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Constraint correlation functions in the O(N) model
Nuclear Physics B361 (1991) 392-414 North-Holland
CONSTRAINT
CORRELATION
FUNCTIONS
IN THE O(N) MODEL*
M. GZiCKELER Institute
for
Theoretical
Physics
E, RWTH
Aachen,
W-5100
Aachen,
Germany
and HLRZ
c/o
KFA Jiilich,
P.O. Box 1913, W-51 70 Jiilich,
Germany
H. LEUTWYLER Institute
for
Theoretical
Physics,
University
of Bern,
Sidlerstrasse
5, CH-3012
Bern,
Switzerland
Received 19 February 1991
For O(N)-symmetric spin systems in the spontaneously broken phase enclosed in a finite volume V, we study two-point correlation functions at a fixed value of the mean field @= I’-‘/ ddx $(x1. Using results from chiral perturbation theory, we investigate their large-volume behaviour. In particular, we derive a large-volume expansion for the correlation functions of the projected field Q%,,(X)= Cp* +(x)/191.
1. Formulation
of the problem
In this paper we continue our study [ll (see also ref. [21) of the large-volume behaviour of O(N)-symmetric scalar field theories in the phase where the symmetry is spontaneously broken down to O(N - 1). Models of this type have been extensively investigated by Monte Carlo methods for various values of N and in different dimensions d, in particular d = 3 and n = 4. The Heisenberg model of a ferromagnet corresponds to the case iV = 3, d = 3, whereas the theory with N = 4, d = 4 is of great interest in connection with the Higgs sector of the standard model of elementary particle physics. The numerical simulations are necessarily performed in a finite volume, although, in the end, only the infinite-volume limit is of interest. The occurrence of Goldstone bosons implies that these models do not contain a mass gap. Consequently, the effects generated by the finite size of the system persist even at large volume. The finite-size effects can be brought under control by introducing an external source which explicitly breaks O(N)-symmetry. The Goldstone bosons then acquire a mass and the finite-size effects become exponentially small, provided only that the volume is large enough, Even if this condition is not satisfied, the *Work supported in part by Schweizerischer Nationalfonds. 0550-3213/91/$03.50
Q 1991 - Elsevier Science Publishers B.V. (North-Holland)
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behaviour of various quantities, like the partition function, correlation functions etc., for large volumes and for weak external sources can be determined by means of chiral perturbation theory [3], which leads to an expansion in inverse powers of the box size. Since the coefficients occurring in this expansion are determined by the low-energy properties of the system at infinite volume, the observation of the volume dependence allows one to extract infinite-volume results from simulations in a finite box (see also ref. [4]). This method is discussed in detail in ref. [5] for the group O(N) and in ref. [61 for the group SUW) X W(N). Applications to the analysis of numerical data are given in refs. [7,8] for the three-dimensional Heisenberg model and for the O(4) model in four dimensions, respectively. A different approach to the problem of Goldstone bosons in a finite volume, which avoids explicit symmetry breaking, starts from the observation that spontaneous symmetry breakdown manifests itself already before the thermodynamic limit is taken: If the volume V is large enough, most of the field configurations are such that the directions of the field +(x> at the various points of space are close to the direction of the mean field
The vanishing of (+(x>) at finite volume is only due to the fact that Q, does not prefer any particular direction. Restricting oneself to field configurations with a fixed value of the mean field [9,10], the various finite-volume observables have a smooth infinite-volume limit. Moreover, the two-point correlation function of the projected field
(1.2) has been studied in the four-dimensional O(4) model [lo] with the intention to suppress the influence of the Goldstone bosons and thus to make the extraction of the Higgs boson mass possible. The present paper is devoted to a study of the finite-size effects occurring in this approach. To be more specific, we work in euclidean space of dimension d > 2 and enclose the system in a periodic box of sides L,, L,, . . . , L,. The mean length L is defined as L = V’ia where V= L i L 2.. . L, is the volume of the box. The action S{+) is taken to be invariant under global O(N) rotations of the N-component scalar field 9(x) = (4’(x), . . . , 4N-1(~>>. The d istribution of the mean field is given by the functional integral
(1.3)
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which, on account of O(N) symmetry, only depends on the magnitude @ = ]@I of 9. The probability to find a field configuration for which the mean field is contained in dN@ is proportional to .&@,)dN@. The logarithm of %(@)-’ is referred to as the constraint effective potential U(G), 2( 0) = const.
X
e-‘@).
(1.4)
The shape of the potential was analyzed in detail in ref. [l]. In the present paper we extend this analysis to the correlation functions of the scalar field. In particular, we examine the correlations at a fixed value of the mean field, , Ww#4Y))
*=
/[d4]ev(
&
-S(e))s(
@ - i/ddx
b(X))4A(X)4B(V).
(1.5) The analysis is based on the observation that the ordinary integral /dN@i(@)(+A(X)4B(Y))
eexp(**.W)
(l-6)
coincides with the quantity
/kWexp( -W
+j.lddx9(x))gn(x)~‘(y)
(l-7)
which, up to a normalization factor, represents the standard two-point function in the presence of a constant external field j. For this two-point function, the expansion in inverse powers of the box size is however known [3,5] from chiral perturbation theory, up to and including terms of order (l/~!,~-~)~. All we need to do, therefore, is to solve (1.6) for the correlation function. After discussing the basic identities in sect. 2 we collect some relations concerning the zero momentum projections of the various two-point functions in sect. 3. Sect. 4 reviews the results from chiral perturbation theory that are needed in the following. These are applied in sects. 5 and 6 to the transverse and longitudinal correlations, respectively. In sect. 7 we study the two-point function of the projected field. Sect. 8 contains a summary and our conclusions. 2. Identities In the presence of a constant external field, the two-point bPY+v(Yh=
&)
function is given by
/P6lew( -444 +j*/+ddx)&‘(x)da(y), (2.1)
M. Gdckeler,
where Z( j> is the corresponding
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partition function,
(2.2) In this notation, field distribution
the relation between the partition Z(Q) becomes Z(j)
function
Z(j) and the mean
=/dN@exp(@.jV)Z(@)
(2.3)
while the equality of the two expressions (1.6) and (1.7) takes the form
In eq. (2.3) the angular integral over the direction of Q, can be carried out with the result [ll Z(j)
=K/gmd~~N-lYN(~jV)i(~),
v=--1,; YN(n)=(;)-“z”(x),
K =
2?TN12,
P-5)
where Z, is the modified Bessel function. To perform the angular integral in eq. (2.41, we exploit O(N) symmetry and decompose the correlation functions as
( 4"(x)4"(Y>)
j= jA]BG,,(~-~,j)
(&-‘(x)@(y))
9=dA&%,,(x-y,@)
+{6AE--J^AJ^B}GI(~-~Y,j),
+ {S”“-&‘~B}~,(x-y,@),
(2.6)
where 1” and &’ are unit vectors in the directions of j and Q, respectively, while the coefficients G,,(x, j), . . . , GL(x,@) only depend on j = ljl or Cp= lcP1. Since C,(&%JA) is direction-independent, we get Z(j>{G,,(x,.i) =K
+ (N-
l)G,(x,i)}
~md~~N-lY~(~jV)i(~)(d,,(x,~)
+ (N-
l)d,(x,@)).
(2.7)
The projection of ( 4A(x)4E(y))j onto the direction of j, on the other hand, leads to an angular integral over the quantity (a *j>2 exp(@ *jV). This represents
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the second derivative of the exponential factor with respect to the volume, and the integral can therefore be expressed in terms of the second derivative of the function Y,(x). The result can be written in the form (N-
l)Z(j)[G,(x,i) =K
-G,(x,j)]
~mdO@N-l[M~(@jV)
-Y,(@jV)]i(@)[G,,(x,@)
-G,(n,@)]. (2.8)
The combination NY; - Y,, which occurs here, can also be expressed in terms of a single Bessel function, m;(x)
-Y,(x)
= (N-
l)( ;)-“L+*(X)
As discussed in detail in ref. [l], the above integral transformations can be inverted explicitly using analytic continuation in j. However, since we do not make use of the inversion formulae, we will not quote them here. The exact relations (2.7) and (2.8) are the starting point of our analysis. Chiral perturbation theory provides an expansion of the left-hand sides in inverse powers of the box size. The structure of this expansion depends on the relative magnitude of the external field j and the box size L. For weak external fields [symmetry restoration region, j = @(Led)], the e-expansion applies while at stronger field [j = &'(L-*)I the p-expansion is relevant. As shown in ref. [ll, the shape of the mean field distribution is controlled by the p-expansion of the partition function. Below, we extend this analysis to the two-point functions and translate the information contained in the p-expansion of G,,(x, j>,G,,(x, j> into a corresponding expansion of the constraint correlations G&x, @P),G, (x, @). 3. Expectation values and susceptibilities Before embarking on our analysis of the constraint correlation functions G&x, @p> and GL (x, @), it is convenient to collect some exact relations concerning the zero momentum projections of the various two-point functions. In the presence of a constant external field, the expectation value of 4 points in the direction of j, <4”)j=.f%(i>,
(3.1)
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with a magnitude given by the first derivative of the partition function with respect to i,
1 i(j) 4td=vz(i). Furthermore, the volume integral of the longitudinal function is related to the susceptibility &j> by
(3.2) component of the correlation
/ @xG,,(x,j)= Jf4(j>’+6(i),
(3.3)
while, for the transverse component, we have
/ ddxGI(x,
4(i) j) = j
V-4)
*
At fixed mean field, the analogous relations are the following. Translation invariance implies that the field expectation value coincides with the mean field,
(+A>*=@A. Moreover,
(3.5)
since the volume integral of the field 4A(~> is equal to VQA, we have
/
d”x$(x,@)
=V@*,
/d&&(x,@)
4. Results from chiral perturbation
=O.
