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K2l and q\K!Mx. All information about the response of the target to the high-energy virtual photon is in the hadronic tensor
= (q, P)& dy e O W1P"J (y)J (0)"P2 , IJ I J
(4.1)
(see Eq. (2.24)). Using q ) y" (q>y\#q\y>)!q ) y Kly\!(Mx/2)y> , , ,
(4.2)
one obtains the following coordinate-space resolutions along the light-cone distances y!"t$y : dy\&1/l and
dy>&1/Mx .
(4.3)
At y\"0 the current correlation function in Eq. (4.1) is not analytic since it vanishes for y>y\!( y )(0 because of causality (see e.g. [7]). Indeed in perturbation theory it turns out to , be singular at y\"0. Assuming that the integrand in (4.1) is an analytic function of y\ elsewhere, this implies that = is dominated for q>PR by contributions from y\"0. Causality implies IJ that, in the transverse plane, only contributions from ( y )K1/Q are relevant: deep-inelastic , scattering is dominated by contributions from the light cone, i.e. y"0. Furthermore, Eq. (4.3) suggests that one probes increasing distances along the light cone as x is decreased. Such a behavior is consistent with approximate Bjorken scaling [118]. The coordinate}space analysis of nucleon structure functions in Section 4.3 con"rms this conjecture. In the
G. Piller, W. Weise / Physics Reports 330 (2000) 1}94
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Fig. 4.1. Two examples of diagrams illustrating the space}time pattern of deep inelastic scattering.
Bjorken limit the dominant contributions to the hadronic tensor at small x come from light-like separations of order y>&1/(Mx) between the electromagnetic currents in (4.1). In the laboratory frame these considerations imply that deep-inelastic scattering involves a longitudinal correlation length y Ky>/2,l
(4.4)
of the virtual photon. Consequently, large longitudinal distances are important in the scattering process at small x. This can also be deduced in the framework of time-ordered perturbation theory (see Section 4.5), where l determines the typical propagation length of hadronic con"gurations present in the interacting photon. The space}time pattern of deep-inelastic scattering is illustrated in Fig. 4.1 in terms of the imaginary part of the forward Compton amplitude: the virtual photon interacts with partons which propagate a distance y> along the light cone. The characteristic laboratory frame correlation length l is one-half of that distance. 4.2. Coordinate-space distribution functions Especially when it comes to the discussion of the relevant space}time scales which govern nuclear e!ects in deep-inelastic scattering, it is instructive to look at quark and gluon distribution functions in coordinate rather than in momentum space. In this section we prepare the facts and return to the underlying dynamics at a later stage. It is useful to express coordinate-space distributions in terms of a suitable dimensionless variable. For this purpose let us introduce the light-like vector nI with n"0 and P ) n"P !P . The hadronic tensor receives its dominant contributions from the vicinity of the light cone, where y is approximately parallel to n. The dimensionless variable z"y ) P then plays the role of a coordinate conjugate to Bjorken x. It is helpful to bear in mind that the value z"5 corresponds to a light-cone distance y>"2z/M+2 fm in the laboratory frame or, equivalently, to a longitudinal distance l,y>/2+1 fm.
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In accordance with the charge conjugation (C) properties of momentum-space quark and gluon distributions, one de"nes coordinate-space distributions by [121]
Q(z, Q),
dx [q(x, Q)#q (x, Q)]sin(zx) ,
(4.5)
dx [q(x, Q)!q (x, Q)]cos(zx) , (4.6) G(z, Q), dx xg(x, Q) cos(zx) , (4.7) where q, q and g are the momentum-space quark, antiquark and gluon distributions, respectively. Flavor indices are suppressed here for simplicity. At leading twist accuracy, the coordinate-space distributions (4.5)}(4.7) are related to forward matrix elements of non-local QCD operators on the light cone [122,123]: Q (z, Q), T
1 1P"tM (y)n/ C(y)t(0)"P2 !(y !y) , Q(z, Q)" / 4iP ) n
(4.8)
1 1P"tM (y)n/ C(y)t(0)"P2 #(y !y) , Q (z, Q)" / T 4P ) n
(4.9)
G(z, Q)"nInJ
1 1P"G (y)C(y)GH(0)"P2 . IH J / 2(P ) n)
(4.10)
Here t denotes the quark "eld and G the gluon "eld strength tensor. The path-ordered IJ exponential (4.11) C(y)"P exp igyI dj A (jy) , I where g denotes the strong coupling constant and AI the gluon "eld, ensures gauge invariance of the parton distributions. Note that an expansion of the right-hand side of Eqs. (4.5)}(4.7) and (4.8)}(4.10) around y"0 (and hence z"y ) P"0) leads to the conventional operator product expansion for parton distributions [6}8]. The functions Q(z), Q (z) and G(z) describe the mobility of partons in coordinate-space. Consider, T for example, the valence quark distribution Q (z). The matrix element in (4.9) has an obvious T physical interpretation: as illustrated in Fig. 4.1a, it measures the overlap between the nucleon ground state and a state in which one quark has been displaced along the light cone from 0 to y. A di!erent sequence is shown in Fig. 4.1b. There the photon converts into a beam of partons which propagates along the light cone and interacts with partons of the target nucleon, probing primarily its sea quark and gluon content.
4.3. Coordinate-space distributions of free nucleons In this section we discuss the properties of coordinate-space distribution functions of free nucleons. Examples of the distributions (4.8)}(4.10) using the CTEQ4L parametrization [124] of momentum-space quark and gluon distributions taken at a momentum scale Q"4 GeV, are shown in Fig. 4.2.
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Fig. 4.2. Coordinate-space quark and gluon distributions resulting from the CTEQ4L parametrization of momentumspace distributions, taken at a momentum transfer Q"4 GeV. A sum over the u and d quarks is implied in the functions Q and Q [117]. T
Some general features can be observed: the C-even quark distribution Q(z) rises at small values of z, develops a plateau at z 9 5, and then exhibits a slow rise at very large z. At z:5, the gluon distribution function zG(z) behaves similarly as Q(z). For z 9 5, zG(z) rises somewhat faster than Q(z). The C-odd (or valence) quark distribution Q (z) starts with a "nite value at small z, then begins T to fall at zK3 and vanishes at large z. Recall that in the laboratory frame, the scale zK5 at which a signi"cant change in the behavior of coordinate-space distributions occurs, represents a longitudinal distance comparable to the typical size of a nucleon. At z(5 the coordinate-space distributions are determined by average properties of the corresponding momentum-space distribution functions as expressed by their "rst few moments [125,126]. For example, the derivative of the C-even quark distribution Q(z) taken at z"0 equals the fraction of the nucleon light-cone momentum carried by quarks. The same is true for the gluon distribution zG(z) (the momentum fractions carried by quarks and by gluons are in fact approximately equal, a well-known experimental fact). At z'10 the coordinate-space distributions are determined by the small-x behavior of the corresponding momentum-space distributions. Assuming, for example, q(x)&x@ for x(0.05 implies Q(z)&z\@\ at z'10. Similarly, the small-x behavior g(x)&x@ leads to zG(z)&z\@\ at large z. For typical values of b as suggested by Regge phenomenology [22] one obtains Q &z\ while Q(z) and zG(z) become constant at very large z. T The fact that Q(z) and zG(z) extend over large distances has a natural interpretation in the laboratory frame. At correlation lengths l much larger than the nucleon size, both Q(z) and zG(z) re#ect primarily the partonic structure of the photon which behaves like a high-energy beam of gluons and quark}antiquark pairs incident on the nucleon. For similar reasons, the valence quark distribution Q (z) has a pronounced tail which extends to distances beyond the nucleon radius. An T antiquark in the `beama can annihilate with a valence quark of the target nucleon, giving rise to long-distance contributions in Q . A detailed and instructive discussion of this frequently ignored T feature can be found in Ref. [127]. Finally we illustrate the relevance of large distances in deep-inelastic scattering at small x. In Fig. 4.3 we show contributions to the nucleon structure function F in coordinate-space, dx F,(x, Q) sin(zx) , (4.12) F (z, Q)" x
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Fig. 4.3. Contributions from di!erent regions in x to the F combination of coordinate-space quark and antiquark distributions at Q"4 GeV [117].
which result from di!erent windows of Bjorken x. This con"rms once more that contributions from large distances &1/(Mx) dominate at small x. 4.4. Coordinate-space distributions of nuclei The implications for scattering from nuclear targets, especially for coherence phenomena, are now obvious. If one compares, in the laboratory frame, the longitudinal correlation length l from Eq. (4.4) with the average nucleon}nucleon distance in the nucleus, dK2 fm, one can clearly distinguish two separate regions: (i) At small distances, l(d, the virtual photon scatters incoherently from the individual hadronic constituents of the target nucleus. Possible modi"cations of the coordinate distribution functions (4.5)}(4.7) in this region are caused by bulk nuclear e!ects such as binding and Fermi motion. (ii) At larger distances, l'd, it is likely that several nucleons participate collectively in the interaction. Modi"cations of the coordinate distribution functions are now expected to come from the coherent scattering on at least two nucleons in the target. Using l&1/(2Mx), this region corresponds to x:0.05. This suggests that the nuclear modi"cations seen in coordinate-space distributions will be quite di!erent in the regions l'2 fm and l(2 fm. This is best demonstrated by studying the ratios of nuclear and nucleon coordinate-space distribution functions: (dx/x) F (x, Q) sin(zx) e Q (z, Q) " D D D , R (z, Q)" $ (dx/x) F,(x, Q) sin(zx) e Q,(z, Q) D D D R (z, Q)"Q (z, Q)/Q,(z, Q) , T T T RG (z, Q)"G (z, Q)/G,(z, Q) .
(4.13) (4.14) (4.15)
The ratios R have been obtained for di!erent nuclei from an analysis of the measured $ momentum-space structure functions [120]. Furthermore, the ratios of valence quark and gluon
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Fig. 4.4. The coordinate space ratio R at Q"5 GeV for He, C, and Ca from Ref. [120]. $ Fig. 4.5. Coordinate-space ratios at Q"4 GeV for gluon distributions, valence-quark distributions, and the F structure function in Ca [117].
distributions have been calculated in [117] as sine and cosine Fourier transforms (4.5)}(4.7) of momentum-space distribution functions which result from an analysis of nuclear DIS and Drell}Yan data [128] (see also Section 5.6). In Fig. 4.4 we show the ratio R for Q"5 GeV taken from [120]. The most prominent $ feature is the pronounced depletion of R at l'2 fm caused by nuclear shadowing. At l:1 fm, $ nuclear modi"cations of R are small, and deep-inelastic scattering proceeds incoherently from $ the hadronic constituents of the target nucleus. The intrinsic structure of individual nucleons is evidently not much a!ected by nuclear mean "elds. In momentum-space, on the other hand, the pronounced nuclear dependence of the structure function F at x'0.1 evidently results from a superposition of long- and short-distance contributions as seen in Fig. 4.3. (For a detailed discussion see Ref. [120].) In Fig. 4.5 we show the valence quark and gluon ratios R and RG for Ca from Ref. [117]. T They behave similarly as the structure function ratio R , where the depletion of gluons $ at large distances is most pronounced. It is interesting to observe that in coordinate-space, shadowing sets in at approximately the same value of l for all sorts of partons. In momentum space, shadowing is found to start at di!erent values of x for di!erent distributions [128]. Finally note that the shadowing e!ect continues to increase for distances larger than the nuclear diameter. The results shown in Fig. 4.5 clearly emphasize the important role of gluons in the shadowing process. Of course, the incident virtual photon does not directly `seea the gluons. In the primary step the photon converts into a quark}antiquark pair. At small Bjorken-x, the subsequent QCD evolution of this pair rapidly induces a cascade of gluons. This cascade propagates along the light cone over distances which can exceed typical nuclear diameters by far: the high energy, high Q photon behaves in part like a gluon beam which scatters coherently from the nucleus. This o!ers interesting new physics. The detailed QCD analysis of nuclear shadowing can, in fact, give information on the `cross sectiona p for gluons incident on nucleons, and a simple eikonal E, estimate using R at asymptotic distances l suggests that this p is indeed large, comparable to E E, typical hadronic cross sections (see also Refs. [129,130]).
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Fig. 4.6. The two possible time orderings for the interaction of a (virtual) photon with a nucleon or nuclear target: (a) the photon hits a quark in the target, (b) the photon creates a qq pair that subsequently interacts with the target.
In summary, a coordinate-space representation which selects contributions from di!erent longitudinal distances, lucidly demonstrates that nuclear e!ects of the structure function F and parton distributions are by far dominated by shadowing and have a surprisingly simple geometric interpretation. 4.5. Deep-inelastic scattering in standard perturbation theory It is instructive to illustrate the previous results by looking at the lab frame space}time pattern of the (virtual) photon}nucleon interaction from the point of view of standard time-ordered perturbation theory. The two basic time orderings are shown in Figs. 4.6a and b: (a) the photon hits a quark or antiquark in the target which picks up the large energy and momentum transfer; (b) the photon converts into a quark}antiquark pair which propagates and subsequently interacts with the target. For small Bjorken-x the pair production process (b) dominates the scattering amplitude, as already mentioned. This can also be easily seen in time-ordered perturbation theory as follows (see e.g. [25] and references therein): the amplitudes A and A of processes (a) and (b) are roughly proportional ? @ to the inverse of their corresponding energy denominators *E and *E . For large energy transfers ? @ l<M one "nds: *E "E (t )!E (t )+!1p2 #(1p2#Q)/2l , (4.16) ? ? ? O O *E "E (t )!E (t )+(k#Q)/2l , (4.17) @ @ @ where 1p2 is the average quark momentum in a nucleon and k is the invariant mass of the O quark}antiquark pair. We then obtain for the ratio of these amplitudes: "A /A "&"*E /*E "+(Mx/1p2 )(1#k/Q) . (4.18) ? @ @ ? O When analyzing the spectral representation of the scattering amplitude one observes that the bulk contribution to process (b) results from those hadronic components in the photon wave function
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Fig. 4.7. Deep-inelastic scattering at small x;1 in the laboratory frame proceeds via hadronic #uctuation present in the photon wave function.
which have a squared mass k&Q (see Section 5.4.1). The ratio in Eq. (4.18) is evidently small for x;0.1. Hence pair production, Fig. 4.6b, is the leading lab frame process in the small-x region. On the other hand, at x'0.1, both mechanisms (a) and (b) contribute. In process (b) the photon couples to a quark pair which forms a complex (hadronic or quark}gluon) intermediate state and then scatters from the target. At small x deep-inelastic scattering can therefore be described in the laboratory frame in terms of the interaction of quark}gluon components present in the wave function of the virtual photon (Fig. 4.7). The longitudinal propagation length j of a speci"c photon-induced quark}gluon #uctuation with mass k is given by the inverse of the energy denominator (4.17): 2l I%/ 1 1 " & , (4.19) j& k#Q 2xM *E @ which coincides with the longitudinal correlation length l of Eq. (4.4). For x(0.05 the propagation length j exceeds the average distance between nucleons in nuclei, j'dK2 fm. For a nuclear target, coherent multiple scattering of quark}gluon #uctuations of the photon from several nucleons in the nucleus can then occur, and this is clearly seen in the coordinate-space analysis discussed in the previous section. For larger values of the Bjorken variable, x'0.2, the propagation length of intermediate hadronic states is small, j(d. At the same time the process in Fig. 4.6a becomes prominent, i.e. the virtual photon is absorbed directly by a quark or antiquark in the target. Now the incoherent scattering from the hadronic constituents of the nucleus dominates. 4.6. Nuclear deep-inelastic scattering in the inxnite momentum frame Let us "nally view the deep-inelastic scattering process in the so-called in"nite momentum frame where the target momentum is large. In this frame the standard parton model applies in which a snapshot of the target at the short time scale of the interaction reveals an ensemble of almost non-interacting partons, i.e. quarks and gluons. Consider the scattering from a nucleus which moves with large longitudinal momentum P +AP PR, where P is the average longitudinal momentum of the bound nucleons , , [131}133]. The average nucleon}nucleon distance in nuclei is now Lorentz contracted as
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G. Piller, W. Weise / Physics Reports 330 (2000) 1}94
compared to the lab frame: d +d M /P +2 fm M/P . On the other hand, the delocalization of , a parton with longitudinal momentum fraction x in the nucleon is given according to the Heisenberg uncertainty principle by dl+1/xP . At small Bjorken-x, x(1/d P +0.1, the wave , , functions of partons from di!erent nucleons have a chance to overlap, i.e. d (dl. Therefore, at x;0.1 we expect an enhanced interaction between partons coming from di!erent nucleons. One can anticipate that, at x;0.1, the parton delocalization extends over the whole nucleus. In the lab frame, this is where the quark and gluon #uctuations of the photon interact simultaneously with the parton content of several nucleons.
5. Shadowing in unpolarized deep-inelastic scattering As outlined in Section 3.3, the most pronounced nuclear e!ect in lepton-nucleus DIS is shadowing. For small values of the Bjorken variable (x(0.1), the nuclear structure functions F are signi"cantly reduced as compared to the free nucleon structure function F,. Equivalently, the virtual photon}nucleus cross section is less than A times the one for free nucleons, p H (Ap H . The analogous behavior is observed for real photons at large energies (l'3 GeV). A, A This reduction of nuclear cross sections is reminiscent of the features seen in high-energy hadron}nucleus collisions. For example, total cross sections for nucleon}nucleus scattering behave as p &A p at center-of-mass energies (s&(10}25) GeV [134]. A simple geometric picture , ,, interprets this e!ect as the hadron projectile interacting mainly with nucleons at the nuclear surface, leading to p &A . , The quantum mechanical description of shadowing in DIS explains this phenomenon by the destructive interference of single and multiple scattering amplitudes. Multiple scattering becomes important as soon as the lab frame coherence length for the hadronic #uctuations of the photon propagator exceeds the average distance between two nucleons in the nuclear target. We have seen in our space}time discussion of Section 4 that this is precisely what happens in the region x(0.1 of the Bjorken variable. The physics issue of nuclear DIS at small x is therefore, roughly speaking, the optics of hadronic or quark}gluon #uctuations of the virtual photon in the nuclear medium. Di!ractive phenomena play an important role in this context, as we shall demonstrate. At extremely small x (i.e. for x(10\) in combination with large Q, the measured free nucleon structure functions indicate a rapidly growing number of partons (mostly gluons). This is the domain of `high-density QCDa where individual partons interact perturbatively, at large Q, but their number increases so strongly that e!ective cross sections can become large (for references see e.g. [135}140]). It is of great interest to investigate the transition of the observed shadowing phenomena into this new domain, accessible by collider experiments, but so far unexplored for nuclear systems. In this section we "rst concentrate on the relationship between di!ractive photo- and leptoproduction from nucleons and shadowing in high-energy photon- and lepton}nucleus interactions. Then we investigate perturbative and non-perturbative QCD aspects of shadowing. After that we summarize existing models which successfully describe data. Finally, we outline implications of shadowing for nuclear parton distributions.
G. Piller, W. Weise / Physics Reports 330 (2000) 1}94
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5.1. Diwractive production and nuclear shadowing In the shadowing region, di!ractive photo- and leptoproduction of high-energy hadrons gives a substantial contribution to the (virtual) photon}nucleon cross section as discussed in Section 2.6.4. This suggests that the di!ractive excitation of hadronic states, cHNPXN, and their coherent interaction with several nucleons inside the target plays an important role for shadowing in high energy photon}nucleus scattering, in a similar way as for hadron}nucleus collisions. For this e!ect to be signi"cant, the following two conditions have to be met in the laboratory frame: (i) The longitudinal propagation length, or coherence length, 2l (5.1) j" M #Q 6 of the di!ractively produced hadronic state of invariant mass M , Eq. (4.19), must exceed the 6 average nucleon}nucleon distance in nuclei: j'dK2 fm .
(5.2)
(ii) In addition, the mean free path l "(op )\ of the di!ractively produced system in the 6 6, nuclear medium must be su$ciently short, at least smaller than the nuclear radius. Note that the mean free path of photons in a nucleus with density o amounts to l +(op )\+550 fm, which is much larger than any nuclear scale. Consequently `barea photons A A, do not scatter coherently from several nucleons and therefore do not contribute to shadowing. Shadowing results from the coherent scattering of a hadronic #uctuation from at least two nucleons in the target. Since the longitudinal propagation length j of a di!ractively produced hadronic state X decreases with its mass M , low mass excitations with M :1 GeV are relevant 6 6 for the onset of shadowing. Eq. (5.2) tells again that shadowing in deep-inelastic scattering at Q<1 GeV should start at x+0.1, in accordance with the observed e!ect and in close correspondence with the space}time picture described in Section 4. For real photons di!ractive processes at low mass are dominated by the excitation of the o- and u-meson. Signi"cant contributions to double scattering and hence to shadowing are therefore expected if the photon energy l exceeds about 3 GeV, in line with the experimental data. Consider now the scattering process in the laboratory frame. Realistic nuclear wave functions are well established only in this frame (with the exception of recent e!orts to construct relativistic nuclear model wave functions on the light front, see e.g. [141}147]). Later, in Section 5.5, we comment on nuclear shadowing as seen in the Breit frame. We "rst neglect e!ects due to nuclear binding, Fermi-motion and non-nucleonic degrees of freedom in nuclei. They are relevant at moderate and large values of the Bjorken variable, x'0.1, as discussed in Section 6. The (virtual) photon}nucleus cross section can be separated into a piece which accounts for the incoherent scattering from individual nucleons, and a correction from the coherent interaction with several nucleons: (5.3) p H "Z p H #(A!Z) p H #dp H . A A A A The single scattering part is the incoherent sum of photon}nucleon cross sections, where Z is the nuclear charge. The multiple scattering correction can be expanded in contributions which account
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for the scattering from n52 nucleons. Expressed in terms of the corresponding multiple scattering amplitudes ALH we have A 1 Im ALH , dp H " (5.4) A A 2M l L where the photon #ux (2.32) is taken in the limit x;1. The leading contribution to nuclear shadowing comes from double scattering. Its mechanism is best illustrated for a deuterium target on which we focus next. 5.1.1. Shadowing in deuterium In this section we review the basic mechanism of shadowing in real and virtual photon}deuteron scattering at high energies l, or equivalently, small x. The cH-deuteron cross section can be written as the sum of single and double scattering parts as illustrated in Fig. 5.1: (5.5) p H "p H #p H #dp H . A A A A The "rst two terms describe the incoherent scattering of the (virtual) photon from the proton or neutron, while 1 (5.6) dp H " Im A AH A 2M l accounts for the coherent interaction of the projectile with both nucleons. For large energies, l'3 GeV, or small values of the Bjorken variable, x(0.1, the double scattering amplitude A is dominated by the di!ractive excitation of hadronic intermediate states AH . At the high energies involved it is a good approxi(Fig. 5.1b) described by the amplitude ¹ H A , 6, :0.15 Im ¹ H mation to neglect the real part of this amplitude. In fact, we expect Re ¹ H A , 6, A , 6, by analogy with high-energy hadron}hadron scattering amplitudes (see e.g. [21]). When including such non-zero real parts, the double scattering contribution changes by less than 10% [148]. We neglect the spin and isospin dependence for unpolarized scattering [71]. Of course, these degrees of freedom play a crucial role in polarized scattering as we will discuss in Section 7.3. Treating the deuteron target in the non-relativistic limit gives [149}152]:
2 dr "t (r)" A H "! A M dk e k r ¹H ¹ (k) (5.7) ; H (k) , (2p) A , 6, (q !k )!k !(q !k )!M #ie 6, A , , 6 6
Fig. 5.1. Single (a) and double (b) scattering contribution to virtual photon}deuteron scattering. The corresponding cross sections are obtained from the imaginary part of the forward scattering amplitude indicated by the dashed line.
G. Piller, W. Weise / Physics Reports 330 (2000) 1}94
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where kI"(k , k) with k"(k , k ) is the four-momentum transferred to the nucleon, and t is the , deuteron wave function normalized as dr "t (r)""1. The sum is taken over all di!ractively excited hadronic states with invariant mass M and four-momentum p "q!k. We write 6 6
5 dp AH, ""64p Ml "¹ H dM A , 6, 6 dM dt 6 KL 6
(5.8)
in terms of the di!ractive production cross section, with t"k. The limits of integration de"ne the kinematically permitted range of di!ractive excitations, with their invariant mass M above the 6 two-pion production threshold and limited by the center-of-mass energy ="(s of the scattering process. We introduce the spin-averaged deuteron form factor,
S (k)" dr e k r "t (r)" ,
(5.9)
perform the integration over the longitudinal momentum transfer in Eq. (5.7) and then take . Actually k is simply "xed by energy}momentum the imaginary part of the amplitude A AH conservation: k +(Q#M )/2l"1/j , 6
(5.10)
which coincides with the inverse of the longitudinal propagation length (4.19) of the intermediate hadronic state. Note that the minimal momentum transfer required to produce a hadronic state di!ractively from a nucleon at rest amounts to t +!k (M ).
6 When all steps are carried out, the result for the double scattering correction is [149,150]
2 5 dp AH, . dM S (k , k +j\(M )) dp H "! dk , 6 , 6 A dM dt p 6 KL
(5.11)
This equation establishes the close relationship between shadowing in deep-inelastic scattering and di!ractive hadron production. It becomes even more transparent for x;0.1, i.e. large j. In this limit the magnitude of shadowing is determined just by the ratio of di!ractive and total cHN cross sections. To verify this let us parametrize the t-dependence of the di!ractive production cross section entering in Eq. (5.11) as
dp dp A H, AH, "e\ R + e\ dM dt dM dt 6 6 R
k ,
dp AH, , dM dt 6 R
(5.12)
neglecting the k dependence of t. Data from FNAL and HERA on di!ractive photo- and leptoproduction of hadrons with mass M '3 GeV give BK(527) GeV\ [54,59,60]. In the 6 di!ractive production of low mass vector mesons (o, u and ) from nucleons, a range of values BK(4210) GeV\ has been found, depending on Q and on the incident photon energy (for reviews and references see e.g. [50,65]). Clearly, the soft deuteron form factor selects momenta such that the double scattering correction in (5.11) is dominated by di!ractive production in the direction of the incident photon.
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Fig. 5.2. Integrated deuteron form factor F from Eq. (5.13) for an average slope B"8 GeV. The dotted curve corresponds to B"0.
In Fig. 5.2 we show the deuteron form factor (5.9) weighted by the exponential t-dependence (5.12) and integrated over transverse momentum,
dk , S (k , k "j\) e\ F (j\), (2p) ,
k ,
,
(5.13)
as obtained with the Paris nucleon}nucleon potential [153] for a slope parameter B"8 GeV\. We observe F +constant as long as the longitudinal propagation length j exceeds the deuteron size 1r2 +4 fm. From Eq. (5.2) one then "nds that hadronic states with an invariant mass =#Q !Q (5.14) M (M " 6
M1r2 contribute dominantly to double scattering. Combining Eqs. (5.11) and (5.14) gives the following approximate expression for the shadowing correction in the limit of large longitudinal propagation length j:
+
dp AH, . (5.15) +!8pF (0)Bp dM AH, 6 dM dt Kp 6 R In the last step we have assumed that contributions to the integrated di!ractive production cross section p from hadronic states with invariant masses M (M (= are small. Since AH,
6 /dM drops strongly for large M as discussed in Section 2.6.4, this approximation is justi"ed dp 6 6 AH, at large center-of-mass energies = or, equivalently, at small x. For the ratio between deuteron and free nucleon structure functions we then obtain dp H +!8p F (j\P0) A
pH p F H (5.16) R " " A +1!4p F (0)B A , . F, 2 p H pH A, A, /p +0.1 for the fraction of di!ractive events in deep-inelastic scattering from free We use p AH, AH, nucleons, as suggested by experiment (see Section 2.6.4). Furthermore, we take B"8 GeV\. One "nds that shadowing at x;0.1 in deuterium amounts to about 2%, i.e. R +0.98. The e!ect is
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small because of the large average proton}neutron distance in the deuteron, but the result agrees well with the experimental data shown in Fig. 3.5. 5.1.2. Shadowing for heavy nuclei The di!ractive production of hadrons from single nucleons also controls shadowing in heavier nuclei for which this e!ect is far more pronounced than in the deuteron. It is an empirical fact that nuclear shadowing increases with the nuclear mass number A of the target (see Section 3.3). For A'2 the hadronic state which is produced in the interaction of the photon with one of the nucleons in the target may scatter coherently from more than two nucleons. However, double scattering still dominates since the probability that the propagating hadron interacts with several nucleons along its path decreases with the number of scatterers. The double scattering contribution to the total photon}nucleus cross section p H is obtained by straightforward generalization of the A deuteron result (5.11) [150,151]:
"!8p db p AH
>
\
dz
>
X
dz o(b, z ; b, z )
5
dp AH, dM cos[(z !z )/j] . 6 dM dt 6 R3 Kp (5.17)
As illustrated in Fig. 5.3 a di!ractive state with invariant mass M is produced in the interaction 6 of the photon with a nucleon located at position (b, z ) in the target. The hadronic excitation propagates at "xed impact parameter b and then interacts with a second nucleon at z . The probability to "nd two nucleons in the target at the same impact parameter is described by the two-body density o(b, z ; b, z ) normalized as dr dr o(r, r )"A. The cos[(z !z )/j] factor in Eq. (5.17) implies that only di!ractively excited hadrons with a longitudinal propagation length larger than the average nucleon}nucleon distance in the target, j'dK2 fm, can contribute signi"cantly to double scattering. Note that nuclear short-range correlations are relevant only if the coherence length of the di!ractively excited states is comparable to the range of the short-range repulsive part of the nucleon}nucleon force, i.e. for j : 0.5 fm. In this case the shadowing e!ect is negligible. Nuclear correlations are therefore not important in the shadowing domain and the target can be considered as an ensemble of independent nucleons with o(r, r )+o (r)o (r ), where o is the nuclear one-body density [154,155]. With increasing photon energies or decreasing x down to x;0.1, the longitudinal propagation length of di!ractively excited hadrons rises and eventually reaches nuclear dimensions. Then
Fig. 5.3. Double scattering contribution to deep-inelastic scattering from nuclei.
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interactions of the excited hadronic states with several nucleons in the target become important. A simple way to account for those is a frequently used equation derived by Karmanov and Kondratyuk [156]:
dp H "!8p db A
>
\
5
dz
>
X
dz o (b, z ) o (b, z )
X dp p A H, dM cos[(z !z )/j] dz o (b, z) . (5.18) exp ! 6, 6 dM dt 2 6 R3 Kp X The exponential attenuation factor describes the elastic re-scattering of the di!ractively produced hadronic states from the remaining nucleons in the target. The hadron}nucleon scattering amplitudes are assumed to be purely imaginary and enter in Eq. (5.18) through the cross sections p . 6, Eq. (5.18) has been applied in several investigations of nuclear shadowing using di!erent models for the di!ractive photoproduction cross section. The more detailed results are discussed in Section 5.4, but we can get a simple estimate of nuclear shadowing at small Bjorken-x already by just looking at the relative amount of di!raction in DIS from free nucleons [67]. We restrict ourselves to the double scattering correction (5.17). For x;0.1, the coherence length j of the hadronic states which dominate di!ractive production in Eq. (5.17), exceed the diameter of the target nucleus. In the limit jPR we "nd ;
> > db dz o (b, z ) o (b, z ) . (5.19) K!8pBp dz p A H, AH \ X have been The slope parameter B and the integrated di!ractive production cross section p AH, introduced as in Eqs. (5.12) and (5.15). For the nuclear densities in Eq. (5.19) we use Gaussian,
3 3r o (r)"A exp ! 2p1r2 2 1r2 and square-well parametrizations,
A
o (r)" 0
3 3 4p 51r22
,
for r((1r2
(5.20)
(5.21)
otherwise ,
with the mean square radius 1r2 "dr ro (r)/A. For both cases the shadowing ratio R " p H /Ap H is easily worked out: A, A B p AH, . R K1!CA (5.22) 1r2 p H A, For Gaussian nuclear densities one "nds C"3, while C"2.7 in the square-well case. Using again typical values for the ratio of di!ractive and total cHN cross sections, p /p K0.1, and for the slope parameter, BK8 GeV\, the magnitude of R comes out in very AH, AH, reasonable agreement with experimental values as shown in Table 1. This estimate may be simple (in fact, higher-order multiple scattering must be included in a more detailed analysis) but it
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Table 1 The shadowing ratio R estimated according to Eq. (5.22) in comparison to experimental data for various nuclei. The data are taken from Refs. [76}78] at the smallest kinematically accessible values of Bjorken-x (namely, xK10\)
R R
Li
C
Ca
Xe
0.93 0.94$0.07
0.84 0.87$0.10
0.73 0.77$0.07
0.65 0.67$0.09
certainly con"rms that shadowing in nuclear DIS is governed by the coherent interaction of di!ractively produced states with several nucleons in the target nucleus. A more detailed investigation of the connection between HERA data on di!raction and shadowing e!ects measured at CERN and FNAL can be found in Ref. [157]. Inelastic transitions between di!erent hadronic states are neglected in Eq. (5.18). They cannot be treated in a model-independent way. Estimates of such higher-order di!ractive dissociation contributions have been performed for high-energy hadron}nucleus scattering [134,158]. In the case of neutron}nucleus collisions at center-of-mass energies s&200 GeV they amount to about 5% of the total reaction cross section. For rising energies the relative importance of inelastic transitions is expected to grow [158]. Given the important role of di!ractive production, we can now enter into a more detailed discussion of the x-dependence of shadowing. The coherence lengths j of hadronic states with small masses become comparable with nuclear dimensions for x(0.1. As j increases with decreasing x, the shadowing e!ect grows steadily for x:0.05. At x;0.1 it is also the energy dependence of the di!ractive production cross section and of the hadron}nucleon cross section p which in#uences 6, the x-dependence of shadowing. Consider the shadowing ratio R "p H /Ap H "1!dp H /Ap H , parametrized as A, A A, A R !1"!c (1/x)C (5.23) with a constant c at small Q where data are actually taken, and a characteristic exponent e. At asymptotically large energies Regge phenomenology suggests eK0.1 (see Section 2.6.4). In Fig. 5.4 we show the quantity log(1!R )"log c!e log x , (5.24) plotted versus log x in comparison with data taken on Pb at small Q. This plot con"rms that, for x(3;10\, the shadowing e!ect indeed approaches the high-energy behavior expected from the Regge limit of di!ractive production. Deviations from this asymptotic behavior at larger values of x indicate how shadowing gradually builds up as the coherence length jJx\ starts to exceed nuclear length scales for low mass di!ractively produced states. At su$ciently high energy or small x, the coherence length becomes comparable to nuclear dimensions even for heavy hadronic intermediate states. Once a major fraction of di!ractively produced states contribute to shadowing
At the typical average center of mass energies = M (25 GeV used at experiments at CERN and FNAL a somewhat stronger energy dependence is expected through the kinematic restriction to di!ractively produced hadronic states with masses M (=. 6
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Fig. 5.4. The quantity log(1!R ) as a function of log x for data taken on lead [78]. The dashed line corresponds to the asymptotic energy dependence (5.24) with e"0.1.
it starts to approach its asymtotic high-energy behavior. Note this asymptotic behavior sets in when the coherence lengths j of low mass hadronic #uctuations of the photon exceed by far the 6 dimension of the nucleus. For example at x"3;10\ and QK0.7 GeV, which corresponds to the onset of the asymptotic behavior in Fig. 5.4, the o meson coherence length becomes j K36 fm. M 5.1.3. Shadowing for real photons Data on the di!ractive production of hadrons in high-energy photon}nucleon interactions have been summarized in Section 2.6.4. They are useful to gain insight into the relative importance of o, u and mesons, as compared to heavier states, for nuclear shadowing with real photons. Di!ractive cNPXN production with M :1 GeV involves primarily the light vector mesons 6 o, u and . Nuclear shadowing at photon energies l up to about 200 GeV is largely determined by the coherent multiple scattering of those di!ractively produced vector mesons. Their propagation lengths jK2l/m easily exceed nuclear dimensions as soon as l'20 GeV. With rising energies 4 additional contributions to shadowing from di!ractively produced states with larger masses, M '1 GeV, become increasingly important. 6 This behavior is illustrated for DIS from deuterium in Fig. 5.5, where we show the ratio of the total photon}deuteron cross section compared to the free photon}nucleon cross section, R "p /2p , from Ref. [148]. The empirical photon}proton cross section from [159] has been A A, used for p . The shadowing correction (5.11) has been calculated using a "t for the di!ractive A, photon}nucleon cross section from Ref. [148]. The observed energy dependence of shadowing in Fig. 5.5 results from two sources as pointed out previously: the dependence of the di!ractive and total photon}nucleon cross sections on energy implies R !1&l for the shadowing ratio. An additional increase of shadowing with rising
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Fig. 5.5. The shadowing ratio R "p /2p as a function of the photon energy. The dashed line shows the vector meson A A, contribution. The experimental data are taken from the E665 collaboration [92]. (The energy values of the data have to be understood as average values which correspond to di!erent x-bins.)
energy l comes from di!ractively produced states with large mass, M '1 GeV, which become 6 relevant at high energies. 5.1.4. Shadowing in DIS at small and moderate Q So far nuclear shadowing has been measured only in "xed target experiments. The kinematic conditions of such experiments imply that the data for x(0.01 had to be taken at small four-momentum transfers, Q:1 GeV, as discussed in Section 3.3.1. The corresponding energy transfers are typically 50 GeV(l(300 GeV. The conclusions just drawn for real photons apply here too: nuclear shadowing as measured by E665 and NMC receives major contributions from the di!ractive production and multiple scattering of vector mesons. In the intermediate range 0.01(x(0.1, on the other hand, the E665 and NMC measurements involve momentum transfers up to Q&30 GeV. At Q'1 GeV vector meson contributions to di!raction and shadowing decrease (Section 2.6.4) and hadronic states with masses M &Q 6 become relevant. The data reveal that the Q-dependence of nuclear shadowing is very weak (Section 3.3.1). This suggests that high-mass hadronic components of the photon which dominate the measured nuclear shadowing at Q'1 GeV, interact strongly with the target, just like ordinary hadrons. The following section gives a schematic view of the scales involved, as outlined in Ref. [160]. 5.2. Sizes, scales and shadowing Consider DIS at small x in the lab frame. In this frame of reference the important feature is the nuclear interaction of hadronic #uctuations of the virtual photon (see Section 4). Since the photon and its hadronic con"gurations carry high energy, the transverse separations and longitudinal momenta of their quark and gluon constituents are approximately conserved during the scattering process. These hadronic con"gurations can be classi"ed as `smalla or `largea, depending on their transverse extension. `Largea con"gurations have hadronic sizes of order K\ &1 fm, whereas /!" `smalla con"gurations are characterized by sizes which scale as Q\. The contribution of a given hadronic #uctuation, h, to the photon}nucleon interaction cross section is determined by its probability weight w H in the photon wave function, multiplied by its A
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Table 2 Relative contributions of small- and large-size hadronic components of a virtual photon to DIS and shadowing at large Q [160] Fluctuation h
wH A
p ,
wH p A ,
w H (p ) A ,
Small size Large size
1 K /Q /!"
1/Q 1/K /!"
1/Q 1/Q
1/Q 1/(K Q) /!"
cross section p . The virtual photon}nucleon cross section is , (5.25) p H " w H p . A , A, The coherent interaction of the virtual photon with several nucleons behaves di!erently. For example, the contribution of a hadronic #uctuation to double scattering, which dominates shadowing, is proportional to its weight in the photon wave function multiplied by the square of its interaction cross section. The double scattering correction to virtual photon-nucleus scattering is: p & w H (p ) . (5.26) AH A , Now, the probability to "nd a quark}gluon con"guration of large size is suppressed (up to possible logarithmic terms) by K /Q as compared to con"gurations with small transverse sizes. On the /!" other hand, the interaction cross sections of hadronic #uctuations are proportional to their squared transverse radii. These properties and their consequences for the cross sections in Eqs. (5.25) and (5.26) are summarized in Table 2. For the scattering from individual nucleons one "nds that both, large- and small-size con"gurations give leading contributions &1/Q to the photon}nucleon cross section (5.25). On the other hand, contributions from small-size components to the shadowing correction p are suppressed by an additional power 1/Q as compared to large-size AH con"gurations (apart from contributions related to di!ractive production from the whole nucleus, not considered in this schematic picture). In view of these scale considerations, we can now understand some of the previously mentioned empirical facts which, on "rst sight, seemed unrelated: E Nuclear shadowing varies only weakly with Q. E The energy dependence of nuclear shadowing for x:0.01, as measured with "xed target experiments at CERN and FNAL, is reminiscent of hadron}nucleus collisions. These features follow from the fact that, to leading order in Q, shadowing is primarily determined by the interaction of large-size hadronic #uctuations of the exchanged photon, even at large Q. These hadronic con"gurations are expected to interact like ordinary hadrons. Note, those observations can be applied to di!raction as well as to shadowing, given that the two phenomena are closely connected as established in the previous sections: di!raction is also a scaling e!ect, i.e. it survives at large Q. Its energy dependence is expected to behave similarly as in hadron collisions. (For limitations to this simple picture see Sections 5.3 and 8.2.)
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Fig. 5.6. Decomposition of a virtual photon into a quark}antiquark pair at large Q.
5.3. Nuclear shadowing and parton conxgurations of the photon The results of the previous sections are elucidated by making contact with the underlying basic QCD and the parton structure of the virtual photon. The photon wave function can be decomposed in a Fock space expansion, (5.27) "c2"c "c 2#c "qq 2#c "qq g2#2 , OOE OO in terms of a `barea photon state "c 2 and partonic (quark}antiquark and gluonic) components. At large Q the minimal Fock component "qq 2 dominates the hadronic part of "c2, higher Fock states enter with powers of the strong coupling a . Let us now have a closer look at this minimal Fock Q component. Consider a virtual photon of four-momentum qI"(l, q), with q"(0, 0, q ) and q '0 de"ning the longitudinal direction, and Q"!q. Let this photon split into a quark}antiquark pair as sketched in Fig. 5.6. The quark has a four-momentum kI"(k , k) with k"(k , k ). The fraction of , the photon light-cone momentum carried by the quark is m"k>/q>"(k #k )/(l#q ) . (5.28) The momentum fraction of the antiquark with kM I"qI!kI is obviously 1!m. For the longitudinally polarized photon, the wave function of its minimal qq #uctuation in momentum space is proportional to the longitudinal component of the quark pair current, multiplied by its propagator [Q#(m#k )/m(1!m)]\ where m is the quark mass [161]. The O , O quantity m#k , M " O 6 m(1!m)
(5.29)
can be interpreted as the squared e!ective mass of the propagating qq pair. It is useful to perform the two-dimensional Fourier transform with respect to the transverse quark momentum k conjugate to the transverse separation b of the qq pair. Neglecting the quark , mass at large Q and using generically one single quark #avor, the squared wave function of the qq component coupled to the longitudinally polarized photon becomes [162}164]: (5.30) "t* (b, m; Q)""(6a/p) Qm(1!m)K (b Q (m(1!m)) , OO where K denotes modi"ed Bessel functions. The cHPqq coupling is proportional to the "ne L structure constant aK . The corresponding result for a virtual photon with transverse polariza tion is "t2 (b, m; Q)""(3a/2p)Qm(1!m)(m#(1!m))K (b Q (m(1!m)) . OO
(5.31)
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Fig. 5.7. Short distance interaction of a color singlet quark}antiquark pair with a nucleon through two-gluon exchange.
Consider now DIS from a free nucleon at small Bjorken-x and large Q. At the high energies involved the photon in the laboratory frame acts like a beam of qq pairs, and one can write the cross section for the longitudinally or transversely polarized virtual photon with the nucleon in the form [163,165]
" db dm"t* 2 (5.32) (b, m)"p (b, x) , p* 2 AH, OO OO, using the wave functions of the leading qq #uctuations. These wave functions as well as the qq nucleon cross section p depend on the transverse separation b of the quark pair. Since the OO, modi"ed Bessel functions in Eqs. (5.30) and (5.31) drop as K (y)Pe\W at large y, the wave functions t* 2 receive their dominant contributions from con"gurations with transverse size OO 1 . (5.33) b& Qm(1!m) Consequently, qq con"gurations at large Q with comparable momenta of the quark and antiquark, m&1!m&, have small transverse size, b&1/k &1/Q, or equivalently, large trans, verse momentum. The interaction of these `non-aligneda con"gurations with the nucleon is therefore determined by the short transverse distance behavior of the cross section p which can OO, be calculated using perturbative QCD. The reasoning goes as follows. At large Q the leading mechanism responsible for the short distance interaction of the qq pair with the nucleon is two-gluon exchange (see Fig. 5.7). The (color singlet) qq pair acts as a color dipole. Its interaction strength with the nucleon or any other (color singlet) hadron is determined by the squared color dipole moment, hence p is proportional to b for small transverse separations b. In the OO, leading-logarithmic approximation valid at large Q one derives [165,166] (5.34) p (b, x)"(p/3)a (Q)bxg (x, Q) Q , OO, with the strong coupling constant a . The Q scale in (5.34) is set by Q&1/b. All non-perturbative Q e!ects are incorporated in the gluon distribution g (x, Q) of the target nucleon. , Small qq con"gurations interact only weakly according to Eq. (5.34). This is the case for the kinematic conditions realized in "xed target experiments at CERN and FNAL (see Sections 2.6.1
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and 3.3.1). The situation is di!erent at x;0.1 and Q<1 GeV, the extreme region accessible at HERA. Here the strong increase of the nucleon structure function F, at very small x is accompanied by a correspondingly strong increase of the gluon distribution function. The gluon density becomes so large with decreasing x that, even for small b&1/Q, the cross section p can OO, eventually reach magnitudes typical for ordinary hadrons [167]. If either the quark or the antiquark becomes soft (that is: if either the momentum fraction m or 1!m tends to zero), large qq separations contribute to the wave functions (5.30) and (5.31). In this limit perturbative QCD is not applicable. The interaction cross section for such large-size con"gurations with small transverse momentum is supposed to be similar to typical hadron} nucleon cross sections [168]. A detailed analysis of the `transversea wave function (5.31) shows that both `smalla (nonaligned) and `largea quark}antiquark con"gurations give leading 1/Q contributions to the transverse photon}nucleon cross section in accordance with our previous discussion. The situation is di!erent for longitudinally polarized photons (see e.g. (5.30)). In this case the contributions from `softa quarks (with mP0 or 1!mP0) are suppressed as 1/Q so that, to leading order in the strong coupling a , only small size qq pairs contribute to p*H . At next to leading order in a , the Q Q A, Fock expansion (5.27) introduces quark}antiquark}gluon states. Large size qq g con"gurations are now important, and they are not suppressed by additional powers of 1/Q [169]. At small momentum transfers, Q:1 GeV, con"gurations of large size dominate the qq wave function. Strong interactions between quark and antiquark now lead to the formation of soft hadronic #uctuations including vector mesons and multi-pion states. Consequently, photon}nucleon cross sections at small x and small Q receive important contributions from the low mass vector mesons. For example, at QK0.5 GeV almost half of the measured nucleon structure function F, comes from o, u and mesons according to the calculation in Ref. [170]. So far we have focused this discussion on free nucleons. Similar considerations apply to deep-inelastic scattering from nuclei which involves the interaction of hadronic components of the virtual photon with the nuclear many-body system. The photon-nucleus cross sections are now obtained from Eq. (5.32) replacing p by the corresponding qq -nucleus cross section p . The OO OO, cross section p of any hadronic #uctuation h interacting with a nucleus at high energies, can be related to the cross section for the scattering from free nucleons by the Glauber}Gribov multiple scattering formalism [150,151]. For a Gaussian nuclear density (5.20) this leads to
3 A!1 1r2 #2 , 1!p exp ! p +Ap , 16p 1r2 , 3j
(5.35)
where j is the propagation length associated with the hadronic #uctuation. Double scattering gives a negative correction proportional to the squared cross section of the hadronic #uctuation. Only those hadronic con"gurations with large interaction cross sections contribute signi"cantly to shadowing. Furthermore, since the nuclear mean square radius behaves approximately as 1r2 &A , the magnitude of the double scattering correction grows for large nuclei with the radius of the target, i.e. proportional to A . The exponential in (5.35) ensures that only hadronic #uctuations with propagation lengths j larger than the target dimension contribute signi"cantly to shadowing. For small-sized #uctuations, interesting e!ects beyond those covered by Eq. (5.35) arise from di!ractive production on the whole nucleus.
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In accordance with our discussion in the previous section we can conclude: (i) In the "xed target experiments at CERN and FNAL, where small values of x(0.01 are accessible only at small average momentum transfers, QM :1 GeV, nuclear shadowing is governed by interactions of con"gurations with large transverse sizes. Contributions from the vector mesons o, u and turn out to be particularly important. (ii) At very small x together with very large Q, the growing number of partons in the photon}nucleus system makes them interact like ordinary hadrons, even if the parton con"gurations have small transverse sizes inversely proportional to Q. One now expects a complex interplay between soft (large-size) and hard (small-size) partonic components of the interacting photon which can no longer be classi"ed by simple book-keeping in powers of 1/Q. 5.4. Models In the following, we sketch several models which have been used quite successfully to describe nuclear shadowing as measured in experiments at CERN and FNAL. As before we restrict ourselves to lab-frame descriptions. We do not aim for completeness but rather emphasize common features of various models and their implications for the underlying mechanism of nuclear shadowing. 5.4.1. Vector mesons and aligned jets As discussed in Section 5.3, the quark}antiquark #uctuation of a virtual photon starts out with a transverse size b&[Qm(1!m)]\ where m is the fraction of longitudinal photon momentum carried by one of the quarks. `Non-aligneda qq con"gurations with mK have small transverse size and interact weakly; `aligneda ones with m&0 or m&1 have large transverse size and are likely, by subsequent strong interactions, to turn into vector mesons if the qq invariant mass matches appropriately. Models which combine aspects of vector meson dominance and the aligned-jet picture [168] are described in Refs. [170,171]. Their starting point is the hadronic spectrum of the virtual photon exchanged in the deep-inelastic scattering process. The spectral function, P(s), is determined by the cross section for electron}positron annihilation into hadrons, where s"q is the squared photon or e>e\ center-of-mass energy: (s) 1 p>\ C C P(s)" 12p p > \ > \ (s) C C I I
(5.36)
with 1 (5.37) P(q)"! (2n)d(q!p )10"J (0)"X21X"JI(0)"02 . 6 I 3q 6 Here JI is the electromagnetic current operator. The sum in Eq. (5.37) is taken over all hadronic #uctuations of the photon with four-momenta p "q and squared invariant masses k,p "q. 6 6 At small center-of-mass energies, s:1 GeV, the spectrum (5.37) is dominated by the vector mesons o, u and as shown in Fig. 5.8. The high-energy spectrum at s'1 GeV is characterized by quark}antiquark continuum plateaus together with isolated charmonium and upsilon resonances.
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Fig. 5.8. Cross section p > \ /p . C C C>C\ I>I\
The lab frame space}time pattern of deep-inelastic scattering (Section 4) suggests that the nucleon structure function at small x can be described by the following expression [150,171]:
Q I kP(k) F,(x, Q)" dk dm p (=, k; m) . , p (k#Q) KL
(5.38)
The basic idea behind this ansatz is the following. For x;1, or large lab frame propagation length j&2l/(Q#k) of a given qq #uctuation of mass k, the vacuum spectrum P(k) remains more or less una!ected by the presence of the target nucleon. The high-energy virtual photon with
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Fig. 5.9. The nucleon structure function F, for small x plotted against Q. The full line has been obtained in Ref. [170] from Eq. (5.38). The dashed line indicates the contribution of the vector mesons o, u and . The data are from the NMC [172].
l
determined by the condition that j must exceed the size RK5M\ of the target nucleon, so that (roughly) k &(lM. The interaction of this beam with the nucleon is described by the cross
section p which depends on k and on the hadron/photon}nucleon center-of-mass energy , ="(2Ml#M!QK(2Ml"Q/(x. For a qq pair treated to leading order in a , it also Q depends on the fraction m of the photon light-cone momentum carried by the quark. The sum in Eq. (5.38) is taken over hadronic #uctuations of the photon with "xed invariant mass. The ansatz neglects contributions to the forward virtual photon scattering amplitude in which the mass k can change during the interaction. The small-x structure function F,(x, Q) as given in Eq. (5.38) is governed by contributions from intermediate hadronic states with an invariant mass k&Q. For small momentum transfers, Q:1 GeV, low mass vector mesons o, u and are of major importance. Their dominance leads to the scale breaking behavior F,(x, Q)&Q for QP0 at small x. For larger momentum transfers, Q'm +1 GeV, the structure function F, is determined ( primarily by the interactions of quark}antiquark pairs from the qq continuum. The color singlet nature of hadronic #uctuations of the virtual photon implies that their interaction cross section is proportional to their transverse size. Quark pairs with momenta `aligneda along the direction of the virtual photon have a large transverse size. Their cross sections should be comparable to typical hadronic cross sections. On the other hand, `non-aligneda quarks are characterized by small transverse size. Their cross sections should therefore be small. With these ingredients, Eq. (5.38) gives a good description of the free nucleon structure function F, for x(0.1 and moderate Q. A comparison with data from NMC is shown in Fig. 5.9. While the vector meson contribution vanishes as 1/Q for large Q, the qq continuum pairs are responsible for scaling, F,(x, Q)&ln(Q), at large Q. Note however the importance of vector mesons at small Q. One "nds that at Q"1 GeV almost half of F, at x"0.01 is due to vector mesons. At Q"10 GeV they still contribute about 15%. In these calculations the vector meson part of the spectrum P(k) is [25]: P4(k)"(m /g )d(k!m ) 4 4 4
(5.39)
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Fig. 5.10. Results from Ref. [170] for the shadowing in He, Li, C, and Ca compared to experimental data from NMC (dots and squares) [72,76] and FNAL (triangles) [78]. The dashed curves show the shadowing caused by the vector mesons o, u and only, the solid curves are the results including the qq continuum.
with V"o, u, , the empirical vector meson masses m and the coupling constants g "5.0, 4 M g "17.0 and g "12.9. The vector meson}nucleon cross sections are p +p +25 mb, S ( M, S, p +10 mb. (, Nuclear structure functions F for x(0.1 are expressed in an analogous way as in Eq. (5.38), with the hadron}nucleon cross sections p replaced by the corresponding hadron}nucleus cross , sections p . The relation between p and p is given by Glauber}Gribov multiple scattering , theory, see Eq. (5.35). In Fig. 5.10 we present typical results for the shadowing ratio R "F /F, from Ref. [170]. Finally, we comment on the observed weak Q-dependence of the shadowing e!ect. In the spectral ansatz (5.38) the given value of Q selects that part of the hadron mass spectrum around k&Q which dominates the interaction, and hence determines which cross sections p (k) , contribute signi"cantly to the multiple scattering series. While the interaction cross sections decrease as 1/k with increasing mass as required by Bjorken scaling, pairs which are aligned with the photon momentum interact with large cross sections, even for large k, and therefore produce strong shadowing. This is the reason for the very weak overall Q-dependence of shadowing in this
It is common practice to normalize F such that it represents the nuclear structure function per nucleon.
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Fig. 5.11. The slope b"d(F1/F! )/d ln Q indicating the Q dependence of the shadowing ratio Sn/C. The calculation is described in [170]. Data are taken from [83]. Fig. 5.12. Shadowing in Xe. Details of the calculation are given in Ref. [155]. The dashed curve shows the contribution of vector mesons o, u and , while the solid curve includes pomeron exchange. The data are from the E665 collaboration [77].
framework. A comparison of results from Ref. [170] with NMC data for the slope b of the ratio F1/F! +a#b ln Q is presented in Fig. 5.11. For a more detailed discussion of these issues including QCD corrections, see Ref. [1]. 5.4.2. Vector meson dominance and pomeron exchange As indicated in Eqs. (5.11) and (5.18), nuclear shadowing is directly related to the di!ractive /dM dt or, equivalently, to the di!ractive structure function F". production cross section dp 6 AH, Di!ractive production at Q:1 GeV is dominated by the excitation of the vector mesons o, u and . Their contributions can be described within the framework of vector meson dominance (see e.g. [25]). Neglecting transitions between di!erent vector mesons and omitting contributions from longitudinally polarized virtual photons one "nds:
dp 4 a P4(M )M AH, 6 6 p . " dM dt 4 (Q#M ) 6, 6 R3 6
(5.40)
Here the vector meson part (5.39) of the photon spectral function enters. Combining Eqs. (5.18) and (5.40) shows that the contribution of vector mesons to nuclear shadowing vanishes indeed as 1/Q. The di!ractive excitation of heavy mass states is commonly parametrized according to the Regge ansatz as in Eq. (2.67). Most descriptions concentrate on the dominant contribution from pomeron exchange. The pomeron structure function F/ is modeled in agreement with available data on di!raction. At large Q it is supposed to scale, i.e. it depends at most logarithmically on Q. This leads to scaling for nuclear shadowing at large Q. On the other hand, at Q;1 GeV one assumes F/ &Q [173] which ensures that the shadowing correction to the nuclear structure function, dF , vanishes at Q"0, just like F itself. Investigations of shadowing e!ects along theses lines can be found in [155,173}177]. In Fig. 5.12 a typical result for shadowing in Xe from [155] is shown.
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5.4.3. Generalized vector meson dominance Generalized vector meson dominance models describe deep-inelastic lepton scattering at small x purely in terms of hadronic degrees of freedom [178}188]. For a free nucleon this leads to the following picture: prior to its interaction with the target the virtual photon #uctuates into a hadronic state with invariant mass k. This #uctuation scatters from the target and converts into a hadronic state with mass k . For transversely polarized photons this translates into a forward Compton amplitude of the form:
o(k, k ; =) A2H (=, Q)& dk dk A, (k#Q)(k #Q)
(5.41)
with a double spectral distribution o depending on the photon}nucleon center-of-mass energy =. Here the integrals over initial and "nal hadronic #uctuations and their propagators are made explicit. The continuum of hadronic intermediate states which determines the double spectral function o(k, k ; =) is commonly approximated by a discrete set of narrow vector meson states V (n"1, 2,2). The resulting transverse photon}nucleon cross section is L e M M e L K R (=) p2H " . (5.42) A, KL g M #Q M#Q g L L L K K K In Refs. [181,186] the vector mesons are assumed to be equally spaced in mass, starting with the o-meson. The photon-vector meson couplings g are chosen to reproduce average scaling in e>e\ L annihilation into hadrons (see Fig. 5.8). R denotes the imaginary part of the vector meson} KL nucleon transition amplitude, V NPV N, in the forward direction. For diagonal terms it is equal K L to the total V -nucleon cross section, R "p(V N), which is taken to be constant. L LL L The next step in simpli"cation is to consider only diagonal (m"n) and nearest o!-diagonal (m"n$1) contributions. A "ne-tuned cancelation between the corresponding amplitudes R leads to a reasonable description of the nucleon structure function F, at moderate momentum KL transfers Q. An extension of this approach to nuclear targets involves multiple scattering of hadronic #uctuations from several nucleons. The multiple scattering process is described by a coupled channel optical model [180,186] which accounts for the shadowing criteria in Eq. (5.2), i.e. only those hadronic #uctuations with longitudinal interaction lengths larger than their mean free path in the nuclear medium contribute signi"cantly to multiple scattering and thus to shadowing. GVMD calculations applied to current nuclear DIS data can be found in Refs. [184,188]. 5.4.4. Vector mesons and quark scattering We add a few remarks and references about approaches dealing with DIS in terms of quark dynamics. The starting point in Ref. [189] is a description of DIS from nucleons at large Q and small x in terms of quark}nucleon scattering [190]. The quark}nucleon scattering amplitude is formulated using Regge phenomenology and constrained by the quark distributions of free nucleons. The interaction strength of quark}nucleon scattering is determined by the quark}nucleon cross section, taken to be about of the nucleon}nucleon cross section. At center of mass energies s&200 GeV one "nds p +13 mb [191]. O,
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An extension to DIS from nuclei at small x involves the quark}nucleus scattering amplitude. Its connection with the amplitude for the scattering from free nucleons is given through the Glauber}Gribov multiple scattering series. In Ref. [191] the interaction of strongly correlated quark}antiquark pairs, i.e. vector mesons, has been added. One "nds that vector mesons carry more than half of the shadowing e!ect measured at E665 and NMC. On the other hand, the interaction of uncorrelated quarks is also important to ensure a weak Q-dependence of shadowing. 5.4.5. Green function methods The previously mentioned models have outlined in di!erent ways the ingredients needed in order to understand the physics of shadowing: the mass spectrum of quark}gluon #uctuations of the virtual photon, and the dynamics of the expanding and strongly interacting quark}gluon con"gurations in the surrounding nuclear system. In most of the models the longitudinal propagation of hadronic #uctuations of the photon is treated by multiple scattering theory, while the transverse degrees of freedom are more or less `frozena during the passage through the nucleus. Several questions are faced in this context. The transverse size of quark}gluon #uctuations needs to be connected with their e!ective mass; the relationship with di!ractive production must be elucidated; higher order terms in the multiple scattering series must be systematically incorporated, at least for heavy nuclei. All those aspects can be uni"ed within a coordinate-space Green function approach. We follow here the presentations in Refs. [192,193] (see also [194}196]). While this approach considers only quark}antiquark #uctuations of the photon, it turns out that some previous approximations can now be identi"ed as limiting cases in a uni"ed picture. Consider the scattering of a virtual photon with high energy l and large squared fourmomentum, Q'1 GeV, through a nucleus as illustrated in Fig. 5.13. The longitudinal (z-) direction is de"ned by the photon three-momentum, as usual. At point z the photon produces a quark}antiquark pair with transverse separation b . Along its passage to point z where it has a transverse separation b , the qq #uctuation experiences multiple interactions with nucleons in the nuclear target. We are interested in the full Green function G(b , z ; b , z ) which describes the propagation of the qq pair from z to z , including its dynamics in the transverse space coordinate.
Fig. 5.13. Propagation of a quark}antiquark #uctuation of the virtual photon cH between points z and z where the pair has transverse separation b and b . The Green function G(b , z ; b , z ) sums all possible paths of the pair through the nucleus.
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This Green function enters in the shadowing part of the cH-nucleus cross section, as follows. One writes dp H (x, Q),p H (x, Q)!Ap H (x, Q) A A, A "!2 db dz o (b, z )=(z , z ) . (5.43) dz o (b, z ) \ X The quantity =(z , z ) has dimension (length) and describes the production of a qq #uctuation in the process cHNPqq N, its propagation from z to z , and its subsequent conversion back to a virtual photon:
=(z , z )"Re db db
dm FH(b , m)G(b , z ; b , z )F(b , m)e GX \X .
(5.44)
It involves the amplitude F(b, m)"t H (b, m)p(b) (5.45) A OO for the cHNPqq N process leading to a qq pair of transverse separation b in which the quark carries the fraction m of the photon light-cone momentum (see also Section 5.3). The color dipole cross section p(b) has the characteristic color screening behavior, i.e. it vanishes as b at bP0 (see also Eq. (5.34)), and the distribution of transverse separations is determined by the amplitude t H . A OO The normalization of F is such that its Fourier transform gives the cHNPqq N di!ractive dissociation amplitude,
f (k )" db F(b, m)e k, b ,
(5.46)
in plane wave impulse approximation. The phase factor e GX \X involves the characteristic wave number of the qq #uctuation: Qm(1!m)#m O , i" 2lm(1!m)
(5.47)
where m is the (constituent) quark mass. For m" the resulting i"(Q#4m)/2l,j\ is just O O the inverse coherence length of a quark and antiquark which travel side by side. (For arbitrary m this coherence length includes the transverse momentum, j\"i#k /[2l m(1!m)].) , Let us now return to the propagation function G. It satis"es a wave equation [192] which can be made plausible by the following considerations. The longitudinal motion along the z-axis is equivalent to the time evolution of the qq #uctuation, represented by the operator i R/Rz . The transverse dynamics has a kinetic term t
@ "! 2lm(1!m)
(5.48)
with the two-dimensional Laplacian acting on the transverse separation coordinate b , and the denominator re#ecting the e!ective mass of the pair. Interactions of the qq pair with the nuclear medium at an impact parameter b are introduced by an absorptive term v(b , b)"!(i/2)p(b )o (b, z ) .
(5.49)
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The wave equation for G is then of the generic form i RG/Rz"(t
#v)G or, more precisely,
R i @ # # p(b )o (b, z ) G(b , z ; b , z )"0 , (5.50) 2lm(1!m) 2 Rz with the initial condition G(b , z ; b , z )"d(b !b ). One can now discuss several interesting limits: i
(i) The `frozena limit: Take the energy lPR so that the kinetic term (5.48) vanishes (with some extra care required at the kinematic corners, m"0 and m"1). Then
p(b ) X G(b , z ; b , z )"d(b !b ) exp ! dz o (b , z) . 2 X Inserting this expression into Eqs. (5.44) and (5.43) with jPR one "nds
p(b ) (5.52) dm "t H (b ,m)" 1!exp ! OO dz o (b, z) A OO OO 2 \ and recovers the shadowing correction dp H as in Glauber}Gribov multiple scattering theory A by expanding the exponential. Note the di!erence compared to the standard Glauber eikonal approximation where the cross section p(b ) is averaged in the exponent. OO (ii) No absorption: Take the limit pP0 in the wave equation (5.50). Then G reduces to the free Green function of the qq pair, p H "2 db db A OO
(5.51)
ik (z !z ) 1 . dk exp ik ) (b !b )# , G(b , z ; b , z )" , , 2lm(1!m) 2p
(5.53)
Inserting this into Eq. (5.44) and using the di!ractive dissociation amplitude (5.46) one "nds
Qm(1!m)#m#k 1 O , (z !z ) . =(z , z )" (5.54) dm dk " f (k )" cos , , 2lm(1!m) 2p We identify the squared e!ective mass, M "(m#k )/m(1!m), of the qq pair as in Eq. (5.29) 6 O , and introduce its coherence length j"2l/(Q#M ). Inserting Eq. (5.54) into Eq. (5.43) one 6 then recovers the double scattering result, Eq. (5.17), with the factorized two-body density o(b, z ; b, z )"o (b, z ) o (b, z ). It is now also apparent how the additional absorption factor in Eq. (5.18) is obtained, introducing an average cross section p in the exponent. 6, (iii) Propagation in uniform nuclear matter: Assume that the qq pair moves in a nuclear medium of uniform density o (b, z)"o "const. (o "0.17 fm\ for normal nuclear matter). Suppose that the color dipole cross section is approximated by p(b )"cb OO OO with a constant parameter c. In this case the wave equation (5.50) reduces to
i
R ic @ # # o b G(b , z ; b , z )"0 . 2lm(1!m) 2 Rz
(5.55)
(5.56)
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This is formally reminiscent of the SchroK dinger equation for a harmonic oscillator with complex frequency. One "nds [192]
a a 2b ) b G(b , z ; b , 0)" exp ! (b #b ) coth(uz)! 2p sinh(uz) 2 sinh(uz) with
u"
co co "i . a lm(1!m)
,
(5.57)
(5.58)
This is a convenient approximation to account for multiple scattering and absorption of the qq #uctuation, still keeping track of its transverse dynamics during its passage through the nuclear medium. Instructive results are discussed in Ref. [192]. 5.4.6. Meson exchange and shadowing Up to now we have concentrated on di!ractive contributions to nuclear shadowing, in which the nucleons interacting with the virtual photon are left unchanged. The coherent interaction of the photon with several nucleons in the target nucleus can also involve non-di!ractive processes, in particular, reactions in which nucleons change their charge. These are commonly described by the exchange of mesons and sub-leading reggeons. Modi"cations to nuclear structure functions at small x through meson exchange have been investigated in Refs. [197,198] for deuterium. In this work signi"cant e!ects come from the interaction of the virtual photon with pions emitted from the target proton or neutron. Here, as in di!raction, a hadronic state X is produced which subsequently re-scatters from the second nucleon. Contributions from the exchange of other mesons, e.g. o and u, turn out to be negligible. For the double scattering contribution through pion exchange one "nds in analogy with Eq. (5.11):
2 5 dppH A, . dM Sp (k) (5.59) dppH " dk , 6 dM dt A p p 6 K Here dppH /dM dt is the cross section for the semi-inclusive production of a hadronic state with A, 6 invariant mass M from a proton or neutron via pion exchange. The form factor in Eq. (5.59) 6 accounts for the spin-dependent response of the deuteron:
1 Sp (k)" dP tKR(P) r ) kK r ) kK tK(P!k) , (5.60) N L 3 K where k is the pion momentum and kK "k/"k". The momentum-space wave function of the deuteron with polarization m is denoted by tK. Furthermore the non-relativistic form of the pion}nucleon coupling is used [199]. Note that the energy of the exchanged pion is determined by k "M !(M#P!(M!(P!k), where P is the momentum of the parent nucleon. We denote the pion four momentum by k"(k , k). For the longitudinal pion momentum one has k +yM with the pion light-cone momentum fraction y"k ) q/P ) q, and we introduce t"k along with the usual Bjorken-x. It is common to factorize the semi-inclusive di!erential cross section: dppH A , (x, Q; y, t)"f (y, t) p H (x/y, Q) , p , Ap dy dt
(5.61)
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where the photon}pion cross section is related to the structure function of the pion by Fp (x, Q)"(Q/4pa) p H (x, Q), and the pion distribution function in the nucleon is given by Ap 3g "F (t)"(!t) f (y, t)" p,, p,, y, (5.62) p , (t!m) 16p p with the pion}nucleon coupling constant g and the pNN form factor F normalized to unity p,, p,, for on-mass-shell pions, i.e. F "1 for t"k"m. p,, p For practical calculations the pion structure function has been approximated by that of the free pion, as parametrized in [200,201] in accordance with Drell}Yan leptoproduction data. Here, however, only the region x'0.1 has been measured. An extraction of the pion structure function at small x from semi-exclusive reactions at HERA has been discussed recently in [202,203]. The resulting pionic correction dppH to double scattering turns out to be positive, i.e. it causes A `anti-shadowinga. The relative weight of dppH decreases with decreasing x. At typical values A Q"4 GeV and 0.001(x(0.1 it amounts to around 30% of the overall shadowing correction [197]. In Ref. [198] the pion correction dppH has been found to be negligible at large Q910 GeV. A Note, however, that the quoted results depend sensitively on the yet unknown pion structure function at small x, the deuteron wave function and the choice for the pion}nucleon form factor. 5.4.7. Discussion The models sketched above give quite reasonable descriptions of the data on nuclear shadowing measured at CERN and FNAL. All of them support the general observation that nuclear shadowing as measured by NMC [72,76,80}83] and E665 [77}79,92] at small x(0.01 receives major contributions from the coherent interaction of the vector mesons o, u and . In fact those experiments are performed at small average momentum transfers QM :1 GeV. On the other hand, the observed weak Q-dependence of the shadowing e!ect originates from the coherent interaction of strongly interacting quark}antiquark #uctuations with large masses, M '1 GeV. 6 5.5. Interpretation of nuclear shadowing in the inxnite momentum frame In this section we brie#y discuss how nuclear shadowing develops in the in"nite momentum frame where the parton model for deep-inelastic scattering can be applied. We found in Section 4.6 that, in this frame, the wave functions of partons from di!erent nucleons in the nucleus start to overlap for x(0.1. One then expects that the interaction of partons belonging to di!erent nucleons increases. Shadowing at x(0.1 is supposed to be due to the fusion or recombination of partons from di!erent nucleons, thereby e!ectively reducing the quark distributions of each nucleon. At the same time parton fusion leads to an enhancement of partons at x'0.1. In Ref. [204] modi"cations of parton distributions due to parton fusion have been derived and found to be proportional to 1/Q. Therefore, parton fusion processes seem to be suppressed at large momentum transfers but can be signi"cant at low Q. Procedures for modeling nuclear parton distributions at small x have been proposed in Refs. [133,204,205]. Recombination e!ects modify these distributions dominantly at a low momentum scale Q where parton fusion is calculated and incorporated in the initial quark and gluon distribution functions. Parton distributions at Q'Q are then derived through the calculation of radiative QCD corrections using DGLAP evolution (see Section 2.4).
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To describe the shadowing as measured by the NMC and E665 collaborations a typical scale Q +0.8 GeV has been used in Refs. [133,205]. As a result the empirical shadowing for F can be described. It should be mentioned that the calculation of the recombination e!ect within perturbation theory is certainly questionable at a low momentum scale Q . The results are strongly sensitive to model parameters, such as the initial scale Q and the input parton distributions. Note that the recombination e!ects discussed here involve parton distributions at a low momentum scale. This `initial-state recombinationa is di!erent from the `radiative recombinationa, discussed in Section 8.3, which modi"es the parton evolution by recombination of radiatively produced partons. 5.6. Nuclear parton distributions at small x Any quantitative QCD analysis of high-energy processes involving nuclei requires a detailed knowledge of nuclear parton distributions. In this section we outline the empirical information on their di!erence with respect to quark and gluon distribution functions of free nucleons. Let us "rst focus on the nuclear gluon distribution. The Q-dependence of deep-inelastic structure functions at small x is dominated by gluon radiation. One can therefore extract nuclear gluon distribution functions from a precise analysis of scaling violations of the structure functions F . In leading order perturbation theory and in the limit x;0.1 the DGLAP equations (2.39,2.40) reduce to the simple form [206]: RF (x, Q) a + Q e xg(2x, Q) . (5.63) D 3p R ln Q D This relation, with further inclusion of small corrections from quark contributions, has been used in an analysis [207] of high statistics NMC data on the Q-dependence of the structure function ratio F1/F! shown in Fig. 3.4. The result for the corresponding ratio of nuclear gluon distributions, g /g , is shown in Fig. 5.14. At x(0.02 the gluon distribution is shadowed, i.e. g /g (1, in 1 ! 1 !
Fig. 5.14. Results from Ref. [207] for the ratio of the Sn and carbon gluon densities, g (x)/g (x), together with the 1 ! measured ratio of structure functions F1(x)/F! (x) [83]. The box represents the extraction of the ratio of gluon distributions from J/t electroproduction data [111].
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a similar way as the structure function F . This is quite natural since F at small x is dominated by contributions from sea quarks. The intimate relation between sea quarks and gluons through DGLAP evolution then also suggests shadowing for gluons. At 0.05(x(0.15 an approximate 8% enhancement of nuclear gluons has been found. This observation is in agreement with an analysis of NMC data for inelastic J/t-lepto-production [111] as indicated in Fig. 5.14. The enhancement of nuclear gluon distributions around xK0.1 is consistent with the fact that the total momentum of hadrons is given by the sum of the momenta of its parton constituents [109,128]. The empirical information on this sum rule applied to quarks has been presented in Section 3.4. It implies that the momentum carried by gluons is, within errors, equal in nucleons and nuclei, i.e.
dx xg (x, Q)+ ,
dx xg (x, Q) .
(5.64)
Consequently, shadowing of nuclear gluon distributions at small x has to be compensated by an enhancement at larger values of x. Assuming the latter to be located in the region 0.05(x(0.15 leads to results similar to the ones shown in Fig. 5.14 [109]. Note that the close relation between shadowing and di!raction allows to estimate gluon shadowing using data on di!ractive charm and dijet production from free nucleons. A corresponding analysis of HERA data has been carried out in Refs. [129,130]. It suggests signi"cantly larger shadowing for gluons than for quarks. Nuclear e!ects in valence and sea quark distributions can be further disentangled using Drell}Yan dilepton production data [109,128]. The E772 collaboration at FNAL has found shadowing for nuclear antiquark distributions at x(0.1 but no enhancement as discussed in Section 3.6.1. Combining this with the fact that the nuclear structure function ratio F /F,91 for 0.05(x(0.2, one concludes that nuclear valence quarks have to be enhanced around x&0.1. From the baryon number sum rule
dx q,(x, Q)" T
dx q (x, Q) , (5.65) T one then "nds [109] that nuclear valence quark distributions, q , must be shadowed at x(0.05. T Typical results from Ref. [128] are shown in Fig. 5.15.
Fig. 5.15. Momentum-space ratios from Ref. [128] relative to the free nucleon, for gluon distributions, valence quark distributions, and the F structure function in Ca at Q"4 GeV.
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To summarize, present data on nuclear shadowing imply that all parton distributions are shadowed at x;0.1, while only valence quarks and gluons are enhanced around x&0.1. The kinematic range where enhancement takes place is related to processes which involve typical longitudinal distances of 1 fm in the laboratory. This is the region where components of the nuclear wave function with overlapping parton distributions should be relevant. In Ref. [109] it was suggested that at such distances inter-nucleon forces are a result of quark and gluon exchange leading to the observed enhancement. In such a picture the enhancement of gluons should increase with the density of the nuclear target. A 10% enhancement of glue at x&0.1 in Sn as compared to C would then imply a 20% increase of the gluon density in Pb as compared to free nucleons [109], indicating a substantial change of gluon "elds in nuclear matter at distances of 1 fm between nucleons. More detailed information on nuclear parton distributions is certainly needed. The shadowing region } where nuclear e!ects are large } is of particular interest. Further constraints on gluon shadowing from deep-inelastic scattering require data on the Q-dependence of nuclear structure functions at smaller values of x as indicated by Eq. (5.63). A more quantitative separation of nuclear e!ects in valence and sea quark distributions could be obtained from Drell}Yan dilepton production or neutrino scattering experiments with high statistics. However, an extraction of nuclear parton distributions in hadron production processes from nuclei, e.g. lepto- or hadroproduction of charmonium or open charm (see e.g. [208,209]), is complicated by possible "nal state interactions and higher twist corrections.
6. Nuclear structure functions at large Bjorken-x Deep-inelastic scattering from nuclei probes the nuclear parton distributions. On the other hand, conventional nuclear physics works well with the concept that nuclei are composed of interacting hadronic constituents, primarily nucleons and pions. For x'0.2 DIS probes longitudinal distances smaller than 1 fm (see Section 4), less than the size of individual hadrons in nuclei. In this kinematic region, incoherent scattering from hadronic constituents of the target nucleus dominates. Such processes explore the quark distributions of nucleons bound in the nucleus. To gain "rst insights suppose that the nucleus is described by nucleons moving in a mean "eld. The quark substructure of bound nucleons may di!er in several respects from the quark distributions of free nucleons. First, there is a purely kinematical e!ect due to the momentum distribution and binding energy of the bound nucleons. This e!ect rescales the energy and momentum of the partonic constituents. To illustrate this recall that for a free nucleon the light-cone momentum fraction of partons cannot exceed x"1. A nucleon bound in a nucleus carries a non-vanishing momentum which adds to the momenta of individual partons in that nucleon. As a consequence light-cone momentum fractions up to x"A are possible in principle, although the extreme situation in which a single parton carries all of the nuclear momentum will of course be very highly improbable. On the other hand, intrinsic properties of bound nucleons, e.g. their size, could also change in the nuclear environment. This may lead to additional, dynamical modi"cations of their partonic structure.
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6.1. Impulse approximation Nuclei are, in many respects, dilute systems. For example, in proton}nucleus scattering the proton mean free path is large in comparison with the average distance between nucleons in nuclei [210]. This observation has motivated the impulse approximation which reduces the nuclear scattering process to incoherent scatterings from the individual nucleons (for reviews and references see [1}5,211]). Final state interactions of the scattered hadron with the residual nuclear system are neglected at high energy. (One should note, however, that the validity of this approximation, illustrated in Fig. 6.1, is still under debate as discussed in [1,5,191,211}215] and references therein.) Given the small average momenta of the weakly bound nucleons, their quark sub-structure is described by structure functions similar to those of free nucleons [191,216]. For a nucleon with momentum p these structure functions depend on the scaling variable x"Q/2p ) q and on the squared momentum transfer Q. However since the energies and momenta of bound nucleons do not satisfy the energy}momentum relation of free nucleons, additional freedom arises. This becomes immediately obvious from the following simple kinematic consideration. In the laboratory frame deep-inelastic scattering from a nucleon bound in a nucleus involves the removal energy, !e , of the struck nucleon: L e "M !ML !M . (6.1) L \ Here M and ML denote the invariant masses of the initial nuclear ground state and of the \ nuclear system, with a nucleon-hole state characterized by its quantum numbers n. The energy of the interacting nucleon is then: p "M#e !¹ , (6.2) L 0 where ¹ "p/2ML is the recoil energy of the residual nuclear system. We "nally obtain for the 0 \ squared four momentum of this interacting nucleon: p"p !pKM#2M(e !¹ !¹)OM L 0
Fig. 6.1. Impulse approximation for deep-inelastic scattering from nuclei at large Q.
(6.3)
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with ¹"p/2M. The squared four momentum of the active nucleon is obviously not restricted by its free invariant mass. It is determined by the nuclear wave function which describes the momentum distributions of bound target nucleons as well as the mass spectrum of the residual nuclear system. Consequently, the structure function of a bound nucleon can in general depend also on p, not just on x and Q. 6.2. Corrections from binding and Fermi motion In the impulse approximation deep-inelastic scattering from a nucleus at large Q proceeds in two steps as shown in Fig. 6.1: the exchanged virtual photon scatters from a quark with momentum k. This quark belongs to a nucleon with momentum p which is removed from the target nucleus. Treating the nucleus as a non-relativistic bound state, the nuclear structure functions factorize into two terms [191]: the information about the quark and gluon substructure of the nucleons is included in the bound nucleon structure functions F,(x/y, p) and F,(x/y, p). They depend on the fraction x/y"k ) q/p ) q+k>/p> of the light-cone momentum of the interacting nucleon carried by the quark, and reduce to the corresponding free nucleon structure functions at p"M. Details about nuclear structure are incorporated in the distribution function of nucleons with squared four-momentum p and a fraction y of the nuclear light-cone momentum:
dp p p > S(p ) 1# d y! d(p!p ) . D (y, p)" , (2p) M M
(6.4)
Here S(p)"2p d(p !M!e #¹ )"W (p)" (6.5) L 0 L L is the spectral function of a nucleon in the nucleus. It is determined by the momentum-space amplitude W (p)"1(A!1) ,!p"WK (0)"A2, with WK (0) representing the non-relativistic nucleon "eld L L operator at the origin r"0. The squared amplitude "W (p)" describes the probability of "nding L a nucleon with momentum p in the nuclear ground state "A2, and the remaining A!1 nucleons in a state n with total momentum !p. In Eq. (6.5) the sum over a complete set of states with A!1 nucleons is taken. Note that the spectral function is normalized to A, the total number of nucleons in the nucleus. This leads to the proper normalization of the nucleon distribution function in Eq. (6.4) (see e.g. Refs. [2,217,218]). The nuclear structure functions are then obtained by a convolution over the squared fourmomentum of the interacting nucleons and their light-cone momentum fraction. For the structure function F per nucleon this gives [191]: (6.6) AF (x)" dy dp D (y, p)F,(x/y, p) , , V where we have suppressed the dependence on Q for convenience.
