Planar master integrals for two-loop NLO electroweak light-fermion contributions to $g g \rightarrow Z H$

For the two-loop next-to-leading-order electroweak (NLO EW) corrections to $gg \rightarrow ZH$, the light-fermion contributions can be classified into eight distinct topologies. Using the canonical differential-equations method, we perform an analytic computation of the master integrals (MIs) associated with the four planar topologies. Canonical bases are constructed using the Magnus-expansion method, and the resulting alphabets consist of algebraic symbol letters involving nontrivial radicals. We develop a systematic framework for identifying the radical structures of the canonical MIs, enabling their organization into suitable subsystems and, whenever possible, their representation in terms of Goncharov polylogarithms (GPLs) up to $\mathcal{O}(\epsilon^4)$. Only a few MIs at $\mathcal{O}(\epsilon^3)$ and $\mathcal{O}(\epsilon^4)$ are instead represented as one-fold integrals over GPLs, due to the presence of nested square roots that obstruct the simultaneous rationalization of all radicals.