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arXiv:hep-lat/0508031v2 27 Apr 2006

KEK-CP-163

UTHEP-509

UTCCS-P-15

HUPD-0506

Nonperturbative O(a) improvement of the Wilson quark action with the

RG-improved gauge action using the Schr¨ odinger functional method

S. Aoki1, M. Fukugita2, S. Hashimoto3,4, K-I. Ishikawa5, N. Ishizuka1,6,

Y. Iwasaki1,6, K. Kanaya1,6, T. Kaneko3,4, Y. Kuramashi1,6, M. Okawa5, S. Takeda1,

Y. Taniguchi1, N. Tsutsui3, A. Ukawa1,6, N. Yamada3,4, and T. Yoshi´ e1,6

(CP-PACS and JLQCD Collaborations)

1Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

2Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan

3High Energy Accelerator Research Organization(KEK), Tsukuba, Ibaraki 305-0801, Japan

4The Graduate University for Advanced Studies, Tsukuba, Ibaraki 305-0801, Japan

5Department of Physics, Hiroshima University,

Higashi-Hiroshima, Hiroshima 739-8526, Japan

6Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan

(Dated: February 7, 2008)

We perform a nonperturbative determination of the O(a)-improvement coefficient cSW and the

critical hopping parameter κc for Nf=3, 2, 0 flavor QCD with the RG-improved gauge action using

the Schr¨ odinger functional method. In order to interpolate cSW and κc as a function of the bare

coupling, a wide range of β from the weak coupling region to the moderately strong coupling points

used in large-scale simulations is studied. Corrections at finite lattice size of O(a/L) turned out to

be large for the RG-improved gauge action, and hence we make the determination at a size fixed

in physical units using a modified improvement condition. This enables us to avoid O(a) scaling

violations which would remain in physical observables if cSW determined for a fixed lattice size L/a

is used in numerical simulations.

I.INTRODUCTION

Fully unquenched simulations of QCD with dynamical up, down and strange quarks have become feasible [1]

thanks to the recent development of algorithms [2] and computational facilities. However, it is still very de-

manding to control discretization errors below a few percent level in dynamical QCD simulations. Thus highly

improved lattice actions are desirable to accelerate the approach to the continuum limit.

The on-shell improvement of the Wilson quark action through O(a) requires only a single additional term,

i.e. the Sheikholeslami-Wohlert (SW) term [3]. In Ref. [4], we determined cSW in three-flavor QCD for the

plaquette gauge action, using the Schr¨ odinger functional method [5, 6, 7, 8]. Applications of the resulting

O(a) improved Wilson-clover quark action in combination with the plaquette gauge action suffer from a serious

problem, however, since it was found in Ref. [9] that this action combination exhibits an unphysical first-order

phase transition at zero temperature in the strong coupling regime (β ≤ 5.0).

We also found in Ref. [9] that such a phase transition weakens, and possibly disappears, when the gauge

action is improved. In this work, motivated by this observation, we extend the determination of cSW for the

case of the RG-improved action [10] for gluons for Nf=3, 2, 0 flavor QCD.

We explore a wide range of β to work out the interpolation formula as a function of the bare coupling. The

critical hopping parameter κcin the O(a)-improved theory is also obtained.

In the Schr¨ odinger functional method, cSW is determined such that the axial Ward-Takahashi identity is

satisfied for a given finite volume. Since the linear extent L of a finite lattice provides an energy scale 1/L, a

determination of cSWgenerally involves corrections of order a/L. We find that this correction is sizable for the

RG improved gauge action. If the determination of cSWis made for a fixed value of L/a, observables calculated

in subsequent simulations using such cSWwould suffer from O(a) scaling violations. To avoid this problem, we

modify the standard improvement condition and determine cSWat a fixed physical size L. Similar considerations

have been made in the determinations of some other O(a) improvement coefficients in Ref. [11, 12].

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This paper is organized as follows. In Sec.II, we briefly recall the Schr¨ odinger functional method, mainly

to fix notations. In Sec. III, corrections at finite lattice size of O(a/L) that affect cSWare discussed, and our

modified method and one-loop calculations relevant for the subsequent analyses are given. Section IV is devoted

to describing our numerical results, and Sec. V to systematic uncertainties in them. Our conclusions are given

in Sec. VI. A preliminary report of this work has been made in Ref. [13].

II.SCHR¨ ODINGER FUNCTIONAL METHOD FOR THE DETERMINATION OF cSW

We briefly introduce the setup of the Schr¨ odinger functional (SF) method and the improvement condition

developed in Refs. [5, 6, 7, 8].

A.SF setup

Consider the SF defined on a four dimensional hypercubic lattice with a volume L3× T and the cylindrical

geometry, i.e., the periodic boundary condition is imposed in the spatial directions and the Dirichlet one in the

temporal direction for both gauge and quark fields. At the temporal boundaries x0= 0 and T, the following

conditions are imposed on the link variables and the quark fields: the spatial link variables on the boundaries

are fixed to the diagonal, constant SU(3) matrices given by

Uk(x,x0)|x0=0= exp[aCk],Uk(x,x0)|x0=T= exp[aC′

k],

,

(1)

Ck=

iπ

6Lk

−1 0 0

0 0 0

0 0 1

,C′

k=

iπ

6Lk

−5 0 0

0 2 0

0 0 3

(2)

while all quark fields on the boundaries are set to zero.

We use the RG-improved gauge action [10] given by,

Sg=

2

g2×

? ?

x

wP

µν(x0) Re Tr(1 − Pµ,ν(x)) +

?

x

wR

µν(x0) Re Tr

?

1 − R(1×2)

µ,ν

(x)

??

,(3)

where Pµ,ν(x) denotes a 1×1 Wilson loop on the µ-ν plane starting and ending at x, and R(1×2)

rectangular loop with the side of length 2 in the ν direction. These terms are added up with proper weights,

wP

µν(x0), respectively. In ordinary simulations with the periodic boundary condition in the temporal

direction, the weights are given by wP

µν=−0.331 independently of x0. In the SF, these weights

are modified. Among several possible choices, we select the choice B defined in Ref. [14] in this work,

µ,ν

(x) a 1×2

µν(x0) and wR

µν=3.648 and wR

wP

µν(x0) =

1

2× (3.648) at t = 0 or T and µ, ν ?=4

3.648otherwise

,(4)

wR

µν(x0) =

0

3

2× (−0.331) at t = 0 or T and µ=4

−0.331otherwise

at t = 0 or T and µ, ν ?=4

. (5)

The O(a)-improved Wilson quark action [3] is given by

Sq =

?

x,y

¯ qxDxyqy,(6)

Dxy = δxy− κ

?

µ

?

(1 − γµ)Ux,µδx+ˆ µ,y+ (1 + γµ)U†

x−ˆ µ,µδx−ˆ µ,y

?

+i

2κ cSWσµνFx,µνδxy,(7)

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with the field strength tensor Fx,µνdefined by

Fx,µν =

1

8{(Pµ,ν(x) + Pν,−µ(x) + P−µ,−ν(x) + P−ν,µ(x)) − (h.c.)},(8)

and σµν = (i/2)[γµ,γν]. The last term in Eq. (7) is the only counter term to get rid of O(a) errors present

for on-shell quantities on the lattice. At tree level, cSW=1. For the O(a)-improvement of the SF, we need to

add extra terms made of the gauge and quark fields at boundaries to the lattice action. However, since these

counter terms affect the PCAC relation used in the following calculations only at O(a2) or higher, they are not

necessary for the determination of cSW.

B.PCAC relation

We determine cSWby imposing the PCAC relation

1

2

?∂µ+ ∂∗

µ

?Aa

imp,µ= 2mqPa,(9)

up to O(a2) corrections. The pseudo-scalar density operator, axial vector current and its O(a)-improved version

are given by

Pa=¯ψγ5τaψ,

Aa

(10)

(11)

µ=¯ψγµγ5τaψ,

µ+ cA1

Aa

imp,µ= Aa

2

?∂µ+ ∂∗

µ

?Pa,(12)

where ∂µ and ∂∗

flavor symmetry acting on the flavor indices of the quark fields¯ψ and ψ.

