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Phase transition over the gauge group center and quark confinement in QCD
Volume 101B, number 6
PHYSICS LETTERS
28 May 1981
PHASE TRANSITION OVER THE GAUGE GROUP CENTER AND QUARK CONFINEMENT IN QCD S.B. KHOKHLACHEV Landau Institute for Theoretical Physics, Moscow, USSR and Yu.M. MAKEENKO Institute for Theoretical and Experimental Physics, Moscow, USSR Received 28 April 1980
A lattice gauge model with the phase transition corresponding to spontaneous breakdown of the group center symmetry is considered. The possible continuum limit in a phase with permanently confined quarks is discussed.
A promising approach to the problem of quark confinement in QCD is based on the symmetry of the gauge group center [1,2]. An adequate tool in attacking this problem are Wilson lattice gauge theories [3]. Although static quarks are confined in the framework of strong-coupling expansions, there remains a serious question - do confinement and asymptotic freedom, observed in perturbation theory, coexist in the same phase of the theory? 1. Recently, there were suggested [4,5] models with a phase transition corresponding to spontaneous breakdown of the symmetry o f the gauge group center. In the present paper, we study a model with the same property. The lattice action of our model consists o f two terms Action(U0p)--ff'~JRe p
,
)
tr Uap +~-,~ltr Uapl 2 . (1)
Here the first one is a standard trace of the product over four elements, U t, of SU(N) along the boundary ap of plaquette p. The sum over p runs over all plaquettes on the lattice. Many properties o f the model (1) can be understood analysing the limit o f large N, i.e. N ~ oo for/3 and X fixed. In the present paper, we find the phase diagram in the 13, X plane. The obtained phase transi-
tion corresponds to spontaneous breakdown of the symmetry o f the gauge group center. 2. Let us first discuss briefly some properties of our model. In the naive local limit (the lattice spacing a --* 0 at/3, X fixed), both terms in the action (1) reduce to the usual continuum Yang-Mills action:
Action(Uap)+-l(--flN+~)fd4xtrFJ,(x
).
(2)
However, they are different in the lattice case. The second term is invariant under the transformation of
UI~ZIUI,
whereZt=exp[(21ri/N)nt] ~ Z N
(3)
takes values in the group Z N. Such a transformation generalizing local gauge invariance is to be called the gauge transformation of the third kind. It is evident that only quantities which are invariant under (3) do not vanish when averaging at 13= 0. For any N, our model can be solved at small X by means of perturbation theory. It is easy to check that U l = Z l S x S x l l is a solution t o the classical lattice equation o f motion. To zeroth order in X, the Wilson loop average
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / S 02.50 © North-Holland Publishing Company
403
Volume 10IB, number 6
PHYSICS LETTERS
w,,(Cl ..... c . ) fDtl ( U ) N - 1 tr UC, ... N - 1t r UCn exp [Action (Uap)]
f Dp (U)exp [Action (Uap ) ] (4) where DO(U) is the Haar measure on SU(N) and Uc stands for the ordered product of UI over l E C, reduces to the sum over these classical solutions:
w(.°)(Cl .... , c . ) = r ( G + ... + c . ; ~ )
EZ I~Z NZcI +... +Cn exp(/3EpRe Zap )
Zc = tPc zt "
(5)
Using the loop equation of motion, we shall show below how eq. (5) gets modified for any X. Equations of motion for the loop averages can be derived by a standard trick of shifting variables in the functional integral (4). The derivation of equations resuiting from the infinitesimal variation U! ~ (I + ~I)UI, with el being the SU(N)-matrix, is quite similar to that in the standard model [6]. One gets
28 May 1981
