The mass gap for Higgs models on a unit lattice

ANNALS

OF PHYSICS

158, 281-319

(1984)

The Mass Gap for Higgs Models TADEUSZ

BALABAN,* Lyman

JOHN IMBRIE,*'+AND

Laboratory Cambridge,

on a Unit Lattice ARTHURJAFFE"

of Physics, Harvard University. Massachusetts 02138

AND

Department

of Mathematics, University of Virginia. Charlottesville, Virginia 22903 Received

March

21, 1984

An isomorphism is established between certain compact and noncompact formulations of Abelian gauge theory on a lattice. For weak coupling, the mass gap predicted by the Higgs mechanism is then established. ‘c\ 1984 Academic Press, Inc.

1. INTRODUCTION The Higgs mechanism in the context of Abelian lattice gauge theories has already been extensively investigated [ 1,2]. We return to the subject with two objectives: (1) to comment on relationships between various different actions, (2) to adapt to this simple context the Glimm-Jaffe-Spencer cluster expansion [3] which is better for investigating continuum limits (weak coupling regime) than the cluster expansion in [ 1,2]. Indeed this paper is to be regarded as a prelude to an analysis of the continuum limit (in preparation). The basic program to study the continuum limit is to perform renormalization transformations which ultimately map an s-lattice model onto a unit lattice model, with properties similar to the model studied here. The two parts of this paper enable us to analyze a different region of the coupling constant space and to analyze a Higgs model which was not included in the previous papers [ 1,2]. We show that at weak coupling, i.e., in the neighborhood of a lattice Gaussian theory, all gauge invariant correlations decay exponentially. Our first result, Theorem 3.1, states an equivalence between noncompact Abelian gauge theory on a lattice (finite difference electromagnetism) and a compact Abelian Higgs theory with an action similar to the Villain model. It contains vortices or * Research supported in part by the National Science Foundation under Grant PHY82-03669. t Junior Fellow, Society of Fellows. $ Alfred P. Sloan Foundation Fellow. Research supported in part by the National Science Foundation under Grant MCS83-02115.

281 0003-4916/84

$7.50

Copyright G 1984 by Academic Press, Inc. All rights of reproduction in any form reserved

282

BALABAN

ET AL.

vortex lines as new fields. There is a remark in [4] about such a relationship but it is not correct as stated. Our second result, Theorem 4.1, exhibits the exponential decay of correlations. The proof can be understood to show that cluster expansions cancel the partition function (and thereby vacuum energy volume divergences) out of expectations. Thus finite volume decay properties can be carried over to infinite volume. There are two standard approximations in which such a cancellation is trivial: (a) independent sites (bonds), (b) Gaussian expectations. High temperature expansions as in Osterwalder and Seiler [ I] or in the Ising model relay on being in a neighborhood in coupling constant space of (a). Field theory cluster expansions [3 1 work in a neighborhood of (b) and therefore have a larger domain of convergence.

2. BASIC DEFINITIONS

AND NOTATION

We consider a complex scalar field 4(x) and a real-valued vector field A, defined on sites and bonds, respectively, of a finite lattice, A, A c Zd;

d> 2.

(2.1)

A bond b c A is an ordered pair of nearest neighbor sites, b = (x, y), X, y E II, and we require that A, = -ASb where -b is the pair b in the opposite order, -b = (y, x}. The set of bonds in /i will be denoted n *. The set of all plaquettes (oriented squares whose corners are four neighboring points in A) is denoted n * *; 3cubes are denoted A***; etc. The choices for Yang-Mills action fall into two classes, compact and noncompact, as summarized below. Compact Formalism

Here exp(ieA,) takes values in U(l), and the Lie algebra variable A, takes values in [-z/e, z/e]. The gauge field action S,, is periodic in A, for each b, and often it is local-namely it has the form

S,(A)= c f(dA(p)),

(2.2)

PCA

for some periodic function f with period 2n/e. Here d denotes the unit lattice exterior derivative or coboundary operator

W)(P)= 2 4, bcap

taking functions on bonds p to functions on plaquettes p. Here ap is the oriented boundary of p. Furthermore, the restriction of the sum (2.2) to plaquettes p lying inside A imposes a Neumann-type boundary condition on &t.

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The Wilson action arises from the choice f(x) = cos x. The Villain corresponds to the choice off(x) as given by the relation e-wg*m)

=

e-(1/2)(x-n)~

p

flE(~l@Z

283 action

(2.3)

But we also require a nonlocal, periodic or compact action S”’ which is not a sum of the form (2.2). This action is defined by the relation

exp

exp

(2.4)

Here v is a map from plaquettes inside /i into the lattice (2x/e) Z, such that

W)(c) = s U(P)

0.5)

PC8C

vanishes for all cubes c cli. The interpretation of (2.5) is a conservation law requiring that no flux be created in any cube c c 4. This means that u can be considered a configuration of vortex lines with conserved vorticity. In dimension 2, we set du = 0 by convention, and in that case ,.F’YM agrees with the Villain action. The partition function is defined by ZFi =

IC

exp(-S$A))

VA,

where

denotes the product Lebesgue measure on the interval [-z/e, z/e]. To be specific. we also use (c to denote integration over this interval. Noncompact Formalism In the noncompact case, A, takes all real values (it is Lie algebra valued) and the action function we choose is quadratic: S%‘(A)

= 4 r

WWP))*.

Again we use Neumann-like boundary conditions. In the noncompact case, VA denotes the product Lebesgue measure on the real line R, and l,, specifies this integration. Since (2.6) is gauge invariant, the function exp(-S$) is not integrable over the noncompact space of A’s. To avoid this difficulty, we introduce a gauge-

284

BALABAN

fixing function G(A). 8,:xEA -4 R. Let

Such a function

ET AL.

does depend on the gauge function

GM* = n de,, XPA where ~59, here denotes Lebesgue measure on R. The gauge fix G(A) must satisfy

e-G(A+d@geAbo=1.

I

(2.7)

In (2.7), d8 denotes the lattice derivative of B and x0 E/i is a point we choose later. After choosing a gauge fix G(A), the partition function is defined by z’N’J

--

exp[-,!?$%‘(A)

- G(A)] L9A,,.

Note that expectations of functions F(A) which are gauge invariant (i.e., functions of aY) have expectations

m (NC)= (ZcNC))-’

IN, F(A) exp[-@?$‘(A)

- G(A)] gA,, ,

which are independent of the gauge fix G(A). An example of an appropriate gauge function is G(A) = G,,(A) + c, with G,(A) = c ; XSA

(d*A)’

(x).

Here A, = 0 for b 65 /i *, and d* denotes the lattice divergence operator (the adjoint of d in the standard I, inner product on lattice forms). The positive constant a can give a one-parameter family of such G’s. The constant c is chosen so (2.7) holds, namely c = In

I

exp [-G,(A

+ de)] @O,, ba.

With our choices, c is easily seen to be finite and independent of A. The Higgs Field The action for the Higgs field minimally

coupled to the gauge field A is defined by

(2.8)

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285

MODELS

Here (xv) denotes the bond from x to y. Also 1 > 0 and p2 > 0. The constant E is chosen so that the minimum of S,($,A) is zero-i.e., there is no classical vacuum energy. Abelian Higgs Theory on a Lattice We define the total action of the Abelian Higgs model by %&A)=

S,,(A)

+ S,($,A)

+ G(A),

(2.9)

where S,, is any of the actions discussed above. In the compact case, we choose G(A) = 0. The total partition function is Z =

exp(-S($, A)] 5’A P#

and expectations of gauge invariant functions F = F(#, A) are (2.10)

(F)=Z-‘/F(#,A)exp[-S(qJA)]QAg#. Here @d denotes the product integral component d(x).

