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Improvement and renormalization constants in O(a) improved lattice QCD
SUPPLEMENTS www.elsevier.nl/locate/n
Nuclear Physics B (Proc. Suppl.) 94 (2001) 599-602
ELSEVIER
Improvement and renormalization constants in O(a) improved lattice QCD* Tanmoy
Bhattacharyaa,
“MS B-285, Los Alamos bPhysics
Department,
Rajan National University
Guptaa,
Weonjong
Leeaand
Lab, Los Alamos, of Washington,
Stephen
Sharpeb
New Mexico 87545, USA
Seattle,
Washington
98195, USA
constants for bilinear We present results at p = 6.0 and 6.2 for the O(a) improvement and renormalization We discuss the extraction of the mass dependence of the operators using axial and vector Ward identities. renormalization constants and the coefficients of the equation of motion operators.
1. INTRODUCTION In quenched Lattice &CD, axial and vector Ward identities can be used to determine, at O(a), all the scale independent renormalization constants for bilinear currents (ZA, ZV, and Zs/Zp), the improvement constants (CA, cy , and CT), the quark mass dependence of all five 20, and the coefficients of the equation of motion operators [l-6]. Here we summarize results at /3 = 6.0 and 6.2 and discuss the highlights of our calculations. Details and notation are given in [6]. We start with the generalized axial Ward identity involving operators improved on and off-shell (6S(‘2) @3,1,(v)
J@i) (0)) = (X$,s$(~)
=
z(ij) o(ij)
=
$j)
=
@i)rj$+(j)
(I)
_‘;“1”,4) oc_@
,
_ $(i)t;lir@j)
on-shell are o(“)
improved
renormalized
operators
E
Zs(l+
b~WfQj)O~‘,
5
Z$(l+
boC377h~j)Ojij)
(AI),
=
A, + aCA&P,
(VI),
=
(Tr)w PI
E
V, + w&Tpv, Tpy + ac~(d,V,, - &V,)
z
P,
R
SF’
(4) (5)
e
(6) (7) ,
SIES,
(8) (9)
JC31) (0))
where SO is the result of the axial variation of 0 (A, +) V,, S +) P, and Tpy + tzpvpoTp,,), and 6.S is the variation in the action. At O(a) there exists only one dimension 4 offshell operator (which vanishes by the equations of motion) for each bilinear that has the appropriate symmetries [7]. Consequently, we define o(ij) &off o(V) r&J Etid 0
proved Dirac operator for quark flavor j. This ensures that the equation-of-motion operator ECJ gives rise only to contact terms, and does not change the overall normalization 20. The O(a)
(2) .
(3)
The 2: are renormalization constants in the chiral limit, mij E (mi + mj)/2 is the average bare quark mass, ami = 1/2&i - l/26,, K~ is the value of the hopping parameter in the chiral limit, and +j is the quark mass defined by the axial Ward identity (AWI) in Eq. 12. Note that m and fi are identical in a discretized theory with chiral symmetry, like staggered fermions. With these definitions, bo = 1, cg = 0, cl0 = 1 at tree level [6]. Since the equation-of-motion operators contribute only contact terms, Eq. (1) can be rewritten in terms of just on-shell improved operators:
Here (ij) (with i # j) specifies the flavor, and $$j = (3 + mj)$j + 0(a2) is the full O(U) im*Presented by Rajan Gupta. Calculations supported by DOE Grand Challenges award at the ACL at Los Alamos and NERSC. the
=
0920-5632/01/$ - see front matter 0 2001 Published by Elsevier Science B.V. PII SO920-5632(01)00935-5
z-g) zy)
g31
+$ -ml2 ++I 2
+
O(a2)
(10)
600
T Bhaftacharya et al. /Nuclear
Physics B (Proc. Suppl.) 94 (2001) 599-602
Table 1 Simulation parameters, statistics, and the time interval in x4 defining the volume V over which the chiral rotation is performed in the AWI. The lattice spacing is fixed using TO = 0.5 fermi, and is thus independent of the fermion action. The source J is placed at t = 0.
