Spontaneous chiral symmetry breaking in quantum chromodynamics

Volume 109B, number 1,2

PItYSICS LETTERS

11 February 1982

SPONTANEOUS CHIRAL SYMMETRY BREAKING IN QUANTUM CHROMODYNAMICS Itzhak BARS 1 J.W. Gibbs Laboratory, Department o f Physics, Yale University, New Haven, CT 06520, USA Received 9 March 1981 Revised manuscript received 20 November 1981

On the basis of certain assumptions a conclusion is reached in this paper that, in SU(3)c-QCD with n flavors, the SU(n) L ® SU(n) R chiral symmetry must break down dynamically for any n, except possibly for one theory containing a special number of flavors n = n*. This result is based on the following assumptions: (1) confinement of light as well as ultra heavy quarks inside baryons, (2) decoupling of ultra heavy quarks in the presence of nonperturbative effects of confinement, (3) validity of 't Hooft's anomaly matching conditions, and (4) absence of phase transitions for the indices of baryons at special values of quark masses, where chiral symmetries may undergo phase transitions. The arguments hold with three colors as well as an arbitrary number of colors.

Recent interest in the possibility of composite quarks and leptons has brought new understanding of spontaneous breakdown in chiral theories. The central ideas are 't Hooft's anomaly and decoupling conditions [1 ] for the existence of massless composite fermions, combined with the Weinberg-Witten observation [2] that, in theories with global symmetries, non-singlet composite massless particles are allowed only for spins 0 and 1/2. In QCD, with n massless quark flavors, the anomaly condition requires that the Adler-Bell-Jackiw anomaly of the chiral flavor SU(n)L ® SU(n)R ® U(1) V currents of the quarks must be reproduced by composite massless hadrons [1] ,1. If the chiral symmetry is unbroken the massless hadrons must be spin-l/2 baryons, while if it is spontaneously broken the massless hadrons are clearly Goldstone bosons such as the pion. One would like to show that QCD chooses only the second alternative, thus explaining what has been assumed since more than two decades.

1 Research supported in part by the US Department of Energy under Contract No. EY-76-C-02-3075. *1 This point has been proven as a consequence of symmetry, analyticity and high energy behaviour by Frishman et al. in ref. [4], and Coleman and Grossmann [3].

By using only the anomaly condition, it is possible to argue that, in a QCD-like theory with an even number o f colors, the chiral symmetry for any number of flavors must spontaneously break down. This is because, with an even number of colors, all color singlets must contain an even number of quarks and therefore they cannot be fermions. The anomaly conditions must then be satisfied by massless Goldstone bosons. While this simple argument illustrates the power of the anomaly conditions, it is thought that, by themselves, they are not sufficiently restrictive for odd numbers of colors, such as the real world of QCD with three colors. 't Hooft [1] and others [4] implemented an additional condition called "decoupling" to show that with these two requirements the only possible alternative for QCD is the Goldstone realization of the chiral symmetry in terms of massless spin-0 particles. However, it is not clear that the way that "decoupling" was implemented is entirely justified. The "decoupling" conditions are simply parity doubling conditions for the baryons containing massive quarks. The baryons appear in nontrivial representations of a residual chiral symmetry after one of the quarks is given an explicit mass in the lagrangian. There is no justification that in QCD baryons satisfy such parity doubling conditions. 73