µare the forward and backward lattice derivatives, and τadenotes the generator of SU(Nf)

We measure two correlation functions,

fA(x0) = −

1

N2

f− 1?Aa

1

N2

0(x)Oa?,(13)

fP(x0) = −

f− 1?Pa(x)Oa?,(14)

where x = (x0,x), and ?···? represents the expectation value after taking trace over color and spinor indices

and summing over spatial coordinate x. The source operator is given by

Oa= a6?

y,z

δ

¯ζ(y)γ5τaζ(z),(15)

ζ(x) =

δ¯ ρ(x),

¯ζ(x) =

δ

δρ(x), (16)

where ρ(x) is the quark field at x0=0 and is set to zero in the calculation of fAand fP. The bare PCAC quark

mass is then calculated using fAand fP through the PCAC relation Eq. (9) as

m(x0) = r(x0) + cAs(x0)

1

4(∂0+ ∂∗

1

2a∂0∂∗

(17)

r(x0) =

0)fA(x0)/fP(x0)(18)

s(x0) =

0fP(x0)/fP(x0).(19)

Using the source operator on the other boundary

O′,a= a6?

y,z

¯ζ′(y)γ5τaζ′(z),(20)

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where ζ′is the boundary field at x0=T, we can calculate another set of quantities m′(x0), r′(x0) and s′(x0)

from the correlation functions defined by

f′

A(T − x0) = +

1

N2

f− 1?Aa

1

N2

0(x)O′,a?, (21)

f′

P(T − x0) = −

f− 1?Pa(x)O′,a?,(22)

A naive improvement condition would be m(x0)=m′(x0). However, this condition requires a nonperturbative

tuning of cAas well as of cSW. To eliminate cAfrom the determination, it was proposed in Ref. [7] to use an

alternative definition of the quark mass given by

M(x0,y0) = m(x0) −m(y0) − m′(y0)

s(y0) − s′(y0)s(x0), (23)

M′(x0,y0) = m′(x0) −m′(y0) − m(y0)

s′(y0) − s(y0)s′(x0). (24)

with which cSWis obtained at the point where the mass difference

∆M(x0,y0) = M(x0,y0) − M′(x0,y0)(25)

vanishes. In principle, we can take an arbitrary choice for (x0,y0), since different choices result only in O(a2)

differences in physical observables. We follow the ALPHA Collaboration and use (x0,y0) = (3T/4,T/4) for

∆M, and (T/2,T/4) for M.In the following, M and ∆M without arguments denote M(T/2,T/4) and

∆M(3T/4,T/4), respectively.

In previous studies, cSWhas been determined through the conditions

?M(g2

∆M(g2

0,L/a)

0,L/a) = ∆M(0,L/a),

= 0,

(26)

at a given g2

massless point, is necessary in order that the resulting cSWreproduces its tree-level value (cSW=1) in the weak

coupling limit. In the next section, we address the issue of corrections at finite lattice size, and propose a new

condition to avoid the problem.

0and L/a. ∆M(0,L/a) on the right hand side, which is the tree-level value of ∆M(g2

0,L/a) at the

III.CORRECTIONS AT FINITE LATTICE SIZE AND MODIFIED IMPROVEMENT

CONDITIONS

A.corrections at finite lattice size

In the standard approach, we first calculate M(g2

The results are fitted as a function of cSWand κ to find cSW(g2

given value of g2

way is expected to be

0,L/a) and ∆M(g2

0,L/a) for a set of values of cSWand κ.

0,L/a) and κc(g2

0,L/a) and κc(g2

0,L/a) satisfying Eq. (26) at a

0,L/a) obtained in such a

0and L/a. The asymptotic a dependence of cSW(g2

cSW(g2

κc(g2

0,L/a) = cSW(g2

0,L/a) = κc(g2

0,∞) + cL· (a/L) + cΛ· (aΛQCD) + O((a/L)2,(a2ΛQCD/L),(aΛQCD)2),

0,∞) + kL· (a/L) + kΛ· (aΛQCD) + O((a/L)2,(a2ΛQCD/L),(aΛQCD)2),

(27)

(28)

where cL, cΛ, kLand kΛare unknown coefficients. (In practice, a logarithmic dependence on a/L also appears,

but it does not alter the following discussion, and hence not written explicitly.)

Consider an on-shell physical quantity Q, and let Qlatt(a) be the value obtained on a lattice with lattice spacing

a using the SW quark action with a choice of the improvement coefficient csim

between Q and Qlatt(a) in the measured value to be

SW. We expect the discrepancy

Q − Qlatt(a) = q ·?csim

SW− cSW(g2

0,∞)?· (aΛQCD) + O(a2Λ2

QCD),(29)

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where q is an unknown constant assumed to be O(1). Hence, if one uses csim

the O(a) error is absent, while if one uses cSW(g2

Q − Qlatt(a) = q · cL· (a/L) · (aΛQCD) + O(a2Λ2

SW= cSW(g2

0,∞) in the simulation,

0,L/a) in Eq. (27), the above expression results in

QCD) + O(aΛQCD(a/L)2).(30)

While the scaling violation appears to start from O(a2), it is actually linear in the lattice spacing if one deter-

mines cSW(g2

values of L/a, e.g. 8, independently of β.

0,L/a) with a fixed value of a/L. Indeed, previous studies determining cSWhave used certain fixed

In Ref. [4], we studied the magnitude of the corrections at finite lattice size in cSWfor the plaquette gauge

action. The coefficient cLdefined in Eq. (27) was evaluated in one-loop perturbation theory in the same SF

setup, and it was found that the effect on cSWdoes not exceed 3% when L/a=8 for β ≥ 5.2. We have repeated

the same perturbative analysis with the RG-improved action, and observed a sizable effect of about 15% at

β=1.9, around which large-scale simulations are carried out. This enhancement of the one-loop correction for

the RG improved action is mainly due to the larger value of the bare coupling compared to that for the plaquette

gauge action for realizing the same value of the lattice spacing.

B.modified improvement condition

We propose to resolve the problem due to the sizable corrections explained above by introducing a fixed

physical length L∗, and determining cSWat the fixed physical volume L∗3× T∗(T∗= 2L∗). If one uses cSW

thus determined, L in (30) is replaced by L∗and scaling violations are O(a2).

The actual procedure we use runs as follows. Instead of Eq. (26), we impose a modified improvement condition

given by

?

M(g2

∆M(g2

0,L/a) = 0,

0,L/a) = 0,

(31)

to calculate cSW(g2

so, we must know the value of L∗/a or 1/a at that value of g2

0,L/a) and κc(g2

0,L/a). The results are converted to cSW(g2

0,L∗/a) and κc(g2

0,L∗/a). To do

0, which we obtain through the two-loop β function,

aΛL = exp

?

−

1

2b0g2

?11

?34

0

?

(b0g2

0)−b1/2b2

0, (32)

b0 =

1

(4π)2

1

(4π)4

3Nc−2

3Nf

?

,(33)

b1 =

3N2

c− Nf

?13

3Nc−

1

Nc

??

.(34)

The transformation from cSW(g2

0,L/a) and κc(g2

cSW(g2

κc(g2

0,L/a) to those at L∗/a are made through

0,L∗/a) = cSW(g2

0,L∗/a) = κc(g2

0,L/a) + δcSW(g2

0,L/a) + δκc(g2

0,L/a;L∗/a),

0,L/a;L∗/a),

(35)

(36)

where

δcSW(g2

δκc(g2

0,L/a;L∗/a) = −cPT

0,L/a;L∗/a) = −κPT

SW(g2

c (g2

0,L/a) + cPT

0,L/a) + κPT

SW(g2

c (g2

0,L∗/a),

0,L∗/a),

(37)

(38)

and cPT

of L/a.

