Spurred by the recent complete determination of the weak currents in two-nucleon systems up to ${\cal O}(Q^3)$ in heavy-baryon chiral perturbation theory, we carry out a parameter-free calculation of the threshold $S$-factors for the solar $pp$ (proton-fusion) and $hep$ processes in an effective field theory that {\it combines} the merits of the standard nuclear physics method and systematic chiral expansion. The power of the EFT adopted here is that one can correlate in a unified formalism the weak-current matrix elements of two-, three- and four-nucleon systems. Using the tritium $\beta$-decay rate as an input to fix the only unknown parameter in the theory, we can evaluate the threshold $S$ factors with drastically improved precision; the results are $S_{pp}(0) = 3.94\times(1 \pm 0.004) \times 10^{-25} {MeV-b}$ and $S_{hep}(0) = (8.6\pm 1.3)\times 10^{20} {keV-b}$. The dependence of the calculated $S$-factors on the momentum cutoff parameter $\Lambda$ has been examined for a physically reasonable range of $\Lambda$. This dependence is found to be extremely small for the $pp$ process, and to be within acceptable levels for the $hep$ process, substantiating the consistency of our calculational scheme.

Authors: T.S. Park, L.E. Marcucci, R. Schiavilla, M. Viviani, A. Kievsky, S. Rosati, K. Kubodera, D.P. Min, M. Rho

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