On the Nf and a Dependence of BK

TL;DR

This paper investigates the dependence of the BK parameter on the number of dynamical flavors using improved lattice operators and multiple ensembles, providing insights into its behavior in different QCD regimes.

Contribution

It introduces a study of BK with tadpole improved operators across various ensembles, highlighting its small dependence on Nf and extrapolation to the continuum limit.

Findings

BK shows a small dependence on Nf.

Extrapolation to zero lattice spacing was performed.

Results contribute to understanding of QCD parameters across different flavors.

Abstract

We present results of a study of BK using tadpole improved gauge invariant staggered operators. Using three ensembles of 16^3x32 configurations with varying numbers of dynamical flavors, we observe a small dependence on Nf. Using seven quenched ensembles between beta=5.7 and beta=6.2 we extrapolate to a=0.

Tables

Table 1. Table 1: Ensembles for N f subscript 𝑁 𝑓 N_{f} Study

$N_f$	$\beta$	$N_{\mathrm{config}}$	$m_{\rho }a$
0	6.05	306	0.384(5)
2	5.7	83	0.384(4)
4	5.4	69	0.391(7)

Table 2. Table 2: Quenched Ensembles for Continuum Extrapolation

$\beta$	Geometry	$N_{\mathrm{config}}$	$m_q$
5.70	${16}^3\times 32$	259	.01 to .08
5.80	${16}^3\times 32$	200	.01 to .04
5.90	${16}^3\times 32$	200	.01 to .04
6.00	${16}^3\times 32$	221	.01 to .04
6.05	${16}^3\times 32$	306	.01 to .05
6.10	${24}^3\times 32$	60	.01 to .04
6.20	${24}^3\times 48$	121	.005 to .035

Equations

$${\mathcal{O}}_i^{cont}=({\delta }_{ij}+\frac{g^2}{16{\pi }^2}({\gamma }_{ij}\log (\frac{\pi }{\mu a})+C_{ij})){\mathcal{O}}_j^{lat},$$

$$a(\beta)=a_{0}{\big{(}{16\pi^{2}\over 11g^{2}}\big{)}}^{51\over 121}\exp({-8\pi^{2}\over 11g^{2}})$$

$$B_K{|}_{a=0,N_f=0}=.573\pm .015.$$

$$B_K(NDR,\mu =2\mathrm{GeV},N_f=3,a=0)=.60\pm .02$$

Full text

1 Introduction

The $B_{K}$ parameter serves to parameterize the weak hadronic matrix element responsible for $K^{0}-\bar{K^{0}}$ mixing. Since this mixing gives us the only CP violation observed to date, $B_{K}$ is a crucial link between the measured quantity $\epsilon$ and the parameters of the Standard Model. Lattice calculations are well suited for the study of $B_{K}$ parameter, and it has by now received much attention. After an early round of calculations[1, 2, 3], the statistics have now been raised to a level which allows one to examine some of the fine points of the calculation, such as checks on the reliability of one-loop lattice perturbation theory [4], the chiral behavior and nondegenerate quark masses [5, 7], the dependence of $B_{K}$ on the lattice spacing [3, 6, 7] and the number of dynamical flavors[8]. In this note we offer more information on these latter two points.

2 Calculational Setup

For the dynamical fermion comparison we use lattices of geometry $16^{3}\times 32$, with parameters as given in table 1. The quenched configurations were generated on the Ohio Supercomputer Center T3D, while the dynamical configurations with two and four flavors of $m_{q}a=0.01$ staggered fermions were generated on the 256-node Columbia machine. The parameters were chosen so as to make the scales of the lattices exceedingly close (and equal to approximately (2 GeV$)^{-1}$), as determined from the $\rho$-meson mass in chiral limit (see Fig. 1 and Ref. [9]). We employ 9 values of (degenerate) valence $d$ and $s$ quark masses from $m_{q}=.01$ to $m_{q}=.05$.

For the continuum limit study we generated 7 ensembles of quenched configurations as listed in table 2, and used 7 to 9 values of $m_{q}$.

For creating kaons (at rest) we use a wall of U(1) noise on timeslice $t=0$, i.e. complex random numbers $\xi_{\vec{x}}$ at each space point such that $\langle\xi_{\vec{x}}\xi^{\dagger}_{\vec{y}}\rangle=\delta_{\vec{x},\vec{y}}$. This is statistically equivalent to computing a collection of delta-function sources. In particular, our wall creates only pseudoscalars.

We use a lattice duplicated in the time direction, with periodic boundary conditions in space and time (see Fig. 2). Computing propagators on the doubled lattice, we obtain forward- and backward-going propagators which we use for computing $B_{K}$. That is, if $G_{\pi}(t)$ is the $\pi$ propagator on the doubled lattice, then our operator correlation functions are schematically of the form $G_{\pi}(t)G_{\pi}(t+N_{t})$, where $N_{t}=32$ or 48.