(3.6)
theory
As is well known, the low-energy structure of the model is controlled by two constants 2 and F, which characterize properties of the theory at infinite volume and in the symmetry limit: 2 represents the expectation value of the field, Z = limj,, lim,,, 4(j), and F* is the residue of the Goldstone boson pole in the current correlation functions. If the external field j is turned on, the Goldstone bosons pick up a mass, which can be expanded in powers of fi (up to logarithms). For weak fields it is proportional to fi. Expressed in terms of the lowest order contribution
(4.1)
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the mass is given by [ll
MpZhys =
N-3 -F 8rr
M
(d=3),
+@(M2)
N-3M2
A,,,,
s$n
z
(d=4).
+@(M’)
(4.2)
In d = 3 the correction only involves the constants 2, F, but in four dimensions a further low-energy constant enters in the form of a (mass- and volume-independent) logarithmic scale AM. We also need the corresponding expansion of the vacuum energy density in powers of j a M2. The vacuum energy manifests itself in the behaviour of the partition function as V+ =J. Denoting the vacuum energy density by -u(j), we have u(j)
= li:-$lnZ(j).
(4.3)
We normalize the partition function such that the vacuum energy vanishes if the external field is turned off, u(O) = 0. The shift generated by the external field is given by the expectation value of the perturbation / ddXj +4(x>. The leading term in the expansion of u(j) in powers of fi is therefore equal to 2j. At first nonleading order one obtains [l]
(d=3), &$$[ln$+$]
+@‘(M4))
(d=4).
Again, for d = 4, the expansion contains a logarithmic term. In this notation, the p-expansion of the logarithm of the partition next-to-next-to-leading order reads [ 11 (N-
l)(N-3) 8F2
(4.4)
function to
M2(g,)2+c9(p3d-4
)I .
(4.5) The p-expansion treats the mass M and the inverse box size l/L as small quantities of order p. The leading term in eq. (4.5) stems from the volume-independent vacuum energy density. According to eq. (4.4) this term is of order p2.
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399
The second term (a g,) represents the free energy density of a free Bose gas of particles with mass Mphys and is of order p”. (For a detailed discussion of the kinematical function g,, =g,(M, L,, L,, . . . , L,) see appendix B of ref. [SJ.) The third term stems from the interaction among the Goldstone bosons. It involves the function g, defined by
and is of order p*“-*. The corresponding from eq. (3.2), 4(j)
expansion of the field expectation
N-l =-Z( 1 + -8rr
M F*
N-lM* --InF167r* F*
value is now obtained
N-l - yp,
(d=J),
+@(P*)
A,
N-l -Tjpl
+@(P4)
(d = 4) .
(4.7)
Note that this is a simultaneous expansion in powers of j = @(p*) and l/L = d(p). The p-expansion of the two-point functions at finite volume was given in ref. [3] for the case of SU(N) x SU(N) symmetry. For d = 3, the straightforward extension of these results to O(N) yields
N-l G,,( x,i)
= 4(i)*
(d = 3))
X2
(4.8)
+ -+(~W*+@(P~)
where G(x, M) is the standard euclidean Green function in a periodic box,
p,=
;+
n, integer.
fi
In the transverse correlation function the mass is needed to first nonleading order, where it contains a volume-dependent contribution a g,, N-3
2F2
g,
+@(p4
(4.10)
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(The same effect also shows up at infinite spatial volume, but finite temperature T: For weak external fields, the pole in the Goldstone boson propagator is shifted to M2[1 + (N - 3)T2/(24F2) + . . . I.) In the longitudinal correlation function, it is irrelevant whether the contribution of the one-loop graph associated with the propagation of two Goldstone bosons between the two vertices is evaluated with M, Mphys, or M,,, because the difference is of order p3. The wave function renormalization constant Z, also depends on the volume,
(d=3).
In four dimensions, the first two terms in the p-expansion of the correlation functions provide a more accurate representation as they account for all contributions up to and including &p”). At this order, short-range contributions originating in the exchange of massive states start showing up in the form of terms proportional to 6(x), G,(x,i)=Z,G(x,M,)
G,,(x,j)
=4(i)‘+
+E~(x)
N-1Z2 77KC-r)
+@(P”)
(d=4).
+b;(i)S(x)
The short-range term appearing in the transverse correlation mined by the logarithmic scales,
z’ (N-I)ln$+(N-3)lng E=16,rrZF4 ( F
(4.12)
+@(P6) function is deter-
.
(4.13)
F 1
The physical significance of A,+, and A, is specified in eqs. (4.2) and (4.4), respectively. The meaning of the third scale, AF, can be seen in the expression for the wave function renormalization constant, which in four dimensions reads
At infinite volume, g, vanishes - the term involving A,,, and AF thus determines the change in the wave function renormalization constant produced by the external field at infinite volume.
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The function K(x) occurring in the representation tion function is defined as K(x)
model
401
of the longitudinal
correla-
(4.15)
=G(x,M)*-6(x)/d4yG(y,M)‘.
Note that in four dimensions the distribution G(x, M)’ is unambiguous only up to a multiple of the S-function. The same regularization dependence however also occurs in the second term and it drops out in the difference: The function K(x) is regularization independent and obeys / d4xK(x) = 0. Using this property and the relation /
d4xG(x,M)=&,
one readily verifies that the representation (4.12) is consistent with the general properties of the zero-momentum projections listed in eqs. (3.3) and (3.4).
5. Transverse correlation
We now translate the information contained in the p-expansion of the correlation function G, (x, j) into a large-volume expansion of the constraint correlation c,(x, a). The calculation parallels the analysis of the mean field distribution Z(a), which runs as follows [l]. One exploits the fact that for large V this distribution is strongly peaked at Q, = Ir; the integral occurring on the r.h.s. of eq. (2.5) receives significant contributions only from the immediate vicinity of this point. Consider the limit L + m, j + 0 keeping the product jL2 fixed such that the quantity
stays constant. In the region which contributes significantly to the integral the argument j@V= j.EV of the function YN is then large and we can use the asymptotic representation [5]
1- (N- 1~~F3’ Furthermore,
+&(x-2)}.
dropping terms of order p d-2, the p-representation
(5.2)
of the partition
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function, eq. (4.51, can be written as
JW* (’-Nv2eYver@)[ Z(j) = (~2 1
1 + a( #-*)I
.
(5.3)
For d = 3, the quantity r(.$) is fixed by the kinematics of free particles, r(s)
N-l = Yj--
L3g, + -&t3/*
+ In 5
(5.4)
(d=3)
and can be expressed in terms of the shape coefficients
p,, introduced
in ref. [5],
(d=3).
In four dimensions, r(t)
(5.5)
involves the logarithmic scale A,,
r(s) = y(L4go+$[in
('r)*
+i)+h&
(d=4).
(5.6)
The coefficient of the term quadratic in LJlogarithmically
depends on the box size,
p- 7
(ln( A,L)
+ 8~*@2}“*.
(5.7)
Using eqs. (5.2) and (5.31, the relation (2.5) simplifies to
er(c)[ 1 + H(J#-*)]
It is convenient to extract a normalization by
factor and to define the potential
e-ww Inserting this definition
. (5.8)
UC@)
(5.9)
in eq. (5.8) and scaling the variable of integration with the
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model
substitution
.=++&)>
(5.10)
we finally obtain (5.11) where Ua(t,!~)is the first term in the p-expansion of the potential,
UC@> = uJ($)+ &d-w
+@((w”-*)*).
(5.12)
The result (5.11) shows that exp{ -U&I,/I)) is related to the explicitly known function r(t) by a Laplace transform, which is readily inverted [I]. We now apply the same analysis to the relations (2.7) and (2.8), which express the correlation functions in the presence of an external field as an integral transform of the constraint correlations. The zero-momentum projections of these quantities are known already (see sect. 3) and it is convenient to remove them. We denote the remainder by a prime:
f’(x) =f(x) - ;jddvfb)-
(5.13)
In this notation, the leading term in the p-expansion of G:(x, j) is given by
G:(x,j)=~~~(~,M)[l+~(p~-*)I’
(5.14)
where G’(x, M) is the Green function associated with the modes with p + 0, G’(x,M)
The p-expansion of the longitudinal hand, only starts at order p*‘“-*I. At large values of the argument, written as
eip..v = $ c p+o M2 +P* *
correlation the function
NY;(X)-YN(X)=(N-l)YN(X)(l-~+C?+-z)).
function
(5.15)
G;,(x, j), on the other
occurring
in eq. (2.8) can be
(5.16)
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Comparison of the two relations (2.7) and (2.8) then shows that the leading term in the p-expansion of &(x, @I is of order pd-* and obeys k/m
dJlexp(t$m
u,($))~~(x,@)
= $G’(x,M)e’@)[l
+d(#-*)I, (5.17)
while d;,(x, @P)is of order p 2(d-2) . On the 1.h.s. of eq. (5.171, the prime can be omitted, because the zero-momentum projection of d,(x, CD>vanishes [cf. eq. (3.611. To solve this relation for G’I (x, @), we represent the &dependence of the euclidean Green function (5.15) as a Laplace transform, G’(x,M)
= &
c
eiP’X/mdAexp(-A(~+L2$)).
