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

TL;DR

The work develops an eikonal phase matrix framework to extract classical observables from scattering amplitudes in gravity with higher-derivative corrections. By computing two-to-two helicity amplitudes at tree and one-loop order across EH, EH+R^3, EH+R^4, and FFR theories, the authors obtain deflection angles and Shapiro time delays to 2PM, including helicity-flip effects that promote the eikonal to a matrix. They show the leading ω terms exponentiate and analyze causality by examining eigenvalues of the eikonal matrix, finding potential time advances for small impact parameter unless positivity or UV-completion constraints are satisfied in certain channels (R^4 and FFR for gravitons/photons), while R^3 generically yields causality issues unless completed. The results illuminate how higher-derivative couplings modify photon/graviton propagation near heavy sources (e.g., black holes) and provide quantitative conditions for avoiding superluminal effects. Together, the paper clarifies the interplay between effective field theory corrections, helicity dynamics, and classical gravitational observables in a coherent amplitude-based framework.

Abstract

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 $ω$ off heavy scalars of mass $m$ in the limit $m\gg ω\gg |\vec{q}\,|$, 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 $ω$ exponentiate à 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.

Figures (3)

• Figure 1: The two-particle cut diagrams for the $R^n$ interaction in the $s=-\vec{q}^{\, \, 2}$-channel. In our conventions external momenta are all outgoing and internal loop momenta flow from left to right in the diagram.
• Figure 2: The two-particle cut diagrams in the $s=-\vec{q}^{\, \, 2}$-channel of the graviton deflection angle in the presence of an ${\rm FFR}$ interaction. The internal lines are photons. The first diagram is proportional to $\kappa^2 \, e^2$ and is only non-vanishing for $h_1=h_2$ for the internal photons. The second diagram is proportional to $\kappa^4$, it is non-vanishing when $h_4 = - h_3$ and $h_2 = - h_1$ thus it contributes solely to the helicity-preserving configuration. Also, it only produces quantum corrections (bubble integrals) with coefficients that vanish in the case of four-dimensional external kinematics.
• Figure 3: The two-particle cut diagrams in the $s=-|\vec{q}\, |^{\, \, 2}$-channel contributing to photon deflection to first order in the $FFR$ interaction. We only show the helicity-flip configuration since the helicity-preserving cuts vanish. The cut diagram on the RHS of the figure only contributes terms which are subleading in the eikonal limit.