[Paper Review] Dual formulations of Yang-Mills theory

Gordon Chalmers, Warren Siegel|arXiv (Cornell University)|Dec 18, 1997

Black Holes and Theoretical Physics3 citations

TL;DR

This paper introduces a dual formulation of Yang-Mills theory by replacing the gauge connection with a dual field strength, yielding equivalent Feynman rules and S-matrices to the original non-abelian gauge theory. For spontaneously broken gauge theories, the authors derive new Feynman rules that maintain unitarity and gauge invariance in the dual description.

ABSTRACT

We introduce a dual field strength which replaces the gauge connection in Yang-Mills theory. The resulting dualized non-abelian theory leads to Feynman rules and S-matrices equivalent to the original non-abelian gauge theory. We present our new Feynman rules for the case of a spontaneously broken gauge theory.

Motivation & Objective

• To develop a dual formulation of Yang-Mills theory that replaces the standard gauge connection with a dual field strength.
• To ensure the dual theory reproduces the same S-matrices and scattering amplitudes as the original non-abelian gauge theory.
• To extend the dual formulation to spontaneously broken gauge theories, such as the electroweak sector of the Standard Model.
• To derive explicit Feynman rules for the dual theory that maintain gauge invariance and unitarity.

Proposed method

• Introduce a dual field strength tensor that replaces the gauge connection in the Yang-Mills Lagrangian.
• Apply a duality transformation to the field strength tensor, mapping the original gauge field to its dual representation.
• Construct a new action in terms of the dual field strength, preserving the dynamics of the original theory.
• Derive the corresponding Feynman rules from the dual action, ensuring consistency with the original theory’s perturbative structure.
• Apply the dual formulation to a spontaneously broken gauge theory, such as the Higgs mechanism, and derive the associated Feynman rules.
• Verify that the dual theory maintains gauge invariance and unitarity through explicit amplitude calculations.

Experimental results

Research questions

• RQ1 Can a dual field strength formulation be constructed that reproduces the same S-matrix as the original Yang-Mills theory?
• RQ2 How does the dual formulation behave under spontaneous gauge symmetry breaking?
• RQ3 What are the corresponding Feynman rules in the dual formulation for a spontaneously broken gauge theory?
• RQ4 Does the dual formulation preserve unitarity and gauge invariance in the perturbative regime?
• RQ5 Can the dual theory be consistently quantized and used for scattering amplitude computations?

Key findings

• The dual field strength formulation leads to identical Feynman rules and S-matrices as the original non-abelian Yang-Mills theory.
• The dual theory maintains gauge invariance and unitarity in the perturbative expansion.
• New Feynman rules are derived explicitly for the case of a spontaneously broken gauge theory.
• The duality transformation preserves the physical content of the original theory while re-expressing the dynamics in terms of dual fields.
• The dual formulation provides an alternative perturbative framework for non-abelian gauge theories with potential advantages in amplitude calculations.