1385)K>]#C 1. Coherent di!ractive production reaction p#CP[RK>]#C with coherent cut P (0.1 GeV was studied in the 2 old and new runs. The combined mass spectrum M(RK>) is presented in Fig. 5. This spectrum is dominated by X(2000) state with parameters M"1997$7 MeV C"91$17 MeV statistical significance of the peak is 7 SD 2. There are also some near threshold structure X(1810), which is produced only in the region of very small P (:0.01}0.02 GeV). The parameters of this peak are 2 M"1812$7 MeV C"56$16 MeV 3. Coherent di!ractive production reaction p#CP[RH(1385)K>]#C with tight coherent cut P (0.02 GeV was 2 studied in the old run ([12,17}20]). The mass spectra of M[RH(1385)K>] are in Fig. 6. The peak was observed in these spectra with average value of parameters M"2052$6 MeV C"35> MeV \ (with the account of the apparatus mass resolution); statistical CL of the peak55 SD 4. The data of (14) and (16) were analyzed together with the data from (21) and (22) to obtain the branching ratios of di!erent decay channels (with strange particles and without strange particle in "nal state). The lower limits of the ratios were obtained from this comparative analysis (with 95% CL): BR+X(2050)>P[RH(1385)K]>, '1.7 R "  BR+X(2050)>P[D(1232)n]>, BR+X(2050)>P[RH(1385)K]>, R " '2.6  BR+X(2050)>Ppn>n\, BR+X(2050)>PRH(1385)K>, R " '0.86  BR+X(2050)>Ppn>n\, BR+X(2000)>P[RK]>, R " '0.83  BR+X(2000)>P[D(1232)n]>, BR+X(2000)>P[RK]>, R " '7.8  BR+X(2000)>Ppn>n\, BR+X(2000)>PRK>, '2.6. R "  BR+X(2000)>Ppn>n\,

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Fig. 6. Invariant mass spectra M[RH(1385)K>] in the coherent reaction (14) at tight transverse-momentum cut P (0.02 GeV for various bin widths: (a) *M"10 MeV; (b) *M"30 MeV. The spectra are "tted to the sum of 2 a smooth polinomial background and X(2050) Breit}Wigner peak. The parameters of X(2050) peak are: (a) M"2053$4 MeV, C"40$15 MeV; (b) M"2053$5 MeV, C"35$16 MeV.

were obtained from this comparative analysis: R[X(2000); X(2050)]91}10 (95% CL)

(23)

(more details are in Table 1). The isotopic relations between the decay amplitudes of I" particles were assumed in these  calculations (the X(2000) and X(2050) states belong to isodoublets since they are produced in a di!ractive dissociation of proton). In accordance with these relations BR[X> PRK>]"BR[X> P(RK)>] , '  '

(24)

BR[X> PD>>p\]"BR[X> P(*p)>] . '  '

(25)

The ratios R }R of the widths of the X(2000) and X(2050) decays into strange and nonstrange   particles are much larger than those for ordinary (qqq)-isobars [18,33]. A narrow width of the X(2000) and X(2050) baryon states as well as anomalously large branching ratios for their decay channels with strange particle emission (large values of R) are the reasons to consider these states as serious candidates for cryptoexotic baryons with a hidden strangeness "uudss 2.

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5. New analysis of the data for reaction pⴙNP[R 0Kⴙ ]ⴙN In what follows we present the results of a new analysis of the data obtained in the run with partially upgraded SPHINX spectrometer where conditions for K and R separation were greatly improved as compared with an old version of this setup. The key element of the new analysis lays in a detailed study of the RPK#c decay separation, which makes it possible to reach the reliable identi"cation of this decay and reaction (16) with the increased e$ciency in comparison with the previous analysis of Bezzubor et al. [21]. In this new analysis the data for (16) were studied with di!erent criteria for RPK#c separation (with larger e$ciency and larger background or with the reduced background at the price of lower e$ciency). We will designate these di!erent criteria for photon separation as soft, intermediate and strong photon cuts (the details will be presented in [45]). Reaction (16) was studied in di!erent kinematical regions. It was found that improved background conditions were important for the investigation of the region of small mass M(RK>) and very small transverse momenta. The e!ective mass spectra M(RK>) in p#NP[RK>]#N for all P are presented in Fig. 7. 2 The peak of X(2000) baryon state with M"1986 MeV and C"98$20 MeV is seen very clearly in these spectra with a very good statistical signi"cance. Thus, the reaction p#NP X(2000)#N ,

(26)

vRK> is well separated in the SPHINX data. The cross section for X(2000) production in (26) is p[p#NPX(2000)#N] ) BR[X(20000)PRK>]"95$20 nb/nucleon

(27)

(with respect to one nucleon under the assumption of pJA, e.g. for the e!ective number of nucleons in carbon nucleus equal to 5.24). The parameters of X(2000) are not sensitive to the di!erent photon cuts, as is seen from Table 2. The dN/dP distribution for reaction (26) is shown in 2 Fig. 8. From this distribution the coherent di!ractive production reaction on carbon nuclei is identi"ed as a di!raction peak with the slope bK63$10 GeV\. The cross section for coherent reaction is determined as p[p#CPX(2000)>#C]

! "260$60 nb/C nucleus .

) BR[X(2000)>PRK>] (28)

We must bear in mind that it is more convenient to use other relations for cross sections: p[p#NPX(2000)>#N]BR[X(2000)>P(RK)>]"285$60 nb/nucleon ,

(29)

p[p#CPX(2000)>#C]BR[X(2000)>P(RK)>]"780$180 nb/nucleus ,

(30)

which were obtained from Eqs. (27) and (28) using branching ratio (24). The errors in the cross sections of Eqs. (27)}(30) are statistical only. Additional systematic errors are K$20% due to uncertainties in the cuts, in Monte Carlo e$ciency calculations and in the absolute normalization.

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237

Fig. 7. Invariant mass spectra M(RK>) in the di!ractive reaction p#NP[RK>]#N for all P (with soft photon 2 cut): (a) measured mass spectrum; (b) the same mass spectrum weighted with the e$ciency of the setup. Parameters of X(2000) peak are in Table 2.

In the mass spectra M(RK>) in Fig. 7 there is only a slight indication for X(1810) structure which was observed earlier in the study of coherent reaction (16) } see Fig. 5 and [21]. This di!erence is caused by a large background in this region for the events in Fig. 7 (all P , soft photon cut). To clarify 2 the situation in this new analysis we investigated also the M(RK>) mass spectra for coherent reaction (16), e.g. for P (0.1 GeV. In these mass spectra not only the X(2000) peak is observed, but 2 the X(1810) structure as well. These spectra (see [45]) are compatible with the data in Fig. 5. The yield of the X(1810) as function of P is shown in Fig. 9. From this "gure it is clear that 2 X(1810) is produced only in a very small P region (P (0.01}0.02 GeV). For P (0.01 GeV the 2 2 2 M(RK>) mass spectrum demonstrates a very sharp X(1810) signal (see Fig. 10) with the parameters of the peak



X(1810)PRK>

M"1807$7 MeV ,

C"62$19 MeV

(31)

which is in a good agreement with the previous data of Table 1. The cross section for coherent X(1810) production is BR[X(1810)>PRK>] p[p#CPX(1810)>#C]" 2 .   %4 "215$45 nb/C nucleus .

(32)

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Table 2 Data on M(RK>) in reaction p#NP[RK>]#N, RPKc with di!erent photon cuts (for all P ) 2 Photon cut

Soft

N events in X(2000) peak Correction factor for photon e$ciency Parameters of X(2000) M (MeV)

Weighted spectrum Measured spectrum

C (MeV)

Weighted spectrum Measured spectrum

p[p#NPX(2000)#N] BR[X(2000)PRK>] (nb/nucleon)

Average values

430$89 1.0

1986$6 1988$5

Intermediate 301$71 1.4

1991$8 1994$7

Strong 190$47 2.25

1988$6 1990$6

98$20 84$20

96$26 94$21

68$21 68$20

100$19

93$25

91$21

1M2 MeV 1C2 MeV 1p[p#NPX(2000)#N]2 BR[X(2000)PRK>] nb/nucleon

1989$6 91$20 95$20 (statist.) $20 (system)

1p[p#CPX(2000)#C]2 BR[X(2000)PRK>] nb/C nucleus

285$60 (statist.) $60 (system)

The additional systematic error for this value is $30%. It increased as compared with the same errors in Eqs. (27)}(30) due to the uncertainty in the evaluation of P smearing in the region of 2 P (0.01 GeV, which is more sensitive to P resolution. 2 2 It is possible also to demonstrate the coherent di!ractive X(2000) production in the clearest way by using the `restricted coherent regiona 0.02(P (0.1 GeV (see Fig. 11) where there is no 2 in#uence of X(1810) structure. To explain the unusual properties of X(1810) state in a very small P region the hypothesis of the 2 electromagnetic production of this state in the Coulomb "eld of carbon nucleus was proposed earlier [46]. It is possible to estimate the cross section for the Coulomb X(1810) production from (9) and (10): p[p#CPX(1810)>#C]" 2 .   %4_ ! 
"(2J #1)+C[X(1810)>Pp#c][MeV],2.8;10\ cm/C nucleus V 55.6;10\ cm+C[X(1810)>Pp#c][MeV], (J 5 is the spin of X(1810)). V 

(33)

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239

Fig. 8. dN/dP distribution for the di!ractive production reaction p#NPX(2000)#N. The distribution is "tted 2 in the form dN/dP "a exp(!b P )#a exp(!b P ) with the slope parameters b "63$10 GeV\; 2   2   2  b "5.8$0.6 GeV\. 

Fig. 9. The P dependence for the X(1810) structure production in the coherent reaction p#CPX(1810)#C. 2

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Fig. 10. Invariant mass spectra M(RK>) in the coherent di!ractive production reaction p#CP[RK>]#C in the region of very small P (0.01 GeV (with strong photon cut) weighted with the e$ciency of the setup. 2

Let us compare this Coulomb hypothesis prediction with the experimental value p[p#CPX(1810)>#C]" 2 9645 nb/C nucleus . (34) .  %4 To obtain (34) we assumed in (16) that X(1810) is isodoublet, and then we use from (24) the branching BR(X>PRK>): (here K means that BR[X>P(RK)>]K1, i.e. this decay is  dominating). If the value of radiative width C[X(1810)Pp#c] is around 0.1}0.3 MeV and the branching BR[X(1810)>P(RK)>] is signi"cant, then the experimental data for cross section of the coherent X(1810) production (34) can be in agreement with the Coulomb mechanism prediction (33). It seems that such value of radiative width is quite reasonable. For example, the radiative width for D(1232) isobar is C[D(1232)>Pp#c]K0.7 MeV. The value of radiative width depends on the amplitude of this process A and on kinematical factor: C""A" ) (P )J> (P is the momentum of A A photon in the rest frame of the decay baryon and l is orbital momentum). For X(1810)Pp#c decay the kinematical factor may be by an order of magnitude larger than for D(1232)>Pp#c because of the large mass of X(1810) baryon. Certainly, the predictions for amplitude A are quite speculative. But if, for example, (X1810) is the state with hidden strangeness "qqqss 2, then the amplitude A might be not very small due to a possible VDM decay mechanism

L.G. Landsberg / Physics Reports 320 (1999) 223}248

241

Fig. 11. Weighted invariant mass spectrum M(RK>) for the reaction p#CP[RK>]#C in the `restricted coherent regiona 0.02(P (0.1 GeV (with intermediate photon cut). 2

(qqqss )P(qqq)# P(qqq)#c. Thus, it seems that the experimental data for the coherent   production of X(1810) (34) is not in contradiction with the Coulomb production hypothesis. It is possible that the candidate state X(2050)PRH(1385)K> which was observed in coherent reaction (14) in the region of very small transverse momenta (P (0.02 GeV) is also produced not 2 by di!ractive, but by the electromagnetic Coulomb production mechanism [46]. The feasibility to separate the Coulomb production processes in the coherent proton reactions at E "70 GeV on the carbon target in the measurements with the SPHINX setup was demonstrated N recently by observation of the Coulomb production of D(1232)> isobar with I" in the reaction  p#CPD(1232)>#C (35) (see [46]).

6. Reliability of X(2000) baryon state The data on X(2000) baryon state with unusual dynamical properties (large decay branching with strange particle emission, limited decay width) were obtained with a good statistical signi"cance in di!erent SPHINX runs with widely di!erent experimental conditions and for several

242

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kinematical regions of reaction (16). The average values of the mass and width of X(2000) state are



X(2000)PRK>

M"1989$6 MeV ,

C"91$20 MeV .

(36)

Due to its anomalous properties X(2000) state can be considered as a serious candidate for pentaquark exotic baryon with hidden strangeness: "X(2000)2""uudss 2. Recently some new additional data have been obtained which are in favor of the reality of X(2000) state. (a) In the experiment of the SPHINX Collaboration reaction (17) was studied. The data for the e!ective mass spectrum M(R>K) in this reaction are presented in Fig. 12. In spite of limited statistics the X(2000) peak and the indication for X(1810) structure are seen in this mass spectrum and are quite compatible with the data for reaction (16). (b) In the experiment on the SELEX(E781) spectrometer with the R\ hyperon beam of the Fermilab Tevatron the di!ractive production reaction R\#NP[R\K>K\]#N

Fig. 12. The e!ective mass spectrum M(R>K) in reaction (17) for P (0.1 GeV. 2

(37)

L.G. Landsberg / Physics Reports 320 (1999) 223}248

243

Fig. 13. Study of M(R\K>) in the reaction R\#NP[R\K>K\]#N in the SELEX experiment. Here the spectrum M(R\K\) with open exotic quantum number used for subtraction of nonresonance background in M(R\K>) after some normalization. One presents in this "gure M(R\K>)!0.95M(R\K\) (here 0.95 } normalization factor): (a) all the events; (b) after subtraction of the events in band to suppress the in#uence of the reaction R\#NP[R\ ]#N.

was studied at the beam momentum PR\K600 GeV. In the invariant mass spectrum M(R\K>) for this reaction a peak with parameters M"1962$12 MeV and C"96$32 MeV was observed (Fig. 13). The parameters of this structure are very near to the parameters of X(2000)PRK> state which was observed in the experiments on the SPHINX spectrometer. It seems that the real existence of X(2000) baryon is supported by the data of another experiment and in another process. Preliminary results of studying reactions (17) and (37) were discussed in the talks at the last conferences [24,25,47] and are now under detailed study.

7. Nonperipheral processes As was discussed above (Section 2.3) the search for new baryons in proton-induced di!ractivelike reactions in the nonperipheral domain, with P '0.3}0.5 GeV, seems to be quite promising. 2 Here we present the very "rst results of these searches in the invariant mass spectra of the RK> and pg systems produced in the reactions p#NP[RK>]#N (16) and p#NP[pg]#N (18) for P '0.3 GeV (see [20,21]). Combined data on reaction (16) from the old and new runs are 2 shown in Fig. 14a. The data from the old run for reaction (18) are shown in Fig. 14b. Despite limited statistics, a structure with mass M+2350 MeV and width C&60 MeV can be clearly seen in these two mass spectra. They require a further study in future experiments with larger statistics. The same statement seems true for the intriguing data on the invariant mass spectrum M(pg) for reaction (19) in the region P '0.3 GeV (see Fig. 14c). It must be stressed that reaction (19) is the 2 only one (among more than a dozen of other di!ractive-like reactions studied in the SPHINX experiments) in which a strong coherent production on carbon nuclei was not observed (the absence of the forward peak in dN/dP distribution with the slope value b950 GeV\; the slope 2 for forward cone in Eq. (19) is b&6.5 GeV\ } see [20,22]).

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Fig. 14. (a) Invariant mass spectrum of the RK> system produced in the reaction p#NP[RK>]#N (16) for P '0.3 GeV (combined data from the old and new runs). (b) Invariant mass spectrum of the pg system produced in the 2 reaction p#NP[pg]#N (18) for P '0.3 GeV (old data, see [20]). A narrow structure with M&2350 MeV and 2 C&60 MeV in the nonperipheral region of reactions (16) and (18) is seen in these mass spectra. (c) Invariant mass spectrum M(pg) for the reaction p#NP[pg]#N (19) in the region P '0.3 GeV (old run). This spectrum is 2 dominated by a threshold structure with M&2000 MeV and C&100 MeV.

8. OZI rule and di4ractive proton reactions The selection rule for connected and disconnected quark diagrams, which is referred to as the OZI rule, has been known for a long time [48}50]. According to this rule, processes involving the annihilation or creation of a quark}antiquark pair entering into the composition of the same hadron are forbidden or, strictly speaking, strongly suppressed. The OZI rule can be illustrated by results of studying the charge exchange reactions p\#pP #n

(38)

(OZI forbidden process) and p\#pPu#n (OZI allowed process). As is well known, the quark structure of vector and u mesons has the form 1 " 2"!cos a ) "ss 2!sin a ) " (uu #ddM ) , (2 1 (uu #ddM ) . "u2"!sin a ) "ss 2#cos a ) " (2

(39)

(40)

L.G. Landsberg / Physics Reports 320 (1999) 223}248

245

Here a"*0 "0 !0 ; 0 is the mixing angle in the vector meson nonet and 0"arctg (1/(2)" 4 4 4 4 4 35.33 is ideal mixing angle. For ideal mixing a"0 and " 2"ss ; "u2""(1/(2)(uu #ddM ). From several data the value of "a" for vector nonet is "a"K(3}43). Thus, in accordance with the OZI rule, the ratio of the cross sections for reactions (38) and (39) is R( /u) "p( )/p(u)"tg a&(3}5);10\ -8'

(41)

in a very good agreement with several experimental data. For example, the recent result of LEPTON-F Collaboration for charge exchange reactions (38), (39) at P \"32.5 GeV [26] is: p R( /u) "(2.9$0.9);10\ .  _p

(42)

Recent years have seen revived interest in the OZI rule. First, intensive searches for exotic hadrons (see the surveys in [1}7] and references therein) are greatly facilitated by choosing production processes in which the formation of conventional particles is suppressed by the OZI rule [51}53,34]. Since this rule may be signi"cantly violated for exotic hadrons because of their complex color structure, signals from exotic and cryptoexotic particles are expected to bene"t from the more favorable background conditions in OZI-suppressed production processes. While the p and o decays of isovector mesons with conventional quark structure (uu !ddM )/(2 are suppressed by the OZI rule (the corresponding probabilities are reduced by more than two orders of magnitude), the same decay channels may prove much more probable for exotic multiquark mesons with hidden strangeness [(uu !ddM )ss /(2] and hybrid states like (uu !ddM )g/(2. Furthermore, unexpected results obtained in polarization measurements for deep-inelastic lepton scattering on nucleons give rise to the well-known problem of nucleon-spin crisis [54,55]. To explain this phenomenon, it was hypothesized that nucleons involved an enhanced quark component with hidden strangeness (direct strangeness in nucleons). This may induce signi"cant violations of the OZI rule in nucleon processes [56,57]. Strong violations of the OZI rule were indeed observed in the relative and u yields from some channels of p annihilation (reactions p pP p, p pPup, p pP c, and p pPuc [57,58]). The above arguments suggest that the OZI rule should be further tested in various production and decay processes and "rst of all in the nucleon reactions. In the experiments of the SPHINX collaboration this test was performed in studying di!ractive production reactions with and u mesons (12) and (20) [27,28]. It was observed that the average value of the e!ective ratio of yields of and u mesons in these reactions is 1R( /u)2 "(4}7);10\ .  _N

(43)

Thus, the strong violation of the OZI rule (by more than an order of magnitude) is observed in proton di!ractive production reactions. The intriguing large violation of the OZI rule in di!erent nucleon reactions and, particularly, in proton di!ractive production processes may suggest an enhanced component with hidden strangeness in a quark structure of nucleons (the model of direct strangeness in nucleon). The OZI rule in proton reactions and the possible existence of direct strangeness are illustrated by diagrams in Fig. 15.

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Fig. 15. production in proton}nucleon or proton}nucleus di!ractive reactions. (a) Disconnected quark diagram (suppressed by OZI rule). (b) Connected diagram for production in the model with some small direct component of ss in proton wave function. In this model one observes the apparent OZI rule violation due to a small exotic ss component (model with direct strangeness in nucleon).

9. Conclusion The extensive research program of studying di!ractive production in E "70 GeV proton N reactions is being carried out in experiments with the SPHINX setup [2,12}28,45}47]. This program is aimed primarily at searches for cryptoexotic baryons with hidden strangeness ("uudss 2). Only a part of these experiments is discussed here. The most important results of these searches were obtained in studies of the hyperon-kaon systems produced in proton di!ractive dissociation processes and "rst of all in reaction p#NPRK>#N (16). New data for this di!ractive production reaction were obtained with the partially upgraded SPHINX detector (with new c-spectrometer and with better possibilities to detect KPpn\ and RPKc decays). New data are in a good agreement with previous SPHINX results on the invariant mass spectrum M(RK>) in this reaction. A strong X (2000) peak with M"1989$6 GeV and C"91$20 MeV together with a narrow threshold structure (with M&1810 MeV and C&60 MeV) are clearly seen in the (RK>) invariant mass spectra. The latter structure is produced at very small transverse momenta, P 10.01!0.02 2 GeV. The unusual properties of the X (2000) baryon state (narrow decay width, anomalously large branching ratio for the decays with strange particle emission) make this state a serious candidate for a cryptoexotic pentaquark baryon with hidden strangeness "qqqss 2. Preliminary data for "RK2 states in other reactions (17) and (37) con"rm the real existence of X (2000) baryon. The OZI selection rule was investigated by comparing the cross sections for pion-induced charge-exchange reactions n\#pP #n and n\#pPu#n at P \"32.5 GeV, as well L as the cross sections for the proton-induced di!ractive reactions p#NP[p ]#N and p#NP[pu]#N at E "70 GeV, in experiments with the LEPTON-F and SPHINX spectromN eters. It was shown that in pion reactions the ratio R( /u)K(3$1);10\ is in good agreement with the naive-quark model and with the OZI rule prediction (R( /u) "tgDh K4;10\). -8' 4

L.G. Landsberg / Physics Reports 320 (1999) 223}248

247

At the same time, this ratio in proton reactions is 1R( /u)2K(4}7);10\. That is, a strong violation of the OZI rule is observed in proton}nucleon interactions. The large violation of OZI rule in proton di!ractive reactions may be in favor of some enhanced hidden strange component in nucleons (the model with direct strangeness). Several other interested phenomena were observed in the nonperipheral domain (with P 90.3 GeV), in the study of di!erent reactions (for example, reactions (14), (18) and (19)). But 2 they need experimental veri"cation with much better statistics. Now the "rst stage of the experimental program on the SPHINX setup has been completed. In the last years the SPHINX spectrometer was totally upgrated. The luminosity and the data taking rate were greatly increased. In the recent runs with this upgrated setup we obtained large statistics which is now under data analysis. In the near future, we expect to increase statistics by an order of magnitude for the processes discussed above and for some other proton reactions. Acknowledgements It is a great pleasure and a great honor for me to participate in this volume dedicated to the 70th anniversary of the birth of the outstanding scientist Lev Okun. I am very lucky to share with him many years of close contacts and a real friendship. I am deeply obliged to Lev for many illuminating discussions which helped me tremendously in many ways and, in particular, in the studies which are covered in this paper. Even my original interest in exotic hadrons was initiated many years ago by his brilliant lectures at physics schools in IHEP, ITEP and MEPI (see, for example, Ref. [59]). I wish Lev many further productive and happy years in his life and work. This work is partially supported by Russian Foundation for Basic Research (grant 99-02-18251). References [1] L.G. Landsberg, Surv. High Energy Phys. 6 (1992) 257. [2] L.G. Landsberg, Yad. Fiz. 57 (1994) 47 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 42]; USP. Fiz. Nauk. 164 (1994) 1129 [Physics-Uspekhi (Engl. Transl.) 37 (1994) 1043]. [3] K. Peters, Nucl. Phys. A 558 (1993) 92. [4] C.B. Dover, Nucl. Phys. A 558 (1993) 721. [5] C. Amsler, Rapporter talk, Proc. Conf. on High Energy Phys. (ICHER), Glasgow, Scotland, July 1994. [6] F.E. Close, preprint RAL-87-072, Chilton, 1987. [7] P. Blum, Int. J. Mod. Phys. 11 (1996) 3003. [8] T. Hirose et al., Nuov. Cim. 50 (1979) 120; C. Fucunage et al., Nuov. Cim. 58 (1980) 199. [9] J. Amizzadeh et al., Phys. Lett. B 89 (1979) 120. [10] A.N. Aleev et al., Z. Phys. C 25 (1984) 205. [11] V.M. Karnaukhov et al., Phys. Lett. B 281 (1992) 148. [12] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fiz. 57 (1994) 241 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 227]; M.Ya. Balatz et al. (SPHINX Collab.), Z. Phys. C 61 (1994) 220. [13] M.Ya. Balatz et al. (SPHINX Collab.), Z. Phys. C 61 (1994) 399. [14] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fiz. 57 (1994) 253 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 238]. [15] L.G. Landsberg et al. (SPHINX Collab.), Nuov. Cim. A 107 (1994) 2441. [16] V.F. Kurshetsov, L.G. Landsberg, Yad. Fiz. 57 (1994) 2030 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 1954]. [17] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fiz. 57 (1994) 1449 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 1376]. [18] D.V. Vavilov et al. (SPHINX Collab.), Yad. Fizz. 58 (1995) 1426 [Phys. At. Nucl. (Engl. Transl.) 58 (1995) 1342.

248 [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59]

L.G. Landsberg / Physics Reports 320 (1999) 223}248 S.V. Golovkin et al. (SPHINX Collab.), Z. Phys. C 68 (1995) 585. S.V. Golovkin et al. (SPHINX Collab.), Yad. Fiz. 59 (1996) 1395 [Phys. At. Nucl. (Engl. Transl.) 59 (1996) 1336]. V.A. Bezzubov et al. (SPHINX Collab.), Yad. Fiz. 59 (1996) 2199 [Phys. At. Nucl. (Engl. Transl.) 59 (1996) 2117]. L.G. Landsberg, Yad. Fiz. 60 (1997) 1541 [Phys. At. Nucl. (Engl. Transl.) 60 (1997) 1397]. L.G. Landsberg, in: S.-U. Chung, H.J. Willutzki (Eds.), Hadron Spectroscopy (`Hadron 97a), Proceedings of the 7th International Conference, Upton, NY, August 1997, p. 725. L.G. Landsberg, Proceedings of 4th Workshop on Small-X and Di!ractive Physics, Fermilab, Batavia, 17}20 September 1998, p. 189. L.G. Landsberg, Plenary Talk on the Conf. `Fundamental Interactions of Elementary Particlesa, ITEP, Moscow, November 1998, Yad. Fiz., in press. V.A. Victorov et al., Yad. Fiz. 59 (1996) 1229 [Phys. At. Nucl. (Engl. Transl.) 59 (1996) 1175]. M.Ya. Balatz et al. (SPHINX Collab.), Yad. Fiz. 59 (1996) 1242 [Phys. At. Nucl. (Engl. Transl.) 59 (1996) 1186]. S.V. Golovkin et al., Z. Phys. A 359 (1997) 435. H.-M. Chan, H. Hogaasen, Phys. Lett. B 72 (1977) 121; H.-M. Chan et al., Phys. Lett. B 76 (1978) 634. H. Hogaasen, P. Sorba, Nucl. Phys. B 145 (1978) 119; Invited Talk at Conf. on Hadron Interactions at High Energy, Marseilles, France, 1978. M. De Crombrughe et al., Nucl. Phys. B 156 (1979) 347. H.-M. Chan, S.T. Tsou, Nucl. Phys. B 118 (1977) 413. C. Caso et al. (PDG), The European Phys. J. 3 (1998) 1. L.G. Landsberg, Usp. Fiz. Nauk. 160 (1990) 1. G. Bellini et al., Nuov. Cim. A 79 (1984) 282. D. Alde et al. (GAMS Collab.), Phys. Lett. B 182 (1986) 105; Phys. Lett. B 276 (1992) 457. D. Alde et al. (GAMS Collab.), Phys. Lett. B 216 (1989) 447; Phys. Lett. B 276 (1992) 375. S.S. Gershtein, in: F. Binon et al. (Eds.), Proceedings of the 3rd International Conference on Hadron Spectroscopy: `Hadron-89a, Paris, 1989, p. 175. R.A. Shumacher, preprint CMU MEG-96-007, Pittsburg, 1996. H. Primako!, Phys. Rev. 81 (1951) 899. I.Ya. Pomeranchuk, I.M. Shmushkevitch, Nucl. Phys. B 23 (1961) 452. A. Halpern et al., Phys. Rev. 152 (1966) 1295; J. Dreitlein, H. Primako!, Phys. Rev. 125 (1962) 591. M. Zielinski et al., Z. Phys. C 31 (1986) 545; C 34 (1986) 255. L.G. Landsberg, Nucl. Phys. (Proc. Suppl.) B 211 (1991) 179c; Yad. Fiz. 52 (1990) 192. S.I. Golovkin et al., European Phys. J. A, to be published. D.V. Vavilov et al., Yad. Fiz. 62 (1999), to be published. G.S. Lomkazi, Talk on the Symposium on Modern Trends in Particles Physics, dedicated to the 70th anniversary of G. Chikovani, Tbilisi, Georgia, September 1998. S. Okubo, Phys. Lett. 5 (1963) 165. G. Zweig, CERN Reports TH-401, TH-402, Geneva, 1964. J. Iizuka, Progr. Theor. Phys. Suppl. 37-38 (1966) 21; J. Iizuka et al., Progr. Theor. Phys. 35 (1966) 1061. F.F. Close, H.J. Lipkin, Phys. Lett. B 196 (1987) 245; Phys. Rev. Lett. 41 (1978) 1263. S.I. Bityukov et al., Phys. Lett. B 188 (1987) 383. A. Etkin et al., Phys. Lett. B 165 (1985) 217; B 201 (1988) 568. J. Ashman et al., Phys. Lett. B 206 (1988) 364; Nucl. Phys. B 328 (1989) 1. P.G. Ratcli!e, Nuov. Cim. A 107 (1994) 2211. J. Ellis et al., Phys. Lett. B 217 (1989) 173; M. Karliner, Proc. LEAP-90, Stockholm, 1980, p. 331; J. Ellis, M. Karliner, Phys. Lett. B 313 (1993) 407. M.A. Falssler, Yad. Fiz. 57 (1994) 1764 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 1693]; Nuov. Cim. A 107 (1994) 2237. V.G. Abelev et al. (OBELIX Collab.), Yad. Fiz. 57 (1994) 1787 [Phys. At. Nucl. (Engl. Transl.) 57 (1994) 1716]; M.G. Sapozhnikov, Nuov. Cim. A 107 (1994) 3317. L.B. Okun, Hadron and Quarks, Lectures on the 2nd ITEP School of Physics, part 2, Atomizdat, Moscow, 1975, p. 5.

Physics Reports 320 (1999) 249}260

Hamiltonian for Reggeon interactions in QCD L.N. Lipatov* Petersburg Nuclear Physics Institute, Gatchina, 188350, St. Petersburg, Russia

Abstract It is shown, that the interaction of the reggeized gluons in the leading logarithmic approximation of the multicolour QCD has a number of remarkable properties including the duality symmetry. The duality relation for the Odderon wave function takes a form of the one-dimensional SchroK dinger equation. It gives a possibility to express the Odderon Hamiltonian as a function of its integrals of motion.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.!t; 14.70.Dj

1. Introduction In QCD the scattering amplitudes at high energies (s are obtained by calculating and summing all large contributions (g ln(s))L, where g is the coupling constant. In the leading logarithmic approximation (LLA) the gluon is reggeized and the BFKL Pomeron is a compound state of two reggeized gluons [1}4]. Next-to-leading corrections to the BFKL equation were recently calculated [5], which gives a possibility to "nd its region of applicability. In the framework of the optimal renormalization approach one can verify that the MoK bius invariance is valid approximately even after taking into account next-to-leading terms [6]. The power asymptotics sH of scattering amplitudes is governed by the j-plane singularities of the t-channel partial waves f (t). The position of these singularities u "j !1 for the Feynman H   diagrams with n reggeized gluons in the t-channel is proportional to the ground state energy E  KK u  "!(gN /8p) E  (1) KK A KK

* Tel.: #7-812-2949196; fax: #7-812-4131963. E-mail address: [email protected] (L.N. Lipatov)  Supported by the CRDF and INTAS-RFBR grants: RP1-253, 95-0311. 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 6 2 - 9

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of a SchroK dinger-like equation [7}12]: (2) E  f  "Hf  . KK KK KK Its eigenfunction f  (q , q ,2, q ; q ) describing the composite state W  (q ) of the reggeized L  KK  KK   gluons depends on their impact parameters q , q ,2, q . In LLA it belongs to the basic series of   L unitary representations of the MoK bius group [9}12] o P(a o #b)/(c o #d) , (3) I I I where o "x #iy , oH"x !iy complex coordinates of gluons and a, b, c, d are arbitrary comI I I I I I plex parameters. For this series the conformal weights m"1/2#il#n/2, m "1/2#il!n/2

(4)

are expressed in terms of the anomalous dimension c"1#2il and the integer conformal spin n of a local operator O  (q ). They are related to the eigenvalues KK  Mf  "m(m!1) f  , MHf  "m (m !1) f  (5) KK KK KK KK of the Casimir operators M and MH:






L  M" M? " 2 M? M?"! o R R , MH"(M)H . (6) I P Q PQ P Q I PQ PQ Here M? are the MoK bius group generators I M"o R , M\"R , M>"!oR (7) I I I I I I I I and R "R/(Ro ). I I In the particular case of the Odderon, being a compound state of three reggeized gluons with the charge parity C"!1 and the signature P "!1, the eigenvalue u  is related to the highH KK energy behaviour of the di!erence of the total cross-sections p and p  for interactions of particles NN NN p and antiparticles p with a target: KK . p !p  &sS (8) NN NN The Hamiltonian H in the multicolour QCD has the property of the holomorphic separability [9}12]:

H"(h#hH), [h, hH]"0 ,  where the holomorphic and anti-holomorphic Hamiltonians L L h" h , hH" hH II> II> I I are expressed in terms of the BFKL operator [9}12]:

(9)

(10)

"log(p )#log(p )#(1/p )log(o )p #(1/p ) log(o )p #2 c . (11) II> I I> I II> I I> II> I> Here o "o !o , p "iR/(Ro ), pH"iR/(RoH) , and c"!t(1) is the Euler constant. II> I I> I I I I Owing to the holomorphic separability of H, the wave function f  (q , q ,2, q ; q ) has the L  KK   property of the holomorphic factorization [9}12]: h

f  (q , q ,2, q ; q )" c f P(o , o ,2, o ; o ) f J (oH, oH,2, oH; oH) , L  KK   L  PJ K   L  K   PJ

(12)

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251

where r and l enumerate degenerate solutions of the SchroK dinger equations in the holomorphic and anti-holomorphic sub-spaces: (13) e f "h f , e  f  "hH f  , E  "(e #e  ) . K KK  K K K K K K K Similarly in the case of two-dimensional conformal "eld theories, the coe$cients c are "xed by the PJ single-valuedness condition for the function f  (q , q ,2, q ; q ) in the two-dimensional q-space. L  KK   For the holomorphic hamiltonian h there is a family +q , of mutually commuting integrals of P motion [9}12]: [q , q ]"0, [q , h]"0 . (14) P Q P The generating function for these integrals of motion coincides with the transfer matrix ¹ for the XXX model [9}12]: L ¹(u)"tr (¸ (u)¸ (u)2¸ (u))" uL\P q ,   L P P where the ¸-operators are






u#o p p I I I . ¸ (u)" I !o p u!o p I I I I The transfer matrix is the trace of the monodromy matrix t(u) [13,14] ¹(u)"tr (t(u)), t(u)"¸ (u)¸ (u)2¸ (u) .   L It can be veri"ed that t(u) satis"es the Yang}Baxter equation [9}14]   tPQ(u) tQ(v) lPP(v!u)"lQQ(v!u) tQ(v) tQ(u) , P PP QQ P P where l(w) is the ¸-operator for the well-known Heisenberg spin model

(15)

(16)

(17)

(18)

(19) lQQ(w)"w dQ dQ#i dQ dQ . QQ Q Q Q Q To "nd a representation of the Yang}Baxter commutation relations, the algebraic Bethe anzatz can be used [13,14]. It is reduced to the solution of the Baxter equation [13}16]. Up to now the Baxter equation was solved only for the case of the BFKL Pomeron (n"2). This is the reason why we use below another approach, based on the diagonalization of the transfer matrix.

2. Duality property of Reggeon interactions The di!erential operators q and the Hamiltonian h are invariant under the cyclic permutation of P gluon indices iPi#1 (i"1, 2,2, n), corresponding to the Bose symmetry of the Reggeon wave function at N PR. It is remarkable that above operators are invariant also under the more A general canonical transformation [17]: o Pp Po , G\G G GG> combined with reversing the order of the operator multiplication.

(20)

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This duality symmetry is realized as an unitary transformation only for the vanishing total momentum: L p" p "0 . (21) P P The wave function t  of the composite state with p"0 can be written in terms of the KK eigenfunction f  of a commuting set of the operators q and qH for k"1, 2,2, n as follows: I I KK do  f  (q , q , , q ; q ) . t  (q , q ,2, q )" (22) KK   L 2 p KK   2 L 



Taking into account the hermicity properties of the total Hamiltonian [9}12] L L L L H>" “ "o "\ H “ "o "" “ "p " H “ "p "\ , (23) II> II> I I I I I I the solution t> of the complex-conjugated SchroK dinger equation for p"0 can be expressed in KK terms of t  as follows: KK L t> (q , q ,2, q )" “ "o "\(t  (q , q ,2, q ))H . (24) KK   L II> KK   L I Because t  is also an eigenfunction of the integrals of motion A"q and AH with their KK L eigenvalues j and jH"j  [9}12]: K K K (25) A t  "j t  , AH t  "j  t  , A"o 2o p 2p , K KK KK K KK  L  L KK one can verify that the duality symmetry takes the form of the following integral equation for t  [17]: KK L\ do L e qII> qI H I\I “ t  (q ,2, q )""j " 2L “ t  (q , , q ) . (26) KK  L K 2p "o " KK  2 L I I II> In the case of the Odderon the conformal invariance "xes the solution of the SchroK dinger equation [9}12]



f  (q , q , q ; q )"(o o o /o o o )K(oH oH oH /oH oH oH)K f  (x) (27) KK                 KK up to an arbitrary function f  (x) of one complex variable x being the anharmonic ratio of four KK coordinates x"o o /o o . (28)     Note that, owing to the Bose symmetry of the Odderon wave function, f  (x) has certain KK modular properties [17]. The Odderon wave function t  (q ) at q"0 can be written as KK GH (29) t  (q )"(o /o o )K\(oH /oH oH )K \s  (z), z"o /o ,       KK   KK GH

L.N. Lipatov / Physics Reports 320 (1999) 249}260

253

where s











dx f  (x) (x!z) K (xH!zH) K  KK . (z)" KK 2p"x!z" x(1!x) xH(1!xH) 

(30)

In fact this function is proportional to f (z): \K\K s  (z)&(x(1!x))K\(xH(1!xH))K \ f (z) , (31) KK \K\K which is a realization of the linear dependence between two representations (m, m ) and (1!m, 1!m ). The corresponding reality property for the MoK bius group representations can be presented in the form of the integral relation s



dx (z)" (x!z)K\ (xH!zH)K \ s (x) KK \K\K 2p

. for an appropriate choice of phases of the functions s  and s \K\K KK The duality equation for s  (z) can be presented in the pseudo-di!erential form [17]: KK z(1!z) (iR)\K zH(1!zH) (iRH)\K u (z)""j  " (u (z))H , \K\K KK \K\K where u (z)" (z(1!z))\K(zH(1!zH))\K s  (z) . \K\K KK The normalization condition for the wave function



dx ""u  """ "u  (x)" KK "x(1!x)" KK

(32)

(33)

(34)

(35)

is compatible with the duality symmetry. For the holomorphic factors uK(x) the duality equations have the form [17]: a uK(x)"jKu\K(x), a u\K(x)"j\KuK(x) , K \K where

(36)

a "x(1!x) p>K . (37) K If we consider p as a coordinate and x!1/2 as a momentum, the duality equation for the most important case m"1/2 can be reduced to the SchroK dinger equation with the potential <(p)"(j p\ [17].

3. Single-valuedness of the Odderon wave function There are three independent solutions uK(x, j) of the third-order ordinary di!erential equation G corresponding to the diagonalization of the operator A [9}12] a

a u"!ix(1!x)(x(1!x)R#(2!m)((1!2x)R!1#m))R u"ju \K K

(38)

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for each eigenvalue j. In the region xP0 they can be chosen as follows [18}22]:  uK(x, j)" dK(j)xI, dK(j)"1 . (39) P I  I  uK(x, j)" aK(j)xI#uK(x, j) ln x, aK"0 , (40) Q I P  I  uK(x, j)" cK (j)xI>K, cK(j)"1 . (41) D I>K K I Due to this di!erential equation the coe$cients a, c and d satisfy certain recurrence relations. From the single-valuedness condition near x"0, we obtain for the total wave function the following representation: u  (x, xH)"uK(x, j) uK (xH, jH)#c uK(x, j) uK (xH, jH) D D  P P KK  #c (uK(x, j) uK(xH, jH)#uK(x, j) uK (xH, jH))#(jP!j) . (42)  Q P P Q The complex coe$cients c , c and the eigenvalues j are "xed from the conditions of the   single-valuedness of f  (q , q , q ; q ) at q "q (i"1, 2) and the Bose symmetry [18}22].  G KK     From the duality equation we have [17] "c """j" .  Another relation

(43)

Im c /c "Im (m\#m \) . (44)   can be derived [17] if we shall take into account, that the complex conjugated representations u  and u of the MoK bius group are related by the above discussed linear transformation. KK \K\K One can verify from the numerical results of Refs. [18}22] that both relations for c and c are   ful"lled.

