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A0 condensate in high-temperature phase of lattice QCD
Volume 264, number 1,2
PHYSICSLETTERSB
25 July 1991
,4o condensate in high-temperature phase of lattice QCD O.A. Borisenko, V.K. Petrov and G.M. Zinovjev Institutefor TheoreticalPhysics,AcademyofSciencesof the UkrainianSSR, SU-252130Kiev 130, USSR Received 27 December 1990; revised manuscript received 18 April 1991
The phase transition generatedby the breakingof the globalgauge invariance is studied. The relation between the C-symmetry breakdown and the (Ao) condensate is also explored.
One believes that a comprehension of gauge-theory physics and further advances are strongly dependent on our ability to find out which global symmetry is spontaneously broken and which is unbroken. The heuristical importance of this statement is illustrated by the example of the BCS theory of superconductivity. In that approach a condensate of Cooper pairs arising due to the nonperturbative interaction between electrons results in the long-range force associated with gauge symmetry of a system acquiring a nonanalytical dependence on some external parameters. Consequently, this force is transformed to a finite-range interaction. Conventional QCD has been already analyzed in this respect quite creatively and also many strict statements have been formulated in this context [ 1 ]. However, a very intensive study of QCD at finite temperature inspired by the problems of the experimental searches of the quark-gluon plasma has opened many new interesting possibilities for gauge theories, too [ 2 ]. It was noticed in the seminal paper [ 3 ] that in QCD a mechanism of spontaneous breaking of global gauge symmetry different from the Higgs one could originate. Just finite-temperature considerations give rise to this possibility as it leads to compactification in the imaginary-time direction. In other words, a transition to the cylinder topology in space-time occurs if the periodic boundary conditions on the gauge fields a u(J~ , "C) = a u ( x , "c-Jvfl) ,
a,u(X , T) : A a u ( x , 7J)~ a
(1)
are imposed with the period length fl= 1/ T of the inverse temperature. An important consequence of this action is that a constant field Ao which can be singled out by a gauge transformation at T = 0 does not permit to perform the same now without coming into conflict with the boundary conditions ( 1 ) for the spatial components A~(x, 7). It is easy to understand that the boundary conditions for A~(x, 7) will be satisfied only for the gauge transformations commuting with all elements of the global gauge group G, i.e., under the transformations from the centre subgroup Z~ [ 4 ]. This ascertainment becomes especially clear in the lattice formulation [ 5 ]. The boundary conditions generate new physical degrees of freedom which can be taken as the eigenvalues of the Polyakov line [ 3 ]
W ( x ) = T e x p ( ! d z A o ( x , 7))
(2)
(the symbol T stands here for time ordering), and Ao =A~2 a transforms under gauge transformations like a matter field in the adjoint representation. Thus, the group of the Polyakov-loop symmetry is the direct product of the group of global gauge transformations acting in colour space and being space independent (it gives us a chance to be not afraid of Elitzur's theorem) and its centre subgroup. The properties of W(x) under Z~ trans166
0370-2693/91/$ 03.50 © 1991 - ElsevierScience Publishers B.V. ( North-Holland )
Volume 264, number 1,2
PHYSICSLETTERSB
25 July 1991
formations were used as the basic ones to study the deconfinement phase transition in pure Yang-Mills theory at finite temperature. It has been argued in ref. [ 3 ] that the symmetry with respect to G is lost at the same transition, and it has been shown by means of Monte Carlo simulations for lattice gluodynamics in unitary gauge that in the deconfining phase a nonzero expectation value for the component of the gauge field in the compactified direction (Ao) appears. An analysis of the effective potential Veff((Ao)) generated by quantum fluctuations ("Higgs potential" ) has displayed the breakdown of the initial SU (3) symmetry to its abelian subgroups U ( 1 ) × U ( 1 ) at the deconfinement-phase-transition temperature. Two-loop analytical calculations of perturbation theory using the background Feynman gauge have also given a nontrivial minimum for the Higgs effective potential supporting the idea of the generation of the Ao condensate and of the spontaneous breaking of the global symmetry in nonabelian theory at high temperature [ 6,7 ] (though the problem of higher perturbative contributions to an effective potential cautions against excessive optimism regarding this point [ 8 ] ). Interesting attempts to clear up the infrared dynamics in the background [ 9 ] and unitary [ 10 ] gauges, combining perturbative results with nonperturbative inputs, have predicted the same phenomena. Lastly, an important result for our motivation comes from ref. [ 11 ], where a simulation of the pure gauge theory fixing the Landau gauge has been carried out. There it was argued that Ao develops a nonvanishing expectation value above the deconfinement transition temperature and a symmetry