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Domain wall fermions and fermion loop observables
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 119 (2003) 407-409
www.clscvicr.com/locatc/npc
Domain wall fermions and fermion loop observables Azusa Yamaguchi a aDepartment of Physics, Columbia University, New York, USA Physical observables which involve fermion loops, such as the chiral condensate, ($+LJ), or matrix elements of the four-fermi operators which enter weak decays, require accurate off-shell fermion behaviour. We investigate such off-shell contributions to (&!I) fo r domain wall fermions (DWF). After removing the unphysical, short-distance contributions to (@lo) by Pauli-Villars subtraction, we compare the mass dependence of the resulting physical quantity, evaluated with both the Wilson and DBW2 gauge actions, since these two actions are known to give quite different 5-dimensional DWF behaviour. We also examine the dependence of the amplitudes entering (g;lcl) on the fifth dimension s, to gain insight into potentially different s-dependence for low- and high-energy modes. Finally, we summarise the present RBC results for conventionally normalised (&/IJ) for the various actions studied. 1.
Introduction
We investigate the accuracy with which observables involving closed fermion loops can be computed using the domain wall fermion formulation by studying the chiral condensate. First a comparison is made of the physically-normalised chiral condensate evaluated directly in the limit of zero fermion mass for three combinations of gauge action and lattice spacing. These results are compared both with each other and with the predictions of the Cell-Mann-Oakes-Renner relation. Second, we make a first attempt at evaluating the mass dependence of (&) after removing the unphysical short distance contributions, using Pauli-Villars subtraction and appropriate multiplicative renormalisation. Finally, we examine the s-dependence of (&,!I) and try to distinguish the dependence of the low- and high-energy modes on this direction. Our simulations were performed on a lattice of size 163 x 32 with L, = 16, using both the Wilson and DBW2 actions, with the lattice spacings u-l N 2 Gev (Wilson and DBW2) and a-l = 1.3(4) Gev (only DBW2). The results from the Wilson gauge action have already been reported in Ref. [l] while those from the DBW2 gauge action are reported at this meeting [2]. 2. (&!J)
in physical units
To compare the ($11) results obtained with the DBW2 actions to those from the Wilson action, we used the Z factor obtained from ratios of input
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 m-f
Figure 1. Values of ($$) for the DBW2 action at 1.3 GeV and 2 Gev compared with those from Wilson gauge action at 2 Gev. Both the ordinate and the absyssa for the DBW2 values have been scaled by the appropriate Z factor. quark masses, mf, which must be used with each action LO give the same value for m,/m,. From the values of m, and mp reported in [2], the Z factor values in each case are: Z(a-’ Z(a-’
=2 GeV) =1.3
GeV)
= 1.118(117)
(1)
= 1.661(204).
(2)
Using the relations, ($$)Wilson
=
z
x ?$DBW2,
(mf)wih~
= z-’
x
(m.f)DBWZ?
(3) (4)
the results for (&/I) in lattice units, but scaled to Wilson gauge action normalisation, are shown in Fig. 1.
0920-5632/03/s - see front matter 0 2003 Published by Elsevier Science B.V. doi: 10.1016/S0920-5632(03)01569-X
A. Yamaguchi/Nuclear
408
Physics B (Proc. Suppl.)
