Numerical prescriptions of early-time divergences of the in-in formalism

TL;DR

This paper introduces two novel numerical methods, Integral Basis and Beta Regulator, to efficiently handle early-time divergences in in-in formalism calculations in quantum field theory, improving accuracy and speed.

Contribution

The paper presents two new numerical techniques for managing early-time divergences in in-in formalism, enhancing computational efficiency and accuracy over existing methods.

Findings

The new methods outperform existing techniques in computation time.

The methods improve the accuracy of in-in formalism integral evaluations.

Benchmark tests demonstrate the effectiveness of the proposed approaches.

Abstract

In quantum field theory, the in and out states can be related to the full Hamiltonian by the $i\epsilon$ prescription. A Wick rotation can further bring the correlation functions to Euclidean spacetime where the integrals are better defined. This setup is convenient for analytical calculations. However, for numerical calculations, an infinitesimal $\epsilon$ or a Wick rotation of numerical functions are difficult to implement. We propose two new numerical methods to solve this problem, namely an Integral Basis method based on linear regression and a Beta Regulator method based on Ces\`aro/Riesz summation. Another class of partition-extrapolation methods previously used in electromagnetic engineering is also introduced. We benchmark these methods with existing methods using in-in formalism integrals, indicating advantages of these new methods over the existing methods in computation time and accuracy.