The finite-range ADWA [14] starts out by considering a full three-body picture for the transfer reaction A ( d , p ) B . As detailed in Ref. [14], it uses Weinberg states to then simplify the T-matrix to T ADWA = 〈 ϕ n A χ p B − | V n p | ϕ n p χ d a d 〉 . (1) In Equation 1, ϕ n p and ϕ n A correspond to the deuteron bound state and single-particle state of the final nucleus B, V n p is the neutron-proton interaction and χ p B is the outgoing distorted wave of the proton relative to the final nucleus B , obtained with the optical potential U p B at the energy of the outgoing proton. The adiabatic distorted wave χ d a d is generated from the effective adiabatic potential: U A p eff = − 〈 ϕ o | V n p U n A + U p A | ϕ o 〉 , where U n A and U p A are the nucleon optical potentials between neutron/proton and the target evaluated at half the deuteron incoming energy. The wave function ϕ o is the first Weinberg basis state, which is directly proportional to the deuteron bound state [14]. The T-matrix of Equation 1 assumes that the remnant term ( U n A − U p B ) is negligible. In ADWA calculations, the sources of parametric uncertainties are therefore the optical potentials used to generate the scattering states and the single-particle potentials used to model bound states.

More details about the ADWA and how to obtain numerical solutions for bound and scattering states can be found in Ref. [31]. In this work, we use the code NLAT [15] to perform all ADWA transfer calculations and the code FRESCO [32] to perform the elastic scattering calculations.

As mentioned in Section 1, we use the global optical potentials KDUQ [25] for all nucleon-nucleus interactions needed in the reaction model of Section 2.1, which is valid for 24 ≤ A ≤ 209 and 1 keV ≤ E ≤ 200 MeV. In this work, we chose the democratic version of KDUQ which weighs every data point equally ¹ . By performing a Bayesian calibration using a large set of reaction data (including nucleon elastic scattering angular distributions and analyzing powers, neutron total cross sections and proton reaction cross sections, all on stable nuclei), the authors of KDUQ obtained parameter posterior distributions and correlations for the 46 parameters of their global optical model. In this work, we use the 416 samples of their posterior distributions, published in Ref. [25], to compute the uncertainties in the transfer cross sections. We quantify the uncertainty in the transfer angular distribution in terms of the relative half-width of the 1 σ credible interval at the peak of the angular distributions (corresponding to a scattering angle θ max ):

ε 68 % = σ max 68 % θ max − σ a v g 68 % θ max σ a v g 68 % θ max (2) with σ a v g 68 % θ max = σ max 68 % θ max + σ min 68 % θ max 2

The test case we focus on is based on the ⁴⁸Ca ( d , p ) 49 Ca (g.s) reaction. Our choice is mainly motivated by the availability of elastic and transfer data. Moreover, since no ⁴⁸Ca data were included in the KDUQ corpus, the analysis performed in this work share similar challenges as the ones on transfer data populating exotic nuclei. All optical potentials are taken consistently from KDUQ and evaluated at the relevant energies, i.e., all three nucleon-nucleus optical potentials are derived from the same KDUQ sample. This consistent treatment hence includes correlations between the various optical potential parameters. For describing the final state of the neutron, we take a standard radius and diffuseness r R = 1.25 fm and a R = 0.65 fm (STD) or we take the geometry of the real part of the KDUQ interaction (KDUQ-real). In both cases, we adjust the depth to reproduce the neutron separation energy and the parameters in the spin orbit term for the final bound-state potential are fixed: the depth is V s o = 6 MeV, the radius is r s o = 1.25 fm and the diffuseness is a s o = 0.65 fm. In the physical ⁴⁹Ca ground state, the neutron is in a 1 p 3 / 2 orbital, bound by S n = 5.146 MeV. We also consider in Section 3.3 other configurations for the final state being populated in the ( d , p ) reaction.

