The sine-Gordon model and the small k^+ region of light-cone perturbation theory

TL;DR

This paper investigates the non-perturbative ultraviolet divergences in the sine-Gordon model, revealing the critical role of the zero light-cone momentum region in vacuum stability and Green's functions, with implications for higher-dimensional gauge theories.

Contribution

It demonstrates the importance of the $k^+=0$ region in light-cone perturbation theory and its impact on divergences and vacuum stability at the critical point.

Findings

Vacuum instability at the critical point due to $k^+=0$ diagrams.

Ultraviolet divergences arise from $k^+=0$ contributions in the energy density.

Proper inclusion of $k^+=0$ diagrams is essential for correct Green's function behavior.

Abstract

The non-perturbative ultraviolet divergence of the sine-Gordon model is used to study the $k^+ = 0$ region of light-cone perturbation theory. The light-cone vacuum is shown to be unstable at the non-perturbative $\beta^2 = 8\pi$ critical point by a light-cone version of Coleman's variational method. Vacuum bubbles, which are $k^+=0$ diagrams in light-cone field theory and are individually finite and non-vanishing for all $\beta$, conspire to generate ultraviolet divergences of the light-cone energy density. The $k^+ = 0$ region of momentum also contributes to connected Green's functions; the connected two point function will not diverge, as it should, at the critical point unless diagrams which contribute only at $k^+ = 0$ are properly included. This analysis shows in a simple way how the $k^+ =0$ region cannot be ignored even for connected diagrams. This phenomenon is expected to occur in higher dimensional gauge theories starting at two loop order in light-cone perturbation theory.