We employ three kinds of operators: Landau gauge, gauge invariant, and tadpole improved. Landau gauge operators are defined by fixing the gauge and omitting explicit links in non-local operators. For gauge-invariant operators we supply the links, averaging over all shortest paths. Tadpole-improved operators are gauge-invariant operators, but with all links rescaled by $u_{0}^{-\Delta}$, where $u_{0}=P^{1/4}$, $P$ is the average plaquette, and $\Delta$ is the number of links needed to connect fermion fields. We opted for tadpole-improved operators on all configurations, using the others on a subset of configurations for checks.

The matching between continuum and lattice operators is of the form

[TABLE]

where $\gamma_{ij}$ is the one-loop anomalous dimension matrix, and $C_{ij}$ are finite coefficients, which can be sizable. We take these from the calculations of Refs. [10, 11]. For the continuum scheme, we choose NDR, quoting results either at scale $\mu=\pi/a$ or at $\mu=2\mathop{\rm GeV}\nolimits$. We use the $\overline{MS}$ coupling constant $g_{\overline{MS}}$, defined as $1/g^{2}_{\overline{MS}}(\pi/a)=P/g_{\rm bare}^{2}+0.02461-0.00704\,N_{f}.$

To check how well the perturbation theory works, we computed all three operators on a subset of the $N_{f}=2$ ensemble, finding that after one-loop corrections are put in, the matrix elements agree within our statistical error. For the bulk of the calculation we used tadpole-improved operators exclusively.

3 Results for $N_{f}$ Dependence

Figs. 4 and 5 show the results for $B_{K}$ on three ensembles of configurations. Values at 9 quark mass points are fitted to the form expected from chiral perturbation theory, $B_{K}=a+bm_{K}^{2}+cm_{K}^{2}\ln{m_{K}^{2}}$. The $N_{f}=4$ and $N_{f}=2$ curves are similar in shape, while the quenched curve crosses between the other two. While this is perfectly allowed, we should also inject a small note of caution—our ensembles have the same $\rho$-masses, but these masses are presumably affected to some degree by the finite volume. If this effect is sizable and depends significantly on $N_{f}$, our curves could shift a little.

Taking the results at face value, we note that the $N_{f}=2$ and $N_{f}=0$ results lie nearly on top of each other at the kaon mass, consistent with our earlier results [8]. Also, most of the $N_{f}=2$ data lie below $N_{f}=0$, consistent with the observation by other groups that quenching seems to increase $B_{K}$ slightly (see, e.g. ref. [12]). However, the $N_{f}=4$ data turn this picture upside down.

Fig. 6 shows our final values for $B_{K}$, obtained at the physical kaon mass and by extrapolation to the chiral limit. We see that the interpolated $N_{f}=3$ result is a few percent higher than quenched.

4 Continuum Extrapolation

Performing the same analysis on the quenched ensembles, we obtain the result shown in figure 7, where we plot $B_{K}(NDR,\mu=2\mathop{\rm GeV}\nolimits)$ versus the scale as determined from $m_{\rho}$. The data are well fit by the quadratic form $B_{K}(a)=B_{K}(a=0)+(a\Lambda_{2})^{2}+(a\Lambda_{4})^{4}$, where the scale of the power correction parameters turns out to be typical of QCD: $\Lambda_{2}\approx 660\mathop{\rm MeV}\nolimits$, $\Lambda_{4}\approx 650\mathop{\rm MeV}\nolimits$. Alternatively, we note that we can avoid making reference to the possibly problematic $m_{\rho}$ by using the scaling form

[TABLE]

where we take $g$ here to be the $\overline{MS}$ coupling. This amounts to shuffling around the $a^{4}$ corrections, and in practice tends to straighten the data out. That is to say, much of the curvature in figure 7 might be ascribed to scaling violations in $m_{\rho}$ itself. To quote a final value we make the conservative choice of a linear fit to the four points with $\beta\geq 6.0$, and obtain

[TABLE]

5 Conclusions

From the dynamical comparison, we find that $B_{K}(N_{f}=3)$ is ($5\pm 2$)% larger than $B_{K}(N_{f}=0)$. Combining with the $a=0$ extrapolation we we quote our current central value $B_{K}$ in the real world:

[TABLE]

Remaining uncertainties include possible finite-size effects in the dynamical ensemble, higher order perturbative corrections in the matching, and higher order chiral ($m_{s}-m_{d}$) effects. A study of hadronic weak matrix elements relevant for $\epsilon^{\prime}/\epsilon$ using the same techniques and ensembles is currently underway.

Acknowledgements. We thank the Columbia collaboration for access to the dynamical configurations. Cray T3D time was supplied by the Ohio Supercomputer Center and the Los Alamos Advanced Computing Laboratory.