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Flavor symmetry breaking and zero mode shift for staggered fermions
Volume 210, number 1,2
PHYSICS LETTERS B
18 August 1988
FLAVOR SYMMETRY BREAKING AND ZERO M O D E SHIFT FOR STAGGERED FERMIONS Jeroen C. V I N K Institute of Theoretical Physics, Valckenierstraat 65, NL 1018 XE Amsterdam, The Netherlands Received 30 April 1988
On the lattice a "fermionic topological charge", Q, can be used to measure the physically relevant fluctuations of the number of "zero modes" of the lattice Dirac operator. It is shown that the mass dependence of Q for small quark mass is related to the flavor symmetry breaking for staggered fermions and an estimate is given for the mass scale below which this breaking of flavor symmetry cannot be neglected. Also the renormalization of(~ is discussed.
In a series of papers [ 1-7] we have investigated a formulation of topological charge which features the lattice Dirac operator, Q = t c p ~ Tr F s ( D + m ) - ' = I C p Q .
(1)
?/f
Here (D + m) - ~ is the fermion propagator for either Wilson or staggered fermions - only the latter will be considered here -/"5 represents the Dirac Ys, rn is the mass parameter for the n r = 4 degenerate flavors of quarks and ~ce is a finite renormalization constant. The motivation for using Q stems from its appearance in a mass formula for the flavor singlet pseudoscalar meson [2,3 ], just like the topological charge in the continuum formulation features in the W i t t e n Veneziano formula [ 8 ]. Similar expressions, without the factor Xp have appeared also in refs. [ 9,10 ]. The alleged topological significance of 0 can be elucidated by rewriting ( 1 ) using the eigenfunctions ofl~, O.g=i~,f,
f~f,=&,.
(2)
Then one arrives at m f~Fsf~ 0=1CP4 - ~ i 2 , + m -
~p m2 4 ~r, ~ 2 2 +,m
(3)
with residues r , = f ~ F s f . In the continuum formulation, where xp-= 1, the same result can be found and there the properties ofy~ imply that either 2 ~ 0 and r , = 0 , or 2~=0 and the eigenfunctions can be chosen such that r~= -+ 1. This leads to the Atiyah-Singer index theorem in the form Q=n+ - n _ , with n+ ( n _ ) 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
the number of zero modes with residue re= + 1 ( - 1 ). On the lattice these properties o f {2,,, r,} cannot be reproduced exactly and we summarize the picture found in refs. [1,4]. For sufficiently smooth gauge fields one can distinguish "zero modes", which pair a small 12ol to a relatively large residue Irol, from non-zero modes, which may or may not have a small eigenvalue but always have I r, I << Irol. Differences with the continuum formulation are that "zero modes" in general have 12ol ¢ 0 and I rol < 1, and that non-zero modes have residues r~¢ 0. This can be seen from the constraint Z & = Tr F5 = 0, which shows that the non-zero mode residues actually balance the "zero m o d e " residues. These lattice artefacts make that O is rn dependent: both for m ~ 0 and for m - ~ it vanishes. The m dependence for small m will be referred to as the "zero mode shift effect". For intermediate mass values, ~ < a m < < 1, O is expected [3,4] to be approximately constant (e is a measure for the typical "zero m o d e " eigenvalue). In this scaling window for m the renormalization constant xr compensates the reduction of the "zero mode" residues: Xp~ ( I r o [ ) 1, tCvro~ -+ 1. An observable that shows both the m dependence mentioned above and the reduction of the "zero m o d e " residues in ( Q(UV) }. Here the gauge field is multiplied with a smooth U ( 1 ) field Vwith (continu u m ) charge Qv. These external fields Vhave a constant plaquette in two orthogonal planes, VI2
=
~'~4=exp(2zfiq/N2),
Vother= 1 ,
(4) 211
Volume 210, number 1,2
