The calculation of radiative corrections to the pion beta decay
rate can be taken from the nuclear independent radiative corrections
to 
transitions in nuclear beta decay. At 
,
these corrections neglect the strong interaction effects in nuclear
beta decay, and can be applied to the purely weak pion beta decay.
This radiative correction function is based on a function
which has been derived by Sirlin [12], and
takes the form
where m is the electron mass, 
is the electron end-point
energy, p is the electron momentum, 
, and L(x) is the
Spence function:
Integrating the Spence function from 0 to 
,
one obtains L=-1.5934.
When applied to pion beta decay and averaged over the electron spectrum,
the function 
becomes [18]
where 
is the 
mass and 
is the 
mass.
To a good approximation, the denominator is found to be
[12].
Integrating the numerators numerically over the electron energy, one
obtains 
. This can be compared with the
asymptotic result for large , which neglects recoil [18],
The radiative correction is derived by Marciano and
Sirlin [13] and takes the form
where 
is a QED short-distance enhancement factor equal
to 1.02256, and 
is a running QED coupling which satisfies
such that 
and
. 
is a small perturbative QCD
correction estimated to be -0.34, and C is a nuclear
structure-dependent correction which is 0 for
transitions. 
is a low energy cutoff applied to the
short-distance part of the 
box diagram, and ranges from
400 MeV to 1600 MeV. Using this range of , the radiative
correction to superallowed Fermi transition rates is
found to be between 1.0334 and 1.0350. [13]