Spontaneous Symmetry Probing

TL;DR

The paper introduces spontaneous symmetry probing (SSP), a framework for time-dependent states that evolve along symmetry directions in relativistic QFTs, leading to a time-independent fluctuation dynamics and a running Goldstone. It shows that SSP implies a gapless Goldstone along the SSP direction, while in non-Abelian settings, Goldstones corresponding to generators that do not commute with the SSP generator can acquire a mass proportional to the SSP speed (higgsed Goldstones). The authors develop a Goldstone theorem for SSP, construct the low-energy P(X)-type effective theory, and demonstrate via semiclassical analysis how the fluctuation spectrum organizes, including the higgsing mechanism in the chiral Lagrangian and SO(N) models. They also discuss cosmological applications, particularly to inflationary perturbations and EFT, and outline how gravity and explicit symmetry breaking would enter in future work.

Abstract

For relativistic quantum field theories, we consider Lorentz breaking, spatially homogeneous field configurations or states that evolve in time along a symmetry direction. We dub this situation "spontaneous symmetry probing" (SSP). We mainly focus on internal symmetries, i.e. on symmetries that commute with the Poincare group. We prove that the fluctuations around SSP states have a Lagrangian that is explicitly time independent, and we provide the field space parameterization that makes this manifest. We show that there is always a gapless Goldstone excitation that perturbs the system in the direction of motion in field space. Perhaps more interestingly, we show that if such a direction is part of a non-Abelian group of symmetries, the Goldstone bosons associated with spontaneously broken generators that do not commute with the SSP one acquire a gap, proportional to the SSP state's "speed". We outline possible applications of this formalism to inflationary cosmology.