In high-energy diffractive scattering of hadrons from nuclei, the eikonal diffraction integral for the elastic amplitude can be approximated, at large momentum transfer and for strong interactions, by a sum over a few amplitudes from complex stationary-phase points in the impact-parameter plane (Bleszynski, Glauber and Osland, 1981). The program recorded from the original paper asked for four developments: uniform validity near the forward direction, consequences and remedies for phase-shift functions lacking the required analyticity, a quantitative delineation of the domain of validity, and systematic incorporation of Coulomb and spin-dependent dynamics. The 2019 open-access monograph by Glauber and Osland delivered the Coulomb component and systematically developed the analytic-structure component, but explicitly declined the near-forward correction and omits spin entirely, and it gives no a priori error theory. The open problem is the completion of this framework: (1) construct and validate the stationary-phase representation of the spin-dependent amplitudes (for spin-1/2 projectiles) with the corresponding spin observables; (2) derive correction terms or a uniform approximation valid down to q→0q\to 0; (3) establish quantitative a priori criteria, in dimensionless parameters of the phase-shift function, delimiting the domain of validity, ideally with error bounds.

Background

When a high-energy projectile scatters elastically from a nucleus, with kinetic energy far exceeding the interaction strength and de Broglie wavelength much shorter than the nuclear radius RR, the scattering is described by eikonal theory (also called Glauber multiple-scattering theory): the projectile follows nearly straight-line paths, and the scattering amplitude at momentum transfer ℏq=ℏ(k′−k)\hbar\boldsymbol{q}=\hbar(\boldsymbol{k}'-\boldsymbol{k}) reduces to a two-dimensional Fourier integral over the impact parameter b\boldsymbol{b} (the transverse position at which the projectile passes the target), f(q)=ik2π∫d2b  e−iq⋅b(1−eiχ(b)),f(\boldsymbol{q})=\frac{ik}{2\pi}\int d^2b\;e^{-i\boldsymbol{q}\cdot\boldsymbol{b}}\left(1-e^{i\chi(\boldsymbol{b})}\right), where kk is the projectile momentum and χ(b)\chi(\boldsymbol{b}) is the phase-shift function, a complex function whose imaginary part expresses absorption (the depletion of the elastic channel by inelastic excitation). Measured differential cross sections dσ/dΩ=∣f∣2d\sigma/d\Omega=|f|^2 at laboratory momenta of order 1 GeV/cc fall by many orders of magnitude over a few tens of degrees while oscillating with an almost fixed period — for example in 800 MeV proton-208{}^{208}Pb elastic scattering — and interpreting these diffraction patterns calls for more than brute-force numerics, which suffers catastrophic cancellations over seven or more orders of magnitude.

Bleszynski, Glauber and Osland (1981) introduced an asymptotic treatment of this integral for large momentum transfer: evaluating it by the method of stationary phase, the amplitude becomes a discrete sum f≃∑sfsf\simeq\sum_s f_s over stationary points bs\boldsymbol{b}_s satisfying q=∇bχ(b)\boldsymbol{q}=\nabla_{\boldsymbol{b}}\chi(\boldsymbol{b}). For absorptive interactions these stationary points lie off the real axis, in the complex impact-parameter plane; the separation of the (typically two) dominant points fixes the oscillation period of the cross section, their imaginary offsets fix its exponential envelope, and at asymptotically large momentum transfer the stationary points migrate toward the singularities of χ\chi. Where two stationary points coalesce (a "rainbow", in analogy with optical rainbows) the leading approximation diverges. The 1981 paper verified the method against numerical integration for three profile families — Gaussian, inverse-cosh (a profile whose analytic continuation has simple poles at finite imaginary impact parameter), and a double-surface (Fermi-like) form — finding accuracy already for quite small momentum transfers (e.g. q>0.25q>0.25 GeV/cc for the Gaussian and q>0.1q>0.1 GeV/cc for the inverse-cosh example), and noted its limits: failure near the forward direction q→0q\to 0; the requirement that χ(b)\chi(\boldsymbol{b}) be analytically continuable into the complex plane so the integration contour can be deformed through the stationary points; only empirical demarcation of the validity domain; and no systematic incorporation of Coulomb or spin-dependent dynamics (spin-12\tfrac{1}{2} projectiles were asserted, in a single sentence, to reduce to "two phase shift functions, one for each spin orientation"). A related closed-form asymptotic formula for the eikonal amplitude dominated by the nuclear edge (its radius controlling the oscillations, its thickness the exponential falloff) had been derived by Amado, Dedonder and Lenz (1980) via a saddle-point expansion.

The program was largely completed by the open-access monograph of Glauber and Osland (2019), which treats Coulomb scattering from point and extended charges and Coulomb-plus-nuclear combinations, maps how stationary points are attracted to poles, essential singularities, and branch-type singularities of model profiles, gives a uniform Airy-function approximation (the standard uniformization for coalescing saddles) at rainbows, pins the asymptotic regime as 1/R≪q≪k1/R\ll q\ll k, and validates the method against exact numerics for a wide variety of profiles — accurate "provided the stationary points are well separated". However, the monograph explicitly declines to correct the method near the forward direction, treats no spin-dependent dynamics beyond passing remarks, and gives no a priori error bounds, so three components of the original program remain open.