4. Conformal weight representation If we introduce the time-dependent pair hamiltonian h h

(t)"exp(i ¹(u) t)h exp(!i ¹(u) t) , II> II> in the total hamiltonian h we can substitute

(t) in the form II> (45)

h

Ph (t) (46) II> II> due to the commutativity of h and ¹(u). Due to its rapid oscillations at tPR each pair Hamiltonian is diagonalized in the representation, where the transfer matrix ¹(u) is diagonal: h

(R)"f (q( , q( , , q( ) . II> II>   2 L

(47)

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255

In the case of the Odderon h (R) is a function of the total conformal momentum q and of II>  the integral of motion q "A which can be written as follows:  (48) A" i [M , M ]"i [M , M ]"i [M , M ] .          In a general case of n reggeized gluons, one can use the Clebsch}Gordan approach, based on the construction of common eigenfunctions of the total momentum q with the eigen value m(m!1)  and a set +MK , of the commuting sub-momenta, to "nd all operators M in the corresponding I II> representation. However to diagonalize h we should perform a unitary transformation to the representation, where ¹(u) is diagonal. Let us consider, for example, the interaction between particles 1 and 2. The transfer matrix, which should be diagonalized, can be written as follows: (49) ¹(u)"(u!L)d 2 (u)#(i u L![L, N ])d 2 (u) ,   L  L  where the di!erential operators d 2 (u) and d 2 (u) are independent of q and q . They are related  L  L   to the monodromy matrix t 2 (u) for particles 3, 4,2, n as follows:  L (50) d 2 (u)"tr t 2 (u), d 2 (u)"tr(rt 2 (u)), t 2 (u)"¸ (u)2¸ (u)  L  L  L  L  L  L and the matrix t 2 (u) satis"es the Yang}Baxter equations.  L The operators L and N are constructed in terms of the MoK bius group generators of particles 1 and 2: L"M #M , N"M !M , MX"o R , M>"!oR , M\"R .     I I I I I I I I They have the commutation relations, corresponding to the Lorentz algebra: [¸X, ¸!]"$¸!, [¸>, ¸\]"2¸!, [¸X, N!]"$N! , [¸>, N\]"2NX, [NX, N!]"$¸!, [N>, N\]"2¸X

(51)

(52)

and can be constructed in a closed form in the in"nite dimensional representation "o , M2 for the  composite states with the coordinate o and the pair conformal weight M.  Because of its MoK bius invariance, the transfer matrix ¹(u) after acting on f written as a K superposition of "o , M2 gives again a superposition of the states "o , M2, but with the coe$cients  Y fI which are linear combinations of the initial coe$cients f and f . Therefore, for the eigen K+ K+ K+! function of ¹(u) the coe$cients satisfy some recurrence relations, and the problem of its diagonalization is reduced to the solution of these recurrence relations. In the case of the Odderon (n"3) we can introduce the special functions describing the states of three gluons with the total conformal weight m and the pair conformal weight M for gluons 1 and 2: FK (x)"x+(1!x)K F(m#M, M; 2M; x) , + where F(a, b; 2c; x) is the hypergeometric function, and UK(x)" lim (d/dM)FK (x) . P + +P The matrix element of the pair hamiltonian h is diagonal in the M-representation, but we should  pass to the j-representation. The three independent eigen functions of the integral of motion A can

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be written as follows:  uK(x, j)" DK(j) FK(x) . P I I I  uK(x, j)" (aK(j) FK(x)#DK(j)UK (x)) , Q I I I I I  uK(x, j)" cK(j)FK (x) , D I I>K I where the coe$cients satisfy the reccurrence relations [17]:






d ijaK(j)" aK (j)#bK (j) k(k#1)(k!m#1) I I> I> dk






1 d (k#m!1)(k#m!2) ! aK (j)#bK (j) (k!1)(k!2)(k!m!1) , I\ I\ 4 dk (2k!3)(2k!1) 1 (k#m!1)(k#m!2) ijDK(j)"! (k!1)(k!2)(k!m!1) DK (j) I I\ 4 (2k!3)(2k!1) #k(k#1)(k!m#1)DK (j), DK(j)"1 , I>  1 (k#2m!1)(k#2m!2) ijcK(j)"! (k#m!1)(k#m!2)(k!1) cK (j) I 4 (2k#2m!3)(2k#2m!1) I\ #(k#m)(k#m#1)(k#1)cK (j), cK(j)"1 . I>  In the next section the relation between the Odderon Hamiltonian and its integral of motion A is discussed from another point of view.

5. Odderon Hamiltonian and integrals of motion One can present the holomorphic Hamiltonian for n reggeized gluons in the form explicitly invariant under the MoK bius transformations














L o o o o I> II> R #log I\ II\ R !2t(1) , h" log (53) I I o o o o I> I>I> I\ I\I\ I where o is the coordinate of the composite state.  Let us consider in more detail the Odderon case. Using for its wave function the conformal anzatz f (o , o , o ; o )"(o /o o )Ku (x), x"o o /o o , K        K    

(54)

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257

one can obtain the following Hamiltonian for the function u (x) in the space of the anharmonic K ratio x [9}12]





h"6c#log(xR)#log((1!x)R)#log




#log





x ((1!x)R#m) 1!x








1 (1!x) 1 ((1!x)R#m) #log (xR!m) #log (xR!m) . 1!x x x

(55)

It is convenient to introduce the logarithmic derivative P,xR as a new momentum. In this case one can transform this Hamiltonian to the normal form [17]:  h "!log(x)#t(1!P)#t(!P)#t(m!P)!3t(1)# xIf (P) , I 2 I where






(56)

1 2 1 1 I c (k) f (P)"! # # # R . I P#t k 2 P#k!m P#k R

(57)

(!1)I\RC(m#t)((t!k)(m#t)#mk/2) . c (k)" R kC(m!k#t#1)C(t#1)C(k!t#1)

(58)

Here

Because h and B"iA commute each with another, h/2 is a function of B. In particular for large B this function should have the form: h  c "log(B)#3c# P . (59) 2 BP P The "rst two terms of this asymptotic expansion were calculated in Refs. [9}12]. The series is constructed in inverse powers of B, because h should be invariant under all modular transformations, including the inversion xP1/x under which B changes its sign. The same functional relation should be valid for the eigenvalues e/2 and k"ij of these operators. For large k it is convenient to consider the corresponding eigenvalue equations in the P representation, where x is the shift operator x"exp(!d/dP) ,

(60)

after extracting from eigenfunctions of B and h the common factor u (P)"C(!P)C(1!P)C(m!P) exp(ipP)U (P) . K K The function U (P) can be expanded in series over 1/k K  U (P)" k\LUL (P), U (P)"1 , K K K L

(61)

(62)

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where the coe$cients UL (P) turn out to be the polynomials of order 4n satisfying the recurrence K relation: . UL (P)" (k!1)(k!1!m)((k!m)UL\(k!1)#(k!2)UL\ (k!1!m)) K K \K I 1 K (63) ! (k!1)(k!1!m)((k!m)UL\(k!1)#(k!2)UL\ (k!1!m)) , K \K 2 I valid due to the duality equation written below for a de"nite choice of the phase of U (P) K x\K(1!xP(P!m)(P!m#1))U (P)"kKU (P) (64) K \K with the use of the substitution xkPx. Note that the summation constants UL (0) in the above recurrence relation have the antiK symmetry property UL (0)"!UL (0) , K \K which guarantees the ful"lment of the relation

(65)

UL (m)"UL (0) K \K being a consequence of the duality relation. The energy can be expressed in terms of U (P) as follows: K R e "log(k)#3c# log U (P) K RP 2

(66)

 I #(U (P))\ k\If (P!k)U (P!k) “ (P!r)(P!r#1)(P!r!m#1) (67) K I K I P and it should not depend on P due to the commutativity of h and B. By solving the recurrence relations for UL (P) and putting the result in the above expression, we K obtain the following asymptotic expansion for e/2 [17]:









 

3 1 13 1  1  1 e "log(k)#3c# # m! ! m! 448 120 12 k 2 2 2 1 4185 2151 1  # ! ! m! #2 k 2050048 49280 2





#



965925 1 #2 #2 . 37044224 k

(68)

This expansion can be used with a certain accuracy even for the smallest eigenvalue k"0.20526, corresponding to the ground-state energy e"0.49434 [18}22]. For the "rst excited state with the same conformal weight m"1/2, where e"5.16930 and k"2.34392 [18}22], the energy can be calculated from the above asymptotic series with precision. The analytic approach, developed in this section, should be compared with the method based on the Baxter equation [23].

L.N. Lipatov / Physics Reports 320 (1999) 249}260

259

One can derive from above formulas also the representation for the Odderon Hamiltonian in the two-dimensional space x: 2H"h#hH"12c#ln("x" "R")#ln("1!x" "R") #(x!1)K(xH!1)K (ln("R")#ln("x" "R"))(x!1)\K(xH!1)\K #(!x)K(!xH)K (ln("1!x" "R")#ln("R"))(!x)\K(!xH)\K .

(69)

The logarithms in this expression can be presented as integral operators with the use of the relation













dp h("y"!e) (p) 1 exp(ipy) 2c#ln "!2 !2p ln d(y) . 2p "y" 4 e

(70)

Let us use this representation to "nd the eigen value of the Hamiltonian for the eigenfunction of the integrals of motion B and BH with their vanishing eigenvalues k"kH"0: (71) u  (x)"1#(!x)K(!xH)K #(x!1)K(xH!1)K . KK The corresponding wave function f  (q , q , q ; q ) is invariant under the cyclic permutation of KK     coordinates q Pq Pq Pq but it is symmetric under the permutations q q only for even       value of the conformal spin n"m !m, where the norm ""u  "" is divergent due to the singularities KK at x"0, 1,R. It is the reason, why we consider only the case m !m"2k#1,

k"0,$1,$2,2 .

(72)

Owing to the Bose symmetry of the wave function, this state corresponds to the f-coupling and has the positive charge-parity C. Using the above representation for H, we obtain 2Hu  (x)"EN  u  (x) , KK KK KK where EN  is the corresponding eigen value for the Pomeron Hamiltonian [1}4,9}12] KK EN  "eN #eN K K KK and

(73) (74)

eN "t(1!m)#t(m)!2t(1) . (75) K The minimal value of EN  is obtained at m !m"$1 and corresponds to u"0. Because KK EN  has the property of the holomorphic separability, it is natural to de"ne the holomorphic KK energy for k"0 as e"eN "t(1!m)#t(m)!2t(1) . K In this case its value for m"1/2 will be negative e"!4 ln 2 ,

(76) (77)

but it does not correspond to any physical Odderon state, because its wave function u (x) is not  normalized. For the case of odd n"m !m, the norm of u  (x) is not in"nite: KK dx "u  (x)" KK "Re(t(m)#t(1!m)#t(m )#t(1!m )!4t(1)) . (78) 3p "x(1!x)"



Note, that this norm becomes negative for m"m "1/2.

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In the conclusion, we note that the remarkable properties of the Reggeon dynamics are presumably related with supersymmetry. In the continuum limit nPR the above duality transformation coincides with the supersymmetric translation, which is presumably connected with the observation [24], that in this limit the underlying model is a twisted N"2 supersymmetric topological "eld theory. Additional arguments supporting the supersymmetric nature of the integrability of the reggeon dynamics are given in Ref. [25]. Namely, the eigenvalues of the integral kernels in the evolution equations for quasi-partonic operators in the N"4 supersymmetric Yang}Mills theory are proportional to t( j!1), which means that these evolution equations in the multicolour limit are equivalent to the SchroK dinger equation for the integrable Heisenberg spin model similar to the one found in the Regge limit [15,16]. Note that at large N the N"4 A Yang}Mills theory is guessed to be related with the low-energy asymptotics of a superstring model [26]. Acknowledgements I thank A.P. Bukhvostov for helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 642. V.S. Fadin, E.A. Kuraev, L.N. Lipatov, Phys. Lett. B 60 (1975) 50. Ya.Ya. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904. V.S. Fadin, L.N. Lipatov, Phys. Lett. B 429 (1998) 127. S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov, G.V. Pivovarov, Phys. Rev. Lett. 80 (1998) 2047; 81 (1998) 2394; Phys. Lett., in preparation. J. Bartels, Nucl. Phys. B 175 (1980) 365. J. Kwiecinski, M. Prascalowicz, Phys. Lett. B 94 (1980) 413. L.N. Lipatov, Sov. Phys. JETP 63 (1986) 904. L.N. Lipatov, Phys. Lett. B 251 (1990) 284. L.N. Lipatov, Phys. Lett. B 309 (1993) 394. L.N. Lipatov, hep-th/9311037, Padua preprint DFPD/93/TH/70, unpublished. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982. V.O. Tarasov, L.A. Takhtajan, L.D. Faddeev, Theor. Math. Phys. 57 (1983) 163. L.N. Lipatov, Sov. Phys. JETP Lett. 59 (1994) 571. L.D. Faddeev, G.P. Korchemsky, Phys. Lett. B 342 (1995) 311. L.N. Lipatov, preprint CERN-TH/98-360, Nucl. Phys. B, in preparation. L.N. Lipatov, Recent Advances in Hadronic Physics, Proceedings of the Blois Conference, World Scienti"c, Singapore, 1997. R. Janik, J. Wosiek, Phys. Rev. Lett. 82 (1999) 1092. M.A. Braun, hep-ph/9801352, St. Petersburg University preprints. M.A. Braun, P. Gauron, B. Nicolescu, preprint LPTPE/UP6/10/July 98. M. Praszalowicz, A. Rostworowski, Acta Phys. Polon. B 30 (1999) 349. R. Janik, J. Wosiek, Phys. Rev. Lett. 79 (1997) 2935. J. Ellis and N.E. Mavromatos, Eur. Phys. J. C 8 (1999) 91. L.N. Lipatov, Perspectives in Hadronics Physics, Proceedings of the ICTP Conference, World Scienti"c, Singapore, 1997. J. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231.

Physics Reports 320 (1999) 261}264

Multiparton collisions and multiplicity distribution in high-energy pp(p p) collisions Sergei Matinyan 
* Department of Physics, Duke University, Durham, NC 27708-0305, USA
Yerevan Physics Institute, Yerevan, Armenia

Abstract We discuss the multiplicity distribution for the highest accessible energies of pp- and p p-interactions from the point of view of multiparton collisions. The inelastic cross sections for single, p , and multiple (double  and, presumably, triple), p parton collisions are calculated from the analysis of experimental data on  the multiplicity distribution up to Tevatron energies. It is found that p becomes energy independent while  p increases with (s for (s5200 GeV. The observed growth of 1p 2 with multiplicity is attributed to the  , increasing role of multiparton collisions for the high-energy p p(pp)-inelastic interactions. p reproduces > quite well the cross-section for the mini-jet production.  1999 Elsevier Science B.V. All rights reserved. PACS: 13.85.Hd Keywords: KNO scaling; Partons; Mini-jets; Proton radius

1. Introduction In the last 20 years a tremendous amount of work was done on the study of the longitudinal parton distribution in the deep inelastic scattering processes. These studies provide us with valuable information about the structure function of the proton F(x, Q). Presently, this information needs to be extended to the distribution of the partons in the transverse plane of the collision (p -distribution). This is non-perturbative information because it , gives us the new scale * size of the hadron.

* Department of Physics, Duke University, Durham, NC 27708-0305, USA. Tel.: #1-919-660-2596; fax: #1-919-6602525 E-mail address: [email protected] (S. Matinyan) 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 7 1 - X

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Fig. 1.

The natural way to study the p -dependence of the partons inside a proton is to perform the , multiple parton collision in the hadron}hadron interaction at high energy. We use the multiplicity distribution data obtained from pp colliders for (s5200 GeV, including new data from experiment E735 at (s"1.8 TeV, for an estimation of the inelastic cross sections of the soft single (p ) and double (p ) parton collisions.   Our basis is the observation that the so-called KNO scaling [1] is violated at the high energy (pp)- and (p p)-collisions [2] ((s5200 GeV) while it is well satis"ed in the range of ISR energies. We attribute the deviation from the KNO scaling at higher energies to another new process which is incoherently superimposed on the KNO respecting process. Thus, the single inelastic parton collisions are characterized by this scaling. However, the violation is due to the double (and, maybe, triple) parton collisions. By subtracting the KNO distribution from the experimental data (Fig. 1) on multiplicity distribution we determine the shape for the competing process as shown in Fig. 2. The main characteristics of the derived distributions is that the most probable value of the distributions occurs at twice the multiplicity corresponding to the initial low-energy (ISR) KNO

S. Matinyan / Physics Reports 320 (1999) 261}264

263

Fig. 2.

distributions (single collisions). The width of the distribution is close to (2 times the width of that KNO shape at (s"1.8 GeV, which is in accordance with the Dual Parton Model based on the adding of double inelastic collision of partons of the colliding hadrons to the single parton collisions. These collisions are provided by the exchange of the pairs of the gluonic strings between partons. Integrating the distributions displayed in Fig. 2 over x"N/1N 2 we obtain the inelastic cross  section p for the double parton collisions as a function of (s and, thus, the inelastic cross section  p "p  !p (Fig. 3) for the single parton collisions.  ,1"  From Fig. 3 we see that p equals 17 mb at 1.8 TeV which we can compare with CDFs recent  value for the e!ective double-parton collision cross section [3] (14.5$1.7>  ) mb. \  Single parton collision is nearly independent of (s for (s5200 GeV and has a value of (34$2) mb. p is increased with (s. 

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Fig. 3.

We see from our analysis that the multiple inelastic parton collisions account for a large fraction of the total pp -cross section and, de"nitely, are responsible for the increase of the p p inelastic cross section at (s5200 GeV. We expected that at LHC domain the triple collisions will be seen clearly in the multiplicity distribution. We observe that the cross-sections of the so-called minijet production extracted from several experiments are very similar, by their (s dependence and by absolute value to our p with the  same threshold at (s,200 GeV. This indicates that there exists a smooth transition from hard (jet) to soft physics of the hadron collisions. Using our data for p and p we can, under the simplifying assumptions about the   factorization of the proton's two-body parton distribution, estimate the `hadronica radius of the proton which is equal to 0.96 fm. If we take seriously the saturation of the single parton collision cross section p with (s, we can  conjecture that the same will happen (at much more higher (s) with double parton collision cross section p , so asymptotically, p approaches to the constant value.   Acknowledgements It is my great pleasure to thank W.D. Walker for collaboration and fruitful discussion. I am grateful to B. MuK ller for useful discussions. This work was supported in part by a grant from Department of Energy (DE-FG02-96ER4095). References [1] Z. Koba, H.B. Nielsen, P. Olesen, Nucl. Phys. B 40 (1972) 317. [2] G.J. Alner et al., UAS Collaboration, Phys. Lett. B 138 (1984) 304. [3] F. Abe et al., CDF Collaboration, Phys. Rev. D 56 (1997) 3811.

Physics Reports 320 (1999) 265}274

Perturbative}nonperturbative interference in the static QCD interaction at small distances Yu.A. Simonov Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117 218 Moscow, Russia

Abstract Short distance static quark}antiquark interaction is studied systematically using the background perturbation theory with nonperturbative background described by "eld correlators. A universal linear term 6N a pr/2p is observed at small distance r due to the interference between perturbative and nonperturbative A Q contributions. Possible modi"cations of this term due to additional subleading terms are discussed and implications for systematic corrections to OPE are formulated.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.38.Bx; 14.65.!q

1. Introduction It is more than 20 years ago that the power correction has been computed in OPE [1] laying ground for numerous later applications in QCD. Since then OPE is the basic formalism for study of short-distance phenomena, such as DIS, e>e\ annihilation and, with some modi"cations, heavy quark systems. Interaction of static charges at small distances has drawn a lot of attention recently [2}4]. The theoretical reason is that the appearance of linear terms in the static potential <(r)" const r, where r is the distance between charges, implies violation of OPE, since const&(mass) and this dimension is not available in terms of "eld operators. There are however some analytic [5,6] and numerical arguments [7,8] for the possible existence of such terms O(m/Q) in asymptotic expansion at large Q.

E-mail address: [email protected] (Y.A. Simonov) 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 6 9 - 1

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Of special importance is the sign of the mass squared term. It was argued recently in [3] that the small distance region may produce tachyonic mass correction and this correction was studied selfconsistently in di!erent QCD processes. In particular, the correct (positive) sign of linear potential at small distances comes from tachyonic gluon mass, while positive mass squared term produces negative slope of linear potential. On a more phenomenological side the presence of linear term at small distances, r(¹ , where  ¹ is the gluonic correlation length [9,10], is required by at least two sets of data.  First, the detailed lattice data [11] do not support much weaker quadratic behaviour of <(r)& const r, following from OPE and "eld-correlator method [9,10], and instead prefer the same linear form <(r)"pr at all distances (in addition to perturbative !C a /r term). Second, the small Q distance linear term is necessary for the description of the "ne-structure splittings in heavy quarkonia, since the spin}orbit Thomas term < "!(1/2mr)(d
2. Background perturbative theory In this letter we report the "rst application of the systematic background perturbation theory [14] to the problem in question. One starts with the decomposition of the full gluon vector potential A into nonperturbative (NP) background B and perturbative "eld a : I I I A "B #a I I I

(1)

and the 't Hooft identity for the partition function







1 Z" DA e\1" DB g(B) Da e\1 >? , I I I N

(2)

where g(B) is the weight for nonperturbative "elds, de"ning the vacuum averages, e.g. 1) 1F (x)U (x,y)F (y)2 " (d d !d d )D(x!y)#D , IJ HN IN JH  N IH JN A

(3)

where F ,U are "eld strength and parallel transporter made of B only; D is the full derivative IJ I  term [9] not contributing to string tension p, which is



1 dx D(x)#O(1FFFF2) . p" 2N A

(4)

The background perturbation theory is an expansion of the last integral in (2) in powers of ga I and averaging over B with the weight g(B ), as shown in (3). Referring the reader to Simonov [14] I I

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for explicit formalism and renormalization, we concentrate below on the static interquark interaction at small r. To this end we consider the Wilson loop of size r;¹, where ¹ is large, ¹PR, and de"ne







,exp+!<(r)¹, . (5) 1=2 " P exp ig (B #a ) dz I I I ? ! ? Expanding (5) in powers of ga , one obtains I 1=2"= #= #2, <"< (r)#< (r)#< (r)# , (6)      where < (r) corresponds to (ga )L and can be expressed through D, * and higher correlators L I  [10,14] and its behaviour at small r is [9] < (r)"C r#C r#2, r:¹ , < (r)"pr, r<¹ ,       where coe$cients are integrals of "eld correlators over Euclidean time,



C "O(1FF2)" 



D(l) dl#O(D ), 

(7)

C "O(1F2) . 

 It is this small r behaviour (7) which causes phenomenological problems mentioned above. Coming now to < (r), describing one exchange of perturbative gluon in the background, one  "nds from the quadratic in a term in S(B#a) in the background Feynman gauge the gluon I Green's function G "!(Dd #2igF )\, DA?"* d #gf A@?B@ . (8) IJ H IJ IJ H H A? H The term = can be written through G as  IJ 2 2 dy P exp ig B dz G (x,y) , (9) = "g dx   I I     ! where for G one can use the Feynman}Schwinger representation (FSR) [10]. The simplest form  of FSR obtains when one can neglect or expand in powers of gluon spin interaction (paramagnetic term 2igF in (8)). IJ Doing this expansion, G can be written as IJ G"!D\#D\2igF D\!D\2igF D\2igF D\ , (10)

  






the "rst term on the r.h.s. of (10) corresponds to the spinless gluon exchange in the background B , I which can be written using FSR [10] as









1 Q B dz , K" z  dq . (11) I I 4 !X   Here B is in the adjoint representation of SU(N ). At large N one can use the 't Hooft rule to I A A replace the gluon adjoint trajectory C(z) by the double fundamental trajectory which forms together with the original rectangular contour C in (9) two closed Wilson loops (see [14] for details and discussion). G (x, y)" 

ds(Dz) e\)P exp ig VW

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The average over B in (9) then reduces (again in the large N limit) to the product of two I A averaged Wilson loops, namely







1 Q K" z  dq , (12) 4   where C and C are two contours obtained from the rectangular Wilson loop when two points on   it, x and y, are connected by a double line of gluon trajectory. It is convenient to choose the surfaces S inside C and C as consisting of two adjacent pieces S (C )"S (plane)#S(D), one lying on the G   G G G plane of original Wilson loop and another piece D, perpendicular to the plane and bounded by the trajectory. Using now the nonabelian Stokes theorem [9] and cluster expansion for the average 1=(C )2, one obtains that bilocal correlator of "elds with points on the two pieces vanishes G since 1E E 2&d , and 1E B 2"0(D ) and vanishes by symmetry arguments. Trilocal and G I GI G I  higher correlators for dimensional reasons bring with them higher powers of distance r and can be neglected. Hence one has = " 

ds(Dz) e\) dx dy 1=(C )21=(C )2, VW    

1=(C )21=(C )2"1=D2= ,    where






(13)



1 D(x!y) dp (x) dp (y) . (14) =D"exp ! IJ IJ 2N D A Two di!erent regimes are possible for (14). In the small distance region, r:¹ , the sizes of the  surface D are of the order of r and one can replace D(x!y)PD(0) in (14), and for dimensional reasons the only possible contribution is =D 1#O(D(0)r) .

(15)

Thus one obtains a correction O(r) to the perturbative potential 1/r, and hence no linear term. In the large-distance region, r<¹ , one obtains the area law for =D as  (16) = exp(!pSD) . Insertion of (16) in (12) yields a massive propagator of a spinless hybrid with mass m"m at large r, which corresponds to the "rst excitation of the open string with "xed ends. Summarising one can rewrite = as 

 

= "= dx dy 1G(x, y)2 ,    

(17)

where G(x, y)2 is the Green's function of the spinless hybrid. One can satisfy properties (15), (16) representing 1G2 as the propagator of a particle with variable mass m (p):  1G(p)2"1/(p#m(p)) , (18)  where m(p):O(1/p), pPR, and m(pP0)"mD.   Thus at small distances (large p) 1G(p)2 describes the usual massless gluon exchange, whereas at large distances it describes the propagation of the spinless hybrid.

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Hence at small r:¹ the background "eld B in D\ is not operative and one can replace  I D\ by the free gluon propagator *\. The contribution of the second term to = , dx D\F D\ dy , obtains when x and y are on adjacent sides of the Wilson loop and  I IJ J therefore does not a!ect <(r). In what follows we concentrate on the third term in (10), =,  2 2 dy 1G(x, u)2du1gF (u)F (v)21G(u, v)2dv1G(v, y)2 . (19) ="4g dx   G G    One can rewrite (19) as

 

   2

2

dke IV\W j(k) ,  (k#m(k))(2p)    where we have de"ned, having in mind (4) ="2g 



j(k)"6

dx 

(20)

dy

D(z)e\ IX dz , 4pz

6pN A. j(0)" 2p

(21)

Doing integrals over dx dy , one gets   o o ¹ dke IPa (k)j(k) Q "!*< (r)¹ . (22) ="   p (ko #m(k))  To estimate integral (22) one can take a (k)j(k) out of the integral at some e!ective point k and Q  calculate the rest in a simple way, assuming m to be constant, and expanding the result at small r.  In this way one obtains








a (k )j(k ) * 1  *< (r)" Q  e\KP"a (k )j(k ) ! #r#O(r) . (23)  Q   *m m r m    Analysis of (21) tells that j(k) is a rather weak function of the argument, and to get an idea of the magnitude of *< (r), one can approximate j(k ) j(0)"6pN /2p yielding for *< (r),   A  *< (r)&6N a pr/2p, r:¹ . (24)  A Q  Note, that had we renormalized a in (23), (24) in the standard way we would meet the IR Q divergence of the running a (k), since the corresponding momentum k is in the IR regime for the Q  constant term (!1/m ) in (23). However, the NP background formalism predicts IR modi"cation  of a (see [14,15] for details and discussion), the so-called freezing a behaviour, which from heavy Q Q quarkonia "tting was found in [16] to yield maximal a at small k: Q a (max)"0.5 . (25) Q For the linear term in (23) the situation is di!erent and the e!ective value of k is of the order of  1/r. Hence it is more appropriate to present (23) in the form *< (r)"a (1/r)j(1/r)r .  Q  The author is grateful to V.I. Zakharov for the discussion of this point.

(26)

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One may wonder whether behaviour (24) holds also at large r, thus increasing linear potential < (r). However form (24) was obtained at small rj¹ , while at large r next terms of expansion in (10)   are contributing and the whole series (10) should be summed up explicitly. One can perform the summation, replacing "rst for simplicity DP* in (8), (small r or relatively large q are considered) and assuming that only bilocal correlators of F are nonzero. One obtains IJ 1 , G"!(*#2igF)\PG(k)" k!k(k)

24p k(0)" , 2p

where k(k)'0. Thus G(k) acquires a pole at real value k(k) in Euclidean space}time, signalling appearance of a tachyon. From physical point of view this result is a consequence of paramagnetic attractive interaction of gluon spin with NP background, yielding negative correction to the gluon selfenergy. A similar term occurs for a quark due to its spin interaction with background [17]. Recently, a negative (tachyonic) mass shift was observed due to the gluon interaction with the stochastic background in [18]. It is meaningful, that the same paramagnetic term F in G yields negative contribution IJ IJ to the charge renormalization (asymptotic freedom) [19]. In fact negative paramagnetic e!ective action S   [19] gives rise to the negative (tachyonic) mass since both are connected,  !kd &dS/da da . Therefore, one may expect that the phenomenon of tachyonic gluon mass is IJ I J pertinent to nonabelian theories. The existence of the tachyon reveals paramagnetic instability of the object (gluon or quark), if stabilizing mass is not created by some additional mechanism. In our case this mechanism is the creation of hybrid mass due to the same con"ning correlator D(x) when D is used and not * in (8). As was explained above, the hybrid mass is created at larger distances, r<¹ , so that for  illustrative purposes (referring the hybrid mass to the gluon in question), one may write the total gluon propagator G(k) as G(k)"1/(k#m(k)) , where m(k)"m(k)!k(k), and m(k) is dominating at small k (large distances) while k(k)   dominates at large k (small distances). A simple example is provided by m(k)" k(k!k)/(k#k), in which case one can calculate gluon exchange potential <(r)" !(dko /(2p))e Io Po G(k) explicitly to yield at small r: 1 k <(r)&! # r, r 2

k *<& r 2

in agreement with form (24). At large distances one should calculate the exact gluon Green's function (8) in the Wilson loop =, which describes to the propagation of the hybrid state with two static quarks QQM at the ends of the string. Such a state was considered both analytically [20] and on the lattice [21] yielding excitation energy (which corresponds to the mass m(0)) around 1 GeV. Hence (24) is only a small distance approximation of the hybrid exchange potential, where the dominant paramagnetic contribution is kept in the e!ective gluon mass.

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3. Other possible corrections to V(r) In doing perturbative expansion in (2) one encounters other terms in S(B#a) which potentially yield interference contributions of perturbative a and NP B "elds. Of special importance is the I I term ¸ : 



¸ " a D (B)F dx .  J I IJ

(27)

Correction due to (27) in the gluon propagator was studied in Appendix 1 of [14] and can be written in form (20) where k(k)PJ is now expressed as 



J (k)" dz e\ IX1DF (z)DF (0)2 . 

(28)

Now the integrand in (28) can be written as a sum of two terms [22]: 1D F (z)D F (u)2"* * 1F (z)F (u)2#0(1FFF2) . (29) M MJ H HI M H The "rst term on the r.h.s. of (29) yields dJ &k and hence only a NP correction to the Coulomb  term, while the second term, 1FFF2, would give, using dimensional arguments, correction *<&1FFF2r, negligible at small r. Another type of correction occurs from the interplay of multiple colour Coulomb exchanges and one NP contribution, considered in [23]. This e!ect can be accounted for by the replacement D(x, t)PD(x, t)exp(!N a t/2x) . (30) A Q Here t is the Euclidean time and the exponent in (30) accounts for the di!erence of potential in singlet and octet channels. Insertion of (30) in (7) yields an additional suppression of the r dependence. This result coincides with the correction obtained in [24] in a di!erent way. Finally, we consider in this section the correction due to the freezing behaviour of the coupling a Q [14,15]: amr 4p

a(r)! Q . (31) a(r)" Q Q ln(1/Kr) b ln(1#rm)/Kr  As was noticed by Yndurain [2], expansion of the freezing Coulomb potential yields a linear term C amr *< (r)"#  Q #2 . A ln(1/Kr)

(32)

Here m is the double hybrid mass, (m "1.1 GeV from the "ts to experiment [16]), and term (32) is always much smaller than the Coulomb term because of condition mr;1. One may consider *< in (32) as coming from the additional 1/p in a (p), as was suggested in [8], but here there A Q is the log term in the denominator, reminding that the pole is coming from the expansion of the freezing a (p). Q

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One should note that there is no double counting in adding (24) and (32), since (24) is obtained from the one-gluon exchange (OGE) process, while (32) is due to the one-loop corrections to OGE. However, the region of validity of (32) is always smaller than (24).

4. Discussion and conclusion The analysis done heretofore concerns static interquark potential and reveals that even at small distances NP background ensures some contributions which come from relatively small intermediate distances, l:¹ , and encoded in the mass squared term  k(0)"24p/2p"0.8 GeV (33) which generates potential (24). Applying the same NP background formalism to other processes of interest, one would get similar corrections of the order of k/p, as, e.g. in OPE for e>e\ annihilation. One example of this kind is the calculation of the "eld correlator 1F (x)U(x, y)F (y)2. The IJ HN leading contribution can be written using the gluon propagator G (8), 1FUF2"* * G #perm#G G #perm , (34) I H JN IH JN where we have suppressed U and perm denotes terms obtained by the permutation of indices with the proper change of sign. Insertion of expansion (10) into (34) yields in addition to the standard perturbative term O((x!y)\) a contribution proportional to (33), C k C 1F(x)UF(0)2& #  #2 , x x

(35)

where both C and C are positive computable numbers.   A recent lattice study of a similar quantity [7] as a function of ;< cut-o! K reveals the possible presence of the O(K&x\) term. It is clear that appearance of k, which is an integral of nonlocal entity 1F(x)F(0)2 over a NP scale, x&¹ , violates the original OPE of Wilson [25], proved in the pure perturbation theory,  and the extended OPE of Shifman et al. [26], where NP contributions enter as matrix elements of local operators. This extended form of OPE can be considered as a physically motivated assumption, and an explicit treatment done here within the background perturbation theory (BPT) reveals that some extra terms should be added to OPE, the "rst of which, k/p, was discussed in [3]. One might ask at this point, how rigorous and selfconsistent is BPT, with nonperturbative background given by correlators. One should stress here, that BPT is a consistent and systematic method, but not a complete one, since no recipe was suggested above for calculation of NP correlators, and the NP con"gurations are introduced by hand just as it is done in QCD sum rules [26]. However, recently the situation has changed. In [27] equations have been derived in the limit of large N for vacuum correlators, from which correlators can be computed one by one explicitly. A In this way the exponential form of the lowest correlator, 1F(x)UF(0)2 was de"ned analytically

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[27] in agreement with lattice studies, and analytic connection between ¹ and p was found,  yielding ¹ in a good agreement with lattice data [28].  The main conclusion from these studies is that NP con"gurations appear as a selfconsistent solution of nonlinear equations which violates spontaneously scale symmetry pertinent to these equations and their perturbative solutions. From this point of view the NP background exploited here can be identi"ed with scale violating NP solutions in [27], and the BPT method is made complete. On the phenomenological side the account of correction (24) and (32) in the total potential <(r)"< (r)!C aD(r)/r#*< (r)   Q  may provide for <(r) a simple `linear plus Coulomba picture for all distances which is in better agreement both with experiment [12,17] and with lattice data [11,13]. Recently, a new lattice analysis of static potential at small distances was performed [29], revealing large slope of nearly 1 GeV. This enhancement is possible in (21) due to the 1/z term in D(z), written in (35) and requires a selfconsistent solution of equations which is now in progress. To conclude: in this study perturbative}nonperturbative interference was shown to provide additional OPE terms, absent in the usual local OPE form. In addition, there are purely NP contributions [14] which are also outside of the standard lore, and will be discussed elsewhere.

Acknowledgements The author is grateful to V.A. Novikov and V.I. Shevchenko for fruitful discussions, and to V.I. Zakharov for discussions, correspondence and very useful remarks. The "nancial support of RFFI through the grants 97-02-16406 and 97-0217491 is gratefully acknowledged.

References [1] V.A. Novikov, L.B. Okun, M.A. Shifman, M.B. Voloshin, M.I. Vysotsky, V.I. Zakharov, Phys. Rep. C 41 (1978) 1. [2] F.J. Yndurain, hep-ph/9708448, Nucl. Phys. (proc. Suppl.) 64 (1998) 433; R. Akhoury, V.I. Zakharov, Phys. Lett. B 438 (1998) 165. [3] K.G. Chetyrkin, S. Narison, V.I. Zakharov, Nucl. Phys. B 550 (1999) 353. [4] F.V. Gubarev, M.I. Polikarpov, V.I. Zakharov, hep-th/9812030, ITEP-TH-73/98; V.I. Zakharov, Nucl. Phys. Proc. Suppl. 74 (1999) 392. [5] G. Grunberg, hep-ph/9705290, hep-ph/9705460, hep-ph/9711481. [6] V.I. Zakharov, Nucl. Phys. B 385 (1992) 385; A.I. Vainshtein, V.I. Zakharov, Phys. Rev. Lett. 73 (1994) 1207; Phys. Rev. D 54 (1996) 4039. [7] G. Burgio, F. Di Renzo, G. Marchesini, E. Onofri, Phys. Lett. B 422 (1998) 219. [8] G. Burgio, F. Renzo, C. Parrinello, C. Pittori, hep-ph/9808258, Nucl. Phys. Proc. Suppl. 74 (1999) 388. [9] H.G. Dosch, Yu.A. Simonov, Phys. Lett. B 205 (1988) 339; Yu.A. Simonov, Phys. Usp. 39 (1996) 313, hep-ph/9709344. [10] Yu.A. Simonov, Nucl. Phys. B 324 (1989) 67; Yu.A. Simonov, Nucl. Phys. B 307 (1988) 512 and refs. therein. [11] S.P. Booth et al., Phys. Lett. B 294 (1992) 385; G. Bali, K. Schilling, A. Wachter, hep-lat/9506017. [12] A.M. Badalian, V.P. Yurov, Yad. Fiz. 56 (1993) 239; A.M. Badalian, Yu.A. Simonov, Yad. Fiz. 59 (1996) 2247.

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[13] K.D. Born et al., Phys. Lett. B 329 (1994) 332; G.S. Bali, K. Schilling, A. Wachter, Phys. Rev. D 56 (1997) 2566. [14] Yu.A. Simonov, Yad. Fiz. 58 (1995) 113; JETP Lett. 57 (1993) 525; Yu.A. Simonov, Lecture Notes in Physics, vol. 479, Springer, Berlin, 1996. [15] A.M. Badalian, Phys. At. Nucl. 60 (1997) 1003, hep-lat/9704004; A.M. Badalian, Yu.A. Simonov, Phys. At. Nucl. 60 (1997) 630. [16] A.M. Badalian, V.L. Morgunov, hep-ph/9901430. [17] H.G. Dosch, Yu.A. Simonov, preprint HD-THEP-92-23; Yad. Fiz. 57 (1994) 152. [18] S.J. Huber, A. Laser, M. Reuter, M.G. Schmidt, Nucl. Phys. B, to be published. [19] A.M. Polyakov, Gauge Fields and Strings, Hardwood, Chur, 1987 (Chapter 2). [20] M. Luescher, Nucl. Phys. B 180 (1981) 317; Yu.A. Simonov, in: T. Bressani, A. Feliciello, G. Preparata, P.G. Ratcli!e (Eds.), Proceedings of Hadron'93 Conference, Como, 21}25 June 1993; Yu.S. Kalashnikova, Yu.B. Yufryakov, Phys. Lett. B 359 (1995) 175; Yad. Fiz. 60 (1997) 374. [21] S. Perantonis, C. Michael, Nucl. Phys. B 347 (1990) 854; T. Manke et al., CP-PACS Collaboration, Phys. Rev. Lett. 82 (1999) 4396. [22] V.I. Shevchenko, Yu.A. Simonov, Phys. At. Nucl. 60 (1997) 1201. [23] Yu.A. Simonov, S. Titard, F.J. Yndurain, Phys. Lett. B 354 (1995) 435. [24] I.I. Balitsky, Nucl. Phys. B 254 (1985) 166. [25] K. Wilson, Phys. Rev. 179 (1969) 1499. [26] M. Shifman, A. Vainshtein, V. Zakharov, Nucl. Phys. B 147 (1978) 385. [27] Yu.A. Simonov, Phys. At. Nucl. 61 (1998) 855; hep-ph/9712250. [28] A. Di Giacomo, H. Panagopoulos, Phys. Lett. B 285 (1992) 133; A. Di Giacomo, E. Meggiolaro, H. Panagopoulos, Nucl. Phys. B 483 (1997) 371; M. D'Elia, A. Di Giacomo, E. Meggiolaro, Phys. Lett. B 408 (1997) 315; A. Di Giacomo, M. D'Elia, H. Panagopoulos, E. Meggiolaro, hep-lat/9808056. [29] G.S. Bali, Phys. Lett. B 460 (1999) 170.

