Near threshold proton-proton fusion in effective field theory

The astrophysical S-factor for proton-proton fusion, S11(E), is obtained with the nuclear matrix element analytically calculated in pionless effective field theory. To the third order, the zero-energy result S11(0) and the first energy derivative S11′(0) are found to be (3.99±0.14)×10−25MeV b and S11(0)(11.3±0.1)MeV−1, respectively; both consistent with the current adopted values. The second energy derivative is also calculated for the first time, and the result S11″(0)=S11(0)(170±2)MeV−2 contributes at the level of 0.5% to the fusion rate at the solar center, which is smaller than 1% as previously estimated.