  The astrophysical S-factor for proton-proton fusion, S_11(E), is obtained
with the nuclear matrix element analytically calculated in pionless effective
field theory. To the third order, the zero-energy result S_11(0) and the first
energy derivative S'_11(0) are found to be (3.99 \pm 0.14)* 10^-25 MeV b and
S_11(0)*(11.3 \pm 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 S"_11(0) = S_11(0)*(170 \pm 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.
