Weak coupling analysis of the finite-temperature twisted Eguchi-Kawai model

Nuclear Physics B244 (1984) 492-498 © North-Holland Publishing Company

WEAK C O U P L I N G ANALYSIS OF T H E F I N I T E - T E M P E R A T U R E T W I S T E D EGUCHI-KAWAI MODEL*

Andrcas GOCKSCH Department of Physics, Brookhaven National Laboratory, Upton, 11973, USA

Filippo NERI Center for Theoretical Physics, University of Maryland, College Park, MD 20742, USA

Pietro ROSSI Department of Physics, New York University, New York, N Y 100003, USA

Received 27 March 1984 We show that with a suitable twist the finite-temperature twisted Eguchi-Kawai model reproduces the planar perturbation expansion of the full Wilson theory in the weak coupling sector.

A b o u t a year ago Eguchi and Kawai [l] introduced an exciting new idea. They suggested that at infinite N the usual Wilson lattice gauge theory is equivalent to a d-matrix model described by the partition function

Z=

I

l-IdU~,exp ~

T r ( U ~ , U ~ U . Uv) p.

.

(1)

=1

Soon afterwards, however, it was pointed out [2] that this is true only at strong c o u p l i n g and that in order to make the equivalence complete one had to m o d i f y (l) in such a way as to prevent the global U ( l ) a symmetry present in (1) from breaking spontaneously. This was achieved by the authors o f ref. [2] by " q u e n c h i n g " the eigenvalues o f the U . ' s in (1). A n o t h e r possibility o f achieving the same thing was pointed out by G o n z a l e z - A r r o y o and O k a w a [3]. They s h o w e d that by twisting the b o u n d a r y conditions in (l) and with a clever choice o f the twist, the weak c o u p l i n g b e h a v i o r o f the Wilson theory could also be reproduced. Recently a finite-temperature version o f the twisted E g u c h i - K a w a i model was studied by the " The submitted manuscript has been authored under contract no. DE-ACO2-76CH00016 with the US Department of Energy. Accordingly, the US Government retains a nonexclusive, royalty-free license to publish or reproduce the published form on this contribution, or allow others to do so, for US Government purposes. 492

A. Gocksch et al. / Eguchi-Kawai model

493

present authors [4]. The model is defined by

z=fHat~.(n)exp{~o"~'(~.

Tr [ Uo(n) Ui(n + 1) Ug(n) U.+,(n)+h.c.]

~,n

n=0

i=1

(2)

+ E Tr[Zi~U~(n)U~(n)U~(n)UT(n)]l. i~j

)

J

Here n is a discrete "time" variable and we have (3)

U , ( n ) = U , ( n + n,vio) ,

where l

T

n~ a.

(4)

Z o = Z ~ is the manifestiation in (2) of having twisted the boundary conditions in the 3 space directions. In [4] Monte Carlo evidence was given that with the simple twist Zij = - 1 the model has the correct weak coupling behavior. Recently, however, it was pointed out [5] that the Z 0 = - l twist is insufficient. They showed explicitly that the free energies of the original Wilson theory and the twisted EKM did not agree at the one-loop level in weak coupling perturbation theory. The deviations, however, are small and well within the statistical uncertainty of Monte Carlo results. In this letter we will show that with an appropriate choice of twist (2) reproduces the free energy of the full Wilson theory at the one-loop level. Let /.~,, denote the classical solution of (2) at flo~OO. Then we have

u ~ t L = z~v * uou~,,

(5)

where Zo~ = Zio = 1. If we now introduce e~ --- (Co = 1, O, O, O) eq. (2) can be rewritten as

xexp

/~o

Y. T r [ Z , , ~ U ~ ( n ) U ~ ( n + e , , ) U ,"( n + e . ) U ~÷( n ) + h . c . ] 0 ~v

/

.

(6)

Let us write

c~(n) = e ' ~ " ) ~ . ,

~.(n) = ~ ( n ) .