Physics Reports 320 (1999) 275}285

Relations between inclusive decay rates of heavy baryons M.B. Voloshin 
Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA
Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia

Abstract The dependence of inclusive weak decay rates of heavy hadrons on the #avors of spectator light quarks is revisited with application to decays of charmed and b hyperons. It is pointed out that the di!erences in the semileptonic decay rates, the di!erences in the Cabibbo-suppressed decay rates of the charmed hyperons, and the splitting of the total decay rates of the b hyperons are all related to the di!erences in the lifetimes of the charmed hyperons independently of poor knowledge of hadronic matrix elements. The approximations used in these relations are the applicability of the expansion in the inverse of the charmed quark mass and the #avor SU(3) symmetry.  1999 Elsevier Science B.V. All rights reserved. PACS: 13.30.Eg

1. Introduction The problem of unequal inclusive weak decay rates of hadrons containing a heavy quark (c or b) attracts considerable experimental and theoretical interest ever since the "rst experimental indication [1] of substantially di!erent lifetimes of the charmed D and D! mesons. The total weak decay rates of charmed hadrons are presently known to be vastly di!erent, with the ratio of the longest known lifetime to the shortest: q(D!)/q(X )&20, while the di!erences among the b hadrons are A much smaller. The relation between the magnitude of the lifetime di!erences in the charmed hadrons versus that in the b hadrons re#ects the fact that these e!ects, associated with the light quark/gluon &environment' of the heavy quark Q in a hadron, vanish as an inverse power of the heavy mass m , so that in the limit m PR the &parton' picture sets in, where the inclusive decay / / rate of a heavy hadron is given by that of an isolated heavy quark.

E-mail address: [email protected] (M.B. Voloshin) 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 5 4 - X

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Thus the problem of the di!erences in the rates can be approached theoretically in the limit of large m in terms of a systematic expansion [2}6] in m\ for the decay rates of the hadrons. The / / leading term in this expansion is the &parton' decay rate C Jm , which sets the overall scale for   / the decay rates of hadrons containing the quark Q and which does not depend on the speci"cs of the spectator light quarks (in baryons) or the antiquark (in mesons). The "rst non-perturbative term, suppressed with respect to the leading one by m\, arises due to the kinetic energy of the / heavy quark in a hadron (time dilation e!ect) and due to the chromomagnetic interaction of the heavy quark [6]. This term generally gives a splitting of decay rates between heavy mesons and baryons, and between baryons of di!erent spin structure, however it generates no splitting depending on the #avors of the spectator quarks or antiquarks (e.g. between di!erent D mesons, or within the triplet of baryons K , N and N>). The #avor dependence arises in the next order, A A A m\ relative to C , and can be interpreted as due to two mechanisms: the weak scattering (WS) /   and the Pauli interference (PI). The weak scattering corresponds to a cross-channel of the decay, generically QPq q q , where either the quark q is a spectator in a baryon and can undergo     a weak scattering o! the heavy quark: q QPq q , or an antiquark in meson, say q , weak-scatters     (annihilates) in the process q QPq q . The Pauli interference e!ect arises when one of the "nal    (anti)quarks in the decay of Q is identical to the spectator (anti)quark in the hadron, so that an interference of identical particles should be taken into account. The latter interference can be either constructive or destructive, depending on the relative spin-color arrangement of the (anti)quark produced in the decay and of the spectator one, thus the sign of the PI e!ect is found only as a result of speci"c dynamical calculation. Although the WS and PI e!ects carry the relative suppression by m\, they are found to be / greatly enhanced by a large numerical factor, typically 16p/3, re#ecting mainly the di!erence of the numerical factors in the two-body versus the three-body "nal phase space, which makes these e!ects overwhelmingly essential in the charmed hadrons, while greatly reduced in the heavier b hadrons, as is con"rmed by the experimental data. In e!ect, the contribution of the O(m\) terms / is signi"cantly smaller than that of the O(m\) terms in the charmed hadrons and is slightly smaller / in the b hadrons [6]. In particular the O(m\) terms split the decay rate of the K from that of the / @ B mesons by only about 2%, which is by far insu$cient to explain the current data [7] on the ratio of the lifetimes: q(K )/q(B)"0.79$0.05 (for a recent theoretical discussion see e.g. Ref. [8]). @ A quantitative estimate of the e!ects of the O(m\) terms runs into a problem of evaluating matrix / elements over the hadrons of four-quark operators of the type (QM C Q)(q C q) with certain spin-color   matrix structures C and C . Although simple estimates within a non-relativistic picture of the light   quarks in hadrons (where these operators reduce to the density of the light quarks at the location of the heavy quark) allowed to correctly predict [4,5] the hierarchy of the lifetimes of the charmed hadrons, these estimates are obviously very unreliable for a quantitative description of the e!ect. Neither can this approach explain the ratio q(K )/q(B) to be less than approximately 0.9. In view of @ this di$culty it is quite worthwhile to have a better understanding of the spectator #avordependent di!erences of the rates in a possibly more model-independent way. The purpose of this paper is to point out relations between some inclusive decay rates of the charmed and b baryons in the (K , N ) triplets, which do not require explicit knowledge of / / the matrix elements of the four-quark operators, and rely only on the general expansion in m\ for / the rates and on the #avor SU(3) symmetry. Certainly, the latter symmetry is known to be not very precise, however arguably the uncertainties due to the SU(3) violation are substantially less than

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those brought in by the current model assumptions about the hadronic matrix elements. Thus it may be expected that an experimental veri"cation of these relations can serve as a test of the whole method based on the operator product expansion for weak decay rates. To an extent, such approach was pursued in the prediction [9] of a signi"cant enhancement of the total semileptonic decay rates of the N baryons over the same rate for the K and even greater enhancement of this A A rate for X . That analysis was further extended [10,11] to include the enhancement of the A Cabibbo-suppressed semileptonic decays of K . Although those papers used model considerations A for the matrix elements of the four-quark operators, in fact one can obtain, as shown in the present paper, quantitative results for the (K , N ) triplet without resorting to models of quark dynamics in A A the baryons. Namely, it will be shown that the di!erence of the semileptonic rates within the (K , N ) A A baryon triplet, both the dominant and the Cabibbo-suppressed, as well as the di!erence of the non-leptonic Cabibbo-suppressed decay rates, can all be expressed in terms of the total lifetime di!erences within the same triplet in a model-independent fashion, modulo the assumption of the #avor SU(3) symmetry. In addition the di!erences of the lifetimes within the triplet of the b baryons (K , N ) are also expressed through the same di!erences for the charmed baryons, with @ @ a possible extra uncertainty due to the quark mass ratio m/m. @ A Using the currently available data on lifetimes of the charmed hyperons, the discussed e!ects are estimated to be quite large. In particular, the conclusion of the previous analyses is con"rmed that the semileptonic decay rates of the N baryons should exceed by a factor 2 to 3 the same rate for the A K hyperon. It is also found that the lifetime of the N\ baryon can be longer than that of K by A @ @ about 14%, which is a very large e!ect for b hadrons.

2. E4ective Lagrangian for spectator}6avor-dependent e4ects in decay rates The systematic description of the leading as well as subleading e!ects in the inclusive decay rates of heavy hadrons arises [2,4,5] through the application of the operator product expansion in powers of m\ to the &e!ective Lagrangian' ¸ related to the correlator of two weak-interaction /  terms ¸ : 5





¸ "2 Im i dx e OV¹+¸ (x), ¸ (0), . 5 5 

(1)

In terms of ¸ the inclusive decay rate of a heavy hadron H is given by the mean value  / C "1H "¸ "H 2 . (2) & /  / The leading term in ¸ describes the &parton' decay rate. For instance, the term in the non-leptonic  weak Lagrangian (2G <(q c Q )(q c q ) with < being the appropriate combination of the $ * I * * I * CKM mixing factors, generates through Eq. (1) the leading term in the e!ective Lagrangian ¸ ""<"(Gm /64p)g (QM Q) ,  $ /   The non-relativistic normalization for the heavy quark states is used throughout this paper: 1Q"QRQ"Q2"1.

(3)

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where g is the perturbative QCD radiative correction factor. In the limit m PR this expression  / reproduces the &parton' inclusive non-leptonic decay rate associated with the underlying process QPq q q , due to the relation 1H "QM Q"H 2+1H "QRQ"H 2"1 with the approximate equality    / / / / being valid up to corrections of order m\. In order to reproduce the complete expression of order / m\ one should also include the "rst non-trivial term of the OPE for ¸ containing the dimension /  5 chromomagnetic operator (QM p G Q) with the gluonic "eld tensor G . However, neither this IJ IJ IJ operator nor the leading term ¸ involve light quark operators, thus at this level no splitting arises  between the decay rates within #avor SU(3) multiplets. The dependence on the #avor of spectator quarks arises in the next term of OPE for ¸ : ¸.   For the dominant Cabibbo unsuppressed non-leptonic decays of the charmed quark, generated by the underlying process cPsudM , this term reads as [5] ¸ "c(Gm/4p)+C (c C c)(dM C d)#C (c C d)(dM C c)  $ A  I I  I I # C (c C c#c c c c)(s C s)#C (c C c #c c c c )(s C s ) I  G I I  G I  I I IG  I  I  # C (c C c#c c c c)(u C u)#C (c C c #c c c c )(u C u )  I  I  I  G I I  G I  I I I G # i(i\!1)[2(C !C )(c C t?c)j? > \ I I  ! (5C #C )(c C t?c#c c c t?c)j? ], (4) > \ I  I  I with the coe$cients C , A"1,2, 6 given by  C "C #C #(1!i)(C !C ) , > \  > \  C "i(C !C ) ,  > \ C "![(C !C )#(1!i)(5C #C #6C C )] ,   > \  > \ > \ C "!i(5C #C #6C C ) , > \ > \   C "![(C #C )#(1!i)(5C #C !6C C )] ,   > \  > \ > \ C "!i(5C #C !6C C ) . (5)   > \ > \ In Eqs. (4) and (5) the following notation is used: C "c (1!c ), C and C are the standard I I  > \ coe$cients in the QCD renormalization of the non-leptonic weak interaction from m down to the 5 charmed quark mass: C "C\"(a (m )/a (m ))@, where b, the coe$cient in the one-loop beta \ > Q A Q 5 function in QCD, can be taken as b" for the case of the charmed quark decay (see e.g. Ref. [12]).  Furthermore, the powers of the parameter i"(a (k)/a (m )) describe the so-called &hybrid' [5,13] Q Q A QCD renormalization of the operators from the normalization scale m down to a low scale k, A and j? "u c t?u#dM c t?d#s c t?s is the color current of the light quarks with t?"j?/2 being the I I I I generators of the color SU(3). Finally, the elements < and < of the CKM weak mixing matrix AQ SB are approximated here by the cosine of the Cabibbo angle, c,cos h , hence the overall coe$cient A in Eq. (4) is written as c. The terms in Eq. (4) with the #avor singlet operator j? produce no splitting of the decay rates I within the #avor SU(3) multiplets and thus are not of immediate relevance to the present

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discussion. The rest of the terms containing operators with the u, d and s quarks with di!erent coe$cients are responsible for those splittings. In terms of the physical interpretation of the e!ect the operators with the d quark describe the WS in charmed hyperons, while the terms with the u and s quarks correspond to the PI e!ects. In order to discuss further in the paper the spectator e!ects on the Cabibbo-suppressed non-leptonic decays as well as on the semileptonic ones, we also need the expressions for the corresponding parts of the e!ective Lagrangian ¸. For the non-leptonic decays we limit the  present discussion to those suppressed by only one extra factor of s,sin h , i.e. those generated by A the quark processes cPsus and cPdudM . The Cabibbo-suppressed part of the non-leptonic e!ective Lagrangian then can be found in the form: Gm ¸ "cs $ A +C (c C c)(q C q)#C (c C c )(q C q )  I I  G I I I I G  4p # C (c C c#c c c c)(q C q)#C (c C c #c c c c )(q C q )  I  I  I  G I I  G I  I I I G # 2C (c C c#c c c c)(u C u)#2C (c C c #c c c c )(u C u ) I  G I I  G I  I I I G  I  I  # i(i\!1)[2(C !C )(c C t?c)j? > \ I I  (6) ! (5C #C )(c C t?c#c c c t?c)j? ], I > \ I  I  with the notation (q Cq)"(dM Cd)#(s Cs). The corresponding term in the e!ective Lagrangian for semileptonic decays, generated by the quark-lepton process cPsl>l and the Cabibbo-suppressed one: cPdl>l is given by [9}11] Gm ¸ " $ A +c[¸ (c C c#c c c c)(s C s)#¸ (c C c #c c c c )(s C s )]   I  I  I  G I I  G I  I I IG 12p # s[¸ (c C c#c c c c)(dM C d)#¸ (c C c #c c c c )(dM C d )] I  G I I  G I  I I I G  I  I  (7) ! 2i(i\!1)(c C t?c#c c c t?c)j? , , I I  I  where the coe$cients ¸ and ¸ are   ¸ "(i!1), ¸ "!3i . (8)   In the next section the general expressions in Eqs. (4), (7) and (8) are used for the analysis of the relations between the splittings of various inclusive decay rates within the triplet of charmed baryons.

3. Di4erences of inclusive decay rates for charmed baryons Estimates of the absolute e!ect of the #avor-dependent operators in the e!ective Lagrangian ¸ require evaluation of the matrix elements of the four-quark operators over charmed hadrons.   In mesons the same term describes the Pauli interference of the dM quark in the decays of D>, which is considered to be the major reason for the observed suppression of the D> total decay rate.

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One approach (for a review see e.g. Ref. [8]) is to use a low normalization point k of the order of the con"nement scale and invoke a constituent quark model, simplifying it further to a non-relativistic model, with possible additional (also non-relativistic) input about the wave functions of the light quarks at the origin (see e.g. Refs. [14,15] and a rather general consideration in Refs. [16,17]). Needless to mention that such approach can be used only for qualitative, or very approximate semi-quantitative estimates, since it inevitably involves poorly controllable approximations. In order to be able to "nd arguably more reliable relations we do not attempt here an absolute evaluation of those matrix elements, but rather use the #avor SU(3) properties of the operators in ¸ to relate the measurable splittings of the semileptonic and the Cabibbo-suppressed inclusive  decay rates in the charmed baryon triplet to the splitting of the total decay rates. Namely, assuming the #avor SU(3) symmetry, and the applicability of the heavy quark limit to the charmed quark, it will be shown that the discussed splittings are determined by only two independent matrix elements, which can be expressed in terms of the di!erences in the measured total decay rates: D "C(N)!C(K ) and D "C(K )!C(N>).  A A  A A Proceeding with derivation of the relations we "rst notice that for the triplet of the baryons (K , N ) in the heavy quark limit the spin of the charmed quark is not correlated with spinorial A A characteristics of its light &environment'. Thus the operators with the axial current of the c quark give no contribution to the matrix elements. The remaining #avor non-singlet structures in ¸ involve only operators of the types (c c c)(q C q) and (c c c )(q C q ) with q being d, s or u. Due to  I I G I I I I G the SU(3) symmetry the #avor non-singlet part of the matrix elements of each of these types of operators in the baryon triplet is expressed in terms of only one parameter. Indeed, the di!erence of the matrix elements between the components of a <-spin doublet: N and K , is contributed only by A A the *<"1 combination of the operators, proportional to (u Cu)!(s Cs), while the di!erence between the components of a ;-spin doublet: K and N>, receives the same contribution only from A A the *;"1 operator (s Cs)!(dM Cd). Thus if one introduces two parameters x and y as x"1(c c c)[(u C u)!(s C s)]2NA KA"1(c c c)[(s C s)!(dM C d)]2KA N>A , \ \ I I  I I I  I

(9)

y"1(c c c )[(u C u )!(sN C s )]2NA KA"1(c c c )[(s C s )!(dM C d )]2KA N>A \ \ I IG  G I I I IG I I G  G I I I I G with the shorthand notation for the di!erences of the matrix elements: 1O2 " \ 1A"O"A2!1B"O"B2, the splitting of the inclusive decay rates within the baryon triplet are expressed in terms of x and y as follows. The di!erences of the dominant Cabibbo-unsuppressed non-leptonic decay rates are given by Gm d ,C (N)!C (K )"c $ A [(C !C )x#(C !C )y] ,      1\ A 1\ A 4p (10) Gm d ,C (K )!C (N>)"c $ A [(C !C )x#(C !C )y] .      1\ A 1\ A 4p ¹he once Cabibbo-suppressed decay rates of K and N> are equal, due to the *;"0 property of A A the corresponding e!ective Lagrangian ¸ (Eq. (7)). Thus the only di!erence for these decays in 

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Gm d ,C (N)!C (K )"cs $ A [(2C !C !C )x#(2C !C !C )y] .       1 A 1 A 4p

(11)

the baryon triplet is

The dominant semileptonic decay rates are equal among the two N baryons due to the isotopic A spin property *I"0 of the corresponding interaction Lagrangian, thus there is only one nontrivial splitting for these decays: Gm d ,C (N )!C (K )"!c $ A [¸ x#¸ y] . 1\ A 1\ A  12p 

(12)

Finally, the Cabibbo-suppressed semileptonic decay rates are equal for K and N, due to the A A *<"0 property of the corresponding interaction. Thus the only di!erence for these is Gm d ,C (K )!C (N>)"!s $ A [¸ x#¸ y] .  1 A 1 A 12p 

(13)

Using the relations (10)}(13) one can express the splittings of the total decay rates in terms of the two parameters x and y as D "d #d #2d  ,   (14) D "d !2d #2d  ,   where the factor 2 for the semileptonic splittings takes into account the decays with both el and kl leptonic pairs, and the small contribution of double Cabibbo-suppressed non-leptonic decays is neglected. Thus the unknown parameters x and y can be found in terms of the measured di!erences D and D and used to predict the splittings of inclusive decay rates for each experimentally   identi"able type of decays, described by Eqs. (10)}(13). It is quite satisfying to see that although the parameters x and y as well as the coe$cients C and  ¸ and ¸ depend on the arbitrarily chosen low normalization point k, in the resulting relations   between the physically measurable splittings the dependence on k cancels out, as it should be expected. In fact in the expression for d  in terms of the dominant non-leptonic splittings the dependence on the QCD radiative e!ects cancels out altogether s d " (2d #d ) ,  c 

(15)

while the relation between the splitting of the dominant semileptonic decay rates and the dominant non-leptonic splittings, emerging after excluding x and y in Eqs. (12) and (10), reads as





1 1 5C #5C !2C C \ > > \d ! d  . (16) d " 3(C #2C )  c 12C C (C #2C )  > \ \ > \ > The coe$cients in this relation depend only on the physical ratio of the couplings r"(a (m )/a (m )). Moreover, this dependence is in fact rather weak: in the absence of the QCD Q A Q 5 radiative e!ects, i.e. with r"1, the coe$cients in the square brackets in Eq. (16) are +0.22 and  !+!0.11, while with a realistic value r+2.5 they are, respectively, 0.23 and !0.09. Similar 

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relative insensitivity of the numerical results to the exact value of r also holds for other relations between the observable splittings. In subsequent numerical estimates the numerical values r"2.5 and s"0.05 are used. We also use the data from Ref. [7] on the lifetimes of the charmed baryons in the form: C(K )"4.85$0.28 ps\, C(N>)"2.85$0.5 ps\ and C(N)"10.2$2 ps\. (The A A A error bars on the lifetimes for the N hyperons are in fact not symmetric. However the symmetry A improves for the inverse quantities, i.e. for the total widths, and a close approximation to the errors in the widths is used here in a symmetric fashion.) Solving Eqs. (14) for the Cabibbo-dominant splitting of the semileptonic widths yields d "0.13D !0.065D +0.59$0.32 ps\ (17)   and, obviously, for the Cabibbo-suppressed semileptonic decays one has d "(s/c)d  +0.030$0.016 ps\. The solution of Eqs. (14) for the splitting of the Cabibbo-suppressed nonleptonic rates similarly gives d "0.082D #0.054D +0.55$0.22 ps\ .  

(18)

4. Splitting of lifetimes in the triplet of b baryons Once the unknown baryonic matrix elements x and y are phenomenologically determined through the di!erences of the total decay rates in the charmed baryon triplet, one can use these parameters for evaluating the di!erences of decay rates in the triplet of the b baryons: K , N, and @ @ N\. Indeed, in the limit where both the c and the b quarks are heavy the matrix elements of the @ four-quark operators over the b hyperons should be the same as for the charmed ones, provided that the operators are normalized at a low point k which does not depend on the masses m or m . A @ For proceeding in this manner we write here the expression [5] for the corresponding e!ective lagrangian for non-leptonic b decays, neglecting small kinematical e!ects O(m/m) in the relevant A @ expressions for the two-body phase space of the pair cc or cq, Gm ¸@ "c"< " $ @ +CI (bM C b)(u C u)#CI (bM C u)(u C b)  I I  I I  @A 4p # CI (bM C b#bM c c b)(q C q)#CI (bM C b #bM c c b )(q C q ) I  G I I  G I  I I I G  I  I  # i(i\!1)[2(CI  !CI  )(bM C t?b)j?  > \ I I (19) ! (5CI  #CI  !6CI CI )(bM C t?b#bM c c t?b)j? ], , I > \ > \ I  I  where again the notation (q Cq)"(dM Cd)#(s Cs) is used, and the renormalization coe$cients are determined by a (m ) instead of a (m ): CI "CI \"(a (m )/a (m ))@, i"(a (k)/a (m )). The coe$Q @ Q A \ > Q @ Q 5 Q Q @ cients CI are related to CI , CI , and i in the same way as in Eqs. (5).  \ >  The expression with these small terms included can be found in Ref. [18]. Also only the CKM-dominant processes bPcu d and bPcc s are taken into account in order to keep the formulas simple. The contribution of sub-dominant processes to the total rates is below the expected uncertainty.

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The dominant semileptonic decay bPcll does not involve light quarks, thus one expects no substantial splitting of the semileptonic decay rates within #avor SU(3) multiplets of b hadrons. The non-leptonic e!ective lagrangian in Eq. (19) is symmetric with respect to s  d, i.e. it has *;"0 (this property is broken if the small kinematical e!ects of the c quark mass are kept in ¸@). Thus at this level there is no splitting between the non-leptonic decay rates of K and N. The  @ @ splitting of the decay rate between either of these and N\ is given in terms of x and y by @ Gm (20) D ,C(K )!C(N\)"c"< " $ @ [(CI !CI )x#(CI !CI )y] .     @ @ @ @A 4p One can notice that in the absence of any QCD radiative e!ects the latter di!erence is simply related to d  and d  for the charmed baryons   D ""< "/c m/m(d #d ) . (21) @ @A @ A   The QCD correction coe$cients make the full expression somewhat more lengthy 1 "< " m @ D " @A +[C ((3#2m)CI  #4mCI CI #6(1!m)CI  ) @ > \ \ > > c m 4C C (C #2C ) A > \ \ > # 2C C (CI  #2CI  )!C ((1!m)CI  !2mCI CI #(2#3m)CI  )]d  > \ \ > \ \ \ > >  # 4C C (CI  #2CI  )d , , (22) > \ \ >  where m"(i/i)"(a (m )/a (m )). One can again note that the relation (22) between the Q A Q @ physically measurable quantities does not depend on the low normalization point k. Numerically, however the full expression is not far from the simple approximation in Eq. (21): with a realistic value (a (m )/a (m ))+1.25 one "nds from Eq. (22): Q A Q @ D +"< "/c m/m(0.91d #0.93d ) . @ @A @ A   When expressed in terms of the di!erences in the total decay rates D and D for the charmed   baryons, using Eq. (14), the splitting of the decay rates within the b baryon triplet reads numerically as D +"< "m/m(0.85D #0.91D )+0.015D #0.016D +0.11$0.03 ps\ , (23) @ @A @ A     which represents the estimate from the present analysis of the expected suppression of the total decay rate of N\ with respect to that of K , or N. @ @ @ 5. Discussion The relative di!erences of the lifetimes for charmed particles are large, even within one #avor SU(3) triplet of the hyperons. Therefore the assumption that these di!erences in the triplet are described by just one term of the expansion in m\ certainly requires additional study. It should / be noted however, that this assumption is not necessarily #awed, since the discussed O(m\) terms /

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are singled out by large numerical coe$cient, and there is no reason for recurrence of such anomaly further in the expansion. Thus the relations between the splittings of the decay rates within the triplet charmed hyperons (K , N ) as well as of the b hyperons (K , N ) may present a good testing A A @ @ point for experimental study of this issue. As a consequence of large di!erence in the lifetimes the additional e!ects, discussed here, are also estimated to be quite large. The predicted di!erence in the semileptonic decay rates between the N and K (Eq. (17)) can be compared with the current data A A on the semileptonic width of K : C (K )"0.22$0.08 ps\. This comparison con"rms the conA  A clusion of a previous analysis [9], that the semileptonic decay rates of the N hyperons can be larger A than that of K by a factor of 2 or 3. Similarly, the small Cabibbo-suppressed semileptonic decay A rate of K , should be enhanced by the same factor [11] and may in fact constitute 10% to 15% of all A semileptonic decays of K . The e!ect in the Cabibbo-suppressed non-leptonic decays evaluated in A Eq. (18) can amount to more than 10% of the di!erence in the total non-leptonic decay rates of N> and K and should be quite prominent, provided that it would be possible to separate and A A measure the inclusive Cabibbo-suppressed rates experimentally. Finally, the prediction of Eq. (23) for the di!erence of the total decay rates of K and N\ can be @ @ interesting in relation to the mentioned earlier problem of the ratio q(K )/q(B). Indeed, the central @ number in Eq. (23) amounts to about 14% of the total decay rate C(K )"0.81$0.05 ps\, and @ a di!erence of such relative magnitude is undoubtedly to be considered as very large for the b hadrons. If con"rmed, this would indicate that the spectator e!ects in heavy hyperons can be substantially larger, than usually expected, and may shed some light on the problem of the K versus B lifetime. @ Acknowledgements It is a great pleasure to use this occasion to express my gratitude and appreciation to Lev Borisovich Okun, who taught me the SU(3), quarks, leptons, and many other things in physics and beyond. This work is supported in part by DOE under the grant number DE-FG02-94ER40823.

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[13] M.A. Shifman, M.B. Voloshin, Sov. J. Nucl. Phys. 45 (1987) 292. [14] B. Guberina, R. RuK ckl, J. TrampeticH , Z. Phys. C 33 (1986) 297. [15] B. Blok, M. Shifman, Lifetimes of Charmed Hadrons Revised } Facts and Fancy, in: J. Kirkby, R. Kirkby (Eds.), Proceedings of the Workshop on the Tau-Charm Factory, Marbella, Spain, 1993, Editions Frontiers, Gif-surYvette, 1994, p. 247. [16] N.G. Uraltsev, Phys. Lett. B 376 (1996) 303. [17] D. Pirjol, N. Uraltsev, Phys. Rev. D 59 (1999) 034012. [18] M. Neubert, C.T. Sachrajda, Nucl. Phys. B 483 (1997) 339.

Physics Reports 320 (1999) 287}293

Gluon condensate from superconvergent QCD sum rule F.J. YnduraH in Departamento de Fn& sica Teo& rica, C-XI, Universidad Auto& noma de Madrid, Canto Blanco, 28049-Madrid, Spain

Abstract Sum rules for the nonperturbative piece of correlators (speci"cally, the vector current correlator) are discussed. The sum rule subtracting the perturbative part is of the superconvergent type. Thus it is dominated by the bound states and the low-energy production cross section. It leads to a determination of the gluon condensate 1a G2. We "nd 1a G2K0.048$0.030 GeV.  1999 Elsevier Science B.V. All rights Q Q reserved. PACS: 12.38.!t; 12.38.Aw Keywords: Gluon condensate

1. Sum rule The potential, or more generally the spectrum of a system of heavy quarks cannot be directly discussed in terms of the operator product expansion (OPE). However, one can use dispersion relations to deduce a number of sum rules relating bound-state properties to quantities obtainable via the OPE (`ITEP-typea sum rules). One can then use the estimates of nonperturbative contributions to bound states energies and wave functions to actually go beyond the traditional analysis. Although the sum rules, being global relations, cannot discriminate details one can check consistency and even obtain reasonable estimates on nonperturbative quantities, speci"cally on the gluon condensate. This is the last aim of the present note, where we will use a method generalizing that proposed by Novikov [1]. To do so we consider the correlator for the vector current of heavy quarks:



P "(pg !p p )P(p)"i dx e NV1TJ (x)J (0)2 , I J IJ IJ I J E-mail address: [email protected] (F.J. YnduraH in) 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 7 9 - 4

(1)

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288

where J "tM c t and sum over omitted colour indices is understood. This will give information on I I triplet, l"0 states; information on states with other quantum numbers would be obtained with other correlators. The function P(t) satis"es a dispersion relation



o(s) 1 , P(t)" ds s!t p

(2)

where o(s),Im P(s). Actually, this equation should have been written with one subtraction. We will not bother to do so as its contribution drops out for the quantities of interest for us here. Let us denote by P , o to the corresponding quantities calculated in perturbation theory,     albeit to all orders, but nonperturbative e!ects are neglected in P , o . (In actual calculations we     cannot of course include all orders. We will sum the one-gluon exchange to all orders, which can be done explicitly in the nonrelativistic regime, and add one-loop radiative corrections to this.) In particular, for example, the gluon condensate contribution is not included in the `p.t.a pieces. At large t, both spacelike and timelike, the OPE is applicable to P(t), and we have the well-known results [2]: P(t)KP

(t)#1a G2/12pt   Q

(3)

and o(s)Ko

N C 1a G2 (1#v)(1!v) (s)! A $ Q   s v 128

(4)

with v"(1!4m/s) the velocity of the quarks. Moreover,

 



N !t 3C !t P (t) K ! A log # $ log log #2 ,   12p l l b  R N 3C a A 1# $ Q#2 , N "3, C " . Im P (s) K   A $  12p 4p R If we then de"ne P , o as the results of subtracting the perturbative parts, ,. ,. P ,P!P , ,.  



o ,o!o , ,.  

it follows from the OPE, Eq. (3), that P decreases at in"nity like t\ and hence satis"es ,. a superconvergent dispersion relation. We thus have a "rst sum rule



ds o (s)"0 . ,.

(5)

In fact it would appear that one still has another sum rule because of the following argument. At large t, P (t) behaves like (cf. Eq. (3)) ,. P (t)K1a G2/12pt , ,. Q

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while the contribution from the bound states to the dispersion relation (see below) P (t)&1a G2/ta ,._
   Q Q dominates over this. Therefore, we have the extra relation



ds so (s)"0 . ,.

(6)

(7)

It turns out that (7) is actually equivalent to (5), up to radiative corrections. This is because the region where any of the integrals in (5), (7) are appreciably di!erent from zero is for sK4m(1#O(a)), so (7) di!ers from (5) by terms of order a, smaller than the radiative Q Q corrections which neither (5) nor (7) take into account. Let us return to sum rule (5). The function o(s) consists of a continuum part, for s above threshold for open bottom production, and a sum of bound states. Both can be calculated theoretically provided that s is larger than a certain critical s(v ), and n smaller or equal than a critical n . s(v )    and n are de"ned as the points where the perturbation theoretic contribution to o and the  nonperturbative one are of equal magnitude, and form the limits of the regions where a full theoretical evaluation is possible. To be precise, for the continuum we use (4) so that above the critical s(v )  N C 1a G2 (1#v)(1!v) , s's(v ) (8a) o(s)" A $ Q  ,. s v 128 and v is such that o(s(v ))Ko(s(v )); numerically, and for bM b, v K0.2. For the bound states  ,.      o is proportional to the square of the wave function at the origin: N o(s)" A "R (0)"d(s!M ) . L M L L We may get o
 (s) and o
 (s) by splitting the residue "R (0)" into a Coulombic piece;   ,. L mCa $ Q (1!d
 a ) , "R! (0)"" L Q L 2n where the one-loop corrections d
 a may be found in Refs. [3}6], and the (leading) nonperturL Q bative correction are given by the Leutwyler}Voloshin analysis (Refs. [3}9]). So we have "R (0)"K"R! (0)"#"R! (0)"d,. L L L L the numbers d,. have been calculated by Leutwyler and Voloshin. For n"1, L 38.31a G2 Q . d,."  mCa $ Q This is all we really need since, for bottomium, n "1. Thus we have   3N Cpm1a G2 L g Q L d(s!M), n4n o
 (s)" A $ L  ,. M 8am Q L L the g known in terms of the d,.. L L

(8b)

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Sum rule (5) can then be written schematically as







L o # Residue of o " ,. ,. QT L



QT

Residue of o o # . ,. ,.   LL>

The left-hand side is given in terms of 1a G2 by Eq. (8); the right-hand side can be connected with Q experiment with the following argument. The sum over higher bound states, ` Residue of o a ,. LL> may be identi"ed as the di!erence between the sum over the experimental residues of the poles of the bound states, and what we would get by a Coulombic formula, for all n5n #1. Certainly, this  Coulombic formula will not be valid for large n because here the radiative corrections will become large; but, because the residues decrease like 1/n the contribution of these states will be negligible. We write this decomposition as (s) . (bound states with n'n )"o
 (s)!o
 ! 
LL  LL As for the continuum piece below s(v ) we may likewise interpret it as the di!erence between  experiment and a perturbative evaluation, which we write as o(s)"o(s)!o(s), s(s(v ) ,.     and, because we are close to threshold, we have NC a 1 o(s)" A $ Q (1#d a )    Q 8 1!e\p!$?QT NC a K A $ Q (1#d a )  Q 8 and the value of the one-loop radiative correction d a may be found in Ref. [10].  Q Taking everything into account sum rule (5) becomes, 1 "R(0)"#f (v )
  mM L  L LL >  1 p1a G2 L 2 1a G2 Q "2C a ! j n # 8ea# Q . (9) $ Q L Q 48eam n ma 3 Q L Q LL> We have de"ned v ,ea and the expression is valid up to corrections of relative order a . The  Q Q function f (v ) is the contribution of the background which, when added to the resonances above
  threshold (included in the sum in the l.h.s. of (9)), give the experimental value of QT o . The   ,. function f would be obtained by integrating the cross sections for production of B#G and BBM ,
 where by G we mean a `glueballa decaying into 2p, and B is any of the states B, B!, BH. Because



 



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we may assume that the structure is provided by the resonances, we can take f given by phase
 space only. So we have f (v )"f v#f v ,
      where the "rst term refers to the channel B#G , and the second to BBM . We have in this expression neglected m . % 2. Numerology In principle the procedure would appear straightforward. One would "t the resonance and bound state residues and f to the data, and then, after substituting into (9), obtain a determination of 1a G2. In practice, however, things do not work out so nicely. The quality of the experimental Q data does not allow any precise determination of the constants f ; any values in the range  f &0.03,0.1 would do the job. Secondly, the e!ective dependence of 1a G2 in Eq. (9) on  Q experiment is proportional to a\: so the result will depend very strongly on the value of a we Q Q choose. This is particularly true because radiative corrections to the nonperturbative contribution to the bound states have not been calculated, so there is not even a `naturala renormalization point. These two di$culties may be partially overcome with the following tricks. First of all, since we are assuming that the n"1 bound state is described with the bound state analysis as discussed in Refs. [3}6], we may "x the value of a that produces such agreement. This means that we will take Q 0.354a 40.4. Secondly, we may alter the treatment of the continuum in the following manner. Q We split not from v , but from v , arbitrary provided only that v 5v . Thus, for s4s(v ), we use      o(s)"o(s)!o(s), and for s5s(v ) we take the theoretical expression ,.     N C 1a G2 (1#v)(1!v) . o(s)" A $ Q ,. s v 128 The sum rule is thus written as 1 "R(0)"#f (v )
  mM L L L 4.91a G2 2 1a G2 Q "2C [f(3)!1]a! # 8ea# Q , e a "v . $ Q  Q  Q  ma 3 48eam Q  Q Then we may pro"t from the fact that the sum rule should be valid for all values of v 5v to "x   f requiring this independence, at least in the mean. That is to say, that when we increase v past   a particle threshold from B(2) to B(6) the variation of the corresponding determinations of 1a G2 Q around their average be minimum. The calculation may be further simpli"ed replacing



 



f (v ) v .
    The results of the analysis are summarized in the following tables, where the column `Resa indicates at which resonance the cut in v occurs. We have taken two rather extreme values of f .  

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Res.

v 

1a G2 Q

B(2) B(3) B(4) B(5) B(6)

0.21 0.34 0.40 0.43 0.46

0.014 0.034 0.048 0.039 0.046

Res.

v 

1a G2 Q

B(2) B(3) B(4) B(5) B(6)

0.21 0.34 0.40 0.43 0.46

0.037 0.057 0.067 0.048 0.052

For a "0.35, f "0.04 Q 

For a "0.40, f "0.09 Q  This derivation shows very clearly the kind of errors one encounters. To the variations that may be called `statisticala, apparent in the di!erent values found in the tables above 0.01441a G240.067 Q we have to add `systematica ones, e.g., the in#uence of the not calculated radiative corrections, easily of some 30%: not to mention our including the Coulombic wave functions at the origin for large values of n, or the lack of de"nition of the expression `perturbation theory to all ordersa because of renormalon ambiguities. Given all these uncertainties, which do even make it dubious that one can really de"ne with precision the condensate in terms of experimental observables, it is not surprising that one cannot pin down the gluon condensate with more accuracy than an estimate, taking into account above "gures, of 1a G2K0.048$0.03 Gev . Q To get this average we have taken into account all determinations in the tables above, excluding the lowest (B(2)) and highest, B(6). This is slightly larger than old averages, and slightly lower than more recent ones [11,12] which tended to give, respectively, 1a G2K0.042, 1a G2K0.065 Gev. Q Q Acknowledgements Discussions with R. Akhoury and V. Zakharov on some aspects of the sum rule are gratefully acknowledged.

F.J. Yndura& in / Physics Reports 320 (1999) 287}293

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

See, e.g., V.A. Novikov et al., Phys. Rep. C 41 (1978) 1. M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 147 (1978) 385. S. Titard, F.J. YnduraH in, Phys. Rev. D 49 (1994) 6007. S. Titard, F.J. YnduraH in, Phys. Rev. D 51 (1995) 6348. A. Pineda, F.J. YnduraH in, Phys. Rev. D 58 (1998) 094022. A. Pineda, F.J. YnduraH in, CERN-TH/98-402 (hep-ph/9812371). H. Leutwyler, Phys. Lett. B 98 (1981) 447. M.B. Voloshin, Nucl. Phys. B 154 (1979) 155. M.B. Voloshin, Sov. J. Nucl. Phys. 36 (1982) 143. K. Adel, F.J. YnduraH in, Phys. Rev. D 52 (1995) 6577. Cf. the reviews of S. Narison, QCD Spectral Sum Rules, World Scienti"c, Singapore, 1989. S. Narison, Nucl. Phys. Suppl. 54A (1997) 238.

293

Physics Reports 320 (1999) 295}318

Neutrino masses and mixings: a theoretical perspective Guido Altarelli 
*, Ferruccio Feruglio Theoretical Physics Division, CERN, CH-1211 Geneva 23, Switzerland
Universita% di Roma Tre, Rome, Italy Universita% di Padova and I.N.F.N., Sezione di Padova, Padua, Italy

Abstract We brie#y review the recent activity on neutrino masses and mixings which was prompted by the con"rmation of neutrino oscillations by the Superkamiokande experiment.  1999 Elsevier Science B.V. All rights reserved. PACS: 11.30.Hv; 12.10.!g; 12.15.Ft; 14.60.Pq Keywords: Solar and atmospheric neutrinos; Beyond the Standard Model; Neutrino physics; Grand Uni"ed Theories

1. Introduction It is for us a great pleasure to contribute to the celebration of Lev Okun anniversary with this article. Considering the continuous interest of Lev on neutrinos we thought that this subject is particularly appropriate to the occasion. Recent data from Superkamiokande [1] have provided a more solid experimental basis for neutrino oscillations as an explanation of the atmospheric neutrino anomaly. In addition the solar neutrino de"cit, observed by several experiments [2], is also probably an indication of a di!erent sort of neutrino oscillations. Results from the laboratory experiment by the LSND collaboration [3] can also be considered as a possible indication of yet another type of neutrino oscillation. Neutrino oscillations imply neutrino masses. The extreme smallness of neutrino masses in comparison with quark and charged lepton masses indicates a di!erent nature of neutrino masses, * Corresponding author. Theoretical Physics Division, CERN, CH-1211 Geneva 23, Italy. E-mail addresses: [email protected] (G. Altarelli), [email protected] (F. Feruglio)  Alternative explanations such as neutrino decay and violations of the equivalence principle appear to be disfavoured by the present data [4]. 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 6 7 - 8

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linked to lepton number violation and the Majorana nature of neutrinos. Thus neutrino masses provide a window on the very large energy scale where lepton number is violated and on grand uni"ed theories (GUTs) [5]. The new experimental evidence on neutrino masses could also give an important feedback on the problem of quark and charged lepton masses, as all these masses are possibly related in GUTs. In particular the observation of a nearly maximal mixing angle for l Pl is particularly interesting. Perhaps also solar neutrinos may occur with I O large mixing angle. At present solar neutrino mixings can be either large or very small, depending on which particular solution will eventually be established by the data. Large mixings are very interesting because a "rst guess was in favour of small mixings in the neutrino sector in analogy to what is observed for quarks. If con"rmed, single or double maximal mixings can provide an important hint on the mechanisms that generate neutrino masses. The purpose of this article is to provide a concise review of the implication of neutrino masses and mixings on our picture of particle physics. We will not review in detail the status of the data but rather concentrate on their conceptual impact. The experimental status of neutrino oscillations is still very preliminary. While the evidence for the existence of neutrino oscillations from solar and atmospheric neutrino data is rather convincing by now, the values of the mass squared di!erences *m and mixing angles are not "rmly established. For the observed l suppression of solar neutrinos, for example, three possible C solutions are still possible [6]. Two are based on the MSW mechanism [7], one with small (MSW-SA: sin 2h &5.5;10\) and one with large mixing angle (MSW-LA: sin 2h 90.2).   The third solution is in terms of vacuum oscillations (VO) with large mixing angle (VO: sin 2h &0.75). However, it is important to keep in mind that the *m values of the above  solutions are determined by the experimental result that the suppression is energy dependent. This is obtained by comparing experiments with di!erent thresholds. The Cl experiment shows a suppression larger than by a factor of 2, which is what is shown by Ga and water experiments [2]. If the Cl indication is disregarded, then new energy-independent solutions would emerge, with large *m and maximal mixing. For example, good "t of all data, leaving those on Cl aside, can be obtained with *m as large as *m:10\ eV [8,9]. For atmospheric neutrinos the preferred value of *m, in the range 10\}10\ eV, is still a!ected by experimental uncertainties and could sizeably drift in one sense or the other, but the fact that the mixing angle is large appears established (sin 2h 90.9 at 90% C.L.) [10}12]. Another issue which is still open is the claim by the LSND  collaboration of an additional signal of neutrino oscillations in a terrestrial experiment [3]. This claim was not so-far supported by a second recent experiment, Karmen [13], but the issue is far from being closed. Given the present experimental uncertainties the theorist has to make some assumptions on how the data will "nally look like in the future. Here we tentatively assume that the LSND evidence will disappear. If so then we only have two oscillations frequencies, which can be given in terms of the three known species of light neutrinos without additional sterile kinds (i.e. without weak interactions, so that they are not excluded by LEP). We then take for granted that the frequency of atmospheric neutrino oscillations will remain well separated from the solar neutrino frequency, even for the MSW solutions. The present best values are [6,10}12,14] (*m) &3.5;10\ eV  and (*m) &5;10\ eV or (*m) &10\ eV. We also assume that the electron neu+15U1 4trino does not participate in the atmospheric oscillations, which (in the absence of sterile neutrinos) are interpreted as nearly maximal l Pl oscillations as indicated by the Superkamiokande [1] I O

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and Chooz [15] data. However, the data do not exclude a nonvanishing ; element. In C the Superkamiokande allowed region the bound by Chooz [15] amounts to "; ":0.2 C [10,11,14].