where
Our calculation is limited to the case 61 = fis (this simplification was used due to limited computer resources), in which case the r.h.s. of Eq. 10 reduces to
0(a2) errors we use two estimates: (i) the difference between our results and those by the ALPHA collaboration [l-3], and (ii) the difference between using 2-point and 3-point discretization of the derivatives [6] in the extraction of cA from &(8,[A,
&(P(ij) =
where fii = 7jiij]mj=mi. Using Eqs. 10 and 11, all the bo (except bT which requires ml # ms), co, cd, and the scale independent normalization constants are determined by making suitable choices for J, 0, and y in Eq. 10 and studying it as a function of tii and &s (Eq. 11). 2. RESULTS The lattice parameters used in our calculation are given in Tab. 1 and 2. Our final results, which supercede those in Ref. [5], are given in Tab. 3. In many cases a given on-shell improvement and normalization constants can be determined in a number of ways as discussed in [6]. Results in Tab. 3 are based on the AWI with the best signal and smallest error. Table 3 also includes results by the ALPHA collaboration [l-3] and the one-loop tadpole improved perturbative results. To simplify _comparison with previous results, we I quote both bv, bA and bv, bA. One of the goals of our calculation is to quantify the residual O(a2) errors and to understand the shortcomings of l-loop perturbation theory. For
+ aCAdpP](i’)(L? t)J@)(O))
2filij,
(2, t) J(ji) (0))
(12)
and the subsequent effect of the difference in CA on other constants. This latter variation is quoted as the second error in Tab. 3. These differences are compared to the expected size of the residual discretization errors: M 0.15 and 0.1 for the improvement (a&~) constants and (aAQcD)2 M 0.02 and 0.01 for the normalization constants at p = 6.0 and 6.2 respectively. A comparison, at p = 6.0, between simulation at csw = 1.4755 (tadpole improved theory) and csw = 1.769 (non-perturbatively O(a) improved theory) shows that all the constants are sensitive to the choice of CSW. It is therefore important to use csw determined non-perturbatively. The most significant comparison is between our results and those of the ALPHA collaboration. The only results which do not agree within 2-a statistical errors are those for Z$, CA and cv at @ = 6, and for 2; at b = 6.2. The differences for 2; are of size 0.01 and 0.005 at /3 = 6 and 6.2 respectively, and are thus consistent with the expected differences of o(a2). The differences for CA and cv are also consistent with the size expected of O(a) differences, but are more notable because they correspond to very large fractional differences (e.g. our CA at B = 6 has less than half the
601
T. Bhaffacharya et al. /Nuclear Physics B (Proc. Suppb) 94 (2001) 59%602
Table 2 Values of IC used in the three simulations, and the corresponding extracted. rii is defined by the AWI in Eq. 12. )cC is the zero of The non-zero value of aM, at K.=is indicative of the inadequacy l/2&, used to extract it, and discretization errors. Of these, the to the need for including quenched chiral logs in the fits [6].
r
60TI
6ONP
l-
Label
aM,
Kl
1.530( 1)
0.
O.lZ(l) 0.118(l) 1) 0.075(l) 0.056( 1) 0.032(l) 0.025(4) 0
K3 K4 K5 n6 K7 &
of that by the we learn is that at tree at @ = 6, to extract
2of the
so = 6.2. The change appears
at
too rapid
to be cv and CT are are
as a small differwe
to design
cv using in zi, in
order terms tively. We
In particuwe obtain in dein [6] CT and dA
we use is N a, 2 M 0.02 and
0.016,
respec-
of renormalization In all as a combination (1 - 2) a:
be understood a 2-100~ correction of of size (1 -
1
62NP
aMr
K2
in cv,
values of aM, and the quark mass arid 771obtained from quadratic fits in l/n. of quadratic fits, a2Mz as a function of first is the dominant cause and points
UAQ~D
O.ln310 0.1321
0.609(l) 0.522( 1) 0.418(l) 0.360(2) 0.307(2) 0.261(2) 0.235(2) ).066( 10)
[for cv , CT and iv], and statistical errors [for &A, bp, and bs]. The only exception is .Z~/.Z$!, for which a very large higher order perturbative contribution of size 4 x a: is needed to reconcile our non-perturbative results with l-loop perturbation theory. In Tab. 4, we present, first results for the equation-of-motion improvement constants CX . The combination cb + CL is extracted by studying the dependence of Eq. 10 on fii once the other constants defined in Eq. 11 have been determined. The errors in the determination of the CL are dominated by two quantities: (i) The uncertainty in CA feeds into the extraction of CL, and (ii) the correlation function from which CL + c’p is determined has a poor signal (the intermediate state is a scalar for J = S, 0 = P and 68 = S in Eq. 10). The uncertainty in c’p then feeds into c:, c’s, and c$. Overall, we find a very significant improvement in the quality of the results with increasing /I, i.e., between ,L?= 6.0 and p = 6.2. Finally, we comment on results presented in two recent papers. Using the Schrodinger functional, Ref. [8] calculates bp, bs Z$/(Z~Z~) for range of 2 6. most striking is that discretizations of lead to different results bA For example, b = this quantity roughly from to -0.17. our number within this our estimate O(a) un-
602
T. Bhattacharya
et al. /Nuclear
Physics B (Proc. Suppl.)