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Therefore, the tentative conclusion of refs. [1,4], that QCD must break its chiral symmetry cannot be accepted without further discussion. This result can be interpreted to mean that the parity doubling alternative for preserving chiral symmetries in QCD cannot be realized. We must still investigate what will happen if the baryons containing heavy quarks do not appear in parity doubled representations of the remaining chiral symmetries. What appears to be a reasonable assumption is that ultra heavy quarks must decouple anyway. This is known to be the case to all orders of perturbation theory, and perhaps also true despite confinement. This effect must become apparent in the baryonic spectrum by having the baryons containing the heavy quarks disappear by becoming ultra heavy. This assumes that heavy quarks remain confined, which seems reasonable according to Wilson loop arguments. We must also argue that the decoupling of these baryons may not occur via vanishingly small effective couplings to hadrons that contain only the massless quarks. This latter scenario would be consistent with unbroken chiral symmetry as well as the Appelquist-Carrazone theorem 15], but we will show that it cannot occur. Thus, if the only remaining possiblity for decoupling is heavy baryons and if there is at least one number of flavors for which SU(n)L ® SU(n)R remains unbroken, then the chiral symmetries must undergo phase transitions at special values of the quark masses as noted by Weinberg [6] Bars and Yankielowicz [7] and elaborated by Preskill and Weinberg [8]. That is, when one or more quarks are given an explicit mass, at a critical value of this mass the residual chiral symmetries can break spontaneously. Below the phase transition the baryons containing massive quarks may remain massless, but after the phase transition they will become massive, even if they are not parity doubled, since chiral symmetry is broken. Thus the decoupling "theorem" will be satisfied without 't Hooft's "decoupling" conditions. Note that this allows the original theory to be unbroken when all the quarks are massless, thus arriving at a conclusion different than refs. [1,4]. We want to investigate in more detail such circumstances. Now let us review what can be learned exclusively from anomalies without any recourse to "decoupling';. Let us consider all the baryonic representations of SU(n)L ® SU(n)R ® U(1)V constructible as color singlets out of three quarks. These are shown in fig. 1 74

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(e,.) (§ , i ) ( e , _ ) ( °, (o, (EP, i )(,e3)

( ( j, ,° ) (B, ( J,

Fig. 1. Baryons constructed out ot three quarks in all possible representations of the chiral group, together with their indices. together with their indices as defined by 't Hooft [1 ]. The index is equal to the number of left.handed minus the number of right-handed helicity states for a given representation. The anomaly of the quarks must be matched by the baryons if the theory is in the Wigner mode. There are two conditions to be imposed; the first is for the [SU(n)L ] 3 anomaly and the second is for the U(1) V [SU(n)L]2 anomaly. The U(1)V quantum numbers correspond to the usual baryon quantum numbers. These conditions, as given by 't Hooft are (n>2) 3 = 21+~_ -

(n +- 3)(n +_ 6)11± + 1 ~ n ( n +- 1)12+ 2+,_ + 4)12_+ + (n 2 - 9 ) l 3 ,

~n(n

(1)

( n + 2 ) ( n + 3)11± + ½ + , ~ n ( n + 1)12±

1 =1 ~ 2+7--

-

-

--

n(n +- 2)12-+ + (n 2 - 3 ) l 3 .

- ~

(2)

+7--

When n = 2 only the second condition is imposed. There are solutions [1 ] to these equations except when n is a multiple of 3. We will n o t assume that the l 1 are independent of n. The most general solution is 1

1

11+=~[7(1 ~n)-nx

+ l 3 - ½(ny + 3x + l)] ,

l 1 _ = ~ [](1 T- n) - nx + 13 + ~(ny + 3x +- 1)] ,

(3)

1

12+ = ~ [~(1 T-n) -- nx - y + 3l 3 -- ~(ny + 9x +-3)] , 12_ = ½ [4(1 T-n) -- nx - y + 3l 3 + ½(ny + 9x +- 3)] , where the upper or lower sign is taken to insure that ](1 T-n) is an integer, and x , y , l 3 are arbitrary integers except as follows: (a) n = even, x = odd, y = even, (b) n = odd, x = even,y = odd, and l 3 must be taken odd or even to insure integer solutions for l 1±, 12±. We must note, however, that the spin-l/2 baryons that we see in nature are correctly classified by taking l 3 = 1 and

