Corrections to nucleon capture cross sections computed in truncated Hilbert spaces

Nucleon capture cross sections enter various astrophysical processes. The measurement of proton capture on nuclei at astrophysically relevant low energies is a challenge, and theoretical computations in this long-wavelength regime are sensitive to the long-distance asymptotics of the wave functions. A theoretical foundation for estimating and correcting errors introduced in capture cross sections due to Hilbert space truncation has so far been lacking. We derive extrapolation formulas that relate the infrared regularized capture amplitudes to the infinite basis limit and demonstrate their efficacy for proton-proton fusion. Our results are thus relevant to current calculations of few-body capture reactions such as proton-proton fusion or proton capture on the deuteron, and they also open the way for the use of ab initio many-body wave functions represented in finite Hilbert spaces in precision calculations of nucleon capture on heavier nuclei.

Figure 1

IR truncation error in the axial-vector matrix element of $pp$ fusion at 50-keV center-of-mass energy (black circles) and in the deuteron binding energy (blue squares). The relative errors for the axial-vector matrix elements and those for the binding energy are expressed in percentage and parts per million (ppm) respectively.

Figure 2

Numerical and analytic results for the correction $\mathrm{\Delta }{\mathcal{I}}_0(k;\eta ;L)$ to the radial overlap between the ${{}^{1}S}_{0}pp$ scattering wave function at 50-keV center-of-mass energy and the deuteron $s$-wave state.

Figure 3

Numerical and analytic results for the correction $\mathrm{\Delta }{\mathcal{I}}_0(k;\eta ;L)$ to the radial overlap between the ${{}^{1}S}_{0}pp$ scattering wave function at 1-MeV center-of-mass energy and the deuteron $s$-wave state.

Figure 4

Numerical and analytic results for the correction $\mathrm{\Delta }{\mathcal{I}}_1(k;\eta ;L)$ to the radial matrix element of the $E1$ operator between the $np{}^3{P}_1$ scattering wave function at 1-MeV center-of-mass energy and the deuteron $s$-wave state.