The chiral phase transition at strong coupling

Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

THE CHIRAL PHASE TRANSITION AT STRONG COUPLING

Frederic GREEN Department of Physics, Northeastern University, Boston, MA 02115, USA

Received 20 June 1983

The transition to a chirally symmetric phase in lattice QCD is investigated with Susskind fermions. In the limit of large coupling and small temperature, the partition function is found to be that of an antiferromagnetic Ising model. The phase transition of the Ising model corresponds to chiral symmetry restoration. It occurs near zero temperature, well below the deconfining transition. The effects of introducing quarks in higher representations and approaching weaker couplings are discussed.

The behavior of QCD with dynamical quarks at finite temperature has received some attention recently [1 ] in light of encouraging numerical calculations of the hadron spectrum [2]. Until now the most impor- tant aspect of this, chiral symmetry restoration (×SR), has been approached numerically [1 ]. As with deconfinement [3-5] , it would be instructive to see how xSR emerges from the strong coupling (SC) expansion of lattice QCD. That is the aim of this paper.

Chiral symmetry breakdown (xSB) at SC and zero temperature is well understood, especially from the hamiltonian viewpoint [6-9] . To zeroth order in SC the vacuum is infinitely degenerate. Higher order terms in the expansion lift this degeneracy such that the lowest energy states, the true vacua, have broken chiral symmetry. The breaking is induced by an effective hamiltonian, derived from a four-fermion interaction, which describes an antiferromagnetic system. The magnetization of that system is identified with the chiral order parameter (~b) .

The finite temperature picture advocated here is a simple extension of the results of refs. [6-9] : ×SR is identified with the loss of spontaneous magnetization at the Curie temperature of the antiferromagnet. The mechanism of the transition is totally unrelated to deconfinement. In fact, in the SC limit xSR is found to occur at a much lower temperature than deconfinement! This is evidently a result of the unphysical limit being taken, in which all the lowest-lying baryon mass-

0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

es go to zero while the excited states attain infinite energy.

Here I will treat the hamiltonian formulation of QCD at SC near zero temperature. Susskind fermions [6,10] are used, mainly because they simplify the physical picture but also because they at least partly solve the doubling problem [11]. Being far from the continuum, I will otherwise ignore that problem.

Thorough explanations of hamiltonian 1 attice gauge theory can be found in refs. [ 10-12 ]. For this paper, the salient points to recall are:

(1) The hamiltonian H is written in terms of SU(N) color electric field variablesEa(r, l)(a = 1, . . . , N 2 - 1) on each site r for each direction l, N X N link matrices U(r, l) E SU(N) and Fermi fields xi(r) (i = 1 . . . . . N) on each site. I will drop the magnetic term since it does not contribute to the order calculated here. For large couplings g2, H is dominated by the electric term (the lattice spacing being set to one).

~2 - g2 - - - f ~ Ea(r, l) 2 • (1)

r,l

The Dirac hamiltonian is treated as a perturbation. (2) At SC the spectrum of physical states is the set

of gauge invariant eigenstates of &2. The lowest lying states have zero flux (&2 = 0), an arbitrary fermionic configuration ~k, and are denoted 10, ~b). They are con- structed [6,10] by defining the state 10) r at each site r such that

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×i(r) lO) r = 0 . (2)

Then defining

i1)r= e . . . . I ×1 . . × b l 0 ) r t l t 2 . . . l N ~ i l i 2 •

(3)

10, (4) ¥

where B (=0 or 1) is the number of times the baryon creation operator has acted at r.

(3) The infinite degeneracy of the states 10, @) is lifted in second order of SC perturbation theory by the effective hamiltonian [6,10],

n e f f = ~ PrPr+l, (5) g2Nc 1 ,

where c 1 = ½(N - l /N) and

Pr = ~. [Xi(r), X!( r)] , PrlB)r =2N(B-})JB)r" (6,7) l

Acting on the states I 0, if), Pr = +-N and their energy E~ is given by

E0; = (0, ¢ lHef f l0 , ~/). (8)

Now consider the finite temperature problem. The thermodynamics of the system is described by the partition function,

Z(/3) = tr e-t~/4 = ~3 (sle-~HIs) = ~ e-~Es , (9) $ g

where/3 is the inverse temperature T -1 and the sum runs over a complete set of physical eigenstates Is) of H. The I 0, ~) degeneracy is present in excited states E s > E~ . This is due to the fact that creation operators [e.g., xt(r)U(r, l)x(r + l)] for these states can be ap- plied to (almost) ,a any of the 10, ff)'s and yield the same unperturbed energy. It is the lifting of such de- generacies which leads to xSB. This occurs in second order in 1/g2. Thus a proper treatment of ×SR would require calculating all the Es'S in eq. (9), i.e. the entire hadron spectrum, to second order. This is an enorm- ously difficult task. However, xSR can be much more easily seen by noting that for large/3 and g2, the ga 2 :~ 0 states have a very large energy. Their Boltzmann factors are proportional to exp(-/3g2cl) , which is very

