Dissertations, Theses, and Capstone Projects

Date of Degree

9-2025

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Physics

Advisor

V. Parameswaran Nair

Advisor

Dimitra Karabali

Committee Members

Robert Pisarski

Alexios Polychronakos

Daniel Kabat

Subject Categories

Elementary Particles and Fields and String Theory | Quantum Physics

Keywords

Non-perturbative methods for gauge theories, complex projective space, 4d Wess-Zumino-Witten theory, gauge-orbit space, gluon propagator mass, Casimir effect

Abstract

This dissertation explores a complex matrix parametrization of four-dimensional non-Abelian gauge theories as a means of gaining analytical insights into (3+1)-dimensional QCD. The low-energy scale of non-Abelian gauge theories remains an elusive area of research even after decades of work. A key challenge lies in identifying physical, gauge-invariant degrees of freedom that are relevant to non-perturbative phenomena. In this dissertation we develop a complex parametrization of gauge fields on which gauge transformations act homogeneously, thereby enabling a manifestly gauge-invariant analytical formulation of the theory.

Chapter one constructs the complex gauge-invariant parametrization for gauge theories living on both complex projective space CP^2 and flat space C^2. Chapter two applies the parametrization to compute the integration measure on the gauge-orbit space—the space of all gauge potentials modulo gauge transformations. Chapter three examines interactions between gauge fields and chiral scalars. The terms appearing in the effective actions that are of particular interest are gauge-invariant, local mass terms for the gauge fields. These terms are significant for supporting the possibility of a soft gluon mass, corroborating results from lattice simulations and Schwinger-Dyson equations. Among the mass terms is a four-dimensional Wess-Zumino-Witten (WZW) action for Hermitian fields, which implies the existence of a kinematic regime in which Yang-Mills theory can be approximated by a 4d WZW model. This has potential implications for confinement, by analogy to the (2+1)-dimensional case.

Chapter four develops a procedure to covariantize the theory by formulating it on the twistor space CP^3 ∼ S^4 × CP^1. The covariantized version is then used in chapter five to compute the Casimir energy for (3+1)-dimensional Yang-Mills theory. The presence of gauge field mass terms in the measure leads to a massive scalar-like behavior of the non-Abelian Casimir effect. These results are compared to recent lattice data, and several open questions are raised.

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