  The proton-proton fusion reaction, $pp\to de^+\nu$, is studied in pionless
effective field theory (EFT) with di-baryon fields up to next-to leading order.
With the aid of the di-baryon fields, the effective range corrections are
naturally resummed up to the infinite order and thus the calculation is greatly
simplified. Furthermore, the low-energy constant which appears in the
axial-current-di-baryon-di-baryon contact vertex is fixed through the ratio of
two- and one-body matrix elements which reproduces the tritium lifetime very
precisely. As a result we can perform a parameter free calculation for the
process. We compare our numerical result with those from the accurate potential
model and previous pionless EFT calculations, and find a good agreement within
the accuracy better than 1%.
