Strong-coupling planar diagrams, random surfaces, and the large-N phase transition in U(N) lattice gauge theories

Nuclear Physacs B265 [FS15] (1986) 223-252 ~ North-Holland Pubhstung Company

STRONG-COUPLING PLANAR DIAGRAMS, RANDOM SURFACES, AND THE LARGE-N PHASE TRANSITION IN U ( N ) LATTICE GAUGE THEORIES I K KOSTOV

Instttute of Nuclear Research and NuJear Ener~,, 72 Boul Lento, 1184 Sofia, Bulgarta Received 20 May 1985

The strong-couphng expansion of U(N) gauge theory on a D-dimensional lattice is reformulated m the hmlt N ~ o0 through a set of dmgrammauc rules directly for the free energy and Wilson loops The strong-couphng planar &agrams are interpreted as surfaces embedded m the lattice The large-N phase transmon ~s related to the entropy of these surfaces It as shown that the strong-coupling phase of the U(oo) gauge theory terminates with a phase transmon of Gross-WJtten type only m 2 and 3 dimensaons When D ~>4 the large-N singularity takes place m a metastable phase because of an earher first-order transxtlon to the weak-coupling phase of the theory

1. Introduction

The possibility of an exact description of confining gauge theories in terms of infinitely thin strings (or random surfaces) has been widely discussed in the last few years. Two rather different mechanisms of the formation of strings are related to the strong coupling and large-N expansions of a gauge theory. In the context of Wilson's strong coupling expansion [1], the gauge theory on a lattice can be viewed as classical statistical mechanics of interacting random surfaces. In the lowest orders of the strong coupling fl, the Wilson loop average is given by the propagator of the free Nambu string (in its lattice regularization) and the free energy is expressed as a sum of closed connected surfaces. Each surface is weighted by flarea. In higher orders, the dominant contribution comes from overlapping surfaces exhibiting complicated interactions and forming new topologies so that the string picture is actually lost. An exception is made by the Z 2 gauge theory in 3 dimensions, to be discussed below. On the other hand, the 1 / N expansion of 't Hooft [2] shares many common features with the dual-loop expansion in relativistic string models of hadrons [3]. By the topology of the Feynman diagrams, the U(N) gauge theory resembles in its large-N limit a free string theory, 1 / N 2 corrections going for string interactions. One may think of a chromodynamical string as the result of condensation of high-order planar diagrams. 223

224

I K Kostov / Strong-couphngplanar dlagram~

Many attempts have been made to find a free-stnng ansatz for the U(oo) gauge theory. In the late 1970's several authors [4] tried to develop a correspondence with the free Nambu string. The Wilson loop average then would be given by the sum of all planar surfaces spanning the loop, weighted by exp(-area). Later it was shown that the Nambu ansatz may represent a solution of the loop equations only for asymptotically large loops [5]. Migdal then argued [6] that an exact solution can be obtained by adding Majorana fermions coupled to the intrinsic geometry of the fluctuating surface. Unfortunately the arising fermlonic string theory is still not well understood. In spite of the impressive ideas of Polyakov [7], no efficient formalism to deal with models of random surfaces in the continuum has yet been developed. One is naturally led to consider U ( N ) gauge theory on the lattice and examine Wilson's mechanism of formation of strings in the limit N ~ ~ . The analogy with the continuum theory suggests that some kind of free string ansatz should emerge. The most commonly used character expansion is not well adapted to the large-N limit. One has to rearrange it as an expansion for the free energy in order to extract the relevant planar diagrams. A lot of work has been done [8] but the results showed no evidence for a free string picture. Recently, the technique used to solve the multicomponent o-model [9] was extended to apply to the U(oo) gauge theory [10-12]. As a result, the Wilson loop average was expressed as a sum over noninteracting planar surfaces on the latuce. The weights of these surfaces were, however, quite complicated, depending on an infinite set of parameters. In the present paper we develop further the methods of ref. [12] to fred a simple model of planar random surfaces reproducing the U(oc) lattice gauge theory in its strong-coupling phase. We define a precise set of diagrammatic rules directly for the free energy and Wilson loops and interpret the strong-coupling planar diagrams as surfaces on the lattice. The weight of each surface depends on the strong coupling fl only through its area. We also investigate the convergent properties and the range of validity of this random-surface ansatz and draw some conclusions about the nature of the phase transition to the weak-coupling phase of the theory. A short version of the results appeared in a letter [13]. The general idea is to convert the U ( N ) integration measure N

dU= H dUi+ ' dU;' 8 (U + U-I)

(l.1)

t,k=l

into a gaussian one by introducing a matrix Lagrange multiplier a = a + for the condition of unitarity N

3(U+U - 1)=

1-I t,k~l

da~exp(Ntr(a(U+U- 1))).

(1.2)

I K Kostov / Strong-couphngplanardtagrams

225

This rather old trick allows one to reduce the integration over the gauge field U (now an unrestricted complex matrix) to a simple set of diagrammatic rules, as has been done in the gaussian gauge model considered by Weingarten [14]. The weights of the resulting strong-coupling diagrams depend on the Lagrange multiplier matnx through its moments t r ( a - " ) , n -- 1, 2 . . . . . The complications come with the integration over the a-field, so we gain nothing if N -- 3, for example. However, in the limit N ~ ~ , which can be considered as a thermodynamical hmlt, the normalized traces of this field freeze at their mean values F, = N - I ( t r a - " ) . These numbers depend only on the strong coupling /3 and can be found by solving a set of self-consistency equations. Further, the planar diagrams surviving in the large-N limit can be identified as planar surfaces on the lattice built up of plaquettes by gluing them along half-bonds. We have therefore a sort of free string ansatz, although rather complicated, for the Wilson loop average. So far the situation is essentially the same as m the multi-component o-model. The Green functions of the latter are those of a free relativistic particle whose mass depends on the bare coupling constant. The next step has no equivalent in the o-model. It turns out that in a resummed version of the surface expansion for the Wilson average, the quantities Fn(/3 ) enter through combinations which are actually/3-independent. As a consequence we have an extremely simple ansatz for the free energy in terms of suitably weighted surfaces. This is another example of an exact exponentiatlon of the partition function, like that reahzed in the Z 2 gauge theory in 3 dimensions [15]. Besides the factor fl .... and the usual symmetry factor, the weight of each surface includes a product of local numerical factors depending on its intrinsic geometry. These factors may have both signs which is a hint for a fermionic structure. Because of the signs there are many cancellations. We identified a large class of surfaces whose contributions are completely compensated. In particular, all surfaces making folds can be dropped out of the sum without affecting the resets. Being a series in the strong coupling/3, our random-surface ansatz reproduces a stable phase of U ( ~ ) gauge theory up to the closest singularity on the positive /3-axis, to be identified below. Let us briefly recall what is known about possible singularities. In D >~4 dimensions U(N) gauge theories (N>~ 2) undergo a first-order phase transition at some value tic of the inverse coupling fl [16]. This phase transition is ignored by the strong coupling expansion which continues to converge until some f12 > /3c' describing for /3c
I K Kostot, / Strong-couphngplanar dtagrams

