# $B$-$L$ global symmetry in the grand unification theories

Does the precise $$B$$-$$L$$ global symmetry present in:

In the standard model lagrangian, B and L are separately conserved global charges, and B-L, a vector like symmetry, is anomaly-free. GUTs, like the G-G SU(5) violate B and L, but preserve B-L.

Wikipedia effectively defines the SU(5)-model U(1) symmetry X as $$X = 5(B − L) -2Y_W,$$ introduced by Wilczek & Zee in 1979. It is not a generator of SU(5), of course.

So you may readily compute for the left-chiral $$\bar {\bf{5}}$$, $$\overline{ d_R} : ~~(2Q=)~~~ Y=2/3 , ~~~~ B-L= -1/3 ~~~ \leadsto X=-3 \\ e^-_L, \nu_L : ~~~~~~~ Y= -1 , ~~~~~~ B-L= -1 \qquad \leadsto X=-3.$$ So the entire multiplet possesses a common X-charge: -3.

Proceed to verify for the left-chiral 10, X= 1. $$\overline{ u_R} : ~~(2Q=)~~~~~ Y=-4/3 , ~~~~ B-L= -1/3 ~~~\leadsto X=1 \\ d_L,u_L : ~~~~~~~~~ Y=1/3 , ~~~~ B-L= 1/3 \qquad \leadsto X=1 \\ e^+_L : ~~(2Q=)~~~~~ Y= 2 , ~~~~~~~ B-L= 1 \qquad \leadsto X=1.$$

Thence, for the $$\langle \phi ^* \rangle$$, X=2, so the Yukawa (mass) term is chargeless.

Note that B-L is vectorlike, but Y is not, so, ipso facto, X is not!

Similarly for SO(10), except here X is now a generator of SO(10) and is then gauged (hence SSBroken), hence violated by small amounts.

Responses to comment questions.

1) B-L is a good global symmetry for the SM and SU(5) and local for SO(10). So, e.g., in SU(5) proton decay to a pion and a positron, it is visibly preserved!

2) X is fine in the SM and SU(5) as a global symmetry, as a linear combination of good quantum numbers. As defined, it has a unique eigenvalue for each SU(5) rep, not necessarily the same for all reps, as you observe. Same for the SM which has smaller reps, several of which entered into each SU(5) rep. That means that, even for SU(5) which mixes baryons and leptons, it is straightforward to match X eigenvalues and monitor the symmetry of the coupling terms, like the Yukawas.

3) SO(10) largely follows suit, paralleling SU(5), and likewise lacks exotic particles, but gauges X, so SSBreaks it. But now the above two reps of SU(5) plus an extra singlet (the R-chiral neutrino) fit into a spinor 16 of SO(10). If your wrote down the multiplet, it might be easier to parse out. This review, eqn. (3.3), has X as a traceless diagonal generator, so you can check its eigenvalue on the fermion 16-plet (4.2).

Now, in most models, even though X is SSBroken and its corresponding gauge boson made massive, this happens at a high scale and results in a GUT with SU(5)-like symmetry, with effective B-L violating operators of dimension 6, i.e. suppressed by the square of such heavy scales, so very small. Thus, in such models, for all intents and purposes, proton decay is still approximately respectful of B-L!

• Thanks, I vote up. So B, L and B-L are all global symmetry in standard model. Agree? Apr 16, 2020 at 21:56
• But B, L are not global symmetry in SU5 or SO10 models. And B-L is a global symmetry in SU5 or SO10 model. Agree? Apr 16, 2020 at 21:57
• Yes, to both, as indicated at the beginning. B,L, and B-L are SM global symmetries, although anomalies violate B and L, preserving B-L. Most GUTs violate B and L, but preserve B-L, and of course Y (in the SM), as in G-G proton decay. Have you clicked the checkmark? Apr 16, 2020 at 22:17
• also can B-L symmetry be adiscrete symmetry in standard models, SU5 or SO10? or some of the Proposed GUTs? en.wikipedia.org/wiki/Grand_Unified_Theory#Proposed_theories Apr 16, 2020 at 22:24
• In SM, SU(5), and SO(10) it is a fine continuous (Lie) symmetry, not just discrete, so why focus on a subgroup, if you have the full enchilada? I am not sure there is a general theory of GUTs --it is always it or miss. They usually have extra fermions for a change, so I don't know how one assigns B and L numbers to those. But, where there is a will, there is a way; recall Gell-Mann--Ramond--Slansky? The ones you link to, however, are truly bad make-work projects. This might be a separate question. Apr 17, 2020 at 0:03