Do gauge fields really contribute negatively to black hole entropy?

TL;DR

This paper investigates the negative contact term in gauge field contributions to black hole entropy, revealing it stems from gauge ambiguities and infrared mode treatment, and shows the true gauge-invariant entropy is positive and finite.

Contribution

It demonstrates that the negative contact term for gauge fields is due to gauge-dependent ambiguities and infrared mode issues, and provides a gauge-invariant calculation showing positive, finite entropy contributions.

Findings

Contact term arises from gauge ambiguity and infrared modes.

Proper gauge-invariant treatment yields positive, finite entropy.

Gauge field entropy equals entanglement entropy in correct framework.

Abstract

Quantum fluctuations of matter fields contribute to the thermal entropy of black holes. For free minimally-coupled scalar and spinor fields, this contribution is precisely the entanglement entropy. For gauge fields, Kabat found an extra negative divergent "contact term" with no known statistical interpretation. We compare this contact term to a similar term that arises for nonminimally-coupled scalar fields. Although both divergences may be interpreted as terms in the Wald entropy, we point out that the contact term for gauge fields comes from a gauge-dependent ambiguity in Wald's formula. Revisiting Kabat's derivation of the contact term, we show that it is sensitive to the treatment of infrared modes. To explore these infrared issues, we consider two-dimensional compact manifolds, such as Euclidean de Sitter space, and show that the contact term arises from an incorrect treatment of zero modes. In a manifestly gauge-invariant reduced phase space quantization, the gauge field contribution to the entropy is positive, finite, and equal to the entanglement entropy.