  Axion inflation coupled to the Standard Model (SM) hypercharge gauge sector
represents an attractive scenario for the generation of primordial
hypermagnetic fields. The description of this scenario is, however, complicated
by the Schwinger effect, which gives rise to highly nonlinear dynamics.
Hypermagnetogenesis during axion inflation in the absence of nonlinear effects
is well studied and known to result in a hypermagnetic energy density that
scales like $H^4\,e^{2\pi\xi}/\xi^5$, where $\xi$ is proportional to the time
derivative of the axion-vector coupling in units of the Hubble rate $H$. In
this paper, we generalize this result to the full SM case by consistently
taking into account the Schwinger pair production of all SM fermions. To this
end, we employ the novel gradient-expansion formalism that we recently
developed in [2109.01651], and which is based on a set of vacuum expectation
values for bilinear hyperelectromagnetic functions in position space. We
parametrize the numerical output of our formalism in terms of three parameters
($\xi$, $H$, and $\Delta$, where the latter accounts for the damping of
subhorizon gauge-field modes because of the finite conductivity of the medium)
and work out semianalytical fit functions that describe our numerical results
with high accuracy. Finally, we validate our results by comparing them to
existing estimates in the literature as well as to the explicit numerical
results in a specific inflationary model, which leads to good overall
agreement. We conclude that the systematic uncertainties in the description of
hypermagnetogenesis during axion inflation, which previously spanned up to
several orders of magnitude, are now reduced to typically less than 1 order of
magnitude, which paves the way for further phenomenological studies.
