Parity-Violating M{\o}ller Scattering at NNLO: Closed Fermion Loops

A complete, gauge-invariant computation of two loop virtual corrections involving closed fermion loops to the polarized M{\o}ller scattering asymmetry is presented. The set of contributions involving two closed fermion loops and the set involving one closed fermion loop are numerically similar in magnitude to the one-loop bosonic corrections, but coincidentally cancel to yield a smaller overall correction of 1% relative to the tree level asymmetry. We estimate sizes of remaining two-loop contributions and discuss implications for the upcoming MOLLER experiment.