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I1+- = 12_+= 0 for all observed numbers of flavors n. This cannot satisfy the anomaly equations (1), (2) and already requires that the chiral symmetry must be broken without any further arguments. We will pretend below that we have no hints from experiment about the possible solutions, since we want a purely theoretical conclusion. It will be useful to generalize 't Hooft's analysis when several quarks become heavy. Consider giving equal intrinsic masses m to a subset of k quarks, leaving the remaining ones massless. The residual symmetry is SU(n - k)L ® SO(n - k)R ® SU(k)v ® [U(1)vl 2. Some baryons containing the massive as well as massless quarks may remain massless for small enough values of m. Let us, in fact, decompose the baryonic representations in fig. 1 with respect to the residual symmetry group and identify the baryons containing one and two massive quarks. This is done in fig. 2. It turns out that the baryons containing three massive quarks come in left-right pairs so that their total index is zero. Similarly, some other possible representations such as (u, n, u) turn out to have total index zero. Note that the indices in fig. 2 are independent of k. If one demands a completely parity doubled set of baryons that contain at least one massive quark then one must set the total indices in fig. 2b and 2c equal to zero. This was 't Hooft's "decoupling" condition. With such conditions it was shown that eqs. (1) and (2) do not admit any solutions. Therefore, to begin with, it is inconsistent to assume that the original SU(n) L × SU(n)R is exact. There is no convincing rea-

a)

(---,i,i ) (.o,.) (§, i,I)(~,_) (o, m, I ) (,~2+) (o,B, I)(£2-)

(I ,~=, I ) (-~,.) (I, I~, I)(-e,_) (m,o, I)(-~2+ ) (fl,o, I)(-,~2_ )

(W,l,I)(~3)

(I,B~,l)(-~3)

(t:l:l, I, O) (,~ I.-,~2 + +,~ 3 )

b) ( B, I, o) (,~_ -,~2_+; 3)

( I, E] ,rn)(-;l_+,~2_-~3)

(o, i, m)( ~, _~z_+e3) c) (o, i, B ) (~a-~z.+~3)

(I .o,~ ) (-e..+~z--e3) (i,o, B )(-~.-+~z+-~3)

Fig. 2. (a) Baryons containing only massless quarks. (b) Baryons containing 1 massive quark. (e) Baryons containing 2 massive quarks.

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son to impose these conditions in QCD and we will not do so in this paper. The set of remaining massless quarks and all the massless baryons in figs. 2a, 2b, 2c must now satisfy the anomaly conditions once again for the residual chiral symmetry. There are now three such conditions: [SU(n - k)L ] 3, U(1)V1 [SU(n - k)L ] 2 and U(1)v 2 [SU(n - k)L ] 2. The first U(1)V 1 is the usual baryon number while the second U(1)v 2 will be taken as the number of massive quarks represented by the matrix

We find that, after using eqs. (1), (2), these conditions become identities and give no further constraints on the indices for any value of k. It is known [see eq. (3)] that in QCD with three colors, it is impossible to satisfy the anomaly condition when the number of flavors is a multiple of 3. Therefore, in these cases chiral symmetry must break. For the remaining numbers of flavors n = 2, 4, 5, 7, 8, 10, 11, 13, 14 .... one can find many solutions, as explicitly given in ref. [8] and eq. (3). Preskill and Weinberg argued that the chiral symmetries may remain unbroken for these values of n, though not for any two consecutive values of n. Elaborating on their argument I will now show that there could possibly be only one critical value of n for which chiral symmetry may remain unbroken; for all other values of n it must spontaneously break down. The simple argument below is independent of the number of colors (N), so it will apply to all SU(N)c QCD-like theories in which left and right handed quarks are in the fundamental representation N of color SU(N). Below, such theories will be referred to as "QCD" *2. In this theory we know that it is possible to satisfy the anomaly equations by certain baryonic representations of the chiral group as in eq. (3). The existence of such solutions for certain numbers of flavors still does not decide whether chiral symmetry remains unbroken for those cases but only that it may escape spontaneous breaking. My approach here will be one of finding contradictions if the chiral sym*2 For large N there are no baryons, therefore chiral symmetry must break, see Coleman and Witten [9].