Here the photon momentum is chosen as qI"(q , 0 , q ) with q (0. ,
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In the following we examine the convolution integral (6.6) in more detail. The momentum distribution of nucleons in the nuclear target,
dp S(p)" "W (p)" , (6.7) L 2p L falls rapidly with increasing "p". This implies that the nucleon light-cone distribution (6.4) is strongly peaked around y+1 and p+M, with a typical width *y&p /M controlled by the Fermi $ momentum p . Expanding the bound nucleon structure function in Eq. (6.6) in a Taylor series $ around y"1 and p"M, and integrating term by term, leads to the following expression for the nuclear structure function per nucleon [191,217], valid in the range 0.2(x(0.7: 1¹2 1e2 xF, (x)# xF, (x) F (x)+F,(x)! 3M M #2
1e2!1¹2 RF,(x, p) p M Rp
,
(6.8)
N + where F, (x) and F, (x) are derivatives of the structure function with respect to x. The mean value of the single-particle energy e"p !M is 1 dp 1e2" S(p)e , (6.9) A (2p)
and
1 dp p 1¹2" S(p) A (2p) 2M
(6.10)
represents the mean kinetic energy of bound nucleons. Except for light nuclei the recoil energy ¹ in Eq. (6.2) can be neglected. Then 1e2 coincides with the separation energy. Corrections to 0 Eq. (6.8) are of higher order in 1e2/M and 1¹2/M. Note that the approximate result for F in Eq. (6.8) is well justi"ed in the region 0.2(x(0.7. Here the kinematic condition x/y(1 in Eq. (6.6) can be ignored in accordance with the underlying expansion. Let us brie#y discuss the physical meaning of the di!erent terms in Eq. (6.8) and their implications. The second term on the right-hand side of Eq. (6.8) involves the average separation energy of nucleons from the target. As such it is determined by nuclear binding. In the range 0.2(x(0.7 it leads to a depletion of the nuclear structure function compared to the structure function of a free nucleon. The third term accounts for the Fermi motion of bound nucleons and yields a strong rise of the structure function ratio F /F, at large x. Finally, the fourth term in (6.8) re#ects the dependence of the structure function of a bound nucleon on its squared fourmomentum. Note that this contribution enters at the same order as binding and Fermi-motion corrections. Information about the p-dependence of bound nucleon structure functions is rare. Nevertheless, such e!ects may lead to signi"cant modi"cations of the EMC ratio F /F, at moderate and large values of x. This has been shown, for example, in the framework of a simple quark}diquark picture for the nucleon [191]. An important and not yet completely solved problem with respect to the binding and Fermimotion corrections in Eq. (6.8) is a reliable calculation of 1e2 and 1¹2. In a simple nuclear shell
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model the separation energy is averaged over all occupied levels. One "nds typical values 1e2+!(20}25) MeV and 1¹2+18}20 MeV. Correlations between nucleons change the simple mean "eld picture substantially and lead to high-momentum components with "p"'p in the $ nuclear spectral function (6.5). This in turn causes an increase of the average separation energy 1e2 [219,220]. In order to see this let us examine the Koltun sum rule [221] 1e2#1¹2"!2k ,
(6.11)
where k is the binding energy per nucleon. This sum rule is exact if only two-body forces are present in the nuclear Hamiltonian. With "xed k +8 MeV, this sum rule tells that an increase of 1¹2 due to high-momentum components is accompanied by an increase of "1e2". We refer in this context to a calculation [222] of the spectral function of nuclear matter based on a variational method. This calculation shows that there is a signi"cant probability to "nd nucleons with high momentum and large separation energies. An integration of the spectral function of Ref. [222] gives 1¹2+38 MeV and 1e2+!70 MeV [220]. In order to estimate these quantities for heavy nuclei one usually assumes [219] that the high-momentum components of the nucleon momentum distribution are about the same as in nuclear matter. Together with Eq. (6.11) this leads to 1e2+!50 MeV. In Fig. 6.2 we show typical results from Refs. [191,218] for iron and gold. We observe that a qualitative understanding of the EMC e!ect can indeed be reached, but at x90.5 a more quantitative description is still lacking. One should note, of course, that the presentation of nuclear e!ects in terms of the ratio F /F, magni"es such e!ects in a misleading manner because F, itself is small in this region (see also the discussion in Section 4.4). The impulse approximation picture of nuclear deep-inelastic scattering can also be maintained in a relativistically covariant way [215]. Here, however, a simple factorization of nuclear structure functions into nuclear and nucleon parts as in Eq. (6.6) is not possible any longer. A relativistic calculation of nuclear structure functions requires relativistic nuclear wave functions as well as
Fig. 6.2. The ratio of nuclear and nucleon structure functions, F /F,, for iron and gold taken from Refs. [218,191]. (a) solid curve: calculation in Ref. [218], dotted curve: calculation in Ref. [217]. (b) results from Ref. [191]: without p-dependence of the bound nucleon structure function (dashed), and including this p-dependence as obtained from a simple quark}diquark picture (full).
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a more detailed knowledge about the structure of bound nucleons. Nevertheless, relativistic e!ects seem to be small: in an explicit model calculation of the deuteron structure function F relativistic corrections to the non-relativistic convolution (6.6) are less than 2% for x(0.9 [223]. In the region x'1, where nuclear structure functions are very small however, larger deviations are expected. In this context a word of caution is in order. A description of nuclear structure functions based on nucleons alone is necessarily incomplete since it violates the momentum sum rule [1]. Non-nucleonic degrees of freedom are brie#y discussed in Section 6.5. 6.3. Beyond the impulse approximation The quality of the impulse approximation has frequently been questioned (see e.g. Refs. [1,5,191,213}215] and references therein). Here we give a brief summary of possible shortcomings in terms of models for nuclear deep-inelastic scattering which go beyond this approximation. 6.3.1. Quark exchange in nuclei The impulse approximation includes only incoherent scattering processes from hadronic constituents of the target nucleus. On the other hand, contributions involving several bound nucleons could also be important, and their role needs to be examined. One such possibility, namely quark exchange between di!erent nucleons, has been investigated in Refs. [213,224]. The nuclear quark wave function which is probed in deep-inelastic scattering must be antisymmetric with respect to permutations of quarks. This however is not realized in the impulse approximation (6.6). Antisymmetrization introduces additional quark exchange terms between di!erent nucleons in the target. Under several simplifying assumptions a softening of the nuclear quark momentum distribution due to quark exchange has been found. For small nuclei the e!ect turned out to be signi"cant. For He approximately 30% of the observed depletion of the structure function ratio F&/F, at x&0.6 has been associated with quark exchange. Only minor modi"cations have been found for heavier nuclei [224]. While the estimates based on a simple quark exchange model may not be reliable at a quantitative level, they certainly point to the fact that the impulse approximation is incomplete as soon as correlations between quarks in several nucleons come into play. 6.3.2. Final state interactions in a mean xeld approach One of the basic assumptions of the impulse approximation is that interactions of the struck, highly excited nucleon with the residual nuclear system can be ignored. In general, there is no solid basis for this assumption since the debris of the struck nucleon includes also low momentum fragments as seen from the target rest frame. A proper treatment of their "nal state interaction requires however a description of the nucleus in terms of quark and gluon degrees of freedom. Investigations in this direction have been made starting out from a quark model for nuclear matter, with nucleons modeled as non-overlapping MIT bags [214,225,226]. The nucleons interact via the exchange of scalar and vector mesons which couple directly to quarks. Within the mean "eld approximation for the meson "elds it is possible to describe several basic properties of nuclear matter, such as its compressibility and the binding energy per nucleon at saturation density.
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This model has been applied to deep-inelastic scattering from "nite nuclei using a local density approximation [214]. The debris of the struck nucleon is represented by a pair of spectator quarks bound in a diquark bag. Its interaction with the remaining nuclear system in the "nal state leads to a non-negligible e!ect on nuclear structure functions: while the full calculation including "nal state interactions allows to reproduce the structure function ratio F /F,, the impulse approximation overestimates nuclear e!ects at x&0.6 by about a factor two. In the framework of this model such a di!erence is expected since the binding of the nucleus is the result of the attractive scalar mean "eld experienced by all three constituent quarks of the interacting nucleon. When ignoring the binding of the spectator quark pair, as done in the impulse approximation, one assumes that the active quark which takes part in the deep-inelastic scattering process behaves as if it carries the binding of all three quarks, a feature which obviously needs to be corrected. The mean "eld approach to nuclear quark distributions is based on several simplifying assumptions, but it nevertheless points to the possible importance of "nal state interactions and, in more general terms, to the relevance of quark degrees of freedom in nuclei. 6.4. Modixcations of bound nucleon structure functions The intrinsic properties of nucleons bound in nuclei can be modi"ed as compared to free nucleons. We summarize below two examples of models which deal with such possible changes in bound nucleon structure functions. 6.4.1. Rescaling At intermediate values of the Bjorken variable, 0.2(x(0.7, the modi"cation of nuclear structure functions F as compared to the free nucleon structure function F, can be described by a shift in the momentum scale which enters the structure functions. We brie#y outline here the basic arguments [227}231]. Consider the moments
M (Q)" L
dx xL\F (x, Q) with n even .
(6.12)
Assume now that the moments of nuclear and nucleon structure functions are related by a shift of their momentum scale: M (k )"M,(k ) . (6.13) L L , At an arbitrary momentum transfer Q the perturbative QCD evolution equations to leading order (see Section 2.4) give M (Q)"M,(m (Q)Q) L L with the rescaling parameter m (Q)"(k /k )?Q I ?Q / . , For simplicity we use the non-singlet part only.
(6.14)
(6.15)
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Of course, Eq. (6.14) can always be satis"ed if one allows di!erent m for di!erent moments n. However, when comparing with data it has turned out that the rescaling parameter is independent of n to a good approximation. Consequently, the scale change for the moments (6.14) can be translated directly into a scale change for the structure functions themselves: F (x, Q)"F,(x, m (Q)Q) . (6.16) Good agreement with experimental data can be achieved at intermediate x. For example, the EMC structure function data on iron suggest m +2 for Q"20 GeV [229]. $ Rescaling gives a reasonable one-parameter description of nuclear structure functions F at intermediate x, but it does not o!er insights into the physical origin of the observed change of scale. One possible suggestion to explain the scale change is a modi"cation of the e!ective con"nement scale for quarks in the nucleus as compared to free nucleons [227}229]. Scale changes are not simply related to possible `swellinga of nucleons inside nuclei which is constrained by inclusive electron-nucleus scattering data in the quasielastic region. The experimentally observed y-scaling indicates a rather small increase of the charge radius for bound nucleons. For example, the study of [232,233] comes to the conclusion that any increase of the nucleon radius in nuclei should be less than 6% of its free radius. Related discussions and a comparison with nuclear DIS data can be found in Refs. [205,219,234}237]. 6.4.2. Color screening in bound nucleons The scenario of Refs. [238,239] assumes that the dominant contribution to the structure function F, at large x&0.5 is given by small size (pointlike) parton con"gurations in the nucleon. In a nuclear environment such con"gurations interact only weakly with other nucleons due to the screening of their color charge. It is argued that the probability for pointlike con"gurations is reduced in bound nucleons. In fact, the probability to "nd parton con"gurations of average size in the nucleon should actually be enhanced in nuclei since they experience the attraction of the nuclear mean "eld. Then the variational principle with normalization condition on the wave function implies that small-size con"gurations should indeed be suppressed. An estimate of such deformations in the wave function of nucleons bound in heavy nuclei gives for x&0.5 [238]: F /F,&1#41;2/EH&0.7}0.8 . (6.17) Here 1;2 is the average potential energy per nucleon, 1;2&!40 MeV, and EH&0.5 GeV is the typical energy scale for excitations of the nucleon. Since 1;2 scales with the nuclear density, the nuclear dependence of the structure function ratio (6.17) is roughly consistent with data. It should be mentioned that the proposed suppression of rather rare pointlike con"gurations in bound nucleons does not necessarily imply a substantial change of average properties of a bound nucleon, such as its electromagnetic radius [1]. 6.5. Pion contributions to nuclear structure functions In conventional nuclear physics meson exchange is responsible for the binding of nucleons in the nucleus. Therefore, deep-inelastic scattering from mesons present in the nuclear wave function
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should give additional contributions to nuclear structure functions. Pions, which are responsible for most of the intermediate- and long-range nucleon}nucleon force, are supposed to play the prominent role (see e.g. [240}244]). The framework is the Sullivan process [245]. Its contribution to the nucleon structure function F, reads dpF,(x)" dy f (y)Fp (x/y) , (6.18) p , V where
R !t"F (t)" 3g p,, dt y (6.19) f (y)" p,, p , (t!m) 16p \ p speci"es the distribution of pions with light-cone momentum fraction y in the nucleon, while Fp is the pion structure function. Eq. (6.18) describes deep-inelastic scattering from a pion emitted from its nucleon source. The nucleon receives a momentum transfer equal to the pion momentum kI"(u, k). The minimal squared momentum transfer t"k required for pion emission is t "!My/(1!y). One "nds that dpF, gets its dominant contributions from pions with
momenta "k"K300}400 MeV. Pions with smaller momenta are suppressed by the explicit factor y in Eq. (6.19), while pions with large momenta are suppressed by the pion propagator and the pNN form factor F [240]. p,, The convolution ansatz in Eq. (6.18) su!ers from similar problems as convolution for nuclear structure functions discussed in Section 6.2: the interacting pion is not on its mass shell, i.e. k+!kOm. Therefore the pion structure function depends also on k. Furthermore, "nal state p interactions of the pion debris with the recoil nucleon are neglected. The detailed treatment of pionic e!ects in nuclei includes the pion propagation in the medium with D resonance excitation, Pauli e!ects and short range spin}isospin correlations. All these e!ects are incorporated in the pion-nuclear response function R(k, u) which determines the spectrum of pionic excitations in the nuclear medium. The resulting distribution function of pions in a nucleus is [240]
k \+W k"F (k)" 3g p,, d"k" R("k", u) , (6.20) du f (y)" p,, y p (t!m) 16p +W p where t"u!k. Using the Sullivan description (6.18) the contribution of excess pions to the nuclear structure function F can be calculated according to dpF (x)" dy ( f (y)!f (y))Fp (x/y) . (6.21) p p , V In the original work in Ref. [240], using the empirical pion structure function, a signi"cant enhancement of the ratio F /F, was found at x(0.3. This observation was in agreement with the early EMC data [75]. Later data on nuclear structure functions showed only a minor enhancement around xK0.15 (see Section 3.3.1). In addition Drell}Yan data from E772 [108] have demonstrated that the antiquark distribution in nuclei is not signi"cantly enhanced as compared to free
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nucleons, in disagreement with the "rst pion model calculations. However, as already emphasized in Ref. [240], the pion contribution to nuclear structure functions is very sensitive to the strength of repulsive short-range spin}isospin correlations in nuclei. Variations of this strength by 15% can easily lead to 30% changes in dpF . While it is not di$cult to accommodate the very small observed pionic enhancement within such uncertainties, it is still a challange to arrive at a consistant overall picture of nuclear DIS which rigorously satis"es the requirements of the momentum sum rule. 6.6. Further notes Related studies of pion "eld e!ects as well as other nuclear medium corrections and their implications on nuclear DIS have been performed in Refs. [246}249]. These studies include calculations within the delta-hole model [248], the role of NN correlations and the energy dependence of nuclear response functions [249], possible e!ects of `Brown}Rho scalinga on nuclear structure functions [246], and implications of low-energy pion-nucleus scattering data for nuclear deep-inelastic scattering and Drell}Yan production [247]. Some further investigations use a relativistic many-body approach to treat mesonic and binding corrections to reproduce nuclear e!ects in the EMC and Drell}Yan measurements [250,251]. From the point of view of nuclear many-body theory, the best nuclear wave functions have been employed in Refs. [252,253], treating both short and long range correlations in nuclear matter and helium at the most advanced level. Two-nucleon correlations turn out to be important in nuclear DIS. It should be mentioned, however, that the results of [250,252,253] have met with some debate concerning the proper choice of the `#ux factora. Questions of rigorous baryon number conservation [1] have also been raised. It has been suggested to investigate intrinsic properties of bound nucleons in semi-inclusive deep-inelastic scattering from nuclei [238,254}256]. Measuring the scattered lepton in coincidence with the residual nuclear system should provide detailed information on changes in bound nucleon structure functions. Possible experiments are discussed at HERMES [257].
7. Deep-inelastic scattering from polarized nuclei Understanding the spin structure of the proton and the neutron is a central issue in QCD. Both the polarized neutron and proton structure functions, g and g , are needed in the investigation of #avor singlet quark spin distributions (see e.g. Ref. [45]), and in the experimental test of the fundamental Bjorken sum rule (2.60). Since free neutron targets are not available one must resort to polarized nuclei, such as the deuteron and He, where the neutron spin plays a well de"ned role in building up the total polarization of the nuclear target. Polarized deep-inelastic scattering from deuterium [33,38,39,44] and He [40,37,42] has been studied with high precision. In order to deduce accurate information about the individual nucleon spin structure functions from these data, it is essential to correct for genuine nuclear e!ects. In addition, the presence of the tensor interaction between nucleons in nuclei creates speci"c spin e!ects which are of interest in their own right.
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7.1. Ewective polarizations As discussed in Section 4, nuclear structure functions at Bjorken-x'0.2 are dominated by the incoherent scattering from bound nucleons. For polarized nuclei, the non-trivial spin}orbit structure of the wave functions causes new e!ects. Bound nucleons can carry orbital angular momentum, so their polarization vectors need not be aligned with the total polarization of the target nucleus: depolarization e!ects occur. In order to describe such nuclear depolarization phenomena it is useful to introduce e!ective polarizations for nucleons bound in the nucleus. Let "A!2 represent a nuclear state polarized in the z-direction. Then the e!ective polarization of protons or neutrons in that nucleus is P "1A!" p (i)"A!2 , X G
(7.1)
P "1A!" p (i)"A!2 . (7.2) X G When the nuclear depolarization e!ects are described entirely in terms of these e!ective polarizations of bound nucleons, the nucleon spin structure functions (say, g and g ) have the following simple additive form: g (x, Q)"P g (x, Q)#P g (x, Q) . (7.3) Nuclear depolarization e!ects are important over the whole kinematic range of recent measurements. Within the impulse approximation these e!ects exceed by far the in#uence of nuclear binding and Fermi motion at 0.2(x(0.7 (see Section 7.4). 7.2. Depolarization in deuterium and He In case of the deuteron the e!ective proton and neutron polarizations are simply determined by the D-state admixture in the deuteron wave function, induced by the tensor interaction between proton and neutron in the spin triplet state. One "nds (7.4) P "P "1!P , " where P is the D-state probability. The numerical values of P range between 0.91 and 0.94 " using deuteron wave functions calculated with the Paris [153] or Bonn [199] nucleon}nucleon potential, respectively. Apart from the interest in neutron spin structure functions the deuteron with its triplet spin structure is of interest all by itself. Its spin-1 property leads to additional structure functions as given in Eq. (3.3). In particular, the new spin structure functions b and b are accessible in deep-inelastic scattering from polarized deuterons and can be investigated in forthcoming HERMES measurements [70]. Polarized He can be viewed, to a "rst approximation, as a polarized neutron target, with the proton}proton subsystem in a spin singlet con"guration and the surplus neutron carrying the spin of the three-body system. Corrections to this picture come from the admixture of S - and D-wave
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components to the He wave function. The consequence is that the e!ective neutron polarization is reduced from unity, and the e!ective proton polarization does not vanish: (7.5) P&"1!P !P "0.86$0.02 , 1Y " (7.6) P&"!(P !P )"!0.056$0.008 . 1Y " These results are obtained from three-body calculations using realistic nucleon}nucleon interactions [258] omitting however e!ects from meson exchange currents (see Section 7.3.4). 7.3. Nuclear coherence ewects in polarized deep-inelastic scattering Coherence phenomena such as shadowing at small Bjorken-x are dominated by the interaction of di!ractively excited hadronic states with several nucleons over large longitudinal distances in the target nucleus. The characteristic space}time properties of DIS are independent of the target or beam polarization. Therefore, nuclear coherence e!ects are also expected in polarized scattering. We explore such e!ects in the following for deuterium and He. 7.3.1. Polarized single and double scattering in the deuteron Consider the deuteron spin structure functions g and b at small values of the Bjorken variable, x(0.1. Following the discussion in Section 3.2, these structure functions can be expressed in terms of virtual photon}deuteron helicity amplitudes. At large Q in the Bjorken limit which we keep throughout this section, only the helicity conserving amplitudes enter. As usual, we choose a right handed, transversely polarized (virtual) photon (index `#a) for reference. We denote the helicity conserving cHd amplitudes by AAH , where H"0,#,! refers to the helicity state of the polarized >& deuteron, and we choose the direction de"ned by the photon momentum q as quantization axis. The spin structure functions of interest are then expressed as (3.7), (3.9) g "(1/4pe) Im(AAH !AAH ) , (7.7) >\ >> H H H b "(1/4pe) Im(2AA !AA !AA ) . (7.8) > >> >\ Let us now decompose AAH into incoherent, single scattering terms and a coherent double >& scattering contribution. We will use the non-relativistic deuteron wave function,
1 u(r) v(r) 1 t (r)" # SK (r( ) s , & & r (8 (4p r
(7.9)
where r""r", and s denotes the S"1 spin wave function of the deuteron. The tensor operator & SK (r( )"3(rp ) r)(r ) r)/r!r ) r , and u(r), v(r) are the S- and D-state radial wave functions normalized as dr[u(r)#v(r)]"1. The D-state probability is P " dr v(r). We have " P K5.8% for the Paris potential [153] and P K4.3% for the Bonn potential [199]. " " In the polarized deuteron, the proton or neutron can have their spins either parallel or antiparallel with respect to the z-axis de"ned by q/"q". Let the corresponding projection operators be P and P , respectively. The amplitude for single scattering of the virtual photon from t s a proton in the polarized deuteron is
AAH " dr tR (r)(P AAH #P AAH )t (r) >& & t >t s >s &
(7.10)
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with the helicity conserving cH-proton amplitudes AAH and AAH . The analogous amplitudes for >t >s single scattering from the neutron are obtained by the replacement [p n]. We then have AAH "AAH #AAH at the single scattering level. Combining with Eqs. (2.30), (7.7) one "nds >& >& >& (7.11) g "(1!P )(g #g )"2Pg, , " where P is the e!ective nucleon polarization (7.4) in the deuteron and g,"(g #g )/2. Of course, b "0 at the single scattering level since nucleons as spin objects do not have a structure function b . Next, we concentrate on the coherent double scattering amplitude dAAH "AAH !(AAH #AAH ) >& >& >& >&
(7.12)
which simultaneously involves both the proton and the neutron. At x(0.1 this amplitude is dominated, as in the unpolarized case, by di!ractive production and rescattering of intermediate hadronic states, but now with the polarized target nucleons excited by polarized virtual photons. We introduce the di!ractive production amplitudes ¹6, (k) and ¹6, (k) which describe the >t >s di!ractive production process cHNPXN for right-handed photons on polarized nucleons, with momentum transfer k. Following steps similar to those described in Section 5.1.1, one "nds [71,259]:
dk i , db e k, b dAAH " dz e X H >& 2Ml (2p) \ 6 ;tR (r)(P ¹6 (k)#P ¹6 (k))(P ¹6 (k)#P ¹6 (k))t (r) & t >t s >s t >t s >s &
(7.13)
with the longitudinal propagation length j"2l(M #Q)\ of the di!ractively produced inter6 mediate system. We recall from Section 4.5 that a hadronic #uctuation of mass M contributes 6 to coherent double scattering only if its propagation length j exceeds the deuteron diameter, 1r2 K4 fm. In the following, we approximate the dependence of the di!ractive production amplitudes on the momentum transfer t"k+!k by , ¹6,(k)+e\
k ,
¹6,
(7.14)
with the forward amplitude ¹6,,¹6,(k"0). Various data on di!ractive leptoproduction at Q:3 GeV suggest an average slope BK(5210) GeV\ (for references see e.g. [50,65]). We then de"ne the integrated (longitudinal) form factor
dk , S (k , j\) e\ F (j\)" & (2p) & ,
k ,
,
(7.15)
where
S (k)" dr"t (r)"e k r & &
(7.16)
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is the conventional helicity-dependent deuteron form factor. Next, we introduce helicity dependent di!ractive production cross sections for transversely polarized virtual photons by
1 5 dpAH, , t ("¹6 "#"¹6 ")"16p dM (7.17) >t >t 6 dM dt 8Ml Kp 6 R 6 and a corresponding expression for dpAH, ,, with the center-of-mass energy = of the cHN system. s The resulting coherent double scattering correction to the spin structure function g becomes [71] dpAH, , dpAH, , 2Q 5 s ! t dM dg (x, Q)"! F (j\) . (7.18) 6 dM dt dM dt > ex Kp R 6 6 Similarly one obtains for b from Eqs. (7.8), (7.13) [71,198,259]: dpAH2 , 2Q 5 dM (F (j\)#F (j\)!2F (j\)) . (7.19) b " 6 dM dt > \ ex Kp 6 R
7.3.2. Shadowing in g The di!erence of polarized di!ractive virtual photoproduction cross sections which enters in Eq. (7.18) has so far not yet been measured. Nevertheless, it is possible to estimate the shadowing correction dg to an accuracy which is su$cient for a reliable extraction of the neutron structure function g from current experimental data. With inclusion of shadowing and the e!ective nucleon polarization in the deuteron one "nds P!dg /(2g,) g !g . g + (P)
(7.20)
The measured spin structure functions have the property "g "("g ", at least for x'0.01 [34]. This implies that, at the present level of data accuracy, shadowing e!ects and uncertainties in the deuteron D-state probability do not play a major role in the extraction of g . To estimate the amount of shadowing in g and its in#uence on the extraction of g one can study the double scattering contribution (7.18) in the framework of a simple model. In the laboratory frame at small x(0.1 the exchanged virtual photon "rst converts to a hadronic state X which then interacts with the target (see Section 4.5), dominant contributions coming from hadronic states with invariant mass M &Q. Consider therefore a single e!ective hadronic state 6 with a coherence length j&1/(2Mx). Comparing shadowing for unpolarized and polarized structure functions gives [71]: dg F (2Mx) dF dF +R +R with R "2 > . (7.21) E E E 2g, F (2Mx) F, F, At small x and B"7 GeV\ one "nds R "2.2 for both the Paris and Bonn nucleon}nucleon E potentials [153,199]. Although shadowing for g turns out to be approximately twice as large as for the unpolarized structure function F , it still leads to negligible e!ects on the extraction of g , at Note that shadowing corrections in unpolarized structure functions, dF "F !(F #F )/2, are de"ned per nucleon, as F themselves.
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least at the present level of experimental accuracy. Using the experimental data on shadowing for F [92] one "nds that the shadowing correction in (7.20) amounts at x&0.01 to less than 5% of the experimental error on g for the SMC analysis [35]. 7.3.3. The tensor structure function b at small x The shadowing correction for the unpolarized structure function dF and the deuteron tensor structure function b are directly related. In order to see this, note again that the propagation lengths (4.19) of di!ractively produced hadrons exceed the deuteron size j'1r2 +4 fm at small x. The deuteron form factors become approximately constant, i.e. F (j\(1/4 fm)+ & F (0), and a comparison with the double scattering correction for the unpolarized structure & function (5.15) gives [71]: F (0)!F (0) > . with R "2 (7.22) b "R dF @ @ F (0) With B"7 GeV\ we "nd from the Paris nucleon}nucleon potential [153] R "!0.66, while @ the Bonn one-boson-exchange potential [199] leads to R "!0.58. Using data for F /F, [92] @ one can estimate b at small x and "nds that it reaches about 2% of the unpolarized structure function F, at x;0.1 [71,260]. In Fig. 7.1 we present b as obtained from Eq. (7.22). The result shown here corresponds to the kinematics of E665 [92]. Estimates of b at large Q<1 GeV and small x;0.1 can be found in [198]. It should be mentioned that the magnitude of b at x(0.1 exceeds estimates from previous model calculations which are applicable at moderate and large x, by several orders of magnitude (see e.g. [261,262]). Unfortunately, the e!ect of b in the observable asymmetry, which is propor tional to b /F , is only of the order of 10\, as already mentioned. 7.3.4. The He case Although He would appear to be an ideal neutron target because of its large e!ective neutron polarization (7.5), the extraction of neutron spin structure functions nevertheless requires dealing
Fig. 7.1. Double scattering contribution to the tensor structure function b from [71]. The dotted and dashed curves correspond to the Bonn [199] and Paris potential [153], respectively.
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with signi"cant nuclear e!ects. We concentrate again on the structure function g at small values of x. As for deuterium two types of corrections are relevant: higher angular momentum components of the He wave function lead to e!ective proton and neutron polarizations (7.5,7.6). Furthermore, the coherent interaction of the virtual photon with several nucleons causes shadowing. Accounting for both e!ects the neutron structure function can be deduced from g& as follows: P& P&!dg&/g g&! g . (7.23) g + P& (P&) Since "P&";"P&" the proton contribution in (7.23) is indeed suppressed. Uncertainties in the e!ective nucleon polarizations which may in#uence the extraction of g , result also from non-nucleonic degrees of freedom, e.g. mesons and *-isobars, present in He. In Ref. [263] this is demonstrated for the non-singlet nucleon and A"3 structure functions: (7.24) *g (x, Q)"g (x, Q)!g (x, Q) , *g (x, Q)"g&(x, Q)!g&(x, Q) , (7.25) where g& is the triton spin structure function. From the e!ective nucleon polarizations (7.5), (7.6) and isospin symmetry of the three-body nuclear wave function one obtains (7.26) *g "(1!P !P )*g . 1Y " Applying the Bjorken sum rule (2.60) to the A"1 and 3 systems and taking their ratio leads to: dx[g&(x, Q)!g&(x, Q)] G (H) " . (7.27) G (n) dx[g (x, Q)!g (x, Q)] The axial vector coupling constants G (H) and G (n) are measured in the b-decay of tritium and the neutron, respectively, with G (n),g "1.26 [264]. If one considers incoherent scattering from individual nucleons and accounts only for the e!ective nucleon polarizations one "nds from Eqs. (7.26, 7.27): (7.28) G (H)/g "1!P !P "0.922$0.006 . 1Y " This result is however at variance with the empirical ratio G (H)/g +0.963$0.003 [264,265]. One concludes that simply using e!ective nucleon polarizations from realistic three-nucleon calculations would lead to a violation of the Bjorken sum rule by approximately 4% [263,266]. On the other hand, it is known from nuclear b-decay and Gamov-Teller transitions that axial coupling constants in nuclei are renormalized by meson exchange currents and *-isobars [267]. The possible in#uence of these non-nucleonic degrees of freedom on the extraction of g has to be carefully investigated [263]. At small values x(0.1 of the Bjorken variable, coherent multiple scattering from several nucleons in the target leads to the shadowing correction dg . Assuming again, as in Section 7.3.2, that the photon-nucleus scattering at small x can be represented by the interaction of one e!ective hadronic #uctuation with invariant mass M &Q, one "nds [263] 6 g&/g +(2 F&/F,!1) , (7.29)
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i.e. shadowing in g& is about twice as large as for the unpolarized structure function F&. This is also true for shadowing in the non-singlet structure function *g [263]. A similar result has been found for the polarized deuterium case, see e.g. (7.21). In Ref. [263] the shadowing correction (7.29) has been combined with the nuclear Bjorken sum rule (7.27). In the nuclear case, shadowing reduces the small-x contribution to the Bjorken integral. Consequently the non-singlet nuclear structure function must be enhanced, i.e. *g '*g , somewhere in the region x90.1. This is suggested to occur around x&0.1, where the projectile may still interact with two nucleons inside the target. As a consequence a signi"cant antishadowing is obtained in this kinematic region. The dynamical origin for such an enhancement is supposed to be independent of the #avor channel considered, so that this is also expected to occur for g&. To summarize this discussion we emphasize that a precise extraction of the neutron spin structure function g from He data at x:0.2 requires a careful analysis of nuclear e!ects due to non-nucleonic (e.g. meson and *-isobar) degrees of freedom in the He wave function and a detailed understanding of shadowing and anti-shadowing e!ects. The use of deuterium as a target may have advantages since non-nucleonic admixtures are supposed to be smaller due to weaker nuclear binding. Furthermore, especially at small x the in#uence of shadowing on the extraction of g is less pronounced for deuterium as compared to He. 7.4. Polarized deep-inelastic scattering from nuclei at x'0.2 At moderate and large values of x, the distances probed by DIS from nuclei are smaller than 2 fm as outlined in Section 4. Incoherent interactions of the virtual photon with hadronic constituents of the nucleus dominate, and the usual starting point is a description based on the impulse approximation including nucleon degrees of freedom only [216,258,268,269]. This approach is not without question, but not much progress has so far been made beyond that level. In the following we "rst discuss general features of the nuclear spin structure functions g and g "g #g in the framework of the impulse approximation. The nuclear target is treated 2 non-relativistically. This allows, as in the unpolarized case, to factorize nucleon and nuclear degrees of freedom, introducing structure functions of nucleons as bound quasi-particles. The nuclear structure functions are then given as two-dimensional convolutions of the bound nucleon structure functions and the nucleon light-cone momentum distributions [216]:
dy x DG (y, p) gG , p , (7.30) g (x)" dp y y V G x x dy g (x)" dp DG (y, p) gG , p #DG (y, p) g , p . (7.31) 2 2 2 y 2 y y V G Here we have suppressed the Q-dependence for simplicity. The expressions in Eqs. (7.30) and (7.31) resemble the result for the unpolarized nuclear structure function in Eq. (6.6): the exchanged virtual photon scatters from quarks which carry a fraction x/y of the light-cone momentum of their parent nucleons, which in turn have a fraction y"p>/M of the nuclear light-cone momentum and a squared four-momentum p.
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The nucleon distribution functions in Eqs. (7.30) and (7.31) are given by
dp p > D (y, p)" tr[S (p )(RK #RK )]d y! d(p!p ) , X (2p) M
dp p > D (y, p)" tr[S (p )RK ]d y! d(p!p ) , 2 , , (2p) M
p > dp D (y, p)" tr[S (p )T K ]d y! d(p!p ) 2 , M (2p)
(7.32) (7.33) (7.34)
with the polarized nucleon spectral function: S (p)"2p d(p !M!e #¹ )t (p)tH (p) . (7.35) NNY L 0 L N L NY L Here the summation is performed over the complete set of states with A!1 nucleons. The functions t (p)"1(A!1) ,!p"t (0)"A2 are the probability amplitudes to "nd a nucleon with L N L N polarization p in the nuclear ground state and the remaining A!1 nucleons in a state n with total momentum !p. The separation energy e and the recoil energy of the residual nuclear system L ¹ enter in Eq. (7.35) as for the unpolarized case (6.5). For g the target nucleus is chosen to be 0 polarized parallel to the photon momentum as indicated in Eq. (7.32) by the subscript "". For the transverse structure function g the target polarization is taken perpendicular to the momentum 2 transfer and denoted by N. The nucleon spin operators which multiply the spectral functions in Eqs. (7.32)}(7.34) refer to the active nucleon. They read: r)p , RK " M
p r)p RK " 1! p# p , H 2M H 2M H
p )S r)p p TK "! , , #p 1! X X M M M
,
(7.36)
where j denotes spatial indices. Furthermore, p is the transverse component of the nucleon , three-momentum, p"(p , p ), and S determines the transverse spin quantization axis relative to , X , the photon momentum which is taken along the z-direction. From Eqs. (7.30) and (7.31) we observe that g is expressed entirely in terms of the corresponding nucleon structure function g,. This is di!erent for g which receives contributions from g, as well 2 2 as from g,. If the bound nucleon structure functions in Eqs. (7.30) and (7.31) are replaced by free ones, one ends up with the conventional one-dimensional convolution ansatz for nuclear structure functions [216,258]. Relativistic contributions which lead beyond the convolution formula (7.30), (7.31) have been investigated in Refs. [270,271], and corrections have been estimated within a quark-diquark model for the bound nucleon. Deviations from non-relativistic convolution were generally found to be small, except at very large x'0.9. Given that the nuclear spectral functions, polarized as well as unpolarized, receive their major contributions from small nucleon momenta, systematic expansions of g and g for x(0.7 can be 2
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performed around y"1 and the mass-shell point p"M, keeping terms of order e/M and p/M . Results and applications to spin structure functions of the deuteron and He are discussed in Refs. [216,258,269,272]. As a general rule, the structure functions at x(0.7 are well described using simply the e!ective nucleon polarizations (7.4)}(7.6).
8. Further developments and perspectives We close this review with a short summary of the key physics points together with an outlook on several selected topics for which investigations are still actively under way. We comment on exclusive vector meson production from nuclei, questions of shadowing at large Q and the issue of high parton densities in nuclear systems. 8.1. Coherence ewects in DIS and in the exclusive electroproduction of vector mesons Nuclear shadowing in inclusive deep-inelastic lepton scattering is a prime source of information on coherently propagating hadronic or quark}gluon #uctuations of the virtual photon in a nuclear medium. By selecting di!erent kinematic cuts in Q and energy transfer l, one can focus on di!erent components of the photon's Fock space wave function. An even more stringent selection of such components can be achieved in exclusive photo- and electroproduction processes, and in particular in high-energy di!ractive vector meson production. Data on vector meson production from nuclei have become available in recent years at FNAL (E665) [273], CERN (NMC) [274,275], and DESY (HERMES) [276], and further experiments are under discussion at TJNAF [277] and DESY (HERA, HERMES) [257]. Depending on energy and momentum transfer, the mechanism of vector meson formation can be quite di!erent. In the range l'3 GeV and Q:1 GeV the production process is well described using the vector meson dominance picture (see e.g. [25]): in the lab frame the photon converts into a vector meson prior to scattering from the target. On the other hand, at large Q<1 GeV perturbative QCD calculations show that the photon}nucleon interaction produces an initially small-sized, color singlet quark}antiquark wave packet [167,278]. At high photon energies the "nally observed vector meson is then formed at a much later stage. The transition from small to large Q interpolates between non-perturbative hadron formation and perturbative quark}antiquark}gluon dynamics, a question of central importance in QCD. Nuclear targets are particular helpful at this point because they serve as analyzers for the coherent interaction of the produced qq -gluon system with several nucleons [279]. The distance between two nucleons provides the `femtometer sticka which can be used to measure the relevant coherence lengths (for reviews and references see [280}282]). The characteristic scales for this discussion have been encountered several times in previous sections. First, there is the typical longitudinal distance (propagation length) j+2l/(m#Q) .
(8.1)
It represents the distance over which a hadronic #uctuation of invariant mass m propagates in the lab frame when induced by a photon of energy l and virtuality Q. At large Q the initially
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produced wave packet is characterized by its transverse size b. For longitudinally polarized photons, b"const./Q .
(8.2)
In perturbative QCD the minimal Fock space component has const.&4}5 at Q95 GeV [167,283]. Thus for Q"5 GeV and const."4, the transverse size of the initial wave packet is b&0.4 fm, a small fraction of the diameter of a fully developed o meson. Recent measurements at HERMES [276] have observed e!ects related to the coherence length j in o electroproduction on hydrogen, deuterium, He, and N. The range of energy and momentum transfers covered by the experiment is 9 GeV(l(20 GeV and 0.4 GeV(Q (5 GeV. This implies coherence lengths in the range 0.6 fm:j(8 fm covering scales from individual nucleons up to and beyond nuclear dimensions. (The interesting upper section of the available Q interval, Q94 GeV, has been accessible only for small energies l with j:1 fm in these measurements.) Given the four-momenta q and k of the incoming virtual photon and the produced o meson, the t-channel momentum transfer to the nucleon is t"(q!k). In Fig. 8.1 the rate of produced o's is plotted against t "t!t , the squared momentum transfer above threshold ("t "Kj\). At "t ";0.1 GeV coherent production dominates, leaving the nucleus as a whole in the ground state. Such coherent processes fall o! rapidly with the nuclear form factor, so that at "t "90.1 GeV mostly incoherent o production from individual nucleons remains. Consider now the incoherent production of vector mesons from nuclei. In the absence of coherent rescattering processes the nuclear o production cross section p would simply be A times the production cross section p on a free nucleon. Nuclear e!ects are conveniently discussed in ,
Fig. 8.1. The rates of produced o's plotted against t "t!t for hydrogen, deuterium, He, and N. The solid lines show "ts to the data, the dashed lines are inferred incoherent contributions (for details see Ref. [276]). Fig. 8.2. The nuclear transparency ¹ as a function of the propagation length j for N. HERMES data [276] are indicated by full dots. The open symbols represent data from previous experiments [273,286]. The dashed curve shows a Glauber calculation from Ref. [284].
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terms of the transparency ratio, ¹ "p /(Ap ). The measured ratio for N is plotted as a function , of the longitudinal propagation length j in Fig. 8.2. The deviation of ¹ from unity for j:1 fm simply re#ects the `triviala "nal state rescattering of the o meson after being produced on one of the nucleons. More interesting e!ects are visible when j exceeds the average nucleon-nucleon distance of about 2 fm. Now the hadronic #uctuations of the photon can scatter coherently on several nucleons also prior to the production of the "nal state vector meson, and the transparency ratio ¹ systematically decreases until it exceeds the nuclear diameter. The dashed curve in Fig. 8.2 shows a theoretical prediction calculated within the vector meson dominance model [284]. Its agreement with data suggests that the production process is dominated, given the relatively low Q involved, by hadronic #uctuations which interact about as strongly as the produced o meson. Further systematic investigations of such coherence length e!ects, especially its detailed dependence on the momentum transfer t, are discussed at TJNAF [277,285]. It is interesting to push these observations to more extreme regions of very large Q and l. Once the energy transfer exceeds several tens of GeV, a further scale enters. At large Q the qq #uctuation of the photon starts out initially as a small-sized wave packet. The time it takes for this wave packet to develop into the "nal vector meson is called the formation time q [280}282]. To be speci"c, let D the observed vector meson again be a o. The small initial wave packet is generally not an eigenstate of the strong interaction Hamiltonian. Now consider expanding the wave packet in hadronic eigenstates. Clearly, for a wave packet with a size small compared to typical hadronic dimensions, several such eigenstates are necessary to represent the wave function of the packet. Let one of those hadronic eigenstates in the expansion be the o meson itself (mass m ), let another one be M a neighboring state with larger mass (say, m "m #dm). The characteristic propagation length of MY M the o component of the qq -gluon wave packet is jK2l/(m#Q), that of the neighboring state is M j K2l/(m#Q#2m dm). The phase di!erence between both states is determined by their wave M M numbers 1/j and 1/j . The time it takes to "lter out all but the o meson component when passing through the nucleus is then l . (8.3) q & D m dm M Thus when l is su$ciently large so that q reaches nuclear dimensions and Q is large, a small-sized D qq wave packet induced by the photon has a chance to travel over large distances inside nuclei and interact weakly, its cross section being proportional to b&1/Q. This phenomenon is commonly named color coherence or color (singlet) transparency [279}282]. It has been addressed in exclusive vector meson production experiments at FNAL (E665) [273] and CERN (NMC) [275]. The interpretation of data in terms of color transparency is still under debate (see e.g. [287,288]). Possible future developments with nuclei at HERA [257] could o!er an enormous extension of the accessible kinematic range. 8.2. Nuclear shadowing in DIS at large Q Experimental information on nuclear shadowing in inclusive DIS is available, up to now, only from "xed target experiments, with the kinematic range restricted to Q:1 GeV at x;0.1. Although there are currently no data on nuclear shadowing at large Q<1 GeV, it is nevertheless instructive to investigate what one should expect in comparison with the previous results at smaller
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Q. Such a study is possible due to recent data on di!ractive production from the HERA collider and has been performed in Refs. [148,289,290]. One "nds that contributions from vector mesons are negligible at large momentum transfers since di!ractive vector meson production is strongly suppressed at Q'10 GeV. Furthermore, the ZEUS data [53] on the ratio of di!ractive to total photon}nucleon cross sections (Fig. 2.8) suggest that di!ractively produced states with large mass are relevant at large Q. Therefore at large Q shadowing probes the coherent interaction of quark}gluon con"gurations with large invariant mass. This is complementary to "xed target experiments at FNAL (E665) and CERN (NMC) where the coherent interaction of low mass vector mesons played a dominant role (see Sections 5.1.4 and 5.4). In Section 5.2 we have argued that to leading order in 1/Q shadowing is dominated by the interaction of large-size hadronic #uctuations of the exchanged photon. This suggests a weak energy dependence of nuclear shadowing. However, at very small x together with very large Q, the steadily growing number of partons in the photon}nucleon system makes quark}gluon con"gurations of the photon interact like ordinary hadrons, even if they have small transverse size proportional to 1/Q (see e.g. Eq. (5.34)). As a consequence one expects a more rapid energy dependence of shadowing as compared to the case of small Q. On similar grounds the di!ractive leptoproduction cross section at x;0.1 and Q<1 GeV should rise more strongly than suggested by Regge phenomenology. Possible indications for this behavior have been found at HERA [51,53], signaling the onset of a new kinematic regime with a complex interplay between soft (large size) and hard (small size) partonic components of the interacting photon. A systematic investigation of strong interaction dynamics in this kinematic region is a major challenge. Here electron}nucleus collider experiments could give important new insights [257]. 8.3. Physics of high parton densities An elementary QCD treatment of radiative corrections in nucleon and nuclear structure functions, or equivalently, in quark and gluon distributions is provided by the DGLAP evolution equations (see Section 2.4). In leading logarithmic approximation one sums over a ln Q/K terms, Q each of which represents a quark radiating a gluon or a gluon splitting into a qq or gluon pair. A contribution (a ln Q/K)L is associated with the radiation from n partons in a physical gauge. Q Due to radiation partons loose momentum. Therefore DGLAP evolution leads to a rise of quark and gluon distribution functions at small x: strongly increasing numbers of partons carry smaller and smaller fractions of the total momentum. For example, consider gluons which dominate the dynamics of parton distributions at small x. Suppose that we are given an initial gluon distribution, prior to radiative QCD corrections, at a momentum scale Q . This initial distribution can be non-singular ("nite) at xP0. Now turn on QCD radiation and DGLAP evolution. The resulting asymptotic behavior of the gluon distribution function at x;1 and large Q is (see e.g. [291]): xg(x, Q)&exp
12 5
ln
ln(Q/K) ln(1/x) . ln(Q /K)
(8.4)
One observes a strong rise of xg(x, Q) at x;1. This increase of the gluon density with decreasing x can however not continue inde"nitely. At very small x the density of gluons becomes so large that
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they interact with each other reducing their density through annihilation. Thus, one enters a new regime of very high parton densities where standard methods of perturbation theory are inappropriate despite small values for the coupling a , and resummation techniques to in"nite order have Q to be applied (for references see e.g. [135}140]). For further illustration let us view the scattering process in a frame where the target momentum is very large, P""P"PR. A measurement at speci"c values of Q and x probes partons over a longitudinal distance *z&1/(Px) and with a transverse size *b&1/(Q. Thus at small values of x and moderate Q parton wave functions overlap, leading to high parton densities, so that the rate of parton}parton annihilation processes increases. This rate involves the probability of "nding at least two partons per unit area in the nucleon [291,292]. In this respect parton recombination can be regarded as a `non-lineara correction to radiation processes described by standard DGLAP evolution, where parton distribution functions enter linearly (see e.g. Eqs. (2.39) and (2.40)). The existing HERA data on free nucleon structure functions at Q'1 GeV and x'10\ do not show a clear sign for the need of additional parton fusion corrections in the evolution equations (see e.g. [9]). However, phenomena related to high parton densities at small x should be magni"ed in nuclei since nuclear parton densities are enhanced. For example, if shadowing is ignored and the nuclear gluon distribution is assumed to be the sum of the gluon distributions of individual nucleons, g (x, Q)"Ag (x, Q), then the ratio of gluon densities in a nucleus and a nucleon per , (transverse) area is [293]:
g (x, Q) g (x, Q) Ar , + , +0.5A , R nr pR ,
(8.5)
where r +0.8 fm and R +r A +1.1 fm A have been used for nucleon and nuclear radii, , respectively. E!ects beyond DGLAP evolution should therefore be ampli"ed in nuclei and set in already at a larger value of x as compared to free nucleons. In Refs. [292,294] corrections to DGLAP evolution for nuclear parton distributions have been calculated in the leading logarithmic approximation taking into account corrections due to parton}parton recombination. The nuclear gluon distribution is written in two parts: xg (x, Q)" xAg (x, Q)#d(xg (x, Q)). The "rst term is associated with independent nucleons and evolves , accordingly. The second term is the correction of interest here and describes the interaction of gluons from di!erent nucleons. Its evolution due to gluon}gluon recombination reads [292,294]:
81 A dx R d(xg (x, Q)) "! a(Q) [x g (x , Q)] . Q , 16 Qr Q A x RQ V
(8.6)
Clearly gluon}gluon recombination is enhanced in nuclei. One "nds [295,296] that the x-range where non-linear e!ects (8.6) become signi"cant, di!ers for heavy nuclei and free nucleons by more than two orders of magnitude, assuming xg (x)&x\ . , Investigations of unitarity constraints in hard two-body amplitudes [167] also suggest that non-linear e!ects at high parton densities can be signi"cant in nuclei at Q&10 GeV and x&10\. While the kinematic bounds for the applicability of `normala DGLAP evolution are quite well de"ned, the dynamical mechanisms responsible for slowing down the rapid increase of parton
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distributions at large Q and very small x are not yet clear. One is entering a new domain of QCD, dealing with partonic systems of high density, which presents new challenges.
Acknowledgements We gratefully acknowledge many discussions and conversations with S. Brodsky, L. Frankfurt, P. Hoyer, B. Kopeliovich, S. Kulagin, L. Mankiewicz, W. Melnitchouk, G.A. Miller, N. Nikolaev, K. Rith, M. Sargsian, M. Strikman, A.W. Thomas and M. VaK nttinen.
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COLD TARGET RECOIL ION MOMENTUM SPECTROSCOPY: A 9MOMENTUM MICROSCOPE: TO VIEW ATOMIC COLLISION DYNAMICS
R. DOG RNER , V. MERGEL , O. JAGUTZKI , L. SPIELBERGER , J. ULLRICH, R. MOSHAMMER, H. SCHMIDT-BOG CKING Insitut f u( r Kernphysik, Universita( t Frankfurt, August Euler Str. 6, D60486 Frankfurt, Germany Fakulta( t fu( r Physik, Universita( t Freiburg, Hermann-Herder-Str. 3, D79104 Freiburg, Germany
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
Physics Reports 330 (2000) 95}192
Cold Target Recoil Ion Momentum Spectroscopy: a &momentum microscope' to view atomic collision dynamics R. DoK rner *, V. Mergel , O. Jagutzki , L. Spielberger , J. Ullrich, R. Moshammer, H. Schmidt-BoK cking Institut fu( r Kernphysik, Universita( t Frankfurt, August Euler Str. 6, D60486 Frankfurt, Germany Fakulta( t fu( r Physik, Universita( t Freiburg, Hermann-Herder-Str. 3, D79104 Freiburg, Germany Received September 1999; editor: J. Eichler
Contents 1. Introduction 2. Kinematics of recoil ion production 2.1. Ion}atom collisions 2.2. Electron}atom collisions 2.3. Photon}atom collisions 3. Experimental technique 3.1. Extended gas-target devices 3.2. E!usive gas targets 3.3. Supersonic gas-jet targets 3.4. Projection spectrometers for slow ions 3.5. Projection spectrometers for electrons 3.6. Position-sensitive detectors 4. Experimental results for charged particle impact 4.1. One-electron processes 4.2. Multiple electron processes in fast collisions
98 101 102 107 107 109 110 112 112 115 119 121 124 124 143
4.3. Multiple electron processes in slow collisions 4.4. Electron impact ionization 5. Experimental results for photon impact 5.1. He single ionization and the ratio of double to single ionization 5.2. Multiple di!erential cross-sections for He photon double ionization 5.3. Circular dichroism in He photon double ionization 5.4. Electron emission from spatially aligned molecules 5.5. Separation of photoabsorption and Compton scattering 6. Outlook Acknowledgements References
158 160 162 163 166 171 173 175 178 182 182
Abstract Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) is a novel momentum space imaging technique for the investigation of the dynamics of ionizing ion, electron or photon impact reactions with atoms or molecules. It allows the measurement of the previously undetectable small three dimensional
* Corresponding author. E-mail address: [email protected] (R. DoK rner) 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 9 - X
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momentum vector of the recoiling target ion created in those reactions with high resolution and 4p solid angle. Combined with novel 4p electron momentum analysers it is a momentum microscope for kinematically complete scattering experiments. We review the technical development, outline the kinematics of atomic reactions from the perspective of the recoil ion, and give an overview of the studies performed with this technique. These studies yield kinematically complete pictures of the correlated motion of the fragments of atomic and molecular breakup processes, unprecedented in resolution, detail and completeness. The multiple-dimensional momentum-space images often directly unveil the physical mechanism underlying the many-particle transitions investigated. The experiments reviewed here include reactions of single and multiple capture and ionization induced by keV proton to GeV/u U> impact, electron and antiproton impact ionization as well as single and double ionization by photoabsorbtion and Compton scattering from threshold to 100 keV. We give an outlook on the exciting future prospects of the method for atomic physics and other "elds of science. 2000 Elsevier Science B.V. All rights reserved. PACS: 34.50.Fa; 34.70.#e; 32.80.Cy; 31.25.!v; 39.30.#w; 39.90.#d Keywords: Recoil ion; Ionization; Imaging; Spectroscopy; Ion}atom collisions; Photoionization
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1. Introduction Stationary systems in which the interaction potential is exactly known, can be described by quantum mechanics with an extremely high precision. For the energy eigenvalues, the central stationary observable, nearly perfect agreement between experiment and theory has been obtained (see, e.g. [1}8]). Dynamical many-body systems on the other hand are still a major challenge for theory and experiment. Today they are the basic issue of many "elds of physics such as solid state, nuclear and atomic physics. The last is in the fortunate situation that in atomic systems the interaction potential is exactly known. Thus all disagreement between theory and experiment in this "eld can be attributed to the many-body aspects and the dynamics of the problem. This makes it even more surprising that such apparently simple problems as the emission of a single electron from an atom by slow charged-particle impact or the emission of two electrons by absorption or Compton scattering of a single photon still challenge theory and experiment. This puzzling lack of understanding of the dynamics of many-body systems is in troubling contrast to the importance of such systems for our everyday world. Structure and evolution of our macroscopic world is to a large extent determined by the dynamics of many-electron processes. They are responsible for many solid state e!ects such as superconductivity but most prominently they govern and fuel chemical reactions and all biological systems. Atomic and molecular many-particle reactions are characterized by fully di!erential crosssections (FDCS), i.e. cross-sections di!erential in all observables of the "nal-state. In an ionization process this typically corresponds to the vector momenta, spins and internal excitation of all reaction products. Such FDCS provide the most stringent test for theory. Any integration over observables often masks important characteristics of the process. In turn, experimental FDCS in the best case directly unveil mechanisms of the many-particle transition. The lack of experimental data on such important details of simple and hence fundamental atomic reactions has for long delayed the development of many-particle collision theory. The goal of Cold Target Recoil Ion Momentum Spectroscopy COLTRIMS is to provide a tool for kinematically complete studies of three- and more-particle atomic collision systems. By kinematically complete we mean that the momenta (and thus angle and energy) of all involved particles are observed in coincidence but the spin is not determined. For electron impact target single ionization (so called (e,2e)-experiments), systematic experimental and theoretical studies of FDCS exist (see [9}11] for reviews). For target double ionization induced by electron impact so far only very few complete di!erential cross sections (see [11] for a review and [12}18]) have been published. These data typically view a very small fraction of the "nal state momentum space. In the "eld of photoionization tremendous progress has been made in the recent years. Following the pioneering work of Schwarzkopf et al. [19], experimental data for FDCS for double ionization of He have been obtained by several groups [19}27]. All these data have been measured using traditional electron-spectroscopy techniques. To yield su$ciently high resolution for the momentum of the ejected electron, the solid angle X of the electron spectrometer is strongly limited, typically to less than 10\ of 4p. In a two- or even
Despite of its importance, we neglect the spin for the rest of this review. The lack of e$cient spin-sensitive electron detectors up to today is prohibitive for spin selective many-particle coincidence experiments.
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three-electron coincidence experiment the total coincidence e$ciency is thus extremely small (10\). This explains why, so far, systematic investigations of FDCS could be performed only for a few many-particle reactions. In addition, the low counting rate and the geometry of the experimental setup mostly restricted those studies to speci"c kinematical conditions (e.g. all emerging particles in one plane). Obtaining a desirable overview over the complete "nal state with this techniques would in general require hundreds of separate experiments (see [28] for such an attempt in (e,2e)-experiments). In this paper we review a rapid development of a new experimental approach, the COLTRIMStechnique, which overcame many of these problems on a basic level (for earlier reviews on this "eld see [29}33]). The experimental solution is an imaging technique using a well localized reaction zone and electromagnetic "elds to guide all charged fragments towards large-area position- and time-sensitive detectors. From the measurement of the time-of-#ight and the position of impact for each particle its momentum vector can be determined. Such an imaging technique has "rst been used for measuring the ion momenta from atomic reactions and has soon been generalized to electron detection as well. For an atomic reaction it is for at least three reasons particularly desirable to determine the vector momentum of the recoiling ion. First, the charge state of the ion gives the multiplicity of the process. Second, there are many reactions of charged particles or photons with atoms where the momentum imparted to the ion gives unique information on the dynamical mechanism of the reaction. The measurement of the ion momentum alone allows, e.g. already the separation of Compton scattering from photo absorption, the identi"cation of the electron}electron interaction between two centers or higher-order processes in transfer ionization or the determination of the "nal state of a capture reaction. These and many more examples of the richness of information inherent in the ionic momentum will be discussed in detail in Sections 4 and 5. Third, for a kinematically complete experiment with n particles in the "nal state, it is necessary to measure 3n!3 momentum components (3n!4 if the Q-value is known). The remaining 3(4) momenta can be inferred from the others by exploiting the momentum- and energy-conservation laws. In fast ion impact atomic reactions the projectile su!ers typically a very small relative momentum change. Only for very selective collisions systems this momentum change could be measured [34}36]. For reactions like 3.6 MeV Ni> on He (see Section 4.2.6) for example the energy loss of the projectile (*E/E) is in the range of 10\ and the typical scattering angle leads to a de#ection in the range of 1 mm over 1 km (lrad). Thus for such fast heavy particle collisions the coincident detection of ion and electron momenta gives, via momentum conservation, the only practical access to the projectile momentum change and thus to a kinematically complete experiment. Historically this problem of the unmeasurably small projectile momentum change motivated the development of the "rst spectrometer for measuring transverse momenta of recoil ions [37]. The idea of such a recoil ion momentum measurement has certainly been considered by many experimental atomic physicists. One immediately recognizes, however, that the target atom at room temperature has already such a large initial momentum spread that typical momenta of the recoil ions gained in the collision are largely covered by the target thermal motion at room temperature (He at room temperature has about 4.6 a.u. mean momentum). This was the reason why the measurement of the momentum of the recoil ion was for a long time not seriously exploited as an alternative high-resolution spectroscopy technique in ion-atom collisions [38]. Using existing techniques for gas target cooling, however, the initial target momentum spread can be reduced
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dramatically. As will be shown below, the target can be cooled to temperatures far below 1 K. With modern laser-cooling techniques, in particular successful for cooling of the alkali atoms, temperatures in the few l K range, i.e. a few neV kinetic energy of the target atoms, are feasible. The use of such cold trapped atoms for ion momentum measurements is currently explored in several laboratories. Using supersonic gas target devices for He a few mK has already been obtained, which is equivalent to a He kinetic energy of below 1 leV. In present COLTRIMS devices based on supersonic gas jet targets a momentum resolution of 0.05}0.2 a.u. for the recoil ion and a detection e$ciency of about 60% of all ions from a reaction (4p solid angle but 60% detection e$ciency) is routinely achieved. Before high-resolution COLTRIMS was developed recoil ion momenta in atomic physics have been measured already in the sixties for slow or small impact parameter collisions, which lead to ion energies of 10}10 eV. The momenta of those ions, whose energies in the present context were extremely high, have been measured for example by Everhard and Kessel [39,40] and Federenko [41]. For fast charged-particle collisions and photoionization, the recoil ion energies are in the meV regime or even below. The charge state distribution of such slow ions has been measured in numerous experiments using a time-of-#ight technique or magnetic de#ection (see [42] for a review). First attempts on measuring angles and momenta of slow recoil ions were reported already in the seventies [43}48]. Ullrich and Schmidt-BoK cking [37] succeeded 1987 in the "rst quantitative measurement of transverse recoil ion momenta in 340 MeV U> on Ne collisions [37,49,50]. They used a static room temperature gas target, time-of-#ight measurement in a "eldfree drift tube and magnetic ion charge state selection. The technical development from this "rst recoil ion momentum spectrometer to todays most advanced momentum microscopes will be discussed in detail in Section 3. The momentum resolution of this gas cell spectrometer was considerably improved using a cooled gas cell (30 K) [51,52], and "rst Multiple-DCS have been measured by coincident detection of recoil ion and projectile transverse momenta [51}61]. Parallel to this development at University Frankfurt Germany, Cocke and coworkers at Kansas State University have used warm e!usive gas-jet targets and an electric projection "eld [62,63]. They measured the "rst recoil ion longitudinal momentum distributions, deducing the Q-value of the collision. Since 1991 at the University Frankfurt (Germany) [64] and at CIRIL/GANIL (Caen, France) [65] recoil ion momentum spectrometers based on supersonic gas-jet targets have been developed. The extremely low internal temperature of such gas-jets (typically below 1 K) yielded a momentum resolution of 0.2 a.u. and below, more than a factor of 10 better than e!usive or static target devices. An even colder gas target has been built using precooling of the target gas to 15 K [66,67]. The ionic momentum resolution in these devices was in most cases not limited by the internal target temperature but by the extension of the target in the ion momentum spectrometer. Spectrometers with electrostatic lenses today eliminate this problem by focussing in all three spatial dimensions and yield an ion-momentum resolution of 0.05 a.u. [31,68}70]. Presently at least in seven laboratories such COLTRIMS devices are in operation (Argonne, GANIL, GSI, Frankfurt, KSU, LBNL, RIKEN) and many more are in preparation. The latest step in this rapid development was the combination of such COLTRIMS spectrometers with novel electron projection spectrometers. These electron imaging systems apply the basic principle of COLTRIMS to electron detection. For very low electron energies (typically (5 eV) the same electric "eld which projects the positive ions onto one detector guides the negative electrons towards another detector [71}75]. Moshammer and coworkers have developed an
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electron spectrometer which superimposes a solenoidal magnetic "eld parallel to the electrostatic "eld. This novel electron analyzer extends the projection technique for electrons towards much higher electron energies [76,77]. Such electron projection spectrometers achieve high resolution and 2p}4p solid angle even for zero energy electrons. A resolution of $5 meV at zero energy and full solid angle up to 300 eV has been reported [77]. In particular the detection of such very low-energy electrons is extremely di$cult by conventional spectrometers. These projection spectrometers can be equipped with fast delayline detectors, capable of handling multiple hits [78]. Thus, all electrons from multi-electron reactions can be analyzed simultaneously [79]. Merging a high resolution recoil ion with such a multi-electron projection spectrometer creates an extremely versatile and powerful tool for atomic and molecular collision physics. Kinematically complete experiments for single ionization in slow, adiabatic collisions, for collisions of fast protons and antiprotons, for highly charged projectile impact, for relativistic collisions, for electron-impact and photon-induced double ionization by linearly and circularly polarized light have been performed already. In pioneering experiments for multiple ionization by fast ion impact, more than two electrons have been detected together with the recoil ion momentum [80]. This review paper is organized as follows: In Section 2 we present the kinematics of atomic reactions induced by heavy projectiles, electrons and photons from the perspective of the momentum transfer to the recoiling ion. The goal is to give guidance for the interpretation of recoil ion momentum spectra and to illustrate the large amount of information on the reaction which can be obtained from the momentum of the recoiling ion. This is followed (Section 3) by a review of the technical development from the "rst recoil ion spectrometers to the most advanced ion-electron momentum imaging systems, termed reaction microscopes. We complete this experimental section by a brief discussion of supersonic gas-jet targets and position sensitive detectors, the two most important ingredients of COLTRIMS. In Section 4 we give an overview on the experimental results obtained by this technique so far for charged particle impact and in Section 5 for photon impact. The overview covers most "elds of atomic collision physics, including single capture and single ionization by ion impact, multiple-electron processes like double capture, transfer ionization and multiple ionization, electron impact ionization, photon-induced double ionization by linearly and circularly polarized light, Compton scattering and electron emission from aligned molecules. The energies of the projectiles range, for charged particles, from a hundred eV electrons to GeV/u bare uranium and, for photons, from threshold to 100 keV. This broad range of topics illustrates the fruitful and wide impact which the still young technique of COLTRIMS has already had. We conclude this review with an outline of some of the future perspectives of this technique and by at least naming some of the possible applications to other "elds of physics.
2. Kinematics of recoil ion production Independent of the dynamics of the ionization process, the observable momenta of collision fragments (recoiling ion, electrons and scattered projectile) are interrelated by the conservation laws of momentum and energy. The "nal state of a reaction with n fragments in the exit channel is described by 3n momentum components (neglecting the spin) plus internal excitation energies (Q-value). Due to momentum and energy conservation, however, only 3n!3 of these momenta are independent. Thus the reaction is kinematically fully determined by (3n!3) linearly independent
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scalars or a (3n!3)-fold di!erential cross section. In the case of complete fragmentation (i.e. no internal excitation energy) the reaction kinematics is fully determined by 3n!4 scalars. In this section we give a short outline of these kinematical relations from the perspective of the recoiling ion. Throughout this paper we use atomic units (a.u.) with m , ,e,1 and c,137.
(1)
2.1. Ion}atom collisions The typical momentum transfer in most ion}atom collisions is in the range of a few atomic units. This is only a small fraction of the initial momentum of the projectile in most collisions. For example in MeV p on He collisions the momentum exchange will be less than 10\ of the initial projectile momentum. Thus observation of momentum transfer in ion}atom collisions by projectile detection (scattering angle and/or energy gain or loss measurements) is restricted in resolution by the fact that in the laboratory system a small change of a huge momentum must be resolved. Much higher resolution can be achieved by measuring quantities like electron or recoil ion momenta since these reaction products are initially nearly at rest in the laboratory frame. Because any momentum change of the projectile must be compensated by the sum momentum of the recoiling ion and all emitted electrons, the complete momentum balance of the reaction can be measured with much improved resolution by detecting recoil ion and electrons (i.e. by the so called inverse kinematics). Utilizing relativistic mechanics, the change of the projectile longitudinal momentum *k for , an ion atom collision in which n target electrons and n projectile electrons are emitted to the continuum and n target electrons are captured to projectile bound states, is given by L >L EH n !n Q v . ! H # *k " v v 1#c\
(2)
Q is the di!erence of the sum of all atomic and ionic binding energies before and after the reaction, (Q(0 denotes endothermic reactions), EH is the kinetic energy of electron j which is emitted to the continuum measured in its parent atom rest frame, v is the projectile velocity and the relativistic factor c"1/(1!v /c. A detailed derivation can be found in [68]. In the nonrelativ istic case Eq. (2) is valid if the projectile energy change is small compared to the projectile kinetic energy. In the ultrarelativistic case, if the energy change is small compared to the relativistic total energy. Additionally Eq. (2) is only valid if the change of mass of the projectile is small compared to its relativistic initial mass. Both approximations are well ful"lled in practically all (distant) ion}atom collisions. From the three terms of the projectile momentum change in Eq. (2) the "rst and second one re#ect the fact that all kinetic and potential energy delivered to electrons has to come from the projectile and thus leads to a projectile momentum loss. The third term re#ects the projectile momentum change due to its change of mass by capturing or losing electrons. For the recoiling ions the momenta in the transverse and longitudinal directions completely decouple within the approximation outlined above. For a polar projectile scattering angle 0 ;1 the recoiling ion momenta in the two directions perpendicular to the beam k V W
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are given by:
L >L "! m v 0 cos U # kH , V V H L >L (3) k "! m v 0 sin U # kH , W W H where U is the azimuthal scattering angle of the projectile with mass m and kH are the V W momentum components of electron j. For pure capture reactions where no electron is emitted to the continuum thus the recoil ion transverse momentum alone re#ects the scattering angle of the projectile. This allows for extremely precise scattering angle measurements even for fast collisions where the typical projectile scattering angles are unmeasurably small. The recoil ion momentum in the longitudinal direction (k ) can be calculated using energy and momentum conservation to be k
"k #k #k , (4) Q n v (5) k "! ! , 1#c\ v L EI #EI !kI , (6) k " v I L E #c\E . (7) k " v J The three terms of the sum originate from contributions of electron transfer to the projectile, ionization of the target and ionization of the projectile (loss). The initial state binding energies E of electrons of the projectile or target are taken to be positive. The kinetic energies of the electrons lost from the projectile E are measured in the projectile frame (not in the laboratory as all other quantities in the equation). Eqs. (4)}(7) are valid for relativistic and nonrelativistic collisions (c"1 for the latter) within the same approximation as above, that the energy and mass change of the projectile are both small compared to the initial energy and relativistic mass. As can be seen from Eq. (6) elastic collisions resulting in small scattering angles do not transfer longitudinal momentum to the recoiling ion. Furthermore the transverse and longitudinal momenta of recoil ion as well as of the projectile are fully decoupled. The recoil ion longitudinal momentum re#ects the inelasticity of the reaction and the electron longitudinal momenta. We will brie#y discuss the physical consequences of Eqs. (4)}(7) for capture (Section 2.1.1), target ionization (Section 2.1.2) and projectile energy loss (Section 2.1.3). A more detailed discussion is given in Section 4 along with the experimental results. k
2.1.1. Kinematics of electron capture reactions For electron-capture reactions (k "k "0) k has discrete values since the Q-value of the reaction is quantized. Thus measuring k is equivalent to traditional translational spectroscopy, with similar resolution at low projectile velocities but decisive advantages for fast
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collisions or heavy projectiles. The Q-value for single capture reactions is related to the recoil ion "nal state longitudinal momentum by v . Q"! !v k 1#c\
(8)
A recoil resolution of 0.05 a.u. results in a Q-value resolution of about 6 eV for a 10 MeV Ne projectile (*E/E(10\). This is already more than factor of 10 better than the best resolution obtainable by projectile energy gain spectroscopy. Even in the slow collision regime the resolution reachable by COLTRIMS for energy gain measurements is in many cases much superior to even the best energy gain measurements performed so far. For example, for 5 keV/u Ar projectiles one reaches a Q-value resolution of 0.6 eV or *E/E"3;10\. In addition to the Q-value COLTRIMS studies yield automatically the transverse momentum distribution of the ion for each "nal state. The recoil ion transverse momentum for pure capture reactions mirrors the projectile scattering angle (see Eq. (3)). For slow collisions thus the impact parameter can be inferred. Examples of applications will be discussed in Section 4.1.1. 2.1.2. Kinematics of target ionization For pure target ionization k can be obtained from the momenta of all electrons. For multiple ionization k re#ects the sum energy and longitudinal momenta of all electrons, allowing for a detailed check of theories of multiple electron processes without detecting all electrons in coincidence. For fast collisions and slow electrons the "rst term in Eq. (6) is small and Eq. (6) simpli"es to L0 (9) "! kI . I The recoil ion longitudinal momentum just mirrors the sum longitudinal momenta of all emitted electrons. This has been demonstrated and widely used in experiments at GSI (Darmstadt) and CIRIL/GANIL (Caen) [32,76,77,81,82]. Two illustrations of Eqs. (9) and (6) are given in Figs. 1 and 2. For single ionization of He by 0.5 MeV protons (v "4.5 a.u.) the term E /v is, for most events, a small contribution. The main contribution to the cross section found close to k "!k #E /v (Fig. 1). For smaller projectile velocities (v "0.77 a.u. in Fig. 2) however, the term E /v cannot be neglected. As a consequence of this a lower threshold for the recoil ion longitudinal momentum becomes visible. From Eq. (6) it follows that this lowest kinematically allowed value of k for ionizing collisions corresponds to electrons travelling with the projectile in forward direction (electron capture to the continuum, ECC) [83] with k
E v (10) k "! ! . v 2 Such electrons are known to produce a singularity in zero degree electron spectra. In the distribution of k the ECC shows up as a step instead of a smooth onset of the cross section at k [83,84]. Another prominent feature in electron spectra is the binary-encounter (BE) peak at an energy of E "2v cos 0 for a polar emission angle of 0 . Inserting this relation in Eq. (6) leads to #
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Fig. 1. Correspondence of He> k and k for 500 keV proton impact single ionization of He. The distribution has , , been integrated over the transverse momenta of all particles. The distribution is shifted forward from the origin by a momentum of E /v , the E /v is small and yields some broadening of the diagonal (from [84]).
Fig. 2. Correspondence of k and He> k for 15 keV proton impact single ionization of He. The distribution has , , been integrated over the transverse momenta of all particles. The threshold at k "0.78 a.u. (dashed dotted line) , corresponding to electrons traveling with the velocity of the projectile (v "0.77 a.u.) is visible (from [90]).
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k # "E /v . Thus, if one neglects this binding energy term, the recoil ion indeed remains a spectator in the production of BE electrons. The relation between electron momentum vector and recoil ion longitudinal momentum (Eq. (6)) has been used by Tribedi and coworkers [85}87] to obtain recoil ion longitudinal momentum distributions by high resolution electron spectroscopy. The features of capture and ionization can be seen in the longitudinal recoil ion momentum distribution for He> ions produced by 15 keV protons (Fig. 3). The discrete peaks correspond to Q-values for capture to excited states of the proton and to capture plus simultaneous excitation of the target. The Rydberg series of capture to bound projectile states converges from the left side towards k corresponding to capture to the continuum. It is followed by the continuous , spectrum of ionization. 2.1.3. Kinematics of projectile ionization For pure projectile ionization the momentum transfer to the target is, in the nonrelativistic limit, given by (E #E )/v where the electron energy E is measured in the projectile system.
Fig. 3. Longitudinal momentum distribution of He> ions from 15 keV proton impact. The dominant peak is due to capture to the projectile ground state. The arrow indicates the position of the capture to the projectile continuum (k of , Eq. (10)). The distribution for ionization has been measured separately by detecting an electron in coincidence with the recoil ion. The momentum resolution is $0.035 a.u., equivalent to an energy gain resolution of $0.7 eV and a recoil ion energy resolution of $4.5 l eV (from [30]).
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Fig. 4. Distribution of recoil ion longitudinal momenta for the reaction O>#HePO>#He>#2e\(from [88]).
Thus, the recoil ion longitudinal momentum can be transformed directly in an energy spectrum of loss electrons. As for target ionization, the longitudinal recoil momentum distribution has a lower threshold. Fig. 4 reported by Wu and coworkers [88,89] shows strong forward emission of the recoil ion in an electron loss reaction. COLTRIMS requires the production of an ion, thus projectile ionization can only be investigated if it is accompanied by simultaneous target ionization. In this case the momentum transfer to the target can be shared between the recoiling ion and the target electron. Observing the recoil ion for such reactions gives a unique "ngerprint to distinguish between projectile ionization by the electron}electron interaction and by the electronnuclear interaction (see Section 4.2.3).
2.2. Electron}atom collisions For electron-impact ionization in general none of the above approximations is valid. The energy loss of the projectile is in many cases not negligible and the projectile scattering is much more likely to lead to larger scattering angles. Both e!ects couple the longitudinal and transverse momenta of the recoil ion to projectile scattering and energy loss. The two components cannot be treated separately. Eqs. (4)}(7) hold only for fast collisions with small momentum transfer. In this case for example the binary peak, as it shows up in (e,2e) experiments, corresponds to the binary-encounter electrons from ion impact discussed in Section 2.1.2.
2.3. Photon}atom collisions For photon impact we consider the two cases of photoabsorption (Section 2.3.1), and Compton scattering (Section 2.3.2). In both cases we restrict the discussion to nonrelativistic photo electrons.
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2.3.1. Kinematics of photoabsorption For absorption of a photon propagating in z direction leading to emission of a single electron and a recoiling ion momentum conservation yields 0"k #k V W V W E /c"k #k A X X combining this with the conservation law of energy one obtains [91]
(11) (12)
m E A k #k # k ! X m #m c V W E m m m m A . (13) "2 (E !E #E )! A m #m (m #m ) c If one neglects the (small) photon momentum E /c, which is a good approximation in many A cases, this equation describes a sphere in momentum space with radius
m m (E !E #E ) A m #m (E being the di!erence in total binding energies of the atom and ion in the initial and "nal state). In this approximation electron and recoil ion emerge antiparallel with the above momentum. Due to the mass of the ion, however, most of the energy is in the electronic motion, only a fraction of m /(m #m ) is in the kinetic energy of the recoiling ion. For single ionization of 80 eV photons by He atoms Fig. 5 shows a two-dimensional slice through the momentum sphere of the ions [92]. The rings result from He> ions in di!erent excited states (di!erent E !E #E ). For a more A detailed discussion see Section 5.1. If one takes the photon momentum into account, two small corrections arise. First, both momentum spheres, of photo electron and ion, are slightly shifted into the forward direction by the photon momentum. Due to the electron/recoil ion mass ratio most of this photon momentum couples to the ionic motion. The ionic momentum sphere given by Eq. (13) is shifted by E m A m #m c while the respective shift for the electron momentum is smaller by m /m . The second correction is a shrinking of the momentum sphere of electrons and ions by the term
m m E A . (m #m ) c This results from the tiny shift of the center of mass of the system by the photon. This correction is of the range of 10\ of the momenta and, thus, irrelevant for all practical purposes. For multiple ionization by photon impact one has to replace the electron momentum in the above equations by the sum momenta of all emitted electrons. Since these electrons can be emitted in all directions the ionic momentum for double ionization for example can range from k "0 to , where E is the energy of the photon minus the electron a maximum of k "2(E
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Fig. 5. (a) Momentum distribution of He> ions emitted after 80 eV photon impact. The x-axis is the direction of the polarization axis of the linear polarized light from an undulator (and at the same time the direction of the electric "eld of the spectrometer), the y-axis is the direction of the gas-jet, perpendicular to the propagation of the light and the polarization axis. The data represent a slice through a three-dimensional distribution and are integrated over a range of $0.1 a.u. in the third direction. The outer ring corresponds to ions in the ground state, the inner rings to excited states (from [90]).
binding energy. The zero value corresponds to equal energy electrons emerging back to back, the latter to two equal energy electrons emitted to the same direction. For the investigation of multiple photoionization processes, the recoil ion momentum measurement yields information on the correlated emission of the electrons. It mirrors the sum momentum of all emitted electrons. 2.3.2. Kinematics of compton scattering Compton scattering can occur for unbound electrons. The momentum transfer takes place between the photon and the emitted electron. Momenta and energies are balanced between photon and electron. The recoil ion momentum re#ects the initial momentum of the bound electron. Thus contrary to photoabsorption Compton scattering can be expected to create slow recoil ions. Spielberger and and coworkers have shown this di!erence between photoabsorption and Compton scattering experimentally [93] (see Section 5.5).
3. Experimental technique Today's recoil ion momentum spectroscopy is the result of 15 yr of experimental development. Historically, the "rst recoil ion momentum spectrometers used extended gas targets and a "eld free drift path for the ions. We will "rst brie#y review these devices (Section 3.1), which were used for
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some of the pioneering experiments in recoil ion momentum spectroscopy. They allowed a measurement of the transverse recoil ion momentum with a solid angle of a few percent. The resolution was restricted by the thermal motion of the target and was 4.2 a.u. for He in the "rst spectrometer operating at room temperature and 1.2 a.u. in the next version operating at 30 K. Similar resolution, but for the "rst time 4p solid angle and information on the longitudinal momentum, was obtained by using an e!usive gas target and projection "elds (Section 3.2). The key to today's high-resolution COLTRIMS systems is a combination of a cold localized gas-jet target, large area position-sensitive detectors for low-energy ion and electron detection and arrangement of electric and in some cases magnetic "elds to guide the ions and electrons from the target region to the detectors. We will therefore give a brief description of the supersonic gas-jet used (Section 3.3). This section naturally cannot review the very active "eld of research on and with supersonic gas-jets but only outline some of the basic properties of such jets which are of importance for their application in COLTRIMS. Di!erent designs for spectrometers, i.e. for creation of the electric "elds, are used at di!erent laboratories. In Section 3.4 we will present these di!erent concepts and will describe the way the momentum information can be obtained from the measured positions and time of #ights. We will in particular discuss the range of accepted momenta and the momentum resolution achievable. The imaging concept for ions has been adopted for electron momentum space imaging as well. We brie#y discuss the concept of such novel electron imaging devices in Section 3.5. Large area position-sensitive detectors have to be used for COLTRIMS. Since the ions and electrons have much too low energies to produce a signal in any kind of detector material or to penetrate even a very thin detector window or dead layer, e$cient and position sensitive secondary electron multipliers have to be used. Again we have to refer the reader to special literature on this subject, e.g. [94}96] for comprehensive information and only give a brief description of the micro-channel-plate detectors with wedge-and-strip read out and with delay-line read out which have mostly been applied in the work reviewed here. 3.1. Extended gas-target devices The goal of the "rst recoil ion momentum spectroscopy (RIMS) experiments was to measure multiple ionization and capture cross-sections for fast (1.4}5.6 MeV/u) heavy-ion-atom collisions di!erential in projectile scattering angle [37]. Since the projectile scattering in those collisions leads only to a de#ection of the projectile of a few lm over 10 m it cannot be measured directly. Thus, the "rst RIMS experiments intended to measure the transverse momentum transfer to the projectile by measuring the transverse momentum of the recoil ion. An extended cylindrical static gas target of 4 cm cylinder length and 1 cm diameter was used. The beam passed along the axis of the cylinder designed as a Faraday cage &free' of electric "elds. The ions, created along the beam path, drifted towards the wall of the cylinder in a time *t inverse proportional to their transverse momentum. The ions left the cylinder through a small aperture, which was a 1 mm hole in a "rst setup and a slit of 20;1 mm in a second version of the spectrometer. They were accelerated over 500 V, pass a focussing einzellens and were charge state analyzed by a magnet before they hit a positionsensitive channel-plate detector. The drift region was shielded from the acceleration "eld by a woven mesh (similar to the spectrometer shown in Fig. 6). The transverse momentum was obtained from the time of #ight measured in coincidence with the projectile and the recoil ion
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charge state is obtained from the position on the channel-plate detector. In the "rst spectrometer of this type the gas cell was at room temperature. The momentum distribution from thermal motion of the target gas before the collision restricted the transverse momentum resolution to 4.2 (8.3) a.u. for He (Ne) targets. Typical recoil ion time of #ights in such "eld free spectrometers were up to 20 ls [37]. A detailed description of this spectrometer can be found in [97]. To compare the experimental result to theoretical predictions in most cases the theory was folded with the thermal momentum distribution. The pioneering experiment yielding the transverse momentum distribution in fast U}Ne collisions are described in Section 4.2.6. A signi"cant improvement in resolution was achieved by precooling the target gas and operating the gas cell on a temperature of 30 K [51}61]. A sketch of this spectrometer is shown in Fig. 6. The cooling reduced the in#uence of thermal motion to 1.2 a.u. for He gas, which corresponds to a recoil ion energy of 2.5 meV. On this level great care has to be taken to avoid any contact potential inside the "eld free drift region. The cylinder was gold plated and a woven copper mesh of 0.12 mm by 0.12 mm spacing covered the inside walls and shielded the acceleration "eld from the drift region. Before each experiment the apparatus was carefully cleaned in a supersonic bath. From the reproducibility of the results the absolute accuracy of the recoil ion energy measurements was estimated to be $5 meV. A more detailed discussion of this spectrometer can be found in [98]. The apparatus was used mainly for coincident measurement of recoil ion transverse momenta with the projectile polar and azimuthal scattering angle (see Section 4 [51}61]).
