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Effective Yukawa couplings in noncompact lattice QED
cm ..__
22June1995
__ EB
PHYSICS
ELSBVIER
LETTERS
B
Physics Letters B 353 (1995) 100-106
Effective Yukawa couplings in noncompact lattice QED M. Giickeler a,b, R. Horsley a, P.E.L. Rakow c, G. Schierholz a,d a Gruppe Theorie der Elementarteilchen, Hiichstleistungsrechenzentrwn HLRZ, c/o Forschungszentrum Jiilich, D-52425 Jiilich, Germany b Institutfiir Theoretische Physik, RWTH Aachen, Physikzentrum, D-52056 Aachen, Germany c Institutfiir Theoretische Physik, Freie Vniversitiit Berlin, Arnimaliee 14, D-14195 Berlin, Germany d Deutsches Elektronen-Synchrotron DES% NotkestraJe 85, D-22603 Hamburg, Germany
Received 10 March 1995 Editor: P.V. Landshoff
Abstract We investigate effective Yukawa couplings of mesons to the elementary fermions in noncompact lattice couplings are extracted from suitable fermion-antifermion-meson three-point functions calculated by Monte Carlo with dynamical staggered fermions. The scaling behaviour is compatible with expectations from perturbation indicating triviality of QED. The lines of constant Yukawa coupling are compared to flows of other quantities. is seen, at most, for weak coupling.
QED. The simulations theory, thus Consistency
In strongly coupled lattice QED chiral symmetry is known to be spontaneously broken, whereas an unbroken chiral symmetry is expected in the weak coupling limit [ 11. This finding immediately raises questions like: Can a nontrivial continuum limit be defined at the corresponding critical point? If so, what is the nature of the associated continuum theory? Several groups have investigated these problems by means of extensive Monte Carlo simulations of lattice QED using various techniques (see, e.g., [ 2-91) . Yet the physical interpretation of the data is still controversial (see also [ lo] ) . Correlation functions of composite fermion-antifermion operators show evidence for at least two fermionantifermion (bound) states (“meson” states) : a pseudoscalar state denoted by P (the Goldstone boson associated with the spontaneous symmetry breaking) and a scalar state (S) [3,11,6]. In this letter we study fermionantifermion-meson three-point functions in order to determine (effective) Yukawa couplings of P and S to the fermions. The investigation should provide additional information about the lattice model. In particular, if it has a trivial continuum limit, the couplings should tend to zero as one approaches the critical point. We have performed simulations with dynamical staggered fermions using the noncompact formulation of the gauge field action. So the total action is given by S = So + SF with
so=tfi c sF = C
{ +C
x
0370-2693/95/$09.50
(1)
F;,,(x), WC”) (Zx>eiAfl(“)x(x + fi>- X(x
P
@ 1995 Elsevier Science B.V. All rights reserved
SSDZO370-2693(95)00555-2
-I- jl)e-iAN(“)X(x))
+
(2) WZ~(X)X(X)} ,
44. Gackeler et al. /Physics
where /3 = 1/e2 (e = bare charge)
and
rlp(X) = (_l)xl+...++l
,
F,,(x)
+ ,%> - A,(x
= A,(x)
+ A,(x
101
Letters B 353 (1995) loo-106
(3) + P) - A,(x).
