The inverse $beta$-decay reaction, $ bar{ u}_e p to e^+ n$, for low-energy anti-neutrinos coming from nuclear reactors is of great current interest in connection with high-precision measurements of the neutrino mixing angle $theta_{13}$. We have deri
ved analytic expressions, up to next-to-leading order in heavy-baryon chiral perturbation theory, for the radiative corrections (RCs) and the nucleon-recoil corrections both for this reaction and for the related neutron $beta$-decay process. We show that the recoil corrections, which include the weak magnetism contribution, are small for neutron $beta$-decay, but for inverse $beta$-decay, the recoil corrections are comparable in size to the RCs for typical energies of reactor anti-neutrinos, and they have opposite signs. The RCs and the recoil corrections exhibit very different dependences on the neutrino energy.
