Scale Anomalies, States, and Rates in Conformal Field Theory

TL;DR

The paper tackles the problem of expressing scale anomaly coefficients $c_{\mathcal{O}}$ in conformal field theories through data intrinsic to the CFT. It develops two complementary sum-rule formulations: a Euclidean-position-space flux integral yielding a weighted OPE sum with nonpositive weights and a Minkowski momentum-space version where the weights are nonnegative due to the anomaly’s link to a logarithm in the forward amplitude and the optical theorem. The authors derive explicit kernel expressions, address convergence and divergence issues (UV/IR) that constrain validity, and validate the framework with concrete checks, including an exact 8-dimensional free-field example. The momentum-space approach opens avenues for analytic bootstrap bounds and a more physical, state-sum perspective on CFT data, while the Euclidean method provides a broader, though sign-indefinite, handle on the same anomaly. Overall, the work connects conformal data to scale anomalies through practical, testable sum rules and lays groundwork for extensions to other conformal anomalies and spinful operators.

Abstract

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.

Figures (7)

• Figure 1: The integration kernel function $\mathcal{K}_4$ in $d=4$ in the complex $z$ plane. The blue lines separate regions related by crossing-symmetry. The maximum of $\mathcal{K}_4$ occurs at the completely crossing-symmetric point $u = v = 1$.
• Figure 2: The fundamental domain of integration defined by the constraints of Eq. (\ref{['eq:fundamentaldomain']}). Crossing symmetry corresponds to reflections across the line at $\mathop{\rm Re}(z)= \frac{1}{2}$ and the two circles centered at $z=0$ and $z=1$.
• Figure 3: Numerical value of the coefficient $C_\Psi$ of Eq. (\ref{['eq:Euclideansum:2']}) in $d = 4$ as a function of the scaling dimension $\Delta_\Psi$ and spin $\ell_\Psi$ of the intermediate operator $\Psi$. The value of the universal contribution $C_\mathds{1}$ is shown as well for reference.
• Figure 4: The two types of diagrams that contributes to the rate $AA \to \text{CFT}$ in the free scalar theory with $\mathcal{O} = \phi^2$, with final states corresponding to the primary operator $\phi^2$ (left), $\phi^4$ (right), or their descendants.
• Figure 5: The coefficient $C_\Psi$ of Eq. (\ref{['eq:CPsiscalar']}) for a scalar intermediate operator $\Psi$ in $d = 4$. The zeroes of $C_\Psi$ when $\Delta_\Psi = 2\Delta_{\mathcal{O}} + 2n$ are visible, as are the UV divergences at $\Delta_\Psi = d/2$ and $(d-2)/2$. At large $\Delta_\Psi$, the coefficient $C_\Psi$ increases exponentially.