The global $B-L$ symmetry in the flavor-unified ${\rm SU}(N)$ theories

TL;DR

This paper investigates the origin and properties of the global $B-L$ symmetry in flavor-unified ${ m SU}(N)$ theories, especially ${ m SU}(8)$, revealing how it can be consistently defined and its implications for Higgs fields and sterile neutrinos.

Contribution

It demonstrates that a global $B-L$ symmetry can always be defined in ${ m SU}(N extgreater=6)$ theories when broken to the SM, and analyzes its role in Higgs and neutrino sectors.

Findings

Global $B-L$ symmetry can be consistently defined in ${ m SU}(N extgreater=6)$ theories.

The symmetry constrains Higgs VEVs and sterile neutrino states.

A reformulation of Georgi's third law aids in understanding the symmetry's origin.

Abstract

We study the origin of the global $B-L$ symmetry in a class of flavor-unified theories with gauge groups of ${\rm SU}(N\geq 6)$. In particular, we focus on the ${\rm SU}(8)$ 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 ${\rm SU}(8)$ theory. We also prove that a global $B-L$ symmetry can always be defined in any ${\rm SU}(N\geq 6)$ theory when it is spontaneously broken to the SM gauge symmetry.