:1-1 There are two states on which all x'~ U× operators give zero: the one with all p's = +Nand the one with all p's = -N.

small. Thus in the limit/3 -~ ~o (small temperature) and g2 ~ oo such that/3/g2 remains fixed, only the zero flux states contribute in eq. (9). In this limit the partition function becomes (up to a constant),

= (_ 2~ ~_lprPr+l ) (10) Zo {p~=+-N) exp g2(N--2-- 1) r,l

( 8 ~t/V2 ~Oror+l) = ~ exp (11) {0=+1/2} g2( - '~ ' - - 1) r,l

with o r = (1/2N)Or. The path sums in eqs. (10), (11) run over all P or o configurations. Clearly eq. (11) is the partition function of an antiferromagnetic Ising model.

Before proceeding, one must examine the approximation leading to eq. (11) more carefully. Ultimately we will be interested in finding a phase transition from Z 0. Incorporating the ga 2 =/: 0 sector may move the transition to higher temperatures, thus invalidating the /3 -+ o~ approximation. However, note that the physical system described by eq. (10) is superconfining. The quarks are not allowed to separate, even by one lattice unit, since I have set exp(-/3g2c 1) = 0. This situation should tend to break chiral symmetry as strongly as possible. If the ~2 =/: 0 modes are excited, the critical temperature for xSR should decrease. Thus it is physically reasonable that the large/3 approximation will not break down. I will return to this question later.

The antiferromagnetic Ising model [eq. (11)] is well known to possess a phase transition. In fact the partition function is identical with that of the ferromagnetic system, and so the critical temperatures are also identical [13]. The order parameter is the staggered magnetization, defined by

• 1. 3 (~r O(rSt')= lhlmo-~-~)(a}( r O ( s t ' )

X exp ( -~ ' r~. l or or+l + h ~ o (st ) ~ (12) , g ,r ]

where

or(St) = (-1)x+Y+Zor , (13)

Z(h)= {~a} exp(-/3' ~ °r°r+l + ho(St) r,, (14)

and

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~' = 81~[V2/g2(N 2 - 1). (15)

Then

/r~O (st)) = f (~ ' ) , ~' ~./3;,

=0, ~' </3' c , (16)

where f (~ ' ) > 0 (<0) for limh ~ 0 (lim)h ~ 0 is some smooth function of ~'. Since the operator :~r °r (st) breaks translation invariance by odd numbers of lattice sites, this symmetry is broken for/~' > t3' c and is restored for t3' < t3' c. In fact we identify [6]

~ o ( s f ) = _ l ~r pr(_l)x+y+z =_ 1_ ~ [ ~ , ~ 1 . ( 1 7 ) r 2N 2N

Thus a zero staggered magnetization means chiral symmetry restoration:

( ~ [ ~ , ~ ] ) : / : 0 , T < T x S R,

= 0 , T > TxSR, (lS)

where

TxS s = SN2/g2(N 2 - 1)/3 c (19) !

(for the Ising model, ~c = 0.22 [13]). Thus asg 2 -+co, Txs R -+ O. This is a result of the leading orde: degeneracy of the states 10, 4). On the other hand, the deconfining temperature Tdc -+ oo like g2 [3,4]. Hence, as promised, Txs R "< Talc in the SC limit. Furthermore, there is apparently no connection between the phenomena of xSR and deconfinement.

The 3D Ising model has a second order phase transition. Hence at SC the ×SR transition is second order, independently of the group. This is not what has been observed (first order) for SU(3) in Monte Carlo simula- tions [ 1 ]. One possible intriguing explanation for this is that the order of the phase transition changes at some intermediate coupling, perhaps where the xSR and deconfining transitions meet. Certainly reconciling this aspect of SC behavior with Monte Carlo is an im- portant and open question.

The above calculation of TxS R to leading order can also be carried out for quarks in higher dimensional representations of SU(N). Let the quarks transform un- der the Rth representation,

X(R) -+ u(R)x(R) , (20)

where U (R) is a dR-dimensional representation. Then it is easy to show, using the usual second order perturbation, that the effective hamiltonian [analogous to eq. (5)] is

_ 1 ~ o (R) a (R) (21) Heft g2d Re R r,l rr Vr+l

e R is the quadratic casimir in the Rth representation, and

dR O (R) = ~ [x}R)(r), X} R)*(r)]. (22)

i=1

The states [0, ~)are defined similarly, except that the baryon operator is replaced by

e i l i 2 . , . id R X~ 1x~ 2 ... X~idR . (23) Clearly the allowed values of pr (R) are +-d R . Then the partition function is precisely that of eq. (14), but with

~'= 8 d R / g 2 c R .