3. EQUIVALENCE

OF THE PERIODIZED

over the complex plane C for each tield

COMPACT

AND NONCOMPACT

FORMALISMS

We consider integer charge observables, which we define as functions F(g), A) of 4 and A which are both gauge invariant and periodic. Gauge invariance means F(e-‘@$, A + de) = F(#, A).

(3.la)

The condition of periodicity means that F is periodic in each component A, with period 2n/e. Thus with 6, denoting the characteristic function of the bond 6, we require that for every b, F(#, A + 6,2n/e) = F(#, A >.

(3.lb)

In general, we restrict attention to polynomial functions of exp(ieA,), b E /i * and $(x) and 4(x), x f /i. Such functions include Wilson loops, W(C)=

fl

exp(ieA,),

bcC

for C a closed curve composed of lattice bonds, as well as string variables,

286

BALABANETAL.

where r,,, is a contour along lattice bonds from x to y. We now consider a finite lattice A and expectations with Neumann boundary conditions in the compact state (. )(‘) and the noncompact state (. )(NC). THEOREM

3.1.

Let F be an integral charge obsemable. Then (I;)“’

= (F)‘NC’.

(3.2)

In addition z(C) = (2z,‘e)l^l - 1 zW'3. ProoJ We consider the expectations as a ratio of numerator to denominator. numerator of (F)(NC) is defined as

The

Z(NC)(F)(NC) = NCFe-’ .@A g# I

where gA denotes the product measure over A, as b ranges over lattice bonds and @# denotes the product integral over 4(x) as x ranges over lattice sites. Write S = S - G + G, where G is the gauge fix. Both S - G and F are gauge invariant, and the integration measure gA!3# is also gauge invariant. Thus the value of (3.3) remains unchanged if we replace exp(-G(A)) in the integrand by its gauge transform exp(-G(A + do)), or by its average over 13. Let c~A,, denote the product Lebesgue integral for each 4(x), x E A. We detine the compact gauge average of exp(-G) by ((exp(-G(A))))

= (2n/e)-lA1 Jc exp(-G(A

+ de)) 819,

where C denotes integration of each O(x) over the compact interval [-z/e, rc/e]. Then Fexp[-S,

Z(F) =I NC

(“?A) - X&h4lWpWkW)) gA@9.

Since S,@, A) is an integral charge observable (and hence periodic in each A, with period 2rr/e) we obtain

(3.4) Here a denotes a function from the bonds b E A to (2x/e) Z. To evaluate the sum over a, break it into two parts:

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MODELS

Since our lattice has trivial cohomology, with u fixed, any two choices for a differ by a coboundary ds, and s is uniquely determined by the restriction s(x,,) = 0. Here x, is the site specified in the gauge fixing conditions (2.7. Thus if a,. is one solution to da = II, any other solution a=a,.+ds

yields s by integration of a - a, from x0 to x, and the integral of ds around any closed loop vanishes. Then Z(F)=xl

0 c

BAjg)Fexp

-;xldA

+vI’-S,(#,A) J

P

x x ((ew(-W + a)>% da=rl

The gauge fixing condition )‘ dz

((exp(-G(A

(2.7) ensures that for fixed U,

-t- a)))) = x ((exp(-G(A

c

+ a,, -t ds))))

s

= 1 exp(-G(A s Ic

= (27r/e)-‘*‘+’

+ a,. + de)) g6J,(21r/e)P””

i

NCexp(-G(A

+ a,, + de)) Qe~, G;,,

= (2n/e)-‘*‘+‘.

Thus Z(NC)(~)‘NC’

=

(2n/e)-1*1+’

Xexp

\‘

f~~dA++~‘--S,(#,A) [

P

1 .

Since our lattice has trivial cohomology, we have replaced the condition that c be exact (u = da) with the equivalent condition that u be closed (du = 0). Hence taking F = 1, we can evaluate ZcNC) = (2n/e)-lA’+’ Z(‘), and therefore (F)‘“” = (F)“‘.

4. THE HIGGS MECHANISM

The Higgs mechanism boils down to the assertion that for certain ranges of the coupling constants the models we have just discussed, despite the massless appearance of their actions, will only have massive spectrum. We address this assertion by proving that the correlations decay exponentially. The method we use

288

BALABANETAL.

applies to any of the, models just discussed, but we only work through the noncompact case. This has not yet been treated in the literature, and indeed it would be difficult without using the equivalence proved in Section 3. The corresponding compact action in the equivalence of Theorem 3.1 is more complicated than the other compact actions, because it is non local. The Model and Results

The coupling constants appearing in our model, which will be defined below, are the electric charge, e, of the boson field and the coupling, L, for the quartic boson self-interaction. Define

This is the classical prediction for the gauge field mass. The classical boson mass is P. For us, the meaning of weak coupling is fix ,u and mA strictly positive and take 1 (and hence e) small depending on ,u, mA. In this paper we make no attempt to analyze carefully the dependencies on ,u and mA. This is one way of describing how our analysis must be improved to discuss the continuum limit. We will see that the weak coupling limit is a lattice Gaussian model describing a vector field with mass mA and a real scalar boson field of mass ,u. Our expansion is in principle able to produce a very detailed analysis of a neighborhood of this limit berturbation theory with symmetry breaking, etc.) but we have settled for a very small part of this. The model is

The theorem is THEOREM 4.1. Let F be any integral charge observable depending only on fields 4, A at sites and bonds in a bounded subsetof Zd, d > 2. Given ,u and m, = pe/fl, let L be suficiently small. Then the infinite volume expectation

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289

exists. Furthermore, let F and G be two such observables, and let G, denote the translate of G by x E Zd. Then with the same conditions on ,a, m,, and 2, it follows that

IWJ - W(G,)I G ew(-m lxlh where m > 0 is independent of F, G. Remark. Gauge invariant observables that are not integral charge observables, such as dA, could be treated by our methods with some additional work. They would

acquire dependence on the vortices in the same manner as the terms in the action. 5. GENERAL

OUTLINE

OF PROOF,NOTATION

The strategy is in large part borrowed from [5]. We start in Section 6 with the standard change of variables into the unitary gauge. This shows, at least formally, that the model is a small perturbation of a massive Gaussian model. It is easiest for the subsequent analysis to work with the compact model, by taking advantage of Theorem 3.1. In Section 7 we use a partition of unity on the space of field configurations to isolate the regions of A where the fields are so large that the Gaussian approximation breaks down. This idea appeared in [6] (using Peierls contours) in the analysis of scalar fields. However, it is important that it is implemented differently in this context, because the deviation from the Gaussian measure in gauge field theories is more drastic than for scalar theories. It is related to having to use several coordinate patches to describe the space in which fields take their values (circles in this simple context). In particular, as in [5], we postpone the integration over the fields which are large. The large field regions set boundary conditions (conditioning) for the complementary region, A, where we carry out a Glimm-Jaffe-Spencer cluster expansion. This is done in Section 9. The cluster expansion has some not-so-standard aspects because the large field regions alter normalization factors and impose nonzero Dirichlet boundary conditions. This will be discussed further in Section 8. Eventually we wind up with a formula for the log of the partition function which displays its analytic properties. We evaluate derivatives with respect to “sources” using Cauchy’s formula and thereby obtain formulas for truncated correlations which exhibit their exponential decay and prove our main theorem. We conclude with a summary of notation; it is generally consistent with (5 ]. Given a bond b’ = (x’y’) with x’, y’ in the coarse lattice (LZ)d we let B(b’) = {(MI) c Zd*: 24E B(x’), u E B( y’)}.