with ~ = ~/co(Sp). It is a remarkable fact that eq. (8) has coincided, after the redefinition of the coupling constant, with the equation for the loop average in the standard model. Consequently, co(C; ~ ) i s equal to the loop average in the standard SU(N)gauge theory with X as coupling constant. The factor I'(C) in eq. (7) is not fixed by eq. (6) and remained up to now arbitrary. To fix F(C), let us derive the equation resulting from the variation of the type (3). In the case of large N, one can substitute the group U(I) for ZN, i.e. eq. (3) takes the foml U/ exp fie/) UI, where eI is a real number. Now, a standard trick may be used and we arrive at the following equation /3 ~ I W n + l ( C 1 :- p -
ap, C/ ..... Cn)
w , + l ( c ! ..... - a p . c i ..... c , ) l
= ~ ~xyT.(l)w,,(Cl ..... c . ) . I~C i
(9)
Substituting herein the factorized ansatz (7), one gets
the equation for F(7)
~X~p IWn+I(CI ..... -Bp, C i + ~p ..... C,,)
L ~ ( x ) r ( C ) = e2t~ec 8 x y r , ( O r ( c ) + O ( N - 2), -- Wn+l(Cl,
...,
ap, Ci
-
~p
(10)
..... Cn)]
= l~_Ci6XYru(l) Wn+l(Cl ..... Cxy, Cyx ..... C n ) + 0(N-2),
(6)
where Ep runs over six plaquettes,p, built on the given link beginning at point x E Ci. The point y is defined as the beginning of I E Ci, if the direction of l is positive, and as its end for the negative one. Eq. (6) is valid as N ~ ~ for fixed/3 and possesses to order O(N -1) a factorized solution
with e -2 =/3" ¢o(~)p). Eq. (10) coincides with the abelian one. Hence, F(C;e -2) is equal to the loop average in the U(1) gauge theory given by eq. (5) (with the redefined coupling constant e - 2). The substitution of ?t = 0 into eq. (7) leads us to eq. (5). It is also straightforward that the factorized solution (7) is a consequence at small ;',, from the factorization within the perturbation theory, valid for large N [7]. However, our method gives the possibility to extend the factorization at any X.
Wn(C 1 ..... Cn) = F(C 1 + ... + Cn; e- 2)l-i co(Ci; X ) ,
3. The results of the previous section make it possible to conclude that there exists a phase transition over/3, the critical value being
providing co(C;X) satisfies the following equation
/3c l(x) -- e~ co(ap,-~),
n
i=I
(7)
1
L,,(x)co(c) - ~ ~ [co(C + ap) - co(C - ap)l p
= - X ~ 8xyrv(1)co(Cxy)co(Cyx) + O ( N - 2 ) , l~C 404
(8)
(l ' )
2 where e c is the critical charge of the U(I)gauge model [8] . The dependence of ¢v(~)p;X) on X is governed by eq. (8). Unfortunately, the above consideration is not a
Volume 101B, number 6
PItYSICS LETTERS
rigorous proof of the phase transition because the asymptotic behavior of co(C) is unknown. To convince that our model does undergo a phase transition at/3 = tic(X) and to establish what symmetry breaks down, one has to consider the asymptotics of the disorder parameter. In our case, it is more convenient to use another criterion of the symmetry, which is closely related to the 't Hooft disorder parameter [2]. Our criterion is based on a phenomenon analogous to that of Higgs for abelian chiral theories. Let us turn on an external field making the symmetry (3) local, i.e. consider Action [Uapexp(iGp)], where the external field, Gp, is attached to the oriented plaquette, p. Then the expansion of the free energy, F(Gp), defined as
exp[-F(Gp)] fO# (U)exp {Action [Uapexp (iGp)]) fD#(U)exp[Action(Uap)]
e x p l - F ( G p ) ] = const. dA I - rt
p
AI+
Gp)1.
\lEap
(13) Thus, the dependence of F(Gp) upon Gp is the same for our model as N ~ ~ and for the U(1) gauge theory with coupling constant e - 2 = ~o)(Op):
F(Gp; 13,X) = FU(1)(Gp;e- 2 ) .
gauge theories is the following. Let the lattice spacing a approach to zero and the bare charges to their critical values, keeping the correlation lengths fixed. If this is the case then the obtained continuum theory will be Lorentz-invariant. In order to discuss the possibility of such a limiting procedure, let us consider the phase diagram in the X, ti- 1 plane. The phase diagram is determined by eq. (1 l), where the constant X = X/o)(~p;X) in turn depends on c o o p ; X), so that tile self-consistency condition arises:
(15)
6o(C;'X )lC=ap = X/X .