94 (2001)
599-602
Table 3 Final results for improvement and renormalization constants. The first error is statistical, and the second, where present, corresponds to the difference between using 2-point and 3-point discretization of the derivative used in the extraction of CA. The ALPHA collaboration results are from 11-31. For the tadpole iml proved perturbative results (labelled P.Th.) see appendix in Ref. I61 and references there. LANL 1.4755 +0.747(1) +0.791(7)(4) +0.811(g)(5) -0.022(6)(l) -0.25(5)(3) ’ +0.09(2)( 1)
B = 6.0 LANL 1.769 +0.770( 1) +0.807(2)(8) +0.842(5)(l) -0.037(4)(8) -0.107(17)(4) +0.06(l)(3)
ALPHA 1.769 0.7809(6) 0.7906(94) N.A. -0.083(5) -0.32(7) N.A.
P. Th. 1.521 +0.810 +0.829 +0.956 -0.013 -0.028 +0.020
LANL 1.614 +0.7874(4) +0.818(2)(5) +0.884(3)(l) -0.032(3)(6) -0.09(2)(l) +0.051(7)(17)
+1.44(3)(2) +1.53(2) -0.51(9)(4) -0.49(9)(4) -0.07(9)(2) -0.126(58)(l)
+1.43(1)(4) +1.52(l) -0.26(3)(4) -0.24(3)(4) -0.06(4)(3) -0.07(4)(5)
N.A. +1.54(2) N.A. N.A. N.A. N.A.
+1.30(1)(1) +1.42(l) -0.11(3)(4) -0.11(3)(4) -0.09(2)(l) -0.09(3)(3)
N.A. +1.4 N.A. N.A. N.A. N.A.
+1.099 +1.255 -0.002 -0.002 -0.062 +0.001
+0.92(10)(6) +1.05(9)(4) +0.80(11)(6) +0.87( 14)(4)
+1.17(4)(8) +1.28(3)(4j +1.10(5)(13) +1.16(6)(11)
N.A. N.A. N.A. N.A.
+1.106 +1.274 -0.002 -0.002 -0.066 +0.002 +1.104 +1.271 +1.105 +1.172
+1.19(3)(5) +i.32(3)(4j
N.A. N.A. N.A. N.A.
+1.097 +1.252 +1.099 +1.161
Table 4 Results for off-shell mixing 60TI
60NPf
coefficients. 60NPb
62NP
c’v +3.72(73)
+2.38(50)
+3.00(37)
+1.72(16)
CL +3.28(94)
+1.99(56)
+2.45(46)
+1.53(20)
clp -0.98(76)
+0.44(49)
-0.33(29)
+0.91(12)
c$ +3.00(73)
+2.00(48)
+2.72(33)
+1.49(14)
cb +3.24(75)
+1.96(49)
+2.60(38)
+1.51(15)
P. Th. 1.481 +0.821 +0.839 +0.959 -0.012 -0.026 +0.019
ral extrapolation these discretization effects are significantly reduced. Nevertheless, once again the large variation should serve as a warning that the O(a) errors in cg can be substantial. REFERENCES 1. 2.
certainties is clearly a substantial underestimate. Reference [9] has determined CA using the same method and similar lattice parameters as here but with significantly more configurations. They study, at one K. (- IE~ at both p = 6.0 and 6.2), the effect of using derivatives that are tree-level improved through O(a2) (our 3-pt), O(a4) and 0(u6). They find a larger dependence than what we get between 2-pt and 3-pt discretizations at ~5. The 0(a2) errors in the two calculations are, however, different due to the choice of source and the fit range in time. Also, we find that after chi-
+1.11(4)(7) +1.19(4)(6)
’ ’ B = 6.2 ‘ALPHA 1.614 +0.7922(4)(9) +0.807(8)(2) N.A. -0.038(4) -0.21(7) N.A.
3. 4. 5. 6. 7. 8. 9.