(7)

Substituting this into (6) gives the following expression for Z:

xex0

Bo~ g Tr[e'%(")O. ~'~("+~ )O* I~ ~ - " ("+~)u' e-'"~(")+h.c.] . ~v

(8)

494

A. Gocksch et al. / Eguchi-Kawai model

In the limit/30-) ~ we can expand (8) which gives to second order in a

z = I II~ I1 . d'~'('|) . exp {/3°L.

v Tr[l-½(D~ct'(n)-D~'av(n"2+h'c']}

=et~°d'd-'m"'fUUda'(n'exp{--½/3°~'...~Tr(D~a'(n)-D'a~(n'):}

(9,

In (9) we have introduced the notation

D.f(n) = (l.f(n + e.) U- .+- f ( n ) ,

(10)

and used the fact that [ d U ] = [ d a ] ( l +O(a2)). To proceed, we will as usual have to fix a gauge. We will choose as a gauge fixing condition

E D.a,,(n) =0,

(l l)

with

D,f(n) =f(n)- O~.f(n - e,) ~-J~.

(12)

The gauge fixing action enforcing the constraint (I I) is SGF = --floY~ E (D,a,(n)) 2 n ~

(13)

which leads to full quadratic action S' (using [ ~ , D~] = 0) S '= S +SGF = --flO~ ~ n

Tr

(D.a~(n)D.a~(n)).

(14)

~,v

BeFore we can integrate we must include the Fadd¢ev-Popov determinant, which is computed as Follows. Let F ( ~ ) be the gauge fixing Function. ~ e n under an' in~nitesimal gauge transFo~ation F(a'(.))

= F(.(.))

+ ~.

~ +0(~),

(~5)

and A~-p= det M. Using ,~ 7.(.) -- , ~ ( . ) - o , . ~ , ( n ) ,

(16)

we obtain

Hence we are led to the eigenvalue problem

2to(n)-to(n+l)-to(n-1)-~.(Oito(n)O*~to(n)O~)=[A-E(d-l)]to(n). i

(18)

495

A. Gocksch et al. / Eguchi-Kawai model

To solve (16) we have to specify the twist. We choose as an explicit solution for (5) ¢01

1

0

O~=t

eia

), /)~ = (U, U3)'

1

.I

(19)

ei(N-I)a

0

corresponding to

0

e -iL" 0

eii~ ) e La

(20)

.

In (19) and (20) we have set a~-2~r/N. Note that det ~ = 1 only for N odd. Fu~hermore, L and N should be relatively prime. It can be shown that the N ~ q 0 ~ p , q ~ N - 1 are linearly independent. In pa~icular this leads matrices U~~ U3, to the conclusion that ~o in (5) should be a multiple of the identity. Now let us make the ansatz:

U~U~

~(n)=expx~.k

(p=q=0excluded),

(21)

where K = 0 . . . . . n ~ - 1. When substituted into (18) this leads to ~.q,k

n,

N

N

The quadratic pa~ of the action leads to the determinant ~ = ~,,

(23)

so that we can finally write down the partition fun~ion

Z = e-0o(d-I) dn,N ad/~ ~FP ~g't S~-1)/2flg,(N~-I)d/2 ~FP

(24)

Hence at d = 4 the free energy per unit volume up to order one is

fl = 12flon,N +~n.(N ~- 1) In Bo+ln d ~ + O ( I / B o ) •

(25)

In the Wilson theory one obtains the same expression for flF except that there the Faddeev-Popov determinant is given by ~w~,~o~= ~ ( 4 s i n ~ k ° ~ + 4 ~ s i n ~ ) .

~,~

~r

"

(26)

However if we choose L such that 0 ~ L ~ N ( L ~ m as N ~ m ) then in the large-N limit (22) and (26) will coincide. ~ i s is because if we take the log of (26) the spatial integration region is topologically a torus. On the toms, however, ~he lines

k~(p-q)~

(p fixed),

(27)

496

A. Goclcsch et aL / Eguchi-Kawai model

are everywhere dense. This is how asymptotically a two-dimensional " m o m e n t u m " integral (22) can become equal to a three-dimensional one (26). To have a feeling of how close the two integrals may be, let us consider the simple case " dkl dk2dk3 k i~).~ f ( I, k2, k3). (28)

f~

Here f is a well-behaved function. Writing .~

f(kl,

k2, k3) =

~

n = -~3

f(n~, n2, n3) e ' k ' ,

(29)

eq. (20) gives f(0, 0, 0). On the other hand Io '' d~}r'~f(Q, P dO

(Q- P)L, P) = f ( 0 ,

0, 0) ÷ f ( - L , l, L) ÷ f ( - E L , 2, EL) ÷ . . . .