Thus B(b’) can be thought of as a face between two L-blocks centered on the ends of b’. We shall sometimes identify b’ with its associated face.

290

BALABANETAL.

Given a subset TC (J~Z)~*, we denote by X(T) the union of L-blocks linked by r,

X(T)=

(J XE someb’sr

B(x)-

We say r is connected if X(F) is connected.

6. CLASSICAL

ASPECTS

Using Theorem 3.1, rewrite the expectation in the form

+ew 1- XSA c b%&d”- fp”’I+Wl’+ElI -FMA). We exhibit the Higgs mechanism on the classical level by using the standard change of variables fd= pe’*,

A+A-id&

e

Then (FL,=+

c A

( gAJorn@p

v:dv=O

c

. exp -4 c w(P)+4m2 I PCA

ie‘4c=)p(y ) - p(x)\ *

+exp -f i

I

- exp - c I

I

XCA

[Ap’(x) - ~,u2p2(x)+ E - log 27~p(x)]

I

F.

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MODELS

The log p comes from the Jacobian of the change of variables. exponent (excluding logp) occurs at

The minimum

of the

A = 0, u = 0.

Po=P/dK

After the translation p -+ p. + p, the action has the quadratic piece

+i

x

@(Y>-P(x))*

++*

x

P'(X)

(6.1)

XE A

WY) = A

and an interaction piece

VI@, A) = pi

x

+po

(1 - cos eA,,,, - ie*Ai.J

=A

(XY)

z (XY)

+

cA

@(Y> +P(x))(~

\' P(Y)P(x)(~ (XYT A

+ 7 z

- ~0s eA~,,J

--~seA~,,,)

(lp4(x> + @~P3(X)-

log ( 1 + 9).

(6.2)

We have altered the normalization by subtracting log(2n po)i /i 1. The expectation is unaffected and is

Here Z = Z, normalizes the expectation to unity. Formally V, tends to zero in the weak coupling limit, so the expectation is expected to become the massive Gaussian with the action (6.1) in the weak coupling limit. Rather than obtaining expansions for (F), we add to the interaction an extra “source” term, V2, where (6.3) Here f runs over some finite set of functions on bonds which are nonzero for finitely many bonds, and the parameters ar, a, are complex numbers subject to the bounds

292

BALABANETAL.

We can usefs that lead to non-gauge-invariant at this point. We let V= V, + V2 and define

Z=

IC

gA

terms in V, because the gauge is fixed

O” GTpepS. f -00

We will find an expansion for log Z and obtain (F) by applying a suitable differential operator in the a’s which are then evaluated at a = 0. The derivatives will be expressed by Cauchy’s formula. The bounds (6.4) ensure the V, is small in the same sense that V, is small, and the combination V is treated as a single entity. This way of getting at expectations does not give bounds on (F) which are asymptotically correct as I + 0, but it is sufficient to prove our main theorem. Our expansion is defined in terms of the coarse lattice /i’. We assume that L, the block size, is chosen so that factors exp(ie CfaAb) in V, do not couple disjoint blocks. This can be achieved for arbitrarily large L.

7. LARGE AND SMALL FIELDS, CONDITIONING An important part of this expansion is to use separate methods on large and small fields respectively. The number p(A) given by p(l)=

IlogLld+’

will be the borderline between large and small; i.e., A, is large if lAbj Xp@), p(x) is large if [p(x)1 Zp(lz). The discrete vortex field v(p) is large iff v(p) # 0. Partition

of Unity of Field ConJigurations

We choose a function x of one variable as an approximate obeying (1)

x E cm, x(t) = x(-t),

(2)

1x1G Lx(t) = 1

(3)

characteristic function,

if 1t I < l/2, =o ifIt < 1 For each n = 1,2,...,

I I -&x

Q cn(n!)p

for some c and p independent of t. Let

C=(l-xx)

(7-l)

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293

so that

1= XEA n W(X)/P(~)) +&4X)/P(~))) =xFA?QX X@(XYP@)) n C@(X)/P(~)) XtX

and similarly

YcA-

bsY

l =zc‘4** c PEZ rI hPw,(~)n 4.(,,.,,04 P@

Here 1, denotes the characteristic function of the event E. We combine these three partitions of unity by multiplying them. For a given term in the resulting partition of unity, let A, be the largest union of L-blocks c A with the property that every site, bond, and plaquette in A,, is selected by x’s to be small field. Set

and resum the partition

of unity holding A, fixed. We write the result in the form

where xA, is the characteristic function for the event that 4, A are small at sites and bonds in A, and u = 0 on plaquettes in A,. x,, Cis the characteristic function for the event: For each L-block B(x) c A:, at least on: field associated with a site touching B(x’), or a bond, or plaquette, intersecting B(x), is large. Finally, it is convenient to include in <,,C the characteristic function of the range of A- and p-integration. We will suppose thil is done and not change notation. (The xfunctions already force p and A to be inside the integration range if A is sufficiently small.) Conditioning We defer the integration over the fields in large field regions as selected by partition of unity, and by multiplying and dividing by a suitable factor, normalize remaining integrals so that the Gaussian part is a probability measure. Define, for any subset XC A, the Gaussian action S,*(X) by throwing away, in action S, defined above, all terms labelled by sites, bonds, or plaquettes which do lie in or intersect X; i.e.,

the the the not

294

BALABANETAL.

+fm:

A:+$

c

bcA,bfW#D

* MY>

Define S*(X)

-P(X))’

c

(XY)CA,(XY)Tw#er +

‘uu’

c

xex

(7.2)

P’(X)-

and V*(X) by applying the same operations to S and V. Then

s = s*(v) + S(F),

(7.3)

so that

Define a normalized

Gaussian measure with nonzero mean by

&, = &

&i?Ax~pxe-s;(x*u=o! 0

Then

(7.6) (7.7)

where E(X) E 1 dpxe-v*‘X’xX.

(7.8)

8. CLUSTER EXPANSIONS

These are analogous to high temperature expansions of are applied to the factors Z,*(A,), E(Ao) produced by the tation is quite complete but nevertheless some review of prelude to this part of the analysis. There are three operations: (1) an expansion for log Z,f

statistical mechanics and conditioning. Our presen[3] is probably a useful ; (2) an expansion for 8;

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(3) the union of (l), (2), and conditioning into one expansion. The expansions of (1) and (2) are only rapidly convergent far away from the break-down of the Gaussian approximation. Accordingly we define, for any union of L-blocks X the set x,

5

U W’) x’:IJx’-X’ll
where r(A) = llog J12. This operation is applied to the large field region /iG to get (A:), =/if incompatible notation which should not cause confusion

and with an

The expansions are localized in /i , . The choice of r(A) is dictated by two requirements: (a) the large action caused by even one large field produces a small factor exp(-O(p2(L))) which dominates a factor O(r(A)d) arising from crude estimates of the vacuum energy in an r(1) neighborhood. (b) the effects of large fields are exponentially damped away from LIP so that an expansion in /i 1 is perturbed by the conditioning at the boundary of ~4‘0 by effects of size of the order ofp*(A) exp(-r(L) minti, m,)) which tends to zero rapidly with 1. Expansion for Z$ We introduce an interpolated Gaussian action S,*(s) and a corresponding Gaussian partition function Z,*(s) where s is a collection of parameters, each in [0, 11, one for each bond b’ E A I”‘, labelling a face between L-blocks. We define this action by multiplying all terms in S$(v = 0) which couple blocks B(x’), B(y’) by s(,,,,) so that s~,,~,) controls the strength of coupling between B(x’) and B(y’). Thus S,*(s) E ; C a,(dA(p))’

