Trace anomaly, massless scalars, and the gravitational coupling of QCD

The anomalous effective action describing the coupling of gravity to a non-Abelian gauge theory can be determined by a variational solution of the anomaly equation, as shown by Riegert long ago. It is given by a nonlocal expression, with the nonlocal interaction determined by the Green’s function of a conformally covariant operator of fourth order. In recent works it has been shown that this interaction is mediated by a simple pole in an expansion around a Minkowski background, coupled in the infrared in the massless fermion limit. This result relies on the local formulation of the original action in terms of two auxiliary fields, one physical scalar and one ghost, which take the role of massless composite degrees of freedom. In the gravity case, the two scalars have provided ground in favor of some recent proposals of an infrared approach to the solution of the dark energy problem, entirely based on the behavior of the vacuum energy at the QCD phase transition. As a test of this general result, we perform a complete one-loop computation of the effective action describing the coupling of a non-Abelian gauge theory to gravity. We confirm the appearance of an anomaly pole which contributes to the trace part of the $TJJ$ correlator and of extra poles in its trace-free part, in the quark and gluon sectors, describing the coupling of the energy-momentum tensor ($T$) to two non-Abelian gauge currents ($J$).

Figure 1

The diagrams describing the anomaly pole in the dispersive approach. Figure (a) depicts the singularity of the spectral density $\rho (s)$ as a space-time process. Figure (b) describes the anomalous pole part of the interaction via the exchange of a pole.

Figure 2

The fermionic contributions with a graviton $h_{\mu \nu }$ in the initial state and two gluons $A_{\alpha }^a$, $A_{\beta }^b$ in the final state.

Figure 3

The gauge contributions with a graviton $h_{\mu \nu }$ in the initial state and two gluons $A_{\alpha }^a$, $A_{\beta }^b$ in the final state.

Figure 4

The ghost contributions with a graviton $h_{\mu \nu }$ in the initial state and two gluons $A_{\alpha }^a$, $A_{\beta }^b$ in the final state.

Figure 5

Higher order contributions to the anomaly pole involved in the covariantization of the graviton/two-gluon amplitude.