# Equivalence Between the Gauge $n\cdot\partial n\cdot A=0$ and the Axial Gauge

## TL;DR

This paper investigates the discontinuity issues in the gauge condition $n ext{ extperiodcentered}\partial n ext{ extperiodcentered} ext{A}=0$ and establishes its nontrivial equivalence to the axial gauge, especially for long-range correlations.

## Contribution

It explicitly solves the Faddeev-Popov determinant for the gauge and analyzes the conditions under which the gauge is equivalent to the axial gauge, highlighting the impact of singularities.

## Key findings

- Discontinuity at $n ext{ extperiodcentered} ext{k}=0$ cannot be regularized by standard methods.
- Perturbation series in the gauge $n ext{ extperiodcentered} ext{ extperiodcentered} ext{A}=0$ matches axial gauge for short-range objects.
- Equivalence is nontrivial for long-range correlations and singular quantities.

## Abstract

Discontinuity of gauge theory in the gauge condition $n\cdot\partial n\cdot A=0$, which emerges at $n\cdot k=0$, is studied here. Such discontinuity is different from that one confronts in axial gauge and can not be regularized by conventional analytical continuation method. The Faddeev-Popov determinate of the gauge $n\cdot\partial n\cdot A=0$, which is solved explicitly in the manuscript, behaves like a $\delta$-functional of gauge potentials once singularities in the functional integral is neglected and the length along $n^{\mu}$ direction of the space tends to infinity. As a sequence, perturbation series in the gauge $n\cdot\partial n\cdot A=0$ returns to that in axial gauge for short-range correlated objects that are free from singularities in path integral. However, the equivalence between the gauge $n\cdot\partial n\cdot A=0$ and axil gauge is nontrivial for long-range correlated objects and quantities that are affected by singularities in path integral. Continuity of gauge links one encounter in perturbation theory and lattice calculation is affected by such discontinuity.