SW(g2

0,L/a) and κPT

c (g2

0,L/a) are calculated at the one-loop level for the same SF setup at the given value

It turned out that the tree and the one-loop coefficients for cSWand κchave a significant a/L dependence.

To describe this dependence precisely we fit them to a Pade or a polynomial-like function of a/L as

c(0)

SW(L/a) =

1 + a1(a/L) + a2(a/L)2+ a3(a/L)3

1 + b1(a/L)

,(39)

c(1)

SW(L/a) = 0.113+ (c1− d1ln(L/a))(a/L) + (c2− d2ln(L/a))(a/L)2,

1

8+ k1(a/L) + k2(a/L)2+ k3(a/L)3+ k4(a/L)4,

κ(1)

(40)

κ(0)

c(L/a) =(41)

c(L/a) = 0.002760894+ (l1− m1ln(L/a))(a/L) + (l2− m2ln(L/a))(a/L)2.(42)

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The coefficients are given in Table I. We note that the one-loop coefficients have an Nf dependence due to the

tadpole diagram, although it vanishes in the large volume limit.

In our actual determination, we define L∗by L∗/a = 6 at β = 1.9, L∗/a = 6 at β = 2.0 and L∗/a = 6 at

β = 2.6 for Nf=3, 2, 0 flavor QCD, respectively. In Table II–IV numerical values of β = 6/g2

our simulations for Nf=3, 2, and 0 cases are summarized. In these tables, we also show the numerical values of

δcSW(g2

hence reliable. On the other hand, if L/a are close to L∗/a, the corrections needed for the conversion from L

to L∗should again be small. Since we fix L∗at strong coupling, the corrections, Eqs. (37) and (38), are small

at both ends of our range of β as one can see in the Tables.

0, L/a and L∗/a in

0,L/a;L∗/a) and δκc(g2

0,L/a;L∗/a). For large values of β, the perturbative corrections are small and

IV.NUMERICAL SIMULATIONS

A. parameters and algorithm

The numerical simulations are performed with Nf=3, 2 and 0 degenerate dynamical quarks on a (L/a)3×

2(L/a) (L/a= 8 or 6) lattice for a wide range of β. The simulation parameters are summarized in Tabs. II–IV

for Nf=3, 2, 0, respectively.

We employ the symmetric even-odd preconditioning introduced in Refs. [15, 16] for the quark matrix D.

Calculation of D−1is made with the BiCGStab algorithm with the tolerance parameter ||Ri||/||B|| < 10−14,

where Ri = DXi− B is the residual vector and Xi is an estimate for the solution X in the i-th BiCGStab

iteration.

We adopt the standard HMC algorithm [17] for the Nf=2 and 0 flavor cases. For the three-flavor case, the

polynomial HMC (PHMC) algorithm [16, 18] is applied to describe the third flavor, employing the Chebyshev

polynomial P[D] to approximate D−1. In order to make the PHMC algorithm exact, the correction factor

Pcorr= det[W[D]] with W[D]=P[D]D is taken into account by the noisy Metropolis method [19]. The square

root of W[D], which is required in the Metropolis test, is evaluated with an accuracy of 10−14using a Taylor

expansion of W[D] [16]. The order of the polynomial Npolyis chosen so that an acceptance rate of about 70%

or higher is achieved for the Metropolis test.

In the calculations of aM and a∆M, fXand f′

are combined to produce aM and a∆M. The bin size dependence of the jackknife error of aM is investigated

in the range Nbin=1–Ntraj/20. We adopt Nbingiving the maximum error in this range in the error analyses

in the following.

X(X=A or P) are first evaluated at every trajectory, and they

B.results

The trial values of cSWand κ at which simulations are made are summarized in Tables. V–VII for Nf=3, 2,

and 0, respectively, together with the results for aM and a∆M and the number of trajectories accumulated.

In order to obtain cSW(g2

the functional forms,

0,L/a) and κc(g2

0,L/a) satisfying Eq. (31) at each β, we make fits of those data using

aM = aM+b(1)

M

κ

+b(2)

M

κ2+ c(1)

+b(2)

κ2

McSW+ c(2)

McSW2+dM

κ

cSW, (43)

a∆M = a∆M+b(1)

∆M

κ

∆M

+ c(1)

∆McSW+ c(2)

∆Mc2

SW+d∆M

κ

cSW.(44)

The results for cSW(g2

tabulated in Tabs. VIII–X. The details of the fit procedure are as follows. In Figs. 1–3 we plot data on the

(aM,a∆M) plane for Nf=3, 2, 0, respectively. For those data for which the origin (0,0) is contained in or close

to the data region, we make a fit leaving only the constant and linear terms in Eqs. (43) and (44). This applies

to all cases except for the three-flavor simulations at β ≤ 2.2, and the dotted lines in the figures show the fit

results.

0,L/a) and κc(g2

0,L/a) obtained with the fits, and the adopted functional form are

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In the three-flavor simulations at β ≤ 2.2, the region of negative aM is not covered, and the origin is missed

by the data. This happens because the PHMC algorithm tends to fail at vanishing or negative PCAC quark

masses at low β due to large quantum fluctuations. Thus, at β ≤ 2.2, we are forced to extrapolate the data.

In the extrapolation, three functional forms are examined: (i) linear, (ii) quadratic without the cross terms,

and (iii) quadratic with the cross terms. At β=2.20 and 2.10, a linear function well fits the data, and we take

this in the following analysis. The data at β=2.00 and 1.90 require the quadratic term, but it turns out that

including the cross terms does not reduce χ2/dof significantly from that without the cross terms, and leads to

cSW(g2

without the cross terms at these β, and dM and d∆M are always set to zero throughout this analysis.

0,L/a) and κc(g2

0,L/a) consistent within one standard deviation. Thus, we adopt the quadratic function

Next, cSW(g2

2(L∗/a), along the line presented in Sec. IIIB. Using Eqs. (35), (36) and the δcSWand δκcgiven in Tables II–

IV, we obtain cSW(g2

three results for β=2.0. The first and second one are obtained by transforming the data with 83× 16 and

63× 12 to those for L∗/a ∼6.805, respectively, and the third one is obtained by simply interpolating the two

raw values at L/a = 8 and 6 in Table VIII to L∗/a ∼6.805, for which the corrections at finite lattice size are

essentially corrected nonperturbatively. The two raw values, 1.670(56) at L/a = 8 and 1.632(45) at L/a = 6,

are very close to each other and consistent within the error, and hence the linear interpolation to L∗/a ∼ 6.805

is more reliable than the perturbative procedure. Similar observations are made at the second smallest β in

each Nf flavor simulation, namely at β=2.10 for Nf = 2 and at β=2.70 for Nf = 0. Thus, at these β the

result interpolated to L∗/a is adopted as our final result, and used in the following analysis. At the same time,

it is worth noting that in all three cases the one-loop corrections have the right sign, which indicates that the

one-loop correction dominates over higher loop corrections. Furthermore, the discrepancy between the results

corrected perturbatively and nonperturbatively is found to be 5%, 3% and less than 1% for the Nf=3, 2 and

0 cases, respectively, while the size of one-loop correction itself at these β is 6–7%, 5% and 2–3%. From this

observation, we expect that the size of the one-loop correction gives a conservative estimate for the unknown

higher loop corrections for all β.

