Automatic generation of Feynman rules in the Schroedinger functional

TL;DR

This paper introduces an algorithm for automatically generating Feynman vertices in the Schr"odinger functional with an abelian background gauge field, facilitating calculations in lattice gauge theory.

Contribution

It presents a novel algorithm and Python implementation for generating vertices with complex color structures in the Schr"odinger functional, enabling efficient perturbative calculations.

Findings

One-loop results for Lambda parameter ratios

Determination of O(a) boundary counter term

Analysis of cutoff effects on step scaling function

Abstract

We provide an algorithm to generate vertices for the Schr\"odinger functional with an abelian background gauge field. The background field has a non-trivial color structure, therefore we mainly focus on a manipulation of the color matrix part. We propose how to implement the algorithm especially in python code. By using python outputs produced by the code, we also show how to write a numerical expression of vertices in the time-momentum as well as the coordinate space into a Feynman diagram calculation code. As examples of the applications of the algorithm, we provide some one-loop results, ratios of the Lambda parameters between the plaquette gauge action and the improved gauge actions composed from six-link loops (rectangular, chair and parallelogram), the determination of the O(a) boundary counter term to this order, and the perturbative cutoff effects of the step scaling function of the Schroedinger functional coupling constant.