226

Revisiting the derivation of the ansatz, we find that the U-integration, the way it was done, is justified only for fl < tic. When fl > tic, one has to expand around nontrivial configurations since U = 0 is no longer a stable classical solution [17]. The reason for the large-N singularity is more subtle. In the limit N ~ oo the moments of the Lagrange multiplier matrix were replaced by their mean values F.(fl) according to the factorization theorem [21]

fiN-itrAk)

fi (N ItrAA)+O(n2/N2),

h=l

/,=1

(1.3)

which is valid for any U(N)-invarIant set of matrix fields A 1. . . . . A,,. It appears that the factorization property of traces can be applied only below the large-N transition point tic. Loosely speaking, for/3 >/~c each link of the lattice is visited more than N times by an average surface and the neglected terms on the RHS of eq. (1.3) become dominant. The r61e of an order parameter for the large-N phase transmon in our formahsm IS played by the moments F,(fl). When fl approaches/~c, all these quantities become singular. By investigating the self-consistency equations for F,,(fl) we were able to evaluate the critical coupling/~¢ in the extremal cases D = 2 (reproducing the result /~c = ½ of refs. [17,18]) and D ~ ~ . When D is large,/~¢ behaves as D 1/2 while the mean field analysis predicts t i c - D - i and r 2 - D x/4 [17]. This means that for sufficiently large D the large-N singularity is sandwiched between the critical coupling tic of the first-order phase transition and the point r2 where the metastable phase terminates. Comparing the result /~c = 0.4 of ref. [20] with the number tic = 0.38 extrapolated from the U ( N ) groups [16,17] for D = 4, we conclude that the large-N singularity is preceded by the first-order phase transition and therefore takes place in a metastable phase for all D >~4. It is well-known that in two dimensions the large-N phase transition can be removed by a careful choice of the lattice action [22]. For the random-surface ansatz this leads to minor complications which we explain in the text. Probably a similar recipe can be applied also for D = 3. In the most interesting case D = 4 variant actions would not help because of the first-order transition. One has to change the rules for the U-integration in order to find the relevant for the weak-coupling phase planar diagrams. This paper is organized as follows. In sect. 2 we derive the diagrammatic rules for the strong coupling expansion. The strong-coupling planar diagrams are interpreted as lattice surfaces in sect. 3. Then we discuss the phase structure of the U ( ~ ) lattice gauge theory (sect. 4) and the possibility of constructing a random-surface ansatz for the weak-coupling regime (sect. 5). In the main body of this paper we restrict ourselves to the case of Wilson action. The general case and some technical details are considered in the appendices.

I K Kostoo / Strong-couphngplanar&agrams

227

2. Strong-couplingplanardiagrams The strong-coupling phase of the U ( ~ ) gauge theory wall not be affected if the infinite hypercubic lattice is replaced by a single hypercube with periodic boundary conditions [23]. In what follows we consider this reduced version of the theory only because it permits shorter notations. The partition function is defined by D

z=fjNa(d~)exp(NBtr ~ V~U~ +U;+),

(2.1)

where the integration over the gauge field U, = U+, is defined by a product of D copies of the unitary measure (1.1). Before introducing a Lagrange multiplier it is convenient to double the degrees of freedom U~ ~---U~'I'u~(2) ,

(dU.)-->(dUJl')(dU~(2)).

(2.2)

The only effect of this operation will be, due to the invariance of the group measure, a multiplication of the partition function (2.1) by the Dth power of the volume of the U ( N ) group. Now we impose the conditions U(1)+ U( 1 ) = 1 and U~(2)U(2)+ = 1 through a set of matrix Lagrange multipliers _,,*c°)=a~o) (o = 1,2; /, = 1,..., D). As a result the partition function takes the following form

z=fl-I dU~")+dU~")da~")exp

l [ ° (a(I,1)(UJ1)+U~(1)-1) Ntr - ~

+ a~2'(u.<2'g.'2,+ - 1))

+ B E u.u.u; u+ ,u,v=l °

1)

(2.3) The integration over the U-field is done by expanding the non-gaussian part of the exponent and applying all possible Wick contractions to each term of the serms. The corresponding Feynman rules are collected in table 1. We have used the double hne notations of 't Hooft to represent the index structure of the propagators and vertmes. The white blobs symbolize a-matrices. A closed contour on the lattice is defined up to translations by the directions of its links taken in a cyclic order. For a contour C = (fitl, ~2 . . . . . ~ L } of length L the Wilson loop functional W(C) = (N-ltrgt~l - . . gut) =- -N tr = U~I)u~if)

(2.4)

is given by the a-average of the sum of all diagrams with L pairs of (double)

228

I K Kostov / Strong-couphngplanar &agrams TABLE 1 Feynman rules for the U-mtegratmn m double-hne notatmn

'~---:--~----- ~

'~. ~,r(~.(t)~ )-ill~,

s............... ' (~)

l}-Ij I IN [ OCt. ]S ~k

Ik

'

e

iU -Q

~

(

)

..... in(

N~lmdex

9

st ructur~]

....

external lines labeled by the indices /~l,/~2,..-,/~L- As an example, the simplest diagram for the plaquette energy W(D) = ( N - i t r U~,U.U+ U+>

(2.5)

is shown in fig. 1. Note that because of the structure of the vertices all propagators involved in a continuous index loop will carry the same spatial index. This was actually the reason for doubling the degrees of freedom. For a fixed a-background, the contribution of each diagram to the Wilson loop average is formed as follows: (i) Vertices enter with their weights Nil, (ii) Dashed index loops yield factors tr I = N, (iii) A continuous index loop carrying spatial index ~t and composed of 2n propagators contributes a factor tr(a~)-" where = ^,(x)=¢2)

r .

r ..... I

I

i

.

.

(2.6)

.

11"1;

I Q)q), I

2UI

I

......

i

I I

i

Fig 1 The lowest-order &agram for the p|aquette energy.

I K Kostov / Strong-couphngplanar diagrams

229

The overall power of N is related to the Euler characteristic of the diagram X #vertices - #propagators + # i n d e x loops - 1 = X - 2 = - 2 H ,

(2.7)

where H is the number of handles. As a consequence, the limit N ---) oo can be taken by restricting the sum to those diagrams which have the topology of a disc. Now we are ready for the second step - the integration over the a-field. Assume that the large-N limit and the sum over planar diagrams are interchangeable; this is certainly true if /~ ls not too large. The factorizatlon theorem (1.3) then can be applied to any individual diagram and the average with respect to the a's is taken simply by replacing the moments of these matrices by their mean values F,(~)=N-

1

tr(a~)

--n

.

(2.8)

By symmetry these numbers should not depend on the index /~. They can be obtained, in principle, by solving a set of self-consistency equations. Before discussing this point, we would like to improve our diagrammatical notation. Continuous index loops will be replaced by a new kind of vertices. We call them link-vertices to be distinguished from the original plaquette-vertices. A link-vertex of order n is produced when a continuous index loop involving 2n propagators is shrinked to a point (see fig. 2). Its 2n legs taken in a cyclic order have spatial indices /~, - # . . . . . . /~, - p . Once the condition of planarity is imposed, the remaining index structure (represented by the dashed lines) can be suppressed. The new propagators will carry only a spatial index /~ = + 1. . . . . + D. For convenience they will be denoted by oriented lines; the orientation takes care of the sign of #. In table 2 we present the graphical notation for the plaquette- and link-vertices which will be used in the rest of this paper. The diagram series for the Wilson loop average, eq. (2.4), is constructed according to the following rules: (i) The sum goes over all planar diagrams with L external lines labeled in a cyclic order by the indices/zz,/L 2. . . . . /~L-

;;2'

I

"" . ' - . 3 , ' I .. ."- f " -~ . c,,

--"

tl 't

"'-

It Fig 2 Correspondence between continuous index loops and hnk-vertaces

230

! K. Kostoo / Strong-couphngplanar dtagrams TABLE2 The improved&agrammatlcnotatmn (hnk- and plaquette-vertmes) F, (/3).

n= 1.2.

