New Modes for Vector Bosons in the Static Patch

TL;DR

This paper investigates the behavior of massive vector bosons in a static patch of de Sitter space, revealing new physical features, stability conditions, and simplifications in three dimensions, extending previous theoretical work.

Contribution

It introduces an effective mass concept for vector bosons in de Sitter space, explores the edge of stability, and generalizes the classical solutions and quasinormal modes to arbitrary dimensions.

Findings

Effective mass differs from naive Lagrangian mass.

Theory remains well-defined in a tachyonic mass range.

In 3D, the vector sector simplifies to a scalar p-wave.

Abstract

We consider a massive vector Boson in a static patch of $D$-dimensional de Sitter space (dS$_D$). We argue that this field is controlled by an effective physical (squared) mass $\mu_{\mathrm{v}}^2 = m_{\mathrm{v}}^2 + 2(D-1)\ell_{\mathrm{dS}}^{-2}$ which differs from the naive "Lagrangian" (squared) mass $m_{\mathrm{v}}^2$ that appears in the usual form of the Proca Lagrangian/action. In particular, we conjecture that the theory remains well-defined in the naively tachyonic Lagrangian mass range $-2(D-1) < m_{\mathrm{v}}^2\ell_{\mathrm{dS}}^2 < 0$. We identify several interesting physical features of the "edge of stability" $m_{\mathrm{v}}^2\ell_{\mathrm{dS}}^2 = -2(D-1)$. Fixing a static patch breaks the $D$-dimensional de Sitter isometries down to a "static patch subgroup", which explains why our theory may continue to be well-defined in the above mass range despite not fitting into a unitary irreducible representation of SO$(D,1)$. We conjecture that for situations such as ours, the usual $\mathrm{SO}(D,1)$ "Higuchi bound" on unitarity is replaced by the concept of the edge of stability. In $D = 3$ spacetime dimensions, the $s$-wave sector of our theory remarkably simplifies, becoming equivalent to the $p$-wave sector of an ordinary massive scalar. In this case we can explicitly check that the $D = 3$ $s$-wave sector remains well-defined -- both classically and quantum mechanically -- in the above mass range. In the course of our analysis, we will derive the general classical solution and the quasinormal frequency spectrum for the massive vector Boson in the static patch of dS$_D$, generalizing previous work by Higuchi [1], which was done for the special case $D = 4$. While this work was being completed, we became aware of upcoming work by Grewal, Law, and Lochab [2] which will contain a similar derivation.