Some approximate calculations in SU(2) lattice mean field theory

Nuclear PhysicsB210[FS6] (1982) 388-406 © North-Holland Pubhshing Company

S O M E A P P R O X I M A T E C A L C U L A T I O N S IN SU(2) L A T T I C E

MEAN FIELD THEORY N.D HARI DASS The Ntels Bohr Instttute, Untoersity of Copenhagen DK-2100 Copenhagen ~, Denmark

P.G LAUWERS NIKHEF, Sectle-H, NL-I O09 DB Amsterdam, The Netherlands

Recetved 20 January 1982 (Revised 9 June 1982) Approximate calculattons are performed for small Wdson loops of SU(2) lattice gauge theory in the mean field approximation. Reasonable agreement Is found with Monte Carlo data. Ways of tmprowng these calculations are discussed.

1. Introduction Mean field techniques in lattice gauge theories have recently generated a lot of interest by their successful description of the phase transition points and plaquette expectation values for a number of gauge theories (abelian and non-abelian) on the lattice [1, 2]. The spirit of mean field theories is the approximation that as far as local measurables are concerned the important field "configurations" are those where all but a few variables assume some average value which is then determined by recourse to some self consistency arguments. These concepts are well understood in spin systems [3]. Obviously such an approximation drastically truncates the nature of the fluctuations and correlations in the system and any attempt at improving the mean field approximation must address itself to the question as to how some of these important fluctuations and correlations may best be incorporated in a systematic way. A n attempt in this direction was undertaken sometime back where the objective was to achieve a calculational scheme that somehow incorporated the successes both of the Monte Carlo techniques used to understand the phase structure of lattice gauge theories as well as the lowest order mean field calculations. Such a scheme was indeed found [4] and nicknamed "mean Carlo". It consisted in fixing most of the lattice variables at their " m e a n " value while variables thought to be specially relevant for the quantities that were being computed were allowed to fluctuate. A Monte Carlo simulation was performed to take into account these 388

N.D Hart Dass, P G. Lauwers /

SU(2) lamce mean field theory

389

fluctuations. Such an approach is considerably more economical than a complete Monte Carlo simulation. In this manner we were able to study the behaviour of small Wilson loops and the numerical agreement with previous Monte Carlo results for the same Wilson loops (if available) was very good. It is obvious that an analytic understanding of the "mean Carlo" prescription would be worthwhile. To this end we have studied a few simple Wilson loops under the approximation that all the link variables on the boundary and inside it are treated as "live" while the rest of the variables are approximated by the mean field value. The resulting partition functions can often be evaluated using character expansion techniques. While the character expansions are expected to be reliable in the strong coupling region, they are rather poor expansions in the crossover and weak coupling regions. Some ways are indicated that may be helpful in summing these character expansions but these ideas are only tentative at the moment. Hence to evaluate the above mentioned Wilson loops in the crossover regions and weak coupling region we evaluate the partition, function in the so-called "quadratic approximation". In this approximation all the integrations become gaussian though they are complicated by the fact that the variables of integration are coupled to each other in accordance with the Wilson action. We compare the results for the expectation value for the few simple Wilson loops under consideration and find their agreement with the previous "'mean Carlo" evaluation as well as the Monte Carlo evaluations rather good. This indicates that the considerations of this paper could form a starting point for a semianalytical understanding of the various features of lattice gauge theories. Since these techniques are well suited to the crossover and weak coupling regions it is hoped that these methods after some improvements will shed some light on the thorny question of the continuum limit of lattice theories. After comparing our quadratic approximation calculations with previous numerical works we indicate briefly how to perform higher order calculations. The surprising success of even the simple quadratic approximation in accounting for the expectation values of these small Wilson loops over a rather wide range of coupling constants indicates that perhaps only a few higher order corrections will yield expressions that may approximate the actual values rather efficiently for the same range of coupling strengths. We conclude our note by studying the expectation value of ladder like structures with a view to set up a framework for studying correlations on the lattice. In the quadratic approximation the evaluations of such expectation values are shown to be equivalent to solving certain recursion relations. We would also like to mention another promising attempt to include higher order corrections to mean field calculations [2]. This method is based on the observation by Br6zin and Droutte [5] that the mean field approximation is equivalent to a saddle-point approximation.

