Prospects for $B_{(s)}^0\to \ddot{I}{€}^0\ddot{I}{€}^0$ and $B_{(s)}^0\to \hat{I}\cdot \hat{I}\cdot$ modes and corresponding $CP$ asymmetries at Tera-$Z$

Abstract

The physics potential of measuring $B_{(s)}^0\to \ddot{I}{€}^0\ddot{I}{€}^0$ and $B_{(s)}^0\to \hat{I}\cdot \hat{I}\cdot$ 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 $\ddot{I}{€}^0$ and $\hat{I}\cdot$ from hadronic final states with high photon multiplicity. The resulting $B$-meson mass resolution is approximately 30 MeV, allowing 3 $\ddot{I}ƒ$ separation between $B^0$ and $B_s^0$. With the assistance of the $b$-jet tagging, the relative sensitivities to $B^0\to \ddot{I}{€}^0\ddot{I}{€}^0$, $B_s^0\to \ddot{I}{€}^0\ddot{I}{€}^0$, $B^0\to \hat{I}\cdot \hat{I}\cdot$, and $B_s^0\to \hat{I}\cdot \hat{I}\cdot$ 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, $B^0\to \ddot{I}{€}^0\ddot{I}{€}^0$ and its two isospin-related modes are paid special attention due to their roles in the determination of the CKM angle $\hat{I}\pm$ ($\ddot{I}{•}_2$). 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 $B^0\to \ddot{I}{€}^0\ddot{I}{€}^0$ $CP$ asymmetry at Tera-$Z$ is complementary to $B$-factory ones. The precision on $\hat{I}\pm$ combining $Z$- and $B$-factory results reaches $0.4^{\circ }$, lower than the systematic uncertainties attached to isospin breaking.