Fig. 6. Recoil ion spectrometer with cooled static gas cell. The spectrometer is mounted on a cryogenic cold head (from [53]).
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3.2. Ewusive gas targets Frohne et al. [62] and Ali et al. [63] measured recoil ion momenta using an e!usive gas jet target. The directed gas #ow was achieved by a multi-capillary array. The collimated projectile beam passed only few mm above this gas outlet. The ions created in this region were extracted by an electric "eld onto a position-sensitive channel-plate detector. The gas-jet was directed onto this detector. The momentum resolution achieved with this technique for a Ne target was about 6 a.u. in the direction transverse to the jet and 8}10 a.u. in the longitudinal direction [99,63], comparable to what was achieved with the cooled gas cell for Ne [60,61]. The collimation of the gas #ow in the channels of the capillary array lead to this e!ective cooling. With this device for the "rst time a measurement of the longitudinal recoil ion momentum could be performed. In addition these devices were the "rst to yield a 4p solid angle for all ions. The recoil ion charge state was determined from the mean time of #ight in the spectrometer, the momentum from the position on the channel plate and the deviation of the time of #ight from the center of the TOF distribution for a given charge to mass ratio. This will be described in more detail in the next section. Wu and coworkers reported on a much improved longitudinal momentum resolution achieved by collimating the e!usive jet with a skimmer leading to a resolution of 1.5 a.u. [100]. 3.3. Supersonic gas-jet targets A decisive step forward in resolution for recoil ion momentum measurement became possible with the use of supersonic gas-jet targets. Such jets provide a dense (local target pressures of more than 10\ mbar can be achieved with standard pumps), well localized and internally cold gas target which makes them ideally suited for recoil ion momentum measurements. They are used in many "elds of physics, for example for providing mono energetic projectiles for atom}atom scattering experiments. A detailed description of free jet sources can be found in [101]. Fig. 7 shows for comparison the momentum distribution of ions from room temperature gas and ions from a supersonic gas jet. If the gas expands through a nozzle and the ratio of pressures on the two sides is larger than typically 2 [101] the #ow reaches supersonic speed. In the ideal case of an adiabatic isochore expansion of an ideal gas, the internal energy ( k¹ , with the Boltzmann constant k and the gas temperature ¹ before the expansion) and the compressional energy are converted to kinetic energy of the gas atoms. For mono-atomic gases the compressional energy is k¹ . Thus under ideal conditions the directed motion of the atoms after supersonic expansion tends towards an asymptotic momentum of k "(5mk¹ (14)
with the mass m of the target gas. For He at room temperature we have k "5.8 a.u. The width of
the velocity distribution around this mean velocity is described by the speed ratio S of the expansion. With the internal temperature ¹ of the jet the speed ratio for an ideal mono atomic gas is de"ned as
S"
5¹ . 2 ¹
(15)
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Fig. 7. Time-of-#ight distribution of recoil ions from room temperature gas (dots) compared to ions from a supersonic gas jet (histogram). The deviation of the TOF from the mean value is proportional to the momentum of the ions in the direction of the electric "eld. The full line shows the thermal momentum distribution (from [65]).
Fig. 8. Speed ratio of a He gas-jet for di!erent temperatures (¹ ) of the nozzle (from [103]).
The speed ratio of an expansion depends on the intra jet scattering cross-sections and, thus, the gas species, the gas temperature and on the product of driving pressure p and nozzle diameter d. The speed ratio for He at various temperatures is shown in Fig. 8 as a function of p d (see also [102]). The region in front of the nozzle where the expansion is still supersonic is called the zone of silence. A typical value for the spatial extension x of this region beyond the aperture is Q
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given by [101] 2 x" Q 3
p d p Q
(16)
with p being the pressure downstream of the nozzle. To achieve a real supersonic gas beam Q a skimmer must be placed inside the zone of silence. The transmitted gas beam follows an unperturbed motion in the high vacuum environment of the scattering chamber. The gas #ow through the nozzle a nd thus the load to the pump is proportional to p d(¹ . For the practical application in COLTRIMS one aims to achieve high density, narrow collimation (typically 1}2 mm at the intersection with the beam, in a distance of 3}5 cm from the nozzle) and narrow internal momentum distribution in jet direction as well as perpendicular to the jet. The width of the momentum distribution in the jet direction is the product of the jet velocity and the inverse speed ratio, both of which improve with cooling the gas. Perpendicular to the jet the momentum distribution is determined by the jet velocity, skimmer diameter and skimmer}nozzle distance. As examples we list three versions of gas jets which have been successfully used in COLTRIMS setups. All are operating with a nozzle diameter of 0.03 mm. Smaller nozzles would improve the speed ratio for a given pumping speed, but are less practical since they tend to get clogged by dust particles. Jagutzki and coworkers [64,93,104,105] used a single stage gas jet with a 8000 l/s oil-di!usion pump backed by a 500m/h roots pump and a 120m/h roughing pump. Due to the large pumping speed this jet can be operated at room temperature still having a good speed ratio and high target density (local target pressure at the beam intersection is several 10\ mbar). Mergel and coworkers used a cryogenic cold head to cool the nozzle to 14 K. This allowed to operate the single stage jet with only one 360 l/s turbomolecular pump. The trade-o! for the small pumping speed is a much lower target density (several 10\ mbar). With the cooling, however, the internal temperature improves in all three dimensions. In particular useful for ion}atom collisions is the low momentum spread in the direction perpendicular to the jet. With the same geometry the cooling improves also the resolution in this direction due to the lower jet velocity. A further advantage of a smaller velocity is a reduced gas load for the same target density. This is of particular interest for the implementation of such jets in the ultra-high vacuum of a storage ring as it is in preparation for the CRYRING (Stockholm) [106]. A third concept of a two-stage jet has been used by Moshammer and coworkers [77] and others. The region between the nozzle and the "rst skimmer is in this system pumped by a 250 m/h roots pump. It can operate at higher pressures than most turbopumps and thus take a higher gas load at a smaller pumping speed. The driving pressure is comparable to the one-stage device with a large di!usion pump but it is much smaller in size. For this setup a second stage with a skimmer (typical skimmer}skimmer distance: 2 cm), pumped by a turbopump is required for two reasons. First the high background pressure of only 10\ mbar leads to a di!usion of warm background He gas through the "rst skimmer. Second the pressure condition achieved with the roots pump leads to a small zone of silence (see Eq. (16)) and thus requires a small nozzle}skimmer distance of typically 2}4 mm resulting in a large divergence of the jet. A two stage jet can additionally by cooled to further improve the jet quality but some thermal shielding is necessary in the "rst stage. In all the setups discussed the gas jet leaves the scattering chamber through an opening of typical 1 cm into a jet dump, pumped by a separate turbo pump. For UHV applications more than one
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stage of jet dump pumping is necessary to reduce back streaming of gas from the jet dump to the chamber [106]. The internal momentum distribution and, thus, the resolution obtainable is di!erent in the three spatial directions. In the direction of the jet it is de"ned by the ratio of mean jet velocity and speed ratio. Typical momentum resolution values in jet direction range in 0.01}0.07 a.u. [102]. In the direction of the projectile beam the resolution is to "rst approximation given by the collimation geometry of the skimmer system and can thus be varied (this also changes the target density). However, as outlined in [101] there is a small non-Gaussian tail extending to large transverse momenta. A typical value for the resolution achieved perpendicular to the jet in beam direction for a cooled jet is 0.05 a.u. In the third direction perpendicular to the jet and the projectile beam the latter is often narrower than the jet which improves the resolution. This can be an important further improvement of resolution for well collimated ion beams (0.1 mm) or well focussed light from third generation synchrotron radiation sources where a focus of below 0.05 mm can be achieved. For a 30 K cooled jet a 0.05 mm focus would correspond to a resolution of 0.002 a.u. or an ion energy of 0.006 leV. 3.4. Projection spectrometers for slow ions All COLTRIMS spectrometers used with supersonic gas-jet targets apply a projection technique. An electric "eld guides the ions from the small overlap volume of the gas jet with the projectile beam (i.e. ion, electron or photon beam) onto a position-sensitive detector. From the position of impact on the detector and the ion time of #ight the starting momentum can be calculated. The various spectrometers used so far di!er in the details of the "eld geometry and "eld direction. An example for the simplest con"guration is shown in Fig. 9. The ions are accelerated over 3 cm by a weak homogeneous static electric "eld (typically 0.3}10 V/cm). They pass a woven mesh (typical wire spacing of 0.25 mm) and enter a "eld free drift region of 6 cm length. After passing a second grid (typically 0.1 mm wires spacing) they are postaccelerated over 2000}3000 V onto a channel-plate detector. The exact ratio of 1 to 2 between the length of acceleration and drift region assures that ions starting at slightly di!erent positions (i.e. potentials) along the electric "eld lines arrive at the same time at the detector [107]. Such McLaren focussing geometry is indispensable since one wants to resolve recoil ion energy di!erences of far below 1 meV, where the typical width of the target of 1 mm would already corresponds to 30 mV di!erence in the starting potential. In the McLaren geometry di!erent starting momenta in direction of the "eld result in di!erent time of #ights. Varying starting momenta in the two directions perpendicular to the electric "eld yields di!erent displacement on the channel-plate detector. Care has to be taken to assure proper "eld conditions in the spectrometer area. In the spectrometer shown in Fig. 9 the extraction region is shielded from external potentials by a carbon "ber (not visible on the "gure). One "ber of 7 lm diameter and 10 m length is wound in a spiral with 1 mm spacing around the four supporting germanium coated insulator screws. The "ber de"nes a very well controlled "eld in the extraction region [108,67,66]. All grids and the inside of the drift tube are germanium coated. Instead of the carbon "ber in other setups wires or rings of thin sheet metal, which are connected by resistors, have been used successfully [64,89,109,110]. Fig. 10 shows a typical time-of-#ight spectrum obtained with such a spectrometer. The width of the
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Fig. 9. COLTRIMSpectrometer. The nozzle is cooled to 14 K. Carbon "bers (not visible in the "gure) are wound around the four rods to create a homogeneous extraction "eld (from [108]).
Fig. 10. Time-of-#ight distribution of He ions from 250 keV/u He> on He collisions, coincident with charge exchanged ejectiles. The width of the peaks re#ects the starting momentum of the ions (from [111]).
time peaks re#ects the di!erent starting momenta in "eld direction. The position distribution of counts on the channel-plate detector is shown in Fig. 11. The narrow peak results from ions created in the overlap of jet and ion beam. The small ridge in front of the peak results from residual gas ions which are created along the ion beam. The width of this band re#ects mainly the thermal momentum distribution of the residual gas before the ionization. It is displaced from the peak ions due to the directed velocity of the jet.
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Fig. 11. Image of He ions created by ionization by a fast heavy ion beam. The electric "eld is perpendicular to the ion beam and the surface of the detector. The small ridge results from warm residual gas ionized along the beam, the peak results from the interaction with the gas jet (from [77]).
To obtain the recoil ion momentum in "eld direction k from the time of #ight (t) it turns out V that it is for all practical purposes su$ciently accurate to account only for the linear term: k
V
;q " (t !t) , s
(17)
where q is the ionic charge, ;/s is the "eld in the spectrometer and t is the time of #ight of ions starting with momentum k "0. Physically, this approximation assumes that the time focussing V is ideal, i.e. that the time of #ight of zero momentum ions is exactly independent of the starting position in the "eld. With an extraction region of 3 cm and a "eld of 0.33 V/cm Eq. (17) is at a momentum of 10 a.u. accurate within 1.6%. For the two directions perpendicular to the extraction "eld there is no force acting on the ions, thus the y and z momenta are simply given by k k
W
*y " m , t
*z " m , X t
(18)
where *y, z is the displacement of the recoil ion from the position where an ion with zero momentum transfer would hit. The jet velocity results in a displacement of the zero point. In practice the zero point in the direction of the jet is simply given by the center of the peak, since the momentum distribution is rotationally symmetric around the ion beam. In the direction of the ion beam (longitudinal momentum) there is no intrinsic symmetry of the ionization process providing the zero point. In principal this zero could be found by varying the extraction "eld. This is,
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however, in most cases not accurate enough. Thus, one can either use reference data or calibrate the spectrometer with capture reactions which result in lines of known longitudinal momenta. Electron impact is well suited for a reference measurement, since an electron gun can be easily implemented in a collision chamber and fast (e.g. 5 keV) electrons yield a maximum in the recoil ion distribution very close to zero [105,64]. For such a calibration measurement no coincidence is necessary, since the time of #ight is not needed. One can simply use the position distribution and a mean time of #ight estimated from the voltages at the spectrometer. An example for a calibration of a spectrometer using di!erent capture reactions can be found in [112]. In spectrometers which exploit homogeneous electric "elds only, the momentum resolution in the two dimensions perpendicular to the electric "eld is determined by the extension of the interaction volume and the ion time of #ight. For a typical time of #ight of 15 ls for He> ions a diameter of the gas jet of 1 mm results in a momentum resolution of 0.2 a.u.. Mergel and coworkers have reported a resolution of 0.26 a.u. FWHM in the direction of the projectile beam obtained with the spectrometer shown in Fig. 1 of [108]. To circumvent these restrictions of the extended reaction volume, electrostatic focussing lenses in the extraction "eld can be used. If the detector is placed in the focal point, ions created at slightly di!erent positions but with the same momentum vector hit the detector at the same position. The "eld geometry of one the "rst three-dimensional focussing spectrometer designed by Mergel following a suggestion by Cocke [68] is shown in Fig. 12. With this "eld geometry the displacement on the detector is still proportional to the starting momentum. The implementation of a lens in the extraction "eld changes, however, the focussing properties in the third, the time-of-#ight direction. Thus, lens and length of the drift tube have to be adjusted such, that the focus length for the time of #ight and the spacial focussing by the lens coincide. In general the implementation of a lens requires a longer drift tube compared to a spectrometer with homogeneous "elds. The spectrometer shown in Fig. 12 focuses an extended target region of 5 mm to a focus spot of below 0.25 mm on the detector. With such focussing spectrometers "nally the grid which shields the "eld free drift region from the high postacceleration "eld (typically 1 kV/mm) in front of the ion detector becomes crucial. The individual pores of the mesh act as strong lenses restricting the achievable position resolution for such low-energy ions to the mesh width. A stack of several meshes produces signi"cant Moire patterns; therefore the use of one narrow grid (typically 100 lm mesh width) is preferable to a stack of wider meshes [70]. The use of electrostatic lenses for imaging of photoelectrons and photofragment ions has recently been reported by Eppkin and Parker [113] who investigated photodissociation of molecular oxygen. Instead of perpendicular to the gas jet and to the projectile beam the ions can alternatively be extracted almost parallel to the beam. This extraction geometry has been widely used by the group at GSI [77] (Fig. 13). In the extremely #exible spectrometer developed there the extraction "eld is
Fig. 12. Field geometry with three-dimensional focussing properties. The three groups of trajectories result from ions starting with di!erent momenta in the direction perpendicular to the "eld (from [68,70]). In such focussing spectrometers the momentum resolution becomes independent of the size of the target.
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Fig. 13. Combined recoil ion electron spectrometer with extraction (almost) parallel to the beam. The electric "eld is generated by resistive plates. A solenoidal magnetic "eld is superimposed to the electric "eld to guide the electrons to the electron detector (from [77]).
generated by two ceramic plates with resistive layers. By adjusting the potentials at the four corners of each plate the ions can be easily steered to any direction. Thus, the extraction can be switched from parallel to perpendicular to the beam without breaking the vacuum. As a further advantage of this con"guration the o!set velocity of the ions from the gas jet can easily be compensated by steering the ions back to the center of the channel-plate detector. Whether perpendicular or parallel extraction of the ions is preferable strongly depends on the physical process to be investigated. In many cases it is advantageous to have extraction parallel to a symmetry axis of the physical process (like the polarization axis for photoionization studies or the beam axis for ion impact). 3.5. Projection spectrometers for electrons Electrons can be detected by a projection technique analogous to the one discussed for ions. The method has "rst been applied to image photoelectrons formed in intense laser "elds [114]. In a COLTRIMS setup the same electric "eld which guides the positively charged ions to one direction accelerates the electrons in the opposite one. The mass di!erence between ions and electrons has, however, some severe consequences for projection techniques. The "rst di!erence is given by the physics of the ionization process; in many ionizing atomic reactions the fragments (e.g. ions and electrons) have momenta of the same order of magnitude. Thus, typical electron energies are of the order of m /m times the recoil ion energies. While the ion energies are mostly small compared to the energy the ions gain in the electric "eld of the spectrometer this approximation is not valid for most of the electrons. Therefore 4p collection e$ciency is harder to achieve for electrons than for ions. The second important di!erence in ion and electron imaging is that for the motion in a homogeneous electric "eld for low-energetic particles the total time-of-#ight scales with (m while the measured displacement on the detector for a given momentum scales linearly with the mass. As a consequence spectrometers which use homogenous electrostatic "elds are much shorter on the electron side than on the recoil side, to achieve large electron detection solid angle.
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Such combined electron-ion-projection spectrometers based on electric "elds only have been used with great success in photoionization studies close to threshold [70,115}117] and for imaging of electron emission in slow ion atom collisions [31,71}75] and fast proton and antiproton}helium collisions [84,118,119]. To extend this 4p electron imaging to higher electron energies a novel spectrometer type has been developed and extensively applied at GSI (see Fig. 13). In this extremely versatile spectrometer a solenoidal magnetic "eld is superimposed parallel to the electric "eld. The electrons travel on spiral trajectories from the reaction volume to the detector. From the two position informations and the measured time of #ight, this trajectory, and the three initial momentum components, can be reconstructed, in most cases uniquely. For illustration Fig. 14 shows the radius on the channel plate versus the measured time of #ight. If the electron time of #ight is an integer multiple of the inverse of the cyclotron frequency in the two dimensions perpendicular to the magnetic "eld all electrons (independent of the momentum in this plane) return to their starting point which leads to the nodes visible in Fig. 14. Therefore in these nodes the information on the momenta perpendicular to the "eld is lost. The resolution is best in the center between the nodes. The existence of these nodes, however, gives a very good control of the experimental conditions. For example the cyclotron frequency and thus the magnetic "eld is measured very precisely via these nodes. Thus, the magnetic "eld is known very well. Details on this spectrometer can be found in [77,120]. The main advantage of this solenoidal magnetic "eld spectrometer is the quasi decoupling of the motion perpendicular and parallel to the "elds. Therefore the electric "eld can be optimized for good resolution in the time-of-#ight direction for electrons and ions. The electron detector can be placed at any distance from the interaction zone to achieve good resolution even for higher energetic electrons which have short time of #ights. The magnetic "eld can be adjusted independently to con"ne electrons up to a certain momentum perpendicular to the "eld. Typical operating conditions are for example a magnetic "eld of 15 G. This "eld results in a revolution time of 24 ns. With an active detector area of 8 cm diameter the spectrometer yields 4p solid angle for electrons up to
Fig. 14. Radial de#ection of electrons on the electron detector in the spectrometer shown in Fig. 13 versus electron time of #ight, for 3.6 MeV/u Ni> impact on He. The frequency of the nodes is given by the cyclotron frequency of the magnetic "eld (from [77]).
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an energy of about 100 eV. Since the recoil ions are accelerated to a very high momentum by the electric "eld the magnetic "eld has only a small in#uence on their trajectories. In most cases it is su$cient to account for this in a global way by rotating the position distribution of the ions on their detector by a few degrees. The GSI spectrometer has been used for kinematically complete measurements of the single and multiple ionization process in fast highly charged ion atom collisions [76,121,122] and (e,3e) studies [17]. A similar setup has been used for investigation of photo double ionization of He [123]. The full solid angle makes such electron spectrometers especially well suited for coincident detection of many electrons from multiple ionization processes. They can be equipped with a multi-hit capable detector and there is no restriction on the number of electrons which can be detected from one single event. Moshammer et al. succeeded in the "rst kinematically complete study of double ionization by fast ion impact [79,81] using the spectrometer shown in Fig. 13 and detecting both electrons. An alternative approach using non-homogeneous electrostatic "elds instead of the magnetic "eld has recently been proposed by Mergel and coworkers [124] for imaging of electrons emitted from surfaces. 3.6. Position-sensitive detectors For recoil ion and electron momentum imaging large-area position-sensitive detectors which combine good position resolution (typically 0.1 mm) with good time resolution (below 1 ns) are essential. Channel-plate detectors with wedge-and-strip or with delay line position encoding have been most widely used for this purpose. For a more complete overview on such detectors see [124,78], for wedge and strip readout see [125]. Channel-plate detectors have a typical detection e$ciency for charged particles of about 60% given by the open area. For imaging purposes chevron (two plates) or z-stack (3 plates) con"gurations of 48 mm or 86 mm diameter are used. The ions are typically accelerated by 2000 V onto the surface of the channel plate. For electrons, maximum e$ciency is reached at about 200}300 eV. A typical pulse height distribution from a channel-plate z-stack and the detection e$ciency of He> ions as function of the acceleration voltage is shown in Figs. 15 and 16. A detector with wedge-and-strip read out is shown in Fig. 17. The electron cloud of about 10}10 electrons created by avalanche ampli"cation in the channels are accelerated onto a highresistivity Ge layer evaporated on a 1.5 mm ceramic plate (see [126]). The image charge is picked up on the backside of the ceramic plate by the three areas of the wedge-and-strip structure. The area of the wedges and stripes grows linearly with the x and y position, respectively. A proper adjustment of the resistivity and the thickness of the ceramic plate assures that the image charge covers more than one structure. This is essential to allow for a determination of the centroid of the charge cloud. The typical period of the structure is 1.4 mm. The charge signals of the wedge, strip and meander structures are ampli"ed by charge-sensitive preampli"ers and main ampli"ers and recorded by analog to digital converters. By normalizing the wedge and the strip signal to the total pulse height one obtains the position of the centroid of the charge cloud. For a 5 cm diameter anode a position resolution of 0.05 mm can be achieved. The position resolution is mainly determined by the signal to noise ratio of the three signals. Therefore a good pulse-height resolution and high gain is desirable. The timing information is picked up either from the front or the back side of the channel-plate stack. Time resolution of 400 ps has been obtained this way [123].
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Fig. 15. Pulse height distribution for He> and He> ions on a z-stack channel-plate detector (i.e. three plates). The ions hit the detector with an energy of 3000 eV;q (from [92]).
Fig. 16. Relative detection e$ciency of a z-stack of channel plates for He> ions as a function of the acceleration voltage to the surface. At 3000 eV the curve is arbitrarily normalized to 1 (from [68]).
A delayline position read out for channel-plate detectors has been suggested by Sobottka [127]. For one dimension a pair of wires is wound around a supporting structure. The spacing between the wires is 0.5 mm. By a potential di!erence of 50 V between the two wires the electrons are collected by only one of the wires (see Fig. 18). The wire pair acts as a Lecher cable. At both ends of the double wire spiral the signals are processed by a di!erential ampli"er. Both wires pick up the same capacitively coupled noise, but their signals di!er by the real electrons from the charge cloud collected by the more positively biased wire. These di!erential ampli"ers yield extremely low noise signals. The time di!erences between a start signal, picked at front or back of the channel plate and the two signals from both ends of the wire pair is measured with two time-to-digital converters. The time di!erence is proportional to the position in the respective direction. For the position information in the perpendicular direction a second pair of wires is wound perpendicular
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Fig. 17. Channel-plate detector with wedge-and-strip read out (from [124]).
Fig. 18. Delay-line anode for multi channel-plate position read out (from [127]). The inner and outer winding pair are for x and y encoding. The two wires of each pair are on di!erent potential such that the electrons are collected on one of them.
to the "rst one. Depending on the TDC used a position resolution of better than 0.1 mm can be achieved. Compared to position encoding by charge division (as in a wedge-and-strip or resistive anode), the delay-line read out has many advantages. Since only fast timing electronics (fast ampli"ers, constant fraction discriminators and TDCs) is used it is much faster, allowing for much higher rates (MHz). Furthermore, the absolute position resolution is constant, thus larger detectors with better relative resolution can be build easily. The position resolution does not directly depend on the gain but stays rather constant, once su$cient gain is achieved. The most important advantage for ion and electron imaging is the capability to handle multiple hits in ns time intervals. The dead time is mainly determined by the electronics. In practice 10}20 ns dead time between double hits has been achieved [79,128,129]. This dead time, however, is no principal limitation for up to two hits. The main reason is that in each spatial direction the arrival time of the signal is measured on both ends of the delay line. To obtain the position information only the arrival time on one of the ends is
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needed. For more than two hits arriving within the delay time of the detector anode, which is 30}100 ns, depending on the size of the detector, some information is lost. Such multi-hit delay-line detectors have been used for detection of two and more electrons from a multiple ionizing collision [79] and for detection of ionic fragments from molecular coulomb explosion [128,129].
4. Experimental results for charged particle impact 4.1. One-electron processes Two types of one-electron processes have been investigated with COLTRIMS: First the transfer of one electron from a bound state of the target to a bound state of the projectile and second the emission of one target electron to the continuum. A third possible one-electron process, target excitation, does not result in a charged recoil ion and is thus in most cases impossible to detect by COLTRIMS. The same is true for projectile ionization or excitation in a one-electron process, since these reactions do not result in a charged target ion either. We will "rst discuss single capture. It is the simplest process from the point of view of the "nal momenta since there are only two particles in the "nal state. We will then deal with single ionization which is more complex due to the three particles in the "nal state. Both processes are most important in ion-atom collisions: electron capture is by far the dominant process at low impact energies, while at high velocities target ionization dominates. 4.1.1. Single capture For single capture the quantized nature of the Q-value (i.e. discrete values for the projectile energy loss or gain) leads to discrete values of k . Thus, as outlined in Section 2 the measurement of k is equivalent to energy loss or gain spectroscopy and gives information about the "nal state to which the electron is captured. The simultaneous determination of k gives the information on , the projectile scattering angle (i.e. on the impact-parameter dependence of the process). While in principle the measurement of the recoil ion momentum is, for single capture, equivalent to measuring the change of momentum (energy gain and scattering angle) of the projectile there are many practical advantages in detecting the recoil ion momentum. Detecting the projectile it is necessary to measure a very small change of a large incoming momentum. For the traditional projectile energy gain measurement this restricts the experiments to relatively low impact energies (keV/u). Typical resolutions are *E/E'10\. For the scattering angle measurement, depending on how well the incoming beam is collimated, *0 '10\ is a practical limit for the resolution (for energy gain spectroscopy see, e.g. [130] and references therein). The recoil ion momentum measurement, however, is almost independent on the preparation of the incoming beam. Therefore it allows a high resolution energy gain measurement even for MeV impact energies, and very high resolution scattering angle measurement without a well collimated beam. Mergel and co-workers have applied COLTRIMS to study the capture reaction 0.25!1 MeV He>#HePHe>#He> . The distribution of k for 0.25 MeV is shown in Fig. 19.
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Fig. 19. Longitudinal momentum distribution for He> recoil ions from 0.25 MeV He>PHe single capture collisions. The di!erent lines correspond to capture to di!erent "nal state con"gurations of target (n) or projectile (n ). The full arrow shows the momentum of !v /2 due to the mass transfer and the dashed arrow the Q-value e!ect with !Q/v (K- and L-shell) (see Eq. (5)) (from [108]).
The left peak is due to capture to the He> ground state, the right peaks to capture to excited states. The experimental resolution of *k "0.26 a.u. was achieved with a spectrometer with extraction perpendicular to the ionic beam. This spectrometer had no electrostatic lens and a "eld of 0.33 V/cm at the target was applied (Fig. 9). Therefore, the limited resolution re#ects approximately the diameter of the gas jet of 1.1 mm. This corresponds to an energy gain or loss resolution of the projectile of about *E/E "1;10\. The full arrow indicates the backward momentum shift N resulting from the mass transfer of the electron (v /2) while the energy gain (for capture to the K shell) or energy loss (for capture to excited states) lead to momentum transfers shown by the dashed arrows. At about 0.5 MeV capture to the K and L is about equally likely. As can be expected from the consideration of velocity matching between electron and projectile, increasing projectile velocity results in a relative increase of K-capture compared to capture to excited states. Fig. 20 shows the recoil ion transverse momentum distribution dp/dk separated for capture to , the K shell and to excited states. The resolution in the transverse direction is much better than in the longitudinal direction, because the source volume in this direction is given by the very well collimated beam (0.1 mm). Thus, a resolution of a few lrad can be achieved, much superior to any direct scattering angle measurement. The L-capture shows a smooth decrease while the K-capture exhibits an oscillatory structure which is well known for K-K vacancy transfer at lower impact energies [131]. Mergel and coworkers found these K-K interference structure up to a projectile velocity of about 2.5 a.u. [132,108]. Wu and coworkers [88] and Kambara and coworkers [133] have employed COLTRIMS to study single electron capture by 0.5}3.7 MeV/u O> and F> on He collisions. They could separate capture to the projectile K-shell from capture to higher excited states (see Fig. 21). In later studies this group achieved even much higher resolution [134] by using supersonic gas and focussing spectrometers. At the lowest velocity of 0.5 MeV/u capture to n"4 and higher dominates by far. K-shell capture plays an increasing role at higher impact energies. The measured ratio of K-capture
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Fig. 20. Recoil ion transverse momentum transfer (i.e. scattering angle) distribution for 0.25 MeV He>#HeP He>#He> for di!erent "nal states of the capture process (from [108]).
to capture to excited states is well described by the Oppenheimer, Brinkmann Cramers theory. Thus, the projectile tends to capture to states which have mean orbital velocities similar to the projectile velocity. At 8.7 MeV/u Kambara and coworkers reported also the transverse momentum distribution of the recoil ions. They found a maximum at around k "1 a.u. but could not con"rm oscillations , in the di!erential cross-section predicted by close coupling calculations [133]. For slow (6.8 keV/u) bare Ne and Ar on He collisions Cassimi and coworkers [109] have studied state selective cross-sections for single capture. With a recoil ion momentum resolution of 0.4 a.u. they obtained a resolution of 5 eV for the Q-value measurements. This allowed to separate capture to the n"4 and n"5 which was found to be the main channel for Ne> impact and n"7 and n"8 for Ar> impact. Their data con"rm the predictions of the over-the-barrier model for the scaling of the average quantum number of the electron capture. In addition to k they measured the transverse momentum, too (see Fig. 22). nCTMC (n-body classical trajectory Monte Carlo, see, e.g. [135,136]) calculation for this process reproduce the "nal-state distributions very well. The results of these calculations were also used to convert the measured transverse momenta to impact parameters. For Ne> impact the maxima in Fig. 22 correspond to impact parameters of 6.5 and 4.5 a.u. This is consistent with quantum mechanical curve crossing calculations. In a subsequent study by the same group Flechard et al. [110] have extended this study to double electron capture. In this work a major improvement in resolution by using a focussing lens was achieved. Single capture for in the comparable collision system of Ar> on He has been investigated by Abdallah and coworkers [137]. They "nd that as the projectile velocity is raised from 0.2 to 1.0 a.u. the reaction window spreads and higher n and l values become the favored capture channels. 4.1.2. Target single ionization The study of the reaction dynamics of single ionization has concentrated mostly on He targets. It is experimentally the easiest accessible target for COLTRIMS, allowing for the highest resolution.
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Fig. 21. Q-value spectra for single capture in O> He collisions. The spectra are obtained from the recoil ion longitudinal momentum. The location of expected Q-values for capture to K,L and M shells is indicated. The lines show Lorentzian "t for capture to K and L-and-higher shells (from [88]).
In general the collision dynamics for single ionization is almost una!ected by electron}electron correlation e!ects. In general single ionization is a three-body momentum exchange process. One can approximate it by splitting it in three separate two-body momentum exchange processes, projectile}electron, projectile}ion and electron}ion. Although all three pairwise interactions are always present, there are paradigmatic cases where one of the three dominates. The fully di!erential studies of single ionization for di!erent collision systems and studies performed in the CTMC model for the "rst time gave complete descriptions of the "nal momentum space of ionizing ion}atom collisions.
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Fig. 22. Momentum distribution of recoil ions from single capture by 6.8 keV/u Ne> (a) and Ar> (b) projectiles. The dashed lines give the positions of capture to speci"c n states according to Eq. (5) (from [109]).
These multi-dimensional images unveil directly which of the momentum exchange processes is most important for the reaction. These studies reach for ion impact the same level of detail as has been reached for (e,2e) reactions using traditional coincident electron spectroscopy (see [9}11] for reviews). They go beyond (e,2e) studies by covering the full "nal state momentum space. Where traditional (e,2e) experiments result in angular distributions for "xed momentum transfer the fully di!erential ion impact studies discussed here give a complete overview in momentum space and therefore highlight the processes which contribute most to the total ionization cross-section (see [17,138,139]). Figs. 23}25 demonstrate three paradigmatic collision systems. For slow p on He collisions the momentum exchange between projectile and target nucleus is by far the dominating one (Fig. 23), for fast proton on He collisions the projectile}electron momentum exchange becomes important (Fig. 24) and for very fast highly charged ion impact the electron-recoil ion momentum exchange dominates (Fig. 25). Each of these momentum exchange patterns suggest simple models for the ionization mechanism. We will group our discussion of target single ionization along this line and discuss single ionization in slow collision in Section 4.1.2.1, in fast proton and antiproton on He collisions in Section 4.1.2.2 and fast highly charged ion collisions in Section 4.1.2.3. For all three collision systems the recoil ion and electron momentum vector has been measured in coincidence event by event. Therefore knowing k and k for each event Dk "!(k #k ) can be calculated.
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Fig. 23. Projection of the momentum transfer vectors of recoil ion (upper) electron (middle) and projectile (lower) in the "nal state onto the plane de"ned by the projectile beam and the momentum vector of the recoil ion for 10 keV/u p#HePHe>#e\#p. The #p axis is parallel to the incoming projectile direction, the #p axis is parallel to the X W "nal transverse momentum component of the recoil ion. The grey scale represents the corresponding doubly di!erential cross section dp/(dk dk ) on a linear scale (similar to [31]). A blowup of the electron distribution is shown in Fig. 26. V Fig. 24. 0.5 MeV/u p#HePHe>#e\#p. Projection of the momentum transfer vectors of recoil ion (upper) electron (middle) and projectile (lower) in the "nal state onto the plane de"ned by the projectile beam and the scattered projectile (not by the recoil ion as Figs. 23 and 25). The #p axis is parallel to the incoming projectile direction, the #p axis point X W in the direction of the scattered projectile. The grey scale represents the corresponding doubly di!erential cross section dp/(dk dk ) on linear scale. The circular arc in the middle "gure shows the location of the binary encounter ridge for V , electrons (from [84]).
4.1.2.1. Slow collisions. In slow collisions, i.e. if the projectile velocity is smaller than the mean electron velocity in a Bohr orbit, promotion of an electron to the continuum is typically less likely then capture to a projectile bound state (see, for example, Fig. 3). The question of which mechanism is responsible for electron emission in such slow collisions is far from answered. The momentum exchange between projectile and target nucleus is generally much larger than the momentum transfer to the emitted electrons in such slow collisions (see Fig. 23). In the transverse direction recoil ion and projectile are scattered oppositely as a result of the internuclear repulsion. From this
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Fig. 25. 1 GeV/u U>#HePHe>#e\#U>. Projection in momentum transfer components of electron and recoil ion in the "nal state onto the plane de"ned by the recoil ion and the beam (as in Fig. 23). The p axis is parallel to , the incoming projectile direction, the !p axis is parallel to the "nal transverse momentum component of the recoil ion. V The momentum change of the relativistic heavy ion is much smaller than the electron and ion momenta. It is not shown, since it re#ects mainly the experimental resolution. The cluster size represents the corresponding doubly di!erential cross section dp/(dk dk ) on logarithmic scale (from [81]). Compare also Fig. 36. V ,
transverse momentum exchange the impact parameter can be inferred, if the e!ective potential is known. The scattering plane is de"ned by the beam direction and the "nal state recoil ion momentum vector. In the longitudinal direction the recoil ions are emitted strongly in forward direction. This large longitudinal momentum transfer to the recoil ion requires a strong coupling of the ionic core to the forward motion of the slow projectile. In the intermediate quasimolecular system the electrons act as a &glue' which allows the recoil ion to follow the forward motion of the projectile. As can be seen in Fig. 3, in most of the cases the electron in the quasimolecular orbital is, however, not emitted to the continuum but captured to projectile orbital (see also [140]). Those few electrons released to the continuum are found mostly with very little momentum in between target and projectile frame (see Fig. 23). Thus in terms of mechanisms (i.e. dynamics) the forward emission of the recoil ions in slow collisions can be seen as a manifestation of the molecular character of the ionization process. From the perspective of kinematics this forward emission follows directly from Eq. (6). E.g. at 10 keV/u impact energy the momentum exchange resulting from the energy transfer of the He target binding energy is already E /v "1.4 a.u. To investigate the mechanism of ionization in slow collisions it is helpful to look for the details of the electron momentum distribution. DoK rner and coworkers [73] have used a spectrometer as shown in Fig. 12. For the reaction 5!15 keVp#HePp#He>#e\
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they have mapped the square of the continuum wave function of the emitted electron in momentum space for fully controlled motion of the nuclei. Figs. 26(a)}(c) show the two-dimensional velocity
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Fig. 26. Projection of the velocity distribution of electrons for single ionization in 5 (a), 10 (b) and 15 (c) keV p}He collisions onto the scattering plane, de"ned by the incoming projectile axis (z) and the "nal momentum vector of the recoil ion, emerging to the -x direction. The target center is at (0,0) the projectile at (1,0) and the saddle at (0.5,0) The data for 10 keV are for a transverse momentum transfer in the interval k "1}5 a.u.. For the other energies this momentum , range is scaled by 1/v in order to sample approximately the same range of impact parameters. (d) sideview to (b), i.e. projection onto the y}z plane perpendicular to the x}z scattering plane (from [73]).
distribution of the electrons from reaction 20 projected onto the scattering plane, de"ned by the beam and the recoil ion momentum vector. The horizontal z-axis is the direction of the incoming projectile. The y-axis points to the direction of the scattered projectile transverse momentum, while
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the recoil ion is emitted into direction-y in Figs. 26(a)}(c). The x-axis in "gure (d) is perpendicular to the beam and the recoil ion momentum. Velocities are scaled to the projectile velocity. Target centered electrons are located at (0,0) while electrons captured to the projectile continuum (ECC) could show up at (1,0). The saddle point of the potential is located at (0.5,0). A sideview onto the distribution of (b) is shown in (d), i.e. a projection onto the y, z plane. The electron momentum distributions shown in Fig. 26 for 10 keV are in coincidence with recoil ions within 1(k (5 a.u. This is the region where most of the ionization cross-sections results from. For all , projectile velocities shown the electrons are strongly forward emitted. While for 5 keV the electrons are emitted preferentially onto the side where the projectile passes by, at 15 keV they are emitted to the direction of the recoiling ion. At 10 keV even a horseshoe-shaped emission pattern is observed. All these distributions are very di!erent from the well-known structures in the electron spectra at high energies. They are a result of quasimolecular promotion via di!erent series of transitions [141}144]. The part of the structure at 10 keV around the saddle point indicates a n state in the continuum on the saddle point. It can be populated via a rotational coupling from the 1s ground state (T001 series of transitions [141]). Such promotion patterns have been predicted by the theory of hidden crossings [141,142]. Recently, Macek and Ovchinnikov showed that an interference between sigma and pi components of the electronic wave function can give rise to the observed rapid oscillation in the electron emission as a function of projectile velocity [145]. On grounds of classical mechanics Olson and coworkers have suggested the mechanism of promotion of the electron on the saddle of the internuclear potential [146}148]. For 15 keV impact energy CTMC calculations yielded very good agreement with the observed k distributions for , ionization and were able to reproduce the main features of the electron emission [73]. The "nal state momentum distribution from these calculations was very sensitive to the initial state momentum and spatial distribution, for which a Wigner distribution of 10 initial binding energies had to be used. This reproduces the quantum mechanical radial distribution very well over 4 orders of magnitude. Abdallah and coworkers reported a study of the electron emission Ne> on Ne [75] collisions and He> and He> on He collisions [75]. Their continuum structures for the Ne and He case di!er signi"cantly. The authors conclude that the continuum momentum distribution of the electrons is determined by the "nal molecular orbit the electron occupied. For He on He collisions the continuum shows n structure similar to the p on He case shown in Fig. 26 while for Ne> on Ne a spiral electron distribution is found together with the fact that ionization occurs in a small impact parameter window. This indicates a promotion of the electron through the 4fp MO. S 4.1.2.2. Fast p and p on He collisions. For fast p-on-He collisions (v several a.u.) the target electrons do not have enough time to adiabaticlly adapt to the rapidly changing two-center potential. On the other hand, now the projectile is fast enough that even a pure two-body collision with the target electron can transmit su$cient energy to ionize the electron. Fig. 24 suggests that in most of the collision this projectile-electron momentum exchange is important. (Note that in Fig. 24 the scattering plane is de"ned by the incoming and scattered projectile momentum vector.) This has been "rst suggested by measurements of the projectile scattering angle dependence (dp/d0 ) for single ionization [149}153] at much larger momentum transfer. The details, however, could only be unveiled in coincidence experiments which measured the projectile scattering (polar
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Fig. 27. Di!erential cross section for 3 MeV p#HePp#He>#e\ as function of the transverse momentum transfer to the projectile k (equivalent to the projectile scattering angle) and to the recoil ion k . Full circles: experiment , , k [59], open circles: experiment k [149], diamonds: experiment k [59]. Lines: nCTMC (from [59]). , , ,
and azimuthal angle) in coincidence with the transverse momentum of the recoiling ions [51}59,98,154,155]. For scattering from an electron at rest the maximum projectile scattering angle 0 is given by arctan 0 "m /m (0 "0.55 mrad for protons). Kamber and coworkers [149,150] have found C N a shoulder at this critical angle in the di!erential cross section dp/d0 for ionizing collision of 3 and N 6 MeV p on He (see Fig. 27). They showed that for scattering angles between 0.2 and 0.55 mrad, the cross-section could be well described by a Rutherford scattering process from a free electron. Over this angular range the transverse momentum exchange would thus be dominated by a hard binary encounter between projectile and target electron. These collisions are dominated by large impact parameters with respect to the target nucleus, but small impact parameters between projectile and emitted electron. Gensmantel and coworkers [59,54] could experimentally manifest this interpretation by measuring the transverse momentum distribution of the recoiling ions dp/dk (Fig. 27) , with a cooled target gas cell (see Fig. 6). They found, as expected from the above interpretation, a smooth dp/dk with no shoulder. , At a projectile scattering polar angle of 0.73 mrad they found two peaks in the transverse recoil ion momentum distribution, which could be associated with the scattering of the projectile at the target nucleus (leading to large recoil momenta) and at the electron (leading to small recoil momenta), respectively (see Fig. 2 in [59]). More recently DeHaven and coworkers [156] con"rmed these "ndings. For 6 MeV p-on-He collisions they measured the scattering angle of the projectile in coincidence with the recoil ion momentum component in the direction of the projectile scattering and found two clear cut ridges originating from projectile scattering at the nucleus and at the electron (see Fig. 28). At smaller impact energies the shoulder in the dp/d0 washes out due to the momentum distribution of the electrons in the initial state [151,153]. For 0.5 MeV p}He collisions DoK rner and coworkers [51] measured the transverse recoil ion energy as a function for the projectile scattering angle (see Fig. 29). They distinguished three regions of projectile scattering. At large scattering angles (0 '0.9 mrad) the projectile scattering is determined by the interaction between the two
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Fig. 28. Single ionization in 6 MeV p-on-helium collisions. The horizontal axis is the recoil ion momentum component transverse to the beam and in the plane de"ned by the scattered projectile (0.39 a.u./channel). The vertical axis show the projectile transverse momentum (0.21 a.u./channel). The cuto! for small scattering angles results from a beam block on the projectile detector. The dashed line locates a projectile scattering angle of 0.55 mrad. The loci of events corresponding to projectile}nucleus and projectile}electron scattering are indicated by full lines (compare also Fig. 2 of [59]).
Fig. 29. Mean transverse energy of He> ions from 0.5 MeV p}He collisions as a function of the projectile scattering angle. Symbols: di!erent experiments (partly from [51,58]). The scatter in the data represents the systematical error due to uncontrolled electric "elds in the cooled gas cell used in this experiment. Dashed line: two-body scattering of the projectile at the target nucleus, full line: nCTMC calculation [98] (see also [58]), these results are di!erent from the nCTMC results in [51], which were found to be in error (see [54]), dotted line: nCTMC folded with the target thermal motion at 30 K, dashed dotted line: Eikonal distorted wave calculation [157,158] (from [53]). Recent experiments with a supersonic gas-jet target by Weber et al. [84] give evidence that the saturation energy of 10$5 meV at small scattering angles is at the upper limit of the error bart including the shift to higher energies due to the target thermal motion. They measured energies down to 1 meV.
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nuclei. At these close collisions the recoil ion energy is given by the nuclear}nuclear two-body collision kinematics and the He> and the proton are emitted back to back [55]. At scattering angles around 0.55 mrad, the recoil ion energy is much smaller than expected from a nuclear two body scattering, giving evidence that the projectile de#ection is mainly caused by a hard collision with the emitted electron. As a consequence the recoil ion and projectile are no longer emitted back to back (see [55]). At small scattering angles the recoil energy becomes independent of the projectile scattering. In this regime the electron}recoil ion momentum exchange dominates. These transverse momentum exchange processes are well described by nCTMC calculations [51,58,159,160] as well as by quantum mechanical calculations in the Continuum Distorted Wave approach [157,158,161,162]. In these early RIMS studies the transverse momenta of the He> ions was measured in coincidence with the projectile scattering angle. Recently Weber et al. [84] investigated the same collision system measuring the three-dimensional momentum vector of the electron and the ion in coincidence using a spectrometer as shown in Fig. 12. They succeeded in imaging the complete 9 dimensional "nal-state momentum space with a resolution of about 0.2 a.u. (see Fig. 24). The high resolution allows to focus on the region of small momentum exchange, which yields by far the dominant contribution to the total cross-section. In this recent study they found transverse recoil ion energies as low as 1 meV for projectile scattering angles below 0.1 mrad. Thus, one can conclude, that the early results from static gas target devices shown in Fig. 29 give a much too high &saturation' energy for the ions at small projectile scattering angles. As outlined above, the coincident recoil ion electron momentum space imaging with todays COLTRIMS systems yields for single ionization the momentum vectors of all particles for each registered event. This multi-dimensional array of data can be looked at from many di!erent perspectives. One perspective on such data for He single ionization by fast projectiles has already been discussed with Fig. 24. Here the three-body momentum distributions are projected onto the plane de"ned by the incoming projectile (z) and the scattered projectile (x). As a second useful perspective at the same data is a projection onto the plane perpendicular to the projectile beam (xy plane) (Fig. 30). For this projection one is free to choose one axis along the transverse momentum of one of the particles and than display the momenta of the others in this coordinate frame. In Fig. 30(a) the vertical axis is given by the transverse momentum vector of the recoil ion (k ) (the , direction of the recoil ion is indicated by the arrow). The horizontal axis is perpendicular to the beam and k . The momentum distributions of the projectile is plotted. Fig. 30(b) shows the , electron momentum distribution in the same coordinate system. In Fig. 30(c) again the electron momenta are shown, but now in a coordinate system where the vertical axis is given by the transverse momentum of the scattered projectile (arrow). The distributions show that the projectile is scattered at both the target nucleus and the electron, and that there are a large number of events where the electron and recoil ion emerge to opposite sides. A more complete view on these data in di!erent coordinate systems can be found in [84]. It is obvious from Fig. 30 that the projectile}electron momentum exchange is important in most of the ionizing collisions. This is supported by Fig. 31 which shows the azimuthal angles between the three transverse momenta. One "nds strong contributions with back-to-back emission between electron and projectile. Moshammer and coworkers have "rst used the azimuthal angular plot of Fig. 31 to illustrate the ionization mechanism in fast highly charged ion}atom collisions (see Section 4.1.2.3 and Fig. 37).
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Fig. 30. 0.5 MeV p#HePp#He>#e\ All three panels show the momentum plane perpendicular to the incoming projectile, which moves into the plane of the paper. (a) shows the momentum distribution of the projectile, the vertical axis (labelled *P ) is along the transverse momentum of the recoil ion. (b) shows the momentum distributions of the VN electrons in the same coordinate frame as (a), in (c) the vertical axis is the direction of the transverse momentum of the scattered projectile, the electron momenta are shown (see text) (from [84]).
They pointed out that the three borders of the triangle have a direct connection to a particular ionization mechanism. The diagonal line
#
"180"
"1803 corresponds to U U U a binary-encounter between projectile and electron,
"1803 to a pure nuclear}nuclear U de#ection as observed for slow collisions (see Section 4.1.2.1) and
"180 is the kinematics U equivalent to photoionization (see Sections 4.1.2.3 and 5). We now focus on the longitudinal momentum balance in fast p}He collisions. For single ionization by fast heavy particle impact the recoil ion longitudinal momentum k for each , collision can be calculated from the electron ejection angle (0 ) and the electron energy (Eq. (6)). Thus, the information about the electron spectra can be obtained from the k distributions and , vice versa. As a consequence of this there are simple formulas to connect di!erential cross sections for electron emission and those for k [83] , dp C> 1 dp " de (21) dk k de d(cos 0) C\ , The integration limits for the electron energy e!"1/2(k!) are k!(0 )"v cos 0 $(v cos 0#2(k v !"e ") . (22)
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Fig. 31. 0.5 MeV p#HePp#He>#e\ Azimuthal angle between recoil ion and electron plotted versus that between recoil ion and projectile. Indicated are the kinematical lines assuming the speci"ed two body interaction and considering the third particle as a spectator. The region below is kinematically forbidden. (from [84]) (compare to same presentation for fast highly charged ion impact, Fig. 37).
Tribedi and coworkers have used these relations to calculate recoil ion longitudinal momentum distributions from measured doubly di!erential electron spectra [85]. From this perspective the advantage of measuring k is that it always samples all electrons (at all angles and energies), which is di$cult with traditional electron spectroscopy. And even more important, since k is , simultaneously measured, the detection of only one particle, provides information about electron emission di!erential in the transverse momentum exchange (i.e. for closer collisions the impact parameter). Besides from its kinematical connection to electron emission, the k distribution can be, and has always been, discussed in terms of forces which act on the nucleus and mechanisms which dominate the ionization process. For 0.25}1 MeV p and He> impact on He DoK rner and coworkers [112] have measured the k distribution. They observed very similar recoil ion longitudinal momentum distribution dp/dk for these projectile velocities and charge states. The mean value of the k distribution is only slightly shifted backward from 0. At 0.5 MeV impact energy it is very close to the Compton pro"le of the He (see Fig. 32). Thus the "nal state momentum distribution re#ects closely the momentum in the initial state. Only for close collisions, i.e. large k the , k distribution gets much broader, indicating faster electrons and more coupling between the projectile, electron and ionic motion. The data are in reasonable agreement with nCTMC calculations which implicitly include the interaction between ion, emerging electron and projectile to all orders. Rodriguez and coworkers have shown [83] that in a quantum mechanical treatment one has to go beyond the Plane Wave Born Approximation to describe the distribution at 0.25 MeV while at 1 MeV a "rst-order treatment gives very good results. They employed the Continuum Distorted Wave Eikonal Initial State (CDW-EIS) approximation to include a post
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Fig. 32. Longitudinal recoil ion momentum distribution for single ionization in 0.5 MeV p}He collision. Open circles: experiment, full line: nCMTC, histogram: nCTMC (from [112]).
collision interaction. Including these higher-order e!ects results in a backward shift of the calculated distribution in agreement with the experiment. Wang and coworkers [163] illustrated the correspondence between the dp/dE d0 and dp/dk by looking at recoil ion momentum distributions for "xed electron emission angle 0 "0. They emphasize that electrons in the binary-encounter peak of the electron spectrum lead to recoil ions with a momentum distribution given by approximately the Compton pro"le, centered at k "0. Weber et al. [84] and Khayyat et al. [118,119] have measured the longitudinal momentum distribution of electrons and recoil ions for single ionization by fast protons and antiprotons. For 1 MeV impact energy they "nd almost identical longitudinal electron and ion momentum distributions for proton and antiproton impact (see Fig. 33). This is in good agreement with CDW-EIS and CTMC calculations at this impact energy. The ionic momentum distributions found in these studies for proton impact are in agreement with the earlier work shown in Fig. 32 [112]. For smaller impact velocities a post-collision e!ect is expected, which separates electrons and ions in momentum space. Two experiments con"rm this prediction and observed a signi"cant forward emission of electrons from proton impact at velocities of 2}4 a.u. [84,71]. A detailed analysis of the charge state dependence of these momentum exchange patterns at an impact velocity of 100 keV/u within the nCTMC approach can be found in Ref. [164]. This post-collision e!ect is much stronger for highly charged ion impact, as illustrated in the following section (see, e.g. [76]). 4.1.2.3. Fast highly charged ion collisions. For collisions of fast highly charged ions with He the momentum exchange patterns are very di!erent from those by fast proton impact (see Fig. 25). Moshammer and coworkers have investigated 1 GeV/u U> impact on He [81]. They found that for relativistic collisions the momentum exchange between recoil ion and emitted electron by far dominates over the momentum exchange with the projectile. The projectile carrying 0.24 TeV of energy induces an &explosion' of the atom by delivering only the energy but almost no momentum. This fragmentation pattern has the characteristics of the photoionization process (see Sections 2.3 and 5). The common nature of fast charged particle and photons interacting with matter was discussed already by Fermi et al. [165,166]. In their approach ionization of an atom by charged
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Fig. 33. Longitudinal momentum distribution for single ionization of Helium by 1 MeV antiproton (data points) in comparison with proton collision (solid line). (a) represents electron momentum data, (b) recoil ion momentum data. The theoretical calculations represent antiproton impact, where the dotted line shows CDW and the dashed line a CTMCresults. (from [119]).
particles is modeled as photoionization by a "eld of equivalent photons of various energies (equivalent photon method). The photon "eld is obtained by a Fourier transformation of the time and impact parameter depended electromagnetic "eld of the passing projectile. A 1 GeV/u U> ion generates a sub attosecond (10\ s) superintense (I'10 W/cm) "eld of virtual photons, shorter and more intense than any laser. The momentum exchange pattern in these relativistic collisions could also be well reproduced in relativistic nCTMC calculations [167]. Fig. 34 shows that the longitudinal momentum distributions of electrons and ions in such collisions are identical and symmetric around zero. The ionization cross-section is very well described by a calculation using the equivalent photon method (see Refs. [81,165,166,168,169]) and by relativistic nCTMC calculations [167]. Going to nonrelativistic collisions (e.g. single ionization by 3.6 MeV/u Se> and Ni> impact) Moshammer et al. have shown that electrons and ions are still emitted mainly back to back [76,121,122,170]. Even though the equivalent photon method is generally not expected to be valid
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Fig. 34. Longitudinal momentum distribution of electrons (dots) and of recoil ions (open circles) for single ionization of He in collisions with 1 GeV/u U>. Full line: equivalent photon method (from [81]).
Fig. 35. Longitudinal momentum distribution of low-energy electrons (full circles) and recoil ions (open circles) for single ionization of He in collisions with 3.6 MeV/u Ni> (from [76]). (The experimental data are distorted by a calibration error and need to be shifted by about #0.2 a.u. for the recoil ions and !0.2 a.u. for electrons (see text).)
in this velocity regime, the experiments show that the projectile momentum exchange is much smaller than the recoil ion and electron momenta. On the outgoing part of the trajectory the post-collision interaction with the potential of the Ni> becomes important pushing the He> ions backward and pulling the electron forward (see Fig. 35). These conclusions are supported by nCTMC calculations. If in the calculations the sign of the projectile charge is changed from #24 to !24, the recoil ions are moved forward while the electrons are pushed backward. More accurate new studies [171] indicate that the experimental electron distribution in Fig. 35 and in Refs. [76,79,122] need to be recalibrated by about 0.2 a.u. into the negative k direction whereas the ,
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recoil ion distribution is shifted forward by the same amount. These corrected data along with recent results on Au> on He collisions [172] are in very close agreement with the CDW calculations. The data of Fig. 34 from the same group are not a!ected by this correction of the early calibration. Thus the CTMC seems to overestimate the post-collision e!ect systematically. Recently, the initial state distribution of the electrons in the CTMC model was improved. Using a Wigner distribution with 10 di!erent binding energies the quantum-mechanical initial state distribution of the electron momentum as well as of its radial distribution was reproduced over many orders of magnitude. The improved model predicts a signi"cantly reduced post-collision e!ect and the theoretical results are in close agreement with the experiment [172]. CDW-EIS calculations by Rodriguez and coworkers [83] and CDW calculations by O'Rourke et al. [173,174] which include the e!ect of post collision interaction did reproduce the electron as well as the recoil ion momentum spectra. In a systematic study of a very similar system (3.6 MeV/u Se> on He) Moshammer et al. have discussed the transverse momentum balance as well. Fig. 36 shows the two-dimensional momentum distributions of all fragments in the plane spanned by the beam and the recoil ion momentum. A comparison of Figs. 36 and 25 strikingly shows the polarization of the fragment distribution by the post collision interaction with the projectile. The similarity to photoionization is again highlighted by the azimuthal angular dependence of the transverse momentum exchange. In the longitudinal direction the approximate compensation of electron and ion longitudinal momenta for large v results already from the conservation laws (see Eq. (6)). In the transverse direction however there are no kinematical restrictions. Thus the transverse momentum balance gives an even clearer probe of the dominating ionization mechanism. Fig. 37 shows the azimuthal angular distribution of the fragments. The main contribution to the cross-section comes from azimuthal angles of close to 1803 between recoil ion and electron as it
Fig. 36. 3.6 MeV/u Se>#HePHe>#e\#Se>. Projection in momentum space of all particles in the "nal state onto the plane de"ned by the beam (horizontal axis) and the recoil ion transverse momentum (negative vertical axis). The cluster size represents the corresponding doubly di!erential cross-section dp/(dk dk ) on logarithmic scale (from [122]). V Compare to Figs. 23}25.
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Fig. 37. 3.6 MeV/u Se>#HePSe>#He>#e\ Relative azimuthal angle between recoil ion and electron ( ) plotted versus that between recoil ion and projectile ( )p. Indicated are the kinematical lines assuming the speci"ed two-body interaction and considering the third particle as a spectator. The region below the line entitled &Binary encounter electron' is kinematically forbidden. (from [122]) (compare to same presentation for fast proton impact, Fig. 31).
is the characteristics of photoionization. This is in striking contrast to the same presentation for single ionization by fast protons shown in Fig. 31. 4.1.3. Connection of single ionization and single capture in the recoil ion kinematics Electrons stripped from the target will either be in the continuum (ionization) or in a bound state of the projectile (electron capture). The transition between capture and ionization can be viewed from the perspective of all particles: the electron, the projectile and the recoil ion. In the doubly di!erential cross-section for electron emission (dp/dE d0 ) the attractive potential of the projec tile leads to a divergence at 0 "0 and E "1/2v . This &cusp'-shaped maximum in the electron spectra, resulting from electrons travelling with the projectile but not being bound has been studied extensively in electron spectroscopy (for a review see [175]). In electron energy space there is a clear cut between capture and ionization. From the perspective of the recoil ion momentum, however, there is a natural and smooth transition from capture to ionization. All recoil ions with k (k (see Eqs. (4)}(7) and (10) for de"nition) result from capture while those with k 'k are due to ionization. The threshold k corresponds to electrons in the &cusp', travelling with the projectile without being bound (see Eq. (10)). Rodriguez and coworkers [83] emphasized that the cusp yields a step (i.e. a "nite cross-section and not a soft onset) in the dp/dk at k . Weber et al. have measured the longitudinal momentum distribution of He> ions from 100 to 250 keV/u p impact. Their results (Fig. 38) clearly show the predicted sudden onset of the cross-section at the kinematical threshold. Ions from capture reactions smoothly continue
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Fig. 38. Longitudinal momentum distribution of He> ion from 150 keV/u p impact on He. The full line with the peaks corresponds to capture to ground state, excited states and capture plus target excitation. The dashed line results from ionization (from [84]).
the k distribution from ionization in this intermediate velocity range where capture and ionization have comparable cross-sections. 4.2. Multiple electron processes in fast collisions In this section we "rst discuss processes involving two active target electrons (double ionization, transfer ionization and double capture), then reactions involving one electron of the target and one of the projectile and "nally multiple electron processes leading to highly charged recoil ions. 4.2.1. Double ionization of He Most of the studies of double ionization of He by charged particle impact are devoted to ratios R "p>/p> of the total cross-sections only. From the dependence of R on the strength of the X X perturbation q/v ln v it was established that in the limit of small perturbations an asymptotic ratio of R"0.26% is approached, independent of the projectile charge and mass (see, e.g. [176,177] for a recent review). For stronger perturbation R increases which has been attributed to X a two-step process of two independent interactions of the projectile with the two electrons [177]. Pioneering multiple di!erential cross-sections for He double ionization by 0.3 MeV proton impact have been reported by Skogvall and Schiwietz [178]. They measured angle and energy resolved electron emission cross-sections for close impact parameters and found strong deviations from predictions of the independent-particle model. The "rst recoil ion momentum studies for double ionization of He determined the transverse recoil ion energy as a function of the projectile scattering angle experimentally as well as theoretically within the dCTMC approach [58,98]. The results di!er from those obtained for single ionization (discussed in detail in Section 4.1.2.2) in two ways: For very small scattering angles ((0.25 mrad) the recoil ion energy is higher than for single ionization. This e!ect could qualitatively be reproduced by dCMTC calculations including
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a dynamical screening for the second electron [58,98]. Second, the region where the transverse recoil ion momenta are much smaller than the projectile transverse momenta extends to larger projectile scattering angles as for single ionization. The latter is caused by the fact that in the two-step process the projectile is scattered sequentially at two electrons. Thus, while for single ionization the maximum scattering angle at an electron at rest is 0.55 mrad, twice this value can be reached for double ionization. This simple fact solves a puzzle originating from single di!erential measurements of Giese and Horsdal [151,152,179,180]. They have measured R as a function of X projectile scattering angle for 0.3}1 MeV p}He collisions and found a distinct peak at a scattering angle of about 0.9 mrad. Taking into account the dynamics of the projectile scattering, this can be interpreted as a manifestation of the double scattering mechanism in the double ionization process. While for single ionization close impact parameter collisions with the nucleus are required to reach a scattering angle of 0.9 mrad, the two-step process for double ionization contributes to all scattering angles up to 1.1 mrad. This explanation has been supported by nCTMC calculations [154,58,52] and has been con"rmed by the measurements of the recoil ion energies of the He> ions [58,181,98] and by quantum mechanical calculations [182], too. The transverse momentum distribution of He> recoil ions has been determined for fast p [98], C> [183] and Xe> projectiles [184,185]. They all show much larger transverse momenta for double ionization than for single ionization indicating closer impact parameters (see Fig. 39). This can be expected from an independent particle model. Unfortunately, no quantum mechanical calculations are available for comparison, since the problem of coupling two electron momenta and the nuclear momentum exchange has not yet been solved. The only quantum mechanical approach which incorporates the electron}electron interaction in the ground state and during the collision (forced impulse methode FIM [179,186}190]) uses the impact-parameter approximation and thus does not provide di!erential cross-sections.
Fig. 39. Transverse momentum distribution of He> and He> ions from 6.7 MeV/u Xe> ion impact (from [185]).
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Recently, Moshammer and coworkers succeeded in determining the "rst fully di!erential cross-section for He double ionization by charged particle impact. They measured the momenta of both emitted electrons in coincidence with the recoil ion momentum for 3.6 MeV/u Se>}He collisions. For this collision system double ionization is completely due to the two-step process (R "15%). In addition, the projectile velocity is fast enough, that the energy loss of the projectile, X given by the sum of both electron energies and their binding energy results in a projectile momentum change *k "*E/v which is very small compared to the electron and recoil , momenta. As outlined in Section 4.1.2.3 the projectile acts as an extremely intense, short and broad band virtual photon pulse. Double ionization under these conditions would be dominated by coupling with two virtual photons. This interpretation within the WeizsaK cker-Williams formulation is supported by a theoretical study by Keller and coworkers [169]. Fig. 40 shows the longitudinal momenta of two electrons for double ionization of He and double and triple ionization of Ne. The distribution is strongly forward shifted as a result of the long-range Coulomb
Fig. 40. Longitudinal momentum of two electrons (k vs. k ) for 3.6 MeV/u Se> on He and Ne double and triple , , ionization. The upper right "gure shows for comparison the correlated He initial state (from [79]).
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potential of the emerging ejectile. More interestingly, the momenta of the two electrons are found to be strongly correlated. Such patterns cannot be explained by any independent particle model or, more precisely, in the absence of the explicit 1/r -interaction between the two electrons. In CTMC calculations the experimentally observed pattern is partially reproduced only if the 1/r potential is included. The basic features of the measured distributions are already present in the correlated He ground state wave function (see Fig. 40). At GSI these experiments have been continued to much higher impact velocities like 1 GeV/u [81], where the classical revolution time of the electrons in the ground state is below 1% of the collision time. Thus, one can hope to achieve a &snapshot' of the correlated initial-state wave function of atoms, molecules or clusters which are multiply ionized in such reactions [191]. For a relativistic nCTMC calculation at this energy see [167]. 4.2.2. Transfer ionization As for double ionization, one might distinguish di!erent mechanisms for transfer ionization (TI), i.e. the simultaneous capture of one target electron and emission of a second one. Within an independent-electron approximation, TI is a result of one step where a target electron is captured, and a second step where by a projectile}electron interaction the second electron is emitted to the continuum (TSTI). If one takes the electron}electron correlation in the initial ground state into account, two new mechanisms are present in theoretical models: shake-o! [177] and shake-over [192]. In the "rst case one electron is captured and the second one is emitted due to the change of the e!ective target potential after the loss of the "rst electron. In the case of shake-over [192] the "rst electron is emitted by a projectile electron interaction and the second one is captured by a relaxation of the target wave function partly to bound projectile states. Finally, if one takes not only the static initial-state correlation into account, but also allows for a dynamic electron}electron correlation during the collision, a fourth mechanism (eeTI) becomes possible. This process of (eeTI) has already been proposed on grounds of classical mechanics by Thomas in 1927 [193]. Assuming the electrons at rest for simplicity, the projectile knocks the "rst electron to a angle of 453 to the beam axis with the velocity (2;v . In a second step with some probability this electron scatters from the second electron into the forward direction with velocity v where it can be captured by the projectile ion. In this case the other electron is emitted perpendicular to the beam with an identical velocity v . This momentum exchange leads to a scattering angle of 0.55 mrad for a proton, independent of the projectile energy. In a pioneering experiment Palinkas and coworkers identi"ed the electrons from this process experimentally [194]. Horsdal and coworkers [195] searched for the (eeTi) in the scattering angle dependence of the TI process (see [182,196,197] for an explanation of the experimental "ndings of [195] a TSTI). Quantum mechanical calculations of the eeTI process in second Born approximation have been reported in [198}200]. Multiple di!erential cross-sections measurable by COLTRIMS allow for a detailed examination of these mechanisms. The determination of the longitudinal momentum distribution alone allows for some qualitative conclusions already. Such experiments have been reported by Wu and coworkers [88] and Kambara and coworkers [133] (a). The "rst kinematical complete experiment for transfer ionization has been reported by Mergel and coworkers [68,69]. They measured the projectile momenta in coincidence with the recoil momentum vector for p}He collisions and obtained complete images of the square of the correlated three body wave function in the "nal state.
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This experiment allows for the "rst time a detailed and quantitative analysis of the two-step and the Thomas double-scattering mechanism (b). (a) Wu and coworkers have measured the longitudinal momentum distributions for single target ionization, single electron capture (Fig. 21) and transfer ionization for 20}60 MeV MeV O> and F> impact on He with a resolution of 0.75 a.u. [88]. For single ionization they found the distribution centered around k "0 with the width of approximately the Compton pro"le in , agreement with the "ndings for fast p impact reported in Section 4.1.2.2. To test the assumption of a two-step process within the independent-electron approach, they calculated the impact parameter dependence for single ionization and single capture and found that the ionization probability is constant and approaches unity in the range of impact parameters where single capture is signi"cant. Thus, within the simplest version of the independent-electron model the k -distribu, tion for transfer ionization is given by the distribution for single capture, folded with the one for single ionization. Fig. 41 shows the measured distribution for transfer ionization together with the results of this independent electron model. Very good agreement is observed. At a much lower impact energy of 8.7 MeV Kambara and coworkers performed the equivalent experiment [133]. They found, that the distribution for transfer ionization is slightly backward shifted compared to the single capture distribution. They concluded that while single capture leads to "nal states n"4 and higher TI leads mainly to n"2!3. This can be understood qualitatively
Fig. 41. Recoil ion longitudinal momentum distribution for F>#HePF>#He>#e\. The full line shows results from the independent electron model (from [88]).
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in a re"ned two-step model [201]. In a two step process it is most likely that ionization is the "rst step due to the lower ionization potential for the "rst electron. In this case the second step would be a capture from a He> ion, which may lead to more tightly bound "nal states as observed in the experiment. (b) A kinematically complete experiment for the reaction 0.3!1.4 MeV p#HePH#He>#e\
(23)
has been reported by Mergel et al. [68,69]. They have measured the three momentum components of the recoiling ion in coincidence with the two momentum components of the scattered hydrogen atom perpendicular to the beam. Since there are three particles in the "nal state and it is known that the capture leads predominately to the ground state, the measurement of 5 momentum components together with momentum and energy conservation provides the complete kinematic information on the process. Thus, even if the electron is not directly measured its momentum can be calculated from the measured quantities. The momentum space of the recoiling He> ion gives a simple criterion to distinguish the (eeTI) from all the other processes (TSTI, Shake-o!, Shakeover). The Thomas mechanism for transfer ionization (eeTI) does not require the nucleus to balance the momentum. In this process the forward momentum of the captured electron is compensated by the second electron. The target nucleus acts only as a spectator. All other TI processes involve one step where one target electron is kinematically captured by the projectile and therefore the target nucleus has to compensate the electron forward momentum yielding a backward momentum transfer of !v /2!Q/v to the recoil ion. In addition the (eeTI) leads to a scattering angle of 0.55 mrad for the projectile. Fig. 42 shows the momentum distribution of the He> recoil ions for projectile polar scattering angles between 0.45}0.65 mrad. The plane shown is de"ned by the beam axis (z direction, vertical) and the momentum vector of the scattered projectile. The projectile is scattered towards positive k . The data are integrated over the recoil ion momenta in the k direction. Contributions from the V W TSTI process are expected along the (dashed) line of k "!v /2!Q/v . While at 0.5 MeV this X is by far the dominant process, at 1 MeV impact energy clearly a second peak from the (eeTI) arises. At 1.4 MeV this correlated Thomas capture already is the dominant contribution at the critical projectile polar scattering angle. This interpretation was further supported by nCTMC calculations. In these calculations the electron}electron interaction can be switched on and o!. They showed that the peak at momentum zero only arises if the electron}electron interaction is taken into account on during the collision. They do, however, not yield the correct cross-section. Mergel and coworkers also obtained the total cross-sections for the contribution of the (eeTI) from the data shown in Fig. 42. They found a scaling of the cross-section with v\ ! where v is the projectile velocity. This is in striking contradiction to the predictions of the classical Thomas model and second-order Born calculations. Both theories predict the same scaling like v\ at asymptotically high velocities. At 1.4 MeV the experimental cross-section is already a factor of 10 larger than predicted by theory. The process of (eeTI) is a unique probe of initial-state correlation in the helium atom. First, the cross-section for this process directly re#ects the spatial distance between the two electrons and, thus, is sensitive to the correlation in coordinate space. Second, the momentum distribution of the left behind He> nucleus re#ects the sum momentum of both electrons in the initial state and, hence, the correlation in momentum space. The (eeTI) at high velocities acts as a very fast knife
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Fig. 42. Momentum distribution of recoil ions for the reaction p#HePH#He>#e\. The plane is de"ned by the incoming beam (z direction) and the momentum vector of the scattered projectile (toward positive k ). All distributions V are for a "xed scattering angle of 0.45}0.65 mrad of the projectile and are integrated over the recoil ion momentum in y direction. The dashed lines show the longitudinal momentum where events from two-step processes and shake-o! processes are expected. The (eeTI) Thomas process is expected close to the origin (from [69]).
which cuts almost instantaneously the bond between the three particles in the He atom. It minimizes all kind of post collision interactions which smear out the signature of the initial state in the "nal state. The fully di!erential cross-sections can be transformed to emission patterns for the electron. For 0.3 MeV it was demonstrated this way that the scattering of the projectile in the region of the critical scattering angle is mainly caused by a hard binary collision with the emitted electron. This is the analog to the "ndings for single ionization reported in Section 4.1.2.2. It explains the sharp peak found in the ratio between transfer ionization to single capture cross-sections at 0.55 mrad by Horsdal and coworkers [195]. This peak is caused by binary-encounter scattering of the projectile at the emitted target electrons, similar to the peak at 0.9 mrad in the ratio of double to single ionization [151] discussed in Section 4.2.1. A puzzling new feature in the electron emission is found for the small scattering angles ((0.3 mrad). At these angles the electrons are found to be emitted backward. No conclusive interpretation has been given for this unusual emission pattern. It has been speculated [68] that it
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may be a result of the initial state correlation, since for capture a forward directed momentum component of the electron in the initial state is needed. In this case the left-over second electron maybe backward directed. 4.2.3. Electron}electron interaction in electron-loss reactions In all processes discussed so far only electrons from the target have been involved. A new and even more complex situation arises if projectile electrons are actively involved in the collision. For this case where a projectile electron is emitted two mechanisms have been discussed in the literature. The projectile electron can either be emitted by an interaction with the target nucleus (ne) or with one of the target electrons (ee) [202}211]. With the use of COLTRIMS a break through in this investigations has been achieved since the measurement of the recoil ion momentum allows an experimental separation of the two mechanisms [66,88,89,212]. The explanation for this is similar to the case of transfer ionization discussed in Section 4.2.2. For the (ne) mechanism the target nucleus must be actively involved in the collision receiving large forward and transverse momenta, while for the (ee) mechanism, the target nucleus acts as a spectator, receiving only little momentum transfer. The longitudinal momentum balance for these reactions can be seen in more details from Eqs. (6) and (7). The loss of kinetic energy of the projectile necessary for ionization corresponds to backward momentum transfer to the projectile. This projectile longitudinal momentum loss has to be compensated by the target atom. If now the target is ionized, the forward momentum can be either transferred to the recoiling ion, which occurs in the case of an (ne) interaction, or to the emitted target electron, which occurs in the case of an (ee) interaction. Thus, measuring k allows , the separation of the two mechanisms. In both cases the recoil ion momentum is smeared out by the momentum transfer caused by the target ionization. In addition to the di!erence in k one , expects larger k for the (ne) than for the (ee) mechanism, since for the (ne) mechanism the impact , parameter between the nuclei needs to be in the range of the projectile electron shell radius [206,207,209,210] and thus smaller than for the (ee) process. Fig. 43 shows the momentum distribution of the recoiling ions for the reaction 0.5!2 MeV He>#HePHe>#He>#2e\
(24)
from Ref. [66]. The two maxima at 1 MeV can be attributed to the contribution of the (ee) interaction (close to the origin) and the (ne) mechanism (forward shifted). The long dashed line shows the position where one expects the contributions of the (ne) according to Eq. (7). In addition Fig. 43(f) shows the He> momentum distribution resulting from electron-impact ionization at velocity equal to 1 MeV He impact. The maximum nicely coincides with the (ee) contribution in (c) illustrating the analogy of the (ee) mechanism with electron-impact ionization. The threshold for electron impact ionization of He> is at a velocity equivalent to 0.4 MeV, thus the contribution of the (ee) mechanism disappears at the lowest energy investigated. For increasing energies the (ee) mechanism is dominating, since it involves only one interaction while the (ne) requires one interaction from target}nucleus and projectile}electron as well as an additional interaction from projectile}nucleus and target}electron. The experimental momentum distributions as well as the absolute cross-sections are very well described by two-center nCTMC calculations (see Refs. [66,212] for a detailed comparison). At the lowest energies investigated here these calculations predict a third mechanism which is a double target ionization followed by a capture of the
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Fig. 43. (a}e) Momentum distributions of recoil ions for the reaction (24) for variable impact energy. The vertical axis shows the recoil ion transverse momentum, the horizontal axis the longitudinal momentum. The contour lines are linear and equally spaced in cross-section. The two maxima for 1 MeV impact result from the (ee) and (ne) interaction, respectively. The long dashed line indicates the position of expected longitudinal momentum for the (ne) contribution from Eq. (7). (f) Momentum distribution of He> ions from 130 eV electron impact, which is equivalent in velocity to 1 MeV He (from [66]).
projectile electron to the target. The expected k for this three-step mechanism (see Eqs. (6) and , (7)) is shown by the short dashed line in Fig. 43. For highly charged ion impact the separation of the (ee) and (ne) mechanism in the longitudinal momentum increases because of the larger binding energy of the projectile electrons and it becomes possible to determine quantitatively the individual cross-sections for both mechanisms. Wu and coworkers have investigated O> on He collisions and extracted the ratio of (ee) and (ne) contributions (see Fig. 44). They found very good agreement with a simple model using a scaled Plane-Wave-Born cross-section for the (ne) mechanism and folding a free-electron impact ionization cross-section with the Compton pro"le of the target (for a more detailed discussion see also [88]). 4.2.4. Double electron capture Capture of both electrons in He> on He collisions is one of the fundamental two electron processes. Ground state capture (resonant) is expected to be by far the dominant channel [213] for this reaction. This transition however cannot be accessed by energy gain spectroscopy or coincident detection of photons or Auger [214] electrons [215], since the ejectile is neutral and in the ground state. Thus, COLTRIMS is the only experimental approach allowing to separate this dominant channel. DoK rner and coworkers [111] found a ratio of about 16$3% for double
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Fig. 44. Ratio of cross-sections for the (ee) and (ne) contribution to simultaneous target and projectile ionization for O>!He collisions, as function of impact energy. Triangles: Experimental data obtained from integrating over the respective range of recoil ion momenta for each process, solid line: Theoretical calculation from [243,244], dashed line theoretical estimate from [89], see text (from [87]).
capture to nonautoionizing exited states to ground state capture for double capture by 0.25}0.75 MeV He> impact on He, by measuring the k distribution of the He> ions. This is in , good agreement with nCTMC predictions by ToK kesi and Hock [216]. Much theoretical e!ort has been put into the calculation of total as well as scattering angle di!erential double capture cross-sections (see, for example [201,217}233]). Most of those calculation depend to some degree on treating the two electrons independently during the collision. Only recently new approaches have been developed which preserve the full 4-body nature of the problem [213,234}239]. E!ects of dynamical electron}electron correlation during the collision can be most sensitively probed in di!erential cross-sections. From the scattering angle dependence of single capture at fast collisions it is well known, that the transverse momentum exchange up to 3 a.u. is mostly mediated through the captured electron [240,241]. Thus, the correlated electronic momenta should be re#ected in the di!erential double capture cross-sections (see [242] for a "rst experiment). State-selective di!erential cross-sections at 0.25 MeV He> impact energy obtained by COLTRIMS (Fig. 45) show that capture to excited states results in larger transverse momentum exchange between projectile and target than ground state double capture. This indicates the necessity of smaller impact parameters for exothermic channels. For higher impact energies signi"cant structure in the di!erential cross-section resulting from three di!erent Thomas type scattering mechanism have been predicted [221,245]. COLTRIMS experiments searching for such structures are under way. 4.2.5. Multiple ionization and capture As for single ionization, the longitudinal and transverse momentum transfers are completely decoupled for multi-electron processes induced by fast ion impact. Contrary to single ionization, however, in multiple electron processes the transverse momenta are often much larger than the longitudinal momenta, since many electron processes require closer impact parameters. The "rst experiments concentrated on this transverse momentum exchange, aiming for determination of the impact parameter dependence of the reactions. Historically, the data were often presented in terms
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Fig. 45. Scattering angle distributions for double electron capture for 0.25 MeV He> on He collisions obtained by measuring the transverse recoil ion momentum. Full circles: All "nal states (data multiplied by 2). Open circles: Ground-state capture. Open squares: Double capture to nonautoionizing excited states (from [111]).
of mean energies of the recoil ions for di!erent reaction channels [46,49,62,99,246}248] partly to get information on the ionization mechanism and partly because of the practical relevance of the recoil ion energies for the brightness and energy spread of heavy-ion beam pumped secondary ion sources [42,249,250]. The general trend for all collision systems is a steeply increasing recoil ion energy with increasing ion charge state. One typical example of such data is shown in Fig. 46. The experiment was performed using an extended gas target at room temperature and the data are thus in#uenced by the target thermal motion at the lower energies. For a detailed comparison with model predictions it is, however, more informative to discuss these data in terms of cross-sections singly di!erential in k . We "rst discuss the results of the , pioneering experiments on k distributions by Ullrich and coworkers for fast U> and , U> impact on Ne and Ar. Second, we discuss results for the collision system F> > > on Ne, for which di!erent groups investigated the transverse momentum balance between projectile, recoil ion and emitted electrons [60}62,99]. All of these studies had low resolution in the recoil ion momentum. Finally, experiments on the longitudinal momentum balance for multi-electron processes involving capture and ionization by three di!erent groups [62,82,99,185] are reviewed. 4.2.6. Transverse momentum transfer in fast multiply ionizing collisions Ullrich and coworkers have performed the "rst experiments to measure the transverse momentum distribution of Ne and Ar recoil ions created by fast U> and U> [50]. These experiments have been performed using a "eld-free gas cell at room temperature. It allowed them to access for the "rst time the transverse momentum exchange in a fast heavy ion collision on the level of accuracy equivalent to scattering angles of 10\ rad. Fig. 47 shows the recoil ion transverse momentum distributions for 1.4 MeV/u U> and 5.9 MeV/u U> on Ne impact summed over all charge states of the ions. At large transverse momentum exchange the di!erential cross-sections are well described by the Rutherford crosssection (i.e. they decrease with 1/k ). At small momenta the data deviate signi"cantly from ,
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Fig. 46. Mean transverse recoil ion energy as a function of recoil ion charge state for 1.4 MeV U>}Ne collisions. Open circles: Experiment. Open squares: nCTMC. Full squares: nCTMC folded with target thermal motion (from [42]).
a Rutherford distribution folded with the thermal motion of the target. nCMTC calculations show that the momenta of the emitted electrons dominate the transverse momentum exchange in this regime. The calculations indicated that the projectile will even be scattered to negative scattering angles for impact parameters greater than 3 a.u. at the attractive negative potential of the strongly polarized electron cloud. This "rst evidence of the strong in#uence of the emitted electrons on the heavy-particle motion was con"rmed by the latest high-resolution studies using supersonic gas-jet targets (for single ionization see Section 4.1.2.3 and Refs. [76,79,121]). While Fig. 47 displays dp/dk summed over all ionic charge states, cross-sections di!erential , in the recoil ion charge state have been obtained simultaneously. These di!erential cross-sections for multiple ionization can either be used directly for a detailed test of theoretical models which calculate the momentum exchange like the nCTMC [159], or they can be used to obtain ionization probabilities. Since the total scattering follows a Rutherford-like shape at larger transverse momentum transfers one can obtain probabilities for each charge state as a function of k , which , is at these large k related to the impact parameter. Horbatsch and coworkers [251,252] have , found that these probabilities can be described in a consistent way by an independent electron model calculating single particle ionization probabilities and neglecting all electron}electron correlation e!ects. For fast (0.5}1 MeV/u) F\>PNe collisions di!erent groups have reported di!erential crosssections for ionization, single and double capture and the emission of up to 6 electrons to the continuum. In these studies the recoil ion transverse momentum and charge state has been
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Fig. 47. Recoil ion transverse momentum distributions for (a) 1.4 MeV/u U> and (b) 5.9 MeV/u U> impact on Ne, summed over all recoil ion and projectile charge states. Full circles: experiment; open squares: nCTMC; full line: nCTMC folded with thermal momentum distribution at room temperature (from [50]).
measured in coincidence with the projectile scattering angle and charge state [60}62,99]. One of the goals of these experiments was to obtain information on the sum momentum of the emitted electrons from the di!erence between projectile and recoil ion momenta. Within the experimental resolution of the projectile momentum measurement of about 20}40 a.u. (which is worse than the k measurement) all the experiments agreed that no deviation is found from a two-body , scattering between projectile and target nucleus. Thus, on this scale even the sum momentum of 6 emitted electrons plays no essential role on the transverse momentum balance. This refutes the "ndings of Gonzales and coworkers [253], who reported a strong &out-of-plane' scattering of projectile and target nucleus caused by very high electron momenta. In general, increasing transverse momenta of projectile and recoil ion are found with increasing number of emitted electrons, indicating smaller impact parameters. Unverzagt and coworkers reported on such data for 5.9 MeV/u U> on Ne collisions. They found excellent agreement between experiment and nCTMC calculations (see Fig. 48). Similar data for Xe> impact on Ar have been reported by Jardin and coworkers [185]. In both experiments also the longitudinal momenta of the ion have been measured (see next sections).
