Eikonal Methods and High-Energy Deflection

Eikonal methods trade a small momentum transfer for a large impact parameter. In that representation, repeated long-range exchanges exponentiate, and the rapidly varying phase acts as a generating function for classical deflection. The method is especially effective when the center-of-mass energy is high but the angle remains small. Those two conditions are independent: high energy strengthens gravity, so the impact parameter must grow with the gravitational radius of the total energy. A real elastic phase gives conservative deflection; radiation, spin, and nonintegrable motion require more structure than that phase alone.

Required background. Events, Intervals, and Minkowski Geometry supplies high-energy invariant kinematics, and Weak Metric Perturbations and Gauge Freedom supplies the weak-field and small-transfer regime.

Helpful background. The Einstein Equation fixes the nonlinear interaction; Presymplectic Potentials and Currents supplies the generating-function viewpoint; and Gravitational Radiation and Waveforms is needed when the elastic channel is not closed.

Impact-parameter phase

Let $\boldsymbol q$ be the momentum transfer to body 1 in the center-of-mass frame. In elastic kinematics it is perpendicular to the mean of that body’s incoming and outgoing spatial momenta. At leading small angle this is the incoming transverse plane. Define the elastic eikonal phase by

$$i\mathcal M_{\rm eik}(s,\boldsymbol q) =4E p_\infty \int d^2\boldsymbol b\, e^{-i\boldsymbol q\cdot\boldsymbol b} \left(e^{2i\delta(s,b)}-1\right), \qquad E=\sqrt{s}.$$

Factors of $2$, $i$, and the flux $4Ep_\infty$ vary in the literature; they must be fixed before comparing phases. The observable relation below is unambiguous once this definition and the direction of $\boldsymbol b$ are fixed. In the classical stationary-phase limit, the exponent is

$$-\boldsymbol q\!\cdot\boldsymbol b+2\delta(s,b),$$

so its saddle obeys

$$\boldsymbol q =\frac{\partial\,2\delta}{\partial\boldsymbol b}.$$

Choose $\boldsymbol b$ to point from body 2 toward body 1 at leading order. Then an attractive interaction requires $\partial_b\delta\lt0$ in this Fourier convention. The exact elastic transfer magnitude obeys

$$\lvert\boldsymbol q\rvert =2p_\infty\sin\frac{\chi}{2}.$$

At leading PM order, the positive tree amplitude on Tree Amplitudes and Leading Gravitational Deflection and its Fourier transform fix the sign:

$$2\delta_1=\frac1{4Ep_\infty} \int\frac{d^2\boldsymbol q}{(2\pi)^2} e^{i\boldsymbol q\cdot\boldsymbol b}\mathcal M_{\rm tree}.$$

In two transverse dimensions, with an infrared regulator and $b\ne0$,

$$\int\frac{d^2\boldsymbol q}{(2\pi)^2} \frac{e^{i\boldsymbol q\cdot\boldsymbol b}}{\boldsymbol q^2} =-\frac1{2\pi}\log\!\left(\frac b{b_0}\right).$$

Consequently,

$$2\delta_1(s,b) =- \frac{2Gm_1m_2(2\gamma^2-1)} {\sqrt{\gamma^2-1}} \log\!\left(\frac{b}{b_0}\right).$$

Its derivative reproduces the attractive 1PM impulse. In the notation of Impulse and Momentum Transfer, $2\delta_1=-W_1$; the positive $W_1$ is a reduced interaction function, not the phase in $S=e^{2i\delta}$. The radial action gives the same positive angle through $-\partial_J I_r^{\rm scatt}$. Changing the infrared length $b_0$ adds a $b$-independent phase and cannot alter that angle.

Beyond leading order, distinguish the Fourier-conjugate impact parameter $b$ from the incoming one $b_J=J/p_\infty$. For elastic planar motion, $b_J=b\cos(\chi/2)$; the difference starts at relative $O(G^2)$ and can affect a 3PM angle. The Fourier normalization and this geometric conversion are given in Di Vecchia et al. 2021, § 2.2, pp. 10–11, eqs. (2.22)–(2.36), PDF.

What exponentiation accomplishes

Expanding $e^{2i\delta}$ generates powers of lower-order exchange. Those terms are the impact-parameter representation of classical trajectory iterations and quantum Born iterations. They must be separated from genuinely new PM information at a given loop order. Exponentiation therefore organizes the answer and enforces important consequences of unitarity, but it does not by itself compute the new coefficient $\delta_n$.

The classical saddle requires a large orbital angular momentum,

$$J=b p_\infty\gg\hbar ,$$

with $\hbar$ temporarily restored. Small momentum transfer requires

$$-t=\boldsymbol q^2\ll s, \qquad \chi\ll1.$$

These conditions allow the sum over partial waves to be approximated by an impact-parameter integral. They also distinguish the eikonal limit from an arbitrary high-energy fixed-angle collision. Porto derives the gravitational eikonal approximation from the amplitude and relates the impact-parameter phase to classical propagation in Porto 2016, § 2.7, p. 14.