Q , = 3 q 2,
PHYSICS LETTERS B
(aN)4=volume,
(5)
where q is an integer and 3 is the number of colors; more details are in ref. [1]. In the continuum (Q(UV))=(O_(U))+O_(V)=Q,, on the lattice the external charge Q,, as measured by (~)(UV) ) is modified by the effects discussed above. Using measurements of ( 0 ( U V ) ) we can estimate the value of xp and the size of the zero mode shift region e. It is important to know these values, since they give information on the extent to which staggered fermions resemble continuum fermions. A large value of Xp emphasizes that the staggered/'5 is a four link, operator. It may be remarked that top only slowly approaches its continuum limit value: 1- xp=clfl-i +O(fl-2) ~ 1/log(a2), a ~ 0 . The value cj = 3.93 can be inferred from results given in refs. [ 11,3 ]. Even though the zero mode shift region is expected to vanish much faster, ~ a z a 2 for a ~ 0 , it is important to know its extent since it puts a lower bound on the quark mass values that can be used meaningfully. In fact we have argued that the zero mode shift effect is related to the well-known breaking of flavor symmetry for staggered fermion. As explained in refs. [3,6 ], the flavor singlet two-quark bound state, the "singlet pion", gives the dominant contribution to ( 0_(UV) ) in quenched QCD. This implies that, for low quark mass, (0(UV))vc
m
- ~ + .... rn~
(6)
with ms the singlet pion mass, which should become degenerate with the Goldstone pion mass for a-~0. Since (O,(UV))~const. for m in the scaling window, we indeed recognize the behaviour that is characteristic for a Goldstone boson: m~ ocm. For rn-,0, inside the zero mode shift region, we know that
18 August 1988
(Q(UV) ) tends to zero and this implies that there are flavor symmetry breaking contributions to m ~ (cf. refs. [3,6]): m 4=m~+Ba2+...,
(7)
where rn~ is the Goldstone pion mass and B is a con6 stant oc flQCD. This motivates the following parametrization for the quark mass dependence of ( Q(UV) ), valid for small am,
(Q_(UV))~Q,,~cff'
(
msv'~ -'/2 l+
m2 /
,
(8)
with ms,, a "scaling violation" mass scale, msv~C a for a o 0 . Apparently rn~,, provides both a measure for the typical zero mode shift, which was denoted above by e, and a measure for the flavor symmetry breaking between m~ and m~. We now turn to the numerical results. We have measured (Q.(UV)) for fl in the range 5.7-6.6, at several mass values in the zero mode shift region. The pseudo fermion method is used to invert []) + m. This works well for the elements o f (I~ + m ) near the diagonal, which are needed to evaluate the trace in ( l ). For small quark mass we are faced with the problem of critical slowing down. Using the overrelaxation technique described in ref. [ 12 ], the autocorrelation time for am <0.05 could be reduced by roughly a factor of four. The lattice sizes used, fl-values and other relevant information are summarized in table 1. Fig. l shows the data where the error bars are estimated from the fluctuations between different gauge fields. In most cases they are o f the same magnitude as the pseudo fermion noise. The full curves in fig. 1 are a least squares fit of
F(am)=Q,,K~; 1
1+ ~ s - j
-a(am)
2
(9)
Table 1 fl value, lattice size, external charge Qv, number of configurations considered, number of pseudo fermion sweeps (for the lowest mass value) and approximate lattice distance.
212
fl
N4
Qv
~configurations
#sweeps
a(fl)2/a(fl=6.0) 2
5,7 6,0 6.3 6.6
84 164 164 164
3 12 12 12
5 3 3 2
8200 7200 7200 7200
4 1 1/2 1/4
Volume 210, n u m b e r 1,2
0.52
F
PHYSICS LETTERS B
6
|
~' =
6.6
0.24 =
.
0.2
W7 °
0.16
+____
0.12
0.08
5.7
0.04 0.
0.
0.04
0. 8
0.12
0.16
02
0.24
mO ---) Fig. 1. The mass d e p e n d e n c e of ( ( ) ( U V )
)/Q,.