+ fm: 1 A i b

P +

+ 1 U(xy,@(Y> (XY)

-

PW

+

iP2

z

P’(XX I

where the sums are over p, b, (xy) which touch .4 0 and x is summed over sites in II 0. Furthermore u P = s(x’y’)

==l G(XY)= S(X,Y’) =z 1 595/158/2-2

ifp couples blocks B(x’), B(f) ifp does not couple L-blocks or ifp intersects four L-blocks if (xu) E B’((x’y’)) if (XJJ) does not couple L-blocks;

(8.1)

296

BALABAN

ET AL.

thus we have left bonds and plaquettes which do not couple L-blocks undisturbed and in addition no bonds near CM,, are altered. In particular, when s = 0 all L-blocks in /1, are decoupled from each other and from n t. Each bond coupling two blocks is decoupled from everything else (except for bonds in a plaquette intersecting four blocks-they are coupled only within that plaquette). Following [3] we interpolate between all s = 1 and s = 0 using the fundamental theorem of calculus logz,*(/i,)=logz,*(s=

1)

= c I ds, a’ log z,* (sr) r where r is summed over all subsets of bonds in Ai, including means that ar and the s-integration are omitted),

the null set (which

yrlfb’ sy

=

(Sb’)

with sb, = 0 if b’ 6ZIY

We define W,(T) =I ds,8

log Z,*(s,)

so that (8.2) becomes Z,*(/i,)=Z,*(/i,,s=O)exp

C (r+*

wl’r’l

*

We divide by the same expansion (i.e., decoupling on faces in /ii’*‘) that Goo)

= ZOV)

for Z,(A), so

zO*(AO~ s =O> ,rrt,w,crj z (A s = 0) 0 3

where I*

= WlV,AO)

- WlV94.

Note that W,(r) = 0 unless r touches A f . Also, if r is not connected, then Z,*(sr) factors and again W,(r) = 0. We substitute this expansion for Z$ into Z(A) as expressed in (7.6) and obtain

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297

MODELS

where p,(llC,) is the function of fields p, A associated with AC, defined by pi(X) E

C

e-s~(x)-y(x)~xZo*(Xc,

s = 0)/Z@,

s = 0).

u:dv=O

Expansion for B

Now we will obtain a polymer expansion for the factor

As in the last section we introduce the Gaussian measure dp, by interpolating the terms in S,* that couple across faces in A;‘*‘. In an analogous way we introduce sdependence in I/*, defining V*(s) by making the replacement

scA;‘c (*.*I-+(xygAc*c (XY) I %Yk..) in the definition of V*(/i,). The u-factors were defined in (8.1). Note that V, is not affected because the coarse lattice is chosen to avoid couplings by V,. By interpolating between s = 1 and s = 0 using the fundamental theorem of calculus we obtain, as in [3],

where, if r=

0, the sr-integration q/i,,

and ar are omitted, and s) = j dp,e-v*(‘)x.

Let us define B,(/i,) to be 8(/1,,0) but with sb = 0 for all b c/i;, conditioning equal to zero. We then put

K,(T) = J dsJqA,,

and with

sr)/~o(Ao).

We call r /1,-connected if either (a) r is connected, (b) the connected components of r may be connected by adjoining connected components of Af’. Then K,(T) factors across connected components:

K,(r) = peAI-connected

rI

components

K,@:) ofr

298

BALABAN

ET

AL.

and

where the sum over r includes the null set. The definition of K, is a ratio of large volume @IO) quantities, but since So, = 0 for all b’ not in r, the numerator and denominator factor into contributions from blocks filling /i ,\{ blocks connected by r}. These blocks cancel out, including integrals over gauge fields associated with bonds in between blocks, so that K,(T) is really a “small” volume (blocks connected by I’ and associated “in between” bonds and plaquettes) quantity. When this expansion for B IS ’ substituted into our expression (8.3) for Z(A) we obtain

*exp (F K(0)

T&(d)

where we have changed r to A in the K, sum to avoid confusion with the W, sum, and p2 is pi with S-factors included: p,(X) e

C

e-s~cx)-ycx)~x

v:dv=O * (Z,*(X’,

s =

O)/Z(A,

s =

0))

E,(XC)/E,(A).

Uniting the Expansions The large field regions represented by the p-factors transmit information via the conditioning at the boundary of rlC, to any quantities W,(T), K,(T) occurring in the expansions for Z$ and S such that r touches /ii. In this section we collect the components of /if along with any ‘s touching /it components into clusters C so that different clusters do not touch directly or indirectly through W, or K, factors. Then there is complete factorization of gp, c~A integrals attached to different C’s. First we expand the exponential in W, in (8.4) according to

exp (cF(T)) =r=(rF..,r”) +!cr w2m where the sum runs over all ordered collections (= ordered sets which are allowed repeated elements) of connected subsets of A I*. In this way Z is expressed as a sum over configurations A E, r, A. A configuration @I;, r, A) is said to form a cluster on C, which is a subset of/i, defined by C=A;uX(I-)UX(A)

iff C is connected.

MASS GAP FORHIGGS

299

MODELS

Not every configuration occurring in the expansion for Z is a cluster on some C but we can uniquely decompose any configuration into clusters on sets C E {C, . c 2,..., C,: n arbitrary} s C. We define K3(C) to be the sum over clusters on C:

KdC) =

~AGP P,(R) fl

ret-

w;OK:kV

(= 0 if sum is empty).

(8.5)

Then Z(A) = Z,,(A) E,(A) 1 n

K3(C).

(8.6)

c CEC

The superscripts R on W, and K, mean that AC, has been changed to R, A; to R, in their definition (8.5). We can do this because W,(T) does not depend on connected components of AT not touched by r, and K, factors across connected components. C is summed over all sets whose elements are connected unions of L-blocks and no two elements touch.

9. TAKING

OF Z

THE LOGARITHM

Z has been expanded in the form

z=z o&c

n KAC) c ccc

where the sum over C ranges over sets whose elements are unions of L-blocks such that no two elements touch. Furthermore K, vanishes unless its argument is connected. We take the logarithm by a now standard procedure ]7] of regarding this expansion without the two multiplicative factors Z,, E, as a grand canonical ensemble of a hard core gas of connected subsets and applying the Mayer expansion. The procedure and our main estimate are given in more detail in the Appendix. The results are log z = log(Z, E,) + \- &(W, xc* where K4(*)

=

x:x,&

,!I,

K3(X)

=s GElconnectedgraphso”

XI

n XYEG

(U(X,

Y> -

1).

300

BALABAN

ET

AL.

