Perturbative Eikonal in Scattering

• Perturbative eikonal is a high-energy, small-angle scattering framework where soft, long-range exchanges exponentiate into an impact-parameter phase.
• It generates the classical scattering map by linking amplitude exponentiation to observable quantities such as deflection angles and Shapiro time delays.
• The method extends across flat and curved spacetimes, incorporating sub-eikonal, spin, and helicity corrections in contexts like perturbative gravity and high-energy QCD.

Perturbative eikonal denotes the high-energy, small-angle organization of scattering amplitudes in which soft long-range exchange dominates and ladder or crossed-ladder contributions exponentiate into an impact-parameter phase; in parallel formulations, the same object appears as a classical scattering generator. In current amplitude theory this structure is developed in flat-space and curved-space gravity, near-horizon black-hole scattering, worldline formalisms, helicity-mixing effective theories, high-energy QCD, celestial amplitudes, and even breakup reaction theory (Ajith et al., 2024, Adamo et al., 2021, Groenenboom, 10 Dec 2025).

1. Kinematic regime and basic representation

The standard perturbative eikonal regime is the high-energy, small-angle limit with soft momentum transfer. In relativistic scattering this is usually expressed as $-t \ll s$, or equivalently by large impact parameter and small transverse momentum exchange. In the post-Minkowskian setting one keeps classical long-range contributions while discarding hard short-distance terms; in the $\hbar$-expansion one holds external massive momenta fixed while exchanged mediator momenta and momentum transfer are soft. Massive propagators then reduce to their particle poles,

$$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$

and the amplitude exponentiates in impact-parameter space (Ajith et al., 2024).

A standard form of the eikonal amplitude is

$$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$

with $\chi(\mathbf{b})$ or $\delta(\mathbf{b})$ denoting the eikonal phase. In loop language one writes

$$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$

with $1{\rm PM}={\rm tree}$, $2{\rm PM}={\rm one\ loop}$, and $3{\rm PM}={\rm two\ loops}$. This impact-parameter representation is the common backbone of perturbative gravity, electrodynamics, and several Wilson-line formulations (Adamo et al., 2021, Vecchia et al., 2021).

2. Phase, exponentiation, and the classical generator

The perturbative eikonal is not only an exponentiated phase in the amplitude; it is also treated as the generator of the classical scattering map. In the classical interaction-picture formulation, the eikonal $\hbar$0 is defined through the canonical transformation from in-data to out-data, and observables evolve as

$\hbar$1

This formulation accommodates both Hamiltonian deformation and symplectic deformation, treats particles and fields on a similar footing, and makes the causality prescription of the propagator essentially the same for non-relativistic and relativistic kinematics (Kim et al., 2 Sep 2025).

The same content is reorganized in worldline quantum field theory. There, the amplitude in impact-parameter space obeys

$\hbar$2

with $\hbar$3 collecting quantum remainders and $\hbar$4 the irreducible classical eikonal. Reducible contributions exponentiate into products such as $\hbar$5, whereas minimally connected worldline diagrams generate $\hbar$6 order by order. This establishes an explicit equivalence between the QFT eikonal and WQFT organizations of the classical limit (Ajith et al., 2024).

Classical observables are extracted directly from the eikonal. The transverse impulse and scattering angle follow from gradients in impact-parameter space,

$\hbar$7

while the Shapiro time delay is obtained from the energy derivative of the real phase,

$\hbar$8

This derivative-based extraction is one reason the eikonal serves as a direct bridge between amplitudes and observables (Vecchia et al., 2022).

3. Post-Minkowskian gravity, radiation, and limits of the PM expansion

In perturbative gravity the eikonal phase encodes classical conservative dynamics at leading orders and, starting at $\hbar$9, radiation-reaction effects as well. A central result of the $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$0 program is that one must retain the full soft region rather than only its potential sub-region: the physical deflection angle includes radiation-reaction contributions, and these are essential for canceling spurious high-energy logarithms and restoring a finite universal ultra-relativistic limit. In this sense, the perturbative eikonal is not exhausted by conservative ladder exchange; it also organizes dissipative information once unitarity cuts are included (Vecchia et al., 2021).

This extension is sharpened by promoting the elastic c-number phase to a soft-radiation operator. In the soft-radiation limit one writes a unitary operator dressing the elastic eikonal with graviton creation and annihilation operators, and the vacuum expectation value generates an imaginary part of the eikonal phase. The resulting framework relates the zero-frequency limit of the radiated spectrum to the infrared-damping of the elastic amplitude and yields Poisson statistics for emitted soft quanta. The same analysis identifies a bound on the validity of the PM expansion: once $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$1, collinear soft radiation becomes sensitive to the $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$2-dependent classical deflection, the usual PM series breaks down, and a different expansion with non-analytic dependence on Newton’s constant is required (Vecchia et al., 2022).

