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Weak coupling phase of the Eguchi-Kawai model
Volume 116B, number 6
PHYSICS LETTERS
28 October 1982
WEAK COUPLING PHASE OF THE EGUCHI-KAWAI MODEL
V.A. KAZAKOV and A.A. MIGDAL L.D. Landau Institute for Theoretical Physics, Academy of Sciences, 117334 Moscow, USSR Received 24 June 1982
The weak coupling behaviour of the recently proposed model of Eguchi and Kawai is studied. The effective distribution of eigenvalues of the link matrices is calculated at X = 0. At d > 2 this distribution exhibits a collapse at coinciding eigenvalues, which leads to spontaneous breaking of the U(1) symmetry. As a consequence this model does not describe asymptotically free QCD.
Recently Eguchi and Kawai [1] found an interesting exact solution of the loop equation [2] * 1 in OCD at infinite N. This solution sums up the strong coupling expansion in lattice gauge theory. The validity of this solution at the relevant small couplings was doubtful, due to the presence of the well known first order phase transitions [4] : In this note we study the properties of this resolution in the small coupling region and diagnose the origin of this disease. The modified solution in the weak coupling domain is given in the next paper by one of the authors [5]. The EK solution for the Wilson loop reads
fled. This matrix product depends upon the form of the loop C because the matrices U 1.... , Ud do not commute with each other. In the paper [1] this expansion was shown to satisfy the loop equation provided the following global U(1) symmetry holds
Uu ~ Uuexp(iau) '
au o: 1.
(3)
WEK(C)=Z-lf #~=IdU#exp(Si,)N-ltr(TUti),(1)
Formally the action (2) as well as the Haar measure (1) are U(1) invariant, but one may wonder about spontaneous symmetry breaking. At large X this symmetry is not broken as follows from the direct strong coupling expansion of (1). This expansion differs from the ordinary expansion for the infinite lattice at finite N, but the limit N = ~ is the same. Let us consider the case of small X. Rewriting (2) as
where
S I, = - ~N ~ tr([Uu ' Uu ] [Uu, Uv] #),
d
N
St=- ~ ~
(4)
~v
tr(UuUvU~Utv - I)
(2)
is the reduced Wilson action (for a 1 X 1 X ... × 1 periodic lattice) and dUu is the usual Haar measure. The product over the loop C involves the running index V which takes on 2d values -+1..... +-d corresponding to the successive links of the loop (U_v = U~). + All the matrices U with the same index # are identiVariations of the loop functionals were also considered in ref. [3].
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
we observe that there is an absolute minimum at commuting matrices [Uu, Uv] = 0, i.e. at the diagonal matrices up to an overall rotation. Hence we parametrize these matrices by eigenvalues exp(i¢~) and by the eigenfunctions ~2u
Uu = ~2~exp(i~bu)~u,
f2?u~2u = I,
(5)
~bu = diag(~u1 ... Cu N )-
The matrices [2u will all coincide up to V~. The group measure reduces to the Weyl form 423
Volume 116B, number 6
PHYSICS LETTERS
N
dU~= i__r]ld~b/ufI<] (S/u/)2d~u,
(6)
S~" = sin ~I (q~i - ~b~).
(7)
field, the determinant (12) goes into the denominator for each i/', which yields an effective measure
d//(~bi) = •=lII i=lI-Id~b/~i<[ill" Strictly speaking there are spurious degrees of freedom in I2~, corresponding to the ambiguity in the left multiplication by diagonal matrices exp(i3u). These degrees of freedom have to be eliminated (see below). We may now omit one of the integrations, say dr21, which corresponds to the gauge fixing (timelike gauge ~21 = / ) . Then all the matrices are close to unity. In leading order we parametrize ik = 8 i k + i ( ~ / N ) I / 2 c ° ~" / S ! "k
/.t= 2 .....
+ O(X),
(8)
d.
The factors S~ in the denominator cancel the Weyl factors for/a = 2 ..... d so that d d~2~-1 1-I dU~ #=1 d =const. 1-1. dO~ . btl
(S~/¢)2
l
dco..
(9)
=
28 October 1982
(Sff)2
(13)
Now we observe the following. At d = 2, when the extra factors are absent, the phases ~/u are distributed uniformly on the unit circle. This corresponds to an unbroken U(1) symmetry. The validity of the EK solution at d = 2 for all coupling was demonstrated in the original paper [ 1] by another method. We checked that the usual diagrams with gluon exchange (in the timelike gauge) arise from higher order terms in the perturbation expansion of the EK model at d = 2. However, at d > 2 there is a collapse. The distribution (13) is peaked at coinciding eigenvalues, which leads to spontaneous symmetry breaking of the U(1) symmetry. As is clear from the note added to ref. [1] the spontaneous breaking of U(1) symmetry in the EK model was already noticed by G. Bhanot et al. and by M. Peskin and K. Wilson. The modification of this solution which preserves the U(1) symmetry at small X is considered in the next paper by one of the authors.
