Yang-Mills beta-function from a large-distance expansion of the Schroedinger functional

TL;DR

This paper develops a method to reconstruct the Yang-Mills Schrödinger functional for varying fields from a local expansion valid for slowly varying fields, using analyticity in a complex scale parameter, and derives the beta-function.

Contribution

It introduces a novel approach leveraging analyticity to connect local expansions with the full functional, enabling calculation of the beta-function from slowly varying field behavior.

Findings

Reproduces the standard perturbative beta-function.

Shows how to reconstruct the functional for arbitrary field variations.

Demonstrates the role of analyticity in field variation analysis.

Abstract

For slowly varying fields the Yang-Mills Schroedinger functional can be expanded in terms of local functionals. We show how analyticity in a complex scale parameter enables the Schroedinger functional for arbitrarily varying fields to be reconstructed from this expansion. We also construct the form of the Schroedinger equation that determines the coefficients. Solving this in powers of the coupling reproduces the results of the `standard' perturbative solution of the functional Schroedinger equation which we also describe. In particular the usual result for the beta-function is obtained illustrating how analyticity enables the effects of rapidly varying fields to be computed from the behaviour of slowly varying ones.