Twisted mass QCD and lattice approaches to the $\Delta I = 1/2$ rule

TL;DR

This paper explores the use of twisted mass lattice QCD with four Wilson quark flavours to compute weak matrix elements related to the $ ext{Δ}I=1/2$ rule, addressing renormalisation challenges.

Contribution

It demonstrates how tmQCD can simplify the renormalisation of four-quark operators in $ ext{Δ}I=1/2$ transitions, especially with an active charm quark and specific twist angle choices.

Findings

tmQCD eliminates unphysical zero modes.

Renormalisation of $K o\pi$ matrix elements is simplified with active charm.

Finite counterterms are sufficient in certain quenched approximations.

Abstract

Twisted mass lattice QCD (tmQCD), generalised to four Wilson quark flavours, can be used for the computation of some weak matrix elements related to $\Delta I=1/2$ transitions. Besides eliminating unphysical zero modes, tmQCD may alleviate long-standing renormalisation problems of the four-quark operators which contribute to CP-conserving $K\to\pi$ transitions. With an active charm quark, the renormalisation of the $K\to\pi$ matrix elements requires at most the subtraction of a linearly divergent counterterm. Furthermore, in the (partially) quenched approximation the twist angles can be chosen so that only a finite counterterm needs to be subtracted.