0,L/a) and κc(g2

0,L/a) are transformed into those for the desired lattice volume, (L∗/a)3×

0,L∗/a) and κc(g2

0,L∗/a) shown in Tables XI– XIII. Notice that in Table XI there are

C.interpolation formula

Our final results for cSW(g2

interpolate cSW, not all available data are used in the fit. As mentioned in Sec. IIIB, the corrections at finite

lattice size estimated perturbatively is small only around the high and low ends of β due to our choice of L∗,

while in the middle range corrections may be significant. Therefore, we use data only if the correction is less

than 5%. In the three flavor case, the data at β=12.0, 8.85, 2.2, 2.1, 2.0, 1.9 are employed. As a consequence,

we obtain the followings interpolation formula,

0,L∗) as a function of g2

0are shown in Fig. 4 for Nf= 3,2,0 flavor QCD. When we

cSW(g2

0,L∗) = 1 + 0.113 g2

0+ 0.0209(72) (g2

0)2+ 0.0047(27) (g2

0)3,(χ2/dof = 0.58).(45)

For κcshown in Fig. 5, the corrections are smaller than 5% for all value of β. Including all data in the fit we

obtain

κc(g2

0,L∗) = 1/8 + 0.003681192 g2

+0.000067(66) (g2

0+ 0.000211(43) (g2

0)3− 0.000038(21) (g2

0)2

0)4.(χ2/dof = 1.1)(46)

When performing the above fits, the tree and one-loop coefficients are fixed to the perturbative values at infinite

volume. This is justified since, as seen in Table II, L∗/a grows very rapidly with β, and hence a/L∗corrections

in Eqs. (39)–(42) are all negligibly small near the continuum limit. We also note that the tree and one-loop

coefficients in the infinite volume limit do not depend on Nf, and hence the same values are used in the analysis

for the Nf= 2 and 0 cases given below.

The interpolation formula for cSWin two-flavor QCD is calculated in the same fashion as in the three-flavor

case. In this case, the sizes of the correction at finite lattice size are acceptable (≤ 5%) at β=12.0, 8.85, 5,0,

2.2, 2.1, 2.0. We first try a polynomial form as before, and obtain

cSW(g2

0,L∗) = 1 + 0.113g2

0+ 0.0158(63)(g2

0)2+ 0.0088(24)(g2

0)3,(χ2/dof = 4.68),(47)

which is denoted by a dashed line in Fig. 4. A sharp rise of the data points near g2

by this polynomial form, while in the three-flavor case the polynomial worked well over the whole range of β

0=3.0 is not described well

Page 8

8

we studied. An alternative is a Pade function, with which we obtain

cSW(g2

0,L∗) =

1 − 0.212(9)g2

0− 0.0108(38)(g2

1 − 0.325(9)g2

0)2− 0.0083(19)(g2

0)3

0

,(χ2/dof = 2.11), (48)

This fit, denoted by solid line in the middle panel of Fig. 4, interpolates our data very well. Since this formula

has a pole at g2

0∼

0=3.08(8), its use is restricted to g2

<3.0. For κc, we use all available data to obtain

κc(g2

0,L∗) = 1/8 + 0.003681192g2

+0.000093(84)(g2

0+ 0.000227(58)(g2

0)3− 0.000049(24)(g2

0)2

0)4,(χ2/dof = 0.98),(49)

for a polynomial, and

κc(g2

0,L∗) =

1/8 − 0.0356(23)g2

0− 0.00089(8)(g2

1 − 0.314(18)g2

0)2− 0.00009(6)(g2

0)3

0

,(χ2/dof = 0.35),(50)

for a Pade function. These results appear in the middle panel of Fig. 5 as dashed and solid line, respectively.

It is interesting that the pole positions for cSWand κcare consistent with each other. This seems to indicate

that above g2

0∼3.0 the Wilson quark action cannot be improved in this fashion consistently for the Nf=2 case.

All in all the Pade fits provide a more satisfactory interpolation of the Nf=2 data, and we take them as the

main result for the Nf=2 case. We have also applied a Pade function for cSWin the Nf=3 case. However, in

this case the resulting fit lies on top of that for a polynomial over the range of β we used, and the position of

pole can be determined only poorly. Hence there seems no reason to favor the Pade fit over the polynomial for

interpolating the data. The difference between the Nf=2 and 3 cases probably arise from the fact, empirically

known, that the Nf=2 lattice is coarser than the Nf=3 lattice at the same value of g2

improvement coefficients was previously seen for the plaquette gauge action toward coarse lattices [7, 8].

0. Indeed, a sharp rise of

In quenched QCD, the size of the correction is smaller than 5% for all availble data, and we use all data to

obtain

cSW(g2

κc(g2

0,L∗) = 1 + 0.113g2

0,L∗) = 1/8 + 0.003681192g2

−0.000053(65)(g2

0+ 0.0371(54)(g2

0)2− 0.0036(26)(g2

0+ 0.000293(37)(g2

0)3+ 0.000008(24)(g2

0)3,(χ2/dof = 4.09), (51)

0)2

0)4,(χ2/dof = 0.46).(52)

In Ref. [20], the authors performed a one-loop determination of c(1)

and reported a very precise value c(1)

to the use of this value in above analyses are expected to be negligibly small.

SWwith conventional perturbation theory,

SW=0.11300591(1) in the infinite volume limit. Changes in our results due

V.SYSTEMATIC ERRORS

There are two sources of systematic errors in our analysis, both related to the conversion to a fixed physical

length scale L∗, one being the use of the two-loop β function to estimate L∗as a function of g2

being the use of one-loop perturbation theory for correcting the value of cSWfrom L to L∗.

0, and the second

In order to examine the magnitude of uncertainties from the first error, we go through the analysis using

the three-loop β function. Since the three-loop term of the lattice β function is not available for the RG-

improved gauge action, we take the value for the plaquette gauge action. Thus the following argument is only

semi-quantitatively valid. In this case, Eq. (32) is replaced with

aΛL = exp

?

−

1

2b0g2

0

?

(b0g2

0)−b1/2b2

0×?1 + qg2

0

?,(53)

where q=0.18960350(1), 0.4529(1), and 0.6138(2) for Nf=0, 2, and 3 [21], respectively. With this function, we

estimate L∗/a, δcSW, and δκcwith Nf=3, which are tabulated in Table XIV. Comparing with Table II, it is

found that L∗/a changes significantly while the changes in δcSWand δκcare at most a few percent and hence

small. Thus we conclude that the uncertainty from scaling violation in the lattice spacing is negligible.

Page 9

9

In order to discuss the uncertainty of one-loop corrections, we write cSW(g2

procedure as

0,L∗) determined through our

cSW(g2

0,L∗) = cSW(g2

0,∞) + c(0)(a/L∗) + g2

c(2)(a/L) − c(2)(a/L∗)

0c(1)(a/L∗) + g4

?

0c(2)(a/L∗)

+g4

0

?

+ O(g6

0). (54)

In other words, Eq. (54) represents the difference between cSW(g2

series with coefficients c(i)(a/L), where c(i)(a/L) vanishes as L → ∞. Since we have corrected the mismatch

between cSW(g2

remains at two-loop and higher. Replacing csim

SWin Eq. (29) with Eq. (54), we obtain

0,L∗) and cSW(g2

0,∞) in terms of perturbative

0,L/a) and cSW(g2

0,L∗/a) only at the tree- and one-loop level, the unwanted a/L dependence

Q − Qlatt(a) =

?

c(0)(a/L∗) + g2

0c(1)(a/L∗) + g4

0c(2)(a/L∗)

?

· (aΛQCD)

+g4

0

?c(2)(a/L) − c(2)(a/L∗)?· (aΛQCD) + O(g6

0aΛQCDa/L) + O(a2Λ2

QCD), (55)

where we omit an unknown O(1) overall coefficient q, because it is not relevant in the following discussion. If

you expand c(i)(a/L(∗)) around a/L(∗)= 0, the first term in Eq. (55) behaves ∼ a2ΛQCD/L∗∼ O(a2) because

L∗is fixed. The second term behaves like ∼ g4

a/L is fixed. As a results, the leading scaling violation could be O(a) rather than O(a2). However it should be

emphasized that when we obtain the interpolation formula we only used the weak coupling and strong coupling

regions because in these regions the perturbative errors are expected to be under control for the following

reasons. In the weak coupling region, L/a and L∗/a are different by several orders of magnitude, but the

coupling is very small, and hence the size of O(g4(a/L − a/L∗)(aΛQCD)) is expected to be as small as the size

of the one-loop corrections. On the other hand, in the strong coupling region, L/a and L∗/a are close to each

other, and again the remaining O(a) scaling violation, O(g4(a/L − a/L∗)(aΛQCD)), should be small. We also

saw in Sec. IVB that the size of perturbative errors is roughly the same as that of the one-loop correction itself.