(4)
The data in this paper were obtained on an Lz x Lt lattice with L, = Lt = 12. The boundary conditions are periodic for the gauge field A,. For the fermion field x,X we have chosen periodic (antiperiodic) spatial (temporal) boundary conditions. For more details on the simulation see [ 61. In order to calculate fermionic observables like the fermion propagator or the fermion-antifermion-meson three-point functions,which are the subject of this paper, we have to fix the gauge. We choose to work with the Landau gauge,
c (&CL(x) - -4,(x - I;)) = o,
c-3
which has the advantage that it can be implemented exactly in the noncompact formulation. Unfortunately, the zero-momentum mode of the gauge field does not average to zero in our ensembles. To cope with this problem we divide our data into blocks of, say, 20 successive configurations, on which this mode is approximately constant, and perform our analysis in each block separately taking the zero-momentum mode into account explicitly (cf. [ 61). The averages and errors are then obtained from these block results. As we have argued in [6], the renormalized charge vanishes as one approaches the critical point PC w 0.19 , m = 0. Moreover, it turns out that the chiral phase transition is reasonably well described by logarithmically improved mean field theory, i.e., the equation of state can be derived from an effective linear a-model [ 6,121. Within such a model, the effective quartic coupling A is driven to zero in the continuum limit according to a (one-loop) renormalization group equation of the form
!!A=(92 dt
’
t=lnu,
where v = (xx) is the chiral condensate and c > 0. From the solution of this equation one sees that A N 1In (+1-t as u + 0. On the other hand, for a Yukawa coupling g, which we are investigating here, one finds in leading order [13]
dg = c’g’
-$
)
c’>O,
so that 1 - = -2c’t + const. g2 Hence the triviality
scenario
leads to the expectation
that g vanishes like
g N 1lna]-‘/2.
(9)
We shall extract our effective Yukawa couplings G:“‘(p,q;te,tt)
@.+-2i7J7(22, =xe x,x’
from three-point
t*)i&(p--
functions
q.o),y(2x’,tl)))c.
of the form (10)
Here (. . .)c denotes the fermion-line connected part of the expectation value, the lattice vectors X, X’ label (a = P) or scalar ( LY= S) meson spatial cubes of size 23, and M,(pa, t) is the standard pseudoscalar operator:
102
M. Giickeler et al. /Physics M&0,
f) = CeiP”~YsanCy)X(y,
t>xty,
Letters B 353 (1995) 100-106
t)
(11)
Y
with q~(y) = ( -l)y’+fi+y3,qs (y) = 1. We have calculated values of the spatial fermion momenta p, q:
(P,4) = (O,O), ($,O),
(Fe3, s
s
the three-point
function
( 10) for the following
$e3).
(12)
s
The spatial meson momentum is then given by p,, = p - q. AS is well known, the operators ( 11) couple not only to the state one is interested in but also to its “parity partner” (and higher excitations). Having computed the three-point function (10) for arbitrary times to, tr we can, however, project onto a single parity by Fourier transforming the to, tt dependences:
=-
-ip&+iq4fo
Ce
j,
Ce2i~x’--Zip’x(~(2x,tl)
Ceicp4-q4)rM,(p
- q,t)~(2x’,tc))~_
(13)
I
x,x’
QJO
Since the Goldstone boson is represented by the component of Mp that “oscillates” in time (see, e.g., [ 141)) we must choose p4 - q4 = s- in order to study the Yukawa coupling of the pseudoscalar. In the case of the scalar, on the other hand, we have to take p4 - q4 = 0. We define effective Yukawa couplings g, by comparing the Monte Carlo data for the three-point functions with tree-level formulas derived from an effective lattice action which describes the interaction between staggered fermions x, y and a (pseudo-)scalar field @ via coupling terms of the form -_gs
C@(xE(x)x(x)
(seal=-),
-gp ~~(x)~(x)~(x)(-l)“‘+“‘+*~ x
(14)
(1-V
(pseudoscalar).
In the tree-level formulas for the three-point functions we replace the free propagators by the full propagators calculated in the simulations. Furthermore, we include the appropriate wave function renormalizations 2~ (for the fermion field) and 2, (for the mesons). In this way we arrive at the following equation defining the momentum-dependent Yukawa couplings g,(p, q) : Ce 21JO x
_!-B,xei(P4-ur)th s
X
=ga(p,q)ZF(p)-*‘2ZF(q)-1’2Za(p-
-ip4t1+iq4foG:“)(p,q;to,tl)
,( M
q-p,t)Ma(p-a@),
t
c
~a(w)e-i(P-_q).~
w
v4(Lr))
X E
(
a
q)-1/2
{C
r14(o)~0ei44foCe2iP(x-x’)
II eKp4+dh iI
(x(2x
+
w, to)H2x’,O))}
x,x’ e2ip’(X-X’)(x(2X
+ w,tr)X(2r’,O))}*,
(16)
BP(a) = 1,
(17)
x,x’
11
where f% = -1,
f3p = 1)
i&(w)
and w runs over the 8 three-vectors
= (-l)~l+~z+~3, with components
0 or 1, i.e. o labels the corners of a spatial unit cube.