breaking pattern was found distinct from that of ref. [ 3 ] which leads to a breakdown of the colour-charge conjugation symmetry. The variety of gauges in the quoted papers is a quite encouraging fact but not finally conclusive [ 12 ]. It has been also conjectured in ref. [ 13 ] that just by virtue of the formation of the temperature-dependent condensate (Ao) the deconfining phase could also be characterized by the dynamical breaking of the colourcharge-reversal (conjugation) symmetry, i.e., it would distinguish between quarks and antiquarks. Such a possibility of the spontaneous breakdown of the colour C-symmetry in the effective model for the Polyakov loops derived in the strong-coupling approximation of lattice QCD has been advocated in refs. [ 14,15 ] and it has been pointed out there that the imaginary part of the Polyakov line appears to be the good order parameter of the associated phase transition. Moreover, an accurate analytical study of the critical region has revealed the practical coincidence of the critical temperature of this transition and the deconfining one. The mechanism of interrelation between the colour C-symmetry breaking and the (Ao)-condensate formation could be simply displayed by the following reasoning [keeping in mind the SU (3) gluodynamics in the unitary gauge]. Then the eigenvalues of the Polyakov lines can be written in the form exp(i{0,) (or = 1, 2, 3 with the condition ~ . {0~= 0 to be imposed), where the substitution
{01=A3o+A~/x//-3, {02=-A3o+A~/x/'3 is implied and the colour conjugation operation is provided with the reversal transformation {0~-, - {0~ [ 14,15 ]. The emergence of the Higgs condensate (Ao) in the deconfining phase [ where both global SU (3) and Z3 symmetries will be broken] could raise three minima of the effective potential for Ao [ 16,3 ]. They can transform one to another by Z3 elements [the product of the 3 × 3 identity matrix and a cubic root of one Z ~ = do exp(~rik), k = 0 , 1, 2] and are located along straight lines where one of the {0~'s is fixed and equal to 0 or _+~n. Let us fix, for instance, {03.Then the two following possibilities occur: case ( 1 ) {01+{01=0. It means that this minimum is C-symmetric under the charge conjugation {0~-{0~ and in accordance with refs. [14,15 ], Im [Tr W] = 0 there; case (2) {01+ {02 -~-2~. These minima are variant under C-conjugation (transform from one to another) and for both Im [Tr W] # 0. Thus, the suggestive possibility ofcolour-charge-conjugation breaking in the high-temperature phase is removed to answer the question whether all three minima of an effective potential (a free energy) as a function of (Ao) are equivalent or that the C-symmetric minimum is deeper and the system gets stuck there. This question together with the others paying attention to the spontaneous breaking of the global gauge symmetry and clarifying the sophisticated nature of the deconfinement phase transition in lattice-QCD thermodynamics are discussed in the present paper. In what follows we are dealing with the staggered fermions in the hamiltonian formulation of lattice QCD at =
167
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finite temperature. As is known, a lattice is implanted in such an approach only for the d spatial directions. The calculations are performed in the temporal gauge Ao(x, z) = 0 with Gauss' law imposed as a constraint on the physical states and the results are equivalent to the ones obtained using the static temporal gauge [ 5 ]. Then the partition function is given as
Z=Tr[Pexp(-flH) ]
(3)
where P is the operator which projects onto the given physical space (the space of locally gauge-invariant states), and the hamiltonian for a lattice with spacing ao in terms of link matrices U and fermion fields ~t is of the form g2
H= ~a~ E E2(x, n)+ ~ x,n
1
E
Tr
p
(Uv+U+)+ ~ ~+Dxy(U)~y+mq ~ ql+qlx x,y
x
(4)
if the quark mass is mq. The second term of eq. (4) representing the chromomagnetic part of gluodynamics indicates the sum over plaquettes Upand D (U) denotes the staggered-fermion matrix (colour indices are omitted) Oxy (U)
= ½
~ ( -- 1 )~' +...+x,( Ux,y(~y,x+n __ Uy,+x6y . . . . ) . n
(5)
The projection operator P can be given by the product of &functions of Gauss' law for the SU (N) gauge group placed at each lattice site:
P=Nfdlz(~)exp[i~o~(~E°'(x,n)-Q~)],
(6)
Q~ = ~+2~q/x.
(7)
Here d/l(~0x) is the invariant measure for SU(N) and 2~ are its diagonal generators, i.e., they belong to the Cartan subgroup. The trace in eq. (3) means the functional integrations and also the summation over the eigenvalues of the o p e r a t o r E 2 and other diagonal operators of the gauge group. Handling some of these operations, we come to the following expression for the partition function:
× ~ -I-[ dxU " (, x ) e. x p (
~ -~~ o ~ Tr( Up + U~-)) Zq,
(8)
where Xt(~), C2(I) are the characters and quadratic Casimir operator eigenvalues of the representation l, respectively, and 7= flg2/2a~. The integration over the quark part of the hamiltonian yields the factor Zq and for the massless quark (for the sake of simplicity let us take it for a short while) the result turns out to read
Zq=Z° det( I+ ~----L'fl no
,
q~=l+exp[iq),~(x)].