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Fig. 48. Recoil ion transverse momentum distribution for 5.9 MeV/uU>#NePU>#NeG>#ie\ (from [82]).
4.2.7. Longitudinal momentum transfer for fast highly charged ion impact The longitudinal momentum balance for multiple ionization has been investigated by Unverzagt and coworkers for 5.9 MeV/u U> on Ne and for Xe> on Ar collisions by Jardin and coworkers. Both groups found backward emission of the ions. The momentum position of the maxima as well as the widths of the distribution increase roughly linear with the ionic charge (see Fig. 49). The data are again in excellent agreement with nCTMC calculations. No electron is captured in these collisions, thus the k is given by k "*E /v ! k C (see Eqs. (4)}(7)). Where *E is the energy loss of the projectile, i.e. the sum over the electron continuum energies, the binding energies and excitation energies of the residual ion. However, since v +16 a.u. the ratio *E /v is in most cases small compared to the sum of the longitudinal electron momenta. Thus, the backward recoil ion momenta prove that the electrons are collectively forward emitted with a mean longitudinal sum energy of 5 eV for single ionization up to 1.1 keV for Ne>. This forward emission is not a trivial result of a binary-encounter between projectile and electron, since it has been shown in Section 2 that binary encounter electrons result in no momentum transfer to the ion. The forward emission of the electrons as well as the backward emission of the ions is a consequence of the strong long-range force of the highly charged projectile on the outgoing path of the trajectory similar to what has been observed for single ionization of He
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Fig. 49. Recoil ion longitudinal momentum distribution for 5.9 MeV/u U>#NePU>#NeG>#ie\ (from [82]).
(see Fig. 35). The same trend has been seen for double ionization of He [185]. Backward emission of highly charged Ne recoil ions by fast highly charged ion impact has "rst been seen in an experiment by Gonzales-Lepra and coworkers [48]. For multiple ionization involving additionally the capture of 1}3 electrons Frohne and coworkers observed an unexpected opposite trend. They investigated 19 MeV F> Ne collisions determining the number of captured (0}3) and continuum electrons (0}5). Using a warm e!usive gas target the mean value of the longitudinal momentum of the recoil ion was measured. Within their resolution they found the distribution centered around zero for the pure ionization channel and up to 5 electrons emitted to the continuum. For capture of n electrons and no emission to the continuum they saw backward emission following the (n;v /2) law as expected from Eq. (5). However, for capture accompanied by emission of m electrons to the continuum they observed increasing forward recoil ion momenta compared to the pure capture channels (see Fig. 50). This is opposite to the prediction of the nCTMC calculations and from what one would expect from the e!ect of post collision interaction. As discussed in Section 4.2.2 Mergel and coworkers [69] have even found mainly backward scattered electrons for transfer Ionization in fast proton on He collisions. No conclusive interpretation of this behaviour has been found.
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Fig. 50. Mean recoil ion longitudinal momentum as function of the ion charge state q for the reaction 19 MeV F>#NePFN>#NeO>#(p#q!9)e\. Full line: nCTMC (from [99]).
4.3. Multiple electron processes in slow collisions Early experiments performed on many-electron transition, at slow impact velocities at Kansas State University have used a warm e!usive gas-jet target. They were therefore not able to resolve the line structure in the k distribution but they did provide important information on the mean Q-value and on the transverse-momentum distribution. Wu and coworkers performed systematic studies on transfer ionization and double capture reactions in O > and N> on He collisions at 0.2}1.7 a.u. impact velocity. At these slow velocities direct ionization is negligible and, thus, transfer ionization proceeds via double capture followed by autoionization. The measurement of the Q-value, the ratio between transfer ionization and true double capture, along with the transverse momentum distributions allows one to obtain information on which states have been populated and whether the transfer of the electrons occurs in one or two steps. Signi"cant di!erences between the O> projectile on one and the N> and O> on the other side have been observed. For O> the ion momentum distributions in k and k direc , tion are similar for transfer ionization and double capture. The authors conclude that asymmetric states, i.e. states where the main quantum number of the two captured electrons is very di!erent, are populated in both cases by a two-step process. In contrary for O> and N> the k and k distributions are di!erent between double capture and transfer ionization (see Fig. 51). For , double capture with successive autoionization the momenta are consistent with the capture channel to (n, n )"(3,3) and (3,4) in a one step process. For double capture very asymmetric states (2,'10) are observed. From the scattering-angle dependence one can conclude that these states are
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Fig. 51. Angular di!erential cross-sections for transfer ionization ("lled circles) and double capture ("lled triangles). The lines are drawn to connect the symbols. The projectile scattering angle has been measured via the transverse recoil ion momentum (from [100]).
populated via a capture plus excitation process which can be described within the independent electron model and a correlated double electron capture as discussed by Stolterfoht for O> impact [254] can be ruled out. The "rst experiment where the longitudinal recoil ion momentum has been measured in order to determine the Q-value of the reaction has been performed by Ali and coworkers [63]. They investigated the capture of up to 5 electrons in 50 keV Ar> on Ar collisions. Up to now it is completely impossible to calculate the details of such a many-electron transition processes from "rst principles. However, a classical over-the-barrier model which has been extended and intensively applied by Niehaus (for a review see [255]) has proven to successfully predict cross-sections, impact-parameter dependences, and "nal-state distributions of such transitions. Since COLTRIMS provides the Q-value and the scattering angle dependence for all reaction channels it allows also for a sensitive test of this model. Ali and coworkers found very good agreement between experimental and predicted mean Q-values for the above collision systems. A later systematical study of multiple capture by slow highly charged ions using the recoil technique has been reported by Raphaelian and coworkers [256].
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Abdallah and coworkers [257] have investigated single and double electron capture in Ar> on He collisions at velocities from 0.3 to 1.5 a.u. This study thus extends from the low velocity regime where energy matching between intial and "nal state are determine the main characteristics of the capture process into a transition region where velocity matching already becomes important. For single capture they "nd a widening of the reaction window (i.e. the energy range of states which are populated) with the square root of the projectile energy and a decreasing energy gain with increasing velocity. The scattering angles for a given Q-value is for the lowest velocity centered around half the coulomb angle (0 "Q/2E with E being the projectile laboratory energy (see A also [109]). This indicates an about equal probability of transfer on the way in and out. If the electron transfer occurs on the way in (out) scattering angles smaller (larger) than 0 can be A expected. At 1.5 a.u. velocity however the peak in the scattering angle distribution is shifted to signi"cantly larger angles than 0 . A The results for double capture at 1 a.u. are shown in Fig. 52. Fig. 52(a) shows the channel of true double capture (TDC), where both electrons are kept by the projectile, (b) is the channel of transfer ionization (TI). Here the capture to a doubly excited state has lead to autoionization. TI contributes about 70% to the total double capture cross-section. All double capture channels are found to peak at much larger scattering angles than 0 . This shows that the double capture does A not occur in a single transition in the crossing of the incident channel with the outgoing channel, but in a two steps of successive single transfers where at least one transition occurs on the way in. 4.4. Electron impact ionization The "eld of electron impact ionization was one of the "rst in which kinematically complete experiments have been performed. For single ionization very successful and comprehensive work by many groups has been reported (for reviews see e.g. [9}11]). In these so called (e,2e) experiments the two outgoing electrons are detected in coincidence. With this technique recently even (e,3e) experiments have been done [12}18]. In this "eld also one of the "rst experiments measuring recoil ion momenta has been performed by McConkey and coworkers for electron impact on He and Ne. They used a gas cell at room temperature and detected the recoil ion through a rotatable slit with a mass spectrometer. With this apparatus they integrated over the ion momenta and measured the angular distribution. In (e,2e) experiments two maxima in the angular distribution of one electron for a "xed angle of the other electron are found at low momentum transfers. One maximum is related to a binary collision of the projectile and target electron and a second from a backscattering of one of the electrons at the target nucleus (recoil peak). McConkey et al. [43] could support this interpretation by measuring the angular distribution of the recoiling ion. Recently, two experiments using COLTRIMS to investigate electron impact ionization have been reported. DoK rner et al. [66] have measured the He> momentum distribution for 130 eV electron impact to compare it to the ionization by electrons which are bound to a projectile. This has been discussed in Section 4.2.3. Jagutzki and coworkers have explored the use of COLTRIMS for (e,3e)-experiments [64,105,258]. They measured the recoil ion momentum distributions for single and double ionization of He by 270}3200 eV electron impact. For single ionization by 500 eV electrons they found the recoil ion momentum distribution (integrated over all emission angles) to be very close to the momentum distribution of the slow emitted electron (see Fig. 53).
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Fig. 52. Double electron capture in Ar> on He collisions at a projectile velocity of 1 a.u.. The horizontal axes shows the Q value of the collision, the vertical axis the projectile scattering angle. Both are measured via the momentum of the recoiling He> ion. (a) shows the true double capture channel, where both electrons are kept by the projectile. (b) shows the channel of transfer ionization, where the doubly excited projectile emits one electron by autoionization. The location of "nal states (n, n ) are shown in the "gure. The dashed line shows the location of 0 "Q/2E assuming a one step A transition (see the text). The open circles and squares indicate the de#ection angles calculated assuming a two step transition with n"7 and n"6, respectively, as the enabling "rst step single capture transition (from [257]).
This is consistent with the "ndings of numerous (e,2e)-experiments that these collisions are dominated by small momentum transfer from the fast projectile. The momentum distribution is very close to the Compton pro"le of the initial state, similar to the case of fast proton impact (see Section 4.1.2.2). For double ionization the He> momentum distribution is found to be broader by about a factor of 2}3. The two-dimensional momentum distribution (k versus k ) shows , a ridge along the kinematic line for a two-body collision of the projectile with the target nucleus. Thus, the He> momentum distribution is signi"cantly forward shifted compared to the He> ions. The distribution along the kinematic line indicates that double ionization is dominated by much closer impact parameters (between projectile and target nucleus) than single ionization. nCTMC calculation yield good agreement for the momentum distribution of the He> ions but does not
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Fig. 53. Recoil ion momentum distribution (integrated over emission angle) for single ionization of He by 500 eV electrons. Line: Experiment. Dashed line: nCTMC calculation (scaled by 2.9). The dots show the electron momentum distribution calculated from (e,2e)-data [259,260] (from [105]).
"nd the ridge along the kinematic line for double ionization. One reason may be that the nCTMC does not include the shake-o! and the interception process for double ionization. Complete (e,3e)-experiments require either the coincident detection of all three electrons [12}18] or the coincident detection of two electrons in addition to the recoiling ion. Such experiments are under way [138,139,17],
5. Experimental results for photon impact The latest "eld to which COLTRIMS has been applied are studies on photon-induced ionization. So far experiments on He and D targets in the energy range from the double ionization threshold (79 eV for Helium) to 90 keV have been reported. The work covers measurements of the ratio of double to single ionization by photoabsorption R"p>/p> from threshold to 400 eV [92], the separation of Compton scattering and photoabsorption between 9 and 100 keV [93,104,261], the determination of fully di!erential cross-sections for photoabsorption [70,115,116,123] and a study of the electron emission in double photoionization from "xed-in-space D molecules. In this section we "rst illustrate the role of the recoil ion for the photoabsorption process for He single ionization and show how this can be used to eliminate systematical errors for the determination of R at low photon energies (Section 5.1). We then show how fully di!erential cross-sections for He double photoionization can be obtained by detecting one electron in coincidence with the ion (Section 5.2). In Section 5.5 we discuss how the detection of the recoil ion momentum allows the separation of photo absorption from Compton scattering processes. The common theme of the COLTRIMS studies of photo double ionization is to explore the role of electron}electron correlations. Their motivation is therefore similar to that of many of the ion and electron impact studies reviewed in the previous sections. One might distinguish the role of correlation in the initial-state, in the "nal-state and dynamical correlation during the double ionization process. The latter has been termed scattering correlation and is often discussed in a simple picture of an electron}electron collision. The studies at low photon energies are mainly aiming at correlation in
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the "nal state and scattering correlation. High photon energies in turn are well suited for the investigation of initial state correlation since the electrons are so fast that any scattering on their way out of the atom is negligible (see [70] for detailed references). For a complete review of the rapidly expanding "eld of photoionization of atoms and molecules using synchrotron radiation the reader is referred to [262]. 5.1. He single ionization and the ratio of double to single ionization Momentum conservation requires, that the recoiling ion from single ionization induced by absorption of a single photon compensates for the continuum momentum of the emitted photoelectron, which carries almost all of the excess energy (see Eq. (13)). COLTRIMS allows for the "rst time to really visualize this basic fact of text book physics. It then exploits this fact to obtain detailed information on the process itself. Fig. 5 shows the momentum distribution of He> ions resulting from absorption of 80 eV linearly polarized photons, measured at the Advanced Light Source at LBNL. The "gure displays a slice through the three-dimensional momentum distribution in the plane perpendicular to the photon beam axis. The light from an undulator is linear polarized with the main axis of the polarization ellipse along the x-axis. The outermost ring results from He> ions in the ground state, the concentric rings result from ions in successive excited states. The momentum resolution in the x direction (which is the direction of the electric "eld of the spectrometer, where the momentum is measured via the TOF) is 0.09 a.u. corresponding to an energy resolution of 5 eV for the electrons of 55.8 eV leading to the n"1 state and about 0.6 eV for nPR at an energy of 1 eV. If one integrates the momentum distribution over all angles one obtains a width of 0.14 a.u. re#ecting not the local resolution but the linearity of the system. Contrary to traditional electron spectroscopy the full angular range (4p solid angle) is detected at one time. There is no need to rotate the spectrometer with respect to the beam axis and there is no range of nonaccessible angles. In the o!-line analysis the data can be sorted according to any angular and energy conditions in any coordinate system. For example Fig. 54 shows the ions sorted versus the cosine of their angle with respect to the polarization axis, for the n"1 and n"2 levels, respectively. The angular distribution of the electrons and hence the one of the recoil ions can be described by
3 p 1 dp(0, ) " 1#b (1!sin sin 0#S cos 0!S cos sin 0)! dX 4 4n 2
.
(25)
In the case of the Stokes parameter S "1 (fully linear polarized light) the equation simpli"es to
dp(0, ) 3 p 1 " 1#b cos 0! dX 2 4p 2
.
(26)
The n"1 level can be used to determine the Stokes parameter S , since the beta parameter is b"2 (we obtain S "0.99$0.01 for the data shown in Fig. 54). Using this value the b parameter for the higher n levels can be determined (see "gure). Qualitatively, the di!erence in the angular distributions for the higher n levels can already be seen in Fig. 5. For nPR the beta parameter becomes negative at 80 eV photon energy.
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Fig. 54. Count rate versus cos 0 of the ion with respect to the direction of the linear polarized light, for He> ions produced by 85 eV photon impact, for the n"1 and n"2 ionic state. The data are integrated over all azimuthal angles (from [70]).
By integrating over the momentum distributions for He> and He> ions one obtains the ratio R of the total cross-sections. Many experimental and theoretical studies have been devoted to the photon energy dependence of R. Close to threshold the double ionization cross-section rises with E as predicted by the Wannier threshold law [263]. R reaches a maximum at around 200 eV and slowly drops to an asymptotic value of 1.7%. The threshold behavior has been experimentally con"rmed by Kossmann and coworkers [264], the asymptotic value has only recently been established [265}267,93], it will be discussed in Section 5.5 in more detail. DoK rner and coworkers [92] have recently used COLTRIMS to determine precise values of R in the energy region from 85}400 eV. The results are shown in Fig. 55 together with most of the available experimental and theoretical results. The data of this work are about 25% lower than most of the previous experimental results, which have been obtained by detecting the time of #ight of the ions only. The application of COLTRIMS allows to eliminate all possible sources of experimental errors which have been discussed in the literature of this subject so far, since the charge state and the three-dimensional momentum vector as well as the pulse height of the channel-plate detector signal are recorded for each ion. For example, admixture of low-energy stray light or higher harmonics would show up as larger diameter rings in Fig. 5 as well as charge exchange by secondary collisions and electron impact ionization by secondary electrons. Also, any H> contribution to the He> would have been seen in the momentum distribution. For more details on this discussion see [92,261,268]. This revision of the established data of R at lower photon energy by the COLTRIMS technique has later been con"rmed by Samson and coworkers [269] (compare also [267]). As can be seen in Figs. 55(b) and (c), the new data have severe impact on the evaluation of the available theoretical results. They clearly favor the most recent calculations by Tang and
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Fig. 55. Ratio of double to single ionization for photoabsorption as a function of photon energy. Full circles: COLTRIMS data taken at the ALS, full squares: COLTRIMS data taken at Hasylab. The open circles in (b) are the same data as in (a) but scaled down by 1.3 (see text). V, A, L stand for results obtained in the velocity, acceleration or length form, respectively (from [92], see there for the references given in the "gure).
Shimamura [270] and Pont and Shakeshaft [271]. They are also in good agreement with calculations by Meyer and Green [272], Kheifets and Bray [273], Pindzola and Robicheaux [274] and Qui et al. [275]. Today for the full range of photon energies from threshold to the asymptotic regime the question of the ratio of total cross-section resulting from photoabsorbtion can be considered as settled. The calculations by Pont and Shakeshaft and those of Kheifets and Bray also provide fully di!erential cross-sections, angular and momentum distributions of the He> ions. This will be discussed in more detail in the following section.
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5.2. Multiple diwerential cross-sections for He photon double ionization For double ionization of He by photon impact a complete kinematical determination of the "nal state requires the coincident detection of "ve momentum components of any of the three particles in the "nal state (the 4 remaining momentum components are then given by energy and momentum conservation laws, see Section 2.3). The "rst pioneering complete experiment on He has been performed by Schwarzkopf and coworkers [19] by detecting the two electrons in coincidence. This kind of experiment can be performed much more e$ciently by measuring the recoil ion momentum in coincidence with one of the electrons. Results have been reported by DoK rner and coworkers [115]. They used a spectrometer as shown in Fig. 12 with a position-sensitive electron detector facing the recoil detector. Similar to many experiments for ion impact discussed above, the data obtained with COLTRIMS di!er from those obtained in coincident photoelectron detection in at least two ways which have more than technical importance. First, the complete momentum space (4p solid angle) for the ion and one of the electrons and are recorded in &event mode'. This means that the "nal momenta of all three particles are determined for every double ionization event, with no necessity to choose &a priori' a particular angle or energy for either electron. Thus, the entire "nal "ve-dimensional momentum space of the escaping 3 particles is sampled without prejudice, and the physical process itself determines which parts of this space are the most important ones. In principle, a similar measurement could be performed by measuring coincident photoelectrons for all angle and energy combinations, but this is an experimentally hopeless task with the solid angles of typical electron spectrometers. The overall data collection e$ciency using the recoil technique is orders of magnitude higher than that realized by the coincidence photoelectron technique. Second, the recoil momentum itself, which is equal and opposite to the center-of-mass momentum of the ejected electron pair, appears to be a particularly convenient coordinate for the description of the physical process at hand. The eventmode sampling of the entire "nal-state momentum space allows one to transform the data into any set of collective coordinates like hyperspherical or Jacobi coordinates. DoK rner and coworkers analyze their data in Jacobi coordinates k "(k #k ) (which is equal and opposite to the recoil ion momentum) and k "1/2(k !k ). k describes the motion of the electrons with respect 0 0 to the center of mass of the electron pair, while k gives the motion of this center of mass (or the Wannier saddle point of the electronic potential) in the laboratory frame (for more details see Refs. [276,277]). Experimental results for a photon energy of 1 eV above threshold are shown in Fig. 56. The recoil ion momentum distribution appears qualitatively dipole like in character, even so close to threshold. The b parameter for the ionic motion is found to be around 0.9 at 1 eV, increasing almost to b"2 at 80 eV above threshold. This is in qualitative agreement with two simple two-step models of double photoionization. In model (a) suggested by Samson [278] a "rst step of photo single ionization is followed by a second step of photo-electron impact ionization of He>. In this case the photon is absorbed in the "rst step by the charge dipole of the nucleus on one side and one electron on the other side. In model (b), which might be more plausible for very low photon energies, the photon is absorbed in the "rst step by the dipole formed by the nucleus on one side and the center of charge of the electron pair on the other side. In a second step this emitted &dielectron' breaks then up. From both models one would qualitatively expect the observed dipole pattern of the ionic emission pattern while the electrons lose this dipolar characteristics in the
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Fig. 56. Density plots of projections of the momentum spectra from double ionization of He by 80.1 eV photons. The z and y components of the momentum are plotted on the horizontal and vertical axes, respectively. The polarization vector of the photon is in the z-direction and the photon propagates in the x-direction. Only events with !0.1(k (0.1 a.u. are projected onto the plane. (a) shows he recoil ion (or !k ) momentum distribution. The outer V circle indicates the maximum calculated recoil ion momentum, and the inner circle is the locus of events for which the k motion has half of the excess energy. (b) shows the distribution of single electron momenta (k or k ). The circle locates the momentum of an electron which carries the full excess energy. (c) shows the relative electron momentum (or k ) 0 distribution. The circle identi"es the maximum possible value for k (from [115], see [123] for similar results 20 eV above 0 threshold).
second step due to the strong electron}electron correlation (see Fig. 56 and the work using traditional electron spectroscopy [279,280]). Fig. 56(c) shows the distribution of the Jakobi-momentum k "1/2(k !k ). This re#ects the 0 motion of the electrons along the interelectronic axis of the electron pair. The electron pair breaks up preferentially perpendicular to the polarization axis of the light which is horizontal in the "gure. This orientation of the interelectronic axis shows directly the preferred population of the state with a quantum number K"1 over the K"0 state, where K is the projection of the angular momentum along the Wannier ridge. The excess photon energy E is split between the k and k motion (E "k#k ). Fig. 57 0 0 demonstrates that as the photon energy approaches threshold a greater fraction of the excess energy goes into the breakup motion of the electron pair (k ), while the recoil ion motion is &cooled' 0 on the Wannier saddle (see [277] for a detailed discussion within 4th order Wannier theory). The
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Fig. 57. Cross-sections di!erential in kinetic energy plotted as a function of the fraction of the excess energy in the k (scaled recoil ion motion) ("lled circles) or k (breakup motion of the &dielectron') (open circles) motion. The solid curve 0 in the upper three "gures is from the fourth-order Wannier calculation. The theoretical curves are normalized to the experiment, which is on an absolute scale. Thick line in (c) calculation by Pont and Shakeshaft [281], which is on absolute scale. The symmetry of the curves is by de"nition, since both fractional energies have to add up to 1 (partly from [115]).
physical reason for this behavior is that for low excess energy only those ions close to the saddle end up in the double ionization channel. Thus, the apparent &cooling' is rather a selection of cold ions in the double ionization channel. In a recent &a priori' calculation Pont and Shakeshaft succeeded in reproducing the relative shape as well as the absolute height of the recoil ion momentum distribution [281]. Their results at 20 eV are shown in Fig. 57. In addition, their calculated b parameter for the ionic motion is in good agreement with the experiment. The COLTRIMS studies of fully di!erential cross-sections have been extended to 20 eV above threshold, by adding a solenoidal magnetic "eld to ensure 4p solid angle for all electrons up to
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10 eV. Surprisingly, BraK uning et al. found no qualitative change in the k -distribution compared to the data 1 eV above threshold [123]. We now change the perspective on double photoionization from collective Jacobi momentum coordinates to the more traditional point of view of single electron momenta. For He double ionization by a single photon the two electrons in the continuum have to be coupled to a PM state. The internal structure of the square of this correlated continuum wave function is shown in Fig. 58. Neglecting the (small) photon momentum the vector momenta of ion and both electrons have to be in one plane. Fig. 58 shows the electron momentum distribution in this plane. The data are integrated over all orientations of the polarization axis with respect to this plane, the x-axis is chosen to be the direction of one electrons. The structure of the observed momentum distribution is dominated by two physical e!ects. First the electron}electron repulsion leads to almost no intensity for both electron in the same half plane. Second, the PM symmetry leads to a node in the square of the wave function at the point k "!k [19,282}284]. At 1 eV this nodal point extends to all energy sharings (see in addition [24,70]). By omitting the integration over the orientation of the polarization, one obtains the conventional "vefold di!erential (FDCS) cross-sections dp/dh dh dU dU dE , where direction and energy of one electron is "xed and the polar angle of the second electron is plotted. Some authors name such cross-sections triply di!erential because they replace dh dh dU dU by dX dX . At low energies the linear momentum of the photon can be neglected (dipole approximation), yielding rotational symmetry around the photon polarization axes. This results in a reduction from a "vefold to a fourfold di!erential cross-section. Two comprehensive set of such FDCS obtained by COLTRIMS have been published [70,286]. Since these experiments are not restricted to any particular plane by the geometry of the detectors one obtains an overview of the correlated electron emission. This is illustrated in Fig. 59 where the
Fig. 58. Photo double ionization of He at 1 and 20 eV above threshold by linear polarized light. Shown is the momentum distribution of electron 2 for "xed direction of electron 1 as indicated. The plane of the "gure is the momentum plane of the three particles. The data are integrated over all orientations of the polarization axis with respect to this plane. The "gure samples the full cross-sections, for all angular and energy distributions of the fragments. The outer circle corresponds to the maximum possible electron momentum, the inner one to the case of equal energy sharing (from [285]).
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Fig. 59. dp/d0 d0 dE dU for photo double ionization of He by 85 eV photons. Horizontal axis: polar angle of fast electron with respect to the polarization axis. Vertical axis: azimuthal angle between the electrons. The energy of the "xed electron is E "0.1}1 eV, 0 "0}203, 0 "30}40, 0 "50}603, 0 "75}903(from top tobottom). The direction of the "xed electron is indicated by the dot. The dashed line shows the location of a node for the case of equal energy sharing. Left column: Experiment. Right column: 4th-order Wannier calculations (from [70]).
evolution of the angular distribution of the second electron for unequal energy sharing for the angle 0 varying from 03 to 903 with respect to the polarization axis for a photon energy 6 eV above the double ionization threshold is shown. Here the energy of the "rst electron is selected to be in 0.1}1 eV (thus the energy of the second electron is between 5 and 5.9 eV). The grey scale of the "gure is linear in the cross-section dp/d0 d0 dE dU. Fig. 59 demonstrates that at all angles 0 the FDCS is maximum for "1803. This correspond to the coplanar geometry to which all experiments by coincident electron}electron detection have been con"ned. The emission of the second electron changes from a cone for 0 close to 03 to one main lobe in the intermediate range of angles, which rotates with 0 . Finally, a second lobe grows as 903 is approached. It has been pointed out by
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several authors [191,282}284] that such FDCS are governed by strict selection rules, especially in the case of equal energy sharing. Maulbetsch and Briggs [283] have shown that for equal energy sharing there is a node at 0 "180!0 . The location of this node is marked with a line in Fig. 59. The node indicated in Fig. 58 is part of this line. For unequal energy sharing this selection rule does not hold strictly. At 6 eV excess energy, however, the selection rules still seem to govern the process even for an energy sharing of 1/5. The multiple di!erential cross-sections for photo double ionization obtained by COLTRIMS can easily be normalized to the absolute total cross-section for photoabsorption. This can be done since the total yield of He> and He> ions are always detected at the same time by the time-of-#ight measurement. Therefore, no knowledge about the total number of photons and the gas pressure is necessary for the normalization procedure [21]. For more details see [70,286]. Such absolute values are of great interest since the di!erent theoretical approaches for double photoionization disagree more on the absolute magnitude of the FDCS than their relative angular shape [287}289]. Fig. 60 shows the variation of the FDCS at 6 eV above threshold with energy sharing and azimuthal angle between the two electrons. The left column shows the coplanar geometry. When the second electron is pointing out of the plane de"ned by the "rst electron and the polarization vector (middle and right columns) the overall cross-section drops dramatically, with the main lobe at !2703 decreasing faster than the smaller lobe. As has been shown already by Lablanquie et al. [24] the shape of the FDCS is very similar for all energy sharings such close to threshold. 5.3. Circular dichroism in He photon double ionization If the double ionization process is induced by absorption of circularly polarized light (instead of linearly polarized), an interesting new questions arises: How is the helicity of the photon, which is absorbed by the spherical symmetric He ground state, transferred to the three-body Coulomb continuum? For photoabsorption by magnetic substances or chiral molecules it is well known that the di!erential cross-section depends on the helicity of the light (see [290] for a recent review). This dependence is termed circular dichroism (CD). Berakdar coworkers [291,292] have "rst theoretically predicted CD for photo double ionization of He. They have argued that in any geometry where the two electrons have unequal energy and are not in one plane with the photon direction, the FDCS can depend on the helicity of the photon. First experimental evidence for the existence of this e!ect has been seen by Viefhaus and coworkers [26]. They detected the two electrons in coincidence using time-of-#ight electron spectrometers. Mergel et al. [117] and Achler [293] have used COLTRIMS to map the angular and energy dependence of CD in detail for 20 and 95 eV excess energy. They have used a spectrometer similar to the one shown in Fig. 12 and detected the fast electron in coincidence with the recoil ion. The fast electron was selected by a retarding "eld in front of the electron detector. Their experiment has used circular polarized light from an elliptical undulator at the Photon Factory (KEK, Tsukuba, Japan). The measured momentum distributions for an excess energy of 20 eV are shown in Fig. 61. The photons propagate into the plane of the paper, the fast electron is "xed to the right, indicated by the arrow. The upper spectra show the He> momentum distribution. The lower spectra show the momentum distribution of the slow electron calculated from the data in the upper row event by event. Without the existence of CD these distributions would be symmetric with respect to the
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Fig. 60. dp/d(cos (0 ))d(cos (0 )) d dE photo double ionization of He by 85 eV photons. The "rst electron is emitted to a polar angle range of 0 "40}653. Each row is for a di!erent energy sharing, each column for varying azimuthal angle between the two electrons as given in the "gure. The dashed lines show a "t with a Gaussian correlation function with F=HM"84.73. Full line: Fourth-order Wannier calculation. The data are on absolute scale in 10\ a.u., the lines are scaled to the data (from [70]).
horizontal axis of k (similar to those shown for linear polarized light in Fig. 58). The data show, however, that CD is a very strong e!ect, the handedness of the photon changes the three-body breakup pattern dramatically.
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Fig. 61. Recoil ion and electron momentum distribution for photo double ionization of He by circularly polarized light at 20 eV excess energy. Shown is the plane perpendicular to the photon propagation (into the plane of the paper). Left row: p> (right helicity) photons, right row p\ (left helicity). The fast electron (11.5}20 eV) is "xed along the arrow (from [117]).
5.4. Electron emission from spatially aligned molecules The investigation of molecular fragmentation is a further domain for the three-dimensional momentum space imaging of RIMS. A special but interesting case occurs when the ionic energies resulting from molecular fragmentation are in the eV regime and, thus, much bigger than those from most of the atomic ionization processes discussed so far. In molecular fragmentation the recoil ion momentum is usually not mainly a result of the many-body momentum exchange between electrons and heavy fragments. In contrary to the atomic case the heavy fragment motion and the electron motion are fully decoupled. Mostly the axial recoil approximation holds [294]. It assumes that one can split the process into two independent subprocesses: First, an electronic transition in which the electrons are emitted or excited while the nuclei can be assumed to stay "xed and second, a much slower process of dissociation of the leftover molecular ion. The ionic momenta measured long after both steps result mainly from the Coulomb repulsion of the nuclei in the second step. Several groups have used position-sensitive detectors capable of handling multiple hits and projection spectrometers to measure the fragmentation pattern from this second step of
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molecular fragmentation. This work is beyond the scope of the present review. The reader is referred to [295,296] and references therein for work concerning ion impact and to [297] and references therein for work concerning photon impact. Only a few studies so far have tried to relate the two steps, i.e. the ejection of the electron(s) and the molecular fragmentation. Those studies combine electron and ionic fragment detection [129,298}302]. If the time scale for fragmentation is short compared to typical rotation times of the molecule, the ion direction can be taken as indication for the orientation of the molecule at the instant of the electronic excitation. Shigemasa et al. [299,301] and Watanabe et al. [302] have used traditional electron spectrometers and channeltrons for ion detection to measure the "rst angular dependence of photo electrons from "xed-in-space molecules. They were able to see direct evidence for a shape resonance in the electron angular distribution from N [299,303}306], CO [302] and CO [301]. Studies on the carbon K electron emission from CO have also been reported by Heiser and coworkers [298]. They used imaging with a position sensitive multihit detector for the fragment-ion detection. Golovin et al. [300] have measured the angular distribution of autoionization electrons from superexcited OH with respect to the molecular axis (see also [307]). They have used a projection spectrometer and a position-sensitive detector for the recoil ion momentum measurement and combined it with a simple time-of-#ight spectrometer without position resolution for electron detection. They found "rst evidence for an intra molecular scattering of the 0.16 eV electron at the second nucleus of the O . The ejection characteristics is shown in Fig. 62. For comparison with the He work reviewed above, double ionization of H , the corresponding molecular two-electron system, is of particular interest. Here, in principle much more complex electron angular distribution becomes possible [308], since angular momentum can be coupled to the nuclear motion. In addition an interference originating from electron emission from the two identical centers has been predicted [309,310]. Pioneering experimental work detecting the two fragment ions, integrating over the electrons can be found in [311,312] and the "rst experiments detecting both electrons in coincidence, but integrating over the ionic momentum distributions has been reported by two groups [27,313,314]. DoK rner et al. [129] have used a COLTRIMS setup with two-dimensional position-sensitive detectors and solenoidal magnetic "eld for 4p momentum
Fig. 62. Distribution of O>-ions coincident with 0.16 eV electrons from photoabsorption of O at a photon energy of 22.36 eV. The light propagates into the plane shown, the slow electron and the polarization are indicated in the "gure (from [300]).
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space mapping of both heavy fragments and one of the two electrons from photo double ionization of D . In this study a supersonic gas-jet target and a delay-line detector as shown in Fig. 18 is used for detection of both d> ions. They illustrated how the total available excess energy is partitioned between the four fragments. This leads to an electron-energy distribution with a smooth upper threshold, signi"cantly di!erent from double photoionization of an atom (e.g. Helium). In turn, the energy deposited in the electronic motion yields ion energies smaller than expected from pure Coulomb repulsion of the two nuclei following instantaneous removal of the electrons. Such behavior has been "rst predicted by LeRouzo [315,316]. The angular distribution of the electrons shows no reminiscence of the broken spherical symmetry in the molecule. Even so the electron angular distribution from spatially aligned molecules can in general not be described by Eq. (25). They found that for D double ionization close to threshold the energy integrated electron angular distributions could be well "tted with a b parameter. This b is found to be more positive for D than for He. In addition, the b parameter depends on the orientation of the molecular axis (see Fig. 63). 5.5. Separation of photoabsorption and Compton scattering It has been predicted as early as 1967 [317] that the ratio R of double to single ionization for photoabsorption will approach an asymptotic value of 1.7% for E PR. In calculating the A process of double photoionization most of the theoretical problems result from the electron}electron correlation in the "nal state. For high photon energy, however, this "nal state
Fig. 63. Polar representation of the angular distribution of one of the two photoelectrons from photo double ionization of helium at 7 eV excess energy and of D also at about 7 eV excess energy (cf. Fig. 1(e)). The data have been integrated over all electron energies. (a) dp/d cos 0 for helium with e along the horizontal. The line shows a "t with b "0$0.04. C C (b) dp/d cos 0 for D with e along the horizontal and the molecular axis held "xed parallel to e. The line shows a "t with C b "0.4$0.1. (c) Similar to (b) but for alignment of the molecule perpendicular to e. The data are integrated over all C azimuthal angles. The line shows a "t with b "0.14$0.08. (d) dp/d for D with 703(0 (1103 (903 is the plane of C C C the paper). The molecule is held "xed perpendicular to e, which now points out of the paper. The full line is a circle to guide the eye (from [129]).
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correlation becomes negligible and, thus, the asymptotic value of R is believed to be a one of the benchmark tests for our understanding electron}electron correlation in the initial ground state of the He. It tests the wave function at the Cusp condition (i.e. for one of the electrons at the nucleus). Only recently, with the availability of modern synchrotron light sources this question has been accessed experimentally. Levin and coworkers have performed the "rst experiment at 2.7 keV in the hope that the asymptotic value might have been reached already [265]. It has been pointed out by Samson and coworkers [318,319] that at 4 keV the contribution of Compton scattering to the He> production is comparable to the contribution from photoabsorption. Therefore, all experiments which detect only the charge state of the ion do not measure R for photoabsorption (R ) but N the mean value of R and R weighted with the corresponding cross-sections. Although agreement N A between various calculation has been reached on the predicted asymptotic value of R (see, e.g. N [320}327,289,177,275]) the calculated values for R di!er by more than a factor of 2 (see, e.g. A [328}331] and Fig. 65). COLTRIMS o!ers a natural way to distinguish for each ion, whether it resulted from a Compton scattering or a photoabsorption process, and, thus, allows to determine R and R separately N A [93,104,268,332,333,319]. As has been outlined in Section 5.1 momentum conservation for photoabsorption requires that the ion compensates the electron (sum-)momentum leading to large recoil ion momenta. In a Compton scattering event, however, the photon delivers momentum and energy to the electron in a binary collision leaving the nucleus a spectator to the process. Thus, the recoil ion will show mainly the initial-state momentum distribution (Compton pro"le). This is very similar to ionization by charged particle impact in the Bethe}Born limit, where the transition matrix is identical to the one obtained for photon scattering as pointed out by Bethe [334] as early as 1930. We have encountered this situation for single ionization by 0.5 MeV p impact (Section 4.1.2.2(a)), the process of projectile ionization by (e}e) interaction (see Section 4.2.3) and the process of (e}e) Thomas scattering in transfer ionization (see Section 4.2.2). All these reaction the maximum contribution to the cross-section results kinematical conditions with a minimum momentum transfer to the nucleus, the so called &Bethe ridge' [334]. The resulting momentum distributions of the recoil ions at around 9 keV photon energy from the pioneering work of Spielberger et al. [93] is shown in Fig. 64. In this experiment &broad-band' light from an undulator had to be used to obtain su$cient counting rate. The He> momentum distribution re#ects this energy spread. From Eq. (13) it can be seen that for photoabsorption the sphere of ion momenta is shifted forward by the momentum of the photon (p "2.4 a.u. at 9 keV), while this does not hold for A Compton scattering (this does not show up in Fig. 64 since both axis plotted are perpendicular to the photon beam propagation). By integrating the respective areas in momentum space one obtains the ratio R and R . Fig. 65 shows the measured ratios for Compton scattering. A N For photoabsorption at 9 keV very good agreement with the predicted value of 1.7% is found. The physical parameter to which the limes applies is not the photon energy, but the velocity with which the primary electron leaves its atom. For photoabsorption at high photon energy, photon and electron energy approach each other. Compton scattering, however, produces a broad distribution of electron energies. Thus, very high photon energies are needed in order to obtain mainly Compton electrons with high energies [320,330,338}341]. In a precision measurement Spielberger et al. explored the photon energy dependence of R up to 100 keV photon energy. They ! found a value of R "0.98$0.09 (see Fig. 65). In a theoretical analysis of these results the authors !
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Fig. 64. Momentum distribution of He ions from photon impact at about 9 keV. Upper figure: Momentum distribution of He> ions. The circular reef is due to photoabsorption, the narrow spike at zero momentum due to Compton scattering. The light propagates into the plane of the "gure, the polarization is along the x-axis. Middle: Distribution of He> ion momenta along the polarization axis, integrated over both other directions. Lower: As middle but for He> ions (from [93]).
showed that this value is still slightly in#uenced by a contribution of slow electrons and not yet the asymptotic value. These studies have clearly proven that the probability for shaking of the second electron di!ers signi"cantly whether the "rst electron is removed by photoabsorption or by Compton scattering.
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Fig. 65. Ratio of He>/He> ions created by Compton scattering as function of the photon energy. Full symbols: COLTRIMS data from [261,104,93] (circles,diamond,square). Open Circle: [266]. Open diamonds: [267]. Open square: [335]. Open triangle pointing up: [336]. Open triangle pointing down: [337]. Dashed line: Calculation (BBK-type wavefunction) [329]. Dotted line: [330], Full line: BBK-type "nal state [261]. Dashed-dotted line: [261]. The arrows indicate the predicted asymptotic values of upper arrow [328], lower arrow [329}331] (adapted from [261]).
The physical di!erences responsible for this can be seen in the recoil ion momentum distribution. While photoabsorption picks only that fraction of the wave function where one electron (and the ionic core) have large momenta (i.e. where the electron is found close to the nucleus), Compton scattering samples the whole momentum space wave function [339]. In both cases the fraction of the initial state wave function which is selected by the process is mirrored in the recoil ion momentum.
6. Outlook The investigation of many-particle atomic collision systems, where small momentum transfers between the collision partners dominate might be the most important application of COLTRIMS. To study the correlated motion of few electron systems in momentum and spin space, wherever the latter is possible too, is of fundamental interest. Here COLTRIMS can provide a momentum resolution which is about a factor hundred better than the mean momentum of the most weakly bound electron in any stable atom. Measuring the sum momentum of two or more ejected electrons or one electron momentum with respect to another given electron momentum in atomic systems, one can obtain detailed information on the correlated motion of electrons in atoms. In this section we will present some ideas of experiments in atomic and molecular physics as well as in other "elds, e.g. neutrino physics and surface science, which so far were not possible with existing detection techniques. This list is just an introduction to a new direction of research and it might stimulate other colleagues to use COLTRIMS in di!erent areas of physics or chemistry. (a) QED ewects in the inner-shell binding energies in H-like uranium ions: To determine QED e!ects, e.g. in the binding energy of the 1s state of H-like uranium ions an absolute energy determination of about 0.1 eV is desirable [5,342}344]. Di!erent attempts have been made, to
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measure these binding energies by detecting the innershell X-ray transition energies. Presently, however, no X-ray detection system is available which can provide the required precision at large enough solid angle. As outlined in Section 2.1.1 the 1s binding energy of U> ions can be determined by measuring the recoil ion longitudinal momentum in the following capture reaction: U>#HePU>(1s)#He>(1s) .
(27)
For decelerated ions with v "14 a.u. the He> longitudinal momentum is k "630 a.u. If this is measured with 0.05 a.u. precision one might obtain an overall Q-value resolution of 10 eV. Thus, the required 0.1 eV resolution for the center of its distribution seems to be feasible. Serious di$culties of the experiment are the very weak capture cross-section into the uranium 1s state, the small impact parameters probably contributing to 1s capture (large transverse recoil momenta) and the second He electron being very likely emitted during the capture. Work is in progress to explore the feasibility of such experiments. (b) Dynamics of correlated multi-electron processes in highly charged ion}atom collision; Resonant Electron Transfer-Excitation Process (RTE): It has been shown that one of the important electron transfer processes between fast ions colliding with atoms is due to the interaction of a projectile electron with a target electron. This process can occur, when both electrons share in a resonant way their energy yielding a resonant excitation of the projectile electron and simultaneous capture of a target electron. Due to the width of the target momentum distribution (Compton pro"le) the resonance as a function of the projectile velocity shows a broad distribution. Since no electron is freed in this process discrete values for the longitudinal recoil ion momenta will be found. The transverse recoil ion momentum distribution, however, will re#ect the full dynamics of this correlated process and allows to access the electron momentum during the transition (see [345}347]). (c) Multidiwerential studies of Compton scattering in momentum space: Di!erential studies of double ionization of He and other atoms and molecules (such as aligned H ) become feasible by measuring the slow shake-o! electron in coincidence with recoil ion momentum. This will provide a much more re"ned test of ground state correlation as total cross-sections. For aligned molecules and laser prepared atoms a coincident measurement of Compton scattered photons with the ion momentum gives access to three-dimensional Compton pro"les. This is analogue to the most advanced coincident Compton scattering studies at solids in which the scattered photon and the Compton electron are measured in coincidence [348]. (d) Molecular fragmentation induced by photons: Momentum space imaging of all ions and all electrons from photoionization and/or excitation is one of the most sensitive probes for molecular structure and chemical dynamics. With the high resolution of the available photon sources highly selective excitation of the molecules can be achieved and the imaging of all the fragment momenta with multi-hit detection devices allows in detail to investigate the internal dynamics in the molecule after the ionization/excitation. (e) Molecular fragmentation following ion impact: Slow highly charged ions are well suited for a soft removal of many electrons of a molecule leading to multiple fragmentation. Relativistic highly charged ion impact can be used also for inducing a complete fragmentation of molecules. Complete momentum space imaging of ionic fragments and electrons (as it has already been demonstrated for atoms by Moshammer and coworkers, see Sections 4.1.2.3 and 4.2.1 and references given there) gives access to electronic and geometric structure of the molecules. The high
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resolution for the ionic momenta achievable with COLTRIMS will also allow to address the fundamental question of the coupling of electronic and nuclear motion in a molecule. (f) Multiphoton double ionization: For double ionization of He in intense laser "elds so far only total rates can be measured (see [349] and references therein). For a sensitive probe of theoretical models, however, di!erential cross-sections for this process are highly desirable. The determination of the momentum distribution of the He> ions is the most promising candidate for di!erential cross-section measurements, since contrary to electron detection, the signal is already discriminated against single ionization. (g) EPR-which-way experiment in momentum space: Photo single ionization of a very cold He atomic beam by an extremely focussed photon beam produced by a free-electron laser (FEL) provides a very monoenergetic electron source with a long coherence length (see Fig. 5 for an example for the emission characteristics). Such electrons can create interference patterns behind a double slit device, due to the superposition of the two di!erent coherent scattering amplitudes. MoK llenstedt and DuK ker [350,351] have "rst demonstrated such interference by using a special electron gun and a biprisma interferometer they invented for the double slit. If the electrons are, however, produced by photoionization of very cold atoms in thin gas phase, each electron emission is accompanied by a recoiling ion which has exactly opposite momentum (see Section 2.3 and Fig. 5). Thus, measuring the recoil ion momentum in coincidence with the electron after the double slit, one can try to use the recoil ion direction to infer through which of the slits the electron was directed. In terms of the EPR paradoxon [352] the ion and the electron form an entangled state and this is an attempt to measure the momentum on one subsystem and the position on the other. This proposed experiment is a realization of a thought experiment which Einstein brought up already in his discussions with Bohr [353}355]. He suggested measuring the recoil momentum on a tiny hole which forms the source of a double slit experiment. An adaption of Bohr's famous refutation of Einstein's thought experiment in the present context shows that one would have to focus the photon beam such that its focal diameter times its transverse momentum are smaller than allowed by the uncertainty principle. A detailed quantum mechanical analysis of such an experiment can be found in [356]. (h) Angular correlation between electron and neutrino in the tritium- b-decay and measurement of the neutrino rest mass: One of the historically "rst recoil ion momentum measurements has been the experimental determination of the recoil e!ect of the neutrino from orbital electron capture decay of Ar to Cl by Rodenback and Allen [357] and its precursors (see for example [358}360]). For a beta decay leading to a free electron a precision measurement of the electron and nuclear recoil momentum together with the precise knowledge of the Q-value allows via energy and momentum conservation a determination of energy, momentum and emission angle of the neutrino for each decay. Since energy and momentum of the neutrino are measured, one can deduce in principle the neutrino rest mass for each single neutrino. This measurement does not depend on a statistical evaluation of a spectrum like measurements of the Curie-plot-endpoint (see, e.g. [361] and references therein) or on model assumptions. The most suitable would be the tritium decay. If a TH or T molecular gas target can be cooled by laser cooling or other techniques to approximately 10 l K temperature the momentum of the recoiling HHe> molecule can be measured by a su$ciently large spectrometer device (about 5 m length) with approximately 0.001 a.u. precision. The absolute momentum vector is obtained by detecting the recoil with a large position-sensitive detector, which measures the TOF and the emission angle. All recoil ions can be projected by
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proper "eld con"gurations on such a detector yielding nearly 4p solid angle. The electron momentum vector has to be determined with equivalent high precision too, which might be achieved by collecting all electrons in a long solenoidal "eld con"guration and projecting them on a second position sensitive channel-plate detector. The TOF of electron and recoil might be obtained by detecting the Lyman photon emission of the excited He>-ion in coincidence with the recoil ion and the electron. To yield the angular e\l correlation only the electron position has to C be measured. For the above given resolution the angular resolution between both particles can be as good as 10\ rad. Estimates for the neutrino mass resolution yield about 10 eV/c for each single detected event (if the electron energy is within 50 eV to the upper limit). (i) Laser controlled targets: The implementation of laser prepared and laser cooled targets for COLTRIMS opens a wide "eld of applications in atomic collision physics. Compared to gas-jet targets much lower internal temperature and thus higher momentum resolution is obtainable with such targets. This was recently demonstrated in pioneering work by Wolf and coworkers [362]. The target density achievable in MOT traps is already today high enough to allow for collision experiments with external beams. In addition to the improved resolution such targets open the way to multi-di!erential studies of charged particle and photon interaction with excited and specially prepared states. For example for reactions with Rydberg atoms highly di!erential cross-sections can be studied. Such experiments are in preparation in several laboratories. ( j) Imaging of surface ionization: The adaption of the COLTRIMS ion imaging technique to electron imaging as outlined in Section 3.5 has a wide application for the study of single and multiple electron emission from surfaces (see Ref. [124] for an imaging spectrometer for surface studies). One example is the correlated emission of two electrons by absorption of one photon from a surface (see, e.g. [363}365] and references therein). Such studies will strongly pro"t from the large solid angle of imaging spectrometers. They can yield information on the correlated motion of electrons in solids. Equally interesting is the electron emission from solids by charged particle impact. In a "rst multi coincidence study Moshammer and coworkers detected up to 10 electrons emitted by impact of one single fast projectile. Besides electrons from surfaces also ions released from surfaces can be observed by such imaging techniques. In particular sputtered ions emitted from slow highly charged ion impact on surfaces are an interesting candidate new types of microscopes. (k) QED ewects in 80 GeV/u U on Au collisions: Theory has predicted that in ultrarelativistic heavy-ion heavy-atom collisions at small internuclear distances (R ) the pair creation probability can exceed unity. Thus, the QED calculations cannot be based on perturbation theory and must include the higher order e!ects in an accurate way. The regime of strong perturbation (i.e. small (R )) can experimentally only be explored, if the pair creation is measured for a given projectile scattering angle. In these encounters the angles of interest are below 10\ rad and, therefore, not resolvable with traditional detection devices. Measuring the recoil ion transverse and longitudinal momentum components complete information on the scattering angle and on the pair creation is obtained. Since for the collisions of interest the recoil momenta are several 10 a.u. even thin solid targets can be used for these measurements. We could only present here a biased and incomplete list of the exciting future prospects of COLTRIMS and related new imaging techniques. We tried to give a #avor of the huge potential of this "eld, which is compared to traditional electron or photon spectroscopy still very young. The recent developments started with drift-time measurements of ions in "eld-free gas cells. In a series
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of innovations cold gas cells, e!usive gas jets, supersonic jets, precooled supersonic jets and in a "rst experiment even laser trapped atoms were used as target. The spectrometer development went from early drift tubes to projection "elds involving three-dimensional focussing for ions and magnetic con"nement for electrons. These latest combined electron-ion imaging devices are equipped with multi-hit detectors for multiple-electron and multiple-ion detection. This rapid technical development in the "eld reviewed here went along with an even more impressive widening of the scope of fundamental physical problems tackled. Today momentum space studies already cover electron impact, photon impact from threshold to 100 keV and ion impact from keV protons to GeV/u U> projectiles. The unprecedented resolution and completeness of many of those investigations allowed the resolution of some long-standing puzzles in atomic collision physics but at the same time raised even more fundamental questions. Similar impact of such imaging techniques can be expected for the future for other "elds in physics, chemistry and related areas.
Acknowledgements Many people have been involved in the development of COLTRIMS reviewed here. Most of the experimental work and technical development has been done, like mostly in todays experimental physics, by collaborators who left academic physics after they received their Diploma or Ph.D. They are to numerous to mention all of them. The work was driven by a longstanding close collaboration between several experimental and theoretical groups. Here we are particularly grateful to our friends and colleagues C.L. Cocke, R.E. Olson, M.H. Prior, J. Feagin, W. Schmitt, Th. Weber, Kh. Khayyat, H. BraK uning, M. Achler, J. McGuire, T. Kambara, Y. Awaya, Y. Azuma, R. Dreizler, S. Keller, H.J. Ast, H.J. LuK dde, A. Cassimi, A. Lahmam-Bennani, R. Ali, U. Buck, V. Schmidt, B. KraK ssig, M. Schulz, D. Gemmel, S. Hagmann, J. BurgdoK rfer, C. Wheelan, R. Shakeshaft, Dz. Belkic, B. Sonntag and to C. Freudenberger for the help with many "gures of this review. We acknowledge "nancial support from BMBF, DFG, GSI, the Max Planck Forschungspreis, Alexander von Humboldt Stiftung, the Habilitandenprogramm der DFG, DFG within SFB276 (project B7,B8), DOE, DAAD, Studienstiftung des deutschen Volkes (V.M.), &Freunde und FoK rderer der UniversitaK t Frankfurt' and the GraduiertenfoK rderung des Landes Hessen.
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GOLDSTONE AND PSEUDO-GOLDSTONE BOSONS IN NUCLEAR, PARTICLE AND CONDENSED-MATTER PHYSICS
C.P. BURGESS Department of Physics, McGill University, 3600 University Ave., Montreal, PQ, M3A 2T8, Canada
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Goldstone and pseudo-Goldstone bosons in nuclear, particle and condensed-matter physics夽 C.P. Burgess Department of Physics, McGill University, 3600 University Ave., Montreal, PQ, M3A 2T8, Canada Received September 1999; editor: J.A. Bagger Dedicated to Caroline, Andrew, Matthew, Ian and Michael
Contents 1. Goldstone bosons 1.1. Introduction 1.2. Noether's theorem 1.3. Goldstone's theorem 1.4. Abelian internal symmetries 1.5. Nonabelian internal symmetries 1.6. Invariant lagrangians 1.7. Uniqueness 1.8. The geometric picture 1.9. Nonrelativistic lagrangians 1.10. Power counting 1.11. The e!ective-lagrangian logic 2. Pions: a relativistic application 2.1. The chiral symmetries of QCD 2.2. The low-energy variables 2.3. Invariant lagrangians 2.4. Explicit symmetry-breaking 2.5. Soft pion theorems
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3. Magnons: nonrelativistic applications 3.1. Antiferromagnetism: ¹ invariance 3.2. Ferromagnetism: ¹ breaking 4. SO(5) invariance and superconductors 4.1. SO(5) symmetry 4.2. The e!ective lagrangian in the symmetry limit 4.3. Symmetry-breaking terms 4.4. Pseudo-Goldstone dispersion relations 4.5. Summary 5. Bibliography 5.1. Review articles 5.2. Particle physics data 5.3. SO(5) invariance in high-¹ superconductors 5.4. Condensed matter physics for particle and nuclear physicists Acknowledgements References
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夽 `E!ective Theories in Mattera Presented at Nuclear Physics Summer School and Symposium, Seoul National University, Korea. E-mail address: [email protected] (C.P. Burgess)
0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 1 1 - 8
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Abstract It is a common feature of many physical systems that their behaviour is relatively simple when examined only at low energies (or temperatures) compared to the system's own characteristic scales. It often happens that there are relatively few states which can participate in low-energy processes, and their interactions can sometimes become less and less important the lower the energies that are examined. Very general theoretical tools exist to exploit this simplicity, when it arises. Systems with spontaneously broken symmetries (i.e. for which a symmetry of the hamiltonian is not also a symmetry of its ground state) form a very important class of examples of this type, due to the Goldstone bosons which inevitably appear in their low-energy spectrum. This review develops the theory of Goldstone bosons, concentrating on their description in terms of an e!ective lagrangian formulation. 2000 Elsevier Science B.V. All rights reserved. PACS: 11.30.Qc; 11.15.Tk; 11.10.!z; 74.20.!z; 75.30.Ds
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1. Goldstone bosons Goldstone bosons are weakly coupled states which appear in the low-energy spectrum of any system for which a rigid (or global) symmetry is spontaneously broken (that is, the symmetry is not preserved by the system's ground state). A great deal is known about the properties of these bosons, since at low energies their properties are largely governed by the nature of the symmetries which are spontaneously broken, depending only weakly on the details of the system itself. This review is devoted to explaining the modern ewective lagrangian method for identifying Goldstone boson properties. These methods are much more e$cient than are the older currentalgebra techniques of yore.
1.1. Introduction It is a common feature of many physical systems that their behaviour is relatively simple when examined only at low energies (or temperatures) compared to the system's own characteristic scales. It often happens that there are relatively few states which can participate in low-energy processes, and their interactions can sometimes become less and less important the lower the energies that are examined. Very general theoretical tools exist to exploit this simplicity, when it arises. One such tool is the technique of e!ective lagrangians. The guiding idea for this method is the belief, "rst clearly enunciated by Weinberg, that there is no loss of generality in using a "eld theory to capture the low-energy behaviour of any system. This is because "eld theory in itself contains very little content beyond ensuring the validity of general &motherhood' properties like unitarity, cluster decomposition, and so on. According to this point of view, if a "eld theory is identi"ed which is the most general consistent with the low-energy degrees of freedom and symmetries of any particular system (together with a few &motherhood' properties) then this "eld theory must provide a good description of the system's low-energy limit. This is a particularly useful observation when the low-energy degrees of freedom are weakly interacting (regardless of how strongly interacting their higher-energy counterparts might be), because then the resulting "eld theory may be simple enough to be used to predict explicitly the system's low-energy properties. This simplicity is somewhat paradoxical since, as we shall see, low-energy e!ective lagrangians are typically very complicated, involving all possible powers of the various "elds and their derivatives. Simplicity is achieved in spite of the complicated e!ective lagrangian because, for weakly coupled theories, general power-counting arguments exist which permit an e$cient identi"cation of the comparatively few interactions which appear at any given order in a low-energy expansion. Remarkably, there turns out to be a very important situation for which very general results are known concerning the existence of very light degrees of freedom whose low-energy interactions are weak. This occurs whenever a continuous global symmetry is spontaneously broken (i.e. which is a symmetry of the hamiltonian but not a symmetry of the ground state), since when this happens Goldstone's theorem guarantees the existence of low-energy Goldstone bosons, as well as determining a great deal about their interactions. These theorems, and their description in terms of an e!ective lagrangian formulation, are the subject of this review.
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1.1.1. A road map This section outlines how the material covered in this review is organized. 1. General formalism: All of the general results may be found in Section 1, starting with a statement of the key theorems } those of Noether and Goldstone } which underlie everything else. This is followed by a motivational discussion of the simplest example to which Goldstone's theorem applies. Although the properties of the Goldstone bosons are guaranteed by general theorems, the moral of the example is that these properties are generally not manifest in a low-energy e!ective lagrangian unless a special choice of variables is used. These variables are identi"ed and exploited "rst for spontaneously broken abelian internal symmetries, and then the process is repeated for nonabelian internal symmetries. Both lorentz-invariant and nonrelativistic systems are considered. For the nonrelativistic case, special attention given to the breaking of time reversal, since this qualitatively a!ects the nature of the low-energy e!ective lagrangian. The spontaneous breaking of spacetime symmetries, like rotations, translations and lorentz transformations, is not discussed in this review. 2. Applications: Sections 2}4 are devoted to speci"c applications of the methods of Section 1 to examples in high-energy/nuclear and condensed-matter physics. Section 2 starts with the classic relativistic example of pions as pseudo-Goldstone bosons, whose study gave birth to many of the techniques described in Section 1. (A pseudo-Goldstone boson is the Goldstone boson for an approximate symmetry, as opposed to an exact symmetry.) This is followed in Section 3 by a study of spin waves (magnons) in both ferromagnets and antiferromagnets. Section 4 then closes with a recent, more speculative, application of these ideas to the SO(5) proposal for the high-temperature superconductors. 3. Bibliography: Finally, Section 5 contains a brief bibiography. It is not meant to be exhaustive, as a great many articles have emerged over the past decades of applications of these methods. I therefore restrict myself to listing those papers and reviews of which I am most familiar. I apologize in advance to the authors of the many excellent articles I have omitted. The review is aimed at upper-year graduate students, or practicing researchers, since it presupposes a familiarity with quantum "eld theory. It was written with an audience of high-energy and nuclear phycisists in mind, and so for the most part units are used for which "c"1. However, I hope it will prove useful to condensed-matter physicists as well. Enjoy! 1.2. Noether's theorem We start with a statement of Noether's theorem, since this plays a role in the statement of Goldstone's theorem, which is the main topic of this section. For a "eld theory Noether's theorem guarantees the existence of a conserved current, jI, for every global continuous symmetry of the action. To low orders in the derivative expansion it is usually enough to work with actions which depend only on the "elds and their "rst derivatives, so we restrict our statement of the theorem to this case. Consider therefore a system governed by an action S"dx L( , R ), where (x) generically I denotes the "elds relevant to the problem. We imagine that S is invariant under a set of transformations of these "elds, d "m ( )u?, where u? denote a collection of independent, ? spatially constant symmetry parameters. Invariance of S implies that the lagrangian density, L,
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must vary at most into a total derivative: dL,R (u?
I I The statement of the theorem follows from this last equation. It states that the quantities
RL jI ,! m #
(1.2.3)
R jI "0 , (1.2.4) I ? when they are evaluated at a solution to the equations of motion for S } i.e. on "eld con"gurations for which the square bracket on the right-hand side of Eq. (1.2.2) vanishes. Even though we have used relativistic notation in this argument, the conclusion, Eq. (1.2.4), is equally valid for nonrelativistic systems. For these systems, if we write o "j for the temporal ? ? component of jI , and denote its spatial components by the three-vector j , then current conserva? ? tion (Eq. (1.2.4)) is equivalent to the familiar continuity equation Ro /Rt# ) j "0 . (1.2.5) ? ? Eq. (1.2.4) or Eq. (1.2.5), are called conservation laws because they guarantee that the charges, Q , ? de"ned by
dr o (r, t)" dr j(x) , (1.2.6) ? ? R are conserved in the sense that they are independent of t. These charges are sometimes called the generators of the symmetry because their commutator with the "elds give the symmetry transformations themselves Q (t)" ?
iu?[Q , (x)]"u?m "d . (1.2.7) ? ? The existence of such a conserved current carries special information if the symmetry involved should be spontaneously broken, as we now describe. 1.3. Goldstone's theorem Whenever the ground state of a system does not respect one of the system's global continuous symmetries, there are very general implications for the low-energy theory. This is the content of
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Goldstone's theorem, which we now present. This theorem is central to the purpose of this section, which is devoted to making its implications manifest in a low-energy e!ective theory. Goldstone's theorem states that any system for which a continuous, global symmetry is spontaneously broken, must contain in its spectrum a state, "G2 } called a Goldstone mode, or Goldstone boson since it must be a boson } which has the de"ning property that it is created from the ground state by performing a spacetime-dependent symmetry transformation. In equations, "G2 is de"ned by the condition that the following matrix element cannot vanish: 1G"o(r, t)"X2O0 .
(1.3.1)
Here, "X2 represents the ground state of the system, and o"j is the density for the conserved charge } guaranteed to exist by Noether's theorem } for the spontaneously broken symmetry. Before turning to its implications, we outline the proof of this result. The starting point is the assumption of the existence of a local-order parameter. This can be de"ned to be a "eld, (x), in the problem which satis"es two de"ning conditions. Firstly, it transforms nontrivially under the symmetry in question: i.e. there is another "eld, t(x), for which: dt,i[Q, t(x)]" (x) .
(1.3.2)
Q is the conserved charge de"ned by integrating the density o(r, t) throughout all of space. Secondly, the "eld must have a nonzero expectation in the ground state: 1 2,1X" (x)"X2,vO0 .
(1.3.3)
This last condition would be inconsistent with Eq. (1.3.2) if the ground state were invariant under the symmetry of interest, since this would mean Q"X2"0, implying the right-hand side of Eq. (1.3.3) must vanish. The proof of the theorem now proceeds from the following steps. (i) Substitute Eq. (1.3.2) into Eq. (1.3.3); (ii) Use Q"o dr in the result, as is guaranteed to be possible by Noether's theorem; (iii) Insert a partition of unity as a sum over energy eigenstates, 1" "n21n", on either side of the L operator o. The resulting expression shows that if no energy eigenstate exists which satis"es the de"ning condition, Eq. (1.3.1), then the right-hand side of Eq. (1.3.3) must vanish, in contradiction with the starting assumptions. This proves the theorem. We next elaborate on its consequences. The de"ning matrix element, Eq. (1.3.1), and the conservation law, Eq. (1.2.5), together imply that Goldstone bosons must have a number of important properties. Besides determining their spin and their statistics, it implies two properties which are of particular importance: 1. The Goldstone boson must be gapless, in that its energy must vanish in the limit that its (three-) momentum vanishes. That is: lim E(p)"0 . N
(1.3.4)
Supersymmetry is an exception to this statement, since spontaneously broken global supersymmetry ensures the existence of a Goldstone fermion, the goldstino. We use nonrelativistic notation here to emphasize that the conclusions are not speci"c to relativistic systems. This will prove useful when nonrelativistic applications are considered in later sections.
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To see why this follows from Eq. (1.3.1), it is helpful to make the dependence on position and time in this equation explicit by using the identities o (r, t)"e\ &Ro (r, 0)e &R and ? ? j (r, t)"e P rj (0, t)e\ P r, together with the energy- and momentum-eigenstate conditions: ? ? H"X2"P"X2"0, P"G(p)2"p"G(p)2 and H"G(p)2"E "G(p)2. Then, di!erentiation of Eq. (1.3.1) N with respect to t, and use of the continuity equation, Eq. (1.2.5), gives: !iE e\ #N R1G"o (r, 0)"X2"1G"(Ro /Rt)(r, t)"X2 N ? ? "!1G" ) j (r, t)"X2 ? "!i p ) 1G" j (r, t)"X2 . ?
(1.3.5)
Eq. (1.3.4) follows from this last equality in the limit pP0, given that the matrix element, 1G"o (r, 0)"X2, cannot vanish by dexnition for a Goldstone boson. ? In relativistic systems, for which E(p)"(p#m where m is the particle mass, the gapless condition, Eq. (1.3.4), is equivalent to the masslessness of the Goldstone particle. 2. More generally, the argument just made can be extended to more complicated matrix elements. One "nds in this way that the Goldstone boson for any exact symmetry must completely decouple from all of its interactions in the limit that its momentum vanishes. Physically, this is because Eq. (1.3.1) states that in the zero-momentum limit the Goldstone state literally is a symmetry transformation of the ground state. As a result it is completely indistinguishable from the vacuum in this limit. These properties have a lot of implications for the low-energy behaviour of any system which satis"es the assumptions of the theorem. The "rst guarantees that the Goldstone boson must itself be one of the light states of the theory, and so it must be included in any e!ective lagrangian analysis of this low-energy behaviour. The second property ensures that the Goldstone mode must be weakly coupled in the low-energy limit, and strongly limits the possible form its interactions can take. The properties of gaplessness and low-energy decoupling can also be useful even if the spontaneously broken &symmetry' in question is really not an exact symmetry. To the extent that the symmetry-breaking terms, H , of the system's Hamiltonian are small, the symmetry may be regarded as being approximate. In this case the violation of the gapless and decoupling properties can usefully be treated perturbatively in H . The Goldstone particles for any such approximate symmetry } called pseudo-Goldstone bosons } are then systematically light and weakly coupled at low energies, instead of being strictly massless or exactly decoupled. The purpose of the remainder of this section is to show in detail how these properties are encoded into the low-energy e!ective lagrangian. By considering simple examples we "nd that although these properties are always true, they need not be manifest in the lagrangian in an arbitrary theory. They can be made manifest, however, by performing an appropriate "eld rede"nition to a standard set of "eld variables. We "rst identify these variables, and use them to extract the implications of Goldstone's theorem for the low-energy e!ective theory in the simplest case, for which the symmetry group of interest is abelian. The results are then generalized in subsequent sections to the nonabelian case.
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1.4. Abelian internal symmetries In order to see the issues which are involved, it is instructive to consider a simple "eld theory for which a symmetry is spontaneously broken. We therefore "rst consider a simple model involving a single complex scalar "eld, . 1.4.1. A toy example The lagrangian density: L"!R HRI !<( H ) , I with <"(j/4)( H !k/j)
(1.4.1)
is invariant with respect to a ;(1) group of symmetries: Pe ? . This is a global symmetry because the term involving derivatives of is only invariant if the symmetry parameter, a, is a constant throughout spacetime. It is called an internal symmetry since the symmetry acts only on "elds and does not act at all on the spacetime coordinate, xI. For later reference, the Noether current for this symmetry is j "!i( HR ! R H) . I I I
(1.4.2)
For small j this system is well approximated by a semiclassical expansion, provided that the "eld
is O(j\ ) in size. This may be seen by rede"ning " I /(j, and noticing that all of the j-dependence then scales out of the lagrangian: L( , k, j)"L( I , k, 1) } for which the limit jP0 H is seen to be equivalent to P0 in the semiclassical limit. The vacuum of the theory is therefore well described, for small j, by the classical con"guration of minimum energy. Since the classical energy density is a sum of positive terms, H" Q H Q #
H )
#<( H ), it is simple to minimize. The vacuum con"guration is a constant throughout spacetime, Q "
"0, and its constant value, "v, must minimize the classical potential: <(vHv)"0. We may use the ;(1) symmetry to choose v to be real, and if k is positive then the solution becomes v"k/(j. Happily this con"guration lies within the conditions of validity of this semiclassical analysis. Since the vacuum con"guration, "vO0, is not invariant under the ;(1) transformations,
Pe ? , the ;(1) symmetry is seen to be spontaneously broken. Goldstone's theorem should apply, and so we now identify the Goldstone degree of freedom. The spectrum may be identi"ed by changing variables to the real and imaginary parts of the deviation of the "eld from its vacuum con"guration. De"ning R,(2 Re ( !v) and I,(2 Im diagonalizes the kinetic and mass terms, and the scalar potential in terms of these variables becomes m g g g g g <" 0 R# R# RI# R# RI# I , 2 3! 2 4! 4 4!
(1.4.3)
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Fig. 1. The Feynman graphs which describe R}I scattering at tree level. Solid lines denote R and dashed lines represent I.
where the couplings and masses in this potential are given in terms of the original parameters, j and k, by g g jv g g g j " " " " " . m "jv, , (1.4.4) 0 3! 16 2 4! 4! 8 2(2 Notice the existence of a massless "eld, I, as is required by Goldstone's theorem. We can verify that I really is the Goldstone boson by writing the Noether current, Eq. (1.4.2), in terms of the mass eigenstates, R and I: j "v(2R I#(RR I!IR R) . I I I I Clearly the matrix element:
(1.4.5)
1I(p)" jI(x)"02Jv(2pIe\ NV
(1.4.6)
does not vanish (unless v"0), as is required of a Goldstone boson. A puzzle with the potential of Eqs. (1.4.3) and (1.4.4) is that the Goldstone boson, I, appears in the scalar potential, and so its couplings do not appear to vanish in the limit of vanishing momentum. This is only an appearance, however, and I really does decouple at low energies, as can be tested by computing Goldstone boson scattering in this limit. For example, the S-matrix at tree level for I}R scattering may be computed by evaluating the Feynman graphs of Fig. 1. The result is iAd(r#s!r !s ) , S[R(r)#I(s)PR(r )#I(s )]" (2p)(16srs r
(1.4.7)
with
1 g g 1 A"!g # #g # . (s!s )#m !ie (s#r)!ie (s!r )!ie 0 In the limit sI, s IP0 this becomes (using the condition r"r "!m ): 0 AP!g #(g g )/m !2g /m , 0 0 1 3 "j ! # !1 "0 . 2 2
(1.4.8)
(1.4.9)
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The scattering amplitude indeed vanishes in the zero-momentum limit, as it must according to Goldstone's theorem. This vanishing is not manifest in the lagrangian, however, and is only accomplished through a nontrivial cancellation of terms in the S-matrix. For many purposes, not least when constructing an e!ective theory to describe the low-energy interactions of the Goldstone bosons, it would be preferable to have this decoupling be manifest in the lagrangian. We will now do so, by making a "eld rede"nition to a new set of variables for which decoupling becomes explicit. 1.4.2. A better choice of variables In order to identify which variables would make the decoupling of Goldstone bosons more explicit in the lagrangian, it is useful to recall the de"nition of what the Goldstone mode physically is. Its de"ning condition, Eq. (1.3.1), can be interpreted to mean that the Goldstone modes are obtained from the ground state by performing a symmetry transformation, but with a spacetimedependent transformation parameter. In the example considered in the previous section the ground state con"guration is "v, and so a local symmetry transformation of this ground state would be "ve FV. If this is substituted into the lagrangian of Eq. (1.4.1), we "nd L( "ve FV)"!vR hRIh. h does not drop out of the I problem because, although the lagrangian vanishes when it is evaluated at "v, the con"guration ve FV is only related to "v by a symmetry when h is a constant. This fact that h parameterizes a symmetry direction when it is restricted to constant "eld con"gurations guarantees that any h-dependence of L must involve at least one derivative of h, thereby dropping out of the problem in the limit of small derivatives } i.e. small momenta, or long wavelengths. All of this suggests that h would make a good representation for the Goldstone mode, since this is precisely what a Goldstone mode is supposed to do: decouple from the problem in the limit of small momenta. We are led to the suggestion of using polar coordinates in "eld space,
(x)"s(x)e FV ,
(1.4.10)
in order to better exhibit the Goldstone boson properties. In this expression both h and s are de"ned to be real. Substituting this into the lagrangian gives L"!R sRIs!sR hRIh!<(s) . (1.4.11) I I It is clear that these variables do the trick, since the fact that h appears in the de"nition, Eq. (1.4.10), in the same way as does a symmetry parameter guarantees that it completely drops out of the scalar potential, as must a Goldstone boson if its low-energy decoupling is to be made explicit. A price has been paid in exchange for making the low-energy decoupling of the Goldstone boson explicit, however. This price is most easily seen once the "elds are canonically normalized, which is acheived by writing s"v#(1/(2)s and h"(1/v(2)u. With these variables the lagrangian is seen to have acquired nominally nonrenormalizable interactions:
s s L "! # R uRIu . I 4v (2v
(1.4.12)
Of course, the S-matrix for the theory in these variables is identical to that derived from the manifestly renormalizable lagrangian expressed in terms of the variables R and I. So the S-matrix
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remains renormalizable even when computed using the variables s and u. (The same is not true for ow-shell quantities like Green's functions, however, since the renormalizability of these quantities need not survive a nonlinear "eld rede"nition.) In this toy model there is therefore a choice to be made between making the lagrangian manifestly display either the renormalizability of the theory, or the Goldstone boson nature of the massless particle. Which is best to keep explicit will depend on which is more convenient for the calculation that is of interest. Since, as we shall see, renormalizability is in any case given up when dealing with e!ective low-energy "eld theories, it is clear that the variables which keep the Goldstone boson properties explicit are the ones of choice in this case. 1.4.3. The general formulation The reason why the above rede"nition works may be seen by asking how the ;(1) symmetry acts on the new variables. The key observation is that the symmetry transformation becomes inhomogeneous: hPh#a, where a is a constant. In terms of the canonically normalized "eld, u, this transformation law becomes uPu#(2va .
(1.4.13)
This kind of transformation rule is the hallmark of a Goldstone boson, since it enforces the explicit nature of all of the Goldstone boson properties in the lagrangian. In fact } as can be expected from the generality of Goldstone's theorem } they can all be derived purely on the grounds of this symmetry transformation, and do not rely at all on the details of the underlying model which motivated its consideration. To show that this is true, imagine writing an arbitrary e!ective theory for a real scalar "eld, u, subject only to the symmetry of Eq. (1.4.13) (and, for simplicity, to PoincareH invariance). The most general lagrangian which is invariant under this transformation is an arbitrary function of the derivatives, R u, of the "eld. An expansion in interactions of successively higher dimension I then gives: L(u)"! R uRIu!(a/4v)R uRIuR uRJu#2 , I J I
(1.4.14)
where we have inserted a power of v as appropriate to ensure that the parameter a is dimensionless. This accords with the expectation that it is the symmetry-breaking scale, v, which sets the natural scale relative to which the low-energy limit is to be taken. In the toy model just considered, integrating out the heavy "eld, s produces these powers of v through the appearance of the inverse of the heavy mass, m . The result must be an e!ective lagrangian of the form of Eq. (1.4.14), but 0 with a speci"c, calculable coe$cient for the parameter a. This, most general, lagrangian automatically ensures that u has all of the Goldstone boson properties. For instance, since the symmetry implies that L can only depend on derivatives of u, it ensures that u cannot appear at all in the scalar potential, and so in particular ensures that u is massless. Similarly, applying Noether's theorem to the kinetic term for u implies that there is a contribution to the Noether current, jI, which is linear in u: jI"(2v[RIu#(a/v)(RJuR u)RIu#2]. J
(1.4.15)
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The ellipses in this expression represent contributions to jI which come from other terms in the lagrangian besides the u kinetic term. Clearly this ensures that the matrix element 1G" jI"02O0 so long as vO0. Such an understanding of the Goldstone nature of a "eld, like u, as an automatic consequence of a symmetry is clearly invaluable when constructing e!ective lagrangians for systems subject to spontaneous symmetry breaking. We next turn to the generalization of these results to the more general case of nonabelian internal symmetries. 1.5. Nonabelian internal symmetries The lesson learned from the abelian example is that half of the art of constructing e!ective lagrangians for Goldstone bosons lies in the choice of a convenient set of variables in terms of which their properties are explicitly displayed in the lagrangian. In this section the above construction is generalized to the case of nonabelian, global, internal symmetries. 1.5.1. A second toy model As guidance towards an appropriate choice of "eld variables we once more start with a simple toy model for which the underlying theory is explicitly known. Consider, therefore, a system of N real scalar "elds, G, i"1,2, N, which for simplicity of notation we arrange into an Ncomponent column vector, denoted by with no superscript. Notice that there is no loss of generality in working with real "elds, since any complex "elds could always be decomposed into real and imaginary parts. We take as lagrangian density (1.5.1) L"! R 2RI !<( ), I where the superscript &T' denotes the transpose, and where <( ) is a potential whose detailed form is not required. The kinetic term of this lagrangian is manifestly invariant under the Lie group, O(N), of orthogonal rotations amongst the N real "elds: PO , where the O's are independent of spacetime position, R O"0, and O2O"1. In general, the potential <( ) need not be also invariant I under these O(N) transformations, but may only preserve some subgroup of these, GLO(N). That is, if g3G, then <(g )"<( ) for all "elds . Suppose now that for some regime of parameters this model is well described by the semiclassical approximation, and further that the potential, <, is minimized for some nonzero value for the "elds:
"vO0. If this is the case, then the symmetry group G may be spontaneously broken to some subgroup, HLG, which is de"ned by: hv"v, for all h3H. 1.5.2. A group-theoretic aside It is important to notice that the current whose existence is guaranteed by Noether's theorem } and so which plays the central role in Goldstone's theorem } arises only if the symmetry of interest is continuous. Continuous here means that the group elements may be parameterized by a continuous parameter (like a rotation angle), as opposed to a discrete label. Groups with continuous labels are called Lie groups provided their labels are su$ciently smooth. Before proceeding it is useful to pause to record some mathematical properties of such Lie groups, and their associated Lie algebras.