It must be underlined that KDUQ posterior distributions contain no constraints from bound state data. Our assumption is to consider that the geometry of the mean field generated by the target nucleus does not change considerably from bound to scattering states. This is consistent with the KDUQ assumption, since it uses an energy-independent parametrization for the radius and diffuseness of the real term, and these parameters are well constrained by the Bayesian calibration. When using 416 samples for the real radius and diffuseness of KDUQ and refitting the real depth to reproduce the correct neutron separation energy (KDUQ-real). The resulting ANC-squared C 2 distribution (not shown) is slightly multimodal and is well constrained: its half-width is about 5%. Interestingly, the value predicted by KDUQ-real C 2 = 28.6 ± 1.3 fm − 1 is consistent with the value C 2 = 32.1 ± 3.2 fm − 1 [33] determined from the analysis of various ⁴⁸Ca ( d , p ) 49 Ca (g.s.) datasets at various energies, i.e., 2 MeV, 13 MeV, 19 MeV, 30 MeV and 56 MeV [34–36]. This surprising agreement seems fortuitous, as ANCs for states of ⁸⁷Kr, ⁸B and ¹⁰B predicted by KDUQ do not match the values extracted from transfer and breakup data [37–39]. The KDUQ-real posteriors obtained in this way will also be used to quantify uncertainties from the neutron singe-particle interaction in ϕ n A .

Optical potentials are determined primarily from observables that are sensitive to the on-shell T-matrix. Transfer observables are also sensitive to properties of the T-matrix off-shell. It is thus not guaranteed, even if the optical potentials reproduce the corresponding elastic channels, that they describe the transfer data. Using the reaction ⁴⁸Ca ( d , p ) 49 Ca (g.s) at 19 MeV, for which there is data [35]. ² We compare predictions using KDUQ with the corresponding data to assess the quality of the uncertainty quantification. We select nucleon elastic scattering data that is close to the energies relevant to the transfer reaction of interest ( E n = 9.5 MeV and E p = 9.5 MeV in the entrance channel and E p = 21.9 MeV in the exit channel) and compare the credible intervals generated from the KDUQ parameter posteriors with the actual data (with the quoted experimental error bars) [40–42]. The corresponding angular distributions are shown in Figure 1: (a) neutron elastic scattering for E lab = 12 MeV, (b) proton elastic scattering for E lab = 14 MeV, (c) proton elastic scattering for E lab = 25 MeV. Note that the KDUQ global optical potential was not fitted on these data sets. The dark (light) shade corresponds to the 68% (95%) credible intervals ³ .

The empirical coverage provides a sanity check for uncertainty quantification. Our empirical coverages for a x% model uncertainty are calculated as the number of data points, including a x% experimental error, that fall into the theoretical x% credible interval divided by the total number of data points in an angular distribution. These are shown in Figures 1D–F for the corresponding three elastic scattering examples. In an ideal situation, the predicted empirical coverage should line up with the black diagonal line ⁴ . In our calculations for proton elastic scattering, empirical coverages calculated at the high-confidence level are only slightly underestimated. However, for neutron scattering, there is a severe mismatch. This suggests that, in this case, the error on the data [40] and/or on the KDUQ parameters are seriously under-reported. To include unaccounted-for uncertainties, we have inflated the width of the posterior distributions of the depths, radii and diffuseness of the neutron-target potential, so that we can reproduce the correct empirical coverage specifically for 1 σ . We do this by approximating the parameter distributions of the neutron-target potential U n A to a multivariate Gaussian. We then rescale the covariance matrix by a factor: it turns out that we need to rescale these by 38, effectively rescaling uncertainties by a factor of 38 ∼ 6 . Note that such approach, although simplistic, allows to keep the correlations between the optical potentials parameters informed by the large KDUQ corpus. We refer to this as KDUQ-n. Replacing KDUQ by KDUQ-n for U n A results in the green bands in Figure 1A and the green dots in Figure 1D. As can be seen, the empirical coverage obtained for the 68% is now exactly 68%.

We now use these parameter posterior distributions and propagate the uncertainties to the ⁴⁸Ca ( d , p ) 49 Ca (g.s.) reaction at beam energy E lab = 19 MeV. In Figure 2, we show the predicted credible intervals for the corresponding transfer angular distributions: we compare the results obtained with the original KDUQ (blue bands) and those obtained when the neutron interaction is replaced by KDUQ-n (green bands). In these calculations, we fix the neutron bound state using the STD geometry (discussed in Section 2.2). Figure 2 already includes the scaling by the SF, taking into account both the optical potential parameter uncertainties propagated in the ADWA model and the experimental error on the transfer data. This is done by adding in quadrature the errors ε opt associated with the optical potential and ε ̄ S F resulting from the fitting procedure to the transfer data. The uncertainty ε opt is the standard deviation of the 416 SFs minimizing the χ 2 obtained from the transfer data and the theoretical predictions, e.g., obtained with each KDUQ sample. The variance ε ̄ S F 2 is calculated by averaging the variances on the SFs associated with each of the 416 fits. Further tests have shown that the total uncertainty on the SF is completely dominated by ε opt for transfer data errors ≲ 10 % , i.e., the total errors on the SFs are the same regardless of we include ε ̄ S F or not.