As for the action (4), it reduces to the Maxwell form
We are grateful to S.B. Khoklachov, R. Mkrtchian, YuaVl. Makeenko and A.M. Polyakov for discussions.
S r = -2-X
R efere n ce s
u~ q
ISu~ v - -~-u
'
w 0 = 0. (10)
Note that w~k is off diagonal. The diagonal elements of co/~k would correspond to the above mentioned spurious degrees of freedom, which have to be eliminated. The calculation of the gaussian integral over co2 ..... t~ d reduces now to the diagonalization of the (d - 1) X (d - 1) matrix (at fixed i , j which we omit) Rab = (S 2 + S2)~ab - SaSb.
(11)
In virtue of the rotational symmetry this problem is trivial. There is one eigenvalue S 2 and (d - 2) eigenvalues S 2 = S 2 + S2,as is clear for S a = (S 2, 0 ..... 0). Hence det(Rab) = $2($2) d - 2 .
(12)
Since each w~" for i < j is an independent complex 424
[1] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063. [2] Yu.M. Makeenko and A.A. Migdal, Phys. Lett. 88B (1979) 135, 89B (1980) 437(E). [3] J. Gervais and A. Neveu, Phys. Lett. 80B (1979) 255; Y. Nambu, Phys. Lett. 80B (1979) 372; A.M. Polyakov, Phys. Lett. 82B (1979) 247; A.A. Migdal, talk seminar on Quantum solitons (Leningrad, 1978), unpublished; T. Eguchi, Phys. Lett. 87B (1979) 91; D. Foerster, Phys. Lett. 87B (1979) 87; Nuel. Phys. B170 (1980) 107; D. Weingarten, Phys. Lett. 87B (1979) 97; 90B (1980) 277. [4] M. Creutz and K. Moriarty, Phys. Rev. D25 (1982) 610; F. Green and S. Samuel, Nuel. Phys. B194 (1981) 107. [5] A.A. Migdal, Phys. Lett. 116B (1982) 425.
PHYSICS LETTERS
28 October 1982
WEAK COUPLING PHASE OF THE EGUCHI-KAWAI MODEL
V.A. KAZAKOV and A.A. MIGDAL L.D. Landau Institute for Theoretical Physics, Academy of Sciences, 117334 Moscow, USSR Received 24 June 1982
The weak coupling behaviour of the recently proposed model of Eguchi and Kawai is studied. The effective distribution of eigenvalues of the link matrices is calculated at X = 0. At d > 2 this distribution exhibits a collapse at coinciding eigenvalues, which leads to spontaneous breaking of the U(1) symmetry. As a consequence this model does not describe asymptotically free QCD.
Recently Eguchi and Kawai [1] found an interesting exact solution of the loop equation [2] * 1 in OCD at infinite N. This solution sums up the strong coupling expansion in lattice gauge theory. The validity of this solution at the relevant small couplings was doubtful, due to the presence of the well known first order phase transitions [4] : In this note we study the properties of this resolution in the small coupling region and diagnose the origin of this disease. The modified solution in the weak coupling domain is given in the next paper by one of the authors [5]. The EK solution for the Wilson loop reads
fled. This matrix product depends upon the form of the loop C because the matrices U 1.... , Ud do not commute with each other. In the paper [1] this expansion was shown to satisfy the loop equation provided the following global U(1) symmetry holds
Uu ~ Uuexp(iau) '
au o: 1.
(3)
WEK(C)=Z-lf #~=IdU#exp(Si,)N-ltr(TUti),(1)
Formally the action (2) as well as the Haar measure (1) are U(1) invariant, but one may wonder about spontaneous symmetry breaking. At large X this symmetry is not broken as follows from the direct strong coupling expansion of (1). This expansion differs from the ordinary expansion for the infinite lattice at finite N, but the limit N = ~ is the same. Let us consider the case of small X. Rewriting (2) as
where
S I, = - ~N ~ tr([Uu ' Uu ] [Uu, Uv] #),
d
N
St=- ~ ~
(4)
~v
tr(UuUvU~Utv - I)
(2)
is the reduced Wilson action (for a 1 X 1 X ... × 1 periodic lattice) and dUu is the usual Haar measure. The product over the loop C involves the running index V which takes on 2d values -+1..... +-d corresponding to the successive links of the loop (U_v = U~). + All the matrices U with the same index # are identiVariations of the loop functionals were also considered in ref. [3].
