A Derivation of the Fermi Function in Perturbative Quantum Field Theory

TL;DR

This paper derives the Fermi function directly from quantum field theory amplitudes, providing a novel approach to calculating beta decay widths and expanding the function to higher orders in the fine-structure constant.

Contribution

It introduces a method to derive the Fermi function from amplitudes rather than solutions of the Dirac equation, with explicit calculations for neutron beta decay.

Findings

Derived the neutron beta decay width to one loop order

Expanded the Fermi function to order α and α^2

Confirmed absence of similar terms in a related process

Abstract

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.