The Trace Anomaly and Massless Scalar Degrees of Freedom in Gravity

TL;DR

The paper demonstrates that the trace anomaly in quantum fields leads to the emergence of massless scalar degrees of freedom in gravity, with implications for long-range gravitational effects and a local effective action formulation.

Contribution

It provides a detailed calculation of the scalar pole associated with the trace anomaly in QED and derives a local effective action with scalar fields representing these degrees of freedom.

Findings

Identifies a massless scalar pole in the stress-energy tensor amplitude due to the trace anomaly.

Derives a sum rule showing the scalar state appears as a delta function in the spectral function.

Expresses the anomaly-induced effective action in local form using scalar auxiliary fields.

Abstract

The trace anomaly of quantum fields in electromagnetic or gravitational backgrounds implies the existence of massless scalar poles in physical amplitudes involving the stress-energy tensor. Considering first the axial anomaly and using QED as an example, we compute the full one-loop triangle amplitude of the fermionic stress tensor with two current vertices, <T^{mn} J^a J^b>, and exhibit the scalar pole in this amplitude associated with the trace anomaly, in the limit of zero electron mass m -> 0. To emphasize the infrared aspect of the anomaly, we use a dispersive approach and show that this amplitude and the existence of the massless scalar pole is determined completely by its ultraviolet finite terms, together with the requirements of Poincare invariance of the vacuum, Bose symmetry under interchange of J^a and J^b, and vector current and stress tensor conservation. We derive a sum rule for the appropriate positive spectral function corresponding to the discontinuity of the triangle amplitude, showing that it becomes proportional to delta function of k^2, and therefore contains a massless scalar intermediate state in the conformal limit of zero electron mass. The effective action corresponding to the trace of the triangle amplitude can be expressed in local form by the introduction of two scalar auxiliary fields which satisfy massless wave equations. These massless scalar degrees of freedom couple to classical sources, contribute to gravitational scattering processes, and can have long range gravitational effects.