to the data points am<,O.2. The term o~(arn) 2 accommodates the leading corrections to (8), coming from the non-zero modes. This renders estimates for ares,, and Kp as listed in table 2. The quoted errors in ams,. and ~Cpare a subjective estimate of the allowed deviations in these parameters, such that the curve still passes through the error bars of the data. One may notice that the number of gauge fields used to estimate am~,, and t% is small. However the number of "zero modes" is large since we have used an external field with a large charge Q,,(for Qv= 12 there are about 48 "zero modes" ). Therefore we may obtain a reasonable estimate for the typical zero mode shift, as measured by am~,,, from even a single gauge field UV. Also the renormalization constant Kp can be estimated reliably since it is determined by the short distance fluctuations of the gauge field U and one benefits from a large lattice volume. There is one catch here: Xp is determined from the reduction of the external charge due to the short-distance fluctuation in the field UV, but there may also be contributions coming from the non-zero charge of the field U, To illustrate this, we assume that approximately Q ( U V ) ~ Q(U) + ( O ( U V ) ). It suggests that for an Table 2 Values ofam~, and Kp o b t a i n e d from the fit. fl
amsv
Kp
5.7 6.0 6.3 6.6
0.070+0.026 0.024+0.005 0.013+0.003 -
9.2_+1.5 5.2_+0.4 4. i_+0.3 3.5_+0.2
18 August 1988
unbiased estimate of ( (~(UV) ) one should use gauge fields such that ~c(~(U) ~ 0. Furthermore the large value of Qv makes ()(UV) less sensitive to contributions of non-zero Q ( U ) . T h e 16 4 lattice is too small at fl=6.6 to expect the infinite volume behaviour for the Goldstone pion mass. This implies that the identification of the zero mode shift effect as exclusively due to flavor symmetry breaking is not valid. Therefore we do not give a value for ams,~ at fl= 6.6. For the renormalization constant Kp on the other hand we expect finite size effects to be small even at fl= 6.6. Comparing the non-perturbative values of ~:p with the lowest order perturbative result, it is clear that fl= 5.7-6.6 is still way out of the regime where these perturbative results are meaningful. Even more worrisome is the large value of the scaling violation mass am~v. One would like the full spin-flavor symmetry to be restored at the fl and am values used in for instance mass spectrum computations. Therefore one should use am>3amsv, say. This amounts to a m > 0 . 0 4 at fl=6.3. Table 1 includes the lattice distance in units of the lattice distance at fl= 6.0. This can be used to check that the expected (asymptotic) behaviour amsv~a 2, approximately holds for fl>~ 6.0. In conclusion: we have measured the size of the zero mode shift effect and the value of the renormalization constant ~Cp.This enables us to monitor the approach towards the continuum limit of the staggered fermions. Our numerical results reveal that •p is significantly different from 1 for fl~<6,6 and we have found an alarming large value for the mass scale below which flavor symmetry breaking cannot be neglected, even at fl= 6.0-6.3. I would like to thank Jan Smit for many discussions and helpful advise. The gauge field configuarations were kindly made available to us by K.C. Bowler. The computations were performed on the Cyber 205 at SARA with financial support by the "Stichting S U R F " from the "Nationaal Fonds gebruik Supercomputers ( N F S ) " . This work is part of the research program o f the "Stichting voor Fundamenteel Onderzoek der Materie ( F O M ) " , which is financially supported by the "stichting voor Zuiver Wetenschappelijk Onderzoek ( Z W O ) " .
213
Volume 210, number 1,2
PHYSICS LETTERS B
References [ 1 ] J. Smit and J.C. Vink, Nucl. Phys. B 284 (1987) 234. [ 2 ] J. Smit and J.C. Vink, Nucl. Phys. B 286 (1987) 485. [ 3 ] J. Smit and J,C. Vink, Nucl. Phys. B 928 ( 1988 ) 557. [4]J. Smit and J.C. Vink, Nucl. Phys. B 303 (1988) 36; in: Lattice gauge theory '86, eds. H. Satz, I. Harrity and J. Potvin (Plenum, New York, 1987). [ 5 ] J. Smit and J.C. Vink, Phys. Lett. B 194 ( 1987 ) 433. [ 6 ] J.C. Vink, ITFA-87-20, in: Proc. Lattice gauge "theory Conf. 1987 (Seillac, France), to be published. [7] J.C. Vink, ITFA-88-2.
214
18 August 1988
[8] E. Witten, Nucl. Phys. B 156 (1979) 269; G. Veneziano. Nucl. Phys. B 159 (1979) 213. [9] F. Karsch, E. Seiler and I.O. Stamatecu, Nucl. Phys. B 271 (1986) 349. [ 10 ] M. Bochicchio, G.C. Rossi, M. Testa and K. Yoshida, Phys. Lett. B 149 (1984) 487. [ 11 ] M.F.L. Golterman and J. Smit, Nucl. Phys. B 245 (1984) 61; D. Daniel and S.N. Sheard, Edinburgh Preprint 87/422 (Septermber 1987). [12] S.L. Adler, Phys. Rev. D 23 (1981) 2901; C. Whitmer, Phys. Rev. D 29 (1984) 306.
