The only frameworks that compute exact coordinate-space Coulomb-distorted scattering waves for nucleon elastic scattering under fully nonlocal optical potentials — the Lippmann–Schwinger-type solver SWANLOP and the finite-difference solver SIDES — are restricted to spin-zero target nuclei with a uniform-charge-sphere Coulomb interaction. The problem is to extend this Coulomb-exact treatment to targets with nonzero ground-state spin J≠0J\neq 0, where the nonlocal optical potential acquires channel-spin structure that couples partial waves, all while keeping the treatment exact: no Coulomb screening and no local-equivalent or Taylor-series reduction of the nonlocality. The deliverables are the generalized formulation, a validated numerical solution yielding scattering waves and observables, and benchmark evidence including recovery of the published spin-zero results.

Background

In the nuclear optical model, the collision of a nucleon with a nucleus is described by a one-body potential UU acting on the nucleon. Microscopic and ab-initio derivations of this potential (folding of in-medium two-nucleon interactions with the target density) yield a nonlocal operator: the potential acting at point r\mathbf{r} depends on the wavefunction at another point r′\mathbf{r}', i.e. U(r,r′)U(\mathbf{r},\mathbf{r}'), while phenomenology classically parametrizes such nonlocality through the Perey–Buck energy-independent form. Treating nonlocality exactly matters physically: local approximations suppress the so-called Perey effect — the systematic reduction of the wavefunction amplitude inside the nucleus — which biases quantities extracted from distorted waves, such as spectroscopic factors in transfer reactions. When the projectile is a proton, the long-range Coulomb interaction must be combined with the short-range nonlocal nuclear potential. Momentum-space methods handle this by screening the Coulomb tail and renormalizing amplitudes, which delivers on-shell observables but not coordinate-space scattering waves. Arellano and Blanchon (Phys. Lett. B 789 (2019) 256) gave the exact coordinate-space solution: the scattering wave satisfies a Lippmann–Schwinger-type integral equation ψ=ϕC+GC US ψ\psi=\phi_C+G_C\,U_S\,\psi, where ϕC\phi_C is the Coulomb wave, GCG_C the Coulomb Green's function, and USU_S any finite-range nonlocal optical potential superposed on the Coulomb field. Two public packages implement this exact treatment for elastic nucleon scattering: SIDES (Comput. Phys. Commun. 254 (2020) 107340), a finite-difference integro-differential solver in the lineage of Raynal's DWBA (distorted-wave Born approximation) code, and SWANLOP (Comput. Phys. Commun. 259 (2021) 107543), which inverts the integral equation by direct matrix inversion over a radial mesh and also accepts potentials given in momentum space, converting them to coordinate representation. Both packages, as released, handle only spin-zero targets (0+0^+ closed-shell nuclei), dimension the calculations for target mass number A≥4A\ge 4 and beam energies up to 1.1 GeV, and in SWANLOP the Coulomb field is that of a uniform-charge sphere (its Appendix B). For a target with ground-state spin J≠0J\neq 0 — the majority of nuclei — the optical potential couples the projectile spin 1/21/2 and the target spin JJ into channel spin SS, producing spin–spin and tensor operator structures that couple different partial waves (L,S)(L,S); the single-channel multipole decomposition Ujl(r′,r)U_{jl}(r',r) used by the existing codes no longer suffices, and the exact Coulomb reference must be carried through the coupled system. A momentum-space alternative for nonzero-spin targets exists (Vorabbi et al., Phys. Rev. C 105 (2022) 014621, using a chiral folding potential — a microscopic potential built by folding two-nucleon interactions from chiral effective field theory with the target ground-state density — and a short-range residual Coulomb term transformed with Coulomb functions), but it computes on-shell amplitudes, not the exact coordinate-space Coulomb-distorted waves. No Coulomb-exact coordinate-space treatment of nonlocal potentials for J≠0J\neq 0 targets existed in the literature as of August 2026.

Problem Statement

Derive, implement, and validate the Coulomb-exact coordinate-space treatment of elastic nucleon–nucleus scattering under fully nonlocal optical potentials for targets whose ground-state spin is J≠0J\neq 0, taking the uniform-charge-sphere Coulomb field of the existing frameworks (SWANLOP's Appendix B) as the reference Coulomb model; generalization to extended, non-uniform nuclear charge distributions — for which the source's Coulomb evaluation subroutine is explicitly user-customizable — is an optional generalization, not required for completion. Concretely: (i) formulate the channel-spin-coupled generalization of the exact Lippmann–Schwinger-type equation ψ=ϕC+GCUSψ\psi=\phi_C+G_C U_S\psi — with channel spin S=J⊕1/2S=J\oplus 1/2, the spin–spin and tensor operator structure of the nonlocal multipoles U(r′,r)U(r',r), and the Coulomb Green's function treatment of the Coulomb interaction — without Coulomb screening and without any local-equivalent or Taylor-series reduction of the nonlocality; (ii) solve the resulting coupled system numerically to obtain the Coulomb-distorted scattering waves and the standard observables (differential and integrated cross sections, analyzing power, spin rotation function) for proton projectiles, where the exact-Coulomb machinery is exercised (for neutron projectiles the Coulomb terms of the framework vanish, making the neutron case an optional consistency check of the channel-spin kernel rather than a completion requirement); and (iii) validate the solution by recovering, at J=0J=0, the published spin-zero SWANLOP/SIDES benchmark results and reproducing analytic limits (pure Coulomb scattering and closed-form separable nonlocal models), and, since those benchmarks probe only the uncoupled limit, additionally by a non-degenerate verification of the J≠0J\neq 0 solution with non-vanishing channel-spin coupling (a nonzero spin–spin or tensor multipole), satisfiable without any independent external benchmark through convergence of the coupled solution in the radial mesh and the coupled (L,S)(L,S) basis and through internal-consistency checks (e.g. a small residual of the coupled Lippmann–Schwinger equation, and agreement between the observables computed from the distorted waves and from the asymptotic scattering amplitudes); given the scarcity of independent high-precision benchmarks for J≠0J\neq 0 targets, cross-comparison of on-shell observables against independent momentum-space calculations (e.g., of the Vorabbi et al. type) is an optional, non-binding validation avenue rather than a completion requirement. Performance engineering of the existing spin-zero code (mesh optimization, parallelization, multipole-runtime improvements above ~50 MeV) and the further extension to few-body or breakup channels with exact Coulomb remain open directions but are not required for completion of this problem.

References

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