(9a,b)
The diagonal part of determinant Z ° resulting from the projection operator is explicitly given [ 17 ] as Nc
Z ° = H I-Iq~ = I-I [l+ReX(~Ox)]. ot=l
x
(10)
x
As is obvious from (9), the high-temperature expansion for the determinant could be readily constructed from the expansion of the logarithmic function. In the course of this simple calculation, one ought to keep in mind that because of the gauge invariance only even powers will remain in the expansion. Then dropping all terms 168
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besides the first one in the high-temperature expansion of Zq, we end up with the following expression for the partition function: 1-I ((,, ~ exp [-yCz(l)] Z = f I~ x d~(~Ox)x,. XZ°exp
-
Tr~ x,n
Xl(~ax)X'[(q~+,)+c.c. )f ~
dU~(x)exp(-
-~o ~ Z Tr (up+Up + ))
~ - 1 gx,x+,,(qx+.) ~p p - l f.... r + ~+. + c . c . . (qx)
(ll)
In the present form eq. ( 11 ) would be viewed as the partition function of an effective lattice theory of the gauge field U,,(x) interacting with the fields ~ , ( x ) defined by the temporal components of the gauge fields Ao3 and A o8 as in the foregoing implicit substitutions. Mathematically, these fields ¢, (x) are analogous to the Higgs ones and realize a mapping of the lattice sites to the vector space where the corresponding abelian subgroup of the initial SU (N) is acting. It means, in particular, that an effective potential for ((p~) could have a minimum in a point away from zero, consequently from finding a non-zero expectation value ((p~). In order to make a real headway in the analytical study of eq. ( 11 ) we employ the strong coupling approximation here. If for ease of calculation and presentation we would employ the partition function ( 11 ) for the SU (2) gauge group, the expectation value ( ~ ) can be found in the strong coupling regime. Following the suggestion of ref. [ 18 ], the expression in the large curly brackets of eq. ( 11 ) can be approximated with high precision by the exponential exp [I(y) cos (ax cos (G+, + [ 0 ' ) sin ~axsin ~ax+, + K ( 7 ) ] , where I, [ a n d K are the well-known functions of 7 given numerically. The value of (~a) is nonzero both for gluodynamics and for the model with quarks in the approximation which is applied here in eq. ( 1 1 ). This result confirms global-gauge-symmetry breakdown declared by ref. [ 3 ], however, it does not bring us nearer to revealing a supposive relation between C-symmetry breaking and the emergence of the ( ~ ) condensate as for SU (2) it is not possible at all. This is the reason why we do not dwell on it here. Starting from SU (3) gluodynamics we find the effective potential V~ff((~)) simplifying the corresponding part of ( 1 1 ) by a mean-field method in the version which is more appropriate for the model under consideration and in fact is very similar to the widely accepted one [ 19 ]. Recalling that now l= (ll,/2) and C2 (l) = ~(ll 1 2+ l 2 -- l~/2) -- 1, we calculate the single-site free energy in the form convenient for the following analysis: - flV~ff= In F ( ( ~ o ) ) =In
[~a(~o)llza-~(~o)] ,
(12)
where for readibility we introduce the denotions #(~o) = sin2( ½(~h -~o2) ) sin2( ½(2~o, +~02) ) sin2( ½(2~z +~Ol) ) , ¢(~a)=exp(9,) ~ c~=l
(--103(~7[O)03(712q~,~)-]O3(ly[(~a)O3(TlO,~)+lO3(ly[q)a)pX> a
+exp(e) O3(}yl0)03 (~,10) + (03 --,02).
(13) 03(1Y[¢,--0~))
(14)
Here Oi(~'l~) are the Jacobi functions and ~,~= ~oa - ~, at Y,~ ~ = 0. Thus ( ~ ) can be found from the local extrema of V~ff. The behaviour of V~ffhas been analyzed numerically at d = 3 and in the interval ),= 1-2 with high precision. In fig. 1 we present the net of the zeroes of the function [/z(~) ]a-1 where the angles between the ~0~, ~2, ~3 axes equal -]n. Everywhere on this net the function In F(~o) becomes infinite. At large ~,> ~,~ (small temperature), the minima of this function are located inside every triangle of this net at the following values: ( 1 ) ((&) =2nk, ( ~ 2 ) = 2 n ( k + ] ) , ( ¢ p 3 ) = 2 z c ( k - ] ) ; (2) ( ( & ) = 2 n k , ( ~ 2 ) = 2 7 r ( k - ~ ) , (tp3)=2zt(k+~), and so on (six combinations are possible). It is obvious that this distribution is invariant under Z3 transformations and we conclude that the system is in the confining phase where ( R e Tr W) = (I.m Tr W) = 0. With y decreasing 169
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Fig. 1.
25 July 1991
Fig. 2.
(temperature increasing) at y = 1.755, three secondary minima of the function In F(fp) are developing, they get deeper and go away. This process is depicted in fig. 2, where we have plotted the ~2 dependence of In F ( ~ ) at fixed (Pl = 3zn and various values of y. One can see that at the same time the initial minimum of In F ( ~ ) is also getting deeper but it develops more slowly and at y= 1.6205 it disappears totally. But still earlier at yc~ = 1.7395 the secondary minima become degenerate with the initial one and the system could undergo the phase transition presumably of the first order (if these minima are stable [ 19] ). Now the eigenvalues of the Polyakov line are not completely degenerate because of the formation of the temperature-dependent "Higgs" condensate ( ~ , ) = f ( y ) signaling in fact the breaking of global gauge symmetry. These newly developed minima just establish the situation when two above discussed possibilities are available and case (2), for example, ~3 = + ] n, ~ + ~2 =-T--]n could lead to the breakdown of the colour-charge-conjugation symmetry or reversal symmetry (a--, - ~ by our definition. It is clear that all phenomena discussed come about at the deconfinement phase transition since above its critical temperature ( T r W) ~ 0 in any minimum. Now one question is to understand the effect of quarks on this strong-coupling picture. Their contribution to an effective potential in the chosen approximation should be given by
