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Commensurate Scale Relations in Quantum Chromodynamics

Stanley J. Brodsky, Hung Jung Lu

TL;DR

The BLM method is used to directly relate perturbatively calculable QCD observables to the commensurate scale relations connecting the effective charges and provides the generalization of the Crewther relation to non-conformally invariant gauge theory.

Abstract

We use the BLM method to show that perturbatively-calculable observables in QCD can be related to each other without renormalization scale or scheme ambiguity. We define and study the commensurate scale relations. We show that the commensurate scales satisfy the renormalization group transitivity rule which ensures that predictions in PQCD are independent of the choice of an intermediate renormalization scheme. We generalize the BLM procedure to higher order. The application of this procedure to relate known physical observables in QCD gives surprisingly simple results. In particular, the annihilation ratio $R_{e^+e^-}$ and the Bjorken sum rule for polarized electroproduction are related through simple coefficients, which reinforces the idea of a hidden symmetry between these two observables.

Commensurate Scale Relations in Quantum Chromodynamics

TL;DR

The BLM method is used to directly relate perturbatively calculable QCD observables to the commensurate scale relations connecting the effective charges and provides the generalization of the Crewther relation to non-conformally invariant gauge theory.

Abstract

We use the BLM method to show that perturbatively-calculable observables in QCD can be related to each other without renormalization scale or scheme ambiguity. We define and study the commensurate scale relations. We show that the commensurate scales satisfy the renormalization group transitivity rule which ensures that predictions in PQCD are independent of the choice of an intermediate renormalization scheme. We generalize the BLM procedure to higher order. The application of this procedure to relate known physical observables in QCD gives surprisingly simple results. In particular, the annihilation ratio and the Bjorken sum rule for polarized electroproduction are related through simple coefficients, which reinforces the idea of a hidden symmetry between these two observables.

Paper Structure

This paper contains 3 sections, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Next-to-Leading-Order BLM Formulas
  3. Conclusion

Figures (1)

  • Figure 1: The scale $\mu/\sqrt{s}$ according to the BLM (dashed-dotted), PMS (dashed), FAC (full) and $\sqrt{y}$ (dotted) procedures for the three-jet rate in $e^+e^-$ annihilation, as computed by Kramer and Lampe KramerLampe. Notice the strikingly different behavior of the BLM scale from the PMS and FAC scales at low $y$. In particular, the latter two methods predict increasing values of $\mu$ as the jet invariant mass ${\cal M} < \sqrt (y s)$ decreases.