M. Giickeler et al. /Physics
Letters B 353 (1995) 100-106
The wave function renormalizations ZF, Z, are obtained contributions in the full propagators with the corresponding particle contribution to the meson propagator
103
in the usual way by comparing the one-particle free propagators. For example, writing the one-
(18)
&M-P.f)M,(piO))c s in the form
A,(P)
(e-Em(P)r+ e--Ea(P)(Lt--r)>,
(19)
we have z,(p)
= 2jAn(P)l
(1 -
e--LtE~(p)) sinh&(p)
(20)
(see, e.g., [ 1.51) . In the case of the fermion wave function renormalization we compare with the propagator in a constant background potential [ 61. Let us now discuss the results. One first observes that gLy(p, 4) is essentially real: The imaginary part is relatively small and noisy, for special values of the momenta it is even exactly zero. Hence we shall disregard it and consider g, as a real quantity. For obtaining a final unique answer it would be desirable to send all momenta to zero. However, the antiperiodic temporal boundary conditions for the fermions would allow us only to give the mesons momentum zero by choosing p = 4, p4 = 44 for the scalar and p4 = q4 + TT (mod 2~) for the pseudoscalar. Admitting for (p, 4) the values (12) we give 1441 the smallest possible value and average over q4 = *z-/L,: ES(PY4)
= $
c
gs((p,q4),(49q4))3
(21)
q‘i=+IrrlL,
For the error we take the largest error of a single value. In Fig. 1 we plot gp(p, q) -2 versus 1lna] as suggested by Eq. (8). (Note that in the preliminary presentation of our results in [ 161 we have erroneously shown the average of gp ( (0, q4), (0, q4) ) .> Although these numbers were obtained at different bare couplings /? (0.17 5 p 5 0.22) and different bare masses m (0.02 5 m 5 0.09), they fall into a narrow band, which with some optimism could even be considered as a single straight line in accordance with the triviality scenario (cf. Eq. (8) ). We consider these data as a qualitative confirmation of the triviality hypothesis. The results for the coupling of the scalar shown in Fig. 2 point into the same direction. However, several caveats have to be kept in mind. We have neglected the fermion-line disconnected contributions to the three-point functions and to the meson propagators, since they are difficult to calculate. These contributions could be especially important in the case of the scalar, which is a flavour singlet. Furthermore it is not clear, how our Yukawa couplings are affected by the fact that for some of our data points the mesons are heavier than two fermions and hence might correspond to resonances rather than stable particles in the continuum limit. Another problem concerns the definition of the wave function renormalizations. Since they are extracted from fits of free propagators to the measured two-point functions, they correspond to on-shell renormalization, whereas the effective Yukawa couplings are defined at fixed momenta (“fixed” in lattice units, the physical scale varies from simulation point to simulation point). The weak momentum dependence of the results may however indicate that this problem is not too severe. With the effective Yukawa couplings at hand we can draw new flow diagrams, i.e. we can construct the lines of constant gs, & in the P-m plane. These are to be compared with the lines of constant renormalized charge
104
M. Giickeler et al./Physics
0.5
1.0
1.5
2.0
2.5
2.0
2.5
Letters B 353 (1995) 100-106
Wlb)
0.5
1.0
1.5 In(l/u)
”
0.5
1.0
1.5
2.0
2.5
Wl/~)
Fig.