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1. Typically, the continuous symmetry groups which arise in physical applications do so as explicit "nite-dimensional unitary matrices. As a result a special role is played by compact groups, for which the parameter space of the group is a compact set. Compact groups are of such special interest since it is only for compact groups that "nite-dimensional, unitary and faithful matrix representations exist. We assume compact groups throughout what follows, and we work explicitly with representations involving "nite-dimensional and unitary matrices, gR"g\. 2. There is also no loss of generality in assuming our representation matrices, g, to be real: g"gH. This is because any complex representation may always be decomposed into its real and imaginary parts. This convention is ensured in the scalar-"eld example we are considering by choosing to employ only real "elds. We do not assume these matrices to be irreducible. Recall that if the matrices are reducible, then there is a basis in which they can be written in a block-diagonal form
g"
g
\ g L
.
(1.5.2)
3. It is useful to phrase much of what follows in terms of the Lie algebra of G and H rather than in terms of the Lie groups themselves. That is, we take advantage of the fact that any group element which is connected to the identity element, g"1, may be written as a matrix exponential: g"exp[ia ¹ ], of a linear combination of a collection of basis matrices, or generators, ? ¹ , a"1,2, d where d is called the dimension of the group. The ¹ 's lie inside what is called the ? ? Lie algebra of G. The unitarity and reality of the group elements, g, imply the matrices ¹ to be ? hermitian and imaginary: ¹ "¹R"!¹H"!¹2 . ? ? ? ?
(1.5.3)
4. Since the generators, ¹ , are "nite dimensional and hermitian, it follows that the matrix ? N "Tr(¹ ¹ ) is positive de"nite. As a result we are free to rede"ne the generators to ensure ?@ ? @ that N "d . With this choice there is no distinction to be made between indices a and b which ?@ ?@ are superscripts and subscripts. We assume this convenient choice to have been made in what follows. 5. Closure of the group multiplication law } i.e. the statement that g , g 3G implies g g 3G } implies commutation relations for the ¹ 's: ¹ ¹ !¹ ¹ "ic ¹ where the c 's are a set of ? ? @ @ ? ?@B B ?@B constant coe$cients which are characteristic of the group involved. From its de"nition it is clear that c is antisymmetric under the interchange of the indices a and b. Whenever the generators ?@B are chosen so that N "d it also turns out that c is completely antisymmetric under the ?@ ?@ ?@B interchange of any two indices.
A representation is faithful if there is a one-to-one correspondence between the group elements and the matrices which represent them. Since the groups of interest are usually dexned by a "nite-dimensional and unitary representation, this representation is, by de"nition, faithful.
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6. For the present purposes it is convenient to choose a basis of generators which includes the generators of the subgroup H as a subset. That is, choose +¹ ,"+t , X ,, where the t 's generate ? G ? G the Lie algebra of H, and the X 's constitute the rest. Since H is de"ned as the group which ? preserves the vacuum con"guration, its generators must satisfy t v"0. The closure of the G subgroup, H, under multiplication ensures that t t !t t "ic t (with no X 's on the rightG H H G GHI I ? hand side). Schematically this can be written c "0. GH? 7. The X 's do not lie within the Lie algebra of H, and so satisfy X vO0. They are said to generate ? ? the space, G/H, of cosets. A coset is an equivalence class which is de"ned to contain all of the elements of G that are related by the multiplication by an element of H. Physically, the X 's ? represent those generators of the symmetry group, G, which are spontaneously broken. 8. The group property of H described above, together with the complete antisymmetry of the c 's ?@B implies a further condition: c "0. This states that t X !X t "ic X (with no t 's on the G?H G ? ? G G?@ @ H right-hand side). This states that the X 's fall into a (possibly reducible) representation of H. ? Once exponentiated into a statement about group multiplication, the condition tX!XtJX implies, for any h3H, that hX h\"¸@X for some coe$cients, ¸@. ? ? @ ? By contrast, X X !X X need not have a particularly simple form, and can be proportional ? @ @ ? to both X 's and t 's. A G 1.5.3. The toy model revisited Returning to the toy model de"ned by the lagrangian, Eq. (1.5.1), we know Goldstone's theorem implies that the assumed symmetry-breaking pattern must give rise to a collection of massless Goldstone bosons, whose interactions we wish to exhibit explicitly. The Goldstone modes are, intuitively, obtained by performing symmetry transformations on the ground state. Since an in"nitesimal symmetry transformation on the ground state corresponds to the directions X v in ? "eld space, we expect the components of in this direction, v2X , to be the Goldstone bosons. It ? is indeed straightforward to verify that the G-invariance of the lagrangian ensures the masslessness of these modes. There is consequently precisely one Goldstone degree of freedom for each generator of G/H. More generally, in order to make low-energy decoupling of these Goldstone bosons manifest we require that they do not appear at all in the scalar potential. Following the example taken for the case of the abelian symmetry, we therefore change variables from (x)"+ G, to s(x)"+sL, and h(x)"+h?,, where
";(h)s ,
(1.5.4)
and ;(h)"exp[ih?(x)X ] is a spacetime-dependent symmetry transformation in the direction of ? the broken generators, X . ? In order for Eq. (1.5.4) to provide a well-de"ned change of variables, s must satisfy some kind of constraint. We therefore require that s be perpendicular (in "eld space) to the Goldstone directions,
If the right-hand side of X X !X X were assumed not to contain any X 's, then the coset G/H would be called ? @ @ ? A a symmetric space, but we do not make this assumption here.
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X v. That is: ? v2X s"0 for all xI and X . (1.5.5) ? ? Notice that this constraint } together with the identity v2X v"0, which follows from the ? antisymmetry of the X 's } is precisely what is required to ensure the vanishing of the cross terms, ? proportional to R h?RIs L, in the quadratic part of the expansion of the kinetic terms about the I ground state con"guration: s"v#s . Since ;(h) is an element of G, the variable h is guaranteed to drop out of the scalar potential. Of course, this is the point of this change of variables, and it happens because G-invariance requires the potential to satisfy <(;s)"<(s). As a result, all of the terms in L which involve the Goldstone bosons, h, vanish when R h?"0, and Eqs. (1.5.4) and (1.5.5) de"ne the change of variables which I makes explicit the low-energy Goldstone-boson decoupling. We pause now to brie#y argue that it is always possible to satisfy Eq. (1.5.5) starting from an arbitary smooth "eld con"guration, (x). That is, we argue that it is always possible to "nd a spacetime dependent group element, ;(h)3G, for which s";\ satis"es Eq. (1.5.5). To this end consider the following function, F[g(a)],v2g(a) , where g(a) is an arbitrary, spacetimedependent element of G. Focus, for a moment, on F as a function of the parameters, a?, of the group for a "xed spacetime position, xI. Since all of the variables, , v and g have been chosen to be real, and since the group, G, is compact, F(a) de"nes a real-valued function having a compact range. It is a theorem that any such function must have a maximum and a minimum, and so there exist group elements, g "g(a), for which (RF/Ra?)" vanishes. Repeating this condition for each ?? point in spacetime de"nes functions a?(x) whose smoothness follows from the assumed smoothness of (x). The "nal step in the argument is to show that the existence of these stationary points of F also give solutions to the problem of "nding a ; for which s";\ satis"es Eq. (1.5.5). This last step follows by explicitly taking the derivative of g with respect to a?: and using parameters a such that (Rg/Ra?)g\"¹ . In this case the vanishing of RF/Ra?, when evaluated at g"g "g(a), implies ? v2¹ g "0. We see that the choice s";\ , with ;"g \, therefore satis"es Eq. (1.5.5), and the ? existence of such a solution follows from the existence of a maximum and minimum of F. This concludes the argument. We next explore the properties of the new variables. 1.5.4. The nonlinear realization Having motivated this choice of variables, we now determine how they transform under the group G of symmetry transformations. The transformation rules which we obtain } and which we show to be unique, up to "eld rede"nitions, below } carry all of the information concerning Goldstone boson properties, and so requiring low-energy lagrangians to be invariant under these transformations automatically encodes these properties into the low-energy theory. The starting point is the transformation rule for : P I "g , where g"exp[ia?¹ ]. The ? transformation rule for the new variables is then hPhI and sPs, where ";(h)s and I ";(hI )s. That is, under any transformation, g3G, h, s, hI and s are related by g;(h)s";(hI )s .
(1.5.6)
This last equation states that the matrix c,;I \g; (where ;I denotes ;(hI )) has the property that cs"s. The central result to be now proven is that this condition implies that c must lie within
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the subgroup, H of unbroken transformations, and so may be written c"exp(iuGt ), for some G function uG(h, g). Once this has been demonstrated, the transformation law therefore becomes: h?PhI ?(h, g) and sPs(h, g, s) ,
(1.5.7)
where ge F?6? "e FI ?6? e SGRG , s"e SGRG s .
(1.5.8)
The "rst of Eqs. (1.5.8) should be read as de"ning the nonlinear functions hI ?(h, g) and uG(h, g). They are de"ned by "nding the element, ge F 63G, and then decomposing this matrix into the product of a factor, eFI 6, lying in G/H times an element, e S R, in H. The second line of Eqs. (1.5.8) then de"nes the transformation rule for the non-Goldstone "elds, s. These rules generally de"ne transformation laws which are nonlinear in the Goldstone "elds, h?. They furnish, nonetheless, a faithful realization of the symmetry group G, in that hI (h, g g )"hI (hI (h, g ), g ), etc. This may either be directly veri"ed using the de"nitions of Eqs. (1.5.8), or by noticing that this property is inherited from the faithfulness of the original linear representation of G on . There is a particularly interesting special case for which Eqs. (1.5.8) can be explicitly solved for c"e S R and ;I "e FI 6. This is when g"h lies in H. In this case, the solution is easily seen to be: c"h and ;I "h;h\. Both s and h therefore transform linearly under the unbroken symmetry transformations, H. That is: h?X PhI ?X "hh?X h\ , ? ? ? sPs"hs ,
(1.5.9)
for all h3H. For the broken symmetries, g3G/H, it is useful to specialize to an in"nitesimal transformation, g"1#iu?X #2 . In this case we have c"1#iuG(h, u)t #2, and ? G ;(hI )";(h)[1#iD?(h, u)X #2], where uG(h, u) and D(h, u) must both also be in"nitesimal ? quantities. Eq. (1.5.8) gives them explicitly to be D "Tr[X e\ F 6(u ) X)e F 6] ? ? +u !c u@hA#O(h), ? ?@A
(1.5.10)
u "Tr[t e\ F 6(u ) X)e F 6] G G +!c u?h@#O(h) , G?@
(1.5.11)
where we used: Tr(X X )"d , Tr(t t )"d and Tr(t X )"0. ? @ ?@ G H GH G ? This last expression for D (h, u) can be re-expressed in terms of the change, ? dh?,m?(h, u),hI ?!h?, of the Goldstone-boson "elds. The relation between D and m? is linear: ? D "M (h)m@, where the matrix, M , of coe$cients may be computed using the following useful ? ?@ ?@
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identity, which holds for any two square matrices, A and B:
e\ e > "1#i
"1#i
ds e\ Q Be Q > ds e\ Q Be Q #O(B) .
(1.5.12)
Using this with A"h ) X and B"m ) X gives
M " ?@
ds Tr[X e\ QF 6X e QF 6]. ? @
(1.5.13)
The transformation rules for the h? with respect to the broken symmetries in G/H have two important properties. The "rst crucial property is that the transformation law is inhomogeneous in the broken symmetry parameters, since dh?"u?!c? u@hA#O(h) . @A
(1.5.14)
As was observed earlier for the abelian example, it is this property which enforces the decoupling of the Goldstone bosons at low energies. The second important property is that, for a nonabelian group, the symmetries in G/H act nonlinearly on the "elds h?. This property is also signi"cant since it ruins many of the consequences which would otherwise hold true for symmetries which are linearly realized. For example, the masses of those particles whose "elds lie in a linear representation of a symmetry group necessarily have equal masses, etc. The same is not true for particles whose "elds are related by nonlinear transformations. There is a corollary which follows from the nonlinearity of the realization of the symmetries in G/H. The fact that the transformation of h? and sL are both "eld dependent implies that the action of these symmetries are spacetime dependent. For example, even though the transformation parameters themselves, u?, are constants } since G is a global symmetry } the transformation matrix c"e S R which appears in the s transformation law is not a constant, R cO0. This fact I complicates the construction of lagrangians which are invariant with respect to these symmetries. 1.6. Invariant lagrangians With the transformation rules for the Goldstone boson "elds in hand we may now turn to the construction of invariant lagrangians which can describe their low-energy interactions. The main complication here is in the construction of the kinetic terms, since the transformation rules for the "elds are spacetime dependent due to their complicated dependence on the "elds. A clue as to how to proceed can be found by reconsidering the toy model of scalar "elds . In this case the kinetic term, proportional to R 2RI is manifestly G invariant. It must therefore remain I so after performing the change of variables to h and s. To see how this is comes about, we notice that after the replacement ";s we have: R ";(R s#;\R ;s). In terms of the I I I new variables, the kinetic term is invariant because the combination D s"R s#;\R ;s I I I
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transforms covariantly: D sPcD s. It does so because ;\R ; transforms like a gauge-potential: I I I ;\R ;P;I \R ;I I I "c(;\R ;)c\!R cc\ . (1.6.1) I I More information emerges if we separate out the parts of ;\R ; which are proportional to I X from those which are proportional to t since the inhomogeneous term, R cc\, is purely ? G I proportional to t . That is, if we de"ne: G ;\R ;"!iAG t #ie? X , (1.6.2) I I G I ? then each of these terms transforms separately under G transformations: !iAG (h)t P!iAG (hI )t "c[!iAG (h)t ]c\!R cc\ , I G I G I G I ie? (h)X Pie? (hI )X "c[ie? (h)X ]c\ . (1.6.3) I ? I ? I ? We see that the quantity AG transforms like a gauge potential. For in"nitesimal transformations, I g+1#iu?X and c(h, g)+1#iuG(h, u)t , we have ? G dAG (h)"R uG(h, u)!cG uH(h, u)AI (h) . (1.6.4) I I HI I Similarly, e? (h) transforms covariantly, with I de? (h)"!c? uG(h, u)e@ (h) . (1.6.5) I G@ I In this last expression, the structure constants de"ne representation matrices, (T ) "c , of G ?@ ?G@ the Lie algebra of H. These are the same matrices which de"ne the representation of H that the generators X form, and it is important for later purposes to recall that these representation ? matrices need not be irreducible. If this representation is reducible then it is possible to de"ne more G-invariant quantities with which to build the low-energy lagrangian than would otherwise be possible. If we extract the overall factor of R h?, so that AG "AG (h)R h? and e? "e? (h)R h@, then the I I ? I I @ I identity, Eq. (1.5.12), gives the following expressions for the coe$cients:
AG (h)"! ?
ds Tr[t e\ QF 6X e QF 6] ?
1 + c h@#O(h) , 2 G?@
(1.6.6)
and
e? (h)" @
ds Tr[X e\ QF 6X e QF 6] ? @
(1.6.7) +d ! c hA#O(h) . ?@ ?@A With these tools it is now clear how to build G-invariant couplings among the h?, and between the h?'s and other "elds, such as s from the scalar-"eld example. It is simplest to build self-interactions for the Goldstone bosons. An invariant lagrangian density may be built by combining the covariant quantity, e? "e? R h@ in all possible H-invariant ways. I @ I
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This is simple to do since this quantity transforms very simply under G: e ) XPce ) Xc\. I I Derivatives of e? can also be included by di!erentiating using the covariant derivative construcI ted from AG t : I G (D e )?"R e? #c? AG e@ , (1.6.8) I J I J G@ I J which transforms in the same way as does e? : d(D e )?"!c? uG(D e )@. I I J G@ I J The lagrangian is L(e , D e ,2), where the ellipses denote terms involving higher covariant I I J derivatives. Provided only that this lagrangian is constrained to be globally H invariant: L(he h\, hD e h\,2),L(e , D e ,2) , (1.6.9) I I J I I J the result is guaranteed to be automatically globally G invariant, as required. For a PoincareH invariant system, the term involving the fewest derivatives therefore becomes: (1.6.10) L "! f gIJe? e@ #(higher-derivative terms) . I J % ?@ In this expression, positivity of the kinetic energy implies that the matrix f must be positive ?@ de"nite. G-invariance dictates that it must also satisfy f cH #f cH "0. It was remarked earlier H@ G? ?H G@ that on general grounds the matrices, X , "ll out a representation, R, of the unbroken subgroup ? H with representation matrices given by (T ) "c , and in terms of these matrices G-invariance G ?@ ?G@ requires the vanishing of all of the commutator, [T , f ], for all of the generators, T . If this G G representation, R, of H is irreducible then, by Schur's lemma, f must be proportional to the unit ?@ matrix, with positive coe$cient: f "Fd . Otherwise, if R is reducible into n irreducible diagonal ?@ ?@ blocks, then f need only be block diagonal, with each diagonal element being proportional to ?@ a unit matrix:
f " ?@
F d ?@
\
,
(1.6.11)
Fd L L L ?@ for n-independent positive constants, F. We see that the lowest-dimension terms in the most L general low-energy Goldstone-boson self-coupling lagrangian is parameterizable in terms of these n constants. If other "elds } denoted here collectively by s } also appear in the low-energy theory then, since the symmetry H is not broken by the ground state, the "elds s must also transform linearly under H : sPhs, where the matrices +h, form a (possibly reducible) representation of H. In this case the starting point for inferring the coupling of the Goldstone bosons is an arbitrary, globally Hinvariant lagrangian: L(s, R s,2) with L(hs, hR s,2),L(s, R s,2), for R h"0. This lagranI I I I gian will be automatically promoted to become G-invariant by appropriately coupling the Goldstone bosons. The promotion to G invariance proceeds by assigning to s the nonlinear G-transformation rule: sPcs, where c"c(h, g)3H is the "eld-dependent H matrix which is de"ned by the nonlinear realization, Eq. (1.5.8). An arbitrary globally H-invariant s-lagrangian then becomes G invariant if all derivatives are replaced by the h-dependent covariant derivative: R sPD s"R s!iAGt s, for I I I G which D sPcD s. I I
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The general lagrangian therefore becomes: L(e , s, D e , D s,2), where G-invariance is ensured I I J I provided only that L is constrained by global H invariance: L(he h\, hs, hD e h\, hD s,2),L(e , s, D e , D s,2) . I I J I I I J I
(1.6.12)
1.7. Uniqueness The previous construction certainly de"nes a G-invariant lagrangian for the interactions of the Goldstone bosons which arise from the symmetry-breaking pattern GPH, given the transformation rules which were derived in earlier sections. Our goal in the present section is to show that this construction gives the most general such invariant lagrangian. That is, we wish to show that the most general lagrangian density which is invariant under the transformation rules of Eqs. (1.5.8) may be constructed using only the quantities e? (h) and AG (h) in addition to any other "elds, s. I I We start with a general lagrangian density, L(h, R h, s, R s), involving the "elds h?, sL and their I I derivatives. We do not include a dependence on second and higher derivatives of these "elds, but this extension is straightforward to make along the lines that are described in this section. It is more convenient in what follows to trade the assumed dependence of L on R h for a dependence on the I combinations e? "e? (h)R h@ and AG "AG (h)R h?. There is no loss of generality in doing so, since I @ I I ? I any function of h and R h can always be written as a function of h, e? and AG . This equivalence is I I I most easily seen in terms of the matrix variable ;(h)"e F 6. Any function of h and R h can equally I well be written as a function of ; and R ;, or equivalently as a function of ; and ;\R ;. But an I I arbitrary function of ;\R ; is equivalent to a general function of e? and AG , as may be seen from I I I expression (1.6.2). The condition that a general function, L(h?, e? , AG , s, R s), be invariant with respect to G transI I I formations is RL RL RL RL RL dh?# de? # dAG # dsL# dR sL"0 . (1.7.1) dL" I ? Re? RAG RsL R(R sL) I Rh? I I I We "rst specialize to the special case where the symmetry transformation lies in H: g"e E R3H. In this case we must use, in Eq. (1.7.1), the transformations: dh?"!c? gGh@, G@
de? "!c? gGe@ , I G@ I
dAG "!cG gGAI , I HI I
and dsL"igG(t s)L, dR sL"igG(t R s)L . (1.7.2) G I G I Requiring dL"0 for all possible transformation parameters, gG, then implies the following identity: RL RL RL RL RL c? h@# c? e@ # cH AI ! i(t s)L! i(t R s)L"0 . (1.7.3) Re? G@ I RAH GI I RsL G R(R sL) G I Rh? G@ I I I This identity simply states that L must be constructed to be an H-invariant function of its arguments, all of which transform linearly with respect to H transformations.
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For the remaining symmetry transformations which do not lie in H, g"e S 63G/H, we instead evaluate Eq. (1.7.1) using the following transformations: dh?"m? u@, @
de? "!c? uGe@ , I G@ I
dAG "R uG!cG uHAI , I I HI I
and dsL"iuG(t s)L, G
dR sL"iuG(t R s)L , I G I
(1.7.4)
where m?"m? (h)u@ and uG"uG (h)u? are the nonlinear functions of h that are de"ned by Eq. (1.5.8). @ ? Using these in Eq. (1.7.1), and simplifying the resulting expression using Eq. (1.7.3), leads to the remaining condition for G invariance:
RL RL RL m? #c? uG hA # R uG # iR uG (t s)L"0 . GA @ Rh? @ RAH I @ R(R sL) I @ G I I
(1.7.5)
This last identity contains two separate pieces of information. The "rst piece can be extracted by specializing to h?"0. In this case, since R uG "R uG R h? vanishes when h?"0, and since I @ ? @ I Eq. (1.5.14) implies m? (h"0)"d? , we "nd @ @ RL Rh?
"0 .
(1.7.6)
F
But, since the group transformation law for h? is inhomogeneous, we may always perform a symmetry transformation to ensure that h?"0 for any point p3G/H. As a result, Eq. (1.7.6) implies the more general statement: RL ,0 throughout G/H . Rh?
(1.7.7)
The rest of the information contained in Eq. (1.7.5) may now be extracted by using RL/Rh?"0 to eliminate the "rst term. One "nds the remaining condition:
RL RL # i(t s)L R uH "0 . I @ RAH R(R sL) H I I
(1.7.8)
This equation has a very simple meaning. It states that L can depend on the two variables, AH and I R sL, only through the one combination: (D s)L,R sL!iAH (t s)L. That is, s can appear di!erentiI I I I H ated in L only through its covariant derivative, D s. I We see from these arguments that the G-invariance of L is equivalent to the statement that L must be an H-invariant function constructed from the covariantly-transforming variables e? , I s and D s. If higher derivatives of h had been considered, then the vanishing of the terms in dL I that are proportional to more than one derivative of uG would similarly imply that derivatives of e? must also only appear through its covariant derivative, (D e )?, de"ned by Eq. (1.6.8). I I J This proves the uniqueness of the construction of invariant lagrangians using these covariant quantities.
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1.8. The geometric picture There is an appealing geometric description of the resulting e!ective lagrangian, which makes available many powerful techniques from di!erential geometry to e!ective lagrangian methods. We pause here to brie#y outline this connection. Consider, for simplicity, only the self-interactions of the Goldstone bosons: L (h). This can be % expanded into terms having increasing numbers of derivatives acting on h. The "rst few terms that are consistent with PoincareH invariance are: L
%
(h)"!<(h)!g (h)R h?RIh@#2 . I ?@
(1.8.1)
Positivity of the kinetic energy for #uctuations about any con"guration, h?, requires the symmetric matrix g to be positive de"nite for all h. ?@ The geometrical interpretation arises once we recall that the "elds h take values in the coset space G/H, and so each Goldstone boson "eld con"guration can be considered to be a map from spacetime into G/H. The function, <, can then be considered to be a real-valued function which is de"ned on the space G/H. Similarly, the positive symmetric matrix, g , de"nes a metric tensor on ?@ G/H. These identi"cations of < and g with geometrical objects on G/H are consistent with their ?@ transformation properties under "eld rede"nitions, dh?"m?(h), which are the analogues of coordinate transformations on G/H. To see this, perform this transformation in the lagrangian of Eq. (1.8.1). The result is to replace < and g by the quantities <#£ < and g #£ g , where the ?@ K ?@ K ?@ linear operator, £ , is known as the Lie derivative in the direction speci"ed by m?, and is given K explicitly for a scalar and a covariant tensor by £ <"m?R < , K ? and £ g "mHR g #g R mH#g R mH . K ?@ H ?@ H@ ? ?H @
(1.8.2)
In these expressions derivatives, like R <, R g or R mH, all represent di!erentiation with respect to ? H ?@ ? h?, and not with respect to spacetime position, xI. Clearly, the G invariance of the "rst few terms of L therefore becomes equivalent to the % problem of "nding a scalar and a metric for which £ <"£ g "0 for each of the m?(h)'s which K K ?@ describe the action of G on G/H. Since every point on G/H can be reached from any other by performing such a G transformation } i.e. G/H is a homogeneous space } it follows that the only possible invariant function, <, is a constant which is independent of h. This expresses the low-energy decoupling of the Goldstone bosons since it shows that G invariance completely forbids their appearance in the scalar potential. Similarly, the condition for the G invariance of the kinetic terms is that the metric g must also ?@ be invariant under the action of all of the m?(h)'s which generate G on G/H. That is, all of these m's must generate isometries of the metric g . The problem of "nding the most general invariant ?@ kinetic term is therefore equivalent to constructing the most general G-invariant metric on G/H. A comparison of the lagrangian of Eq. (1.8.1), with our earlier result, Eq. (1.6.10), gives
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a representation of the metric g
?@
in terms of the quantities e? (h). We have @
L g "f eA eH " Fd P P eAP eHP ?@ AH ? @ P AH ? @ P L + F[d P P !c P P hA#O(h)] . ?@A P ?@ P
(1.8.3)
Eq. (1.8.3) also has a geometric interpretation. It shows that the object e? can be interpreted as @ a G-covariant vielbein for the space G/H. In Riemannian geometry a vielbein is the name given to a set of N linearly independent vectors, e? , a"1,2, N, which are tangent to an N-dimensional @ space. (The name is German for &many legs', with viel meaning &many', and so such vectors are also called zweibein in two dimensions } with zwei " &two' } or vierbein in four dimensions } with vier "&four'.) Part of the utility of identifying such a set of vectors is that it is always possible to reconstruct from them the space's metric, using: g "d e? e@ . ?@ ?@ ? @ The geometrical interpretation of e? as a vielbein, and the uniqueness of the construction of @ invariant lagrangians proven in the previous section, gives the general solution to the geometrical problem of constructing G-invariant metrics on the space G/H. We see that there is an n-parameter family of such metrics, where n counts the number of irreducible representations of H which are formed by the generators, X , of G/H. The n parameters are given explicitly by the constants ? F, r"1,2, n. P For many physical applications the representation of H that is furnished by the X is irreducible, ? and in this case the G-invariant metric is uniquely determined up to its overall normalization: g "Fg( , with g( "d #O(h). For such systems there is precisely one constant in the e!ective ?@ ?@ ?@ ?@ lagrangian for Goldstone bosons which is undetermined by the symmetries of the problem, if we include only the fewest possible (two) derivatives: L "!(F/2)g( (h)R h?RIh@#(higher derivative terms) . % ?@ I
(1.8.4)
Once the one constant, F, is determined, either by calculation from an underlying theory, or by appeal to experiment, the lowest-order form for all of the Goldstone boson interactions are completely determined by the symmetry breaking pattern. The resulting model-independent predictive power has wide applications throughout physics, as we shall see when we consider examples in subsequent sections. 1.9. Nonrelativistic lagrangians Before turning to examples, we pause to outline the changes in the above analysis which become necessary when it is applied to nonrelativistic systems, for which the spacetime symmetry is not PoincareH invariance. This is what is appropriate, for example, in condensed-matter applications for which there is a preferred frame, de"ned by the centre-of-mass of the medium of interest. So long as our attention is restricted to internal symmetries, most of the considerations of this section apply just as well to nonrelativistic problems as they do to relativistic ones. In particular,
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the expressions obtained for the nonlinear realization of broken symmetries on the Goldstone boson "elds does not depend at all on the spacetime symmetries which are assumed. We assume for simplicity here that the system remains invariant with respect to translations and rotations, although the generalization to di!erent spacetime groups is straightforward in principle. In practice, the results we obtain also apply to some lattice systems, for which translations and rotations are not symmetries. This is because, at least for the "rst few derivatives in a derivative expansion of the lagrangian, the lattice group for some lattices, such as a cubic lattice for example, implies the same restrictions as do translation and rotation invariance. There are two cases, which we consider separately below, depending on whether or not time reversal is a good symmetry of the system. 1.9.1. When time-reversal is a good symmetry For unbroken time reversal, the main di!erence from the relativistic situation lies in the fact that the time- and space-derivatives become independent since they are unrelated by any symmetries. For instance, assuming unbroken translation and rotation invariance, the most general selfcouplings amongst the Goldstone bosons for the symmetry-breaking pattern GPH are: 1 L L "# [F g( P (h)hQ ?hQ @!F g( P (h) h? ) h@]#(higher derivative terms) . P R ?@ P Q ?@ % 2 P
(1.9.1)
Compared to the relativistic case, twice as many (that is, 2n) constants, F and F , are required P R P Q to parameterize the terms containing the fewest number of derivatives. Once these constants are determined, all other interactions at this order of the derivative expansion are clearly completely determined. 1.9.2. When time-reversal is broken New possibilities arise for the lagrangian when time-reversal symmetry is broken, as is the case for a ferromagnet, for example. In this case, it is possible to write down terms in L which involve % an odd number of time derivatives. In particular, the dominant, lowest-dimension term involving time derivatives involves only one: *L "!A (h)hQ ? . (1.9.2) % ? The coe$cient function, A (h), can be considered to de"ne a vector "eld on the coset space G/H. It ? is to be chosen to ensure the G invariance of the low-energy theory. If *L is required to be invariant under G transformations, then it must be built using the % quantity e? (h)"e? (h) * h@ and its covariant derivatives. If it is to involve only a single time I @ I derivative, then it must be proportional only to e (h). But the only such G-invariant quantity is: k e? hQ @, where the constants, k , must satisfy k c? "0 for all indices i and b. Such a k exists only if ? @ ? ? G@ ? the corresponding generator, X , commutes with all of the generators of the unbroken subgroup, ? H. This quite restrictive condition is not ever satis"ed in many situations of physical interest, and for these systems it appears (at least super"cially) that no terms involving only a single time derivative are consistent with G-invariance. This conclusion would be too strong, however, because it is too restrictive a condition to demand the G-invariance of the lagrangian density, L. We are only required by G symmetry to demand the
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invariance of the action. The lagrangian density need not be invariant, provided that its variation is a total derivative. It is therefore worth re-examining the time-reversal-violating term of Eq. (1.9.2) in this light. Once we drop total derivatives in *L , it is clear that the coe$cient A (h) is only de"ned up to % ? the addition of a gradient. That is, any two choices A and AI ,A #* X(h), di!er in their ? ? ? ? contribution to *L only by the total derivative: % *L (A#*X)!*L (A)"!* X(h)hQ "!XQ (h) . (1.9.3) % % ? In geometrical terms we may therefore regard the coe$cient function, A (h), as de"ning a gauge ? potential on the coset space G/H. The condition that *L contribute a G-invariant term to the action therefore only requires the % coe$cient A (h) to be G-invariant up to a gauge transformation. In equations, G-invariance of the ? action only requires: £ A ,m@* A #A * m@"* X , (1.9.4) K ? @ ? @ ? ? K for each generator dh?"m? of the action of G on G/H, and for some scalar functions, X (h), on G/H. K This last condition is equivalent to demanding the invariance of the gauge-invariant quantity: F "* A !* A . That is, ?@ ? @ @ ? £ F ,mH* F #F * mH#F * mH"0 . (1.9.5) K ?@ H ?@ H@ ? ?H @ We shall "nd that this condition does admit solutions in many cases of interest } most notably for the example of a ferromagnet. 1.10. Power counting Before proceeding to real-life applications, a "nal important issue must be addressed. Since the lagrangian expressed using Goldstone boson variables is typically nonrenormalizable, it is necessary to know how to use nonrenormalizable lagrangians when making quantitative calculations. The key to doing so is to consider the Goldstone boson lagrangians to which we have been led in previous sections to be &e!ective theories' which describe only the low-energy behaviour of the system of interest. For instance, in our toy models the Goldstone bosons (h?) are massless while the other degrees of freedom (s) are not. (Although the pseudo-Goldstone bosons for an approximate symmetry are not exactly massless, they may nonetheless appear in the low-energy theory so long as their mass, m, is su$ciently small.) A lagrangian involving only Goldstone bosons or pseudoGoldstone bosons can only hope to describe physics at energies, q, below the mass threshhold, M, for producing the heavier particles. (It is often the case that M is proportional to the symmetrybreaking scale(s), v.) The predictions of such a lagrangian are to be regarded as reproducing, in powers of q/M, whatever the &underlying' (or µscopic') theory } i.e. the theory involving the heavy s states } might be. In order to make this concrete, consider one such a lagrangian, having the form c L"f L O ( /v) , MBL L L
(1.10.1)
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where denotes a generic boson "eld, c are a set of dimensionless coupling constants which we L imagine to be at most O(1), and f, M and v are mass scales of the underlying problem. (For example, in the application to pions which follows we will have f"(F K , M"K and v"F , where Q L L Q F and K are scales which characterize the strength of the appropriate symmetry breaking in the L Q strong interactions.) d is the dimension of the operator O , in powers of mass, as computed by L L counting only the dimensions of the "eld, , and derivatives, * . Imagine using this lagrangian to compute a scattering amplitude, A (q), involving the scattering # of E particles whose four-momenta are collectively denoted by q. We wish to focus on the contribution to A due to a Feynman graph having I internal lines and < vertices. The labels i and GI k here indicate two characteristics of the vertices: i counts the number of lines which converge at the vertex, and k counts the power of momentum which appears in the vertex. Equivalently, i counts the number of powers of the "elds, , which appear in the corresponding interaction term in the lagrangian, and k counts the number of derivatives of these "elds which appear there. 1.10.1. Some useful identities The positive integers, I, E and < , which characterize the Feynman graph in question are not all GI independent since they are related by the rules for constructing graphs from lines and vertices. This relation can be obtained by equating two equivalent methods of counting the number of ways that internal and external lines can end in a graph. On the one hand, since all lines end at a vertex, the number of ends is given by summing over all of the ends which appear in all of the vertices: i < . GI GI On the other hand, there are two ends for each internal line, and one end for each external line in the graph: 2I#E. Equating these gives the identity which expresses the &conservation of ends': 2I#E" i< (Conservation of Ends) . (1.10.2) GI GI A second useful identity dexnes of the number of loops, ¸, for each (connected) graph: ¸"1#I! < (De"nition of ¸) . (1.10.3) GI GI For simple planar graphs, this de"nition agrees with the intuitive notion what the number of loops in a graph means. 1.10.2. Dimensional estimates We now collect the dependence of A (a) on the parameters in L. # Reading the Feynman rules from the lagrangian of Eq. (1.10.1) shows that the vertices in the Feynman graph contribute the following factor:
p I f 4GI , (Vertices)" i(2p)d(p) M vG GI where p generically denotes the various momenta running through the vertex. Similarly, each internal line in the graph contributes the additional factors:
(Internal Lines)" !i
dp Mv 1 ' , (2p) f p#m
(1.10.4)
(1.10.5)
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where, again, p denotes the generic momentum #owing through the line. m denotes the mass of the light particles which appear in the e!ective theory, and it is assumed that the kinetic terms which de"ne their propagation are those terms in L involving two derivatives and two powers of the "elds, . As usual for a connected graph, all but one of the momentum-conserving delta functions in Eq. (1.10.4) can be used to perform one of the momentum integrals in Eq. (1.10.5). The one remaining delta function which is left after doing so depends only on the external momenta, d(q), and expresses the overall conservation of four-momentum for the process. Future formulae are less cluttered if this factor is extracted once and for all, by de"ning the reduced amplitude, A I , by A (q)"i(2p)d(q)A I (q) . (1.10.6) # # The number of four-momentum integrations which are left after having used all of the momentum-conserving delta functions is then I! < #1"¸. This last equality uses the de"niGI GI tion, Eq. (1.10.3), of the number of loops, ¸. We now wish to estimate the result of performing the integration over the internal momenta. In general, these are complicated integrals for which a simple result is not always possible to give. Considerable simpli"cations arise, however, if all of the masses and energies of the particles in the low-energy theory are of the same order of magnitude, since in this case much can be said about the order of magnitude of the momentum integrals purely on dimensional grounds. (Although this is often the situation of interest when employing e!ective theories, it must be borne in mind that it does not always apply. For instance, it excludes a situation of considerable practical interest, where the low-energy theory includes very massive but slowly moving, nonrelativistic particles. Power counting for such systems is beyond the scope of this review.) In order to proceed with a dimensional argument it is most convenient to regulate the ultraviolet divergences which arise in the momentum integrals using dimensional regularization. For dimensionally regularized integrals, the key observation is that the size of the result is set on dimensional grounds by the light masses or external momenta of the theory. That is, if all external energies, q, are comparable to (or larger than) the masses, m, of the light particles whose scattering is being calculated, then q is the light scale controlling the size of the momentum integrations, so dimensional analysis implies that an estimate of the size of the momentum integrations is
2
dLp p 1 & qL > \! , (2p)L (p#q)! 4p
(1.10.7)
with a dimensionless prefactor which carries all of the complicated dependence on dimensionless ratios like q/m. The prefactor also depends on the dimension, n, of spacetime, and may be singular in the limit that nP4. With this estimate for the size of the momentum integrations, we "nd the following contribution to the amplitude A I (q): # dp * p GI I4GI 1 * q*\'> GI I4GI , 2 & (1.10.8) (2p) (p#q)' 4p
We ignore here any logarithmic infrared mass singularities which may arise in this limit.
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which, with liberal use of the identities (1.10.2) and (1.10.3), gives as estimate for A I (q): #
1 # M * q >*> GI I\4GI AI (q)&f . # v 4pf M
(1.10.9)
This expression is the principal result of this section. Its utility lies in the fact that it links the contributions of the various e!ective interactions in the e!ective lagrangian, (1.10.1), with the dependence of observables on small mass ratios such as q/M. As a result it permits the determination of which interactions in the e!ective lagrangian are required to reproduce any given order in q/M in physical observables. Most importantly, Eq. (1.10.9) shows how to calculate using nonrenormalizable theories. It implies that even though the lagrangian can contain arbitrarily many terms, and so potentially arbitrarily many coupling constants, it is nonetheless predictive so long as its predictions are only made for low-energy processes, for which q/M;1. (Notice also that the factor (M/f )* in (1.10.9) implies, all other things being equal, the scale f cannot be taken to be systematically smaller than M without ruining the validity of the loop expansion in the e!ective low-energy theory.) Before stating more explicitly the e!ective-lagrangian logic, which Eq. (1.10.9) summarizes, we pause to generalize it to include fermions in the low-energy e!ective theory. 1.10.3. Including fermions It is straightforward to extend these results to include light fermions in the e!ective theory, although once again subject to the important assumption that all masses and energies are small in the e!ective theory. To this end, "rst generalize the starting form assumed for the lagrangian to include fermion "elds, t, in addition to boson "elds, :
t c , . L"f L O MBL L v v $ L
(1.10.10)
An important di!erence between fermion and boson propagators lies in the way each falls o! for large momenta. Whereas a boson propagator varies like 1/p for large p, a fermion propagator goes only like 1/p. This leads to a di!erence in their contribution to the power counting of a Feynman graph. It is therefore important to keep separate track of the number of fermion and boson lines, and we therefore choose to now label vertices using three indices: k, i and i . As before, k labels the $ numbers of derivatives in the corresponding interaction, but now i and i separately count the $ number of bose and fermi lines which terminate at the vertex of interest. The number of vertices in a graph which carry a given value for k, i and i we now label by < $ . $ GGI Consider now computing an amplitude which has E external bosonic lines, E external fermion $ lines, and I and I internal bose and fermi lines. Repeating, with the lagrangian of Eq. (1.10.10), $ the power counting argument which led (using dimensional regularization) to Eq. (1.10.9) now gives instead the following result: (q)&f (1/v )# (1/v )#$ (M/4nf )*(q/M). , AI $ # #$
(1.10.11)
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where the power P can be written in either of the following two equivalent ways:
3 3 P"4!E ! E # k#i # i !4 < $ , GGI 2 $ 2$ G G$ I 1 1 "2#2¸! E # k# i !2 < $ . GGI 2 $ 2$ G G$ I
(1.10.12)
1.11. The ewective-lagrangian logic The powercounting estimates just performed show how to organize calculations using nominally nonrenormalizable theories, considering them as e!ective "eld theories. They suggest the following general logic concerning their use. Step I: Choose the accuracy (e.g. one part per mille) with which observables, such as A (q), are to # be computed. Step II: Determine the order in the small mass ratios q/M or m/M that must be required in order to acheive the desired accuracy. Step III: Use the power counting result, Eq. (1.10.9), to "nd which terms in the e!ective lagrangian are needed in order to compute to the desired order in q/M. Eq. (1.10.9) also determines which order in the loop expansion is required for each e!ective interaction of interest. Step IVa: Compute the couplings of the required e!ective interactions using the full underlying theory. If this step should prove to be impossible, due either to ignorance of the underlying theory or to the intractability of the required calculation, then it may be replaced by the following alternative: Step IVb: If the coe$cients of the required terms in the e!ective lagrangian cannot be computed then they may instead be regarded as unknown parameters which are to be taken from experiment. Once a su$cient number of observables are used to determine these parameters, all other observables may be unambiguously predicted using the e!ective theory. A number of points bear emphasizing at this point. 1. The possibility of treating the e!ective lagrangian phenomenologically, as in Step IVb above, immeasurably broadens the utility of e!ective lagrangian techniques, since they need not be restricted to situations for which the underlying theory is both known and calculationally simple. Implicit in such a program is the underlying assumption that there is no loss of generality in working with a local "eld theory. This assumption has been borne out in all known examples of physical systems. It is based on the conviction that the restrictions which are implicit in working with local "eld theories are simply those that follow from general physical principles, such as unitarity and cluster decomposition. 2. Since Eq. (1.10.9) [or Eqs. (1.10.11) and(1.10.12)] states that only a "nite number of terms in L contribute to any "xed order in q/M, and these terms need appear in only a "nite number of loops, it follows that only a "nite amount of labour is required to obtain a "xed accuracy in observables. Renormalizable theories represent the special case for which it su$ces to work only to zeroeth order in the ratio q/M. This can be expected to eventually dominate at su$ciently low energies
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compared to M, which is the reason why renormalizable theories play such an important role throughout physics. 3. An interesting corollary of the above observations is the fact that only a "nite number of renormalizations are required in the low-energy theory in order to make "nite the predictions for observables to any "xed order in q/M. Thus, although an e!ective lagrangian is not renormalizable in the traditional sense, it nevertheless is predictive in the same way that a renormalizable theory is.
2. Pions: a relativistic application We now present a relativistic application of these techniques to the low-energy interactions of pions and nucleons. This example provides a very useful, and experimentally successful, description of the low-energy limit of the strong interactions, and so illustrates how e!ective lagrangians can remain predictive even if it is impossible to predict their e!ective couplings from an underlying theory. This example is also of historical interest, since the study of low-energy pion scattering comprises the context within which the above Goldstone-boson formalism initially arose. 2.1. The chiral symmetries of QCD The modern understanding of the strong interactions is based on the theory of mutually interacting spin-half quarks and spin-one gluons that is called quantum chromodynamics (QCD). This theory is described by the lagrangian density 1 (2.1.1) L "! G? GIJ! q (D. #m L )q , L O L /!" 4 IJ ? L where G? "* G? !* G? #gf ? G@ gA is the "eld strength tensor for the gluon "elds, G? . Here IJ I J J I @A I J I a"1,2,8 labels the generators of the gauge symmetry group of the theory, S; (3), for which the f ? are the structure constants. The subscript &c' of S; (3) stands for &colour', which is the name @A given to the strong charge. The quarks are represented by Dirac spinors, q , where n"1,2,6 counts the six kinds of L quarks. In order of increasing mass, these are: u, d, s, c, b and t. For the purposes of later comparison we list here the quark masses, m L , in GeV: m "0.0015}0.005, m "0.003}0.009, m "0.06}0.17, S B Q O m "1.1}1.4, m "4.1}4.4 and m "173.8$5.2. All of these quarks are assumed to transform in @ R the de"ning three-dimensional representation of the colour symmetry group, S; (3), and so their covariant derivative (which appears in the combination D. "cID in the lagrangian) is I D q "* q !(i/2)gG? j q . The eight matrices, j , denote the 3;3 Gell}Mann matrices, which I L I L I ? L ? act on the (unwritten) colour index of each of the quarks. The explicit form for these matrices is not required in what follows. In all of these expressions g represents the coupling constant whose value controls the strength of the quark}gluon and gluon}gluon couplings. The strong interactions as given by the above lagrangian density are believed to bind the quarks and gluons into bound states, which correspond to the observed strongly interacting particles (or, hadrons), such as protons (p), neutrons (n), pions (n), kaons (K), etc. Table 1 lists the masses
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Table 1 Masses and quantum numbers of the lightest hadrons Particle
Quark content
Mass (GeV)
Spin
Isospin
n\(n>)[n] K>(K) K\(KM ) g o\(o>)[o] u KH>(KH) KH\(KM H) g f a p(n)
du (udM )[uu , ddM ] us (ds ) su (sdM ) uu , ddM , ss du (udM )[uu , ddM ] uu , ddM , ss us (ds ) su (sdM ) uu , ddM , ss uu , ddM , ss uu , ddM , ss uud(ddu)
0.140 0.494 0.494 0.547 0.770 0.782 0.892 0.892 0.958 0.980 0.980 0.938
0 0 0 0 1 1 1 1 0 0 0
1 0 1 0 0 0 1
[0.135] (0.498) (0.498)
(0.896) (0.896)
(0.940)
and some of the quantum numbers for all of the hadrons whose masses are less than 1 GeV. Considerably more states have masses above 1 GeV. For the present purposes the most signi"cant feature about this particle spectrum is that the lightest two quarks, u and d, have masses which are much smaller than all of the masses of the states which make up the particle spectrum. This suggests that the QCD dynamics may be well approximated by taking m , m +0, and working perturbatively in these masses divided by a scale, S B K , which is typical of the strong interactions. From the observed bound-state spectrum we expect Q K to be roughly 1 GeV. Q The approximation for which m and m vanish turns out to be a very useful one. This is because S B the QCD lagrangian acquires the very useful symmetries
u
u P(; c #; c ) , * * 0 0 d d
(2.1.2)
in this limit, where ; and ; are arbitrary 2;2 unitary matrices having unit determinant. The * 0 Dirac matrix c " (1#c ) projects onto the left-handed part of each of the quarks, u and d, while * c " (1!c ) projects onto their right-handed part. The group of symmetries which is obtained in 0 this way is G"S; (2);S; (2), with the subscripts &L' and &R' indicating the handedness of the * 0 quarks on which the corresponding factor of the group acts. A symmetry such as this which treats left- and right-handed fermions di!erently is called a chiral symmetry. These transformations are exact symmetries of QCD in the limit of vanishing m and m , but are only approximate symmetries S B when these masses take their real values. Because the approximate symmetry involved is chiral, the technique of expanding quantities in powers of the light-quark masses is called Chiral perturbation theory. If this symmetry, G, were not spontaneously broken by the QCD ground state, "X2, then we would expect all of the observed hadrons to fall into representations of G consisting of particles having approximately equal masses. This is not seen in the spectrum of observed hadrons, although
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the known particles do organize themselves into roughly degenerate representations of the approximate symmetry of isospin: S; (2). The isospin quantum number, I, for the observed S; (2) ' ' representations of the lightest hadrons are listed in Table 1. (The dimension of the corresponding representation is 2I#1.) Isospin symmetry can be understood at the quark level to consist of the diagonal subgroup of G, for which ; "; . That is, the approximate symmetry group which is * 0 seen to act on the particle states is that for which the left- and right-handed components of the quarks u and d rotate equally. This suggests that if QCD is to describe the experimentally observed hadron spectrum, then its ground state must spontaneously break the approximate symmetry group G down to the subgroup H"S; (2), for which: ' u u P; . (2.1.3) d d
There is indeed good theoretical evidence, such as from numerical calculations, that the ground state of QCD really does behave in this way. Given this symmetry-breaking pattern, we know that the low-energy spectrum of the theory must include the corresponding Goldstone bosons. If G had been an exact symmetry, then the corresponding Goldstone bosons would be exactly massless. But since G is only a real symmetry in the limit that m and m both vanish, it follows that the Goldstone bosons for spontaneous S B G breaking need only vanish with these quark masses. So long as the u and d quarks are much lighter than the natural scale (K+1 GeV) of the strong interactions, so must be these Goldstone bosons. Indeed, the lightest hadrons in the spectrum, n! and n, have precisely the quantum numbers which are required for them to be the Goldstone bosons for the symmetry-breaking pattern S; (2);S; (2)PS; (2). Particles such as these which are light, but not massless, * 0 ' because they are the Goldstone bosons only of an approximate symmetry of a problem are called pseudo-Goldstone bosons. Since the low-energy interactions of Goldstone bosons are strongly restricted by the symmetrybreaking pattern which guarantees their existence, it is possible to experimentally test this picture of pions as pseudo-Goldstone bosons. The remainder of this section is devoted to extracting some of the simplest predictions for pion interactions which can be obtained in this way. The fact that these predictions successfully describe the low-energy interactions of real pions gives support to the assumed symmetry-breaking pattern for the ground state of the strong interactions. 2.2. The low-energy variables In order to proceed, we must "rst construct the nonlinear realization for the case G"S; (2);S; (2) and H"S; (2). To do so, we "rst write out the representation we shall use * 0 '
In fact, the next-lightest particles, K and g, together with the pions have the quantum numbers to be the Goldstone bosons for the pattern S; (3);S; (3)PS; (3), which would be appropriate in the limit that the lightest three quarks, * 0 4 u, d and s, were all massless. The unbroken subgroup here, S; (3), again is the diagonal, handedness-independent, 4 subgroup.
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for the elements of each of these groups. Denoting the Pauli matrices, s"+q ,, L 0 1 0 !i 1 0 by q " ,q " and q " , 1 0 i 0 0 !1
we write:
g"
3S; (2);S; (2) , * 0
(2.2.1)
3S; (2) . '
(2.2.2)
e x* s
0
0
e x0 s
e x' s
0
0
e x' s
and
h"
We adopt here, and throughout the remainder of the section, an obvious vector notation for the three-component quantities uL, hL, uL, etc. In this representation, the Goldtone boson "eld becomes
;(h)"
e h s
0
0
e\ h s
,
(2.2.3)
and the nonlinear H transformation, c, is
c(h, g)"
e u s
0
0
e u s
.
(2.2.4)
2.2.1. A notational aside Before passing to the nonlinear realization, we brie#y pause to make contact between the variables as de"ned here, and those that are often used in the literature. We have de"ned the elements, g3G, the matrices ;(h), and c(h, g) in a block-diagonal form which emphasizes the leftand right-handed parts of the transformations:
g"
g * 0
0
, ;(h)"
; (h) * 0
0
,
c(h, g)"
h(h, g)
0
. (2.2.5) g ; (h) 0 h(h, g) 0 0 In terms of these quantities, the transformation law ;P;I "g;cR becomes ; P;I "g ; hR * * * * and ; P;I "g ; hR. It is common practice to work with the composite quantity, N, for which 0 0 0 0 the transformation rule does not depend on the implicitly de"ned matrix h. That is, if N,; ;R , * 0 then NPNI "g NgR . This transformation law has the advantage of involving only explicit, con* 0 stant matrices. In terms of the Goldstone boson "elds, h, we have ; "e h ) s";R , so 0 * N"; ;R "e h s. * 0 It is possible, and often convenient, to reformulate all of the Goldstone boson self-couplings that are obtained elsewhere in this section in terms of this "eld N. It is not possible to express the Goldstone-boson couplings to other "elds, s, in this way since the matrix c cannot be removed from the transformation law for these other "elds.
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2.2.2. The nonlinear realization The nonlinear realization is now obtained by constructing both hI (h, g) and u"u(h, g), using the condition g;(h)";(hI )c. For the groups under consideration this construction may be performed in closed form by using the identity: exp[ia ) s]"cos a#ia( ) s sin a ,
(2.2.6)
where a"(a ) a, and a( "a/a. Using this identity to multiply out both sides of the de"ning equation g;(h)";(hI )c, and equating the coe$cients of 1 and s, separately for the left- and right-handed parts of the matrices, gives explicit expressions for dh"n and u. If g "exp[(i/2)x ) s], and de"ning * 0 * 0 x , (x #x ) and x ,(x !x ), then: 0 * 0 ' * h n"h;x # (tan h/2#cot h/2)[x !hK (hK ) x )]#hK (hK ) x ) , 4 2 "x #h;u #O(h) ; (2.2.7) 4 u"x #(hK ;x )tan h/2 4 "x #[(h;x )/2]#O(h) . (2.2.8) 4 For future reference we notice that the transformation law for h implies that the three broken generators of G"S; (2);S; (2) form an irreducible, three-dimensional representation of the * 0 unbroken subgroup, H"S; (2). ' Similarly, evaluating the combination
s i ;R* ;" e ) I I 2
s i # A ) , I 2 !s s
(2.2.9)
gives the quantities with which the invariant lagrangian is built: e "((sin h)/h)* h!((sin h!h)/h)(h ) * h)h , I I I "* h(1! h)# (h ) * h)h#O(h) ; (2.2.10) I I A "!2(sin(h/2)/h)(h;* h) I I "! h;* h#O(h) . (2.2.11) I Notice that e is odd, and A is even, under the interchange hP!h. The low-energy Goldstone I I boson lagrangian will be required to be invariant under such an inversion of h, since this is a consequence of the parity invariance of the underlying QCD theory. It is useful to also record here the G-transformation rules for the other "elds which can appear in the low-energy theory. Of particular interest are the nucleons (neutrons and protons) since low-energy pion}nucleon interactions are amenable to experimental study. The nucleon transformation rules under G"S; (2);S; (2) are completely dictated by their transformations under * 0 the unbroken isospin subgroup, H"S; (2). Since the nucleons form an isodoublet, N"(N), they ' L
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transform under isospin according to dN"(i/2)x ) sN. The rule for the complete G tranforma' tions is therefore simply dN"(i/2)u ) sN .
(2.2.12)
We therefore see that the appropriate covariant derivative for nucleons is D N"* N!(i/2)A (h) ) sN . I I I
(2.2.13)
2.3. Invariant lagrangians We may now turn to the construction of the invariant lagrangian which governs the low-energy form for pion interactions. The lagrangian describing pion self-interactions involving the fewest derivatives is uniquely determined up to an overall normalizing constant. As was discussed in detail in the previous section, this is because of the irreducibility of the transformation rules of the broken generators, X" sc , under the unbroken isospin transformations. The most general G-invariant lagrangian density involving only two derivatives is F L "! g( (h)* hK*IhL#(higher-derivative terms) , I LL 2 KL
(2.3.1)
where the metric, g( , on G/H is KL g (h)"d eP eQ "d ((sin h)/h)#h h ((h!sin h)/h) KL PQ K L KL K L (2.3.2) "d (1!h)#h h #O(h) . K L KL For applications to pion scattering it is useful to canonically normalize the pion "elds, that is, to ensure that their kinetic terms take the form: !* p ) *Ip. This requires the rescaling: h"p/F. I With this choice we have
p!F sin(p/F) 1 F sin(p/F) L "! * p ) *Ip# (p ) * p)(p ) *Ip) LL I I p p 2 #(higher-derivative terms) 1 1 "! * p ) *Ip! (p ) * p)(p ) *Ip)#O(n)#2 . I I 2 2F
(2.3.3)
An integration by parts has been performed in writing the n term of the expansion of the lagrangian. The couplings between nucleons and pions to lowest order in the derivative expansion involve only one derivative. The most general form for these that is consistent with the nonlinearly-realized G-invariance, and with parity invariance, is L "!NM (R. !(i/2)A . (h) ) s#m )N!(ig/2)(NM cIc sN)e (h) L, , I "!NM (R. #m )N!(ig/2F)(NM cIc sN) ) * p , I i (NM cIsN) ) (p;* p)#2 . ! I 2F
(2.3.4)
The ellipses here represent terms which involve either three or more powers of the pion "eld, more than two powers of the nucleon "eld, or involve more than one derivative.
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Clearly, only the one constant F need be determined in order to completely "x the dominant low-energy pion self-interactions, and a second constant, g, is also required to determine the lowest-order pion}nucleon couplings. (The constant g should not be confused with the QCD coupling constant of the lagrangian (2.1.1), which plays no role in the formulas which follow.) 2.3.1. Conserved currents For future reference it is instructive to compute the Noether currents for the symmetry group G"S; (2);S; (2) in both the underlying theory (i.e. QCD), and in the e!ective low-energy * 0 pion}nucleon theory. In QCD, the symmetry transformation under G is given by dq"(i/2)(x c #x c ) ) sq, where * * 0 0 q"(S) denotes the two-component quantity containing the two lightest quarks. The corresponding B Noether currents that are obtained from the QCD lagrangian, Eq. (2.1.1) are: I"(i/2)q cIc sq and I"(i/2)q cIc sq . (2.3.5) * * 0 0 The current, I, which corresponds to the unbroken S; (2) isospin symmetry is therefore: ' ' I" I# I"(i/2)q cIsq. The current for the broken symmetry is similarly: I " ' * 0 I! I"(i/2)q cIc sq. * 0 In the e!ective pion}nucleon theory the corresponding current may also be constructed using the known action of G on p and N, and using the lagrangian, whose lowest-derivative terms are given by Eqs. (2.3.3) and (2.3.4). Keeping only the terms involving a single pion or only two nucleons, at the lowest order in the derivative expansion, then gives: I"!(p;*Ip)#(i/2)NM cIsN#2 , ' (2.3.6) I "F*Ip#(ig/2)NM cIc sN#2 . There are an in"nite number of higher-order terms in these currents corresponding to the in"nite number of interactions in the e!ective pion}nucleon lagrangian. All of the terms not written explicitly above involve additional factors of the "elds p or N, or involve more derivatives of these "elds than do the terms displayed. 2.3.2. Determining F and g These expressions for the Noether currents for G turn out to furnish a handle for experimentally determining the constants F and g. This is because, as is made explicit in the following section, experimental information exists concerning the value of some of the matrix elements of the broken current I. This experimental information exists because it is precisely the current I which appears in that part of the weak-interaction lagrangian which describes transitions from d quarks to u quarks. Since these transitions are responsible for many reactions, including all nuclear b-decays, free-neutron decay, and n! decay, the corresponding matrix elements of this current can be measured. The terms in the underlying lagrangian which describe these decays are obtained by supplementing the QCD interactions of Eq. (2.1.1) with the weak-interaction term: G cos h ! u cJ(1#c ) dll c (1#c )l#h.c . L " $ J (2
(2.3.7)
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Here the Dirac spinor "eld l and the Majorana "eld ll , respectively, represent a charged lepton (in practice, the electron and muon) and the corresponding neutrino. G is the Fermi coupling constant, $ which is determined from the muon decay rate to be G "1.16649(2);10\ GeV\. The angle $ h is called the Cabbibo angle, and it parameterizes the fact that the size of the coupling constant, ! G cos h , as seen in superallowed nuclear b-decays is smaller than G as is measured in muon $ $ decay. Numerically, cos h "0.9753(6). ! The main feature to be noticed from Eq. (2.3.7) is that the quark combination which appears is a linear combination of the conserved S; (2);S; (2) currents: * 0 iu cI(1#c )d"iq cIc (q #iq )q * "[( j )I #i( j )I ]#[( j )I #i( j )I ] . ' '
(2.3.8)
In preparation for using Eqs. (2.3.6) we have re-expressed the left-handed currents which appear in the weak interactions in favour of the axial and vector currents using: I"( I# I). * ' To compute the decay rate for the reaction n>Pk>l we require the following matrix element: I "n>2. The part of this matrix element which involves strongly interacting particles 1k>,l "L I is 1X" I "n>2, where "X2 is the QCD ground state. The isospin current, I, does not appear in ' n> decay because its matrix element vanishes due to the parity invariance of the strong interactions. The most general form for this matrix element which is consistent with PoincareH and isospin invariance is given by iF qIe OV d , 1X"( j )I (x)"n (q)2" p L K ((2p)2q KL
(2.3.9)
where it is conventional to extract the numerical factor 1/((2p)2, and the pion states are labelled here as members, "n 2 (n"1,2,3), of an isotriplet. These are related to the physical states, having L de"nite electric charge, by: "n!2"( ("n 2Gi"n 2), and "n2""n 2. The only unknown quantity in this matrix element is the constant F , which is inferred to be L F "92 MeV by comparing the prediction, 1/q "(G cos h Fmm /4p)(1!m/m), with the L $ A L I L I L observed mean lifetime, q "2.6030(24);10\ s, for the decay n>Pk>l . I Now, to lowest order in the derivative expansion, the matrix element of Eq. (2.3.9) can be directly evaluated as a function of the parameter F using the second of Eq. (2.3.6). Comparing these results permits the inference F"F "92 MeV . L
(2.3.10)
With this constant in hand, we may now use the low-energy e!ective lagrangian to predict the low-energy pion self-interactions. Before proceeding to these predictions, we "rst repeat these steps for another matrix element in order to infer the value of the constant, g, which governs the size of the pion}nucleon coupling. We once again consider the weak interaction, Eq. (2.3.7), but this time consider its prediction for the decay rate of a free neutron into a proton, an electron and an antineutrino: nPpel . In this case, C the most general PoincareH -, parity-, time-reversal- and isospin-invariant form for the desired matrix
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element is ie OV 1N(k, p)" I(x)"N(l, f)2" u (k, p)[F (q)cI#F (q)cIJq ]su(l, f) , ' J 2(2p) ie OV u (k,p)[G (q)cIc #G (q)c qI]su(l, f) . 1N(k, p)" I (x)"N(l, f)2" 2(2p)
(2.3.11)
Here, lI and kI are the four-momenta of the initial and "nal nucleons, and qI"(l!k)I is their di!erence. f and p similarly represent the spins of the initial and "nal nucleons. u(k, p) is the Dirac spinor for a free particle having momentum kI, with k#m "0, and spin p. (Our normalization , is: u (k, p)u(k, p )"(m /k)d .) Finally, cIJ stands for the commutator [cI, cJ]. , NNY The unknowns in this matrix element are the four Lorentz-invariant functions, F , F , G and G , of the invariant momentum transfer, q. These functions are not completely arbitrary, however, since they must encode the fact that we are working in a limit where G"S; (2);S; (2) is taken * 0 to be a symmetry of the QCD lagrangian. The implications of G-invariance are easily extracted by demanding current conservation, * I"* I "0, for all of the currents. Keeping in mind that I ' I the nucleons have equal mass in the G-invariant limit in which we are working, this implies no conditions on the functions F and F , but implies for the others: 2im G (q)"qG (q) . (2.3.12) , In the rest frame of the decaying neutron, the components of the momentum transfer, qI, are at most of order 1 MeV. Since this is much smaller than the typical strong-interaction scale, K&1 GeV, which characterizes the matrix element, for neutron decay it su$ces to simplify Eqs. (2.3.11) using qI+0. In this approximation the neutron decay rate depends only on the two unknown constants, F (0) and G (0). Since the constants F (0) and G (0) correspond to the low-energy limit of these current matrix elements, they may be related to the constants which appear in the dominant terms of the low-energy e!ective lagrangian. This may be done by using Eqs. (2.3.6) to directly evaluate the matrix elements of Eqs. (2.3.11). Doing so, we "nd contributions from the two Feynman graphs of Fig. 2. The "rst of these gives the direct matrix element of Eqs. (2.3.6), and contributes to the form factors F and G . The second graph uses the NNn interaction of the e!ective lagrangian, Eq. (2.3.4), together with the vacuum-pion matrix element of Eq. (2.3.9). It contributes only to the form factor G . Evaluating these graphs, we "nd F "1, G "g and G "2igm /q , (2.3.13) , from which we see F (0)"1 and G (0)"g. The factor 1/q in G comes from the massless pion propagator in the second of Fig. 2. Notice that this result for G is precisely what is required to satisfy the current-conservation condition of Eq. (2.3.12). The "nding that F (0)"1 states that this part of the matrix element is not renormalized by the strong interactions, since this value for F (0) is the same as would have been obtained if the matrix elements of I were taken using the underlying quark states rather than with the composite nucleon ' states. F (0) is the same for both quarks and nucleons because F (0) is the quantity which determines the matrix elements in these states of the conserved isospin charges, I"dr . But ' these have matrix elements which depend only on the S; (2) transformation properties of the '
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Fig. 2. The Feynman graphs which give the dominant nucleon matrix elements of the Noether currents in the low-energy e!ective theory. Solid lines represent nucleons, and dashed lines represent pions.
states whose matrix elements are taken. Since both quarks and nucleons are isodoublets, and since inspection of Eq. (2.3.5) shows that quarks have F (0)"1, the same must be true for nucleons. The same argument does not hold for the axial current because this is a current for a symmetry which is spontaneously broken. This turns out to imply that the corresponding conserved charge is not well de"ned when acting on particle states, and so G (0) need not be unity. We "nally arrive at the desired conclusion: the neutron decay rate, which is completely determined by the constants F (0)"1 and G (0)"g, can be used to experimentally infer the numerical value taken by the remaining constant, g, of the e!ective lagrangian. The measured neutron mean life (which is q "887(2) s) then implies g"1.26. Having determined from experiment the values taken by F and g, we are now in a position to use the e!ective pion}nucleon lagrangian to predict the low-energy properties of pion}pion and pion}nucleon interactions. 2.3.3. The Goldberger}Treiman relation Historically, the trilinear N}N}n interaction has been written with no derivatives, as a Yukawa coupling: L "ig (NM c sN) ) p , (2.3.14) ,,L ,,L with the constant g found from phenomenological studies to be close to 14. But the value of this ,,L constant can be predicted in terms of the constant g, and this prediction serves as the "rst test of the low-energy pion}nucleon lagrangian. The prediction starts with the trilinear N}N}n interaction of Eq. (2.3.4): L "!(ig/2F )(NM cIc sN) ) R p , (2.3.15) ,,L L I and performs an integration by parts to move the derivative to the nucleon "elds. One then uses the lowest-order equations of motion for N: i.e. (R. #m )N"0, to simplify the result. One obtains , a result of the form of Eq. (2.3.14), but with g
"gm /F . (2.3.16) ,,L , L Using the experimental values: g"1.26, m "940 MeV and F "92 MeV gives the prediction , L g "12.8, which agrees well with the phenomenologically inferred value. This prediction, ,,L Eq. (2.3.16), is known as the Goldberger}Treiman relation. We turn now to one last dangling issue which remains to be addressed before we can compute low-energy pion}pion and pion}nucleon scattering.
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2.4. Explicit symmetry-breaking Notice that the e!ective lagrangian, Eqs. (2.3.3) and (2.3.4), has very de"nite implications for the masses of the pions and nucleons. It states that the pion multiplet must be exactly massless, and that the nucleon masses must be equal. Since these predictions rely only on the assumption of unbroken G invariance, and since G-invariance only holds for QCD in the limit that m and S m vanish, corrections to the pion and nucleon mass predictions can only be inferred by including B the e!ects of the symmetry-breaking quark mass terms for the low energy e!ective theory. We do so, in this section, to lowest order in the light-quark masses. The quark mass terms in the QCD lagrangian are proportional to q Mc q#h.c., where * m 0 M" S 0 m B is the light-quark mass matrix. Under the G"S; (2);S; (2) symmetry, qP(g c #g c )q, this * 0 * * 0 0 transforms into:
q Mc qPq gR Mg c q#h.c . (2.4.1) * 0 * * Although this is not invariant, it would have been invariant if the mass matrix had been a "eld which had also transformed under G according to: MPg MgR . 0 * We imagine the e!ective pion}nucleon theory having an expansion in the light quark masses, M: L "L #L #2, where the subscript indicates the power of M it contains. Each of these terms may be separately expanded in powers of the derivatives, and of the "elds n and N. The construction to this point has given the lowest-derivative terms which can appear in L . Our goal now is to determine the most general form which may be taken by L , and which contains no derivatives of any "elds. This will give the dominant symmetry-breaking contribution at low energies. 2.4.1. Pions only: vacuum alignment We start by focussing on the part of L which depends only on the pion "elds. The form taken by L may be obtained from the following argument. We require that L be G-invariant, but only if we take MPg MgR in addition to transforming the "elds n in their usual way. 0 * It is straightforward to construct one such a term involving only the pion "elds. The simplest construction is to use the quantity N,u uR "e h s"cos h#ihK ) s sin h, de"ned in Section 2.2.1, * 0 which transforms according to NPNI "g NgR . (Recall here that h and hK are de"ned by h"(h ) h * 0 and hK "h/h.) A possible lagrangian therefore is: L "!A Re Tr[MN]!B Im Tr[MN] (2.4.2) LL "!A(m #m )cos h!B(m !m )h (sin h)/h . (2.4.3) S B S B Clearly this generates a potential energy which is a function of h, as is possible because of the explicit breaking of the S; (2);S; (2) symmetry by the quark masses. As a result, all values * 0 for h are not equally good descriptions of the vacuum, and it is necessary to minimize the potential in order to determine the vacuum value for h. This choosing of the vacuum value for the
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pseudo-Goldstone "elds after the introduction of explicit symmetry-breaking is a process known as vacuum alignment. In the present instance the potential is minimized by h "h "0, and has the schematic form <(h)"!L (h)"a cos h#b sin h"!"a"cos(h!h ), for h"h and a and b (or, equivalently, a and h ) constants. This, once minimized (giving h "h ) and expanded about the minimum
(with h"h #h ), the potential becomes <(h )"!"a"cos h , leaving our lagrangian density of
the form L "(M/2)Tr[M(N#NR)] LL "(m #m )M cos h S B (2.4.4) "m [F! p ) p!(1/4!F)(p ) p)#O(n)] , L L L with the constant M positive. Eq. (2.4.4) gives the required symmetry-breaking interaction, where the p's are chosen so that the vacuum is at p"0, and, in the last line, we have also eliminated the arbitrary parameter, M, which has the dimensions of mass, in terms of the common mass, m , we "nd for all three pions: L m"(m #m )M/F . (2.4.5) L S B L There are several features here worth highlighting. Firstly, notice that all of the pion selfinteractions necessarily preserve isospin to this order in the derivative and quark-mass expansions. This implies, among other things, degenerate masses for all three pions. This preservation of isospin does not rely on the isospin-breaking di!erence, m !m , being small in comparison with m or m . S B S B Rather, it relies only on m and m both being small compared to the characteristic scale of QCD. S B We must look elsewhere for an understanding of the observed mass di!erence between the charged and neutral pions, such as to the isospin-breaking electromagnetic interactions. Secondly, the lagrangian of Eq. (2.4.4) necessarily implies a quark-mass-dependent contribution to the vacuum-energy density, !o "L (p"0)"mF"(m #m )M. This contribution T LL L L S B permits a physical interpretation for the parameter M, as follows. In the underlying theory the derivative of the total vacuum energy, o , with respect to any quark mass is given by T Ro T "1X"q q"X2 , (2.4.6) Rm O where "X2 is the QCD ground state. We see, by comparison with the pion scalar potential, Eq. (2.4.4), that 1X"u u"X2"1X"dM d"X2"!M#2 ,
(2.4.7)
where the ellipses here denote the contributions due to quantum e!ects in the low-energy pion}nucleon theory. Evidently, M+[mF/(m #m )] gives the size of the expectation value L L S B which is responsible for the spontaneous breaking of the chiral S; (2);S; (2) symmetry. * 0 Using the values m "140 MeV, F "92 MeV and 4.5 MeV(m #m (14 MeV then gives L L S B 230 MeV(M(330 MeV for this scale. Next, we remark that since none of the terms which appear in L depend on derivatives of p, LL they do not at all a!ect expressions (2.3.6) for the conserved Noether currents of the theory. We
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therefore need not at all change the above analysis which determined the experimental values for the constants F and g from pion and nucleon weak decays. Finally, we note that the mass term given in Eq. (2.4.4) is the only possible term (up to normalization) which is linear in M and depends only on h, and not on its derivatives. This uniqueness follows from the impossibility of building a G-invariant scalar potential. To see this, we write L "Tr(MO)#h.c., for O(h) a 2;2 matrix function of the Goldstone boson "elds. O must transform under G according to OPg OgR . O "N satis"es these conditions, but suppose O"O * 0 were a second, independent solution. In this case, the combination <(h)"Tr[O OR ] or <(h)"det[O OR ] would be a G-invariant scalar potential, as would any of the eigenvalues of the matrix O OR . Since we know from the previous section no such potential is possible, it follows that an independent quantity, O , also cannot exist. 2.4.2. Including nucleons We next consider the part of L which involves precisely two factors of the nucleon "eld, and no derivatives. That is: L "!NM f (h, M)c N#h.c., where the transformation laws: ,L * hPhI and NPNI "h(h, g)N imply that the matrix-valued function, f (h, M), must satisfy: f(hI , g MgR )"hf (h, M)hR. 0 * The solution, unique up to normalization, to this condition is f"uR Mu , where 0 * u "uR "exp[(i/2)h ) s]. We therefore "nd: * 0 L "!jNM [e h sMe h s]N#h.c. ,L sin h NM +h ) s, M,c N "!jNM MN!ij 2h
#j
sin h/2 NM [h ) sMh ) s#hM]N h
"!jNM MN!(ij/2F )NM +p ) s, M,N L #(j/4F)NM [p ) sMp ) s#p ) pM]N#2 . (2.4.8) L As usual, curly braces denote the anticommutator of the corresponding matrices: +A, B,"AB#BA. Notice that besides providing nonderivative pion}nucleon couplings, this term also splits the neutron and proton masses by an amount: d m "j(m !m ) . (2.4.9) H , B S Even though they do not contribute to the pion mass splittings, the di!ering u and d quark masses do act to split the masses of the nucleon isodoublet. Now, m and m may be determined by S B repeating the above analysis for the masses of the lightest eight mesons, n, K, g, under the assumption that these are all pseudo-Goldstone bosons for the symmetry group S; (3);S; (3) * 0 which is appropriate when the s quark is assumed to be light in addition to the u and d quarks. In principle, once this has been done, Eq. (2.4.9) permits the constant j to be extracted from the experimental di!erence, m !m "1.293318(9) MeV. It is important in so doing to include also the L N
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contributions of the electromagnetic interactions to this mass di!erence, since these are similar in size to Eq. (2.4.9). 2.5. Soft pion theorems We may now proceed to work out some of the implications of the e!ective lagrangian for low-energy pion}pion scattering. As usual, the "rst question must be to ask which interactions need be considered in which Feynman graphs in order to properly mimic the low-energy expansion of the underlying QCD theory. To this end we use the powercounting results of Section 1. 2.5.1. Power counting For simplicity, we consider here only the case where there are no nucleons in the low-energy theory, since only in this case we can directly use the power-counting results obtained in Section 1. As has already been emphasized, these results cannot be directly applied to nucleons, because they were derived under the assumption of very light, relativistic fermions, and the nucleons in the low-energy pion}nucleon theory are very massive and nonrelativistic at the energies of interest. 2.5.1.1. Power counting in the symmetric limit. We start by omitting all symmetry-breaking terms of the pion lagrangian which are proportional to the quark masses. These are considered in the next section. In this case the pions are massless, and their Goldstone-boson lagrangian has the form given in Eq. (2.3.1): (h)R hKRIhLR hNRJhO#2] , (2.5.1) L "!F[ g( (h)R hKRIhL#KA h I I J LL L KL Q KLNO where c is a dimensionless number which is, in principle, calculable from QCD. The ellipses represent an in"nite sequence of additional terms, including several others which also have four derivatives, as does the displayed term proportional to c. Eq. (2.5.1) has the form of Eq. (1.10.1), with: f"(F K , v"F , and M"K &4pF &1 GeV. L Q L Q L The powercounting estimate of Eq. (1.10.9) for the scattering of pions becomes
1 # K * q >*> GI I\4GI Q , (2.5.2) AI (q)&FK # L Q F 4pF K L L Q which is a famous result, due "rst to Weinberg. For a given observable, A (q), the number, E, of external particles is "xed. In this case it is only # the last two factors of Eq. (2.5.2) which di!erentiate di!erent types of contributions. We remark that in practical applications for pion scattering, it happens that K &4pF &1 GeV. As a result, Q L the second-last factor, (K /4pF )*, turns out to be O(1) for realistic pion scattering. This means Q L that it is only the last factor which controls the importance of various interactions. According to Eq. (2.5.2), the contribution of higher-derivative interactions is clearly only suppressed by the ratio q/K , which limits us to considering only low-energy pion dynamics near Q threshhold. The dominant term in the expansion in powers of q/K corresponds to choosing the Q I thank John Donoghue for reminding me of the importance of this electromagnetic contribution.
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smallest possible value for the quantity P"2#2¸# (k!2)< . It is noteworthy, when using GI GI this expression, to remark that all of the interactions have at least two derivatives (we temporarily ignore pion masses, etc.), and so k52. Furthermore, it is only the "rst term in the derivative expansion which has k"2, and so the k"2 interaction is unique. As a result the lowest value possible for P is P"2, and this is only possible if ¸"0 and if < "0 GI for all k'2. This implies that the dominant contribution to pion scattering is computed by using only the "rst term in the e!ective lagrangian, Eq. (2.5.1), and working only to tree level with these interactions. The next-to-leading terms in q/K then have P"4, which can arise in either one of two ways. Q (i) We can have ¸"1 and < "0 for all k'2; or (ii) we can have ¸"0 and < "1 for some i, GI G while < can take any value. This states that the subleading, O(q), contribution is obtained by G either working to one loop order using only the interactions of the "rst term of the lagrangian of Eq. (2.5.1), or using tree graphs having exactly one vertex taken from the four-derivative interactions in the lagrangian, as well as any number of interactions from the "rst term in the lagrangian. In this way, it is clear how to compute any given order in the expansion in powers of q/K . Q 2.5.1.2. Symmetry-breaking terms. Before proceeding to calculations, we must also include one other feature. We must track the appearance of the explicit symmetry-breaking terms, of which we only keep those which are proportional to a single power of the light-quark masses, m &m , m . O S B These vertices, which come from the symmetry-breaking term L , can be very simply included into the power-counting results of Section 1 by considering all of the non-derivative interactions to be suppressed by a dimensionless coupling, c &(m /K )&(m/K). I O Q L Q With these points in mind, Eq. (2.5.2) becomes
1 # q . m O AI (q)&K F # Q L F K K L Q Q
4
I GI
,
(2.5.3)
where P can be written in either of two equivalent ways: P"4!E# (k#i!4)< "2#2¸# (k!2)< . GI GI GI GI
(2.5.4)
Using the "rst of these forms for P we see that the contribution to the powercounting estimate due to the insertion of the symmetry-breaking terms with k"0 is
q G\4GI m 4GI O . K K Q Q I
(2.5.5)
The dangerous interactions are clearly those for which i(4. For example, the pion mass term, &mp, has i"2, and so contributes the factor L I
m K 4GI O Q . q
(2.5.6)
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Fig. 3. The Feynman graphs which give the dominant contributions to pion}pion scattering in the low-energy pion}nucleon theory. The "rst graph uses a vertex involving two derivatives. The second involves the pion mass, but no derivatives.
Now, we are interested in applications for which the external momenta, q, are of order several hundred MeV, and so q&m &(m K . For these momenta the factor (m K /q)& L O Q O Q (m/q)&O(1). It follows that it is not a good approximation to perturb in the pion mass term, L and so we should include this term in the unperturbed lagrangian. That is, we should include the pion mass explicitly into the pion propagator so that G (q)"!i/(q#m!ie). It is legitL L imate to perturb in all of the other symmetry-breaking interactions of the scalar potential, however, since for these k"0 and i54. We now turn to speci"c interactions, starting with pion}pion scattering, for which E"4. In this case the above powercounting shows that there are precisely two dominant contributions. The "rst of these consists of the tree graph of Fig. 3, using the four-point vertex from the G-invariant term which involves two derivatives, Eq. (2.3.3). The second contribution is also obtained using the graph of Fig. 3, but this time takes the four-point pion self-interaction from the symmetry-breaking scalar potential of Eq. (2.4.4). Although the "rst term is unsuppressed by the light-quark masses, it gives a contribution which is down relative to the second term by two powers of external momenta, q. Both are therefore comparable in size for pions near threshhold, q&m. All other graphs are L smaller than these two by powers of either m or q. O 2.5.2. Pion}pion scattering We now compute pion}pion scattering by evaluating the graphs of Fig. 2 using the e!ective pion self-couplings of Eqs. (2.3.3) and (2.4.4). A straightforward calculation gives the following S matrix element for the scattering n n Pn n : ? @ A B id(p #p !p !p ) ? @ A B A , (2.5.7) S(n n Pn n )" ?@ AB ? @ A B (2p)(pppp ? @ A B with A "(1/F)[d d (s!m )#d d (t!m)#d d (u!m)] , (2.5.8) ?@ AB L ?@ AB L ?A @B L ?B @A L where the Lorentz-invariant Mandelstam variables, s"!(p #p ), t"!(p !p ) and ? @ ? A u"!(p !p ) are related by the identity: s#t#u"4m. In the CM frame s, t and u have ? B L simple expressions in terms of the pion energy, E, and three-momentum, q: s"4E, t"!2E#2qcos 0 and u"!2E!2qcos 0. Here 0 denotes the scattering angle, also in the CM frame.
