Toward the Continuum Limit of $B_K$ with the Quenched Kogut-Susskind Quark Action

TL;DR

This paper reports on precise lattice QCD calculations of the kaon B parameter using quenched Kogut-Susskind quarks, revealing the lattice spacing dependence and providing a continuum limit estimate.

Contribution

It presents new results at different lattice spacings, analyzes the curvature change in B_K dependence, and refines the continuum limit estimate with an $O(a^2)$ fit.

Findings

B_K in the continuum limit is 0.587(7)(17).

The lattice spacing dependence of B_K changes curvature at finer lattices.

Finite-size effects and theoretical behavior of $O(a^2)$ are analyzed.

Abstract

We present new results of our ongoing project toward a precision determination of the kaon $B$ parameter with the Kogut-Susskind quark action in quenched QCD. New results taken at $\beta$=6.4 and $\beta=5.7$ suggest that an apparently linear $a$ dependence of $B_K$ previously observed for $\beta=5.85-6.2$ arises from a change of curvature from convex to concave as the lattice spacing is reduced. Fitting data for $\beta\geq 5.93$ with an $O(a^2)$ form yields $B_K$(NDR,2 GeV)=0.587(7)(17) in the continuum limit. We also describe a finite-size study of $B_K$ at $\beta=$ 6.0 and 6.4, and a reanalysis of the theoretical argument for $O(a^2)$ behavior.

Tables

Table 1. Table 1: Run parameters of our simulation. New runs since Lattice 95 are marked with ∗ .

$\beta$	5.7	5.85	5.93	6.0			6.2	6.4
$L^{3}T$	${12}^3{24}^{\ast }$	${16}^{3}32$	${20}^{3}40$	${24}^{3}64$	${18}^3{64}^{\ast }$	${32}^3{48}^{\ast }$	${32}^{3}64$	${40}^3{96}^{\ast }$	${32}^3{96}^{\ast }$	${48}^3{96}^{\ast }$
#conf.	150	60	50	50	50	40	40	40	40	20
skip	1000	2000	2000	2000			5000	5000
$a^{-1}$(GeV)	0.806(14)	1.36(3)	1.59(3)	1.88(4)			2.65(9)	3.47(7)
$La$(fm)	2.9	2.3	2.5	2.5	1.9	3.4	2.4	2.3	1.8	2.7
$m_{q}a$	0.02-0.08	0.01-0.04	0.01-0.04	0.02-0.04			0.005-0.02	0.005-0.02
$m_{s}a/2$	0.0563	0.0202	0.0160	0.0126			0.0089	0.0070
fit	6-16	10-20	12-26	20-42	20-42	14-32	20-42	25-69	25-69	25-69

Equations

$$S_{KS}$$

$$Q=(1-\sum _{\mu }{P}_{\mu }{D}_{\mu }/2)q,\bar{Q}=\bar{q}(1-\sum _{\mu }{P}_{\mu }{\overleftarrow{D}}_{\mu }/2)$$

$$S_{KS}$$

$$⟨P|\mathcal{O}(\bar{Q}Q)|P^{\prime }⟩$$

$$=$$

$$B_K(\mathrm{NDR},2GeV)=0.587\pm 0.007\pm 0.017$$

Full text

1 Introduction

Lattice QCD determination of the kaon $B$ parameter with the quenched Kogut-Susskind quark action has been pursued over a number of years. Last year we initiated a renewed effort toward this goal using VPP500/80 at KEK, improving upon a number of points of calculation of the previous pioneering studies[1, 2, 3]. In particular, with a systematic scaling study over a wide range of lattice spacing carried out with improved statistics, we attempted to achieve a verification of the $O(a^{2})$ scaling violation theoretically suggested[3] and then a precision determination of the continuum value of $B_{K}$.

Our initial results taken at $a^{-1}=1.3-2.6$ GeV ($\beta=5.85-6.2$), reported at Lattice 95[4], however, did not exhibit the expected $O(a^{2})$ behavior but were apparently more consistent with an $O(a)$ behavior. In order to understand this result, we have since extended our study by a new calculation both at a larger and a smaller value of $a^{-1}$. We have also carefully examined the question of finite-size effects in $B_{K}$, which could jeopardize the correct scaling behavior. In addition a reexamination of the theoretical argument for the $O(a^{2})$ behavior has been made. Here we present results of these analyses.

In this article we mostly concentrate on data taken with an equal mass for $d$ and $s$ quarks. Toward the end we briefly touch upon the question of quenched chiral logarithms that could affect the case of unequal quark masses.