Here O)(C;X)is the loop averageir; the standard model with coupling constant X. For small X, o)(C; ~ ) is determined by perturbation theory expansion. For large X, the strong-coupling expansion yields o ) ( a p ; ~ ) = l/2X +
(12)
in powers of Gp is different for the phase with spontaneous breakdown of the center group symmetry and for the symmetric one. The calculation of F(Gp) for our model can be performed in the large N limit. Expanding the rhs of eq. (12) in/3 and using eq. (7), one finds
X f /[-I +T-~-zrzr exp[tio)(ap)~cos(~
28 May 1981
I/8~ s + O ( ~ - 9 ) ,
(16)
in accordancewith the generalrequirement,do) (ap; X) x d~ ~<0. Given o)(ap;X), the self-consistency condition (15) allows one to find the function ~(X). For small X the perturbation solution holds, whereas for ~ ~< k ~ k c one more solution appears, which is, however, unstable because d~(X)/dk < 0. Thus, the phase diagram in the X, ti- 1 plane has a shape as shown in fig. 1. All the above results were obtained in the limit N oo at ti, X fixed. One can show that for X > Xc the correlation length becomes dependent on N at finite N and vanishes as N ~ oo. This implies that there exist such field configurations only when a quark and its antipartner reside in the same site on the lattice, which we call absence of quarks. This phase transition
(14)
Since the U(1) gauge theory undergoes the phase transition at e 2 = e 2 the same is true for model (1) at /3 = l~c(h) and, according to our criterion, this phase
transition corresponds to spontaneous breakdown of the gauge group center symmetry. 4. Our above results were based on the lattice gauge model. It is interesting to discuss the possibility of a continuum limit in the confining phase. A ususal way of constructing the local limit of lattice
Phase 1
e~! t.
Phase~ " ~ .
.
.
.
.
.
f l
~_¢
Absence of quarks L
½ xc
Fig. 1. Phase diagram corresponding to spontaneous breakdown of the center of SU (N) for large N.
405
Volume IOIB, number 6
PItYSICS LETTERS
depicted in fig. 1 by the vertical dashed line, is an artifact of the limit N = 0% like the Gross-Witten phase transition in the two-dimensional model. The obtained phase diagram o f fig. 1 allows us to come to some conclusions about the possibility of a Lorentz-invariant continuum limit in the confining phase. One should let ), tend to zero according to the asymptotic freedom law and/3 to its critical value e c 2 -- 0, keeping fixed the correlation lengths in F(C) and ~ ( C ) when a ~ 0. Since the U ( I ) lattice gauge theory has a second-order transition, confirmed by computer calculations, the Lorentz-invariance of the continuum limit is guaranteed. The local limit obtained in this way has two correlation lengths, their relation being a free parameter. The correlation length in P ( C ) i s the radius of quark confinement due to ZN-excitations of the center of SU(N) and that in w(C) is the radius of charge screening due to gluon excitations. Our local theory with confined quarks guarantees asymptotic freedom over ~. At present the problem con. cerning the renormalization properties of the Z N gauge theory when 13~ ec 2 - 0 has not been studied. It is
406
28 May 1981
evident that both this problem and the properties of QCD with two correlation lengths are of great interest and deserve future investigation. In our opinion, mesons composed from quarks and antiquarks in octet representation (whose colour is screened by gluons) could give information about the confinement mechanism. This will be considered elsewhere. We would like to acknowledge helpful discussions with A.A. Migdal, V.L. Pokrovskii, A.M. Polyakov, K.A. Ter-Martirosyan, S.N. Vergeles and A.B. Zamolodchikov.
References [1] [2] [3] [4] [5] [6]
A.M. Polyakov, Phys. Lett. 72B (1978) 477. G. 't Hooft, Nucl. Phys. B138 (1978) 1; B153 (1974) 141. K. Wilson, Phys. Rev. DI0 (1974) 2445. T. Yoneya, Nucl. Phys. B153 (1979) 431. G. Mack and V.B. Petkova, Ann. Phys. 123 (1979) 442. Yu.M. Makeenko and A.A. Migdal, Phys. Lett. 88B (1979) 135; 89B (1980) 437(E). [71 G. 't Hooft, Nucl. Phys. B72 (1974) 461. [8] J.M. Drouffe and C. Itzykson, Phys. Rep. 38C (1978) 133.