-N/3Vq~tr=lnZ°+ln
1 - I d U . ( x ) exp - ~ - ~ a 2 T r E ( q ) - ~ U ( q * ) - ~ U + + c . c . + . . . . x,n
(15)
x,n
The dots represent here contributions of the next even terms of the logarithmic-function expansion. All these terms are negative, therefore we can regard the maximal values of Z ° as corresponding to the minimal value of an effective potential. As above, there are six solutions; however, due to the evidently broken Z3 symmetry they are not equivalent. Now the basic maximum of Z ° is developing at ( ~ ) = (~2) = 0, which allows to settle that C-symmetry is unbroken and the Higgs condensate is absent in this situation for any temperature. It is clear that evaluating the effect of massive.quarks h la Banks and Ukawa [ 17 ], i.e., making a substitution 170
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qx~ -~ exp (flmq) + exp [ i~,~ (x) ] ,
25 July 1991 (16)
we come to the same conclusion. If we consider the system with nonzero chemical potential/t which breaks down the C-symmetry evidently ( ~ - - , ~ _ + i / t ) the main minima appear to be ( ~ ) = 2 n k , (q)2) = 2 n ( k + ~) (the sign here is defined by the sign of/z). Performing the invariant integration in (15) we have up to an irrelevant constant -flVeqff ~ In [ 1 + R e X(~p) ] -
~d(fl/ao)E[Im X(~) ]2/[1
+ R e X(~o) ]2,
(17)
and then combining (12) and (17) with the substitution (16 ) we analyzed the effective potential Vcff(~)= V~ff(~0) + Vqff(~o) at different values o f 9' and mq, and found the following picture. At larger 9' there is one Csymmetrical minimum. With 9' decreasing a phase transition occurs at 7q which is somewhat larger than for pure gluodynamics. Below 9'¢the effective potential develops two minima where colour-charge-conjugation symmetry is broken owing to the appearance o f two condensates ( ~ ) and with mq increasing 9'cq --'9'¢G • Unfortunately, the approximation developed to calculate the fermionic determinant does not give reliable results for the limit mq-,0. It is clear from eq. (17) that in the temperature region ~6d(fl/aa) 2< 1 the second term contributes negligibly to the final result. This fact generates the belief that the higher terms of the high-temperature determinant expansion lead to the same conclusion. We have, thus, found that the chromoelectric sector o f the theory (both without matter fields and with them being switched on) generates a Higgs condensate ( A o ) appearing in the deconfinement phase. Since the field Ao and the relevant Polyakov loop transform nontrivially in the centre-group transformations the condensate carries out the charges o f the centre (as the matter fields). This leads to a quark-colour-charge screening and to a vanishing of the long-range forces, which explains, in fact, the deconfinement mechanism. Concluding, we would like to suggest a quite possible source for the discrepances between our results and the calculations o f ref. [20 ], where ( Im W ) = 0 was found. The latter treats the non-compact finite-difference Q C D formulation obtained from a compact lattice version by expanding Up around the unit element of the group centre only. It could seem that the contribution to the continuum limit comes from a single solution o f (Ao) corresponding to this unit element and the unbroken C-symmetry. However, it was shown [21 ] that there exist gauge-field singular configurations (vortices) such that ZpOp= I, where Zp is a nontrivial element o f the group centre and providing, hence, the m i n i m u m o f S. Thus the continuum limit is given by the contribution o f all elements o f the SU (N) centre and even for the expansion around Zp, the smallness of the coupling constant is not required. Besides, all three minima of the free energy will survive in the limit and, consequently, the possibility o f a spontaneous charge-symmetry breaking will be preserved in continuum. One of us (G.M.Z.) thanks very much F. Karsch, L. McLerran, B. Petersson, J. Polonyi and especially K. Redlich and H. Satz for many enlightening discussions.
References [ 1] C. Vafa and E. Witten, Nucl. Phys. B 234 (1984) 173; Phys. Rev. Lett. 53 (1984) 535. [ 2 ] H. Satz, in: Proc. fourth Intern. Conf. on Ultrarelativistic nucleus-nucleus collisions, ed. K. Kajantie ( 1984); F. Karsch, CERN preprint TH-5498 (August 1989). [ 3 ] J. Polonyi, in: Advances in lattice gauge theory, eds. D. Duke and J. Owens ( 1985); J. Polonyi and H.W. Wyld, University of Illinois preprint ILL-85-83 (April 1985). [ 4 ] K. Kajantie, Helsinki preprint HU-TFT-89-49 (November 1989). [ 5 ] L.A. Averchenkova, O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Kiev prcprint ITP 90-61E (1990). [6] R. Anishetty, J. Phys. G 10 (1984) 439. [7] V.M. Belyaev and V.L. Eletzky, Z. Phys. C 45 (1989) 355. [8] V.M. Belyaev, Phys. Lett. B 241 (1990) 91. [9] K.J. Dahlem, Z. Phys. C 29 (1985) 553. 171
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[10] S. Nadkarni, Phys. Rev. Lett. 60 (1988) 491. [ 11 ] J. Mandula and M. Ogilvie, Phys. Lett. B 201 (1988) 117. [ 12 ] P. Landshoff, Cambridge preprint DAMTP 89-31 ( 1989 ). [ 13 ] G.M. Zinovjev, in: Proc. NATO Advanced Studies Institutes on The nuclear equation of state, ed. W. Greiner (1989). [ 14] O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Phys. Lett. B 221 ( 1989 ) 155. [ 15 ] O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Phys. Lett. B 236 (1990) 349. [ 16] N. Weiss, Phys. Rev. D 24 ( 1981 ) 475. [ 17] T. Banks and A. Ukawa, Nucl. Phys. B 225 (1983) 145. [ 18 ] O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Teor. Mat. Fiz. 80 (1989) 381. [ 19 ] F. Green and F. Karsch, Nucl. Phys. B 238 (1984) 297. [ 20] K. Enqvist, K. Kajantie, L. Karkkainen and K. Rummukainen, Helsinki preprint HU-TFT-90-21 (April 1990). [21 ] R.L. Stuller, Brookhaven preprint BNL-41677 (1989).
