KDUQ: Uncertainty-Quantified Optical Potential

• The paper introduces a Bayesian recalibration of the Koning–Delaroche optical potential, converting point estimates into posterior ensembles with uncertainty quantification.
• It employs Markov-chain Monte Carlo sampling with robust outlier handling to accurately propagate uncertainties in elastic scattering and reaction cross sections.
• The model reliably predicts forward-angle observables within its calibration range while highlighting limitations in larger-angle and high-energy extrapolations.

Searching arXiv for papers on KDUQ and closely related global optical potentials. Search query: "KDUQ Koning Delaroche uncertainty quantified optical potential" KDUQ denotes the uncertainty-quantified Koning–Delaroche family of global nucleon optical-model potentials, that is, a Bayesian recalibration of the canonical Koning–Delaroche global optical potential for single-nucleon scattering on near-spherical nuclei. In the later literature, the acronym is also expanded as “Uncertainty-Quantified phenomenological optical potentials for single-nucleon scattering.” Its defining feature is not a replacement of the Koning–Delaroche functional form, but the promotion of that form from a point estimate to a posterior ensemble with parameter covariances, nuisance-uncertainty terms, outlier handling, and predictive credible intervals over observables such as elastic angular distributions, total and reaction cross sections, and analyzing powers (Pruitt et al., 2022, Smith et al., 2024).

1. Definition, scope, and historical placement

KDUQ emerged from the recognition that global optical-model potentials in widespread use, especially Koning–Delaroche and Chapel Hill ’89, encode substantial parametric uncertainty that is invisible in deterministic parameter tables. The uncertainty-quantified program therefore retained the original Koning–Delaroche domain of applicability—near-spherical nuclei with $24 \leq A \leq 209$ and laboratory energies $0.001$–$200$ MeV—while replacing the original best-fit parameters by posterior distributions inferred with Markov-chain Monte Carlo and accompanied by full covariance information (Pruitt et al., 2022).

Within this usage, KDUQ is both a specific object and a broader methodological template. In the narrow sense, it is the Bayesian update of the Koning–Delaroche global optical potential used through posterior samples, often 416 published draws from either a “democratic” or a “federal” posterior. In the broader sense, it has become a reference model for uncertainty-aware reaction calculations, including transfer, charge exchange, eikonal observables, and compound de-excitation (Sargsyan et al., 30 Mar 2026, Shiu et al., 17 Jul 2025).

The designation should be distinguished from other next-generation optical-potential efforts. The non-local, exactly dispersive optical-model program presented in “Towards next-generation optical potentials for nuclear reactions and structure calculations” does not define or use the term “KDUQ”; it is not a KD-derived uncertainty band construction, but a different framework aimed at a non-local, exact-dispersive, bound-state-constrained successor or complement to KD-like parameterizations (Perrotta et al., 2024).

2. Optical-potential structure retained by KDUQ

KDUQ preserves the canonical local Koning–Delaroche coordinate-space structure. A representative form used in later summaries is

$$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$

with Woods–Saxon shapes

$$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$

The model therefore contains real central volume, imaginary volume, imaginary surface, spin–orbit, and Coulomb terms with explicit energy and mass dependence, all in a local Woods–Saxon form (Sargsyan et al., 30 Mar 2026).

In the original Koning–Delaroche parametrization retained by KDUQ, the energy dependence is organized through $\Delta E^{n,p} = E - E_f^{n,p}$, with separate neutron and proton Fermi-energy shifts, and selected strengths carry explicit asymmetry dependence through $\alpha=(N-Z)/A$ (Pruitt et al., 2022). The isovector content is often discussed through a Lane decomposition,

$$U(r,E) = U_0(r,E) + \frac{U_1(r,E)}{A}\,\boldsymbol{\tau}\cdot\mathbf{T},$$

so that

$$U_p(r,E) = U_0(r,E) + U_1(r,E)\frac{(N-Z)}{A},\qquad U_n(r,E) = U_0(r,E) - U_1(r,E)\frac{(N-Z)}{A},$$

but later assessments emphasize that KD/KDUQ is not strictly Lane-consistent because some parameters are fit separately for protons and neutrons (Smith et al., 2024, Sargsyan et al., 30 Mar 2026).

This retained form also fixes the principal physical limitations of KDUQ. It is phenomenological, local, and not constrained by dispersion relations; it does not include explicit nonlocal corrections. Those omissions are central to later comparisons with dispersive, non-local, or microscopic alternatives (Sargsyan et al., 30 Mar 2026).

3. Bayesian calibration and uncertainty representation

The statistical novelty of KDUQ lies in its calibration pipeline. The posterior is defined by a likelihood that augments reported experimental uncertainties with data-type-specific “unaccounted-for” uncertainty nuisance parameters and a covariance ansatz designed to represent the limited independence of data points. In compact form,

$$p(\boldsymbol{\theta},\boldsymbol{\delta_t}\mid D) \propto L(\mathbf{y}\mid\mathbf{x},\boldsymbol{\delta_y}, \boldsymbol{\theta}, \boldsymbol{\delta_t}) \, p(\boldsymbol{\theta})\, p(\boldsymbol{\delta_t}),$$

with residual covariance built from

$0.001$0

The priors on optical-model parameters are truncated Gaussians centered on canonical KD values, and the nuisance parameters $0.001$1 are also given positive-support priors (Pruitt et al., 2022).