(5.18)
0
PfO
On the 1.h.s. of eq. (5.17), the factor exp( -At)
corresponds to the shift # + I,!I+ A,
Hence the leading order term in the p-expansion correlation function is given by
of the transverse constraint
dh exp( -AL*p*)exp( X[l +a(JY)].
- Uo( $ + A) + Uo( JI)) (5.20)
This shows that if the mean field is constrained to the value Q, = Z[l + +I/ (F2Ld-*>1, the correlation function explores the shape of the effective potential to the right of this point. Approximating the difference U,(I,!J + A) - Uo($) by A d&($)/d+, the relevant effective mass becomes
M2 = L duo(@) eff
and the correlation
L*
dl(l
function takes the approximate
(5.21)
' form
(5.22)
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To the right of the minimum, MS, is positive and grows with $. To the left, MS, is negative. For a box of equal sides, U&J/) tends to -4r2JI as JI + -03, such that the square of the effective mass approaches -(27r/L)2. This implies that the denominators MA, +p2 which occur in the representation (5.15) of the propagator remain positive. Far to the left of the minimum, the correlation function is dominated by the contribution from the lowest momenta, which grows proportional to I$]. (A detailed discussion of the function U,J$> is given in ref. [l].)
6. Longitudinal
correlation
To calculate the leading term in the p-expansion of the longitudinal constraint correlation function we need to evaluate the integrals (2.7) and (2.8) to first nonleading order. The factor 1 -N/x occurring in eq. (5.16) then generates mixing between the transverse and longitudinal correlations:
1
Gl (x, j) er(*) x[l Inserting the representations 1
!G
m
/
dJIexp([$--m
+b(pd-2)].
(6.1)
given in eq. (4.8) for d = 3, this leads to
U,(JI))G;l(x,@)
N-l Z2 = FFK’(x)ercf’+@(P3)
(d = 3)) (6.2)
where K’(x) is the projection of G’(x, M)2 onto non-zero momenta, K’(x)
=G’(x,M)~-
;jd3yG’(yJ4)2
(d=3).
(6.3)
In four dimensions, the expansion of the correlation functions G&X, j> and G,(x, j) to order p4 contains short-range contributions. Moreover, the square of the propagator is singular at short distances and the straightforward extension of eq. (6.3) .to d = 4 leads to a regularization-dependent quantity. In the definition (4.15) of the function K(x) we have removed the regularization dependence by subtracting a contact term. We define the function K’(x) in an analogous manner, K’(x)
=G’(x,M)2--(x)/d4yG’(y,M)2
(d=4).
(6.4)
Note that K’(x) differs from the projection of K(x) onto nonzero momenta by a
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term proportional to G’(x, M) and by a contribution eq. (6.2) in four dimensions then becomes
N-l = 2
X2 FK’(
model
a 6(x) - l/V.
The analog of
x)ercc) + c#i(j) (d=4).
(6.5)
The relations (6.2) and (6.5) can again be solved explicitly for d;,(x, @p>,invoking the Laplace transform of the kinematical function K’(x) with respect to the mass. We do not go into this here, but instead analyze the correlation function of the projected field, which is of more immediate interest in connection with numerical simulations and is of a more simple structure.
7. Correlation
The projected field #,(x) function is given by
function of the projected field
is defined in eq. (1.2). The corresponding
= &jd%Pi(o)C,,(x,@). The denominator
correlation
(7.1)
occurring here is the partition function at j = 0, Z(0) = /d%Pg(@).
(7.2)
Note that the value of the field 4,(x) at the point x depends on the field configuration throughout the volume: The projection C/B(X)+ 4,(x) is not a local operation. This implies, in particular, that the transfer matrix formalism cannot be applied to the correlation functions of 4,(x). Nevertheless, there is an exact formula which allows one to express the quantity (4,(x)~$,(O)) in terms of the local correlation functions G&x, j) and G,(x, j). The formula reads (cb,W4,W)
= G,,W9
+ (N-
1)1(x).
(7.3)
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The first term is the local correlation function at j = 0 where there is no distinction between longitudinal and transverse components, G&x, 0) = G, (x, 0). The nonlocality inherent in ( c#J,(x)c#J,,(~))manifests itself in the second term, which involves an integral over nonzero values of the external source, -G,,(x,i)],
(7.4)
the integration extending along the imaginary axis. To establish the result (7.3), we evaluate the integral (7.4) with eq. (2.8). Using the property 2 dXXYN+‘,( ix) = (7.5) / 0 IyiN + 1) this leads to I(x) Furthermore,
1 = -/d’?Z’~(@)[G,,(x,@) NZ(O)
-6,(x,@)].
(7.6)
setting j = 0, the relation (2.7) implies
1 G,,( x-0) = -jd”@i(@)[G,,(x,@) N-V)
+ (N-
1)6,(x,@)].
The result (7.3) immediately follows from these two identities. Moreover, obtain an analogous representation for the mean transverse correlation, &ld%@S)d,(x,@)
=G,,(x,O)
-I(X).
(7.7) we
(7.8)
Let us now work out the leading term in the large-volume expansion of the correlation function (4,(x)+,(O)). To this end, we consider eq. (7.1) and observe that, at leading order in the p-expansion, the mean field distribution is proportional to exp(- V,>. Removing the contribution from the zero-momentum mode, we therefore obtain
(4,(x)4p(o))‘=~e-‘“‘~~-cad$exp(-Uo($))G;l(x,@)[l
+~(~‘-~)l’ (7.9)
where we have made use of the fact that the integral (7.2) is proportional to exp &$)lsCo [compare eq. (5.8)]. The remaining integral in eq. (7.9) is a special case of eq. (6.2) or (6.5): It suffices to take the limit 5 ---)0 there. In this limit, the
M. Giickeler,
408
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/
O(N)
model
mass M tends to zero such that the kinematical function K’(x) reduces to the corresponding quantity of the massless theory. In the notation of ref. [5], lim K’(x)
=K(x).
(7.10)
M-0
For d = 3, we therefore obtain
(&(x)4,(0))‘=(N;Fyr”K_(Jz) +@(P3) (d=3).
(7.11)
The zero-momentum projection of ( +,(x)c#J,,(O)) coincides with the expectation value of the square of the mean field,
/ ddx(cb,(x)+,(O)) = v(@‘>,
(7.12)
which was worked out in ref. [l] to order L 4- 2d. Putting things together, we finally arrive at
(f$,(x)q6,(0))
= (cv)
(@2>=s2
+ (N;F;)r2qx)
+a(
L-3))
N-l N-l 1+=&+m[@;+(2N-3)P2]
+8(L-3) (d = 3).
(7.13)
The short-range term occurring in four dimensions [see eq. (6.91 is readily worked out with the representation (4.7) of the expectation value, which implies
b(j) =
(N-1)X2 2F4
1 [s(ln$-f)-$+B(p’)]
The expansion of dg,/dM2 ities [5],
dg, -=--AdM2 The term
1 (ML)4
(d=4).
(7.14)
in powers of the mass contains infrared
singular-
1 8rr2
ln(ML)+i
a Mb4 cancels against the contribution
-
c= F,l+2(ML)2n. n=O n! a jm2 contained
(7.15) in eq. (6.5)
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/
O(N)
409
model
while the logarithmic singularity amalgamates with the term a ln(A,/M) tained in eq. (7.14). The coefficient of the short-range term thus becomes
con-
1
. (7.16)
In the limit 5 + 0, the relation (6.5) therefore implies
( f$,(x)r#J,(O)) = (G2>+ (N2F4 - 1)z2 1 8,rr2 14 A,L)
1 i
x K(x)+
+
N-l 16rr2F4L4
+
P2
)(w-;)]+a(L-6),
(N-3)ln$+(2N-3)ln(AzL) L
1 I +b(L-“)
(d=4),
(7.17)
where the expression for (CD’> is again taken from ref. [ll. Note that the leading term in eqs. (7.13) and (7.17) is a constant given by the expectation value of the square of the mean field: The dependence of the correlation function (4,(x)4,(y)) on the distance lx - yl is a small effect, which starts showing up only at second nonleading order. In three dimensions, the space dependence is dominated by the exchange of two Goldstone bosons and is of order Lm2. In four dimensions the space-dependent part is of order Lm4 and contains both a term from the exchange of two Goldstone bosons and a short-range contribution which accounts for the exchange of massive scalar particles (for d = 3, short-range contributions start manifesting themselves only at order Lm3, which is beyond the accuracy of our representation). Next, we determine the large-volume expansion of the two quantities G,,(x,O) and I(X) which occur in the exact formula (7.3) for the projected correlation function. The term G&x,0) represents the correlation function of the local field +(x> in the limit where the symmetry breaking external source is turned off. In this limit, symmetry restoration phenomena occur, which are connected with the fact that, at finite volume, symmetries do not break down spontaneously. These phenomena are controlled by the e-expansion, which is analyzed in detail in ref. [51. We quote the explicit expression for the first three terms in the large-volume
M. Giickcler,
410
expansion of G&x,0) G&x,0)
= ;(cD2)
H. Leutwyler
/
O(N)
model
from this reference,
+ Z2~~1)[(l+~)C(r)+~~(x)+~~(x)]
1 1
+e 6(x) -;
(7.18)
+lqP”“).
