Leading Log Solution for Inflationary Yukawa

TL;DR

This work extends Starobinskiĭ’s stochastic inflation framework to a massless, minimally coupled scalar interacting with a massless fermion in de Sitter space, treating the fermion as a passive field that can propagate infrared logs and induce new interactions. By integrating out the fermion and evaluating the fermion propagator in the constant-scalar background, the authors derive a scalar effective potential $V(\varphi)$ that is unbounded below and capture the leading infrared logarithms nonperturbatively via a stochastic, Starobinskiĭ‑type approach. They validate the stochastic formulation with a explicit two-loop calculation of a coincident vertex, confirming the leading-log predictions, and they derive a nonperturbative leading-log stress tensor $V_{\rm s}(\varphi)$ showing how backreaction can arise from the same infrared physics. The results suggest that backreaction in this Yukawa model could dynamically counteract a large cosmological constant, though the unbounded potentials signal an eventual instability of the inflationary regime.

Abstract

We generalize Starobinskii's stochastic technique to the theory of a massless, minimally coupled scalar interacting with a massless fermion in a locally de Sitter geometry. The scalar is an ``active'' field that can engender infrared logarithms. The fermion is a ``passive'' field that cannot cause infrared logarithms but which can carry them, and which can also induce new interactions between the active fields. The procedure for dealing with passive fields is to integrate them out, then stochastically simplify the resulting effective action following Starobinskiĭ. Because Yukawa theory is quadratic in the fermion this can be done explicitly using the classic solution of Candelas and Raine. We check the resulting stochastic formulation against an explicit two loop computation. We also derive a nonperturbative, leading log result for the stress tensor. Because the scalar effective potential induced by fermions is unbounded below, back-reaction from this model might dynamically cancel an arbitrarily large cosmological constant.