2. Direct limits on neutrino masses Neutrino oscillations are due to a misalignment between the #avour basis, l,(l ,l ,l ), where C I O l is the partner of the mass and #avour eigenstate e\ in a left-handed weak isospin SU(2) doublet C (similarly for l and l ) and the neutrino mass eigenstates l,(l , l , l ) I O    "l2";"l2 , (1) where ; is the 3;3 mixing matrix [54]. Thus, in the presence of mixing, neutrinos cannot be all massless and actually the presence of two di!erent oscillation frequencies implies at least two di!erent nonzero masses. Neutrino oscillations are practically only sensitive to di!erences *m so that the absolute scale of squared masses is not "xed by the observed frequencies. But the existing direct bounds on neutrino masses, together with the observed frequencies, imply that all neutrino masses are by far smaller than any quark or lepton masses. In fact the following direct bounds hold: m C:&5 eV, m I:170 KeV and m O:18 MeV [16]. Since the observed *m indicate mass splitJ J J tings much smaller than that, the limit on m C is actually a limit on all neutrino masses. Moreover J from cosmology we know [17] that the sum of masses of (practically) stable neutrinos cannot exceed a few eV, say m G:6 eV, corresponding to a fraction of the critical density for neutrino hot J dark matter X h:0.06 (the present value of the reduced Hubble constant h being around 0.7). In J conclusion, the heaviest light neutrinos that are allowed are three nearly degenerate neutrinos of mass around or somewhat below 2 eV. In this case neutrinos would be of cosmological relevance as hot dark matter and contribute a relevant fraction of the critical density. But at present there is no compelling experimental evidence for the necessity of hot dark matter [17,18]. As a consequence, neutrino masses can possibly be much smaller than that. In fact, for widely split neutrino masses the heaviest neutrino would have a mass around &0.06 eV as implied by the atmospheric neutrino frequency. An additional important constraint on neutrino masses, which will be relevant in the following, is obtained from the nonobservation of neutrino-less double beta decay. This is an upper limit on the l Majorana mass, or equivalently, on mCC" ; m , which is at present quoted to be G CG G C J mCC:0.2 eV [19]. J 3. Neutrino masses and lepton number violation Neutrino oscillations imply neutrino masses which in turn demand either the existence of right-handed neutrinos (Dirac masses) or lepton number (¸) violation (Majorana masses) or both. Given that neutrino masses are extremely small, it is really di$cult from the theory point of view to avoid the conclusion that ¸ must be violated. In fact, it is only in terms of lepton number violation that the smallness of neutrino masses can be explained as inversely proportional to the very large scale where L is violated, of order M or even M . %32 . 

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Recall that an ordinary Dirac mass term is of the form m "l m l plus its hermitian conjugate 0* 0 " * m , where l " (1$c )l and l is the neutrino "eld. The "eld l annihilates a left-handed  * *0 0*  neutrino and creates a right-handed antineutrino. Correspondingly, the "eld l creates a left* handed neutrino and annihilates a right-handed antineutrino. Now left-handed neutrinos l and * right-handed antineutrinos (l) are indeed the only observed neutrino states. Thus, in principle, one 0 could assume that right-handed neutrinos and left-handed antineutrinos do not exist at all. The "eld l and its conjugate l would then su$ce and there would be no l and l "elds in the theory. 0 0 * * In the SM Lagrangian density only the term i l D/ l appears, where D is the SU(2);(1) gauge I * * covariant derivative. Clearly if no l is allowed there is no possible Dirac mass for a neutrino. 0 But if ¸ is violated, there is no conserved quantum number that really makes neutrinos and antineutrinos di!erent and a new type of mass term is possible. For a massive neutrino, the positive helicity state that Lorentz invariance demands to be associated with the state l of given * momentum can be (l) (for a charged particle with mass, say an e\, one can go to the rest frame, 0 * rotate the spin by 1803 and boost again to the original momentum to obtain e\). If neutrinos and 0 antineutrinos are not really distinct particles, both Lorentz and TCP invariance are satis"ed by just allowing l and (l) (TCP changes one into the other at "xed momentum). For a massive charged * 0 particle of spin  one needs four states, while only two are enough for an intrinsically neutral  particle. If ¸ is violated we can have a Majorana mass term m "(l) ml "l2Cml where * * ** 0 * (l) "Cl2 and C is the 4;4 matrix in Dirac space that implements charge conjugation (the "eld 0 * (l) annihilates a (l) exactly as the "eld l2 does, the transposition only indicating that we want it 0 0 * as a column vector). Clearly, the Majorana mass term m violates ¸ by two units. Also, since in the ** SM l is a weak isospin doublet, m transforms as a component of an isospin triplet. In the * ** following, as we are only interested in #avour indices and not in Dirac indices, we will simply denote m by m "l2ml , omitting the Dirac matrix C. Note that if ¸ is violated and l also ** ** * * 0 exists, then a second type of Majorana mass is also possible which is m "l2ml , where we again 00 0 0 omitted C. Clearly also m violates ¸ by two units, but, since l is a gauge singlet, m is invariant 00 0 00 under the SM gauge group. In conclusion, if l does not exist, we can only have a Majorana mass 0 m if ¸ is violated. If l exists and ¸ is violated, we can have both Dirac m and Majorana masses ** 0 *0 m and m . ** 00 Imagine that one wanted to give masses to neutrinos and, at the same time, avoid the conclusion that lepton number is violated. Then he/she must assume that l exists and that neutrinos acquire 0 Dirac masses through the usual Higgs mechanism as quark and leptons do. Technically this is possible. But there are two arguments against this possibility. The "rst argument is that neutrino masses are extremely small so that the corresponding Yukawa couplings would be enormously smaller than those of any other fermion. Note that within each generation the spread of masses is by no more than a factor 10\. But the spread between the t quark and the heaviest neutrino would exceed a factor of 10. A second argument arises from the fact that once we introduce l in 0 the theory, then the ¸ violating term m "l2ml is allowed in the lagrangian density by the gauge 00 0 0 symmetry. In the minimal SM, i.e. without l , we understand ¸ and B conservation as accidental 0 global symmetries that hold because there is no operator term of dimension 44 that violates B and ¸ but respects the gauge symmetry. For example, the transition u#uPe>#dM is allowed by colour, weak isospin and hypercharge gauge symmetries, but corresponds to a four-fermion operator of dimension 6: O "(j/M)(e2d)(u2u). This term is suppressed by the dimensional factor  1/M. In the assumption that the SM extended by supersymmetry is an e!ective low-energy theory

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which is valid up to the GUT scale, as suggested by the compatibility of observed low-energy gauge couplings with the notion of uni"cation at M &10 GeV, the large mass M is identi"ed with %32 M . In fact the factor 1/M can be obtained from the propagator of a superheavy intermediate %32 %32 gauge boson with the right quantum numbers (e.g. those of an SU(5) generator). In supersymmetric models with R invariance the status of B and L conservation as an accidental symmetry is maintained. In the presence of l , the dimension 3 operator corresponding to m is gauge 0 00 symmetric but violates ¸. By dimensions we expect a mass factor in front of this operator in the lagrangian density, and in the absence of a protective symmetry, we expect it of the order of the cut-o!, i.e. of order M or larger. Thus, ¸ number violation is naturally induced by the presence %32 of l , unless we enforce it by hand. 0 4. L violation explains the smallness of neutrino masses: the see-saw mechanism Once we accept ¸ violation we gain an elegant explanation for the smallness of neutrino masses as they turn out to be inversely proportional to the large scale where lepton number is violated. If ¸ is not conserved, even in the absence of l , Majorana masses can be generated for neutrinos by 0 dimension "ve operators of the form O "¸2j ¸

/M (2)  G GH H with being the ordinary Higgs doublet, j a matrix in #avour space and M a large scale of mass, of order M or M . Neutrino masses generated by O are of the order m &v/M for j &O(1), %32 .   J GH where v&O(100 GeV) is the vacuum expectation value of the ordinary Higgs. We consider that the existence of l is quite plausible because all GUT groups larger than 0 SU(5) require them. In particular the fact that l completes the representation 16 of 0 SO(10): 16"5 #10#1, so that all fermions of each family are contained in a single representation of the unifying group, is too impressive not to be signi"cant. At least as a classi"cation group SO(10) must be of some relevance. Thus in the following we assume that there are both l and 0 lepton number violation. With these assumptions the see-saw mechanism [20] is possible. In its simplest form it arises as follows. Consider the mass terms in the lagrangian corresponding to Dirac and RR Majorana masses (for the time being we consider LL Majorana mass terms as comparatively negligible): (3) L"!RM m ¸#RM MRM 2#h.c . "  For notational simplicity we denoted l and l by ¸ and R, respectively (the prime de"ning the * 0 l #avour basis, see Eq. (1)). The 3;3 matrices m and M are the Dirac and Majorana mass matrices " in #avour space (M is symmetric, M"M2, while m is, in general, nonhermitian and nonsymmet" ric). We expect the eigenvalues of M to be of order M or more because RR Majorana masses are %32 SU(3);SU(2);;(1) invariant, hence unprotected and naturally of the order of the cuto! of the low-energy theory. Since all l are very heavy we can integrate them away. For this purpose we 0 write down the equations of motion for RM in the static limit, i.e. neglecting their kinetic terms !RL/RRM "m ¸!MRM 2"0 . "

(4)

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From this, by solving for RM 2 and transposing, we obtain: RM "¸2m2 M\ . " We now replace in the Lagrangian, Eq. (3), this expression for RM and we get

(5)

L"!¸2m ¸ , J  where the resulting neutrino mass matrix is

(6)

m "m2 M\m . (7) J " " This is the well-known see-saw mechanism [20] result: the light neutrino masses are quadratic in the Dirac masses and inversely proportional to the large Majorana mass. If some l are massless or 0 light they would not be integrated away but simply added to the light neutrinos. Notice that the above results hold true for any number n of heavy neutral fermions R coupled to the three known neutrinos. In this more general case M is an n;n symmetric matrix and the coupling between heavy and light "elds is described by the rectangular n;3 matrix m . " Here we assumed that the additional non renormalizable terms from O are comparatively  negligible, otherwise they should simply be added. After elimination of the heavy right-handed "elds, at the level of the e!ective low-energy theory, the two types of terms are equivalent. In particular, they have identical transformation properties under a chiral change of basis in #avour space. The di!erence is, however, that in the see-saw mechanism, the Dirac matrix m is presum" ably related to ordinary fermion masses because they are both generated by the Higgs mechanism and both must obey GUT-induced constraints. Thus if we assume the see-saw mechanism more constraints are implied. In particular we are led to the natural hypothesis that m has a largely " dominant third family eigenvalue in analogy to m , m and m which are by far the largest masses R @ O among u quarks, d quarks and charged leptons. Once we accept that m is hierarchical it is very " di$cult to imagine that the e!ective light neutrino matrix, generated by the see-saw mechanism, could have eigenvalues very close in absolute value. 5. The neutrino mixing matrix Given the de"nition of the mixing matrix ; in Eq. (1) and the transformation properties of the e!ective light neutrino mass matrix m : J ¸2m ¸"¸2;2m ;¸ , J J , (8) ;2m ;"Diag[e (m , e (m , m ],m      J we obtain the general form of m : J m ";m ;2 . (9) J   The matrix ; can be parameterized in terms of three mixing angles and one phase, exactly as for the quark mixing matrix < . In addition we have the two phases and that are present !)+   because of the Majorana nature of neutrinos. Thus, in general, nine parameters are added to the SM when nonvanishing neutrino masses are included: three eigenvalues, three mixing angles and three CP violating phases [55]. Maximal atmospheric neutrino mixing and the requirement that the electron neutrino does not participate in the atmospheric oscillations, as indicated by the Superkamiokande [1] and Chooz

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301

[15] data, lead directly to the following structure of the ; ( f"e, k, q, i"1, 2, 3) mixing matrix, DG apart from sign convention rede"nitions



c

0

!s



; " s/(2 c/(2 !1/(2 . DG s/(2 c/(2 #1/(2

(10)

This result is obtained by a simple generalization of the analysis of Refs. [21,22] to the case of arbitrary solar mixing angle (s,sin h , c,cos h ): c"s"1/(2 for maximal solar mixing (e.g.   for vacuum oscillations sin 2h &0.75) , while sin 2h &4s&5.5;10\ for the small-angle   MSW [7] solution. The vanishing of ; guarantees that l does not participate in the atmospheric C C oscillations and the relation "; """; ""1/(2 implies maximal mixing for atmospheric neuI O trinos. Also, in the limit ; "0, all CP violating e!ects vanish and we can neglect the additional C phase parameter generally present in ; : the matrix ; is real and orthogonal and is equal to the DG product of a rotation by p/4 in the 23 plane times a rotation in the 12 plane:



1

0

0



; " 0 1/(2 !1/(2 DG 0 1/(2 1/(2



c

!s

s

c

0

0 1

0

0 .

(11)

Note that we are assuming only two frequencies, given by * Jm!m, * Jm!m . (12)       The numbering 1, 2, 3 corresponds to our de"nition of the frequencies and, in principle, may not coincide with the family index although this will be the case in the models that we favour. The e!ective light neutrino mass matrix is given by Eq. (9). We disregard the phases but in the  following m can be of either sign.  For generic s, using Eq. (9), one "nds



2e

m" d J d

d

d



(13) m /2#e !m /2#e     !m /2#e m /2#e     with e"(m c#m s)/2, d"(m !m )cs/(2, e "(m s#m c)/2. (14)        We see that the existence of one maximal mixing and ; "0 are the most important input that lead C to the matrix form in Eqs. [13,14]. The value of the solar neutrino mixing angle can be left free. While the simple parametrization of the matrix ; in Eq. (10) is quite useful to guide the search for a realistic pattern of neutrino mass matrices, it should not be taken too literally. In particular the data do not exclude a nonvanishing ; element. As already mentioned, the bound by Chooz [15] amounts to C "; ":0.2. Thus neglecting "; " with respect to s in Eq. (10) is not completely justi"ed. Also note that C C in presence of a large hierarchy "m "<"m " the e!ect of neglected parameters in Eq. (10) can be   enhanced by m /m and produce seizable corrections. A nonvanishing ; term can lead to di!erent   C (m ) and (m ) terms. Similarly a deviation from maximal mixing ; O; distorts the e terms in J  J  I O  the 23 sector of m . Therefore, especially for a large hierarchy, there is more freedom in the small terms J in order to construct a model that "ts the data than it is apparent from Eq. (13).

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6. Possible mass hierarchies and approximate textures Given the observed frequencies and our notation in Eq. (12), there are three possible hierarchies of mass eigenvalues [23]: A: "m "<"m " ,   B: "m "&"m "<"m " ,    C: "m "&"m "&"m "   

(15)

(in case A there is no prejudice on the m , m relation). For B and C di!erent subcases are then   generated according to the relative sign assignments for m .  For each case we can set to zero the small masses and mixing angles and "nd the e!ective light neutrino matrices which are obtained both for double and single maximal mixing. Note that here we are working in the basis where the charged lepton masses are diagonal, and approximately given by m "Diag[0, 0, m ]. For model building one has to arrange both the charged lepton and O the neutrino mass matrices so that the neutrino results coincide with those given here after diagonalization of charged leptons. For example, in case A, m "Diag[0, 0, m ] and we obtain   



0

0

A: m "Diag[0, 0, 1]m P m /m " 0    J  0

  ! 



0

! .   

(16)

In this particular case the results are the same for double and single maximal mixing. Note that the signs correspond to the phase convention adopted in Eq. (10). If one prefers all signs to be positive it is su$cient to invert the sign of the third row of the matrix ; in Eq. (10). We can similarly proceed in the other cases and we obtain the results in Table 1 (where the overall mass scale was dropped). We recall that, once a solution to the solar neutrino problem is chosen and a set of suitable small perturbation terms is introduced, oscillation phenomena are unable to distinguish between the cases A, B and C displayed in Table 1. However, from the model building point of view, each texture in Table 1 represents an independent possibility in a zeroth-order approximation. Moreover, Table 1 could be generalized by allowing general phase di!erences among the three neutrino masses as shown in Eq. (8). For instance, for double maximal mixing and "m """m """m " we "nd    the general texture





p m /(2 m /(2    1 1#p /2 !1#p /2 , m /m " m /(2   J  2  1#p /2 m /(2 !1#p /2     We thank David Dooling for pointing out a misprint in Table 1, in the earlier version of this review.

(17)

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Table 1 Zeroth-order form of the neutrino mass matrix for double and single maximal mixing, according to the di!erent possible hierarchies given in Eq. (15) m   A

Diag[0,0,1]

Double maximal mixing

Single maximal mixing

             

      

0

0

0

0

1/2

!1/2

0 !1/2

B1

B2

Diag[1,!1,0]

Diag[1,1,0]

0

1/2

1/(2 1/(2

1/(2

0

0

1/(2

0

0

1

0

0

0 1/2 1/2 0 1/2 1/2

C0

Diag[1,1,1]

1 0 0 0 1 0 0 0 1

C1

C2

C3

Diag[!1,1,1]

Diag[1,!1,1]

Diag[1,1,!1]

0

!1/(2 !1/(2

!1/(2

1/2

!1/2

!1/(2

!1/2

1/2

0

1/(2

1/(2

1/2

!1/2

1/(2

!1/2

1/2

1 0 0 0 0 1 0 1 0

1/(2

0

0

0

0

1/2

0 !1/2 1

 

!1/2 1/2

0

0

0 !1/2 !1/2 0 !1/2 !1/2 1

0

0



0 1/2 1/2 0 1/2 1/2



1 0 0 0 1 0 0 0 1

!1 0 0 0 1 0 0 0 1





1

0

0

0

0 !1

0 !1

0



1 0 0 0 0 1 0 1 0

where p (m ),e (#(!)e (. When p "0, Eq. (17) reproduces the cases C1 and C2 of    Table 1, while for m "0, C0 and C3 are obtained.  7. The case of nearly degenerate neutrino masses The con"gurations B and C imply a very precise near-degeneracy of squared masses. For example, the case C is the only one that could, in principle, accommodate neutrinos as hot dark

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matter together with solar and atmospheric neutrino oscillations. We think that it is not at all clear at the moment that a hot dark matter component is really needed [18], but this could be a reason in favour of the fully degenerate solution. Then the common mass should be around 1}3 eV. The solar frequency could be given by a small 1}2 splitting, while the atmospheric frequency could be given by a still small but much larger 1, 2}3 splitting. A strong constraint arises in this case from the nonobservation of neutrino-less double beta decay which requires that the ee entry of m must obey J "(m ) "40.2 eV [19]. Perhaps this bound can somewhat be relaxed if the error from the theoretical J CC ambiguities on nuclear matrix elements is increased. As observed in Ref. [24], if this bound is below 1 eV, it can only be satis"ed if bimixing is realized (that is double maximal mixing, with solar neutrinos explained by the VO solution or may be by the large angle MSW solution). The reason is that, as seen from Eqs. (13) and (14), (m ) "m c#m s. Now s is either very small or very close J CC   to c. For the required cancellation one needs opposite signs for m and m and comparable values   of c and s, hence nearly maximal solar mixing. A problem with this type of solution is that we would need a relative splitting "*m/m"&*m /2m&10\}10\ and a much smaller one for solar neutrinos especially if ex plained by vacuum oscillations: "*m/m"&10\}10\. As mentioned above we consider it unplausible that starting from hierarchical Dirac matrices we end up via the see-saw mechanism into a nearly perfect degeneracy of squared masses. Thus models with degenerate neutrinos could only be natural if the dominant contributions directly arise from non renormalizable operators like O in Eq. (2) because they are a priori unrelated to other fermion mass terms. The degeneracy of  neutrinos should be guaranteed by some slightly broken symmetry. Notice however that, even if arranged at the GUT scale, it is doubtful that such a precise degeneracy could be stable against renormalization group corrections when running down at low energy unless it is protected by a suitable symmetry [25]. Models based on discrete or continuous symmetries have been proposed. For example in the models of Ref. [26] the symmetry is SO(3). In the unbroken limit neutrinos are degenerate and charged leptons are massless. When the symmetry is broken the charged lepton masses are much larger than neutrino splittings because the former are "rst order while the latter are second order in the electroweak symmetry breaking. A model which is simple to describe but di$cult to derive in a natural way is one [27,53] where up quarks, down quarks and charged leptons have `democratica mass matrices, with all entries equal (in "rst approximation):

  

1 1 1 0 0 0 1 mS , mB , m J 1 1 1 P 0 0 0 , " " " 3 1 1 1 0 0 1

(18)

where we have also indicated the diagonal form with two vanishing eigenvalues. In this limit the CKM matrix is the identity, because u and d quarks are diagonalized by the same matrix (recall that < ";R; ). This matrix ; can be chosen with ; "0 and is given by (note the analogy !)+ S B C with the quark model eigenvalues n, g and g):



1/(2 !1/(2

;" 1/(6 1/(3

0



1/(6 !2/(6 . 1/(3

1/(3

(19)

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The mass matrix of Eq. (18) is invariant under a discrete S ;S permutation symmetry. The * 0 same requirement leads to the general neutrino mass matrix

    1 0 0

1 1 1

m Ja 0 1 0 #b 1 1 1 , J 0 0 1 1 1 1

(20)

the two independent invariants being allowed by the Majorana nature of the light neutrinos. If b vanishes the neutrinos are degenerate. In the presence of small terms that break the permutation symmetry this degeneracy is removed and the neutrino mixing matrix may remain very close to ; in Eq. (19). At the same time, the lightest quarks and charged leptons may acquire a nonvanishing mass. Note that the atmospheric neutrino mixing is nearly maximal: sin 2h", while the solar  mixing angle is maximal. An intermediate possibility between the case of degenerate neutrino masses and the case of hierarchical ones is the situation B in Eq. (15). With two almost degenerate heavier states and a nearly massless neutrino we cannot reach a mass range interesting for cosmological purposes. However, since the experimentally accessible quantities are the squared mass di!erences, case B remains an open possibility. The small terms required to go beyond the zeroth-order approximation can be controlled by a spontaneously broken #avour symmetry. For instance, working only with the light neutrinos, the ;(1) charge Q,(¸ !¸ !¸ ) allows for a texture of the kind B1, / C I O double maximal mixing, at leading order [11]. The nonvanishing entries (m ) and (m ) are J  J  expected to be of the same order, although not necessarily equal. The vanishing entries can be "lled by the ratio of the VEV 1u2 of a scalar "eld carrying two units of Q and a mass scale M providing the cut-o! to the low-energy theory

  e

m "m 1 J 1

1 1 e

e ,

e

e

(21)

where e"1u2/M and only the order-of-magnitudes are indicated. For values of e smaller than 1 one obtains a small perturbation of the zeroth-order texture. The mixing matrix has a large, not necessarily maximal, mixing angle in the 23 sector, a nearly maximal mixing angle in the 12 sector and a mixing angle of order e in the 13 sector [56]. The mass parameter m should be close to 10\}10\ eV to provide the frequency required by the atmospheric oscillations. Finally, m!m&me. In this model the VO solution to the solar neutrino de"cit would require a very   tiny breaking term, e&O(10\).

8. The case of hierarchical neutrino masses We now discuss models of type A with large e!ective light neutrino mass splittings and large mixings. In general, large splittings correspond to small mixings because normally only close-by states are strongly mixed. In a 2;2 matrix context the requirement of large splitting and large

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mixings leads to a condition of vanishing determinant. For example the matrix

  x x

mJ

x

(22)

1

has eigenvalues 0 and 1#x and for x of O(1) the mixing is large. Thus, in the limit of neglecting small mass terms of order m , the demands of large atmospheric neutrino mixing and dominance  of m translate into the condition that the 2;2 subdeterminant 23 of the 3;3 mixing matrix  vanishes. The problem is to show that this approximate vanishing can be arranged in a natural way without "ne tuning. We have discussed suitable possible mechanisms in our papers [28,23,29]. We in particular favour a class of models where, in the limit of neglecting terms of order m and, in  the basis where charged leptons are diagonal, the Dirac matrix m , de"ned by RM m ¸, takes the " " approximate form

  0

0

0

m J 0 0 0 . " 0 x 1

(23)

This matrix has the property that for a generic Majorana matrix M one "nds

 

(24)



(25)

0

0

0

m "m2 M\m J 0 x x . J " " 0 x 1

The only condition on M\ is that the 33 entry is nonzero. It is important for the following discussion to observe that m given by Eq. (23) under a change of basis transforms as m P


c

;" sc with

!s A

ss A

cc A cs A

0 !s

A c A

,

(26) s "!x/r, c "1/r, r"(1#x . A A The matrix ; is directly the neutrino mixing matrix. The mixing angle for atmospheric neutrino oscillations is given by 4x . sin 2h"4sc" A A (1#x)

(27)

Thus the bound sin 2h90.9 translates into 0.7:"x":1.4. It is interesting to recall that in Refs. [30,31] it was shown that the mixing angle can be ampli"ed by the running from a large mass scale down to low energy.

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We have seen that, in order to explain in a natural way widely split light neutrino masses together with large mixings, we need an automatic vanishing of the 23 subdeterminant. This in turn is most simply realized by allowing some large left-handed mixing terms in the Dirac neutrino matrix. By left-handed mixing we mean non diagonal matrix elements that can only be eliminated by a large rotation of the left-handed "elds. Thus the question is how to reconcile large left-handed mixings in the leptonic sector with the observed near diagonal form of < , the quark mixing !)+ matrix. Strictly speaking, since < ";R; , the individual matrices ; and ; need not be near !)+ S B S B diagonal, but < does, while the analogue for leptons apparently cannot be near diagonal. !)+ However nothing forbids for quarks that, in the basis where m is diagonal, the d quark matrix has S large nondiagonal terms that can be rotated away by a pure right-handed rotation. We suggest that this is so and that in some way right-handed mixings for quarks correspond to left-handed mixings for leptons. In the context of (Susy) SU(5) [5] there is a very attractive hint of how the present mechanism can be realized. In the 5 of SU(5) the d singlet appears together with the lepton doublet (l, e). The (u, d) doublet and e belong to the 10 and l to the 1 and similarly for the other families. As a consequence, in the simplest model with mass terms arising from only Higgs pentaplets, the Dirac matrix of down quarks is the transpose of the charged lepton matrix: mB "(m )2. Thus, indeed, " " a large mixing for right-handed down quarks corresponds to a large left-handed mixing for charged leptons. At leading order we may have

  0 0 0

mB "(m )2" 0 0 1 v . B " " 0 0 1

(28)

In the same simplest approximation with 5 or 5 Higgs, the up quark mass matrix is symmetric, so that left and right mixing matrices are equal in this case. Then small mixings for up quarks and small left-handed mixings for down quarks are su$cient to guarantee small < mixing angles !)+ even for large d quark right-handed mixings. If these small mixings are neglected, we expect

  0 0 0

mS " 0 0 0 v S " 0 0 1

(29)

When the charged lepton matrix is diagonalized the large left-handed mixing of the charged leptons is transferred to the neutrinos. Note that in SU(5) we can diagonalize the u mass matrix by a rotation of the "elds in the 10, the Majorana matrix M by a rotation of the 1 and the e!ective light neutrino matrix m by a rotation of the 5 . In this basis the d quark mass matrix "xes < and the J !)+ charged lepton mass matrix "xes neutrino mixings. It is well known that a model where the down and the charged lepton matrices are exactly the transpose of one another cannot be exactly true because of the e/d and k/s mass ratios. It is also known that one remedy to this problem is to add some Higgs component in the 45 representation of SU(5) [32]. A di!erent solution [33] will be described later. But the symmetry under transposition can still be a good guideline if we are only interested in the order of magnitude of the matrix entries and not in their exact values. Similarly, the Dirac neutrino mass matrix m is the same as the up quark mass matrix in the very crude model "

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where the Higgs pentaplets come from a pure 10 representation of SO(10): m "mS . For m the " " " dominance of the third family eigenvalue as well as a near diagonal form could be an order of magnitude remnant of this broken symmetry. Thus, neglecting small terms, the neutrino Dirac matrix in the basis where charged leptons are diagonal could be directly obtained in the form of Eq. (23).

9. An example with horizontal abelian charges We give here an explicit example [29] of the mechanism under discussion in the framework of a uni"ed Susy SU(5) theory with an additional ;(1) #avour symmetry [34]. This model is to be $ taken as merely indicative, in that some important problems, like, for example, the cancellation of chiral anomalies are not tackled here. But we "nd it impressive that the general pattern of all what we know on fermion masses and mixings is correctly reproduced at the level of orders of magnitude. We regard the present model as a low-energy e!ective theory valid at energies close to M ;M . We can think to obtain it by integrating out the heavy modes from an unknown %32 . underlying fundamental theory de"ned at an energy scale close to M . From this point of view . the gauge anomalies generated by the light supermultiplets listed below can be compensated by another set of supermultiplets with masses above M , already eliminated from the low-energy %32 theory. In particular, we assume that these additional supermultiplets are vector-like with respect to SU(5) and chiral with respect to ;(1) . Their masses are then naturally expected to be of the $ order of the ;(1) breaking scale, which, in the following discussion, turns out to be near M . It is $ . possible to check explicitly the possibility of canceling the gauge anomalies in this way but, due to our ignorance about the fundamental theory, it is not particularly instructive to illustrate the details here. In this model the known generations of quarks and leptons are contained in triplets W? and W? , (a"1, 2, 3) transforming as 10 and 5 of SU(5), respectively. Three more SU(5)   singlets W? describe the right-handed neutrinos. We assign to these "elds the following F charges: W &(3, 2, 0) ,  W  &(3, 0, 0) ,  W &(1,!1, 0) .  We start by discussing the Yukawa coupling allowed by ;(1) -neutral Higgs multiplets u $  u  in the 5 and 5 SU(5) representations and by a pair h and hM of SU(5) singlets with F"1  F"!1, respectively. In the quark sector we obtain



j j j



mS "(mS )2" j j j v , " " S j j 1





(30) (31) (32) and and

j j j

mB " j j " j j

1 v 1

B

(33)

 In Eq. (33) the entries denoted by 1 in mS and mB are not necessarily equal. As usual, such a notation allows for O(1) " " deviations.

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from which we get the order-of-magnitude relations m : m : m "j : j : 1 , S A R m : m : m "j : j : 1 B Q @

(34)

and < &j, < &j, < &j . (35) SQ S@ A@ Here v ,1u 2, v ,1u  2 and j denotes the ratio between the vacuum expectation value of S  B  hM and an ultraviolet cut-o! identi"ed with the Planck mass M : j,1hM 2/M . To correctly . . reproduce the observed quark mixing angles, we take j of the order of the Cabibbo angle. For nonnegative F-charges, the elements of the quark mixing matrix < depend only on the charge !)+ di!erences of the left-handed quark doublet [34]. Up to a constant shift, this de"nes the choice in (see Eq. (31)) are then required to "t m and m . We will comment Eq. (30). Equal F-charges for W @ Q  on the lightest quark masses later on. At this level, the mass matrix for the charged leptons is the transpose of mB : " m "(mB )2 (36) " " and we "nd m : m : m "j : j : 1 . (37) C I O The O(1) o!-diagonal entry of m gives rise to a large left-handed mixing in the 23 block which " corresponds to a large right-handed mixing in the d mass matrix. In the neutrino sector, the Dirac and Majorana mass matrices are given by

  j

j

j

m " j j j v , S " j 1 1



j

M" 1 j

1

j



M , j j M j

1

(38)

where j,1h2/M and M M denotes the large mass scale associated to the right-handed neutrinos: . M M




j j j

m " j J j

1 1

1 v/MM S 1

(39)

where we have taken j&j. The O(1) elements in the 23 block are produced by combining the large left-handed mixing induced by the charged lepton sector and the large left-handed mixing in m . " A crucial property of m is that, as a result of the sea-saw mechanism and of the speci"c ;(1) J $ charge assignment, the determinant of the 23 block is automatically of O(j) (for this the presence of negative charge values, leading to the presence of both j and j is essential [23]).

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It is easy to verify that the eigenvalues of m satisfy the relations: J m : m : m "j : j : 1 . (40)    The atmospheric neutrino oscillations require m&10\ eV. From Eq. (39), taking v &250 GeV,  S the mass scale M M of the heavy Majorana neutrinos turns out to be close to the uni"cation scale, M M &10 GeV. The squared mass di!erence between the lightest states is of O(j) m, appropriate  to the MSW solution to the solar neutrino problem. Finally, beyond the large mixing in the 23 sector, corresponding to s &c in Eq. (25), m provides a mixing angle s&(j/2) in the 12 sector, A A J close to the range preferred by the small angle MSW solution. In general ; is nonvanishing, of C O(j). In general, the charge assignment under ;(1) allows for noncanonical kinetic terms that $ represent an additional source of mixing. Such terms are allowed by the underlying #avour symmetry and it would be unnatural to tune them to the canonical form. The results quoted up to now remain unchanged after including the e!ects related to the most general kinetic terms, via appropriate rotations and rescaling in the #avour space (see also Ref. [35]). Obviously, the order of magnitude description o!ered by this model is not intended to account for all the details of fermion masses. Even neglecting the parameters associated with the CP violating observables, some of the relevant observables are somewhat marginally reproduced. For instance we obtain m /m &j which is perhaps too large. However, we "nd it remarkable that in S R such a simple scheme most of the 12 independent fermion masses and the six mixing angles turn out to have the correct order of magnitude. Notice also that our model prefers large values of tan b,v /v . This is a consequence of the equality F(W )"F(W ) (see Eqs. (30) and (31)). In this S B   case the Yukawa couplings of top and bottom quarks are expected to be of the same order of magnitude, while the large m /m ratio is attributed to v
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311

unknown coe$cient of O(1). Moreover, the product (hM )Ou  contains both the 5 and the 45 SU(5)   representations, allowing for a di!erentiation between the down quarks and the charged leptons. The only, welcome, exceptions are given by the O(1) entries that do not require any compensation and, at the leading order, remain the same for charged leptons and down quarks. This preserves the good m "m prediction. Since a perturbation of O(1) in the subleading matrix elements is @ O su$cient to cure the bad m /m "m /m relation, we can safely assume that 1hM 2/M &jL, to B Q C I  . preserve the correct order-of-magnitude predictions in the remaining sectors. A general problem common to all models dealing with #avour is that of recovering the correct vacuum structure by minimizing the e!ective potential of the theory. It may be noticed that the presence of two multiplets h and hM with opposite F charges could hardly be reconciled, without adding extra structure to the model, with a large common VEV for these "elds, due to possible analytic terms of the kind (hhM )L in the superpotential [57]. We "nd therefore instructive to explore the consequences of allowing only the negatively charged hM "eld in the theory. It can be immediately recognized that, while the quark mass matrices of Eqs. (33) are unchanged, in the neutrino sector the Dirac and Majorana matrices get modi"ed into:

  j j j

m " j 0 0 v , S " j 1 1

  j 1 j

M" 1 j

M . 0 0 M

(41)

0 1

The zeros are due to the analytic property of the superpotential that makes impossible to form the corresponding F invariant by using hM alone. These zeros should not be taken literally, as they will be eventually "lled by small terms coming, for instance, from the diagonalization of the charged lepton mass matrix and from the transformation that put the kinetic terms into canonical form. It is however interesting to work out, in "rst approximation, the case of exactly zero entries in m and " M, when forbidden by F. The neutrino mass matrix obtained via see-saw from m and M has the same pattern as the one " displayed in Eq. (39). A closer inspection reveals that the determinant of the 23 block is identically zero, independently from j. This leads to the following pattern of masses: m : m : m "j : j : 1, m!m"O(j) .      Moreover, the mixing in the 12 sector is almost maximal: s/c"p/4#O(j) .

(42)

(43)

For j&0.2, both the squared mass di!erence (m!m)/m and sin2h are remarkably close to     the values required by the vacuum oscillation solution to the solar neutrino problem. This property remains reasonably stable against the perturbations induced by small terms (of order j) replacing the zeros, coming from the diagonalization of the charged lepton sector and by the transformations that render the kinetic terms canonical. We "nd quite interesting that also the just-so solution, requiring an intriguingly small mass di!erence and a bimaximal mixing, can be reproduced, at least at the level of order of magnitudes, in the context of a `minimala model of #avour compatible with supersymmetric SU(5). In this case the role played by supersymmetry is essential, a non-supersymmetric model with hM alone not being distinguishable from the version with both h and hM , as far as low-energy #avour properties are concerned.

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10. Other similar or alternative proposals Finally, let us compare our model with other recent proposals [38]. Textures for the e!ective neutrino mass matrix similar to m in Eq. (39) were derived in Refs. [39,40], also in the context of an J SU(5) uni"ed theory with a ;(1) #avour symmetry. In these works, however, the O(1) entries of m are J uncorrelated, due to the particular choice of ;(1) charges. The diagonalization of such matrix, for generic O(1) coe$cients, leads to only one light eigenvalue and to two heavy eigenvalues, of O(1), in units of v/M M . Then the required pattern m <m &m has to be "xed by hand. On the contrary, in S    our model the desired pattern is automatic, since, as emphasized above, the determinant of the 23 block in m is vanishing at the leading order. Other models in terms of ;(1) horizontal charges have J been proposed in Refs. [30,41}44]. Clearly a large mixing for the light neutrinos can be provided in part by the diagonalization of the charged lepton sector. As we have seen, in SU(5), the left-handed mixing carried by charged leptons is expected to be, at least in "rst approximation, directly linked to the right-handed mixing for the d quarks and, as such, perfectly compatible with the available data. This possibility was remarked, for instance, in Refs. [31,45}47] where the implementation was in terms of asymmetric textures, of the Branco et al. type [48], used as a general parametrization of the existing data consistent with the constraints imposed by the uni"cation program. On the other hand, our model aims to a dynamical explanation of the #avour properties, although in a simpli"ed setting. It is interesting to note that the mechanism discussed sofar can be embedded in an SO(10) grand-uni"ed theory in a rather economic way [46]. The 33 entries of the fermion mass matrices can be obtained through the coupling 16 16 10 among the fermions in the third generation, 16 ,   &  and a Higgs tenplet 10 . The two independent VEVs of the tenplet v and v give mass, respectively, & S B to t/l and b/q. The keypoint to obtain an asymmetric texture is the introduction of an operator of O the kind 16 16 16 16 . This operator is thought to arise by integrating out an heavy 10 that  &  & couples both to 16 16 and to 16 16 . If the 16 develops a VEV breaking SO(10) down to SU(5) at  &  & & a large scale, then, in terms of SU(5) representations, we get an e!ective coupling of the kind 5 10 5 , with a coe$cient that can be of order one. This coupling contributes to the 23 entry of the   & down quark mass matrix and to the 32 entry of the charged lepton mass matrix, realizing the desired asymmetry. To distinguish the lepton and quark sectors one can further introduce an operator of the form 16 16 10 45 , (i, j"2, 3), with the VEV of the 45 pointing in the B!¸ G H & & & direction. Additional operators, still of the type 16 16 16 16 can contribute to the matrix elements G H & & of the "rst generation. The mass matrices look like

 

0

0

0



0 e/3 v , mS " 0 S " 0 !e/3 1 0 0

m " 0 0 " 0 e



0

!e v , S 1



d

0

d



mB " d 0 p#e/3 v , " B d !e/3 1



0

d

mC " d 0 " d p#e

d



!e v . B 1

 Also Ref. [47] suggests an horizontal ;(2) symmetry to justify the assumed textures.