The critical/~' is of course the same, so one finds for quarks in the Rth representation,

2 T(R) = 8dR/g CR/~e (24) ×SR

The variation of T(R)~ with R is certainly not as dra- X~K matic as in weak coupling [1,11 ]. It also does not necessarily increase with R. For example, in SU(3), in SC,

T(8)xSR > T(3S)R but T (6)XSR < T(3)xSR • (25)

This, like Txs R "~ Tdc , is just another oddity of the 1 (R SC lim't. At least T t¢~ depends on R, and the phase

transition occurs fo)~t'uarks in any representation, regardless of whether or not they are confined.

How does the rest of the partition function (i.e., &2 =/= 0 states) affect this picture of xSR? Suppose we try to include the first excited state in Z:

Z : ~ e x p ( - t S ' ~ PrPr+l) {o} \ r,l

+ exp(-/3g2cl) {p~) exp[(~/g2)f(p)] . (26)

(1/g2)f(p) lifts the degeneracy of the first excited states, and depends on p since we can apply the X t UX

101


operators to (almost ,2) any of the 10, ~)'s. One can then calculate a new effective hamiltonian for p by per- forming a low temperature expansion in exp(-/3g2c 1):

Z ~ ~ exp [-13 'geff(p) ]

where

Ee ff (P) = ~ Pr Pr +l

1 I f(P) + [J' ~, PrPr+l) (27) 13' exp(--13gZcl) exp ~ ~ r , , -

The second term is a small perturbation, and should affect the critical temperature in a small way. The form o f f ( p ) is rather complicated, and its calculation will be described in a subsequent publication [14]. It turns out that the second term in eq. (27) represents a small ferromagnetic interaction. Such an interaction does not break odd-site translation invariance, so its effect is to decrease Txs R as we approach weaker couplings. On the other hand, the increasing mass of the baryons will serve to increase TxS R.

In conclusion, we have seen that in the 13 -+ oo, g2 oo limit with [j/g2 held fixed, the QCD parti t ion func-

tion with Susskind fermions is precisely tha t of an antiferromagnetic Ising model. It has a phase transition corresponding to chiral symmetry restoration. The most surprising aspect of this transition is that it is deep in the confining region, essentially because baryon masses are nearly zero. Obviously chiral symmetry breakdown, as defined here, is not an inevitable con- sequence of confinement. This contradicts some well- known intuitive arguments [ 15,16]. However, as pointed out in ref. [16] such arguments are valid only if helicity and chirality can be identified, which not the case here. For similar reasons, the inequality TxS R />/ 'de [17] does not hold.

It would be interesting to apply this approxima-

t2 See footnote 1. The fact that two states are left out of the second term in eq. (26) does not affect the sequel.

tion to other fermion methods, especially those in which pions are realized as the goldstone bosons of ×SB. It is quite possible that the relevant antiferromagnetic Heisenberg models [ 7 - 9 ] also have phase transitions.

I am indebted to W. Celmaster, R. Cordery, H. Goldberg, R. Gupta and F. Krausz for helpful con- versations. This work is supported in part by a grant from the National Science Foundation.

Note added. After this paper was typed, I received a preprint by Banks and Ukawa [18] which contains many of the same conclusions.

References

[1] J. Kogut et al., Phys. Rev. Lett. 48, 1140 (1982); J. Kogut et al., University of llfinois preprint ILL(TH)- 82-39 (November 1982).

[2] H. Hamber and G. Parisi, Phys. Rev. Lett. 47 (1981) 17921 E. Marinari, G. Parisi and C. Rebbi, Phys. Rev. Lett. 47 (1981) 1795; D. Weingarten, Phys. Lett. 109B (1982) 57.

[3] A.M. Polyakov, Phys. Lett. 72B (1978) 477. [4] L. Susskind, Phys. Rev. D20 (1979) 2610. [5] F. Green, Nucl. Phys. B215 [FS7] (1983) 83. [6] L. Susskind, Phys. Rev. D16 (1977) 3031. [7] B. Svetitsky, S.D. DreU, H.R. Quinn and M. Weinstein,

Phys. Rev. D22 (1980) 490. [8] J. Smit, Nucl. Phys. B175 (1980) 307. [9] J. Greensite and J. Primack, Nucl. Phys, B180 [FS2]

(1981) 170. [10] T. Banks et al., Phys. Rev. DI5 (1977) 1111. [11 ] J. Kogut, University of Illinois preprint ILL(TH)-82-46

(December 1982), and references therein. [12] J. Kogut and L. Susskind, Phys. Rev. Dl l (1975) 395. [13] C. Domb, in: Phase transitions and critical phenomena,

Vol. 3, ed. C. Domb and M.S. Green (Academic Press, New York, 1974).

[14] F. Green,in preparation. [15] A. Casher, Phys. Lett. 83B (1979) 395. [16] T. Banks and A. Casher, Nucl. Phys. B169 (1980) 103. [17] E.V. Shuryak, Phys. Lett. 107B (1981) 103;

R.D. Pisarski, Phys. Lett. l l0B (1982) 155. [18] T. Banks and A. Ukawa, University of Tokyo preprint

Ins. Rep. 465 (t983).

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The chiral phase transition at strong coupling

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