Bayesian Global Optical Potential (KDUQ)

• Bayesian Global Optical Potential (KDUQ) is a probabilistic nucleon–nucleus model calibrated using Bayesian methods, yielding a posterior distribution over optical-model parameters from extensive scattering data.
• The model enables uncertainty propagation through both exact and approximate reaction formalisms, providing credible intervals for observables across a wide range of nuclei and energies.
• KDUQ’s applications in eikonal, semiclassical, and transfer reaction analyses demonstrate its strength in preserving posterior correlations and improving uncertainty quantification in nuclear reaction modeling.

Searching arXiv for KDUQ and closely related optical-potential papers to ground the article with current citations. arXiv search query: KDUQ optical potential Bayesian nucleon nucleus Bayesian Global Optical Potential, commonly denoted KDUQ, is a global nucleon–nucleus optical potential with quantified uncertainties, developed through Bayesian calibration to a large body of scattering data. In current applications it is treated not merely as a single best-fit interaction, but as a posterior distribution over optical-model parameters that can be propagated through reaction calculations. This makes KDUQ a probabilistic interaction model for nucleon-induced observables, and recent studies have used it as a common input for exact optical-model calculations, eikonal and semiclassical reaction formalisms, and transfer-reaction analyses on the same uncertainty-quantified footing (Shiu et al., 17 Jul 2025, Hebborn et al., 17 Jul 2025).

1. Definition and calibration scope

KDUQ is described as a global nucleon–nucleus optical model calibrated in a Bayesian framework and distributed through a posterior ensemble rather than a point estimate. Related studies characterize the model as global over a broad mass and energy domain, stating that it is intended to work across $24 \le A \le 209$ and $1~\text{keV} \le E \le 200~\text{MeV}$, while also describing the calibration corpus as consisting of hundreds of scattering datasets on various targets, with beam energies between 0 and 200 MeV and multiple observables (Hebborn et al., 17 Jul 2025).

The calibration data are reported to include nucleon elastic scattering angular distributions, analyzing powers, neutron total cross sections, and proton reaction cross sections on stable nuclei. One recent study further specifies that it employs the democratic version of KDUQ, in which every data point in the calibration corpus is given equal weight in the likelihood. In that usage, KDUQ functions as an uncertainty-quantified global optical potential whose posterior samples can be reused consistently across distinct reaction channels (Hebborn et al., 17 Jul 2025).

This scope is central to the meaning of “global” in the KDUQ context. The term refers not only to a broad intended range of nuclei and energies, but also to a single statistically calibrated parameterization meant to support forward uncertainty propagation across multiple observables. A plausible implication is that KDUQ is designed as an infrastructural model for reaction theory rather than as a nucleus-by-nucleus refit.

2. Optical-model ansatz and posterior structure

In the studies that use it, KDUQ is written in the standard optical-model form

$$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$

where $V_N$ is the real nuclear part, $W$ is the imaginary absorptive part, and $V_C$ is the Coulomb interaction (Shiu et al., 17 Jul 2025).

The central statistical distinction is that KDUQ is an uncertainty-quantified global potential: it provides a posterior distribution over optical-model parameters rather than only a best-fit parameter set. One study states that the original KDUQ work produced posterior distributions and correlations for 46 optical-model parameters, and current uncertainty-propagation applications use 416 posterior samples from that analysis (Hebborn et al., 17 Jul 2025).

A salient parameterization issue arises outside the fitted energy range. For energies above 200 MeV, the real-depth energy dependence becomes poorly constrained. The coefficient of

$\Delta E^4=(E-\epsilon_f)^4$

in the KDUQ parameterization is reported as

$-4.30^{+25.60}_{-20.30}\times 10^{-9},$

and this large uncertainty is explicitly linked to the rapid growth of predictive uncertainty above the calibration range (Shiu et al., 17 Jul 2025).

This behavior defines an important limitation. KDUQ is uncertainty-quantified in the Bayesian sense, but the posterior remains conditional on the chosen parameterization and on the data-supported energy interval. The extrapolative regime above 200 MeV is therefore statistically represented, yet only weakly constrained.

3. Bayesian propagation and credible intervals

The operational use of KDUQ in recent reaction studies is based on posterior sampling. The uncertainty propagation procedure is described as follows: for each of the 416 samples of the KDUQ democratic posterior distribution, a nucleon-target optical potential is generated, the observable is computed, and the distribution over all 416 results is used to define credible intervals (Shiu et al., 17 Jul 2025).