Table 2 The Gell-Mann-Oakes-Renner relation equates the results shown in the right-most two columns. The quantity b is the slope mi(mf) versus mf. b &f,lO4 (++!J)104 Action Wilson 2 GeV 3.14(9) 3.3(5) 2.80(9) DBW2 2 GeV 2.3(2) 2.0(3) 2.58(2) D B W 2 1 . 3 GeV 5.02(41) 10.0(2) 9.87(5)
0.003 0.0025 0.002 0.0015 0.001 o.cQO5 0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 d&l
subtraction; Figure 2. Linearm p a r t Z(a-‘)3{ ($$J)~ - $($+!J)~~} i n D B W 2 a t 1.3 GeV and 2 GeV with Z factor and that in Wilson at 2 GeV, with ml wilson = 0.04 fixed. Since for domain wall fermions, the fermion zero modes which come from instanton-like gauge fields give @T/I) a pole at rnf = -mres, we have to take this fact into account, using the nonnegligible mres = 1.24 . 10e3 for the Wilson action and but neglecting the much smaller mres = 1.7. 10m5 for the DBW2 action at o-l N 2 Gev. For the DBW2 action at u-l = 1.3 Gev the large volume suppresses zero modes to such an extent that this pole term can be omitted. Thus, we use the fitting function for (&J),
iwmr> =
a-1
mf + mTes
+ao+a1
Xrnf,
(5)
for u-l = 2.0 Gev and drop the pole term for the o-l = 1.3 Gev case. Taking the (&!I) value at mf = -mres and converting to conventional, MS normalisation [ 11, we obtain the results in Tab. 1. Table 1 Results for ($$I) normalised at 2 GeV in the MS scheme together with the corresponding 2 factor. Action w?w)1’3 Z(MS) 0.619(25) 0.245(6) GeV Wilson (2 GeV) DBW2 (2 GeV) 0.712(14) 0.235(13) GeV D B W 2 ( 1 . 3 GeV) 0.699(16) 0.265(6) GeV Another method determines ($,!J) from pion observables using the Gell-Mann-Oakes-Renner relation:
f,
119 (2003) 407-409
m2, 48(mf + mres)
=
(6)
Replacing (m,~~ves) with the slope, b, of rn: versus mf, we can simply compare the left- and the right-hand sides of Eq. 6, as is done in the Tab. 2, which shows good consistency. 3. Higher energy contribution to (&j) To explore higher energy contributions to (&!I), we would like to examine the dependence of ($$J) on mf for AQCD 5 mf < l/u. However, a meaningful comparison of the mass dependence of (@,!I) requires that the unphysical, divergent contributions to ($+) must first be removed. This can be done by performing a series of Pauli-Villars subtractions and evaluating the renormahsed operator:
(7) where mc = mZ and CO = 1. The subtraction constants {ci}r
A. Yamaguchi/Nuclear Physics B (Proc.
Suppl.)
409
119 (2003) 407-409
lattice action and cutoff l/u. In order to satisfy these conditions, four masses are required. We simplified these conditions and studied twomass case:
DBW2
1.3GeV
DBW22GeV
--f-- + .
x
Wilson 2GeV x
as a function of m with the fixed ml = 0.04. Since this satisfies only the second condition, the other divergences contaminate this quantity. Fig. 2 shows this quantity subtracted at the same physical value of ml for our three cases. The deviations between the three curves suggest possibly differing contributions to ($$) from intermediate energies. 4. S-dependence of($$) How the various domain wall fermion states are localised along the fifth dimensional direction is a very interesting question. To see it, we investigated the s-dependence of ($$) for both gauge actions. Fitting ($$) as a function of mf for fixed s we obtain, iw%
mf 1 = ,fail,,
+ Q(S) + Q(S)
x rnf
.(lO)
We can then examine the s-dependence of the coefficients se(s) and al(s) which describe the low and high energy contributions to ($7/j): (-Jo(s)
=
CO exp(P x s)
al(s)
=
cl exp(Q x s) + c2 exp(R x s).
(11) (12)
The coefficients P, Q, and R are given in Tab. 3. This table suggests that DWF localisation for the low energy part of ($$J) is improved by the DBW2 action, see Fig. 3, while the high energy contribution is not much affected by the change of action. Table 3 Fitting parameters descripting the s dependence of the low energy_(P) and high energy (Q and R) contribution to ($$). Action P Q R Wilson 2 GeV 1.70 1.87(7) 0.55(l) DBW2 2 GeV 7.74 1.57(7) 0.69(2) D B W 2 1 . 3 GeV 3 . 6 6 1.84(17) 0.60(3)
13
13.5
‘ii
145
15
s
Figure 3. Low energy contribution: as(s) = co exp(P x s) for DBW2 and Wilson actions. 5. Conclusion From our comparison of the values of ($$) on a 163 x 32 L, = 16 lattice for the DBW2 and Wilson gauge gauge actions are nicely consistent. Values obtained from the Gell-Mann-Oakes-Renner relation also agree. With proper subtraction, ($$L) shows a mass behaviour that is also very consistent between the DBW2 and Wilson cases. With greater accuracy, this may be an even more interesting quark loop observable to study. Our study of the s-dependence of ($$) suggests that the DBW2 action improves the chiral properties of low energy physics observables. 6. Acknowledge This work was supported in part by the U.S. Department of Energy. The author thanks RIKEN, Brookhaven National Laboratory and the U.S. Department of Energy for providing the facilities essential for carrying out, this work. And I would like to thank Dr. Chris Dawson for helping data analysis and giving me helpful advice. I would like also thank to all BNL and Columbia collaborators for DWF and DBW2 data and very helpful advice. I would like to thank Prof. Norman Christ sincerely for helpful discussion and advice and clarifying my talk and this proceeding. REFERENCES 1. T. Blum, et al., hep-lat/0007038 2. Contributions of Y. Aoki, C. Dawson, J. Noaki and K. Orginos, to these proceedings.
