The astrophysical S -factor for the proton weak capture on ^ { 3 } He is calculated with correlated-hyperspherical-harmonics bound and continuum wave functions corresponding to realistic Hamiltonians consisting of the Argonne v _ { 14 } or Argonne v _ { 18 } 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 ^ { 3 } He S- and P-wave channels to the ^ { 4 } He bound state are considered .
The S -factor at a p ^ { 3 } He center-of-mass energy of 10 keV , close to the Gamow-peak energy , is predicted to be 10.1 \times 10 ^ { -20 } keV b with the AV18/UIX Hamiltonian , a factor of \simeq 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 \times 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 \times 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 ) \times 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 .