PHYSICS LETTERS B
18 August 1988
FLAVOR SYMMETRY BREAKING AND ZERO M O D E SHIFT FOR STAGGERED FERMIONS Jeroen C. V I N K Institute of Theoretical Physics, Valckenierstraat 65, NL 1018 XE Amsterdam, The Netherlands Received 30 April 1988
On the lattice a "fermionic topological charge", Q, can be used to measure the physically relevant fluctuations of the number of "zero modes" of the lattice Dirac operator. It is shown that the mass dependence of Q for small quark mass is related to the flavor symmetry breaking for staggered fermions and an estimate is given for the mass scale below which this breaking of flavor symmetry cannot be neglected. Also the renormalization of(~ is discussed.
In a series of papers [ 1-7] we have investigated a formulation of topological charge which features the lattice Dirac operator, Q = t c p ~ Tr F s ( D + m ) - ' = I C p Q .
(1)
?/f
Here (D + m) - ~ is the fermion propagator for either Wilson or staggered fermions - only the latter will be considered here -/"5 represents the Dirac Ys, rn is the mass parameter for the n r = 4 degenerate flavors of quarks and ~ce is a finite renormalization constant. The motivation for using Q stems from its appearance in a mass formula for the flavor singlet pseudoscalar meson [2,3 ], just like the topological charge in the continuum formulation features in the W i t t e n Veneziano formula [ 8 ]. Similar expressions, without the factor Xp have appeared also in refs. [ 9,10 ]. The alleged topological significance of 0 can be elucidated by rewriting ( 1 ) using the eigenfunctions ofl~, O.g=i~,f,
f~f,=&,.
(2)
Then one arrives at m f~Fsf~ 0=1CP4 - ~ i 2 , + m -
~p m2 4 ~r, ~ 2 2 +,m
(3)
with residues r , = f ~ F s f . In the continuum formulation, where xp-= 1, the same result can be found and there the properties ofy~ imply that either 2 ~ 0 and r , = 0 , or 2~=0 and the eigenfunctions can be chosen such that r~= -+ 1. This leads to the Atiyah-Singer index theorem in the form Q=n+ - n _ , with n+ ( n _ ) 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
the number of zero modes with residue re= + 1 ( - 1 ). On the lattice these properties o f {2,,, r,} cannot be reproduced exactly and we summarize the picture found in refs. [1,4]. For sufficiently smooth gauge fields one can distinguish "zero modes", which pair a small 12ol to a relatively large residue Irol, from non-zero modes, which may or may not have a small eigenvalue but always have I r, I << Irol. Differences with the continuum formulation are that "zero modes" in general have 12ol ¢ 0 and I rol < 1, and that non-zero modes have residues r~¢ 0. This can be seen from the constraint Z & = Tr F5 = 0, which shows that the non-zero mode residues actually balance the "zero m o d e " residues. These lattice artefacts make that O is rn dependent: both for m ~ 0 and for m - ~ it vanishes. The m dependence for small m will be referred to as the "zero mode shift effect". For intermediate mass values, ~ < a m < < 1, O is expected [3,4] to be approximately constant (e is a measure for the typical "zero m o d e " eigenvalue). In this scaling window for m the renormalization constant xr compensates the reduction of the "zero mode" residues: Xp~ ( I r o [ ) 1, tCvro~ -+ 1. An observable that shows both the m dependence mentioned above and the reduction of the "zero m o d e " residues in ( Q(UV) }. Here the gauge field is multiplied with a smooth U ( 1 ) field Vwith (continu u m ) charge Qv. These external fields Vhave a constant plaquette in two orthogonal planes, VI2
=
~'~4=exp(2zfiq/N2),
Vother= 1 ,
(4) 211
Volume 210, number 1,2
Q , = 3 q 2,
PHYSICS LETTERS B
(aN)4=volume,
(5)