(ii) The contribution of a planar diagram is due to its vertices: a plaquette-vertex yields a factor 13 and a link-vertex of order n - a factor Fn(/3). (iii) Direct connections between vertices of the same type are forbidden; the external lines grow only from link-vertices. The exact weights of the hnk-vertices can be found from the unitarity condition U~U~ = 1 applied to the loop functional (2.4). Namely, the Wilson loop average will not change if a backtracking part of the loop is removed. In terms of planar diagrams this reads

(2.9) The backtracking condition (2.9) produces an infinite set of equations for the weights Fn(/3 ). The first iteration in fl yields F , ( B ) =f~ - 2B2(D - 1)(8x, + 2nf,,+ 1) + O ( / 3 4 ) ,

(2.10)

where

i.= (-1)"

+1 ( 2 n - 2)!

n = 1,2 . . . . .

(2.11)

Due to the complexity of the equations an exact solution is possible only in the extremal cases D = 2 and D ~ ~ (see sect. 4). As was mentioned in the Introduction, the diagram series for W(C) can be rearranged in a particularly s~mple and elegant form. It turns out that the backtracking condition (2.9) can be solved immediately in terms of dressed link-vertices. A

I K. Kostov / Strong-couphngplanardtagrams

231

dressed link-vertex represents the sum of all connected planar diagrams with the same configuration of external lines. To find the weights °2-, of the dressed linkvertices, consider the diagrammatic expansion of the quantity

O

=

(N_ltr(l_tU, U_, ) -1)

= 1/(1 - t)

(2.12)

and write down the corresponding planar Schwmger-Dyson equations (shaded blobs represent dressed link-vertices)

(2.13) In terms of the generating function °-$(t) = 1 +

,6~lt q- 6"~212 q- • • •

eq. (2.13) reads

°$(t/(1 - t) 2) = 1/(1 - t).

(2.14)

Inspecting the solution i f ( t ) = ½(1 + ¢g+ 4t) we see that the weights fin of the dressed lank-vertices are given by the fl = 0 approximation fn of the original link-vertices, eq. (2.11). These numbers satisfy a set of recurrence equations

fl=l;

f~+ ~ f k f ~ ~ = 0 ,

n=2,3 .....

(2.15)

k=l

which are equivalent to eq. (2.14). Thus the diagram technique will simplify substantially if the link vertices are taken with weights fn and the sum is restricted to skeleton diagrams. The new diagram technique is defined by the rules (i)-(iii) with Fn(/3) replaced by f, plus: (iv) Diagrams dressing hnk-vertices have to be omitted. Such diagrams have nontrivial connected parts with the structure of link-vertices and will be called one-link-reducible. The corresponding planar Dyson-Schwinger equations are equivalent to the well-known contour equation for the Wilson loop functional [6, 24]. Let us demonstrate this.

232

LK Kostov / Strong-couphngplanar dzagrams

Consider an external line/~ and draw explicitly the dressed link-vertices which are associated with it

(2.16)

The lines connected a link-vertex with the rest of the diagram are drawn as slightly penetrating the white blob. By the rule (iii) they have to be distinguished from the usual external lines. Inserting eq. (2.15) m the RHS of eq. (2.16), we obtain

#

(2.17) Now trace the possible evolutions of the penetrating line (to meet a plaquette-vertex or to grow into another external line)

_

~=~I

,~ -

7__

..

~ -

~

t*

_

(2.18)

I K Kostot, / Strong
233

Eq. (2.18) is identical to the loop equaUon for W(C) if the external lines of the diagrams are interpreted as oriented links on the lattice. The diagrammatic rules for the free energy Eo(fl) = N-2 log Z(fl) can be obtained by integrating the series for the plaquette energy, eq. (2.5), with respect to ft. It follows that Eo(fl ) is given by the sum of all one-link-irreducible (1LI) vacuum diagrams with the topology of a sphere, taken with appropnate symmetry factors (their explicit form wdl be specified in the next section). In this section we restricted ourselves to the Wilson action S(U)= Nfl tr(U + U +) since it produces the simplest diagrammatic rules. The general case is considered in appendix A.

3. Random surfaces

The results of the previous section can be readily reformulated in terms of random surfaces, as it was already discussed in the Introduction. Let us first recall the definition of the simplest model of planar random surfaces (the Nambu string on the lattice) proposed by Weingarten [14] and thoroughly studied in recent years [25]. The underlying large-N gauge theory is defined by the partition function (2.1), with the 8-function in the Haar measure (1.1) replaced by a gaussian distribution e x p ( - N t r U* U). The strong-coupling expansion of this theory corresponds to a particular choice ( F 1 = 1, F 2 = F 3 . . . . . 0) of the weights of the link-vertices. The Wilson loop average is given by the lattice Nambu ansatz W(C) =

y"

fllsl

(3 1)

S OS=C

where the sum goes over all planar surfaces on the D-dimensional hypercubic lattice having as a boundary the contour C, and ]SJ denotes the area ( = number of plaquettes) of the surface S. The free energy is equal to the sum of all closed connected surfaces with the topology of a sphere flIll

Eo(fl)=

E

S 0S=0

k(s)=D(D-1)

flIll+l

E

S 0S=t2

IS[+I

(3.2)

The symmetry factor 1/k(S) arises when the surface S wraps k times around a simpler one (see Itzykson [15]). The continuum limit of the Weingarten model describes a single massless excitation as a result of the dolmnance of tree-like surfaces. This is in fact the mean-field critical behaviour which is common for all random-surface models on the lattice m the large-D limit [26]. Now turn back to the U(oc) gauge theory. The planar diagrams of sect. 2 will be identified as planar surfaces on a lattice A D which is obtained from the D-dimen-

234

I K Kostot, / Strong-~ouphngplanar diagrams

b

,,

,

b

,,

q

~

Fig 3 The lattice A D ( D = 2)

n=l

n~.2

n=3

Fig 4 Link-vertices as cychc contractions of plaqucttes

sional hypercubic lattice by adding a new site at the middle point of each link (fig. 3). The surfaces embedded in this lattice consist of oriented plaquettes glued together along half-bonds. These surfaces possess new kinds of vertices corresponding to cyclic contractions of 2n plaquettes (n = 1,2 .... ) along a common link (fig. 4). The correspondence between the planar diagrams and planar surfaces on A o maps plaquette-vertices into oriented plaquettes and link-vertices into cyclic contractions of plaquettes. A planar surface on A D is called 1-1ink-irreducible if such is the corresponding planar diagram. A 1LI surface cannot be split into two nontrivial pieces by cutting it along all copies of a given link on the lattice (trivial surfaces are those which have only boundary and no area). The Wilson loop average W(C) is equal to the sum of all 1PI surfaces with boundary C, weighted by a product of local factors (fl from plaquettes and f, from link-vertices of order n). Symbolically one can write W(C) =

E'

fllsl H f , , , v ,

S 0S=C

~ES

(3.3)

where the prime means that the sum is restricted to 1LI surfaces and n(v) stands for the order of the link-vertex v of the surface S. The expression for the free energy E0(fl)= N-21og Z follows from the surface expansion (3.3) of the plaquette energy W(H) = ( D ( D - 1))-1 O E o ( f l ) / O f l .

