Scale anomalies, states, and rates in conformal field theory

• Published in: The journal of high energy physics 2017-04, Vol.2017 (4), p.1-33, Article 171
• Main Authors: Gillioz, Marc, Lu, Xiaochuan, Luty, Markus A.
• Format: Article
• Language: English
• Subjects: Anomalies; Anomalies in Field and String Theories; Classical and Quantum Gravitation; Conformal Field Theory; Elementary Particles; Euclidean geometry; Euclidean space; Field theory; Formalism; High energy physics; Lattice theory; Logarithms; Mathematical analysis; Mathematical models; Momentum; Operators; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Sum rules
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP04(2017)171

A bstract 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.