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metries were indeed unbroken. The argument does not depend on the anomalies and could be given without any reference to them. Let us assume that for some number of flavors n = n* chiral s y m m e t r y is indeed unbroken (Wigner mode). Then the hadrons include massless spin-l/2 "baryons" in various representations of an unbroken SU(n*)L × SU(n*)R ® U ( I ) v symmetry (fig. 1). I will argue that for any other number of flavors n > n* or n < n*, "QCD" can exist only in the Goldstone mode and therefore must break its chiral symmetries dynamically: In the theory with n = n* let k quarks be given equal intrinsic masses m while the rest remain massless. When this mass parameter is small (compared to AC) it is not ruled out that some of the baryons containing the massive quarks may still remain massless due to the residual chiral symmetry SU(n* - k)L ® SU(n* - k)R ® SU(k)v ® [U(1)V]2 (fig. 2). (As an example, think of the charmed baryons containing a massive charmed quark but that cannot get a mass if the SU(3) X SU(3) chiral symmetry of the u, d, s quarks remains unbroken.) In this theory (n = n*) we assume to have a non parity doubled set of such massless baryons that contain massive quarks. (If the set is parity doubled the remainder of the argument cannot be carried out. Then we must rely on anomalies to argue that in "QCD" with any number of colors, if the set is parity doubled the chiral symmetry has to be in the Goldstone mode anyway [1,4], as we have seen above.) The left-right symmetry also requires that these baryons occur in left and right pairs that would tend to make mass terms, but the mass terms are prevented by the residual chiral symmetry. As we let the mass parameter m increase beyond the strong interaction scale A c and move to infinity, the k massive quarks will decouple and the underlying "QCD" theory will reduce to a n = n* - k flavor theory with a SU(n* - k)L × SU(n* - k)R chiral symmetry. For this decoupling to be manifest in the composite hadronic sector, the mesons and baryons containing the infinitely massive quarks should also become infinitely massive. This can happen only if there is a phase transition that allows the remaining SU(n* - k)L ® SU(n* - k)R chiral symmetry to break spontaneously down to e.g. SUv(n* - k}. But this tells us that in the remaining theory with n = n* - k quarks, all chiral symmetries must be broken dynamically. Note that this breaking is not induced by the presence of the massive quarks, it is decided by the n* - k 76

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quarks all by themselves. Therefore, in "QCD" with n < n* only the Goldstone mode is allowed, since k is arbitrary. Now, we ask whether, for some number of flavors n** larger than n*, "QCD" with unbroken chiral symmetry is allowed? Let's argue that this cannot be true: If it were true, we could apply the above analysis to this theory and conclude that "QCD" withn = n* < n** must be in the Goldstone mode. This contradicts our starting assumption that for n = n* "QCD" exists with unbroken chiral symmetry. Therefore, we must conclude that for both n > n * and n < n * the chiral flavor symmetries of "QCD" are spontaneously broken by a dynamical mechanism. The theory may exist in the unbroken mode f o r only one number o f flavors n = n* ( i f at all) and no other. We emphasize that this argument applies to "QCD" with any number of colors. Returning to QCD with three colors, we have already mentioned that n* could not be a multiple of 3. What we have shown is that among the remaining possibilities n = 2, 4, 5, 7, 8 .... only one value is allowed to exist with unbroken chiral symmetry (Wigner mode). Thus, e.g. if for two flavors chiral symmetry is not broken, then it must be broken for all other numbers of flavors. Of course, n* need not exist al all. In fact, our arguments make it unlikely to exist. The conclusions arrived at above may not hold if the phase transition that must occur as a function of the k massive quarks is of first rather than second or higher order. In this case there may be a lack o f continuity for the indices of the bound states at the transition point. Beyond the transition point the baryons of figs. 2a, b, c may completely dissociate and new ones may form, involving only the massless quarks just as in fig. 2a, but with completely new indices l'l_+, 1½_+,l~ that satisfy the anomaly equations. While this possibility cannot be completely ruled out at present, it seems unlikely for the following two steps. First, we know that when n* - k is a multiple of 3 there will be no solution for any It.',and the symmetry must break as anticipated by the use of the Appelquist-Carrazone theorem, as above. Second, it is hard to imagine that the phase transition and/or the dynamics of decoupling would depend on the number k of massive quarks, as to be sensitive to whether n* - k is a multiple of 3 or not. In any case, since we could not be sure of such arguments, we have listed as one of our assumptions that there is no phase transition for the indices.