PHYSICS LETTERS
28 May 1981
PHASE TRANSITION OVER THE GAUGE GROUP CENTER AND QUARK CONFINEMENT IN QCD S.B. KHOKHLACHEV Landau Institute for Theoretical Physics, Moscow, USSR and Yu.M. MAKEENKO Institute for Theoretical and Experimental Physics, Moscow, USSR Received 28 April 1980
A lattice gauge model with the phase transition corresponding to spontaneous breakdown of the group center symmetry is considered. The possible continuum limit in a phase with permanently confined quarks is discussed.
A promising approach to the problem of quark confinement in QCD is based on the symmetry of the gauge group center [1,2]. An adequate tool in attacking this problem are Wilson lattice gauge theories [3]. Although static quarks are confined in the framework of strong-coupling expansions, there remains a serious question - do confinement and asymptotic freedom, observed in perturbation theory, coexist in the same phase of the theory? 1. Recently, there were suggested [4,5] models with a phase transition corresponding to spontaneous breakdown of the symmetry o f the gauge group center. In the present paper, we study a model with the same property. The lattice action of our model consists o f two terms Action(U0p)--ff'~JRe p
,
)
tr Uap +~-,~ltr Uapl 2 . (1)
Here the first one is a standard trace of the product over four elements, U t, of SU(N) along the boundary ap of plaquette p. The sum over p runs over all plaquettes on the lattice. Many properties o f the model (1) can be understood analysing the limit o f large N, i.e. N ~ oo for/3 and X fixed. In the present paper, we find the phase diagram in the 13, X plane. The obtained phase transi-
tion corresponds to spontaneous breakdown of the symmetry o f the gauge group center. 2. Let us first discuss briefly some properties of our model. In the naive local limit (the lattice spacing a --* 0 at/3, X fixed), both terms in the action (1) reduce to the usual continuum Yang-Mills action:
Action(Uap)+-l(--flN+~)fd4xtrFJ,(x
).
(2)
However, they are different in the lattice case. The second term is invariant under the transformation of
UI~ZIUI,
whereZt=exp[(21ri/N)nt] ~ Z N
(3)
takes values in the group Z N. Such a transformation generalizing local gauge invariance is to be called the gauge transformation of the third kind. It is evident that only quantities which are invariant under (3) do not vanish when averaging at 13= 0. For any N, our model can be solved at small X by means of perturbation theory. It is easy to check that U l = Z l S x S x l l is a solution t o the classical lattice equation o f motion. To zeroth order in X, the Wilson loop average
0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / S 02.50 © North-Holland Publishing Company
403
Volume 10IB, number 6
PHYSICS LETTERS
w,,(Cl ..... c . ) fDtl ( U ) N - 1 tr UC, ... N - 1t r UCn exp [Action (Uap)]
f Dp (U)exp [Action (Uap ) ] (4) where DO(U) is the Haar measure on SU(N) and Uc stands for the ordered product of UI over l E C, reduces to the sum over these classical solutions:
w(.°)(Cl .... , c . ) = r ( G + ... + c . ; ~ )
EZ I~Z NZcI +... +Cn exp(/3EpRe Zap )
Zc = tPc zt "
(5)
Using the loop equation of motion, we shall show below how eq. (5) gets modified for any X. Equations of motion for the loop averages can be derived by a standard trick of shifting variables in the functional integral (4). The derivation of equations resuiting from the infinitesimal variation U! ~ (I + ~I)UI, with el being the SU(N)-matrix, is quite similar to that in the standard model [6]. One gets
28 May 1981
