Transversality in the Coupling of Gravity to Gauge Theories

TL;DR

This paper investigates the transversality properties of quantum gravity coupled to the Standard Model, providing detailed propagator and vertex rules, and identifying optimal gauge fixings for these theories.

Contribution

It introduces a comprehensive analysis of transversality in effective quantum gravity coupled to matter, including identities and decompositions analogous to Yang--Mills theory, and proposes optimal gauge fixings.

Findings

De Donder gauge is optimal for quantum gravity.

Lorenz gauge is optimal for Yang--Mills theory.

Derived identities facilitate transversality analysis in quantum gravity.

Abstract

We consider (effective) Quantum General Relativity coupled to the Standard Model and study its transversality. To this end, we provide all propagator and three-valent vertex Feynman rules. Then we examine the longitudinal, identical and transversal projection tensors for the de Donder gauge fixing and the Lorenz gauge fixing. In particular, we recall several identities from Quantum Yang--Mills theory and introduce their counterparts in (effective) Quantum General Relativity: This includes decompositions of the longitudinal projection tensors as well as expressions of the corresponding propagators in terms of their transversal structure, together with longitudinal contraction identities for all three-valent vertex Feynman rules. In addition, we introduce the notion of an optimal gauge fixing as the natural choice for a given gauge theory: In particular, we find that this is the de Donder gauge fixing in General Relativity and the Lorenz gauge fixing in Yang--Mills theory.