Posterior inference is performed with the affine-invariant ensemble sampler emcee, together with iterative robust outlier handling. At intervals during the MCMC run, a datum is flagged as an outlier when its residual exceeds a $0.001$2 criterion built from both its quoted error and the current model predictive variance. The fit is then repeated on the updated inlier set until both parameter means and outlier fractions stabilize (Pruitt et al., 2022).

Two posterior-weighting conventions became standard in later applications. The “democratic” posterior weights each datum equally in the likelihood; the “federal” posterior weights each observable class equally irrespective of its number of points. Both are distributed as posterior ensembles in the supplementary material of the original KDUQ work, and later calculations commonly propagate 416 published draws from one of these posteriors (Shiu et al., 17 Jul 2025, Sargsyan et al., 30 Mar 2026).

The calibration corpus includes neutron and proton elastic angular distributions, analyzing powers, proton reaction cross sections, and neutron total cross sections. Comparison against both the original Koning–Delaroche training corpus and a later “test” corpus showed that the uncertainty-quantified ensembles improve residual coverage relative to deterministic KD and expose the dominant sources of overconfidence in the earlier fit, namely underreported experimental uncertainties and lack of outlier rejection (Pruitt et al., 2022).

4. Propagation to observables and reaction theory

Because KDUQ is distributed as posterior samples rather than a single parameter set, observable uncertainties are obtained by repeated forward solves of the optical model. For single-nucleon scattering this means sampling $0.001$3, constructing the potential, solving the Schrödinger equation, extracting the $0.001$4-matrix, and forming distributions of $0.001$5, $0.001$6, $0.001$7, or $0.001$8 (Pruitt et al., 2022).

Its most direct use beyond elastic scattering has been in charge-exchange and transfer theory. For $0.001$9 reactions to isobaric analogue states, the two-body distorted-wave formalism isolates the isovector sector through

$200$0

so that uncertainty in the KDUQ isovector optical field enters both distorted waves and transition operator. For $200$1Ca$200$2 to the IAS, full two-body propagation of KDUQ parameter samples produces 68% relative half-widths at the angular-distribution peak of approximately $200$3 at 25 MeV, $200$4 at 35 MeV, $200$5 at 45 MeV, $200$6 at 135 MeV, and $200$7 at 160 MeV, whereas distorted-wave-only propagation stays at or below about $200$8. The dominant uncertainty source is therefore the transition operator, not the distorted waves alone (Smith et al., 2024).

For transfer reactions, KDUQ has been propagated through finite-range ADWA. In $200$9Ca$$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$0Ca(g.s.) at $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$1 MeV, the half-width of the 68% credible interval at the peak is approximately $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$2 with the nominal KDUQ posterior and approximately $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$3 when the neutron posterior covariance is empirically inflated to restore elastic-scattering coverage. Across the surveyed beam-energy range, typical transfer-observable uncertainties are roughly $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$4–$$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$5, increase with beam energy, and do not depend strongly on the properties of the final state; the study also shows that uncertainties associated with the bound-state potential and with the optical potentials do not, in general, add in quadrature (Hebborn et al., 17 Jul 2025).

The same posterior-sampling logic has been used in approximate high-energy reaction theories. In systematic tests of the eikonal approximation for nucleon–nucleus reactions, 416 KDUQ samples were propagated through both eikonal and exact optical-model solvers. Within the KDUQ calibration range, the $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$6 uncertainty intervals on integrated observables are generally less than $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$7, but they increase rapidly above $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$8 MeV, especially for neutron elastic and total cross sections (Shiu et al., 17 Jul 2025).

5. Empirical performance and application domains

Applications to magnesium isotopes illustrate both the utility and the extrapolative limits of KDUQ. In comparisons with ab initio nonlocal optical potentials for $$\begin{aligned} U(r,E) &= V_V(E)\, f_V(r) + i\, W_V(E)\, f_W(r) \ &\quad + i\, 4\, a_S(E)\, W_S(E)\, \frac{d}{dr} f_S(r) \ &\quad + \left[\, V_{\mathrm{SO}}(E) + i\, W_{\mathrm{SO}}(E)\,\right]\, \frac{1}{r}\, \frac{d}{dr} f_{\mathrm{SO}}(r)\, \mathbf{L}\cdot\mathbf{S} \ &\quad + V_C(r)\,\delta_{N,p} + U_{\mathrm{Lane}}(r,E), \end{aligned}$$9Mg, KDUQ was used via posterior sampling in jitR. Its training set includes elastic $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$0Mg and natMg data up to about 20 MeV and neutron total cross sections up to about 50 MeV, so predictions in the 65–200 MeV range, especially for $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$1Mg, are extrapolations. In that regime KDUQ agrees well with data and with the ab initio calculation at forward angles, while discrepancies grow at larger angles; for proton reaction cross sections, KDUQ centroids lie about $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$2–$$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$3 below the ab initio results across the Mg chain (Sargsyan et al., 30 Mar 2026).

