Toward Precision Measurement of $B_K$ with Quenched Kogut-Susskind Quarks

TL;DR

This paper reports on the progress of precisely measuring the kaon $B$ parameter using quenched Kogut-Susskind quarks, highlighting unexpected linear lattice spacing dependence contrary to theoretical expectations.

Contribution

It provides new lattice QCD results for $B_K$ at multiple lattice spacings and discusses the observed deviation from expected $O(a^2)$ scaling.

Findings

Results do not show expected $O(a^2)$ behavior

Data suggests an $O(a)$ dependence

Progress toward high-precision $B_K$ measurement

Abstract

We present a status report of our ongoing effort toward a precision determination of the kaon $B$ parameter with the Kogut-Susskind quark action in quenched QCD. Results for $B_K$ so far accumulated at $\beta =5.85$, 5.93, 6.0 and 6.2 corresponding to $a^{-1}=1.3-2.6$ GeV do not exhibit the theoretically expected $O(a^2)$ behavior, but are apparently more consistent with an $O(a)$ behavior.

Tables

Table 1. Table 1: Run parameters

$\beta$	5.85	5.93	6.0	6.2
$L^{3}T$	${16}^{3}32$	${20}^{3}40$	${24}^{3}40$	${32}^{3}64$
#conf.	60	50	50	40
$a^{-1}$	1.34(3)	1.58(4)	1.88(5)	2.62(9)
$m_{q}a$	0.01-0.04	0.01-0.04	0.01-0.03	0.005-0.02
$m_{s}a/2$	0.0197	0.0156	0.0125	0.0089
fit	10-20	12-26	14-24	20-42

Equations

$$B_K=B\left[1-3c(m_{q}a)\log (m_{q}a)+b(m_{q}a)\right],$$

Full text

1 Introduction

The kaon $B$ parameter is one of the crucial hadron weak matrix elements for constraining the $CP$ violation phase of the Cabibbo-Kobayashi-Maskawa matrix from experiments. A pioneering study was carried out by Gupta, Kilcup, Patel and Sharpe (GKPS)[1, 2, 3], which laid a foundation for a lattice QCD determination of this quantity with the Kogut-Susskind quark action. A number of problems, however, are posed to warrant viability of the lattice result in the continuum. While some aspects have been examined previously [4, 5], further studies are needed to establish the continuum value of $B_{K}$. The statistics of GKPS are only 15 ($\beta$=6.0) to 24 ($\beta$=6.4) configurations: as a consequence the error of $B_{K}$ at $\beta=6.4$ is so large that an $O(a^{2})$ dependence of $B_{K}$ on the lattice spacing inferred from a theoretical analysis[3] can not be ascertained. Furthermore the lattice size of $32^{3}\times 48$ employed at $\beta=6.2$ and 6.4 may be too small in the temporal direction. We have therefore decided to carry out a new set of simulations with improved statistics, incorporating advance in methods of analyses. Here we present a status report of this program which is being carried out on VPP500/80 newly installed at KEK.

2 Simulation method and run parameters

We basically employ the formalism developed by GKPS for calculating the $B$ parameter. Following improvements are included in our simulations, however.

On each configuration we calculate $2^{3}=8$ quark propagators corresponding to wall sources with a unit source placed at one of the corners of each spatial cube. This allows us to construct a wall source which creates only the pseudoscalar meson in the Nambu-Goldstone channel[6]. In contrast the even-odd source employed in previous studies[1, 2, 3, 4] also creates vector mesons that contaminate signals. The sources are placed at the two edges of the lattice, and the Dirichlet boundary condition is employed in the temporal direction. Gauge configurations are fixed to the Landau gauge.

Four-quark operators written in terms of Kogut-Susskind quark fields are extended over a $2^{4}$ hypercube. We employ both gauge-invariant and non-invariant operators[4], which differ by an insertion of appropriate gauge link factors for the former. The one-loop result for the lattice renormalization factors[7, 8] is used to convert lattice values of $B_{K}$ to those in the continuum, renormalized with the naive dimensional regularization(NDR) scheme.

