Four quark operators for kaon bag parameter with gradient flow

TL;DR

This paper calculates matching coefficients for four-quark and quark bilinear operators using the gradient flow method, enabling more accurate lattice QCD evaluations of the kaon bag parameter relevant to CP-violation studies.

Contribution

It introduces a method to compute matching coefficients for operators related to the kaon bag parameter using the gradient flow, improving lattice QCD calculations.

Findings

Calculated matching coefficients for four-quark operators.

Provided a framework for accurate renormalization in lattice QCD.

Enhanced the precision of kaon bag parameter evaluations.

Abstract

To study the CP-violation using the $K_0-\bar{K}_0$ oscillation, we need the kaon bag parameter which represents QCD corrections in the leading Feynman diagrams. The lattice QCD provides us with the only way to evaluate the kaon bag parameter directly from the first principles of QCD. However, a calculation of relevant four quark operators with theoretically sound Wilson-type lattice quarks had to carry a numerically big burden of extra renormalizations and resolution of extra mixings due to the explicit chiral violation. Recently, the Small Flow-time eXpansion (SFtX) method was proposed as a general method based on the gradient flow to correctly calculate any renormalized observables on the lattice, irrespective of the explicit violations of related symmetries on the lattice. To apply the SFtX method, we need matching coefficients, which relate finite operators at small flow-times in the gradient flow scheme to renormalized observables in conventional renormalization schemes. In this paper, we calculate the matching coefficients for four quark operators and quark bi-linear operators, relevant to the kaon bag parameter.