Those Mg studies also clarify the character of KDUQ uncertainties. The bands are generally tight at forward angles and widen at larger angles; for analyzing powers, the uncertainty grows with energy and can span much of the measured range at selected angles. At the same time, the KDUQ uncertainty bands do not expand markedly away from stability for $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$4Mg, which suggests that the linear $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$5 isovector structure likely underestimates epistemic uncertainty at extreme asymmetries (Sargsyan et al., 30 Mar 2026).

A different application is the propagation of optical-model uncertainty into Monte Carlo Hauser–Feshbach calculations of fission-fragment de-excitation. There, KDUQ was sampled to generate neutron transmission coefficients and then propagated to prompt-neutron observables in $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$6Cf(sf) and $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$7U(n$$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$8,f). The resulting analysis found that parametric optical-model uncertainty is significant for neutron-fragment correlated observables involving neutron energy, but features such as neutron-fragment correlations near the $$f_X(r) = \left[1 + \exp\left(\frac{r - R_X}{a_X}\right)\right]^{-1},\qquad R_X = r_X\, A^{1/3}.$$9Sn shell closure and the high-energy component of neutron spectra are unlikely to be explained by the optical potential alone (Beyer et al., 2024).

Across these applications a consistent pattern emerges. Within its calibration regime, KDUQ is most reliable for forward-angle elastic scattering, total cross sections, and moderate extrapolations on stable or near-stable nuclei. Larger-angle diffraction structure, strongly isovector observables, and high-energy or far-from-stability extrapolations expose the limits of a local KD-type functional form (Sargsyan et al., 30 Mar 2026, Shiu et al., 17 Jul 2025).

6. Limitations, controversies, and successor frameworks

Several limitations are now well established. First, KDUQ inherits the locality and phenomenology of Koning–Delaroche; it is not exactly dispersive and does not enforce the nonlocality required by the Feshbach formalism. Second, its isovector dependence is comparatively rigid: later work emphasizes the linear $\Delta E^{n,p} = E - E_f^{n,p}$0 structure and the absence of strict Lane consistency. Third, extrapolations beyond the fitted energy range require caution. In the eikonal study, uncertainties remain small within $\Delta E^{n,p} = E - E_f^{n,p}$1 MeV but rise rapidly above that range, reaching approximately $\Delta E^{n,p} = E - E_f^{n,p}$2 for neutron elastic and total cross sections at 400 MeV (Shiu et al., 17 Jul 2025).

These limitations motivate two distinct successor directions. One is phenomenological but more flexible in the isovector sector. The East Lansing Model is a Bayesian global optical potential for neutrons and protons with a novel asymmetry component, calibrated to $\Delta E^{n,p} = E - E_f^{n,p}$3, $\Delta E^{n,p} = E - E_f^{n,p}$4, and $\Delta E^{n,p} = E - E_f^{n,p}$5 angular distributions on spherical nuclei with $\Delta E^{n,p} = E - E_f^{n,p}$6 and $\Delta E^{n,p} = E - E_f^{n,p}$7–$\Delta E^{n,p} = E - E_f^{n,p}$8 MeV. It enforces Lane symmetry and allows independent geometries for the isoscalar and isovector components, so that $\Delta E^{n,p} = E - E_f^{n,p}$9 data can encode neutron skins directly into the optical potential; in extrapolations toward the driplines, its uncertainties are reduced relative to KDUQ and CHUQ (Beyer et al., 30 Mar 2026).

The second direction is microscopic or dispersive. The WLH global microscopic optical potential provides quantified uncertainties from a chiral-EFT ensemble for approximately 1800 nuclei in the range $\alpha=(N-Z)/A$0 and $\alpha=(N-Z)/A$1 MeV, offering a theory-driven alternative to phenomenological KDUQ (Whitehead et al., 2020). The non-local, exact-dispersive program of 2024 goes further by combining nonlocality, exact dispersion relations, bound-state constraints, and planned Bayesian uncertainty quantification in a single optical-model framework; however, it is explicitly not a KD-derived UQ refit and the paper does not use the term KDUQ (Perrotta et al., 2024).

KDUQ therefore occupies a specific place in the evolution of optical-model methodology. It was the first broadly deployed uncertainty-quantified global reanalysis of the Koning–Delaroche optical model, and it remains a practical reference for posterior-based uncertainty propagation in reaction theory. At the same time, later work increasingly treats it as a baseline against which nonlocal, dispersive, microscopic, or improved-isovector global optical potentials are judged (Pruitt et al., 2022, Beyer et al., 30 Mar 2026).