0(a/L−a/L∗)(aΛQCD), which gives O(a) scaling violation because

Most importantly, at our strongest and the second strongest couplings around which large-scale simulations

are performed, there are no perturbative errors in cSWdue to our choice of L∗and interpolation to L∗at the

second strongest couplings. Thus we believe O(a) scaling violations are well below O(a2), though we need to

check this in future work.

VI. CONCLUSION

In this work, we have performed a nonperturbative determination of the O(a)-improvement coefficient cSWof

the Wilson quark action with the RG-improved gauge action for Nf=3, 2, and 0 flavor QCD. The corrections at

finite lattice size turn out to be sizable, and are taken into account by modifying the improvement condition and

carrying out the determination at a fixed physical length scale of L∗. While we have to resort to perturbation

theory to incorporate the corrections, we have attempted to choose L∗at a moderately strong coupling, close

to the range of lattice sizes of order a−1∼ 2 GeV where physics simulations are practically made, so that their

magnitude are reasonably under control.

Using the data for cSWthus obtained over a wide range of β, we have determined the interpolation formulas,

given in Eqs. (45), (48) and (51), which represent the main results of this work. These results do depend on L∗

chosen, but the removal of O(a) scaling violations in physical observables hold independent of the value of L∗.

As a byproduct, we have also obtained the interpolation formula for κc, Eqs. (46), (50) and (52), which may

be useful to locate simulation points.

The three-flavor results reported here are already being used in a large-scale simulation aiming to carry out

a systematic evaluation of hadronic observables for the realistic quark spectrum incorporating the dynamical

up, down and strange quarks. The preliminary results have been reported in Refs. [22].

Page 10

10

Acknowledgments

This work is supported by the Supercomputer Project No.132 (FY2005) of High Energy Accelerator Research

Organization (KEK), and also in part by the Grant-in-Aid of the Ministry of Education (Nos. 13135204,

14740173, 15204015, 15540251, 16028201, 16540228, 17340066, 17540259).

[1] For a recent review on hadron spectrum, see, for example, K. I. Ishikawa, Nucl. Phys. Proc. Suppl. 140, 20 (2005)

[arXiv:hep-lat/0410050].

[2] For a recent review on algorithm, see, for example, A. D. Kennedy, Nucl. Phys. Proc. Suppl. 140, 190 (2005)

[arXiv:hep-lat/0409167].

[3] B. Sheikholeslami and R. Wohlert, Nucl. Phys. B 259, 572 (1985).

[4] N. Yamada et al. [CP-PACS and JLQCD Collaboration], Phys. Rev. D 71, 054505 (2005) [arXiv:hep-lat/0406028].

[5] M .L¨ uscher, R. Narayanan, P. Weisz and U. Wolff, Nucl. Phys. B 384, 168 (1992).

[6] M. L¨ uscher, S. Sint, R. Sommer and P. Weisz, Nucl. Phys. B 478, 365 (1996).

[7] M. L¨ uscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, Nucl. Phys. B 491, 323 (1997).

[8] K. Jansen and R. Sommer (ALPHA Collaboration), Nucl. Phys. B 530, 185 (1998).

[9] S. Aoki et al. [JLQCD Collaboration], arXiv:hep-lat/0409016.

[10] Y. Iwasaki, Nucl. Phys. B 258, 141 (1985); Univ. of Tsukuba report UTHEP-118 (1983), unpublished.

[11] M. Luscher, S. Sint, R. Sommer and H. Wittig, Nucl. Phys. B 491, 344 (1997) [arXiv:hep-lat/9611015].

[12] M. Guagnelli, R. Petronzio, J. Rolf, S. Sint, R. Sommer and U. Wolff [ALPHA Collaboration], Nucl. Phys. B 595,

44 (2001) [arXiv:hep-lat/0009021].

[13] S. Aokietal.[CP-PACSandJLQCDCollaborations],

[arXiv:hep-lat/0211034]; K. I. Ishikawa et al. [CP-PACS and JLQCD Collaborations], Nucl. Phys. Proc. Suppl.

129, 444 (2004) [arXiv:hep-lat/0309141].

[14] S. Aoki, R. Frezzotti and P. Weisz, Nucl. Phys. B 540, 501 (1999) [arXiv:hep-lat/9808007].

[15] K. Jansen and C. Liu, Comput. Phys. Commun. 99, 221 (1997).

[16] S. Aoki et al. (JLQCD Collaboration), Phys. Rev. D 65, 094507 (2002).

[17] S. Duane, A.D. Kennedy, B.J. Pendleton, and D. Roweth, Phys. Lett. B 195, 216 (1987); S. Gottlieb, W. Liu,

D. Toussaint, R.L. Renken and R.L. Sugar, Phys. Rev. D 35, 2531 (1987).

[18] T. Takaishi and Ph. de Forcrand, hep-lat/0009024; Nucl. Phys. B (Proc.Suppl.) 94, 818 (2001).

[19] A. D. Kennedy and J. Kuti, Phys. Rev. Lett. 54, 2473 (1985).

[20] S. Aoki and Y. Kuramashi, Phys. Rev. D 68, 094019 (2003) [arXiv:hep-lat/0306015].

[21] A. Bode and H. Panagopoulos, Nucl. Phys. B 625, 198 (2002) [arXiv:hep-lat/0110211].

[22] T. Kaneko et al. [CP-PACS Collaboration], Nucl. Phys. Proc. Suppl. 129, 188 (2004) [arXiv:hep-lat/0309137];

T. Ishikawa et al. [CP-PACS Collaboration], Nucl. Phys. Proc. Suppl. 140, 225 (2005) [arXiv:hep-lat/0409124];

T. Ishikawa et al. [CP-PACS and JLQCD Collaborations], PoS LAT2005, 057 (2005) [arXiv:hep-lat/0509142].

Nucl.Phys.Proc.Suppl.

119,433(2003)

Page 11

11

TABLE I: Finite-size coefficients in Eqs. (39)–(42).

c(0)

SW

c(1)

SW

Nf = 0

−4.5736

−3.3402

−1.1681

−8.9448

Nf = 2

−6.2641

−8.0488

−1.5466

−14.306

κ(1)

c

Nf = 2

Nf = 3

−7.1094

−10.403

−1.7359

−16.987

a1

a2

a3

b1

−3.4415

−5.0248

11.1475 d1

−3.9702 d2

c1

c2

κ(0)

c

Nf = 0Nf = 3

k1

k2 −0.845333×10−5

k3 −0.103610×10−1m1 0.547826×10−3−0.507665×10−3−0.155835×10−3

k4

0.751742×10−2m2 0.882220×10−2−0.136413×10−2−0.645729×10−2

0.260982×10−6

l1 0.101302×10−2−0.224650×10−2−0.387626×10−2

l2 0.162496×10−1

0.862878×10−2

0.481835×10−2

TABLE II: Inverse coupling β and lattice size L/a chosen for the three-flavor QCD simulation. L∗/a is estimated by the

two-loop β function assuming L∗/a = 6 at β=1.9. Finite-size corrections δcSW and δκc calculated with Eqs.(37) and(38)

are also shown.