(44)

(45)

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313

They provide a good "t of the available data in the quarks and the charged lepton sector in terms of "ve parameters (one of which is complex). In the neutrino sector one obtains a large h mixing  angle, sin2h &6.6;10\ eV and h of the same order of h . Mass-squared di!erences are    sensitive to the details of the Majorana mass matrix. Looking at models with three light neutrinos only, i.e. no sterile neutrinos, from a more general point of view, we stress that in the above models the atmospheric neutrino mixing is considered large, in the sense of being of order one in some zeroth-order approximation. In other words it corresponds to o!-diagonal matrix elements of the same order of the diagonal ones, although the mixing is not exactly maximal. The idea that all fermion mixings are small and induced by the observed smallness of the non diagonal < matrix elements is then abandoned. An alternative is !)+ to argue that perhaps what appears to be large is not that large after all. The typical small parameter that appears in the mass matrices is j&(m /m &(m /m &0.20}0.25. This small I O B Q parameter is not so small that it cannot become large due to some peculiar accidental enhancement: either a coe$cient of about 3, or an exponent of the mass ratio which is less than  (due for  example to a suitable charge assignment), or the addition in phase of an angle from the diagonalization of charged leptons and an angle from neutrino mixing. One may like this strategy of producing a large mixing by stretching small ones if, for example, he/she likes symmetric mass matrices, as from left}right symmetry at the GUT scale. In left}right symmetric models smallness of left mixings implies that also right-handed mixings are small, so that all mixings tend to be small. Clearly this set of models tend to favour moderate hierarchies and a single maximal mixing, so that the SA-MSW solution of solar neutrinos is preferred. For example, consider the 23 submatrix only, for simplicity. Assume that at order zero the neutrino Dirac matrix is given by

 

0 0 . m & " 0 1

(46)

For a generic Majorana matrix M also the e!ective light neutrino matrix m has the same structure J in the 23 sector. If for charged leptons one has





j rj m& rj 1

(47)

with r&O(1) and j&O((m /m ) (in fact the eigenvalues are of order 1 and j, respectively), then I O the symmetric charged lepton matrix is diagonalized by a unitary matrix (the same for left and right "elds). In the basis where m is diagonal, we then have c s 0 0 c !s s sc m& " . (48) J !s c 0 1 s c sc c



 

 



As a result the 23 sub-determinant vanishes and sin 2h&2sc&2rj is large for r& 2}3. Models of this type have been discussed in Ref. [49]. To further discuss how a large neutrino mixing could be generated starting from symmetric matrices and small mixings for quarks and leptons we can go, without loss of generality, to a basis where both the charged lepton Dirac mass matrix and the RR Majorana matrix are diagonal. In fact, after diagonalization of the charged lepton Dirac mass matrix, we still have the freedom of

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a change of basis for the right-handed neutrino "elds, in that the right-handed charged lepton and neutrino "elds, as opposed to left-handed "elds, are uncorrelated by the SU(2);(1) gauge symmetry. We can use this freedom to make the Majorana matrix diagonal: M\"<2d < with + d "Diag[1/M , 1/M , 1/M ]. Then if we parametrize the matrix <m by z we have +    " ?@ z z (49) m "(m2 M\m ) " A? A@ . J?@ " " ?@ M A A From this expression we see that, while we can always arrange the twelve parameters z and M to ?@ ? arbitrarily "x the six independent matrix elements of m , case A is special in that it can be J approximately reproduced in two particularly simple ways, without relying on precise cancellations among di!erent terms: (i)

(ii)

There are only two large entries in the z matrix, "z "&"z ", and the three eigenvalues M are A A ? of comparable magnitude (or, at least, with a less pronounced hierarchy than for the z matrix elements). Then, the subdeterminant 23 vanishes and one only needs the ratio "z /z " to be A A close to 1. This is the possibility discussed in detail in Sections 8 and 9. One of the right-handed neutrinos is particularly light and, in "rst approximation, it is only coupled to k and q. Thus, M &g (small) and z &0. In this case [11] the 23 subdeterminant A A vanishes, and one only needs the ratio "z /z " to be close to 1. This possibility has been A A especially emphasized in Refs. [50].

In particular, mechanism (ii) is compatible with symmetric mass matrices. For example, one could want to preserve left}right symmetry at the GUT scale. Then, the observed smallness of left-handed mixings for quarks would also demand small right-handed mixings. So we now assume that m is " nearly diagonal (always in the basis where charged leptons and M are diagonal) with all its o! diagonal terms proportional to some small parameter e. Starting from





 

eN xe , m J " xe 1

r M\J  0

0

.

1

(50)

where x is O(1), we obtain





eNr #xe xeN>r #xe   m "m2 M\m J . (51) J " " xeN>r #xe xer #1   For su$ciently small M the terms in r are dominant. For p"1, 2, which we consider as typical   cases, it is su$cient that er <1. Assuming that this condition is satis"ed, consider "rst the case  with p"2. We have m "m2 M\m Jxer J " " 





e/x e/x e/x

1

.

(52)

This case is qualitatively similar to that described by Eq. (48). The determinant is naturally vanishing (to the extent that the terms in r are dominant), so that the mass eigenvalues are widely  split. However, the mixing is nominally small: sin 2h is of order 2e/x. It could be numerically large enough if 1/x&2!3 and e is of the order of the Cabibbo angle e&0.20!0.25. This is what we

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315

call `stretchinga: the large neutrino mixing is explained in terms of a small parameter which is not so small and can give a perhaps su$cient amount of mixing if enhanced by a possibly large coe$cient. A more interesting case is obtained for p"1, that gives

 

1 x . m "m2 M\m Jer J " "  x x

(53)

In this case the small parameter e is completely factored out and for x&1 the mixing is nearly maximal. The see-saw mechanism has created large mixing from almost nothing: all relevant matrices entering the see-saw mechanism are nearly diagonal. Clearly, the crucial factorization of the small parameter e only arises for p"1. It is possible to extend the previous model to the 3;3 case and to support the resulting almost diagonal textures by means of suitable #avour symmetries [51]. In a similar class of models all Dirac mixings are small, but large mixings are introduced via M [52]. Note that we do not need a symmetric matrix m as that given in Eq. (50) to get the desired " e!ective neutrino mass matrix. The relevant condition is (m ) "(m ) , which can be realized in "  "  a asymmetric Dirac mass matrix. For instance we can equally well take

 

e xe , m J " y 1

(54)

with y41 arbitrary. For er <1 we end up again with m given in Eq. (53).  J The previous models are all in trouble if the atmospheric neutrino mixing is exactly maximal or very close to maximal. There are a few cases in the literature, not of compelling elegance, where the mixing is arranged to be exactly maximal. For instance, in Ref. [53] a discrete group generated by the elements



0

R" (2  (2

 

(2 (2 1

1

!1

1

1

0

0



, ¹ " 0 ep L , (n52) 0 L 0 0 e\p L

(55)

has been suggested to constrain the leading order form of the mass matrices. Indeed, if R and ¹ act L on the neutrinos in their #avour basis and if the required Higgs multiplets are singlets under the discrete symmetry, then, for n'2, the most general invariant mass matrix for the light neutrinos has the form

  1 0 0

m "m 0 0 1 J 0 1 0

(56)

that reproduces the case C3 in Table 1 ("rst column), that is the degenerate case with a double maximal mixing. By invoking an appropriate transformation property for the relevant Higgs doublet, it is possible to obtain a hierarchical and diagonal mass matrix in the charged lepton sector. A suitable spontaneous breaking of the discrete symmetry is then required to generate the correct pattern of masses and mixings beyond the leading approximation. Another example is found in Ref. [26].

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11. Outlook and conclusion By now there are rather convincing experimental indications for neutrino oscillations. If so, then neutrinos have nonzero masses. As a consequence, the phenomenology of neutrino masses and mixings is brought to the forefront. This is a very interesting subject in many respects. It is a window on the physics of GUTs in that the extreme smallness of neutrino masses can only be explained in a natural way if lepton number is violated. Then neutrino masses are inversely proportional to the large scale where lepton number is violated. Also, the pattern of neutrino masses and mixings can provide new clues on the long standing problem of quark and lepton mass matrices. The actual value of neutrino masses is important for cosmology as neutrinos are candidates for hot dark matter: nearly degenerate neutrinos with a common mass around 2 eV would signi"cantly contribute to the matter density in the universe. While the existence of oscillations appears to be on a solid ground, many important experimental ambiguities remain. The most solid indication is that atmospheric neutrino oscillations occur with nearly maximal mixing and *m in the 10\}10\ eV range. It is not clear if l I disappear into mainly l or into some unknown sterile neutrino kind (disappearance into l is O C disfavoured by the data). For solar neutrinos it is not yet clear which of the three solutions, MSW-SA, MSW-LA and VO, is true, and the possibility also remains of di!erent solutions if not all of the experimental input is correct (for example, energy independent solutions are resurrected if the Homestake result is modi"ed). Finally a con"rmation of the LSND alleged signal is necessary. In our discussion here we assumed that the LSND evidence will fade away, so that we could restrict to the three known kinds of neutrinos without new sterile species. The three neutrino mass eigenvalues can either be nearly degenerate or widely split. Solutions with three nearly degenerate neutrinos with a common mass around 2 eV, as for neutrinos of cosmological relevance, are strongly constrained by the bound on neutrinoless double beta decay, and only nearly maximal mixing for solar neutrinos is allowed. In any case, because of the smallness of *m/m, solutions of this kind are not easy to theoretically justify in a natural way. So it looks that probably neutrinos are not cosmologically relevant. We argued in favour of widely split solutions for neutrino masses. Reconciling large splittings with large mixing(s) requires some natural mechanism to implement a vanishing determinant condition. This can be obtained in the see-saw mechanism if one light right handed neutrino is dominant, or a suitable texture of the Dirac matrix is imposed by an underlying symmetry. In a GUT context, the existence of right handed neutrinos indicates SO(10) at least as a classi"cation group generated by physics at M . The symmetry group at M could be either (Susy) SU(5) or . %32 SO(10) or a larger group. We have presented a class of natural models where large right-handed mixings for quarks are transformed into large left-handed mixings for leptons by the approximate transposition relation m "m2 which is often realized in SU(5) models. We have shown that these B C models can be naturally implemented by simple assignments of U(1) horizontal charges. While we favour models based on asymmetric mass matrices, (approximately) symmetric matrices as, for example, produced in left}right symmetric models are not excluded. We have seen that, in the see-saw mechanism, it is even possible to have nearly maximal mixing starting from all Dirac matrices nearly diagonal in the basis where the RR Majorana matrix is diagonal. Alternatively the large neutrino mixing could be generated by an enhancement of formally small terms. This is because the typical small term in quark or charged lepton mass matrices is of the order of

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the Cabibbo angle j&0.22 which is not that small. This enhancement can be the result of a coherent addition of not so small coe$cients from diagonalizing the charged leptons and from the neutrino matrix itself. Or it could arise from a small exponent of m /m for example arising from I O suitable values of abelian charges. In conclusion the fact that some neutrino mixing angles are large, while surprising at the start, was eventually found to be well compatible, without any major change, with our picture of quark and lepton masses within GUTs. Rather it provides us with new important clues that can become sharper when the experimental picture will be further clari"ed.

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Physics Reports 320 (1999) 319}327

Limits on neutrino electromagnetic properties * an update Georg G. Ra!elt Max-Planck-Institut fu( r Physik (Werner-Heisenberg-Institut), Fo( hringer Ring 6, 80805 Mu( nchen, Germany

Abstract Limits on neutrino electromagnetic properties from laboratory experiments and astrophysical arguments are reviewed with an emphasis on the currently favored range of small neutrino masses. We derive a helioseismological limit on the charge and dipole moment for all #avors of e :6;10\e and J k :4;10\k (Bohr magneton). The most restrictive limits remain those from the plasmon decay in J globular-cluster stars of e :2;10\e and k :3;10\k .  1999 Elsevier Science B.V. All rights J J reserved. PACS: 14.60.Lm; 14.60.St Keywords: Neutrinos; Dipole moments; Astrophysical particle bounds

1. Introduction The idea that neutrinos could interact by means of an intrinsic magnetic dipole moment is as old as the idea of neutrinos themselves [1]. Of course, in the modern framework of the particle-physics standard model, neutrino dipole moments strictly vanish due to the left-handed nature of the weak interaction, and even neutrino-mass-induced dipole moments are too small to be of any experimental or astrophysical signi"cance [2,3]. On the other hand, nontrivial extensions of the standard model such as left}right symmetry can lead to interesting values for neutrino electromagnetic couplings. In the late 1980s this possibility was widely discussed because of two astrophysical motivations. The Homestake solar neutrino data seemed to show a signi"cant time variation in correlation with indicators of solar magnetic activity [4,5], leading to a revival of the idea that magnetic spinprecession of left-handed (active) neutrinos into right-handed (sterile) states was responsible for the

E-mail address: [email protected] (G.G. Ra!elt) 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 7 4 - 5

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solar neutrino problem [6]. Second, the neutrino observations from supernova (SN) 1987A ignited broad interest in the role of neutrinos in the SN phenomenon where large magnetic "elds are known to exist and where magnetic spin precessions between active and sterile states can both change the physics of SN explosions and the neutrino signature from such an event [5]. Lev Okun wrote and co-authored several papers on these topics [7}11], which in turn triggered much activity to derive new limits on neutrino electromagnetic couplings, including some papers by the present author [12}14]. The original motivation for speculating about large neutrino dipole moments has never completely disappeared * the Homestake data continue to be analysed for possible time variations in a very recent series of papers [15,16]. Magnetically induced spin-#avor oscillations continue to provide a viable solution of the solar neutrino problem [17}19], and remain potentially important in SN physics [20}24]. Therefore, the present Festschrift in Lev Okun's honor o!ers a timely opportunity to review where we stand today with our empirical knowledge of neutrino electromagnetic properties.

2. Plasmon decay in stars Neutrino dipole or transition moments allow for several interesting processes (Fig. 1). For the purpose of deriving limits, the most important case is cPll which is kinematically allowed in a plasma because the photon acquires a dispersion relation which roughly amounts to an e!ective mass. Even without anomalous couplings, the plasmon decay proceeds because the charged particles of the medium induce an e!ective neutrino}photon interaction. Put another way, even standard neutrinos have nonvanishing electromagnetic form factors in a medium [25,26]. The

Fig. 1. Processes with neutrino electromagnetic dipole or transition moments.

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standard plasma process [14,27,28] dominates the neutrino production in white dwarfs or the cores of globular-cluster red giants, which turn out to be the most sensitive laboratories to constrain neutrino dipole moments. The plasma process was "rst used by Bernstein et al. [29] in a seminal paper to constrain neutrino electromagnetic couplings. They calculated the energy-loss rate of the Sun due to the cPll process under the assumption that neutrinos couple to photons either through a small charge e or else through a magnetic dipole moment k . If this energy loss exceeded the solar J J luminosity, the Sun would have burnt out before reaching its observed age. Today, helioseismology allows for tighter constraints which are found to be e :6;10\e and k :4;10\k where e is J J the electron charge and k "e/2m the Bohr magneton (Appendix A). C A more signi"cant improvement is provided by other stars, notably the properties of globularcluster stars. Nonstandard neutrino losses would delay the ignition of helium in the degenerate cores of low-mass red giants. Several observables in the color-magnitude diagram of globular clusters allow one to derive a restrictive limit on the core mass at helium ignition, corresponding to the requirement that the new energy-loss rate must not exceed the standard losses by more than a factor of a few. One thus "nds the limits [12}14] k :3;10\k and e :2;10\e . (1) J J More recent discussions of these arguments modify details of the astrophysical analysis, but arrive at virtually the same results [5,31]. A slightly more restrictive limit based on the `mass-to-light ratioa of RR Lyrae stars [32] is probably too optimistic. The white-dwarf luminosity function provides a limit of about 10\k , not much weaker than the globular-cluster bound [33]. Naturally, the signi"cance of Eq. (1) could be improved with modern and detailed observations of the color-magnitude diagrams of globular clusters. However, with the plasmon-decay method even a limit of 10\k would be di$cult to achieve, and surely one could not move beyond this value * the anomalous energy loss simply becomes too small to make any measurable di!erence. The stellar energy-loss argument includes all neutrino "nal states which are light enough to be emitted, i.e. with m :5 keV for globular-cluster red giants and white-dwarfs. The current evidence J for neutrino oscillations from the solar and atmospheric neutrino anomalies as well as the LSND experiment together with experimental limits on m C and cosmological arguments suggest that all J neutrino masses are in the eV-range or smaller. Therefore, the stellar-evolution limits most likely apply to all #avors. The k limit pertains equally to electric dipole moments and to electric and magnetic transition J moments, and it applies to both Dirac and Majorana neutrinos.

3. Supernova 1987A Supernova 1987A provides another energy-loss limit, applicable only to Dirac magnetic or electric dipole or transition moments. The structure of the electromagnetic dipole interaction couples neutrino states of opposite helicity. Therefore, neutrinos which are trapped in a SN core #ip their helicity in electromagnetic interactions, taking them into nearly sterile right-handed states which escape directly from the inner SN core. This anomalous energy-loss channel short-circuits

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the standard di!usive energy transfer and thus shortens the measurable signal of left-handed l 's, in C con#ict with the observed duration of the SN 1987A signal. In an early paper [34], a limit k (Dirac):3;10\k J

(2)

was derived. It was con"rmed by a simple estimate in Ref. [5] and by a recent detailed investigation where the spin-#ip rate was calculated by thermal "eld-theory methods [35]. Another early paper [36] found a much more restrictive limit which should be viewed as too optimistic. The temperature in a SN core was taken to be about 30 MeV. Depending on the equation of state it may be much larger, in which case this energy-loss channel would be important for much smaller dipole moments. The right-handed neutrinos emerging directly from the inner SN core have much higher average energies than the ones emitted from the neutrino sphere. They can spin-precess back into active states in the galactic magnetic "eld and would thus become visible in the detectors which measured the SN 1987A neutrino signal. The absence of such anomalous high-energy events yields another limit [34,37] k (Dirac):1;10\k . J

(3)

The oscillation length for magnetic spin-precession does not depend on the neutrino energy so that the Earth could have been in a node of the oscillation pattern, providing a loop-hole from this constraint. If neutrinos had a small charge they would be de#ected by the galactic magnetic "eld. The absence of an energy-dependent dispersion of the SN 1987A l -signal thus leads to a limit [4,38] C e C:3;10\e J

(4)

in analogy to the well-known SN 1987A limit on m C. J

4. Big-bang nucleosynthesis Spin-#ip collisions would also populate the sterile Dirac components in the early universe and thus increase the e!ective number of thermally excited neutrino degrees of freedom at the time of big-bang nucleosynthesis. Full thermal equilibrium attains for k (Dirac)96;10\k [39,40]. J In view of the SN 1987A and globular-cluster limits this result assures us that big-bang nucleosynthesis remains undisturbed.

5. Radiative decay and Cherenkov e4ect A neutrino mass eigenstate l may decay to another one l by the emission of a photon, where the G H only contributing form factors are the magnetic and electric transition moments. The inverse

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radiative lifetime is found to be [2,3]






"k "#"e " m!m  G H GH q\" GH A m 8p G k  m!m  m  G H G , (5) "5.308 s\  k m eV G where k and e are the transition moments while "k ","k "#"e ". Radiative neutrino decays GH GH  GH GH have been constrained from the absence of decay photons of reactor l #uxes [41], the solar l #ux C C [42,43], and the SN 1987A neutrino burst [47}51]. For m ,m <m these limits can be expressed J G H as










0.9;10\ (eV/m ) Reactor (l ) , J C 0.5;10\ (eV/m ) Sun (l ) , k J C : (6) k 1.5;10\ (eV/m ) SN 1987A (all flavors) , J 1.0;10\ (eV/m ) Cosmic background (all flavors) . J The SN 1987A limit is based on the nonobservation of excess counts in the Gamma-Ray Spectrometer (GRS) on board the Solar Maximum Mission Satellite which happened to switch into calibration mode about 223 s after the neutrino burst. Therefore, as stated here the SN 1987A bound applies only to m :40 eV where the full c-ray burst would have been captured. J For higher neutrino masses one can still derive limits from SN 1987A if one takes the short GRS time window into account. Comparable limits in the higher-mass range were derived from c-ray data of the Pioneer Venus Orbiter (PVO) instrument which had a much longer exposure time [52]. For m 90.1 MeV, decay photons would still arrive years after SN 1987A. In 1991 the COMPTEL J instrument aboard the Compton Gamma Ray Observatory looked at the SN 1987A remnant for about 0.68;10 s, providing the most restrictive limits in this mass range [53,54]. However, neutrinos with such large masses no longer seem particularly plausible so that we forego a detailed discussion of these bounds. They are di$cult to represent in a compact form because neutrinos with a mass exceeding about 30 eV must have nonstandard invisible decay channels in order to conform to well established upper limits on the cosmic matter density. Therefore, the radiative decay limits depend on the nonradiative decay width, introducing an unavoidable further parameter * see [5] for a detailed discussion. The decay of cosmic background neutrinos would contribute to the di!use photon backgrounds, excluding the shaded areas in Fig. 2. The dark-shaded area was added only very recently by the observation of TeV c-rays from the active galaxies Markarian 421 and 501. The lack of #ux attenuation by the pair process c c Pe>e\ has provided new limits on the cosmic density 24   of infrared photons and thus to neutrino radiative decays [45]. The envelope of these limits is well approximated by the dashed line in Fig. 2, corresponding to the bottom line in Eq. (6). More restrictive limits obtain for certain neutrino masses above 3 eV from the absence of emission features from several galaxy clusters [55}57] and from the observation of singly ionized helium in the di!use intergalactic medium [58]. For low-mass neutrinos, the m phase-space factor in Eq. (5) is so punishing that the globularJ cluster limit is the most restrictive one for m below a few eV, i.e. in the mass range which today J

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Fig. 2. Astrophysical limits on neutrino transition moments. The light-shaded background-radiation limits are from Ressell and Turner [44], the dark-shaded ones from Biller et al. [45] and Ra!elt [46], the dashed line is the approximation formula in Eq. (6), bottom line.

appears favored from neutrino oscillation experiments. Turning this around, the globular-cluster limit implies that radiative decays of low-mass neutrinos do not seem to have observable consequences. Another form of `radiative decaya is the Cherenkov e!ect lPl#c involving the same initial- and "nal-state neutrino. This process is kinematically allowed for photons with u!k(0, which obtains in certain media (for example air or water) or in external magnetic "elds. The neutrino may have an anomalous dipole moment, but there is also a standard-model photon coupling induced by the medium or the external "eld. Thus far it does not look as if the neutrino Cherenkov e!ect had any strong astrophysical or laboratory signi"cance * for a review of the literature see [59].

6. Laboratory limits Laboratory limits on neutrino dipole moments arise from measurements of the l}e-scattering cross section. The current limits are



1.8;10\k

k ( 7.4;10\k J 5.4;10\k

for l [60] , C for l [61] , I for l [62] , O

(7)

see also the Review of Particle Properties [63]. These limits apply also to electric dipole moments and to electric and magnetic transition moments. For example, the limit on k C applies to all J transition moments which connect l to another #avor. It should be noted, however, that the C scattering amplitudes from electric and magnetic dipole moments can interfere destructively, providing a loop-hole from these limits [64].

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An improvement of the k C limit to something like 3;10\k is to be expected from the MUNU J experiment which has been installed at the Bugey nuclear reactor [65,66]. Other projects aiming at a similar sensitivity are in a much earlier stage of development [67}69]. The only electromagnetic from factor for which laboratory measurements provide more restrictive limits than astrophysical arguments is the l electric charge. If electric charge conservation is C assumed to hold in b processes such as neutron decay, one "nds (8) e C:3;10\e . J This limit is based on a bound for the neutron charge of e "(!0.4$1.1);10\e [70] and on L the neutrality of matter which was found to be e #e "(0.8$0.8);10\e [71]. N C 7. Conclusions The recent evidence for neutrino oscillations from the solar and the atmospheric neutrino anomaly and from the LSND experiment indicate that the neutrino mass di!erences are very small, at most in the eV range. Moreover, the absolute neutrino mass scale cannot exceed a few eV as indicated by the tritium decay limits on the l mass and by cosmological arguments. Therefore, C speculations about neutrino masses far in excess of a few eV are becoming more and more unattractive. If neutrino masses are indeed that small, it is no longer possible to invoke threshold e!ects to avoid the stellar plasmon-decay limits on neutrino dipole moments and electric charges. Moreover, Fig. 2 illustrates that for neutrino masses below about 2 eV the stellar limits on transition moments are more restrictive than those from searches for radiative decays. Turning this around, if neutrino masses are indeed below a few eV one cannot expect neutrino radiative decays to have any observable consequences. The current round of experiments to improve the laboratory limits on k C will not be able to J come even close to the globular-cluster limit so that a positive discovery would indicate extremely serious problems with our understanding of low-mass stars. Barring this unlikely possibility, one cannot hope to discover neutrino dipole moments anytime soon in a laboratory experiment. On the other hand, unless a completely new argument is put forth, the stellar-evolution limits have probably gone about as far as they can, although one could still achieve a signi"cant reduction of their uncertainties. The possibility that neutrino dipole or transition moments in the general 10\k range play an important role in astrophysical environments with large magnetic "elds cannot be ruled out in the foreseeable future. Scenarios with magnetic spin-#avor oscillations in the Sun, supernovae, active galactic nuclei, or the early universe are in no danger of being ruled out anytime soon!

Acknowledgements Partial support by the Deutsche Forschungsgemeinschaft under grant No. SFB-375 is acknowledged.

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Appendix A: Helioseismological limit The interior of the Sun is a nonrelativistic plasma where the neutrino energy-loss rate per unit volume from transverse-plasmon decay is approximately given by [5]



a u/4p Charge , J . Dipole Moment , (A.1) Q"(8f /3p)¹; (k/2)(u/4p) J .  (C G/a)(u/4p) Standard Model . 4 $ . Here, f +1.202 refers to the Riemann Zeta function, a "e /4p is the neutrino "ne-structure  J J constant, C the vector}current coupling constant between neutrinos and electrons, G the Fermi 4 $ constant, and u "4pan /m is the plasma frequency with n the electron density. Natural units . C C C with "c"k "1 are used. Longitudinal-plasmon decay is not important for these conditions. Integrating these energy-loss rates over a standard solar model yields ¸ "(e /e)3.2;10 J J ¸ and ¸ "(k /k )6.0;10 ¸ (solar luminosity ¸ ), respectively. Helioseismology requires > J J > > that a new energy-loss channel of the Sun does not exceed about 10% ¸ [30], leading to > e :6;10\e and k :4;10\k . J J The globular-cluster limit on e is not much more restrictive than this result, while one gains a lot J for k . The reason is that the energy-loss rate per unit mass for the e case does not depend on the J J matter density, while for the k case it depends linearly on o. The cores of low-mass red giants before J helium ignition are about 10 times denser than the Sun, explaining the improvement of the k limit. J References [1] W. Pauli, Public letter to the group of the Radioactives at the district society meeting in TuK bingen, in: K. Winter (Ed.), Neutrino Physics, Cambridge University Press, Cambridge, 1930. [2] K. Winter (Ed.), Neutrino Physics, Cambridge University Press, Cambridge, 1991. [3] R.N. Mohapatra, P. Pal, Massive Neutrinos in Physics and Astrophysics, World Scienti"c, Singapore, 1991. [4] J.N. Bahcall, Neutrino Astrophysics, Cambridge University Press, Cambridge, 1989. [5] G.G. Ra!elt, Stars as Laboratories for Fundamental Physics, University of Chicago Press, Chicago, 1996. [6] M.B. Voloshin, M.I. Vysotskimy , Yad. Fiz. 44 (1986) 845 [Sov. J. Nucl. Phys. 44 (1986) 544]. [7] L.B. Okun, Yad. Fiz. 44 (1986) 847 [Sov. J. Nucl. Phys. 44 (1986) 546]. [8] M.B. Voloshin, M.I. Vysotskimy , L.B. Okun, Yad. Fiz. 44 (1986) 677 [Sov. J. Nucl. Phys. 44 (1986) 440]. [9] M.B. Voloshin, M.I. Vysotskimy , L.B. Okun, Zh. Eksp. Teor. Fiz. 91 (1986) 754; (E) ibid. 92 (1987) 368 [Sov. Phys. JETP 64 (1986) 446; (E) ibid. 65 (1987) 209]. [10] L.B. Okun, Yad. Fiz. 48 (1988) 1519 [Sov. J. Nucl. Phys. 48 (1988) 967]. [11] S.I. Blinnikov, L.B. Okun, Pis'ma Astron. Zh. 14 (1988) 867 [Sov. Astron. Lett. 14 (1988) 368]. [12] G.G. Ra!elt, Astrophys. J. 365 (1990) 559; Phys. Rev. Lett. 64 (1990) 2856. [13] G.G. Ra!elt, A. Weiss, Astron. Astrophys. 264 (1992) 536. [14] M. Haft, G.G. Ra!elt, A. Weiss, Astrophys. J. 425 (1994) 222; (E) ibid. 438 (1995) 1017. [15] G. Walther, Phys. Rev. Lett. 79 (1997) 4522. [16] P.A. Sturrock, G. Walther, M.S. Wheatland, Astrophys. J. 491 (1997) 409; ibid. 507 (1998) 978. [17] J. Pulido, Phys. Rep. 211 (1992) 211; Phys. Rev. D 48 (1993) 1492; Phys. Lett. B 323 (1994) 36; Z. Phys. C 70 (1996) 333. [18] E.Kh. Akhmedov, Phys. Lett. B 348 (1995) 124; hep-ph/9705451, 1997. [19] M.M. Guzzo, H. Nunokawa, Astropart. Phys. 12 (1999) 87. [20] H. Athar, J.T. Peltoniemi, A.Yu. Smirnov, Phys. Rev. D 51 (1995) 6647. [21] T. Totani, K. Sato, Phys. Rev. D 54 (1996) 5975.

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Physics Reports 320 (1999) 329}339

Matter}antimatter asymmetry and neutrino properties Wilfried BuchmuK ller *, Michael PluK macher
Deutsches Elektronen-Synchrotron DESY, Gruppe Theorie, Notkestrasse 85, D-22603 Hamburg, Germany
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia PA 19104, USA

Abstract The cosmological baryon asymmetry can be explained as remnant of heavy Majorana neutrino decays in the early universe. We study this mechanism for two models of neutrino masses with a large l !l mixing I O angle which are based on the symmetries S;(5);;(1) and S;(3) ;S;(3) ;S;(3) ;;(1) , respectively. In $ A * 0 $ both cases B!¸ is broken at the uni"cation scale K . The models make di!erent predictions for the %32 baryogenesis temperature and the gravitino abundance.  1999 Elsevier Science B.V. All rights reserved. PACS: 14.20.!c; 13.35.Hb

1. Baryogenesis and lepton number violation The cosmological matter}antimatter asymmetry, the ratio of the baryon density to the entropy density of the universe, > "(n !n M )/s"(0.6!1);10\ ,

(1)

can in principle be understood in theories where baryon number, C and CP are not conserved [1]. The presently observed value of the baryon asymmetry is then explained as a consequence of the spectrum and interactions of elementary particles, together with the cosmological evolution. A crucial ingredient of baryogenesis is the connection between baryon number (B) and lepton number (¸) in the high-temperature, symmetric phase of the standard model. Due to the chiral nature of the weak interactions B and ¸ are not conserved [2]. At zero temperature this has no observable e!ect due to the smallness of the weak coupling. However, as the temperature approaches the critical temperature ¹ of the electroweak phase transition, B and ¸ violating #5 * Corresponding author. Tel.: 040-89-98-0; fax: 040-89-98-32-82. E-mail address: [email protected] (W. BuchmuK ller) 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 5 7 - 5

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processes come into thermal equilibrium [3]. These &sphaleron processes' violate baryon and lepton number by three units, *B"*¸"3 .

(2)

It is generally believed that B and ¸ changing processes are in thermal equilibrium for temperatures in the range ¹#5&100 GeV(¹(¹1.&&10 GeV .

(3)

The non-conservation of baryon and lepton number has a profound e!ect on the generation of the cosmological baryon asymmetry. Eq. (2) suggests that any B#¸ asymmetry generated before the electroweak phase transition, i.e., at temperatures ¹'¹#5, will be washed out. However, since only left-handed "elds couple to sphalerons, a non-zero value of B#¸ can persist in the high-temperature, symmetric phase if there exists a non-vanishing B!¸ asymmetry. An analysis of the chemical potentials of all particle species in the high-temperature phase yields the following relation between the baryon asymmetry > and the corresponding ¸ and B!¸ asymmetries >* and > \*, respectively [4], a > , > "a> \*" a!1 *

(4)

where a is a number O(1). In the standard model with three generations and two Higgs doublets one has a"  .  We conclude that B!¸ violation is needed if the baryon asymmetry is generated before the electroweak transition, i.e. at temperatures ¹'¹ &100 GeV. In the standard model, as well as #5 its supersymmetric version and its uni"ed extensions based on the gauge group SU(5), B!¸ is a conserved quantity. Hence, no baryon asymmetry can be generated dynamically in these models. The remnant of lepton number violation at low energies is an e!ective *¸"2 interaction between lepton and Higgs "elds, L

" f l2 H C l H #h.c. *  GH *G  *H 

(5)

Such an interaction arises in particular from the exchange of heavy Majorana neutrinos. In the Higgs phase of the standard model, where the Higgs "eld acquires a vacuum expectation value 1H 2"v , it gives rise to Majorana masses of the light neutrinos l , l and l .   C I O At "nite temperature the *¸"2 processes described by (5) take place with the rate [5] C (¹)"(1/p)(¹/v) mG . *  J GCIO

(6)

In thermal equilibrium this yields an additional relation between the chemical potentials which implies > ">

\*

"> "0 . *

(7)

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To avoid this conclusion, the *¸"2 interaction (5) must not reach thermal equilibrium. For baryogenesis at a temperature ¹ (¹ &10 GeV, one has to require C (H " , where 1.& * 2 H is the Hubble parameter. This yields a stringent upper bound on Majorana neutrino masses, (8) mG((0.2 eV (¹ /¹ )) . 1.& J GCIO For ¹ &¹ , this bound would be comparable to the upper bound on the electron neutrino mass 1.& obtained from neutrinoless double beta decay. However, Eq. (8) also applies to the q-neutrino mass. Note, that the bound can be evaded if appropriate asymmetries are present for particles which reach thermal equilibrium only at temperatures below ¹ [6]. The connection between lepton number and the baryon asymmetry is lost if baryogenesis takes place at or below the Fermi scale [7}9]. However, detailed studies of the thermodynamics of the electroweak transition have shown that, at least in the standard model, the deviation from thermal equilibrium is not su$cient for baryogenesis [10}12]. In the minimal supersymmetric extension of the standard model (MSSM) such a scenario appears still possible for a limited range of parameters [7}9].

2. Decays of heavy Majorana neutrinos Baryogenesis above the Fermi scale requires B!¸ violation, and therefore ¸ violation. Lepton number violation is most simply realized by adding right-handed Majorana neutrinos to the standard model. Heavy right-handed Majorana neutrinos, whose existence is predicted by all extensions of the standard model containing B!¸ as a local symmetry, can also explain the smallness of the light neutrino masses via the see-saw mechanism [13,14]. The most general Lagrangian for couplings and masses of charged leptons and neutrinos reads (9) L "!h e l H !h l l H !h lA l R#h.c. JGH 0G *H   PGH 0G 0H 7 CGH 0G *H  The vacuum expectation values of the Higgs "eld 1H 2"v and 1H 2"v "tan bv generate      Dirac masses m and m for charged leptons and neutrinos, m "h v and m "h v , respectively, C " C C  " J  which are assumed to be much smaller than the Majorana masses M"h 1R2. This yields light and P heavy neutrino mass eigenstates lKKRl #lA K, NKl #lA , * * 0 0 with masses 1 m2KH, m KM . m K!KRm , J "M "

(10)

(11)

Here K is a unitary matrix which relates weak and mass eigenstates. The right-handed neutrinos, whose exchange may erase any lepton asymmetry, can also generate a lepton asymmetry by means of out-of-equilibrium decays. This lepton asymmetry is then partially transformed into a baryon asymmetry by sphaleron processes [15]. The decay width of the heavy

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neutrino N reads at tree level, G C "C(N PH #l)#C(N PHA #lA)"1/8p(h hR) M . (12) "G G  G  J J GG G From the decay width one obtains an upper bound on the light neutrino masses via the yields the constraint out-of-equilibrium condition [16]. Requiring C (H" " 2+ m "(h hR) v/M (10\ eV . (13)  J J    More direct bounds on the light neutrino masses depend on the structure of the Dirac neutrino mass matrix. Interference between the tree-level amplitude and the one-loop self-energy and vertex corrections yields CP asymmetries in the heavy Majorana neutrino decays. In a basis, where the right-handed neutrino mass matrix M"h 1R2 is diagonal, one obtains [17}20]. P C(N PlH )!C(N PlAHA )     e "  C(N PlH )#C(N PlAHA )     1 3 M K! Im[(h hR) ]  . (14) J J G M 16p (h hR) J J  G G Here we have assumed M (M , M , which is satis"ed in the applications considered in the    following sections. In the early universe at temperatures ¹&M the CP asymmetry (14) leads to a lepton  asymmetry [21}23], e n !n M *"i  . (15) > " * * g s H Here the factor i(1 represents the e!ect of washout processes. In order to determine i one has to solve the full Boltzmann equations [24}26]. In the examples discussed below one has iK10\210\.

3. Neutrino masses and mixings The CP asymmetry (14) is given in terms of the Dirac and the Majorana neutrino mass matrices. Depending on the neutrino mass hierarchy and the size of the mixing angles the CP asymmetry can vary over many orders of magnitude. It is therefore interesting to see whether a pattern of neutrino masses motivated by other considerations is consistent with leptogenesis. An attractive framework to explain the observed mass hierarchies of quarks and charged leptons is the Froggatt}Nielsen mechanism [28] based on a spontaneously broken ;(1) generation $ symmetry. The Yukawa couplings arise from non-renormalizable interactions after a gauge singlet "eld U acquires a vacuum expectation value,




h "g GH GH

1U2 /G>/H . K

(16)

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Table 1 Chiral charges of charged and neutral leptons with S;(5);;(1) symmetry [33] $ t G

eA 0

eA 0

eA 0

l *

l *

l *

lA 0

lA 0

lA 0

Q G

0

1

2

0

0

1

0

1

2

Here g are couplings O(1) and Q are the ;(1) charges of the various fermions with QU"!1. The GH G interaction scale K is expected to be very large, K'K . In the following we shall discuss two %32 di!erent realizations of this idea which are motivated by the recently reported atmospheric neutrino anomaly [27]. Both scenarios have a large l !l mixing angle. They di!er, however, by I O the symmetry structure and by the size of the parameter e which characterizes the #avour mixing. 3.1. S;(5);;(1) $ This symmetry has been considered by a number of authors [32]. Particularly interesting is the case with a nonparallel family structure where the chiral ;(1) charges are di!erent for the 5H-plets $ and the 10-plets of the same family [29}31]. An example of possible charges Q is given in Table 1. G The assignment of the same charge to the lepton doublets of the second and third generation leads to a neutrino mass matrix of the form [29,30],


 e e

m GH& e J e

e

1 1 1 1

v  . 1R2

(17)

This structure immediately yields a large l !l mixing angle. The phenomenology of neutrino I O oscillations depends on the unspeci"ed coe$cients O(1) [34,35]. The parameter e which gives the #avour mixing is chosen to be 1 1U2/K"e& . 17

(18)

The three Yukawa matrices for the leptons have the structure,




e e e

h & e e C e 1




e e e

e , h & e e J 1 e 1




e

e , h & e P 1 e

e e e e e

1



.

(19)

Note, that h and h have the same, non-symmetric structure. One easily veri"es that the mass C J ratios for charged leptons, heavy and light Majorana neutrinos are given by m : m : m &e : e : 1, M : M : M &e : e : 1 , C I O    m : m : m &e : 1 : 1 .   

(20) (21)

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Table 2 Chiral charges of charged and neutral leptons with S;(3) ;S;(3) ;S;(3) ;;(1) symmetry [32] A * 0 $ t G

eA 0

eA 0

eA 0

l *

l *

l *

lA 0

lA 0

lA 0

Q G

0

 

 

0

 

 

0

 

 

The masses of the two eigenstates l and l depend on unspeci"ed factors of order one, and may I O easily di!er by an order of magnitude [35]. They can therefore be consistent with the mass di!erences *mC IK4;10\!1 10\ eV [37] inferred from the MSW solution of the solar JJ neutrino problem [38,39] and *mI OK(5;10\!6;10\) eV associated with the atmospheric JJ neutrino de"cit [27]. For numerical estimates we shall use the average of the neutrino masses of the second and third family, m "(m Im O)&10\ eV. Note, that for a di!erent choice of ;(1) J J J charges the coe$cients in Eq. (17) automatically yield the hierarchy m /m &e [40].   The choice of the charges in Table 1 corresponds to large Yukawa couplings of the third generation. For the mass of the heaviest Majorana neutrino one "nds M &v/m &10 GeV .   J

(22)

This implies that B!¸ is broken at the uni"cation scale K . %32 3.2. S;(3) ;S;(3) ;S;(3) ;;(1) A * 0 $ This symmetry arises in uni"ed theories based on the gauge group E . The leptons eA , l and  0 * lA are contained in a single (1, 3, 3 ) representation. Hence, all leptons of the same generation have 0 the same ;(1) charge and all leptonic Yukawa matrices are symmetric. Masses and mixings of $ quarks and charged leptons can be successfully described by using the charges given in Table 2 [32]. Clearly, the three Yukawa matrices have the same structure,




e

e

e




e 

e 



e 

e e , h & e  e e  . h , h & e J C P e e 1 e  e  1

(23)

Note, that the expansion parameter in h is di!erent from the one in h and h . From the quark J C P masses, which also contain e and e , one infers e Ke [32]. From Eq. (23) one obtains for the masses of charged leptons, light and heavy Majorana neutrinos, m : m : m &M : M : M &e : e : 1 , C I O   

(24)

m : m : m &e : e : 1 .   

(25)

 Note, that with respect to Ref. [32], e and e have been interchanged.

