The NLAT code (Titus, Ross and Nunes, 2016) computes cross sections for deuteron-induced single-nucleon transfer reactions A(d,N)B with explicitly nonlocal nucleon-target optical potentials, within the adiabatic distorted wave approximation (ADWA), by iteratively solving the nonlocal Schrodinger integro-differential equations in coordinate space. Its authors left a bundle of acknowledged gaps: no nonlocal deuteron optical potential parameterization existed; the remnant term of the transfer T-matrix was dropped without systematic quantification; the iterative solver's discretization parameters required user tuning with no convergence study; validation beyond Perey-Buck-type phenomenological potentials was absent; and dynamics beyond ADWA was not included. The headline gap has since been largely filled (global nonlocal deuteron-nucleus optical potential parameterizations appeared in 2022 and were applied to (d,p) transfer in 2024), but the remaining core is open - quantify the neglected remnant term (especially for light targets) and establish controlled and user-independent discretization/convergence methodology for the iterative solution of the nonlocal equations.

Background

When a nucleon scatters elastically off a nucleus, the many-body problem is replaced by a complex effective interaction, the optical model potential (OMP). Such potentials are inherently nonlocal in coordinate space - they encode exchange terms from antisymmetrization and the coupling to eliminated reaction channels - but nearly all global parameterizations used in data analysis are local, for computational convenience. The prototype nonlocal OMP is the Perey-Buck potential: a Woods-Saxon-shaped function of the mean radius multiplied by a Gaussian nonlocality factor exp⁡[−(r2+r′2)/β2]\exp[-(r^2+r'^2)/\beta^2] connecting source radius r′r' and field point rr. With such a potential the Schrodinger equation becomes an integro-differential equation, whose solution is suppressed in the nuclear interior relative to a phase-equivalent local potential (the Perey effect). This suppression matters whenever a reaction amplitude samples the interior - as transfer reactions do. Deuteron-induced single-nucleon transfer, A(d,p)B, is the standard tool for extracting single-particle nuclear structure: spectroscopic factors (the norm of the A+B overlap function) and asymptotic normalization coefficients (ANCs, which fix the tail of the transferred neutron's bound-state wave function and feed astrophysical capture-rate estimates). In the distorted wave Born approximation (DWBA) the transfer amplitude is built from distorted waves generated by optical potentials in the entrance (d+A) and exit (p+B) channels, with deuteron breakup absorbed only implicitly into the deuteron OMP. The adiabatic distorted wave approximation (ADWA) of Johnson and Tandy instead constructs the deuteron-target distorting potential by folding the neutron-target and proton-target OMPs over the deuteron wave function, thereby including deuteron breakup to all orders at leading order in the neutron-proton relative kinetic energy; it relies on the much better constrained nucleon OMPs and is the workhorse for (d,p) analyses at deuteron energies of roughly 10-70 MeV. Titus, Nunes and Potel extended ADWA to explicitly nonlocal nucleon OMPs (Perey-Buck-type and the Tian-Pang-Ma systematic nonlocal nucleon potential) and showed that the local approximation can bias extracted spectroscopic factors by up to 40%. The companion code paper by Titus, Ross and Nunes released NLAT, which solves the nonlocal integro-differential equations for scattering and bound states iteratively in coordinate space and computes (d,p), (p,d), (d,n) and (n,d) cross sections with a target accuracy of about 4%. In the post-form transfer T-matrix, the exact transition operator contains, besides the proton-target interaction, a remnant term - the difference between the exit-channel optical potential and its entrance-channel counterpart - which NLAT drops, citing local-potential studies that found it negligible for intermediate and heavy targets; for light targets this claim is unquantified. The paper also documents that the iterative solver's mesh step, an integration cut-off parameter (CutL), quadrature point count and partial-wave-dependent iteration counts require user tuning, and it expresses the hope that the method will serve future studies beyond Perey-Buck-type potentials. Since 2016 the landscape has moved: global nonlocal deuteron-nucleus OMP parameterizations have been published and applied to (d,p) in DWBA; exact momentum-space three-body calculations with nonlocal and microscopic optical potentials have been pursued in parallel; and uncertainty-quantification studies of (d,p) observables have matured. No systematic quantification of the remnant term and no controlled discretization/convergence methodology for the nonlocal transfer equations have appeared - the surviving open problem stated below.

Problem Statement

Within the scope of the source work - deuteron-induced single-nucleon transfer reactions A(d,N)B and their inverses at deuteron energies of roughly 10-70 MeV - complete the explicit inclusion of nonlocality in the reaction theory so that transfer predictions are exact to controlled, documented accuracy. Two linked gaps must be closed. (1) Remnant term: quantify the effect of the neglected remnant term of the post-form transfer T-matrix (the difference between the exit-channel optical potential and its entrance-channel counterpart) on transfer observables within nonlocal ADWA and nonlocal DWBA, systematically across target mass and beam energy, with particular attention to light targets (1p-shell, i.e. nuclei with valence nucleons in the second shell, up to about mass 16) where existing local-potential evidence does not establish negligibility; either include the term or provide justified bounds on the induced error in cross sections and in extracted spectroscopic factors and ANCs. (2) Numerics: establish a controlled, user-independent discretization and convergence methodology for the iterative coordinate-space solution of the nonlocal integro-differential equations for scattering and bound states - mesh step, integration cut-off parameters such as CutL, quadrature, partial-wave and iteration-count convergence - demonstrating convergence of transfer observables across wide partial-wave ranges. These components jointly serve one objective: making the nonlocality correction to structure quantities extracted from transfer reactions (spectroscopic factors, ANCs) exact and auditable rather than approximate. The original 2016 headline gap - the absence of any nonlocal deuteron optical potential parameterization - is treated as resolved by later literature and excluded from the open core.

References

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