2 New runs and analysis procedures

In Table 1 we list the parameters of all our runs carried out so far. Lattice sizes marked with $*$ signify new runs since Lattice 95. Our main sequence of runs taken at a physical spatial lattice size of $La\approx 2.5$fm now includes those at $\beta=6.4(40^{3}\times 96)$ and at $\beta=5.7(12^{3}\times 24)$. Finite-size studies are carried out at $\beta=6.4$, and also at $\beta=6.0$ to examine possible lattice spacing dependence of finite-size effects.

Gauge configurations are generated with the 5-hit pseudoheatbath algorithm. Four values of quark masses, equally spaced in the interval given, are employed at each $\beta$.

The method of calculation of $B_{K}$ is the same as reported at Lattice 95[4]. We only summarize the main points here, referring to ref. [4] for details: (i) The physical scale of lattice spacing is set by the $\rho$ meson mass in the VT channel calculated on the same set of configurations. (ii) For the case of an equal quark mass taken for $d$ and $s$ quark, we estimate half the strange quark mass $m_{s}a/2$ from $m_{K}/m_{\rho}=0.648$. Interpolation of data for $B_{K}$ to $m_{s}a/2$ is made by fitting results for $B_{K}$ to the form predicted by chiral perturbation theory. (iii) We employ tadpole-improved operators using the quartic root of plaquette as the mean link $u_{0}=P^{1/4}$. The one-loop renormalization factors, which include finite parts for the continuum $\overline{MS}$ scheme with the naive dimensional regularization (NDR), are evaluated with $g^{2}_{\overline{MS}}(1/a)$ evolved from $g^{2}_{\overline{MS}}(\pi/a)=P/g^{2}_{0}+0.02461$. (This point differs from the choice $g^{2}_{\overline{MS}}(\pi/a)$ adopted at the time of Lattice 95[4].) Setting $\mu=2$GeV in the logarithm of the renormalization factors, we extract $B_{K}$ in the continuum $\overline{MS}$ scheme with NDR, which we denote as $B_{K}($NDR, 2GeV). (iv) We estimate errors by a single elimination jackknife procedure.

3 Lattice spacing dependence of $B_{K}$

In Fig. 1 we present $B_{K}$(NDR, 2 GeV) from our main sequence of runs as a function of $m_{\rho}a$. The left and rightmost points are the new data taken at $\beta=6.4$ on a $40^{3}\times 96$ lattice and at $\beta=5.7$ on a $12^{3}\times 24$ lattice.

As we observed at the time of Lattice 95, the middle four points are consistent with a linear $a$ dependence. Viewed more closely, the fourth point on the right lies below, while the other three are on a straight line. This suggests a convex shape of the curve of $B_{K}$ toward larger lattice spacings, which is confirmed by the new data at $\beta=5.7$ (the rightmost point in Fig. 1).

On the other hand, the new data at $\beta=6.4$ at the leftmost lies higher than the straight line, which is suggestive of an onset of an $O(a^{2})$ behavior for smaller lattice spacings. The solid line shows a fit of form $B_{K}=c_{0}+c_{1}(m_{\rho}a)^{2}$ to the four points, excluding the two points on the right at $\beta=5.85$ and 5.7.

We note, however, that the data by themselves still do not allow a distinction between an $O(a)$ and $O(a^{2})$ dependence; five points for $m_{\rho}a\leq 0.6$ can also be fitted with the form $B_{K}=c_{0}+c_{1}m_{\rho}a$ (dashed lines).

Let us also note that the discrepancy of values between gauge invariant and non-invariant operators, even in the continuum limit, represents a systematic uncertainty of $O(g^{4})$ arising from the renormalization factors which are correctly incorporated only to the order of one loop.

4 Lattice size dependence of $B_{K}$

We should emphasize that a distinction between an $O(a)$ or $O(a^{2})$ dependence rests on a very tiny effect in the range of lattice spacing being examined; a 2% decrease of the point at $\beta=6.4$ would render the $O(a)$ behavior strongly favored. Clearly a reexamination of finite-size effects is needed, especially since our physical spatial size decreases by about 10% from $La=2.5$fm to 2.3fm between $\beta=6.0$ and 6.4.

In Fig. 2 we summarize results of our finite-size study at $\beta=6.0$ and 6.4, plotting $B_{K}$(NDR, 2GeV) as a function of physical spatial size $La$. For $La\geq 2.3$fm, we do not observe a size-dependent shift of $B_{K}$ beyond the statistical error of at most 1%. Therefore, we conclude that the data in Fig. 1, taken with $La$=2.3-2.9 fm, are not affected by finite size effects. In particular, the rise of $B_{K}$ at $\beta=6.4$ is not a finite size effect.

An unexpected feature in Fig. 2 is that finite-size effects below $La\approx 2$fm is markedly enhanced at a weaker coupling of $\beta=6.4$ than at 6.0. Origin of this behavior is not clear to us.

