Uncertainty quantification due to optical potentials in models for ($d,p$) reactions

Background: Recent work has studied the uncertainty in predictions for $A(d,p)B$ reactions using the distorted-wave Born approximation (DWBA), coming from the parametrization of the effective $dA$ interactions [Lovell et al. Phys. Rev. C 95, 024611 (2017)]. There are different levels of sophistication in reaction theories for one-nucleon transfer reactions, including the adiabatic wave approximation (ADWA) which takes deuteron breakup into account to all orders.

Purpose: In this paper, we quantify the uncertainties associated with the ADWA method that come from the parametrization of the NA interactions and compare ADWA with DWBA.

Method: Similar to Lovell et al. [Phys. Rev. C 95, 024611 (2017)], we use nucleon elastic-scattering data on a wide variety of targets at the appropriate incoming and outgoing energies to constrain the optical potential input to the ADWA theory. Pulling from the ${\chi }^2$ distribution, we obtain 95% confidence bands for the elastic distributions. From the resulting parameters, we predict 95% confidence bands for the $(d,p)$ transfer cross sections. Results obtained with the standard uncorrelated ${\chi }^2$ are compared to those using the correlated ${\chi }^2$ of Lovell et al. [Phys. Rev. C 95, 024611 (2017)]. We also repeat the DWBA calculations for the exact same reactions for comparison purposes.

Results: We find that NA elastic-scattering data provide a significant constraint to the interactions, and, when the uncertainties are propagated to the transfer reactions using ADWA, predictions are consistent with the transfer data.

Conclusions: The angular distributions for ADWA differ from those predicted by DWBA, particularly at small angles. As in Lovell et al. [Phys. Rev. C 95, 024611 (2017)], confidence bands obtained using the uncorrelated ${\chi }^2$ function are unrealistically narrow and become much wider when the correlated ${\chi }^2$ function is considered. For most cases, the uncertainty bands obtained in ADWA are narrower than DWBA when using elastic data of similar quality and range. However, given the large uncertainties predicted from the correlated ${\chi }^2$ function, at this point, the transfer data cannot discriminate between these two methods.

Figure 1

Elastic scattering on ${}^{90}\mathrm{Zr}$: (a) protons at 12.7 MeV, (b) neutrons at 10 MeV, and (c) protons at 22.5 MeV; best-fit predictions using the uncorrelated ${\chi }^2$ (red-dashed line) and correlated ${\chi }^2$ (green-dotted line); the 95% confidence bands using the uncorrelated (red-dashed-hashed band) and correlated ${\chi }^2$ (green-dotted-hashed band). Data from Refs. [25, 26, 27].

Figure 2

ADWA angular distribution for ${}^{90}\mathrm{Zr}(d,p)$ at 22.7 MeV: uncorrelated and correlated 95% confidence band predictions. Best-fit predictions are shown by the red-dashed line (uncorrelated), green-dotted line (correlated with uncorrelated initialization), and blue-dot-dashed line (correlated with BG initialization). Data from Ref. [37].

Figure 3

Elastic scattering for ${}^{90}\mathrm{Zr}(d,d)$ at 23.2 MeV, best-fit predictions using the uncorrelated ${\chi }^2$ (red-dashed line) and correlated ${\chi }^2$ (green-dotted line), and the corresponding 95% confidence bands. Data from Ref. [24].

Figure 4

Comparison of the DWBA correlated and uncorrelated confidence band predictions for ${}^{90}\mathrm{Zr}(d,p)$ at 22.7 MeV: DWBA using the uncorrelated ${\chi }^2$ (red-dashed line) and $\mathrm{DWBA}\text{-C}1$ using the correlated ${\chi }^2$ (green-dotted line). $\mathrm{DWBA}\text{-C}1$ corresponds to the correlated fit using the global potential AC [38] as initial parameters. Data from Ref. [37].

Figure 5

Comparison of the ADWA and the DWBA confidence band predictions for ${}^{90}\mathrm{Zr}(d,p)$ at 22.7 MeV: (a) uncorrelated and (b) correlated. Data from Ref. [37].

Figure 6

Comparison of the ADWA and the DWBA confidence band predictions for ${}^{48}\mathrm{Ca}(d,p)$ at 19.3 MeV: (a) uncorrelated and (b) correlated. Data from Ref. [39].

Figure 7

Comparison of the ADWA and the DWBA confidence band predictions for ${}^{208}\mathrm{Pb}(d,p)$ at 32.9 MeV: (a) uncorrelated and (b) correlated.