A note on the power divergence in lattice calculations of $\Delta I = 1/2$ $K\to\pi\pi$ amplitudes at $M_{K}=M_{\pi}$

TL;DR

This paper clarifies that the expected power divergence in lattice calculations of certain kaon decay amplitudes is absent at specific kinematic points, confirmed through explicit leading-order Chiral Perturbation Theory calculations, aligning with CPS symmetry predictions.

Contribution

It demonstrates, using leading-order Chiral Perturbation Theory, that the power divergence does not appear at the specific kinematic point where $M_K = M_\pi$, confirming CPS symmetry predictions.

Findings

Power divergence is absent at $M_K = M_\pi$ with all mesons at rest.

Explicit calculation confirms theoretical predictions.

Supports the validity of lattice calculations at this kinematic point.

Abstract

In this note, we clarify a point concerning the power divergence in lattice calculations of $\Delta I = 1/2$ $K\to\pi\pi$ decay amplitudes. There have been worries that this divergence might show up in the Minkowski amplitudes at $M_{K}=M_{\pi}$ with all the mesons at rest. Here we demonstrate, via an explicit calculation in leading-order Chiral Perturbation Theory, that the power divergence is absent at the above kinematic point, as predicted by CPS symmetry.