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Table 2 Theory vs. experiment for low-energy pion scattering Parameter
Leading order
a b a a b
7m/32nF L L m/4nF L L m/24nF L L !m/16nF L L !m/8nF L L
0.16 0.18 0.030 !0.044 !0.089
Next order
Experiment
0.20 0.26 0.036 !0.041 !0.070
0.26(5) 0.25(3) 0.038(2) !0.028(12) !0.082(8)
Comparison with the data is made using channels having de"nite angular momentum and isospin. If we decompose A into combinations, A', having de"nite initial isospin: ?@ AB A "Ad d #A(d d !d d )#A[(d d #d d )!d d ] , (2.5.9) ?A @B ?B @A ?A @B ?B @A ?@ AB ?@ AB ?@ AB then 2s!m t!u s!2m L, L . A" A" , A"! AB F F F L L L The next step is to resolve these amplitudes into partial waves:
(2.5.10)
1 (2.5.11) d cos 0Pl (cos 0)A' , A' l , 64p \ where Pl (cos 0), as usual, denote the Legendre polynomials (so P (x)"1 and P (x)"x). Since all of the dependence on 0 appears through the variables t and u, and since Eqs. (2.5.10) give A and A as functions of s only, it is clear that only the partial wave l"0 is predicted at lowest order for the even isospin con"gurations. Also, since A is strictly linear in cos 0, it only involves the partial wave l"1. The actual comparison with the data is made by expanding the (real part of ) A' l in powers of the squared pion momentum: q/m"E/m!1"(s!4m)/4m. That is, writing L L L L l A' #b'l q/m#2) , (2.5.12) l "(q/m) (a' L l L de"nes the pion scattering lengths, a'l , and slopes, b'l . Applying these de"nitions to Eqs. (2.5.10) gives the predictions of the second and third columns of Table 2. Column three gives the numerical value corresponding to the analytic expression which is given in column two. The predictions including the next-order terms in the q/K expansion have also been worked out, and are given in Q the fourth column of this table. Comparison of these predictions with experiment is not straightforward, since it is not feasible to directly perform pion}pion scattering experiments. Instead, the pion}pion scattering amplitudes
I have taken these values from the excellent book Dynamics of the Standard Model by Donoghue, Golowich and Holstein [16].
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at low energies are inferred from their in#uence on the "nal state in other processes, such as KPnnel or nNPnnN. The experimental results, as obtained from kaon decays, for those C quantities which are predicted to be nonzero at lowest order are listed in the right-hand most column of Table 2. Data also exist for other partial waves which are predicted to vanish at lowest order, such as I"0, l"2, and these are found to be in good agreement with the nonzero predictions which arise at next-to-leading order in the low-energy expansion. This example nicely illustrates the predictive power which is possible with a low-energy e!ective lagrangian, even if it is impossible to predict the values for the couplings of this lagrangian in terms of an underlying theory. This predictive power arises because many observables (e.g. the pion scattering lengths and slopes) are all parameterized in terms of a single constant (the decay constant, F ) which can be extracted directly from experiment. We emphasize that this predictive L power holds regardless of the renormalizability of the e!ective theory. Computing to higher orders involves the introduction of more parameters, but predictions remain possible provided that more observables are computed than there are parameters to "x from experiment. The information underlying these predictions comes from the symmetries of the underlying theory, as well as the restrictions due to the comparatively small number of possible interactions which can appear at low orders of the low-energy expansion.
3. Magnons: nonrelativistic applications We now turn to a second application, this time to a nonrelativistic system. Besides once again illustrating the utility of the e!ective-lagrangian techniques, this example shows how the analysis can be applied to more complicated condensed-matter systems. It also illustrates how e!ective lagrangians permit the separation of the generic predictions which follow only from general properties such as the symmetry-breaking patterns, from the details of the models which may be used to establish these symmetry-breaking patterns from the underlying physics. We take as our application the macroscopic behaviour of ferromagnets and antiferromagnets. These systems exhibit a transition at low temperatures to a phase which is characterized by a bulk order parameter, which we call S for the ferromagnet and N for the antiferromagnet, which transforms under rotations as a vector. For ferromagnets this order parameter can be taken to be the overall magnetization of the sample. Because this order parameter spontaneously breaks the rotational symmetry, Goldstone bosons must exist and so must appear in any low-energy (or long-wavelength) description of these systems. It is the low-energy interaction of these Goldstone bosons which is described in this section. The distinction between a ferromagnet and an antiferromagnet requires more information concerning the underlying material. As for most condensed-matter systems, the underlying microscopic system consists of an enormous number of electromagnetically interacting electrons and atomic nuclei. One picture of what is going on in a ferromagnet or antiferromagnet consists of imagining the electrons being reasonably localized to their corresponding atoms, with these atoms carrying a net magnetic moment due to its having a net electronic spin. The electron in each atom which carries the net spin interacts with its counterparts on neighbouring atoms, resulting in (among other things) an e!ective spin}spin interaction between these atoms. This spin}spin interaction can come about due to the exchange part of the Coulomb interaction, which arises due
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Fig. 4. The distinction between ferromagnets (F) and antiferromagnets (AF).
to the antisymmetrization of the wave function which is required because of the statistics of the electrons. Under varying circumstances one might suppose this spin}spin interaction to either favour the mutual alignment of neighbouring spins, or their antialignment (where the spins line up to point in opposite directions). The behaviour of such mutually interacting electronic or atomic spins may then be investigated by abstracting out just this spin dynamics into a simplifying model. For example, the interacting electrons can be replaced by a system of spins which are localized to each of the lattice sites which de"ne the nuclear positions in the solid. The mutual interactions of the atoms can be reduced to a spin}spin coupling having a phenomenological sign and magnitude, according to whether it is energetically favourable for neighbouring spins to be aligned or antialigned. Such models show that at low temperatures macroscopic numbers of these spins tend to either align or antialign, according to which of these takes less energy. A system for which neighbouring spins tend to align, prefers to acquire a net magnetization since the magnetic moment of each atom adds to give a macroscopically large total. This is a ferromagnet. If neighbouring spins prefer to antialign, then the order of the ground state consists of spins which alternate in their alignment, with every other spin pointing in a "xed direction, and the others pointing in the opposite direction. Such an arrangement is called antiferromagnetic. These two alternative arrangements are pictured in Fig. 4. The statistical mechanics of such spin models can successfully describe many features of real ferromagnets and antiferromagnets. So long as calculations are based on models, however, it is di$cult to quantitatively assess their accuracy. One of the purposes of the present chapter is to show that some predictions for these systems are very robust, since they do not rely on more than the qualitative features of the models. The robust predictions are those which can be formulated completely within the framework of a low-energy e!ective theory, and which therefore rely only on the spectrum and symmetries which dominate at low energies. The accuracy of this kind of prediction can be quantitatively assessed since this accuracy is controlled by the domain of validity of the e!ective theory itself. The role played in this kind of calculation by the details of an underlying model is simply the prediction of the quantum numbers and symmetries of the low-energy degrees of freedom, and so the model need only get these qualitative features right in order to accurately reproduce the proper low-energy behaviour. This line of reasoning, in which some quantitative predictions can be justi"ed as general low-energy features of a given system can be of great practical importance. For example, some very high-precision measurements are now based on the macroscopic behaviour of complicated condensed matter systems. Examples are the Josephson e!ect, or the Integer Quantum Hall e!ect, both of which have been used to "x the best measured value for the electromagnetic "ne structure constant, a. These determinations are accurate to within very small fractions of a percent. We should only believe such a determination of a if we can equally accurately justify the theoretical predictions of the e!ects on which the determination is based } a very tall order if the prediction is to be based on
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a model of the underlying system. Happily, such an accuracy is possible, and is one of the fruits of an e!ective lagrangian analysis of these systems. We now turn to the application of these e!ective-lagrangian techniques to the description of the long-distance, low-energy behaviour of the ordered spin systems. 3.1. Antiferromagnetism: ¹ invariance We start with applications to the low-energy properties of antiferromagnets. We do so because antiferromagnets preserve a type of time-reversal symmetry, which makes the analysis of its low-energy behaviour fairly similar to what would apply for relativistic systems. For an antiferromagnet, the order parameter, N, can be taken to be the staggered sum of the spins, s , for each lattice site, &i': G N" (!)Gs , (3.1.1) G G where the sign, (!)G, is positive for one sublattice for which all spins are parallel in the ordered state, and is negative for the other sublattice for which all spins are antiparallel with those of the "rst lattice. We shall refer to these sublattices in what follows as the &even' and &odd' sublattice, respectively. By taking such an alternating sum we "nd the expectation 1N2O0 in the system's ground state, "X2. The action of time-reversal invariance, ¹, is to reverse the sign of the spin of every site: s P!s . G G Although this transformation also reverses the order parameter, NP!N, it may be combined with another broken symmetry, S, to obtain a transformation, ¹I "¹S, which is a symmetry of N. This other symmetry, S, consists of a translation (or shift) of the whole lattice by a single lattice site, taking the entire &even' sublattice onto the &odd' sublattice, and vice versa. Since both S and ¹ act to reverse the direction of N, they preserve N when they are performed together. We next turn to the construction of the general low-energy lagrangian for the Goldstone bosons for the breaking of rotation invariance (called magnons) for these systems. 3.1.1. The nonlinear realization The "rst step is to identify the symmetry breaking pattern, GPH. At "rst it is tempting to assume that the role of G should be played by the spacetime symmetries, since these include rotations. This is not correct, however, for several reasons. Firstly, the spacetime symmetries of a lattice do not consist of the full group of translations and rotations since these symmetries are broken by the lattice itself. The unbroken subgroup consists only of the group of lattice symmetries: i.e. those translations and rotations which take the lattice to itself. There are indeed Goldstone bosons for the spontaneous breaking of translational and rotational symmetry down to this lattice group, but these are the phonons and are not the focus of the present analysis. In fact, the rotations of the spins on the lattice can be taken to be an internal S;(2), or SO(3), symmetry, rather than a spacetime symmetry. This is because the action of rotations on the intrinsic spin of a particle becomes an independent internal symmetry, separate from spacetime rotations, in the limit that the particle involved is nonrelativistic. This is because all of the interactions which couple the orbital angular momentum with the spin angular momentum vanish in the limit that the particle mass tends to in"nity. Since the spins of interest for real systems are
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those for nonrelativistic electrons or atoms, we may consider the broken symmetry group to be an internal symmetry, G"S;(2) (which equals G"SO(3), locally). In real life, the electron mass is not in"nite, so there are small &spin-orbit' e!ects which really do break the internal spin symmetry. These introduce small corrections to predictions based on this symmetry, such as the exact gaplessness of the Goldstone mode. We ignore any such symmetry-breaking e!ects in what follows. The order parameter for the symmetry breaking is the vector, N, itself, and so the group of unbroken transformations is H";(1) (or, SO(2)), consisting of rotations about the axis de"ned by 1N2. The coset space which is parameterized by the Goldstone bosons is therefore the space G/H"S;(2)/;(1), or SO(3)/SO(2). This last way of writing G/H identi"es it as a two-sphere, S , since this describes the space swept out by the action of rotations on a vector, 1N2, of "xed length. We now have two ways to proceed. We could, on the one hand, follow the steps outlined in Section 1 to construct the nonlinear realization of G/H and its invariant lagrangian. Instead we choose here to take a simpler route. As discussed in Section 1, the most general possible low-energy Goldstone boson lagrangian must necessarily take the form of Eq. (1.9.1): L "(F/2)g( (h)hQ ?hQ @!(F/2)g( (h) h? ) h@#(higher-derivative terms) , (3.1.2) $ R ?@ Q ?@ where g( (h) is an SO(3)-invariant metric on the two-sphere. This form is the most general ?@ consistent with the nonlinearly realized SO(3) invariance, as well as with invariance with respect to translations, rotations and the time-reversal-like symmetry, ¹I , described above. The ¹I invariance rules out interactions having an odd number of time derivatives, such as the term linear in time derivatives which was constructed in Section 1.9. The main point to be made is that the lagrangian given in Eq. (3.1.2) is unique, a result which follows from the uniqueness of the SO(3)-invariant metric on the two-sphere. The uniqueness of this metric is a consequence of the fact that the two broken generators of SO(3)/SO(2) form an irreducible representation of the unbroken subgroup SO(2) } a condition which was shown to imply a unique metric in Sections 1.6 and 1.7. Since it is unique, any representation of it is equally good and we choose here to use the familiar polar coordinates, (h, ), for the two-sphere, in terms of which the invariant metric has the usual expression: ds"dh#sin h d . With this choice the above lagrangian becomes L "(F/2)(hQ #sin h Q )!(F/2)( h ) h#sin h
)
) $ R Q #(higher-derivative terms).
(3.1.3)
Alternatively, we can equally well parameterize S using a unit vector, n, where n "sin h cos , V n "sin h sin and n "cos h, so n ) n"1. Then W X L "(F/2)n ) n !(F/2) n ) n#(higher-derivative terms). (3.1.4) $ R Q This variable, n(r, t), makes most clear the physical interpretation of the Goldstone modes: they describe long-wavelength variations in the direction of the order parameter 1N2. It has the
We use translation and rotational invariance for simplicity, even though these are too restrictive for real solids, for which only the lattice symmetries should be imposed. For some lattices, such as cubic ones, the implications of the lattice group turn out to be the same as what is obtained using rotation and translation invariance, at least for those interactions involving the fewest derivatives which are studied here.
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drawback of hiding the self-interactions which are implied by L , since the lagrangian of $ Eq. (3.1.4) is purely quadratic in n. The self-interactions, which are manifest in expression (3.1.3), are nonetheless present, and are hidden in the constraint n ) n"1. The nonlinear realization of SO(3) transformations on these variables is straightforward to work out, starting with the transformation rule for n: dn"x;n, where the vector x represents the three SO(3) transformation parameters. This implies the transformations: dh"u cos !u sin , W V d "u !u cot h cos !u cot h sin . (3.1.5) X V W With these transformation laws we may immediately write down the "rst terms in a derivative expansion of the Noether currents, jI"(q, j), for the SO(3) invariance in the low-energy e!ective theory. They may be most compactly written: q"F(n ;n)#2 and j"!F( n;n)#2 , (3.1.6) R Q where the dots are a reminder of the unwritten higher-derivative contributions. The other quantities which arise in the general nonlinear realization may also be constructed in terms of these variables. For example, the four independent components of the covariant quantity (zweibein), e? (h, ), are most easily constructed, following the geometrical picture of Section 1, as @ the components of any two orthogonal vectors which are tangent to the two sphere. These may be found by di!erentiating the unit vector, n, because the identity n ) n"1 implies n ) dn"0, for any variation, dn. Denoting these two vectors by e "Rn/Rh and e "Rn/R , we have in cartesian F ( components: (e ) "cos h cos , (e ) "cos h sin , (e ) "!sin h , FV FW FX (e ) "!sin h sin , (e ) "sin h cos , (e ) "0 . (3.1.7) (V (W (X Clearly these vectors satisfy e ) e "e ) n"e ) n"0, and e ) e "1, e ) e "sin h, so e ) e F ( F ( F F ( ( ? @ "g( , as required. The 2;2 matrix, e? , of components may be found by expressing the two vectors, ?@ @ e , de"ned by Eqs. (3.1.7), as linear combinations of any two orthonormal basis vectors, t?, which lie @ tangent to the sphere: i.e. e? "t? ) e . Using the basis vectors e themselves for this purpose leads to @ @ @ the result:
eF eF 1 0 F ( " . e( e( 0 sin h F (
(3.1.8)
3.1.2. Physical applications Any physical question that could be asked of the low-energy limit of the underlying theory can equally well be addressed using the low-energy e!ective lagrangian. In particular, the lagrangian just derived for the Goldstone modes for a ferromagnet may be used to describe the response (at zero or nonzero, but small, temperature) of the system to probes which couple to the spin degrees of freedom. In order to interpret the constants F and F it is convenient to expand the "eld n, or equivalently Q R h and , about its vacuum con"guration, n "1N2. We are free to perform an SO(3) rotation to choose the direction of n arbitrarily, and so we use this freedom to ensure that n points up the
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positive x-axis. This implies that h and take the vacuum values, h "p/2 and "0. Writing the canonically normalized #uctuation "elds by h"p/2#0/F and "u/F , the lagrangian becomes R R L " (0Q !v 0 ) 0)# cos(0/F )(u !v u ) u)#(higher-derivative terms) R $ " (0Q !v 0 ) 0#u !v u ) u)!(0/2F)(u !v u ) u)#2 . (3.1.9) R The constant, v, here represents the ratio v"F /F . The ellipses denote terms that involve at least Q R six powers of the "elds, or which involve more than two derivatives with respect to either position or time. The terms quadratic in the "elds describe two real modes which propagate according to the linear dispersion law: E(p)"vp. These modes physically correspond to spin waves: small, longwavelength precessions of the vector n about its vacuum value, n . They carry $1 unit of the conserved SO(2) spin in the direction parallel to 1N2. This gives the physical interpretation of the parameter v to be the velocity of propagation of these modes. The condition that this velocity must be smaller than the velocity of light is v4c"1 (in fundamental units), or, equivalently, F 4F . Q R F is similarly seen to govern the strength of the interaction terms in Eq. (3.1.9). R These modes and their interactions are amenable to experimental study through their electromagnetic couplings. Although magnons carry no electric charge, they do couple to magnetic "elds, B, due to the interactions of the microscopic magnetic moments which participate in the longwavelength spin waves. This gives a coupling of the magnetic "eld to the medium's spin density. This coupling can be probed, for example, by scattering neutrons which are also electrically neutral but which carry an intrinsic magnetic moment which interacts with magnetic "elds. The interaction between magnons and electromagnetic "elds is therefore given by a term of the form: L "!l s ) B, where s is the system's spin density. The lowest-dimension e!ective interac tion between magnons and electromagnetic "elds is now obtained by expressing the spin density in a derivative expansion, using Noether's result, Eq. (3.1.6), for s"q. The result is L "!l s ) B "!lF[B (hQ sin # Q sin h cos h cos ) R V #B (!hQ cos # Q sin h cos h sin )!B Q sin h)] W X (3.1.10) "lF (B 0Q #B u )#2 , X R W where l is an e!ective coupling parameter having the dimensions of magnetic moment (or: inverse mass, in fundamental units). Notice that the time derivative in this interaction ensures invariance with respect to ¹I transformations, under which sP!s and BP!B, A nonrelativistic neutron couples to the magnetic "eld with strength L"!k nRr n ) B , (3.1.11) L where n(x) denotes the two-component neutron "eld, and r denotes the Pauli matrices acting in the two-component neutron spin space. The constant k is the neutron magnetic moment which is, in , order of magnitude, k &e/m . As usual e is the electromagnetic coupling constant (i.e. the proton , , charge) and m is the neutron (or nucleon) mass. , Using these interactions, Eqs. (3.1.10) and (3.1.11), the cross section per-unit-volume for neutron scattering from the medium can be computed. For slowly moving neutrons, and under the
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assumption that only the momentum, p , of the scattered neutron is measured we "nd dp/
S (u, q)" dt dr 1s (r, t) s (0)2 e SR\ q r . G H GH
(3.1.14)
The quantity 1s (r, t) s (0)2"Tr[o s (r, t) s (0)] appearing in here is the expectation de"ned by the G H G H density matrix, o, which characterizes the initial state of the medium with which the neutron scatters. Notice that it is not a time-ordered product of operators which appears in this expectation. This correlation function may be explicitly computed at low energies using the previously derived e!ective lagrangian which describes the low-energy magnon self-interactions. The simplest case is that for which the medium is initially in the no-magnon ground state, and where the energies involved are low enough to neglect the magnon self-interactions. In this case the spin density may be well-approximated by the "rst terms of Eq. (3.1.6). Then: S (u, q)"(puF/v"q")d d(u!v"q") , (3.1.15) GH R GH where v"F /F is the speed of magnon propagation. Q R We see that the cross-section has sharp peaks when the neutron energy and momentum transfers are related by the magnon dispersion relation: E!E "v" p!p ". This corresponds to inelastic scattering in which the neutron transfers its energy and momentum to the medium by creating a magnon. According to Eq. (3.1.15) the resulting peaks in the cross-section are in"nitely sharp, but in real systems they have a "nite width due to processes which cause the produced magnons to scatter or decay. If the lifetime for undisturbed magnon propagation, q"1/C, is much longer than the other interaction times of interest in the neutron scattering, then the delta function in Eq. (3.1.15) becomes replaced by the line shape: 1 C . (3.1.16) d(u!v"q")P 2p (u!v"q")#C Measurements of the positions and widths of these peaks as functions of the scattered neutron energy and momentum can be used to measure the magnon dispersion relation (and so the constant v) and its decay rate, C, for the scattering medium. The predicted linear spectrum is indeed found when neutrons are scattered from antiferromagnets. The situation when neutrons scatter from ferromagnets is di!erent, as we shall now see.
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3.2. Ferromagnetism: T breaking For ferromagnets, the order parameter is simply the total magnetization, or the total spin, of the system. Since this de"nes a vector in space, the spin symmetry, G"SO(3), is spontaneously broken to H"SO(2) just as for an antiferromagnet. The low-energy behaviour of ferromagnets and antiferromagnets are nevertheless quite di!erent, and this di!erence is due to the fact that time-reversal symmetry is broken in a ferromagnet but not in an antiferromagnet. We denote by S the order parameter for a ferromagnet, which is related to the spins, s , of the G underlying spin model by S" s . (3.2.1) G G Ferromagnets are characterized by having ground states for which there is a nonzero expectation for this quantity: 1S2O0. The action of time-reversal invariance, ¹, is to reverse the sign of the spin of every site, s P!s , G G and so it does the same for the order parameter, SP!S. The di!erence with the antiferromagnetic case arises because for a ferromagnet it is not possible to "nd another broken symmetry which combines with time reversal to preserve S. The low-energy e!ective theory can therefore contain ¹-violating terms, and this changes the properties of its Goldstone bosons in an important way. 3.2.1. The nonlinear realization Since the symmetry-breaking pattern for both ferromagnets and antiferromagnets is SO(3)PSO(2), the nonlinear realization of this symmetry on the Goldstone bosons is identical for these two systems. We therefore use the same polar coordinates in this case, h and , as in the previous sections. As before it is convenient to use these to de"ne a unit vector, denoted by s, with components s "sin h cos , s "sin h sin and s "cos h, so that s ) s"1. The "eld, s(r, t), again V W X describes long-wavelength oscillations in the direction of 1S2. The action of SO(3) on these variables is once more given by Eq. (3.1.5), and the term in the e!ective lagrangian which involves the fewest spatial derivatives is again determined to be L "!(F/2) ( h ) h#sinh
)
) . (3.2.2) $ Q Q The new features appear once the term with the fewest time derivatives is constructed. As is discussed in some detail in Section 1.9, this involves only a single time derivative because of the broken time-reversal symmetry. It has the form given by Eq. (1.9.2): L "!A (h)hQ ? , (3.2.3) $ R ? where the coe$cient function, A (h), may be considered to be a gauge "eld de"ned on the coset ? space G/H. In Section 1 it was determined that the condition that this term be G invariant is that A must only be G-invariant up to a gauge transformation, in the sense that ? £ A ,m@R A #A R m@"R X , (3.2.4) K ? @ ? @ ? ? K for each generator dh?"m? of G on G/H, where X (h) are a collection of scalar functions on G/H. K This last condition is equivalent to the invariance of the "eld strength for A : £ F "0. Our ? K ?@
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problem is to explicitly construct such a gauge potential for the example of interest, G/H"SO(3)/SO(2),S . This construction is quite simple. Since our coset space is two dimensional, it is always possible to write the "eld strength in terms of a scalar "eld: F "B(h)e , where e is the antisymmetric ?@ ?@ ?@ tensor which is constructed using the coset's G-invariant metric. The condition that F be ?@ G invariant is then equivalent to the invariance of B. That is: £ B,m?R B"0 , (3.2.5) K ? which is only possible for all G transformations if B is a constant, independent of h?. Our solution for L for a ferromagnet therefore simply boils down to the construction of $ R a gauge potential for which F "Be on the two-sphere, S "SO(3)/SO(2). But such a gauge ?@ ?@ potential is very familiar } it is the gauge potential for a magnetic monopole positioned at the centre of the two-sphere. The result may therefore be written (locally) as: A dh?"B cos h d , and ? so the corresponding lagrangian is given by L "!B cos h Q , (3.2.6) $ R where B is a constant. In terms of the vectors s, e "Rs/Rh and e "Rs/R this may be written: F ( L "!Bs ) (e ;e ) . (3.2.7) $ R F F The complete Goldstone boson lagrangian containing the fewest time and space derivatives is found by combining the contributions of Eqs. (3.2.2) and (3.2.6), giving: L "!B cos h Q !(F/2) ( h ) h#sin h
)
) . (3.2.8) $ Q It is instructive to compute the Noether currents for the SO(3) symmetry that are implied by this lagrangian density. The conserved current density is the same as was found for the antiferromagnet: j"F(s; s)#2 . (3.2.9) Q In computing the corresponding expression for the charge density, it is necessary to keep in mind that under these transformations L is not invariant, but instead transforms into a total derivative: $ dL "!dX/dt"!(B/sin h) (u cos #u sin ) . (3.2.10) $ V W Using this in the general expression, Eq. (1.2.3), for the Noether current gives the conserved charge density: q"Bs#2 .
(3.2.11)
The ellipses in this equation, and in Eq. (3.2.9), represent more complicated terms which are suppressed by additional derivatives. As is easily veri"ed, the classical equations of motion for the lagrangian, (3.2.8), are equivalent to the conservation condition for this current: s #k(s; s)"0 .
(3.2.12)
This equation has long been known to describe long-wavelength spin waves in ferromagnets, and is called the Landau}Lifshitz equation. The constant, k, here is given in terms of F and B by Q k"F/B . (3.2.13) Q
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Eq. (3.2.11) brings out a feature of time-reversal breaking systems which is qualitatively di!erent from those which preserve time reversal. It states that it is the conserved charge density itself, q, which acquires a vacuum expectation value and breaks the SO(3) symmetry: 1q2"B1s2"Bs O0 . (3.2.14) Clearly, the breaking of time reversal (and lorentz invariance) are prerequisites for the acquisition of a nonzero ground-state expectation value for q, which is the time component of a current. 3.2.2. Physical applications The propagation of small-amplitude, long-wavelength spin waves is therefore seen to be completely determined by the underlying pattern of spontaneous symmetry breaking: SO(3)PSO(2) together with ¹ violation. Linearizing the Landau Lifshitz equation, Eq. (3.2.12), shows the resulting propagating modes to have the quadratic dispersion relation: E(p)"kp .
(3.2.15)
This dispersion relation, and the value of the constant k, can be measured by neutron scattering, in a manner that is similar to what was found for antiferromagnets. We highlight here only the di!erences which arise from the antiferromagnetic example. The lowest-dimension e!ective interaction which couples the "eld s to electromagnetic "elds in the ferromagnetic case is L "!ks ) B "!kB(B sin h cos #B sin h sin #B cos h) V W X "!kBB !kB(B d !B dh)#2 , (3.2.16) V W X where k is an e!ective coupling parameter and we have taken the expectation value, 1S2 to point in the positive x direction, so h"p/2#dh and "d . The constant term, independent of dh and d , in the "nal line of Eqs. (3.2.16) gives the interaction energy between the magnetic "eld and the expectation value, 1S2. This permits the physical interpretation of the constant k as the magneticmoment density of the material. This interaction between s and B breaks ¹ invariance, and has the following puzzling feature. It does not involve any derivatives of the Goldstone boson "elds, h and , in apparent contradiction with the general results of Section 1. In fact, the absence of derivatives in Eq. (3.2.16) is very much like the absence of derivatives in the pion mass term. This is because the coupling between s and B relates the internal spin SO(3) symmetry to ordinary rotations in space, and so destroys the freedom to consider both as separate symmetries. But because rotation invariance is a spacetime symmetry, the derivations of Section 1 do not directly apply, since these assumed the action of internal symmetries from the outset in the transformation rules of the "elds. Once more taking the neutron coupling to the magnetic "eld as in Eq. (3.1.11), we may compute the cross-section for inelastic neutron scattering. For slowly moving neutrons, and under the assumption that only the momentum, p , of the scattered neutron is measured we "nd dp/< dp "(k k/4pv )< ( p!p )S (E!E , p!p ) . , , GH GH
(3.2.17)
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The variables are the same as for the antiferromagnetic example: v and k are the speed and , , magnetic moment of the slow incoming neutron, < is the volume of the medium, E (E ) is the energy of the initial ("nal) neutron, and p ( p ) are the corresponding momenta. The magnetic-moment interaction, < (q), is as given in Eq. (3.1.13). GH The medium-dependent quantity, S (u, q), once more represents the spin correlation function, GH de"ned by Eq. (3.1.14). As was the case for the ferromagnet this is dominated by sharp peaks when the neutron scatters to produce a magnon, and so has an energy and momentum transfer related by the magnon dispersion relation. For a ferromagnet this is: E!E "k( p!p ). Measurements of these peaks as functions of the scattered neutron energy and momentum indeed veri"es the quadratic dispersion relation, and can be used to measure the constant k. A second consequence of the quadratic magnon dispersion relation is the temperature dependence of the magnetization, M""M", of a sample at very low temperatures. Since the magnon "eld describes the long-wavelength deviations of the net magnetization from its ground state value, the net magnetization at very low temperatures is simply proportional to the average magnon occupation number. That is:
M(0)!M(¹)JM(0) dp n
E , ¹
(3.2.18)
where n(E/¹)"(exp[E/¹]!1)\ is the Bose}Einstein distribution. The temperature dependence of this result can be determined by changing integration variables from p to the dimensionless quantity x"E/¹. If E(p)JpX, for some power, z, then dp p dp"p dEJE XE\X\ X dEJ¹ X . dE
(3.2.19)
For z"2 this predicts [M(0)!M(¹)]/M(0)J¹ , in agreement with low-temperature observations.
4. SO(5) invariance and superconductors The techniques described herein have recently proven useful to analyse the consequences of a remarkable proposal for the existence of an SO(5) invariance amongst the cuprates which exhibit high-temperature superconductivity. This section presents a bare-bones outline of this proposal, together with a brief summary of the Goldstone-boson properties which emerge. 4.1. SO(5) symmetry Although a proper presentation of the arguments for } and against, since the subject remains controversial } the SO(5) proposal is beyond the scope of this review, the form of the proposed symmetry itself is easy to state. The starting point is the following experimental fact: by performing small adjustments to any of the high-¹ cuprates, it is possible to convert them from superconduc tors into antiferromagnets. This adjustment is typically accomplished in practice by altering the &doping', which means that atoms having di!erent valences are randomly substituted into a portion
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Fig. 5. A typical Temperature-vs-Doping phase diagram for a high-¹ system.
of the unit cells of the material of interest. For example, the element Sr might be substituted for the element La in some fraction, x, of the unit cells. Physically, this substitution has the e!ect of changing the number of charge carriers in the band from which the superconducting electrons are taken (Fig. 5). This basic association of superconductivity and antiferromagnetism suggests a fundamental connection between the two. Zhang's proposal is that } at least over part of the theory's parameter space } these two phases are related by an approximate SO(5) symmetry. The action of the symmetry is simplest to state for the order parameters for the two phases. As we have seen, in Section 3, the order parameter for the antiferromagnetic (AF) phase is simply the direction in space, n, into which the alternating aligned spins point. A nonzero value of this order parameter spontaneously breaks the SO(3)KS;(2) symmetry of rotations amongst electron spins:
n n (4.1.1) n PO n , n n where O denotes an n;n real, orthogonal matrix. L For the superconducting (SC) phase, on the other hand, the order parameter, t, is a quantity which carries the quantum numbers of a pair of electrons (or holes). Speci"cally, it has electric charge q"$2e, if e is the proton charge, and is usually taken to have no spin. (The sign of this charge depends on whether the charge carriers are electrons or holes.) A nonzero value for t signals the breaking of the symmetry of electromagnetic phase rotations: tPe OSt. This symmetry forms the group SO(2)K;(1), as is easily seen by grouping the real and imaginary parts of t into a two-component vector:
t t 0 PO 0 , t t ' ' where t"t #it . 0 '
(4.1.2)
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The SO(5) proposal is that the system is approximately invariant under 5;5 rotations of all "ve components of the order parameter,
t t 0 0 t t ' ' (4.1.3) n PO n . n n n n This symmetry contains electron-spin rotations, Eq. (4.1.1), and electromagnetic phase transformations, Eq. (4.1.2), as the block-diagonal SO(2);SO(3) subgroup which acts separately on the t and I the n . ? As of this writing, there is a controversy over whether such an approximate symmetry actually exists for high-¹ superconductors and, if so, to how large a portion of the phase diagram it might apply. Regardless of how this controversy ultimately becomes resolved, two points on which everyone must agree are: 1. For any part of the phase diagram for which approximate SO(5) symmetry holds, and which lies within the ordered (AF or SC) phases, there must be a total of four Goldstone (or pseudoGoldstone) bosons. This corresponds to one each for the four SO(5) generators which are broken by a nonzero value for one of the t or n . These bosons include the usual Goldstone bosons for I ? SO(3) or SO(2) invariance (e.g., the magnons), plus some new pseudo-Goldstone bosons which are consequences only of the assumed SO(5) symmetry. 2. The low-energy properties of these Goldstone and pseudo-Goldstone bosons are completely dictated by the assumed symmetry-breaking pattern, and are independent of the details of whatever microscopic electron dynamics gives rise to the symmetry in the "rst place. These low-energy properties may be e$ciently described using an e!ective lagrangian along the lines of those described in Chapter 1. These two properties taken together make an unambiguous detection of the SO(5) pseudoGoldstone bosons a particularly attractive test of the SO(5) proposal. Their detection would be a &smoking gun' for the existence of an extended symmetry like SO(5). Better yet, their properties are unambiguously predicted theoretically, without the usual complications which arise when complicated electron dynamics is squeezed into a simple theoretical model. We now construct the low-energy e!ective lagrangian describing these pseudo-Goldstone bosons, following the general techniques of the previous sections. 4.2. The ewective lagrangian in the symmetry limit We start with the e!ective lagrangian in the (idealized) limit where SO(5) is not just approximate, but is instead a bona xde symmetry of the system. In this case the lagrangian symmetry is G"SO(5), and this is spontaneously broken (by the order parameters we are considering) to the subgroup, H"SO(4). We require in this limit the nonlinear sigma model for the quotient space G/H"SO(5)/SO(4). As discussed in general in Section 1, the lowest terms in the derivative
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expansion of the lagrangian for this system are therefore completely determined up to a small number of constants. (Precisely how many constants depends on how much symmetry } including crystallographic symmetries } the system has. As we describe shortly, more possibilities also arise once explicit SO(5)-breaking interactions are introduced.) In this particular case, the space SO(5)/SO(4) is an old friend: the four-sphere, S , which is de"ned as the points swept out by an arbitrary "ve-dimensional vector, N, which has unit length: N ) N" N,1. In terms of such a "eld, the invariant lagrangian obtained using the techG G niques of previous sections is L "( f )/2)R N ) R N!( f /2) N ) N (4.2.1) R R R Q (For simplicity of this and later expressions, Eq. (4.2.1) assumes rotational invariance, which is not appropriate for real cuprates. For real systems, the electrons believed responsible for superconductivity and antiferromagnetism move preferentially along planes made up of copper and oxygen atoms. The low-energy lagrangian for such systems is better written either in two space dimensions (for Goldstone bosons con"ned to the planes), or in three dimensions with separate coe$cients, f for each spatial direction, . These complications are ignored here, but are discussed in more ? ? detail in the original references. As is also discussed in these references, in speci"c dimensions it is sometimes also possible to write more invariants than are considered here, such as those which depend on the completely antisymmetric tensor, e .) GHI
A convenient parameterization for the four-sphere, and so of our Goldstone bosons, is given by polar coordinates: sin a cos b n cos
, n "sin h sin a sin b . (4.2.2) N" / , where n "cos h / 1 n sin
1 cos a
(As usual, care is required to properly handle those points where these coordinates are singular). Using the standard expression for the round metric on S then gives the unique Goldstone boson e!ective lagrangian for a rotational- and time-reversal-invariant system in the SO(5) symmetry limit: f L" R [(R h)#cosh(R )#sinh((R a)#sina(R b))] R R R 2 R f ! Q [( h)#cosh(
)#sinh(( a)#sina( b))] . 2 4.3. Symmetry-breaking terms Next, consider how small SO(5)-breaking e!ects can change the low-energy lagrangian. We may do so by following the same steps as were taken in Section 2 to describe the implications of quark masses on low-energy pion properties. We do so here in two steps. We "rst classify the types of violation of SO(5) symmetry which can arise in real systems. We then examine which consequences follow only from SO(2);SO(3) invariance, in order to be able to disentangle these from predictions which are speci"c to SO(5). Finally, we perturb in the various SO(5) symmetry-breaking parameters to obtain the predictions of approximate SO(5) invariance. By contrasting what is obtained with
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the result assuming only SO(2);SO(3) invariance, the consequences of approximate SO(5) invariance may be found. 4.3.1. Kinds of explicit symmetry breaking (qualitative) There are several kinds of SO(5) symmetry breaking which are worth distinguishing from one another. These are: 1. Electromagnetic interactions: One source of explicit SO(5) symmetry breaking is any couplings to applied macroscopic electromagnetic "elds. Electromagnetic gauge invariance requires these to be incorporated into the lagrangian using the usual precedure of minimal substitution: R tP(R !iqA )t,
tP( !iqA)t . (4.3.1) R R Such couplings necessarily break SO(5) because they treat the electrically charged components of N di!erently than the electrically neutral ones. Although such couplings do not pose any complications of principle, for simplicity's sake we imagine no macroscopic electromagnetic "elds to be applied in what follows. 2. Doping: One of the physical variables on which the phase diagram of the cuprates crucially depends is the doping. Since changes in the doping correspond to changes in the density of charge carriers amongst the electrons which are relevant for both the antiferromagnetism and the superconductivity, it may be described within the e!ective theory by using a chemical potential, k, coupled to electric charge. In this way adjustments in k may be chosen to ensure that the system has any given experimental charge density. Mathematically a chemical potential is introduced by replacing the system Hamiltonian, H, with the quantity H!kQ, where Q is the electric charge. Within the Lagrangian formulation in which we are working, this amounts to simply making the replacement A PA #k in the electrostatic scalar potential, A . 3. Intrinsic breaking: The third, and "nal, category of symmetry-breaking consists of everything apart from the previous two. It is known that SO(5) is not an exact symmetry, even with no chemical potential, and in the absence of any applied electromagnetic "elds. It is the interactions (involving the fewest derivatives) in this last class of symmetry-breaking terms which we now wish to classify. 4.3.2. General SO(2);SO(3)-invariant interactions If SO(5) were not a symmetry at all, then there would be no guarantee that the low-energy spectrum should contain particles described by all of the "elds a, b, h and . For the purposes of later comparison, it is nevertheless useful to ask what kinds of low-energy interactions among such states are permitted by SO(2);SO(3) invariance. The most general such lagrangian involving these four states may be written in terms of the "elds n and n , where these "elds satisfy the constraint n ) n #n ) n ,1, to the extent that we are 1 / 1 1 / / interested in only those modes which would be Goldstone or pseudo-Goldstone modes in the SO(5) limit. The most general such result, which involves at most two derivatives, supplements the invariant expression, Eqs. (4.2.1) or (4.2.3), with the following terms: L "!<#f [AR n ) R n #BR n ) R n #C(n ) R n )] R R / R / R 1 R 1 / R / !f [D n ) n #E n ) n #F(n ) n )] , Q ? / ? / ? 1 ? 1 / ? /
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where f and f are the constants appearing in the invariant lagrangian. The quantities R Q <, A, B, C, D, E and F are potentially arbitrary functions of the unique SO(2);SO(3) invariant which involves no derivatives: n ) n . (Recall n ) n is not independent due to the constraint / / 1 1 n ) n #n ) n "1.) 1 1 / / In terms of polar coordinates, and inserting a chemical potential as just described, the total e!ective lagrangian becomes f L"!<# R [(1#2A sinh#2B cosh#2C sinh cosh)(R h) R 2 #(1#2A)cosh(R #qk)#(1#2B)sinh((R a)#sina(R b))] R R R f ! Q [(1#2D sinh#2E cosh#2F sinh cosh)( h) 2 #(1#2D)cosh(
)#(1#2E)sinh(( a)#sina( b))]#2 , where all coe$cient functions, <, A,2 etc., are now to be regarded as functions of cosh. 4.3.3. Kinds of explicit symmetry breaking (quantitative) Eq. (4.3.2) does not yet use any information concerning the nature or size of the explicit symmetry breaking (apart from the inclusion of k). This we must now do if we are to quantify the predictions of approximate SO(5) invariance. We do so by making an assumption as to how the symmetry-breaking terms transform under SO(5). In Section 2 we saw, for pions, that the quark masses were responsible for explicitly breaking the would-be chiral symmetry of the underlying microscopic theory (QCD). Although the same reasoning can be applied to SO(5) breaking due to electromagnetic interactions and chemical potential dependence, incomplete understanding of the dynamics of the microscopic theory so far precludes a similar identi"cation of the other symmetry-breaking parameters within some underlying condensed-matter system. For these, we instead are forced to make an assumption. We therefore assume all SO(5)-breaking terms of the e!ective lagrangian to be proportional to one of two possible quantities: 1. Chemical potential: Since we know how the chemical potential appears in the lagrangian, we know in detail how it breaks SO(5). It does so by an amount which is proportional to the electric charge. For the "elds appearing in N, this is represented by the "ve-by-"ve electric charge matrix: Q"diag(!q, !q, 0, 0, 0). 2. Intrinsic symmetry breaking: In the absence of more information, we make the simplest assumption for the form taken in the e!ective lagrangian by all other microscopic e!ects which explicitly break SO(5). Since these break SO(5) to SO(3);SO(2) we take them to be proportional to a 5;5 matrix, M, where M"e diag(3, 3,!2,!2,!2). Here e;1 is a measure of the quality of the approximation that SO(5) is a symmetry. With these choices the lagrangian is then the most general function of the "elds N"(L/1 ), kQ and L M, subject to the following SO(5) transformation property L(O N, O kQO2, O MO2)"L(N, kQ, M) ,
(4.3.2)
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where O is an SO(5) transformation. The implications of the approximate SO(5) invariance may then be extracted by expanding L in powers of the small quantities e and k. Since M and Q always appear premultiplied by these small numbers, this expansion restricts the kinds of symmetry breaking which can arise order by order, which in turn constrains the possible h-dependence of the coe$cient functions in L. For example, a term in the scalar potential involving 2n powers of N must have the following form: 2 C 2 L L [N ) (eM)I (kQ)J N]2[N ) (eM)IL (kQ)JL N ] . (4.3.3) < " I J I J L IL JL $ I J $ Only even powers of Q enter here due to its antisymmetry, and the term k "l "0 is excluded G G from the sums due to the constraint N2N"1. Clearly, expanding L to low order in the SO(5)-breaking parameters e and k necessarily also implies keeping only the lowest powers of n ) n "cos h in <. / / Similar conclusions may be obtained for the other coe$cient functions in the lagrangian of Eq. (4.3.2). Working to O(e, ek, k) in <, and to O(e,k) in the two-derivative terms then gives: 1 <"< #< cos h# < cos h , 2
(4.3.4)
and A"A #A cos h, B"B #A cos h, C"C , D"D #D cos h, E"E #D cos h, F"F , (4.3.5) for the coe$cient functions in Eq. (4.3.2). Notice that the terms proportional to cos h in A and B are identical, as are the corresponding terms in D and E. Expanding in powers of e and k, the constants in Eqs. (4.3.4) and (4.3.5) start o! linear in e and k: A "Ae#Ak#2 etc. The G G G only exceptions to this statement are: B , E Je (no k term), C , F Jk (no e term), and < "<e#<ek#<k. Furthermore, since the kn ) n term in < arises from substitu / / ting R PR !ikQ in the kinetic term for n , we have: <"! f q to leading order. Higher R R / R powers of k originate from terms in L which involve more than two derivatives. 4.4. Pseudo-Goldstone dispersion relations We now turn to the calculation of the pseudo-Goldstone boson dispersion relations. The scalar potential of Eq. (4.3.2) has three types of extrema: (1) (2)
h "0 or n; 3p p or ; h " 2 2
h where c"cos h satis"es < (c)"0 . (4.4.1) This leads to the four possible phases: (i) SC phase: extremum (1) is a minimum, and (2) is a maximum; (ii) AF phase: (2) is a minimum, and (1) is a maximum; (iii) MX phase: both (1) and (2) are (3)
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maxima, and (3) is a minimum; or (iv) metastable phase: both (1) and (2) are minima, and (3) is a maximum. We focus here purely on the AF and SC phases. 4.4.1. Superconducting phase An expansion about the superconducting mimimum, h "0, gives the dispersion relations in this phase for the four bosons. Three of these (h, a and b) form a spin triplet of pseudo-Goldstone modes for which E(k)"[ck#E] ,
(4.4.2)
with the phase speed, c(SC), and gap, E , given to lowest order in SO(5)-breaking parameters by: ? 1! c(SC)"( f /f )[1#2(E(1)!B(1))]"( f /f )[1#2(E !B )#2(D !A )], ? Q R Q R E "!2< (1)/f "!2(< #< )/f . 1! R R
(4.4.3)
In both of these results the "rst equation uses the general e!ective theory, Eq. (4.3.2), while the second equality incorporates the additional information of Eqs. (4.3.4) and (4.3.5). An important part of the SO(5) proposal is that these states have been seen in neutron-scattering experiments in the superconducting phase of the high-¹ cuprates, even quite far away from the antiferromagnetic regime. The remaining "eld, , would have been a bona xde gapless Goldstone mode in the absence of electromagnetic interactions. Its dispersion relation, E(k) is a more complicated function of ck and eqk, whose form is not required here. The quantity c which appears with k throughout its dispersion relation is given explicitly by c (SC)"( f /f )[1#2(D(1)!A(1))] ( Q R "( f /f )[1#2(D !A )#2(D !A )] . Q R
(4.4.4)
4.4.2. Antiferromagnetic phase Expanding about the AF minimum gives the usual two magnons, as in Section 3, satisfying dispersion relation of Eq. (4.4.2) with c (AF)"( f /f )[1#2(E(0)!B(0))]"( f /f )[1#2(E !B )], % Q R Q R E (AF)"0 . %
(4.4.5)
The remaining two states group into an electrically charged pseudo-Goldstone state satisfying: E (k)"[ck#E] $qk , !
(4.4.6)
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with: c (AF)"( f /f )[1#2(D(0)!A(0))] N% Q R "( f /f )[1#2(D !A )], Q R E "E (AF)"(2< (0)/f ) $ N% R "(2< /f ) . R These expressions imply a simple dependence of the gap on the chemical potential:
(4.4.7)
E "m!ik, $ E "!m#ik!mk, (4.4.8) 1! where m " : 2<e/f #O(e), i " : !2</f #O(e)"q#O(e) and m " : 2</f #O(e). R R R Within the AF phase the pseudo-Goldstone boson gap is predicted to fall linearly with k: E +E (0)[k !k] , (4.4.9) $ $ $ where k represents the doping for which one leaves the AF regime. Similarly e varies qua $ 1! dratically with k. Robust consequences of SO(5) invariance are obtained from expressions such as these by eliminating the free parameters to obtain relations amongst observables. For example, if one eliminates parameters in favour of properties of the gap as a function of k we "nd E (0) E (k)" $ [k !k], $ $ k $ E (opt) E (k)" 1! (k!k )(2k !k), 1! 1!\ k E (0) E (opt) $ "2 1! , k k $ k "k #O(e). (4.4.10) $ 1!\ Here k denotes the chemical potential corresponding to the maximum gap, E . 1! Similarly, the phase velocities for all modes in both SC and AF phases are equal to one another, and to f /f , in the limit of strict SO(5) invariance. (The parameters f and f may be related to other R Q R Q observables, such as the electric and magnetic screening lengths.) It turns out that the O(e) corrections to this limit are not arbitrary, but also satisfy some model-independent relations, which follow by eliminating parameters from the above expressions: c (SC)!c (AF)"c(SC)!c(AF)"O(e) . ( ( ? ?
(4.4.11)
4.5. Summary Approximate SO(5) invariance clearly carries real implications for the low-energy excitations of the system. It predicts, in particular, the existence of a spin-triplet pseudo-Goldstone state in the SC
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phase, and an electrically charged state in the AF phase. Furthermore, SO(5) invariance unambiguously relates the properties of these states, like their gap and phase velocity, to one another. Better yet, these properties are claimed to have been measured in the SC phase, since the spin-triplet state is argued to have been observed, with a gap (at optimal doping) of 41 meV. If true, this permits the inference of the size of the rough order of the SO(5)-breaking parameter e, and hence to predictions for the properties of the hitherto undetected boson in the AF phase. It is extremely unlikely that an electrically charged state having a gap of only &40 meV can exist deep within the AF phase. Among other things it would make its presence felt through the electromagnetic response of these systems in the AF phase. At the very least, one can therefore conclude that SO(5) invariance cannot penetrate very far into the AF part of the phase diagram, despite its appearance fairly deep in the SC phase. Being based purely on pseudo-Goldstone boson properties, this conclusion comes independent of the details of how the underlying electrons are interacting on more microscopic scales. There is not yet a consensus as to how uncomfortable this conclusion should make one feel about the remarkable SO(5) hypothesis. Either way, robust predictions based on the low-energy consequences of symmetries are likely to play a key role in forming any such "nal consensus.
5. Bibliography The classic early papers which founded this subject are [1}3]. My thinking has been strongly in#uenced by graduate courses given on this subject during the early 1980s in the University of Texas at Austin, by Steven Weinberg. This review is, in its presentation, very similar with the logic of his textbook [4,5]. The papers and reviews [6}13] have also shaped how I understand this subject. 5.1. Review articles There have been a number of well-written review articles on e!ective "eld theories, most of which concentrate on chiral perturbation theory. A partial list is in Refs. [14}22]. I have inevitably omitted several good ones, for which I give my apologies now. 5.2. Particle physics data I have taken the experimental values for the section on pions and nucleons from Ref. [23]. 5.3. SO(5) invariance in high-¹ superconductors It is not the intention of these notes to provide more than a super"cial discussion of the SO(5) proposal for the high-¹ cuprates, which was "rst made by Zhang, in Ref. [24]. This remarkable proposal immediately stimulated much interest, with an associated literature. I do not attempt a literature survey here, apart from listing three highlights in which the &big picture' is addressed by several of the "eld's major players. These are Refs. [25}27].
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The applications of the technique of nonlinear realizations to identify the model-independent properties of the resulting Goldstone and pseudo-Goldstone bosons, which is followed in Section 4, is performed in [28}30]. 5.4. Condensed matter physics for particle and nuclear physicists There are several sources for condensed matter physics which I have found useful. Some are [31}35]. A treatment of spin waves which is in the spirit of this review has appeared in a preprint by Roman and Soto [36] and by Hofmann [37].
Acknowledgements This review is my personal presentation of the theory of Goldstone bosons, and (apart from the SO(5) section) were "rst given as part of a lecture course for the Swiss Cours de Troisie% me Cycle, in Lausanne and Neucha( tel in June 1994. I am indebted to the University of Neucha( tel, the University of Oslo, Seoul National University, and my colleagues at McGill University for providing the forum in which to work out the presentation of these ideas. I thank Patrick Labelle and Oscar Hernandez for constructively criticizing them as they have evolved. My thanks also to John Donoghue for his constructive criticism of the "nal written version. Needless to say, any errors that remain are mine alone.
References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11] [12] [13] [14] [15]
S. Weinberg, Phys. Rev. Lett. 18 (1967) 188. S. Weinberg, Phys. Rev. 166 (1968) 1568. C.G. Callan, S. Coleman, J. Wess, B. Zumino, Phys. Rev. 177 (1969) 2247. S. Weinberg, The Quantum Theory of Fields, Vol. I, Cambridge University Press, Cambridge, 1995. S. Weinberg, The Quantum Theory of Fields, Vol. II: Modern Applications, Cambridge University Press, Cambridge, 1996. S. Weinberg, Phenomenological Lagrangians, Physica 96A (1979) 327}340. S. Weinberg, What is quantum "eld theory, and what did we think it is?, Talk given at Conference on Historical Examination and Philosophical Re#ections on the Foundations of Quantum Field Theory, Boston, MA, 1}3 March 1996 (hep-th/9702027). J. Polchinski, E!ective theory of the fermi surface, in the Proceedings of the 1992 Theoretical Advanced Study Institute, Boulder, Colorado (hep-th/9210046). W.E. Caswell, G.P. Lepage, Phys. Lett. B 167 (1986) 437. T. Kinoshita, G.P. Lepage, in: T. Kinoshita (Ed.), Quantum Electrodynamics, World Scienti"c, Singapore, 1990, pp. 81}89. J. Gasser, H. Leutwyler, Chiral perturbation theory to one loop, Ann. Phys. (NY) 158 (1984) 142. N. Isgur, M.B. Wise, Weak decays of heavy mesons in the static quark approximation, Phys. Lett. B 232 (1989) 113. S. Weinberg, Nuclear forces from chiral lagrangians, Phys. Lett. B 251 (1990) 288}292. G.S. Guralnik, C.R. Hagen, T.W.B. Kibble, in: R.L. Cool, R.E. Marshak (Eds.), Advances in Particle Physics, Vol. 2, Wiley, New York, 1968. H. Georgi, Weak Interactions and Modern Particle Theory, Benjamin/Cummings, Menlo Park, 1984.
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[16] J.F. Donoghue, E. Golowich, B.R. Holstein, Dynamics of the Standard Model, Cambridge University Press, Cambridge, 1992. [17] A. Manohar, E!ective "eld theories, in Schladming 1996, Perturbative and Nonperturbative Aspects of Quantum Field Theory, pp. 311}362 (hep-ph/9606222). [18] M. Rho, Acta Phys. Polon. B 29 (1998) 2297. [19] Antonio Pich, E!ective "eld theory: Course in the Proceedings of the Les Houches Summer School in Theoretical Physics, Session 68: Probing the Standard Model of Particle Interactions, Les Houches, France, 1997 (hep-ph/ 9806303). [20] D. Kaplan, E!ective "eld theories, in the Proceedings of the 7th Summer School in Nuclear Physics Symmetries, Seattle, WA, 1995 (nucl-th/9506035). [21] H. Georgi, E!ective "eld theory, Ann. Rev. Nucl. Part. Sci. 43 (1995) 209}252. [22] H. Georgi, Heavy quark e!ective "eld theory, in the Proceedings of the 1991 Theoretical Advanced Study Institute, Boulder, Colorado, 1991. [23] Particle Data Group, The review of particle properties, The European Phys. J. C 3 (1998) 1. [24] S.C. Zhang, Science 275 (1997) 1089. [25] G. Baskaran, P.W. Anderson, On an SO(5) uni"cation attempt for the cuprates, cond-mat/9706076. [26] R.B. Laughlin, A critique of two metals, cond-mat/9709195. [27] P.W. Anderson, G. Baskaran, A critique of &A critique of two metals', cond-mat/9711197. [28] C.P. Burgess, C.A. LuK tken, Phys. Rev. B 57 (1998) 8642 (cond-mat/9705216). [29] C.P. Burgess, C.A. LuK tken, cond-mat/9611070. [30] C.P. Burgess, J.M. Cline, C.A. LuK tken, Phys. Rev. B 57 (1998) 8549. [31] R. Shankar, E!ective "eld theory in condensed matter physics, lecture given at Boston Colloquium for the Philosophy of Science, Boston, Mass., 1996 (cond-mat/9703210). [32] R. Shankar, Renormalization group approach to interacting fermions, Rev. Mod. Phys. (cond-mat/9307009). [33] P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [34] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1972. [35] Michael Tinkham, Introduction to Superconductivity, 2nd Edition, McGraw-Hill, New York, 1996. [36] J.M. Roman, J. Soto, cond-mat/9709298. [37] C. Hofmann, cond-mat/9805277.
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QUARKS AND LEPTONS BEYOND THE THIRD GENERATION
Paul H. FRAMPTON!, P.Q. HUNG", Marc SHER# !Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA "Physics Department University of Virginia, Charlottesville, VA 22901, USA #Physics Department, College of William and Mary, Williamsburg, VA 23187, USA
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Quarks and leptons beyond the third generation Paul H. Frampton , P.Q. Hung, Marc Sher * Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA Physics Department, University of Virginia, Charlottesville, VA 22901, USA Physics Department, College of William and Mary, Williamsburg, VA 23187, USA Received July 1999; editor: J. Bagger Contents 1. Introduction 2. Quantum numbers 3. Masses and mixing angles 3.1. Precision electroweak constraints 3.2. Vacuum stability bounds 3.3. Perturbative gauge uni"cation 3.4. Mixing angles 4. Lifetime and decay modes 4.1. Leptons 4.2. Quarks 5. Dynamical symmetry breaking 6. CP violation 6.1. CP violation in the standard model
266 268 271 272 274 283 294 299 299 304 308 314 314
6.2. Strong CP and the standard model 6.3. Strong CP and extra quarks 6.4. CP and a fourth generation 6.5. CP in the Aspon Model 7. Experimental searches 7.1. Search for long-lived quarks 7.2. Lepton searches 8. Conclusions Note added in proof Acknowledgements References
316 317 318 319 330 330 338 341 342 342 342
Abstract The possibility of additional quarks and leptons beyond the three generations already established is discussed. The make-up of this Report is (1) Introduction: the motivations for believing that the present litany of elementary fermions is not complete; (2) quantum numbers: possible assignments for additional fermions; (3) masses and mixing angles: mass limits from precision electroweak data, vacuum stability and perturbative gauge uni"cation; empirical constraints on mixing angles; (4) lifetimes and decay modes: their dependence on the mass spectrum and mixing angles of the additional quarks and leptons; the possibility of exceptionally long lifetimes; (5) dynamical symmetry breaking: the signi"cance of the top quark and other heavy fermions
* Corresponding author. E-mail address: [email protected] (M. Sher) 0370-1573/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 9 5 - 2
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for alternatives to the elementary Higgs Boson; (6) CP violation: extensions to more generations and how strong CP may be solved by additional quarks; (7) experimental searches: present status and future prospects; (8) conclusions. 2000 Elsevier Science B.V. All rights reserved. PACS: 12.15.Ff ; 14.65.!q; 11.30.!j
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1. Introduction The elementary spin-half fermions as we now know them are the quarks and leptons. The principal constituents of normal atoms and normal matter are the electron, as well as the up and down quarks which comprise the valence quarks of the protons and neutrons in the atomic nucleus. In addition, there is the electron neutrino which was "rst postulated by Pauli [1] in 1931 to explain conservation of energy and angular momentum in nuclear b-decay, and which was eventually discovered in 1956 by Reines and Cowan [2]. These quarks and leptons } the up and down quarks, the electron and its neutrino } comprise what is now called the "rst generation. The "rst intimation that Nature is more complex came with the discovery of the muon in 1937 [3,4]. The muon appears identical to the electron except for its mass which is &200 times heavier. The muon appeared so surprising that there was a famous comment by I.I. Rabi [5]: `Who ordered that?a The fact that the muon neutrino di!ers from the electron neutrino was established in 1962 [6]. The strange quark had already been discovered implicitly through the discovery of strange particles beginning in 1944 [7}9]. The completion of the second family with the charmed quark, predicted in 1970 [10], was accomplished in 1974 [11}14]. At "rst only hidden charm was accessible but two years later explicit charm was detected [15]. By this time, a renormalizable gauge theory was available [16}20] based on the "rst two generations and incorporating the Cabibbo mixing [21] between the two generations. The situation became even more challenging to theorists when experimental discovery of the third generation of quarks and leptons began with the tau lepton, discovered in 1975 [22] in e>e\ scattering at SLAC. Next was the bottom quark in 1977 [23,24]. The top quark, at &175 GeV much heavier than originally expected, was "nally discovered in 1995 [25,26]. Together with the q neutrino which presumably participates (its distinct identity } while not questioned } is not fully demonstrated) in tau decay, this completed the third generation. Since the present review is dedicated to the premise of further quarks and leptons beyond the third generation, it is worthwhile to recall to what extent and how the third generation was anticipated from the existence of the "rst two generations, why it is regarded as the end of the litany of quarks and leptons and what loopholes there are in the latter arguments. One early theoretical anticipation of a third generation was the paper of Kobayashi and Maskawa [27] who pointed out, at a time (1973) when only three #avors u, d, s of quark were established, that the existence of six #avors in three generations would allow the standard model naturally to accommodate CP violation. Study of the formation of the light elements (hydrogen, deuterium, helium, and lithium) in the early universe was started earlier in the 1960s [28,29]. In the 1970s tighter constraints were found based on the steadily improving estimates of the primordial abundances of these light isotopes. Since the expansion rate of the universe in this era of Big-Bang Nucleosynthesis, and hence the abundances, depends sensitively on the number of light neutrinos it was then possible to limit the acceptable number. The group of Schramm et al. [30}32] found in this way that the number of generations should not be greater than four [30], or in some analyses not greater than three (see e.g. footnote 4 on p. 242 of [31]); it is surely remarkable that such a strong constraint was found from early universe considerations already in 1979, a decade before the situation was clari"ed using colliders.
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Fig. 1. Helium-4 production for N "3.0, 3.2, 3.4. The vertical band indicates the baryon density consistent with J (D/H) "(2.7$0.6);10\ and the horizontal line indicates a primeval Helium-4 abundance of 25%. The widths of the . curves indicate the two-sigma theoretical uncertainty. Figure from Ref. [33].
A current plot of the primordial He abundance (whose exact value is still controversial in 1998) versus neutrino number n(l) for n(l)"3.0, 3.2, 3.4 is given in Fig. 1. The main point is that the neutrino number from cosmology is by now tied very closely to the high-energy experimental value in what is the strongest known link between particle theory and cosmology. In 1989, there came an experimental epiphany concerning the number of generations, or more precisely, the number of light neutrinos. This arose from the measurement of the Z width at SLAC [34,35] and especially at CERN [36}39]. The answer from this source is indisputably equal to three. The argument is simple: one can measure the total width of the Z to high accuracy, and then subtract the visible width to get the invisible width. Identifying the invisible width with neutrino decays leads to [40] n(l)"3.00$0.025 .
(1)
This provides compelling proof that there are only three conventional neutrinos with mass below M /2K45 GeV, and, by extrapolation, it leads to the idea that there are only three quark-lepton 8
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families. Since this ties in nicely with the KM mechanism of CP violation and with the Big-Bang Nucleosynthesis, the overall picture looks very attractive. However, this "nality was not universally accepted [41,42]. There are other reasons for entertaining the possibility of further quarks and leptons: E In many grand-uni"ed theories (GUTs), there naturally occur additional fermions. Although the minimal S;(5) GUT can contain only the basic three families, extension to any higher GUT such as SO(10) or E(6) inevitably adds new fermions. In SO(10) this may be only right-handed neutrinos (leptons) but in E(6) there are also non-chiral color triplets (quarks) and color singlet (leptons). Although there is no direct evidence for GUTs they are attractive theoretically and suggestive of how the standard nodel is extended. E Models of CP violation which solve the strong CP problem without axions generically require additional quarks. E For some models of dynamical symmetry breaking [43}46] the top quark mass, although higher than originally expected, is still not quite high enough to play its role in electroweak symmetry breaking. This might be interpreted as evidence for even heavier quarks. E It has been shown that a non-supersymmetric model with four generations can have successful uni"cation of the gauge couplings at the uni"cation scale. E In recently popular models of gauge-mediated supersymmetry breaking, additional vector-like quarks and leptons arise automatically. In addition, in models in which higher dimensions arise at the TeV scale [47], it has been shown [48] that if some standard model "elds live in the higher-dimensional space, low-scale gauge uni"cation can be obtained } the Kaluza}Klein excitations of these "elds, which could be rather light, must be vector-like. None of these reasons is fully compelling but each is suggestive that one should keep alive the study of this issue. Our hope is that this review will play a role in encouraging further thought about this open question. The present review contains the following subsections: Section 2 is on the possible quantum numbers of additional quarks and leptons; Section 3 discusses their masses and mixing angles; Section 4 deals with lifetimes and decay modes; dynamical symmetry breaking is in Section 5; CP violation is treated in Section 6, and "nally in Section 7 there is a treatment of the experimental situation and in Section 8 are the conclusions.
2. Quantum numbers When we add fermions to the standard model, there are choices in the possible quantum numbers. Under the color S;(3) group we refer to color triplets as quarks, color antitriplets as antiquarks. ! Color singlets which do not experience the strong interaction are generically referred to as leptons. Higher representations of color such as 6, 6 , 8,2 may be called quixes, antiquixes, queights, and so on. Such exotic color states are necessary in some models to cancel chiral anomalies. For example, in chiral color [49,50] one version (called Mark II in [49]) involves three conventional fermion generations, an extra Q" quark, and an S;(3) sextet fermion or quix. The extended gauge group of chiral color is S;(3) ;S;(3) ;S;(2) ;;(1) and we may list the fermions by * 0 * 7
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their (S;(3) , S;(3) , Q) quantum numbers. There are three colored weak doublets * 0 3[(3, 1, )#(3, 1,!)] , eight colored weak doublets 4(1, 3 ,!)#3(1, 3 ,#)#(3, 1, ) , a weak singlet quix
269
(2)
(3)
(4) (6 , 1,!)#(1, 6, ) , and three charged leptons and their neutrinos. The quix plays an essential role in anomaly cancellation. But in this review we restrict our attention only to quarks and leptons because while more exotic color states are a logical possibility it is one which is di$cult to categorize systematically. Quarks and leptons may be either chiral or non-chiral. The latter are sometimes alternatively called vector-like. Let us therefore de"ne the meaning of these adjectives. Chiral fermions are, for present purposes, spin- paricles where the left and right components transform di!erently under the electroweak gauge group S;(2) ;;(1). All the fermions of the * standard model are chiral. This means that they are strictly massless before the electroweak symmetry is broken. The simplest generalization of the standard model is surely to add a fourth sequential family. Of course, a fourth light neutrino is an immediate phenomenological problem with the invisible Z width, but the addition of a right-handed neutrino can resolve this. More generally, we may add a chiral doublet quark or lepton where the left-handed components transform as a doublet of S;(2) and the right-handed components as singlets. A chiral doublet of * quarks is
(5)
(6)
; *
(7)
;
; ; ; D 0 0 D * while a chiral doublet of leptons is N
; N , E . 0 0 E * Equally possible are chiral singlets such as or
D * for quarks, or for leptons or
(8)
N *
(9)
E . *
(10)
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Of course, with chiral doublets or singlets, there is a constant danger of chiral anomalies. In the standard model, there is a spectacular cancellation in each generation between the anomalies of the chiral doublets of quarks and leptons. In other models, one sometimes adds mirror chiral doublets to cancel anomalies. For example, a mirror chiral doublet of quarks is ; , D , * *
;
. (11) D 0 There can also be non-chiral (also known as vector-like) fermions, where the right and left components transform similarly under the electroweak S;(2);;(1) group. For example, a vectorlike quark doublet is
;
,
;
. (12) D * D 0 A vector-like doublet is not to be confused with the doublets occurring in the left}right model [51}53] where the gauge group is extended to S;(2) ;S;(2) ;;(1) . The ; and D quarks * 0 \* transform there chirally as in (5) above under S;(2) , but the right-handed singlets ; and D of * 0 0 S;(2) are then assumed to transform as a doublet under the additional gauge group S;(2) . * 0 General patterns for adding anomaly-free charge-vectorial chiral sets of fermions which acquire mass by coupling to the Higgs doublet of the standard model have been studied in [54] and further developed in [55,56]. An extensive analysis of the possible quantum numbers for additional fermions can be found in Ref. [57]. They looked at the general structure of exotic generations given the gauge and Higgs structure of the standard model. A similar analysis for left}right symmetric models was carried out in Ref. [58]. In grand unixed theories (GUTs) all types of additional fermions are possible. For example, in S;(5) there may be no additional fermions. But in SO(10) there must be at least an additional chiral right-handed neutrino in each family. In E(6) each 27 of fermions contains not only two extra neutrino-like states but also a 5#5 of S;(5) which contains a non-chiral singet of quarks and a non-chiral doublet of leptons. In superstring models, and M-theory models, the additional fermions are even less constrained. For example, E(6) and its content has been a familiar superstring possibility since the beginning [59], and any gauge group contained in at least O(44) is possible. Families and anti (mirror) families often occur in superstrings. Some extensions of the standard model require extra chiral fermions to cancel anomalies e.g. E As already mentioned, chiral color [49], anomaly cancellation dictates addition of quixes. E In some GUTs, e.g. S;(15) anomaly cancellation requires [60] inclusion of mirror fermions. More interesting are the types of additional quarks and leptons appearing in extensions of the standard model motivated by attempting to explain shortcomings of the model itself. Examples are: E Attempts to solve the strong CP problem without an axion [61,62] can introduce a non-chiral doublet of quarks, or non-chiral singlets (but not both), E In trying to explain the three families through anomaly cancellation, the 331-model [63,64] extends the individual families of the standard model by adding non-chiral singlets of quarks.
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The charges of the additional quarks di!er between the families. In a sense, this is not `Quarks and Leptons beyond the Third Generationa since the quarks are being added to the discovered generations. Nevertheless, our framework is su$ciently general to accommodate this possibility } as additional non-chiral singlets of quarks. E In the non-supersymmetric standard model, gauge uni"cation of the couplings fail } the three couplings do not meet at a point. It has been noted [65] that extending the model to allow a fourth generation introduces enough #exibility that a successful uni"cation of couplings can occur. E A potential problem of the minimal supersymmetric standard model is that the scalar quark masses must be very nearly degenerate to avoid large tree-level #avor-changing neutral currents. This degeneracy is very unnatural, using the conventional mechanism of gravitationally mediated supersymmetry breaking (where the supersymmetry breaking part of the Lagrangian is transmitted to the known sector via gravitational interactions). However, in gauge-mediated supersymmetry breaking, the supersymmetry breaking is transmitted to the known sector via the gauge interactions of `messengera "elds. Since the gauge interactions are #avor-blind, the degeneracy of squark masses is natural. The messenger "elds are, in the simplest case, composed of a 5#5 of S;(5) (or of several such "elds), which constitute a vector-like Q"! quark and a vector-like lepton. E There has been recent excitement about the possibility that additional space-time dimensions could be compacti"ed at (or even well below) the TeV scale [47]. In such models, the Kaluza}Klein excitations must be vector-like (to avoid having, for example, too many light neutrinos). In this Report, we will primarily concentrate on chiral quarks and leptons, or vector-like quarks and leptons, since they appear in the majority of models. However, when appropriate, we will note how our various constraints and bounds will apply to mirror quarks.
3. Masses and mixing angles One of the most unsatisfactory features of the standard model is the apparent arbitrariness of the masses and mixing angles of the known fermions. Although masses and mixing angles can be accommodated in the standard model (with addition of right-handed neutrino "elds, if necessary), there is no understanding of their values. An entire industry of model building has developed in an attempt to provide some theoretical guidance, ranging from #avor symmetries to relationships from grand uni"cation, but no model seems particularly compelling. In the case of additional fermions, the masses and mixing angles also are arbitrary. Nonetheless, some general features can be found. Phenomenological bounds can be obtained from high precision electroweak studies, theoretical bounds can be obtained from requiring the stability of the standard model vacuum and from requiring that perturbation theory be valid (up to some scale). In this section, these bounds are explored in some detail. We will start with a discussion of the phenomenological constraints from precision electroweak studies. Then we will consider bounds from vacuum stability and perturbation theory. In the vast majority of analyses of these bounds, the authors focused on bounds to the top quark mass, since its mass was unknown until
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relatively recently, thus we will "rst look at constraints on the top quark mass, and then generalize the results to "nd bounds on masses of additional quarks and leptons. Finally, we will discuss plausible models for the mixing angles. 3.1. Precision electroweak constraints 3.1.1. Chiral fermions In the past decade, high-precision electroweak measurements have led to remarkable constraints on potential physics beyond the standard model. The most important of these is the o parameter. As originally pointed out by Veltman [66,67], the tree level mass relation o,M /M cos h "1 5 8 5
(13)
is very sensitive to non-standard model physics (ruling out, for example, signi"cant vacuum expectation values for Higgs triplets). Since the relation is good to better than 1%, it is assumed that deviations from the relation are due to electroweak radiative corrections, which are sensitive to new particles in loops. An extensive, detailed analysis of electroweak radiative corrections can be found (with a long list of references) in the work of Peskin and Takeuchi [68,69]. They de"ne the S and ¹ parameters as aS,4e[P (0)!P (0)] , / a¹,(e/sin h M )[P (0)!P (0)] , 5 5
(14)
where a is the "ne-structure constant, P (q) is the vacuum-polarization amplitude with 67 (X>)"(11), (22), (33), (3Q), (QQ), and P "(P(q)!P(0))/q. Roughly, ¹ is a measure of the deviation of the o parameter from unity, coming from isospin violating contributions. S is an isospin symmetric quantity which measures the momentum dependence of the vacuum polarization; it is roughly the `sizea of the new physics sector. As an example, Peskin and Takeuchi consider the case of a chiral lepton doublet, E and N. In the limit M , M <M , they show that if the mass splitting is small, then the contribution for S is just , # 8 1/6p, and the contribution for ¹ is ("*M"/12p sin h M ). For a chiral quark doublet, these 5 5 contributions are tripled. As stated above, one can see that ¹ is a measure of the isospin splitting, while S is a measure of the size of the new sector. Thus, for a complete degenerate fourth generation, the contribution to S is S"2/3p&0.21 ,
(15)
while the contribution to ¹ is "*M " "*M " / * # . ¹" 4p sin h M 12p sin h M 5 5 5 5
An alternative representation, using parameters e , e and e can be found in Ref. [70].
(16)
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It is important to note that these results were obtained under the assumption that the extra fermion masses were much greater than that of the Z, and that the mass splitting was small. Peskin and Takeuchi give more exact expressions. What are the experimental bounds? The values can be determined from the data on Z-pole asymmetries, M , M , R , m , R and Z-decay widths. The contribution from new physics, 8 5 J @ S "S!S and ¹ "¹!¹ , can be determined [40,71,72]. The value of this contribution 1+ 1+ depends on the Higgs and top quark masses (which a!ect S , for example). For a top mass of 1+ 175 GeV and a Higgs mass between M and 1 TeV, Erler and Langacker [72] "nd 8 S "!0.20> (3p) . \
(17)
One can immediately see a apparent con#ict between Eqs. (15) and (17). Independent of the mass, a fourth-generation chiral multiplet is roughly 3 standard deviations o!. This led Erler and Langacker to claim that a fourth sequential family is excluded at the 99.8% con"dence level. However, there are several reasons why we believe it is premature to exclude a fourth sequential family. First, of course, is the fact that many 3p e!ects in recent years have disappeared. Second, the results for S are based on virtual heavy fermion loops. Any additional new physics will likely make a similar contribution. For example, Erler and Langacker show that the allowed range for S in minimal SUSY is !0.17> , where the error bars are 1p, and this is only in con#ict by 2.2 \ standard deviations. Third, it has been noted [73] that Majorana neutrino masses (which may be needed to give neutrino masses in the right range) lower S, as do models which involve mixing of two scalar doublets [74], and this could reduce the discrepancy further. Finally, the result above assumed a degenerate family. If one uses the exact expressions, takes M "180 GeV, & M "M "150 GeV, M "100 GeV and M "200 GeV, for example, one "nds the contribu" 3 , # tion to S to be approximately 0.11, rather than 0.21. This also would lower the discrepancy to slightly more than 2p. For S&0.2, the 2p upper bound on ¹ is approximately 0.2, leading to bounds on the mass splitting from Eq. (15). For quarks, the splitting must be less [72] than 54(M /M ) GeV and for 5 " leptons must be less than 162(M /M ) GeV, at the 2p level. Note that for quark masses above 5 # 200 GeV, the splitting must be less than 10%. Of course, the extra fermions must not contribute signi"cantly to the width of the Z, and their masses are thus bounded from below by M /2. Other experimental bounds will be discussed 8 in Section 7. 3.1.2. Non-chiral doublets Vector-like fermions do not contribute in leading order to S and ¹, and thus the values of these parameters do not constrain their masses. However, since vector-like doublets do not couple to the Higgs boson, the mass terms involving the E and the N cannot violate isospin invariance, and thus the masses must be degenerate at tree level, as must the masses of the ; and D. Even if one adds a singlet Higgs "eld, the degeneracy will remain. Only a Higgs triplet can split this degeneracy, but Higgs triplet vacuum expectation values are severely constrained by the o parameter. We conclude (in the absence of sizeable mixing with lighter generations) that the masses of states in a vector-like doublet are degenerate at tree level. The masses will be split by a few hundred MeV due to electroweak radiative corrections } this calculation will be done in the next section.
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3.1.3. Other fermions Non-chiral singlets will have arbitrary mass terms and arbitrary couplings to any Higgs singlets. No constraints can be placed on their masses. 3.2. Vacuum stability bounds Upper bounds on fermion masses can be obtained from the requirement that fermionic corrections to the e!ective potential do not destabilize the standard model vacuum. We will "rst discuss the e!ective potential, and its renormalization group improvement. Then, the bounds from the requirement of vacuum stability will be discussed, "rst for the top quark mass, and then for additional quarks and leptons. 3.2.1. The ewective potential An extensive review of the e!ective potential and bounds from vacuum stability appeared in 1989 [75]. Since then, the potential has been improved, including a proper renormalization-group improvement of scalar loops, and the bounds have been re"ned to much higher precision. In addition, the discovery of the top quark has narrowed the region of parameter space that must be considered. In this section, we discuss the e!ective potential and its renormalization-group improvement. It is easy to see how bounds on fermion masses can arise. The one-loop e!ective potential, as originally written down by Coleman and Weinberg [76] can be written, in the direction of the physical "eld, as <( )"< #< , where
and
< "!k #j A A
(18)
(19)
M( ) 1 A (!1)$g M( )ln (20) <" G A 64p M G in which the sum is over all particles in the model, F is the fermion number, g is the number of G degrees of freedom of the "eld i, and M( ) is the mass that the "eld has in the vacuum in which A the scalar "eld has a value . In the expression for < , we have ignored terms which can be A absorbed into < * these will be "xed by the renormalization procedure. In the standard model, we have for the W-boson, M( )" g , for the Z-boson, M( )"(g#g ) , for the Higgs A A A A boson, M( )"!k#3j , for the Goldstone bosons, M( )"!k#j and for the A A A A top quark M( )"h . For very large values of , quadratic terms are negligible and the A A potential becomes <"j #B ln( /M) , where B"(3/64p)[4j# (3g#2gg #g )!h] .
(21)
(22)
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One can see that if the top quark is very heavy, then h is large and thus B is negative. In this case, the potential is unbounded from below at large values of . This is the origin of the instability of the vacuum caused by a heavy quark. Although this form of the e!ective potential is well known, it is not useful in determining vacuum stability bounds. The reason is as follows. Suppose one denotes the largest of the couplings in a theory by a, in the standard model, for example, a"[max(j, g, h)]/(4p). The loop expansion is an expansion in powers of a, but is also an expansion in powers of logarithms of /M, since each A momentum integration can contain a single logarithmic divergence, which turns into a ln( /M) A upon renormalization. Thus, the n-loop potential will have terms of order aL>[ln( /M)]L .
(23)
In order for the loop expansion to be reliable, the expansion parameter must be smaller than one. M can be chosen to make the logarithm small for any speci"c value of the "eld, but if one is interested in the potential over a range from to , then it is necessary for a ln( / ) to be smaller than one. In examining vacuum stability, one must look at the potential at very large scales, as well as the electroweak scale, and the logarithm is generally quite large. Thus, any results obtained from the loop expansion are unreliable; in fact, the bound on the top quark mass can be o! by more than a factor of two. A better expansion, which does not have large logarithms, comes from solving the renormalization group equation (RGE) for the e!ective potential. This equation is nothing other than the statement that the potential cannot be a!ected by a change in the arbitrary parameter, M, i.e. d
R R R M #b(g ) !c
<"0 , (24) G RM Rg R
G where b"Mdg /dM and there is a beta function for every coupling and mass term in the theory. G The c function is the anomalous dimension. It is important to note that the renormalization group equation is exact and no approximations have been made. If one knew the beta functions and anomalous dimensions exactly, one could solve the RGE exactly and determine the full potential at all scales. Although we do not know the exact beta functions and anomalous dimensions, we do have expressions for them as expansions in couplings. Thus, by only assuming that the couplings are small, the beta functions and c can be determined to any level of accuracy and <( ) can be found. The resulting potential will be accurate if g ;1 and will not require g ln( /M);1. G G For example, in massless j theory, the RGE can be solved exactly to give <"j (t, j)G(t, j) , where t"ln( /M) and j (t, j) is de"ned to be the solution of the equation dj /dt"b(j )/(1#c(j ))
(25)
(26)
with the boundary condition being determined by the renormalization condition. G(t, j) is de"ned as exp(!4 R dt (c(j )/(1#c(j )))). Note that this potential gives the same result as before in the
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limit that c"0 and b"constant. Then G"1 and j "bt#constant. With t"ln( /M) this gives the ln( /M) terms as above. What about the massive case? The RGE is given by
R R R R R M #b #b(g ) #b k !c
<"0 . H Rj G Rg I RM Rk R
G
(27)
One is tempted to reduce this equation to a set of ordinary di!erential equations as before, giving <( )"k(t)g(t) #j(t)G(t) ,
(28)
where the coe$cients are running couplings obeying "rst-order di!erential equations as in the massless case. However, this is not correct. By considering small excursions in "eld space, one does not, as in the massless case, reproduce the unimproved one-loop potential. This is not surprising. In the massless theory, the only scale is set by , and thus all logarithms must be of the form t"ln( /M). In the massive theory, there is another scale, and there will be logarithms of the form ln((!k#3j )/M). Thus, one cannot easily sum all of the leading logarithms. In addition, the scale dependence of the constant term in the potential (the cosmological constant) can be relevant. In earlier work (and in the review of Ref. [75]), it was argued that the bounds only depend on the structure of the potential at large , and thus the mass term and constant term are irrelevant. However, in going from j to the Higgs mass, the structure of the potential near its minimum is important, and thus using the naive expression above is not as accurate (although it is fairly close). This will be discussed more in the next section. More recently, Bando et al. [77] and Ford et al. [78], following some earlier work by Kastening [79], found a method of including the additional logarithms found in the massive theory. In general, they showed that if one considers the ¸-loop potential, and runs the parameters of that potential using ¸#1 beta and gamma functions, then all logarithms will be summed up to the ¸th-to-leading order. The standard model potential, including all leading and next-to-leading logarithms, is then (in the 't Hooft gauge)
1 1 3 = 5 3 Z 5 1 = ln ! # Z ln ! <( )"! k # j # 4 16p 2 M 6 4 M 6 2
1 H 3 3 G 3 ¹ 3 # H ln ! # G ln ! !3¹ ln ! 4 M 2 4 M 2 M 2
(29)
with =,g /4, Z,(g#g ) /4, H,!k#3j , G,!k#j and ¹"h /2. All of the couplings in this potential run with t"ln /M. Use of two-loop beta and gamma functions will then give a potential in which all leading and next-to-leading logarithms are summed over. It was shown by Casas et al. [80] that the resulting minima and masses are relatively independent of the precise choice of M, as long as this potential is used (use of earlier potentials was inaccurate due to a sensitive dependence on the choice of scale). It is this potential that will be used to determine bounds on the top quark and Higgs masses in the next section.
