Higher partial wave contamination in finite-volume formulae for 1-to-2 transitions

TL;DR

This paper investigates the impact of neglecting higher partial waves in finite-volume formulas for 1-to-2 particle transitions, highlighting potential systematic errors in high-precision lattice calculations.

Contribution

It provides a comparison between S-wave-only results and those including the next partial wave, assessing the significance of higher partial wave contamination.

Findings

Higher partial waves can introduce systematic errors in finite-volume calculations.

Neglecting higher partial waves may be justified at lower precision but becomes problematic as accuracy improves.

The study quantifies the size of contamination from higher angular momentum states.

Abstract

It is common practice to truncate the finite-volume formula for $K\to\pi\pi$, and other one-to-two transitions, to only include the lowest partial wave, as in the original derivation by Lellouch and L\"uscher. However, as the precision of lattice calculations increases, it may become important to asses the systematic effect of this approximation. With this motivation, we compare the $S$-wave-only ($\ell=0$) results with those truncated at the next lowest value of angular momentum.