Eradication of singularities in the next-to-leading order RG evolution for the \Delta S = 1 effective Hamiltonian with 3 quark flavours

TL;DR

This paper addresses and resolves the issue of singularities in the next-to-leading order renormalization group evolution of the $ riangle S=1$ effective Hamiltonian with three quark flavors, improving the accuracy of lattice QCD calculations.

Contribution

It introduces a method of analytic continuation to eliminate singularities and rectifies the approach to ensure singularity-free solutions from the start in the three-flavor case.

Findings

Singularities in the NLO RG evolution are removable via analytic continuation.

The origin of singularities is linked to the breakdown of previous methods in the three-flavor case.

A corrected approach avoids singularities from the outset.

Abstract

We consider the renormalization group evolution for the operators in the $\Delta S=1$ effective Hamiltonian with 3 active quark flavors, which is needed in the numerical analysis of data sets for $\epsilon'/\epsilon$ calculated in lattice QCD. Singularities are present in the original solution of Buras et al. at next-to-leading order. We show how these can be eradicated through a method of analytic continuation to obtain the correct finite solution in this case. Furthermore, we trace the origin of the singularities to a breakdown of the approach of Buras et al. in the 3 flavour case, and show how it can be rectified so that singularitites are absent from the beginning.