# Optimization for factorized quantities in perturbative QCD

## TL;DR

This paper revisits the optimization of scheme choices in perturbative QCD calculations, correcting previous deficiencies and simplifying the process by identifying proper scheme variables and invariants.

## Contribution

It corrects and clarifies the application of the principle of minimal sensitivity in optimizing factorized quantities in perturbative QCD, simplifying the optimization procedure.

## Key findings

- Recovered earlier results of Nakkagawa and Niegawa.
- Showed that optimized coefficient C^opt=1, simplifying calculations.
- Identified proper scheme variables, RG equations, and invariants.

## Abstract

Perturbative calculations of factorized physical quantities, such as moments of structure functions, suffer from renormalization- and factorization-scheme dependence. The application of the principle of minimal sensitivity to "optimize" the scheme choices is reconsidered, correcting deficiencies in the earlier literature. The proper scheme variables, RG equations, and invariants are identified. Earlier results of Nakkagawa and Niegawa are recovered, even though their starting point is, at best, unnecessarily complicated. In particular, the optimized coefficients of the coefficient function C are shown to vanish, so that C^opt=1. The resulting simplifications mean that the optimization procedure is as simple as that for purely-perturbative physical quantities.