(3.4)

The integration of eq. (3.4) with respect to fl gives the exact symmetry factors for

LK Kostov / Strong-couphngplanar dtagrams

235

closed connected surfaces with the topology of a sphere fllSl

Eo( ) =

E'

FI

S 0S=0

v~S

(3.5)

This nice formula reflects the fact that the 1LI surfaces do not interact and therefore the partition function exponentiates trivially (no cumulant expansion is needed). Another example of a gauge theory with this property is the D = 3 Islng model [15]. The main difference between the surface expansions of U(oo) and Z 2 gauge theories is that here the surface amplitude depends only on the intrinsic geometry of the surface and not on the way it is embedded in the lattice, as it does in the Ising model. We also remark that in the second case there is no restriction on the topology of the surfaces which makes the free-string interpretauon impossible. The price for the exponentiation of the partition function m both cases is that the weights of the surfaces are not always positive. This rules out the direct interpretation of these expansions in terms of statistical mechanics of surfaces. However there remains the possibility that the signs are manifestations of an underlying fermlonic structure. An interesting discussion on this point in the context of the Ising gauge theory is given in ref. [27]. Due to the signs there are many cancellations. In practical calculations it is therefore convenient to arrange the enumeration of surfaces so that the irrelevant terms cancel as soon as possible. This can be achieved by evaluating first the contribution of the surfaces w~th given configuration of plaquettes and then taking the sum over all possible configurations. The procedure is described in detail in appendix B where the calculation of the free energy is carried out up to the 16th order m 13. Below we prove two statements which will enable us to reduce severely the amount of surfaces entering the RHS of eqs. (3.3), (3.5). Let the planar 1LI surfaces S' and S" have as a bound a loop C forming a tree (t.e., enclosing no area)" OS t

= 0 S 't = C = C

1.

(3.6)

Consider the class of all closed 1LI surfaces whwh can be obtamed by contractmg the edges of OS' and OS". The total contnbutton of these surfaces to the R H S of eq. (3.5) is zero. Proof. One-hnk-irreducible closed surfaces will be obtained only if the contour C occupies at least two distinct links on the lattice. Consider first the case when C is contained in exactly two links £1 and £2. We say that an edge belonging to aS' or 0S" is of type l if it occupies the link £, (t = 1, 2). The same classtfication can be applied to the link-vertices contracting these edges. Now take the sum over all contractions of the edges of type 1 which are consistent with a given contraction of the rest of the edges. A constraint on this sum is zmposed by the 1LI condition: only

236

L K. Kostov / Strong-couphngplanar dtagrams

(a)

(b)

Fig 5 A surface dwlded into two components by a backtracking loop (a) The unconstraaned sum over type 1 contrachons (b) The sum over the contractions which have to be suppressed as violating the 1LI condatlon All white blobs yteld a factor 1 and the two terms cancel

t h o s e c o n t r a c t i o n s are p e r m i t t e d which include at least one link-vertex of b o t h types c o n n e c t i n g 0S' with 0S". The u n i t a r i t y c o n d i t i o n in the f o r m (2.12) can be a p p l i e d to c a l c u l a t e s e p a r a t e l y the u n c o n s t r a i n e d sum a n d the c o n t r i b u t i o n of the f o r b i d d e n c o n t r a c t i o n s (see fig. 5). T h e results are identical which proves the s t a t e m e n t in this case. I n g e n e r a l the c o n t o u r C occupies m links £ 1 , - - - , £ , , (2 ~< m ~< L ) . T h e contractions to b e m a d e m a y be classified into m types. A s b e f o r e we first p e r f o r m the sum o v e r the c o n t r a c t i o n s of t y p e 1 k e e p i n g the rest fixed. T h e r e are two possibilities. If a s u b t r a c t i o n has to be m a d e (this d e p e n d s on the w a y the rest of the edges are c o n t r a c t e d ) the result will be zero. If the s u m is u n c o n s t r a i n e d , it yields a factor 1 a n d we are left with a tree-like closed c o n t o u r o c c u p y i n g m - 1 hnks. T h e n we take the s u m o v e r the edges of type 2 a n d p r o c e e d in the s a m e w a y until the n u m b e r of o c c u p i e d links is r e d u c e d to 2. This c o m p l e t e s the proof. Let the planar 1LI surfaces S 1. . . . . S 2n (n >12) have boundartes 8S 1 . . . . .

8S n = C ,

8S n+l . . . . .

8S 2n = C - I ,

(3.7)

where C is an arbitrary oriented loop. Then the total contribution to the R H S of eq. (3.5) for all connected closed 1LI surfaces obtained by contracting the edges of 8S 1. . . . . 8S 2n is zero. T h e p r o o f goes in the same w a y as before. E v i d e n t l y sirmlar statements can b e f o r m u l a t e d for the surfaces involved m the e x p a n s i o n (3.3) of the W i l s o n l o o p average. N o w we c a n give the following recipe for reducing the e n u m e r a t i o n of surfaces: All surfaces whose hnes of self-lntersectton form a one-dtmenstonal complex with no more than one loop have to be omttted.

I K Kostov / Strong-couphngplanar diagrams

237

Another class of irrelevant surfaces includes those having folds. These surfaces correspond to the longitudinal modes of the string. We checked that their contributions cancel in some particular cases but the general proof is missing (it would need elaborate combinatorial arguments). An indirect proof is based on a comparison with the usual character expansion. The above considerations make sense only if the surface expansion is absolutely convergent. This can be proved immediately. Indeed, the surfaces with half-bond contractions can be embedded in a D-dimensional hypercubical lattice with spacing ½ (this lattice contains all points of A D). Further, the (dressed) link-vertices f , grow as ( - 4)" when n is large. The weight of each surface is therefore bounded by (16/3) area. It follows that the sum over surfaces in our model is absolutely convergent for 1-- 4 /3 < (~/3c) where/~c is the critical coupling of the Weingarten model in D dimensions. Let us compare the critical properties of the surface expansion of the U ( ~ ) gauge theory and the Weingarten model. In the limit D ~ ~ both models exhibit the same critical behaviour similar to the condensation of branched polymers [26]. The dominating configurations in the Weingarten model are trees of double plaquettes and the relevant expansion parameter is 2Dfl 2. In the U ( ~ ) model these configurations give no contribution as we proved above and the critical behaviour is governed by tree-like surfaces made up of 3-dimensional cubes. The relevant parameter in this case fs 2 D/3 4. The mean-field critical behaviour characterizing the large-D limit actually starts at some finite critical dimension D c. For the Weingarten model Durhuus et al. [25] showed that Dc -- 2. In our case the arguments they used are not applicable since the weights of the surfaces are not positively defined. It is most likely that here D c = 4. Finally, let us say a few words about the strong coupling methods developed In refs. [10, 11] and how they are related to our results. Using the Lagrange-multiplier formalism in a somewhat different context, V. Kazakov [10] obtained an elegant algorithm for performing U ( N ) integrations in the large-N limm As a result the partition function of the U ( N ) gauge theory has been expanded in terms of closed surfaces of the type we considered throughout this section. The weights of the link-vertices were given by the numbers fn (eq. (2.11)). However these surfaces were still interacting in the limit N ~ oe. For each surface there is a factor N 2 and when a number of surfaces touch at a link they interact with a weight - 1 / N z. The exact form of the interaction is quite complicated but does not depend on /3; it has been calculated by O'Brien and Zuber [11]. Thus the free energy is equal to the sum of tree-like clusters of connected surfaces. In the expansion of the Wilson loop average one can leave only one of the surface components (the one which is bounded by the loop) and take into account the other components by dressing the local factors in the surface amplitude. We therefore obtain a random-surface ansatz with B-dependent link-vertices F~(/3), i.e., recover the rules (i)-(nl) of the previous section. The meaning of the rule (iv) in terms of the

238

LK Kostov/ Strong-couphngplanardiagrams

Kazakov-O'Brien-Zuber expansion is that the total contribution of the trees of surfaces with more than one component and the one-link-reducible surfaces is zero.

