Abstract
4D CFTs have a scale anomaly characterized by the coefficient c, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for c in terms of TT mathcal{O} OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the 〈TTTT〉 correlation functions that contribute. The finiteness of these limits is checked explicitly for free scalar, fermion, and vector CFTs. The sum rule gives c as a sum of positive terms, and therefore implies a lower bound on c given any lower bound on TT mathcal{O} OPE coefficients. We compute the coefficients to the sum rule for arbitrary operators of spin 0 and 2, including the energy-momentum tensor.
Highlights
In this paper we consider two related problems in 4D conformal field theory (CFT)
The first is the computation of physical rates for processes involving particles coupled to a CFT defined by its operator spectrum and operator product expansion (OPE) coefficients. (This has been studied in the phenomenology literature as “unparticle physics” [1, 2].) Here we assume that the coupling of the ordinary particles to the CFT is sufficiently weak that it does not affect the dynamics of the CFT
We first review the properties of the pseudo-amplitudes that enter into the sum rule
Summary
In this paper we consider two related problems in 4D conformal field theory (CFT). The first is the computation of physical rates for processes involving particles coupled to a CFT defined by its operator spectrum and operator product expansion (OPE) coefficients. (This has been studied in the phenomenology literature as “unparticle physics” [1, 2].) Here we assume that the coupling of the ordinary particles to the CFT is sufficiently weak that it does not affect the dynamics of the CFT. (This has been studied in the phenomenology literature as “unparticle physics” [1, 2].) Here we assume that the coupling of the ordinary particles to the CFT is sufficiently weak that it does not affect the dynamics of the CFT. This is similar in spirit to the study of electromagnetism as a probe of QCD, for example in processes like e+e− → hadrons or deep inelastic scattering. The final result of this work is eq (1.13), a sum rule that gives c as a positive sum over T T O OPE coefficients, for all primary operators O other than 1 and T μν itself
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