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 quantum field theory. Then, we obtain the following results:(1) We give the amplitude and width of the neutron beta decay, n -> p + e(-) + (nu) over bar (e), to the first order in alpha. We evaluate it using Feynman parameters.(2) As the result, we confirm the terms, which can be interpreted as the Fermi function expanded to order alpha.(3) We give the same result using the contour integral.(4) We check that there are no such terms in a similar process, (nu) over bar (e) + p -> e(+) + n. (5) We perform the Fermi function expanded to the second order in alpha using contour integral. (6) The conventional Fermi function affects the convergence of perturbation theory. DOI: 10.1103/PhysRevC.86.065502