Complex-energy analysis of proton-proton fusion

Abstract

An analysis of the astrophysical $S$ factor of the proton-proton weak capture ($\mathrm{p}+\mathrm{p}\to {}^2\mathrm{H}+{\mathrm{e}}^{+}+{\nu }_{\mathrm{e}}$) is performed on a large energy range covering solar-core and early Universe temperatures. The measurement of $S$ being physically unachievable, its value relies on the theoretical calculation of the matrix element $\Lambda$. Surprisingly, $\Lambda$ reaches a maximum near $0.13\mathrm{MeV}$ that has been unexplained until now. A model-independent parametrization of $\Lambda$ valid up to about $5\mathrm{MeV}$ is established on the basis of recent effective-range functions. It provides an insight into the relationship between the maximum of $\Lambda$ and the proton-proton resonance pole at $(-140-467\mathrm{i})\mathrm{keV}$ from analytic continuation. In addition, this parametrization leads to an accurate evaluation of the derivatives of $\Lambda$, and hence of $S$, in the limit of zero energy.

13 pages, 9 figures, 43 references