Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT

TL;DR

The paper derives a positive sum rule for the 4D CFT Weyl c anomaly by examining graviton–graviton scattering inside a CFT, reframing the problem in terms of a momentum-space TT four-point function and an optical-theorem–like relation. It expresses c as a weighted sum over TT$ o ext{O}$ OPE data with momentum-space conformal blocks computed via conformal Ward identities, focusing on spin-0 and spin-2 intermediate operators. The authors verify IR finiteness and normalization in free theories (scalar, fermion, vector), including an explicit free-scalar example that demonstrates the block structure and convergence. They discuss the bootstrap-like implications, potential holographic interpretations, and the need to understand IR subtleties more fully, outlining avenues for extending the approach to other anomalies and dimensions.

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 $\langle TTTT \rangle$ 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.

Figures (8)

• Figure 1: Contributions to the physical graviton-graviton scattering amplitude at order $1/M_\text{Planck}^4$. Curvy double lines denote gravitons, and the blob denotes a correlation function of energy-momentum tensors in the CFT. The pseudo-amplitude corresponds to the contribution (a). The contributions (b) and (c) are important for unitarity of the physical graviton amplitude, but are not included in the pseudo-amplitude.
• Figure 2: Numerical value of the conformal blocks \ref{['eq:fscalar']} for an operator of spin zero and \ref{['eq:conformalblock:spin2']} for spin two, showing the eigenvalues of the $2 \times 2$ matrix in the latter case. The dashed lines correspond to the amplitude of the asymptotic expressions \ref{['eq:conformalblock:scalar:asymptotics']} and \ref{['eq:conformalblock:spin2:asymptotics']} with the $\sin^2$ factored out.
• Figure 3: Upper bound on the size of the OPE coefficient for a scalar operator of dimension $\Delta$ entering the $T \times T$ OPE, for a given value of the $c$ anomaly. The bound is weaker for operators of dimension close to 2. This can be alternatively be understood as a lower bound on $c$ for a fixed value of the OPE coefficient.
• Figure 4: Feynman diagrams for the three-point function $\langle T^{\beta_1\beta_2} | \text{T}[ \tilde{T}_i(p_1) \tilde{T}_j(p_2)] | 0 \rangle$ in the free scalar theory. Here we use straight double lines for $T$ operators and dashed singled lines for scalars. Red lines indicate Wightman propagators, black lines ordinary Feynman propagators.
• Figure 5: Feynman diagram for two-point function $\langle 0 | \tilde{T}^{\alpha_1\alpha_2}(-p_1-p_2) \tilde{T}^{\beta_1\beta_2}(p_1+p_2) | 0 \rangle$ in the free scalar theory. Straight double lines are for the $T$ operator and dashed singled lines for scalars. Red lines indicate Wightman propagators.