Prospects for $B^0_{(s)}\to\pi^0\pi^0$ and $B^0_{(s)}\to\eta\eta$ modes and corresponding $CP$ asymmetries at Tera-$Z$

The physics potential of measuring $B^0_{(s)}\to\pi^0\pi^0$ and $B^0_{(s)}\to\eta\eta$ decays via four-photon final states at Tera-$Z$ phase of CEPC or FCC-ee is investigated in this paper. We propose an electromagnetic calorimeter (ECAL) with both high energy resolution and excellent separation power to efficiently reconstruct $\pi^0$ and $\eta$ from hadronic final states with high photon multiplicity. The resulting $B$-meson mass resolution is approximately 30 MeV, allowing 3 $\sigma$ separation between $B^0$ and $B_s^0$. With the assistance of the $b$-jet tagging, the relative sensitivities to $B^0\to\pi^0\pi^0$, $B^0_s\to\pi^0\pi^0$, $B^0\to\eta\eta$, and $B^0_s\to\eta\eta$ signal strengths at Tera-$Z$ are projected as 0.45%, 4.5%, 18%, and 0.95%, respectively. Their dependence on various detector performances is also discussed. In addition, $B^0\to\pi^0\pi^0$ and its two isospin-related modes are paid special attention due to their roles in the determination of the CKM angle $\alpha$ ($\phi_2$). The anticipated precisions of their branching-ratio and $CP$-asymmetry measurements at Tera-$Z$ are evaluated. We show that the measurement of the time-integrated $B^0\to\pi^0\pi^0$ $CP$ asymmetry at Tera-$Z$ is complementary to $B$-factory ones. The precision on $\alpha$ combining $Z$- and $B$-factory results reaches $0.4^\circ$, lower than the systematic uncertainties attached to isospin breaking.