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We compute the three-loop QCD corrections to the massivequark-anti-quark-photon form factors $F_1$ and $F_2$ involving a closed loop ofmassless fermions. This subset is gauge invariant and contains both planar andnon-planar contributions. We perform the reduction using FIRE and compute themaster integrals with the help of differential equations. Our analytic resultscan be expressed in terms of Goncharov polylogarithms. We provide analyticresults for all master integrals which are not present in the large-$N_c$calculation considered in Refs. [1,2].

Three-loop massive form factors: complete light-fermion corrections for the vector current