The credible interval is defined as the smallest interval containing $x\%$ of the predicted cross sections from the posterior ensemble. For 68% intervals, one study defines the relative half-width by

$$\varepsilon_{68}=\frac{\sigma^{68}_{\max}-\sigma^{68}_{\min}}{2\sigma^{68}_{\mathrm{avg}}}, \qquad \sigma^{68}_{\mathrm{avg}}=\frac{\sigma^{68}_{\max}+\sigma^{68}_{\min}}{2}.$$

These intervals are computed not only for exact partial-wave solutions, but also for the standard eikonal model and for the eikonal model with semiclassical correction (Shiu et al., 17 Jul 2025).

A closely related transfer-reaction study uses the same “smallest interval containing $1~\text{keV} \le E \le 200~\text{MeV}$0” definition, but evaluates the relative half-width at the peak of the angular distribution. In that context, the statistical significance of KDUQ lies not only in marginal parameter uncertainties but in the preservation of posterior correlations: the sampled parameter sets are joint posterior draws, so the optical-model components are not varied independently (Hebborn et al., 17 Jul 2025).

This posterior-sampling workflow distinguishes KDUQ from deterministic optical-potential usage. The uncertainty bands are not ad hoc error bars; they are derived from an explicit posterior ensemble and can therefore be propagated through nonlinear reaction formalisms without relinearizing the model.

4. Exact optical-model calculations and the eikonal benchmark

In the eikonal-validity study, KDUQ enters both exact and approximate reaction calculations through the same optical interaction. The exact reaction dynamics are obtained from

$1~\text{keV} \le E \le 200~\text{MeV}$1

with the $1~\text{keV} \le E \le 200~\text{MeV}$2-matrix extracted from partial-wave solutions. In the eikonal approximation, the wave function is factorized as

$1~\text{keV} \le E \le 200~\text{MeV}$3

leading to the eikonal $1~\text{keV} \le E \le 200~\text{MeV}$4-matrix

$1~\text{keV} \le E \le 200~\text{MeV}$5

The semiclassical correction is implemented through an impact-parameter shift $1~\text{keV} \le E \le 200~\text{MeV}$6, where $1~\text{keV} \le E \le 200~\text{MeV}$7 is the distance of closest approach in the classical trajectory. The complex-distance correction is explicitly not used because its interpretation is unclear and it can overestimate absorption (Shiu et al., 17 Jul 2025).

The calculations cover neutron and proton reactions on

$1~\text{keV} \le E \le 200~\text{MeV}$8

for beam energies up to 400 MeV. The integrated observables include proton absorption cross sections, neutron total cross sections, neutron absorption cross sections, and neutron elastic cross sections; elastic angular distributions are also studied for both proton and neutron scattering at selected energies between about 30 and 136 MeV (Shiu et al., 17 Jul 2025).

The resulting validity thresholds are strongly observable-dependent. For proton absorption cross sections, the standard eikonal model is reported to be usable down to around 60 MeV, while the semiclassical correction extends its use to around 30 MeV; for $1~\text{keV} \le E \le 200~\text{MeV}$9Pb the more specific range quoted is about 25–30 MeV. By contrast, neutron total cross sections are much less favorable: for $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$0Pb the standard eikonal model is valid only down to about 110–120 MeV, extended to about 50 MeV by the semiclassical correction. The standard eikonal model tends to underestimate neutron total cross sections at low energies. For elastic angular distributions, the eikonal model reproduces the exact diffraction pattern only at sufficiently high energies, fails at low energy, and the semiclassical correction is not very helpful (Shiu et al., 17 Jul 2025).

KDUQ’s propagated uncertainties behave differently from the validity of the reaction approximation itself. A central result is that KDUQ uncertainties propagate similarly through the eikonal and exact models in most cases. Below 200 MeV, the uncertainty intervals are generally small and typically below 5%; above 200 MeV they grow rapidly, especially for neutron elastic cross sections, reaching about 25% at 400 MeV. Proton and neutron absorption cross sections remain much more stable, staying below about 5%, because they depend mainly on the imaginary part of the optical potential rather than the poorly constrained real part (Shiu et al., 17 Jul 2025).

This separation between interaction uncertainty and model validity corrects a common misconception. Bayesian uncertainty quantification of the optical potential does not by itself validate the eikonal approximation. The uncertainty bands can remain modest while the approximation itself fails, especially for neutron observables and angular distributions.

5. Transfer reactions, channel correlations, and KDUQ-n

KDUQ has also been used to propagate nucleon optical-potential uncertainty into transfer observables within finite-range ADWA. In the $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$1 study, the transfer amplitude is written as

$$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$2

with the adiabatic entrance-channel potential

$$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$3

The dependence on $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$4, $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$5, and $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$6 makes posterior correlation structure especially consequential (Hebborn et al., 17 Jul 2025).