1. &(p,q)-’
Ee3).
eR
(ges,O)
versus ln(l/o)
(top to bottom).
for (p,q)
= (O,O), (Ee3,
Fig. 2.
gs(p,q)-z
versus In(l/a)
$e3),
(ges,O)
(topto bottom).
for (p,q)
= (O,O),
(fJe3,
or constant mass ratios [4,6]. In Fig. 3 we show the curves of constant gp(0, 0) together with the lines of constant en. We recall that the curves of constant en > 0 end on the line of first-order chiral phase transitions m = 0,/3 < PC, while only the curve en = 0 flows into the critical point [ 61, in accordance with triviality. The lines of constant gp (0,O) look rather similar, although truly parallel flows seem to be possible only in the lower right hand corner, i.e. for small values of en and gp. This suggests that also in the case of the Yukawa couplings only the curve of vanishing coupling flows into the critical point as perturbation theory would predict. In Fig. 4 we compare with the lines of constant ma/q where mn is the renormalized fermion mass and mp the pseudoscalar mass. Here we observe a completely different flow pattern in the left part of the diagram,
M. Giickeler et al. /Physics
105
Letters B 353 (1995) 100-106
m
m 0.08
0.08
0.06
0.06
O.OY
0.011
0.02
0.02
0.0 0.16
0.17
I
I
0.18
0.19
0.20
I
I
0.21
0.22
0.0
0.23
-
’ 0.16
0.17
P
Fig. 3. Curves of constant gp(0,0) (full lines) and curves of constant renormalized charge (dashed lines) with gP(O,O) ranging from 2.0 (lower right-hand comer) to 3.6 (upper left-hand comer) in steps of 0.2 and ei ranging from 2.8 (lower right-hand comer) to 4.6 (upper left-hand comer) in steps of 0.2.
0.18
I
I
0.19
0.20
0.21
0.22
0.23
P Fig. 4. Curves
of constant gp(O.0) (full lines) and curves of constant mu/mp (dashed lines) with &(O,O) ranging from 2.0 (lower right-hand comer) to 3.6 (upper left-hand comer) in steps of 0.2 and rnR/mp ranging from 0.4 (lower right-hand comer) to 1.6 (lower left-hand comer) in steps of 0.1.
whereas there is a tendency towards parallel flows in the region of small eR. The corresponding plots for look qualitatively similar. In conclusion, these results are in agreement with our previous findings [4,6] indicating that lines (approximately) constant physics can only be expected for small couplings. To achieve renormalizability larger couplings it would seem to be necessary to include further relevant operators in the action, such four-fermion terms. This work was supported in part by the Deutsche Forschungsgemeinschaft. performed on the Cray Y-MP in Jtilich with time granted by the Scientific thank both institutions for their support.
gs of for as
The numerical computations were Council of the HLRZ. We wish to
References f 11 f2] 131 141 [51 [6] [71 [8] [9] [ 101 [II I
M. Salmhofer and E. Seiler, L&t. Math. Phys. 21 (1991) 13; Commun. Math. Phys. 139 (1991) 395; 146 (1992) 637(E). Sl? Booth, R.D. Kenway and B.J. Pendleton, Phys. Left. B 228 (1989) 115. M. GBckeler, R. Horsley, E. Laermann, P Rakow, G. Schierholz, R. Sommer and U.-J. Wiese, Nucl. Phys. B 334 (1990) 527. M. Gockeler, R. Horsley, E. Laermann, P Rakow, G. Schierholz, R. Sommer and U.-J. Wiese, Phys. Lett. B 251 (1990) 567; B 256 (1991) 562(E). S.J. Hands, J.B. Kogut, R. Renken, A. Kocic, D.K. Sinclair and KC. Wang, Phys. Len. B 261 (1991) 294; A. KociC, J.B. Kogut and K.C. Wang, Nucl. Phys. B 398 (1993) 405. M. Giickeler, R. Horsley, P. Rakow, G. Schierholz and R. Sommer, Nucl. Phys. B 371 ( 1992) 713. V. Azcoiti, G. Di Carlo and A.F. Grillo, Mod. Phys. Len. A 7 (1992) 3561; Int. J. Mod. Phys. A 8 (1993) 4235. A. Kocic, Nucl. Phys. B (Proc. Suppl.) 34 (1994) 123. M. Gockeler, R. Horsley, P Rakow, G. Schierholz and H. Stiiben, Nucl. Phys. B (Proc. Suppl.) 34 (1994) 527. A.M. Horowitz, Phys. Rev. D 43 (1991) 2461. A. Kocic, J.B. Kogut, M.-P Lombard0 and KC. Wang, Nucl. Phys. B 397 (1993) 451; S.J. Hands, A. Kocic, J.B. Kogut, R.L. Renken, D.K. Sinclair and K.C. Wang, Nucl. Phys. B 413 (1994) 503.
