2026/07/20 by Emil Mottola
#hep-th #gr-qc
A compact form of the stress tensor ⟨ TabTcd⟩ correlation function of a quantum scalar field of arbitrary mass m and curvature coupling ξ is presented, with particular emphasis on decoupling in the m→∞ limit and infrared origin of the conformal anomaly as m → 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 ⟨ TabTcd⟩ 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 ξ≠ 1/6, but can be simply amended to do so. Particular interest attaches to the ξ=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 m2 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 δ-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.