172
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,4o condensate in high-temperature phase of lattice QCD O.A. Borisenko, V.K. Petrov and G.M. Zinovjev Institutefor TheoreticalPhysics,AcademyofSciencesof the UkrainianSSR, SU-252130Kiev 130, USSR Received 27 December 1990; revised manuscript received 18 April 1991
The phase transition generatedby the breakingof the globalgauge invariance is studied. The relation between the C-symmetry breakdown and the (Ao) condensate is also explored.
One believes that a comprehension of gauge-theory physics and further advances are strongly dependent on our ability to find out which global symmetry is spontaneously broken and which is unbroken. The heuristical importance of this statement is illustrated by the example of the BCS theory of superconductivity. In that approach a condensate of Cooper pairs arising due to the nonperturbative interaction between electrons results in the long-range force associated with gauge symmetry of a system acquiring a nonanalytical dependence on some external parameters. Consequently, this force is transformed to a finite-range interaction. Conventional QCD has been already analyzed in this respect quite creatively and also many strict statements have been formulated in this context [ 1 ]. However, a very intensive study of QCD at finite temperature inspired by the problems of the experimental searches of the quark-gluon plasma has opened many new interesting possibilities for gauge theories, too [ 2 ]. It was noticed in the seminal paper [ 3 ] that in QCD a mechanism of spontaneous breaking of global gauge symmetry different from the Higgs one could originate. Just finite-temperature considerations give rise to this possibility as it leads to compactification in the imaginary-time direction. In other words, a transition to the cylinder topology in space-time occurs if the periodic boundary conditions on the gauge fields a u(J~ , "C) = a u ( x , "c-Jvfl) ,
a,u(X , T) : A a u ( x , 7J)~ a
(1)
are imposed with the period length fl= 1/ T of the inverse temperature. An important consequence of this action is that a constant field Ao which can be singled out by a gauge transformation at T = 0 does not permit to perform the same now without coming into conflict with the boundary conditions ( 1 ) for the spatial components A~(x, 7). It is easy to understand that the boundary conditions for A~(x, 7) will be satisfied only for the gauge transformations commuting with all elements of the global gauge group G, i.e., under the transformations from the centre subgroup Z~ [ 4 ]. This ascertainment becomes especially clear in the lattice formulation [ 5 ]. The boundary conditions generate new physical degrees of freedom which can be taken as the eigenvalues of the Polyakov line [ 3 ]
W ( x ) = T e x p ( ! d z A o ( x , 7))
(2)
(the symbol T stands here for time ordering), and Ao =A~2 a transforms under gauge transformations like a matter field in the adjoint representation. Thus, the group of the Polyakov-loop symmetry is the direct product of the group of global gauge transformations acting in colour space and being space independent (it gives us a chance to be not afraid of Elitzur's theorem) and its centre subgroup. The properties of W(x) under Z~ trans166
0370-2693/91/$ 03.50 © 1991 - ElsevierScience Publishers B.V. ( North-Holland )
Volume 264, number 1,2
PHYSICSLETTERSB
25 July 1991
formations were used as the basic ones to study the deconfinement phase transition in pure Yang-Mills theory at finite temperature. It has been argued in ref. [ 3 ] that the symmetry with respect to G is lost at the same transition, and it has been shown by means of Monte Carlo simulations for lattice gluodynamics in unitary gauge that in the deconfining phase a nonzero expectation value for the component of the gauge field in the compactified direction (Ao) appears. An analysis of the effective potential Veff((Ao)) generated by quantum fluctuations ("Higgs potential" ) has displayed the breakdown of the initial SU (3) symmetry to its abelian subgroups U ( 1 ) × U ( 1 ) at the deconfinement-phase-transition temperature. Two-loop analytical calculations of perturbation theory using the background Feynman gauge have also given a nontrivial minimum for the Higgs effective potential supporting the idea of the generation of the Ao condensate and of the spontaneous breaking of the global symmetry in nonabelian theory at high temperature [ 6,7 ] (though the problem of higher perturbative contributions to an effective potential cautions against excessive optimism regarding this point [ 8 ] ). Interesting attempts to clear up the infrared dynamics in the background [ 9 ] and unitary [ 10 ] gauges, combining perturbative results with nonperturbative inputs, have predicted the same phenomena. Lastly, an important result for our motivation comes from ref. [ 11 ], where a simulation of the pure gauge theory fixing the Landau gauge has been carried out. There it was argued that Ao develops a nonvanishing expectation value above the deconfinement transition temperature and a symmetry breaking pattern was found distinct from that of ref. [ 3 ] which