The theoretical predictions for d σ d Ω agree well with the data. At the 1 σ level, the relative half-width ε 68 % Equation 2 is 5% (16%) when using KDUQ (KDUQ-n). These uncertainties can be compared with the 20% full-width uncertainty shown in green in Figure 3 of [21] for the same reaction, at slightly higher beam energy. Note that the work in [21] uses local potentials with mock data (with 10% error) and systematically renormalizes the likelihood, effectively increasing the error on both proton and neutron optical potentials. These results demonstrate the benefit of a global parametrization: although there are no ⁴⁸Ca elastic angular distributions in the data corpus used to calibrate the KDUQ parameters, the uncertainties on the transfer predictions are reliable (of the Ca isotopes, the KDUQ data corpus includes only ⁴⁰Ca elastic angular distributions).

To complement our analysis, we also study the uncertainties associated with the single-particle potential used to generate the final bound state. We quantify its uncertainties using the geometry of the real part of the KDUQ interaction (KDUQ-real), as discussed in Section 2.2. We consider various cases in Table 1 for which we compute the relative half-widths of the transfer angular distribution ε 68 % and of the extracted SF, which also account for the errors on the experimental transfer data. Specifically, we investigate if the geometry of the single-particle potentials, used to generate the final state, has an impact on the relative uncertainties due to the optical potentials in transfer observables. First, we include only uncertainties in the optical potential, for two choices of single-particle potentials. In the first two lines of Table 1, we compare the results obtained when using the STD single-particle potential ( r R = 1.25 fm and a R = 0.65 fm) and the mean values of the real radius and diffuseness of KDUQ-real ( r R = 1.17 fm and a R = 0.689 fm). One can see that the relative uncertainties stay rather constant, showing that the magnitude of relative uncertainties due to the optical potentials do not depend strongly on the geometry of the single-particle potential in the final state.

Then, we evaluate the uncertainties due to the choice of single-particle potentials, using the geometry of the real volume term of 416 KDUQ and KDUQ-n samples. In the KDUQ case, the uncertainty half-widths ε 68 % are below 4% for both the peak of the distribution and the extracted SF, indicating that the KDUQ posterior distributions for r R and a R are quite constrained. The magnitude of the uncertainties on the peak of the transfer cross sections are similar to those on the ANC squared (see Section 2.2), suggesting that this reaction is quite peripheral. As expected, the uncertainties when using the KDUQ-n samples are larger. Interestingly, they are scaled by roughly the same factor 38 ∼ 6 , as the one used to scale the KDUQ uncertainties (compare left and right column of the third line). This suggests that the uncertainties in the single-particle radius and diffuseness seem to propagate linearly to transfer observables. Finally, we include both the uncertainties due to the single-particle potential and the optical potentials (last line). Contrary to what was found in [21], in both cases KDUQ original and KDUQ-n, the total uncertainties cannot be deduced simply by summing in quadrature the two source uncertainties, hinting at the presence of strong correlations. These correlations are due to the interplay between the extension of the single-particle wavefunction, and the range of the real part of the neutron-target optical potential. Although in [21] the ANC-squared is explicitly used as a constrain, the single-particle potential parameter sampling is independent of the sampling of the optical potential parameters, whereas in this work, KDUQ-real used for the single-particle potential is perfectly correlated to KDUQ (or KDUQ-n) used for the optical potentials.

Having established a realistic foundation for the uncertainty estimates of the angular distributions of the ⁴⁸Ca ( d , p ) 49 Ca (g.s.) reaction at E lab = 19 MeV, for which we can compare to experimental data, we now explore how these uncertainties change with beam energy and how they evolve with various properties of the final bound state. In this exploration, no uncertainty quantification is included for the bound state interactions - the mean field that binds the neutron in the final state is kept fixed. We vary either the kinematics or the structure of the final state, and take the optical potential parameters from the same original KDUQ posterior distributions. This is done in order to show how the same parameter posteriors for optical potentials propagate through the model differently, depending on the details of the reaction. Obviously, because we are not including the additional error in KDUQ-n, nor the uncertainty in the bound state interaction, the overall magnitude of the uncertainty estimates shown in Sections 3.2 and 3.3 are underestimated. It is their variation with beam energy or single-particle properties that matters.

