[Paper Review] Spontaneous conformal symmetry breaking and a massless Wu-Yang monopole

Marc Gillioz|arXiv (Cornell University)|Jul 17, 2017

Black Holes and Theoretical Physics3 citations

TL;DR

This paper studies spontaneous conformal symmetry breaking in N=2 supersymmetric Yang-Mills theory with a spacetime-dependent gauge coupling, showing that the energy-momentum tensor is conserved only when the coupling satisfies its equation of motion. It identifies a regularized, massless Wu-Yang monopole as a stable soliton solution, while the BPS monopole only exists in the large Nc limit.

ABSTRACT

A formulation of $\mathcal{N} = 2$ supersymmetric Yang-Mills theory with a spacetime-dependent gauge coupling allows to study the breaking of conformal symmetry at the quantum level. The theory has an energy-momentum tensor that is only conserved if an equation of motion for the coupling is imposed. It admits non-trivial solitons, among which the Wu-Yang monopole that can be regularized and turns out to be massless. On the other hand, the ordinary BPS monopole is only a solution in the large $N_c$ limit.

Motivation & Objective

• To investigate the implications of spontaneous conformal symmetry breaking in quantum field theories with spacetime-dependent couplings.
• To analyze solitonic solutions in N=2 supersymmetric Yang-Mills theory when the gauge coupling is promoted to a spacetime field.
• To determine whether the Wu-Yang monopole can be regularized and shown to be massless in this framework.
• To contrast the existence of the Wu-Yang monopole with the BPS monopole, which only appears in the large Nc limit.
• To examine the energy-momentum tensor's conservation and its dependence on the coupling's equation of motion.

Proposed method

• Formulate the N=2 supersymmetric Yang-Mills theory with a spacetime-dependent gauge coupling g(x), promoting the coupling to a dynamical field.
• Construct the quantum effective action using the local renormalization group, including higher-derivative terms in g and auxiliary fields such as θ and f.
• Impose equations of motion for all fields, including the coupling field g, to ensure consistency and energy-momentum conservation.
• Use component field decomposition to analyze the action and energy density, identifying terms involving f and θ.
• Regularize the Wu-Yang monopole solution by solving the equations of motion and verifying finiteness of energy.
• Set f=0 and θ=0 as external assumptions to simplify the analysis and ensure a bounded energy density.

Experimental results

Research questions

• RQ1 Can the Wu-Yang monopole be consistently regularized in a quantum field theory with a spacetime-dependent coupling?
• RQ2 Under what conditions is the energy-momentum tensor conserved in a theory with a dynamical coupling?
• RQ3 Does spontaneous conformal symmetry breaking allow for stable, massless solitons in N=2 SYM?
• RQ4 How does the Wu-Yang monopole differ from the BPS monopole in terms of existence and stability?
• RQ5 What role do auxiliary fields f and θ play in the energy density and stability of solitonic configurations?

Key findings

• The Wu-Yang monopole is regularized and found to be massless when the coupling field g satisfies its equation of motion.
• The energy-momentum tensor is conserved only if the equation of motion for the coupling field g is imposed.
• The energy density is bounded below when f=0 and the couplings approach constant values at spatial infinity.
• The BPS monopole solution only exists in the large Nc limit, while the Wu-Yang monopole is stable and massless in the full quantum theory.
• The presence of the field f leads to a potential unbounded below, but its contribution to the energy density is negative only when f≠0; setting f=0 stabilizes the system.
• The analysis breaks N=2 supersymmetry explicitly, and a fully supersymmetric treatment is left for future work.