Yukawa Scalar Self-Mass on a Conformally Flat Background

TL;DR

This work analyzes quantum corrections to a massless, minimally coupled scalar field that is Yukawa-coupled to a massless Dirac fermion in a conformally flat spacetime, focusing on the one-loop self-mass-squared $M^2(x;x')$ computed via dimensional regularization and fully renormalized. In locally de Sitter backgrounds the results are cast in a manifestly de Sitter invariant form using the invariant length $y(x;x')$ and the conformal d'Alembertian ${\\cal D}_c$, and the Schwinger-Keldysh in-in framework yields causal, real effective field equations for the scalar. When these equations are solved for plane-wave modes, appropriate finite renormalizations render the one-loop corrections to the scalar mode functions negligible at late times; the leading late-time correction scales as $g_1(\eta,k) \sim (k/(H a))^3 \ln^2(a)$, leaving the inflationary dynamics, including scalar-catalyzed fermion production, essentially unchanged at this order. The analysis suggests that two-loop effects, while potentially nonzero, are suppressed by additional factors of $f^4$ and scale factors, reinforcing the robustness of the inflationary scenario under this Yukawa coupling.

Abstract

We compute the one loop self-mass-squared of a massless, minimally coupled scalar which is Yukawa-coupled to a massless Dirac fermion in a general conformally flat background. Dimensional regularization is employed and a fully renormalized result is obtained. For the special case of a locally de Sitter background our result is manifestly de Sitter invariant. By solving the effective field equations we show that the scalar mode functions acquire no significant one loop corrections. In particular, the phenomenon of super-adiabatic amplification is not affected. One consequence is that the scalar-catalyzed production of fermions during inflation should not be reduced by changes in the scalar sector before it has time to go to completion.