Problem Statement

Complete the asymptotic stationary-phase (asymptotic diffraction) theory of diffractive hadron-nucleus scattering along the three directions in which the framework remains incomplete after the 2019 Glauber-Osland monograph. The common objective is to make the stationary-phase representation of the eikonal diffraction integral f(q)=(ik/2π)∫d2b e−iq⋅b(1−eiχ(b))f(\boldsymbol{q})=(ik/2\pi)\int d^2b\,e^{-i\boldsymbol{q}\cdot\boldsymbol{b}}(1-e^{i\chi(\boldsymbol{b})}) uniformly applicable and rigorously characterized. The three requested results, which jointly serve this objective, are:

(1) Spin-dependent amplitudes. Construct the stationary-phase representation of the spin-dependent elastic scattering amplitudes — for spin-12\tfrac{1}{2} projectiles with central and spin-orbit phase functions (the reduction, asserted without derivation in the 1981 paper, to two spin-orientation-dependent phase-shift functions). This requires identifying the generally different complex stationary points of the spin-dependent phase functions, deriving the corresponding asymptotic amplitudes with their prefactors and phase conventions, extracting the interference structure of at least one spin observable such as the analyzing power (the left-right scattering asymmetry generated by the spin-orbit part of the interaction), and validating the construction against direct numerical evaluation of the spin-dependent diffraction integrals.

(2) Near-forward uniformity. Derive the correction terms, or a uniform approximation, that renders the stationary-phase representation valid down to q→0q\to 0, joining smoothly onto the exact diffraction integral in the forward region where the stationary-phase expansion breaks down.

(3) Quantitative domain of validity. Establish a priori criteria — expressed in dimensionless parameters of the phase-shift function (such as qβq\beta with β\beta the transverse range of the profile, qRqR with RR the nuclear radius, the peak magnitude ∣χ0∣|\chi_0| of χ\chi, and the separation of the stationary points), and covering profiles with varied singularity content — that delimit the domain in which the stationary-phase sum approximates the diffraction integral to a specified accuracy, ideally with error bounds; this formalizes the source's informal conditions of "sufficiently large momentum transfer" and "strong interactions".

The Coulomb-incorporation component of the original 1981 question is excluded from the surviving core because it has been delivered for spinless amplitudes by the 2019 monograph; Coulomb phases in combination with spin-dependent dynamics fall under component (1).

References

  1. M. Bleszynski, R.J. Glauber, P. Osland, "Asymptotic theory of diffractive scattering", Physics Letters B 104 (1981) 389-393. DOI: 10.1016/0370-2693(81)90705-X. https://doi.org/10.1016/0370-2693(81)90705-x
  2. Roy J. Glauber, Per Osland, "Asymptotic Diffraction Theory and Nuclear Scattering", Cambridge University Press, 2019 (open access). DOI: 10.1017/9781316219256. https://doi.org/10.1017/9781316219256
  3. Per Osland, "Roy Glauber and asymptotic diffraction theory", SciPost Physics Proceedings 3, 016 (2020). DOI: 10.21468/SciPostPhysProc.3.016, arXiv:1910.01408. https://arxiv.org/abs/1910.01408
  4. R.D. Amado, J.P. Dedonder, F. Lenz, "Explicit formula for hadron-nucleus elastic scattering in the eikonal approximation", Physical Review C 21 (1980) 647-661. DOI: 10.1103/PhysRevC.21.647. https://doi.org/10.1103/PhysRevC.21.647
  5. M. Bleszynski, P. Osland, "Noneikonal effects in spin-dependent proton-nucleus interactions", Physics Letters B 84 (1979) 157-161. DOI: 10.1016/0370-2693(79)90272-7. https://doi.org/10.1016/0370-2693(79)90272-7
  6. G. W. Hoffmann, L. Ray, M. Barlett, J. McGill, G. S. Adams, G. J. Igo, F. Irom, A. T. M. Wang, C. A. Whitten, R. L. Boudrie, J. F. Amann, C. Glashausser, N. M. Hintz, G. S. Kyle, G. S. Blanpied, "0.8-GeV p + Pb-208 elastic scattering and the quantity Delta r_np", Physical Review C 21 (1980) 1488-1494. DOI: 10.1103/PhysRevC.21.1488. https://doi.org/10.1103/PhysRevC.21.1488
  7. G. Fäldt, A. Ingemarsson, "Analytic proton-nucleus scattering amplitudes", Nuclear Physics A 393 (1983) 372-384. DOI: 10.1016/0375-9474(83)90148-3. https://doi.org/10.1016/0375-9474(83)90148-3
  8. J.-F. Germond, M.B. Johnson, J.A. Johnstone, "Diffraction theory analysis of pion elastic scattering from the calcium isotopes", Physical Review C 32 (1985) 983-994. DOI: 10.1103/PhysRevC.32.983. https://doi.org/10.1103/PhysRevC.32.983
  9. J.-F. Germond, R.J. Lombard, "Asymptotic behaviour of backward elastic scattering", Physics Letters B 206 (1988) 543-545. DOI: 10.1016/0370-2693(88)91625-5. https://doi.org/10.1016/0370-2693(88)91625-5
  10. S.M. Troshin, N.E. Tyurin, "Reflective scattering at the LHC and two-scale structure of a proton", EPL 129 (2020) 31002. DOI: 10.1209/0295-5075/129/31002, arXiv:2001.06260. https://arxiv.org/abs/2001.06260
  11. Pierre Capel, "The eikonal model of reactions involving exotic nuclei; Roy Glauber's legacy in today's nuclear physics", SciPost Physics Proceedings 3, 017 (2020). DOI: 10.21468/SciPostPhysProc.3.017, arXiv:1911.11446. https://arxiv.org/abs/1911.11446
  12. Yu. M. Shabelski, A. G. Shuvaev, "High energy nucleus-nucleus collisions with accounting for Coulomb interaction", Modern Physics Letters A 41 (2026) 2650102. DOI: 10.1142/S0217732326501026, arXiv:2405.11842. https://arxiv.org/abs/2405.11842