On the Trace Anomaly as a Measure of Degrees of Freedom

TL;DR

This paper investigates the trace anomaly coefficient as a potential measure of degrees of freedom in quantum field theories, analyzing its behavior across different spins and dimensions, and its relation to the c-theorem.

Contribution

It provides explicit calculations of the trace anomaly coefficient for various free fields in arbitrary dimensions and explores its implications for the c-theorem and degrees of freedom.

Findings

The trace anomaly coefficient varies with spin and dimension.

In high dimensions, the coefficient exceeds the classical count of field components.

The behavior aligns with c-theorem conjectures and renormalization-group patterns.

Abstract

Recent conjectures of the c-theorem in four and higher dimensions have suggested that the coefficient of the Euler characteristic in the trace anomaly could measure the degrees of freedom in field theory and decrease along the renormalization-group flow. We compute this quantity for free massless scalar, fermion and antisymmetric tensor fields in any dimension, and analyse its dependence on spin and space-time dimension. In the limit of large number of dimensions, where the theories become semiclassical, we find that this quantity does not approach the classical number of field components, but is enhanced for spinful particles. This seemingly strange behaviour is found to be consistent with known renormalization-group patterns and a specific c-theorem conjecture.