Worldline Formalism in Snyder Spaces

TL;DR

This paper applies the Worldline Formalism to a scalar field model in Snyder space, deriving a master equation for 1-loop functions, analyzing renormalization, and exploring implications of divergences and effective metrics.

Contribution

It introduces a novel application of the Worldline Formalism to Snyder space, deriving a master equation and analyzing renormalization properties of the scalar field model.

Findings

Derived a master equation for 1-loop n-point functions.

Identified a $\,phi^6$ divergence affecting renormalizability.

Observed effective metric contributions from the renormalized action.

Abstract

We study the $\phi_{\star}^4$ model for a scalar field in a linearization of the Snyder model, using the methods of the Worldline Formalism. Our main result is a master equation for the 1-loop n-point function. From this we derive the renormalization of the coupling parameters of the theory and observe the appearance of a $\phi^6$ divergent contribution that opens the question of whether this theory is renormalizable or not. Additionally, we observe that some terms in the renormalized action can be interpreted as coming from an effective metric proportional to the square of the field.