106
M. Giickeler et al. /Physics
Letters 3 353 (1995) 100-106
[ 121 M. GSckeler, R. Horsley, V. Linke, P Rakow, G. Schierholz and H. Stiiben, preprint ZIB-SC 94-31, PUB-W/94-13,
hep-laU9412064. [ 131 A. Zee, Phys. Rev. D 7 (1973) 3630. [ 141 MEL. Golterman and J. Smit, Nucl. Phys. B 255 (1985) 328; Nucl. Phys. B 273 (1986) 663; R. Altmeyer, K.D. Born, M. GSckeler, R. Horsley, E. Laermann aad G. Schierholz, Nucl. Phys. B 389 (1993) 445. [ 151 S. Aoki, J. Shigemitsu aad J. Sloan, Nucl. Phys. B 372 (1992) 361. [ 161 M. GBckeler, R. Horsley, P.E.L. Rakow and G. Schierholz, Nucl. Phys. B (Proc. Suppl.) 34 ( 1994) 531.
HLRZ 94-53.
22June1995
__ EB
PHYSICS
ELSBVIER
LETTERS
B
Physics Letters B 353 (1995) 100-106
Effective Yukawa couplings in noncompact lattice QED M. Giickeler a,b, R. Horsley a, P.E.L. Rakow c, G. Schierholz a,d a Gruppe Theorie der Elementarteilchen, Hiichstleistungsrechenzentrwn HLRZ, c/o Forschungszentrum Jiilich, D-52425 Jiilich, Germany b Institutfiir Theoretische Physik, RWTH Aachen, Physikzentrum, D-52056 Aachen, Germany c Institutfiir Theoretische Physik, Freie Vniversitiit Berlin, Arnimaliee 14, D-14195 Berlin, Germany d Deutsches Elektronen-Synchrotron DES% NotkestraJe 85, D-22603 Hamburg, Germany
Received 10 March 1995 Editor: P.V. Landshoff
Abstract We investigate effective Yukawa couplings of mesons to the elementary fermions in noncompact lattice couplings are extracted from suitable fermion-antifermion-meson three-point functions calculated by Monte Carlo with dynamical staggered fermions. The scaling behaviour is compatible with expectations from perturbation indicating triviality of QED. The lines of constant Yukawa coupling are compared to flows of other quantities. is seen, at most, for weak coupling.
QED. The simulations theory, thus Consistency
In strongly coupled lattice QED chiral symmetry is known to be spontaneously broken, whereas an unbroken chiral symmetry is expected in the weak coupling limit [ 11. This finding immediately raises questions like: Can a nontrivial continuum limit be defined at the corresponding critical point? If so, what is the nature of the associated continuum theory? Several groups have investigated these problems by means of extensive Monte Carlo simulations of lattice QED using various techniques (see, e.g., [ 2-91) . Yet the physical interpretation of the data is still controversial (see also [ lo] ) . Correlation functions of composite fermion-antifermion operators show evidence for at least two fermionantifermion (bound) states (“meson” states) : a pseudoscalar state denoted by P (the Goldstone boson associated with the spontaneous symmetry breaking) and a scalar state (S) [3,11,6]. In this letter we study fermionantifermion-meson three-point functions in order to determine (effective) Yukawa couplings of P and S to the fermions. The investigation should provide additional information about the lattice model. In particular, if it has a trivial continuum limit, the couplings should tend to zero as one approaches the critical point. We have performed simulations with dynamical staggered fermions using the noncompact formulation of the gauge field action. So the total action is given by S = So + SF with
so=tfi c sF = C
{ +C
x
0370-2693/95/$09.50
(1)
F;,,(x), WC”) (Zx>eiAfl(“)x(x + fi>- X(x
P
@ 1995 Elsevier Science B.V. All rights reserved
SSDZO370-2693(95)00555-2
-I- jl)e-iAN(“)X(x))
+
(2) WZ~(X)X(X)} ,
44. Gackeler et al. /Physics
where /3 = 1/e2 (e = bare charge)
and
rlp(X) = (_l)xl+...++l
,
F,,(x)
+ ,%> - A,(x
= A,(x)
+ A,(x
101
Letters B 353 (1995) loo-106
(3) + P) - A,(x).