Here U is defined by U(X, Y) = 0 = 1

if X, Y touch otherwise

and X is summed over ordered collections of connected unions of L-cubes. To discuss convergence we define the c-norm by

The result is, for any c > 0,

Furthermore when the right-hand side is convergent, the analyticity properties of K, as a function of parameters in its action extend to K,-because it is a uniform limit of analytic functions. Proof of Theorem 4.1. norm of K, which will be We obtain a formula derivatives with respect to example,

We can now give the proof, assuming an estimate on the cproved in the rest of this paper. for the expectation of F by applying a suitable set of the a-parameters occurring in V, to log 2. (See (6.3).) For

KIWI Id(Y>l)- (I4(~IXl~(YM

(9-l)

as allowed in (6.4). The restrictions X 2 {x, y} Here we take (a,l-‘, la,/-‘=A-“, may be imposed because X4(X) will vanish when the a-derivatives are imposed if

MASS GAPFOR

301

HIGGS MODELS

X 5 {x, y}. We assume X, y are in different blocks so that the derivatives vanish when applied to log 5,. In the next section we will prove that, for any c if L is large enough, as 1--t 0.

llK3 I/c + 0

Thus exponential decay for this correlation is proved for A small, uniformly in A. Other correlations may be handled by variations of the same argument. To see that the infinite volume limit exists we note that in (9.1) there is a non explicit constraint XC A on the sum and this is the only dependence on /i. It is clear that a change in /i affects the sum by adding or subtracting terms of the form K&Y), with

x= tx,JJ1 These additional use

terms are exponentially -5x,xax.

