Constraining transfer cross sections using Bayes' theorem

Abstract

Being able to rigorously quantify the uncertainties in reaction models is crucial to moving this field forward. Even though Bayesian methods are becoming increasingly popular in nuclear theory, they are yet to be implemented and applied in reaction theory problems. The purpose of this work is to investigate, using Bayesian methods, the uncertainties in the optical potentials generated from fits to elastic scattering data and the subsequent uncertainties in the transfer predictions. We also study the differences in two reaction models where the parameters are constrained in a similar manner, as well as the impact of reducing the experimental error bars on the data used to constrain the parameters. We use Bayes' Theorem combined with a Markov Chain Monte Carlo to determine posterior distributions for the parameters of the optical model, constrained by neutron-, proton-, and/or deuteron-target elastic scattering. These potentials are then used to predict transfer cross sections within the adiabatic wave approximation or the distorted-wave Born approximation. We study a number of reactions involving deuteron projectiles with energies in the range of $10-25$ MeV/u on targets with mass $A=48-208$. The case of $^{48}$Ca(d,p)$^{49}$Ca transfer to the ground state is described in detail. A comparative study of the effect of the size of experimental errors is also performed. Five transfer reactions are studied, and their results compiled in order to systematically identify trends. Uncertainties in transfer cross sections can vary significantly (25-100\%) depending on the reaction. While these uncertainties are reduced when smaller experimental error bars are used to constrain the potentials, this reduction is not trivially related to the error reduction. We also find smaller uncertainties when using the adiabatic formulation than when using distorted-wave Born approximation.