A defining feature of this application is that all relevant nucleon–nucleus interactions are taken consistently from the same posterior sample. Thus the correlations among entrance-channel neutron and proton potentials and the exit-channel proton potential are retained. The study emphasizes that this differs from older procedures in which different optical potentials were sampled independently (Hebborn et al., 17 Jul 2025).

The baseline case is the $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$7 ground state, described as a neutron in a $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$8 orbital bound by $$U(\mathbf r)=V_N(\mathbf r)+iW(\mathbf r)+V_C(\mathbf r),$$9. Two bound-state prescriptions are used: a standard geometry with $V_N$0 and $V_N$1, and a KDUQ-real prescription in which the real-part geometry is taken from KDUQ samples. The spin-orbit term is fixed at $V_N$2, $V_N$3, and $V_N$4 (Hebborn et al., 17 Jul 2025).

The reported transfer uncertainties are generally small. The study summarizes the typical half-width of the 68% credible interval as roughly 5–10%, comparable to the experimental error on the transfer data. For $V_N$5 at $V_N$6 MeV, the original KDUQ propagation gives $V_N$7 at the peak. However, neutron elastic scattering on $V_N$8 is found not to be well covered by the nominal KDUQ uncertainty bands, and a modified distribution denoted KDUQ-n is introduced by inflating the neutron-target covariance matrix by a factor of 38, increasing the uncertainty by roughly $V_N$9. With KDUQ-n, the transfer half-width at the peak becomes $W$0 (Hebborn et al., 17 Jul 2025).

One of the main results is that optical-potential and bound-state uncertainties do not add in quadrature. The study explicitly attributes this to strong correlations between the spatial extension of the bound-state wave function and the range and sensitivity of the neutron-target optical potential. It also reports that the relative uncertainty in transfer observables increases with beam energy but does not depend strongly on the properties of the final state; across variations in $W$1, $W$2, $W$3, orbital angular momentum $W$4, and node number $W$5, the relative uncertainty remains below 10% in the explored configurations (Hebborn et al., 17 Jul 2025).

6. Position within the broader optical-potential landscape

KDUQ belongs to a broader movement toward uncertainty-quantified optical potentials, but related work indicates that it is not the final endpoint of that development. Two nearby directions are especially relevant: first, Bayesian global optical potentials with modified isovector structure for rare isotopes; second, next-generation optical-model potentials that incorporate exact non-locality and dispersive constraints while remaining uncertainty-aware (Beyer et al., 30 Mar 2026, Perrotta et al., 2024).

Framework	Defining properties	Relation to KDUQ
KDUQ	Global nucleon–nucleus optical potential; Bayesian calibration; posterior sampling; standard optical-model form $W$6	Reference uncertainty-quantified global potential in recent eikonal and transfer studies
ELM	Global, uncertainty-quantified optical potential for neutrons and protons; Lane form with independent isoscalar and isovector geometries; calibrated on $W$7, $W$8, and $W$9 data for spherical targets with $V_C$0 and $V_C$1 MeV	Explicitly compared with KDUQ and CHUQ for rare-isotope extrapolation
Next-generation prototype	Phenomenological nucleon–nucleus OMP that is non-local, exactly dispersive, and uncertainty-quantified; trained on scattering and bound-state data	Presented as aligned with the KDUQ vision but not yet a full KDUQ implementation

In the ELM comparison, the authors state that extrapolations toward the proton and neutron driplines lead to reduced uncertainties when compared to other global optical potentials in use, and they attribute part of that difference to the statistical treatment of the likelihood: unlike KDUQ and CHUQ, which used a rescaled likelihood corresponding to a power posterior, ELM uses the standard Bayesian posterior. This suggests that extrapolative behavior in global optical potentials depends not only on the optical-model ansatz but also on the calibration formalism (Beyer et al., 30 Mar 2026).

The next-generation non-local dispersive program points in a complementary direction. That work argues that a global OMP should combine reaction data, bound-state observables, exact physical constraints, and statistically sound uncertainty quantification; its present implementation remains preliminary, but it is explicitly framed as a move toward a Bayesian-style global optical-model potential. In that sense, it can be read as a possible successor architecture to present local phenomenological UQ frameworks such as KDUQ (Perrotta et al., 2024).

Taken together, these developments clarify both the importance and the limitation of KDUQ. KDUQ established a practically reusable Bayesian global optical potential with posterior sampling that can be propagated through complex reaction calculations. At the same time, recent work indicates two fronts on which the field continues to evolve: improved extrapolation in the isovector sector, and tighter theoretical constraints through non-local and exactly dispersive formulations.