We first analyze the dependence of the uncertainties on the beam energy. For this study, we keep the final bound state fixed using the STD single-particle geometry as described in Section 3.1, and take all optical potential posteriors from the original KDUQ parametrization. The relative half-width ε 68 % Equation 2, evaluated at the peak of the transfer angular distribution for ⁴⁸Ca ( d , p ) 49 Ca (g.s.), are shown in Figure 3. There is no convolution with the experimental error on the transfer data in this plot; only the theoretical uncertainties are considered.

We first consider ADWA calculations using all three optical potentials derived consistently from the same KDUQ sample (blue line). We find that the relative half-width ε 68 % Equation 2 increases with the beam energy ⁵ . This can be explained by the fact that at higher beam energy, transfer observables become more sensitive to the short-range part of scattering wave functions, which are typically less constrained by observables used to calibrate optical potentials. This explanation was verified by comparing the relative uncertainties obtained when computing the short-range contribution to the radial integral of the T-matrix Equation 1, i.e., considering only d -⁴⁸Ca distances smaller than R < r R ∗ 4 8 1 / 3 , to the uncertainties associated with the long-range contribution to the T-matrix.

To investigate the importance of correlations in the uncertainties of the optical potentials, we consider ADWA calculations using optical potentials derived from different KDUQ samples (black line). For almost all beam energies, the relative uncertainties are slightly larger than in the previous results, where all potentials were derived consistently from the same KDUQ sample. At the highest beam energies studied, the shift in ε 68 % is about 25%.

Next, we consider the effects of different properties of the final bound state, namely, the dependence of the uncertainty with the r.m.s Radius squared ⟨ r 2 ⟩ , the angular momentum l , the number of nodes n r and the separation energy S n .

We first consider the effect of the single-particle potential radius r R used to generate the final bound state wave function on the reaction ⁴⁸Ca ( d , p ) 49 Ca (g.s) at 23 MeV. We take the original n -⁴⁸Ca bound wave function, in a 1 p 3 / 2 orbital, and vary the mean field radius parameter in STD in the range r R = 1.0 − 1.4 fm, along with the depth to reproduce the same separation energy. The results are shown in Figure 4: the transfer angular distributions for a range of single-particle potential radii are shown (on the left, panels a–e) and the relative half-width ε 68 % Equation 2 as a function of the r.m.s Single-particle radius squared (on the right, panel f). The same message is relayed when plotting the uncertainty estimates as a function of the ANC squared.

We find that the diffraction pattern of the transfer angular distributions do not change significantly with radius. Expectedly, the magnitude of the transfer cross section increases with the single-particle potential radius r R . Since these reactions are primarily peripheral, they scale with the ANC squared, which in turn is directly related to the r.m.s radius squared. However, the percent uncertainty remains roughly constant and small, similar to what was observed in Table 1. Further tests have shown that the same conclusions, i.e., independence of the shape, larger magnitude, small and constant uncertainties for the transfer cross sections, can be drawn when increasing the diffuseness a R .

Next, we consider the dependence on the separation energy of the final state, of the uncertainty for the transfer cross section due to the optical potentials. We fix the STD geometry for the neutron single-particle potential to the original values ( = 1.25 fm and a R = 0.65 fm) and adjust the depth of this interaction to reproduce the neutron separation energies S n = 1.146 -15.146 MeV in the 1 p 3 / 2 wave. We repeat the procedure considering a bound state in a 0 p 3 / 2 orbital. The corresponding wave functions are shown in Figures 5A,B. The lower the separation energy, the more extended is the single-particle wave function. The resulting ( d , p ) angular distributions in Figure 6, panels (a-h) (resp. (i-n)) are obtained with a final bound state in the 1 p 3 / 2 (resp. 0 p 3 / 2 ) wave. In both cases, the peak of the angular distribution shifts to large angles for larger separation energy, as this directly increases the Q-value for the reaction and the momentum matching. The magnitude of the cross section is also affected: the cross sections are larger for bound states with smaller separation energies. This is due to the spatial extension of the final bound-state wave function.

The resulting relative half-width ε 68 % Equation 2 are summarized in Figure 7. Here we include not only the uncertainties due to the optical potentials for transfer observables populating a bound state in the 1 p 3 / 2 orbital (blue) and in the 0 p 3 / 2 orbital (red), but also in the 1 s 1 / 2 orbital (magenta). The uncertainties remain small (below 10%), regardless of separation energy, the number of nodes n r and the angular momentum l of the neutron orbital in the final state.