A complementary development treats the eikonal as a scattering generator and generalizes it to a radiation eikonal by including mediator field degrees of freedom. In that language, the waveform, radiated momentum, and the time-asymmetric radiation-loss contribution to the impulse arise from nested Poisson brackets involving $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$3 and the retarded-minus-advanced graviton bracket. This construction reproduces the known leading-order waveform, the radiated momentum, and the missing radiation-loss contribution to the $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$4 impulse (Kim, 13 Jan 2025).

At next-to-leading power in high-energy perturbative gravity, unequal-mass scalar scattering provides a particularly explicit result: in four dimensions, only gravitational corrections contribute to the exponentiated phase in impact-parameter space, and the leading saddle-point impact parameter is shifted by a multiple of the Schwarzschild radius independent of the light-particle energy. This sharpened the statement that the first power correction is suppressed by a single power of the ratio of momentum transfer to the light-particle energy (Akhoury et al., 2013).

4. Curved backgrounds and the black-hole eikonal phase

Perturbative eikonal methods extend beyond flat-space two-body scattering. For a scalar on a stationary curved background, the on-shell action reduces to a boundary term at spatial infinity,

$$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$5

and the associated $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$6 amplitude is obtained from the bilinear part of that boundary term. In the eikonal/WKB limit the outgoing wave takes the form $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$7 with

$$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$8

so the amplitude again reduces to an exponentiated impact-parameter phase. Applied to Schwarzschild, this reproduces the known massive-scalar eikonal. Applied to Kerr, it yields exponentiation together with a Kawai-Lewellen-Tye-like factorization in momentum space (Adamo et al., 2021).

Near a Schwarzschild horizon, the perturbative eikonal acquires a specifically black-hole form. In Kruskal–Szekeres coordinates with $$\frac{1}{(p_a + k)^2 + m_a^2 - i\epsilon} \;\xrightarrow[\text{eikonal}]{\hbar\to 0}\; \frac{1}{2 p_a\!\cdot\!k \pm i\epsilon},$$9, the black-hole eikonal phase is defined by

$$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$0

and spherical harmonics on $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$1 induce an effective angular mass $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$2. The basic transverse Green’s function satisfies

$$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$3

with $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$4 for graviton exchange and $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$5 for photon exchange. The elastic $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$6 amplitude exponentiates into

$$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$7

and the phase contains both $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$8 and $$A_{\rm eik}(s,t)=2s\int\mathrm{d}^2\mathbf{b}\;e^{i\,\mathbf{q}_\perp\cdot\mathbf{b}}\left(e^{i\,\chi(\mathbf{b})}-1\right),\qquad t=-\mathbf{q}_\perp^2,$$9 contributions. The formalism resums all partial waves, restores transverse-separation effects, treats particle masses correctly in the pole structure, and extends the eikonal amplitude to arbitrarily many particles in both flat space and on the black-hole background (Groenenboom, 10 Dec 2025).

The many-particle Schwarzschild result is

$\chi(\mathbf{b})$0

and matches precisely the horizon $\chi(\mathbf{b})$1-matrix derived by ’t Hooft when written in terms of momentum and charge distributions on the sphere. Within the stated approximations, this is identified as the most general elastic contribution achievable in the eikonal phase (Groenenboom, 10 Dec 2025).

5. Spin, helicity, higher-derivative corrections, and sub-eikonal structure

Once spin or non-minimal couplings are included, the perturbative eikonal need not remain a scalar phase. In effective gravity with $\chi(\mathbf{b})$2, $\chi(\mathbf{b})$3, and $\chi(\mathbf{b})$4 interactions, helicity-flip processes survive the eikonal limit and promote the phase to a $\chi(\mathbf{b})$5 matrix in helicity space. Its eigenvalues $\chi(\mathbf{b})$6 determine the deflection angles and Shapiro time delays or advances through

$\chi(\mathbf{b})$7

This framework isolates a concrete causality issue: whenever the eigenvalues are non-degenerate, one eigenchannel can exhibit time advance at small impact parameter. The data show that time advance is unavoidable for graviton scattering in the $\chi(\mathbf{b})$8 theory and for photon scattering in the $\chi(\mathbf{b})$9 theory, while for graviton scattering in the $\delta(\mathbf{b})$0 and $\delta(\mathbf{b})$1 theories it is avoided if $\delta(\mathbf{b})$2, $\delta(\mathbf{b})$3, or $\delta(\mathbf{b})$4, respectively (Huber et al., 2020).

