Renormalization group dependence of the QCD coupling

TL;DR

This paper derives an exact relation between expansion coefficients and the beta function in QCD, revealing lost logarithmic terms in finite expansions and demonstrating that a new coupling improves convergence.

Contribution

It provides a mathematically rigorous derivation of the relation between expansion coefficients and the beta function in QCD, and introduces a new coupling that enhances convergence.

Findings

Standard finite expansions omit infinite logarithmic terms.

The new coupling with a four-loop beta function converges faster.

Lost logarithmic terms can be expressed in closed form.

Abstract

The general relation between the standard expansion coefficients and the beta function for the QCD coupling is exactly derived in a mathematically strict way. It is accordingly found that an infinite number of logarithmic terms are lost in the standard expansion with a finite order, and these lost terms can be given in a closed form. Numerical calculations, by a new matching-invariant coupling with the corresponding beta function to four-loop level, show that the new expansion converges much faster.