The Fermi Function Beyond The Second Order Perturbation Theory

The Fermi function is historically derived from the Dirac equation or the Schr\"odinger equation. However, we claim that the Fermi function should be derived from quantum field theory. Then, we obtain the following results: (1) We give the electromagnetic correction of the beta decay to the fourth order in $\alpha/v$, where $v$ is the electron velocity. (2) Our result is consistent with the ordinary non-relativistic Fermi function. (3) We give the iterated integral representation of the Fermi function for arbitrary order in $\alpha/v$. (4) This representation is related to the multiple L values, and they enable us to calculate the amplitude in the algebraic approach.