Gravitational Vacuum Polarization: Decoupling and the Conformal Anomaly

A compact form of the stress tensor $\langle T^{ab}T^{cd}\rangle$ correlation function of a quantum scalar field of arbitrary mass $m$ and curvature coupling $\xi$ is presented, with particular emphasis on decoupling in the $m\to\infty$ limit and infrared origin of the conformal anomaly as $m \to 0$. The Ward Identity (WI) of covariant conservation is verified for the full second order metric variation of the one-loop effective action, of which $\langle T^{ab}T^{cd}\rangle$ is part, including local contact terms. This WI is satisfied by two tensors, one spin-2 (traceless) and the second spin-0 (traceful), each multiplied by a Lorentz invariant form factor computed in $n$-dimensional regularization. Minimal subtraction of pole terms at $n =4$ produces form factors that fail to satisfy decoupling for general $\xi \neq 1/6$, but can be simply amended to do so. Particular interest attaches to the $\xi=1/6$ case, in which the spin-0 form factor at $n =4$ is completely finite, requires no UV regularization or subtractions, and satisfies decoupling directly. Its imaginary part defines a spectral function that obeys a UV finite sum rule for any $m$, while its real part unambiguously determines the $\square R$ in the massless limit. Its $m^2$ dependence describes a Wilsonian renormalization group flow to an effective field theory limit at large distances. At $m= 0$ the spin-0 spectral function becomes a Dirac $\delta$-function at zero energy, demonstrating the existence of a massless scalar Goldstone collective excitation due to the conformal anomaly in the low energy effective theory of gravity.