Quantum gauge boson propagators in the light front

TL;DR

This paper investigates the gauge boson propagators in light-front gauge theories, addressing residual gauge freedoms, non-local singularities, and proposing a causal prescription to handle zero modes without breaking causality.

Contribution

It introduces a singularity-softening prescription for gauge boson propagators in light-front gauge, improving the treatment of zero modes and non-local singularities beyond the Mandelstam-Leibbrandt approach.

Findings

The ML prescription does not eliminate all zero mode pathologies.

A new causal prescription effectively handles singularities and zero modes.

Physical degrees of freedom propagate without causality violations.

Abstract

Gauge fields in the light front are traditionally addressed via the employment of an algebraic condition $n\cdot A=0$ in the Lagrangian density, where $A_{\mu}$ is the gauge field (Abelian or non-Abelian) and $n^\mu$ is the external, light-like, constant vector which defines the gauge proper. However, this condition though necessary is not sufficient to fix the gauge completely; there still remains a residual gauge freedom that must be addressed appropriately. To do this, we need to define the condition $(n\cdot A)(\partial \cdot A)=0$ with $n\cdot A=0=\partial \cdot A$. The implementation of this condition in the theory gives rise to a gauge boson propagator (in momentum space) leading to conspicuous non-local singularities of the type $(k\cdot n)^{-\alpha}$ where $\alpha=1,2$. These singularities must be conveniently treated, and by convenient we mean not only matemathically well-defined but physically sound and meaningfull as well. In calculating such a propagator for one and two noncovariant gauge bosons those singularities demand from the outset the use of a prescription such as the Mandelstam-Leibbrandt (ML) one. We show that the implementation of the ML prescription does not remove certain pathologies associated with zero modes. However we present a causal, singularity-softening prescription and show how to keep causality from being broken without the zero mode nuisance and letting only the propagation of physical degrees of freedom.