(4)
The data in this paper were obtained on an Lz x Lt lattice with L, = Lt = 12. The boundary conditions are periodic for the gauge field A,. For the fermion field x,X we have chosen periodic (antiperiodic) spatial (temporal) boundary conditions. For more details on the simulation see [ 61. In order to calculate fermionic observables like the fermion propagator or the fermion-antifermion-meson three-point functions,which are the subject of this paper, we have to fix the gauge. We choose to work with the Landau gauge,
c (&CL(x) - -4,(x - I;)) = o,
c-3
which has the advantage that it can be implemented exactly in the noncompact formulation. Unfortunately, the zero-momentum mode of the gauge field does not average to zero in our ensembles. To cope with this problem we divide our data into blocks of, say, 20 successive configurations, on which this mode is approximately constant, and perform our analysis in each block separately taking the zero-momentum mode into account explicitly (cf. [ 61). The averages and errors are then obtained from these block results. As we have argued in [6], the renormalized charge vanishes as one approaches the critical point PC w 0.19 , m = 0. Moreover, it turns out that the chiral phase transition is reasonably well described by logarithmically improved mean field theory, i.e., the equation of state can be derived from an effective linear a-model [ 6,121. Within such a model, the effective quartic coupling A is driven to zero in the continuum limit according to a (one-loop) renormalization group equation of the form
!!A=(92 dt
’
t=lnu,
where v = (xx) is the chiral condensate and c > 0. From the solution of this equation one sees that A N 1In (+1-t as u + 0. On the other hand, for a Yukawa coupling g, which we are investigating here, one finds in leading order [13]
dg = c’g’
-$
)
c’>O,
so that 1 - = -2c’t + const. g2 Hence the triviality
scenario
leads to the expectation
that g vanishes like
g N 1lna]-‘/2.
(9)
We shall extract our effective Yukawa couplings G:“‘(p,q;te,tt)
@.+-2i7J7(22, =xe x,x’
from three-point
t*)i&(p--
functions
q.o),y(2x’,tl)))c.
of the form (10)
Here (. . .)c denotes the fermion-line connected part of the expectation value, the lattice vectors X, X’ label (a = P) or scalar ( LY= S) meson spatial cubes of size 23, and M,(pa, t) is the standard pseudoscalar operator:
102
M. Giickeler et al. /Physics M&0,
f) = CeiP”~YsanCy)X(y,
t>xty,
Letters B 353 (1995) 100-106
t)
(11)
Y
with q~(y) = ( -l)y’+fi+y3,qs (y) = 1. We have calculated values of the spatial fermion momenta p, q:
(P,4) = (O,O), ($,O),
(Fe3, s
s
the three-point
function
( 10) for the following
$e3).
