$U$-spin sum rules for two-body decays of bottom baryons

$U$-spin symmetry, which reflects the symmetry between the down-type $d$ and $s$ quarks, is a powerful tool for analyzing heavy hadron weak decays. Motivated by recent experimental achievements in the bottom baryon sector, we study the $U$-spin sum rules for bottom baryon decays. The effective Hamiltonian for $b$ quark decay is zero under the $U$-spin lowering operators $U_-^n$, permitting us to derive $U$-spin sum rules involving only the $b\to d$ transition or $b\to s$ transition. Moreover, a new operator, $S_b=U_++rU_3-r^2U_-$, is proposed to generate $U$-spin sum rules involving both the $b\to d$ and $b\to s$ transitions. The proof that the effective Hamiltonian for $b$ quark decay is zero under $U_-^n$ and $S_b$ is presented. The master formulas for generating $U$-spin sum rules for the two-body decays of bottom baryons involving $b\to c\overline cd/s$, $b\to c\overline ud/s$, $b\to u\overline ud/s$, and $b\to u\overline cd/s$ transitions are derived. Numerous $U$-spin sum rules for the two-body decays of bottom baryons are obtained through these master formulas, which provide hints for new decay modes and enable the extraction of dynamical information. As a phenomenological analysis, some branching fractions are predicted according to $U$-spin symmetry. Several rate and decay parameter sum rules beyond the $U$-spin limit are found, providing a more precise test of flavor symmetry in the bottom baryon sector. Moreover, some $CP$ asymmetry relations for $U$-spin conjugate pairs in heavy baryon decays are derived for the first time by taking partial-wave amplitudes into account.