where q is an integer and 3 is the number of colors; more details are in ref. [1]. In the continuum (Q(UV))=(O_(U))+O_(V)=Q,, on the lattice the external charge Q,, as measured by (~)(UV) ) is modified by the effects discussed above. Using measurements of ( 0 ( U V ) ) we can estimate the value of xp and the size of the zero mode shift region e. It is important to know these values, since they give information on the extent to which staggered fermions resemble continuum fermions. A large value of Xp emphasizes that the staggered/'5 is a four link, operator. It may be remarked that top only slowly approaches its continuum limit value: 1- xp=clfl-i +O(fl-2) ~ 1/log(a2), a ~ 0 . The value cj = 3.93 can be inferred from results given in refs. [ 11,3 ]. Even though the zero mode shift region is expected to vanish much faster, ~ a z a 2 for a ~ 0 , it is important to know its extent since it puts a lower bound on the quark mass values that can be used meaningfully. In fact we have argued that the zero mode shift effect is related to the well-known breaking of flavor symmetry for staggered fermion. As explained in refs. [3,6 ], the flavor singlet two-quark bound state, the "singlet pion", gives the dominant contribution to ( 0_(UV) ) in quenched QCD. This implies that, for low quark mass, (0(UV))vc
m
- ~ + .... rn~
(6)
with ms the singlet pion mass, which should become degenerate with the Goldstone pion mass for a-~0. Since (O,(UV))~const. for m in the scaling window, we indeed recognize the behaviour that is characteristic for a Goldstone boson: m~ ocm. For rn-,0, inside the zero mode shift region, we know that
18 August 1988
(Q(UV) ) tends to zero and this implies that there are flavor symmetry breaking contributions to m ~ (cf. refs. [3,6]): m 4=m~+Ba2+...,
(7)
where rn~ is the Goldstone pion mass and B is a con6 stant oc flQCD. This motivates the following parametrization for the quark mass dependence of ( Q(UV) ), valid for small am,
(Q_(UV))~Q,,~cff'
(
msv'~ -'/2 l+
m2 /
,
(8)
with ms,, a "scaling violation" mass scale, msv~C a for a o 0 . Apparently rn~,, provides both a measure for the typical zero mode shift, which was denoted above by e, and a measure for the flavor symmetry breaking between m~ and m~. We now turn to the numerical results. We have measured (Q.(UV)) for fl in the range 5.7-6.6, at several mass values in the zero mode shift region. The pseudo fermion method is used to invert []) + m. This works well for the elements o f (I~ + m ) near the diagonal, which are needed to evaluate the trace in ( l ). For small quark mass we are faced with the problem of critical slowing down. Using the overrelaxation technique described in ref. [ 12 ], the autocorrelation time for am <0.05 could be reduced by roughly a factor of four. The lattice sizes used, fl-values and other relevant information are summarized in table 1. Fig. l shows the data where the error bars are estimated from the fluctuations between different gauge fields. In most cases they are o f the same magnitude as the pseudo fermion noise. The full curves in fig. 1 are a least squares fit of
F(am)=Q,,K~; 1
1+ ~ s - j
-a(am)
2
(9)
Table 1 fl value, lattice size, external charge Qv, number of configurations considered, number of pseudo fermion sweeps (for the lowest mass value) and approximate lattice distance.
212
fl
N4
Qv
~configurations
#sweeps
a(fl)2/a(fl=6.0) 2
5,7 6,0 6.3 6.6
84 164 164 164
3 12 12 12
5 3 3 2
8200 7200 7200 7200
4 1 1/2 1/4
Volume 210, n u m b e r 1,2
0.52
F
PHYSICS LETTERS B
6
|
~' =
6.6
0.24 =
.
0.2
W7 °
0.16
+____
0.12
0.08
5.7
0.04 0.
0.
0.04
0. 8
0.12
0.16
02
0.24
mO ---) Fig. 1. The mass d e p e n d e n c e of ( ( ) ( U V )
)/Q,.