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To eliminate the possibility of unbroken symmetry for n = n* (including n* = 2) is not easy, but may not be impossible: Let us consider now a theory with n = n* + k flavors. According to the arguments above the SU(n* + k)L ® SU(n* + k)R symmetry must be dynamically broken. Therefore, the baryons and mesons in this theory now generalyy appear in mixed representations of the chiral group and the baryons can become massive. However, one can still ask the question of how decoupling will be satisfied as the k quarks become ultra massive and the SU(n*)L ® SU(n*)R theory is recovered. If this latter symmetry is exact, there will probably occur a phase transition at a special value of the quark mass in which the SU(n*)L X SU(n*)R subgroup of SU(n* + k)L ® SU(n* + k)R goes from a broken phase to an exact one. Then the baryons must rearrange themselves from mixed representations to pure representations of the chiral group. At this point there will be again baryons containing the heavy quarks, which may not be parity doubled (nonzero index), and thus being prevented from becoming massive by the unbroken SU(n*)L ® SU(n*)R symmetry. This will be contrary to decoupling and thus we would have to conclude that the symmetry must have been broken even for n = n*. To eliminate the possibility of parity doubling, one can try to repeat the anomaly matching argument of 't Hooft for the subgroup SU(n*)L® SU(n*)R after it undergoes the phase transition. What is missing, so far, in completing this analysis is the lack of a relationship between the indices of baryons containing the heavy quarks and those containing only the massless quarks (similar to figs. 2b, c versus 2a). Although one might still expect some relation, the information may be harder to obtain because of the representation mixing that prevailed prior to the phase transition to the unbroken phase. Thus, the argument to eliminate n = n* remains incomplete because the parity doubling possibility has not been discarded. There is an apparent loophole in the arguments of this paper as well as in all previous discussions of this subject: All that is required according to the Appelquist-Carrazone theorem is that in low energy processes of light hadrons the composites containing the ultra heavy quarks should not contribute as intermediate states. Then why doesn't the theory satisfy decoupling by developing vanishingly small couplings between hadrons containing heavy quarks and hadrons containing only light quarks? Then the baryons in figs. 2b, c could

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remain massless without violating either chiral symmetry or decoupling. This would resolve the impass and all previous arguments would collapse. Fortunately, it is possible to show that this scenario could not occur. Consider the anomaly matching equations. If the baryons containing heavy quarks were massless, they would necessarily contribute to the anomaly. In fact, then, the SU(n - k)L ® SU(n - k)R anomaly conditions would automatically be satisfied because they correspond to a rearrangement of the SU(n)L ® SU(n)R anomaly equations. Thus, the contribution of these baryons isnecessarily non-zero if the SU(n - k)L X SU(n - k)R symmetry is to remain unbroken. However, it will be recalled that the value of the anomalyA is obtained from the three-current (axial X vector X vector) Green's functions Fuvx, with momenta k~, k~, qX = _ ( k l + k2)X, as the limit of an absorptive part of the amplitude. That is, there is an imaginary part of the amplitude p(k 2, q2), in terms of which A is completely determined, as can be seen in e.g. Frishman et al. [4]. When k 2 = k22 = k 2, it satisfies

lim p(k 2 q2)=A6(q2).