M. Liischer et al., Nucl. Phys. B478 (1996) 365. M. Liischer et al., Nucl Phys. B491 (1997) 323. M. Liischer et al., Nucl Phys. B491 (1997) 344. M. Bochicchio et al., Nucl. Phys. B262 (1985) 331. T. Bhattacharya et al., Phys. Lett. B461 (1999) 79. T. Bhattacharya et al., hep-lat/0009038. G. Martinelli et al., Phys. Lett. B411 (1997) 141. M. Guagnelli et al., hep-lat/0009021. S. Collins and C. Davies, hep-lat/0010045.
Nuclear Physics B (Proc. Suppl.) 94 (2001) 599-602
ELSEVIER
Improvement and renormalization constants in O(a) improved lattice QCD* Tanmoy
Bhattacharyaa,
“MS B-285, Los Alamos bPhysics
Department,
Rajan National University
Guptaa,
Weonjong
Leeaand
Lab, Los Alamos, of Washington,
Stephen
Sharpeb
New Mexico 87545, USA
Seattle,
Washington
98195, USA
constants for bilinear We present results at p = 6.0 and 6.2 for the O(a) improvement and renormalization We discuss the extraction of the mass dependence of the operators using axial and vector Ward identities. renormalization constants and the coefficients of the equation of motion operators.
1. INTRODUCTION In quenched Lattice &CD, axial and vector Ward identities can be used to determine, at O(a), all the scale independent renormalization constants for bilinear currents (ZA, ZV, and Zs/Zp), the improvement constants (CA, cy , and CT), the quark mass dependence of all five 20, and the coefficients of the equation of motion operators [l-6]. Here we summarize results at /3 = 6.0 and 6.2 and discuss the highlights of our calculations. Details and notation are given in [6]. We start with the generalized axial Ward identity involving operators improved on and off-shell (6S(‘2) @3,1,(v)
J@i) (0)) = (X$,s$(~)
=
z(ij) o(ij)
=
$j)
=
@i)rj$+(j)
(I)
_‘;“1”,4) oc_@
,
_ $(i)t;lir@j)
on-shell are o(“)
improved
renormalized
operators
E
Zs(l+
b~WfQj)O~‘,
5
Z$(l+
boC377h~j)Ojij)
(AI),
=
A, + aCA&P,
(VI),
=
(Tr)w PI
E
V, + w&Tpv, Tpy + ac~(d,V,, - &V,)
z
P,
R
SF’
(4) (5)
e
(6) (7) ,
SIES,
(8) (9)
JC31) (0))
where SO is the result of the axial variation of 0 (A, +) V,, S +) P, and Tpy + tzpvpoTp,,), and 6.S is the variation in the action. At O(a) there exists only one dimension 4 offshell operator (which vanishes by the equations of motion) for each bilinear that has the appropriate symmetries [7]. Consequently, we define o(ij) &off o(V) r&J Etid 0
proved Dirac operator for quark flavor j. This ensures that the equation-of-motion operator ECJ gives rise only to contact terms, and does not change the overall normalization 20. The O(a)
(2) .
(3)
The 2: are renormalization constants in the chiral limit, mij E (mi + mj)/2 is the average bare quark mass, ami = 1/2&i - l/26,, K~ is the value of the hopping parameter in the chiral limit, and +j is the quark mass defined by the axial Ward identity (AWI) in Eq. 12. Note that m and fi are identical in a discretized theory with chiral symmetry, like staggered fermions. With these definitions, bo = 1, cg = 0, cl0 = 1 at tree level [6]. Since the equation-of-motion operators contribute only contact terms, Eq. (1) can be rewritten in terms of just on-shell improved operators:
Here (ij) (with i # j) specifies the flavor, and $$j = (3 + mj)$j + 0(a2) is the full O(U) im*Presented by Rajan Gupta. Calculations supported by DOE Grand Challenges award at the ACL at Los Alamos and NERSC. the
=
0920-5632/01/$ - see front matter 0 2001 Published by Elsevier Science B.V. PII SO920-5632(01)00935-5
z-g) zy)
g31
+$ -ml2 ++I 2
+
O(a2)
(10)
600
T Bhaftacharya et al. /Nuclear
Physics B (Proc. Suppl.) 94 (2001) 599-602
Table 1 Simulation parameters, statistics, and the time interval in x4 defining the volume V over which the chiral rotation is performed in the AWI. The lattice spacing is fixed using TO = 0.5 fermi, and is thus independent of the fermion action. The source J is placed at t = 0.