to the data points am<,O.2. The term o~(arn) 2 accommodates the leading corrections to (8), coming from the non-zero modes. This renders estimates for ares,, and Kp as listed in table 2. The quoted errors in ams,. and ~Cpare a subjective estimate of the allowed deviations in these parameters, such that the curve still passes through the error bars of the data. One may notice that the number of gauge fields used to estimate am~,, and t% is small. However the number of "zero modes" is large since we have used an external field with a large charge Q,,(for Qv= 12 there are about 48 "zero modes" ). Therefore we may obtain a reasonable estimate for the typical zero mode shift, as measured by am~,,, from even a single gauge field UV. Also the renormalization constant Kp can be estimated reliably since it is determined by the short distance fluctuations of the gauge field U and one benefits from a large lattice volume. There is one catch here: Xp is determined from the reduction of the external charge due to the short-distance fluctuation in the field UV, but there may also be contributions coming from the non-zero charge of the field U, To illustrate this, we assume that approximately Q ( U V ) ~ Q(U) + ( O ( U V ) ). It suggests that for an Table 2 Values ofam~, and Kp o b t a i n e d from the fit. fl
amsv
Kp
5.7 6.0 6.3 6.6
0.070+0.026 0.024+0.005 0.013+0.003 -
9.2_+1.5 5.2_+0.4 4. i_+0.3 3.5_+0.2
18 August 1988
unbiased estimate of ( (~(UV) ) one should use gauge fields such that ~c(~(U) ~ 0. Furthermore the large value of Qv makes ()(UV) less sensitive to contributions of non-zero Q ( U ) . T h e 16 4 lattice is too small at fl=6.6 to expect the infinite volume behaviour for the Goldstone pion mass. This implies that the identification of the zero mode shift effect as exclusively due to flavor symmetry breaking is not valid. Therefore we do not give a value for ams,~ at fl= 6.6. For the renormalization constant Kp on the other hand we expect finite size effects to be small even at fl= 6.6. Comparing the non-perturbative values of ~:p with the lowest order perturbative result, it is clear that fl= 5.7-6.6 is still way out of the regime where these perturbative results are meaningful. Even more worrisome is the large value of the scaling violation mass am~v. One would like the full spin-flavor symmetry to be restored at the fl and am values used in for instance mass spectrum computations. Therefore one should use am>3amsv, say. This amounts to a m > 0 . 0 4 at fl=6.3. Table 1 includes the lattice distance in units of the lattice distance at fl= 6.0. This can be used to check that the expected (asymptotic) behaviour amsv~a 2, approximately holds for fl>~ 6.0. In conclusion: we have measured the size of the zero mode shift effect and the value of the renormalization constant ~Cp.This enables us to monitor the approach towards the continuum limit of the staggered fermions. Our numerical results reveal that •p is significantly different from 1 for fl~<6,6 and we have found an alarming large value for the mass scale below which flavor symmetry breaking cannot be neglected, even at fl= 6.0-6.3. I would like to thank Jan Smit for many discussions and helpful advise. The gauge field configuarations were kindly made available to us by K.C. Bowler. The computations were performed on the Cyber 205 at SARA with financial support by the "Stichting S U R F " from the "Nationaal Fonds gebruik Supercomputers ( N F S ) " . This work is part of the research program o f the "Stichting voor Fundamenteel Onderzoek der Materie ( F O M ) " , which is financially supported by the "stichting voor Zuiver Wetenschappelijk Onderzoek ( Z W O ) " .
213
Volume 210, number 1,2
PHYSICS LETTERS B
References [ 1 ] J. Smit and J.C. Vink, Nucl. Phys. B 284 (1987) 234. [ 2 ] J. Smit and J.C. Vink, Nucl. Phys. B 286 (1987) 485. [ 3 ] J. Smit and J,C. Vink, Nucl. Phys. B 928 ( 1988 ) 557. [4]J. Smit and J.C. Vink, Nucl. Phys. B 303 (1988) 36; in: Lattice gauge theory '86, eds. H. Satz, I. Harrity and J. Potvin (Plenum, New York, 1987). [ 5 ] J. Smit and J.C. Vink, Phys. Lett. B 194 ( 1987 ) 433. [ 6 ] J.C. Vink, ITFA-87-20, in: Proc. Lattice gauge "theory Conf. 1987 (Seillac, France), to be published. [7] J.C. Vink, ITFA-88-2.
214
18 August 1988
[8] E. Witten, Nucl. Phys. B 156 (1979) 269; G. Veneziano. Nucl. Phys. B 159 (1979) 213. [9] F. Karsch, E. Seiler and I.O. Stamatecu, Nucl. Phys. B 271 (1986) 349. [ 10 ] M. Bochicchio, G.C. Rossi, M. Testa and K. Yoshida, Phys. Lett. B 149 (1984) 487. [ 11 ] M.F.L. Golterman and J. Smit, Nucl. Phys. B 245 (1984) 61; D. Daniel and S.N. Sheard, Edinburgh Preprint 87/422 (Septermber 1987). [12] S.L. Adler, Phys. Rev. D 23 (1981) 2901; C. Whitmer, Phys. Rev. D 29 (1984) 306.