  The inverse $\beta$-decay reaction, $ \bar{\nu}_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 derived 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.