leads to a breakdown of the colour-charge conjugation symmetry. The variety of gauges in the quoted papers is a quite encouraging fact but not finally conclusive [ 12 ]. It has been also conjectured in ref. [ 13 ] that just by virtue of the formation of the temperature-dependent condensate (Ao) the deconfining phase could also be characterized by the dynamical breaking of the colourcharge-reversal (conjugation) symmetry, i.e., it would distinguish between quarks and antiquarks. Such a possibility of the spontaneous breakdown of the colour C-symmetry in the effective model for the Polyakov loops derived in the strong-coupling approximation of lattice QCD has been advocated in refs. [ 14,15 ] and it has been pointed out there that the imaginary part of the Polyakov line appears to be the good order parameter of the associated phase transition. Moreover, an accurate analytical study of the critical region has revealed the practical coincidence of the critical temperature of this transition and the deconfining one. The mechanism of interrelation between the colour C-symmetry breaking and the (Ao)-condensate formation could be simply displayed by the following reasoning [keeping in mind the SU (3) gluodynamics in the unitary gauge]. Then the eigenvalues of the Polyakov lines can be written in the form exp(i{0,) (or = 1, 2, 3 with the condition ~ . {0~= 0 to be imposed), where the substitution
{01=A3o+A~/x//-3, {02=-A3o+A~/x/'3 is implied and the colour conjugation operation is provided with the reversal transformation {0~-, - {0~ [ 14,15 ]. The emergence of the Higgs condensate (Ao) in the deconfining phase [ where both global SU (3) and Z3 symmetries will be broken] could raise three minima of the effective potential for Ao [ 16,3 ]. They can transform one to another by Z3 elements [the product of the 3 × 3 identity matrix and a cubic root of one Z ~ = do exp(~rik), k = 0 , 1, 2] and are located along straight lines where one of the {0~'s is fixed and equal to 0 or _+~n. Let us fix, for instance, {03.Then the two following possibilities occur: case ( 1 ) {01+{01=0. It means that this minimum is C-symmetric under the charge conjugation {0~-{0~ and in accordance with refs. [14,15 ], Im [Tr W] = 0 there; case (2) {01+ {02 -~-2~. These minima are variant under C-conjugation (transform from one to another) and for both Im [Tr W] # 0. Thus, the suggestive possibility ofcolour-charge-conjugation breaking in the high-temperature phase is removed to answer the question whether all three minima of an effective potential (a free energy) as a function of (Ao) are equivalent or that the C-symmetric minimum is deeper and the system gets stuck there. This question together with the others paying attention to the spontaneous breaking of the global gauge symmetry and clarifying the sophisticated nature of the deconfinement phase transition in lattice-QCD thermodynamics are discussed in the present paper. In what follows we are dealing with the staggered fermions in the hamiltonian formulation of lattice QCD at =
167
Volume 264, number 1,2
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25 July 1991
finite temperature. As is known, a lattice is implanted in such an approach only for the d spatial directions. The calculations are performed in the temporal gauge Ao(x, z) = 0 with Gauss' law imposed as a constraint on the physical states and the results are equivalent to the ones obtained using the static temporal gauge [ 5 ]. Then the partition function is given as
Z=Tr[Pexp(-flH) ]
(3)
where P is the operator which projects onto the given physical space (the space of locally gauge-invariant states), and the hamiltonian for a lattice with spacing ao in terms of link matrices U and fermion fields ~t is of the form g2
H= ~a~ E E2(x, n)+ ~ x,n
1
E
Tr
p
(Uv+U+)+ ~ ~+Dxy(U)~y+mq ~ ql+qlx x,y
x
(4)
if the quark mass is mq. The second term of eq. (4) representing the chromomagnetic part of gluodynamics indicates the sum over plaquettes Upand D (U) denotes the staggered-fermion matrix (colour indices are omitted) Oxy (U)
= ½
~ ( -- 1 )~' +...+x,( Ux,y(~y,x+n __ Uy,+x6y . . . . ) . n
(5)
The projection operator P can be given by the product of &functions of Gauss' law for the SU (N) gauge group placed at each lattice site:
P=Nfdlz(~)exp[i~o~(~E°'(x,n)-Q~)],
(6)
Q~ = ~+2~q/x.
(7)
Here d/l(~0x) is the invariant measure for SU(N) and 2~ are its diagonal generators, i.e., they belong to the Cartan subgroup. The trace in eq. (3) means the functional integrations and also the summation over the eigenvalues of the o p e r a t o r E 2 and other diagonal operators of the gauge group. Handling some of these operations, we come to the following expression for the partition function:
× ~ -I-[ dxU " (, x ) e. x p (
~ -~~ o ~ Tr( Up + U~-)) Zq,
(8)
where Xt(~), C2(I) are the characters and quadratic Casimir operator eigenvalues of the representation l, respectively, and 7= flg2/2a~. The integration over the quark part of the hamiltonian yields the factor Zq and for the massless quark (for the sake of simplicity let us take it for a short while) the result turns out to read
Zq=Z° det( I+ ~----L'fl no
,
q~=l+exp[iq),~(x)].