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N D. H a d Dass, P G Lauwers / $U(2) lattice mean field theory

2. Mean field approximation In what follows we restrict our attention to SU(2) lattice gauge theories with the Wilson action. The partition function is given by Z = I ~ {dUt} e-S{Uk ,

(1)

where

/ where/7 is related to the square of the inverse coupling constant and x , ( U ) represents the character of the group element U in the representation r. In what follows we shall write Z as

The partition function in (2) differs from that in (1) by an irrelevant overall multiplicative factor. In (1) and (2) dUt is the Haar measure for the group integration over the variable fit. Up is the ordered product of four link variables around a unit square. The mean field approximation consists in writing (Ui) = m • 1

(3)

so that (XII2(UI))

m.

061/2(1))

m is determined self-consistently by evaluating the l.h.s, of eq. (3) in such a way that all the link variables except one are set equal to their " m e a n " value m. In this way it is straightforward to obtain m = dln

Z"~(z),

where t ZCl~ = j d U exp

[~zxl/2(U)]

(4)

and z = 2(d - 1)m 3, d being the dimensionality of space-time. The integral in eq. (4) is the so-called one-link integral and is easy to evaluate. For the SU(2) gauge group one gets

Z~l~_2II(z) z

N.D. Hart Dash, P.G Lauwers / SU(2) lattice mean field theory

391

and hence the self-consistency equation for m becomes //I-

12(2(d - 1)m3/3) I1(2(d - 1)m3/3) "

(5)

Solving eq. (5) (ignoring the trivial solution m = 0) yields m as a function of 3. For/3 less than some critical value/3~ eq. (5) has no solutions. In analogy with the Weiss theory in statistical mechanics Greensite and Lautrup [1] have interpreted this/3c as the phase-transition point. It should be appreciated that this criterion for locating phase-transition points is quite different from the ones advocated by Brdzin and Drouffe [5]. Even though the criterion adopted by Br6zin and Drouffe is more in the spirit of thermodynamics, the criterion adopted by Greensite and Lautrup seems to locate phase-transition points more accurately (except for Z , groups). The reasons for this are not well understood [6]. Using the former criterion, Cvitanovi6, Greensite and Lautrup have calculated the phase-transition points for a number of lattice gauge theories [1]. A better understanding of the consistency equation (5) is provided by Brdzin and Drouffe [5] where they interpret the mean field approximation as a sort of saddle-point approximation and eq. (5) is precisely the equation for locating the saddle points. Their treatment also elucidates the connection between mean field approximation for gauge theories and the so-called Elitzur [6] theorem which simply states that for systems like lattice gauge theories with local continuous symmetry (U) = 0. Eq. (3) seems to contradict this theorem. But according to Brdzin and Drouffe m is a solution of the equation locating the saddle points and in particular, if m is a solution, then any other m ' = g(x)mg'(x +a) is also a solution. Thus if we average U over all possible saddle-point configurations one indeed gets zero in agreement with Elitzur's theorem. On the other hand if we are only interested in calculating the expectation values of gauge invariant quantities then all the different saddle point solutions give the same value so it does not matter which particular saddle point one uses for their evaluation. Thus there is no conflict between mean field theory and Elitzur's theorem. The expectation value of the plaquette (X1/2(U))/X1/2(1) in the lowest order mean field approximation is simply m 4. As was noted by Greensite and Lautrup even this crude estimate agrees with the Monte Carlo data to within 15%, the agreement becoming poorer as one approaches/3c. However, these authors noted that even a minimal improvement obtained by keeping the four links of the concerned plaquette "live" while all the rest were put at the mean field value dramatically improved the agreement to within a few percent all the way from the weak coupling region (/3 - 5) to the "transition" point (/3c = 2.2). Below this value the self consistency equation has only the trivial solution m = 0 and the plaquette expectation value resembles the strong coupling expansion. The expectation value

N D. Hart Dass, P G. Lauwer~ / SU(2) latnce mean field theory

392

of the plaquette is given by

(X~2(Uv)~ 1 ,~, ~x,/2(Uv) ~/=Z-I

exp [½3 Y.

x(Uv,)]

(6)

under the improved approximation scheme this could be evaluated as

[Ht~.dUt½x,/2(U.)

exp [½(2d-3)m3fl

_.