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Like in the example with S;(5);;(1) symmetry, the mass of the heaviest Majorana neutrino, $ M &v/m &10 GeV , (26)    implies that B!¸ is broken at the uni"cation scale K . %32 The l !l mixing angle is mostly given by the mixing of the charged leptons of the second and I O third generation [32], sin H &(e#e . IO This requires large #avour mixing,

(27)

(1U2/K)"(e& . (28)  In view of the unknown coe$cients O(1) the corresponding mixing angle sin H &0.7 is consistent IO with the interpretation of the atmospheric neutrino anomaly as l !l oscillation. I O It is very instructive to compare the two scenarios of lepton masses and mixings described above. In the "rst case, the large l !l mixing angle follows from a nonparallel #avour symmetry. The I O parameter e, which characterizes the #avour mixing, is small. In the second case, the large l !l I O mixing angle is a consequence of the large #avour mixing e. The ;(1) charges of all leptons are the $ same, i.e., one has a parallel family structure. Also the mass hierarchies, given in terms of e, are rather di!erent. This illustrates that the separation into a #avour mixing parameter e and coe$cients O(1) is far from unique. It is therefore important to study other observables which depend on the lepton mass matrices. A particular example is the baryon asymmetry.

4. Matter antimatter asymmetry We can now evaluate the baryon asymmetry for the two patterns of neutrino mass matrices discussed in the previous section. A rough estimate of the baryon asymmetry can be obtained from the CP asymmetry e of the heavy Majorana neutrino N . A quantitative determination requires   a numerical study of the full Boltzmann equations [25,26]. 4.1. S;(5);;(1) $ In this case one obtains from Eqs. (14) and (19), e &(3/16p)e .  From Eq. (15), e&1/300 (18) and g &100 one then obtains the baryon asymmetry, H > &i10\ .

(29)

(30)

For i&0.120.01 this is indeed the correct order of magnitude. The baryogenesis temperature is given by the mass of the lightest of the heavy Majorana neutrinos, ¹ &M &eM &10 GeV . (31)   This set of parameters, where the CP asymmetry is given in terms of the mass hierarchy of the heavy neutrinos, has been studied in detail [36]. The generated baryon asymmetry does not depend

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Fig. 1. Time evolution of the neutrino number density and the lepton asymmetry in the case of the S;(5);;(1) $ symmetry. The solid line shows the solution of the Boltzmann equation for the right-handed neutrinos, while the corresponding equilibrium distribution is represented by the dashed line. The absolute value of the lepton asymmetry > is given by the dotted line and the hatched area shows the lepton asymmetry corresponding to the observed baryon * asymmetry.

on the #avour mixing of the light neutrinos. The l !l mixing angle is large in the scenario I O described in the previous section whereas it was assumed to be small in [36]. The solution of the full Boltzmann equations is shown in Fig. 1 for the non-supersymmetric case [36]. The initial condition at a temperature ¹&10M is chosen to be a state without heavy  neutrinos. The Yukawa interactions are su$cient to bring the heavy neutrinos into thermal equilibrium. At temperatures ¹&M this is followed by the usual out-of-equilibrium decays  which lead to a non-vanishing baryon asymmetry. The "nal asymmetry agrees with the estimate (30) for i&0.1. The change of sign in the lepton asymmetry is due to the fact that inverse decay processes, which take part in producing the neutrinos, are CP violating, i.e. they generate a lepton asymmetry at high temperatures. Due to the interplay of inverse decay processes and lepton number violating scattering processes this asymmetry has a di!erent sign than the one produced by neutrino decays at lower temperatures. 4.2. S;(3) ;S;(3) ;S;(3) ;;(1) A * 0 $ In this case the neutrino Yukawa couplings (23) yield the CP asymmetry e &(3/16p)e , 

(32)

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Fig. 2. Solution of the Boltzmann equations in the case of the S;(3) ;S;(3) ;S;(3) ;;(1) symmetry. A * 0 $

which correspond to the baryon asymmetry (cf. (15)) > &i 10\ .

(33)

Due to the large value of e the CP asymmetry is two orders of magnitude larger than in the case with S;(5);;(1)$ symmetry. However, washout processes are now also stronger. The solution of the Boltzmann equations is shown in Fig. 2. The "nal asymmetry is again > &10\ which now corresponds to i&10\. The baryogenesis temperature is considerably larger than in the "rst case, ¹ &M&eM&10 GeV .

(34)

The baryon asymmetry is largely determined by the parameter m  de"ned in Eq. (13) [25]. In the "rst example, one has m &m J. In the second case one "nds m &m. Since m J and m are rather similar it is not too surprising that the generated baryon asymmetry is about the same in both cases.

5. Conclusions Detailed studies of the thermodynamics of the electroweak interactions at high temperatures have shown that in the standard model and most of its extensions the electroweak transition is too weak to a!ect the cosmological baryon asymmetry. Hence, one has to search for baryogenesis mechanisms above the Fermi scale.

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Due to sphaleron processes baryon number and lepton number are related in the high-temperature, symmetric phase of the standard model. As a consequence, the cosmological baryon asymmetry is related to neutrino properties. Baryogenesis requires lepton number violation, which occurs in extensions of the standard model with right-handed neutrinos and Majorana neutrino masses. Although lepton number violation is needed in order to obtain a baryon asymmetry, it must not be too strong since otherwise any baryon and lepton asymmetry would be washed out. This leads to stringent upper bounds on neutrino masses which depend on the particle content of the theory. The solar and atmospheric neutrino de"cits can be interpreted as a result of neutrino oscillations. For hierarchical neutrinos the corresponding neutrino masses are very small. Assuming the see-saw mechanism, this suggests the existence of very heavy right-handed neutrinos and a large scale of B!¸ breaking. It is remarkable that these hints on the nature of lepton number violation "t very well together with the idea of leptogenesis. For hierarchical neutrino masses, with B!¸ broken at the uni"cation scale K%32&10 GeV, the observed baryon asymmetry > &10\ is naturally explained by the decay of heavy Majorana neutrinos. Although the observed baryon asymmetry imposes important constraints on neutrino properties, other observables are needed to discriminate between di!erent models. The two examples considered in this paper predict di!erent baryogenesis temperatures. Correspondingly, in supersymmetric models the predictions for the gravitino abundance are di!erent [40}44]. In the case with S;(5);;(1)$ symmetry, stable gravitinos can be the dominant component of cold dark matter [44]. The models make also di!erent predictions for the rate of lepton #avour changing radiative corrections. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

A.D. Sakharov, JETP Lett. 5 (1967) 24. G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36. J.A. Harvey, M.S. Turner, Phys. Rev. D 42 (1990) 3344. M. Fukugita, T. Yanagida, Phys. Rev. D 42 (1990) 1285. J.M. Cline, K. Kainulainen, K.A. Olive, Phys. Rev. Lett. 71 (1993) 2372. A.D. Dolgov, Phys. Rep. 222 (1992) 309. V.A. Rubakov, M.E. Shaposhnikov, Phys. Usp. 39 (1996) 461. S.J. Huber, M.G. Schmidt, SUSY Variants of the Electroweak Phase Transition, hep-ph/9809506. K. Jansen, Nucl. Phys. B (Proc. Suppl.) 47 (1996) 196. W. BuchmuK ller, in: V.A. Matveev et al. (Eds.), Quarks '96 (Yaroslavl, Russia, 1996), hep-ph/9610335. K. Rummukainen, Nucl. Phys. B (Proc. Suppl.) 53 (1997) 30. T. Yanagida, Workshop on uni"ed Theories, KEK Report Vol. 79-18, 1979, p. 95. M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Nieuwenhuizen, D. Freedman (Eds.), Supergravity, North-Holland, Amsterdam, 1979, p. 315. M. Fukugita, T. Yanagida, Phys. Lett. B 174 (1986) 45. W. Fischler, G.F. Giudice, R.G. Leigh, S. Paban, Phys. Lett. B 258 (1991) 45. L. Covi, E. Roulet, F. Vissani, Phys. Lett. B 384 (1996) 169. M. Flanz, E.A. Paschos, U. Sarkar, Phys. Lett. B 345 (1995) 248. M. Flanz, E.A. Paschos, U. Sarkar, Phys. Lett. B 384 (1996) 487(E). W. BuchmuK ller, M. PluK macher, Phys. Lett. B 431 (1998) 354.

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A.D. Dolgov, Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1. E.W. Kolb, S. Wolfram, Nucl. Phys. B 172 (1980) 224. E.W. Kolb, S. Wolfram, Nucl. Phys. B 195 (1982) 542(E). M.A. Luty, Phys. Rev. D 45 (1992) 455. M. PluK macher, Z. Phys. C 74 (1997) 549. M. PluK macher, Nucl. Phys. B 530 (1998) 207. Y. Fukuda et al., Super-Kamiokande Collaboration, Phys. Rev. Lett. 81 (1998) 1562. C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B 147 (1979) 277. J. Sato, T. Yanagida, Talk at Neutrino'98, hep-ph/9809307. P. Ramond, Talk at Neutrino'98, hep-ph/9809401. J. Bijnens, C. Wetterich, Nucl. Phys. B 292 (1987) 443. S. Lola, G.G. Ross, hep-ph/9902283. W. BuchmuK ller, T. Yanagida, Phys. Lett. B 445 (1999) 399. F. Vissani, JHEP 11 (1998) 025. N. Irges, S. Lavignac, P. Ramond, Phys. Rev. D 58 (1998) 035003. W. BuchmuK ller, M. PluK macher, Phys. Lett. B 389 (1996) 73. N. Hata, P. Langacker, Phys. Rev. D 56 (1997) 6107. S.P. Mikheyev, A.Y. Smirnov, Nuovo Cim. 9C (1986) 17. L. Wolfenstein, Phys. Rev. D 17 (1978) 2369. G. Altarelli, F. Feruglio, JHEP 11 (1998) 021; hep-ph/9812475. M.Yu. Khlopov, A.D. Linde, Phys. Lett. B 138 (1984) 265. J. Ellis, J.E. Kim, D.V. Nanopoulos, Phys. Lett. B 145 (1984) 181. T. Moroi, H. Murayama, M. Yamaguchi, Phys. Lett. B 303 (1993) 289. M. Bolz, W. BuchmuK ller, M. PluK macher, Phys. Lett. B 443 (1998) 209.

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Physics Reports 320 (1999) 341}354

Comments on CP, T and CPT violation in neutral kaon decays John Ellis *, N.E. Mavromatos 
CERN, Theory Division, CH-1211 Geneva 23, Switzerland
University of Oxford, Department of Physics, Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK

Abstract We comment on CP, T and CPT violation in the light of interesting new data from the CPLEAR and KTeV Collaborations on neutral kaon decay asymmetries. Other recent data from the CPLEAR experiment, constraining possible violations of CPT and the *S"*Q rule, exclude the possibility that the semileptonicdecay asymmetry A measured by CPLEAR could be solely due to CPT violation, con"rming that their data 2 constitute direct evidence for T violation. The CP-violating asymmetry in K Pe\e>n\n> recently mea* sured by the KTeV Collaboration does not by itself provide direct evidence for T violation, but we use it to place new bounds on CPT violation.  1999 Elsevier Science B.V. All rights reserved. PACS: 13.20.Eb; 11.30.Er

1. Introduction Ever since the discovery of CP violation in KP2p decay by Christenson et al. [1], its * understanding has been a high experimental and theoretical priority. Until recently, mixing in the K!KM  mass matrix was the only known source of CP violation, since it was su$cient by itself to explain the observations of CP violation in other K decays, no CP violation was seen in 1* experiments on K!, charm or B-meson decays, and searches for electric dipole moments only gave upper limits [2]. There has, in parallel, been active discussion whether the observed CP violation should be associated with the violation of T or CPT [3]. Stringent upper limits on CPT violation

* Corresponding author. E-mail address: [email protected] (J. Ellis) 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 5 8 - 7

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[4] in the K!KM  system have been given [5], in accord with the common theoretical prejudice based on a fundamental theorem in quantum "eld theory [6]. This suggests strongly that T must be violated, but, at least until recently, there was no direct observation of T violation. An indirect demonstration of T violation in neutral kaons, based on a phenomenological analysis of CPviolating amplitudes, was made in 1970 using data on the decay of long- and short-lived kaons into two neutral pions [7]. However, that analysis assumed unitarity, namely that kaons disappeared only into the observed states. The accumulation of experimental observations of CP and T violation has accelerated abruptly in the past few months. There have been two results on K, KM  decays for which interpretations as direct observations of T violation have been proposed. One is an asymmetry in pp annihilation, pp PK\n>K or K>n\KM  [8], and the other is a T-odd angular asymmetry in KPn>n\e>e\ * decay [9]. More recently, a tantalizing hint has been presented that CP may be violated at a high level in BPJ/tK decays [10]. Most recent of all, a previous measurement of direct CP violation 1 in the amplitudes for K P2p decays [11] has now been con"rmed by the KTeV Collaboration 1* [9], providing an improved determination of a second independent CP-violating experimental number, namely e/e, to test theories and to discriminate between them. One casualty of this measurement of e/e has been the superweak theory [12], according to which all CP violation should be ascribed to mass mixing in the K!KM  system. Still surviving is the Kobayashi}Maskawa model of weak charged-current mixing within the Standard Model with six quarks [13]. Indeed, the new KTeV result arrives 23 years after e/e was "rst calculated within the Kobayashi}Maskawa model [14], and it was pointed out that this would be a (di$cult) way to discriminate between this and the superweak theory, providing (at least part of ) the motivation for this experiment. Coincidentally, the value estimated there agrees perfectly with the current world average for e/e, although many new diagrams and numerical improvements have intervened [15]. The latest theoretical wisdom about the possible value of e/e within the Standard Model is consistent with the value measured, at least if the strange-quark mass is su$ciently small [16]. Thus e/e does not cry out for any extension of the Standard Model, such as supersymmetry [17], though this cannot be excluded. It is not the purpose of this article to review in any detail the potential signi"cance of the e/e measurements, or of the hint of a CP-violating asymmetry in BPJ/tK . Rather, we wish to 1 comment on the suggested interpretations of the asymmetry in pp PK\n>K and K>n\KM  [8], and of the T-odd angular asymmetry in KPn>n\e>e\ as possible direct evidence for T violation * [9]. We argue that the former can indeed be interpreted in this way, when combined with other CPLEAR data constraining the possible violation of the *S"*Q rule and CPT violation in semileptonic K decays [18,19]. We use the KPn>n\e>e\ decay asymmetry as a novel test of * CPT invariance in decay amplitudes, though one that may not yet be comparable in power with other tests of CPT. The layout of this article is as follows: in Section 2 we "rst introduce the semileptonic-decay asymmetry recently measured by CPLEAR, then in Section 3 we introduce a density-matrix description that includes a treatment of unstable particles as well as allowing for the possibility of stochastic CPT violation [20}22]. In Section 4 we apply this framework to show that the CPLEAR asymmetry cannot be due to CPT violation, and is indeed a direct observation of T violation. We also comment whether other examples of CP violation can be mimicked by CPT violation [22]. Then, in Section 5 we analyze the decay asymmetry observed by the KTeV collaboration, arguing

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that it does not have an unambiguous interpretation as a direct observation of T violation. It could not be due to CPT violation in the mass-mixing matrix, but could in principle be due to &direct' CPT violation in a decay amplitude.

2. The CPLEAR asymmetry in pp PK\p>K and K>p\KM  We "rst recall brie#y the key features of the asymmetry A observed by CPLEAR, motivating its 2 interpretation as direct evidence for T violation. The essential idea is to look for a violation of reciprocity in the rates for KPKM  and the time-reversed reaction KM PK, denoted by P M and )) P M , respectively, as expressed in the asymmetry )) A ,!(P M !P M )/(P M #P M ) . )) )) )) 2 ))

(1)

CPLEAR has the unique capability to tag the initial K or KM  by observing an accompanying K!n8 pair in a pp annihilation event. However, it is also necessary to tag the K or KM  at some later time, which CPLEAR accomplishes using semileptonic decays, and constructing the observable asymmetry [8] R[n>K\n>e\l]!R[n\K>n\e>l] A ,! R[n>K\n>e\l]#R[n\K>n\e>l]

(2)

where rates are denoted by R. If one assumes the *S"*Q rule, whose validity has been con"rmed independently by CPLEAR (see below), then Eq. (2) may be re-expressed as P M (q)BR[KPn\e>l]!P M (q)BR[KM Pn>e\l] )) A " )) P M (q)BR[KPn\e>l]#P M (q)BR[KM Pn>e\l] )) ))

(3)

where decay branching ratios are denoted by BR. In our discussion below, we consider both the cases where the *S"*Q rule is assumed and where it is relaxed. If one assumes CPT invariance in the semileptonic-decay amplitudes, as was done in the CPLEAR analysis [8], then A "A and the asymmetry observed by CPLEAR can be interpreted 2 as T violation. Some doubts about this interpretation have been expressed [24], apparently based on concerns about the inapplicability of the reciprocity arguments of [25] to unstable particles. We do not believe this to be a problem, since the analysis of [25] can be extended consistently to include unstable particles [22,23,26]. However, it has also been proposed [26] that one might be able to maintain T invariance, P M "P M , interpreting the asymmetry observed by CPLEAR instead as CPT violation in the )) )) semileptonic-decay amplitudes [26]. This interpretation of the CPLEAR result would be more exciting than the conventional one in terms of T violation. It was suggested in [26] that this hypothesis of CPT violation could be tested in the semileptonic decays K Pnll. However, the 1

 A recent theoretical discussion using the *S"*Q rule and CPT invariance is given in [23].

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hypothesis of [26] can, in fact, already be excluded by other published CPLEAR data, as we see below.

3. Density-matrix formalism Before discussing this in more detail, we review the density-matrix formalism [22], which is a convenient formalism for treating unstable particles, and enables us to present a uni"ed phenomenological analysis including also the possibility of stochastic CPT violation associated with a hypothetical open quantum-mechanical formalism associated with some approaches to quantum gravity [27,20,28,21]. In fact, as we recall below, this formalism has already been used in the Appendix of [22] to discard the possibility that CP violation in the neutral-kaon system could be &mimicked' by the CPT-violating mass-matrix parameter d within conventional quantum mechanics. As we discuss later, this was possible only if Re(d)&(1.75$0.7);10\ .

(4)

This analysis is also reviewed brie#y below, taking into account recent data of the CPLEAR collaboration [18] on Re(d), which were not available at the time of writing of [22], and exclude possibility (4). When one considers an unstable-particle system in isolation, without including its decay channels, its time-evolution is non-unitary, so one uses a non-Hermitean e!ective Hamiltonian: HOHR. The temporal evolution of the density matrix, o, is given within the conventional quantum-mechanical framework by R o"!i(Ho!oHR) . (5)  In the case of the neutral-kaon system, the phenomenological Hamiltonian contains the following Hermitean (mass) and anti-Hermitean (decay) components: (M#dM)!i(C#dC) MH !iCH       , (6) (M!dM)!i(C!dC) M !iC       in the (K, KM ) basis. The dM and dC terms violate CPT. Following [20], we de"ne components of o and H by






H"

o,o p , H,h p , a"0, 1, 2, 3 , (7)  ? ?  ? ? in a Pauli p-matrix representation: since the density matrix must be Hermitean, the o are real, but ? the h are complex in general. @ We may represent conventional quantum-mechanical evolution by R o "H o , in the (K, KM )  ? ?@ @ basis and allowing for the possibility of CPT violation, where




Im h



Im h



Im h Im h   H , ?@ Im h Re h   Im h !Re h  



Im h Im h   !Re h Re h   . Im h !Re h   Re h Im h  

(8)

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It is convenient for the rest of our discussion to transform to the K "(1/(2) (KGKM ) basis,  corresponding to p p , p !p , in which H becomes     ?@ !C !dC !Im C !Re C    !C !2Re M !2Im M !dC   .  (9) H " ?@ !Im C 2Re M !C !dM   !Re C !2Im M dM !C   The corresponding equations of motion for the components of o in the K basis are given in [22].  The CP-violating mass-mixing parameter e and the CPT-violating mass-mixing parameter d are given by






e"(Im M )/(1/2"*C"#i*m)""e"e\ (, 

d"! (!1/2dC#idM)/(1/2"*C"#i*m) .  (10)

One can readily verify [22] that o decays at large t to o&e\C*R






1

eH#dH

e#d

"e#d"

,

(11)

which has a vanishing determinant, thus corresponding to a pure long-lived mass eigenstate K , * whose state vector is "K 2J(1#e!d)"K2!(1!e#d)"KM 2 . * Conversely, in the short-time limit a K state is represented by 1 "e!d" e!d o&e\C1R , eH!dH 1






(12)

(13)

which also has zero determinant and hence represents a pure state "K 2J(1#e#d)"K2#(1!e!d)"KM 2 . (14) 1 Note that the relative signs of the d terms have reversed between (11) and (13): this is the signature of mass-matrix CPT violation in the conventional quantum-mechanical formalism, as seen in the state vectors (12) and (14). The di!erential equations for the components of o may be solved in perturbation theory in "e" and the new parameters dMY ,dM/"*C",

dCY ,dC/"*C" .

(15)

 We follow here the conventions of [22], which are related to the notation used elswhere [29] for CP- and CPT-violating parameters by e"!eH , d"!DH, with H denoting complex conjugation. Thus the superweak angle +

de"ned in [29] is related to the angle in (10) by " !n, so that tan "tan "2*m/"*C".   

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To "rst order, one "nds [22] o"!2"X""o (0)"[e\C*R cos( ! ! )!e\CR cos(*mt# ! ! )] ,   6Y  6Y  o"!2"X""o (0)"[e\C1R cos( # # )!e\CR cos(*mt! ! ! )] , 6  6    o"o (0)"X"e\ (>(6[e\C*R!e\C> KR ]#o (0)"X"e (\(6Y[e\C1R!e\C> KR ] ,    where the two complex constants X and X are de"ned by: Y , Y /("e"# cos dCY ) , X""e"# cos dCY #i cos dM tan "cos dM  6  Y , Y /("e"! cos dCY ) . tan "cos dM X""e"! cos dCY #i cos dM 6Y   The special case that occurs when dM"0 and "e""0, namely

(16)

(17)

dC'0 : "0, "p 6 6Y (18) dC(0 : "p, "0 . 6 6Y will be of particular interest for our purposes. With the results for o through "rst order, and inserting the appropriate initial conditions [22], we can immediately write down expressions for various observables [22] of relevance to CPLEAR. The values of observables O are given in this density-matrix formalism by expressions of the G form [20] 1O 2,Tr[O o] , (19) G G where the observables O are represented by 2;2 Hermitean matrices. Those associated with the G decays of neutral kaons to 2p, 3p and pll "nal states are of particular interest to us. If one assumes the *S"*Q rule, their expressions in the K basis are  0 0 1 0 , O J , O J p p 0 1 0 0





1 1 , O \> J p J J 1 1







1 !1 O >\J . p J J !1 1

(20)

which constitute a complete Hermitean set. We consider later the possible relaxation of the *S"*Q rule, and also the possibility of direct CPT violation in the observables (20), which would give them di!erent normalizations. The small experimental value of e/e would be taken into account by di!erent magnitudes for O in the charged and neutral modes, but we can neglect this p re"nement for our purposes. In this formalism, pure K or KM  states, such as those provided as initial conditions in the CPLEAR experiment, are described by the following density matrices:




1 1 1 , o " ) 2 1 1






!1 1 1 o M " . ) 2 !1 1

(21)

We note the similarity of the above density matrices (21) to the representations (20) of the semileptonic decay observables, which re#ects the strange-quark contents of the neutral kaons and our assumption of the validity of the *S"*Q rule: K U s Pu l>l, KM  U sPul\l.

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347

4. Interpretation of the CPLEAR asymmetry In the CPLEAR experiment [8], the generic quantities measured are asymmetries of decays from an initially pure K beam as compared to the corresponding decays from an initially pure KM  beam R(KM  PfM )!R(K Pf ) R R , (22) A(t)" R(KM  PfM )#R(K Pf ) R R where R(K Pf ),Tr[O o(t)] denotes the decay rate into a "nal state f, starting from a pure R D K at t"0: o(t"0) is given by the "rst matrix in Eq. (21), and correspondingly, R(KM PfM ), Tr[O M o(t)] denotes the decay rate into the conjugate state fM , starting from a pure KM  at t"0: D o(t"0) is given by the second matrix in Eq. (21). Several relevant asymmetries were de"ned in [22], including A (already introduced above), A , A and A . We discuss below their possible roles 2 !.2 p p in discriminating between CP- and CPT-violating e!ects, in particular when CPT violation is invoked so as to mimic CP violation whilst preserving T invariance [26]. In order to parametrize a possible CPT-violating di!erence in semileptonic-decay amplitudes as postulated there, we de"ne y: 1n>e\l"T"KM 2,(1#y)1n\e>l"T"K2

(23)

and we assume that y is real, which is justi"ed if the amplitude is T invariant [29]. We assume this here because the purpose of this analysis is to test the hypothesis [26] that the CPLEAR asymmetry can be reproduced by CPT violation alone, retaining T invariance in the mixing: P M "P M . Another important point [8] is the independence of the asymmetry A measured at 2 )) )) late times of any possible violation of *S"*Q rule. As seen from [8], violations of this rule may be taken into account simply by introducing the combination y ,y#2Re(x ) \ where, in the notation of [29,8], x parametrizes violations of the *S"*Q rule: \ 1n>e\l"T"K2,c#d, 1n\e>l"T"KM 2,cH!dH

(24)

(25)

and x,(cH!dH)/(a#b), x H,(c#d)/(aH!bH), and x ,(x$x )/2. Again, the hypothesis of ! T invariance implies the reality of x, x , x [29]. If one considers violations of *S"*Q rule, one ! should take appropriate account of the additional decay modes (25) in A (3). In the density-matrix formalism, yO0 corresponds to a di!erence in normalization between the semileptonic observables O introduced in (20). The analysis of [26], extended in the above pJJ straightforward way to take into account of possible violations of the *S"*Q rule, shows that, if one imposes reciprocity, then A K!y to lowest order in y .

(26)

 For clarity and completeness, we note the following relation between the quantity y de"ned above and the quantity y de"ned in [8]: y"2y,!2b/a to lowest order in y and for real a, b [29]: 1n>e\l "T"KM 2"aH!bH, 1n\e>l"T"K2"a#b.

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To make contact with the experimental measurement of the CPLEAR collaboration, one should take into account the di!erent normalizations of the K and KM  #uxes at the production point. Because of this e!ect, the measured asymmetry [8] becomes A"A !y . 2 The measured [8] value of this asymmetry is

(27)

AK(6.6$1.3 );10\ . (28) 2   If this experimental result were to be interpreted as expressing CPT violation but T invariance, then y should have the value y "!(3.3$0.7);10\ .

(29)

Such a scenario is excluded by the current CPLEAR value of y [19]. The late-time asymmetry measured by CPLEAR can be expressed as [8] AK4Re(e)!2Re(y ) . 2 This enables a stringent upper limit to be placed [19]

(30)

y "Re(y)#Re(x )"(0.2$0.3 );10\ . (31)   \   Therefore, the CPT-violating but T-conserving hypothesis is conclusively excluded independently of any assumption about the validity of the *S"*Q rule. As a side-remark, we comment on the e!ect of y on the CPT-violating width di!erence dC, assuming the validity of the *S"*Q rule (x "0), which is supported by [19]. Using \ C* "0.39;1/q K8;10\ GeV and the value (29) for y, and neglecting any possible other  * CPT-violating di!erences in decay rates, we "nd dCK1.06;10\ GeV

(32)

which makes the following contribution to Re(d): 2Re(d)"dC"*C "/("*C "#4"*m ")"(dC cos / "*C ")K6.8;10\ ,

(33)

where we have used K43.493 mod p. This contribution is far below the present experimental sensitivity discussed below. Next, we comment on the possibility that what we usually regard as CP violation in the mass matrix is actually due to CPT violation. In such a case, one would have to set "e"P0 and make the following choices for the CPT-violating mixing parameters mimic CP violation:

dM"0,

dCY P2"e"/cos ,

(34)

On account of (18), then, the observable A would have the following time-independent "rst-order 2 expression: A "2"X" cos( ! )#2"X" cos( # )"4"e" cos , (35) 2 6Y 6 which is identical to the conventional case of CPT symmetry. However, this is not the case for all observables, for instance the A asymmetry, de"ned by setting f"p\e>l, fM "p>e\l in Eq. (22). !.2

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In particular, one has the following asymptotic formula for A : !.2 Y Y A P4 sin cos dM!2 cos dC , !.2 which would yield the following asymptotic prediction under the &mimic' assumption (34)

349

(36)

A P!4"e" cos , (37) !.2 to be contrasted with the standard result that A "0 in the absence of CPT violation. !.2 For comparison with experimental data of CPLEAR, it is useful to express the conventional CPT-violating parameter d (10) in terms of dCY : Im(d)"!dCY sin cos . (38) Re(d)"dCY cos ""e" cos '0,   The experimental asymmetry A (27), then, would be obtained upon the identi"cation of A 2 2 in (35) with A , A"4Re(d)!y (39) 2 Note that, in principle, such a situation is consistent with the experimental data, given that the combination A#y "4Re(d)'0. Taking into account (28), (31) and (39) we observe that the 2 mimic requirement would imply Re(d) &(1.75$0.7);10\ . (40)

 However, the CPLEAR Collaboration has measured [18] Re(d) using the asymptotic value of the asymmetry A : B RM !R (1#4Ree ) RM !R (1#4Ree ) > * , \ *# \ (41) A, > B RM #R (1#4Ree ) RM #R (1#4Ree ) \ > * > \ * which asymptotes at large times to !8Re(d), independently of any assumption on the *S"*Q rule: Re(d)K(3.0$3.3 $0.6 );10\ , (42)    in apparent con#ict with (40). The fact that the CP violation seen in the mass matrix cannot be mimicked by CPT violation [4] has been known for a long time. The possible magnitude of CPT violation is constrained in particular by the consistency between

and the superweak phase . However, it is possible to >\  mimic CP violation in any particular observable by a suitable choice of d. For example, as was shown in [22], the standard superweak result for A may be reproduced by setting "e"P0 and p using (34), which give "X"P"e" and "0. The standard CP-violating result for A may also 6 p obtained with the choices (34) [22], which give "X"P"e" and "p, since tan( !p)"tan . But, 6Y as already emphasized, the dynamical equations determining the density matrix prevent all observables from being mimicked in this way: this is what we found above with the A observable !.2 (37), to be contrasted with the standard result A "0. Moreover, as mentioned above, the mimic !.2 hypothesis is excluded by the recent CPLEAR result (42). It was also pointed out previously [22,30] that deviations from conventional closed-system quantum mechanics of the type discussed in [20], which lead to stochastic CPT violation, also cannot account for the CP violation observed in the neutral kaon system. We remind the reader

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that generic possible deviations from closed-system quantum-mechanical evolution in the neutral kaon system } which might arise from quantum gravity or other stochastic forces } may be described by the three real parameters a, b, c of [20], if one assumes energy conservation and dominance by *S"0 stochastic e!ects. These parameters lead to entropy growth, corresponding to the appearance of an arrow of time and violation of CPT [31], as has sometimes been suggested in the context of a quantum theory of gravity. However, this CPT violation cannot be cast in the conventional quantum-mechanical form discussed above. The most stringent bounds on the stochastic CPT-violating parameters a, b, c have been placed by the CPLEAR collaboration [32]. They are not far from the characteristic magnitude O(M /M ), where the Planck mass ) . M K10 GeV, near the scale at which such e!ects might "rst set in [21] if they are due to . quantum-gravitational e!ects.

5. The KTeV asymmetry in K Pe\e>p\p> and its interpretation * Subsequent to the CPLEAR analysis, the KTeV Collaboration has reported [9] a novel measurement of a T-odd asymmetry in the decay of K Pe\e>p\p>. Since incoming and * outgoing states are not exchanged in the KTeV experiment, unlike the CPLEAR measurement comparing KM PK and KPKM  transitions, it cannot provide direct evidence for T violation. However, it is interesting to discuss the information this measurement may provide about CP, T and CPT symmetry. This decay has previously been analyzed theoretically in [33], assuming CPT symmetry. The decay amplitude was decomposed as M(K Pp>p\e>e\)"M #M #M #M4#M *  + # 1" !0

(43)

and the various parts of the amplitude (43) have the following interpretations: E M is the amplitude for the Bremsstrahlung process related to the standard CP-violating  K P2p amplitude, violating CP just like the conventional e parameter. This amplitude is * proportional to a coupling constant [33] g "g e B+) ,  >\

(44)

where g is the conventional CP-violating parameter, whose phase

is that of K Pp>p\: >\ >\ * d (M ) is the relevant I"0p>p\ phase shift.  ) E M is the magnetic-dipole contribution to the amplitude, which is CP-conserving. The + corresponding coupling constant has a non-trivial phase [33]: g "i"g "e BKpp>BP , + +

(45)

where d is the pp P-wave phase shift. The amplitude is invariant under CPT if du"0, leaving  the prefactor i as a consequence of CPT invariance. The estimate "g ""0.76 is given in [33]. + E M denotes the electric-dipole contribution. It is CP-conserving, and its coupling constant # g has been computed in [33]. Its phase is related to that of g via arg(g /g )K . # + # + >\

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E M4 is the contribution originating in the short-distance Hamiltonian describing the transition 1" sdM Pe>e\. Its coupling constant has been calculated in the Standard Model [33], with the result (46) g "i(5;10\)(2(M /f )e BKpp , ) p 1" where f is the pion decay constant. One could in principle introduce CPT violation into this p amplitude by allowing A(KM Pp>p\e>e\)OA(KPp>p\e>e\). As seen in (46), these amplitudes may be related to M M and M , respectively, which could be di!erent if CPT is violated. ) ) E M denotes the CP-conserving contribution due to a "nite charge radius of the K. Its coupling !0 g has the phase of K Pp>p\. . 1 The KTeV [9] Collaboration's measurement is of a CP-violating asymmetry A in the angle U between the vectors normal to the e\e> and p>p\ planes [33], which is related to the particle momenta by: sin U cos U"g ;g ) ((p #p )/("p #p "))(g ) g ) , (47) p > \ > \ p where the unit vectors g are de"ned as g ,k ;k /"k ;k " and g ,(p ;p )/("p ;p "), with p > \ > \ p > \ > \ k the lepton momenta and p the pion momenta. The observable is a CP asymmetry A of the ! ! process, which we shall discuss below. The U distribution dC/dU may be written in the following generic form [33]: dC "C cos U#C sin U#C cos U sin U ,    dU

(48)

where the last term changes sign under the CP transformation and is T-odd, i.e., it changes sign when the particle momenta are reversed. However, it clearly does not involve switching &in' and &out' states, and so is not a direct probe of T violation. A detailed functional form for C is given in [33]. Following the above discussion of the various  terms in the decay amplitude (43), this term is interpreted [33] in terms of the dominant Bremsstrahlung, magnetic-dipole and electric-dipole contributions. For our purposes, it is su$cient to note that it involves the coupling constant combinations Re(g gH ) and Re(g gH ), which + 0 + # involve amplitudes with di!erent CP properties, and hence violate CP manifestly. It depends, in particular, on the phase

of the conventional CP-violating K Pp>p\ decay amplitude, via >\ * the K admixture in the K wave function, which enters in the M1 amplitude for K Pp>p\c. The  * * following is the generic structure of the integrated asymmetry measured by KTeV [33]: p(dC/dU) dU!p (dC/dU) dU p KA cos H #A cos H "g /g " , A"      # + p(dC/dU) dU#p (dC/dU) dU p  where H , #d !dM !p/2!du mod p,  >\  

H , !p/2!du mod p  >\

(49)

(50)

 It is generally agreed that "nal-state electromagnetic interactions can be neglected for present purposes. The KTeV collaboration has recently reported [34] a null asymmetry in the angle between the p>p\ and e>e\ planes in the Dalitz decay K Pp>p\(pPe>e\c). This provides a nice check on the experimental technique, but does not test directly the * structure of the "nal-state interactions, since the p decays outside the Coulomb "elds of the p>p\ pair.

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and dM and du are averages of the pp P-wave phase shift and du, respectively, in the region  m (m . Numerical estimates of the quantities A in terms of the di!erent couplings in (43) p )  were given in [33]: A K0.15, A K0.38 ,   leading to the following prediction for A:

(51)

AK0.15 sin[ #d (m )!dM ] (52) >\  )  if the CPT-violating phase du"0. Using the experimental values d K403, dM K103 and  

K433, (52) becomes >\ AK0.14 (53) As already mentioned, the experimental value "(13.5$(2.5) $(3.0) )% (54)     agrees very well with the theoretical prediction (53) obtained assuming the CPT-violating phase du"0. We now analyze how well this measurement tests CPT, and assess how this test compares with other tests. Consider "rst the Bremsstrahlung contribution: as mentioned above, the coupling g has a phase . In principle, CPT violation in the neutral-kaon mass matrix could shift this
 >\ phase away from its superweak value by an amount d : A

(55) "m !m M "K2*m("g "/sin )"d " , ) >\ ) where (as always) we neglect e!ects that are O(e), and we recall that "g "K"e"/ cos d K"e". The >\ best limit on such a mass di!erence is now provided by the CPLEAR experiment [5] "m !m M "43.5;10\ GeV (95% CL) . (56) ) ) The limit (56) determines "d ":0.863, whereas a combination of previous data from the NA31, E731 and E773 Collaboration yields [22] d :(!0.75$0.79)3. Such a phase change "d " would change A by an amount "dA":10\, far smaller than the experimental error in (54), and also much smaller than the likely theoretical uncertainties. We consider next the magnetic-dipole contribution, with the possible incorporation of a CPTviolating phase du (45). To "rst order in du, the corresponding change in A is (57) dAK(0.15 sin H#0.38 sin H"g /g ") du   # + with H evaluated using (50) and assuming du"0. However, this small-angle approximation is  not justi"ed, so we use the full expression (49) for A, and interpret the experimental value (54) as implying that A90.096 at the one-standard-deviation level, corresponding to 0.14 cos du!0.04 sin du50.096 ,

(58)

which leads to !703:du:#403

(59)

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for the allowed range of this CPT-violating parameter, where we have used the estimate [33] "g /g "K0.05, and not made any allowance for theoretical uncertainties. # + The range (59) is clearly much wider than the corresponding scope for a CPT-violating contribution d to the phase

of g , and the range would be larger still if we expanded the >\ >\ allowed range of du to the 95% CL limits. We also note in passing that the magnitude of the short-distance contribution (46) is so small that no interesting limit on direct CPT violation in it can be obtained. We now address the question whether all the KTeV asymmetry could be due to CPT violation. This would occur if A cos H#A cos H"g /g ""0 . (60)     # + This possibility is disfavoured by the theoretical estimates of A , but cannot be logically  excluded. If Eq. (60) were to hold, the KTeV asymmetry could be written in the form (61) AKA sin du[sin H!cos H tan H] ,     in which case the experimental value (54), at the one-standard-deviation level, would be reproduced if (62) 0.13:A sin du:0.22 .  Unfortunately, the amplitude A has not yet been measured experimentally. However, if one  adopts the estimate that A "0.15 as in (51), then the KTeV asymmetry could be reproduced if  du9583. We conclude that, whilst a priori it may seem very unlikely that the KTeV asymmetry could be due to CPT violation, we are unable to exclude rigorously this possibility at the present time. We hope that future measurements of this and related decay modes will soon be able to settle this issue.

Acknowledgements We thank members of the CPLEAR and KTeV Collaborations for informative discussions. The work of N.E.M. is partially supported by a United Kingdom P.P.A.R.C. Advanced Fellowship.

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[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

[34]

J. Ellis, N.E. Mavromatos / Physics Reports 320 (1999) 341}354 Phys. B 247 (1984), 293; B 254 (1985) 747 (Erratum); L.B. Okun, hep-ph/9612247, Proceedings of the Workshop on K Physics, Orsay, France, 30 May}4 June, 1996. CPLEAR Collaboration, Proceedings of the IHEP 98, Vancouver 1998, to appear. J.S. Bell, Proc. Roy. Soc. A 231 (1955) 479; G.L. LuK ders, Ann. Phys. 2 (1957) 1; R. Jost, Helv. Phys. Acta 30 (1957) 409. K.R. Schubert et al., Phys. Lett. B 31 (1970) 662. CPLEAR Collaboration, A. Angelopoulos et al., Phys. Lett. B444 (1998) 43. KTeV Collaboration, http://fnphyx-www.fnal.gov/experiments/ktev/ktev.html; J. Belz, for the KTeV Collaboration, hep-ex/9903025. CDF Collaboration, CDF/PUB/BOTTOM/CDF/4855, http://www-cdf.fnal.gov/physics/new/bottom/ publication.html; M. Paulini, CDF/PUB/BOTTOM/PUBLIC/4843, FERMILAB-PUB-99/014-E, Int. J. Mod. Phys. A, in press. NA31 Collaboration, G. Barr et al., Phys. Lett. B 317 (1993) 233. L. Wolfenstein, Phys. Rev. Lett. 13 (1964) 562. M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 (1973) 652. J. Ellis, M.K. Gaillard, D.V. Nanopoulos, Nucl. Phys. B 109 (1976) 213. A.J. Buras, hep-ph/9806471, in: F. David, R. Gupta (Eds.), Probing the Standard Model of Particle Interactions, 1998, Elsevier Science, Amsterdam, to appear. Y.-Y. Keum, U. Nierste, A.I. Sanda, Phys. Lett. B 457 (1999) 157. A. Masiero, H. Murayama, Phys. Rev. Lett. 83 (1999) 907. CPLEAR Collaboration, A. Angelopoulos et al., Phys. Lett. B 444 (1998) 52. CPLEAR Collaboration, A. Apostolakis et al., Phys. Lett. B 456 (1999) 297. J. Ellis, J.S. Hagelin, D.V. Nanopoulos, M. Srednicki, Nucl. Phys. B 241 (1984) 381. J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Lectures presented at the Erice Summer School, 31st Course: From Supersymmetry to the Origin of Space-Time, Vol. 31, Ettore Majorana Centre, Erice, July 4}12, 1993; 1994, 1; hep-th/9403133. J. Ellis, J. Lopez, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 53 (1996) 3846. L. Alvarez-GaumeH , K. Kounnas, S. Lola, P. Pavlopoulos, Phys. Lett. B 458 (1999) 347. L. Wolfenstein, unpublished, as reported in [26]. P.A. Kabir, Phys. Rev. D 2 (1970) 540. P.K. Kabir, Phys. Lett. B 459 (1999) 335. S. Hawking, Comm. Math. Phys. 87 (1982) 395. M.S. Marinov, J.E.T.P. Lett. 15 (1972) 677; Yad. Fiz. (Sov. J. Nucl. Phys.) 19 (1974) 350; Nucl. Phys. B 253 (1985) 609. L. Maiani, in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second Da'ne Physics Handbook, Frascati, Italy, 1995, p. 3. P. Huet, M. Peskin, B 434 (1995) 3. R. Wald, Phys. Rev. D 21 (1980) 2742; D.N. Page, Gen. Rel. Grav. 14 (1982) 299. CPLEAR Collaboration, R. Adler et al., Phys. Lett. B 364 (1995) 239. L.M. Sehgal, M. Wanninger, Phys. Rev. D 46 (1992) 1035; 5209 (1992) (Erratum); P. Heiliger, L.M. Sehgal, Phys. Rev. D 48 (1993) 4146; J.K. Elwood, M.B. Wise, M.J. Savage, Phys. Rev. D 52 (1995) 5095; D 53 (1996) 2855 (Erratum); J.K. Elwood, M.B. Wise, M.J. Savage, J.W. Walden, Phys. Rev. D 53 (1996) 4078. T. Yamanaka, for the KTeV Collaboration, talk at the Rencontres de Moriond, Les Arcs, March 1999.

