Modified L\"uscher zeta-function and the modified effective range expansion in the presence of a long-range force

TL;DR

This paper introduces an efficient numerical method for calculating the modified L"uscher zeta-function with long-range forces, demonstrating improved accuracy and stability over standard approaches, and discusses regularization and renormalization techniques.

Contribution

It develops a new numerical algorithm for the modified L"uscher zeta-function in long-range force scenarios, with detailed regularization and renormalization procedures, and shows minimal partial wave truncation effects.

Findings

Truncation of higher partial waves has little impact on results.

The proposed renormalization scheme yields natural size parameters.

The algorithm improves the calculation of finite-volume energy levels.

Abstract

An efficient numerical algoritm is proposed for the calculation of the modified L\"uscher zeta-function in the presence of a long-range force. Using the formalism developed in Ref.~\cite{Bubna:2024izx} for the analysis of synthetic data on the finite-volume energy levels in a toy model, it is demonstrated that, in contrast to the standard L\"uscher approach, the truncation of the higher partial waves has very little effect on the final result. Furthermore, the regularization and renormalization of the modified L\"uscher zeta-function is discussed in detail, as well as the problems arising within the cutoff regularization. It is shown that, using the renormalization scheme proposed in the present paper, one obtains modified effective range expansion parameters of natural size in all partial waves.