(9a,b)
The diagonal part of determinant Z ° resulting from the projection operator is explicitly given [ 17 ] as Nc
Z ° = H I-Iq~ = I-I [l+ReX(~Ox)]. ot=l
x
(10)
x
As is obvious from (9), the high-temperature expansion for the determinant could be readily constructed from the expansion of the logarithmic function. In the course of this simple calculation, one ought to keep in mind that because of the gauge invariance only even powers will remain in the expansion. Then dropping all terms 168
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besides the first one in the high-temperature expansion of Zq, we end up with the following expression for the partition function: 1-I ((,, ~ exp [-yCz(l)] Z = f I~ x d~(~Ox)x,. XZ°exp
-
Tr~ x,n
Xl(~ax)X'[(q~+,)+c.c. )f ~
dU~(x)exp(-
-~o ~ Z Tr (up+Up + ))
~ - 1 gx,x+,,(qx+.) ~p p - l f.... r + ~+. + c . c . . (qx)
(ll)
In the present form eq. ( 11 ) would be viewed as the partition function of an effective lattice theory of the gauge field U,,(x) interacting with the fields ~ , ( x ) defined by the temporal components of the gauge fields Ao3 and A o8 as in the foregoing implicit substitutions. Mathematically, these fields ¢, (x) are analogous to the Higgs ones and realize a mapping of the lattice sites to the vector space where the corresponding abelian subgroup of the initial SU (N) is acting. It means, in particular, that an effective potential for ((p~) could have a minimum in a point away from zero, consequently from finding a non-zero expectation value ((p~). In order to make a real headway in the analytical study of eq. ( 11 ) we employ the strong coupling approximation here. If for ease of calculation and presentation we would employ the partition function ( 11 ) for the SU (2) gauge group, the expectation value ( ~ ) can be found in the strong coupling regime. Following the suggestion of ref. [ 18 ], the expression in the large curly brackets of eq. ( 11 ) can be approximated with high precision by the exponential exp [I(y) cos (ax cos (G+, + [ 0 ' ) sin ~axsin ~ax+, + K ( 7 ) ] , where I, [ a n d K are the well-known functions of 7 given numerically. The value of (~a) is nonzero both for gluodynamics and for the model with quarks in the approximation which is applied here in eq. ( 1 1 ). This result confirms global-gauge-symmetry breakdown declared by ref. [ 3 ], however, it does not bring us nearer to revealing a supposive relation between C-symmetry breaking and the emergence of the ( ~ ) condensate as for SU (2) it is not possible at all. This is the reason why we do not dwell on it here. Starting from SU (3) gluodynamics we find the effective potential V~ff((~)) simplifying the corresponding part of ( 1 1 ) by a mean-field method in the version which is more appropriate for the model under consideration and in fact is very similar to the widely accepted one [ 19 ]. Recalling that now l= (ll,/2) and C2 (l) = ~(ll 1 2+ l 2 -- l~/2) -- 1, we calculate the single-site free energy in the form convenient for the following analysis: - flV~ff= In F ( ( ~ o ) ) =In
[~a(~o)llza-~(~o)] ,
(12)
where for readibility we introduce the denotions #(~o) = sin2( ½(~h -~o2) ) sin2( ½(2~o, +~02) ) sin2( ½(2~z +~Ol) ) , ¢(~a)=exp(9,) ~ c~=l
(--103(~7[O)03(712q~,~)-]O3(ly[(~a)O3(TlO,~)+lO3(ly[q)a)pX> a
+exp(e) O3(}yl0)03 (~,10) + (03 --,02).
(13) 03(1Y[¢,--0~))
(14)
Here Oi(~'l~) are the Jacobi functions and ~,~= ~oa - ~, at Y,~ ~ = 0. Thus ( ~ ) can be found from the local extrema of V~ff. The behaviour of V~ffhas been analyzed numerically at d = 3 and in the interval ),= 1-2 with high precision. In fig. 1 we present the net of the zeroes of the function [/z(~) ]a-1 where the angles between the ~0~, ~2, ~3 axes equal -]n. Everywhere on this net the function In F(~o) becomes infinite. At large ~,> ~,~ (small temperature), the minima of this function are located inside every triangle of this net at the following values: ( 1 ) ((&) =2nk, ( ~ 2 ) = 2 n ( k + ] ) , ( ¢ p 3 ) = 2 z c ( k - ] ) ; (2) ( ( & ) = 2 n k , ( ~ 2 ) = 2 7 r ( k - ~ ) , (tp3)=2zt(k+~), and so on (six combinations are possible). It is obvious that this distribution is invariant under Z3 transformations and we conclude that the system is in the confining phase where ( R e Tr W) = (I.m Tr W) = 0. With y decreasing 169
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Fig. 1.
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Fig. 2.
(temperature increasing) at y = 1.755, three secondary minima of the function In F(fp) are developing, they get deeper and go away. This process is depicted in fig. 2, where we have plotted the ~2 dependence of In F ( ~ ) at fixed (Pl = 3zn and various values of y. One can see that at the same time the initial minimum of In F ( ~ ) is also getting deeper but it develops more slowly and at y= 1.6205 it disappears totally. But still earlier at yc~ = 1.7395 the secondary minima become degenerate with the initial one and the system could undergo the phase transition presumably of the first order (if these minima are stable [ 19] ). Now the eigenvalues of the Polyakov line are not completely degenerate because of the formation of the temperature-dependent "Higgs" condensate ( ~ , ) = f ( y ) signaling in fact the breaking of global gauge symmetry. These newly developed minima just establish the situation when two above discussed possibilities are available and case (2), for example, ~3 = + ] n, ~ + ~2 =-T--]n could lead to the breakdown of the colour-charge-conjugation symmetry or reversal symmetry (a--, - ~ by our definition. It is clear that all phenomena discussed come about at the deconfinement phase transition since above its critical temperature ( T r W) ~ 0 in any minimum. Now one question is to understand the effect of quarks on this strong-coupling picture. Their contribution to an effective potential in the chosen approximation should be given by