βL/a

8

8

8

8

8

8

8

8

8

6

6

L∗/a

7.51×106

8.46×104

3.81×102

2.50×101

1.48×101

1.14×101

8.78

7.73

6.81

6.81

6

δcSW(g2

0,L/a;L∗/a)

5.51×10−3

1.42×10−2

5.14×10−2

1.14×10−1

1.08×10−1

8.70×10−2

3.42×10−2

−1.59×10−2

−9.36×10−2

1.10×10−1

δκc(g2

0,L/a;L∗/a)

6.35×10−5

7.95×10−5

1.23×10−4

6.80×10−5

1.34×10−5

−8.82×10−6

−9.84×10−6

5.70×10−6

3.85×10−5

−5.08×10−5

12.00

8.85

5.00

3.00

2.60

2.40

2.20

2.10

2.00

2.00

1.9000

TABLE III: Same as Table II, but for two-flavor QCD.

βL/a

8

8

8

8

8

8

8

6

6

L∗/a

2.35×106

3.66×104

2.45×102

1.98×101

1.22×101

7.58

6.74

6.74

6

δcSW(L/a;L∗/a)

2.43×10−3

1.01×10−2

4.63×10−2

9.51×10−2

7.84×10−2

−2.11×10−2

−8.24×10−2

8.37×10−2

δκ(L/a;L∗/a)

5.93×10−5

7.38×10−5

1.08×10−4

2.69×10−5

−1.81×10−5

1.41×10−5

6.14×10−5

−7.04×10−5

12.00

8.85

5.00

3.00

2.60

2.20

2.10

2.10

2.0000

Page 12

12

TABLE IV: Same as Table II, but for quenched QCD.

βL/a

8

8

8

8

8

8

6

6

L∗/a

3.09×1011

2.41×105

6.33×103

8.04×101

9.12

6.66

6.66

6

δcSW(L/a;L∗/a)

−1.11×10−2

−3.70×10−3

2.24×10−3

3.80×10−2

2.07×10−2

−4.81×10−2

3.98×10−2

δκ(L/a;L∗/a)

3.47×10−5

5.08×10−5

6.18×10−5

6.29×10−5

−2.68×10−5

8.79×10−5

−8.40×10−5

24.00

12.00

8.85

5.00

3.00

2.70

2.70

2.6000

Page 13

13

TABLE V: Results for aM and a∆M for three-flavor QCD. The accep-

tance rates for the MD and the noisy Metropolis test are shown together

with the number of MD steps per trajectory and the order of the poly-

nomial Npoly used in the noisy Metropolis test. The final column gives

the number of trajetories accumulated.

cSW

κaMa∆MPacc[NMD]Pcorr[Npoly]Ntraj

β = 12.00, L/a = 8

1.00 0.12659

0.12676

0.12693

0.12709

0.12659

0.12676

0.12693

0.12709

0.12659

0.12676

0.12693

0.12709

0.01235(13)

0.006906(91)

0.00149(13)

−0.00368(13)

0.008565(98) −0.00009(18) 0.72(2)[100]

0.00283(11)

−0.00003(13) 0.74(1)[100]

−0.00221(11)0.00004(17)

−0.007708(89) 0.00023(10)

0.00460(12)

−0.00076(13) 0.72(2)[100]

−0.00097(15) −0.00069(22) 0.71(3)[100]

−0.00625(21) −0.00073(16) 0.74(2)[100]

−0.01161(17) −0.00070(13) 0.73(2)[100]

0.00101(13)

0.00072(13)

0.00087(14)

0.00092(16)

0.73(2)[100]

0.75(1)[100]

0.73(2)[100]

0.75(1)[100]

0.983(5)[100]

0.979(4)[100]

0.973(5)[100]

0.970(5)[100]

0.984(4)[100]

0.968(6)[100]

0.969(6)[100]

0.955(5)[100]

0.981(4)[100]

0.972(4)[100]

0.953(9)[100]

0.94(2)[100]

1600

1600

1600

1600

1600

1600

1600

1600

1600

1600

1600

1600

1.05

0.72(3)[100]

0.74(2)[100]

1.10

β = 8.85, L/a = 8

1.01410.12698

0.12730

0.12762

0.12826

0.12698

0.12730

0.12762

0.12826

0.12698

0.12730

0.12762

0.12826

0.12719

0.12753

0.12713

0.12747

0.02316(13)

0.01311(11)

0.00309(12)

−0.01734(10)

0.02121(11)

0.01119(16)

0.00080(13)

−0.01948(13)

0.01903(14)

0.00896(11)

−0.00130(11)

−0.02166(13)

0.00983(15)

−0.00078(11)

0.00982(35)

−0.00121(11) −0.00039(14)

0.00094(15)

0.00104(18)

0.00102(15)

0.00092(16)

0.00060(12)

0.00057(17)

0.00060(16)

0.00055(15)

0.00013(14)

0.00033(10)

0.00024(20)

0.00039(10)

−0.00013(15)

0.00007(12)

−0.00041(19)

0.68(2)[80]

0.66(2)[80]

0.71(2)[80]

0.70(1)[80]

0.70(2)[80]

0.71(1)[80]

0.70(2)[80]

0.71(2)[80]

0.69(2)[80]

0.68(1)[80]

0.69(2)[80]

0.70(2)[80]

0.69(2)[80]

0.68(2)[80]

0.68(2)[80]

0.69(1)[80]

0.989(3)[100]

0.988(4)[100]

0.969(5)[100]

0.943(6)[110]

0.990(2)[100]

0.985(3)[100]

0.972(4)[100]

0.937(6)[110]

0.987(4)[100]

0.975(4)[100]

0.964(5)[100]

0.928(7)[110]

0.990(2)[120]

0.989(3)[130]

0.990(3)[110]

0.986(3)[120]

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

1.0350

1.0559

1.0800

1.1000

β = 5.00, L/a = 8

1.08 0.12958

0.12974

0.12989

0.13004

0.12932

0.12948

0.12963

0.12978

0.12907

0.12922

0.12937

0.12952

0.01031(33)

0.00553(17)

0.00049(35)

−0.00377(20)

0.01027(26)

0.00541(24)

0.00030(24)

−0.00428(29)

0.01002(22)

0.00489(25)

0.00043(43)

−0.00441(28) −0.00089(21)

0.00073(22)

0.00082(29)

0.00060(25)

0.00070(19)

0.00039(22)

0.00043(27)

0.00024(21)

0.00042(37)

−0.00071(20)

−0.00105(22)

−0.00074(21)

0.72(1)[64]

0.77(2)[64]

0.74(2)[64]

0.74(2)[64]

0.73(1)[64]

0.73(1)[64]

0.77(1)[64]

0.74(1)[64]

0.76(2)[64]

0.73(2)[64]

0.75(1)[64]

0.76(1)[64]

0.982(3)[100]

0.968(4)[100]

0.970(4)[100]

0.962(5)[100]

0.976(6)[100]

0.975(4)[100]

0.964(7)[100]

0.950(6)[100]

0.982(3)[100]

0.974(4)[100]

0.970(4)[100]

0.958(6)[100]

2200

2200

2200

2200

2200

2200

2200

2200

2200

2200

2200

2200

1.13

1.18

β = 3.00, L/a = 8

1.200.13281

0.13311

0.13341

0.13370

0.13235

0.13265

0.13294

0.13324

0.13190

0.13219

0.13248

0.13278

0.13145

0.13174

0.13203

0.13232

0.02798(33)

0.01769(47)

0.00813(46)

−0.00036(41)

0.02774(42)

0.01820(52)

0.00940(38)

−0.00027(40) −0.00029(34) 0.773(9)[50]

0.02742(65)

−0.00052(28)

0.01713(64)

−0.00040(36)

0.00915(54)

−0.00066(67)

−0.00008(41)0.00031(64)

0.02697(35)

−0.00098(46)

0.01643(70)

−0.00075(44) 0.770(8)[50]

0.00800(43)

−0.00092(58)

−0.00079(38) −0.00109(38)

0.00099(26)

0.00036(57)

0.00044(33)

0.00104(40)

−0.00015(61)

0.00003(40)

0.00004(36)

0.770(9)[50]

0.77(1)[50]

0.78(1)[50]

0.77(1)[50]

0.78(1)[50]

0.76(1)[50]

0.77(1)[50]

0.988(2)[100]