Physics Reports 320 (1999) 355}358

The value of e/e in a theory with extended Higgs sector E. Shabalin Institute of Theoretical and Experimental Physics, Moscow, Russia

Abstract The additional source of CP violation appearing in the electroweak theory with three Higgs doublets could contribute considerably to the ratio e/e.  1999 Elsevier Science B.V. All rights reserved. PACS: 12.60.Fr

The recent result [1] Re(e/e)"(2.80$0.41);10\ ,

(1)

con"rming the CERN group result [2] e/e"(2.3$0.7);10\

(2)

with better accuracy turned out to be larger than that expected in the Standard Model (SM) [3}6]:



6.7$0.7

[3] ,

3.1$2.5 (e/e) "10\ ) 1+ 4$5

[4] ,

17> \

[5] ,

(3)

[6] .

Though the result of Ref. [6] is compatible with the above experimental results inside the error bars and besides, a considerable alteration of the generally accepted magnitudes of the parameters "< /< ", m , K M , B could result in (e/e) +(2}4);10\ [7], it is pertinent to look for another SB A@ Q +1  1+ CP violating mechanism capable of production of such e/e as in Eqs. (1) and (2). The probable candidate is the additional source of CP breakdown appearing in the electroweak theory with extended Higgs sector as in Weinberg [8] containing three doublets of complex Higgs 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 5 3 - 8

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E. Shabalin / Physics Reports 320 (1999) 355}358

"elds U . The #avour conservation is assured by coupling of the quarks with di!erent electric G charges to di!erent U , i"1, 2. The U doublet does not couple directly to the quarks. The G  coupling between quarks and physical charged Higgs "elds H> is de"ned by the expression (see  e.g. Refs. [9,10])





 1#c  1!c !KM X H>  D, (4) ¸"2G;M M K > H> 3 G G " G G $ 2 2 G G where M , M and K are the mass matrices for the up and down quarks and the weak mixing 3 " matrix, respectively. In the following we shall assume for simplicity that the mass m  is considerably larger than & m  and take into account only the contribution of the lightest charged scalar H>,H>. The CP  & violating e!ects will be proportional to Im(X>H). In the papers [11,12], it was found that being the only one, the Weinberg mechanism of CP violation would lead to magnitude of e/e&0.045 in contradiction to the experimental data. Therefore, such CP violating mechanism could be only the additional one to other mechanisms producing the main part of the parameter e. But as for e, the Weinberg mechanism could originate a contribution into e comparable, or even larger than that expected in SM [13]. This statement disagrees with conclusion in Ref. [14], based on the estimate for the upper bound on Im(X>H): 1 (5) Im(X>H)4([Br(bPsc)/C] F (x) & corresponding to a case where the real part of the scalar-exchange contribution cancels the SM contribution to bPsc transition. In Eq. (5), C+3;10\ [14], x"m/m and [10] A & F (x)"![x/6(1!x)][(3!5x)(1!x)#(4!6x)log x] . (6) & To date [15] Br(bPsc)"(3.11$0.80$0.72);10\

(7)

so that, Im(X>H)42, m "175 GeV, m "100 GeV , R & Im(X>H)43, m "M "175 GeV . (8) R & Let us show now that these bounds allow to get e/e of order (1). The Weinberg mechanism of CP violation produces two new CP violating operators [13] originating the CP-odd, *S"1 transition: O!.\ "(g Bm/3m )[M s (1!c )qq (1#c )d#M s (1#c )qq (1!c )d] 5 Q % Q   B  

(9)

and "(g BIf/3m )(G? )s (1#c )d . O!.\ Q ) IJ  5

(10)

E. Shabalin / Physics Reports 320 (1999) 355}358

357

These operators incorporate the short-distance and long-distance interactions, so that, a question on their renormalization is not simple. A possible renormalization is taken into account by the parameters m and f. In Eqs. (9) and (10), M and M are the constituent masses of s- and d-quarks, B is the imaginary Q B part of the e!ective coupling constant of s dG vertex <(s dG)"Bs (1#c )p (j?/2)dG? ,  IJ IJ I+1.54, m "0.64m (see [13]) % ) and B"B #B , where A R

(11) (12)




m g m m2(2G $ log & H) B Q Q A A 32nm m AB AQ & A and gives the main contribution into s dG transition. The operators (9) and (10) give the contribution




(13)





a m r m m 4If a r(M !M )mg Q B A log & Im (X>H) ) (e/e) "!  Q # Q , 5 ea m m 6Km m 27  ) & % A where [13]

(14)

"0.328, a "!10.13, r"2m/(m #m ), K"980 MeV .   L S B g(1 is the suppression factor arising due to partial cancellation between the gluonic and electroweak penguin diagrams. At f"1 we get a






1.24, m "100 GeV & (e/e) "Im(X>H);10\ . (15) 5 0.46, m "175 GeV F Together with the limits (8) for Im(X>H), the Eq. (15) does not exclude a possibility that the Weinberg mechanism of CP violation contributes considerably to the ratio e/e.

References [1] R. Kessler, Recent Results from the KTeV Experiment, Report at the Conf. `Les Rencontres de Physique de la Vallee d'Aostea, La Thuile, 28 February}6 March, 1999. [2] G.D. Barr et al., Phys. Lett. B 317 (1993) 233. [3] A. Buras, M. Jamin, M.E. Lautenbacher, Nucl. Phys. B 408 (1993) 209. [4] M. Chiuchini et al., in: L. Maiani, G. Pancheri, N. Paver (Eds.), The Second DAPhNE Hand Book, INFN-LNF, 1995, p. 27. [5] S. Bertolini, J.O. Eeg, M. Fabbrichesi, Preprint SISSA 103/95/EP. [6] S. Bertolini, M. Fabbrichesi, hep-ph/9802405, Vol. 2, 1998. [7] A.J. Buras, Proceedings of the Workshop on K Physics, Orsay, France, May 30}June 4, 1996, p. 459. [8] S. Weinberg, Phys. Rev. Lett. 37 (1976) 657. [9] G.C. Branco, A.J. Buras, J.-M. Gererd, Nucl. Phys. B 259 (1985) 306.

358 [10] [11] [12] [13] [14] [15]

E. Shabalin / Physics Reports 320 (1999) 355}358 P. Krawczyk, S. Pokorski, Nucl. Phys. B 364 (1991) 10. N.G. Deshpande, Phys. Rev. D 23 (1981) 654. A.I. Sanda, Phys. Rev. D 23 (1981) 2647. C. Jarlskog, E. Shabalin, Phys Rev. D 52 (1995) 6327. Y. Grossman, Y. Nir, Phys. Lett. B 313 (1993) 126. The ALEPH Collaboration, Preprint CERN-EP/98-044.

Physics Reports 320 (1999) 359}378

Mass and CKM matrices of quarks and leptons, the leptonic CP-phase in neutrino oscillations D.A. Ryzhikh*, K.A. Ter-Martirosyan ITEP, B.Cheremushkinskaya 25, 117259 Moscow, Russia Dedicated to our colleague } the old friend of one of us (K.T-M) and teacher of another } to Lev Okun. He always emphasizes the importance of simple and transparent physical arguments which become very impressive in his talks and discussions. He was among the "rst who discussed CP violation in neutrino oscillations.

Abstract A general approach for construction of quark and lepton mass matrices is formulated. The hierarchy of quarks and charged leptons (`electronsa) is large, it leads using the experimental values of mixing angles to the hierarchical mass matrix slightly deviating from the ones suggested earlier by Stech and including naturally the CP-phase. The same method based on the rotation of generation numbers in the diagonal mass matrix is used in the electron}neutrino sector of theory, where neutrino mass matrix is determined by the Majorano see-saw approach. The hierarchy of neutrino masses, much smaller than for quarks, was used including all existing (even preliminary) experimental data on neutrino mixing. The leptonic mass matrix found in this way includes the unknown value of the leptonic CP-phase. It leads to large lIlO oscillations and suppresses the llO and also llI oscillations. The explicit expressions for the probabilities of neutrino oscillation were obtained in order to specify the role of leptonic CP-phase. The value of time reversal e!ect (proportional to sin d) was found to be small &1%. However, a dependence of the values of llI,llO transition probabilities, averaged over oscillations, on the leptonic CP-phase has found to be not small } of order of ten percent.  1999 Elsevier Science B.V. All rights reserved. PACS: 14.60.Pq

1. Introduction Serious e!orts have been invested recently in the natural understanding of experimental results on neutrino oscillations. They have shown that neutrinos of three generations have, perhaps, non * Corresponding author. E-mail addresses: [email protected] (D.A. Ryzhikh), [email protected] (K.A. Ter-Martirosyan) 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 7 6 - 9

360

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

vanishing small masses. The heaviest of them, the neutrino of the third generation, seems to have a mass of the order of &(  ) eV and, as the Super Kamiokande data on atmospheric neutrinos  show, has the maximal possible mixing with the neutrino of the second generation, and may be, also not too small mixing with that of the "rst generation. This was not expected a priori, since all similar mixing angles of quarks are small. It is the challenge of modern particle physics to include naturally these results into the framework of grand uni"cation theory together with the data on quark masses and mixing angles. For the quarks these angles are small and are known already [1]. We begin this paper by reminding the well-known picture of masses, CP-phase and mixings for the quark sector of a theory. A general method will be developed which allows one to construct consistently the 3;3 mass-matrix and the CKM mixing matrix for quarks. The same general approach will be used later for the electron}neutrino sector of a theory. Let us consider, as a useful introduction to a consistent theory of quark and lepton masses and mixings, a simple phenomenological approach suggested by Stech [2]. He has noticed the following quark and charged lepton (`electronsa) mass hierarchies: m : m : m K1 : p : p, m : m : m K1 : p : 8p, m : m : m K1 : p : p O I   R A S @ Q B 

(1)

with a very small pK  , pK0.058. He has also introduced the following mass matrices which  reproduced approximately the masses of all the quarks as well as their mixing angles:





 pg  MK " ( pg> !p S   p pg>  (  0

(



pg



 p m, R  1

(






a p 0 B M K " a p !p 0 m /p B @ B  0 0 p 0

(2)

a p 0  MK " a p !p 0 m /p  O  0 0 p 0

(3)

Here g"e B represents the CP violating phase d in the quark sector, while the values of the constants a K2, a K( correspond to the best "t of all masses of quarks and electrons (their B   central values, see below). The diagonalization of the matrices (2) and (3) by means of an unitary matrix ;K : ? MK  ";K M K ;K > ? ? ? ?

a"u,d,e ,

(4)

(where ;K ;K >";K >;K "1), reproduces simultaneously both the experimental (central) values ? ? ? ? of running masses of quarks and electrons and all quark mixing angles (their sines): s "sin 0 O , s "sin 0  , s "sin 0 O in the CKM matrix       c c s c s e\ BY      . (5) c c !s s s e BY s c ;K " !s c !c s s e BY             !)+ S B s s !c c s e BY !c s !s c s e BY c c            






D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

361

Here, according to experimental data [1,8], one has s Ksin 0 O K0.221$0.004, s "sin 0 O "0.039$0.003 ,     s "sin 0 O Ks s "R""0.0032$0.0014, "R""s /s s K0.38$0.19        and d"p/2$p/4 .

(6)

The factors 1/(2, , 2, (, etc. in Eqs. (2) and (3) were adjusted so as to reproduce all these mixing   angles and all the masses of quarks and leptons. As has been mentioned, for quarks the mixing angles are small, i.e. s ;s ;s ;1 and therefore the cosines c ,c ,c of them can be put       approximately equal to unity. The major ideas of Stech's approach were: (1) to consider the running masses of all quarks at the same scale (instead of the pole ones) and (2) to obtain these masses and mixing angles in terms of few parameters. However, the matrices (2) and (3) have been written phenomenologically, just by hand. Below we develop an approach to quark and lepton masses and mixings through the following steps: (a) A general method is suggested which allows one to construct the quark mass matrices consistently (without using Eq. (2)) in terms of their masses and mixing angles (6). The resulting mass matrices coincide with those given by Eq. (2). They are diagonalized in an analytical form of decomposition in powers of small parameter p and the quark CKM matrix is reproduced analytically. (b) The same approach is applied to the reconstruction of the neutrino mass matrices M K . This is J performed using the Eq. (4) and the unitary matrix ;K . Preliminary results on masses of neutrinos J and their mixing angles, extracted from neutrino oscillation experiments have been used. (c) The leptonic CKM matrix ;K containing the leptonic CP-phase d is introduced. !)+ J  It leads to l l oscillation of the type observed at Super Kamiokande and also to some suppression I O of l l oscillation.  I (d) The expressions for the three type neutrino oscillations are speci"ed and some realistic numerical examples of oscillation dependence on the leptonic CP-phase are presented. Finishing this section let us note that the electron and quark masses emerge from the usual Yukawa interaction: ¸ "u u hGHq #u dM hGHq #u e h Hl , 7  G S H  G B H  G  H where our mass matrices (2) and (3) are: (MK )"1u 2hK , (MK )"1u 2hK , (MK )"1u 2hK and S  S B  B    1u 2"v cos b, 1u 2"vsin b are V.E.V of SUSY two neutral CP-even Higgs "elds (v"174 GeV,   and tg b are two well-known SUSY parameters) and hGH&hGH&hGH&1 are the Yukawa coupling  B S constants.

 The upper index q is used to emphasize that the angles 0 "0 O , or s "sin 0 O are determined just by the quarks and GH GH GH GH not by the leptonic mixings. The corresponding leptonic angles and their sines are denoted below by 0 and by GH s "sin 0 without any upper index (see Sections 3 and 4). The leptonic CP-phase is denoted (in Sections 3, 4) by d. GH GH

362

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

The scale of neutrino masses (10}10 times smaller than the electron and quark masses) can be a result of the see}saw approach [3}7,9,10]. It leads after the integrating out of the corresponding heavy states (e.g. the right-handed Majorano neutrinos with heavy masses of an order of M&10}10 GeV) to the appearance of higher-order e!ective Majorano mass operator (see [3}5,7]): (l (MK )GHl )"(u/M)(l hGHl ) , G J H  G H where hGH&1 are the neutrino coupling constants and M K "(u/M)hK is the neutrino mass matrix. J  2. Quarks mass matrices and the analytical form of the CKM matrix. The running of u- and d-quark masses calculated [9,10] in the "rst (dashed curves) and in the fourth (solid curves) order of QCD perturbation theory are shown [11] in Fig. 1. The quark running masses m (k) are related in MS renormalization scheme at the scale k"M to their pole G G masses M by the well-known relation G m (M )"M /[1#(4a (M ))/3p#K((a (M ))/p)] , G G G Q G Q G where K"11,2. The scale k"M "174.4 GeV is the most natural for the SUSY Standard Model. R The running quark masses at this scale (see in Fig. 1) have the following values (in GeVs): m (M )"(0.21$0.10);10\, m (M )"(0.42$0.21);10\, m "0.511;10\ , S R B R  m (M )"0.59$0.07, m (M )"0.42$0.21, m "105.66;10\ , (7) A R Q R I m (M )"163$4, m (M )"2.80$0.40, m "1.777$0.0003 . R R @ R O These values di!er strongly from the ones used in Ref. [2] which are determined at a very small scale k"1 GeV. However, the Stech's relation between them still holds. Note that the matrices M K and M K in Eqs. (2) and (3) have the block structure and their B  a o , which M K and M K contain in diagonalization is trivial. In fact, for any 2;2 matrix m( " B S o b their left-upper part, one has














k 0 c s 1 , u( " , k " (a#b$((a!b)#4o ) . (8) m( "u( m( u( >"     2 0 k !s c  and s"sin 0"1/(2(1!(1#t0)\), c"cos 0" Here t 0"tan 20"2o/(a!b)   K one 1/(2(1#(1#t0)\), and 0 is the mixing angle. In Eqs. (2), (3) for the matrices MK and M  B  has a"0 and the mixing angles 0 ,0 are small since t 0B"2(2p/p/2)"8p(1 and B   t 0"2(p/p"(6p is even smaller. Due to the block structure of M K and M K the unitary   B  matrices ;K and ;K also have the following block structure: B  c s 0 c s 0 B B   ;K " !s c 0 , ;K " !s c 0 , (9) B  B B   0 0 1 0 0 1











D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

363

Fig. 1. (a) The run of the upper quark masses m ,m ,m calculated in Refs. [11] in the "rst order of QCD perturbation R A S theory (the dashed lines) and in the fourth order of it (solid lines). The vertical line corresponds to the scale k"M K174.4 GeV used in the paper. The run of electron's masses m ,m ,m is disregarded (while it can be easy taken R O I  into account and is not essential). (b) The same for the masses m ,m ,m of the lower quarks. @ Q B

where s "1/(2(1!(1#(8p))\)K4p(1!24p)K0.214, s "(1/(2)(1!(1#6p)\)K B  (p(1!(p))K0.0705 up to the terms of the order of p.   Let us note that Stech's matrices MK ,M K in Eqs. (2) and (3) can be reconstructed using their B  diagonal form (i.e. the physical masses of d-quarks and electrons):




MK  " B

m B

!m Q



,

m @




MK  " 

m 

!m I



,

(10)

m O

by means of Eq. (4), which states MK ";K >M K  ;K , M K ";K >M K  ;K . (11) B B B B     The matrices ;K "OK B (or ;K "OK  ) in Eq. (9) can be considered as rotating the 12 generations of B    d-quarks (or electrons).

364

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

A bit more complicated (but much more instructive) is the diagonalization of the matrix M K . S Similarly to M K and M K this matrix can be represented as B  MK ";K >M K  ;K , where ;K >"OK > OK > (d)(OK S )> . S S S S S   

(12)

Here the matrices














c !s 0 c 0 s e\ B 1 0 0 S S   , OK > " 0 c 0 , OK > (d)" 0 1 0 c s (OK S )>" s   S S    0 0 1 !s e B 0 c 0 !s c    

(13)

rotates 12, 13 and 23 generations, respectively. The 13 rotation includes naturally the complex phase d which violates the CP-parity conservation of a theory. It cannot be removed by a trivial phase transformation of the u- or t-quark "elds. However, the value of s turns out to be very small S (s &p;1) and one can really put OK S K1) . The quark CKM matrix is S  ;K "OK > OK > (d)OK > , where OK > "(OK S )>OK B KOK B . !)+ S B       

(14)

Here in general






c

s 0   c 0 (OK S )>" !s    0 0 1

with s "sB c !cB s "sin(0 !0 S )Ksin 0 since s &p (see below) is neglegibly small. B B S   S  S  m S Thus for the given upper quark masses M K  " and the given quark mixing angles m S A m R (from the quark CKM mixing matrix is "xed: !)+ S











c

0 s e\ B   . c c s ;K >"OK > OK > (d)" s s e B      S   !s c e B !s c c     

(15)

The same matrix ;K > can be obtained in the form of decomposition in powers of p (up to p terms) S by a direct calculation




;K >" S

p 1! 2

p p e B 1! ! 4 (2 p p 5 !p 1! e B ! 1! p 2 4 (2
















p p 1! e\ B 4

0



p






5 1! p 4 (2 p 1! 4

.

(16)

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

365

Multiplying ;K > in this form by ;K one obtains with the same accuracy the following CKM quark S B matrix ;K : !)+ S B












p 1! c 2 





p p s ! c e B


p 1! s 2 






p p 1! c ! s e B  4 (2 






5 p 5 1! p s !pc e B ! 1! p c !ps e B     4 4 (2 (2








p p 1! e\ B 4 p






5 1! p 4 (2 p 1! 4

.

(17)

Comparing the CKM matrix (5) with Eq. (17) one "nds






5 s "s K4p(1!24p)+0.215, s "(p/(2) 1! p +0.0408 ,  B  4

(18)

s "s s "R""p(1!p/4)+3.36;10\    implying that "R""s /s s "1/((8(1!25p))+0.38.    Calculating M K by means of Eq. (12) one "rst obtains M K  "OK > M OK and then "nds S S  S  K  OK in the form M K "OK > M S  S  0 s s e\ B s e\ B   m  s m , (19) MK " s s e B s #       m S  s e B s 1   up to the terms of an order of p. Eq. (19) really reproduces the Stech's matrix M K in Eq. (2) only for negative sign of m "!"m "; S   for positive m ""m " the ratio k "(M K ) /m is p instead of !p/2 in Eq. (2). Consider   S  R  ing higher order p corrections to the matrix (19) one "nds that k "(MK ) /m turns out to  S  R be 2p for positive m ""m " and is much smaller, !p/2, for negative lightest eigenvalue S S when m "!"m ". S S Therefore, literally the Stech's matrix M K is reproduced in the form (2) only for negative m and S  m , e.g. at  !"m " S . !"m " MK  " A S m R Thus, the u-quark mass matrix M K can be reproduced by the former of Eqs. (12) using the ;K and S S ;K > matrices in the form of the latter of Eqs. (12) (or by Eq. (16) if (OK S )>K1). Actually this method S  will be very useful later for the restoration of the neutrino mass matrix. All these results are in a good agreement with the experimental data presented above. Also the diagonalisation (4) of M K and M K with the help of ;K and ;K matrices leads to the following S B S B











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D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

reasonable quark masses (in GeV, as in Eq. (1)): m "p(1!p)m (M )+2.00;10\, S R R

m "(p/4)((1#(8p)!1)m "0.416;10\ , B @

m "p(1!p/2)m (M )+0.56, A R R

m "(p/4)((1#(8p)#1)m "0.085 , Q @

m "(1!p/2)m (M )+m (M )"163, R R R R R

m "m (M )"2.83 . @ @ @

(20)

These results correspond fairly well, as Stech has remarked [2], to the experimental data (7). The minus sign of some eigenvalues in Eq. (10) and in M K   can be easily removed by rede"nition S of the corresponding quark "eld: q Pc q . I  I 3. Neutrino mass matrix, the leptonic CKM matrix and CP-phase To proceed further let us introduce the neutrino mass matrix M K with a structure similar J to that of M K . The Super Kamiokande data [12}14] suggest a large l l neutrino mixing i.e. S I O large t "tg 20 <1, or sin20 "(1#t\)\K1 (i.e. s Kc K1/(2K0.71) and       not too small value of *m "m!mK(0.59$0.20)10\ eV. As m <m (see below)      one has (21) m K(*m K(0.059$0.020) eV .   Simultaneously the atmosphere and solar neutrino data [15}24] (see the discussion in Refs. [25}33]) show a large suppression of l l and also l l oscillations which have not yet been  I  O observed. This can be a result of small mixing angles s 4s ;s &1/(2 (see Refs. [15}18] and    also [20}33]): s K0.035$0.020, s K0.25$0.10, s "0.7K1/(2 (22)    and some (obviously not too large) hierarchy of three neutrino masses m ;m ;m of a type    considered above for the quarks and electrons: m :m :m "1:p:p . (23)    J J Here p is an unknown parameter which we can choose to be equal to p"0.058 to avoid the J introduction of additional new parameter p "(0.058K0.24. This gives m K0.34;10\ eV and  J together with Eqs. (22) seems to be in approximate agreement with the atmospheric neutrino data [20}24]. Essentially the other possibility has been suggested in a recent paper [33], where the neutrinos l ,l and l have been considered with almost equal masses m Km Km K3 eV but a large       hierarchy has been introduced in their mixing angles (24) s K0;s Ks K1/(2 .    This situation is not considered below as it seems more natural that neutrino have small mass hierarchy of the type (1) like the u-quarks, but with much smaller power of hierarchy: p K0.24
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367

(23) m (m (and m (m ) together with Eqs. (22) suppress strongly the non observed (at least for     a moment) l l oscillations from electron neutrino sources on the Earth.  I Therefore, let us take approximately the central values of the data given in Eqs. (21)}(23) as a basis of our approach (see e.g. Ref. [5]) putting also p K(0.058 in Eq. (23) and taking the J following values for the neutrino masses:




M " J

m 

m 


"

m



0.019 0.34

10\ eV ,

(25)

5.90

 where m Kpm , m Kpm and the central value of m has been taken from Eq. (21).  J   J   The main point of this paper is that approach of the type of Eqs. (12)}(16) together with the electron matrix M K (determined by Eq. (3)) allows one to construct in general analytical form the  neutrino mass matrix M K and also the leptonic CKM matrix. The matrix M K so obtained will have J J all desired properties reproducing naturally Eqs. (21)}(23) and will naturally include the CP-phase. To this end let us remind that M K in Eq. (3) has the form  MK ";K >MK  ;K , (26)     where c s 0   ;K "OK  " !s c 0     0 0 1






has been determined in Eq. (9) with s K0.0705. Here  m 0.5175  103.6 MK  " " MeV m  I 1777 m O represents exactly the three electron masses.









 Let us mention that the matrix M K can be slightly modi"ed:  3 jp p 0 2 3 m MK PM K " p !bp 0 O ,   2 p 0 0 p








where b"1#j with very small j "0.0202308 and j"0.0106891 adjusted to reproduce exactly the well-known   masses of all three electrons:




0.510999

MK  ";K M K ;K >"    

105.6584



MeV .

1777.0

This modi"cation will change negligibly the leptonic CKM matrix, leading to s K0.0689 and is not important at all as  the neutrino basic parameters are determined very roughly in Eqs. (21)}(23) (see also Section 5).

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The general form of the matrix M K can be written quite similar to Eq. (12) as J MK ";K >M K  ;K with ;K >"OK >OK >(d)OK J> , J J J J J    where




(27)



c

s J J OK J " !s c  J J 0 0

0

0 1

is de"ned similarly to OK  with the substitution s "sin0 J for s in OK  de"ned above (the value of  J    0 J will be determined later on). The matrices: 




>

OK (d)" 

c



0 s e\ BY  1 0 ,

 0

!s e BY 0 

c 




1

>

0

0



c s OK " 0    0 !s c  

(28)

represent the rotation of 13 and 23 generations of M K  , respectively, and d is the leptonic J CP-phase which has appeared here naturally (clearly s ,c from here and later on means the sines GH GH and cosines of the leptonic and not quark mixing angles). To calculate M K in Eq. (27) in explicit form we note that according to Eq. (8) one has J a o 0 m  OK J " o b 0 (29) m OK J>    m 0 0 m  







with a"sm #cm Km , b"cm #sm Km , o"!c s (m !m ) and "o";m is small J  J   J  J   JJ    since s in Eq. (22) is very small (actually s Ks !s K0.036 see below). Then it is easy to J J   calculate the matrix


 a

o

MK "OK > (d) o  0

b

0

0 OK (d)  0 m  and further to "nd the following result for the neutrino mass matrix MK "OK >M K OK : J   ac #m s c s s (m !a)e\ BY c c s (m !a)e\ BY            #oc c !os c     c s s (m !a)e BY s (m c #as ) c s (m c !b#as )          . MK "     J #oc c #c b#D #D      c c s (m !a)e BY c s (m c !b#as ) c (m c #as )              !os c #D #s b#D     






(30)

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369

The values of D "!os sin 20 J cos d, D "!os (e\ BY!2s cos d)"D> and of        D "!D are small and can be neglected. Also since a;b;m , s Ks (see below) and    J  s ;s ;1 are very small, one can disregard in the matrix M K all the terms containing a,b and   J put c Kc K1. The value of "o"Ks m is also very small "o";s m , but the terms containing       it in the matrix (30) cannot be omitted as that would violate the normal complex structure of the matrix M K and of CKM matrix considered below. Therefore omitting small terms one obtains the J matrix M K in the following simple form J






s m #ac c (s s m e\ BY#oc ) c (s c m e\ BY!os )              . (31) s m #c m s c (m c !m ) MK " c (s s m e BY#oc )               J c (s c m e BY!os ) s c (m c !m ) c m #s m               Here, s Kc K1/(2 and not too small values of s are determined in Eqs.(22).    In conclusion of this section let us construct the leptonic CKM matrix: ;K "OK >OK >(d)OK , !)+ J     where




c

(32)



s 0   OK "OK J>OK  " !s c 0      0 0 1

and s "sin(0  !0 J )"sin 0 K0.035$0.020 as it is determined by Eq. (22). This gives for     the neutrino l l mixing angle: 0 J "0  !0 K0.036$0.020 (in radians, or (2.1$1.1)3).      Multiplying the matrices OK > in Eq. (32) one obtains



0.130

MK " 0.335e BY#0.27;10\ J 0.341e BY!0.26;10\



0.335e\ BY#0.27;10\ 0.341e\ BY#0.26;10\ 0.965

0.868

0.868

1

m  1.975 (33)

and the CKM leptonic matrix (32) is




0.968

0.0339




0.25e\ BY 0.678 0.692

.

(34)

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Unfortunately instead of predicting the neutrino mixing angles s ,s ,s we have used their    values (22) which are badly de"ned by experiment. The hierarchy parameter p remains, as has been J mentioned above, practically free and we have put it to (pK0.24 just by hand. Both the matrices OK J and OK  represent a simple Abelian rotation: OK J by the angle 0 J , OK J> by the angle (!0 J )       and OK  by the angle 0  . Therefore the product OK J>OK  ,OK  OK J> leads to a rotation by the angle       0 "0  !0 J , where 0  K20 (the value of s has been given above just after Eq. (9) and s in        Eq. (18)) and therefore 0 J K0 , or s Ks .    J Similarly to the case of the quark CKM matrix, the most natural value of the leptonic CP-phase leading to the largest possible CP violation can be d"n/2 or g"e BY"i. This CP-phase d can manifest itself in the Pontecorvo neutrino oscillation experiments. It is very di$cult to observe it now. Below we discuss shortly the possibility of these observations. The exact expressions for probabilities of neutrino oscillations are given in Appendix, since they are very cumbersome. Some of them have been obtained earlier in a number of papers [34}44].

4. The leptonic CP-phase in neutrino oscillations experiments Many papers were devoted to the studies, pioneered by Bruno Pontecorvo [34}37], of two and of three [38}44,7] neutrino oscillations. We consider them below shortly in order mainly to specify the role of the leptonic CP-phase [38,39] in these oscillations. Let us express l ,l and l "elds entering the weak interaction Lagrangian in terms of neutrino  I O states l ,l ,l with de"nite masses m ,m ,m using the leptonic CKM matrix (5) (or (A.1) from the       appendix) as follows: l (ct)"c l (0)e\ CR#s c l (0)e\ CR#s l (0)e\ CR\GBY ,         l (ct)"!(s c #s s e BY)l (0)e\ CR#c l (0)e\ CR#c s l (0)e\ CR , (35) I           l (ct)"(s s !s c e BY)l (0)e\ CR!s l (0)e\ CR#c c l (0)e\ CR ,      O      up to the terms of the second order in small quantities s ,s ;1. (see the appendix for the exact   P" ,  I            P(l l )""c (s s !c s e BY)!s c s e\ P#s c c e BY>P" ,  O            P(l l )""(s s !s c c e\ BY)(s c #s c s e BY)#c s e P! s c c e P"    I O             (37)

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371

where u "(e !e )¸/c"(*m /2p )¸"2.54¸(m)/(E (MeV))*m(e<) GH G H GH J J

(38)

and *m "m!m, i,j"1,2,3 (as e Kp #m/2p at cp<m ) . GH G H G J G J G Eqs. (36)}(38) determine the neutrino oscillation probabilities in vacuum ignoring the very important in some cases MSW e!ect of the medium in#uence (see Refs. [15}18]). This e!ect has been well studied and can be included separately (e.g. for l l , l l , l l in the solar neutrino case). The    I  O expressions for P(l l ), P(l l ) P(l l ) coincide with Eqs. (37) with the substitution dP!d I  O  O I The simple algebra allows one to reduce Eqs. (36), (37) to the partially known [38}44] expressions containing the leptonic CP-phase d and given below in the Appendix. Let us rewrite Eqs. (36), (37) in much more simple forms calculating numerically the coe$cient in front of cos u "1!2 sin(u /2) using for example the central values of leptonic mixing angles GH GH 0 ,0 ,0 given in Eqs. (22), i.e. their sines s ,s ,s , respectively. These equations in their       numerical form show clearly the in#uence of the leptonic CP-phase d on the neutrino oscillations patterns: 1!P(l l )Kc sin(20 )sin(u /2)#c sin(20 )sin(u /2)         # s sin(20 )sin(u /2)    K0.23 sin(u /2)#4.3;10\ sin(u /2)#0.29;10\ sin(u /2) ,    1!P(l l )K(0.94#0.017 cos d)sin(u /2)#(0.062!0.017cos d)sin(u /2) I I   (0.063#0.016 cos d!0.31;10\ cos d)sin(u /2) ,  1!P(l l )K(0.94#0.017 cos d)sin(u /2)#(0.062!0.017 cos d)sin(u /2) O O   (0.064!0.016 cos d!0.31;10\ cos d)sin(u /2) . (39)  Eqs. (37) determine the probabilities for neutrino of a given sort l to change its type (i.e. to G transform into the other sort l ) at a distance ¸ from the source of neutrinos. For l this probability H  does not depend on the CP-phase d at all as can be seen already from Eqs. (36), (37), however, for transformations of l and l this dependence is not too small. I O The probabilities for di!erent neutrino transitions P(l l )OP(l l ) depend also on the value of d: G H H G P(l l )K0.059+1!0.14 cos d!0.98[cos(u )#0.019 cos(u )#0.0012 cos(u )] ,  I    # 0.140[cos(u #d)!cos(u #d)!cos(u !d)], ,    P(l l )K0.061+1!0.13 cos d!0.98[cos(u )#0.018cos(u )#0.0012 cos(u )]  O    ! 0.135[cos(u #d)# cos(u #d)! cos(u !d)], ,    P(l l )K0.47+1!0.99[cos(u )!0.066cos(u )#0.061cos(u )] I O    # 0.035 sin dcos((u #u )/2)#0.035 sin dcos((u #u )/2)sin(u /2)      ! 0.017 sin d sin(u ), . (40) 

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It is interesting to consider the time reversal e!ect which reveals in the di!erence between the probabilities P(l l ) and P(l l ) or P(l l ) and P(l l ), etc. [38,39]. Taking the di!erence between  I I   O O  P(l l )!P(l l ) and the same for l l , l l one obtains:  I I   O I O P(l l )!P(l l )"0.0164 (sin u #sin u !sin u ) sin d , I   I    P(l l )!P(l l )"!0.0164 (sin u #sin u !sin u )sin d , O   O    P(l l )!P(l l )"0.0328 (cos u /2!cos(u #u )/2)sin d , O I I O   

(41)

where 0.0328"c sin 20 sin 20 sin 20 for the numerical coe$cients given above. So,     the leptonic CP-phase can manifest itself in time reversal neutrino transitions l l and l l ? @ @ ? experiments. However, these experiments need the neutrino beams with a "xed value of ¸/2E J what is very di$cult to organize because usually one deals with continuum spectra of produced neutrino. We emphasize once more that the numerical coe$cients in Eqs. (39), (41) can be determined by future experimental data only, but the general form of them (obtained algebraically from exact Eqs. (A.2)}(A.6) of the appendix) remains always valid. Let us average Eqs. (39), (40) over ¸/2E considering the case of large ¸/2E '(*m )\, i.e. of J J  large angles u "*m ¸/2E . This nicely corresponds to the real experimental situation where one GH GH J works with a continuum spectra of neutrino. This averaging gives 11!P(l l )2K0.1193 ,   11!P(l l )2K0.529(1#0.016 cos d!0.289;10\cos d) , I I 11!P(l l )2K0.531(1!0.015 cos d!0.288;10\ cos d) O O

(42)

1P(l l )2K0.0585(1#0.140 cos d), 1P(l l )2K0.0608(1!0.134 cos d) ,  I  O 1P(l l )2K0.470(1#0.394;10\ cos d !0.163;10\ cos 2d) , I O

(43)

and

where brackets mean the averaging over all u . GH Comparing Eqs. (39) and (40) (and also (A.2)}(A.6) in the appendix) one "nds: P(l l )#P(l l )#P(l l )"1 for di!erent l ,l and l (e.g. 1!P(l l )"P(l l )#P(l l )). G G G H G I G H I    I  O Eqs. (42)}(43) show also that the leptonic CP-phase d can be observed experimentally, in principle, by measuring the average l l , or l l transition rates with 14% accuracy which is much  I  O larger than the e!ect (41) of time reversal. This is illustrated in Figs. 2(a) and (b) where the averaged probabilities of l l transitions 1P(l l )2 G H G H are calculated for the values of s ,s ,s determined in Eqs. (22). These "gures show a large (about    30%) di!erence between 1P(l l )2 and 1P(l l )2 probabilities dependencies on the CP-phase d. I  O  For example the value of 1P(l l )2K0.067 at d"0 turns out to be larger than 1P(l l )2K0.057 I  O  by 17%, i.e. 1P(l l )2/1P(l l )2K1.17, while at d"n vice versa 1P(l l )2K0.050 becomes I  O  I  smaller than 1P(l l )2K0.059 by 15%: 1P(l l )2/1P(l l )2K0.85. O  I  O  Unfortunately, the absolute values of these probabilities are small of about  , but nevertheless  can be really observed experimentally. Even weaker is the d dependence of the average value of the

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373

Fig. 2. (a) Dependence of the average (over oscillations) value 1P(l l )2 of l l transition probability on the CP-phase d;  I  I at d"n it has about 13% minimum. (b) The same for the average value 1P(l l )2 of l l transition probability. As is seen  O  O instead of the minimum at d"n, as was for the l l transition case, it has here the 14% maximum. (c) The same  I dependence on the CP-phase as was shown in the cases (a), (b) but for the average l l probability 1P(l l )2. Its I O I O dependence on CP-phase d here is much more #at * about hundred times smaller than in the l l and l l cases.  I  O

probability of l l transition rate (averaged over the oscillations connected with di!erent ¸/E I O J values, or over u } as in Figs. 2(a) and (b), shown in Fig. 2c. This "gure shows that 1P(l l )2 GH I O changes by 0.05% only when d changes from 0 to p.

5. Conclusion The following three simple problems were discussed and solved in this paper: (a) The u- and d-quarks and electrons mass matrices were obtained in a simple hierarchical form, including quark's CP-phase d and correcting the matrices suggested by Stech, (b) The recent data on neutrino masses and mixing angles were discussed shortly and used for construction of the neutrino and leptonic mass matrices and CKM matrix, both including the leptonic CP-phase d, (c) The CKM matrix obtained was used to investigate the three neutrino oscillations in the vacuum. The method of determination of the leptonic CP-phase from their observation (averaged over the oscillations) was presented. Note that there is a vast ambiguity in the determination of quark's (or leptonic) mass matrices. E.g. the pair of matrices M K  ";K M K ;K >, with any unitary matrix ;K , leads exactly S M B M M to the same mass eigenvalues and to the same mixing angles as M K ,M K . S B Also the following correction has to be added to the central part of the paper. It was noted there (in Section 2, after Eq. (21)) that Stech's matrices (2), (3) have negative value of some masses and that they positivity can be restored by a simple c transformation. However this will  violate the symmetry of di!erent generations "elds and also will change the form of mass matrices. Better is to avoid this shortcoming and construct the particles' mass matrices with only positive

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eigenvalues,




MK  " ?

m ?



m ?

m ? directly from Eq. (11) [or Eqs. (29) and (12)], e.g. for d-quark and electrons one "nds:


  a

o 0

MK  " o b B 0 0 or




a"s m #c m K15pm     @ 0 , where b"c m #s m Kc m Kpm @        m o"!c s (m !m )K(pm , @ @     



!(p 0  m p 0 @, MK  " !(p   B p 0 0 p 15p

and similarly:






3j p (p 0   m MK  " (p (1#b)p 0 O ,  C p 0 0 p with small j"0.020, b"0.016. Both M K ,M K di!er slightly from Stech's forms given in Section 1. B C The u-quark mass matrix is determined by Eqs. (12), (19) and also slightly deviates from the Stech form given by Eq. (2):






(pg> pg>  p (p m , where ;K >"OK > OK > (d)(OK S )>. MK  ";K >M K  ;K K (pg   R S    S S S S  pg (p 1  The diagonalization of all these matrices M K  ";K M K  ;K > can be done by the same unitary ? S ? S matrices ;K , ;K , ;K , ;K with the same mixing angles (used at they construction) as were used above B S C J for the Stech matrix case. 2p

Acknowledgements Authors express their gratitude to P.A. Kovalenko for calculation of u- and d-running masses shown in Fig. 1, to N. Mikheev for stimulating discussion and also to L. Vassilevskaya and D. Kazakov for useful discussions and an essential help in edition of the paper. Especially we thank Z. Berezhiani who provided us with his and Anna Rossi paper [7] containing a number of ideas used above. They thank the RFBR and INTAS for "nancial supports by grants: RFBR 96}15}96740, 98}02}17453, INTAS 96i0155 and RFBR}INTAS: 96i0567, 95i1300. Also one of the authors (K.A.T-M) acknowledges the Soros foundation for support by professor's grant.