~I~pX1/2(Ut)+~3X( 1 "U

v)]

I 1-I,~pdUt exp [½(2d - 3)rn 33 Y.~p x(U~) + ½fix(Up)] (7) Eq. (3) can be recast as (5X,/2( v))=

In

11 dUt exp l~p

~z lF. ,~(Ut)+~3,~(Up) ~p

(8)

where z = (2d - 3)m 3. Eq. (8) can be evaluated using character expansion techniques. The character expansion for exp [ax(U)/x(1)] is: e (1/2~(u)= ~.

d,br(o~)xr(U).

(9)

r=0

In eq. (9) r is the representation label, X, the character of U in that representation and d, is the dimension of the representation and dr = 2 r + l for SU(2). The coefficients b,(a) for SU(2) are given by br(a) =

212~+1(a)

,

(10)

where the I are Bessel functions of imaginary argument. The characters satisfy the orthonormality conditions

I dUx,(U))¢~(U) = 6r~.

(11)

Now

I(z,B)= I ,~v11dU, exp [½z ~x(Ut'+½Bx(Up)] I IEp [I dUt ~.d,d,~b,,(z)x,,(Ut,)... Zd,,b,,(z)x,,(U,,) r1 r4

xE d~b,(B)x,(Ut,U~2U,+~U, +,).

(12)

Actually we need eq. (11) in a somewhat different form in order to evaluate (12) and that is

dUx,(U))r,(U'V)

= x,(V) d--~"

(13)

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393

Using (13) it is easy to show that

I(z, fl)=Ed2fl4(z)fl,(z'),

z'=/3,

and hence 2 4 z' d z ' (~Xt/2(Uv))= ~''d2flT(z)dfl'(z')fi E,d2fl~(z)fl,(z ')

] I z,=B

(14)

We have recapitulated this calculation as we shall be using similar techniques later on.

3. "Mean Carlo" calculations [4]. As was emphasised in the last section even a minimal inclusion of live links dramatically improves the predictions of the mean field theory. This numerical success suggested to us a natural generalisation of this scheme: keep a small set of variables alive in the functional integrals and fix the remainder at the mean field value. As was pointed out in ref. [4], this technique is analogous to the strong coupling expansion in the sense that while in the strong coupling expansion a certain set of variables are kept alive and the rest set to zero, in our way of calculation the remainder is fixed to be at a self-consistently determined mean field value. Clearly as the volume of the link variables kept alive is increased the results converge towards the Monte Carlo data except for some surface effects, neglecting the fact that the periodic boundary condition of the usual Monte Carlo studies are replaced by mean field boundary conditions. In this way the set of small Wilson loops shown in fig. 1 was studied. For the results of these calculations (numerical) and their comparison with Monte Carlo data the reader is referred to ref. [4].

4. Analytic treatment oi mean Carlo techniques For the plaquette expectation value with all four of the link variables kept alive and coupled to each other the result based on the character expansion is given in eq. (14). Now we repeat the analysis for figs. l b and le also. The idea is to integrate out all the variables not contained in the Wilson loop under consideration to obtain an effective action under which the final expectation value is evaluated.

(o)

(b)

(c) Fig 1

(0)

(e)

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N D. Hari Dass, P.G. Lauwers / SU(2) lattice mean field theory

Fig. 2

We label the various links in fig. lb as shown in fig. 2. The Wilson loop under consideration is then x(Up) =x(blb2bja~a~at). After putting all the other link variables on their mean field value it is easy to see that the part of the action involving the variables al, a2, a3, c, bl, b2, b3 is 3 S' = ½(2d - 3)m 3fl y. {X(a,) + X (b,)} + ~(2d - 4)m 3fix (c) I=I

1 + + + + +~fl{x(ca3azai)+x(cb3b2bl )}.

(15)

Clearly c is the only link not belonging to the Wilson loop and will be integrated out. In order to do so we define

I dUxa(U)xb(UV+)xc(UW +) = ~.a N~bcaddxd(VW4-).

(16)

It is straightforward to show, using eq. (16), that the effective partition function after integrating out c is

Z'= I I-Ida, I-Idb, exp [~z{~x(a,)+ ~x(b,)} ] X ~

rstu

d,d~d~l,N,~,,b~ (z 3)b, (z 3)b, (z 2)X, (a l b i b2b 3+ a 3+ a 24- ),

where Z2 =

(2d - 4 ) m 3 f l ,

z3 = fl,

zl = (2d - 3)m3fl.

