Arellano and Blanchon (Phys. Lett. B 789 (2019) 256) constructed an exact coordinate-space solution for elastic scattering waves off a nonlocal nucleon-nucleus potential superposed to the long-range Coulomb interaction, expressing the wavefunction through a Coulomb Green's function kernel solved by direct matrix inversion with no restriction on the type of nonlocality or the beam energy. They stated that the approach is well suited for coupled channels, with an extension to inelastic processes underway. Coupled-channel calculations with nonlocal microscopic optical and transition potentials have so far been developed and validated only for neutron scattering off spin-zero targets (Nasri et al., 2021), with the charged-projectile case explicitly excluded; the exact method's realization (SWANLOP) covers elastic scattering only; and a 2025 review still lists coupled-channel calculations with nonlocal potentials as an ongoing development. The open problem is therefore to formulate, implement, and validate the coupled-channel and inelastic generalization of this exact method for charged projectiles, in which fully nonlocal optical and transition potentials are treated together with the long-range Coulomb interaction without screening, iteration, or perturbative approximation.

Background

In the nuclear optical model, the elastic scattering of a nucleon off a nucleus is described by a complex effective potential, the optical potential, which absorbs the flux lost to unobserved reaction channels. Microscopic derivations — folding an effective two-nucleon interaction (a Brueckner g-matrix or t-matrix, i.e. the effective in-medium two-nucleon scattering operator computed in infinite nuclear matter) with the target density — yield a potential that is nonlocal: U(r,r′)U(\mathbf{r},\mathbf{r}') depends on two position vectors, so the Schrödinger equation becomes an integro-differential equation in which the potential acting at r\mathbf{r} depends on the wavefunction at all points r′\mathbf{r}'. Phenomenological fits, by contrast, mostly use local Woods-Saxon forms (the standard radially symmetric well shape of phenomenological nuclear potentials) with, at best, the ad hoc Perey-Buck Gaussian nonlocality. For inelastic scattering to specific excited states (direct reactions), one works either in the distorted-wave Born approximation (DWBA) — transition potentials acting between distorted waves generated by the optical potential, treated to first order — or in a coupled-channels formulation, in which the equations for the elastic and inelastic partial waves are solved simultaneously; this is required when the coupling to collective states is strong. When the projectile is charged, the nuclear interaction is superposed on the long-range Coulomb repulsion, which cannot be truncated at finite radius. Momentum-space treatments must then screen the Coulomb potential and extrapolate the screening to zero (the Vincent-Phatak procedure), while standard coordinate-space solvers either iterate the nonlocal kernel (with convergence that cannot be guaranteed beforehand) or restrict the admitted nonlocality. Arellano and Blanchon (2019) closed this gap for the single elastic channel: writing the scattering wave as ψ=ϕC+GCUSψ\psi=\phi_{C}+G_{C}U_{S}\psi, where ϕC\phi_{C} is the incoming Coulomb wave, GCG_{C} the Coulomb Green's function, and USU_{S} the nonlocal nuclear potential superposed to the Coulomb-screened interaction, they showed that the resulting integral equation has a continuous, non-singular kernel that yields the exact scattering waves by direct matrix inversion, with no restriction on the type of nonlocality or the beam energy. They benchmarked it against Raynal's DWBA98 code — an established exact solver of the integro-differential Schrödinger equation — for elastic proton scattering off 40Ca at 30.3 and 300 MeV with a microscopic nonlocal potential. The method, however, was formulated and demonstrated only for a single elastic channel off a spin-zero target. In their concluding remarks the authors stated that the approach is well suited for coupled channels, with an extension to inelastic processes underway, referencing their earlier two-channel charge-exchange work (Arellano and Love, 2007) and the CEA program on microscopic nonlocal potentials for deformed nuclei. Whether that extension — coupled channels and inelastic transitions, with nonlocal optical and transition potentials and an exact treatment of the long-range Coulomb interaction — has since been completed is the subject of this problem.

Problem Statement

Formulate, implement, and validate the coupled-channel generalization of the exact coordinate-space scattering-wave method of Arellano and Blanchon (2019), extending it from a single elastic channel to inelastic scattering of a charged projectile (e.g., a proton) off a nucleus, in which the nonlocal optical potentials, the nonlocal transition potentials that couple the elastic and inelastic channels, and the long-range Coulomb interaction are all treated exactly — without screening of the Coulomb force, without perturbative truncation of the channel coupling, and without iterative solution of the nonlocal kernel. A solution requires a coupled-channel integral equation whose kernel combines the Coulomb Green's function with nonlocal optical and transition potentials and reduces in the single-channel limit to the published elastic equation ψ=ϕC+GCUSψ\psi=\phi_{C}+G_{C}U_{S}\psi, together with coupled-channel scattering waves and transition amplitudes obtained by direct matrix inversion of a continuous, non-singular discretized kernel, with numerical convergence with respect to mesh size, partial-wave truncation, and channel space. Validation is part of the completion standard and follows the practice of the underlying works: the required check is that, with the couplings switched off, the method reproduces the published elastic benchmarks (e.g., the DWBA98 benchmarks and the public SWANLOP elastic-solver results for 40Ca(p,p)). In the neutral-projectile limit, consistency with existing neutron coupled-channel results such as those of Nasri et al. (2021), and comparability of inelastic observables for charged projectiles (differential cross sections exciting discrete states) with established coupled-channel/DWBA calculations or experimental data, are encouraged additional checks and are not required for completion. The unresolved case is the charged-projectile case with exact Coulomb treatment, which the existing literature explicitly leaves untreated.

References

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