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.Comment: 39 pages, 7 figure
