Complete quantification of parametric uncertainties in $(d,p)$ transfer reactions

Background: Deuteron-induced transfer reactions are a popular probe in nuclear structure and nuclear astrophysics studies. The interpretation of these transfer measurements relies on reaction theory that takes as input effective interactions between the nucleons and the target nucleus.

Purpose: Previous work quantified the uncertainty associated with the optical potentials between the nucleons and the target. In this study, we extend that work by also including the parameters of the mean field associated with the overlap function of the final bound state, thus obtaining the full parametric uncertainty on transfer observables.

Method: We use Bayesian Markov chain Monte Carlo simulations to obtain parameter posterior distributions. We use elastic-scattering cross sections to constrain the optical potential parameters and use the asymptotic normalization coefficient of the final state to constrain the bound-state interaction. We then propagate these posteriors to the transfer angular distributions and obtain confidence intervals for this observable.

Results: We study $(d,p)$ reactions on ${}^{14}\mathrm{C}$, ${}^{16}\mathrm{O}$, and ${}^{48}\mathrm{Ca}$ at energies in the range $E_d=7-24$ MeV. Our results show a strong reduction in uncertainty by using the asymptotic normalization coefficient as a constraint, particularly, for those reactions most sensitive to ambiguities in the mean field. For those reactions, the importance of constraining the bound-state interaction is equal to that of constraining the optical potentials. The case of ${}^{14}\mathrm{C}$ is an outlier because the cross section is less sensitive to the nuclear interior.

Conclusions: When minimal constraints are used on the parameters of the nucleon-target interaction, the $1\sigma$ uncertainties on the differential cross sections are large ($\approx 50\text{–}100\%$). However, if elastic-scattering data and the asymptotic normalization coefficient are used in the analysis, with an error of 10% (5%), this uncertainty reduces to $\approx 30\%$ ($\approx 15\%$).

Figure 1

Parameter posteriors and correlations for single-particle parameters of (a) the ${}^{15}\mathrm{C}$ bound state, (b) the ${}^{17}\mathrm{O}$ bound state, and (c) the ${}^{49}\mathrm{Ca}$ bound state, using Bayesian optimization to compare two different experimental errors on $C^2$, ${\varepsilon }_{10}(\varepsilon =10\%$) shown in brown and ${\varepsilon }_{100}(\varepsilon =100\%)$ shown in teal. Histograms on the diagonal correspond to the posterior distributions of the given parameters; the priors are shown in green.

Figure 2

A comparison of results for transfer assuming two $C^2$ experimental data errors, $10\%$ error in brown (${\varepsilon }_{10}$) and $100\%$ error in teal (${\varepsilon }_{100}$). ${}^{14}\mathrm{C}(d,p)$ at 17-MeV $68\%$ confidence intervals (a) and percentage uncertainty (b). ${}^{16}\mathrm{O}(d,p)$ at 15-MeV $68\%$ confidence intervals (c) and percentage uncertainty (d). ${}^{48}\mathrm{Ca}(d,p)$ at 24-MeV $68\%$ confidence intervals (e) and percentage uncertainty (f).

Figure 3

A comparison of results using only ANC ($C^2$) data in brown, only elastic data (el) in bright green, and both types of data simultaneously ($C^2+\text{el}$), all at $10\%$ error (${\varepsilon }_{10}$) in blue and unconstrained in gray. (a) and (b) ${}^{14}\mathrm{C}(d,p)$ at 17-MeV $68\%$ confidence intervals and percentage uncertainty plot. (c) and (d) ${}^{16}\mathrm{O}(d,p)$ at 15-MeV $68\%$ confidence intervals and percentage uncertainty plot. (e) and (f) ${}^{48}\mathrm{Ca}(d,p)$ at 24-MeV $68\%$ confidence intervals and percentage uncertainty plot.

Figure 4

A comparison of results using different combinations of ANC ($C^2$) and elastic (el) constrains, $10\%$ error on both the ANC and the elastic data in blue, $10\%$ error on the ANC and $100\%$ error on the elastic in red, $10\%$ error on the elastic and $100\%$ error on the ANC in green, and unconstrained ANC and elastic data ($100\%$ error on both) in gray. (a) and (b) ${}^{14}\mathrm{C}(d,p)$ at 17-MeV $68\%$ confidence intervals and percentage uncertainty plot. (c) and (d) ${}^{16}\mathrm{O}(d,p)$ at 15-MeV $68\%$ confidence intervals and percentage uncertainty plot. (e) and (f) ${}^{48}\mathrm{Ca}(d,p)$ at 24 MeV $68\%$ confidence intervals and percentage uncertainty plot.

Figure 5

A comparison of results using different errors on both ANC ($C^2$) and elastic (el) data, $5\%$ error in magenta, $10\%$ error in blue, and unconstrained data ($100\%$ error) in gray. (a) and (b) ${}^{14}\mathrm{C}(d,p)$ at 17-MeV $68\%$ confidence intervals and percentage uncertainty plot. (c) and (d) ${}^{16}\mathrm{O}(d,p)$ at 15-MeV $68\%$ confidence intervals and percentage uncertainty plot; (e) and (f) ${}^{48}\mathrm{Ca}(d,p)$ at 24-MeV $68\%$ confidence intervals and percentage uncertainty plot.