4. The large-N phase transition As we discussed in the Introduction, there are two types of singularities of rather different origin blocking the access to the continuum limit fl ~ oo. The critical coupling for the first-order phase transition can be evaluated by comparing the weak- and strong-coupling expansions around the mean-field solution [17]. On the contrary, the large-N singularity can be located using only strong-coupling methods. Within the ordinary character expansion the relevant order parameter is the string tension for the determinant of the Wilson loop [20]. In our surface expansion the quantities which are sensitive to this singularity are the mean values F,,(fl) of the moments of the Lagrange-multiplier field. Let us illustrate this on the simplest example of the one-plaquette model defined by the partition funcUon

Z(fl) = f dUexp(flNtr(U+ U+)).

(4.1)

After doubling the degrees of freedom and introducing the Lagrange multipliers a (1) and a (2) the pamtion function takes the form Z(fl)=

f o=l--~L2da°exp(Ntr(a(O)-lna(O)))exp( Ntr

n= ~1 (atl)a(2))-"flz"/n) " (4.2)

In this simplest case the plaquette-vemces are tadpoles (the "plaquette" has only one edge). Expanding the Wdson loops

IV, = ((1/N)trU")

= ~

(4.3) v n

m terms of 1LI diagrams we readily obtain the strong-couphng result W1 = fl, W2 = W3 . . . . . 0 of ref. [18]:

(~ = ~ =(3f1"0 (4.4) * f2)=O

etc.

I K Kostov / Strong-couphngplanardiagrams

239

Let us find the weights of the bare link-vertices. The sum of planar diagrams dressing a link-vertex

(4.5)

can be evaluated explicitly:

f=

~Bz~ k=0

n+k k

F,+k(fl )

n = l , 2 .... '

(4.6) "

In terms of the generating functions

f ( t ) = 1 + f i t +fzt2 + . . . .

1(1 + ~ - + 4 t ) ,

F ( t ; f l ) = 1 + F l ( f l ) t + F2(fl)t 2 + " .

(4.7)

(4.8)

eq. (4.6) reads f ( t ) = F(t + 132,/3). The solution F(t; 13) = ½(1 + ~/1 - 4132 + 4t)

(4.9)

can be expanded as a series m t only for fl < ½. As expected, the singularity occurs exactly at the point/~c = ½ where the large-N phase transition takes place [18]. When fl approaches/3c the quantities Fn(fl) tend to _+ oo and for fl >t/~c eq. (4.6) ,s not satisfied for any choice of Fn(fl). This means that in the weak-coupling region the moments of the matrix a 1= (ao)a(2))-i cannot be treated as c-numbers. In this region the factorization theorem (1.3) is not applicable since the dominant contribution to the expansion of the Wilson loops in the large-N limit comes from terms containing more than N a-matrices. We conclude that m general the large-N phase transition takes place at the point /~c where the quantities F,(fl) become singular, i.e. the expansion (2.11) starts to diverge. To find qualitatively how the critical coupling/]~ depends on the dimension D let us consider the large-D limit which can be treated analytically. As we already mentioned, the sum over surfaces in this limit is saturated by tree-like configurations [26]. The corresponding planar diagrams may contain

240

I K Kostoo/ Strong-¢ouphngplanardiagrams

plaquette-vertices only in combinations of self-energy type

~Z~_~

=

2(O-ll~2tl 3

:

(4.10)

The relevant parameter to be kept fixed is therefore X = 2(D - 1)/~ 2.

(4.11)

An equation for the generating function (4.8) can be found as in the one-plaquette case by summing up the diagrams dressing link-vertices. After simple but tedious calculation we obtain

1-F(t2;X)

1

=

~/f

~t~dyF(y-2;Xll/t_y _

_

t +

2

),

(4.12)

where the functions ~(y) and ~(t) are defined by

1 X 2 - =~+ -~-f(~ ),

t=~+

X 2 -fff(~ )

(4.13)

Y under the condition that ~(y) is regular at y ~ 0 and ~(t) is regular at t ~ 0. It is quite a job to solve this equation but the critical point can be easily located at X = 1. For this value of X the function ~(t) behaves a s t 1/3 when t approaches 0. The critical coupling in the large-D limit is therefore given by 2(D - 1)/82 = 1,

D --* oo.

(4.14)

When extrapolated to D = 4 this number fits unexpectedly well with the value 2(4 - 1)132 = 0.96 found in ref. [20]. The large-N phase transition has to be distinguished from the first-order phase transition experienced by U ( N ) gauge theories in D >/4 dimensions. The corresponding critical coupling Bc behaves as 1/D in the limit D ~ ~ [17]. It follows that above some critical dimension the large-N singularity is preceded by the first-order transition and therefore takes place in a metastable phase. In four dimensions the result/3c = 0.40 of ref. [20] can be compared with the value tic = 0.38 extrapolated from the mean-field and Monte Carlo calculations for the U ( N ) groups ( N ~< 6) [16,17]. As far as these numbers can be trusted* we can conclude that in * Since the order parameter used m ref [20] is non-local it may be influenced by the roughening transmon (see ref [17])

I K Kostot, / Strong-couphngplanardmgram6

241

D >i 4 dimensions/~c >/3c and the strong-coupling phase terminates at the point/3c of the first-order phase transition.

5. D i s c u s s i o n

We have seen in this paper that there is no substantial difference between the weak- and strong-coupling expansions in the planar limit. This leads us to believe that a convergent diagram expansion filling the gap between random surfaces and planar Feynman diagrams can be constructed. It might supply an efficient instrument for studying infrared phenomena. Below we survey what we know or suspect (depending on the dimension D) about such a possibility. (i) D = 2: In this case we can eliminate the large-N singularity by an appropriate choice of the lattice action [22]. In the space of parameters [3 = (fl:,/32 .... ) characterizing the most general form of the one-plaquette action (see appendix A) there are many points where the continuum limit can be reached. One of them, 13= (1,1 . . . . ), can be achieved without leaving the strong-coupling phase. An explicit construction is given in appendix C. Thus in two dimensions there is an exact correspondence between random surfaces and Feynman planar diagrams. (ii) D = 3: The large-N transition remains the only singularity we encounter. We believe that it can be removed as in the previous case, in spite of the objections made m ref. [28]. Indeed, the (2 + 1)-dimensional one-plaquette problem considered in [28] exhibits no large-N singularity if the action (1 - U ) ( 1 - U +) S ( U ) = - N i l tr (1 + U)(1 + U +)