5 Reexamination of the theoretical argument for $O(a^{2})$ behavior

A subtle point in the theoretical analysis of $a$ dependence of $B_{K}$ is that numerical simulations employ 4-component hypercubic fields $q(y),\overline{q}(y)$ defined on a coarse lattice of lattice spacing $2a$, in terms of which the Kogut-Susskind quark action has an apparent $O(a)$ term:

[TABLE]

However, a redefinition of fields given by

[TABLE]

with $P_{\mu}=(1-\gamma_{\mu}\gamma_{5}\otimes\xi_{\mu}\xi_{5})/2$ removes the $O(a)$ terms at tree-level so that the action becomes

[TABLE]

For bilinear and four-quark operators it is also possible to show that

[TABLE]

Therefore $B_{K}$ evaluated with $q(y)$’s with $P=P^{\prime}$ has no $O(a)$ corrections at tree level.

The rest of the argument essentially follow that of ref. [3]. In the presence of interactions the action expressed in terms of $Q$ fields may contain $O(a)$ terms. However, using symmetry of the original action under translation, $Q(y)\rightarrow(I\otimes\xi_{\mu})(1+\partial_{\mu}/2)Q(y^{\prime})$, rotation, $Q(y)\rightarrow((1-\gamma_{\mu}\gamma_{\nu})\otimes(1-\xi_{\mu}\xi_{\nu}))/2Q(y^{\prime})$, and $U(1)_{A}$ symmetry, $Q(y)\rightarrow{\rm e}^{i\alpha(\gamma_{5}\otimes\xi_{5})}Q(y)$ performed simultaneously with a mass rotation, $m\rightarrow{\rm e}^{-i\alpha(\gamma_{5}\otimes\xi_{5})}m{\rm e}^{-i\alpha(\gamma_{5}\otimes\xi_{5})}$, one can show that there are no dimension five operators giving rise to an $O(a)$ term. Therefore, the original action is itself improved up to $O(a^{2})$. One can also check that the $B_{K}$ operator is similarly improved to $O(a^{2})$ since there are no dimension seven operators with the same symmetry properties as the original one.

6 Extrapolation to the continuum limit

Our theoretical reexamination supports the original assertion[3] that there should be no $O(a)$ corrections in $B_{K}$ including gauge interactions. This leads us to attempt a continuum extrapolation of our data assuming the $O(a^{2})$ dependence.

Fitting data in Fig. 1 to the purely quadratic form $B_{K}=c_{0}+c_{1}(m_{\rho}a)^{2}$, excluding the two points on the right at $\beta=5.85$ and 5.7, we find

[TABLE]

where the first error is statistical and the second is the systematic error of $O(g^{4})$ estimated from the difference of extrapolated values for gauge non-invariant and invariant operators. Changing the number of points and/or including higher order terms in $m_{\rho}a$, we find that the extrapolated values stay consistent with the above value.

7 $B_{K}$ with unequal quark masses

We measure $B_{K}$ also for unequal quark masses for $d$ and $s$ quark in our simulation. In this case, a potential problem of quenched chiral logarithm may arise in the extrapolation to the chiral limit in $m_{d}$.

We examine this problem through fits of our data for $m_{d}\neq m_{s}$ to the form predicted by quenched chiral perturbation theory[5]. Generally we find very good fits. However, the parameter $\delta=m_{0}^{2}/48\pi^{2}f_{\pi}^{2}$ multiplying the quenched chiral logarithms turns out negative albeit with a large error, $\delta\approx-0.3\pm 0.3$ (we note that the sign of the $\delta$ term in ref. [5] is incorrect[6]). Also fits excluding the logarithm are quite acceptable, only slightly worse than those with it. Thus we cannot conclusively conclude the presence of quenched chiral logarithms.

Let us add the remark that the mass inside the quenched chiral logarithm should theoretically be that of the non-Nambu-Goldstone pion[6]. Fits in this case differ from those with the Nambu-Goldstone pion mass only very close to $m_{d}=0$, however, so that the conclusion above does not change.

8 Conclusions

Our new data for $B_{K}$ extending the results of last year to smaller and larger lattice spacings suggest that the curve of $B_{K}$ changes curvature from convex to concave as the lattice spacing is reduced from $m_{\rho}a\approx 0.9$ to 0.2. This behavior explains an apparently linear dependence we observed in our previous data, and is also consistent with an $O(a^{2})$ dependence theoretically expected toward the continuum limit. To confirm if $B_{K}$ stays on an $O(a^{2})$ curve, we plan to carry out one more run on a $56^{3}\times 96$ lattice at $\beta$=6.65 with yet smaller lattice spacing of $m_{\rho}a\approx 0.16$.