With very mild regularity properties of f we can show that for L - ~ C ~ the extra terms vanish faster than any power of L.) In figs. 1-4 we show numerically the convergence of the two integrals. In order to prove the equivalence of the model to the full Wilson theory to all orders in l//~o we must look at every single vertex generated by the reduced theory. It is easy to

f(-mL, ra, mL)-*O. (For f - -

0.4

I

~

I

I

I

I

I

I

I

a 0.001 •

0.3

÷

~

6

e

•

•

•

•

•

+ 0.2

0.1

0.0 • -0.

10

I

I

20

I

40

30

l

50

I

I

60

70

I

80

I

90

I

100

II0

Fig. 1. The circles represent the value of ~1 ~e

~:-~ ( 2¢rP~ 2 o r ( P"Q - ~) N N . 2wQ'~ log ~,.~=o _ a + sin + sin +sm N~]"

s t a ~ represent the value of

!N '

log

?6

~-o

X

+ sin = ~

+ s,n ~ + sin = ~ . N" N'/

Here we made the replacement N = N '2, 1 = ~ N , L = N.

A . G o c k s c h et al. / E g u c h i - K a w a i

0.4

I

I

I

1

I

I

497

model

I

I

a=0.OI 0.3

+ '4"

&

•

•

•

•

•

•

•

0.2

0.1

I0

I

I

I

I

I

I

I

I

1

20

30

40

.50

60

70

80

90

I00

I10

Fig. 2. The same as fig. I but for a = 01. show that the vertices of the r e d u c e d theory can be o b t a i n e d by the vertices o f the full W i l s o n theory m a k i n g the following r e p l a c e m e n t : h-~Q,(Q-P)L,P,

Tr ( A a t ' ' '

Aa,)=exp

{indp(P,

. . . P,,

Q,

. . . Q,)),

and the same holds for the gluon and the ghost propagator. The roles of the term Tr ( A a . • • Aa,) a n d the phase factor exp { i n d p ( P i • • • P , , Q ~ • • • Qm)} are to select only p l a n a r graphs in both theories. It is a tedious b u t straightforward c a l c u l a t i o n to show that the total phase o f a p l a n a r d i a g r a m is o n e a n d for a n o n p l a n a r d i a g r a m 0.39

f

I

I

I

I

I

I

I

I

a~OA +

O.37

+

~

~

•

•

•

•

•

1 40

I 50

I 60

I 70

1 80

I 90

I 1(30 I10

•

0.35

0.31 •

0.29 I0

I 20

I ~

Fig. 3. The same as fig. 1 but for a = 0.1.

498

A. Gocksch et al. / Eguchi-Kawai model

0.890

=

=

~

~

1

~

~

~

a~l

O. 886

+

+ •

Q

•

•

Q

Q

•

I

I 40

I 50

I 60

I 70

I 80

I 90

0.882

O. 87~

O. 874

•

O. 87C I0

1

20

~0

I

I00

I0

Fig. 4. The same as fig. I but for a = I.

it is a factor of the form exp {inG(plq~)} oscillating rapidly and going to a zero in the limit N -~ ~ . Therefore the proof that two topologically equivalent graphs give the same contribution rests on the fact that if a given graph in the full Wilson theory gives the contribution f o :'~ d3kl

d3k,

(2rr)' (27r)3 f ( k , , k2. . . . . k , ) ,

the reduced model will generate the integral

fo:" dP' do' d~2 d~._-TQ~ f(p,, o,; .. " • P~, Q~) (2¢r):

where Pi, Qi means

(

)

'

'

(Q, (Q-P)L, P). References

[I] [2] [3] [4] [5]

T. Eguchi and H. Kawai, Phys. Rev. Lett. 48 (1982) 1053 G. Bhanot, U. Heller and H. Neuberger, Phys. Left. 113B (1982)47 A. Gonzalez-Arroyo and M. Okawa, Phys. Lett. 120B (1982) 237; Phys. Rev. D27 (1983) 2397 A. Gocksch, F. Ned and P. Rossi, NYU preprint NYU/TR2/83 Y. Brihaye and P. Rossi, CERN preprint (January 1983)