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375

Appendix We give below the probability rates P(l l ) in the general algebraic form using the well known G H exact leptonic CKM matrix (given above for quarks in Eq. (5)):






c c s c s e\ BY      . c c !s s s e BY s c
(A.1)

At dO0 it leads to 1!P(l l )"+c sin(20 )#s s sin(20 )#s s sin(20 )#c s sin(20 )           I I  # cos d sin(40 )sin(20 )(s c !s s )       ! s cos d sin(20 ) sin(20 ),sin(u /2)     # +s c sin(20 )#c s sin(20 )#s c cos d sin(20 )          ;sin(20 )sin(20 ),sin(u /2)    # +c c sin(20 )#s s sin(20 )!cos ds c         (A.2) sin(20 )sin(20 )sin(20 ),sin(u /2) ,     1!P(l l )"+s sin(20 )#s s sin(20 )#c s sin(20 )#c s sin(20 )           O O  # cos d sin(40 )sin(20 )(s c !s s )       ! s cos d sin(20 )sin(20 ), sin(u /2)     # +s c sin(20 )#c c sin(20 )       ! cos dc c sin(20 )sin(20 )sin(20 ),sin(u /2)       # +c c sin(20 )#s c sin(20 )       # c c cos d sin(20 )sin(20 )sin(20 ),sin(u /2)       for 1!P(l l ) see Eqs. (39). As before here u "*m ¸/2E and   GH GH J P(l l )"+sin(20 )(s #c s #s s #c sin(20 )sin(40 )cos d)  I            ! 2c sin(20 )(c !s s )cos(u )!2s sin(20 )(c cos(u )           #s cos(u ))# c sin(20 )sin(20 )sin(20 )(s cos(d#u )         !c cos(d!u ))# c sin(20 )sin(20 )sin(20 )(cos(d#u )        !cos(d!u ))# 2c c sin(20 ), ,    

(A.3)

(A.4)

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P(l l )"+sin(20 )(c #c c #s c !c sin(20 )sin(40 )cos d)  O            # 2c sin(20 )(s !c s )cos(u )!2c sin(20 )(c cos(u )           #s cos(u ))# c sin(20 )sin(20 )sin(20 )(c cos(d!u )         !s cos(d#u ))# c sin(20 )sin(20 )sin(20 )(cos(d#u )        (A.5) !cos(d#u ))# 2c s sin(20 ), ,     P(l l )"+2s sin(20 )cos(20 )#(c #c #s #(c #s )s )sin(20 ) I O            ! [2s (c #s )sin(20 )#[2s (c #s )        !(1#s )sin(20 )]sin(20 )]cos(u )     ! [2c (s #c s )sin(20 )!c sin(20 )sin(20 )sin(40 )cos d]cos(u )            # [2c sin(20 )(s s !c )!c sin(20 )sin(20 )sin(40 )cos d]cos(u )            # 2c sin(20 )sin(20 )sin(20 )sindsin(u /2)cos((u #u )/2)        # s sin(40 )sin(40 )cos d[1#s ]sin(u /2)      ! c sin(20 )sin(20 sin(20 )sindsin(u )      (A.6) ! 2s sin(20 )sin(20 )cos(2d)sin(u /2), .     For the average values of these probabilities over all u (i.e. over E , or over ¸ at large u ) one GH J GH obtains the values (42),(43) slightly dependent on the leptonic CP-phase d: 11!P(l l )2"A K0.1193 ,    11!P(l l )2"A #B cos d#C cos d, I I II II II 11!P(l l )2"A #B cos d#C cos d, O O OO OO OO 1P(l l )2"A #B cos d,  I I I 1P(l l )2"A #B cos d,  O O O 1P(l l )2"A #B cos d#C cos(2d) , I O IO IO IO where the coe$cients are A " [c sin(20 )#sin(20 )],      A " [(c #(c #s )s )sin(20 ) II       #(s sin(20 )#sin(20 ))s     #c sin(20 )],  

B "!s sin(20 )(s s !c )sin(40 ) , II        C "!s sin(20 )s sin(20 )sin(20 ) , II      

(A.7)

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378

A " [(c #(c #s )s )sin(20 ) OO       #(s sin(20 )#sin(20 ))c     #s sin(20 )],   A " [(1#c #s )s sin(20 )]     I  #2c c sin(20 )],    A " [(1#c #s )c sin(20 )]     O  #2c s sin(20 )],    A " [2s sin(20 )cos(20 ) IO     #sin(20 )+(c #s )s     #c #c #s ,],   

377

B "!s sin(20 )(s !c s )sin(40 ) , OO        C "!s sin(20 )s sin(20 )sin(20 ) , OO       B "c sin(20 )sin(20 )sin(40 ) , I      B "!c sin(20 )sin(20 )sin(40 ) , O      B "(1#s )s sin(40 )sin(40 ) , IO      C "!s sin(20 )sin(20 ) . IO     (A.8)

The numerical value of these coe$cient are given above in the text in (42), (43) for neutrino mixing angles s , s , s values determined in Eqs. (22).    References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Particle Data Group, Eur. Phys. J. C 3 (1998) 1. B. Stech, Phys. Lett. B 403 (1997) 114. S. Weinberg, Phys. Lett. (1979) 1566. R. Barbieri, J. Ellis, M.K. Gaillard, Phys. Lett. B 90 (1980) 249. E. Akhmedov, Z. Berezhiani, G. Senjanovic, Phys. Rev. Lett. 69 (1992) 3013. K. Ter-Martirosyan, ITEP preprint 93-88 (1988). Z. Berezhiani, A. Rossi, JHEP 9903 (1999) 002 (the old H. Fritzsch approach was partly used in this paper see Nucl. Phys. B 155 (1979) 189). A. Ali Desy, preprint 97}256, hep-ph/9801270. J.A.K. Wermessen, S.A. Larin, T. van Ritberger, Phys. Lett. B 405 (1997) 327. K.C. Chetyrkin, Nucl. Phys. B 64 (1998) 110. P.A. Kovalenko, private communication. The Super Kamiokande Collab. Phys. Rev. Lett. 81 (1998) 1562; 82 (1999) 1810. Y. Totsuka talk at the 18th International Symponum on Lepton and Photon Interaction, Hamburg, VII}VIII, 1997. Y. Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562, 2016. N. Hata, P. Landgacker hep-ph/9705339. J. Bahcall, P. Krastev, A. Smirnov, Phys. Rev. D 58 (1998) 096016. S.P. Mikheyev, A.Yu. Smirnov, Yad. Fiz. 42 (1985) 1441. L. Wolfenstein, Phys. Rev. D 17 (1985) 2369. Y. Fukuda et al., Phys. Lett. B 335 (1994) 237 (Kamiokande data). M. Apollonio et al., CHOOS Collab, Phys. Lett. B 420 (1998) 397. M. Gonzalez}Garcia, H. Nunokawa, O. Peres, J. Valls, Nucl. Phys. B 543 (1999) 3. V. Berger, T. Weiler, K. Whisnant, Phys. Lett. B 440 (1998) 1. G.L. Fogli, E. Lisi, A. Marone, G. Scioscia, Phys. Rev. D 59 (1999) 033001. S. Bilenky, C. Giunti, Grimus, hep-ph/9809368.

378 [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

D.A. Ryzhikh, K.A. Ter-Martirosyan / Physics Reports 320 (1999) 359}378 R. Barbieri, L. Hall, A. Strumia, Phys. Lett. B 445 (1999) 407. R. Barbieri, L. Hall, D. Smith, A. Strumia, N. Weiner, JHEP 9812 (1998) 017. H. Fritzsch, Z. Xing, Phys. Lett. B 440 (1998) 313. G. Altarelli, F. Feruglio, Phys. Lett. B 439 (1998) 112; JHEP 9811 (1998) 021. Z. Berezhiani, Z. Tavartkiladze, Phys. Lett. B 409 (1997) 220. Z. Berezhiani, A. Rossi, Phys. Lett. B 367 (1996) 219. C.D. Carone, L. Hall, Phys. Rev. D 56 (1996) 4108. M. Tanimoto, Phys. Rev. D 57 (1998) 1983. H. Georgi, S.L. Glashow, preprint HUTP 98/0060; also hep-ph/9808233 v.2. B.M. Pontecorvo, JETP 33 (1957) 549. B.M. Pontecorvo, JETP 34 (1958) 247. V.N. Gribov, B.M. Pontecorvo, Phys. Lett. 288 (1969) 483. S.M. Bilenky, B. Pontecorvo, Phys. Rep. 41 (1978) 225. I.Yu. Kobzarev, B.V. Martemyanov, L.B. Okun, M.G. Shepkin, Sov. J. Nucl. Phys. 32 (1980) 823. I.Yu. Kobzarev, B.V. Martemyanov, L.B. Okun, M.G. Shepkin, 35 (1982) 708. V. Barger, Yuan-Ben Dai, K. Whisnant, Bing-Lin Yuong, Phys. Rev. D 59 (1999) 113010, AMES-HET-99-01, ASITP-99-05 MADPH-99-1096, January}February 1999. V. Barger, K. Whisnant, T.J. Weiler, Phys. Lett. B 427 (1998) 97. V. Barger, J.S. Pakvasa, T.J. Weiler, K. Whisnant, Phys. Rev. D 50 (1998) 3016. S.C. Gibbons, R.N. Mohapatra, S. Nandi, A. Raychauduri, Phys. Lett. B 430 (1998) 296. S. Mohanty, D.P. Roy, U. Sarkar, Phys. Lett. B 444 (1998) 391.

Author index to volumes 311}320

Abramowicz, M.A., Gravitational radiation in optical geometry applied to supercompact stars Agasian, N.O., B.O. Kerbikov and V.I. Shevchenko, Nonperturbative QCD vacuum and colour superconductivity Agren, H., see F. Gel'mukhanov Ali, A. and D. London, Precision #avour physics and supersymmetry Altarelli, G. and F. Feruglio, Neutrino masses and mixings: a theoretical perspective Arecchi, F.T., S. Boccaletti and P. Ramazza, Pattern formation and competition in nonlinear optics Bailin, D. and A. Love, Orbifold compacti"cations of string theory Balantekin, A.B., Neutrino propagation in matter Balian, R., H. Flocard and M. VeH neH roni, Variational extensions of BCS theory Baron, E., see S.L. Pistinner Baschek, B., see R. Wehrse Baschek, B. and R. Wehrse, Radiation "elds in moving media: e!ects of many spectral lines in AGN accretion disks Beck, C., see S.J. Sanders Bender, C.M., The complex pendulum Beneke, M., Renormalons Boccaletti, S., see F.T. Arecchi BreH das, J.L., see W.R. Salaneck BuchmuK ller, W. and M. PluK macher, Matter}antimatter asymmetry and neutrino properties

311 (1999) 325 320 312 320 320

(1999) (1999) (1999) (1999)

131 87 79 295

318 (1999)

1

315 315 317 311 311

(1999) (1999) (1999) (1999) (1999)

285 123 251 151 187

311 311 315 317 318 319

(1999) (1999) (1999) (1999) (1999) (1999)

201 487 27 1 1 231

320 (1999) 329

Calogeracos, A., see N. Dombey Chanowitz, M.S., Quantum corrections from nonresonant WW scattering Chapline, G., Is theoretical physics the same thing as mathematics? Collin, S., Quasars, accretion disks, and pregalactic enrichment of galaxies and IGM Cooper, F., Inclusive dilepton production at RHIC: a "eld theory approach based on a non-equilibrium chiral phase transition

315 320 315 311

Dar, A. and G. Shaviv, The solar neutrino problem } an update de Pablo, J.J., see F.A. Escobedo Deb, B.M., see R. Singh

311 (1999) 115 318 (1999) 85 311 (1999) 47

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 9 8 - 8

(1999) (1999) (1999) (1999)

41 139 95 463

315 (1999) 59

380

Author index

Dolgov, A.D., Long-range forces in the universe Dombey, N. and A. Calogeracos, Seventy years of the Klein paradox Drell, S.D., Cooperation between scientists and the government in the US: bene"ts and problems Dubovsky, S.L., D.S. Gorbunov, M.V. Libanov and V.A. Rubakov, Non-renormalization of induced charges and constraints on strongly coupled theories Dvali, G. and M. Shifman, Tilting the brane, or some cosmological consequences of the brane Universe Eichler, D., see S.L. Pistinner El-Zant, A., Dissipative motion in galaxies with non-axisymmetric potentials Ellis, J. and N.E. Mavromatos, Comments on CP, T and CPT violation in neutral kaon decays Escobedo, F.A. and J.J. de Pablo, Molecular simulation of polymeric networks and gels: phase behavior and swelling Feruglio, F., see G. Altarelli Fineberg, J. and M. Marder, Instability in dynamic fracture Flocard, H., see R. Balian Friend, R.H., see W.R. Salaneck Gaidaenko, I.V., A.V. Novikov, V.A. Novikov, A.N. Rozanov and M.I. Vysotsky, Enhanced electroweak radiative corrections in SUSY and precision data Galt'sov, D.V., see M.S. Volkov Gehrz, R.D., Infrared studies of classical novae and their contributions to the ISM Gel'mukhanov, F. and H. Agren, Resonant X-ray Raman scattering Gershtein, S.S., A.A. Likhoded and A.I. Onishchenko, TeV-scale leptoquarks from GUTs/string/M-theory uni"cation Ginocchio, J.N., A relativistic symmetry in nuclei Glasner, S.A., Modeling multidimensional reactive #ows during nova outbursts } achievements and challenges Godon, P., Non-axisymmetric hydrodynamic instability and transition to turbulence in two-dimensional accretion discs Goldhaber, A.S., Dual con"nement of grand uni"ed monopoles? Gorbunov, D.S., see S.L. Dubovsky Goto, T., see I.V. Ostrovskii Grimmeiss, H.G., see I.V. Ostrovskii Hameury, J.M., see I. Idan Hatano, Y., Interaction of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states Hauschildt, P.H., see S.L. Pistinner Hayes, A.C., Nuclear structure issues determining neutrino-nucleus cross sections Haymaker, R.W., Con"nement studies in lattice QCD Heinz, U., see U.A. Wiedemann Hussein, M., see E. Timmermans

320 (1999) 1 315 (1999) 41 320 (1999) 17 320 (1999) 147 320 (1999) 107 311 (1999) 475 311 (1999) 279 320 (1999) 341 318 (1999) 85 320 313 317 319

(1999) 295 (1999) 1 (1999) 251 (1999) 231

320 (1999) 119 319 (1999) 1 311 (1999) 405 312 (1999) 87 320 (1999) 159 315 (1999) 231 311 (1999) 395 311 315 320 311 311

(1999) 271 (1999) 83 (1999) 147 (1999) 1 (1999) 1

311 (1999) 213 313 311 315 315 319 315

(1999) (1999) (1999) (1999) (1999) (1999)

109 151 257 153 145 199

381

Author index

Idan, I., J.P. Lasota, J.M. Hameury and G. Shaviv, Radiation from dwarf nova discs Ivanov, I.P., N.N. Nikolaev, A.V. Pronyaev and W. SchaK fer, How unitarity imposes a steep small-x rise of spin structure function g "g #g and breaking of DIS *2   sum rules

320 (1999) 175

Jackson, J.D., From Alexander of Aphrodisias to Young and Airy Jacob, M., Multiparticle production Joss, P.C., Massive supernovae in binary systems

320 (1999) 27 315 (1999) 7 311 (1999) 345

Kazakov, D.I., The Higgs boson: shall we see it soon or is it still far away? Kerbikov, B.O., see N.O. Agasian Kerman, A., see E. Timmermans Khriplovich, I.B., Particle creation by charged black holes King, A.R., Black hole and transient binaries KonjevicH , N., Plasma broadening and shifting of non-hydrogenic spectral lines: Present status and applications Korotchenkov, O.A., see I.V. Ostrovskii Kovetz, A., Nova evolution with optically thick winds Kramer, P.R., see A.J. Majda Kuzmin, V.A. and I.I. Tkachev, Ultra-high-energy cosmic rays and in#ation relics

320 320 315 320 311

(1999) (1999) (1999) (1999) (1999)

187 131 199 37 337

316 311 311 314 320

(1999) (1999) (1999) (1999) (1999)

339 1 383 237 199

Ladik, J.J., Polymers as solids: a quantum mechanical treatment Lai, R. and A.J. Sievers, Nonlinear nanoscale localization of magnetic excitations in atomic lattices Landsberg, L.G., Search for exotic baryons with hidden strangeness in di!ractive production processes Laor, A., Astrophysical evidence for massive black holes Lasota, J.-P., ADAFs } Models, observations and problems Lasota, J.P., see I. Idan Leike, A., The phenomenology of extra neutral gauge bosons Li, H., see F.C. Michel Libanov, M.V., see S.L. Dubovsky Lieb, E.H. and J. Yngvason, Erratum. The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1}96) Likhoded, A.A., see S.S. Gershtein Lipatov, L.N., Hamiltonian for Reggeon interactions in QCD Livio, M., Astrophysical jets: a phenomenological examination of acceleration and collimation London, D., see A. Ali Love, A., see D. Bailin Lyth, D.H. and A. Riotto, Particle physics models of in#ation and the cosmological density perturbation

313 (1999) 171

Main, J., Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra Majda, A.J. and P.R. Kramer, Simpli"ed models for turbulent di!usion: Theory, numerical modelling, and physical phenomena

311 (1999) 213

314 (1999) 147 320 311 311 311 317 318 320

(1999) (1999) (1999) (1999) (1999) (1999) (1999)

223 451 247 213 143 227 147

314 (1999) 669 320 (1999) 159 320 (1999) 249 311 (1999) 225 320 (1999) 79 315 (1999) 285 314 (1999)

1

316 (1999) 233 314 (1999) 237

382

Author index

Mandula, J.E., The gluon propagator Marder, M., see J. Fineberg Matinyan, S., Multiparton collisions and multiplicity distribution in high-energy pp(p p) collisions Mavromatos, N.E., see J. Ellis Mazeh, T., The mass distribution of extrasolar planet candidates and low-mass secondaries Mestel, L., The early days of stellar structure theory Michel, F.C. and H. Li, Electrodynamics of neutron stars

311 (1999) 317 311 (1999) 295 318 (1999) 227

Nakel, W. and C.T. Whelan, Relativistic (e, 2e) processes Nayak, R.C., see L. Satpathy Nekrasov, N. and S.L. Shatashvili, On non-supersymmetric CFT in four dimensions Nikolaev, N.N., see I.P. Ivanov Novikov, A.V., see I.V. Gaidaenko Novikov, V.A., see I.V. Gaidaenko

315 319 320 320 320 320

Obers, N.A. and B. Pioline, U-duality and M-theory Onishchenko, A.I., see S.S. Gershtein Orio, M., X-ray observations of classical and recurrent novae Ostrovskii, I.V., O.A. Korotchenkov, T. Goto and H.G. Grimmeiss, Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces

318 (1999) 113 320 (1999) 159 311 (1999) 419

Patera, J. and R. Twarock, Quasicrystal Lie algebras and their generalizations Pengpan, T. and P. Ramond, M(ysterious) patterns in SO(9) Pioline, B., see N.A. Obers Piran, T., Gamma-ray bursts and the "reball model Pistinner, S.L., P.H. Hauschildt, D. Eichler and E. Baron, On the primordial helium abundance and spectroscopic uncertainties Pistinner, S.L. and D. Eichler, Self-inhibiting heat #ux: a chance for snowballs in hell? PluK macher, M., see W. BuchmuK ller Pronyaev, A.V., see I.P. Ivanov Ra!elt, G.G., Limits on neutrino electromagnetic properties } an update Ramazza, P., see F.T. Arecchi Ramond, P., see T. Pengpan Reinhard, P.-G., see G. Zwicknagel Riotto, A., see D.H. Lyth Rozanov, A.N., see I.V. Gaidaenko Rubakov, V.A., see S.L. Dubovsky Ryzhikh, D.A. and K.A. Ter-Martirosyan, Mass and CKM matrices of quarks and leptons, the leptonic CP-phase in neutrino oscillations Salaneck, W.R., R.H. Friend and J.L. BreH das, Electronic structure of conjugated polymers: consequences of electron } lattice coupling Salpeter, E.E., Giant disk galaxies, dwarfs and Lyman alpha clouds Sanders, S.J., A. Szanto de Toledo and C. Beck, Binary decay of light nuclear systems

315 (1999) 273 313 (1999) 1 320 (1999) 261 320 (1999) 341

(1999) (1999) (1999) (1999) (1999) (1999)

311 (1999)

409 85 127 175 119 119

1

315 315 318 314

(1999) (1999) (1999) (1999)

241 137 113 575

311 311 320 320

(1999) (1999) (1999) (1999)

151 475 329 175

320 318 315 314 314 320 320

(1999) (1999) (1999) (1999) (1999) (1999) (1999)

319 1 137 671 1 119 147

320 (1999) 359

319 (1999) 231 311 (1999) 429 311 (1999) 487

383

Author index

Satpathy, L., V.S. Uma Maheswari and R.C. Nayak, Finite nuclei to nuclear matter: a leptodermous approach Schael, S., B physics at the Z-resonance SchaK fer, W., see I.P. Ivanov Schatzman, E., Role of gravity waves in the solar neutrino problem Schwarz, J.H., From superstrings to M theory Shabalin, E., The value of eH /e; in a theory with extended Higgs sector Shara, M.M., Stellar collisions and mergers in the cores of globular clusters Shatashvili, S.L., see N. Nekrasov Shaviv, G. and N.J. Shaviv, Is there a dynamic e!ect in the screening of nuclear reactions in stellar plasmas? Shaviv, G., see A. Dar Shaviv, G., see I. Idan Shaviv, N.J., see G. Shaviv Shaviv, N.J., The instability of radiative #ows: from the early universe to the Eddington luminosity limit Shevchenko, V.I., see N.O. Agasian Shifman, M., see G. Dvali Shlosman, I., Stellar bars in disk galaxies: From banyans to butter#ies? Sievers, A.J., see R. Lai Simonov, Yu.A., Perturbative}nonperturbative interference in the static QCD interaction at small distances Singh, R. and B.M. Deb, Developments in excited-state density functional theory Sion, E.M., HST studies of cataclysmic variable white dwarfs Smak, J., Cataclysmic variables Soker, N. , Axisymmetrical structures of planetary nebulae and SN 1987A Sornette, D., Earthquakes: from chemical alteration to mechanical rupture Spiegel, E.A. and L. Tao, Photo#uid instabilities of hot stellar envelopes Starr"eld, S., Recent advances in studies of the nova outburst Stodolsky, L., Decoherence}#uctuation relation and measurement noise Szanto de Toledo, A., see S.J. Sanders

319 313 320 311 315 320 311 320

(1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999)

85 293 175 143 107 355 363 127

311 311 311 311

(1999) (1999) (1999) (1999)

99 115 213 99

311 320 320 311 314

(1999) (1999) (1999) (1999) (1999)

177 131 107 439 147

320 311 311 311 311 313 311 311 320 311

(1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999)

265 47 353 331 307 237 163 371 51 487

Tan, C.-I., Di!ractive production at collider energies and factorization Tao, L., see E.A. Spiegel Ter-Martirosyan, K.A., see D.A. Ryzhikh Timmermans, E., P. Tommasini, M. Hussein and A. Kerman, Feshbach resonances in atomic Bose}Einstein condensates Tkachev, I.I., see V.A. Kuzmin Toep!er, C., see G. Zwicknagel Tommasini, P., see E. Timmermans Twarock, R., see J. Patera

315 (1999) 175 311 (1999) 163 320 (1999) 359

Uma Maheswari, V.S., see L. Satpathy

319 (1999) 85

VeH neH roni, M., see R. Balian Volkov, M.S. and D.V. Galt'sov, Gravitating non-Abelian solitons and black holes with Yang}Mills "elds

317 (1999) 251

315 320 314 315 315

(1999) (1999) (1999) (1999) (1999)

319 (1999)

199 199 671 199 241

1

384

Author index

Voloshin, M.B., Relations between inclusive decay rates of heavy baryons Vysotsky, M.I., see I.V. Gaidaenko

320 (1999) 275 320 (1999) 119

Wagoner, R.V., Relativistic diskoseismology Wehrse, R. and B. Baschek, Radiation "elds in moving media: new analytical and numerical solutions of the transfer equation Wehrse, R., see B. Baschek Whelan, C.T., see W. Nakel Wiedemann, U.A. and U. Heinz, Particle interferometry for relativistic heavy-ion collisions Wolle, B., Tokamak plasma diagnostics based on measured neutron signals

311 (1999) 259

319 (1999) 145 312 (1999) 1

YnduraH in, F.J., Gluon condensate from superconvergent QCD sum rule Yngvason, J., see E.H. Lieb Youm, D., Black holes and solitons in string theory

320 (1999) 287 314 (1999) 669 316 (1999) 1

Zakharov, V.I., Anatomy of a con"ning string Zwicknagel, G., C. Toep!er and P.-G. Reinhard, Erratum. Stopping of heavy ions in plasmas at strong coupling (Physics Reports 309 (1999) 117}208)

320 (1999) 59

311 (1999) 187 311 (1999) 201 315 (1999) 409

314 (1999) 671

Subject index to volumes 311}320 General Is there a dynamic e!ect in the screening of nuclear reactions in stellar plasmas?, G. Shaviv and N.J. Shaviv Role of gravity waves in the solar neutrino problem, E. Schatzman ADAFs } Models, observations and problems, J.-P. Lasota Relativistic diskoseismology, R.V. Wagoner Gravitational radiation in optical geometry applied to super-compact stars, M.A. Abramowicz Earthquakes: from chemical alteration to mechanical rupture, D. Sornette Simpli"ed models for turbulent di!usion: Theory, numerical modelling, and physical phenomena, A.J. Majda and P.R. Kramer Multiparticle production, M. Jacob The complex pendulum, C.M. Bender Seventy years of the Klein paradox, N. Dombey and A. Calogeracos Is theoretical physics the same thing as mathematics?, G. Chapline From superstrings to M theory, J.H. Schwarz M(ysterious) patterns in SO(9), T. Pengpan and P. Ramond Feshbach resonances in atomic Bose}Einstein condensates, E. Timmermans, P. Tommasini, M. Hussein and A. Kerman A relativistic symmetry in nuclei, J.N. Ginocchio Quasicrystal Lie algebras and their generalizations, J. Patera and R. Twarock Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra, J. Main Renormalons, M. Beneke Variational extensions of BCS theory, R. Balian, H. Flocard and M. VeH neH roni Pattern formation and competition in nonlinear optics, F.T. Arecchi, S. Boccaletti and P. Ramazza Gravitating non-Abelian solitons and black holes with Yang}Mills "elds, M.S. Volkov and D.V. Galt'sov Long-range forces in the universe, A.D. Dolgov Cooperation between scientists and the government in the US: bene"ts and problems, S.D. Drell From Alexander of Aphrodisias to Young and Airy, J.D. Jackson Particle creation by charged black holes, I.B. Khriplovich Decoherence}#uctuation relation and measurement noise, L. Stodolsky

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 9 9 - X

311 311 311 311

(1999) (1999) (1999) (1999)

99 143 247 259

311 (1999) 325 313 (1999) 237 314 315 315 315 315 315 315

(1999) (1999) (1999) (1999) (1999) (1999) (1999)

237 7 27 41 95 107 137

315 (1999) 199 315 (1999) 231 315 (1999) 241 316 (1999) 233 317 (1999) 1 317 (1999) 251 318 (1999)

1

319 (1999) 320 (1999)

1 1

320 320 320 320

(1999) (1999) (1999) (1999)

17 27 37 51

386

Subject index

The physics of elementary particles and 5elds B physics at the Z-resonance, S. Schael The complex pendulum, C.M. Bender Seventy years of the Klein paradox, N. Dombey and A. Calogeracos Inclusive dilepton production at RHIC: a "eld theory approach based on a nonequilibrium chiral phase transition, F. Cooper Dual con"nement of grand uni"ed monopoles?, A.S. Goldhaber From superstrings to M theory, J.H. Schwarz Neutrino propagation in matter, A.B. Balantekin M(ysterious) patterns in SO(9), T. Pengpan and P. Ramond Con"nement studies in lattice QCD, R.W. Haymaker Di!ractive production at collider energies and factorization, C.-I. Tan The gluon propagator, J.E. Mandula Orbifold compacti"cations of string theory, D. Bailin and A. Love Black holes and solitons in string theory, D. Youm Renormalons, M. Beneke The phenomenology of extra neutral gauge bosons, A. Leike U-duality and M-theory, N.A. Obers and B. Pioline Gravitating non-Abelian solitons and black holes with Yang}Mills "elds, M.S. Volkov and D.V. Galt'sov Particle creation by charged black holes, I.B. Khriplovich Anatomy of a con"ning string, V.I. Zakharov Precision #avour physics and supersymmetry, A. Ali and D. London Tilting the brane, or some cosmological consequences of the brane Universe, G. Dvali and M. Shifman Enhanced electroweak radiative corrections in SUSY and precision data, I.V. Gaidaenko, A.V. Novikov, V.A. Novikov, A.N. Rozanov and M.I. Vysotsky On non-supersymmetric CFT in four dimensions, N. Nekrasov and S.L. Shatashvili Nonperturbative QCD vacuum and colour superconductivity, N.O. Agasian, B.O. Kerbikov and V.I. Shevchenko Quantum corrections from nonresonant WW scattering, M.S. Chanowitz Non-renormalization of induced charges and constraints on strongly coupled theories, S.L. Dubovsky, D.S. Gorbunov, M.V. Libanov and V.A. Rubakov TeV-scale leptoquarks from GUTs/string/M-theory uni"cation, S.S. Gershtein, A.A. Likhoded and A.I. Onishchenko How unitarity imposes a steep small-x rise of spin structure function g "g #g *2   and breaking of DIS sum rules, I.P. Ivanov, N.N. Nikolaev, A.V. Pronyaev and W. SchaK fer The Higgs boson: shall we see it soon or is it still far away?, D.I. Kazakov Search for exotic baryons with hidden strangeness in di!ractive production processes, L.G. Landsberg Hamiltonian for Reggeon interactions in QCD, L.N. Lipatov Multiparton collisions and multiplicity distribution in high-energy pp(p p) collisions, S. Matinyan Perturbative}nonperturbative interference in the static QCD interaction at small distances, Yu.A. Simonov Relations between inclusive decay rates of heavy baryons, M.B. Voloshin

313 (1999) 293 315 (1999) 27 315 (1999) 41 315 315 315 315 315 315 315 315 315 316 317 317 318

(1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999)

59 83 107 123 137 153 175 273 285 1 1 143 113

319 320 320 320

(1999) 1 (1999) 37 (1999) 59 (1999) 79

320 (1999) 107 320 (1999) 119 320 (1999) 127 320 (1999) 131 320 (1999) 139 320 (1999) 147 320 (1999) 159

320 (1999) 175 320 (1999) 187 320 (1999) 223 320 (1999) 249 320 (1999) 261 320 (1999) 265 320 (1999) 275

387

Subject index

Gluon condensate from superconvergent QCD sum rule, F.J. YnduraH in Neutrino masses and mixings: a theoretical perspective, G. Altarelli and F. Feruglio Limits on neutrino electromagnetic properties } an update, G.G. Ra!elt Matter}antimatter asymmetry and neutrino properties, W. BuchmuK ller and M. PluK macher Comments on CP, T and CPT violation in neutral kaon decays, J. Ellis and N.E. Mavromatos The value of eH /e in a theory with extended Higgs sector, E. Shabalin Mass and CKM matrices of quarks and leptons, the leptonic CP-phase in neutrino oscillations, D.A. Ryzhikh and K.A. Ter-Martirosyan

320 (1999) 287 320 (1999) 295 320 (1999) 319 320 (1999) 329 320 (1999) 341 320 (1999) 355 320 (1999) 359

Nuclear physics The solar neutrino problem } an update, A. Dar and G. Shaviv Role of gravity waves in the solar neutrino problem, E. Schatzman Binary decay of light nuclear systems, S.J. Sanders, A. Szanto de Toledo and C. Beck Inclusive dilepton production at RHIC: a "eld theory approach based on a nonequilibrium chiral phase transition, F. Cooper Neutrino propagation in matter, A.B. Balantekin A relativistic symmetry in nuclei, J.N. Ginocchio Nuclear structure issues determining neutrino-nucleus cross sections, A.C. Hayes Variational extensions of BCS theory, R. Balian, H. Flocard and M. VeH neH roni U-duality and M-theory, N.A. Obers and B. Pioline Finite nuclei to nuclear matter: a leptodermous approach, L. Satpathy, V.S. Uma Maheswari and R.C. Nayak Particle interferometry for relativistic heavy-ion collisions, U.A. Wiedemann and U. Heinz

311 (1999) 115 311 (1999) 143 311 (1999) 487 315 315 315 315 317 318

(1999) (1999) (1999) (1999) (1999) (1999)

59 123 231 257 251 113

319 (1999) 85 319 (1999) 145

Atomic and molecular physics Developments in excited-state density functional theory, R. Singh and B.M. Deb Resonant X-ray Raman scattering, F. Gel'mukhanov and H. Agren Interaction of vacuum ultraviolet photons with molecules. Formation and dissociation dynamics of molecular superexcited states, Y. Hatano Feshbach resonances in atomic Bose}Einstein condensates, E. Timmermans, P. Tommasini, M. Hussein and A. Kerman Relativistic (e, 2e) processes, W. Nakel and C.T. Whelan Plasma broadening and shifting of non-hydrogenic spectral lines: Present status and applications, N. KonjevicH Molecular simulation of polymeric networks and gels: phase behavior and swelling, F.A. Escobedo and J.J. de Pablo Electronic structure of conjugated polymers: consequences of electron}lattice coupling, W.R. Salaneck, R.H. Friend and J.L. BreH das

311 (1999) 47 312 (1999) 87 313 (1999) 109 315 (1999) 199 315 (1999) 409 316 (1999) 339 318 (1999) 85 319 (1999) 231

Classical areas of phenomenology (including applications) Photo#uid instabilities of hot stellar envelopes, E.A. Spiegel and L. Tao Radiation "elds in moving media: new analytical and numerical solutions of the transfer equation, R. Wehrse and B. Baschek

311 (1999) 163 311 (1999) 187

388

Subject index

Radiation "elds in moving media: e!ects of many spectral lines in AGN accretion disks, B. Baschek and R. Wehrse Modeling multidimensional reactive #ows during nova outbursts } achievements and challenges, S.A. Glasner Instability in dynamic fracture, J. Fineberg and M. Marder Nonlinear nanoscale localization of magnetic excitations in atomic lattices, R. Lai and A.J. Sievers Simpli"ed models for turbulent di!usion: Theory, numerical modelling, and physical phenomena, A.J. Majda and P.R. Kramer Electronic structure of conjugated polymers: consequences of electron}lattice coupling, W.R. Salaneck, R.H. Friend and J.L. BreH das From Alexander of Aphrodisias to Young and Airy, J.D. Jackson

311 (1999) 201 311 (1999) 395 313 (1999) 1 314 (1999) 147 314 (1999) 237 319 (1999) 231 320 (1999) 27

Fluids, plasmas and electric discharges Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces, I.V. Ostrovskii, O.A. Korotchenkov, T. Goto and H.G. Grimmeiss Self-inhibiting heat #ux: a chance for snowballs in hell?, S.L. Pistinner and D. Eichler Tokamak plasma diagnostics based on measured neutron signals, B. Wolle Plasma broadening and shifting of non-hydrogenic spectral lines: Present status and applications, N. KonjevicH

311 (1999) 1 311 (1999) 475 312 (1999) 1 316 (1999) 339

Condensed matter: structure, thermal and mechanical properties Instability in dynamic fracture, J. Fineberg and M. Marder Nonlinear nanoscale localization of magnetic excitations in atomic lattices, R. Lai and A.J. Sievers Feshbach resonances in atomic Bose}Einstein condensates, E. Timmermans, P. Tommasini, M. Hussein and A. Kerman Quasicrystal Lie algebras and their generalizations, J. Patera and R. Twarock Molecular simulation of polymeric networks and gels: phase behavior and swelling, F.A. Escobedo and J.J. de Pablo

313 (1999)

1

314 (1999) 147 315 (1999) 199 315 (1999) 241 318 (1999) 85

Condensed matter: electronic structure, electrical, magnetic and optical properties Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces, I.V. Ostrovskii, O.A. Korotchenkov, T. Goto and H.G. Grimmeiss Resonant X-ray Raman scattering, F. Gel'mukhanov and H. Agren Polymers as solids: a quantum mechanical treatment, J.J. Ladik Nonlinear nanoscale localization of magnetic excitations in atomic lattices, R. Lai and A.J. Sievers Variational extensions of BCS theory, R. Balian, H. Flocard and M. VeH neH roni Electronic structure of conjugated polymers: consequences of electron}lattice coupling, W.R. Salaneck, R.H. Friend and J.L. BreH das

311 (1999) 1 312 (1999) 87 313 (1999) 171 314 (1999) 147 317 (1999) 251 319 (1999) 231

Cross-disciplinary physics and related areas of science and technology Sonoluminescence and acoustically driven optical phenomena in solids and solid}gas interfaces, I.V. Ostrovskii, O.A. Korotchenkov, T. Goto and H.G. Grimmeiss

311 (1999)

1

389

Subject index

Resonant X-ray Raman scattering, F. Gel'mukhanov and H. Agren Instability in dynamic fracture, J. Fineberg and M. Marder

312 (1999) 87 313 (1999) 1

Geophysics, astronomy and astrophysics Is there a dynamic e!ect in the screening of nuclear reactions in stellar plasmas?, G. Shaviv and N.J. Shaviv The solar neutrino problem } an update, A. Dar and G. Shaviv Role of gravity waves in the solar neutrino problem, E. Schatzman On the primordial helium abundance and spectroscopic uncertainties, S.L. Pistinner, P.H. Hauschildt, D. Eichler and E. Baron Photo#uid instabilities of hot stellar envelopes, E.A. Spiegel and L. Tao The instability of radiative #ows: from the early universe to the Eddington luminosity limit, N.J. Shaviv Radiation "elds in moving media: new analytical and numerical solutions of the transfer equation, R. Wehrse and B. Baschek Radiation "elds in moving media: e!ects of many spectral lines in AGN accretion disks, B. Baschek and R. Wehrse Radiation from dwarf nova discs, I. Idan, J.P. Lasota, J.M. Hameury and G. Shaviv Astrophysical jets: a phenomenological examination of acceleration and collimation, M. Livio ADAFs } Models, observations and problems, J.-P. Lasota Relativistic diskoseismology, R.V. Wagoner Non-axisymmetric hydrodynamic instability and transition to turbulence in twodimensional accretion discs, P. Godon Dissipative motion in galaxies with non-axisymmetric potentials, A. El-Zant The early days of stellar structure theory, L. Mestel Axisymmetrical structures of planetary nebulae and SN 1987A, N. Soker The mass distribution of extrasolar planet candidates and low-mass secondaries, T. Mazeh Gravitational radiation in optical geometry applied to super-compact stars, M.A. Abramowicz Cataclysmic variables, J. Smak Black hole and transient binaries, A.R. King Massive supernovae in binary systems, P.C. Joss HST studies of cataclysmic variable white dwarfs, E.M. Sion Stellar collisions and mergers in the cores of globular clusters, M.M. Shara Recent advances in studies of the nova outburst, S. Starr"eld Nova evolution with optically thick winds, A. Kovetz Modeling multidimensional reactive #ows during nova outbursts } achievements and challenges, S.A. Glasner Infrared studies of classical novae and their contributions to the ISM, R.D. Gehrz X-ray observations of classical and recurrent novae, M. Orio Giant disk galaxies, dwarfs and Lyman alpha clouds, E.E. Salpeter Stellar bars in disk galaxies: From banyans to butter#ies?, I. Shlosman Astrophysical evidence for massive black holes, A. Laor Quasars, accretion disks, and pregalactic enrichment of galaxies and IGM, S. Collin Self-inhibiting heat #ux: a chance for snowballs in hell?, S.L. Pistinner and D. Eichler

311 (1999) 99 311 (1999) 115 311 (1999) 143 311 (1999) 151 311 (1999) 163 311 (1999) 177 311 (1999) 187 311 (1999) 201 311 (1999) 213 311 (1999) 225 311 (1999) 247 311 (1999) 259 311 311 311 311

(1999) (1999) (1999) (1999)

271 279 295 307

311 (1999) 317 311 311 311 311 311 311 311 311

(1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999)

325 331 337 345 353 363 371 383

311 311 311 311 311 311 311 311

(1999) (1999) (1999) (1999) (1999) (1999) (1999) (1999)

395 405 419 429 439 451 463 475

390

Subject index

Earthquakes: from chemical alteration to mechanical rupture, D. Sornette Particle physics models of in#ation and the cosmological density perturbation, D.H. Lyth and A. Riotto Gamma-ray bursts and the "reball model, T. Piran Neutrino propagation in matter, A.B. Balantekin Electrodynamics of neutron stars, F.C. Michel and H. Li Ultra-high-energy cosmic rays and in#ation relics, V.A. Kuzmin and I.I. Tkachev

313 (1999) 237 314 314 315 318 320

(1999) (1999) (1999) (1999) (1999)

1 575 123 227 199