(5.1)

is taken, for example. Thus it is quite possible that the scaling limit can be achieved by an appropriate tuning of the parameters fli,/32 .... entering the surface amphrude. Another possible approach has been advocated by V. Kazakov [10]. It prescribes performing the a-integration by the saddle point method. The corresponding effective action will depend on the eagenvalues of the matrices a~;/~ = 1. . . . . D, through a sum of closed connected surfaces [10,12]. If the factorization is assumed to take place we obtain relatively simple saddle-point equations. But as we have seen in sect. 4, this is justified only in the strong-coupling phase which casts some doubts on the relevance of this approach. (iii) D >~ 4: Now it is the first-order phase transition which causes the trouble. The weak-coupling regime in this case should be achieved by doing the U-integration in a more clever way. (Recall what happens with the multlcomponent o-model on the lattice above 2 dimensions: in the weak-couphng phase the vector field acquires non-zero v.e.v, and the random walk representing the two-point Green function

I K. Kostov / Strong-couphngplanardiagrams

242

should be allowed to break.) One can imagine that the random-surface ansatz can be adjusted to the weak-coupling phase by allowing the surface to have "worm-holes". They will condensate in the asymptotically free region converting the fluctuating surface into a network of thin strips. We would like to be able to report some concrete results in a future publication.

Appendix A T H E R A N D O M - S U R F A C E A N S A T Z FOR A G E N E R A L L A T T I C E A C T I O N

In the large-N limit we can restrict ourselves to one-plaquette achons of the form

[29] S(U) = N ~. fl.tr(U" + U+")/n.

(A.1)

n= 1

The corresponding diagrammatic rules will include an infinite number of plaquettevertices. A plaquette-vertex of order n posesses 4n legs labeled in a cychc order by the indices /~, v, - # , - v repeated n times. The weight of such a vertex is /3,, (the factor 1/n in the action counts for the cyclic symmetry). On the lattice a plaquettevertex of order n is associated with a planar surface covering n times an elementary plaquette and having a branching point of order n - 1 (fig. 6). We call such a surface a multiplaquette of order n or simply an n-plaquette. The surface expansions (3.3) and (3.5) should be replaced by

W(C)=

~'

A(S),

Eo(13)=

S 3S=C

~-~[ A(S)/k(S),

(A.2)

S OS=O

where S this time stands for a surface made with multlplaquettes by gluing them along half-edges. The surface amplitude A(S) is defined through the numbers L n and Pn of link-vertices and multiplaquettes of order n, n -- I, 2 .... :

A(S) = H(fl.)v"(f~) L"

(A.3)

As before the prime means that the sum is restricted on one-link-irreducible (1LI) planar surfaces and the symmetry factor k(S) arises when the surface S is a k-covering of a simpler one.

n=l

n=2

nz3

Fig 6 Examples of multlplaquettes

LK. Kostot, / Strong-couphngplanar &agrams

243

The area of a surface containing P, multiplaquettes of order n (n = 1, 2 . . . . ) is defined as ISI = Z n e . .

(A.4)

Appendix B CALCULATION OF THE FREE ENERGY UP TO 16TH ORDER

In this appendix we consider the random-surface ansatz in its most general form (A.2) and evaluate the contribunon to the free energy of all surfaces with area IS I < 16.* As explained in sect. 3 it is convenient to classify the surfaces according to their plaquette configurations. By plaquette configuration of a surface we mean the set of the occupied plaquettes of the lattice. Let E be a possible plaquette configuration, 1.e. a connected set of plaquettes, each plaquette appearing only once. It defines a two-dimensional complex on the lattice whose regular components E, are surfaces with the topology of a sphere with a number of holes. We will supply each of these surfaces with an orientation. Let S ( E ) be the set of all closed 1LI surfaces with the same plaquette configuration E. Each surface S ~ $ ( E ) is characterized by a set of multiplaquettes located in ~, (each multiplaquette can appear in several copies) and the way their edges are contracted. Denote by P,(p) the number of n-plaquettes located on the plaquette p and having its orientation. Then the set of multiplaquettes associated with the surface S can be defined in terms of the "occupation numbers"

{P)={P.(p),P.O)lp~ff;

n = l , 2 .... },

(B.1)

where ~ denotes the plaquette obtained from p by changing its orientation. Further, it is convenient to define the numbers a -+(p) = Y'~n ( Pn (P) + P. (P))-

(B.2)

n

By eq. (A.4), the area of the surface S is [SI= Ea+(P) •

(B.3)

p~t~

In sect. 3 we mentioned that the surfaces with folds do not contribute to the free energy. This means that the following two restrictmns can be ~mposed: (i) A configuration should not have free edges, i.e. it should represent a pure two-dimensional complex. * The author is indebted to J -B Zuber for useful commumcatmnconcerningthe enumerationof these surfaces

244

L K Kostov / Strong-couphngplanar dtagrams

(ii) The numbers a-+(p) should not change along the regular components of a configuration. (The restriction on a-(p) is actually needed in order to'obtain closed surfaces.) As a consequence, we can divide the relevant closed surfaces into classes, each of them described by a configuration E and the numbers a/+ characterizing the way its regular components E, are covered with multiplaquettes. An additional restriction follows from the two statements proved m sect. 3: (iii) The class of surfaces described by a configuration whose singular lines form an one-dimensional complex w~th no more than one loop, and such that a f = 1 for all t, does not contribute to the free energy. The surface amplitude A(S) (eq. (A.3)) depends on the intrinsic geometry of the surface and not on the way it is embedded in the lattice. It is therefore convenient to introduce the notion of a diagram of a configuration. The diagram gives the topological description of a configuration; the number of inequivalent plaquette configurations with the same diagram (divided by the volume of the lattice) is called its configuration number. The sum over surfaces for the free energy can be arranged in the form E0([3) = E ( @ } A ( @ ) ,

(B.4)

®

where @ stands for a diagram, {@} is its configuration number, and A(@) is the contribution of the abstract surfaces associated with this diagram. The amplitude A ( ® ) in turn can be calculated as the sum over all possible values of the numbers a, -+ associated with the regular components ®, of @ (a + = 1 , 2 .... ; a~-= - a +, - a + + 2 . . . . . a+): A(@)=

E A[@;(a+,a;)]/k(a+, {.,-+}

a ).

(B.5)

The symmetry factor k is the maximal &visor of the numbers a, +-. In what follows we will enumerate the surfaces with area IS I < 16. The diagrams of these surfaces can be divided into two classes (see table 3): (1) Sp-sphere with p plaquettes (p = 6,10,12,14,16) (2) Opzp2p3 -- diagram made of the discs D, with p, plaquettes (i = 1, 2, 3) soldered along a 3-fold singular line (( PiP2 P3) = (551), (842), (951)). The configuration numbers of these diagrams have been already calculated [17]. It remains to evaluate the amplitudes A(®) which will be done m the rest of this appendix. It ~s convenient to introduce new notations for the coupling constants fin: /31=/3;

/3.=/3"fl.,

n = 1 , 2 ....

(B.6)

Then the contribution of the surface S will depend on its area through the factor/31st.

LK Kostot~ / Strong-couphngplanar diagrams

245

Up to the 16th order in fl we can restrict ourselves to a~+ = 1, 2. Then the diagram (1) produces three classes of surfaces (la)

Sp;

a += 1,

a-= +1.