0.978(3)[100]

0.960(4)[100]

0.937(5)[100]

0.986(4)[100]

0.980(3)[100]

0.960(3)[100]

0.948(5)[100]

0.990(2)[100]

0.980(2)[100]

0.962(3)[100]

0.945(6)[100]

0.988(3)[100]

0.979(4)[100]

0.972(3)[100]

0.947(4)[100]

4300

4000

4000

3800

4200

3800

4200

3900

4200

4200

3900

4000

4300

4100

4100

4000

1.25

1.300.76(1)[50]

0.77(2)[50]

0.78(2)[50]

0.77(1)[50]

0.77(1)[50]1.35

0.76(1)[50]

0.77(1)[50]

Page 14

14

β = 2.60, L/a = 8

1.20 0.13531

0.13550

0.13574

0.13594

0.13454

0.13473

0.13496

0.13516

0.13378

0.13420

0.13440

0.13473

0.13303

0.13322

0.13344

0.13364

0.13277

0.13301

0.13202

0.13226

0.02110(64)

0.01528(44)

0.00810(69)

0.00140(72)

0.02061(53)

0.01512(73)

0.00721(52)

0.00327(73)

0.02177(75)

0.00830(57)

0.0018(11)

−0.00968(93) −0.00007(34) 0.880(6)[64]

0.02040(50)

−0.00055(43) 0.874(7)[64]

0.01375(64)

−0.00042(67) 0.874(7)[64]

0.00792(81)

−0.00064(37) 0.873(9)[64]

0.00116(66)

−0.00022(50) 0.882(6)[64]

0.0036(10)

−0.00109(49) 0.883(6)[64]

−0.00316(79) −0.00085(34) 0.872(6)[64]

0.00476(58)

−0.00218(36) 0.873(7)[64]

−0.00270(73) −0.00195(48) 0.879(10)[64]

0.00168(38)

0.00142(85)

0.00170(43)

0.00158(62)

0.00073(91)

0.00218(47)

0.00102(48) 0.883(10)[64]

0.00039(53)

−0.00006(57) 0.883(6)[64]

−0.00004(45) 0.872(6)[64]

0.00031(44)

0.878(9)[64]

0.879(6)[64]

0.87(1)[64]

0.882(7)[64]

0.870(6)[64]

0.881(6)[64]

0.979(3)[110]

0.972(3)[110]

0.966(6)[120]

0.965(8)[130]

0.983(2)[110]

0.978(2)[110]

0.971(5)[120]

0.972(3)[130]

0.984(2)[110]

0.972(3)[120]

0.975(2)[130]

0.903(5)[110]

0.983(2)[110]

0.979(3)[110]

0.975(3)[120]

0.964(3)[130]

0.978(3)[130]

0.979(2)[150]

0.980(2)[130]

0.973(3)[140]

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

4500

5000

5000

5000

5000

1.27

0.883(6)[64]

1.34

0.876(9)[64]

1.41

1.48

1.55

β = 2.40, L/a = 8

1.3 0.135917

0.136152

0.136387

0.134882

0.135113

0.133410

0.133636

0.133862

0.132400

0.132680

0.131230

0.131510

0.0211(39)

0.01146(50)

0.00276(49)

0.01207(50)

0.00466(45)

0.0206(10)

0.01300(56)

0.00584(49)

0.0151(12)

0.00567(70)

0.01385(71)

0.0047(13)

0.00162(74)

0.00011(100) 0.819(5)[50]

0.00100(29)

0.00022(60)

0.00034(50)

−0.00040(30) 0.823(4)[50]

0.00027(68)

−0.00095(39) 0.827(5)[50]

−0.00210(81)

−0.00194(55) 0.825(5)[50]

−0.00236(54) 0.888(5)[64]

−0.00321(28) 0.886(7)[64]

0.819(7)[50] 0.974(2)[110]

0.971(2)[120]

0.947(2)[120]

0.974(2)[120]

0.954(2)[120]

0.983(2)[110]

0.979(2)[120]

0.966(2)[120]

0.980(2)[120]

0.966(2)[120]

0.983(1)[120]

0.978(1)[130]

10000

10000

10000

10000

10000

10000

10000

10000

11900

11900

10700

10700

0.82(1)[50]

0.815(5)[50]

0.828(5)[50]

1.4

1.5

0.828(5)[50]

1.60.82(1)[50]

1.7

β = 2.20, L/a = 8

1.30.138247

0.138487

0.138729

0.135400

0.135654

0.135885

0.132712

0.132934

0.133156

0.130170

0.130370

0.130570

0.01685(90)

0.0100(15)

0.00128(62)

0.01877(64)

0.01083(46)

0.00285(55)

0.01913(70)

0.01226(45)

0.00494(49)

0.02045(54)

0.01248(76)

0.00561(55)

0.00161(63)

0.00144(36)

0.00149(45)

−0.00028(38) 0.844(4)[50]

−0.00004(36) 0.844(4)[50]

0.00079(51)

−0.00148(29) 0.846(5)[50]

−0.00198(45) 0.844(8)[50]

−0.00205(50) 0.834(5)[50]

−0.00385(32) 0.843(6)[50]

−0.00342(30) 0.840(4)[50]

−0.00322(32) 0.841(4)[50]

0.840(5)[50]

0.836(3)[50]

0.843(4)[50]

0.958(2)[120]

0.946(3)[130]

0.919(3)[140]

0.973(1)[120]

0.965(2)[130]

0.951(2)[140]

0.983(1)[120]

0.977(2)[130]

0.968(2)[140]

0.984(1)[120]

0.984(2)[130]

0.977(1)[140]

16500

16500

16500

16000

16500

16500

16500

16500

16500

16100

16100

16100

1.5

0.841(4)[50]

1.7

1.9

β = 2.10, L/a = 8

1.50.1355

0.1358

0.1360

0.1362

0.1340

0.1344

0.1346

0.1326

0.1329

0.1331

0.1333

0.1335

0.1315

0.1318

0.1321

0.1324

0.0579(12)

0.0500(15)

0.04380(77)

0.0386(10)

0.06167(75)

0.04811(59)

0.04179(72)

0.05964(58)

0.04952(50)

0.04393(48)

0.03752(54)

0.02976(87)

0.05054(44)

0.04101(59)

0.03149(47)

0.01956(59)

0.00066(57)

0.00011(44)

0.00033(44)

0.00047(56)

−0.00027(49) 0.854(5)[50]

0.00039(83)

−0.00035(76) 0.850(4)[50]

−0.00132(47) 0.853(5)[50]

−0.00128(33) 0.850(4)[50]

−0.00145(32) 0.854(3)[50]

−0.00018(31) 0.849(3)[50]

−0.00096(31) 0.856(3)[50]

−0.00192(33) 0.850(4)[50]

−0.00182(35) 0.844(3)[50] 0.9883(10)[100] 14500

−0.00176(41) 0.847(4)[50]

−0.00148(33) 0.850(4)[50]

0.850(4)[50]

0.847(4)[50] 0.9877(10)[100] 14500

0.855(4)[50] 0.984(1)[100]

0.851(6)[50] 0.978(2)[100]

0.986(2)[80]

0.848(5)[50] 0.9888(10)[100] 14500

0.985(1)[100]

0.987(1)[80]

0.987(2)[90]

0.988(1)[100]

0.982(2)[100]

0.974(1)[100]

0.9889(9)[90]

0.979(3)[80] 14500

14400

14500

145001.6

14500

14500

14500

14500

14500

14500

14500

1.7

1.8

0.983(2)[110]

0.980(1)[120]

14500

14500

β = 2.00, L/a = 8

1.50.1383550

0.1386672

0.0276(15)

0.0176(12)

0.00020(44)

0.00082(56)

0.863(3)[50]

0.857(3)[50]

0.961(2)[130]

0.928(2)[140]