A source of uncertainty in the conversion is the choice of the gauge coupling constant used for the renormalization factors. We take the mean-field improved $\overline{MS}$ coupling constant at the cut-off scale $\mu=\pi/a$[9]. Correspondingly we improve the operators[10] through the replacement $\chi\rightarrow\sqrt{u_{0}}\chi$ and $U_{\mu}\rightarrow u_{0}^{-1}U_{\mu}$, where $\chi$ is the Kogut-Susskind quark field, $U_{\mu}$ the link variable and $u_{0}=P^{1/4}$ with $P$ the average plaquette. Modifications of renormalization factors due to this replacement[7, 8] are also taken into account.

We note that we have ignored the mixing of non-Nambu-Goldstone four-quark operators in the present analysis. Studies of this mixing effect, which is believed to vanish in the continuum limit, are left for future work.

Parameters of runs we have so far carried out are listed in Table 1. Gauge configurations are generated with the 5-hit heatbath algorithm at 2000($\beta\leq 6.0$) or 5000($\beta=6.2$) sweep intervals. The physical scale of lattice spacing is determined from the $\rho$ meson mass in the $VT$ channel. Values of $\beta$ are chosen so that the lattice spacing is roughly equally spaced, and the lattice size is increased for weaker coupling to keep the physical lattice size approximately constant at $La\approx 2.4-2.5$ fm. A similar consideration is made for the choice of the fitting interval in extracting $B_{K}$ from kaon Green’s functions. Quark mass is taken in steps of $0.01$ for $\beta\leq 6.0$ and 0.005 at $\beta=6.2$. While we have evaluated $B_{K}$ for non-degenerate strange and down quark masses, we shall restrict ourselves to the degenerate case in the present analysis. The strange quark mass yielding the observed ratio $m_{K}/m_{\rho}=0.648$ for this case is also given in Table 1. We estimate errors by a single elimination jackknife procedure.

3 Comparison with previous work

In Fig. 1 lattice values of $B_{K}$ for the gauge non-invariant operator at $\beta=6.0$ and $6.2$ are plotted. At $\beta=6.2$ our results are consistent with those of GKPS[2, 3] at the level of one standard deviation. On the other hand, our values at $\beta=6.0$ are systematically lower than those of GKPS and Ishizuka et al.(IFMOSU)[4] obtained on the same size of lattice.

In order to understand the discrepancy, we note that GKPS and IFMOSU respectively used 15 and 10 configurations as compared to our 50, and that both groups employed the even-odd wall source, while we used the $2^{3}$ source. For a comparison under the same condition, we divided our ensemble into 5 sets of 10 configurations and recalculated $B_{K}$ for each set. In addition we made a new calculation for 3 sets of 10 configurations using the even-odd wall source. The eight values of $B_{K}$ thus obtained are plotted in Fig. 2 and are compared with those of IFMOSU and GKPS. Solid lines represent values obtained for the full ensemble for each type of source.

We note that, while results for the full ensemble are consistent between the two types of sources, both errors and fluctuations of central values are significantly reduced with the $2^{3}$ source, demonstrating the advantage of this method. Since values of GKPS and IFMOSU are consistent with ours for the even-odd source within one to two standard deviations, we conclude that previous results have overestimated the value of $B_{K}$ at $\beta=6.0$ due to insufficient statistics.

