[Paper Review] Precision computation of B_K in quenched lattice QCD

P. Dimopoulos, Jochen Heitger+3 more|ArXiv.org|Sep 8, 2004

Quantum Chromodynamics and Particle Interactions2 references3 citations

TL;DR

This paper presents a precision computation of the $B_K$ parameter in quenched lattice QCD using Wilson fermions and twisted mass QCD (tmQCD) regularisation, enabling direct computation at the physical kaon mass and eliminating mass extrapolations. The study achieves a continuum limit result of $\hat{B}_K = 0.817(22)$, with nonperturbative renormalisation via the Schr"odinger Functional scheme, significantly reducing systematic uncertainties in $B_K$ determinations for CP violation analyses.

ABSTRACT

We present the results of a precision computation of B_K with Wilson fermions. Simulations are performed at different lattice spacings, enabling continuum limit extrapolations. Two different twisted mass QCD (tmQCD) regularisations are considered for the computation of bare matrix elements. In both cases the relevant four-fermion operator renormalises multiplicatively. In one regularisation it is possible to perform the computation directly at the physical kaon mass value, thus avoiding extrapolations in the mass. Nonperturbative renormalisation is carried out using available Schroedinger Functional results.

Motivation & Objective

• To reduce systematic uncertainties in $B_K$ determinations within quenched lattice QCD, particularly those from exceptional configurations and operator mixing.
• To eliminate mass extrapolations by employing a tmQCD regularisation with a $(\pi/4)$-twisted $(s,d)$ doublet, enabling direct computation at the physical kaon mass.
• To achieve a precise continuum limit for $\hat{B}_K$ using multiple lattice spacings and nonperturbative renormalisation via the Schr"odinger Functional framework.
• To validate the reliability of the method through finite-volume and SU(3)_V breaking tests, ensuring systematic errors are well-controlled.

Proposed method

• Simulations are performed at three values of the lattice coupling $\beta$ ($6.0$, $6.2$, $6.3$) with varying quark masses, using $16^3 \times 48$, $24^3 \times 64$, and $24^3 \times 72$ lattices.
• The $B_K$ parameter is extracted from ratios of correlation functions, with the bare matrix element computed using a $(\pi/4)$-twisted $(s,d)$ doublet to ensure physical kaon mass access.
• Nonperturbative renormalisation is applied using Schr"odinger Functional results, ensuring accurate removal of lattice artefacts and correct operator mixing.
• Continuum limit extrapolation is performed using $a/r_0$ as the cutoff variable, with fits to a constant due to negligible cutoff dependence within statistical errors.
• Finite-volume effects are tested by comparing results on $L \approx 1.5~\text{fm}$ and $L \approx 2.2~\text{fm}$ lattices, showing no significant volume dependence at lightest masses.
• SU(3)_V breaking effects are studied by varying the $\epsilon$ parameter, with no significant dependence observed up to $\epsilon \sim 0.4$.

Experimental results

Research questions

• RQ1 Can the $B_K$ parameter be computed directly at the physical kaon mass using tmQCD, eliminating the need for mass extrapolations?
• RQ2 Does the use of nonperturbative renormalisation in the Schr"odinger Functional scheme suppress systematic errors in $B_K$ computations?
• RQ3 What is the continuum limit value of $\hat{B}_K$ when using $\text{O}(a)$-improved actions and tmQCD regularisation?
• RQ4 How do finite-volume effects and SU(3)_V breaking influence the $B_K$ determination in this framework?

Key findings

• The continuum limit value of the renormalisation group invariant $\hat{B}_K$ is determined to be $0.817(22)$, with negligible cutoff dependence observed in the data.
• The $B_K$ parameter at the physical kaon mass is obtained via linear interpolation in $M_{\text{PS}}^2$, with results from the $(\pi/4)$-twisting approach showing good consistency with the main method.
• No significant finite-volume effects are observed on $L \approx 1.5~\text{fm}$ and $L \approx 2.2~\text{fm}$ lattices at the lightest pseudoscalar mass, supporting the reliability of the results.
• No significant dependence of $B_K$ on the SU(3)_V breaking parameter $\epsilon$ is found up to $\epsilon \sim 0.4$, indicating robustness of the computation.
• The final result for the $\overline{\text{MS}}$-scheme $B_K$ at $2~\text{GeV}$ is $0.592(16)$, consistent with the continuum limit value.
• The use of tmQCD with a $(\pi/4)$-twisted doublet enables direct access to the physical kaon mass, eliminating a major source of systematic uncertainty in previous computations.