The global $B-L$ symmetry in the flavor-unified $\mathrm{S}\mathrm{U}(N)$ theories

We study the origin of the global $B-L$ symmetry in a class of flavor-unified theories with gauge groups of $\mathrm{S}\mathrm{U}(N\ge 6)$. In particular, we focus on the $\mathrm{S}\mathrm{U}(8)$ theory which can minimally embed three-generational SM fermions non-trivially. A reformulation of the third law for the flavor sector proposed by Georgi is useful to manifest the underlying global symmetries. The 't Hooft anomaly matching and the generalized neutrality conditions for Higgs fields play the key roles in defining the $B-L$ symmetry. Based on the global $B-L$ symmetry, we count the Higgs fields that can develop the VEVs and the massless sterile neutrinos in the $\mathrm{S}\mathrm{U}(8)$ theory. We also prove that a global $B-L$ symmetry can always be defined in any $\mathrm{S}\mathrm{U}(N\ge 6)$ theory when it is spontaneously broken to the SM gauge symmetry.