(17)

Now the expectation value of the Wilson loop Up can be evaluated and with the following definitions: ~. Nrsmd~lsdtbr( z 2)bs (z 3)bt( z 3) = fu (z2, z3), r$I

X1/2( U ) x , ( U ) = ~. C,1/2,'d,,xr,( U ) ,

(18)

r'

we obtain ~'rr' (X (Uw))

=

2 drdr,Crl/2r,f,(z2, z3)b 6.(z1) 2

Z,d,fr(z2,

6

z3)br (zl)

Similar calculations can be performed for the other Wilson loops of fig. 1.

(19)

N.D Hart Da$s, P.G. Lauwers / SU(2) lattice mean field theory

395

In the vicinity of /3~ m becomes small and hence z~, z2 are small and it is straightforward to compute (18) and (20) as only a few representations enter the relevant summations. Yet if/3 is large, the characters b,(z3) do not fall off with r and hence the above expressions are not useful. Below/3~ we have to set m = 0 and eqs. (19) and (21) represent the first term in the strong coupling expansion. We wish to r e m a r k at this stage that even though character expansions are not of much help in large-/3 regions we may still be able to extract information from them owing to certain sets of "sum rules" a m o n g the characters. For example, if we have e ~t/2~''2~rs)= ~. d ~ b , ( a ) x • ( U ) , •= 0

we consider e {'+¢J)~l/z)~(t'}= ~. d~b,(a + 3 ) x , ( U ) .

(20)

r=O

But the l.h.s, of eq. (20) can be evaluated alternatively as e(a +~)l/2xl[Y)

:

e(l/2)t,x(u) e(l/2)Bx(u) = Y. d ~ l u b , ( a ) b , ( f l ) x , ( U ) x , ( U ) ,

(21)

I.u

on using a generalisation of the second of eq. (18), i.e.

x,(u)x,(u) = E c,,d,x,(u),

(22)

t

we can recast eq. (21) as e ('~+t3)(1/2)~U)= ~ d,d~d~bt(a)b,(/3)~(~(U). t.u.v

C o m p a r i n g with (21) and using the orthogonality of the characters we get b,(a +/3) = 2 C,,,drd, br(ot )b~ (fl ) .

(23)

r,s

Likewise we can derive b,(a +/3 +3') = Y Cs, u,dsd, d ~ b , ( y ) b u ( a ) b ~ ( 3 ) ,

(24)

sub

etc., where we have defined C,,.,~, = E Crs,C,~o .

(25)

Relations like (23) and (24) are valid even when the arguments of the Bessei functions b e c o m e large. By a judicious choice of equations like (23) and (24) it may be possible to extract the behaviour of the sums over Bessel functions like in

396

N.D. Hari Dass, P G. Lauwers / SU(2) lattice mean field theory

(17) even when we are in the weak coupling regime. Factors like C,,, N,~,,, C,,..o though complicated can be computed from the properties of the irreducible representations of SU(2). It is the purpose of this p a p e r to consider approximation schemes for large/3 which are simpler than the ones presented in this section.

5. Quadratic approximation The basis of this approximation is the observation that when/3 is very large, the link variables U fluctuate predominantly around unity as away from unity the contribution to the functional integral for Z becomes very small. We have tacitly assumed in our arguments that/3 is positive. The unitary matrices U are represented as (for SU(2)) U = e '~ ,

where s¢ = A • nr,

(26)

the r ' s being the generators in the appropriate representation (for our discussions the r ' s are in the fundamental representation). If/3 is very large, the important contributions come from A ~ 0 and we can expand U as s/2 U = 1 + is~ - - - +. • •, 2~ XI/2( U)=

2-~dI

(27)

2+ . . . ,

where we have used X1/2(r,~'j) = ½8,r T h e measure for A integration can be written as

d U = c J ( A ) d3A,.

(28)

For SU(2) J takes the very simple form: J(A)=

sin2 ½1AI (½1AI) 2 ,

1 c

16tr 2.

(29)

For more complicated S U ( N ) groups the form of J is [7]

J(A)=

sin ½A ½~ adj

(30)

AS long as we are only interested in the quadratic approximation we can set

J ( A ) equal to unity. But if we wish to go beyond the quadratic approximation we have to expand J ( A ) also. We evaluated the expectation values of the Wilson loops la, l b and le. These calculations are tedious but can be simplified considerably by some standard techniques. As a good example of these calculations we will show the full derivation of the expectation value of fig. le. The link variables that are kept alive for this are shown in fig. 3. If g, is the link variable its expansion is given by g, = exp iG, • ~r,, etc.

