  The astrophysical S-factor for the proton weak capture on 3He is calculated
with correlated-hyperspherical-harmonics bound and continuum wave functions
corresponding to realistic Hamiltonians consisting of the Argonne v14 or
Argonne v18 two-nucleon and Urbana-VIII or Urbana-IX three-nucleon
interactions. The nuclear weak charge and current operators have vector and
axial-vector components, that include one- and many-body terms. All possible
multipole transitions connecting any of the p 3He S- and P-wave channels to the
4He bound state are considered. The S-factor at a p 3He center-of-mass energy
of 10 keV, close to the Gamow-peak energy, is predicted to be 10.1 10^{-20} keV
b with the AV18/UIX Hamiltonian, a factor of about 4.5 larger than the value
adopted in the standard solar model. The P-wave transitions are found to be
important, contributing about 40 % of the calculated S-factor. The energy
dependence is rather weak: the AV18/UIX zero-energy S-factor is 9.64 10^{-20}
keV b, only 5 % smaller than the 10 keV result quoted above. The model
dependence is also found to be weak: the zero-energy S-factor is calculated to
be 10.2 10^{-20} keV b with the older AV14/UVIII model, only 6 % larger than
the AV18/UIX result. Our best estimate for the S-factor at 10 keV is therefore
(10.1 \pm 0.6) 10^{-20} keV b, when the theoretical uncertainty due to the
model dependence is included. This value for the calculated S-factor is not as
large as determined in fits to the Super-Kamiokande data in which the hep flux
normalization is free. However, the precise calculation of the S-factor and the
consequent absolute prediction for the hep neutrino flux will allow much
greater discrimination among proposed solar neutrino oscillation solutions.