Ultrarelativistic gravitational bending

When $\gamma\gg1$ at fixed masses,

$$s=m_1^2+m_2^2+2m_1m_2\gamma \simeq2m_1m_2\gamma,$$

and the magnitude of the coefficient of $\log b$ approaches $2Gs$. Thus

$$2\delta_1\simeq-2Gs\log\!\left(\frac{b}{b_0}\right), \qquad \lvert\boldsymbol q\rvert\simeq\frac{2Gs}{b}.$$

Since $p_\infty\simeq\sqrt{s}/2$,

$$\chi_{\rm eik}^{(1)} \simeq\frac{4G\sqrt{s}}{b}.$$

The controlling gravitational radius is therefore $G\sqrt{s}$, not $Gm_1$ or $Gm_2$ separately. Weak deflection requires $b\gg G\sqrt{s}$. In the probe limit, where a null projectile scatters from a much heavier mass $M$ without appreciable recoil, the same relation becomes the familiar $\chi=4GM/b$.

This leading universality should not be overextended. Subleading terms can depend on mass, spin, tidal structure, the order in which the massless limit is taken, and radiation. Di Vecchia and collaborators show how the elastic eikonal and radiative cuts combine through third PM order and discuss the high-energy limit in Di Vecchia et al. 2021, §§ 2–4 and 6.

Elastic phase, generator, and radiation

For planar integrable conservative scattering, the classical eikonal phase, the on-shell action, and a scattering generator are closely related—often by a Legendre transformation. They are not synonymous in general. A 2026 analysis proves that the phase of an $S$-matrix element tends to the classical on-shell action, whereas the classical limit of the operator logarithm of the full $S$-matrix generates the canonical scattering map; the two are inequivalent for generic dynamics and coincide only in special integrable cases Kim, Patil, Scheopner, and Steinhoff 2026, §§ 1–3 and § 5.3.

Radiation makes the distinction operational. Probability leaves the two-body elastic channel, so a single real phase cannot encode the emitted waveform and momentum. One may introduce an inelasticity or enlarge phase space to include radiation modes. A radiation-eikonal construction has reproduced the leading waveform, radiated momentum, and radiation-loss part of the impulse, but its demonstrated domain is a controlled leading-order PM test rather than a universal replacement for retarded dynamics Kim 2025, §§ III–VIII.

Domain and breakdown

The eikonal approximation is valid for large $J/\hbar$, small $-t/s$, weak deflection, and an impact parameter large compared with gravitational radii and relevant finite-size scales. It becomes unreliable near a capture threshold, near a strong-field critical orbit, or when multiple saddles and caustics compete. If the radiated fraction is not perturbatively small, the elastic saddle alone is incomplete. For extended bodies, tidal excitation and absorption add channels even before capture.

There are also order-of-limits hazards. Taking $\gamma\to\infty$ inside a formula derived for timelike worldlines need not agree with a calculation that starts with null sources. Infrared-divergent phases may cancel from impulses while still affecting how soft radiation states are defined. Finally, an exponentiated answer may be unitary and still use the wrong classical iteration subtraction. Independent checks should therefore include the mass shell, momentum balance, the low-energy test-body limit, and agreement between the phase derivative and a direct force or radial-action calculation.

Exercises

Derive the saddle-point deflection

With the Fourier convention used in the first equation, take $2\delta=-A\log(b/b_0)$ with $A\gt0$ and derive the momentum transfer.

Solution

Stationary phase gives

$$\boldsymbol q =\frac{\partial\,2\delta}{\partial\boldsymbol b} =-A\frac{\boldsymbol b}{b^2}.$$

Thus $\lvert\boldsymbol q\rvert=A/b$, and the vector points toward body 2. The sign also follows independently from $\nabla_b^2[-\log b/(2\pi)]=-\delta^{(2)}(\boldsymbol b)$, the defining Green-function equation for the transform of $1/\boldsymbol q^2$. At leading small angle, $b=b_J+O(\chi^2b)$ and

$$\chi=\frac{\lvert\boldsymbol q\rvert}{p_\infty} +O(\chi^3) =\frac{A}{b p_\infty}+O(\chi^3) =\frac{A}{J}+O(\chi^3).$$

This matches $-\partial_J[-A\log(J/J_0)]$ from the radial action.

Find the high-energy validity parameter

Show that the leading ultrarelativistic angle is small precisely when $G\sqrt{s}/b\ll1$.

Solution

At high energy, $\lvert\boldsymbol q\rvert=2Gs/b$ and $p_\infty=\sqrt{s}/2$. Hence

$$\chi\simeq\frac{\lvert\boldsymbol q\rvert}{p_\infty} =\frac{4G\sqrt{s}}{b}.$$

Thus $\chi\ll1$ is equivalent, up to the numerical factor four, to $G\sqrt{s}/b\ll1$. Raising the energy at fixed $b$ eventually violates the eikonal weak-deflection domain even though $b$ itself has not changed.

References

• Di Vecchia, Paolo, Carlo Heissenberg, Rodolfo Russo, and Gabriele Veneziano. “The Eikonal Approach to Gravitational Scattering and Radiation at $\mathcal O(G^3)$.” Journal of High Energy Physics 2021, 169 (2021). DOI. Open PDF.
• Kim, Jung-Wook, Raj Patil, Trevor Scheopner, and Jan Steinhoff. “Magnusian: Relating the Eikonal Phase, the On-Shell Action, and the Scattering Generator.” Journal of High Energy Physics 2026, 241 (2026). DOI. Open PDF.
• Kim, Jung-Wook. “Radiation Eikonal for Post-Minkowskian Observables.” Physical Review D 111, L121702 (2025). DOI. Open PDF.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633, 1–104 (2016). DOI. Open PDF.