Spinning observables fit naturally into the eikonal formalism through the KMOC framework. A key ingredient is the covariant impact parameter

$\delta(\mathbf{b})$5

which packages the polarization-dependent shift induced by generic spin. At leading order the momentum impulse and spin kick are

$\delta(\mathbf{b})$6

and the paper derives next-to-leading-order covariant formulas for the linear-in-spin impulse and spin kick, matching previous QFT and worldline results even when no spin supplementary condition is imposed (Gatica, 2023).

Beyond leading eikonal order, high-energy QCD replaces scalar impact-parameter phases by Wilson lines. The projectile crosses a shockwave background and acquires the lightlike Wilson line

$\delta(\mathbf{b})$7

Sub-eikonal corrections are organized by field-strength and covariant-derivative insertions between Wilson lines, yielding gauge-invariant operators that carry spin dependence. This is the mechanism needed for polarized small-$\delta(\mathbf{b})$8 observables and for extending BK/JIMWLK-type evolution beyond pure eikonal accuracy (Chirilli, 2018).

Higher-derivative renormalizable gravity provides another non-GR correction to the eikonal. There the modified propagator produces Yukawa-type $\delta(\mathbf{b})$9 terms, and in certain pole structures Struve–Bessel combinations $$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$0, in the impact-parameter phase. Different from the GR case, the next-to-leading eikonal order is generically non-vanishing and requires explicit Riesz and Hadamard regularization (Lanosa et al., 20 Apr 2025).

6. Celestial, QCD, and reaction-theory implementations

Eikonal exponentiation has a distinctive role in celestial amplitudes because the ordinary Mellin transform of perturbative gravitational amplitudes is generically divergent. Dressing the amplitude by the oscillating eikonal phase produces celestial amplitudes that are analytic apart from poles at integer negative conformal dimensions. For $$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$1 flat-space gravitational scattering, the celestial eikonal amplitude develops poles at

$$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$2

with increasing maximal order at each location, and admits a large-$$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$3 saddle-point asymptotic controlled by the same high-energy impact-parameter physics that defines the eikonal regime. Related dispersion and monodromy relations can be derived by contour methods, and analogous constructions exist for shockwave, Schwarzschild, and Kerr eikonal-probe amplitudes (Adamo et al., 2024).

In perturbative QCD, the eikonal approximation organizes soft-gluon radiation through Wilson lines and dipole radiators. For multiple energy-ordered gluon emissions at finite $$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$4, the squared matrix element can be written algorithmically in terms of antenna functions and ordered color operators, and explicit results have been given up to five loops. Starting at four loops, genuine finite-$$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$5 structures proportional to $$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$6 appear and break Bose and mirror symmetries, vanishing in the large-$$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$7 limit (Delenda et al., 2015).

The same Wilson-line logic underlies event-shape resummation. In the eikonal version of dressed gluon exponentiation, soft emissions are encoded in lightlike Wilson lines and the dominant power corrections follow from elementary eikonal phase-space integrals. The method reproduces the leading large-angle soft term, isolates half-integer renormalon poles, and yields the standard rigid $$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$8 shift of two-jet distributions through the leading $$\delta(b)=\delta_0(b)+\delta_1(b)+\delta_2(b)+\cdots,$$9 pole (Agarwal et al., 2020).

A distinct, non-relativistic implementation appears in halo-nucleus breakup. There the adiabatic eikonal approximation fails for Coulomb-dominated breakup because the tidal Coulomb phase diverges. The Coulomb-corrected eikonal replaces the divergent first-order term by the finite first-order phase

$1{\rm PM}={\rm tree}$0

while the simplified dynamical eikonal approximation combines this with a first Magnus or Fer truncation and an absorptive treatment of imaginary nuclear phases. In the benchmark systems studied, this improves energy distributions in both Coulomb- and nuclear-dominated breakup and enhances the physically observed asymmetry of parallel-momentum distributions for nuclear-dominated breakup (Hebborn et al., 2019).

Taken together, these developments show that perturbative eikonal is not a single formula but a tightly connected set of resummation, generator, and Wilson-line constructions. Its central object is the exponentiated phase or generator, but its modern scope includes many-particle horizon $1{\rm PM}={\rm tree}$1-matrices, radiative observables, helicity-space phase matrices, sub-eikonal spin operators, celestial Mellin regularization, and finite-$1{\rm PM}={\rm tree}$2 multi-loop soft radiation.