A compact form of the stress-tensor $⟨T^{ab}{T}^{cd}⟩$ 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 the IR 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 $⟨T^{ab}{T}^{cd}⟩$ 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 $\xi \ne 1/6$, but can be simply amended to do so. Particular attention is given 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 $\square R$ term in the conformal anomaly 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.

Close

PrevNext

Reuse & PermissionsView figure in article

• 
• 
• 
• 
• 

• Anomalies
• Effective field theory
• Quantum fields in curved spacetime
• Sum rules