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First, one should comment on the gauge dependence of the potential. Bounds on the masses of the top quark and Higgs boson are physical quantities, so how can one draw conclusions based on a gauge-dependent potential? It has long been known [75] that the existence (or non-existence) of minima of the potential are gauge-independent; an early calculation of the mass of the Higgs boson in the Coleman}Weinberg model [81] to two loops in the R gauge showed that the gauge K dependence drops out in the "nal result. A detailed analysis of the gauge dependence of the bounds on the Higgs and top quark masses has been carried out by Loinaz et al. [82}84]. They "nd a gauge-invariant procedure for determining the bounds, and "nd that the "nal result is numerically very close to the procedure discussed below. In a model with stronger gauge couplings, however, the gauge invariant method might give signi"cantly di!erent results. 3.2.2. Bounds on the top quark and Higgs masses The "rst paper to notice that fermionic one-loop corrections could destabilize the e!ective potential was by Krive and Linde [85], working in the context of the linear sigma model. Later, independent investigations by Krasnikov [86], Hung [87], Politzer and Wolfram [88] and Anselm [89] all looked at the one-loop, non-renormalization group improved potential of Eqs. (19) and (21), and required that the standard model vacuum be stable for all values of . The "rst of these was that of Krasnikov [86] who noted that the bound would be of O(100) GeV, rising to O(1000) GeV if scalar loops were included. The works of Politzer and Wolfram [88] and Anselm [89] gave much more precise numerical results, but ignored scalar loop contributions } thus they obtained upper bounds of 80}90 GeV on the top quark mass. Hung [87] gave detailed numerical results and did include scalar loops, thus his upper bound ranged from 80 to 400 GeV as the Higgs mass ranged from 0 to 700 GeV. All of these results are unreliable because the potential used is not valid for large values of . In these papers, the instability would occur for large values of , and thus ln( /p) is large enough that only a renormalization group improved potential is reliable. The "rst attempt to use an improved potential was the work of Cabibbo et al. [90]. They included the scale dependence of the Yukawa and gauge couplings, and required that the e!ective scalar coupling be positive between the weak scale and the uni"cation scale. Although they did not use the language of e!ective potentials, this procedure turns out to be very close to that used by considering the full renormalization group improved e!ective potential. Similar results, using the language of e!ective potentials, was later obtained by Flores and Sher [91]. Use of the renormalization-group improved potential will weaken the bounds. The beta function for the top quark Yukawa coupling is negative, and thus the coupling falls as the scale increases. Thus, the e!ects of fermionic corrections will decrease at larger scales. Compared with the bounds that one would obtain by ignoring the renormalization-group improvement, the decrease in the Yukawa coupling at large scales will weaken the upper bounds. This e!ect is not small; the Yukawa coupling for a quark will fall by roughly a factor of three between the weak and uni"cation scales. Note that for additional leptons, the Yukawa coupling does not fall signi"cantly, thus the bounds obtained by the non-renormalization-improved potential will not be greatly changed. The "rst attempt to bound fermion masses using the full renormalization group improved e!ective potential (earlier works, for example, never mentioned anomalous dimensions) was the 1985 work of Duncan et al. [92]. Their results, however, used tree-level values for the Higgs and top
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masses, in terms of the scalar self-coupling and the `MS Yukawa couplinga, and found a bound which, to within a couple of GeV, can be "t by the line M
(80 GeV#0.54M . (30) & As we will see shortly, however, corrections to the top quark mass can be sizeable, as much as 10 GeV. A much more detailed analysis, using two-loop beta functions and one-loop corrections to the Higgs and top quark masses (de"ned as the poles of the propagator), was carried out in 1989 by Lindner et al. [93], and followed up with more precise inputs in 1993 by Sher [94]. In all of these papers, the allowed region in the Higgs-top mass plane was given } the allowed region was always an upper bound on the top mass for a given Higgs mass, or a lower bound on the Higgs mass for a given top mass. The allowed region depended on the cuto! K at which the instability occurs. For example, if the instability occurs for values of above 10 GeV, then one concludes that the standard model vacuum is unstable iw the standard model is valid up to 10 GeV (should the lifetime of the metastable vacuum be less than the age of the Universe, one would conclude that the standard model cannot be valid up to 10 GeV). Thus, all of the bounds depend on the value of K. In the above papers, the e!ective potential used was the renormalization group improved tree-level potential, Eq. (28). As discussed in the previous section, this would be as precise as the precision of the beta functions and anomalous dimensions (two loop were used) if the only logarithms were of the form ln( /M); the resulting potential is exact in terms of the beta and gamma functions. However, when scalar loops are included, terms of the form ln((k#j )/M) arise, and these terms are not summed over. In the earlier papers, it was argued that when is large, the scalar terms are e!ectively of the form ln( /M), and thus the di!erence is irrelevant. But, in determining the Higgs boson mass in terms of the potential, the structure of the potential at the electroweak scale is relevant, and thus the di!erence in the form of the scalar loops is relevant. It turns out that this di!erence is especially crucial when the value of K is relatively small (1}10 TeV), and less important when K is large (10} GeV), thus the results of the above papers are valid in the large K case. To include the proper form of the scalar loops, one must use the form of Ford et al. [78], discussed in the last section. This analysis was carried out very recently by Casas et al. [95] and by Espinosa and Quiros [96]. A very pedagogical review of the analysis can be found in Espinosa's Summer School Lectures [99]. We now brie#y review that analysis and present their results. Consider the tree-level renormalization group improved potential, Eq. (28). At large values of , the quadratic term becomes negligible, and the question of whether the standard model vacuum is stable is essentially identical to the question of whether j(t) ever goes negative. If j(t) goes negative at some scale K, then the instability will occur at that scale. Casas et al. [80,95] analyzed the question using the full one-loop renormalization group improved potential, with two-loop beta and gamma functions, of Eq. (29). They showed that the
See Altarelli and Isidori [98] for a similar and independent analysis. For a discussion of the relationship between the location of the instability and the required onset of new physics, see Ref. [100].
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instabilty sets in when jI becomes negative, where jI is slightly di!erent from j:
h 3 g 1 3 g#g 1 1 3h ln !1 ! g ln ! ! (g#g ) ln ! jI "j! 2 8 4 3 16 4 3 16p
.
(31)
All that remains is to relate the parameters in the potential to the physical masses of the Higgs boson and of the top quark. It is not a trivial matter to extract the Higgs and top quark masses from the values of h(t) and j(t) used in the potential. One can write 1 h(k) , m (k)"m (1#d (k))" (2(2G $
m (k)"m (1#d (k))" & & &
(32)
(2 j(k) , G $
where the pole masses are the physical masses of the top and Higgs, and d (k) is the radiative corrections to the MS top quark mass. Note that the physical Higgs mass is not simply the second derivative of the e!ective potential, since the potential is de"ned at zero external momentum and the pole of the propagator is on-shell; d (k) accounts for the correction. & The correction d (k) receives contributions from QCD, QED and weak radiative e!ects, with the QCD corrections being the largest. The QCD corrections have been calculated to O(g ) in Ref. [101] and to O(g ) in Refs. [102,103], the other corrections were determined in Refs. [104}106]. The correction d (k) can be found in Refs. [80,107]. The detailed expressions for these & quantities, which correct several typographical errors in the published works, are summarized in an extensive review article by Schrempp and Wimmer [108]. The largest correction is to the top quark mass; the leading order term is a /p, which is 5%, or almost 10 GeV. All of these corrections were included in Refs. [95,96] and reviewed in Ref. [97]. If one requires stability of the vacuum up to a scale K, then there is an excluded region in the Higgs mass-top mass plane. The result, for various values of K, is given in Fig. 2. This "gure, in addition, also includes the region excluded by the requirement that the scalar and Yukawa couplings remain perturbative by the scale K; these bounds will be discussed in the next section. The lower part of each curve is the vacuum stability bound; the upper part is the perturbation theory bound. The excluded region is outside the solid lines. Thus, for a top quark mass of 170 GeV, we see that a discovery of a Higgs boson with a mass of 90 GeV would imply that the standard model vacuum is unstable at a scale of 10 GeV, i.e. if we live in a stable vacuum, the standard model must break down at a scale below 10 GeV. The curves in Fig. 2 are approximately straight lines in the vicinity of M &170 GeV, thus the top mass dependence can be given analytically [97]. For K"10 GeV, we must have a (M )!0.12 M (GeV)'133#1.92(M (GeV)!175)!4.28 8 & 0.006
(33)
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Fig. 2. Perturbativity and stability bounds on the SM Higgs boson. K denotes the energy scale where the particles become strongly interacting.
and for K"1 TeV, a (M )!0.12 . M (GeV)'52#0.64(M (GeV)!175)!0.5 8 & 0.006
(34)
It is estimated [95,97] that the error in the result, primarily due to the two-loop correction in the top quark pole mass and the e!ective potential, is less than 5 GeV. In Fig. 3, the stability and perturbation theory bounds are given explicitly as a function of K for M "175 GeV. Of course, it is not formally necessary that we live in a stable vacuum. Should another deeper vacuum exist, it is only necessary that the Universe goes into our metastable vacuum and then stay there for at least 10 billion years. A detailed discussion of the "nite temperature e!ective potential and tunnelling probabilities is beyond the scope of this review; the reader is referred to Refs. [95,97] for the details, as well as a comprehensive list of references. In short, the bound in the above paragraph for K"10 GeV weakens by 8 GeV, and for K"1 TeV, weakens by about 25 GeV. In all cases, the bound obtained by requiring that our vacuum have a lifetime in excess of 10 billion years is weaker than the bound obtained by requiring that the Universe arrive in our metastable vacuum.
This was "rst pointed out in Ref. [109].
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Fig. 3. Perturbativity and stability bounds on the SM Higgs boson as a function of K for M
281
"175 GeV.
We now turn to the question of vacuum stability for models with additional quarks and leptons. 3.2.3. Vacuum stability bounds and four generations In this review, we are concentrating on two possibilities for quarks and leptons beyond the third generation: the chiral case and the vector-like case. In the latter case, the quarks and leptons cannot couple to Higgs doublets. They will thus have no e!ect on the vacuum stability bounds (except very weakly through their e!ects on the two-loop beta functions of the gauge couplings). We conclude that there are no vacuum stability bounds on the masses of vector-like quarks and leptons in models with Higgs doublets. Should one include a Higgs singlet, of course, then the vector-like quarks and leptons would couple, and a bound could be found on their masses which depends on the singlet Higgs mass as well as the fraction of the mass which comes from the singlet vev (since a bare mass is possible). There is one exception to this conclusion. As noted by Zheng [110], if one adds a vector-like doublet and one or more vector-like singlets, then Yukawa couplings can exist. He studied the stability bounds in that case (using the tree-level potential and one-loop beta functions), assuming that the Yukawa couplings were unity, and found that the bounds are much more stringent than in the three generation case } the lower bound from vacuum stability and the upper bound from perturbation theory come together at a scale well below the uni"cation scale. In the chiral case, more speci"c bounds can be found. It is clear from Eq. (22) that one can naively just replace the h term with a summation over all quark and lepton Yukawa couplings. This amounts to replacing M with (M #M #M #M ) , 3 " # ,
(35)
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where ;, D, E and N are additional ;-type quarks, D-type quarks, charged leptons and neutrinos, respectively. The factor is a color factor. This replacement was noted by Babu and Ma [111] who simply rewrote Eq. (30) by substituting M with the above expression (their paper was written when the top quark mass was believed to be 40 GeV, so the top mass was not included in the above). As discussed above, however, this procedure is not particularly accurate. A more detailed analysis, using one-loop beta functions, was carried out by Nielsen et al. [112]. For simplicity, they assumed that the fourth-generation fermions all had a common mass, M ; since the quarks must have very similar mass, relaxing this assumption will not signi"cantly a!ect the results. If one assumes that the standard model is valid up to K&10 GeV, then they found that there is an upper bound on M of only 110 GeV. It is easy to see why this bound is so stringent. Suppose that M were equal to M . Then the expression in the above paragraph(ignoring the leptons) would be 2M , and thus the quark contribution would be 3M . One might expect the bounds to be smaller by a factor of roughly 3. In fact, it is not quite that severe since the upper line in the allowed region of Fig. 2 is not signi"cantly a!ected by the presence of additional generations. Nonetheless, the bound is quite stringent. Nielsen et al. [112] argued that CDF bounds [113,114] on stable, color triplets, as well as results from the successful description of top quark decays (which occur at the vertex), rule out M up to 139 GeV, and thus the standard model cannot be valid up to 10 GeV. They found that new physics had to start below approximately 10 GeV. However, this argument has a #aw. It is true that the CDF bounds on stable, color triplets rule out heavy quarks with decay lengths greater than a meter or so, and that the successful description of top quark decays rule out heavy quarks which decay at the vertex (at least up to about 150 GeV), but there is still a window for decay lengths in which the quarks would have evaded detection. Depending on mixing angles, these decay lengths are quite plausible. Nonetheless, theirs is the most detailed analysis to date of the case in which M is above 140 GeV (and thus the scale of new physics is well below the uni"cation scale). A much more detailed analysis was carried out by Dooling et al. [115,116]. They performed a complete analysis, using the full, two-loop analysis of Casas, Espinosa and Quiros (discussed in the last section). They only consider the case in which the standard model is valid up to the uni"cation scale, and thus only look at the case in which M is very light, typically less than 120 GeV. Their work is thus complimentary to that of Nielsen et al. They "nd that the point where the triviality bound and vacuum stability bound come together (see Fig. 3) is (for K&10 GeV) M (110 GeV. Thus, if the standard model is valid up to the uni"cation scale, only a narrow region of masses still exists between the LEP lower bound (roughly 1/2 the center-of-mass energy) and the vacuum stability bound of 110 GeV. These works did assume that the quark and lepton masses were all degenerate with a mass M . If one relaxes this assumption, then one approximately can replace (8/3) M with Eq. (35). Clearly, it is easier to accommodate heavy leptons and neutrinos than heavy quarks. Hung and Isidori [117] relaxed the assumption of a common M and simply assumed a doublet of degenerate quarks with mass M and a doublet of degenerate leptons with mass M . They found / * that with M &M , M can be extended to 150 GeV before a Landau pole appears at the Planck * 5 / mass. As M is raised, M should correspondingly decrease if one requires that the Landau pole * / appear at or above the Planck mass.
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3.3. Perturbative gauge unixcation The use of the RG equations as a tool to set bounds on and, in particlular, to `predicta particle masses is an `olda subject. The discussion on the bounds has been carried out in previous subsections. This subsection concentrates instead on the use of the RG equations to `predicta various masses. In particular, we shall pay special attention to the masses of any extra family of quarks and leptons. This analysis only works for chiral fermions. Vector-like fermions, having no Yukawa coupling to the SM Higgs "eld, will not have the desired in#uence on the evolution of the couplings as we shall see below. In order to use the RG equations to make `predictionsa on the masses, one has to invoke either some experimental necessities or some theoretical expectations } or rather prejudices } such as "xed points, gauge uni"cation, etc. We shall describe below these concepts along with their consequences. To begin, we shall list the RG equations at two loops for the minimal SM with three generations [118]: 16p
dj "24j#4j(3g!2.25g !0.45g )!6g#(16p)\ R R dt ;+30g![3g#2g!80g g]j!144jg!312j!32g g, , R R J R R R
16p
(36a)
dg R "g+9g!16g !4.5g !1.7g # (8p)\ R R dt ;[!12g#6j#g(!12j#36g )!108g ], , R R
(36b)
16p
dg "g +(41/5)#(16p)\[(199/25)g #(27/5)g # (88/5)g !(17/5)g], , R dt
(36c)
16p
dg "g +!(19/3)#(16p)\[(9/5)g #(35/3)g #24g !3g], , R dt
(36d)
16p
dg "g +!14#(16p)\[(11/5)g #9g !52g !4g], . R dt
(36e)
To set the notations straight, our de"nition of g uses the S;(5) convention and is related to the ;(1) coupling g by g "(3/5g . In the evolution of these couplings we will neglect the 7 contributions coming from the lighter fermions. In the above RG equations, clearly the important couplings are those of the top quark Yukawa and of the Higgs quartic couplings, and, to a certain extent, also the QCD coupling. As we have seen earlier, one important use of such equations is by following the evolution of j with the initial value of g "xed by its experimental value. Requiring j to be positive (for vacuum stability reason) R at least up to the Planck scale allows us to set a lower limit on the Higgs mass to be &136 GeV. This use of the RG equations is rather solid in the sense that it relies only on the vacuum stability criterion of quantum "eld theory. Other uses which are discussed below are more speculative but are quite interesting in that several predictions can be made and can be tested.
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In dealing with RG equations, a natural question that comes to mind is whether or not there exist stable "xed points. Basically, a stable "xed point is a point in coupling space to which various couplings converge regardless of their initial values. This is an attractive idea that has wide applications in many "elds of physics, such as critical phenomena } to mention just one of many. In particle physics, there were many speculations concerning the nature of such "xed points if they truly exist. For example, Gell-Mann and Low [119], and subsequently, Johnson et al. [120] have speculated that quantum electrodynamics might possess an ultraviolet stable "xed point which will render QED "nite. Other more `recenta speculations dealt with the very interesting subject of the origin of particle masses } at least of the heavy one(s). In general, a stable "xed point appears as a zero of the b function which would be meaningful only if one has a full knowledge of such a function. In the absence of such a knowledge, one might have to resort to approximations allowed by perturbation theory. In regions where various couplings can be considered to be `smalla enough so that the use of one- or two-loop b functions might be justi"ed, one would try to `runa the couplings over a large region of energy and see if they converge to a point for arbitrary initial values. If such a point is found, say by a numerical study of the RG equations, one would qualify this as a "xed point. Such an approach has been pioneered by Pendlenton and Ross [121], and, in particular by Hill [122], where the "xed points are of the infrared nature. Of relevance to this report is the suggestion by various authors that a fourth generation might be needed for the existence of such a "xed point. Let us "rst summarize what has been done for the top quark mass and subsequently describe works related to the masses of a fourth generation. Pendleton and Ross [121] were the "rst to suggest a relationship between the top quark Yukawa coupling and the QCD coupling as a result of an infrared (IR) stable "xed point. To see this, one can combine the one-loop RG equations for g and the QCD coupling g ("rst terms on the R right-hand side of Eq. (38)) to form a RG equation for the ratio g /g , namely R 16p(d(g /g )/dt)"g!g !(3g#g )!g . R R
(37)
Ignoring the electroweak contributions in Eq. (39), there is an IR "xed point obtained by setting the right-hand side to zero. Pendleton and Ross obtained a relation g"g "g . R R
(38)
It turns out that the above relation gives too low of a mass for the top quark. In fact, the original prediction [121] using Eq. (40) and a value of a "g /4p&1/7 (at a scale of &2M ) gives a mass 5 of &110 GeV. Using the current value of a +0.12 (at the Z mass), the prediction would have been even lower, even after electroweak corrections are included. It goes without saying that this cannot be true for we already know that the top quark mass is &175 GeV. As pointed out by Hill [122] long before the discovery of the top quark, the Pendleton}Ross "xed point is only a quasi-"xed point in the sense that it can never be reached at the scale of interest &100 GeV. Hill proposed an intermediate "xed point that can be found by setting the b function in the RG for g to zero with R a `slowly varyinga g replaced by a constant taken to be some average value between two scales: 100 and 10 GeV. This translates into an approximate relation g+8g . R
(39)
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With g &0.7, Hill made the prediction for the top quark mass to be +240 GeV. This is now known to be much too large, although at the time the prediction was made, it appeared to be a plausible value. In the above discussion for the top quark mass as a result of an IR stable "xed point, one feature clearly emerges: a heavy fermion is needed to drive the evolution towards a "xed point. This point was made even clearer in a detailed study of Bagger et al. [123]. These authors made two assumptions: the "rst one is the existence of a desert between the weak scale M and some Grand 5 Uni"ed scale M &10 GeV and the second one being that of perturbative uni"cation. The 6 question asked in Ref. [123] was the following: what should the initial values of various Yukawa couplings at M be in order for those couplings to reach the "xed in a `physical timea 6 t "(1/16p) ln(M /M )&1/5? Again, it turns out that `largea initial values (at M ) of Yukawa 5 6 5 6 couplings guarantee that the "xed point is reached in `physical timea. What is this "xed point and what does it say about masses of possible extra generations if they exist? We shall describe below the salient points of the analysis of Ref. [123]. We begin the discussion of Ref. [123] with the following one-loop RG equations for the quark and lepton Yukawa couplings: d¹ / "2(G !¹)¹ !3¹r(S ) , / / 3 dt
(40a)
d¹ * "2(G !¹)¹ !3¹r(S ) , * * # dt
(40b)
where various Yukawa factors are de"ned as ¹ "¹r(>R>) with >";, D, E or N, 7 ¹ "¹ #¹ , ¹ "¹ #¹ , ¹"3¹ #¹ , S ";R;!DRD and S "ERE!NRN. / 3 " * # , / * 3 # Finally, the gauge factors G's are de"ned as G "G "G "8g #g and G "G "G "g # , * 3 " / where the contribution from g has been neglected. Notice that t"(1/16p) ln(M /M). 6 To simplify the discussion, Ref. [123] "rst assumed degenerate quarks and degenerate leptons so that S "0 as well as S "0. If the gauge couplings can be approximated as constant (or very 3 # slowly varying), G and G in Eqs. (42) can be replaced by some averages similar to the procedure / * used by Hill [122]. Let us denote these averages by GM and GM . It is then easy to see that Eqs. (42) / * have the following two distinct "xed points: GM "¹ (quark radial "xed point) and GM "¹ (lepton / * radial "xed point). Whether or not these "xed points are reached will depend on the initial values of ¹ or ¹ at the Grand Uni"ed scale M . If ¹(M ) is below some critical value ¹(M ) , the "xed / * 6 6 6 point GM will not be reached in `physical timea t&1/5: the value of ¹ at M will be less than the 5 "xed point value GM "¹. This fact has allowed Ref. [123] to set an upper limit on heavy fermion masses with the upper limit being the IR stable "xed point. Setting S "¹ "0, Ref. [123] plotted 7 * ¹ at the weak scale as a function of its value at M . This is shown in Fig. 4a. The result for the / 6 lepton case is shown in Fig. 4b. The "gure shows the result for eight families. We are, of course, concerned only with four families which are still allowed. within error, by precision electroweak results. For four families, Ref. [123] gave the following upper bounds on ¹: ¹ :2.7, /
¹ :3.4 . *
(41a,b)
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Fig. 4. (a) The trace ¹ at M as a function of the trace ¹ and M for N "8 and ¹ "0. The dotted line denotes the / 5 / 6 $ * radial quark "xed point. For ¹ (M )'0.1, the "xed point is reached in physical time. (b) ¹ (M ) as a function of / 6 * 5 ¹ (M ) for N "8 and ¹ "0. * 6 $ /
(The above numbers used values of gauge couplings which are now outdated.) This translates into the following distinct bounds on fermion masses for four families: RM :(290 GeV), RM :(325 GeV) . (42a,b) / * (These bounds would be slightly less if recent values of the gauge couplings are used.) The above bound for the quarks, for example, would translate roughly into a bound on the mass of a degenerate fourth-generation quark (after subtracting out the top quark) as M :164 GeV. Is 3 " this what one should be aiming for when one tries to look for fourth-generation quarks? Hill et al. [124] made an extensive study of the RG "xed points and their connections with the mass of the Higgs boson(s) for the one- and the two-Higgs doublet cases, and for up to "ve generations. In this work, the Higgs quartic coupling(s) is run simultaneously with the various
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Fig. 5. Flow of j and g towards "xed points in the standard one-Higgs doublet model. Open circles denote initial points. R Crosses denote "nal "xed points.
Yukawa couplings and, as a result, one clearly sees again the interplay between heavy fermions and the Higgs "eld. This is shown, for example, in Figs. 1 and 2 of Ref. [124] for the one-Higgs doublet case with three generations, which are reproduced in Figs. 5 and 6. Other attempts of using the RG "xed points to construct fermion mass matrices have been made, for example in Ref. [125]. However the `predicteda value for the top quark mass is now outdated. A di!erent approach was taken by one of us (P.Q.H.) [65] concerning the in#uence of a possible fourth generation on the evolution of all couplings of the SM and not merely the Yukawa couplings. In particular, the question that was asked was whether or not there can be gauge uni"cation in the nonsupersymmetric SM and under which conditions this can be achieved. As we shall see below, it turns out that a fourth family of chiral fermions will be needed and that their masses are found to be fairly constrained. The possibility of coupling-constant uni"cation of the three gauge interactions of the Standard Model (SM) is, without any doubt, one of the most important issues in particle physics. Couplingconstant uni"cation is a necessary, but not su$cient, condition for a Grand Uni"cation of the SM [126}128]. Such a possibility is particularly attractive since it would provide a uni"ed explanation for a number of puzzling features of the SM such as electric charge quantization for example.
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Fig. 6. Relation between the Higgs mass and the top quark mass in the standard one-Higgs model.
There are various ways that the three gauge couplings can get uni"ed. The simplest way is to assume that there is a `deserta (i.e. no new physics) between the electroweak scale and the scale at which uni"cation occurs. Simply speaking, the three gauge couplings are left to evolve beyond the electroweak scale under the assumption that there is no additional gauge interactions of the type which would modify the evolution of some of the gauge couplings. A more complicated way would be to assume that there is one or several intermediate scales where partial uni"cation among two of the three couplings occurs. This might well be the case. However, we shall restrict ourselves, in this report, to the simplest scenario of uni"cation with a `deserta and search for conditions under which this can be achieved. This was the approach taken by one of us (P.Q.H.) [65]. We will proceed in two steps. First, we will present the evolution of the gauge couplings and show the places where they cross, ignoring any heavy threshold e!ects that might be } and should be if there truly is uni"cation } present. In this discussion, we will show both the minimal SM with three generations and the one with an extra fourth generation. We shall see that, under certain restrictions on the masses (fourth generation and Higgs masses), the latter possibility provides a better `convergencea of the three gauge couplings. By convergence under the quotation marks, we mean that they do not precisely meet at the same point. The `truea convergence will be shown to be accomplished by the inclusion of heavy threshold e!ects. In fact, it would be senseless to claim uni"cation without taking into account such e!ects. We "rst summarize the situation with three families. The "rst task is to integrate Eq. (38) numerically and look for the places where the couplings meet, disregarding for the moment the possibility that there might be uni"cation. We then have to set up some kind of criteria to decide on how close to each other all three couplings have to be in order for them to have a chance of actually converging to a single scale, once heavy particle thresholds, such as those of the X and > bosons of S;(5) for instance, are taken into account. Once these criteria are satis"ed and heavy particle threshold e!ects are included, one can put an error on the uni"cation scale and, consequently, an error on the proton lifetime. Fig. 7 shows the evolution, without heavy threshold e!ects, of g , g , and g of S;(3) S;(2) ;(1) for the case with three generations. Clearly these three couplings do not converge.
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Fig. 7. Evolution of couplings in the three generation case.
The question is: How far apart are they from each other and at what scales? As far as the scales are concerned, we will be interested only in those which are above some minimum value implied by the lower bound on proton decay. A rough estimate of that lower bound is obtained by noticing that q > (yr)+10(M /4.6;10). This gives M 91.3;10 GeV (corresponding to % % N C p ln(E/175)"29.64 on the graph) for q > (yr)95.5;10. The next question is the following: N C p Starting from M &1.3;10 GeV, how far apart are the three gauge couplings from each other % at a given energy scale? As stated above, the reason for asking such a question stems from the fact that, if the SM were to be embedded at M in a Grand Uni"ed model such as S;(5) [127] (for % instance), the decoupling of various heavy GUT particles would shift the three couplings from a common a to (possibly) di!erent values. As shown below, for a wide range of `reasonablea heavy % particle masses, such an e!ect produces not more than &5% shift from the common value and in the same direction. It turns out that the modi"ed couplings can di!er by not more than &4%. From this a reasonable criterion would be to require that, at a scale M 5m , the three gauge % % couplings are within 4% of each other. We shall take S;(5) [127] as a prototype of a Grand Uni"ed Theory. Let us assume the following heavy particle spectrum: (X, >)"(3 , 2, 5/6)#c.c. with mass M , real scalars 4 (8, 1, 0)#(1, 3, 1)#(1, 1, 0) (belonging to the 24-dimensional Higgs "eld) with mass M , and the complex scalars (3, 1,!) (belonging to the "ve-dimensional Higgs "eld), with mass M . (The quantum numbers are with respect to S;(3) S;(2) ;(1).) The heavy threshold corrections are then [129]
M M 1 35 % ! % #D,0- , ln D " ln 4p M M 30p 4
(43a)
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M M 1 21 1 % ! ln % #D,0- , D "! # ln M M 6p 4p 6p 4
1 7 1 1 M M M % ! % ! ln % #D,0- , D "! # ln ln 4p 2p 12p 4p M M M 4 where
2 M % , D,0-"!gk G G 25pa M % .
(43b)
(43c)
(44)
with k "1/2, 3/2,!1 for i"1, 2, 3, is the correction coming from possible dimension 5 operators G present between M and M . The modi"ed gauge couplings can be expressed in terms of the % . uni"ed coupling a (at M ) as % % a (M )"a /(1!a D ) , G % % % G
(45)
where i"1, 2, 3. We then de"ne the fractional di!erence between the modi"ed gauge couplings as d "(a !a )/a , GH G H G
(46)
for i, j"1, 2, 3 and the de"nition refers to a as being the larger of the two couplings. For a wide G range of heavy particle masses (in relation with M ) and the parameter g appearing in D , and for % G a &0.024!0.028, it is straightforward to see that d can be at most 4% [65]. From this simple % GH analysis, one can reasonably set a criterion for a given scenario to have a chance of having gauge coupling uni"cation: the fractional di!erence among the three gauge couplings at some scale M 5M should not exceed 4%. % % For the SM with three generations and taking into account the presence of M &1.3; % 10 GeV, one "nds the following trend: d decreases from 3% as one increases the energy scale beyond M , while d increases from 4% and d also increases from 7%. (For example, at % M &3.3;10 GeV, d &1.4%, d &8.4% and d &9.7%). From these considerations % } and not from just `eyeballinga the curves } one might conclude that the minimal SM with three generations does indeed have some problem with uni"cation of the gauge couplings. There is a drastic change to the whole scenario when one postulates the existence of a fourth generation of quarks and leptons [65]. The main reason is the fact that the Yukawa contributions to the running of the gauge couplings appear at two loops. In the three generation case, the top Yukawa coupling actually decreases sligthly with energy because its initial value is partially cancelled by the QCD contribution (at one loop). As a result, the presence of a heavy top quark is insigni"cant in the evolution of the gauge couplings at high energies when there are only three generations. The presence of more than three generations drastically modi"es the evolution of the Yukawa, Higgs quartic self-coupling, and the three gauge couplings. For example, with a fourth generation which is su$ciently heavy, all Yukawa couplings grow with energy, signi"cantly a!ecting the evolution of the gauge couplings. It turns out, as we shall see below, that the Yukawa couplings can develop Landau poles below the Planck scale. If there were any possibility of gauge uni"cation, one would like to ensure that it occurs in an energy region where perturbation theory is
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still valid. Furthermore, the uni"cation scale will have to be greater than M (as discussed above). % As we shall see, the validity of perturbation theory plus the lower bound on the proton lifetime put a severe constraint on the masses of the fourth generation. The two-loop renormalization group equations applicable to four generations are given by [118,65] 16p
dj "24j#4j(3g#6g#2g!2.25g !0.45g )!2(3g#6g#2g)#(16p)\ R O J R O J dt ;+30g#48g#16g![3g#6g#2g!80g (g#2g)]j R O J R O J R O !6j(24g#48g#16g)!312j!32g (g#2g), , R O J R O
16p
(47a)
dg R "g+9g#12g#4g!16g !4.5g !1.7g #(8p)\ R R O J dt ;[1.5g!2.25g(6g#3g#2g)!12g!(27/4)g!3g#6j#g R R O R J O R J R ;(!12j#36g )!(892/9)g ], ,
16p
dg O "g+6g#12g#4g!16g !4.5g !1.7g #(8p)\ O R O J dt ;[3g!g(6g#3g#2g)!12g!(27/4)g!3g#6j O O O R J O R J #g(!16j#40g )!(892/9)g ], , O
16p
(47c)
dg J "g+6g#12g#4g!4.5(g #g )#(8p)\[3g!g(6g#3g#2g) J R O J O O O R J dt !12g!(27/4)g!3g#6j!16jg], , O R J J
16p
(47b)
(47d)
dg "g +(163/15)#(16p)\[(787/75)g #6.6g #(352/15)g dt !3.4g!4.4g!3.6g], , R O J
(47e)
16p
dg "g +!(11/3)#(16p)\[2.2g #(133/3)g #32g !3g!3g!2g], , (47f) R O J dt
16p
dg "g +!(34/3)#(16p)\[(44/15)g #12g !(4/3)g !4g!8g], . R O dt
(47g)
For simplicity, we have made the following assumptions: a Dirac mass for the fourth neutrino and the quarks and leptons of the fourth generation are degenerate S;(2) doublets. The respective * Yukawa couplings are denoted by g and g , respectively. Also, in the evolution of the quartic O J coupling j and the Yukawa couplings, we will neglect the contributions of q and bottom Yukawa
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Fig. 8. Couplings as a function of energy for the set of masses given in the text.
couplings, as well as the electroweak gauge couplings, g and g , to the two-loop b functions since they are not important. Also, as long as the mixing between the fourth generation and the other three is small, one can neglect such a mixing. In the numerical analysis given below we shall "x the mass of the top quark to be 175 GeV. We shall furthermore restrict the range of masses of the fourth generation so that the Landau poles lie comfortably above 10 GeV, in such a way that uni"cation occurs at a scale which would guarantee the validity of perturbation theory as well as satisfying the lower bound on the proton lifetime. Concerning the former requirement, it basically says that one should look at uni"cation scales where the values of the Higgs quartic and Yukawa couplings are still su$ciently perturbative that one can neglect contributions coming from three-loop (and higher) terms to the b functions. Fig. 8 shows g , g and g as a function of energy for a particular set of masses: m "151 GeV, / m "95.3 GeV, where m and m denote the fourth-generation quark and lepton masses, respec* / * tively. It is already well known, from the discussion in the previous section, that, by adding more heavy fermions, the vacuum will tend to be destabilized unless the Higgs mass is large enough. As we have seen, the vacuum stability requirement is equivalent to the restriction j'0. Furthermore, the heavier the Higgs boson is, above a minimum mass that ensures vacuum stability, the lower (in energy scale) the Landau pole turns out to be. It turns out that this Landau pole should not be too far from M , otherwise g , g and g would not come close enough to each other. On the other % hand, it should not be too close either because of the requirement of the validity of perturbation theory. These considerations combine to give a prediction of the Higgs mass, namely m "188 GeV for the above values of the fourth-generation masses [65]. The dependence of the & Higgs mass on the fourth-generation mass in this analysis is obviously striking.
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Following the criteria that we have set for taking into account the heavy threshold e!ects, the modi"ed couplings a (M ) expressed in terms of a (M ) (which can be read o! from the graph) G % G % and the threshold correction factors D are given by 1/a (M )"1/a (M )#D . The choice of the G G % G % G mass scales M , M , M , and the parameter g is arbitrary and is only "xed to a certain extent by 4 the requirement that a (M )'s should be as close to each other as the precision allows. As an G % example, the choice M "M , M "M , M "0.5M and g"10 (where we have picked % % 4 % M +3.5;10 GeV) transforms a (M )"0.0278, a (M )"0.0273 and a (M )"0.0285 % % % % (values that can be read o! from Fig. 8) to a (M )"0.02735, a (M )"0.02662 and % % a (M )"0.02705. From these values, one can conclude that the couplings are practically the same % with all three equal to a +0.027 or 1/a +37. % % The above simple exercise simply shows that, with just an additional fourth generation having a quark mass m +151 GeV, a lepton mass m +95.3 GeV and a Higgs mass m +188 GeV, / * & uni"cation of all three gauge couplings in the nonsupersymmetric SM can be achieved after one properly takes into account threshold e!ects from heavy GUT particles [65]. Other combinations of masses are possible for gauge uni"cation but their values will not be much di!erent from the quoted ones, the reason being the requirement that the mass range of the fourth generation be restricted to one that will have Landau poles only above 10 GeV. Do the masses given above satisfy the requirement of perturbation theory? In fact, at the uni"cation point M "3.5;10 GeV, one has (with a ,g/4p): a "0.4, a "0.16, a "0.48 and j/4p"0.19. % G G R O J Although these values are not `smalla, they nevertheless satisfy the requirements of perturbation theory, namely a :1 and j/4p:0.4. (The latter requirement comes from lattice calculations R O J which put an upper bound on the Higgs mass of &750 GeV.) For comparison, a in the R three-family SM has a value of 0.016 at a comparable scale and this explains why it is unimportant in the evolution of the SM gauge couplings. An important consequence of a fourth generation in bringing about gauge uni"cation is the value of the uni"cation scale itself. In the example given above, it is M "3.5;10 GeV [65]. In % the nonsupersymmetric SU(5) model, the dominant decay mode of the proton is pPe>p and the mean partial lifetime is q > (yr)+10(M /4.6;10). Taking into account various uncertain% N C p ties such as heavy threshold e!ects, hadronic matrix elements, etc., the predicted lifetime is q > (yr)+3.3;10! to be compared with q > (yr)'5.5;10 [65]. Notice that the N C p N C p central value is within reach of the next generation of SuperKamiokande proton decay search. Another hint on the masses of a fourth generation comes from considerations of models of dynamical symmetry breaking a` la top-condensate [46]. This will be discussed in Section 5 where one can see how the original idea of using the top quark as the sole agent for electroweak symmetry breaking (in the form of ttM condensates) led to a prediction for the top quark mass (before its discovery) to be much larger than its experimental value. The original form of this attractive idea obviously has to be modi"ed, most likely by the introduction of new fermions such as a fourth generation or S;(2)-singlet quarks. In the above discussion on perturbative gauge uni"cation, as well as in the subsequent related discussion in Section 5, the issue of the gauge hierarchy problem is not considered. Such an issue is beyond the scope of the largely phenomenological approach that we are taking. This point was alluded to in the Introduction where we stressed that none of the reasons given for considering quarks and leptons beyond the third generation is fully compelling, but each, including the one on perturbative gauge uni"cation, is suggestive. It is certainly possible that the `solutiona of the gauge
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hierarchy problem will not a!ect the above arguments; the recently developed alternative to supersymmetry and technicolor, TeV-scale gravity, for example, may not appreciably change results on gauge uni"cation. A full consideration of the gauge hierarchy problem is beyond the scope of this review. 3.4. Mixing angles In previous sections, we have seen that the masses of quarks and leptons, although arbitrary, are constrained by phenomenological considerations as well as vacuum stability and perturbation theory. The mixing angles of quarks and leptons are also arbitrary, however there are no constraints from vacuum stability and perturbation theory (and only weak phenomenological constraints). Thus, a much wider range of mixing angles can be accommodated, and one can only be guided by considering various models for these angles. In this section, we will discuss plausible models for mixing angles. Since we know that the quark sector has nonzero mixing angles, but that the lepton sector may not, we will "rst look at the lepton sector, and then the quark sector. 3.4.1. Leptons The only phenomenological indication of any mixing in the lepton sector comes from neutrino oscillations. At the time of this writing, there are three indications of oscillations: solar neutrinos [130], atmospheric neutrinos [131] and LSND [132]. It is di$cult, although not quite impossible, to reconcile all three of these in a three-generation model. If there are four light neutrinos, in this case, the fourth neutrino must be sterile (an isosinglet) in order to avoid the bounds from LEP. Such a neutrino could exist without requiring the existence of any additional fermions. It is likely that the situation will be clari"ed within a year or so at Superkamiokande and the Solar Neutrino Observatory. A detailed discussion of neutrino oscillations and their phenomenology, including the recent strong evidence for atmospheric neutrino oscillations at SuperKamiokande, can be found in Ref. [133]. We will defer to that review in this paper, and will not discuss the possibility of light isosinglet neutrinos further. We certainly will, however, discuss the case in which a fourth generation neutrino is very heavy. This will automatically occur if the fourth generation is vector-like. Even if it is chiral, models exist that can give such a mass. Recently, one of us [134] has considered a model of neutrino masses with four generations where one can obtain dynamically one heavy fourth generation and three light, quasi-degenerate neutrinos. Ref. [135] has also considered a scenario with four generations which has similar consequences. Suppose that the heavy leptons form a standard chiral family, with a right-handed neutrino. The bounds from the Z width obtained at LEP force the mass of the N and E to be greater than 45 GeV. Are there any phenomenological bounds on the mixing? In analogy with the quark sector (as well as the prejudice from most models), one expects the mixing to be the greatest between the third and fourth generations. This will a!ect the ql = vertex, multiplying it by cos h, where h is the mixing O angle. Since all q decays occur through this vertex, the result will be a suppression in the overall rates. For some time, it was believed that the mass of the q was 1782$2 MeV, and the measured rate was too low; mixing with a fourth generation was a straightforward explanation [136}138]. However, the q mass has now been measured to much higher precision at BES [139] to be 1776.96$0.2$0.2 MeV, and the measured rate is now in agreement with theoretical expectations.
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This has been analyzed by Swain and Taylor [140,141], who "nd a model-independent bound on the mixing of sinh(0.007. A similar bound can be obtained for mixing between the fourth generation and the "rst two, although one expects those angles to be smaller. What values of the mixing might one expect? There are four plausible (in the view of the authors, of course) values of the mixing angle between the third and fourth generations: (a) sin h"m /m , (b) sin h"m O /m , (c) sin h"m /m , (d) sinh"0. 5 .J O # J , The "rst of these occurs in typical see-saw models. The second occurs in models in which the mixing occurs only in the neutrino mass matrix. The third occurs in models with a global or discrete lepton-family symmetry broken by Planck scale e!ects, and the fourth occurs when the symmetry is not broken by Planck scale e!ects. We now discuss each of these. The "rst relation, sinh"m /m , occurs in models in which the 2;2 mass sub-matrices are of O # the form
0
A
A B
.
If the neutrino and charged lepton mass matrices are of that form, then the mixing angle is given by O (m /m !(m /m , which gives sin h"m /m for realistic values of the l mass. Models of this J , O # O O # type were pioneered by Weinberg [142] and Fritzsch [143], who noticed that they will give the successful relation for the Cabibbo angle: sinh "m /m . Fritzsch also showed [144] that there are A B Q some very simple symmetries which automatically give this relation. When the rate for leptonic decays was believed to be too low, Fritzsch [145] used this relation to propose that a fourthgeneration lepton of 100}200 GeV could account for the discrepancy. As noted above, there is a lower bound on the mixing between the q and the E given by sin h(0.007. Using the Fritzsch relation, this becomes a lower bound on m , which is given # by m '250 GeV. This is very near the bounds from perturbation theory. We conclude that a very # slight improvement in the uncertainties in the q decay rate will rule out the very general relationship sin h"(m /m (or discover the e!ect!). O # The second relationship, sin h"m O /m , will occur in models in which, because of some discrete J , or global symmetry, the charged lepton mass matrix is diagonal. The Fritzsch relationship will then give sin h"m O /m . Given the cosmological bound on the l mass, this gives a value of sinh O J , which is less than 10\. The E or N lifetime (whichever is the lighter) will then be in the picosecond}nanosecond range, with extremely interesting phenomenological consequences. Suppose that one simply assumes that a discrete symmetry forbids any mixing at all between the E and N and the other three generations. This is simply an extension of the familiar electronnumber, muon-number and tau-number conservations laws. In this case, the mixing angle vanishes and the lighter (the E or the N) is absolutely stable. As will be seen in the next section, this would be cosmologically disastrous if E is stable, but not if N is stable. Finally, one can assume the discrete symmetry which forbids mixing, but note that Planck mass e!ects are expected to violate all discrete and global symmetries. This means that higher dimension operators, suppressed by the Planck mass, will violate these symmetries. Two such examples are given by Kossler et al. [271]. The mixing angle is then given by sin h&M /M . This gives 5 .J a lifetime for the lighter of the E or N of approximately ten years, which is very near the bound for charged leptons, discussed in the next section.
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3.4.2. Quarks In the lepton case, one could obtain stringent bounds on mixing with a fourth generation by considering precise measurements of leptonic decays with theoretical expectations. Here, such precision (both theoretical and experimental) is impossible. One can still obtain bounds on mixing between the "rst two generations and a fourth from the unitarity of the CKM matrix. As noted in the Particle Data Group Tables [40], the mixing angle between the "rst and fourth generations, < must be less than 0.08. However, other bounds are much weaker [269,146] } the mixing angle S" between the second and fourth generations, < , is only bounded by sinh(0.5. In the top sector, A" one can use constraints [269] from K Pk>k\ to "nd that Re(
d * s * b * D *
< < < <
SB AB RB B
< SQ < AQ < RQ < Q
< S@ < A@ < R@ < @
< S" < A" < R" < "
d * s * . b * D *
(48)
In the basis where the Q"2/3 mass matrix is diagonal, the "rst three rows and columns of the < matrix are just the usual CKM matrix. The fourth row is not relevant for the weak interactions with the = and Z. Since the entire matrix is unitary, the CKM matrix will not be, leading to a suppression of #avor-diagonal couplings. The Z couplings are given by L
$!,!
"g z q cIZ q , 8 GH G* I H* G$H
(49)
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where, using the unitarity of the 4;4 matrix z "d !
(50)
The FDNC couplings are given by L
$",!
1 1 "g q cIZ z (1!c )! sin h q . 8 G I 4 GG 5 G 3 G
(51)
Thus, mixing with the D quarks reduces the direct left-handed FDNC couplings of the light quarks, while leaving the right-handed quark couplings unchanged. For a Q"2/3 isosinglet, the results are very similar. Barger, Berger and Phillips (BBP) then analyze a large number of processes, constraining elements of the 4;4 matrix. Their paper (written in early 1995) has a huge number of references to papers dealing with isosinglet quarks; the reader is referred to it for earlier references. Since the top quark was discovered, calculation of the e!ects of isosinglets became much more precise, since a major uncertainty in the calculations had disappeared. This was the motivation of BBP for reanalyzing all of the constraints on the elements of the matrix. BBP examine Z-decays, meson}antimeson oscillations, K , BM, DMPk>k\, B, DPXl>l\, and radiative * B decays, for both the Q"! and Q" cases. They list bounds on the o!-diagonal matrix elements. The above argument that < could be very small, while < could be very large, does not work in R@ R" the isosinglet Q"! case. The reason is that the mixing, by reducing the left-handed FDNC couplings of the left-handed quarks, can be constrained from high-precision SLC and LEP results. In the case where one just adds a chiral family, the FDNC couplings are not reduced. This is discussed in detail by Nardi et al. [149], who show that the precision data gives "< "(0.0023, B "< "(0.0036, "< "(0.0020. Unitarity of the matrix then gives "< "'0.996, which in terms Q @ " gives "< "(0.09 for q"u, c, t. Thus, the mixing between the top quark and the D cannot be very O" large. In the Q"! case, BBP "nd that the o!-diagonal matrix elements (in the fourth row or column) have upper bounds ranging from 0.045 to 0.09. Tighter bounds on the geometric mean of two couplings are also found. In the Q" case, the bounds are weaker. In that case, in fact, the "< " R and "< " mixings are completely unconstrained, thus one could have large mixings between the 3@ third and fourth generations. The bounds on "< " and "< " are very weak, given by 0.15 and 0.56, 3B 3Q respectively, while the bounds on "< " and "< " are as strong as the corresponding terms in the S A Q"! case. Other papers discuss certain particular processes in more detail, and look at constraints in more speci"c models. Let us "rst consider the case in which the isosinglet quark has Q". This possibility was discussed in detail by Branco et al. [146]. They showed that if one has an isosinglet Q" quark, then the strongest signal will come from DM!DM M mixing. Suppose the ;u, ;c and ;t elements of the Q" mass matrix are given by J /M , J /M and J /M , respectively. They show S 3 A 3 R 3 that if one assumes that the J are equal, then the current bound on DM!DM M mixing gives an upper G bound on J/M of 0.033. This bound could easily be saturated in realistic models, and thus 3 DM!DM M mixing gives the strongest constraint. If one were to take the J to be hierarchical, with G
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a value of km , then the current limits give an upper bound on k of 2, and thus the well-motivated G k&1 is well within reach of the next round of experiments. The Q"! case is more strongly motivated (since such states appear naturally in representa tions of E ). These models have been analyzed in great detail in papers by Silverman and collaborators. The bounds on the Zds vertex coming from K Pk>k\, e, and the K !K mass * * 1 di!erence were considered in early papers by Shin et al. [150] and by Nir and Silverman [151,152]; bounds on the Zbd and Zbs vertices arising from B !BM mixing and rare B decays were O O discussed in Refs. [151,152], and followed up in Refs. [153,154]. Bhattacharya et al. [155] analyzed radiative B-decays in detail, and in another paper, Bhattacharya [156] analyzed the bounds from Z-decays. Although these processes are also discussed by BBP, they are described in much more detail in the above papers. More recently, Silverman [157] has analyzed the mixing constraints using the latest data from B physics, and looks at the constraints that will be reached in upcoming B factories. Lavoura and Silva [158] extended the analysis to the case of multiple isosinglets. They also pointed out that a very strong bound comes from K>Pp>ll, a process which was not considered by BBP. As noted by Branco et al. [146], using the realistic assumptions on the mixing mentioned above, the bound from K>Pp>ll is the most stringent for the Q"! case. The resulting bounds on the J/M are more stringent than in the Q" case, J/M(0.008. It is interesting that the rate for K>Pp>ll, which gives the strongest bound, has been measured and may be high (one event seen and a quarter of an event expected), but drawing conclusions on the basis of a single event is certainly premature. The results of above two paragraphs apply to the case of an isosinglet vector-like fourth generation. The results are quite di!erent in the case of a vector-like isodoublet. As shown in the analysis of the Aspon Model by Frampton and Ng [159], the #avor changing ZdM d vertex will be G H suppressed relative to the isosinglet case by a factor of m m /m . This is because, in the isodoublet G H " case, the mismatch with light quarks occurs in the right-handed sector, forcing a helicity #ip of the usual quarks. (In the isosinglet case, the mismatch is in the left-handed sector, thus no helicity #ip is required.) This extra factor eliminates any signi"cant constraints from #avor changing neutral currents. What about more exotic states? Recently, del Aguila et al. [160] looked at the constraints on anomalous top quark couplings in models with exotic quarks. They look at chiral and non-chiral singlets and doublets, including mirror quarks, and "nd some very general inequalities which allow one to go from LEP bounds on diagonal Z couplings to stringent bounds on the o!-diagonal couplings. Thus, the bounds on mixing of a chiral fourth generation with the third are virtually nonexistent, as are bounds on an isodoublet fourth generation, but the bounds on an isosinglet vector-like fourth generation are getting near the `interestinga range } and may be improved signi"cantly with more measurements on K>Pp>ll and on DM!DM M mixing. It should also be pointed out that the fourth-generation model does have many additional phases. A detailed analysis of CP violation in the isosinglet Q"! case has been carried out by Silverman [157,161]. This is also discussed in BBP. The entire `unitarity quadranglea is analyzed. We will discuss CP violation in more detail in Section 6. What are the theoretical expectations for the mixings? The Fritzsch ansatz (for the 3;3 quark mass matrices) fails [162}164] for a 174 GeV top quark, although the generic expressions sin h"(m /M or (m /M could easily be accommodated in other models. As noted above in @ " R 3
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the lepton case, many models with #avor symmetries will have the 2;2 third}fourth generation mass sub-matrices of the form
0
A
A B
.
These will have a 3}4 mixing angle of O((m /M ), which is of order unity. Thus, one should keep R 3 in mind that the mixing angle between the third and fourth generation could be very large. As in the lepton case, one could imagine a symmetry in which the Q" quarks are diagonal, and then the 3-4 mixing angle would be of order (m /M . @ " The possibility that there is a symmetry prohibiting mixing altogether cannot be excluded. With such a symmetry, the lighter of the ; or D would be stable, leading to a cosmological disaster; however one could assume that Planck mass e!ects violate the symmetry, giving a long (but possibly acceptable) lifetime of O(10!100) yr. In the vector-like case, the mixing angles are also related to the J discussed above, and the G expectations are not very di!erent. It should be noted that in the Aspon Model, the mixing angles are typically 10\}10\ in order to account for the appropriate amount of CP violation. 4. Lifetime and decay modes In the previous section, we discussed the masses and mixing angles of additional quarks and leptons. Now, we consider the lifetime and decay modes of such fermions. In the standard model, the lifetime and decay modes of the most recently discovered fermion, the top quark, were not particularly interesting } it was known that the top quark would decay very quickly (quickly enough that the width is large enough to obscure any structure in the toponium system) and that it would decay almost entirely into a b and a =. However, there are several interesting possibilities for the case of additional fermions. In the chiral case, the N could be heavier than the E, forcing the E to decay only via mixing; if the mixing angles are small (as discussed in the last section), the lifetime could be very long. In the quark case, the mass of the D is likely less than the sum of the masses of the top quark and =, and thus the D will decay only via the doubly-Cabibbo suppressed c#= mode or the one-loop b#Z mode; either could give a long lifetime, especially if mixing angles are very small. In the non-chiral case, the GIM mechanism will not be operative, leading to tree-level #avor-changing neutral decays, such as EPqZ, NPl Z and DPbZ; these decays could give very unique and interesting O phenomenological signatures. In addition, we will see that the mass splitting of the leptons and quarks in the non-chiral doublet case is calculable, and gives lifetimes with potentially observable decay lengths. We will begin by discussing the lepton sector, "rst for the chiral case and then for the non-chiral case, and then turn to quarks. 4.1. Leptons 4.1.1. Chiral leptons Much of the early work on the phenomenology of heavy leptons [165] assumed that m is less , than m , however, as discussed in the last section, there is no particular reason for that assumption. #
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Fig. 9. Relative branching ratios of the N into E=H vs. q=H for various values of the mixing angle and the E to N mass ratio. One sees that the decay into q=H will dominate unless the mixing angle is very small. sin 2h is the mixing between the third and fourth generations.
Let us "rst assume the opposite } m (m (and both greater than M /2). This case has been # , 8 discussed in detail by Hou and Wong [166]. The E will only be able to decay via mixing, EPl =H, where =H is a real or virtual =. The N will decay via either NPE=H or NPq=H. O The decay rates are C(NPE=H)"9 cos h (G m /192p) f (m /m , m /m )) , $ , , 5 # , C(NPq=H)"9 sin h (G m /192p) f (m /m ,0)) , $ , , 5 where f (a, b) is given by [167]
(52)
\(@dx[(1!b)#(1!b)x!2x](1#b#x!2(b#bx#x)) . (53) [(1!xa)#C /M ]] 5 5 This function accounts for both real and virtual ='s. The rate for EPl =H is identical to the O second of these equations with m Pm . Since the angle is expected to be small, one might expect , # that NPE=H would be favored over NPq=H; however, one must recall that the S and ¹ bounds discussed above imply that the N and E must be fairly close in mass, and thus the decay might be signi"cantly phase space suppressed. These rates are plotted in Fig. 9. f (a, b)"2
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We see that the decay of N into q=H tends to dominate, unless the mixing angle is extremely small. This leads to interesting phenomenology, as will be discussed in Section 7. If the E is heavier than the N, the results are the same with NE and l q. O Note that the decay rate of EPl =H is proportional to sinh . In the previous section, we O noted that this angle could be very small. Simplifying the expression for the decay, and assuming that the mass of the E is greater than the = (thus the = is real), the width of the E is given by C(EPl =)"(180 MeV)sin h (m /m ) . O # 5
(54)
Consider the four plausible values of sin h discussed in the previous section. If sin h "m /m , then the decay is very rapid and would occur at the vertex. If sin h "m O /m , O # J , then the lifetime is of the order of a few picoseconds, which has very interesting phenomenological consequences } it might be possible to see the charged track. If sin h "0, then the E is stable; this would be cosmologically disastrous, since they would bind with protons to form anomalously heavy hydrogen. If sin h "m /m , then the lifetime is approximately 10}100 yr. We now 5 .J address whether such a lifetime would pass cosmological muster. The bounds on the lifetime of a fourth-generation charged lepton were "rst discussed several years ago [168]. They considered charged lepton masses ranging from 50 GeV to 50 TeV. Stable leptons are ruled out by searches for heavy hydrogen. Since any decay of the E will result in photon emission, failure to observe such emission in the di!use photon background implies that the lifetime must be less than 10 s (time of the cosmic background radiation (CMBR) production). Using COBE data on the CMBR, and requiring that the radiation in the decay does not distort the CMBR, they found a bound on the lifetime which ranges from 10 to 10 s as the mass ranges up to 1 TeV. Very recently, an analysis by Holtmann et al. [169] looked at the radiative decay of a long-lived particle, X, and the e!ects on big-bang nucleosynthesis (the EPl = is not `radiativea; however, a signi"cant fraction of the energy will O eventually turn into photons, and the results are the same). The photons emitted in the decay may photodissociate deuterium (lowering its abundance) and helium (which raises the deuterium abundance), destroying the agreement between theory and observation. They give bounds on the lifetime as a function of m > , where > ,n /n is the relative abundance of the X. For heavy 6 6 6 6 A leptons, the abundance as a function of mass was calculated in Ref. [168]. For heavy lepton masses between 100 and 500 GeV, the contribution to Xh varies from 0.05 to 0.01, leading to a value of m > which varies from 6;10\ to 1.2;10\ GeV (for a Hubble constant of 65 km/s/Mpc). 6 6 From Tables 3}5 of Holtmann et al. [169], one can see that this correponds to an upper bound on the lifetime of between 10 and 10 s. Given the uncertainties in the abundance calculation, nucleosynthesis calculation, deuterium and helium abundance observations, etc., this is not inconsistent with a lifetime of 10}100 yr. Thus, the possibility that sin h "m /m is marginally 5 .J allowed. To summarize, if the mixing angle h is not very small, then both the N and E will decay via Cabibbo-suppressed decays: NPq=H and EPl =H. If the angle is very small (of the order of O m O /m or less), then the heavier of the two will decay into the lighter, while the lighter decays via J , the Cabibbo-suppressed decay. In this case, the latter lifetime could be quite long, as long as a few picoseconds for sinh "m O /m and as long as 10 yr for sinh "m /m . 5 .J J ,
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4.1.2. Non-chiral leptons As noted in the previous sections, an interesting feature of models with a vector-like doublet (with small mixing with light generations) is that the two members of the doublet are degenerate in mass, at tree level. This degeneracy will be split by radiative corrections, and the size of this splitting is crucial in understanding the lifetimes and decay modes of the heavy leptons. The splitting is due to the diagrams in Fig. 10. The size of the splitting was "rst calculated by Dimopoulos et al. [170] (DTEH), and later calculated by Sher [171] (S) and even later by Thomas and Wells [172] (TW). The result is that the charged lepton is heavier than the neutrino, with a mass splitting of dm"am f (m /m ) , # 8 8 where
(55)
(x x f (x)" . (56) dx (2!x)ln 1# p r(1!x) For small x, f (x)P0, but for large x, f (x)P1, and thus the splitting reaches an asymptotic value of am K350 MeV for m <m . The splitting is plotted in Fig. 11. # 8 8 In DTEH [170], the authors considered very heavy leptons (of the order of several TeV) and looked at the question of whether such leptons could constitute the dark matter (they cannot). In
Fig. 10. Diagrams contributing to the E}N mass di!erence. Fig. 11. Mass di!erence between the E and the N as a function of the E mass.
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Fig. 12. Decay length for the E as a function of its mass. The lines are labelled with the center of mass energy of the collider in GeV.
S [171], the decay width of EPNel and EPNkl was calculated } the inverse of these widths corresponded to a lifetime of 1}2 ns, which is obviously of great phenomenological interest. In S, it was also pointed out that the value of the splitting is very robust, and that supersymmetric contributions to the splitting turn out to be very small, since for most of parameter space, the contributions turn out to be proportional to (!sinh ). In TW [172], it was pointed out that the 5 dominant decay of the E will be into Np (a decay neglected in S), resulting in a much shorter (by roughly a factor of 10) lifetime. The decay length then is of the order of centimeters, rather than tens of centimeters, greatly complicating detection. In Fig. 12, we plot the decay distance in the lab frame as a function of m for a variety of center of mass energies. TW do propose an interesting # signature involving triggering on an associated hard radiated photon; this will be discussed in Section 7. Thus, in the vector-like doublet case, without signi"cant mixing with the lighter generations, the charged member of the doublet is heavier and decays primarily into Np (with a VERY soft pion) with a decay length given in Fig. 12. The fully leptonic decays, even though they have branching ratios of only a few percent, might be easier to detect, although even in that case, the very soft electron or muon would be di$cult to separate from backgrounds. If there is signi"cant mixing, then both the E and the N will decay into the lighter generations. We now consider this possibility (which also applies to vector-like singlets). We now consider the case in which there is a signi"cant mixing of vector-like leptons. By `signi"canta, we mean that the mixing angle, sin h , is greater than about 10\, so that the decay (see the expression for the lifetime above) occurs at the vertex. The results will be very similar to the
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chiral case, with one extremely important di!erence. Due to the breakdown of the GIM mechanism, the decays EPqZ and NPl Z will occur. O For m 'M , the branching ratios of the E are given by [173] # 8 (m !2m #m /m )(m !m ) "; " C(EPqZ) # 8 # 8 # 8 . #O " C(EPl =) 2cos h "; O " (m !2m #m /m )(m !m ) # 5 # 5 # 5 5 #J O
(57)
There is no particular reason to believe that this branching ratio should be small. One might expect "; O " to be of the order of m /m , as discussed in detail in the last section. As discussed O # #J in the last section, in the isosinglet heavy lepton case, one "nds that "; " is of order of the #O ratio of M to M in the leptonic mass matrix which would be similar to "; O "; the resulting #J branching ratio would be very large. In the isodoublet heavy lepton case, there is an additional suppression of m /m , which results in a branching ratio of about 0.1%. Thus, one expects O # a branching ratio for EPqZ to be a fraction of a percent in the isodoublet case and very large in the isosinglet case. It is important to note that even if the branching ratio is as low as a fraction of a percent, the background for a particle decaying into qZ would be extremely small. A major problem with conventional heavy-lepton detection has been backgrounds; so the EPqZ signature, even with a branching ratio of a fraction of a percent or so, might very well be easiest to detect. This will be discussed in more detail in Section 7. 4.2. Quarks The discussion of the lifetime and decay modes of the quarks follows the general discussion of those of leptons, with a few crucial di!erences. While the E and N are certainly much heavier than the q, it is not necessarily the case that the ; and D quarks are much heavier than the top. In addition, the mass splitting in the vector-like quark case is a factor of three smaller than that for vector-like leptons, which can drastically a!ect the decay modes and lifetime. 4.2.1. Chiral quarks Due to constraints from the o parameter, the ; and D quarks cannot have masses which are too di!erent, and thus the decay ;PD#= or DP;#= cannot occur in real ='s. Suppose, for the moment, that m 'm . Then the allowed decays of the ; will be ;P(D or q)#=H, 3 " ;Pq#=, where =H refers to a virtual =. The allowed decays of the D are DP(t or q)#(=H or =). In addition, one can have a #avor-changing neutral current decay DPb#Z, which (in the chiral case) can occur through one loop. The fact that the #avor-changing neutral decay of a fourth generation quark could be signi"cant was "rst pointed out by Barger et al. [174], and followed up by Hou and Stuart [175,176]. In the latter works, they noted that the decay DPb#Z dominates over other #avor-changing decays, such as DPb#c and DPb#g. The possible DPb#Z mode, like the EPq#Z mode, is very important phenomenologically due to the very clear signatures [177,178]. Precise analytical formulae for the various decays, and a discussion of the decays of the D (from which much of the discussion below is taken), can be found in the recent work of Frampton and Hung [179].
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The two- and three-body decay widths are given by the "rst of Eq. (52), with the obvious substitutions of "< " or "< " for cos h . The #avor-changing neutral current decay is "O 3O given by G (m ) $ " C(DPbZ)""< " "R 4p(2cos h 5
g D(m , m ) I (m /m , m /m ) , 3 R @ " 8 " 16p
(58)
where we have assumed, for simplicity, that < "!< so there will be a GIM suppression when 3@ "R m "m (this occurs in most models). We have also assumed that < is approximately unity. 3 R 3" Here,
D(m , m )" 3 R
M !m m 3 R ln 5 !1 m m 5
(59)
refers to the heavier of ; and t. The factor I is the standard two-body phase space and m factor, which is unity in the limit m <m . " 8 The ; decays very rapidly, at the vertex (unless the masses are unusually close together, which is unlikely in the chiral quark case, since they arise from di!erent terms); however the D can be long-lived. The decay modes of the D depend crucially on the mass. First, suppose the D is lighter than the top quark. The decay can only occur into c#= or b#Z (one expects the decay into u#= to be highly suppressed). Two important issues arise: which of these has a larger branching ratio (since their phenomenological signatures are very di!erent), and is it possible that the decay length might be large enough to be detected? We now address both of these questions. From the above formulae, the ratio of the widths can be calculated. The only dependence of the D-mass in this ratio is in the ratio of phase spaces } unless the D is fairly close in mass to the Z this ratio is near unity. The only mass dependence is then in the ; mass dependence in D(m , m ) above. 3 R The result is given in Fig. 13, for various values of the ; mass (recall the fact that the ; and D, in this range, cannot be di!erent in mass by more than 20 GeV due to o parameter constraints). The results depend on the ratio of "< " to "< ". Since the former is `doubly-Cabbibo-suppresseda "A "R (crossing two generations), one expects the ratio to be small. We see that the DPbZ decay mode dominates if the ratio is small, whereas DPc= dominates if it is large (unless the ; mass is very near the top mass). Since there is little theoretical guidance as to the size of this ratio, both signatures should be looked for. Could a vertex be seen? From the expression for C(DPbZ), one can see that the width of this mode [175] is between 10\"< " and 10\"< " MeV over the mass range of interest. This implies "R "R that the lifetime will be at most 6;10\/"< " s. This would mean that one must have "R "< ";10\ in order to detect a vertex. Although such a small angle is not expected in the chiral "R sequential quark case (the Aspon Model involves non-chiral quarks), it is not excluded, if one has a #avor symmetry with an almost unbroken 3#1 structure. For m between 177 and 256 GeV, the D can decay into the top quark via the three-body decay " into a virtual =. In this case, DPt#=H will be competetive with DPc#= and DPb#Z } the latter are suppressed by the doubly Cabibbo suppressed mixing angle or the extra loop, the former is suppressed by three-body phase space. If we compare the rate for DPt#=H with
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Fig. 13. Ratio of width of DPb#Z to that of DPc#=. Note the GIM suppression of the FCNC decay when the ; mass equals the top mass.
DPb#Z, the mixing angle cancels out. In Fig. 14, we have plotted the ratio of these two decays. Thus, as pointed out in Ref. [179], the three-body decay of the D is irrelevant for D masses below about 230 GeV (depending on M ), and thus the arguments of the previous two paragraphs are 3 unchanged. As the mass increases beyond 230 GeV, the three body decay becomes more important, and as soon as the mass exceeds 256 GeV, the two-body decay DPt#= becomes accessible. At that point, DPt#= becomes the dominant decay. Again, very small mixing would be needed to see the vertex, "< "(10\. "R What about the decay of the ; quark (again, assuming that m 'm )? The ; can decay into 3 " D and a virtual =, ;PD#=H, or into a light quark (most likely a b) and a real =, ;Pb#=. Which decay mode dominates depends on how close the ; and D are in mass and on the "< " 3@ mixing angle. This was discussed in Ref. [179]. As m varies from 180 to 250 GeV, the width for 3 ;Pb#= varies from 1.75"< " GeV to 4.7"< " GeV. For ;PD#=H, the width (assuming 3@ 3@ "< " is nearly unity) is 5.2;10\ GeV for m /m "1.1, and drops to 3;10\ GeV for 3" 3 " m /m "1.02. For moderate mixing angles, "< "'0.003, the ;Pb#= decay mode will always 3 " 3@ dominate. For mixing angles in the range between 10\ and 10\, which dominates depends sensitively on the mass ratio between the ; and the D. For mixing angles below 10\, the ;PD#=H will dominate. In no cases will the width be small enough so that a vertex can be seen: the ; will decay at the vertex. If m 'm , everything we have said above will carry through, exchanging t with b, etc. " 3 A principle di!erence is that one can consider lighter long-lived ; quarks (since one can have m (m ) than for D quarks. 3 R
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Fig. 14. Ratio of width of DPt#=H to that of DPb#Z for various values of M . The non-chiral line corresponds to 3 the vector-like doublet case, which is independent of M . For the non-chiral isosinglet case, the ratio is too small to be 3 seen on the graph.
To summarize, if m 'm , then the primary decays mode of the D will be DPc#= and 3 " DPb#Z if the D mass is below about 230 GeV. The former will dominate if "< "/"< " is greater A" "R than 0.01, the latter will dominate if it is less than 0.001. As the D mass increases above 230 GeV, the decay DPt#=H begins to dominate. In all cases, one must have very small mixing angles ("< "(10\ or 10\, depending on the mode) in order for the decay to occur a measureable "R distance from the vertex. The ; will always decay at the vertex, into either D#=H or b#=, depending on the precise masses and mixing angles. 4.2.2. Non-chiral quarks For vector-like quarks, either isosinglet or isodoublet, many of the above results are unchanged. As noted in Ref. [179], the decay widths into real or virtual ='s will not be signi"cantly changed. As in the nonchiral lepton case, there are two major di!erences: the mass di!erence between the ; and the D in the isodoublet case is calculable (if mixing is small), and the #avor-changing neutral decay DPbZ can occur at tree level. In the isodoublet case, the mass di!erence between the ; and the D can be shown [171] to be 1/3 that of the lepton case, and will thus be between 70 and 110 MeV, if the mixing with lighter generations is small. This means that the hadronic decay is forbidden, and the only decay would be ;PDel, with a lifetime of the order of microseconds. Thus, in the absence of mixing, both the ; and the D would be absolutely stable, as far as accelerators are concerned, and the mass di!erence is irrelevant. We thus must consider both ; and D decays. The D can decay, as in the chiral case, into either c#=, t#=H, t#= or b#Z. The only di!erence in the above discussion is the rate for DPb#Z. In the expression for C(DPb#Z) in
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the last section, one must replace the factor of gD(m , m )/64p from the loop with m/m , in the 3 R @ " isodoublet case, and with unity in the isosinglet case (the factor of "< " is the same in each case). In "R the isodoublet case, this does not change the result for the width by more than an order of magnitude, but in the isosinglet case, it increases the width by roughly three orders of magnitude. First, if the D is lighter than the top quark, then the decay modes in the isodoublet case are very similar to the chiral case * the results depend sensitively on the ratio of "< to "< ", and either the "A "R DPc#= and DPb#Z will be dominant. A displaced vertex can only be seen if "< "(10\. "R This is expected in the Aspon Model case, and thus one can expect a signi"cantly displaced vertex in the model (as emphasized in Ref. [179]). In the isosinglet case, the absence of the m/m @ " suppression makes the DPb#Z mode dominant, and also requires that "< ";10\ in order for "R a displaced vertex to be seen (which is not expected in the Aspon Model). If the D is heavier than the top quark, then the argument in the previous section carries through without signi"cant modi"cation in the isodoublet case; the cross-over where the three body decay becomes relevant is closer to 210 GeV than to 230 GeV. In the isosinglet case, however, the DPb#Z mode dominates, even over the two-body top quark decay, until m is well over " 300 GeV, and remains signi"cant even to much higher masses. Thus, we see that in either the chiral or non-chiral case, the decay DPb#Z is very important. It was pointed out in Ref. [179] that, if this mode were detected, then one could look at the chirality of the Z to determine which of the two cases applies. This might be the quickest way to determine the chirality of the heavy quarks. Unless the ; is very heavy, above 310 GeV, the #avor-changing neutral decay ;Pt#Z is forbidden or highly suppressed by phase space. Since ;Pc#Z is suppressed by small mixing angles, one has the ;Pb#= decay mode dominating. Again, one can only detect the vertex if the angle is very small, less than 10\.
5. Dynamical symmetry breaking It is fair to say that perhaps one of the most important discoveries that can be made in the future will be that of the Higgs boson. In the absence of any alternative plausible explanation for particle masses, the concept of spontaneous breakdown of a gauge symmetry via the Higgs "eld as the origin of all masses is universally accepted. The only problem is that it has not been found. Not only does one not know its mass, but one also does not know in what shape or form it should be. One fact that we do know however, regardless of how massive and in what form the Higgs boson may be, is the scale of electroweak symmetry breaking: v"246 GeV. All masses in the SM are expressed in terms of that scale, e.g. M "(1/2)gv and m "g G v/(2, where g G are Yukawa 7 5 G 7 couplings. There is a rather intriguing fact: with v/(2&174 GeV, it follows that the top quark Yukawa coupling, g , is of order unity, unlike all other fermions. That the top quark is so heavy and R its mass is so close to the electroweak breaking scale is the cause to wonder about any relationship that it might have with the mechanism of symmetry breaking itself. If the top quark mass is so close to the electroweak scale, is it possible that it itself is reponsible for the breaking of the electroweak symmetry? This fascinating possibility was "rst entertained in Refs. [43}46]. It is now commonly referred to as `top-condensate modelsa. In these models, the Higgs boson is generally viewed as a ttM composite "eld generated by some unknown dynamics at
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a scale K<246 GeV. In other words, the electroweak symmetry breaking is dynamical in these scenarios, i.e. it is broken by a ttM condensate or something similar. One interesting feature of these models is the prediction of a heavy fermion mass (e.g. the top quark mass or a fourth-generation mass) as a function of the Higgs boson mass through the so-called compositeness conditions. Before describing these models and their variants, let us "rst summarize the results of the simplest version, that of Bardeen et al. [46]. In Ref. [46], the "rst scenario including only a heavy top quark predicted a top mass of order 230 GeV. This is now excluded. Ref. [46] also presented results involving a heavy, degenerate fourth-generation quark doublet, assuming that the top quark and other fermions are much lighter. This result combined with the known top quark mass indicates that something else other than the top quark alone must exist if this picture has any chance of being correct. The subject of dynamical symmetry breaking a` la top-condensate deserves a whole review; a book on the topic already exists [180]. The best we could do here is to review salient points and results, especially those pertaining to the subject of this review. We "rst state the so-called compositeness conditions used in the top-condensate type of models. As we shall see below, these conditions allow us to relate the masses of the heavy fermions to that of the Higgs boson. Simply speaking, it is the requirement that the Higgs quartic coupling j and the relevant Yukawa couplings diverge at the same scale K (the Landau poles) while the ratio of j to the square of the ! Yukawa coupling(s) remains "nite, namely K !
P R j(k), g (k) I R K
P const. j(k)/g(k) I R !
(60) (61)
where g can be either the top or just a generic Yukawa coupling. The above boundary conditions R modify the structure of the SM Lagrangian at the scale K in the following way. Let us consider ! a simple toy model where there is a degenerate heavy quark doublet Q"(;, D) coupled to the Higgs "eld U (light fermions will be ignored in this discussion). Let us rescale U as follows: (62) UPU /g , D where g is the Yukawa coupling of the degenerate quark doublet. The SM Lagrangian (with only D that degenerate doublet present) becomes L"L (;, D)#ZU D UR DIU #m UR U !jI (UR U ) I #QM U D #QM U! ; #h.c. , * 0 * 0 where
(63)
(64) U! "ip UH, ZU "1/g , m "ZU m and jI "ZU j . D The "rst remark one can make when one looks at the above expressions is the compositeness condition itself: the vanishing of the wave function renormalization constant. Indeed, as can be seen from the expression for ZU , one immediately notices that, if g has a Landau singularity at some D scale K , the wave function renormalization constant for the Higgs "eld, Z U , vanishes at that same !
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scale. In other words, the Landau pole is identi"ed with the compositeness scale. Furthermore, if j and g develop a singularity at the same scale in such a way that conditions (60) are satis"ed, the D Lagrangian at the compositeness scale becomes L"L (;, D)#QM U D #QM U! ; #m UR U #h.c. (65) * 0 * 0 U is now just an auxiliary "eld and can be integrated out, resulting in a Nambu-Jona-Lasinio form for the Lagrangian, namely L"L (;, D)#G QM (; ;M #D DM )Q , (66) * 0 0 0 0 * where G "!1/m . In this picture, the Higgs boson becomes a fermion}antifermion composite particle below the scale K . ! What might be even more interesting is the relationship between the Higgs boson mass and that of the heavy fermions in the top-condensate type of scenario. In particular, the Higgs mass, m , can & be seen to be bounded from above by 2m and from below by m , where m is a heavy fermion D D D mass. The search for a heavy fermion is intimately tied to the search for the Higgs boson } a feature already seen in the discussion of gauge coupling uni"cation [65]. To see this in the context of the top-condensate type of model, let us look at the RG equations for j and g , the heavy fermion D Yukawa coupling, at one loop level (the one-loop terms of Eqs. (47a)) in the toy model with one doublet of heavy quarks. Let us de"ne x"j/g . The one-loop RG for x is then D dx (67) 16p "24g (x!x )(x!x ) , D > \ dt where x "(!1$3)/4, if both members of the quark doublet are degenerate in mass, or ! x " (!1$(65), if one member is much heavier than the other one (e.g. the third-generation ! case). The boundary conditions (60) are satis"ed if x is one of the two "xed points, x . The solution ! x (always negative) is ruled out by vacuum stability. This leaves us with x . From the de"nition \ > of m and m , one can then obtain a relationship between these two masses as follows: D & m "4m x . (68) & D > In the large N limit, the right-hand side of Eq. (67) becomes 4N (x!1) with the "xed point being A A x"1 which implies m "2m , a familiar result found in the Nambu}Jona}Lasinio model. Since & D x is always greater than , one can easily see that, for "nite N , one obtains the bound A > m (m (2m . (69) D & D As we have mentioned above, a necessary condition for this scenario to work is the boundary (60). The minimal SM with three generations unfortunately does not satisfy these boundary conditions: the top quark with a mass of 175 GeV is simply too `lighta. One needs either a heavier quark (216}230 GeV), which cannot be the case with the minimal SM, or more heavy quarks or leptons such as in the fourth-generation scenario. The fourth-generation case was studied by Hill et al. [181] in the context of Majorana neutrinos. Later, Hung and Isidori [117] also examined the fourth-generation scenario as part of an overall `anatomya of the Higgs mass spectrum, starting form m of 65 GeV to m 52m . It was in this last mass range, m 52m , that the focus on & & 8 & 8
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Fig. 15. RG evolution of the Yukawa couplings g (full and dash-dotted lines) and g (dashed lines). The top mass is R O always "xed to be 175 GeV and the heavy-lepton mass is assumed to be 90 GeV. The dash-dotted line is the evolution of g without extra fermions. Near each dashed line is indicated the value of m and the corresponding value of m obtained R O & by the requirement K "K (the error on both m and m is about $10 GeV). * O & O
top-condensate types of models was diverted to. In their analysis, Ref. [117] assume a Dirac mass for the fourth neutrino and degenerate quarks and degenerate leptons for the fourth generation. It turns out that the inclusion of such a fourth generation drastically modi"es the evolution of the couplings even if these fermions were lighter than the top quark. In the analysis of Ref. [117], Eqs. (47a) were used at the one-loop level. The results are shown in Fig. 15 below, where the mass of the top quark is "xed at 175 GeV and that of the fourth lepton doublet is "xed at 90 GeV. A few remarks are in order concerning the above "gure. As the "gure caption already indicates, the dash-dotted line represents the top Yukawa coupling squared, g, as a function of energy for the R minimal SM. One can see that g actually decreases in value with energy and remains "nite at the R Planck scale. This is so because a mass of 175 GeV is `too smalla to provide a large enough initial value plus the fact that the contribution from the QCD coupling to the b function for g occurs R with the opposite sign. When a fourth generation is added, the evolution of the couplings change drastically, partly due to the fact that there are more degrees of freedom than the minimal case. So, as long as the initial Yukawa couplings of the fourth generation are not too small, there will be Landau poles that appear below the Planck scale. As one can see from Fig. 15, this occurs when the fourth-generation quark mass m 9150 GeV. Also an interesting phenomenon occurs: all Yukawa / couplings `draga each other in such a way that they all `blow upa at the same point. The top quark Yukawa coupling, which by itself decreases with energy, is now `draggeda in such a way as to develop a Landau pole at the same time as the fourth-generation quarks. These results } the
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existence of Landau poles below the Planck scale in the presence of a fourth generation } encourage a reconsideration of the top-condensate type of model. Since the quartic coupling j is a free parameter, one can now choose its initial value, i.e. its mass, in such a way that the boundary conditions (60) are satis"ed. This procedure results in a relationship between the fourth-generation quark mass and the Higgs mass. This is shown in Fig. 15 (m is D the notation used in the "gure to refer to the fourth-generation quark mass instead of m which is / used here). The relationship for low fourth-generation quark mass (e.g. 150 GeV) is strikingly similar to the one obtained in the discussion of the gauge coupling uni"cation of Ref. [65]. The range of mass used in Ref. [117] for m is however considerably larger than that of Ref. [65] / because, in that study of a top-condensate type of model, no constraint on where the Landau poles should be has been used. In fact, for masses larger than 150 GeV, the Landau poles move down in energy. For example, if m "230 GeV which corresponds to m "300 GeV, the Landau pole is / & situated at approximately 100 TeV. As remarked in Ref. [117], the relationship between the Higgs mass and the fourth-generation quark mass approaches more and more the "xed point value for a degenerate quark doublet, m "(2m , as the Landau pole `approachesa the electroweak scale. / & This can be seen in the "gure shown above. The scenario described above intimately links the search for the fourth-generation quarks and leptons to that of the Higgs boson, and vice versa. It is not clear at this point how the higher mass values ('160 GeV) for the fourth-generation quarks would a!ect the evolution of the gauge couplings, except for the fact that perturbation theory breaks down above the Landau poles and it is not legitimate to evolve those couplings beyond that point. For the purpose of this report, we shall however leave open the possibility of such top-quark condensate type of models as a possible mechanism for electroweak breaking. It is partly for this reason that the range of masses considered in the search for long-lived quarks by Frampton and Hung [179] is larger than the mass range used in the study of gauge coupling uni"cation of Hung [65]. Recently [182], there was an analysis of the two-loop RG equations in the SM with three and four generations. Although many results presented there [182] were already discussed in [65,117], there were statements that are not correct. In order to clarify the issues, we repeat the statements of Ref. [182] and explain why they are misleading. Ref. [182] chose the masses of the leptons and the quarks of the fourth generation as follows: m /m "1/2 and 1, and restricted m to be greater than * / / 180 GeV which was referred to by the authors as the direct experimental constraint. Furthermore, an upper bound of 200 GeV for m was used. Using these constraints, Ref. [182] claimed that & a fourth generation is ruled out by plotting the allowed regions in the m }m plane (Fig. 11 of & / [182]). First, m less than 180 GeV is not ruled out by direct experiment if a long-lived quark / decays in the detector at a distance between 100 lm and 1 m, a subject discussed by Frampton and Hung [179]. As we shall discuss in the section on experiments, there is and will be such a search at the Tevatron. Second, the constraints on m at the present time from precision experiments are & rather loose at best. A much larger bound than 200 GeV is possible [183]. For example, Refs. [184}186] gave an upper bound as high as &280 GeV, while Ref. [187] gave an upper bound (within 95% CL) from 340 GeV to 1 TeV. In summary, the mass ranges used by [182] to rule out a fourth generation are not warranted. In fact, the values used by Hung [65], m "151 GeV and / m "188 GeV, were shown to lead to a better uni"cation of the SM gauge couplings, and higher & masses (such as 180 GeV) for m were not used because precisely of the fact that the Landau poles / were much too low to trust perturbation theory in the evolution of the gauge couplings.