0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
we observe that there is an absolute minimum at commuting matrices [Uu, Uv] = 0, i.e. at the diagonal matrices up to an overall rotation. Hence we parametrize these matrices by eigenvalues exp(i¢~) and by the eigenfunctions ~2u
Uu = ~2~exp(i~bu)~u,
f2?u~2u = I,
(5)
~bu = diag(~u1 ... Cu N )-
The matrices [2u will all coincide up to V~. The group measure reduces to the Weyl form 423
Volume 116B, number 6
PHYSICS LETTERS
N
dU~= i__r]ld~b/ufI<] (S/u/)2d~u,
(6)
S~" = sin ~I (q~i - ~b~).
(7)
field, the determinant (12) goes into the denominator for each i/', which yields an effective measure
d//(~bi) = •=lII i=lI-Id~b/~i<[ill" Strictly speaking there are spurious degrees of freedom in I2~, corresponding to the ambiguity in the left multiplication by diagonal matrices exp(i3u). These degrees of freedom have to be eliminated (see below). We may now omit one of the integrations, say dr21, which corresponds to the gauge fixing (timelike gauge ~21 = / ) . Then all the matrices are close to unity. In leading order we parametrize ik = 8 i k + i ( ~ / N ) I / 2 c ° ~" / S ! "k
/.t= 2 .....
+ O(X),
(8)
d.
The factors S~ in the denominator cancel the Weyl factors for/a = 2 ..... d so that d d~2~-1 1-I dU~ #=1 d =const. 1-1. dO~ . btl
(S~/¢)2
l
dco..
(9)
=
28 October 1982
(Sff)2
(13)
Now we observe the following. At d = 2, when the extra factors are absent, the phases ~/u are distributed uniformly on the unit circle. This corresponds to an unbroken U(1) symmetry. The validity of the EK solution at d = 2 for all coupling was demonstrated in the original paper [ 1] by another method. We checked that the usual diagrams with gluon exchange (in the timelike gauge) arise from higher order terms in the perturbation expansion of the EK model at d = 2. However, at d > 2 there is a collapse. The distribution (13) is peaked at coinciding eigenvalues, which leads to spontaneous symmetry breaking of the U(1) symmetry. As is clear from the note added to ref. [1] the spontaneous breaking of U(1) symmetry in the EK model was already noticed by G. Bhanot et al. and by M. Peskin and K. Wilson. The modification of this solution which preserves the U(1) symmetry at small X is considered in the next paper by one of the authors.
As for the action (4), it reduces to the Maxwell form
We are grateful to S.B. Khoklachov, R. Mkrtchian, YuaVl. Makeenko and A.M. Polyakov for discussions.
S r = -2-X
R efere n ce s
u~ q
ISu~ v - -~-u
'
w 0 = 0. (10)
Note that w~k is off diagonal. The diagonal elements of co/~k would correspond to the above mentioned spurious degrees of freedom, which have to be eliminated. The calculation of the gaussian integral over co2 ..... t~ d reduces now to the diagonalization of the (d - 1) X (d - 1) matrix (at fixed i , j which we omit) Rab = (S 2 + S2)~ab - SaSb.
(11)
In virtue of the rotational symmetry this problem is trivial. There is one eigenvalue S 2 and (d - 2) eigenvalues S 2 = S 2 + S2,as is clear for S a = (S 2, 0 ..... 0). Hence det(Rab) = $2($2) d - 2 .
(12)
Since each w~" for i < j is an independent complex 424
[1] T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1063. [2] Yu.M. Makeenko and A.A. Migdal, Phys. Lett. 88B (1979) 135, 89B (1980) 437(E). [3] J. Gervais and A. Neveu, Phys. Lett. 80B (1979) 255; Y. Nambu, Phys. Lett. 80B (1979) 372; A.M. Polyakov, Phys. Lett. 82B (1979) 247; A.A. Migdal, talk seminar on Quantum solitons (Leningrad, 1978), unpublished; T. Eguchi, Phys. Lett. 87B (1979) 91; D. Foerster, Phys. Lett. 87B (1979) 87; Nuel. Phys. B170 (1980) 107; D. Weingarten, Phys. Lett. 87B (1979) 97; 90B (1980) 277. [4] M. Creutz and K. Moriarty, Phys. Rev. D25 (1982) 610; F. Green and S. Samuel, Nuel. Phys. B194 (1981) 107. [5] A.A. Migdal, Phys. Lett. 116B (1982) 425.