The only surface belonging to this class yields (B.7)

A(Sp; (1, +_I)) = fl p.

(lb)

Sp;

a += 2,

a-= +2.

Each plaquette of Se is occupied either by two 1-plaquettes with same orientations, or by a 2-plaquette. Planar surfaces will be produced only if there are no more than two 2-plaquettes. All these surfaces have two branching points. A branch point can be located (c0 at a site, (fl) at the middle of a link (representing a link-vertex of order 2 and therefore producing a factor - 1 ) , (7) at the center of a plaquette (producing a factor f12). Let us denote by s, £ and p the numbers of sites, links and plaquettes of the sphere Sp. Then the number of realizations of each of the possible combinations is

#aa = ½s(s- 1 ) - £ ,

#tiff= ½ £ ( £ - 1),

# 7 7 = ½P(P- 1)

# a B = £ s - 2£,

# a y = sp,

#By = £p

(B.8)

(we took into account that, according to the 1LI condition, the two branch points cannot be situated on the same link). Summing up all contributions and applying the Euler relation s - £ + p = 2 we find A(Sp;(2,+2))=fl2P{½(p -1)(p-2)-p(p-2)fl2+½p(p-1)/~2}.

(B.9)

All surfaces of this class have a nontrlvlal symmetry factor k = 2. (lc)

Sp;

a +=2,

a =0.

In this case each plaquette of the sphere 1s occupied by two 1-plaquettes with opposite orientations. Below we shall see that the corresponding amplitude is A(Sp: (2,0)) = ( p - 2)fl 2p.

(B 10)

S,4

S14

141

14 2

Sl2

2,814

~2D(D- 1 ) ( D - 2)(2D- 5) 2

2D(D-1)(D

2)(4D 2

22D+31)

2~8t4

2B 12

13D(D- 1)(D - 2)(D - 3)

@

122

2f112

43D(D- 1 ) ( D - 2 ) ( D - 3)

St2

@

121

56fl 16

0ss~

111

~D(D- 1 ) ( D - 2)(2D

~zD(D- 1 ) ( D - 2)(2D- 5)

Sto

101

2B ~o

2/~6 +/~12(14 - 24fi2 + 15fi~)

~D(D- 1 ) ( D - 2)

S6

61

5)

Amphtude A(@ )

Configuration number { o~ }

Diagram Q

Type of the dtagram

Number of occuDed plaquettes and diagram number

TABLE 3 Dxagrams for the closed surfaces contributing to the free energy up to [S I = 16

N,

o,

0842

095i

0951

S16

Sl6

S16

144

151

152

161

162

163

164

S14

143

2)(D-3)

4D(D-1)(D

~2D( D - 1)( D - 2)(4D2 - 24D + 37)

16D(D

1 ) ( D - 2 ) ( D - 3) 2

2f116

2f116

2B 16

16D(D

8D(D- I ) ( D - 2 ) ( D - 3) 2

0

2B 16

I ) ( D - 2)(4D 2 - 22D + 31)

2fl16( 1 - ~2 )2

2314

1 ) ( D - 2 ) ( D - 3) 2

4D(D-

D ( D - 1)(D - 2 ) ( 2 D - 5) 2

2 ) ( D - 3)

D(D-1)(D

TABLE 3 (continued)

".,3

O~

g,

248

I K Kostov / Strong-~ouphngplanar &agrams

,

.:" ....../

¢ .....

'//#'" "'l " •

....... :~-.

:

"" : " "2.~

,/

(a)

'1

~

"'"

(b) Fig 7 Typmal cylindrical surfaces associated with the disc D l

The total contribution of the diagram Sp is therefore A(Sp) = 2A(Sp; (1,1)) + 2A(Se; (2,2)) + A(Sp; (2,0))

1 ( p - 1)j~22)~ 2p. (B.11) =2flP+(½(P+ I)(p-2)-p(p-2)~2 +Sp (2a)

8p,p2p,; a?=2,

a~-=a~-=l,

a { = +_2,

a 2=a;-=_+l.

The corresponding closed 1LI surfaces are made of two discs D 2, D 3 glued to a cylindrical surface associated with the disc D 1. Such a surface should have two branch points (fig. 7a). Applying the same arguments as in the case (lb) we obtain from eq. (B.8) and the Euler relation s I - £ 1 + P l -- 1 A(0plp2p,; (2,_2),(1,-I-1),(1,-I-1)) = (2b)

Oplp2p3; a?=2,

a1=0,

½p(p- 1)(1-fl2)afl 2p'+p2+p3. (B.12) +

a 2 ---a~-=l,

a 2 = a 3 =___1.

The surfaces of this class have the same structure as in the case (la), the cylindncal surface this time made of two sheets with opposite orientations joined along a set of links and half-links forming a connected tree on D 1 (fig. 7b). Each half-link is a part

I K Kostov / Strong-~ouphngplanardtagrams

249

of a cyclic contraction of order 2 and therefore yields a factor f2 = - 1. A connected tree on the disc D1 corresponds to an 1LI surface under the following conditions: (i) Adding a new link or half-link cannot produce a dosed loop, (i0 The tree involves at least two distinct links of Dv We have to perform the sum over all trees on D 1 satisfying these two conditions, each tree taken with a factor ( - 1) :~(halr4'r~s}.Denote by T{~(D1) the sum of the trees satisfying only the condition (1) and by T0I(D1) the contribution of the trees violating the condition (2). Then A(Op,p~p,;(2,0),(1, ± 1),(1, + 1))=/3 2p~+P'-+,,(T'I'(DI)-

(B.13)

The sum in T{2)(D1) includes the trees corresponding to sites, half-links and links of the disc DI: TO)(D1) = s l - 2£, +£a = 1 - p , .

(B.14)

Further, we will show that T{I'(D1) = 0

(B.15)

using the following fact. The contribution to T(1)(D1) of the trees containing a connected piece of the boundary of D I is zero. Proof. Let C be an oriented contour belonging to the boundary 0D 1 and let C"be its complement to 0D x. The trees containing C and satisfying the condition (1) are in 1-1 correspondence with the planar 1LI surfaces made of 2pl plaquettes and bounded by the backtracking loop CC 1 On the other hand, the contribution of these surfaces vanishes due to the backtracking condition (2.9). (We have to consider the Wilson loop W(CC-1) in the context of a two-dimensional U ( ~ ) gauge theory defined on the disc D1. ) Now we can successfully remove the links forming the boundary until the &sc is reduced to a single plaquette for which eq. (B.15) is evidently true. Combining the contributions of the classes (2a) and (2b) we have for the diagram

OplPzP3: A(0p,p=p3) = 2A(0e,p~e,; (2, 2), (1,1), (1,1))+ 2A( 8p~p2p3;(2,0), (1,1), (1,1))

Xj~Pl+P2+P'{(pl- 1)(2 Jr-p l ( l - ~2)2)~8 pl "b permutations}. (B.16) Eq. (B.10) follows from the identification {Sp; (2,0)} ~ { 0p_,,la; (2,0),(1,1), (1,1)}.

(B.17)

I K Kostov / Strong-couphngplanardtagrams

250

The amplitudes and the configuration numbers of the relevant diagrams are hsted in table 3. Collecting all terms we finally obtain Eo(13) = D( D - 1){ £~8,2,/n + (D - 2) [~/~ 6 + (2D - 5)fl m --

t!