with ~ = ~/co(Sp). It is a remarkable fact that eq. (8) has coincided, after the redefinition of the coupling constant, with the equation for the loop average in the standard model. Consequently, co(C; ~ ) i s equal to the loop average in the standard SU(N)gauge theory with X as coupling constant. The factor I'(C) in eq. (7) is not fixed by eq. (6) and remained up to now arbitrary. To fix F(C), let us derive the equation resulting from the variation of the type (3). In the case of large N, one can substitute the group U(I) for ZN, i.e. eq. (3) takes the foml U/ exp fie/) UI, where eI is a real number. Now, a standard trick may be used and we arrive at the following equation /3 ~ I W n + l ( C 1 :- p -
ap, C/ ..... Cn)
w , + l ( c ! ..... - a p . c i ..... c , ) l
= ~ ~xyT.(l)w,,(Cl ..... c . ) . I~C i
(9)
Substituting herein the factorized ansatz (7), one gets
the equation for F(7)
~X~p IWn+I(CI ..... -Bp, C i + ~p ..... C,,)
L ~ ( x ) r ( C ) = e2t~ec 8 x y r , ( O r ( c ) + O ( N - 2), -- Wn+l(Cl,
...,
ap, Ci
-
~p
(10)
..... Cn)]
= l~_Ci6XYru(l) Wn+l(Cl ..... Cxy, Cyx ..... C n ) + 0(N-2),
(6)
where Ep runs over six plaquettes,p, built on the given link beginning at point x E Ci. The point y is defined as the beginning of I E Ci, if the direction of l is positive, and as its end for the negative one. Eq. (6) is valid as N ~ ~ for fixed/3 and possesses to order O(N -1) a factorized solution
with e -2 =/3" ¢o(~)p). Eq. (10) coincides with the abelian one. Hence, F(C;e -2) is equal to the loop average in the U(1) gauge theory given by eq. (5) (with the redefined coupling constant e - 2). The substitution of ?t = 0 into eq. (7) leads us to eq. (5). It is also straightforward that the factorized solution (7) is a consequence at small ;',, from the factorization within the perturbation theory, valid for large N [7]. However, our method gives the possibility to extend the factorization at any X.
Wn(C 1 ..... Cn) = F(C 1 + ... + Cn; e- 2)l-i co(Ci; X ) ,
3. The results of the previous section make it possible to conclude that there exists a phase transition over/3, the critical value being
providing co(C;X) satisfies the following equation
/3c l(x) -- e~ co(ap,-~),
n
i=I
(7)
1
L,,(x)co(c) - ~ ~ [co(C + ap) - co(C - ap)l p
= - X ~ 8xyrv(1)co(Cxy)co(Cyx) + O ( N - 2 ) , l~C 404
(8)
(l ' )
2 where e c is the critical charge of the U(I)gauge model [8] . The dependence of ¢v(~)p;X) on X is governed by eq. (8). Unfortunately, the above consideration is not a
Volume 101B, number 6
PItYSICS LETTERS
rigorous proof of the phase transition because the asymptotic behavior of co(C) is unknown. To convince that our model does undergo a phase transition at/3 = tic(X) and to establish what symmetry breaks down, one has to consider the asymptotics of the disorder parameter. In our case, it is more convenient to use another criterion of the symmetry, which is closely related to the 't Hooft disorder parameter [2]. Our criterion is based on a phenomenon analogous to that of Higgs for abelian chiral theories. Let us turn on an external field making the symmetry (3) local, i.e. consider Action [Uapexp(iGp)], where the external field, Gp, is attached to the oriented plaquette, p. Then the expansion of the free energy, F(Gp), defined as
exp[-F(Gp)] fO# (U)exp {Action [Uapexp (iGp)]) fD#(U)exp[Action(Uap)]
e x p l - F ( G p ) ] = const. dA I - rt
p
AI+
Gp)1.
\lEap
(13) Thus, the dependence of F(Gp) upon Gp is the same for our model as N ~ ~ and for the U(1) gauge theory with coupling constant e - 2 = ~o)(Op):
F(Gp; 13,X) = FU(1)(Gp;e- 2 ) .
gauge theories is the following. Let the lattice spacing a approach to zero and the bare charges to their critical values, keeping the correlation lengths fixed. If this is the case then the obtained continuum theory will be Lorentz-invariant. In order to discuss the possibility of such a limiting procedure, let us consider the phase diagram in the X, ti- 1 plane. The phase diagram is determined by eq. (1 l), where the constant X = X/o)(~p;X) in turn depends on c o o p ; X), so that tile self-consistency condition arises:
(15)
6o(C;'X )lC=ap = X/X .