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As the above discussion and earlier ones made it crystal clear, there are several theoretical reasons for looking at quarks and leptons beyond the third generation. In particular, our primary motivation in this review is to examine those reasons which `predicta fermion masses which are within reach either of present experiments such as the ones performed at the Tevatron or at future machines such as the LHC, NLC, etc. By `within reacha, we mean masses which are close to the top quark mass (&150}230 GeV) for the quarks. It goes without saying that there exists also plenty of reasons for considering fermions which are much heavier than the top quark, some of which are even primary focus of the topics discussed in this section: the techniquarks and leptons of Technicolor models [188}191]. It is outside of our goal to review such topics but for completeness, we shall give a brief description of one of such scenarios: the topcolor-assisted Technicolor model. Our discussion of top-condensate type of models above relies on one crucial assumption: The entire electroweak symmetry breaking is due to the condensate of the top quark and/or that of the fourth-generation quark. The topcolor-assisted Technicolor model of Hill [192] did away with that assumption. But then how does one explain the fact that the top quark mass is so much larger than all other fermion masses and so close to the electroweak breaking scale? The salient points of the topcolor model are basically the assumptions that the electroweak symmetry is still broken by some form of Technicolor, there exists an extra (`topcolora) group, S;(3) ;(1) , which couples 7 preferentially to the third family and which triggers a dynamical condensate for the top quark, giving rise to a large top mass. This new `topcolora group is assumed to be spontaneously broken by Technicolor at a scale &1 TeV. An extended Technicolor interaction (ETC) is also assumed so that quarks and leptons can obtain some mass which is much smaller than the top mass. The top quark itself obtains most of its mass from the condensate with a small contribution coming from ETC. Variations of the model include cases in which there are S;(2) singlet quarks which are however very massive (&1 TeV). In any case, all fermions beyond the third generation is these models have masses around 1 TeV. Although there is a rich phenomenology involving objects such as top-pions, etc., it is beyond the scope of this review to discuss it here. We end this section by mentioning that there are interesting recent developments on the subject of electroweak symmetry breaking involving the so-called seesaw mechanism of quark condensation [193,194]. This is yet another variant of the topcolor model. In this particular scheme, the top quark mass obtains its `observeda value dynamically by a mass mixing with a new SU(2) singlet quark s of the form k s t #m tM s #k s s with m &0.6 TeV, k &0.9 TeV and QR * 0 RQ * 0 QQ * 0 RQ QR k &2 TeV. The physical top mass is found by diagonalizing that 2;2 matrix giving QQ m +k m /k . The other mass eigenstate would be &2 TeV and it might not be easy to be R QR RQ QQ observed directly. The model predicts a number of pseudo-Nambu}Goldstone bosons, some of which are s-bound states. How this new degree of freedom manifest itself experimentally is a subject which is under active investigation. One last comment which is worth mentioning here is the mass of the Higgs boson in the di!erent variants of the topcolor model. Generically, the physical Higgs scalar would have a mass of the order of a TeV just like standard Technicolor models. However, in the seesaw version of the topcolor model, there appears an extra `lighta Higgs scalar (in addition to the charged ones) as one would have in a two-Higgs doublet model. Depending on a delicate cancellation of some parameters of the model, one could have a `truly lighta scalar with mass of O(100) GeV. These remarks are meant to emphasize the importance of the search of Higgs scalar(s) in addition to that of new quarks and leptons.
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6. CP violation 6.1. CP violation in the standard model At "rst sight, there may appear to be no connection between additional quarks and leptons and the violation of CP symmetry. The object of this subsection is therefore to convince the reader that the better understanding of CP violation may necessitate the incorporation of additional fermions. This is not an inevitable conclusion but it is a suggestive one. To set the scene, we need to describe the status of CP symmetry and its violation in the context of the unadorned standard model. This discussion will be in two separate parts: weak CP violation (the KM mechanism), and the strong CP problem. The former is not really a problem for the standard model, merely one that is not yet veri"ed unambiguously by experiment. The latter, the strong CP problem, is de"nitely a di$culty, a shortcoming, for the standard model and one whose solution (we shall mention the axion possibility) is still unknown. The gauge group of the standard model is S;(3) ;S;(2) ;;(1) , broken at the weak scale to ! * 7 S;(3) ;;(1) . Under the standard group the "rst generation transforms as ! 7 u l , u , dM ; ¸ " C , e> (70) Q " * * d * * * e\ * *
and the second (c, s, l , k) and third (t, b, l , q) generations are assigned similarly. I O The quarks acquire mass from the vacuum expectation value (VEV) of a complex S;(2) doublet * of scalars
"
>
giving rise to up and down quark mass matrices: M(;)"j3 1 2; M(D)"j" 1 2 (71) GH GH which are arbitrary matrices that may, without loss of generality, be chosen to be hermitian. The matrices M(;), M(D) of Eq. (71) are de"ned so that the Yukawa terms give, e.g. Q M(;)u #h.c., * 0 and can be diagonalized by a bi-unitary transformation: K(;) M(;)K(;)\"diag(m , m , m ) , (72) * 0 S A R K(D) M(D)K(D)\"diag(m , m , m ) . (73) * 0 B Q @ These mass eigenstates do not coincide with the gauge eigenstates of Eq. (70) and hence the charged = couple to the left-handed mass eigenstates through the 3;3 CKM matrix < de"ned by !)+ < "K(;) K(D)\ . (74) !)+ * * This is a 3;3 unitary matrix which would in general have nine real parameters. However, the "ve relative phases of the six quark #avors can be removed to leave just four parameters comprising three mixing angles and a phase. This KM phase underlies the KM mechanism of CP violation.
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With N generations and hence a N;N mixing matrix there are N(N!1)/2 mixing angles and (N!1) parameters in the generalized CKM matrix. The number of CP violating phases is therefore (N!1)!N(N!1)/2"(N!1)(N!2)/2. This is zero for N"2, one for N"3, three for N"4, and so on. In particular, as Kobayashi and Maskawa [27] pointed out, with three generations there is automatically this source of CP violation arising from the 3;3 mixing matrix. This is the most conservative approach to CP violation. This source of CP violation is necessarily present in the standard model; the only question is whether it is the only source of CP violation. Since the only observation of CP violation remains in the neutral kaon system, there is not yet su$cient experimental data to answer this question de"nitively. There are various equivalent ways of parametrizing the CKM matrix. The one proposed [27] by KM involved writing
cos h !sin h cos h !sin h sin h < M" sin h cos h cos h cos h cos h !sin h sin h e B cos h cos h sin h #sin h cos h e B . !) sin h sin h cos h sin h cos h #cos h sin h e B cos h sin h sin h !cos h cos h e B (75) Another useful parametrization [195] writes
< " !)+
1!j !j
j
1!j jA(1!o!g) !jA
jA(o!ig) jA 1
.
(76)
In Eq. (76), j is the sine of the Cabibbo angle sin h in Eq. (75) and CP violation is proportional to g. If we write the CKM matrix a third time as
< < < SB SQ S@ < " < , (77) < < !)+ AB AQ A@ < < < RB RQ R@ then the unitarity equation (< )R< "1 dictates, for example, that !)+ !)+
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predictions and hence be easily distinguishable from the KM mechanism when accurate measurements on B-meson decays are made in B Factories. 6.2. Strong CP and the standard model Next we turn to a brief outline of the strong CP problem in the standard model. (More detailed reviews are available in [198}200].) The starting observation is that one may add to the QCD lagrangian an extra term ¸" q (ic D !m)q !HG GI , (79) I I I I IJ IJ I where the sum over k is for the quark #avors and D is the partial derivative for gauged color I S;(3) . The additional term proportional to H violates P and CP symmetries. This term is a total ! divergence of a gauge non-invariant current but can contribute because of the existence of classical instanton solutions. It turns out that chiral transformations can change the value of H via the color anomaly but cannot change the combination: HM "H!argdet M(;)!argdet M(D)
(80)
where det M(;, D) are the determinants of the up, down quark mass matrices, respectively. Thus, HM which is an invariant under chiral transformations measures the violation of CP symmetry by strong interactions. A severe constraint on HM arises from the neutron electric dipole moment d L which has been measured to obey d 410\e. cm. [201,202]. A calculation of d [203,204] leads L L to an estimate that HM 410\. This "ne tuning of HM is unexplained by the unadorned standard model and raises a serious di$culty thereto. A popular approach (which does not necessitate additional fermions) involves the axion mechanism which we brie#y describe, although since only a relatively narrow window remains for the axion mass, and since the mechanism is non-unique, it is well worth looking for alternatives to the axion for solving the strong CP problem. In the axion approach, one introduces a color-anomalous global ;(1) Peccei-Quinn symmetry [205,206] such that di!erent Higgs doublets couple to the up- and down-type quarks. The e!ective potenetial now becomes a function of the two Higgs "elds and HM (x) regarded as a dynamical variable. An analysis then shows that the potential acquires the form <"<(H , H )!cos HM (81) and hence the minimum energy condition relaxes HM to zero. Because a continuous global symmetry is spontaneously broken, there is a pseudo-Goldstone boson [207,208], the axion, which acquires a mass through the color anomaly and instanton e!ects. The simplest model predicts an axion with mass of a few times 100 keV, but this particle was ruled out phenomenologically. Extensions of the axion model [209}212] lead to an axion mass which becomes a free parameter. Empirics constrain the mass to lie between about a microelectronvolt and a milli-electronvolt, and searches are underway for such an axion. In one variant of the axion approach, new heavy quarks are necessary [209]; in fact, this was the "rst-proposed model to avoid the experimentally excluded `visiblea axion in favor of an `invisiblea axion. Alternative versions of the invisible axion [211,212] do not involve extra quarks.
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A second solution of the strong CP problem is to assume that m "0, but this seems to be at S variance with the successes of chiral perturbation theory [213}216]. 6.3. Strong CP and extra quarks For present purposes, let us believe neither the axion nor the massless up quark. In this case we are inevitably led to the existence of further fermions beyond the standard model. Such a conclusion can have the additional bonus of connecting the strong CP solution to the occurrence of weak CP violation. The appearance of new fermions in this context was "rst suggested by Nelson [217] and Barr [218,219] who looked within the framework of GUTs. Their additional fermions have gigantic masses &10 GeV, well beyond accessible energy, but their basic idea involving the quark mass matrix texture is one that will reappear in the models where the new fermions are at accessible masses. Nelson [217] invented a GUT based on S;(5) ;(SO(3);;(1)) . The fermions are assigned to the representations [10.3.0]#[5 ,3,0]#[10,1,1]#[10 ,1,!1]#[5,1,1]#[5 ,1,!1] .
(82)
Note, in particular, the additional fermion representations in the last four terms of Eq. (82). The scalars are the (5,1,0) which is complex and contains the standard Higgs doublet together with a superheavy color triplet; then there are (r , 3, 0) where r are S;(5) representations containing G G singlets of S;(3);S;(2);;(1) such as 1, 24, 75. If we write the most general Yukawa terms we "nd that the couplings of the quarks to the light Higgs is complex but has a real determinant. Thus, if the lagrangian respects CP symmetry, the value of HM vanishes at tree level. Looking at the loop corrections to HM , it turns out that the additional fermions must be lighter than the GUT scale by three or four orders of magnitude to suppress HM adequately. The secret of the Nelson model lies in the arrangement of additional fermions such that a basis may be chosen where complex entries in the quark mass matrix are always multiplied by zero in the evaluation of the determinant. Barr [218,219] examined what are the general circumstances under which this suppression of HM happens. Barr was led to the following rules for a GUT in which HM "H #H "0 at tree level. Let /!" /$" the gauge group be G and let CP be a symmetry of the Lagrangian. Then H "0 and the /!" couplings have no CP violating phases. Let the fermion representation be divided into two sets F and R where F contains the fermions of the three families and R is a real non-chiral set. Let R be composed of a set C and its conjugate set CM . Then the following two conditions are su$cient to ensure that HM "0 at tree level: E At tree level there are no Yukawa mass terms coupling F fermions to CM fermions, or C fermions to CM fermions. E The CP violating phases appear at tree level only in those Yukawa terms which couple F fermions to R"C#CM fermions.
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In such Nelson}Barr GUT models, strong CP is solved by physics (the additional fermions) close to the GUT scale. The unrelated (in this approach) physics of weak CP violation arises from the usual KM mechanism. Non-GUT models which adopt the Nelson}Barr mechanism have been discussed especially in the papers [220}224]. The model proposed in [220] by Bento, Branco and Parada introduces a non-chiral charge ! quark together with a complex singlet scalar S. The "eld S develops a complex VEV1S2"<e ? while the standard Higgs doublet has VEV 1 2"v which is real. The KM phase d is generated from a in an unsuppressed manner. HM is zero at tree level and its loop )+ corrections are suppressed by powers of (1 2/1S2)"(v/<). Consequently, HM can be naturally su$ciently small; e.g. if <'100 TeV, the Yukawa can be &10\ while even if <'1 TeV, the Yukawa coupling can still be as large as &10\. Further papers examine other consequences of such a model. In [221] the impact for B!BM mixing is found, while in [222] D!DM is also found to di!er from the standard model predictions. One general feature of this type of model is that HM is suppressed by the Nelson}Barr-type mechanism and, as in Nelson}Barr, the CP violation arises from d . In the Aspon Model described below, the CP violation necessarily arises from an )+ additional mechanism because there the generation of d is highly suppressed. )+ 6.4. CP and a fourth generation Before moving to that, let us mention a number of valuable papers which discuss the parametrization of the CKM matrix when the number of generations of quarks and leptons is increased from three to four or more [225}231]. In particular, Harari and Leurer [229] claim to have an optimal parametrization for general generation number. Let us here merely quote [231] an example of a parametrization for < in terms of six mixing angles and three CP-violating phases: !)+ < " !)+
c !s c
s c c c c #s s c e B
!s s c
c s c c !c s c c e B !s s s e B
!s s s
c s c s !c s s c e B #s c s e B
s s c c c s s !s c s c e B #s s s e B >B c s s c c #c c c c c e B !c c s s e B >B #c s c s e B #s s c e B >B c s s s c #c c s c e B !c s s s e B >B !c c c s e B !c s c e B >B
s s s c c s s !s c s c e B !s s s e B >B c s s c c #c c c c c e B #c c s s e B >B #c s c s e B !s s c e B >B c s s s c #c c s c e B #c s s s e B >B !c c c s e B #c s c e B >B
.
(83)
The large number of phases and mixing angles makes a thorough analysis of CP violation in the model with four chiral generations impractical. However, if one considers the case of an additional
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isosinglet quark, then the discussion of CP violation becomes tractable, and this case has all of the essential features of the more general case. An extremely extensive and detailed analysis of CP violation in models with a Q"! isosinglet quark has been given in a series of papers by Silverman [152,153,157,161]; the Q" case was discussed brie#y by Barger et al. [148]. Using the notation of Barger et al. (see the last subsection of Section 3), one can write the unitarity relations of the 4;4 matrix as or
(84)
k\, * B !BM mixing, B !BM mixing, BPk>k\X, R in ZPbbM , and the recent event in K>Pn>ll. B B Q Q @ The results are presented as plots (one for the standard model and one for the isosinglet model) in the (o, g) plane, in the (sin(2a), sin(2b)) plane, and in the (x , sin c) plane. In each case, he shows the Q current bounds (with 1p, 2p, 3p contours), and then shows projected results from the upcoming B-factories. There are several interesting features of the isosinglet model case: the presently observed CP violation in e can come entirely from new phases, and the value of sin c can take on any value whatsoever. The range of A Q is just as large for its negative values as for its positive values. It is clear from the plots how the upcoming B-factories will drastically narrow the allowed parameter space, can could easily rule out standard model CP-violation. 6.5. CP in the Aspon Model Here we shall concentrate speci"cally on the Aspon Model [61,62,179,232}237] and its requirement of additional quarks to solve strong CP. The "rst generation of the standard model contains quarks with the following (¹ , >) values: (86) (!, )"d ; (0, )"dM , * * (, )"u ; (0,!)"u . (87) * *
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We introduce [61] a ;(1) symmetry and assign Q "0 to all of the above quark states and to the leptons, although the latter do not play a signi"cant role in solving strong CP. The second and third families have parallel assignments under the same ;(1) . In the model there is also a real representation of exotic heavy quarks corresponding to a complex representation C and its conjugate CM . In CM the exotic heavy quarks have quantum numbers exactly like some of the usual quarks; for example, in CM there may be one doublet (!, )"D ; (, )"; . * * These have charges Q "#h. In representation C we shall then have (,!)"D! ; (!,!)";! . * * These have Q "!h. The Higgs sector has one complex doublet
(#,!), Q "0 , and two complex singlets
(88)
(89)
(90)
s
(0, 0), Q "#h . (91) The gauge group is S;(3) ;S;(2) ;;(1) ;;(1) , where we gauge ;(1) to avoid an ! * 7 unwanted Goldstone boson when it is spontaneously broken. In breaking the symmetry, we give a real vacuum expectation value to and complex VEVs to s with a non-vanishing relative phase. The Lagrangian contains bare mass terms M(;! ; #D! D ) for the extra quarks. The allowed * * * * Yukawa coupling include u G uH , dM G dH for the families and ;! uG s , DA dH s coupling light quarks * * * * * * ? * * ? to C heavy quarks (a"1, 2 and i"1, 2, 3). Because the families have no couplings to CM exotics, the quark mass matrix determinant arising from spontaneous symmetry breaking is real at lowest order; it has the required texture. We do not allow terms in the Higgs potential which explicitly break ;(1) . Disallowed terms include M s, s, s and s. If any of these terms are present, ;(1) is explicitly broken and the model can have H"0 at tree level only in very special cases where, e.g. we choose particular representations of a grand uni"ed group such that the quark matrix is real. Without explicit breaking of the ;(1) , there is correct texture at tree level; the mass matrix has the tree-level texture (F"family)
real
0
(FCCM ) complex real
0
0
real
0
complex
F
C .
(92)
CM
Thus H "0 at tree level. If we assume CP symmetry of the Lagrangian then H "0 also. /!" /$" In this case HM "0 at tree level and will be non-zero by a small amount through radiative corrections; this can be consistent with experiment if the Yukawa couplings are within a certain window, as will be shown below.
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Because it is anomaly-free, we may gauge the ;(1) symmetry and the Higgs mechanism will lead to a massive gauge boson, the aspon, which couples only to the exotic quarks and indirectly via nondiagonal mixing to quarks and leptons. The Yukawa interactions of the model are given by !¸ "q m d 7 * B 0
(2 (2 (2 U #q m u UI #lM m e U #h? q Q s #h.c. * S 0 v * C 0 v * 0 ? v
(93)
where v/(2 is de"ned as the VEV of and UI as ( M ,! \)2. The generation indices are implicit. Usual quarks and leptons acquire their masses through spontaneous symmetry breaking (SSB) induced by the VEV of the doublet Higgs scalar. The new quarks acquire their mass through a gauge-invariant mass term of the form MQM Q . Hence, ; and D quarks are degenerate in mass. * 0 m , m , m , v, h , and M are real by the assumption of CP invariance. The VEVs of s and s are B S C chosen to be 1s 2"(1/(2)i e F,
1s 2"(1/(2)i .
(94)
Hence CP is broken spontaneously. [CP can be broken softly by i(sHs !sHs ).] The up- and down-quark mass matrices linking the right-handed sector to the left-handed sector are of the form
m M " S 0
F
m ; M " B M 0
F
M
(95)
where F"h1s 2#h1s 2 .
(96)
The Kobayashi}Maskawa (KM) matrices will be generalized to 4;4. From the constraint "< "#"< "#"< ""0.9979$0.0021, we "nd "F "/M and "F "/M to be less than 10\ and SB SQ S@ 10\, respectively. Although F is a complex column matrix, the determinants of M and M are real. All entries become complex and, therefore a non-vanishing value of HM arises through radiative corrections. The calculation of HM at one loop level will be done below. First, we discuss how the #avor-changing neutral currents (FCNC) in the presence of new quarks are suppressed at the tree level. After introducing the new vector-like quark doublet, we "nd that there are FCNCs induced by Z coupling because of the mismatch of the new and usual quarks in the right-handed sector. Therefore, the #avor-changing Z couplings are induced by the terms
1 g DM c D ZI#(; contributions) , ¸$!,!" ! 0 I 0 0 8 2 cos h 5
(97)
where the factor ! is the isospin of D and g is the S;(2) gauge coupling constant. Let us 0 consider the down sector "rst. Without loss of any generality, we assume the down-quark mass
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matrix is in the partially diagonalized form
(98)
¸$!,!(down)"b dM G c dH Z 8 GH 0Y I 0Y I
(99)
m
M
0
0
m
0
0
Q 0
0
0
m @ 0
B 0
"
F F . F M
This mass matrix can be diagonalized by a biunitary transformation, K R M K . The transforma* 0 tion matrices are given in Ref. [62]. Thus, Eq. (97) can be written in terms of mass eigenstates dYG as
for iOj, where
1 g (K )H (K ) b " 0 G 0 H GH 2 cos h 5 "
1 g m Gm H B B x xH , 2 cos h M G H 5
(100)
where x ,F /M. Therefore, the FCNC induced by Z coupling is highly suppressed by the small G G mass ratio of usual to new quarks. It is because the mixings of right-handed quarks require a helicity #ip of the usual quarks. For example, b K544.8;10\x xH(10\ for M"100 GeV, GH while the experimental limits on FCNCs require only that b (10\. GH FCNC can also be induced by aspon (A) couplings which are given by ¸$!,!(down)"a DM G c d H AI GH * I *
(101)
for iOj, where a "!g x xH . GH G (
(102)
Therefore, FCNCs induced by A will be important if A is not too heavy compared to Z. Consider the K!KM mixing matrix element M . e(M ) is expected to be dominated by standard 2=-exchange box diagrams, while Im(M ) receives its largest contribution from the A exchange shown in Fig. 16. We obtain f m 1 Im(M )" ) ) Im(x xH) , 6 i
(103)
where i"i #i . The color factor has been taken into account in Eq. (103). Im(M ) receives contributions from the new 2=-exchange box diagrams shown in Fig. 17, but these contributions are negligible. We will consider the CP-violating parameters, such as Im(M )/*M in more detail ) below.
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Fig. 16. Contributions to Im(*M ) by aspon exchange (circles mean mixings). Fig. 17. Contributions to Im(*M ) by new quarks and two-= box diagrams (circles mean mixings).
Next we consider the up sector for completeness. We can also choose states such that M in Eq. (95) is replaced by
0
0
FI
M " 0
m A 0
0
FI
m
FI
0
0
m S 0
with
R 0
FI "C F , G GH H
M
(104)
(105)
where C is the real standard 3;3 KM matrix. The transformation matrices J and J that * 0 diagonalize M in Eq. (104) can be related to K and K by changing x into x ("FI /M) and * 0 G G G m , m and m into m , m and m . The generalized 4;4 KM matrix is given by B Q @ S A R
C 0 K . V "JR )+ * 0 1 *
(106)
Before proceeding, we note that #avor-changing Z-coupling can be induced by the one-loop diagram shown in Fig. 18.
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Fig. 18. Flavor-changing Z coupling induced by new quarks at one loop. Fig. 19. One-loop diagram of which the imaginary part contributes to HM .
By naive dimensional arguments, the e!ective coupling constant b is GH 1 g hh M G H . (107) b " ! GH 2 cos h 16p m 5 Q Using h K0.01 and (M/m )K0.1, we conservatively estimate b to be less than 10\. Q GH Therefore, we expect these FCNCs to be smaller than those in the standard model. Consider now the one-loop corrections to HM . Although the mass matrices in Eq. (95) are complex, their determinants are real. Therefore, HM de"ned in Eq. (80) is zero at tree level. HM will be non-zero when the mass matrices receive radiative corrections. For example, the contributions to hM from the up sector are given by
HM (up)"Arg[det(M #dM )] "Im ¹r ln[M (1#M\dM )] KIm ¹r(M\dM ) , (108) where we have used the fact that M is real and that the corresponding radiative corrections dM are small. The last line in Eq. (108) is valid at one-loop order. De"ning the one-loop corrections dM by dm dF S (109) dM " dm dM 3S
and combining with Eq. (108), we obtain HM (up)"Im ¹r(m\dm !m\FM\dm #M\dM) . S S S 3S
(110)
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Notice that dF will not contribute to HM at one-loop order because of the structure of M . The expression for HM (down) is strictly analogous to Eq. (110). By studying all possible one-loop diagrams [62,237] we "nd that the only contribution to HM comes through dm in Eq. (110) and, in particular, from the diagram depicted in Fig. 19. The S imaginary part gives a contribution i 1 1 ? J h?Im[xH]j HM (up)" J J ? M (2 (4p) ? J
(111)
which gives the estimate (for convenience we shall write x to denote x""x ", the modulus, and G taken to be generation independent since the limits on x are not sensitive to the generation considered) HM "jx/16p .
(112)
Here j is the coe$cient of the quartic interaction between the two types of Higgs " ""s ", and ? ? j with no subscript is an average value. (Actually, there are three independent j corresponding to indices 11, 12#21, 22 but our estimates will not distinguish these.) As mentioned earlier, the neutron electric dipole moment d has been calculated [204,205] in L terms of HM long ago with the result that d K10\HM e.cm. L
(113)
and so we know from d 410\ e.cm. empirically that HM 410\, from which it follows by L Eq. (112) that jx is less than 10\. In the kaon system, the CP violation parameter "e " is given [62, 236] by ) f 2 1 m I x . "e "" ) (2*m ) 3 i )
(114)
Using (*m /m )"7.0;10\, f "0.16 GeV gives the relationship between x and the ;(1) ) ) ) breaking scale i/x"2.9;10 GeV .
(115)
Thus, if we insist that the ;(1) is broken above the electroweak breaking scale (&250 GeV) then x910\ .
(116)
From Eq. (112), this means that j(10\. In [235], it was argued plausibly that j'10\ on the basis of naturalness; this would imply that HM '10\ and hence d '10\ e.cm. L But we can "nd a yet more solid and conservative lower bound on d . It follows from the fact that L the " ""s " interaction receives a one-loop correction from the quark loop box diagram where ? three sides are the top quark and the fourth is the heavy ;-quark (see Fig. 20).
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Fig. 20. One loop " ""s" counterterm for the coupling j.
The full j is given by (including counterterm)#higher loops (117) j"j #j } and the one-loop "nite contribution, for the dominant diagram, Fig. 20, neglecting quark masses and taking h? , g as the respective Yukawa couplings to s and of the third generation by R ? dk 1 "h? ""g " # counterterm (118) j K \ (2p) R k
which imples that the lowest value for j (without accidental cancellations) is j9x/16p .
(119)
Combining Eqs. (112) and (119) then gives the estimate for HM of HM 9x/16p
(120)
which implies that x410\ (incidentally in full agreement with [235]) and that 10\5 HM 510\. From Eq. (113), this then gives a lower limit on the neutron electric dipole moment of d 510\ e.cm. (121) L This is more than two orders of magnitude greater than the prediction of the KM mechanism and thus provides another distinguishing feature of spontaneous CP violation. Before addressing other consequences of the Aspon Model, let us point out that the production of the predicted heavy quarks and the aspon in a hadron collider is discussed in Ref. [232]. A promising approach to "nding new quark #avors is through searching for heavy quark bound states. This fails for the top quark since the single t quark decay to b= is more rapid than the
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formation time of toponia. However, the Q of the Aspon Model has a very much longer lifetime because of the small mixing with the three light generations. Such production of the exotic quarkonia is discussed in [32]. The experimental information regarding CP violation still comes primarily from the neutral kaon system and is inadequate to determine whether the KM mechanism is the correct underpinning of CP violation. In dedicated B studies, with more than 10 samples of B(BM ) decay, it will be possible [238}242] to test this assumption stringently by measuring the angles of the well-known unitarity triangle whose sides correspond to the complex terms of Eq. (78). If CP is spontaneously broken, as in the Aspon Model, the outcome of these measurements will be di!erent from the predictions of the standard model. The three angles of the unitarity triangle are conventionally de"ned as a, b, and c between the "rst and second, second and third, third and "rst sides in Eq. (78), respectively. These angles can be separately measured for the standard model by the time-dependent CP asymmetry C(B(t)Pf )!C(BM Pf ) , a (t)" D C(B(t)Pf )#C(BM Pf )
(122)
where the "nal state f is a CP eigenstate. We de"ne q, p in B!BM mixing by the mass eigenstates B : "B 2"p"B2$q"BM 2
(123)
and similarly for K in the kaon system. Also, A, AM are the decay amplitudes A, AM "1 f "H"B, BM 2 .
(124)
Let us consider the speci"c cases of f"n>n\, tK from B decay and f"oK from B decay. 1 B 1 Q We de"ne j( f ) by
j(n>n\)"
q p
q j(tK )" 1 p
B
B
AM A
AM A
,
B
(125)
> \
L L
q p
,
(126)
q H . p Q M) )
(127)
B
R)
)
and
q j(oK )" 1 p
Q
AM A
The complex conjugate appears in Eq. (127) because BPKM while BPK. If to a su$ciently Q B good approximation "q/p""1 and "AM /A""1, as we shall show for the Aspon Model below, then j( f ) is related to the CP asymmetry through the B !B mass di!erence *M by a (t)"!Im j( f ) sin(dMt) . D
(128)
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In the standard model the angles of the unitarity triangle are related to the j( f ) by sin 2a"Im j(n>n\); sin 2b"Im j(tK ); sin 2c"Im j(oK ) . (129) 1 1 Such relations are no longer valid in the Aspon Model because Im(q/p) B has a major contribution from aspon exchange and Im(q/p) is dominated by aspon exchange. ) To evaluate the CP asymmetries in B decays for the Aspon Model one needs to evaluate the di!erent factors in the j( f ) given in Eqs. (125)}(127) above. More precisely we need, from Eq. (128), the imaginary part of the j( f ). The Aspon Model adds new Feynman diagrams involving aspon exchange to those already present in the standard model. Because CP is only spontaneously broken, the W-exchange amplitudes are predominantly real and have very small phases while the aspon exchange has a much smaller magnitude but an unpredicted arbitrary phase. As a result, the "Im j( f )" appearing in Eq. (128) are of order 0.002 or less, compared to the standard model expectation that "Im j( f )" be of order of, although less than, unity. Thus, CP asymmetries in B decays are predicted to be correspondingly smaller than in the standard model. The technical details can be found in Ref. [234]. It is found, setting M "300 GeV, that "Im j(n>n\)"41;10\ , (130) "Im j(tK )"42;10\ , (131) 1 "Im j(oK )"42;10\ . (132) 1 The resulting asymmetries a (t) are much smaller than those predicted by the standard model D where these imaginary parts are all of order unity. It is also interesting to observe from Eqs. (125)}(127) that
q j(tK )j(oK ) 1 1 " p j(n>n\) "
q p
Q
Q
AM A AM A
B
R)
Q
">Q"\Q
"j(D>D\) . (133) Q Q In the Aspon Model, where the j( f ) have unit moduli and "Im j( f )";1, this relation implies a linear relation for the imaginary parts: Im j(tK )#Im j(oK )!Im j(n>n\)!Im j(D>D\)"0 , (134) 1 1 Q Q which provides an additional test of the Aspon Model. In conclusion, our result is that, if the Aspon Model is correct, CP asymmetries in B decays would be much smaller than predicted by the standard model and the relation (134) would be satis"ed. One may ask [235] whether the present experimental situation of B decay is compatible with the Aspon Model?
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In the model, the quark mixing matrix is a complex unitary 4;4 matrix C . Mixings of the IJ conventional six quarks with one another are speci"ed by the 3;3 matrix C , whose indices GH run from 1 to 3. C is neither real nor unitary because of s-induced mixings to the undiscovered GH quark doublet. However, these terms are &x. Thus, since x410\, C is, to a precision of GH at least 0.1%, a real orthogonal matrix. It is a generalized Cabibbo matrix rather than a Kobayashi}Makawa matrix. This is unfortunate for the search for CP violation in the beauty sector, but has other observable consequences. In the standard model, the Kobayashi}Maskawa matrix < is complex and unitary. The sides of the unitarity triangle are unity and
< < < < (135) R " SB S@ , R " RB R@ . R @ v < v < AB A@ AB A@ The value of R has been measured. According to [252], R "0.35$0.09. The value of R cannot @ @ R be extracted from experimental data alone. Appeal must be made to a theoretical evaluation of the neutral B-meson mass di!erence using the standard model. The analysis in [242] yields R "0.99$0.22. These results suggest a rather large value of the CP-violating angle b, namely, R 0.344b40.75. In the Aspon Model, the matrix C is orthogonal up to terms of order x arising from mixings GH with unobserved quarks. Thus, we anticipate no readily observable manifestations of CP violation in the beauty sector. Furthermore, the unitarity triangle must degenerate into a straight line: "R $R ""1. In this case, we cannot appeal to a theoretical calculation of the neutral B-meson @ R mass di!erence since it depends on unknown parameters. On the other hand, the matrix C with GH neglect of terms &x involves only three parameters. From data at hand, in this context, we obtain R "1!o in the Wolfenstein [243] parametrization, whence R "0.637$0.09. We note that R R present data yields R #R "0.99$0.13. This result is compatible with an approximately @ R orthogonal mixing matrix and hence with the Aspon Model. As a "nal CP violation parameter in the Aspon Model, we shall discuss the value predicted for Re(e /e) * a measure of direct CP violation in K decay. To evaluate Re(e /e) requires the study of several Feynman diagrams [244}249], and their comparison to the standard model. Recall that the most recent evaluations completed at CERN (NA31) [250] and FNAL (E371) [251,252] give results Re(e /e)"(23$3.6$5.4);10\ and Re(e /e)"(7.4$5.2$2.9);10\, respectively, where the "rst error is statistical and the second is systematic. These results are consistent within two standard deviations; the error is expected to be reduced to 1;10\ in foreseeable future experiments. A detailed analysis of e /e in the Aspon Model was carried out by Frampton and Harada [236,237]. They showed that penguin diagrams involving the additional quarks give the dominant contribution. The outcome of these considerations is that Re(e /e) is not larger than 10\ in the Aspon Model. While it does not vanish exactly, it does correspond closely to a superweak model prediction [254].
A new result from KTeV [253] at Fermilab gives Re(e /e)"(28$4.1);10\; it will be interesting to see whether CERN con"rms this result.
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This discussion of the Aspon Model is presented as a motivation for additional quarks beyond the six discovered #avors. Because of the smallness of x which characterizes the mixing of the additional quarks with the known ones, the new quarks have a long lifetime. This is important in their experimental detection, as discussed in the next section.
7. Experimental searches 7.1. Search for long-lived quarks 7.1.1. Present searches The search for long-lived quarks is an ongoing process at the Fermilab Tevatron. The "rst search at the Tevatron was made by the D0 collaboration [255] which looked for signals b Pbc, where b is a charge ! quark, setting a limit m 'M #m . The second and most recent search was made @Y 8 @ by the CDF collaboration [256] which looked for a displaced vertex for ZPe>e\ coming from b PbZ resulting in m '148 GeV (for cq"1 cm). The latter search will be described below. @Y Eventually, such a search will be carried out at the LHC which has a much greater CM energy, with
Fig. 21. Physical cross section for pp PttM X at (s"1.8 TeV as a function of the top mass. This cross section applies to a heavy quark Q as well with t changed to Q. The two data points are from CDF and D0, respectively, for the top quark.
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Fig. 22. Physical cross section for pp PttM X at (s"2.0 TeV as a function of the top mass. This cross section applies to a heavy quark Q as well with t replaced by Q.
a much larger production cross section. In this section, we will concentrate on the current limits coming out of two operating facilities: LEP2 and Fermilab. In the next section, we will discuss the prospects for future searches, both at the Fermilab Tevatron and at the LHC. We will only brie#y discuss prospects for such a search at facilities which are under discussion, such as the NLC, etc. Also in this section, we will focus primarily on hadron colliders such as the Tevatron since these are the machines which can explore the mass range that was discussed earlier. In the search for a new particle, there are two principal activities to attend to: how to produce the particle and how to detect it. For a particle which is somewhat `exotica, such as supersymmetric particles, the production process would be highly model-dependent. Fortunately, for a heavy quark, this is rather standard: it proceeds through the qq and gg channels. For the range of heavy quark masses considered in this Report, the qq process via the electroweak channels =, c, Z is completely negligible compared with the QCD process with gluons. The production cross section, at the Tevatron and at the LHC, for the top quark as a function of its mass has been computed up to the next-to-leading order in QCD [257}262]. (The qq channel is dominant at the Tevatron while the gg channel is dominant at the LHC.) This can be directly applied to the present search for long-lived quarks whose production mechanism should be similar to that of the top quark. The
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Fig. 23. Cross section for ppPttM X at (s"10 and 14 TeV as a function of the top mass. Notice that p has been mislabeled as p in the "gure. The same prediction applies to a heavy quark Q.
production cross sections as a function of the heavy quark mass are shown in Figs. 21 and 22 for the Tevatron at (s"1.8 and 2 TeV respectively, and in Fig. 23 for the LHC at (s"10,14 TeV. Here `m a will stand for a generic heavy quark mass, for both the Tevatron and the LHC. R As can be seen above, the predicted cross section at the LHC, for a given heavy quark mass, exceeds that at the Tevatron by more than two orders of magnitude, which will facilitate the search for such an object. The next task is to de"ne the detection capability of various detectors. Since the latest constraint on long-lived quarks come from CDF [256], we shall use it as a prototype of detectors dedicated to such a purpose. Other detectors such as D0 are very similar in layout. Needless to say, the CDF detector is a complicated, multipurpose one whose speci"cations can be found in [263]. A generic detector for hadron colliders generally consists of a (silicon) vertex detector immediately surrounding the beam pipe for high-precision determination of the location of the tracks. Next comes a central tracking chamber which measures charged tracks and momenta of charged particles. Surrounding these two units are generally hadron calorimeters which measure the energy deposited by hadrons. Next comes the muon chambers which detect the location of the particles which penetrate the calorimeters. As described below, the "rst two parts (vertex detector and central tracking chamber) were used to search for displaced vertices coming from the decay of a long-lived
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Fig. 24. The ¸ distribution of the Z's after applying all cuts. The data are represented by the circles. The histogram is VW the expected ¸ distribution for prompt Z's based on the measured ¸ uncertainty in the event sample. The inset shows VW VW the distribution after the 2 jet requirement is applied. The vertical dashed lines separate the prompt and non-prompt regions.
particle. For stable or very long-lived particles, the muon chambers are used in conjunction with the ionization energy loss in the tracking chambers to make such a search. The current search at CDF can be divided into two categories: the search for those quarks whose decay lengths, l"cbcq with q being the proper decay time, is (1) less than 1 meter, and (2) greater than 1 meter. Let us "rst concentrate on the "rst category (l(1 m) [256]. The parts of the detector which are relevant here consist of two components: a silicon vertex detector immediately surrounding the beam pipe for precision tracking and a central tracking chamber embedded in a 1.4 T solenoid magnetic "eld which measures the momenta and trajectories of charged particles. In the search for a new particle, a crucial task would be the identi"cation of a characteristic signature which would distinguish it from background. In the present case, that characteristic signature is the decay ZPe>e\ with the e>e\ vertex displaced from the pp interaction point. This Z boson could come from the decay of a charge ! quark (denoted by b in Ref. [256] and by D in Ref. [179]) in the process DPb#Z with a subsequent decay ZPe>e\. In this case, D would be the long-lived parent of the Z boson. The search at CDF concentrated on events containing an electron}positron pair whose invariant mass is consistent with the Z mass and whose vertex is displaced from the pp interaction point. The data used was from the 1993}1995 Tevatron run with an integrated luminosity of 90 pb\ of pp collisions at (s"1.8 TeV.
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Fig. 25. The 95% con"dence level upper cross section limit for p.Br times the acceptance for an electron}positron pair to be within the detector as a function of "xed j ,cb cq. Cross sections above the curve have been excluded at the 95% VW VW con"dence level. The inset shows the exclusion curve and the theoretical prediction for a b quark of mass 110 GeV as a function of its lifetime, assuming 100% decay into bZ.
In the search for a long-lived parent of the Z, the CDF collaboration focused on the measurement of ¸ which is the distance in the transverse (r! ) plane between the pp interaction point VW and the e>e\ vertex. Notice that ¸ "cb cq with b being the transverse component of the VW VW VW parent particle divided by c. As de"ned by the CDF collaboration, ¸ can be either positive or VW negative. For prompt Z's coming from the SM process qq PZPe>e\, one would expect ¸ +0 VW because of the short lifetime of the Z. The ¸ distribution with appropriate cuts taken into account VW is shown in Fig. 24. As emphasized in [256], this distribution is consistent with that for prompt Z's where one would expect less than one event for "¸ "(0.1 cm. Ref. [256] also pointed out that the number of events VW with ¸ signi"cantly less than zero is an e!ective measure of the background. The CDF VW collaboration observed one event for ¸ '0.1 cm and three events for ¸ (!0.1 cm. As stated, VW VW there is no evidence for a long-lived parent of the Z. This is shown in Fig. 25 where the constraint is expressed in terms of the 95 level upper limit on the product of the production cross section for the long-lived parent, p , its branching ratio, Br(XPZ), the branching ratio, Br(ZPe>e\), and the 6 e>e\ acceptance for pseudorapidity "g"(1. The above discussions and "gures deal with limits on the production of a single parent with its subsequent decay into a Z. We are, however, most interested in the detection of a long-lived quark.
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Fig. 26. The hatched areas in this plot represent the 95% con"dence-level regions of b mass and lifetime that have been excluded. For cq"1 cm, CDF excluded up to a mass of 148 GeV.
As we have mentioned earlier, this long-lived quark would be produced in pair. The CDF search for a long-lived D (or b ) which is pair-produced can be summarized as follows. The kind of events that are searched for would include, besides the e>e\ pair coming from the Z, two or more jets. For instance, this could come from the reaction: qq PDDM PbZbM ZPbe>e\bM qq . The ¸ distribution VW is shown in the inset of Fig. 25. There one would expect less than one event for ¸ 40.01 cm. VW The CDF collaboration found one event. This was then translated into a cross section limit as a function of cq, where q is the lifetime of D. Assuming that Br(DPbZ)"100%, the exclusion curve for the DDM production cross section as a function of cq is shown as an inset of Fig. 25 for a D quark of mass 110 GeV. The theoretical prediction for the cross section for such a mass is shown as a horizontal line. Clearly, this is ruled out for a wide range of lifetimes. For other masses, the exclusion curves for the cross section are not shown but are instead translated into exclusion regions in the mass-lifetime plane. (This is because the production cross section can be calculated in QCD as a function of the D mass as mentioned above.) The plot shown in Fig. 26 assumes the above branching ratio. In Fig. 26, three forbidden regions are presented: the LEP, the D0, and the CDF constraints. The most stringent constraint comes, of course, from the CDF results. As can be seen from Fig. 26, for every cq, there is a range of forbidden masses represented by the shaded region. The largest forbidden range is for cq"1 cm corresponding to a lifetime q+3.3;10\ s. This rules out the mass of the D quark up to 148 GeV. For smaller or larger cq, we can see that the lower bounds on the D mass become somewhat smaller than 148 GeV. If the D quark happens to have a mass larger that 148 GeV, it could escape detection for a large range of lifetimes as can be seen in Fig. 26. The
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mere fact that there exists unexplored regions of the detector, as shown in the unshaded areas of Fig. 26, is reason to believe that there are plenty of opportunities for future searches. What does this result tell us about the long-lived quarks with the mass range that we have discussed in Section 4? First, as we have mentioned earlier, we are especially concerned with the mass of the heavy quarks larger than 150 GeV. From Fig. 26, one can see that the CDF search based on the decay mode DPbZ does not set any constraint on long-lived quarks with a mass greater than 150 GeV. In other words, if this long-lived quark exists and if it decays at a distance ¸ '0.01 cm, it has yet to be discovered. This statement is, of course, based on the assumption VW that the branching ratio for DPbZ is 100%. As discussed in Section 4 and in Ref. [179], for m 4m , there is another possible decay mode for D, namely DP(c, u)= (the c quark channel in " R any reasonable scenario dominates over the u quark channel). Whether or not DPbZ dominates over DPc= will depend on a particular model for the mixing element "< ". As discussed in Ref. "A [188], even with a very naive assumption "< "&x , with x being the mixing parameter between "A the third generation and the heavy quark, DPbZ can dominate over DPc= for a certain range of mass and mixing x. For "< "&x (or less), the bZ mode will almost always be the dominant one. "A Since this is a rather model-dependent statement, one should, in principle, look for both modes when m 4m . Unfortunately, the mode DPc= would be rather di$cult to detect. We shall come " R back to this issue and others in the discussion of future searches. The next question concerns the limits on a charged `stablea or very long-lived massive particle. Such a search is being carried out a CDF. Basically, this search focuses on decay lengths larger than 1 m, i.e. larger than the radius of the Central Tracking Chamber. As of this writing, preliminary results have only appeared in conference talks [113,114]. Therefore, what is described below will be considered preliminary. A stable massive quark moving at a low velocity will leave an ionization track in the tracking chamber because the energy loss dE/dxJ1/b (the Bethe}Bloch equation) and a low b would imply a large energy loss. Furthermore, for such a stable massive quark to be detected, one would look for signals in the muon detector after mesons formed from this particular quark have traversed the calorimeters and reached the muon detector. To distinguish it from a muon, one would have to correlate this signal with the large energy deposited in the tracking chamber. The measurement of dE/dx as a function of bc"p/M, combined with the momentum measurement would allow for a determination of the mass of the particle. The mass limits from CDF for such a very long-lived quark are approximately 200 GeV. In summary, the most stringent limits, so far on long-lived quarks come from the CDF collaboration [256]. It excludes a long-lived, charge ! quark of mass up to 148 GeV for a lifetime q+3.3;10\ s (cq"1 cm). For other values of lifetimes, the excluded mass ranges are weaker as can be seen in Fig. 26. This constraint was based on the search for a displaced vertex for the decay ZPe>e\ which could come from the decay DPbZ. Furthermore, all of these constraints come from the search for the decay of the D quark inside the Central Tracking Chamber. If the D quark lives long enough to enter the calorimeters and subsequently trigger a signal in the muon chamber, the constraint (which is preliminary) is much stronger: a D quark mass below approximately 200 GeV is excluded. In short, under what circumstances will a D quark escape detection so far? First, if its mass is above 148 GeV (as referred to in Section 4) and if it decays inside the Central Tracking Chamber. If the mass is above &200 GeV, D is no longer required to decay in the tracking chamber: it simply escapes detection regardless of where it decays. Most of the discussion in Section 4 concerned these possibilities.
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What kind of improvements should be made in order to be able to search for these quarks heavier than 148 GeV which could either decay inside the Central Tracking Chamber or, if heavier than 200 GeV, could also travel through the calorimeters? What about the ; quark? How could one detect it? What if the dominant decay mode of the D is DPc=? These are the kinds of questions that one would like to address, at least qualitatively, in the next section. 7.1.2. Future searches The "rst kind of future searches would be based on present facilities such as the Tevatron. In particular, one might ask what kind of improvement one can make by exploiting the present CDF RunI data with up to 120 pb\. One can then ask what light RunII with a large improvement in luminosity and detector might shed on the search for long-lived quarks. The above search has focused on the discovery of displaced vertices for decay lengths greater than 100 lm, with the resulting constraints as described above. What if the decay length is less than 100 lm? The appropriate kind of experiment would be a counting experiment which is not based on the search for displaced vertices [264]. How feasible this kind of experiment might be will probably depend on the improvements planned for RunII. These improvements include: (a) a detector upgrade with, among several things, added layers of silicon; (b) an increase by a factor of 20 in the luminosity. The radius of the silicon vertex detector is roughly of the order of 22.3 cm. Added silicon layers would increase that radius to about 28 cm [264] and consequently the tracking ability of the detector. For decay lengths between 100 lm and 1 m, one of the most important tasks would be to improve the tracking e$ciency of the Central Tracking Chamber by adding, for instance, more silicon layers to the vertex detector. Decay lengths of a few tens of cm might be hard, although possible, to detect because of poor tracking e$ciency in such a region. This would require new reconstruction algorithm. The search for very long-lived or `stablea quarks will also be improved by the detector upgrade and the increase in luminosity. One might ask what else could be done at CDF and D0 in the next run beside those issues discussed above. In particular, one would like to know how feasible might the detection of a signal such as DPc= be if it happens to be the dominant decay mode of the D. Needless to say, such a task would be much more daunting than the detection of DPbZ. Nevertheless, a feasibility study would probably be extremely useful. As we have discussed at length in Section 4 and in Ref. [179], there is also the partner of the D, namely the charge quark denoted by ;, which should not be forgotten. If ; is heavier than D } but not by much because of the o-parameter constraint } it will decay into D via ;PD#(l>l, q q !1/3), where particles inside the parantheses denote light quarks or leptons or it can decay via ;Pb=, depending on how degenerate ; and D are and how large "< " is. For the "rst mode, it was shown in Ref. [179] that ; practically decays near the 3@ interaction point, with a decay length typically of the order of 10\ lm. The D will subsequently decay between 100 lm and 1 m. It will be a challenge to be able to identify such a signal. For the second mode ;Pb=, one has to be able to distinguish it from a signal coming from top decay. It would be extremely hard, if not impossible, to be able to resolve the decay vertex to distinguish ; from t. However, by comparing the predicted number of t's with the observed ones, one might rule out the mode ;Pb= with a ; mass close to the top mass.
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Turning our attention to the upcoming experiments at the LHC, we would like to brie#y describe the two main detectors which will be crucial to the search for long-lived quarks (if such a search would be carried out). They are the Compact Muon Solenoid (CMS) and a Toroidal LHC Apparatus (ATLAS) detectors [265]. The layout for both detectors is generically very similar to CDF and D0. In the search for long-lived quarks, the components which are crucial would be the central tracking system of CMS and the inner detector of ATLAS. The central tracking system of CMS consists of silicon pixels, silicon and gas microstrip detectors with high resolutions. (The resolution of the silicon pixels is about 11}17 lm while the outermost part of the detector, namely the gas microstrip detector, has a resolution of approximately 2 mm!) The silicon pixels and silicon microstrips cover a radial region up to about 40 cm, a marked improvement over the CDF vertex detector. The microstrip gas chamber covers a radial region to approximately 1.18 m which is roughly similar to that covered by the Central Tracking Chamber of CDF. The inner detector of ATLAS consists of a Semi-Conductor Tracker (SCT): pixel detectors, silicon microstrips and GaAs detector, and Microstrip Gas Counters (MSGC). The pixel detectors have a spatial resolution of about 14 lm while the MSGC have a resolution of about 1.8 mm, very comparable to CMS. The SCT part covers a radial region of up to 60 cm while the MSGC covers a radial region of up to 1.15 m. Again one sees a marked improvement over CDF in the region of interest. In addition, as can be seen from Fig. 23, the production cross section for a given mass is now increased by at least two orders of magnitude at the LHC because the center of mass energy is now 14 TeV. Such an increase in the cross section combined with the increase in the radial distance covered by silicon detectors would, in principle, help in the search for long-lived quarks. One might wonder if a similar analysis as the one preformed by CDF could be carried over to the LHC experiments. Considering the fact that, with a much higher energy and consequently larger cross section, the number of events and background will be signi"cantly higher as well. This would probably require a di!erent search algorithm. Finally, concerning proposed but not yet approved colliders such as the Next Linear Collider (NLC) with (s"500 GeV, the long-lived quarks with the mass range discussed in Section 4, would be produced copiously and with little background. What kind of signal would one search for will depend on the kind of detectors involved. Whether or not the existence (or non-existence thereof) of these long-lived quarks will be established by CDF, ATLAS, or CMS by the time the NLC operates (if approved) remains an open question. 7.2. Lepton searches Earlier in this Report, indirect bounds on the masses of heavy leptons arising from violations of e!k!q universality were discussed. The bounds were very sensitive to the mixing angle between the third and fourth generations. In this section, we discuss direct detection of heavy leptons. All current experimental bounds on heavy leptons come from experiments at electron-positron colliders. This is not surprising; the cross section for heavy lepton production at hadron colliders is small and backgrounds are large. Of course, once LEP200 shuts down in a couple of years, the only available colliders for searching for heavy leptons will be the Tevatron and the LHC. We will "rst examine the current bounds on heavy lepton masses, and then turn towards the future.
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It is generally believed that charged heavy leptons can be excluded up to the approximate kinematic limit of LEP. This is not necessarily the case however; the charged heavy leptons of many of the most interesting models have not been excluded for masses above approximately 45 GeV. Below 45 GeV, heavy leptons (charged or neutral) would contribute to the decay width of the Z, and such leptons can be excluded. The strongest bounds on heavy lepton masses reported by LEP have been reported by OPAL [266] and by L3 [267]. Both experiments assume that the heavy leptons decay via the charged current decay } the decay EPqZ, which can be large in vector-like models, is not considered (since it seldom dominates the charged current decay, their bounds are not a!ected). In the OPAL analysis, they exclude charged leptons which decay via EPl = with masses below 80.2 GeV, and J those which decay via EPN= with masses below 81.5 GeV. Unfortunately, this latter decay assumes that the mass di!erence between the E and the N is greater than 8.4 GeV. As we have seen in earlier sections, vector-like models have mass splittings on the order of a few hundred MeV, and even in chiral models, the splitting could also be small. Note, however, that if the mixing angle between the third and fourth generation is bigger than 10\, then the EPl = would occur near J the vertex, and the OPAL bound would apply. The bound was obtained for LEP at (s"170}172 GeV, and can be improved somewhat for the later runs. In the L3 analysis, the mass splitting was assumed to be larger than in the OPAL case, greater than 10 GeV, and similar bounds were obtained. The L3 analysis also looked for long-lived charged leptons, which would exist if the mixing angles with lighter generations were small (typically less than 10\) and the charged lepton is lighter than its neutral partner. Of course, such leptons must eventually decay, for cosmological reasons, but we discussed a variety of such scenarios earlier in this Report. L3 excludes such leptons up to a mass of 84.2 GeV. Both experiments also looked for heavy neutrinos, which decay at the vertex (mixing angle greater than 10\ or so) into a charged lepton and a =. For both experiments, the bounds for Dirac (Majorana) neutrinos are approximately 78 (66) GeV for decays into electrons or muons and 70 (58) for decays into taus. So, summarizing the current situation, the bounds on the charged heavy lepton are approximately at the kinematic limit of the collider if and only if this lepton is either stable (i.e. with a lifetime greater than tens of nanoseconds), has a large (8 GeV or greater) splitting with its neutrino partner, or has a relatively large mixing angle (10\ or greater) with lighter generations. Note that one of the most interesting models is the E motivated model with a vector-like doublet with very small mixing, and this lepton satis"es none of the above conditions. The mass bound for such a lepton is still only given by Z decays. (A search for a nearly degenerate lepton doublet was reported [268] many years ago by the Mark II detector, but only applied for leptons lighter than 10 GeV.) Could more analysis at LEP "nd such a charged heavy lepton? In vector-like models, the principal decay of the E is into the N plus a very soft pion. It appears to be impossible to pick this pion out of the background from soft tracks from beam}beam interactions. Recently, Thomas and Wells [172] proposed a new signature } triggering on an associated hard radiated photon. This is similar to proposals for counting neutrino species through e>e\Pllc. At LEP, one would look for e>e\P¸>¸\c. There are backgrounds from the above neutrino process, but they can be reduced by looking for a displaced vertex (the decay length is of the order of centimeters) and for the soft pions. Thomas and Wells plot the cross section as a function of the ¸ mass and the minimum photon energy. With an integrated luminosity of 240 pb\ at (s"183 GeV, and a minimum
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photon energy of 8 GeV, they estimate that a doublet mass of up to 70 GeV could be detected. Presumably, this reach will be considerably higher for the more recent higher energy runs. They note that this signature is unusual for e>e\ machines since it is not only limited by the machine energy, but also by the luminosity, and that higher luminosity can signi"cantly extend the reach. In the chiral case, there remains a `holea to be "lled. If the E is either very close in mass to or lighter than the N, it will primarily decay via mixing. If the mixing angle is greater than about 10\, then it decays near the vertex and can be detected at LEP up to the kinematic limit. If the mixing angle is smaller than about 10\, it is e!ectively stable and can be detected at LEP up to the kinematic limit. For intermediate angles, the decay length is of the order of tens of centimeters to a meter. Of course, some will still decay near the vertex, and some will decay well within or outside the detector, and so it is possible that a complete analysis could close this hole. Doing so would be useful, since one of the plausible values for the mixing angle discussed in Section 3 is given by (m O /M , which, using 0.01 eV for the l mass, gives 3;10\ for the mixing angle. , O J In future, similar analyses to the above will give similar bounds at lepton colliders near the kinematic limit of the colliders. We have noted, however, that the detection of the vector-like doublet leptons remains problematic and can best be attacked looking for an associated hard photon. There also seems to be a window for decay lengths of the order of tens of centimeters which has yet to be closed. The decay mode EPqZ is generally smaller than the charged current decay } although it is a much cleaner mode, backgrounds are not the problem for EPl=, thus the latter would be detected "rst. This neutral current decay mode, however, provides a much cleaner signature for hadron colliders, which we now discuss. A study of searches for heavy charged leptons at hadron colliders was performed by Frampton et al. [173]. They considered charged lepton production at the SSC and at the LHC (at 17 TeV). There are two main production mechanisms for heavy leptons at hadron colliders. The "rst is gluon fusion, through a triangle graph, into a Higgs boson or a Z-boson. The second is quark fusion directly into a Z-boson (the e!ects of photon exchange are much smaller than those of the Z). The cross section for lepton production through quark fusion falls o! very rapidly as the lepton mass increases, but the cross section through gluon fusion does not fall o! as rapidly, since the matrix elements increases as the square of the lepton mass. First consider the chiral case. Here, gluon fusion dominates for lepton masses above about 150 GeV, and the total cross section for masses between 100 and 800 GeV drops from 0.5 to 0.05 pb. This will lead to many thousands of events per year at the LHC. The signature would be a conventional heavy lepton signature. For those masses, and for chiral leptons, one can expect a reasonably large splitting between the N and the E, leading to standard single lepton and missing momenta signatures; even if there was an unexpected degeneracy (or if the N were heavier) mixing would lead to clear signatures (note, as discussed above, the importance of closing the window for mixing angles near 10\). What about the vector-like case? Here, gluon fusion does not contribute, since the leptons do not couple to the Higgs and the vector-like coupling to the Z gives no contribution due to Furry's theorem. Thus, the contributions are only through quark fusion, which fall o! much faster. As the lepton mass increases from 100 to 800 GeV, the cross section falls from 1 to 0.001 pb. For a 400 GeV heavy lepton, this will give only 1000 events annually at the LHC. This makes detection more di$cult; however one should recall that these leptons can decay via the neutral current: EPqZ
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which has a branching fraction of at least a few percent (and in some models much larger). This would give a very clear signature with very low background. Even if the decay is suppressed by very small mixing angles, and thus the E passes through the detector, stable lepton searches should see it (it can be readily distinguished from a muon by time-of-#ight using the velocity distributions as given in Ref. [173]).
8. Conclusions There are still several reasons to believe that further quarks and leptons remain to be discovered. Although the fourth or further quark}lepton generations cannot be exactly similar and sequential to the "rst three generations, there are plenty of alternative possibilities which avoid the experimental embarassment to the invisible Z partial width of a fourth light neutrino. The additional quarks and leptons may be chiral as in the "rst three generations or non-chiral and vector-like. The allowed masses are constrained by the precise electroweak data particularly at the Z pole where the data now agree with the minimal standard model at an astonishing 0.1% level. The S, ¹, ; parameters then restrict what states may be added as discussed above in Section 3. Also the stability of the observed vacuum places constraints on additional fermions as does the (optional) requirement of grand uni"cation of the three gauge couplings. Mixing angles for the new quarks and leptons are relatively unconstrained, except by unitarity, without new experimental data. The lifetime and decay modes (see Section 4) of a heavy lepton depend critically on whether the N or E state is the more massive. A similar dependence occurs for heavy quarks which may have such small mixing with the known quarks that at least one new quark may have an exceptionally long lifetime. In Section 5 we have considered the fascinating possibility that the Higgs boson is not elementary but rather some bound state of additional fermions which transform under the standard gauge group. The heaviness of the top quark has suggested to some that it plays a special role in electroweak symmetry breaking, but even heavier fermions are more attractive candidates to participate in dynamical symmetry breaking. CP symmetry violation has two disparate but likely related aspects in the Standard Model: the strong CP problem and the weak CP violation in kaon decay. Strong CP can be addressed by addition of extra quarks as explained in our Section 6. Weak CP violation by the KM mechanism requires at least three generations, and acquires even more CP violating phases in the presence of additional quarks. We have illustrated this with the Aspon Model which invokes spontaneous CP violation to relate solution of the strong CP problem by extra vector-like quarks to the violation of CP symmetry in kaon decay. The new vector-like quarks may have long lifetime as mentioned in Section 4. Experiment is the "nal arbitor of everything we have reviewed. Long-lived quarks are being sought at collider facilities. To some extent, detectors have not been designed for such a possibility and this review may encourage further thought in detector design. Similarly heavy leptons are being, and will be, investigated at existing and future colliders. Discovery of a further quark or lepton would be revolutionary and propel high-energy physics in a new and exciting direction.
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Note added in proof The NA-48 experiment at CERN has reported [270] a result for e /e which is consistent with the KTeV measurement. This rules out superweak models, and poses interesting challenges for the aspon model. In our discussion of the decays of the neutral current decays of a b quark, we concentrated on b Pb#Z. CDF has just reported [272] an improved result on the decay of a &&short-lived'' b : if the branching ratio is 100%, then the lower bound on the b mass is 199 GeV. They also give results as a function of the branching ratio. Another neutral current decay that we did not discuss is b Pb#H. Although small in the standard model, this can become competitive with the b Pb#Z mode for many extensions. This decay mode is discussed in detail in Ref. [273].
Acknowledgements This work was supported in part by the US Department of Energy under Grants No. DEFG02-97ER41036 and DE-A505-89ER40518, and by NSF Grant No. PHY-9600415. We would like to thank David Stuart for many useful discussions about the CDF Collaboration.
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Author index to volumes 321}330 A.Ya. Ender, H. Kolinsky, V.I. Kuznetsov and H. Schamel, Collective diode dynamics: an analytical approach AchuH carro, A. and T. Vachaspati, Semilocal and electroweak strings Afanasjev, A.V., D.B. Fossan, G.J. Lane and I. Ragnarsson, Termination of rotational bands: disappearance of quantum many-body collectivity Aharony, O., S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Large N "eld theories, string theory and gravity Beck, M.H., A. JaK ckle, G.A. Worth and H.-D. Meyer, The multicon"guration timedependent Hartree (MCTDH) method: a highly e$cient algorithm for propagating wavepackets Bender, H., see J. Heil Bertulani, C.A., V.Yu. Ponomarev, Microscopic studies on two-phonon giant resonances Bhattacharjee, P. and G. Sigl, Origin and propagation of extremely high-energy cosmic rays BoK hm, A., see J. Heil Boccaletti, S., C. Grebogi, Y.-C. Lai, H. Mancini and D. Maza, The control of chaos: theory and applications Brown, G.E., see H.K. Lee Burgess, C.P., Goldstone and pseudo-Goldstone bosons in nuclear, particle and condensed-matter physics Bychkov, V.V. and M.A. Liberman, Dynamics and stability of premixed #ames
328 (2000) 1 327 (2000) 347 322 (1999)
1
323 (2000) 183
324 (2000) 1 323 (2000) 387 321 (1999) 139 327 (2000) 109 323 (2000) 387 329 (2000) 103 325 (2000) 83 330 (2000) 193 325 (2000) 115
Chowdhury, D., L. Santen and A. Schadschneider, Statistical physics of vehicular tra$c and some related systems
329 (2000) 199
DoK rner, R., V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer and H. Schmidt-BoK cking, Cold target recoil ion momentum spectroscopy: a &momentum microscope' to view atomic collision dynamics Dytman, S.A., see T.P. Vrana
330 (2000) 95 328 (2000) 181
Elsaesser, T. and M. Woerner, Femtosecond infrared spectroscopy of semiconductors and semiconductor nanostructures
321 (1999) 253
Fedorovich, R.D., A.G. Naumovets and P.M. Tomchuk, Electron and light emission from island metal "lms and generation of hot electrons in nanoparticles Fossan, D.B., see A.V. Afanasjev
328 (2000) 73 322 (1999) 1
0370-1573/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 5 1 - X
350
Author index
Frampton, P.H., P.Q. Hung, M. Sher, Quarks and leptons beyond the third generation Giudice, G.F. and R. Rattazzi, Theories with gauge-mediated supersymmetry breaking Gleiser, R.J., C.O. Nicasio, R.H. Price and J. Pullin, Gravitational radiation from Schwarzschild black holes: the second-order perturbation formalism Grebogi, C., see S. Boccaletti Grenet, G., see P. Politi Grill, W., see J. Heil GroK ger, A., see J. Heil GruK newald, M.W., Experimental tests of the electroweak standard model at high energies Gubser, S.S., see O. Aharony Heil, J., A. BoK hm, A. GroK ger, M. Primke, P. Wyder, P. Keppler, J. Major, H. Bender, E. SchoK nherr, H. Wendel, B. Wolf, K.U. WuK rz, W. Grill, H. Herrnberger, S. Knauth and J. Lenzner, Electron focusing in metals and semimetals Heiselberg, H. and M. Hjorth-Jensen, Phases of dense matter in neutron stars Herrnberger, H., see J. Heil Hjorth-Jensen, M., see H. Heiselberg HuK tt, M.-Th., A.I. L'vov, A.I. Milstein and M. Schumacher, Compton scattering by nuclei Hung, P.Q., see P.H. Frampton
330 (2000) 263
322 (1999) 419 325 329 324 323 323
(2000) (2000) (2000) (2000) (2000)
41 103 271 387 387
322 (1999) 125 323 (2000) 183
323 328 323 328
(2000) (2000) (2000) (2000)
387 237 387 237
323 (2000) 457 330 (2000) 263
Inoue-Ushiyama, A., see K. Takatsuka
322 (1999) 347
JaK ckle, A., see M.H. Beck Jagutzki, O., see R. DoK rner
324 (2000) 1 330 (2000) 95
Keppler, P., see J. Heil Knauth, S., see J. Heil Kolinsky, H., see A.Ya. Ender KroK ger, H., Fractal geometry in quantum mechanics, "eld theory and spin systems Kuznetsov, V.I., see A.Ya. Ender
323 323 328 323 328
L'vov, A.I., see M.-Th. HuK tt Lai, Y.-C., see S. Boccaletti Landa, P.S. and P.V.E. McClintock, Changes in the dynamical behavior of nonlinear systems induced by noise Lane, G.J., see A.V. Afanasjev Lee, H.K., R.A.M.J. Wijers and G.E. Brown, The Blandford}Znajek process as a central engine for a gamma-ray burst Lee, T.-S.H., see T.P. Vrana Lenzner, J., see J. Heil Liberman, M.A., see V.V. Bychkov
323 (2000) 457 329 (2000) 103
(2000) 387 (2000) 387 (2000) 1 (2000) 81 (2000) 1
323 (2000) 322 (1999) 325 328 323 325
(2000) (2000) (2000) (2000)
1 1 83 181 387 115
351
Author index
Major, J., see J. Heil Maldacena, J., see O. Aharony Mancini, H., see S. Boccaletti Marty, A., see P. Politi Maza, D., see S. Boccaletti McClintock, P.V.E., see P.S. Landa Mergel, V., see R. DoK rner Meyer, H.-D., see M.H. Beck Milstein, A.I., see M.-Th. HuK tt Mirlin, A.D., Statistics of energy levels and eigenfunctions in disordered systems Moshammer, R., see R. DoK rner
323 323 329 324 329 323 330 324 323 326 330
Naumovets, A.G., see R.D. Fedorovich Nicasio, C.O., see R.J. Gleiser
328 (2000) 73 325 (2000) 41
Ooguri, H., see O. Aharony Oz, Y., see O. Aharony
323 (2000) 183 323 (2000) 183
Piller, G., W. Weise, Nuclear deep-inelastic lepton scattering and coherence phenomena Politi, P., G. Grenet, A. Marty, A. Ponchet and J. Villain, Instabilities in crystal growth by atomic or molecular beams Ponchet, A., see P. Politi Ponomarev, V.Yu., see C.A. Bertulani Price, R.H., see R.J. Gleiser Primack, H. and U. Smilansky, The quantum three-dimensional Sinai billiard } a semiclassical analysis Primke, M., see J. Heil Pullin, J., see R.J. Gleiser Ra"i-Tabar, H., Modelling the nano-scale phenomena in condensed matter physics via computer-based numerical simulations Ragnarsson, I., see A.V. Afanasjev Rattazzi, R., see G.F. Giudice Rosenberg, L.J, K.A. van Bibber, Searches for invisible axions Santen, L., see D. Chowdhury Schadschneider, A., see D. Chowdhury Schamel, H., see A.Ya. Ender Schmidt-BoK cking, H., see R. DoK rner SchoK nherr, E., see J. Heil Schumacher, M., see M.-Th. HuK tt Shabanov, S.V., Geometry of the physical phase space in quantum gauge systems Sher, M., see P.H. Frampton Sieniutycz, S., Hamilton}Jacobi}Bellman framework for optimal control in multistage energy systems Sigl, G., see P. Bhattacharjee
(2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000)
330 (2000) 324 324 321 325
(2000) (2000) (1999) (2000)
387 183 103 271 103 1 95 1 457 259 95
1 271 271 139 41
327 (2000) 1 323 (2000) 387 325 (2000) 41
325 322 322 325
(2000) 239 (1999) 1 (1999) 419 (2000) 1
329 329 328 330 323 323 326 330
(2000) (2000) (2000) (2000) (2000) (2000) (2000) (2000)
199 199 1 95 387 457 1 263
326 (2000) 165 327 (2000) 109
352
Author index
Singh, S., Phase transitions in liquid crystals Smilansky, U., see H. Primack Sorokin, D., Superbranes and superembeddings Spielberger, L., see R. DoK rner Struck, C., Galaxy collisions
324 327 329 330 321
(2000) 107 (2000) 1 (2000) 1 (2000) 95 (1999) 1
Takatsuka, K., H. Ushiyama, A. Inoue-Ushiyama, Tunneling paths in multi-dimensional semiclassical dynamics Tomchuk, P.M., see R.D. Fedorovich
322 (1999) 347 328 (2000) 73
Ullrich, J., see R. DoK rner Ushiyama, H., see K. Takatsuka
330 (2000) 95 322 (1999) 347
Vachaspati, T., see A. AchuH carro van Bibber, K.A., see L.J. Rosenberg Villain, J., see P. Politi Vrana, T.P., S.A. Dytman and T.-S.H. Lee, Baryon resonance extraction from nN data using a unitary multichannel model
327 (2000) 347 325 (2000) 1 324 (2000) 271
Weiner, R.M., Boson interferometry in high-energy physics Weise, W., see G. Piller Wendel, H., see J. Heil Wijers, R.A.M.J., see H.K. Lee Woerner, M., see T. Elsaesser Wolf, B., see J. Heil Worth, G.A., see M.H. Beck WuK rz, K.U., see J. Heil Wyder, P., see J. Heil
327 330 323 325 321 323 324 323 323
328 (2000) 181 (2000) (2000) (2000) (2000) (1999) (2000) (2000) (2000) (2000)
249 1 387 83 253 387 1 387 387
Subject index to volumes 321}330 General Tunneling paths in multi-dimensional semiclassical dynamics, K. Takatsuka, H. Ushiyama and A. Inoue-Ushiyama Changes in the dynamical behavior of nonlinear systems induced by noise, P.S. Landa and P.V.E. McClintock Fractal geometry in quantum mechanics, "eld theory and spin systems, H. KroK ger Large N "eld theories, string theory and gravity, O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz The multicon"guration time-dependent Hartree (MCTDH) method: a highly e$cient algorithm for propagating wavepackets, M.H. Beck, A. JaK ckle, G.A. Worth and H.-D. Meyer Hamilton}Jacobi}Bellman framework for optimal control in multistage energy systems, S. Sieniutycz Statistics of energy levels and eigenfunctions in disordered systems, A.D. Mirlin The quantum three-dimensional Sinai billiard } a semiclassical analysis, H. Primack and U. Smilansky The control of chaos: theory and applications, S. Boccaletti, C. Grebogi, Y.-C. Lai, H. Mancini and D. Maza Statistical physics of vehicular tra$c and some related systems, D. Chowdhury, L. Santen and A. Schadschneider
322 (1999) 347 323 (2000) 1 323 (2000) 81 323 (2000) 183
324 (2000)
1
326 (2000) 165 326 (2000) 259 327 (2000)
1
329 (2000) 103 329 (2000) 199
The physics of elementary particles and 5elds Experimental tests of the electroweak standard model at high energies, M.W. GruK newald Theories with gauge-mediated supersymmetry breaking, G.F. Giudice and R. Rattazzi Large N "eld theories, string theory and gravity, O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz Compton scattering by nuclei, M.-Th. HuK tt, A.I. L'vov, A.I. Milstein and M. Schumacher Searches for invisible axions, L.J Rosenberg and K.A. van Bibber Geometry of the physical phase space in quantum gauge systems, S.V. Shabanov Boson interferometry in high-energy physics, R.M. Weiner Semilocal and electroweak strings, A. AchuH carro and T. Vachaspati Baryon resonance extraction from nN data using a unitary multichannel model, T.P. Vrana, S.A. Dytman and T.-S.H. Lee 0370-1573/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 0 5 2 - 1
322 (1999) 125 322 (1999) 419 323 (2000) 183 323 325 326 327 327
(2000) 457 (2000) 1 (2000) 1 (2000) 249 (2000) 347
328 (2000) 181
354
Subject index
Phases of dense matter in neutron stars, H. Heiselberg and M. Hjorth-Jensen Superbranes and superembeddings, D. Sorokin Goldstone and pseudo-Goldstone bosons in nuclear, particle and condensed-matter physics, C.P. Burgess Quarks and leptons beyond the third generation, P.H. Frampton, P.Q. Hung, M. Sher
328 (2000) 237 329 (2000) 1 330 (2000) 193 330 (2000) 263
Nuclear physics Microscopic studies on two-phonon giant resonances, C.A. Bertulani, V.Yu. Ponomarev Termination of rotational bands: disappearance of quantum many-body collectivity, A.V. Afanasjev, D.B. Fossan, G.J. Lane and I. Ragnarsson Compton scattering by nuclei, M.-Th. HuK tt, A.I. L'vov, A.I. Milstein and M. Schumacher Boson interferometry in high-energy physics, R.M. Weiner Phases of dense matter in neutron stars, H. Heiselberg and M. Hjorth-Jensen Nuclear deep-inelastic lepton scattering and coherence phenomena, G. Piller and W. Weise
321 (1999) 139 322 (1999)
1
323 (2000) 457 327 (2000) 249 328 (2000) 237 330 (2000)
1
Atomic and molecular physics Tunneling paths in multi-dimensional semiclassical dynamics, K. Takatsuka, H. Ushiyama and A. Inoue-Ushiyama The multicon"guration time-dependent Hartree (MCTDH) method: a highly e$cient algorithm for propagating wavepackets, M.H. Beck, A. JaK ckle, G.A. Worth and H.-D. Meyer Modelling the nano-scale phenomena in condensed matter physics via computerbased numerical simulations, H. Ra"i-Tabar Electron and light emission from island metal "lms and generation of hot electrons in nanoparticles, R.D. Fedorovich, A.G. Naumovets and P.M. Tomchuk Cold target recoil ion momentum spectroscopy: a &momentum microscope' to view atomic collision dynamics, R. DoK rner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer and H. Schmidt-BoK cking
322 (1999) 347
324 (2000)
1
325 (2000) 239 328 (2000) 73
330 (2000) 95
Classical areas of phenomenology (including applications) Femtosecond infrared spectroscopy of semiconductors and semiconductor nanostructures, T. Elsaesser and M. Woerner Hamilton}Jacobi}Bellman framework for optimal control in multistage energy systems, S. Sieniutycz Collective diode dynamics: an analytical approach, A.Ya. Ender, H. Kolinskyu, V.I. Kuznetsov and H. Schamel Statistical physics of vehicular tra$c and some related systems, D. Chowdhury, L. Santen and A. Schadschneider
321 (1999) 253 326 (2000) 165 328 (2000)
1
329 (2000) 199
Fluids, plasmas and electric discharges Dynamics and stability of premixed #ames, V.V. Bychkov and M.A. Liberman
325 (2000) 115
355
Subject index
Collective diode dynamics: an analytical approach, A.Ya. Ender, H. Kolinskyu, V.I Kuznetsov and H. Schamel
328 (2000)
1
Condensed matter: structure, thermal and mechanical properties Electron focusing in metals and semimetals, J. Heil, A. BoK hm, A. GroK ger, M. Primke, P. Wyder, P. Keppler, J. Major, H. Bender, E. SchoK nherr, H. Wendel, B. Wolf, K.U. WuK rz, W. Grill, H. Herrnberger, S. Knauth and J. Lenzner Phase transitions in liquid crystals, S. Singh Modelling the nano-scale phenomena in condensed matter physics via computerbased numerical simulations, H. Ra"i-Tabar
323 (2000) 387 324 (2000) 107 325 (2000) 239
Condensed matter: electronic structure, electrical, magnetic and optical properties Femtosecond infrared spectroscopy of semiconductors and semiconductor nanostructures, T. Elsaesser and M. Woerner Electron focusing in metals and semimetals, J. Heil, A. BoK hm, A. GroK ger, M. Primke, P. Wyder, P. Keppler, J. Major, H. Bender, E. SchoK nherr, H. Wendel, B. Wolf, K.U. WuK rz, W. Grill, H. Herrnberger, S. Knauth and J. Lenzner Statistics of energy levels and eigenfunctions in disordered systems, A.D. Mirlin Electron and light emission from island metal "lms and generation of hot electrons in nanoparticles, R.D. Fedorovich, A.G. Naumovets and P.M. Tomchuk Goldstone and pseudo-Goldstone bosons in nuclear, particle and condensed-matter physics, C.P. Burgess
321 (1999) 253
323 (2000) 387 326 (2000) 259 328 (2000) 73 330 (2000) 193
Cross-disciplinary physics and related areas of science and technology Instabilities in crystal growth by atomic or molecular beams, P. Politi, G. Grenet, A. Marty, A. Ponchet and J. Villain Dynamics and stability of premixed #ames, V.V. Bychkov and M.A. Liberman Modelling the nano-scale phenomena in condensed matter physics via computerbased numerical simulations, H. Ra"i-Tabar Collective diode dynamics: an analytical approach, A.Ya. Ender, H. Kolinskyu, V.I. Kuznetsov and H. Schamel
324 (2000) 271 325 (2000) 115 325 (2000) 239 328 (2000)
1
321 (1999)
1
Geophysics, astronomy and astrophysics Galaxy collisions, C. Struck Gravitational radiation from Schwarzschild black holes: the second-order perturbation formalism, R.J. Gleiser, C.O. Nicasio, R.H. Price and J. Pullin The Blandford}Znajek process as a central engine for a gamma-ray burst, H.K. Lee, R.A.M.J. Wijers and G.E. Brown Origin and propagation of extremely high-energy cosmic rays, P. Bhattacharjee and G. Sigl Phases of dense matter in neutron stars, H. Heiselberg and M. Hjorth-Jensen
325 (2000) 41 325 (2000) 83 327 (2000) 109 328 (2000) 237
