Classical vs Quantum Eikonal Scattering and its Causal Structure

TL;DR

The work analyzes high-energy, large-ℓ gravitational scattering in the eikonal regime, revealing that eikonal exponentiation follows from a group contraction $SU(2)\to ISO(2)$ and continuous-spin ISO$(2)$ representations encode the classical limit. It develops an all-orders eikonal framework, with impact-parameter and momentum-space transforms linking the phase $\delta(s,\mathbf{b})$ to observable deflection and time delay, and demonstrates that quantum (non-gravitational) corrections can dominate over PM terms in transplanckian settings. Through a QED+gravity case (photon–scalar scattering) and a robust causality/dispersive analysis, the paper derives infinite families of non-linear positivity bounds on EFT coefficients via eikonal arcs, connecting analyticity, unitarity, and causality to infrared observables. The findings reveal that quantum corrections are not universally subdominant and provide a precise framework to constrain EFTs of gravity with gauge interactions, offering a path to test causality in quantum-gravitational settings.

Abstract

We study the eikonal scattering of two gravitationally interacting bodies, in the regime of large angular momentum and large center of mass energy. We show that eikonal exponentiation of the scattering phase matrix is a direct consequence of the group contraction $SU(2)\to ISO(2)$, from rotations to the isometries of the plane, in the large angular momentum limit. We extend it to all orders in the scattering angle, and for all masses and spins. The emergence of the classical limit is understood in terms of the continuous-spin representations admitted by $ISO(2)$. We further investigate the competing classical vs quantum corrections to the leading classical eikonal scattering, and find several interesting examples where quantum corrections are more important than Post-Minkowskian's. As a case of study, we analyse the scattering of a photon off a massless neutral scalar field, up to next-to-leading order in the Newton constant, and to leading order in the fine structure constant. We investigate the causal structure of the eikonal regime and establish an infinite set of non-linear positivity bounds, of which positivity of time delay is the simplest.

Figures (5)

• Figure 1: Scale lengths of the system. Bottom: tidal relative corrections $(L_\odot/b)^n$ are largest and dominate the modifications to the leading eikonal. Top: the most important corrections to leading eikonal arise from higher-derivative operators generated by particles of Compton wavelength ${\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}=1/m_e$ that are running in loops, as long as $\alpha {\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}^2\gg R_s$ and $b\gg {\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$. For $b\ll {\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}$ resummation to all orders in ${\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}/b$ is needed, corresponding to work with a new EFT where new degress of freedom are propagating. A typical example of $\delta\theta/\theta$, to first order in the coupling constant, is $\sim \alpha \log^2 {\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}/b$ in this regime.
• Figure 2: Pictorial representation of the R.H.S. of Eq. \ref{['eq:ImM1']}. In this picture, the grey blobs represent effective vertices of order $\alpha$. Recall that with in-coming momenta $p_1^\mu + p_3^\mu = q^\mu$, and $s = (p_1 + p_2)^2$.
• Figure 3: Topologies arising from the contraction ${\cal M}_{5\,\lambda_1}{\cal M}^*_{5\,\lambda_3}$. One must add the symmetric contributions w.r.t. the cut for (d), (e) and (f).
• Figure 4: The eight MIs needed to solve the two-loop integral. (a) -- (e) are the planar MIs, while (a) -- (c) and (f) --(h) are the non planar one. A dot on the propagator line means that it is squared in the integral.
• Figure 5: Thick blue lines represent the contour integral defining the arc \ref{['eq:arcDef']}. Lighter blue lines correspond to the contour deformation giving rise to the UV representation \ref{['arcDisc1']}. Orange lines on the real axis represent $s$ and $u$ channel branch-cuts.