Disparity in sound speeds: implications for elastic unitarity and the effective potential in quantum field theory theory

TL;DR

This paper investigates scalar field theories with anisotropic sound speeds, deriving unitarity relations, bounds, and effective potentials, revealing how anisotropy influences scattering and renormalization-group behavior.

Contribution

It provides exact unitarity relations, bounds, and one-loop effective potential calculations for anisotropic scalar theories, extending understanding of anisotropic quantum field dynamics.

Findings

Anisotropy induces s-d mixing in scattering amplitudes.

Verified anisotropic optical theorem at one loop for a two-scalar model.

Derived RG flow invariants and analyzed scalon mass modifications due to anisotropy.

Abstract

We study interacting scalar field theories in which different fields propagate with inequivalent spatial kinetic tensors, corresponding to different sound speeds in different directions. We derive the exact elastic two-body unitarity relation and show that the phase space defines a positive kernel on the sphere, so that the scattering amplitude acts as an operator in angular-momentum space. The corresponding unitarity bounds constrain the eigenvalues of the phase-space-rescaled amplitude. In the weak-anisotropy regime, we obtain the leading correction explicitly and show that it induces $s-d$ mixing. For a two-scalar quartic model, we verify the anisotropic optical theorem at one loop and derive coupled channel elastic unitarity bounds. We also compute the local one-loop effective potential and analyze the corresponding one-loop renormalization-group structure. In the classically scale-invariant limit, the Gildener-Weinberg flat direction is unchanged, whereas anisotropy modifies the radiatively generated scalon mass. In the isotropic but unequal-velocity limit, several results become analytic and the RG flow exhibits an additional invariant ray.