Models, measurements, and effective field theory: Proton capture on Be 7 at next-to-leading order

We employ an effective field theory (EFT) that exploits the separation of scales in the $p$-wave halo nucleus $^{8}\mathrm{B}$ to describe the process $^{7}\mathrm{Be}(p,\ensuremath{\gamma})^{8}\mathrm{B}$ up to a center-of-mass energy of 500 keV. The key leading-order (LO) and next-to-leading-order (NLO) results appeared in our earlier papers. Here we first present full details of the EFT calculation. We develop the lagrangian and power counting in terms of velocity scaling, thereby making manifest that the Coulomb force between $^{7}\mathrm{Be}$ and proton plays a major role in both scattering and radiative capture at these energies: Coulomb interactions must be included to all orders in ${\ensuremath{\alpha}}_{\mathrm{em}}$. The EFT calculation of the capture reaction is then carried out using Feynman diagrams computed in time-ordered perturbation theory, so we recover existing quantum-mechanical technology such as the Lippmann-Schwinger equation and the two-potential formalism for the treatment of the Coulomb-nuclear interference. Meanwhile, the strong interactions and the E1 operator are dealt with via EFT expansions in powers of momenta, with a breakdown scale set by the size of the $^{7}\mathrm{Be}$ core, $\mathrm{\ensuremath{\Lambda}}\ensuremath{\approx}70$ MeV/c. This is worked out up to NLO in the EFT expansion; at this order the relevant physics in the different channels that enter the radiative capture reaction is encoded in ten different EFT couplings. The result is a model-independent parametrization for the reaction amplitude in the energy regime of interest. In the second part of the paper we consider other approaches that have been used to describe $^{7}\mathrm{Be}{(p,\ensuremath{\gamma})}^{8}\mathrm{B}$ in this energy range. We discuss the relationship of EFT to each of these approaches in qualitative terms and then make the connection quantitative by determining what the ten NLO EFT coefficients are in five different calculations that we consider representative. The EFT parameters are of natural size in all five cases, which shows that each of these earlier calculations corresponds to a particular point in the EFT parameter space. This understanding of the relationship between EFT and other ways of computing $^{7}\mathrm{Be}{(p,\ensuremath{\gamma})}^{8}\mathrm{B}$ allows us to update earlier results for the dependence of $S(0)$ on asymptotic normalization coefficients and scattering lengths, since EFT separates dependence on these asymptotic quantities from dependence on shorter-distance contributions to the matrix element. We also summarize the fit to experimental capture data presented in our earlier work and explain why we obtain an extrapolated $S(0)$ with a markedly smaller error bar than that of the previous standard evaluation. Finally, we demonstrate that the only ${\mathrm{N}}^{2}\mathrm{LO}$ corrections in $^{7}\mathrm{Be}(p,\ensuremath{\gamma})^{8}\mathrm{B}$ come from an inelasticity that is practically of $\mathrm{N}^{3}\mathrm{LO}$ size in the energy range of interest, and so the truncation error in our calculation is effectively $\mathrm{N}^{3}\mathrm{LO}$.