The kinematical functions C(X) and I?(X) occurring here are associated with the propagation of a single Goldstone boson, eip..v qx)+~,
H(x)
=
(7.19)
-$P
(P2)2
-
The accuracy of the representation includes short-range contributions only in four dimensions, where the constant e can be expressed in terms of the three logarithmic scales introduced in sect. 4, 85~~/3,+ln(nJL)+(N-3)ln~-2(N-2)ln~
,r
c
1 .
(7.20) In d = 3, e vanishes. The corresponding large-volume expansion of the quantity Z(x) is obtained by inserting the above representations for (4,,(x)4,(0)) and for G&x,0) in eq. (7.3). We quote the result for the mean transverse correlation function, 1 -/dN@@P)Gl(x,@) Z(O)
= 41 F2 [( 1+ +)‘b)
+ $h)]
(7.21) Again, the short-range term is nonzero only for d = 4, where it is given by z=
(N-3)In$-2(N-2)ln$ 2
1 .
I
(7.22)
Note that the space dependence of the local correlation function G,,(x,O) is more pronounced than that of the correlation function of the projected field, due to the
M. Giickeler,
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411
exchange of one Goldstone boson, which contributes at first nonleading order. Moreover, the representation (7.21) for the mean transverse correlation function shows that, in this quantity, a constant term does not occur - the expansion starts with a Goldstone boson exchange term of order L*-“, which exhibits a pronounced dependence on the distance. We add a remark concerning the interchange of the two limits L --f 03, j + 0. As is well known, the expectation value of the field Cptends to zero if the source is switched off at fixed L, while if the volume is sent to infinity first (4) tends to a finite value, which persists if one subsequently takes the limit j + 0. In contrast, the expectation value and the correlation function of the projected field are not sensitive to the perturbation generated by a weak external source - for these quantities the two limits are interchangeable. This offers a welcome check on our calculation. The above derivation of the representations (7.13) and (7.17) for the correlation function ( c$,(x)$,(O)) is based on the p-expansion formulae quoted in sect. 4: We first analyzed the relations (2.7) and (2.8) in terms of a series in inverse powers of the box size at fixed 5 = M*L* to arrive at the formulae (6.2) and (6.5) and then performed the limit 5 + 0. In this calculation we are approaching the limit L + 03, j + 0 from a direction where jLd is large. Alternatively, one may also evaluate the integral Z(X) defined in eq. (7.4) directly. Here, the dominating contributions arise from the symmetry restoration region, where jLd is of order 1. This region is controlled by the E-expansion. Using the explicit representations for the partition function and for the correlation functions given in ref. [5], one verifies that, to the accuracy required here, the quantity Z(j) [G,(x, j> - G,,(x, j>]/j can be written as a derivative with respect to j, such that the integral I(X) is easily worked out. The result indeed agrees with the representation given above. In effect, the two methods of evaluation amount to an interchange of the limits L + 03,j + 0, and the fact that the results for the correlation function (4,(x)4,(y)) agree provides a rather thorough consistency check of our machinery. Finally, we consider the projection of the correlation function (~$,(x)4,(0)) onto zero (spatial) momentum,
G,(t)= ;/dd-‘X( 4,(x,+b,(o)), with L, =L,
(7.23)
and t =xd. Using the notation of ref. [5] one has
(7.24)
where 7 = t/L,.
The explicit expression of the kinematical
function
h&r)
and a
412
M. Gijckeler,
H. Leutwyler
/
O(N)
model
discussion of its properties are given in ref. [5]. Now G,,(t) can be written as
GJf)
= (@*) +
G,(t)
= (Q2> +
h3(T) +b(L-3) (N-1)2’2 2F4
(8(r)
- 1) +@(L-‘)
1
(d = 4).
(7.25)
8. Summary and discussion We have studied correlation functions in O(N)-invariant field theories of an N-component scalar field transforming according to the fundamental representation of O(N). The system was considered in a periodic box of volume V and in a phase where, for I/+ w, the O(N) symmetry is spontaneously broken down to OUV- 1). From chiral perturbation theory we know how to expand various quantities, in particular two-point correlation functions, in powers of a symmetry breaking external source j and l/L = V- ‘Id . On the other hand, the correlation functions in the presence of the source j are connected by an integral transformation with the correlation functions for a fixed value of the mean field @= ;jddX+(X)
(8.1)
but with j = 0. Hence, the expansions provided by chiral perturbation theory can be translated into corresponding large-volume expansions of these constraint correlation functions. In particular, we have calculated the first three terms in the large-volume expansion of the two-point function of the projected field
In the application to numerical results the following two limitations of our method should be kept in mind. First, our calculation is done in the continuum, so lattice effects are not accounted for. In fact, in quantities which are sensitive to the
M. Giickeler,
H. Lelrtwyler
/ O(N)
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413
regularization, the finite-size effects are not controlled by symmetry alone and, to our knowledge, a framework which would allow one to calculate these effects outside the scaling region does not exist. To illustrate the problem, let us compare the O(4)-model and QCD with two massless flavours. Since the groups involved in the spontaneous symmetry breakdown of these two theories are locally isomorphic, the corresponding effective Lagrangians are of the same form and differ only in the values of the coupling constants Z, F, A,, AF, A,,,, . . . . Accordingly, the finite-size formulae given above apply in either case. The regularization effects are however different in the two cases, because the short-distance behaviour of the two theories is different. Presumably, in the case of the O(4)-model, the main effects generated by the regularization can be accounted for by simply replacing the continuum propagators occurring in our formulae by the corresponding propagators on the lattice*, but we see no reason why this recipe should also work in the context of QCD. The second limitation concerns the short-range contributions such as those generated by the exchange of a massive scalar particle. In our framework, these contributions manifest themselves only in the form of a-function contact terms - our machinery is tailored to analyze long wavelengths and does not have the resolution needed to investigate the structure at short distance. A perturbative analysis of the A(+2)2-interaction shows that the mass and the width of the unstable scalar particle (position of the pole in the correlation function) as well as the corresponding wave function renormalization constant (residue of the pole) also depend on the volume. A systematic study of these effects, however, yet remains to be carried out. M.G. wishes to thank the Institut fur Theoretische Physik der Universitat Bern for its kind hospitality. Useful discussions with I. Dimitrovid, P. Hasenfratz, K. Jansen, J. Jersak, J. Nager and T. Neuhaus are also gratefully acknowledged.
References [l] M. Gockeler and H. Leutwyler, Nucl. Phys. B350 (1991) 228; Phys. Lett. B253 (1991) 193 [2] M. Gockeler, Nucl. Phys. B (Proc. Suppl.1 17 (1990) 347 [3] J. Gasser and H. Leuhvyler, Ann. Phys. (N.Y.) 158 (1984) 142; Phys. Lett. B184 (1987) 83; B188 (1987) 477 [4] H. Neuberger, Phys. Rev. Lett. 60 (1988) 889; Nucl. Phys. B300 [FS22] (1988) 180; U.M. Heller and H. Neuberger, Phys. Lett. 8207 (1988) 189 [5] P. Hasenfratz and H. Leuhvyler, Nucl. Phys. B343 (1990) 241 [6] F.C. Hansen, Nucl. Phys. B345 (1990) 685; F.C. Hansen and H. Leutwyler, Nucl. Phys. B350 (1991) 201 [7] I. Dimitrovid, P. Hasenfratz, J. Nager and F. Niedermayer, Nucl. Phys. B350 (1991) 893 l
