Production and Backreaction of Fermions from Axion-$SU(2)$ Gauge Fields
  during Inflation
  $SU(2)$ gauge fields and axions can have a stable, isotropic and homogeneous
configuration during inflation. However, couplings to other matter species lead
to particle production, which in turn induces backreaction on and
destabilization of the non-abelian and axion background. In this paper, we
first study the particle production by a $SU(2)$ gauge field coupled to a
massive Dirac doublet. To carry out this calculation we have made two technical
improvements compared to what has been done in the literature. First, we apply
the anti-symmetrization of the operators to treat particles and anti-particles
on equal footing, second, to deal with the UV divergences, we apply
instantaneous subtraction. We find that, the backreaction of produced fermions
on the $SU(2)$ background is negligible for model parameters of observational
interest. Next, we consider production of fermions due to coupling to the
axion. The tree-level backreaction on the gauge fields, as well as on the
axion, is vanishingly small. We also provide an estimate for the loop effects.
