On the c-theorem in more than two dimensions

TL;DR

This paper discusses the c-theorem in higher dimensions, providing insights into the c-number for free theories and the stress tensor's three-point function in four dimensions, aiding the search for a rigorous proof.

Contribution

It offers explicit values of the c-number for free theories in even dimensions and characterizes the three-point stress tensor function in four dimensions, advancing understanding of the c-theorem.

Findings

Values of the c-number for free theories across even dimensions

Form of the three-point stress tensor function in four dimensions

Implications for the physical interpretation of trace anomalies

Abstract

Several pieces of evidence have been recently brought up in favour of the c-theorem in four and higher dimensions, but a solid proof is still lacking. We present two basic results which could be useful for this search: i) the values of the putative c-number for free field theories in any even dimension, which illustrate some properties of this number; ii) the general form of three-point function of the stress tensor in four dimensions, which shows some physical consequences of the c-number and of the other trace-anomaly numbers.