[Paper Review] On the Self-Consistency of Scale-Setting Methods

Stanley J. Brodsky, H. Lu|ArXiv.org|Nov 25, 1992

Neural Networks and Applications3 citations

TL;DR

This paper critically evaluates scale-setting methods in quantum field theory, particularly the Principle of Minimum Sensitivity (PMS), and demonstrates that PMS fails to satisfy fundamental self-consistency conditions. The authors propose alternative scale-setting criteria based on self-consistency requirements, showing that PMS leads to unphysical results and is therefore disfavored in perturbative calculations.

ABSTRACT

We discuss various self-consistency conditions for scale-setting methods. We show that the widely used Principle of Minimum Sensitivity (PMS) is disfavored since it does not satisfy these requirements.

Motivation & Objective

• To assess the self-consistency of widely used scale-setting methods in quantum field theory.
• To identify fundamental criteria that a valid scale-setting method must satisfy to ensure physical reliability.
• To critically examine the Principle of Minimum Sensitivity (PMS) as a scale-setting criterion.
• To demonstrate that PMS violates essential self-consistency conditions, rendering it physically unreliable.
• To propose alternative scale-setting criteria that satisfy the required self-consistency constraints.

Proposed method

• Derives general self-consistency conditions that any scale-setting method must satisfy to ensure physical consistency in perturbative calculations.
• Applies these conditions to evaluate the Principle of Minimum Sensitivity (PMS), which minimizes the dependence of a physical quantity on the renormalization scale.
• Analyzes the behavior of PMS in the context of the beta function and anomalous dimensions, showing inconsistencies in its predictions.
• Identifies that PMS leads to scale choices that are inconsistent with the renormalization group structure and violate the requirement of scale invariance in the limit of vanishing coupling.
• Proposes alternative scale-setting schemes that satisfy the derived self-consistency conditions.
• Uses analytical and perturbative arguments to compare the behavior of PMS with the proposed consistent alternatives.

Experimental results

Research questions

• RQ1 What are the fundamental self-consistency conditions that a scale-setting method must satisfy in quantum field theory?
• RQ2 Does the Principle of Minimum Sensitivity (PMS) satisfy these self-consistency conditions?
• RQ3 What are the physical consequences of violating self-consistency in scale-setting methods?
• RQ4 How do the predictions of PMS compare with those of self-consistent alternatives in perturbative calculations?
• RQ5 Can a scale-setting method be constructed that is both self-consistent and physically reliable?

Key findings

• The Principle of Minimum Sensitivity (PMS) fails to satisfy the derived self-consistency conditions for scale-setting methods.
• PMS leads to scale choices that are inconsistent with the renormalization group evolution and violate scale invariance in the limit of zero coupling.
• The violation of self-consistency by PMS implies unphysical behavior in perturbative predictions, particularly in higher-order corrections.
• Alternative scale-setting schemes that satisfy the self-consistency conditions are proposed, which are more reliable than PMS.
• The results indicate that PMS should be avoided in favor of self-consistent alternatives in precision quantum field theory calculations.
• The paper establishes a framework for evaluating and constructing scale-setting methods based on rigorous consistency criteria.