4 Quark mass dependence

Extraction of the physical value of $B_{K}$ requires an interpolation of results to the quark mass corresponding to the physical kaon. We find a suggestive result that the quark mass dependence of our data is well described by the form predicted by chiral perturbation theory, which we employ for the above purpose. For the case of degenerate strange and down quark treated in the present analysis, the formula is given by[11]

[TABLE]

where $c=\alpha/(4\pi f_{\pi})^{2}$ with $\alpha=(m_{K}a)^{2}/m_{q}a$ and $b$ is an unknown constant. A representative example of the fit taking $B,c$ and $b$ as parameters is plotted in Fig. 3 together with the quark mass corresponding to $m_{K}/m_{\rho}=0.648$ (dashed line). Several uncertainties exist, however, to take the agreement as evidence for the presence of chiral logarithm in our data. The pion decay constant $f_{\pi}$ estimated from the fitted values of $c$ and the spectrum results for the slope $\alpha$ turned out to have a large value of $150-300$ MeV with $10-20$% error as compared to the experimental value of 93 MeV. Also a quadratic form without logarithm fits our data equally well.

5 Results for $B_{K}$

We plot our results for $B_{K}$ renormalized at $\mu=2$ GeV in the NDR scheme as a function of $m_{\rho}a$ in Fig. 4 both for gauge invariant (filled circles) and non-invariant (open circles) operators. We observe that the data do not exhibit the theoretically predicted $O(a^{2})$ dependence[3], and are apparently more consistent with a linear behavior, at least over the range $m_{\rho}a=0.57-0.29$ (or $1/a=1.3-2.6$ GeV from $m_{\rho}$) corresponding to $\beta=5.85-6.2$. If we assume a linear form, we obtain $B_{K}(a)=0.4825(83)+0.489(17)m_{\rho}a$ and $0.5097(83)+0.464(17)m_{\rho}a$ for the gauge invariant and non-invariant operators.

We have varied the analysis procedure to see to what extent the values of $B_{K}$ are modified: (i) Instead of $m_{K}/m_{\rho}$ we examined the choice $m_{\phi}/m_{\rho}$ for estimating the strange quark mass. This only resulted in a common downward shift of the four values of $B_{K}$ by about $0.02$, and hence does not change a linear dependence on the lattice spacing. (ii) We also replaced the mean-field improved $\overline{MS}$ coupling constant $g_{\overline{MS}}(\pi/a)$ in the renormalization factors by the running coupling constant at $\mu=2$ GeV evaluated by the two-loop renormalization group evolution starting from $g_{\overline{MS}}(\pi/a)$. This increases the value of $B_{K}$ by about 0.03-0.05 without, however, changing the linear behavior.

Let us comment on the fact that our results for $B_{K}$ obtained with the two types of operators do not agree with each other and that the discrepancy does not become smaller toward the continuum limit. Possible sources of the discrepancy are an operator dependence of finite lattice spacing effects, which disappears in the continuum limit, and $O(g^{4})$ corrections in the renormalization factors which are not included in our analysis. Naively one would expect the magnitude of $O(g^{4})$ corrections to also diminish toward the continuum limit. This is not the case, however, for $B_{K}$ renormalized at a fixed scale $\mu=2$ GeV, since the gauge coupling constant $g(\mu)$ governing these corrections is independent of the lattice spacing. We have checked that increasing the scale $\mu$ leads to a smaller discrepancy. Thus the difference of $B_{K}$ for the two types of operators extrapolated to the continuum limit provides an estimate of the magnitude of $O(g^{4})$ corrections.

Finally we remark that an $O(a^{2})$ dependence is not completely ruled out from our data if there exists an appropriate $O(a^{3})$ or $O(a^{4})$ term which just brings the curve to behave as if it is linear over a range of lattice spacing including that explored in our simulation. If this rather exceptional case happens, the value of $B_{K}$ extrapolated to the continuum limit $a=0$ would increase by about 0.05.

6 Summary

In this report we have described our effort toward improving previous work on the kaon $B$ parameter with the Kogut-susskind quark action in quenched QCD. The results obtained at four values of $\beta$ show that $B_{K}$ falls linearly with the lattice spacing over the range $1/a=1.3-2.6$ GeV. If the theoretical prediction of an $O(a^{2})$ dependence is valid, this means that such a behavior will become apparent only for smaller values of lattice spacing. We plan to carry out a simulation at $\beta=6.4-6.5$ employing the lattice size of $40^{3}\times 96$.