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Anomalous Gravitational TTT Vertex, Temperature Inhomogeneity, and Pressure Anisotropy

The conformal anomaly in curved spacetime generates a nontrivial anomalous vertex, given by the three-point correlation function $TTT$ of the energy-momentum tensor $T^{μν}$. We show that a temperature inhomogeneity in a gas of charged massless particles generates, via the $TTT$ vertex, a pressure anisotropy with respect to the axis of the temperature variation. This very particular signature may provide an experimental access to the elusive gravitational coefficient $b$ which determines the anomaly contribution of the Weyl tensor to the trace of the energy-momentum tensor in curved spacetime. We present an estimate of the pressure anisotropy both for fermionic quasiparticles in the solid-state environment of Dirac semimetals as well as for a quark-gluon plasma in relativistic heavy-ion collisions. In both cases, the pressure anisotropy is small compared to the mean thermal pressure.