[Paper Review] Infrared scaling solutions beyond the Landau gauge: The maximally Abelian gauge and Abelian infrared dominance

Markus Q. Huber, Reinhard Alkofer+1 more|arXiv (Cornell University)|Mar 1, 2011

Quantum Chromodynamics and Particle Interactions31 references4 citations

TL;DR

This paper investigates infrared scaling solutions in Yang-Mills theory beyond the Landau gauge, focusing on the maximally Abelian gauge (MAG), where the diagonal (Abelian) gluon propagator exhibits infrared dominance—similar to ghost dominance in the Landau gauge. Using functional equations and the DoFun tool for automated equation derivation, it identifies two solution types: decoupling (finite diagonal propagator) and scaling (infrared divergent diagonal propagator), with the latter supporting Abelian infrared dominance and a full tower of infrared exponents.

ABSTRACT

Functional equations like exact renormalization group and Dyson-Schwinger equations have contributed to a better understanding of non-perturbative phenomena in quantum field theories in terms of the underlying Green functions. In Yang-Mills theory especially the Landau gauge has been used, as it is the most accessible gauge for these methods. The growing understanding obtained in this gauge allows to proceed to other gauges in order to obtain more information about the relation of different realizations of the confinement mechanism. In the maximally Abelian gauge first results are very encouraging as a variant of Abelian infrared dominance is found: The Abelian part of the gauge field propagator is enhanced at low momenta and thereby dominates the dynamics in the infrared. Its role is therefore similar to that of the ghost propagator in the Landau gauge, where one denotes the corresponding phenomenon as ghost dominance. Also the ambiguity of two different types of solutions (decoupling and scaling) exists in both gauges. Here we present how the two solutions are related in the maximally Abelian gauge. The intricacy of the system of functional equations in this gauge required the development of some new tools and methods as, for example, the automated derivation of the equations by the program DoFun. We also present results for linear covariant and ghost anti-ghost symmetric gauges.

Motivation & Objective

• To investigate non-perturbative dynamics in Yang-Mills theory beyond the Landau gauge, particularly in the maximally Abelian gauge (MAG).
• To test the hypothesis of Abelian infrared dominance, where the diagonal gluon sector dominates the infrared physics.
• To determine whether scaling solutions—previously observed in the Landau gauge—also exist in the MAG and how they relate to decoupling-type solutions.
• To extend functional renormalization group and Dyson-Schwinger equation techniques to non-Landau gauges, including linear covariant and ghost-anti-ghost symmetric gauges.

Proposed method

• The study employs functional equations, specifically the exact renormalization group and Dyson-Schwinger equations, to analyze Green functions in the MAG.
• It uses the DoFun package for automated derivation of the system of functional equations, enabling handling of the complex structure in the MAG.
• Infrared exponent relations (IREs) are derived for all Green functions, determining scaling behavior via critical exponents.
• The analysis distinguishes between transverse and longitudinal components of the gluon propagator, particularly in linear covariant gauges.
• The role of the ghost-gluon vertex is identified as key for determining infrared scaling behavior.
• A comparison between the MAG and other gauges is performed by analyzing the behavior of the diagonal gluon two-point function at zero momentum.

Experimental results

Research questions

• RQ1 Does the maximally Abelian gauge support an infrared scaling solution analogous to the one found in the Landau gauge?
• RQ2 How are the decoupling and scaling solutions in the MAG related, particularly through the zero-momentum value of the diagonal gluon propagator?
• RQ3 Can the hypothesis of Abelian infrared dominance be realized via an infrared divergent diagonal gluon propagator in the MAG?
• RQ4 What is the role of the ghost-gluon vertex in determining the infrared scaling behavior in the MAG?
• RQ5 Why do linear covariant and ghost-anti-ghost symmetric gauges fail to support a non-trivial scaling solution, and what are the implications?

Key findings

• A scaling solution is found in the maximally Abelian gauge where the diagonal gluon propagator diverges in the infrared, indicating Abelian infrared dominance.
• The two solution types—decoupling and scaling—are related by the zero-momentum value of the diagonal gluon two-point function: vanishing at zero momentum leads to the scaling solution.
• The infrared exponent of the diagonal gluon propagator is found to be δ_A = -2δ_c, consistent with scaling behavior.
• The ghost-gluon vertex is identified as the dominant vertex for determining the infrared scaling behavior in the MAG.
• In linear covariant gauges, the longitudinal part of the gluon propagator remains proportional to the gauge parameter, leading to trivial scaling (δ_A = 0), suggesting that standard IR analysis may not apply.
• Ghost-anti-ghost symmetric gauges feature an additional quartic ghost interaction that prevents a consistent non-trivial scaling solution, resulting in δ_A = δ_c = 0.