Scale anomalies, states, and rates in conformal field theory

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...

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Bibliographic Details
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
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Summary: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.
ISSN:1029-8479
1029-8479
DOI:10.1007/JHEP04(2017)171