xn/ic

and

# 0.

small in the distance from {x,y} to 8/i. We

IKIW G e- cdist(lx.yt,AC)lL

~~~~~~~

;;r;xnAc+!z

to see that the infinite volume limit exists, for 1 small. The same type of argument demonstrates exponential decay for correlations of Wilson loops. At first one obtains exponential decay only for Wilson loops translated by multiples of L because the coarse lattice must be chosen to contain each Wilson loop within single blocks. However, the argument can be made for a finite number of different L’s and thereby extended to arbitrary translates of Wilson loops.

10. COMBINATORIC

ASPECTS

In this section we will give the combinatoric

OF CONVERGENCE

aspects of the proof that with A,

IlK3llc+ 0

deferring until a later section some estimates which require a detailed grasp of Gaussian integrals. Given a function f on subsets of A’* we define its c-norm by

If f is a function on unions of L-blocks then we define its c-norm by

Ilfllc- sup I’

If(X)1 eclx”.

c X:X

touches

x’

302

BALABANETAL.

The following two lemmas contain the basic combinatorial ideas, then we state the three propositions which contain the more technical analysis and combine them to prove that )IK, Jlc-+ 0 with 1. PROPOSITION

10.1 [3,p. 2181.

There are constants c,, c2 depending only on

dimension such that

(a) (b)

xx eWC1’*” < 00 Cr ePc21rl < 00

where X is summedover all connected unions of L-blocks touching the origin, and I is summedover all connected sets of bonds b’ c (LZ)d touching somefixed site x’ in the coarse lattice. In part (b), III can be replaced by IX’(I)1 with a dtJ2rent c2. LEMMA

10.2.

Let f (X) be a function on unions of L-blocks and let

(=l if X = 0),

g(X) = n f VI; xex

where X is a collection of setsX. Then (4

CX:XtoucheSy Ifml~llflloI~‘I

(b) C X:eachXtouchesY cc> c

(‘/lxl!)

X#O:eachXtouchesY

I gcx)I (l/ix!!)

<

1 g(x)i

exp(iif ilOI y’l>
where A > 1. Analogous statements with X, X replaced by I, r hold Proof.

iff =f (I’).

(a) is trivial and implies (b) because

For (c), we apply (b) with f replaced by Af: PROPOSITION

10.3. P3V)

For all L and all c, p, which is defined by c

=

R~(R:R~=xt;;~connected)

sC -@A,w~PRP~R)

tends to zero in c-norm as I + 0. PROPOSITION

10.4.

Given c,, for L suflciently large and 1 suflciently small, it

follows that

II w; llc,< C*Ld-‘.

MASS GAP FORHIGGS

For the next proposition

MODELS

303

we define K, by if r is connected otherwise

K;(T) = K:(T)

=o

PROPOSITION 10.5. For all c, ifL is suQj7cientlylarge, there are constants c,(k), cZ where c,(n) tends to zero with J and such that for 1 small

where r is summedso that all its connected components touch RI. If R = 0 then this reduces to IlK~llc G c,(l)If r = 0 then for r small, depending on L, /K;(T)1
With the aid of these lemmas we now prove that for any c, if L is large enough,

IIK, IL-+0 Define R:,

as A -+ 0.

@: by R;“(r) = /K:(T)1 eclx’(“’

together with an analogous formula for @. Also define x3 and p” by z3(C) = IK3(C)I eC’(“’ p’(R) = p,(R) ectRti.

The definition

of K,, and in particular

the connectedness of C, ensures that

We estimate the r sum using Lemma 10.2. It is less than exp(/l W:II, IR; I), because every r must touch RI or else W!(r) = 0. By Proposition 10.4 this in turn is less than exp(c,Ld-’ IR; I) provided L is large and A is small (depending on c). We hold R and the components of d which touch RI fixed and estimate the sum over A subject to these constraints. The sum over A is, equivalently, the sum over the collection of connected components of A. By Lemma 10.2 it is less than

exr41K211c IW

304

BALABANETAL.

(We have replaced R by the null set in K, because if r does not touch R; then Kfj(r> = K:(T).) By Proposition 10.5 this is less than exp(c,(A)/ C’ I). Next we use Proposition 10.5 to bound the sum over the connected components of A that do touch R; . There is the possibility that this part of d is null. Thus we obtain the bound [ 1 + c3(A)] ec41Rt’. If 1 is sufficiently small this is less then 2eC41Ri’, so that R,(C) < 2

c

1 gAC3p p,(R) eYiRi’

RcC

provided L is large and y=c,Ld-‘+c4+c. This bound is improved R = 0, R # 0 separately. r = 0 and X(A) = C. By

R,(C)Gc A,X(A)

=

1 is small, depending on c. Here y = y(L, c) is given by by considering, in the foregoing analysis, the possibilities If R = 0 then the only way to form a cluster on C is to let separating out these terms we obtain Z??(A) + 2

c ( GiFACSpp,(R) eyiRi’

=C

A *;),=,

ec2(‘)ic” 1

Rf0

m4

+2

x XCC.X#0

n

b3(y)

eYl”l)

ec2wc’I,

Y~(conn.cPts.ofXl

where pj is the quantity that appears in Proposition 10.3. By Lemma 10.2 applied to the sum over X, rewritten as a sum over the collection of connected components of X, R,(C)<

c

Rfyd) + 2A -1

A,X(A)=C

If Iz is suffkiently

small we can achieve

so that

fdc) <

C

@(A) + 2 lip3 llye21c” ;

A,X(A)=C

i.e.,

IKACI Q

c A,X(A)=C

IKWl + 2 llp3Ily~(z-c)~c”t

MASS GAPFORHIGGS

MODELS

305

and therefore for any c, if L is large, IIK3ll,+ by Lemma 10.1 and Propositions 11. LARGE

0

asA+

10.3 and 10.5.

FIELD REGIONS;

PROOF OF PROPOSITION

10.3

In this section we quantify the idea that large fields are suppressed by action density by proving Proposition 10.3. LEMMA Il. 1 (Stability). There exists a constant y(u) > 0 independent of m, , A, e such that on the range of integration,

Proof: It is easiest to work with the untranslated p and A. Insert into S, + V, the elementary bounds (a) (b)

Ap4- $*p* + E 2 +,u’@ -ppo)*, t leieAw P(Y) -PW > t@W -P(X))'

+ frlp(y)p(x)(eA,,,,)',

where y, is independent of ,u, m, , II, e. Case 1.

If p(y) p(x) > fpg, then we continue (b) with 2 fb4.d

-P@))'

=%AY>-P(x))~

+ b~~~p~e*A&,, + fan&-&.

Case 2.

p,,/@

If on the other hand, p(‘)p(x) < +pi, then p = p(y) or p(x) is less than and we continue (a) with the bounds

valid on the range of A, and in either case the requisite lower bound follows. LEMMA

11.2.

Let X be a union of L-blocks in A. Then there exists c, , cz so that r gpxgA,,.,p2(X)

< c\xl’e-c~2(*)‘x”.

306 Proof:

BALABAN

By the Stability Re WI

ET AL.

Lemma 11.1,

2 ~01) S,(x) + Re v2(x) > cl fMx) - c2 1x1

for some cr, c2 > 0 and A small. For any c Le- (lI2kS~W < e-c’wPwwI since [ forces each L-block B(x’), x’ E X’, together with its nearest neighbor sites to contain at least one large field. (When v(p) # 0 then either (Q!A + 0)’ (p) > cp2(J) or else A(b)’ > q’(A) f or some b Ep.) We substitute this bound into the definition of p2 along with, for 1 small depending on L, IZ(Ao =Xc, s = 0, cond. = O)/E(A, s = O)l < exp(c IX; I). To obtain this estimate we note that regions outside X, cancel exactly. Inside X, we use

x Iew(-VI G ew(c IT I> in the numerator and bound the denominator

using u&xc Re ePvcc’

components

C

The last type of inequality is discussed more fully at the end of Section 14, below. Now we are reduced to proving that for each c, > 0 there exists c, such that e-c~socx)Z~(Xc,

s = O)/Z,(A,

s

= 0) < cLxll.

(11.1)

v:dv=O

We drop the constraint du = 0. As before each u(p) # 0 produces a factor O(M:/e’) or O(v(p)‘/ e’) in the exponent. After summing over u(p) # 0, the exponent is still bounded below by a constant times the u = 0, A-field form. Thus

W(P) + U(P))’- hmfi c bcX

-%h>

c (d410)~ PCX

- fcmi

c bcX

which holds at sufficiently weak coupling in the range of A-integration. the left-hand side of (11.1) is less than 9pxG9Axcsce

-c(ma)So(X.“=O)Zg*(XC,

,q =

O)/Z,(/i,

s =

Consequently

0).

MASS

GAP

FOR

HIGGS

307

MODELS

When we substitute in the fact that Z,*, Z are integrals over exp(-S$), exp(-S,) and make use of the decoupling caused by s = 0 to cancel out contributions from XT we obtain a bound 21x1

which is less than cixl’ for some c2 by scaling p and A in the numerator. Proof of Proposition

10.3. By Lemma 11.2,

<

eeCIX’

+

r

eclR,le-c~2(~)lR’l

.

Re(R:R,=X)

By the definition

of R, ,

for any E > 0 if A is small enough. Thus for a decreased constant c2 PI(x)

< e-~lX’l

C

e-c~*(~)lR’l~

RcX

We estimate the sum over R by rewriting it as a sum over the set of connected components of R and applying Lemma 10.2 with f(R)

=

e-czpZ(~)IW,

= 0,

if R is connected, otherwise,

so that

