Pervasiveness of the breakdown of self-interacting vector field theories

TL;DR

This paper demonstrates that self-interacting vector field theories can break down even at low initial amplitudes due to wave packet growth, indicating a widespread pathology affecting their well-defined evolution.

Contribution

It provides a general proof that small initial data in these theories can still lead to breakdown, highlighting the pervasive nature of their instabilities.

Findings

Low amplitude wave packets grow and reach high amplitudes

Breakdown occurs in more than one spatial dimension

Self-interacting vector theories are broadly unstable

Abstract

Various groups recently argued that self-interacting vector field theories lack a well-defined time evolution when the field grows to large amplitudes, which has drastic consequences for models in gravity and high energy theory. Such field amplitudes can be a result of an external driving mechanism, or occur intrinsically, due to large values of the field and its derivatives in the initial data. This brings a natural question: is small amplitude initial data guaranteed to evolve indefinitely in these theories in the absence of an outside driving term? We answer this question in the negative, demonstrating that arbitrarily low amplitude initial data can still lead to the breakdown of the theory. Namely, ingoing spherically symmetric wave packets in more than one spatial dimensions grow as an inverse power of the radius, and their amplitudes can generically reach high enough values where time evolution ceases to exist. This simple example further establishes the pervasiveness of the pathology of self-interacting vector field theories.