T. Neuhaus, private communication.
414
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/ O(N)
tnodel
[8] T. Neuhaus, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 21; A. Hasenfratz, K. Jansen, J. Jersak, C.B. Lang, H. Leutwyler and T. Neuhaus, Z. Phys. C46 (1990) 257; A. Hasenfratz, K. Jansen, J. Jersak, H.A. Kastrup, C.B. Lang, H. Leutwyler and T. Neuhaus, Nucl. Phys. B356 (1991) 332 [9] J. Kuti and Y. Shen, Phys. Rev. Lett. 60 (1988) 85; J. Kuti, L. Lin and Y. Shen, Phys. Rev. Lett. 61 (1988) 678; irr Lattice Higgs Workshop, ed. B. Berg et al. (World Scientific, Singapore, 19881 p. 140; Y. Shen, J. Kuti, L. Lin and S. Meyer, in Lattice Higgs Workshop, ed. B. Berg et al. (World Scientific, Singapore, 19881 p. 216; M.M. Tsypin, in Lattice Higgs Workshop, ed. B. Berg et al. (World Scientific, Singapore, 19881 p. 258; J. Kuti, L. Lin and Y. Shen, Nucl. Phys. B (Proc. SuppI. 4 (19881 397; L. Lin, J. Kuti and Y. Shen, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 26; J. Kuti, irl Proc. XXIV Int. Conf. on High energy physics, Munich 1988, ed. R. Kotthaus and J.H. Kuhn (Springer, Berlin, 1989) p. 814 [lo] A. Hasenfratz, K. Jansen, C.B. Lang, T. Neuhaus and H. Yoneyama, Phys. Lett. B199 (19871531; K. Jansen, Nucl. Phys. B (Proc. SuppI. 4 (1988) 422; C.B. Lang, in Lattice Higgs Workshop, ed. B. Berg et al. (World Scientific, Singapore, 19881 p. 158; H. Neuberger, in Lattice Higgs Workshop, ed. B. Berg et al. (World Scientific, Singapore, 19881 p. 197; A. Hasenfratz, K. Jansen, J. Jersak, C.B. Lang, T. Neuhaus and H. Yoneyama, Nucl. Phys. B317 (1989181
CONSTRAINT
CORRELATION
FUNCTIONS
IN THE O(N) MODEL*
M. GZiCKELER Institute
for
Theoretical
Physics
E, RWTH
Aachen,
W-5100
Aachen,
Germany
and HLRZ
c/o
KFA Jiilich,
P.O. Box 1913, W-51 70 Jiilich,
Germany
H. LEUTWYLER Institute
for
Theoretical
Physics,
University
of Bern,
Sidlerstrasse
5, CH-3012
Bern,
Switzerland
Received 19 February 1991
For O(N)-symmetric spin systems in the spontaneously broken phase enclosed in a finite volume V, we study two-point correlation functions at a fixed value of the mean field @= I’-‘/ ddx $(x1. Using results from chiral perturbation theory, we investigate their large-volume behaviour. In particular, we derive a large-volume expansion for the correlation functions of the projected field Q%,,(X)= Cp* +(x)/191.
1. Formulation
of the problem
In this paper we continue our study [ll (see also ref. [21) of the large-volume behaviour of O(N)-symmetric scalar field theories in the phase where the symmetry is spontaneously broken down to O(N - 1). Models of this type have been extensively investigated by Monte Carlo methods for various values of N and in different dimensions d, in particular d = 3 and n = 4. The Heisenberg model of a ferromagnet corresponds to the case iV = 3, d = 3, whereas the theory with N = 4, d = 4 is of great interest in connection with the Higgs sector of the standard model of elementary particle physics. The numerical simulations are necessarily performed in a finite volume, although, in the end, only the infinite-volume limit is of interest. The occurrence of Goldstone bosons implies that these models do not contain a mass gap. Consequently, the effects generated by the finite size of the system persist even at large volume. The finite-size effects can be brought under control by introducing an external source which explicitly breaks O(N)-symmetry. The Goldstone bosons then acquire a mass and the finite-size effects become exponentially small, provided only that the volume is large enough, Even if this condition is not satisfied, the *Work supported in part by Schweizerischer Nationalfonds. 0550-3213/91/$03.50
Q 1991 - Elsevier Science Publishers B.V. (North-Holland)
M. Giickcler,
H. Leutwyler
/ O(N)
model
393
behaviour of various quantities, like the partition function, correlation functions etc., for large volumes and for weak external sources can be determined by means of chiral perturbation theory [3], which leads to an expansion in inverse powers of the box size. Since the coefficients occurring in this expansion are determined by the low-energy properties of the system at infinite volume, the observation of the volume dependence allows one to extract infinite-volume results from simulations in a finite box (see also ref. [4]). This method is discussed in detail in ref. [5] for the group O(N) and in ref. [61 for the group SUW) X W(N). Applications to the analysis of numerical data are given in refs. [7,8] for the three-dimensional Heisenberg model and for the O(4) model in four dimensions, respectively. A different approach to the problem of Goldstone bosons in a finite volume, which avoids explicit symmetry breaking, starts from the observation that spontaneous symmetry breakdown manifests itself already before the thermodynamic limit is taken: If the volume V is large enough, most of the field configurations are such that the directions of the field +(x> at the various points of space are close to the direction of the mean field
The vanishing of (+(x>) at finite volume is only due to the fact that Q, does not prefer any particular direction. Restricting oneself to field configurations with a fixed value of the mean field [9,10], the various finite-volume observables have a smooth infinite-volume limit. Moreover, the two-point correlation function of the projected field
(1.2) has been studied in the four-dimensional O(4) model [lo] with the intention to suppress the influence of the Goldstone bosons and thus to make the extraction of the Higgs boson mass possible. The present paper is devoted to a study of the finite-size effects occurring in this approach. To be more specific, we work in euclidean space of dimension d > 2 and enclose the system in a periodic box of sides L,, L,, . . . , L,. The mean length L is defined as L = V’ia where V= L i L 2.. . L, is the volume of the box. The action S{+) is taken to be invariant under global O(N) rotations of the N-component scalar field 9(x) = (4’(x), . . . , 4N-1(~>>. The d istribution of the mean field is given by the functional integral
(1.3)
394
M, Giickeler,
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O(N)
model
which, on account of O(N) symmetry, only depends on the magnitude @ = ]@I of 9. The probability to find a field configuration for which the mean field is contained in dN@ is proportional to .&@,)dN@. The logarithm of %(@)-’ is referred to as the constraint effective potential U(G), 2( 0) = const.
X
e-‘@).
(1.4)
The shape of the potential was analyzed in detail in ref. [l]. In the present paper we extend this analysis to the correlation functions of the scalar field. In particular, we examine the correlations at a fixed value of the mean field, , Ww#4Y))
*=
/[d4]ev(
&
-S(e))s(
@ - i/ddx
b(X))4A(X)4B(V).
(1.5) The analysis is based on the observation that the ordinary integral /dN@i(@)(+A(X)4B(Y))
eexp(**.W)
(l-6)
coincides with the quantity
/kWexp( -W
+j.lddx9(x))gn(x)~‘(y)
(l-7)
which, up to a normalization factor, represents the standard two-point function in the presence of a constant external field j. For this two-point function, the expansion in inverse powers of the box size is however known [3,5] from chiral perturbation theory, up to and including terms of order (l/~!,~-~)~. All we need to do, therefore, is to solve (1.6) for the correlation function. After discussing the basic identities in sect. 2 we collect some relations concerning the zero momentum projections of the various two-point functions in sect. 3. Sect. 4 reviews the results from chiral perturbation theory that are needed in the following. These are applied in sects. 5 and 6 to the transverse and longitudinal correlations, respectively. In sect. 7 we study the two-point function of the projected field. Sect. 8 contains a summary and our conclusions. 2. Identities In the presence of a constant external field, the two-point bPY+v(Yh=
&)
function is given by
/P6lew( -444 +j*/+ddx)&‘(x)da(y), (2.1)
M. Gdckeler,
where Z( j> is the corresponding
H. Leutwyler
/
O(N)
model
395
partition function,
(2.2) In this notation, field distribution
the relation between the partition Z(Q) becomes Z(j)
function
Z(j) and the mean
=/dN@exp(@.jV)Z(@)
(2.3)
while the equality of the two expressions (1.6) and (1.7) takes the form
In eq. (2.3) the angular integral over the direction of Q, can be carried out with the result [ll Z(j)
=K/gmd~~N-lYN(~jV)i(~),
v=--1,; YN(n)=(;)-“z”(x),
K =
2?TN12,
P-5)
where Z, is the modified Bessel function. To perform the angular integral in eq. (2.41, we exploit O(N) symmetry and decompose the correlation functions as
( 4"(x)4"(Y>)
j= jA]BG,,(~-~,j)
(&-‘(x)@(y))
9=dA&%,,(x-y,@)
+{6AE--J^AJ^B}GI(~-~Y,j),
+ {S”“-&‘~B}~,(x-y,@),
(2.6)
where 1” and &’ are unit vectors in the directions of j and Q, respectively, while the coefficients G,,(x, j), . . . , GL(x,@) only depend on j = ljl or Cp= lcP1. Since C,(&%JA) is direction-independent, we get Z(j>{G,,(x,.i) =K
+ (N-
l)G,(x,i)}
~md~~N-lY~(~jV)i(~)(d,,(x,~)
+ (N-
l)d,(x,@)).