24500

21500

Page 15

15

0.1388000

0.1364310

0.1367500

0.1370700

0.1348740

0.1354309

0.1354980

0.1356000

0.1333520

0.1336570

0.1339620

0.1320578

0.1323422

0.1326278

0.1308300

0.1311100

0.1293800

0.1296600

0.0138(13)

0.0360(10)

0.02618(98)

0.0156(12)

0.03723(61)

0.01588(70)

0.01435(93)

0.01043(82)

0.03533(61)

0.02441(57)

0.01374(65)

0.02843(57)

0.01918(68)

0.01016(60)

0.0208(13)

0.01208(82)

0.02589(84)

0.01483(96)

0.00034(58)

0.00036(60)

−0.0011(10)

0.00049(52)

−0.00134(41) 0.856(3)[50]

−0.00073(30) 0.861(3)[50]

−0.00037(46) 0.859(3)[50]

−0.00072(58) 0.861(4)[50]

−0.00095(40) 0.859(3)[50]

−0.00129(33) 0.863(3)[50]

−0.00050(51) 0.853(5)[50]

−0.00237(38) 0.860(3)[50]

−0.00166(42) 0.860(2)[50]

−0.00183(39) 0.858(3)[50]

−0.00357(75) 0.853(6)[50]

−0.00310(40) 0.860(7)[50]

−0.00391(65) 0.859(8)[50]

−0.00406(79) 0.855(5)[50]

0.853(3)[50]

0.858(4)[50]

0.857(4)[50]

0.857(7)[50]

0.914(7)[160]

0.9912(9)[140] 20000

0.982(1)[150]

0.961(2)[160]

0.9932(9)[140] 20000

0.950(2)[130]

0.971(1)[160]

0.951(2)[150]

0.9936(6)[140] 20000

0.9899(8)[150] 20000

0.978(1)[160]

0.977(1)[110]

0.976(1)[130]

0.945(2)[130]

0.989(2)[150]

0.983(2)[160]

0.996(2)[150]

0.988(2)[160]

21200

1.6

20000

20000

1.7

24500

20000

20300

1.8

20000

24500

24500

24500

5400

7300

5400

7300

1.9

2.0

2.1

β = 2.00, L/a = 6

1.300.1400

0.1400

0.1405

0.1405

0.1410

0.1410

0.1415

0.1415

0.1380

0.1380

0.1385

0.1385

0.1390

0.1390

0.1395

0.1395

0.1355

0.1355

0.1360

0.1360

0.1365

0.1365

0.1370

0.1370

0.1330

0.1330

0.1335

0.1335

0.1340

0.1340

0.1345

0.1345

0.1002(43)

0.1017(33)

0.0871(30)

0.0890(29)

0.0669(28)

0.0721(30)

0.0505(26)

0.0508(24)

0.0752(22)

0.0784(17)

0.0564(17)

0.0568(24)

0.0401(20)

0.0417(21)

0.0231(17)

0.0245(23)

0.0673(20)

0.0698(16)

0.0530(17)

0.0535(14)

0.0340(19)

0.0347(15)

0.0165(14)

0.0171(16)

0.0682(37)

0.0683(20)

0.05035(96)

0.05242(96)

0.0339(10)

0.0350(11)

0.0164(15)

0.0197(15)

0.0055(15)

0.00346(89)

0.00390(95)

0.0030(11)

0.0033(11)

0.0045(15)

0.0037(11)

0.0048(12)

0.00153(95)

0.00126(66)

0.00224(73)

0.00094(76)

−0.0001(12)

0.00173(83)

0.00128(80)

0.00238(83)

0.0021(16)

−0.00090(78) 0.924(3)[50]

−0.00053(56) 0.920(3)[50]

−0.0004(14)

0.0008(10)

0.00078(62)

−0.00013(69) 0.922(3)[50]

−0.0004(12)

−0.00223(95) 0.920(7)[50]

−0.0016(10)

−0.00293(66) 0.925(3)[50]

−0.00187(64) 0.923(3)[50]

−0.00088(59) 0.927(3)[50]

−0.0025(13)

−0.0026(10)

−0.00095(70) 0.920(3)[50]

0.925(3)[50]

0.926(3)[50]

0.923(2)[50]

0.926(2)[50]

0.922(3)[50]

0.922(2)[50]

0.922(4)[50]

0.923(2)[50]

0.925(3)[50]

0.925(3)[50]

0.921(3)[50]

0.922(4)[50]

0.921(2)[50]

0.921(3)[50]

0.923(5)[50]

0.926(3)[50]

0.925(4)[50]

0.9998(2)[120] 11200

1.0000(0)[120] 11200

0.9993(2)[120] 14500

0.9991(5)[120] 14500

0.9974(6)[120] 15000

0.9974(8)[120] 15000

0.990(1)[120]

0.990(1)[120]

0.9995(2)[120] 14900

0.9994(3)[120] 14900

0.9985(4)[120] 14900

0.9979(5)[120] 14900

0.994(1)[120]

0.993(1)[120]

0.982(2)[120]

0.981(1)[120]

0.9994(2)[120] 11200

0.9998(2)[120] 11200

0.9983(6)[120] 15000

0.9990(3)[120] 15000

0.9957(8)[120] 15000

0.9971(5)[120] 15000

0.988(1)[120]

15100

15100

1.45

15000

15000

15100

15100

1.60

0.925(3)[50]

0.926(2)[50]

0.925(3)[50]

15000

0.923(2)[50] 0.9888(10)[120] 15000

1.0000(0)[120]

0.927(7)[50]0.9997(3)[120]

0.9996(3)[120] 14500

0.9995(2)[120] 14500

0.9985(5)[120] 15000

0.924(3)[50]0.9983(5)[120] 15000

0.920(3)[50]0.9948(6)[120] 15000

0.994(1)[120]

1.753400

3400

15000

β = 1.90, L/a = 6

1.4 0.1410

0.1410

0.1415

0.1415

0.1420

0.1420

0.1340

0.1340

0.1345

0.1345

0.1350

0.1350

0.1355

0.1355

0.1280

0.1280

0.1285

0.1285

0.1290

0.1315(62)

0.1338(41)

0.1103(64)

0.1136(36)

0.0857(25)

0.0876(31)

0.0878(25)

0.0900(21)

0.0710(21)

0.0733(14)

0.0520(24)

0.0535(21)

0.0307(16)

0.0309(14)

0.06146(84)

0.06303(92)

0.04376(79)

0.04434(83)

0.02540(87)

0.00112(92)

0.0008(13)

0.0006(12)

0.00144(94)

0.0028(10)

0.00166(90)

−0.00021(84) 0.929(3)[50]

−0.00174(58) 0.928(3)[50]

−0.00159(69) 0.926(3)[50]

−0.00193(55) 0.929(2)[50]

−0.00136(59) 0.929(2)[50]

−0.00045(76) 0.928(2)[50]

−0.00102(68) 0.926(2)[50]

−0.00108(55) 0.926(2)[50]

−0.00601(53) 0.928(2)[50] 0.99991(9)[120] 22700

−0.00699(67) 0.926(2)[50]

−0.00734(52) 0.928(2)[50]

−0.00594(51) 0.928(2)[50]

−0.00681(84) 0.929(2)[50]

0.930(2)[50]

0.923(2)[50]

0.928(2)[50]

0.927(2)[50]

0.930(3)[50]

0.923(2)[50]

0.9997(1)[120] 24000

0.9997(1)[120] 23900

0.9985(4)[120] 24000

0.9989(2)[120] 24000

0.9930(8)[120] 24100

0.9945(5)[120] 24100

1.0000(0)[120] 15100

1.0000(0)[120] 15100

0.9994(2)[120] 24000

0.9995(3)[120] 24000

0.9973(3)[120] 24100

0.9976(6)[120] 24100

0.9888(7)[120] 24100

0.9895(8)[120] 24100

1.8

2.2

0.9998(1)[120] 22700

0.9993(2)[120] 23600

0.9996(1)[120] 23600

0.9976(6)[120] 24100