397

N D Hart Dass, P.G Lauwers SU(2) lattice mean field theory g~

g2

e l ~j e;,

g"

V,e3

c]~

g5

F~g 3 The action for this system of links with all the rest put on the mean field is S=½(2d-3)m3/3

8

4

E X(g,)+½(2d-4)m3/3

~. x(e,)

I:l

I=l

+ ~/3{x(elg2g3e2 )+ x(e2gngse3 ) + x(e3g6g7e ~ ) + x(e4g8g ~e~ )} .

(31)

After the quadratic approximation S can be written as (terms not involving the variables E,, Gj have been left out): S=-~(2d-3)m3/3

8

4

E G,.G,-~(2d-4)m3/3

Y. E , . E ,

t=l

I=1

+/3[-~ Y{G, " G , - ~ Y . E , " E,+~ ~.E, " I'Ij !

1

+~{Ei • E 2 + E 2 " E 3 + E 3 " E4 + E 4 ' El} (32)

+~{GI" G s + G 6 " G 7 + G 4 . G s + G 2 " G3}] , where we have defined Hi

G2 + G8 - G 1 ,

//3

=

//2 = G2 - G3 + G5 - G 4 ,

//4

= G6 - G7 +

= G3

-

G4

-

G5 +

G7

-

G6,

(33) G1 - G s .

We integrate out the E variables first in order to get an effective action for the G variables. T h e action containing E variables is

S'=-z*~

E, "E,

-~

E,'H,+EI'Ez+Ez'E3+E3"E4+E4"Et

,

(34)

t=l

calling z * = l[(2d - 4)m 3 + 2]/3,

,~/3 = 2z * a * .

(35)

We write S' as: (36)

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N D. Hart Dass, P.G Lauwer~ / SU(2) lamce mean field theory

/ 1

0 :(-0" \-a*

-a*

0

1 -a* 0

-a* 1 -a*

-

*i

0

/

(37)

--0/* J

1

det O = 1 - 4 a .2 and /1 - - 2 a .2

¢~-1 = (det ¢~)-1 t 2aa *.2

a*

2 a .2

Or*

1 -a* 2a*

1 - 2a* a .2

2a* ~'2

2a .2

a*

1 - 2 a .2

a*

\

.~"

(38)

/

Using a linear transformation of the integration variables, the expression ~ I-I dE, e -s' is evaluated to be 6

71"

(~-~) ( 1 - - 4 O t ' 2 ) - 3/2

exp z,c~,2H. (~-1. H ,

where 1-2a'2 (~ ) 2a* 1 - -4 a .2 H, • H, + 1 - a a .2 (H1 •/'/'2+/'/2" H3

H.

(~-I . H = -

4or .2

+H3" Ha +Ha • HI)+ 1 - 4 a .2 (H1 "H3 +H2 " 1t4) = f ( H ) .

(39)

We can see from (33) that the 14, are not linearly independent as ~, 1t, = 0. Using this linear dependence of the H ' s it is easy to show that 2(H1' 1"13+1"12"/'/4) = -~/'/~ • H, - 2 ( H I • H2+H2" H 3 + H 3 " ['!4+I"[4" Hx). I

(40) Therefore, H.

0-~

• H =~. H,.

l

1-1, +

2or* 1 + 2 a * (Ht • H 2 + H 2 ' H 3 + H 3 • !"!4+1"14. H1)

(41)

and is thus singularity-free for positive a*. The expectation value of the Wilson loop under consideration is: (x{Uw)) = (x(g2g~gag~g6g~gsg~f)) If we introduce the notation: G2-G3

=At

,

G2+G3 =al,

Ga-G5

=A2,

G6-G7

=..43 ,

Gs-GI

--Aa,

G6+GT=a3; G8+GI = a 4 ,

Ga+G5 =a2 ,

(42)

it is clear that 1

2

(43)

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399

It is also clear that H, d e p e n d only on At, and not on a,. T h e effective action for G is now rewritten as: 8

Sc~(G) = - ~ [ ( 2 d - 3 ) m 3 +

1]/3 ~, G, • G, + z * o t * 2 f ( H ) I=l

B

(44)

+'-(GI"G8+G6"G7+Ga'Gs+Gz'G3) 4 Using (42) this is rewritten as: 4

4

St.(At, a ) : - ~ [ ( 2 d - 3)m 3 +1]/3(,~, At," At, + ,=~ a,. a,)

+z*a*Zf(A)+~[3

a, . a , - Y . A , .

