Scale Anomalies, States, and Rates in Conformal Field Theory

TL;DR

This paper introduces two methods to compute scale anomaly coefficients in conformal field theories using position and momentum space, revealing positivity properties and conditions for validity.

Contribution

It develops novel sum rules for scale anomalies in CFTs in both Euclidean and Minkowski spaces, highlighting differences in positivity and applicability.

Findings

Weights in position space are not positive.

Weights in momentum space are positive due to the optical theorem.

Explicit check in 8D free field theory confirms the formalism.

Abstract

This paper presents two methods to compute scale anomaly coefficients in conformal field theories (CFTs), such as the c anomaly in four dimensions, in terms of the CFT data. We first use Euclidean position space to show that the anomaly coefficient of a four-point function can be computed in the form of an operator product expansion (OPE), namely a weighted sum of OPE coefficients squared. We compute the weights for scale anomalies associated with scalar operators and show that they are not positive. We then derive a different sum rule of the same form in Minkowski momentum space where the weights are positive. The positivity arises because the scale anomaly is the coefficient of a logarithm in the momentum space four-point function. This logarithm also determines the dispersive part, which is a positive sum over states by the optical theorem. The momentum space sum rule may be invalidated by UV and/or IR divergences, and we discuss the conditions under which these singularities are absent. We present a detailed discussion of the formalism required to compute the weights directly in Minkowski momentum space. A number of explicit checks are performed, including a complete example in an 8-dimensional free field theory.