Non-perturbative $\langle \phi \rangle$, $\langle \phi^2 \rangle$ and the dynamically generated scalar mass with Yukawa interaction in the inflationary de Sitter spacetime

TL;DR

This paper investigates the non-perturbative quantum effects on a scalar field with Yukawa interaction in de Sitter space, revealing how quantum fluctuations influence the scalar's vacuum expectation value and generate a dynamical mass.

Contribution

It introduces a non-perturbative approach to compute scalar field expectation values and mass in inflationary spacetime with Yukawa coupling, including resummation of secular logarithms up to three loops.

Findings

Vacuum expectation value of scalar is significantly suppressed by quantum fluctuations.

Dynamically generated scalar mass is computed at late times.

Quantum fluctuations can screen the inflationary cosmological constant.

Abstract

We consider a massless minimally coupled self interacting quantum scalar field coupled to fermion via the Yukawa interaction, in the inflationary de Sitter background. The fermion is also taken to be massless and the scalar potential is taken to be a hybrid, $V(\phi)= \lambda \phi^4/4!+ \beta \phi^3/3!$ ($\lambda >0$). The chief physical motivation behind this choice of $V(\phi)$ corresponds to, apart from its boundedness from below property, the fact that shape wise $V(\phi)$ has qualitative similarity with standard inflationary classical slow roll potentials. Also, its vacuum expectation value can be negative, suggesting some screening of the inflationary cosmological constant. We choose that $\langle \phi \rangle\sim 0$ at early times with respect to the Bunch-Davies vacuum, so that perturbation theory is valid initially. We consider the equations satisfied by $\langle \phi (t) \rangle$ and $\langle \phi^2(t) \rangle$, constructed from the coarse grained equation of motion for the slowly rolling $\phi$. We then compute the vacuum diagrammes of various relevant operators using the in-in formalism up to three loop, in terms of the leading powers of the secular logarithms. For a closed fermion loop, we have restricted ourselves here to only the local contribution. These large temporal logarithms are then resummed by constructing suitable non-perturbative equations to compute $\langle \phi \rangle$ and $\langle \phi^2 \rangle$. $\langle \phi \rangle$ turns out to be at least approximately an order of magnitude less compared to the minimum of the classical potential, $-3\beta/\lambda$, owing to the strong quantum fluctuations. For $\langle \phi^2 \rangle$, we have computed the dynamically generated scalar mass at late times, by taking the appropriate purely local contributions. Variations of these quantities with respect to different couplings have also been presented.