Nuclear Physics B (Proc. Suppl.) 119 (2003) 407-409
www.clscvicr.com/locatc/npc
Domain wall fermions and fermion loop observables Azusa Yamaguchi a aDepartment of Physics, Columbia University, New York, USA Physical observables which involve fermion loops, such as the chiral condensate, ($+LJ), or matrix elements of the four-fermi operators which enter weak decays, require accurate off-shell fermion behaviour. We investigate such off-shell contributions to (&!I) fo r domain wall fermions (DWF). After removing the unphysical, short-distance contributions to (@lo) by Pauli-Villars subtraction, we compare the mass dependence of the resulting physical quantity, evaluated with both the Wilson and DBW2 gauge actions, since these two actions are known to give quite different 5-dimensional DWF behaviour. We also examine the dependence of the amplitudes entering (g;lcl) on the fifth dimension s, to gain insight into potentially different s-dependence for low- and high-energy modes. Finally, we summarise the present RBC results for conventionally normalised (&/IJ) for the various actions studied. 1.
Introduction
We investigate the accuracy with which observables involving closed fermion loops can be computed using the domain wall fermion formulation by studying the chiral condensate. First a comparison is made of the physically-normalised chiral condensate evaluated directly in the limit of zero fermion mass for three combinations of gauge action and lattice spacing. These results are compared both with each other and with the predictions of the Cell-Mann-Oakes-Renner relation. Second, we make a first attempt at evaluating the mass dependence of (&) after removing the unphysical short distance contributions, using Pauli-Villars subtraction and appropriate multiplicative renormalisation. Finally, we examine the s-dependence of (&,!I) and try to distinguish the dependence of the low- and high-energy modes on this direction. Our simulations were performed on a lattice of size 163 x 32 with L, = 16, using both the Wilson and DBW2 actions, with the lattice spacings u-l N 2 Gev (Wilson and DBW2) and a-l = 1.3(4) Gev (only DBW2). The results from the Wilson gauge action have already been reported in Ref. [l] while those from the DBW2 gauge action are reported at this meeting [2]. 2. (&!J)
in physical units
To compare the ($11) results obtained with the DBW2 actions to those from the Wilson action, we used the Z factor obtained from ratios of input
0
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 m-f
Figure 1. Values of ($$) for the DBW2 action at 1.3 GeV and 2 Gev compared with those from Wilson gauge action at 2 Gev. Both the ordinate and the absyssa for the DBW2 values have been scaled by the appropriate Z factor. quark masses, mf, which must be used with each action LO give the same value for m,/m,. From the values of m, and mp reported in [2], the Z factor values in each case are: Z(a-’ Z(a-’
=2 GeV) =1.3
GeV)
= 1.118(117)
(1)
= 1.661(204).
(2)
Using the relations, ($$)Wilson
=
z
x ?$DBW2,
(mf)wih~
= z-’
x
(m.f)DBWZ?
(3) (4)
the results for (&/I) in lattice units, but scaled to Wilson gauge action normalisation, are shown in Fig. 1.
0920-5632/03/s - see front matter 0 2003 Published by Elsevier Science B.V. doi: 10.1016/S0920-5632(03)01569-X
A. Yamaguchi/Nuclear
408
Physics B (Proc. Suppl.)
