Quantum correction to the diffusion term in stochastic inflation from composite-operator matching in Soft de Sitter Effective Theory

TL;DR

This paper develops a formalism for composite-operator renormalisation in Soft de Sitter Effective Theory, enabling the calculation of next-to-leading order quantum corrections to stochastic inflation's diffusion term.

Contribution

It introduces a systematic approach for operator mixing and matching beyond leading order, leading to the first calculation of two-loop quantum corrections in stochastic inflation.

Findings

Renormalised the one-loop bispectrum and two-loop one-point function of composite operators.

Matched these computations onto full-theory results.

Determined the two-loop correction to the diffusion term in the Fokker-Planck equation.

Abstract

In the framework of Soft de Sitter Effective Theory (SdSET), the Fokker-Planck equation for the late-time dynamics of the massless minimally coupled scalar field and its extension to the Kramers-Moyal equation are obtained from operator mixing of composite operators of the effective superhorizon field. We construct the formalism for composite-operator renormalisation, mixing and matching in dimensional regularisation, allowing for computations beyond the leading order. The general formalism is illustrated in free SdSET, which already features non-trivial structures including the well-known diffusion coefficient for stochastic inflation. As explicit examples in the interacting theory, we renormalise the one-loop bispectrum and the two-loop one-point function of the composite operator $\varphi_+^2$, and match them onto their full-theory counterparts. These results allow us to determine the next-to-leading order (two-loop) correction to the diffusion term of the Fokker-Planck equation of stochastic inflation for the first time.