+ ( ~ ( 1 0 D - 2 3 ) - 4 / ~ 2 + ~2,{/~2~/312 + ( 20D2-106D+143)]314 + (84D2 - 432D + 569 + 4 ( 1 2 D - 31)/32(/32- 2))/316 + " - ] }

(B.a8) The result for the Wilson action (132 = 0) up to this order can be found in ref. [17].

Appendix C A FREE-STRING ANSATZFOR THE CONTINUUM LIMIT OF U(oc) GAUGE THEORY IN TWO.DIMENSIONS It is known [30] that the strong-coupling phase for the general action (A.1) in two dimensions is characterized by the solutmn W,,(13) =/3,,

n = 1 , 2 ....

(C.1)

for the one-plaquette Wilson loops (4.3). Thus for any action reproducing correctly the continuum hmit we can define through eq. (C.1) an eqmvalent action of the form (A.1) with the desired properties. We can take, for example, the heat-kernel action which leads to a trivial scaling m two dimensions. Then the LHS of eq. (C.1) should be replaced by W . ( X ) = e -'x/2 ~ nml ,,=0

(C2) (m+l)!

'

where h is the weak couphng [29]. This defines a line 13(X) in the space of parameters 13; the continuum limit is reached at the point 13(0) = (1,1 .... ). Specifying in this way the parameters/3 n in the surface amphtude (A.3), we should obtain the same answers for the Wilson loops as those predicted by the continuum theory w~th an action S(U)= ~trfd2xF~(x).

(C.3)

Below we demonstrate th~s m simple examples. The gluons m two dimensions have no dynamical degrees of freedom. This is reflected by the fact that the longitudinal fluctuations of surfaces (the only possible

251

I K Kostot, / Strong-couphngplanar dlagrams

in this case) are spurious. As a consequence, the surface expansion degenerates into a finite sum. The simplest example of a Wilson loop is given by a contour C without self-intersections. There is only one non-folding surface bounded by C so that W(C) = (~1) p = e -xp,

(C.4)

where p denotes the area ( = number of plaquettes) enclosed by C. As a non-trivial example consider a contour C with one self-intersection such that the two subcontours C 1 and C 2 have the same orientation. Let s,, £,, and p, are correspondingly the numbers of sites, links, and plaquettes enclosed by the contour C, (including the boundary); t = 1, 2. The area of a non-folding surface bounded by C will be always equal to Pl +P2- All these surfaces consist of two overlapping sheets and have a branching point located at a site, or at the middle-point of a link, or at the centre of a plaquette somewhere in the area enclosed by the smaller contour C 2. Adding the contributions of these surfaces we obtain (using the Euler relation $2 -- ~ 2 "1"-P2 = 1)

W(C)

~- ( ~ 1 ) pl+p2

s 2 nt- f 2

2+

P2 f l ~ ' + ~

=flP'+P~(1-(f12/f121 - 1 ) p 2 ) = ( 1 - X p 2 ) e x p ( - X ( P l

+P2))-

(C.5)

The general form of the Wilson loop functional for the continuum theory has been found in ref. [31]. The result was interpreted there as a sum over non-folding surfaces whose amplitude is composed of the usual Nambu factor exp(-X(area)) and a polynormal of the areas of the overlapping parts of the surface. Now we see that this polynomial can be explained as the result of the integration over the positions of a number of branching points on the surface. A branching point of order n should be taken with a weight ( - ~ ) " .

References [1] [2] [3] [4]

[5] [6] [7] [8]

K Wxlson, Phys Rev D10 (1974)2455 G 't Hooft, Nucl Phys B72 (1974) 461 S Mandelstam, Phys Reports 13 (1974) 259 J L Gervms and A Neveu, Phys Lett 80B (1979)255, Nucl Phys B153 (1979)445, Y Nambu, Phys Lett 80B (1979) 372, A M Polyakov, Phys Lett 82B (1979) 247; E Comgan and B Hasslacher, Phys Lett 81B (1979 181 Y u M Makeenko and A A Magdal, Nucl Phys B188 (1981)269 A A Mlgdal, Nucl Phys B189 (1981) 253, Phys Reports 102 (1983) 201 A M Polyakov, Phys Lett 103B (1981) 207 I Bars, J Math Phys. 21 (1980) 2678, S Samuel, J Math Phys 21 (1980)2695,

252

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

[22]

[23] [24]

[25]

[26] [27] [28] [29] [30] [31]

I K Kostov / Strong-couphngplanardtagrams R.C Brower and M Nauenberg, Nucl Phys B180 [FS2] (1981) 221, D Wemgarten, Phys Lett. 90B (1980) 280; D Foerster, Nucl Phys B170 (1980) 107 H.E Stanley, Phys Rev. 176 (1968) 718 V A Kazakov, Phys Lett 128B (1983) 316; JETP (Russian edition) 85 (1983) 1887 K H O'Brlen and J -B Zuber, Nucl Phys. B253 (1985) 621; Orsay preprant LDTHE/84/28 I K Kostov, Phys Lett 138B (1984)191 I K Kostov, Phys Lett 147B (1984) 445 D Wemgarten, Phys Lett 90B (1980)285 V G Dotsenko and A M Polyakov, unpubhshed; C Itzykson, Nucl Phys B210 [FS6] (1982) 477 M Creutz and K J M Monarty, Phys Rev D25 (1982) 610 J -M. Drouffe and J.-B. Zuber, Phys Reports 102 (1983) 1 D Gross and E Wltten, Phys Rev D21 (1980) 446 S Wad~a, Chacago prepnnt EFI 80/15, unpubhshed F Green and S Samuel, Nucl Phys B194 (1982)107 A A Magdal, Ann of Phys 126 (1980)279, E Wxtten, an Recent developments m gauge theories, 1979 Carg~se lectures, ed G 't Hooft (Plenum 1980) N S Manton, Phys Lett 96B (1980)328, P Menotta and E Onofn, Nucl Phys B190 [FS3] (1984) 288, C B Lang, P Salomonson and B S. Skagerstam, Phys Lett 107B (1981) 211; Nucl Phys B190 (1981) 337 T Eguchl and H Kawai, Plays Rev Lett 48 (1982) 1063 D Foerster, Phys Lett 81B (1979)87, T. Eguchi, Phys Lett 87B (1979) 81; D Wemgarten, Phys Lett 87B (1979) 97 T Eguchl and H Kawal, Phys Lett. ll0B (1982) 143; 114B (1982) 247, H. Kawax and N Okamoto, Phys Lett 130B (1983) 415, B Durhuus, J Fr5hllch and T Jonsson, Nucl Phys. B225 (1983) 185 J-M Drouffe, G Pans1 and N Sourlas, Nucl Phys B161 (1979) 397 A Kavalov and A Sedrakyan, Erevan preprlnt EPI-695(10)-84, A Casher, D F~Srsterand P Wldney, Nucl Phys B251 [FS13] (1985) 29 J Rodngues, Phys Rev D26 (1982)2833, O Tnnhammer, Phys Lett 129B (1983) 234 P Rossx, Ann Phys 132 (1981) 463 J Jurkievlch and K Zalewskl, Nucl Phys B220 [FS 8] (1983) 167 V Kazakov and I Kostov, Nucl Phys B176 (1980) 199