More on the bending of light in quantum gravity

We reconsider the long-range effects of the scattering of massless scalars and photons from a massive scalar object in quantum gravity. At the one-loop level, the relevant quantum mechanical corrections could be sorted into the graviton double-cut contributions, massless-scalar double-cut contributions and photon double-cut contributions. In Reference [N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Planté, and P. Vanhove, Phys. Rev. Lett. 114, 061301 (2015), N. E. J. Bjerrum-Bohr, J. F. Donoghue, B. R. Holstein, L. Planté, and P. VanhoveJ. High Energy Phys. 11 (2016) 117] N. E. J. Bjerrum-Bohr et al. have considered explicitly the implications of the graviton double-cut contributions on the gravitational bending of light and some classical formulations of the equivalence principle, using the modern double-copy constructions and on-shell unitarity techniques. In this article, instead we consider all three contributions and redo the analysis using the traditional Feynman diagrammatic approach. Our results on the graviton double-cut contributions agree with the aforementioned references, acting as a nontrivial check of previous computations. Furthermore, it turns out that the massless-scalar double-cut contributions and the photon double-cut contributions do leave nonvanishing quantum effects on the scattering amplitudes and the gravitational bending of light. Yet, we find that the general structure of the gravitational amplitudes and the quantum discrepancy of the equivalence principle suggested in the aforementioned references remain intact.

Figure 1

Graviton double-cut diagrams for the one-loop scattering between the photon $\gamma$ and the massive scalar $\mathrm{\Phi }$. Unlike Refs. [23, 24], in this article we take the time direction to be horizontal. In the last diagram, we have the graviton FP ghost running in the loop.

Figure 2

Photon double-cut diagrams for the one-loop scattering between the photon $\gamma$ and the massive scalar $\mathrm{\Phi }$.

Figure 3

The one-loop Feynman diagrams for the scattering of the massless scalar $\phi$ from the massive scalar $\mathrm{\Phi }$. The first eleven diagrams constitute the graviton double-cut contributions, while the last two diagrams constitute the massless-scalar double-cut contributions.