A,

.

T h u s the a, A completely d e c o u p l e f r o m each other and since the Wilson loop u n d e r consideration is also i n d e p e n d e n t of a the a variables b e c o m e irrelevant. W e can then write the effective action for At in the following c o m p a c t form: se.(a

) = -~,f , (.4) - .f2(a

) - ~f3(a

) ,

(45)

where Z*0/.2

t~ = ~ [ ( 2 d - 3)m 3 + 2]/3

1 + 2 a * (3 + 2 a * ) ,

Z*O~ . 2

1 -- 201~*

1 +2a* '

1 + 2a*

(46)

Z*O¢ . 2

where

f,(A) =~.A, . A,, i

/3(A) = 2(A2" At4 + A 1 " A 3 ) .

(47)

It is clear that ~(X(Uw)) = 1 +1_ 0__ In Z + . • • . 8 0v

(48)

Thus we have r e d u c e d the p r o b l e m f r o m one of 8 coupled variables to one of f o u r c o u p l e d variables. A g a i n the remaining integral can be rewritten as a p r o d u c t of gaussian integrals. T h e partition function we obtain is

Z = (/z +4u

+A)-3/2F,

(49)

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N D. Hart Dass, P G. Lauwers / SU(2) latttce mean field theory

where F is a function independent of u. Using (48) and (49) we get (½h'(Uw)) = 1

3 1 4 tx + 4 u + A

+.

• •.

(50)

If we take m ~ 1 and d = 4 we get, using (72) and (61) (½h'(Uw)) = 1

12 1

7

(51)

An analogous but simpler calculation gives, for figs. l a and b; fig. la:

1 U 21 (~h'(w)) = 1-3

fig. lb:

(½x(Uw)) = 1 _ 9 - - + . . •.

(52) 1

(53)

8/3

These results should be contrasted with the naive mean field calculations. In this approximation we have l U (~X( w))=

m"

= 1 - - -n+l . . . 4/3

(54)

where n is the number of links in W. Thus the predictions of naive mean field theory for the Wilson loops of la, b and e are: la:

(ix(Uw))= 1 - 1 / B

lb:

1 U (~X( w ) ) = l - - - -3+1" "

(53')

le:

1 U (~X( w ) ) = l - - - +2 ' " .

(51')

+'"

2/3

/3

',

(52')

It is instructive to compare these results to what one would obtain by performing the calculations with gauge fixing as in ref. [2]:* la:

lb: le:

(½(Uw)) . .1 . 3. 1 +. 4/3 1

(~'(

U

• •

91 w))=l----+'",

* A s i m d a r c a l c u l a t t o n for SU(3) ts m a d e m ref [8].

(53")

8/3

(½(Uw)) . .1 . 3. 1 +.

2/3

(52")

• •.

(51")

N.D. Hart Dass, P G. Lauwers/ SU(2) latttce mean field theory

401

It is clear that this naive mean field theory always predicts a perimeter law. It is also clear that correlations show themselves significantly even in quadratic approximations. These calculations are c o m p a r e d with the Monte Carlo evaluations. The predictions of eqs. (51), (52) and (53) are denoted by quadratic approximation I and those of (51'), (52'), (53') are denoted by MF0 to indicate that these are naive mean field results. We have also plotted the results of eq. (50) with d = 4 and m = 1 - 1/4/3 instead of m = 1 and the equivalent expressions for fig. l a and lb. This we have denoted as quadratic approximation II. In a strict sense quadratic approximation II has contributions that go beyond the quadratic approximation and is not self-consistent. But it gives us some idea as to how part of the corrections to the quadratic approximation behave. We discuss the comparison of these calculations with Monte Carlo data in the last section. Before that we briefly discuss how the quadratic approximation may be applied to more complicated Wilson loops and also discuss briefly the technique needed to carry out the calculations of this p a p e r beyond the quadratic approximation.