(2.7)
The projection of ( 4A(x)4E(y))j onto the direction of j, on the other hand, leads to an angular integral over the quantity (a *j>2 exp(@ *jV). This represents
396
M. Giickeler,
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/
O(N)
model
the second derivative of the exponential factor with respect to the volume, and the integral can therefore be expressed in terms of the second derivative of the function Y,(x). The result can be written in the form (N-
l)Z(j)[G,(x,i) =K
-G,(x,j)]
~mdO@N-l[M~(@jV)
-Y,(@jV)]i(@)[G,,(x,@)
-G,(n,@)]. (2.8)
The combination NY; - Y,, which occurs here, can also be expressed in terms of a single Bessel function, m;(x)
-Y,(x)
= (N-
l)( ;)-“L+*(X)
As discussed in detail in ref. [l], the above integral transformations can be inverted explicitly using analytic continuation in j. However, since we do not make use of the inversion formulae, we will not quote them here. The exact relations (2.7) and (2.8) are the starting point of our analysis. Chiral perturbation theory provides an expansion of the left-hand sides in inverse powers of the box size. The structure of this expansion depends on the relative magnitude of the external field j and the box size L. For weak external fields [symmetry restoration region, j = @(Led)], the e-expansion applies while at stronger field [j = &'(L-*)I the p-expansion is relevant. As shown in ref. [ll, the shape of the mean field distribution is controlled by the p-expansion of the partition function. Below, we extend this analysis to the two-point functions and translate the information contained in the p-expansion of G,,(x, j>,G,,(x, j> into a corresponding expansion of the constraint correlations G&x, @P),G, (x, @). 3. Expectation values and susceptibilities Before embarking on our analysis of the constraint correlation functions G&x, @p> and GL (x, @), it is convenient to collect some exact relations concerning the zero momentum projections of the various two-point functions. In the presence of a constant external field, the expectation value of 4 points in the direction of j, <4”)j=.f%(i>,
(3.1)
hi. Giickeler,
H. Leutwyler
/
O(N)
model
397
with a magnitude given by the first derivative of the partition function with respect to i,
1 i(j) 4td=vz(i). Furthermore, the volume integral of the longitudinal function is related to the susceptibility &j> by
(3.2) component of the correlation
/ @xG,,(x,j)= Jf4(j>’+6(i),
(3.3)
while, for the transverse component, we have
/ ddxGI(x,
4(i) j) = j
V-4)
*
At fixed mean field, the analogous relations are the following. Translation invariance implies that the field expectation value coincides with the mean field,
(+A>*=@A. Moreover,
(3.5)
since the volume integral of the field 4A(~> is equal to VQA, we have
/
d”x$(x,@)
=V@*,
/d&&(x,@)
4. Results from chiral perturbation
=O.
(3.6)
theory
As is well known, the low-energy structure of the model is controlled by two constants 2 and F, which characterize properties of the theory at infinite volume and in the symmetry limit: 2 represents the expectation value of the field, Z = limj,, lim,,, 4(j), and F* is the residue of the Goldstone boson pole in the current correlation functions. If the external field j is turned on, the Goldstone bosons pick up a mass, which can be expanded in powers of fi (up to logarithms). For weak fields it is proportional to fi. Expressed in terms of the lowest order contribution
(4.1)
hf. Giickeler,
398
H. Leutwyler
/
O(N)
model
the mass is given by [ll
MpZhys =
N-3 -F 8rr
M
(d=3),
+@(M2)
N-3M2
A,,,,
s$n
z
(d=4).
+@(M’)
(4.2)
In d = 3 the correction only involves the constants 2, F, but in four dimensions a further low-energy constant enters in the form of a (mass- and volume-independent) logarithmic scale AM. We also need the corresponding expansion of the vacuum energy density in powers of j a M2. The vacuum energy manifests itself in the behaviour of the partition function as V+ =J. Denoting the vacuum energy density by -u(j), we have u(j)
= li:-$lnZ(j).
(4.3)
We normalize the partition function such that the vacuum energy vanishes if the external field is turned off, u(O) = 0. The shift generated by the external field is given by the expectation value of the perturbation / ddXj +4(x>. The leading term in the expansion of u(j) in powers of fi is therefore equal to 2j. At first nonleading order one obtains [l]
(d=3), &$$[ln$+$]
+@‘(M4))
(d=4).
Again, for d = 4, the expansion contains a logarithmic term. In this notation, the p-expansion of the logarithm of the partition next-to-next-to-leading order reads [ 11 (N-
l)(N-3) 8F2
(4.4)
function to
M2(g,)2+c9(p3d-4
)I .
(4.5) The p-expansion treats the mass M and the inverse box size l/L as small quantities of order p. The leading term in eq. (4.5) stems from the volume-independent vacuum energy density. According to eq. (4.4) this term is of order p2.
M. G&k&r,
H. Lcutnyler / O(N) model
399
The second term (a g,) represents the free energy density of a free Bose gas of particles with mass Mphys and is of order p”. (For a detailed discussion of the kinematical function g,, =g,(M, L,, L,, . . . , L,) see appendix B of ref. [SJ.) The third term stems from the interaction among the Goldstone bosons. It involves the function g, defined by
and is of order p*“-*. The corresponding from eq. (3.2), 4(j)
expansion of the field expectation
N-l =-Z( 1 + -8rr
M F*
N-lM* --InF167r* F*
value is now obtained
N-l - yp,
(d=J),
+@(P*)
A,
N-l -Tjpl
+@(P4)
(d = 4) .
(4.7)
Note that this is a simultaneous expansion in powers of j = @(p*) and l/L = d(p). The p-expansion of the two-point functions at finite volume was given in ref. [3] for the case of SU(N) x SU(N) symmetry. For d = 3, the straightforward extension of these results to O(N) yields
N-l G,,( x,i)
= 4(i)*
(d = 3))
X2
(4.8)
+ -+(~W*+@(P~)
where G(x, M) is the standard euclidean Green function in a periodic box,
p,=
;+
n, integer.
fi
In the transverse correlation function the mass is needed to first nonleading order, where it contains a volume-dependent contribution a g,, N-3
2F2
g,
+@(p4
(4.10)
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(The same effect also shows up at infinite spatial volume, but finite temperature T: For weak external fields, the pole in the Goldstone boson propagator is shifted to M2[1 + (N - 3)T2/(24F2) + . . . I.) In the longitudinal correlation function, it is irrelevant whether the contribution of the one-loop graph associated with the propagation of two Goldstone bosons between the two vertices is evaluated with M, Mphys, or M,,, because the difference is of order p3. The wave function renormalization constant Z, also depends on the volume,
(d=3).
In four dimensions, the first two terms in the p-expansion of the correlation functions provide a more accurate representation as they account for all contributions up to and including &p”). At this order, short-range contributions originating in the exchange of massive states start showing up in the form of terms proportional to 6(x), G,(x,i)=Z,G(x,M,)
G,,(x,j)
=4(i)‘+
+E~(x)
N-1Z2 77KC-r)
+@(P”)
(d=4).
+b;(i)S(x)
The short-range term appearing in the transverse correlation mined by the logarithmic scales,
z’ (N-I)ln$+(N-3)lng E=16,rrZF4 ( F
(4.12)
+@(P6) function is deter-
.
(4.13)
F 1
The physical significance of A,+, and A, is specified in eqs. (4.2) and (4.4), respectively. The meaning of the third scale, AF, can be seen in the expression for the wave function renormalization constant, which in four dimensions reads
At infinite volume, g, vanishes - the term involving A,,, and AF thus determines the change in the wave function renormalization constant produced by the external field at infinite volume.
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The function K(x) occurring in the representation tion function is defined as K(x)
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401
of the longitudinal
correla-
(4.15)
=G(x,M)*-6(x)/d4yG(y,M)‘.
Note that in four dimensions the distribution G(x, M)’ is unambiguous only up to a multiple of the S-function. The same regularization dependence however also occurs in the second term and it drops out in the difference: The function K(x) is regularization independent and obeys / d4xK(x) = 0. Using this property and the relation /
d4xG(x,M)=&,
one readily verifies that the representation (4.12) is consistent with the general properties of the zero-momentum projections listed in eqs. (3.3) and (3.4).
5. Transverse correlation
We now translate the information contained in the p-expansion of the correlation function G, (x, j) into a large-volume expansion of the constraint correlation c,(x, a). The calculation parallels the analysis of the mean field distribution Z(a), which runs as follows [l]. One exploits the fact that for large V this distribution is strongly peaked at Q, = Ir; the integral occurring on the r.h.s. of eq. (2.5) receives significant contributions only from the immediate vicinity of this point. Consider the limit L + m, j + 0 keeping the product jL2 fixed such that the quantity
stays constant. In the region which contributes significantly to the integral the argument j@V= j.EV of the function YN is then large and we can use the asymptotic representation [5]
1- (N- 1~~F3’ Furthermore,
+&(x-2)}.
dropping terms of order p d-2, the p-representation
(5.2)
of the partition
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function, eq. (4.51, can be written as
JW* (’-Nv2eYver@)[ Z(j) = (~2 1
1 + a( #-*)I
.
(5.3)
For d = 3, the quantity r(.$) is fixed by the kinematics of free particles, r(s)
N-l = Yj--
L3g, + -&t3/*
+ In 5
(5.4)
(d=3)
and can be expressed in terms of the shape coefficients
p,, introduced
in ref. [5],
(d=3).
In four dimensions, r(t)
(5.5)
involves the logarithmic scale A,,
r(s) = y(L4go+$[in
('r)*
+i)+h&
(d=4).
(5.6)
The coefficient of the term quadratic in LJlogarithmically
depends on the box size,
p- 7
(ln( A,L)
+ 8~*@2}“*.
(5.7)
Using eqs. (5.2) and (5.31, the relation (2.5) simplifies to
er(c)[ 1 + H(J#-*)]
It is convenient to extract a normalization by
factor and to define the potential
e-ww Inserting this definition
. (5.8)
UC@)
(5.9)
in eq. (5.8) and scaling the variable of integration with the
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substitution
.=++&)>
(5.10)
we finally obtain (5.11) where Ua(t,!~)is the first term in the p-expansion of the potential,
UC@> = uJ($)+ &d-w
+@((w”-*)*).