k2~0

Since A gets a contribution from baryons containing heavy quarks, so does p(k 2, q2) and, therefore, these baryons occur as intermediate states in the three- current Green's function, where the currents are those of SU(n - k)L X SU(n - k)R contructed only from massless quarks. By considering the meson states with the same quantum numbers as the currents, occuring as poles in the variables k 2, q2 at various masses, we deduce that there must be non-vanishing vertices between mesons containing only light hadrons and baryons containing ultra heavy quarks. This is contrary to the hypothesis and we conclude again that decoupling must be satisfied by heavy baryons which require dynamically broken chiral symmetries. An objection to this argument may be that while we are insured that these baryons contribute to p(k2,q 2) at k 2 = q2 = 0, they may not contribute at the meson poles at nonzero values o f k 2, q2. In order to dispel such doubts, it is useful to consider an enlarged theory consisting of QCD, plus gauge particles for SU(n)L X SU(n)R X U(1)v and the necessary "leptons" to cancel the anomalies of this gauge group to maintain renormalizability and decoupling, as in 't Hooft [1]. In this setting, the baryons that contri77

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bute to the anomaly A can and will occur as intermediate states, via the usual triangle and their unavoidable couplings to the SU(n - k)L ® SU(n - k)R gauge bosons. Therefore, the baryons containing the heavy quarks will certainly not decouple from low energy processes involving the low mass hadrons, "leptons" and gauge bosons. This is sufficient to demonstrate that they will conflict with decoupling as a principle. Thus, we should abandon the hypothesis that such baryons could remain massless. The present results have implications for composite theories of quarks and leptons. If one wants to explain the near masslessness of these particles in terms of an underlying unbroken chiral symmetry realized on preons, then one should construct an underlying theory which is somewhat different from multicolor "QCD". Many models, including the Rishon model [10] are on unsafe grounds, since the chiral symmetries, which are defined by neglecting all but hypercolor interactions, are of the type we considered in this paper. However, models which avoid the present arguments by satisfying both the anomaly as well as the parity doubling (or "decoupling") conditions have been constructed [7]. It is encouraging that one finds just the correct quarks and leptons as the only massless composite fermions when SU(3)C × SU(2)W × U(1)W is exact in a rather unique solution based on supergroups [11].

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It is a pleasure to acknowledge many useful discussions with T. Appelquist, A.B. Balantekin, D. Gross, F. Giirsey, Y. Kazama, P. Mannheim, J. Oliensis, M. Peskin, A. Schwimmer, R. Shankar, C. Sommerfeld and S. Weinberg.

References [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11]

G. 't Hooft, Lecture at Cargese Summer Institute (1970). S. Weinberg and E. Witten, Phys. Lett. 96B (1980) 59. S. Coleman and B. Grossmann, private communication. S. Dimopoulos, S. Raby and L. Susskind, Stanford preprint ITP-662 (1980); Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, Weizmann preprint WLS-80-27(1980); A. Zee, Penn preprint (1980); G. Ferrar, CERN preprint TH.2909(1980); R. Barbieri, L. Maiani and R. Petronzio, CERN preprint TH.2900(1980); R. Chanda and P. Roy, CERN preprint TH2923(1980). T. Appelquist and J. Carrazone Phys. Rev. Dll (1975) 2856. S. Weinberg, private communication. I. Bars and S. Yankielowicz, Phys. Lett. 101B (1981) 159. J. PreskiU and S. Weinberg, Texas preprint (Jan. 1981). S. Coleman and E. Witten, Phys. Rev. Lett. 45 (1980) 100. H. Harari and N. Seiberg, Phys. Lett. 98B (1981) 269. I. Bars, Phys. Lett. 106B (1981) 105.