Field-theoretic derivation of bubble-wall force

TL;DR

This paper develops a quantum field theoretic framework to calculate the force on expanding bubbles during a first order phase transition, revealing conditions for bubble runaway or stationary behavior in the early Universe.

Contribution

It introduces a general QFT-based formula for bubble-wall force, analyzing effects of thermal equilibrium and ballistic limits, with applications to the Standard Model and scalar fields.

Findings

Force proportional to entropy increase in thermodynamic limit

Strong friction in thermal equilibrium prevents bubble runaway

Ballistic limit allows bubbles to accelerate indefinitely

Abstract

We derive a general quantum field theoretic formula for the force acting on expanding bubbles of a first order phase transition in the early Universe setting. In the thermodynamic limit the force is proportional to the entropy increase across the bubble of active species that exert a force on the bubble interface. When local thermal equilibrium is attained, we find a strong friction force which grows as the Lorentz factor squared, such that the bubbles quickly reach stationary state and cannot run away. We also study an opposite case when scatterings are negligible across the wall (ballistic limit), finding that the force saturates for moderate Lorentz factors thus allowing for a runaway behavior. We apply our formalism to a massive real scalar field, the standard model and its simple portal extension. For completeness, we also present a derivation of the renormalized, one-loop, thermal energy-momentum tensor for the standard model and demonstrate its gauge independence.