(5.12)
The result (5.11) shows that exp{ -U&I,/I)) is related to the explicitly known function r(t) by a Laplace transform, which is readily inverted [I]. We now apply the same analysis to the relations (2.7) and (2.8), which express the correlation functions in the presence of an external field as an integral transform of the constraint correlations. The zero-momentum projections of these quantities are known already (see sect. 3) and it is convenient to remove them. We denote the remainder by a prime:
f’(x) =f(x) - ;jddvfb)-
(5.13)
In this notation, the leading term in the p-expansion of G:(x, j) is given by
G:(x,j)=~~~(~,M)[l+~(p~-*)I’
(5.14)
where G’(x, M) is the Green function associated with the modes with p + 0, G’(x,M)
The p-expansion of the longitudinal hand, only starts at order p*‘“-*I. At large values of the argument, written as
eip..v = $ c p+o M2 +P* *
correlation the function
NY;(X)-YN(X)=(N-l)YN(X)(l-~+C?+-z)).
function
(5.15)
G;,(x, j), on the other
occurring
in eq. (2.8) can be
(5.16)
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Comparison of the two relations (2.7) and (2.8) then shows that the leading term in the p-expansion of &(x, @I is of order pd-* and obeys k/m
dJlexp(t$m
u,($))~~(x,@)
= $G’(x,M)e’@)[l
+d(#-*)I, (5.17)
while d;,(x, @P)is of order p 2(d-2) . On the 1.h.s. of eq. (5.171, the prime can be omitted, because the zero-momentum projection of d,(x, CD>vanishes [cf. eq. (3.611. To solve this relation for G’I (x, @), we represent the &dependence of the euclidean Green function (5.15) as a Laplace transform, G’(x,M)
= &
c
eiP’X/mdAexp(-A(~+L2$)).
(5.18)
0
PfO
On the 1.h.s. of eq. (5.17), the factor exp( -At)
corresponds to the shift # + I,!I+ A,
Hence the leading order term in the p-expansion correlation function is given by
of the transverse constraint
dh exp( -AL*p*)exp( X[l +a(JY)].
- Uo( $ + A) + Uo( JI)) (5.20)
This shows that if the mean field is constrained to the value Q, = Z[l + +I/ (F2Ld-*>1, the correlation function explores the shape of the effective potential to the right of this point. Approximating the difference U,(I,!J + A) - Uo($) by A d&($)/d+, the relevant effective mass becomes
M2 = L duo(@) eff
and the correlation
L*
dl(l
function takes the approximate
(5.21)
' form
(5.22)
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To the right of the minimum, MS, is positive and grows with $. To the left, MS, is negative. For a box of equal sides, U&J/) tends to -4r2JI as JI + -03, such that the square of the effective mass approaches -(27r/L)2. This implies that the denominators MA, +p2 which occur in the representation (5.15) of the propagator remain positive. Far to the left of the minimum, the correlation function is dominated by the contribution from the lowest momenta, which grows proportional to I$]. (A detailed discussion of the function U,J$> is given in ref. [l].)
6. Longitudinal
correlation
To calculate the leading term in the p-expansion of the longitudinal constraint correlation function we need to evaluate the integrals (2.7) and (2.8) to first nonleading order. The factor 1 -N/x occurring in eq. (5.16) then generates mixing between the transverse and longitudinal correlations:
1
Gl (x, j) er(*) x[l Inserting the representations 1
!G
m
/
dJIexp([$--m
+b(pd-2)].
(6.1)
given in eq. (4.8) for d = 3, this leads to
U,(JI))G;l(x,@)
N-l Z2 = FFK’(x)ercf’+@(P3)
(d = 3)) (6.2)
where K’(x) is the projection of G’(x, M)2 onto non-zero momenta, K’(x)
=G’(x,M)~-
;jd3yG’(yJ4)2
(d=3).
(6.3)
In four dimensions, the expansion of the correlation functions G&X, j> and G,(x, j) to order p4 contains short-range contributions. Moreover, the square of the propagator is singular at short distances and the straightforward extension of eq. (6.3) .to d = 4 leads to a regularization-dependent quantity. In the definition (4.15) of the function K(x) we have removed the regularization dependence by subtracting a contact term. We define the function K’(x) in an analogous manner, K’(x)
=G’(x,M)2--(x)/d4yG’(y,M)2
(d=4).
(6.4)
Note that K’(x) differs from the projection of K(x) onto nonzero momenta by a
M, Giickeler,
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term proportional to G’(x, M) and by a contribution eq. (6.2) in four dimensions then becomes
N-l = 2
X2 FK’(
model
a 6(x) - l/V.
The analog of
x)ercc) + c#i(j) (d=4).
(6.5)
The relations (6.2) and (6.5) can again be solved explicitly for d;,(x, @p>,invoking the Laplace transform of the kinematical function K’(x) with respect to the mass. We do not go into this here, but instead analyze the correlation function of the projected field, which is of more immediate interest in connection with numerical simulations and is of a more simple structure.
7. Correlation
The projected field #,(x) function is given by
function of the projected field
is defined in eq. (1.2). The corresponding
= &jd%Pi(o)C,,(x,@). The denominator
correlation
(7.1)
occurring here is the partition function at j = 0, Z(0) = /d%Pg(@).
(7.2)
Note that the value of the field 4,(x) at the point x depends on the field configuration throughout the volume: The projection C/B(X)+ 4,(x) is not a local operation. This implies, in particular, that the transfer matrix formalism cannot be applied to the correlation functions of 4,(x). Nevertheless, there is an exact formula which allows one to express the quantity (4,(x)~$,(O)) in terms of the local correlation functions G&x, j) and G,(x, j). The formula reads (cb,W4,W)
= G,,W9
+ (N-
1)1(x).
(7.3)
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The first term is the local correlation function at j = 0 where there is no distinction between longitudinal and transverse components, G&x, 0) = G, (x, 0). The nonlocality inherent in ( c#J,(x)c#J,,(~))manifests itself in the second term, which involves an integral over nonzero values of the external source, -G,,(x,i)],
(7.4)
the integration extending along the imaginary axis. To establish the result (7.3), we evaluate the integral (7.4) with eq. (2.8). Using the property 2 dXXYN+‘,( ix) = (7.5) / 0 IyiN + 1) this leads to I(x) Furthermore,
1 = -/d’?Z’~(@)[G,,(x,@) NZ(O)
-6,(x,@)].
(7.6)
setting j = 0, the relation (2.7) implies
1 G,,( x-0) = -jd”@i(@)[G,,(x,@) N-V)
+ (N-
1)6,(x,@)].
The result (7.3) immediately follows from these two identities. Moreover, obtain an analogous representation for the mean transverse correlation, &ld%@S)d,(x,@)
=G,,(x,O)
-I(X).
(7.7) we
(7.8)
Let us now work out the leading term in the large-volume expansion of the correlation function (4,(x)+,(O)). To this end, we consider eq. (7.1) and observe that, at leading order in the p-expansion, the mean field distribution is proportional to exp(- V,
(4,(x)4p(o))‘=~e-‘“‘~~-cad$exp(-Uo($))G;l(x,@)[l
+~(~‘-~)l’ (7.9)
where we have made use of the fact that the integral (7.2) is proportional to exp &$)lsCo [compare eq. (5.8)]. The remaining integral in eq. (7.9) is a special case of eq. (6.2) or (6.5): It suffices to take the limit 5 ---)0 there. In this limit, the
M. Giickeler,
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mass M tends to zero such that the kinematical function K’(x) reduces to the corresponding quantity of the massless theory. In the notation of ref. [5], lim K’(x)
=K(x).
(7.10)
M-0
For d = 3, we therefore obtain
(&(x)4,(0))‘=(N;Fyr”K_(Jz) +@(P3) (d=3).
(7.11)
The zero-momentum projection of ( +,(x)c#J,,(O)) coincides with the expectation value of the square of the mean field,
/ ddx(cb,(x)+,(O)) = v(@‘>,
(7.12)
which was worked out in ref. [l] to order L 4- 2d. Putting things together, we finally arrive at
(f$,(x)q6,(0))
= (cv)
(@2>=s2
+ (N;F;)r2qx)
+a(
L-3))
N-l N-l 1+=&+m[@;+(2N-3)P2]
+8(L-3) (d = 3).
(7.13)
The short-range term occurring in four dimensions [see eq. (6.91 is readily worked out with the representation (4.7) of the expectation value, which implies
b(j) =
(N-1)X2 2F4
1 [s(ln$-f)-$+B(p’)]
The expansion of dg,/dM2 ities [5],
dg, -=--AdM2 The term
1 (ML)4
(d=4).