(12)
s
The spatial meson momentum is then given by p,, = p - q. AS is well known, the operators ( 11) couple not only to the state one is interested in but also to its “parity partner” (and higher excitations). Having computed the three-point function (10) for arbitrary times to, tr we can, however, project onto a single parity by Fourier transforming the to, tt dependences:
=-
-ip&+iq4fo
Ce
j,
Ce2i~x’--Zip’x(~(2x,tl)
Ceicp4-q4)rM,(p
- q,t)~(2x’,tc))~_
(13)
I
x,x’
QJO
Since the Goldstone boson is represented by the component of Mp that “oscillates” in time (see, e.g., [ 141)) we must choose p4 - q4 = s- in order to study the Yukawa coupling of the pseudoscalar. In the case of the scalar, on the other hand, we have to take p4 - q4 = 0. We define effective Yukawa couplings g, by comparing the Monte Carlo data for the three-point functions with tree-level formulas derived from an effective lattice action which describes the interaction between staggered fermions x, y and a (pseudo-)scalar field @ via coupling terms of the form -_gs
C@(xE(x)x(x)
(seal=-),
-gp ~~(x)~(x)~(x)(-l)“‘+“‘+*~ x
(14)
(1-V
(pseudoscalar).
In the tree-level formulas for the three-point functions we replace the free propagators by the full propagators calculated in the simulations. Furthermore, we include the appropriate wave function renormalizations 2~ (for the fermion field) and 2, (for the mesons). In this way we arrive at the following equation defining the momentum-dependent Yukawa couplings g,(p, q) : Ce 21JO x
_!-B,xei(P4-ur)th s
X
=ga(p,q)ZF(p)-*‘2ZF(q)-1’2Za(p-
-ip4t1+iq4foG:“)(p,q;to,tl)
,( M
q-p,t)Ma(p-a@),
t
c
~a(w)e-i(P-_q).~
w
v4(Lr))
X E
(
a
q)-1/2
{C
r14(o)~0ei44foCe2iP(x-x’)
II eKp4+dh iI
(x(2x
+
w, to)H2x’,O))}
x,x’ e2ip’(X-X’)(x(2X
+ w,tr)X(2r’,O))}*,
(16)
BP(a) = 1,
(17)
x,x’
11
where f% = -1,
f3p = 1)
i&(w)
and w runs over the 8 three-vectors
= (-l)~l+~z+~3, with components
0 or 1, i.e. o labels the corners of a spatial unit cube.
M. Giickeler et al. /Physics
Letters B 353 (1995) 100-106
The wave function renormalizations ZF, Z, are obtained contributions in the full propagators with the corresponding particle contribution to the meson propagator
103
in the usual way by comparing the one-particle free propagators. For example, writing the one-
(18)
&M-P.f)M,(piO))c s in the form
A,(P)
(e-Em(P)r+ e--Ea(P)(Lt--r)>,
(19)
we have z,(p)
= 2jAn(P)l
(1 -
e--LtE~(p)) sinh&(p)
(20)
(see, e.g., [ 1.51) . In the case of the fermion wave function renormalization we compare with the propagator in a constant background potential [ 61. Let us now discuss the results. One first observes that gLy(p, 4) is essentially real: The imaginary part is relatively small and noisy, for special values of the momenta it is even exactly zero. Hence we shall disregard it and consider g, as a real quantity. For obtaining a final unique answer it would be desirable to send all momenta to zero. However, the antiperiodic temporal boundary conditions for the fermions would allow us only to give the mesons momentum zero by choosing p = 4, p4 = 44 for the scalar and p4 = q4 + TT (mod 2~) for the pseudoscalar. Admitting for (p, 4) the values (12) we give 1441 the smallest possible value and average over q4 = *z-/L,: ES(PY4)
= $
c
gs((p,q4),(49q4))3
(21)
q‘i=+IrrlL,
For the error we take the largest error of a single value. In Fig. 1 we plot gp(p, q) -2 versus 1lna] as suggested by Eq. (8). (Note that in the preliminary presentation of our results in [ 161 we have erroneously shown the average of gp ( (0, q4), (0, q4) ) .> Although these numbers were obtained at different bare couplings /? (0.17 5 p 5 0.22) and different bare masses m (0.02 5 m 5 0.09), they fall into a narrow band, which with some optimism could even be considered as a single straight line in accordance with the triviality scenario (cf. Eq. (8) ). We consider these data as a qualitative confirmation of the triviality hypothesis. The results for the coupling of the scalar shown in Fig. 2 point into the same direction. However, several caveats have to be kept in mind. We have neglected the fermion-line disconnected contributions to the three-point functions and to the meson propagators, since they are difficult to calculate. These contributions could be especially important in the case of the scalar, which is a flavour singlet. Furthermore it is not clear, how our Yukawa couplings are affected by the fact that for some of our data points the mesons are heavier than two fermions and hence might correspond to resonances rather than stable particles in the continuum limit. Another problem concerns the definition of the wave function renormalizations. Since they are extracted from fits of free propagators to the measured two-point functions, they correspond to on-shell renormalization, whereas the effective Yukawa couplings are defined at fixed momenta (“fixed” in lattice units, the physical scale varies from simulation point to simulation point). The weak momentum dependence of the results may however indicate that this problem is not too severe. With the effective Yukawa couplings at hand we can draw new flow diagrams, i.e. we can construct the lines of constant gs, & in the P-m plane. These are to be compared with the lines of constant renormalized charge
104
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0.5
1.0
1.5
2.0
2.5
2.0
2.5
Letters B 353 (1995) 100-106
Wlb)
0.5
1.0
1.5 In(l/u)
”
0.5
1.0
1.5
2.0
2.5
Wl/~)
Fig.