where
Our calculation is limited to the case 61 = fis (this simplification was used due to limited computer resources), in which case the r.h.s. of Eq. 10 reduces to
0(a2) errors we use two estimates: (i) the difference between our results and those by the ALPHA collaboration [l-3], and (ii) the difference between using 2-point and 3-point discretization of the derivatives [6] in the extraction of cA from &(8,[A,
&(P(ij) =
where fii = 7jiij]mj=mi. Using Eqs. 10 and 11, all the bo (except bT which requires ml # ms), co, cd, and the scale independent normalization constants are determined by making suitable choices for J, 0, and y in Eq. 10 and studying it as a function of tii and &s (Eq. 11). 2. RESULTS The lattice parameters used in our calculation are given in Tab. 1 and 2. Our final results, which supercede those in Ref. [5], are given in Tab. 3. In many cases a given on-shell improvement and normalization constants can be determined in a number of ways as discussed in [6]. Results in Tab. 3 are based on the AWI with the best signal and smallest error. Table 3 also includes results by the ALPHA collaboration [l-3] and the one-loop tadpole improved perturbative results. To simplify _comparison with previous results, we I quote both bv, bA and bv, bA. One of the goals of our calculation is to quantify the residual O(a2) errors and to understand the shortcomings of l-loop perturbation theory. For
+ aCAdpP](i’)(L? t)J@)(O))
2filij,
(2, t) J(ji) (0))
(12)
and the subsequent effect of the difference in CA on other constants. This latter variation is quoted as the second error in Tab. 3. These differences are compared to the expected size of the residual discretization errors: M 0.15 and 0.1 for the improvement (a&~) constants and (aAQcD)2 M 0.02 and 0.01 for the normalization constants at p = 6.0 and 6.2 respectively. A comparison, at p = 6.0, between simulation at csw = 1.4755 (tadpole improved theory) and csw = 1.769 (non-perturbatively O(a) improved theory) shows that all the constants are sensitive to the choice of CSW. It is therefore important to use csw determined non-perturbatively. The most significant comparison is between our results and those of the ALPHA collaboration. The only results which do not agree within 2-a statistical errors are those for Z$, CA and cv at @ = 6, and for 2; at b = 6.2. The differences for 2; are of size 0.01 and 0.005 at /3 = 6 and 6.2 respectively, and are thus consistent with the expected differences of o(a2). The differences for CA and cv are also consistent with the size expected of O(a) differences, but are more notable because they correspond to very large fractional differences (e.g. our CA at B = 6 has less than half the
601
T. Bhaffacharya et al. /Nuclear Physics B (Proc. Suppb) 94 (2001) 59%602
Table 2 Values of IC used in the three simulations, and the corresponding extracted. rii is defined by the AWI in Eq. 12. )cC is the zero of The non-zero value of aM, at K.=is indicative of the inadequacy l/2&, used to extract it, and discretization errors. Of these, the to the need for including quenched chiral logs in the fits [6].
r
60TI
6ONP
l-
Label
aM,
Kl
1.530( 1)
0.
O.lZ(l) 0.118(l) 1) 0.075(l) 0.056( 1) 0.032(l) 0.025(4) 0
K3 K4 K5 n6 K7 &
of that by the we learn is that at tree at @ = 6, to extract
2of the
so = 6.2. The change appears
at
too rapid
to be cv and CT are are
as a small differwe
to design
cv using in zi, in
order terms tively. We
In particuwe obtain in dein [6] CT and dA
we use is N a, 2 M 0.02 and
0.016,
respec-
of renormalization In all as a combination (1 - 2) a:
be understood a 2-100~ correction of of size (1 -
1
62NP
aMr
K2
in cv,
values of aM, and the quark mass arid 771obtained from quadratic fits in l/n. of quadratic fits, a2Mz as a function of first is the dominant cause and points
UAQ~D
O.ln310 0.1321
0.609(l) 0.522( 1) 0.418(l) 0.360(2) 0.307(2) 0.261(2) 0.235(2) ).066( 10)
[for cv , CT and iv], and statistical errors [for &A, bp, and bs]. The only exception is .Z~/.Z$!, for which a very large higher order perturbative contribution of size 4 x a: is needed to reconcile our non-perturbative results with l-loop perturbation theory. In Tab. 4, we present, first results for the equation-of-motion improvement constants CX . The combination cb + CL is extracted by studying the dependence of Eq. 10 on fii once the other constants defined in Eq. 11 have been determined. The errors in the determination of the CL are dominated by two quantities: (i) The uncertainty in CA feeds into the extraction of CL, and (ii) the correlation function from which CL + c’p is determined has a poor signal (the intermediate state is a scalar for J = S, 0 = P and 68 = S in Eq. 10). The uncertainty in c’p then feeds into c:, c’s, and c$. Overall, we find a very significant improvement in the quality of the results with increasing /I, i.e., between ,L?= 6.0 and p = 6.2. Finally, we comment on results presented in two recent papers. Using the Schrodinger functional, Ref. [8] calculates bp, bs Z$/(Z~Z~) for range of 2 6. most striking is that discretizations of lead to different results bA For example, b = this quantity roughly from to -0.17. our number within this our estimate O(a) un-
602
T. Bhattacharya
et al. /Nuclear
Physics B (Proc. Suppl.)