Table 2 The Gell-Mann-Oakes-Renner relation equates the results shown in the right-most two columns. The quantity b is the slope mi(mf) versus mf. b &f,lO4 (++!J)104 Action Wilson 2 GeV 3.14(9) 3.3(5) 2.80(9) DBW2 2 GeV 2.3(2) 2.0(3) 2.58(2) D B W 2 1 . 3 GeV 5.02(41) 10.0(2) 9.87(5)
0.003 0.0025 0.002 0.0015 0.001 o.cQO5 0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 d&l
subtraction; Figure 2. Linearm p a r t Z(a-‘)3{ ($$J)~ - $($+!J)~~} i n D B W 2 a t 1.3 GeV and 2 GeV with Z factor and that in Wilson at 2 GeV, with ml wilson = 0.04 fixed. Since for domain wall fermions, the fermion zero modes which come from instanton-like gauge fields give @T/I) a pole at rnf = -mres, we have to take this fact into account, using the nonnegligible mres = 1.24 . 10e3 for the Wilson action and but neglecting the much smaller mres = 1.7. 10m5 for the DBW2 action at o-l N 2 Gev. For the DBW2 action at u-l = 1.3 Gev the large volume suppresses zero modes to such an extent that this pole term can be omitted. Thus, we use the fitting function for (&J),
iwmr> =
a-1
mf + mTes
+ao+a1
Xrnf,
(5)
for u-l = 2.0 Gev and drop the pole term for the o-l = 1.3 Gev case. Taking the (&!I) value at mf = -mres and converting to conventional, MS normalisation [ 11, we obtain the results in Tab. 1. Table 1 Results for ($$I) normalised at 2 GeV in the MS scheme together with the corresponding 2 factor. Action w?w)1’3 Z(MS) 0.619(25) 0.245(6) GeV Wilson (2 GeV) DBW2 (2 GeV) 0.712(14) 0.235(13) GeV D B W 2 ( 1 . 3 GeV) 0.699(16) 0.265(6) GeV Another method determines ($,!J) from pion observables using the Gell-Mann-Oakes-Renner relation:
f,
119 (2003) 407-409
m2, 48(mf + mres)
=
(6)
Replacing (m,~~ves) with the slope, b, of rn: versus mf, we can simply compare the left- and the right-hand sides of Eq. 6, as is done in the Tab. 2, which shows good consistency. 3. Higher energy contribution to (&j) To explore higher energy contributions to (&!I), we would like to examine the dependence of ($$J) on mf for AQCD 5 mf < l/u. However, a meaningful comparison of the mass dependence of (@,!I) requires that the unphysical, divergent contributions to ($+) must first be removed. This can be done by performing a series of Pauli-Villars subtractions and evaluating the renormahsed operator:
(7) where mc = mZ and CO = 1. The subtraction constants {ci}r
A. Yamaguchi/Nuclear Physics B (Proc.
Suppl.)
409
119 (2003) 407-409
lattice action and cutoff l/u. In order to satisfy these conditions, four masses are required. We simplified these conditions and studied twomass case:
DBW2
1.3GeV
DBW22GeV
--f-- + .
x
Wilson 2GeV x
as a function of m with the fixed ml = 0.04. Since this satisfies only the second condition, the other divergences contaminate this quantity. Fig. 2 shows this quantity subtracted at the same physical value of ml for our three cases. The deviations between the three curves suggest possibly differing contributions to ($$) from intermediate energies. 4. S-dependence of($$) How the various domain wall fermion states are localised along the fifth dimensional direction is a very interesting question. To see it, we investigated the s-dependence of ($$) for both gauge actions. Fitting ($$) as a function of mf for fixed s we obtain, iw%
mf 1 = ,fail,,
+ Q(S) + Q(S)
x rnf
.(lO)
We can then examine the s-dependence of the coefficients se(s) and al(s) which describe the low and high energy contributions to ($7/j): (-Jo(s)
=
CO exp(P x s)
al(s)
=
cl exp(Q x s) + c2 exp(R x s).
(11) (12)
The coefficients P, Q, and R are given in Tab. 3. This table suggests that DWF localisation for the low energy part of ($$J) is improved by the DBW2 action, see Fig. 3, while the high energy contribution is not much affected by the change of action. Table 3 Fitting parameters descripting the s dependence of the low energy_(P) and high energy (Q and R) contribution to ($$). Action P Q R Wilson 2 GeV 1.70 1.87(7) 0.55(l) DBW2 2 GeV 7.74 1.57(7) 0.69(2) D B W 2 1 . 3 GeV 3 . 6 6 1.84(17) 0.60(3)
13
13.5
‘ii
145
15
s
Figure 3. Low energy contribution: as(s) = co exp(P x s) for DBW2 and Wilson actions. 5. Conclusion From our comparison of the values of ($$) on a 163 x 32 L, = 16 lattice for the DBW2 and Wilson gauge gauge actions are nicely consistent. Values obtained from the Gell-Mann-Oakes-Renner relation also agree. With proper subtraction, ($$L) shows a mass behaviour that is also very consistent between the DBW2 and Wilson cases. With greater accuracy, this may be an even more interesting quark loop observable to study. Our study of the s-dependence of ($$) suggests that the DBW2 action improves the chiral properties of low energy physics observables. 6. Acknowledge This work was supported in part by the U.S. Department of Energy. The author thanks RIKEN, Brookhaven National Laboratory and the U.S. Department of Energy for providing the facilities essential for carrying out, this work. And I would like to thank Dr. Chris Dawson for helping data analysis and giving me helpful advice. I would like also thank to all BNL and Columbia collaborators for DWF and DBW2 data and very helpful advice. I would like to thank Prof. Norman Christ sincerely for helpful discussion and advice and clarifying my talk and this proceeding. REFERENCES 1. T. Blum, et al., hep-lat/0007038 2. Contributions of Y. Aoki, C. Dawson, J. Noaki and K. Orginos, to these proceedings.