Since, by Lemma 10.1, llfll o -+ 0 with 1, we can choose A so that A ll$l/o = 1. We find p&Y) < jlfllo e-‘C-l’lX’l which, by Lemma 10.1, proves the proposition.

308

BALABAN

ET AL.

12. PROPERIES OF COVARIANCES; DIFFERENTIATING

GAUSSIAN

EXPECTATIONS

FORMULAS FOR WITH RESPECT TO S-FACTORS

This section is very close in spirit to parts of [3]. We want to establish analogues to the formulas of [3] for a conditional Gaussian measure. The covariance for the gauge field also needs discussion. We establish here the basic estimates used to prove Propositions 10.4 and 10.5. We will use (. ) to denote Gaussian expectations with respect to the Gaussian measures (7.5) or one of its s-dependent analogs. Estimates on Covariances.

We shall set

c&G u) = @(xl P(Y)) - @(X)MY))~ CA& cl = @@)-4(c))- (A@))@(c)). Since ( ) is Gaussian, we can evaluate

qxLY)= ~P’-4)1-’ (&.Y> C,(b, c) = [m: + 6d(s)] -’ (b, c), where d(s) is the linear operator on r’(A,) defined by

(4-4) 6 = f c qxy,tw - W)‘, (X-Y) c A

with 6 = 0 on Ai. The u’s depend on s; see (8.1). Likewise &f(s) is the linear operator on I*@,*) defined by

(w9Ws) w) = f 2 ~pvW)2(PI PCA

with v(b) = 0 if b & A,. LEMMA

12.1.

There exists m = m(u, m,), nonzero if m, , ,u > 0, and c = c(j~, mA)

such that

C,(x, y) < cepm Ilx-yll c,(b, c) < cepm dist(bvc) uniformly ProoJ

in s, A 0. The easiest method to prove this seems to be an idea of Combes and

MASS

GAP

FOR

HIGGS

309

MODELS

Thomas. (See [8] and also [9].) We will work through the CA case. C, is the same with obvious variations. Define a multiplication operator V = V(a) on l*(A,*) by f(b) Y(a)

@‘““f(b)

where xb is the midpoint of bond b. Then, letting 6, denote the element of f*(A$) which equals one on bond b, and equals zero everywhere else,

IC,(b, cl = I@,, bfi + Ws)l -’ J,)l = I(VS,, v-‘(mfi + 6d(s))-’ VP6,)l -‘II. Next we use the fact that if A is an operator on 12(A$), such that for all v/ E 12(A ,f ),

hww>clIv11*~ then (IA -‘)I < c-‘. Thus (WY[m:, - v-‘Jd(s) = rn: c 1y(b)l’ b

VI w) + 1 up

1

e-“by(b)

bsap

P

c

e%y(b)

,

bcap

which is a strictly positive operator if JlaJI is not too large. This proves the lemma. Derivatives

of Covariances

We evaluate derivatives of C,, C, using

& [P’-4W

= [P’-&sr’

(g-/(s))

[P’-4)1-’

repeatedly. The result can be wriiten in the form @C,)(X~ v> =

2,

steFEu CJstep)

where w is summed over all walks from x to y which consist of a sequence of steps of the form (x3 b,), (b, 7 bJ, (b, 3Wv.., (b,, y> where each bi, I = l,..., n, belongs to a face in r, so that each face is visited once and only once. Thus n = Irl. To define C, (step), let b, c be the bonds b = (tu), c = (VW) and set C&3 c) = C,(x, v> - C,(x, WI, C,(b, c) = C,(t, c) - CJu, c), C,(b, x) = C,(t, x) - CO@, x).

310

BALABAN

ET AL.

This sufftces to define C, (step) since a step is either a pair of bonds or a bond and a site. An analogous formula holds for C, except that sites get replaced by bonds and bonds by plaquettes which link L-blocks linked by IY These formulas enable us, in conjunction with Lemma 12.1, to read off the following estimate: LEMMA

12.2.

There exist ci = c&u, m,), i = 1,2, m = m(u, m,) such that

where d(x, y, r) is the length of the shortest path which joins x toy and visits a bond in each face in K Define d(b, c, r) analogously.

By increasing L, the exponential decay dominates clr’. Finally note that

@(x>)=

2

c&Ax,Y) P(Z)*

(YGEaA,,rdA,

Together with an analogous formula for (A(b)),

we can estimate derivatives

la’@(x))l

< c,cir’p(A)

e-md(x,anl*r),

la’@(b))1

< c,cir’p(12)

e-md(b9aA19r),

(12.1)

where ci = c,(m, p), i = 1,2 and d(x, &I r, r) is the distance to the boundary &l r from x, via I’. Here d(b, &I r, r) is defined analogously. The formula for differentiating an s-parameter appearing in the Gaussian measure is

p

(P) = ((Ki’ + Kg P) b’

where the K’s are differential operators

MASS GAP FORHIGGS

MODELS

311

These formulas are the same as the standard one in [3], except for the first order terms which arise because the measure does not have mean zero. They may be proved by first translating so that the measure does have mean zero, using the standard formula, and translating back again. Following [3] we can write a formula for a multiple s-derivative: (12.2)

KY=~;+~;,

(12.3)

and P(r) is the set of partitions

of lY

13. PROOF OF PROPOSITION

10.4

We begin by estimating I = ar log Z,*(sr). If we pick one bond b’ in r and differentiate with respect to sbI , we obtain

Now we perform the remaining tiating the expectation:

derivatives, a’, using the formula (12.2) for differen-

Since S,* is quadratic, the only partitions 71which can contribute are those with one or two elements. There are less than 2”‘-’ such partitions, so

Lemma 12.2 and (12.1) tell us that there exists m > 0 such that

595/158/2-3

312

BALABAN

ET AL.

where d(T) is the shortest path connecting faces in IY Since p exp(-mr) with A, for 1 small and a new c,,

tends to zero

111< cl c~r’Ld-le-md(r). By considering faces in r perpendicular prove that

to each coordinate axis it is not difficult to

d(T)> ($rl-

1) L.

Therefore, by Lemma 10.1, if L is large enough, II exp(-W~Nll < oo, and Proposition 10.4 is proved. 14. PROOF OF PROPOSITION

depending

on

c, then

10.5

If r is the null set, then K,(T) has the form B(R’,

s = 0 in Rt)/8,(RC).

Since s = 0, the couplings across boundaries of (connected components of R,)\R and L-blocks in Rf are zero, and numerator and denominator factor. They would cancel exactly, except for the conditioning in the numerator and the extra s = 0 bonds. Consequently contributions from (connected components of R ,)\R do not cancel. We bound the uncancelled factors in the numerator above and the factors in the denominator below, just as in the proof of Lemma 11.2, and obtain, for 1 small depending on L,

The definition

of K, implies eclr’ Q c r+0

sup (8E(Rc, sr)/E,(RC)I ecirI, sr

where the sum over r is constrained so that each connected component of r touches of sr, all coupling across faces not in r or in R r is absent, if we set

R. Since, by the definition

X= (R,\R)UX(I’),

then both 8’s in K,(T) factor across aK and the factors corresponding to Xc (and 8X) cancel in the ratio so that arE(Rc, s,)/E,(A,)

= ar(e-“*(“,Y‘)X~)/B,(X).

MASS

GAP

FOR

HIGGS

Furthermore E,(X) factors across all boundaries of L-blocks in x\R, connected components of X. As in the proof of Lemma 11.2, we find

for ,I sufficiently

313

MODELS

and across

small, depending on L. Set I,

ZE z,(r)

= Jar(e-““x~B~‘X.Y>I

so that (14.1) Here x\R 1 is a disjoint union of sets of the form 2 = X(f) where f is a connected component of r. Accordingly Ip 1< 2 Ir”i and (X’I < IRi / + 2 /rl. This estimate is used to eliminate IX’1 in (14.1). We find that in order to prove the proposition, it is sufftcient to prove that given c2, if L is sufficiently large and i is sufficiently small, then Z,(r)

,< c,(A) eCc2tr’ec~‘R~i.

(14.2)

Here c,(d) -t 0 with A. By Lemmas 10.1 and 10.2, if c, is large enough, then ‘T e-‘l”’ 7


for some c2. Besides the s-dependence of ( ), the V is also a function of s; each derivative in P“ can act on either dependence, so we write

where the subscripts (), V specify where i3 acts. The expectation is analytic as a function of the s-dependence in V because x limits the size of the fields. We estimate the 8, derivatives by Cauchy’s formula: For any A > 0 (and we choose A large for /I small) I, < 21r’ sup A-lrvl’ r,cr

sup Iar;(e-“x)l,

(14.3)

y.EilDA

where 21r’ is the number of terms in the expansion of (a, + ac,)r, and D,d is the contour corresponding to the polydisc of radius A centered at the origin in complex sspace. Now we concentrate on estimating

314

BALABAN

ET AL.

for arbitrary Z. By the formula for differentiating

expectations (12.2) we find

Each K in I, is split into its first and second order parts and into its a/aA, a/+ parts. Let I, be the largest term in the resulting sum I, < 41r’z

4’

Next we write each partial differential operator as a sum over the positions (x, b) of its p or A derivatives. We take absolute values inside these sums and bound sdifferentiated covariances and @), (A) factors using Lemma 12.2, and (12.1):

