Super-Logarithmic Entanglement Scaling in a Monitored Superconducting Chain

We develop a Keldysh-replica non-linear sigma model (NLSM) for the entanglement dynamics of a monitored one-dimensional spinful $s$-wave BCS chain in the rare-measurement regime, $\gamma \ll J,\Delta$. Although the clean spinful $s$-wave BCS Hamiltonian belongs to symmetry class CI, spin-resolved measurements and projection to a conserved $f$-sector reduce the effective problem to class C. Starting from the corresponding parent symplectic saddle, we show that measurement backaction and the pairing amplitude impose complementary mass constraints that gap out different fluctuation channels. Their interplay dynamically projects the surviving massless modes onto an $\textrm{SO(R)}$ target manifold in replica space. A one-loop renormalization group analysis of this $\textrm{SO(R)}$ NLSM shows that, in the replica limit $R\to1$, the beta function becomes negative, producing a weak-anti-localization flow. This flow yields a super-logarithmic steady-state entanglement scaling $S(L)\sim \ln^2 L$ in the rare-measurement regime. Our field-theoretic result explains the numerical evidence reported in the companion Letter [arXiv:2604.04375] and shows that a topologically trivial monitored $s$-wave superconductor can realize an $\textrm{SO(R)}$ weak-anti-localizing critical phase without relying on a Wess-Zumino-Witten term.