Here O)(C;X)is the loop averageir; the standard model with coupling constant X. For small X, o)(C; ~ ) is determined by perturbation theory expansion. For large X, the strong-coupling expansion yields o ) ( a p ; ~ ) = l/2X +
(12)
in powers of Gp is different for the phase with spontaneous breakdown of the center group symmetry and for the symmetric one. The calculation of F(Gp) for our model can be performed in the large N limit. Expanding the rhs of eq. (12) in/3 and using eq. (7), one finds
X f /[-I +T-~-zrzr exp[tio)(ap)~cos(~
28 May 1981
I/8~ s + O ( ~ - 9 ) ,
(16)
in accordancewith the generalrequirement,do) (ap; X) x d~ ~<0. Given o)(ap;X), the self-consistency condition (15) allows one to find the function ~(X). For small X the perturbation solution holds, whereas for ~ ~< k ~ k c one more solution appears, which is, however, unstable because d~(X)/dk < 0. Thus, the phase diagram in the X, ti- 1 plane has a shape as shown in fig. 1. All the above results were obtained in the limit N oo at ti, X fixed. One can show that for X > Xc the correlation length becomes dependent on N at finite N and vanishes as N ~ oo. This implies that there exist such field configurations only when a quark and its antipartner reside in the same site on the lattice, which we call absence of quarks. This phase transition
(14)
Since the U(1) gauge theory undergoes the phase transition at e 2 = e 2 the same is true for model (1) at /3 = l~c(h) and, according to our criterion, this phase
transition corresponds to spontaneous breakdown of the gauge group center symmetry. 4. Our above results were based on the lattice gauge model. It is interesting to discuss the possibility of a continuum limit in the confining phase. A ususal way of constructing the local limit of lattice
Phase 1
e~! t.
Phase~ " ~ .
.
.
.
.
.
f l
~_¢
Absence of quarks L
½ xc
Fig. 1. Phase diagram corresponding to spontaneous breakdown of the center of SU (N) for large N.
405
Volume IOIB, number 6
PItYSICS LETTERS
depicted in fig. 1 by the vertical dashed line, is an artifact of the limit N = 0% like the Gross-Witten phase transition in the two-dimensional model. The obtained phase diagram o f fig. 1 allows us to come to some conclusions about the possibility of a Lorentz-invariant continuum limit in the confining phase. One should let ), tend to zero according to the asymptotic freedom law and/3 to its critical value e c 2 -- 0, keeping fixed the correlation lengths in F(C) and ~ ( C ) when a ~ 0. Since the U ( I ) lattice gauge theory has a second-order transition, confirmed by computer calculations, the Lorentz-invariance of the continuum limit is guaranteed. The local limit obtained in this way has two correlation lengths, their relation being a free parameter. The correlation length in P ( C ) i s the radius of quark confinement due to ZN-excitations of the center of SU(N) and that in w(C) is the radius of charge screening due to gluon excitations. Our local theory with confined quarks guarantees asymptotic freedom over ~. At present the problem con. cerning the renormalization properties of the Z N gauge theory when 13~ ec 2 - 0 has not been studied. It is
406
28 May 1981
evident that both this problem and the properties of QCD with two correlation lengths are of great interest and deserve future investigation. In our opinion, mesons composed from quarks and antiquarks in octet representation (whose colour is screened by gluons) could give information about the confinement mechanism. This will be considered elsewhere. We would like to acknowledge helpful discussions with A.A. Migdal, V.L. Pokrovskii, A.M. Polyakov, K.A. Ter-Martirosyan, S.N. Vergeles and A.B. Zamolodchikov.
References [1] [2] [3] [4] [5] [6]
A.M. Polyakov, Phys. Lett. 72B (1978) 477. G. 't Hooft, Nucl. Phys. B138 (1978) 1; B153 (1974) 141. K. Wilson, Phys. Rev. DI0 (1974) 2445. T. Yoneya, Nucl. Phys. B153 (1979) 431. G. Mack and V.B. Petkova, Ann. Phys. 123 (1979) 442. Yu.M. Makeenko and A.A. Migdal, Phys. Lett. 88B (1979) 135; 89B (1980) 437(E). [71 G. 't Hooft, Nucl. Phys. B72 (1974) 461. [8] J.M. Drouffe and C. Itzykson, Phys. Rep. 38C (1978) 133.