1. &(p,q)-’
Ee3).
eR
(ges,O)
versus ln(l/o)
(top to bottom).
for (p,q)
= (O,O), (Ee3,
Fig. 2.
gs(p,q)-z
versus In(l/a)
$e3),
(ges,O)
(topto bottom).
for (p,q)
= (O,O),
(fJe3,
or constant mass ratios [4,6]. In Fig. 3 we show the curves of constant gp(0, 0) together with the lines of constant en. We recall that the curves of constant en > 0 end on the line of first-order chiral phase transitions m = 0,/3 < PC, while only the curve en = 0 flows into the critical point [ 61, in accordance with triviality. The lines of constant gp (0,O) look rather similar, although truly parallel flows seem to be possible only in the lower right hand corner, i.e. for small values of en and gp. This suggests that also in the case of the Yukawa couplings only the curve of vanishing coupling flows into the critical point as perturbation theory would predict. In Fig. 4 we compare with the lines of constant ma/q where mn is the renormalized fermion mass and mp the pseudoscalar mass. Here we observe a completely different flow pattern in the left part of the diagram,
M. Giickeler et al. /Physics
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Letters B 353 (1995) 100-106
m
m 0.08
0.08
0.06
0.06
O.OY
0.011
0.02
0.02
0.0 0.16
0.17
I
I
0.18
0.19
0.20
I
I
0.21
0.22
0.0
0.23
-
’ 0.16
0.17
P
Fig. 3. Curves of constant gp(0,0) (full lines) and curves of constant renormalized charge (dashed lines) with gP(O,O) ranging from 2.0 (lower right-hand comer) to 3.6 (upper left-hand comer) in steps of 0.2 and ei ranging from 2.8 (lower right-hand comer) to 4.6 (upper left-hand comer) in steps of 0.2.
0.18
I
I
0.19
0.20
0.21
0.22
0.23
P Fig. 4. Curves
of constant gp(O.0) (full lines) and curves of constant mu/mp (dashed lines) with &(O,O) ranging from 2.0 (lower right-hand comer) to 3.6 (upper left-hand comer) in steps of 0.2 and rnR/mp ranging from 0.4 (lower right-hand comer) to 1.6 (lower left-hand comer) in steps of 0.1.
whereas there is a tendency towards parallel flows in the region of small eR. The corresponding plots for look qualitatively similar. In conclusion, these results are in agreement with our previous findings [4,6] indicating that lines (approximately) constant physics can only be expected for small couplings. To achieve renormalizability larger couplings it would seem to be necessary to include further relevant operators in the action, such four-fermion terms. This work was supported in part by the Deutsche Forschungsgemeinschaft. performed on the Cray Y-MP in Jtilich with time granted by the Scientific thank both institutions for their support.
gs of for as
The numerical computations were Council of the HLRZ. We wish to
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