The Fermion Self-Energy during Inflation

TL;DR

This work computes and renormalizes the one-loop quantum gravitational self-energy of massless Dirac fermions in a locally de Sitter background using dimensional regularization and BPHZ counterterms. A noninvariant counterterm is required because the chosen gauge breaks de Sitter invariance, yet all divergences can be absorbed consistently; the graviton propagator is decomposed into A, B, and C scalar components, with the A-type encoding inflationary infrared effects. The renormalized self-energy contains both local logarithmic terms proportional to $\ln(a a')$ and nonlocal terms involving $\ln(\mu^2 \Delta x^2)/\Delta x^2$, signaling inflation-enhanced gravitational corrections to the quantum-corrected Dirac equation. The leading infrared behavior is dominated by these logarithms, yielding corrections of order $\kappa^2 H^2$ times logarithms, which are universal and largely independent of scalar sector details, and set the stage for potential further resummations and phenomenological implications for inflationary cosmology.

Abstract

We compute the one loop fermion self-energy for massless Dirac + Einstein in the presence of a locally de Sitter background. We employ dimensional regularization and obtain a fully renormalized result by absorbing all divergences with BPHZ counterterms. An interesting technical aspect of this computation is the need for a noninvariant counterterm owing to the breaking of de Sitter invariance by our gauge condition. Our result can be used in the quantum-corrected Dirac equation to search for inflation-enhanced quantum effects from gravitons, analogous to those which have been found for massless, minimally coupled scalars.