6. The ladder As an example of a m o r e complicated system to which the quadratic approximation is to be applied consider the arrangement in fig. 4 which we shall call a ladder. Such ladders may be helpful in understanding correlations within the mean field f r a m e w o r k (contrary to popular belief it is not correct to assert that in mean field theory correlations are absent). The wavy lines in fig. 4 indicate links that have been set to their mean field value. U. indicates the n th link in the ladder. The action relevant for this system is N-1

S = ~ ( 2 d - 3 ) m 3/3[g(U~)+x(UN)]+½(2d-4)m3/3

E x(U.) rt=2

N-I

+½m 2 Y. /3.x(U.U+~-I).

(54)

n=l

In (54) we have introduced coupling constants that vary from plaquette to plaquette. This enables us to extract plaquette-plaquette correlation functions as appropriate derivatives w.r.t./3.. We will now show that the partition function can be solved in the quadratic approximation by means of recursion relations.

~....... N - ~ Fig4

402

N.D. Hari Dass, P.O. Lauwers / SU(2) lattice mean field theory

In the quadratic approximation the action expressed in eq. (54) can be written as: 1

S=-~m

3

2

2

1

jS{Al+AN}-~(2d-4)rn

3

N

j8 ~'. A .

2

n=l

- ~1m

2

N--1

y. ~ . ( a ~ + A .~ + t - 2 A . • A . + 1) (constant terms must be included).

hE1

(55) Let us write I-I dA, exp

= Z . - t exp ( - C . _ t A 2 . ) ,

(56)

where S,., = -st- (2d - 4 ) m 3BA 2, - X m 2 B , . , I A , . , - A,.,+al 2 .

(57)

The partition function for the action S is given by N

z = l-I z , .

(58)

I=l

The way to solve for Z is to obtain recursion relations for Z . and C._1. This is done as follows: Z . exp (-c,,A,,+x) =

dA, exp

s,.

I=1

=Z._~ J dam exp ( - C . _ I A ~ ) x exp {-~(2d - 4)m 1

-~m

2

2

3/3A2n

2

B.(A.+A.+I

-2A.

• A.+I)}.

(59)

If we call m2/3. = "0., ~: = (2d -4)rn3B, C._1+1f f + ~n. 1 = ~X.,

(60)

it is straightforward to p e r f o r m the integration in (59) and comparing to the l.h.s. we get the recursion relations: Zn =

(8T/.)3/2/~ ~312

e~+ra2-ti.Zn_l ,

Ca ='0--2" 8C"-1 +so 8 8C._1 +~:+'0.

(61)

Even though it is hard to solve eqs. (87) analytically they are solvable on a computer.

N.D. H a d Dass, P.G Lauwers / SU(2) lamce mean field theory

403

7. Beyond the quadratic approximation

Let us take the most complicated case we have considered, namely, fig. l e and see what additional techniques will be needed to push the calculations beyond the quadratic approximation. When we expand the action in eq. (31) by one order (in /3) beyond the quadratic approximation we typically generate terms of the type (this includes the expansion of the determinant also) E a ~ , b ~ . c ~,d

f-2_a~,_bf2_c~2d

~.a~,bl.-2_cl..2_d

Eaf-2bf2c f2d l~.a~.b~,c f 2 d , v~..l, ~..Ikl...I I . . . . ,Zd, kW,.-II ,

(62)

where the superscript refers to the SU(2) component while the subscript refers to the space-time index for the vectors. In the quadrati~ approximation integrals over E, s were of the form [see (36)]

I= I ~ dE, e - Z ' [ E . O ' E - 2 a * E E , . H , ] . l=1

t

(63)

Because of eq. (62) we have to evaluate integrals of the type

I I-IdE, f(E,, G,) exp (-z *[ E "D "E - 2a * • E, "H,]) ,

(64)

i

where f is a term of the type occurring in (62). It is clear that a

1

0

(f(E' ) ) = f ( 2 a ~ z * OI-I~) and thus all the integrals of the type (63) can simply be obtained from a knowledge of I. After this step the typical integrals that occur are of the following type I

8

l-I dG, G~ e s~6~ = O,

I=1

(65)

I ~ dG, G~G~ e se"~a~

°ab

z=l f

8

H

dG, G,a G IbG kc

eSee(G~

= O, (66)

f FI8 dG, G a, G IbG ckG ,d eSee(G) _-- ~abcd " ,lkt • tzl

N D. Hart Dass, P.G. Lauwers / SU(2) lamce mean field theory

404

But now the a, variables are no longer irrelevant; instead, the set of integrals in eqn. (66) d e c o m p o s e into: I ,l~i =1 da,a,ba,CeS.,(~ , = 8,flb~ ~1 Io,

f [Ida,ah, aC,a aka ~e t so.(~' = (6,fikt6b~6d~ +" " " + cyclic permutations) f-~ , I=|