r(y) is, depending on y, either a pair of sites, a pair of bonds, a single site or a single bond. The r(y) label positions of field derivatives which are collectively denoted by D. Here d@(y), y) is the distance along the shortest path that links the element(s) of r(y) and the faces in y. We have used I@(X))/, /(A(b))/ < 1 for 1 sufficiently small. This follows from (12.1) and p(A) exp(-r(A)) + 0, as A--+ 0. By Leibnitz’ rule, I(DW”*x)>l<

41r’ I@,@-“*)

D,x)I,

where the derivatives in D are assigned to either D, or D, so as to maximize. We estimate derivatives of exp(-V*) with, for small fields, and A sufficiently small depending on L, )D, e-‘*

I< n (n(~‘)!)~ X’

(q(A))“”

dx’I,

where n(Y) is the number of derivatives localized in B(x’); N, is the total number of derivatives in D, ; p, c are constants and

with /3 > 0 a constant. This estimate is easy and the proof is omitted. The derivatives of x are bounded with W,xI)

<

n

x

t~tx>Y

n

WNp

~LqW”-“’

(14.4)

b

where p, c are constants; q(l) = O(JO); N, is the order of D, ; n(x), n(b) are, respectively, the number of derivatives at site x, bond b; and M is the number of derivatives within distance r(A)/2 of R,. The proof of this estimate is deferred to the end of our argument.

MASS

GAP

FOR

HIGGS

315

MODELS

We combine the last two estimates with, for E > 0,

n (n(x’)!)“lfl x’

(n(x)!)“* n (n(b)!)P3< c:

x

fl

b

e-‘m-“d”‘y)*

y).

Y

This estimate can be found on p. 240 of [3]. It is not hard. The idea is that if one of the factorials is large then most of the corresponding derivatives are attached by covariances to distant locations. By the last live estimates we obtain (with different m > 0. and different c)

where N is the total number of derivatives. By decreasing M some more and increasing c, we may drop the (q()l))-M factor because exp(-sr(A))p(A) -+ 0 with 1 faster than any power of L for all E > 0. Let d(y) be the length of the shortest path connecting the faces in y. We estimate the sums over r(y), y fixed, using some of the exponential decay: with different m, c

We have eliminated N using ]r] > N > ] rc1. We now obtain an estimate on I, by doing the sum over 7cE P(r). Let Q(y) s q(3L)Iy12 eCmdcy) eriYir-’

where v < 1, r > 0. Then from our estimate on I,, 12

< w IX'I

t

irie-llri

2 n Q(Y) . I REP(r) YE= 1

We enlarge the sum over rc to all collections (sets with repeated elements) of subsets of r so that it corresponds to an exponential:

< vc ‘x’l+‘r’e-rirlexp(q(l)

9-l (11.I2 e-mdlll ITi).

For L large enough depending on < the <-norm is bounded by a constant, c, say. Consequently if we choose q = q(A), 12 G 4(k) c

i~‘l+lri~(-~+~,~rl

G 4(A) c l~it t 3irie(-l+c,)iri which yields (14.2) because, by taking L large, we can achieve any value for r.

316

BALABANETAL.

The proof is complete except for the bound (14.4) on derivatives of x. Since the expectation and the right-hand side of (14.4) factor into an A-field and a p-field part, we suppose without loss of generality that D, only contains p-derivatives. The M pderivatives that take place at sites x closer than r(J)/2 to R, are bounded by (7.1). Let 0” be the remaining N2 - M derivatives. Let the set of sites where these take place be denoted by S. Since x is constant except on the borderline between small and large fields we have

Ifix& n ((coWW/W-’

cosW4)l~xI

XES

so that by (7.1), (Ifixl)

<

n

,-Pu,Isl/2

c”‘~‘(~(x)!)~

( n

XES

Let ( ),, denote the expectation with zero conditioning n

cash p(x)).

XES

on R, then by translation

coshp(x) ) = ( n cash@(x) + @W)

XES

0

< 21S’ n

cosh(@(x)))

( n

XSS

cash/W)

XES

0

As A+O, @(x))+O uniformly in x by (12.1), together with p(J) exp(-r@)/2) + 0. Furthermore, by splitting each cash into exponentials and explicitly evaluating the expectation, we find (

n

cash p(x)

XPS

)0

< e@’

for some constant c. All these estimates collect to give

(I&l) Q n c”(“)(n(x)!)” e-p(n)lsl/* XES

for Iz small. This implies (14.4).

APPENDIX:

ON

THE

MAYER

EXPANSION

Let 9 be an abstract set with a reflexive binary relation “x touches y”, x,y E 9. We will say a subset XC g touches x E 9 if some element in X touches x. Likewise two subsets X, Y c n touch if they contain elements x, y which touch. Given f(X) a function on subsets of/i set

MASS

GAP

FOR

HIGGS

317

MODELS

DEFINITION. X is used to denote an ordered collection of subsets of Y’. A collection is a set which is allowed repeated elements. A- graph G on X is a set of unordered pairs XY, called lines, with X, YE X. Define f(X), where XC 9 by

-F-

X

GEco”“ect~maphs

n (ww- 1)

on X XYEG

where U(XY) = 0 if X touches Y and is 1 otherwise. The relationship

between f and Jis that if X is a set of subsets of /i c 2. and if

Z--Y-

l

-

;zT

n Xs~

IX/!

is the grand canonical partition

n X,YEX

f(X)

ww

function of a hard core gas, then formally (A.1 1

and furthermore, for all c > 0, E > 0, (A.21

A proof of (A.l) is given on p. 33 of [lo]. Our proof of (A.2) is mostly extracted from 1111. Proof of the estimate.

We will appeal to the following fact:

\‘ GEconnected <

graphs -c Getrgoon

[I

~WY) - 1)

on X XYEG

rI II

X XYEG

- 1I.

(A-3 1

Here a free is a connected graph which becomes disconnected if any line is removed. This estimate is proved in [ 121. The proof of a more powerful theorem allowing many-body graphs is implicit in [ 131. Let X E 4p be arbitrary, then

..tr;

on X XII,

I u(xy)

-

’ I

318

BALABAN

ET AL.

*

fi 1f(Xi)l c = nzl (n -! 1)’* x ,,..., x,cIP,someX13x i=l

n

z:

Gstreesonll,...,nl

I"(xfxj)-

eclxr’ (A-4)

II*

ijeG

Let di denote the coordination number of graph G at Xi. This is the number of lines that contain Xi. Cayley’s theorem says that the number of trees on n vertices with prescribed coordination numbers d, ,..., d, (adding up to 2(n - 1)) is equal to (+2)!/[

fi

(d;-

I)!]

(n>2).

(A-5)

i=l

In (A.4) we resum over Xi,..., X, and G while holding n and the coordination numbers of G fixed. We estimate the sum subject to these constraints by the number of graphs times the supremum over graphs, so that (A.4) is less than to (n-2)!

azl (?I - l)! ,,,z,,, Xl,...,X,

GEtrees

cY,x,3c

on [ l,p!,a1

i=l

with

d(fixed

1 (di - l)! Iftxi)l

ec’xi’

JJ 1U(XiXj) ijeG

- 11.

(A*6)

We perform the sums over the X, starting with the “outer branches,” i.e., those Xi for which di = 1, and working inwards using

v lf(m1 ecixiIW IWu) - 1I G I Yl Ilft *)I. IplIc

L XCY

and thereby obtain the bound O” (n-2)! #Z, (n - l)! ,,,T,,”

n 1 ,T=I d(i)! Ilf(*)l * IdCi)IIc

64.7)

where d(i) = di - 1 for all but one i =.i, say, for which d(i) = dj, j is arbitrary. Next we use IXld < d! ($--)

d e(‘+ 8)1X1

and sum over the dis to continue with

< F (n--V ,fel (n - l)! (

1+e “,,J,,” E 1

c+1+u-

When n = 1, (n - 2)! is to be interpreted as 1 so this implies our claimed result (A.2).

MASS GAP FOR HIGGS MODELS

319

REFERENCES 1. K. OSTERWALDER, AND E. SEILER, Ann. Phys. (NY) 110 (1978), 440471. (See also [lo].) 2. R. ISREAL, AND C. NAPPI, Commun. Math. Phys. (1979), 177-188. 3. J. GLIMM, A. JAFFE, AND T. SPENCER, “Constructive Quantum Field Theory,” Lecture Notes in Physics No. 25, Proceedings of the 1973 Ettore Majorana International School of Mathematical Physics (G. Velo and A. Wightman, Eds.) Springer-Verlag, New York, 1973. (See also J. Glimm, and A. Jaffe, “Quantum Physics,” Springer, New York, 1981.) 4. D. BRYDGES.J. FRGHLICH, AND E. SEILER, Ann. Phys. (NY) 121. (1979), 227-284. 5. T. BALABAN, Commun. Math. Phys. 86 (1982), 555-594. 6. J. GLIMM, A. JAFFE, AND T. SPENCER,Ann. Phys. (NY) 101 (1976), 610-669. I. T. BALABAN, AND K. GAWEDZKI, Ann. Inst. Henri Poincark 36 (1982), 271-400. 8. J. M. COMBES, AND L. THOMAS, Commun. Math. Phys. 34 (1973). 251-270. 9. J. FR~~HLICH,AND T. SPENCER,Commun. Math. Phys. 88 (1983) 15 1-184. IO. E. SEILER, “Gauge Theories as a Problem of Constructive Field Theory and Statistical Mechanics.” Lecture Notes in Physics No. 159, Springer-Verlag, New York, 1982. Il. C. CAMMAROTA, Commun. Math. Phys. 85 (1982) 517-528. 12. 0. PENROSE,“Statistical Mechanics” (T. Bak, Ed.), Benjamin, New York, 1967. 13. G. BATTLE, AND P. FEDERBUSH,Ann. Phys. (NY) 142 (1982) 95-139, (Appendix B).