Pole-skipping for massive fields and the Stueckelberg formalism

TL;DR

This paper investigates pole-skipping phenomena in holographic Green's functions for massive fields, revealing how mass and gauge invariance breaking influence pole-skipping points and their relation to Stueckelberg fields.

Contribution

It introduces a detailed analysis of pole-skipping for massive p-form and gravitational fields, connecting gauge invariance breaking with additional pole-skipping points and employing the Stueckelberg formalism.

Findings

Pole-skipping points double when mass is non-zero.

Breaking gauge invariance introduces extra pole-skipping points.

Some pole-skipping points shift from non-physical to physical regions as mass varies.

Abstract

Pole-skipping refers to the special phenomenon that the pole and the zero of a retarded two-point Green's function coincide at certain points in momentum space. We study the pole-skipping phenomenon in holographic Green's functions of boundary operators that are dual to massive $p$-form fields and the dRGT massive gravitational fields in the AdS black hole background. Pole-skipping points for these systems are computed using the near horizon method. The relation between the pole-skipping points of massive fields and their massless counterparts is revealed. In particular, as the field mass $m$ is varied from zero to non-zero, the pole-skipping phenomenon undergoes an abrupt change with doubled pole-skipping points found in the massive case. This arises from the breaking of gauge invariance due to the mass term and the consequent appearance of more degrees of freedom. We recover the gauge invariance using the Stueckelberg formalism by introducing auxiliary dynamical fields. The extra pole-skipping points are identified to be associated with the Stueckelberg fields. We also observe that, as the mass varies, some pole-skipping points of the wave number $q$ may move from a non-physical region with complex $q$ to a physical region with real $q$.