A Derivation of the Fermi Function in Perturbative Quantum Field Theory
  We postulate that the Fermi function should be derived from the amplitude, not from the solution of the Dirac equation, in the quantum field theory.   Then, we obtain the following results.   1, We give the amplitude and the width of the neutron beta decay, $n \to p + e^- + \bar \nu_e $ to one loop order. It is carried out by the Feynman parameter integration.   2, As the result, we find the terms which can be interpreted as the Fermi function expanded to order $\alpha$.   3, We also give the same result using complex analysis.   4, We check that there are no such terms in the similar process, $\bar \nu_e + p \to e^+ + n$.   5, We perform the Fermi function expanded to order $\alpha^2$ using complex analysis. 
Akihiro Matsuzaki and Hidekazu Tanaka
hep-ph nucl-th