(7.14)
in powers of the mass contains infrared
singular-
1 8rr2
ln(ML)+i
a Mb4 cancels against the contribution
-
c= F,l+2(ML)2n. n=O n! a jm2 contained
(7.15) in eq. (6.5)
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while the logarithmic singularity amalgamates with the term a ln(A,/M) tained in eq. (7.14). The coefficient of the short-range term thus becomes
con-
1
. (7.16)
In the limit 5 + 0, the relation (6.5) therefore implies
( f$,(x)r#J,(O)) = (G2>+ (N2F4 - 1)z2 1 8,rr2 14 A,L)
1 i
x K(x)+
+
N-l 16rr2F4L4
+
P2
)(w-;)]+a(L-6),
(N-3)ln$+(2N-3)ln(AzL) L
1 I +b(L-“)
(d=4),
(7.17)
where the expression for (CD’> is again taken from ref. [ll. Note that the leading term in eqs. (7.13) and (7.17) is a constant given by the expectation value of the square of the mean field: The dependence of the correlation function (4,(x)4,(y)) on the distance lx - yl is a small effect, which starts showing up only at second nonleading order. In three dimensions, the space dependence is dominated by the exchange of two Goldstone bosons and is of order Lm2. In four dimensions the space-dependent part is of order Lm4 and contains both a term from the exchange of two Goldstone bosons and a short-range contribution which accounts for the exchange of massive scalar particles (for d = 3, short-range contributions start manifesting themselves only at order Lm3, which is beyond the accuracy of our representation). Next, we determine the large-volume expansion of the two quantities G,,(x,O) and I(X) which occur in the exact formula (7.3) for the projected correlation function. The term G&x,0) represents the correlation function of the local field +(x> in the limit where the symmetry breaking external source is turned off. In this limit, symmetry restoration phenomena occur, which are connected with the fact that, at finite volume, symmetries do not break down spontaneously. These phenomena are controlled by the e-expansion, which is analyzed in detail in ref. [51. We quote the explicit expression for the first three terms in the large-volume
M. Giickcler,
410
expansion of G&x,0) G&x,0)
= ;(cD2)
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from this reference,
+ Z2~~1)[(l+~)C(r)+~~(x)+~~(x)]
1 1
+e 6(x) -;
(7.18)
+lqP”“).
The kinematical functions C(X) and I?(X) occurring here are associated with the propagation of a single Goldstone boson, eip..v qx)+~,
H(x)
=
(7.19)
-$P
(P2)2
-
The accuracy of the representation includes short-range contributions only in four dimensions, where the constant e can be expressed in terms of the three logarithmic scales introduced in sect. 4, 85~~/3,+ln(nJL)+(N-3)ln~-2(N-2)ln~
,r
c
1 .
(7.20) In d = 3, e vanishes. The corresponding large-volume expansion of the quantity Z(x) is obtained by inserting the above representations for (4,,(x)4,(0)) and for G&x,0) in eq. (7.3). We quote the result for the mean transverse correlation function, 1 -/dN@@P)Gl(x,@) Z(O)
= 41 F2 [( 1+ +)‘b)
+ $h)]
(7.21) Again, the short-range term is nonzero only for d = 4, where it is given by z=
(N-3)In$-2(N-2)ln$ 2
1 .
I
(7.22)
Note that the space dependence of the local correlation function G,,(x,O) is more pronounced than that of the correlation function of the projected field, due to the
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411
exchange of one Goldstone boson, which contributes at first nonleading order. Moreover, the representation (7.21) for the mean transverse correlation function shows that, in this quantity, a constant term does not occur - the expansion starts with a Goldstone boson exchange term of order L*-“, which exhibits a pronounced dependence on the distance. We add a remark concerning the interchange of the two limits L --f 03, j + 0. As is well known, the expectation value of the field Cptends to zero if the source is switched off at fixed L, while if the volume is sent to infinity first (4) tends to a finite value, which persists if one subsequently takes the limit j + 0. In contrast, the expectation value and the correlation function of the projected field are not sensitive to the perturbation generated by a weak external source - for these quantities the two limits are interchangeable. This offers a welcome check on our calculation. The above derivation of the representations (7.13) and (7.17) for the correlation function ( c$,(x)$,(O)) is based on the p-expansion formulae quoted in sect. 4: We first analyzed the relations (2.7) and (2.8) in terms of a series in inverse powers of the box size at fixed 5 = M*L* to arrive at the formulae (6.2) and (6.5) and then performed the limit 5 + 0. In this calculation we are approaching the limit L + 03, j + 0 from a direction where jLd is large. Alternatively, one may also evaluate the integral Z(X) defined in eq. (7.4) directly. Here, the dominating contributions arise from the symmetry restoration region, where jLd is of order 1. This region is controlled by the E-expansion. Using the explicit representations for the partition function and for the correlation functions given in ref. [5], one verifies that, to the accuracy required here, the quantity Z(j) [G,(x, j> - G,,(x, j>]/j can be written as a derivative with respect to j, such that the integral I(X) is easily worked out. The result indeed agrees with the representation given above. In effect, the two methods of evaluation amount to an interchange of the limits L + 03,j + 0, and the fact that the results for the correlation function (4,(x)4,(y)) agree provides a rather thorough consistency check of our machinery. Finally, we consider the projection of the correlation function (~$,(x)4,(0)) onto zero (spatial) momentum,
G,(t)= ;/dd-‘X( 4,(x,+b,(o)), with L, =L,
(7.23)
and t =xd. Using the notation of ref. [5] one has
(7.24)
where 7 = t/L,.
The explicit expression of the kinematical
function
h&r)
and a
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M. Gijckeler,
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discussion of its properties are given in ref. [5]. Now G,,(t) can be written as
GJf)
= (@*) +
G,(t)
= (Q2> +
h3(T) +b(L-3) (N-1)2’2 2F4
(8(r)
- 1) +@(L-‘)
1
(d = 4).
(7.25)
8. Summary and discussion We have studied correlation functions in O(N)-invariant field theories of an N-component scalar field transforming according to the fundamental representation of O(N). The system was considered in a periodic box of volume V and in a phase where, for I/+ w, the O(N) symmetry is spontaneously broken down to OUV- 1). From chiral perturbation theory we know how to expand various quantities, in particular two-point correlation functions, in powers of a symmetry breaking external source j and l/L = V- ‘Id . On the other hand, the correlation functions in the presence of the source j are connected by an integral transformation with the correlation functions for a fixed value of the mean field @= ;jddX+(X)
(8.1)
but with j = 0. Hence, the expansions provided by chiral perturbation theory can be translated into corresponding large-volume expansions of these constraint correlation functions. In particular, we have calculated the first three terms in the large-volume expansion of the two-point function of the projected field
In the application to numerical results the following two limitations of our method should be kept in mind. First, our calculation is done in the continuum, so lattice effects are not accounted for. In fact, in quantities which are sensitive to the
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413
regularization, the finite-size effects are not controlled by symmetry alone and, to our knowledge, a framework which would allow one to calculate these effects outside the scaling region does not exist. To illustrate the problem, let us compare the O(4)-model and QCD with two massless flavours. Since the groups involved in the spontaneous symmetry breakdown of these two theories are locally isomorphic, the corresponding effective Lagrangians are of the same form and differ only in the values of the coupling constants Z, F, A,, AF, A,,,, . . . . Accordingly, the finite-size formulae given above apply in either case. The regularization effects are however different in the two cases, because the short-distance behaviour of the two theories is different. Presumably, in the case of the O(4)-model, the main effects generated by the regularization can be accounted for by simply replacing the continuum propagators occurring in our formulae by the corresponding propagators on the lattice*, but we see no reason why this recipe should also work in the context of QCD. The second limitation concerns the short-range contributions such as those generated by the exchange of a massive scalar particle. In our framework, these contributions manifest themselves only in the form of a-function contact terms - our machinery is tailored to analyze long wavelengths and does not have the resolution needed to investigate the structure at short distance. A perturbative analysis of the A(+2)2-interaction shows that the mass and the width of the unstable scalar particle (position of the pole in the correlation function) as well as the corresponding wave function renormalization constant (residue of the pole) also depend on the volume. A systematic study of these effects, however, yet remains to be carried out. M.G. wishes to thank the Institut fur Theoretische Physik der Universitat Bern for its kind hospitality. Useful discussions with I. Dimitrovid, P. Hasenfratz, K. Jansen, J. Jersak, J. Nager and T. Neuhaus are also gratefully acknowledged.
References [l] M. Gockeler and H. Leutwyler, Nucl. Phys. B350 (1991) 228; Phys. Lett. B253 (1991) 193 [2] M. Gockeler, Nucl. Phys. B (Proc. Suppl.1 17 (1990) 347 [3] J. Gasser and H. Leuhvyler, Ann. Phys. (N.Y.) 158 (1984) 142; Phys. Lett. B184 (1987) 83; B188 (1987) 477 [4] H. Neuberger, Phys. Rev. Lett. 60 (1988) 889; Nucl. Phys. B300 [FS22] (1988) 180; U.M. Heller and H. Neuberger, Phys. Lett. 8207 (1988) 189 [5] P. Hasenfratz and H. Leuhvyler, Nucl. Phys. B343 (1990) 241 [6] F.C. Hansen, Nucl. Phys. B345 (1990) 685; F.C. Hansen and H. Leutwyler, Nucl. Phys. B350 (1991) 201 [7] I. Dimitrovid, P. Hasenfratz, J. Nager and F. Niedermayer, Nucl. Phys. B350 (1991) 893 l
T. Neuhaus, private communication.
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