-N/3Vq~tr=lnZ°+ln
1 - I d U . ( x ) exp - ~ - ~ a 2 T r E ( q ) - ~ U ( q * ) - ~ U + + c . c . + . . . . x,n
(15)
x,n
The dots represent here contributions of the next even terms of the logarithmic-function expansion. All these terms are negative, therefore we can regard the maximal values of Z ° as corresponding to the minimal value of an effective potential. As above, there are six solutions; however, due to the evidently broken Z3 symmetry they are not equivalent. Now the basic maximum of Z ° is developing at ( ~ ) = (~2) = 0, which allows to settle that C-symmetry is unbroken and the Higgs condensate is absent in this situation for any temperature. It is clear that evaluating the effect of massive.quarks h la Banks and Ukawa [ 17 ], i.e., making a substitution 170
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qx~ -~ exp (flmq) + exp [ i~,~ (x) ] ,
25 July 1991 (16)
we come to the same conclusion. If we consider the system with nonzero chemical potential/t which breaks down the C-symmetry evidently ( ~ - - , ~ _ + i / t ) the main minima appear to be ( ~ ) = 2 n k , (q)2) = 2 n ( k + ~) (the sign here is defined by the sign of/z). Performing the invariant integration in (15) we have up to an irrelevant constant -flVeqff ~ In [ 1 + R e X(~p) ] -
~d(fl/ao)E[Im X(~) ]2/[1
+ R e X(~o) ]2,
(17)
and then combining (12) and (17) with the substitution (16 ) we analyzed the effective potential Vcff(~)= V~ff(~0) + Vqff(~o) at different values o f 9' and mq, and found the following picture. At larger 9' there is one Csymmetrical minimum. With 9' decreasing a phase transition occurs at 7q which is somewhat larger than for pure gluodynamics. Below 9'¢the effective potential develops two minima where colour-charge-conjugation symmetry is broken owing to the appearance o f two condensates ( ~ ) and with mq increasing 9'cq --'9'¢G • Unfortunately, the approximation developed to calculate the fermionic determinant does not give reliable results for the limit mq-,0. It is clear from eq. (17) that in the temperature region ~6d(fl/aa) 2< 1 the second term contributes negligibly to the final result. This fact generates the belief that the higher terms of the high-temperature determinant expansion lead to the same conclusion. We have, thus, found that the chromoelectric sector o f the theory (both without matter fields and with them being switched on) generates a Higgs condensate ( A o ) appearing in the deconfinement phase. Since the field Ao and the relevant Polyakov loop transform nontrivially in the centre-group transformations the condensate carries out the charges o f the centre (as the matter fields). This leads to a quark-colour-charge screening and to a vanishing of the long-range forces, which explains, in fact, the deconfinement mechanism. Concluding, we would like to suggest a quite possible source for the discrepances between our results and the calculations o f ref. [20 ], where ( Im W ) = 0 was found. The latter treats the non-compact finite-difference Q C D formulation obtained from a compact lattice version by expanding Up around the unit element of the group centre only. It could seem that the contribution to the continuum limit comes from a single solution o f (Ao) corresponding to this unit element and the unbroken C-symmetry. However, it was shown [21 ] that there exist gauge-field singular configurations (vortices) such that ZpOp= I, where Zp is a nontrivial element o f the group centre and providing, hence, the m i n i m u m o f S. Thus the continuum limit is given by the contribution o f all elements o f the SU (N) centre and even for the expansion around Zp, the smallness of the coupling constant is not required. Besides, all three minima of the free energy will survive in the limit and, consequently, the possibility o f a spontaneous charge-symmetry breaking will be preserved in continuum. One of us (G.M.Z.) thanks very much F. Karsch, L. McLerran, B. Petersson, J. Polonyi and especially K. Redlich and H. Satz for many enlightening discussions.
References [ 1] C. Vafa and E. Witten, Nucl. Phys. B 234 (1984) 173; Phys. Rev. Lett. 53 (1984) 535. [ 2 ] H. Satz, in: Proc. fourth Intern. Conf. on Ultrarelativistic nucleus-nucleus collisions, ed. K. Kajantie ( 1984); F. Karsch, CERN preprint TH-5498 (August 1989). [ 3 ] J. Polonyi, in: Advances in lattice gauge theory, eds. D. Duke and J. Owens ( 1985); J. Polonyi and H.W. Wyld, University of Illinois preprint ILL-85-83 (April 1985). [ 4 ] K. Kajantie, Helsinki preprint HU-TFT-89-49 (November 1989). [ 5 ] L.A. Averchenkova, O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Kiev prcprint ITP 90-61E (1990). [6] R. Anishetty, J. Phys. G 10 (1984) 439. [7] V.M. Belyaev and V.L. Eletzky, Z. Phys. C 45 (1989) 355. [8] V.M. Belyaev, Phys. Lett. B 241 (1990) 91. [9] K.J. Dahlem, Z. Phys. C 29 (1985) 553. 171
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[10] S. Nadkarni, Phys. Rev. Lett. 60 (1988) 491. [ 11 ] J. Mandula and M. Ogilvie, Phys. Lett. B 201 (1988) 117. [ 12 ] P. Landshoff, Cambridge preprint DAMTP 89-31 ( 1989 ). [ 13 ] G.M. Zinovjev, in: Proc. NATO Advanced Studies Institutes on The nuclear equation of state, ed. W. Greiner (1989). [ 14] O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Phys. Lett. B 221 ( 1989 ) 155. [ 15 ] O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Phys. Lett. B 236 (1990) 349. [ 16] N. Weiss, Phys. Rev. D 24 ( 1981 ) 475. [ 17] T. Banks and A. Ukawa, Nucl. Phys. B 225 (1983) 145. [ 18 ] O.A. Borisenko, V.K. Petrov and G.M. Zinovjev, Teor. Mat. Fiz. 80 (1989) 381. [ 19 ] F. Green and F. Karsch, Nucl. Phys. B 238 (1984) 297. [ 20] K. Enqvist, K. Kajantie, L. Karkkainen and K. Rummukainen, Helsinki preprint HU-TFT-90-21 (April 1990). [21 ] R.L. Stuller, Brookhaven preprint BNL-41677 (1989).
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