B − L
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This quantum number is the charge of a global/gauge U(1) symmetry in some Grand Unified Theory models, called U(1)B−L. Unlike baryon number alone or lepton number alone, this hypothetical symmetry would not be broken by chiral anomalies or gravitational anomalies, as long as this symmetry is global, which is why this symmetry is often invoked. If B − L exists as a symmetry, it has to be spontaneously broken to give the neutrinos a nonzero mass if we assume the seesaw mechanism. In the case of a gauged B − L, the gauge bosons associated with this symmetry are commonly called X and Y bosons.
The anomalies that would break baryon number conservation and lepton number conservation individually cancel in such a way that B − L is always conserved. One hypothetical example is proton decay where a proton (B = 1; L = 0) would decay into a pion (B = 0, L = 0) and positron (B = 0; L = −1).
Weak hypercharge Y
W is related to B − L via
- X + 2Y
W = 5(B − L),