Eikonal phase matrix, deflection angle and time delay in effective field theories of gravity

The eikonal approximation is an ideal tool to extract classical observables in gauge theory and gravity directly from scattering amplitudes. Here we consider effective theories of gravity where in addition to the Einstein-Hilbert term we include non-minimal couplings of the type $R^3$, $R^4$ and $FFR$. In particular, we study the scattering of gravitons and photons of frequency $\omega$ off heavy scalars of mass $m$ in the limit $m\gg \omega \gg |\vec{q}\,|$, where $\vec{q}$ is the momentum transfer. The presence of non-minimal couplings induces helicity-flip processes which survive the eikonal limit, thereby promoting the eikonal phase to an eikonal phase matrix. We obtain the latter from the relevant two-to-two helicity amplitudes that we compute up to one-loop order, and confirm that the leading-order terms in $\omega$ exponentiate \`{a} la Amati, Ciafaloni and Veneziano. From the eigenvalues of the eikonal phase matrix we then extract two physical observables, to 2PM order: the classical deflection angle and Shapiro time delay/advance. Whenever the classical expectation of helicity conservation of the massless scattered particle is violated, i.e. the eigenvalues of the eikonal matrix are non-degenerate, causality violation due to time advance is a generic possibility for small impact parameter. We show that for graviton scattering in the $R^4$ and $FFR$ theories, time advance is circumvented if the couplings of these interactions satisfy certain positivity conditions, while it is unavoidable for graviton scattering in the $R^3$ theory and photon scattering in the $FFR$ theory. The scattering processes we consider mimic the deflection of photons and gravitons off spinless heavy objects such as black~holes.