Graviton Vertices and the Mapping of Anomalous Correlators to Momentum Space for a General Conformal Field Theory

TL;DR

This work reconciles position-space conformal Ward identities with momentum-space perturbative calculations for the $TOO$, $TVV$, and $TTT$ correlators. By computing one-loop free-field realizations in general $d$, it identifies the independent tensor structures and their coefficients, establishing an explicit inverse mapping between Osborn–Petkou results and diagrammatic momentum-space amplitudes. It clarifies how trace anomalies arise both from counterterms in dimensional regularization and from finite parts of renormalized amplitudes, and demonstrates the appearance of anomaly poles in $TVV$ while detailing the renormalization of $TTT$ in $d=4$. The paper then introduces a general, diagram-independent algorithm based on differential regularization and tensor-uniqueness to transform massless configuration-space correlators into well-defined momentum-space expressions, applicable to a broad class of correlators. The results provide a consistent bridge between conformal field theory in position space and perturbative quantum field theory in momentum space, with implications for the anomaly action and potential dilaton-like degrees of freedom in gravity–gauge interactions.

Abstract

We investigate the mapping of conformal correlators and of their anomalies from configuration to momentum space for general dimensions, focusing on the anomalous correlators $TOO$, $TVV$ - involving the energy-momentum tensor $(T)$ with a vector $(V)$ or a scalar operator ($O$) - and the 3-graviton vertex $TTT$. We compute the $TOO$, $TVV$ and $TTT$ one-loop vertex functions in dimensional regularization for free field theories involving conformal scalar, fermion and vector fields. Since there are only one or two independent tensor structures solving all the conformal Ward identities for the $TOO$ or $TVV$ vertex functions respectively, and three independent tensor structures for the $TTT$ vertex, and the coefficients of these tensors are known for free fields, it is possible to identify the corresponding tensors in momentum space from the computation of the correlators for free fields. This works in general $d$ dimensions for $TOO$ and $TVV$ correlators, but only in 4 dimensions for $TTT$, since vector fields are conformal only in $d=4$. In this way the general solution of the Ward identities including anomalous ones for these correlators in (Euclidean) position space, found by Osborn and Petkou is mapped to the ordinary diagrammatic one in momentum space. We give simplified expressions of all these correlators in configuration space which are explicitly Fourier integrable and provide a diagrammatic interpretation of all the contact terms arising when two or more of the points coincide. We discuss how the anomalies arise in each approach [...]

Figures (6)

• Figure 1: The fermion and the scalar sectors contributing to the conformal VVV vertex in any dimension.
• Figure 2: One loop expansion of the 3-graviton vertex. Shown here are the general topologies, i.e. the triangle and the self-energy type (T-bubble) contributions for the fermion case. The general correlator for any CFT in $d=4$ can be obtained by adding to these diagrams similar ones where the fermion is replaced by a scalar and a photon in the loops. Ghost corrections follow the same topologies.
• Figure 3: List of the vertices used in the Lagrangian mapping of the conformal correlators
• Figure 4: The fermion/scalar sectors in the TVV vertex.
• Figure 5: $TTT$ and its counterterms generated with the choice of the square of the Weyl ($F$) tensor in 4 dimensions and the Euler density ($G$).