Mass hierarchies and non-decoupling in multi-scalar field dynamics

TL;DR

This work develops a covariant formalism for multi-scalar field dynamics with a mass hierarchy, showing that curved trajectories in field space induce non-decoupling between heavy and light modes. By introducing a moving frame and a canonical basis, the authors derive a low-energy effective theory in which heavy dynamics imprint a β-parameter that rescales spatial gradients; the key result is the action with $ e^{β} $ or $ e^{-β} $ factors and the explicit expression $ β = \ln\left( 1 + \frac{4 \dot{φ}_0^2}{ κ^2 M_H^2} \right) $. They provide an exact solution for constant radius of curvature and a general procedure to integrate out very heavy modes, with applications to inflation and supergravity decoupling. The findings imply potentially observable consequences in the inflationary power spectrum and non-Gaussianities when heavy fields are non-negligibly mixed into light dynamics, and offer criteria for when low-energy effective theories are valid in curved field-space settings.

Abstract

In this work we study the effects of field space curvature on scalar field perturbations around an arbitrary background field trajectory evolving in time. Non-trivial imprints of the 'heavy' directions on the low energy dynamics arise when the vacuum manifold of the potential does not coincide with the span of geodesics defined by the sigma model metric of the full theory. When the kinetic energy is small compared to the potential energy, the field traverses a curve close to the vacuum manifold of the potential. The curvature of the path followed by the fields can still have a profound influence on the perturbations as modes parallel to the trajectory mix with those normal to it if the trajectory turns sharply enough. We analyze the dynamical mixing between these non-decoupled degrees of freedom and deduce its non-trivial contribution to the low energy effective theory for the light modes. We also discuss the consequences of this mixing for various scenarios where multiple scalar fields play a vital role, such as inflation and low-energy compactifications of string theory.

Figures (2)

• Figure 1: A curved trajectory with a sudden bend, where a scalar potential presents a turning flat direction with a massive mode remaining perpendicular to it
• Figure 2: The difference between $\phi_{*}$ and $\phi_{0}$.