Renormalization in theories with modified dispersion relations: weak gravitational fields

TL;DR

This paper investigates how modified dispersion relations affect the renormalization of quantum scalar fields in curved spacetime, revealing that divergence behavior depends on the order of the dispersion relation and differs from previous homogeneous background results.

Contribution

It provides a detailed analysis of divergences in <> and <T_{}> for scalar fields with modified dispersion relations in weak gravitational fields, highlighting new dependence on the dispersion order.

Findings

<> becomes finite for high enough dispersion order s

Divergences in <T_{}> increase with s

Differences from previous homogeneous background results

Abstract

We consider a free quantum scalar field satisfying modified dispersion relations in curved spacetimes, within the framework of Einstein-Aether theory. Using a power counting analysis, we study the divergences in the adiabatic expansion of <\phi^2> and <T_{\mu\nu}>, working in the weak field approximation. We show that for dispersion relations containing up to $2s$ powers of the spatial momentum, the subtraction necessary to renormalize these two quantities on general backgrounds depends on $s$ in a qualitatively different way: while <\phi^2> becomes convergent for a sufficiently large value of $s$, the number of divergent terms in the adiabatic expansion of <T_{\mu\nu}> increases with $s$. This property was not apparent in previous results for spatially homogeneous backgrounds.