# Complex-energy analysis of proton-proton fusion

## TL;DR

This paper provides a model-independent parametrization of the astrophysical S factor for proton-proton fusion, revealing a maximum near 0.13 MeV and linking it to the resonance pole, with implications for solar and early universe conditions.

## Contribution

It introduces a new, model-independent parametrization of the matrix element Lambda for proton-proton fusion, connecting the S factor maximum to the resonance pole and enabling precise derivative calculations.

## Key findings

- Lambda reaches a maximum near 0.13 MeV
- The parametrization is valid up to about 5 MeV
- Accurate derivatives of S at zero energy are obtained

## Abstract

An analysis of the astrophysical $S$ factor of the proton-proton weak capture ($\mathrm{p}+\mathrm{p}\rightarrow {}^2\mathrm{H}+\mathrm{e}^++\nu_{\mathrm{e}}$) 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.