94 (2001)
599-602
Table 3 Final results for improvement and renormalization constants. The first error is statistical, and the second, where present, corresponds to the difference between using 2-point and 3-point discretization of the derivative used in the extraction of CA. The ALPHA collaboration results are from 11-31. For the tadpole iml proved perturbative results (labelled P.Th.) see appendix in Ref. I61 and references there. LANL 1.4755 +0.747(1) +0.791(7)(4) +0.811(g)(5) -0.022(6)(l) -0.25(5)(3) ’ +0.09(2)( 1)
B = 6.0 LANL 1.769 +0.770( 1) +0.807(2)(8) +0.842(5)(l) -0.037(4)(8) -0.107(17)(4) +0.06(l)(3)
ALPHA 1.769 0.7809(6) 0.7906(94) N.A. -0.083(5) -0.32(7) N.A.
P. Th. 1.521 +0.810 +0.829 +0.956 -0.013 -0.028 +0.020
LANL 1.614 +0.7874(4) +0.818(2)(5) +0.884(3)(l) -0.032(3)(6) -0.09(2)(l) +0.051(7)(17)
+1.44(3)(2) +1.53(2) -0.51(9)(4) -0.49(9)(4) -0.07(9)(2) -0.126(58)(l)
+1.43(1)(4) +1.52(l) -0.26(3)(4) -0.24(3)(4) -0.06(4)(3) -0.07(4)(5)
N.A. +1.54(2) N.A. N.A. N.A. N.A.
+1.30(1)(1) +1.42(l) -0.11(3)(4) -0.11(3)(4) -0.09(2)(l) -0.09(3)(3)
N.A. +1.4 N.A. N.A. N.A. N.A.
+1.099 +1.255 -0.002 -0.002 -0.062 +0.001
+0.92(10)(6) +1.05(9)(4) +0.80(11)(6) +0.87( 14)(4)
+1.17(4)(8) +1.28(3)(4j +1.10(5)(13) +1.16(6)(11)
N.A. N.A. N.A. N.A.
+1.106 +1.274 -0.002 -0.002 -0.066 +0.002 +1.104 +1.271 +1.105 +1.172
+1.19(3)(5) +i.32(3)(4j
N.A. N.A. N.A. N.A.
+1.097 +1.252 +1.099 +1.161
Table 4 Results for off-shell mixing 60TI
60NPf
coefficients. 60NPb
62NP
c’v +3.72(73)
+2.38(50)
+3.00(37)
+1.72(16)
CL +3.28(94)
+1.99(56)
+2.45(46)
+1.53(20)
clp -0.98(76)
+0.44(49)
-0.33(29)
+0.91(12)
c$ +3.00(73)
+2.00(48)
+2.72(33)
+1.49(14)
cb +3.24(75)
+1.96(49)
+2.60(38)
+1.51(15)
P. Th. 1.481 +0.821 +0.839 +0.959 -0.012 -0.026 +0.019
ral extrapolation these discretization effects are significantly reduced. Nevertheless, once again the large variation should serve as a warning that the O(a) errors in cg can be substantial. REFERENCES 1. 2.
certainties is clearly a substantial underestimate. Reference [9] has determined CA using the same method and similar lattice parameters as here but with significantly more configurations. They study, at one K. (- IE~ at both p = 6.0 and 6.2), the effect of using derivatives that are tree-level improved through O(a2) (our 3-pt), O(a4) and 0(u6). They find a larger dependence than what we get between 2-pt and 3-pt discretizations at ~5. The 0(a2) errors in the two calculations are, however, different due to the choice of source and the fit range in time. Also, we find that after chi-
+1.11(4)(7) +1.19(4)(6)
’ ’ B = 6.2 ‘ALPHA 1.614 +0.7922(4)(9) +0.807(8)(2) N.A. -0.038(4) -0.21(7) N.A.
3. 4. 5. 6. 7. 8. 9.
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