'~

(67)

where

Io=I

[ ] d a , eSe,(a) ,

t=l

f ~ ~a A , ,Abac a , A , e S°~(A)= I

*b~ ,

(68)

t=l

f~

d A , A , bCdeleSeet(A)=t*bcde A,AkA ",jkt •

t=l

Sen(A) is of the form - A • O* • A. Let V be the matrix that diagonalises O*, i.e. O* = V O * V -1 where O * is diagonal. We can define new variables A ' = V - 1 A . Then eq. (68) becomes I~ bc = f l-I dAI(V-1),,'(V-1),,'AI'hA; ~' e s~n'A') = 8bc(V -1V'-1), I I] d A I ( A I A I ) e s'n(a') , etc.

(69)

The integral in eq. (69) can be solved in a manner similar to (68) (but not identically as the A~ are weighted differently viz according to the eigenvalues of O*). The system of equations (62)-(69) are then the required ingredients to go beyond the quadratic approximation.

8. Conclusion We have indicated in this p a p e r some steps towards an analytic treatment of the properties of lattice gauge theories in the weak coupling and crossover regions. We have calculated some simple Wilson loops in the quadratic approximation and c o m p a r e d them to the Monte Carlo data. The results are displayed in figs. 5-7; we see that the predictions of the naive mean field theory are not so good. But even a minimal extension improves considerably the predictions of the quadratic approximation in mean field theory. The quadratic approximation II agrees surprisingly well with Monte Carlo data over the whole coupling constant region up to /3c except for fig. le. This indicates that for larger Wilson loops quadratic approxima-

N D. Harz Dass, P O. Lauwers / S U ( 2 ) lamce mean [ield I

08

theory

405

I

I



06

(a) 04 [a) MF', [b) MO~te Carlo clcxta (c) Ouadrat,capprox- n" (cl) Ouadrot~col:~rox- T

02

[] I

I 4

I

I~c

3

I 5

Ftg 5.

tion II should be trusted only after the next order corrections have been included. Quadratic approximation I again agrees quite well with Monte Carlo data even for small/3 - the error being about 6 to 7% near/3c for l a and l e and s o m e w h a t larger error of 15% for lb. The errors are quite small in the large/3c region. H e n c e it is hoped that the next order corrections can significantly improve these results and

08


06

OL. (a)

(o) (b) [c) (d)

02

0 0

t

1

MF. Ouaclrahc opprox '[ Ouodrahc approx - "IT Monte Carlo data

i

2

~c

3

Fig. 6.

4

5

406

N.D. Hari Dass, P.G. Lauwers / SU(2) lattice mean field theory 1

0.8

I

!



06

06

02

/ /

~c

t 3

z,

5

Fig. 7

make it possible for the existing Monte Carlo data to be analysed analytically. Methods for going beyond the quadratic approximation are also discussed. It is a pleasure to thank many members of the high energy group at the Niels Bohr Institute and NORDITA for innumerable discussions pertaining to the subject of this paper. Our thanks are especially due to J.P. Greensite for many discussions and to T.H. Hansson for participating in the early stages of this work. References [1] J. Greensite and B. Lautrup, Phys Len. 104B (1981) 41; P. Cvitanovi~, J. Greensite and B. Lautrup, Phys. Lett. 105B (1981) 197, D. Pritchard, Phys. Lett. 106B (1981) 193 [2] H. Flyvbjerg, B. Lautrup and J.B. Zuber, NBI-HE-81-52 [3] R. Brout, in Phase transmons (Benjamin, New York, 1965) [4] J. Greenslte, T.H Hansson, N.D. Hart Dass and P.G. Lauwers, Phys. Lett. 105B (1981) 201 [5] E. Brgzin and J.N. Drouffe, Nucl. Phys. B200[FS4] (1982) 93 [6] S. Elitzur, Phys. Rev. D12 (1975) 3978 [7] D. Pritchard, unpublished [8] N. Kawamoto and K Shlgemoto, Phys. Lett. l14B (1982) 42