Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,352 answers , 22,785 comments
1,470 users with positive rep
820 active unimported users
More ...

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

+ 2 like - 0 dislike
4788 views
  • What are the precise $B$-$L$ (baryon - lepton) global symmetry in the standard model?

  • Is this a $U(1)$ global symmetry or a discrete finite group $\mathbb{Z}/N$ global symmetry for the following examples?

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

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user annie marie heart
asked Apr 14, 2020 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]

1 Answer

+ 7 like - 0 dislike

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!

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user Cosmas Zachos
answered Apr 15, 2020 by Cosmas Zachos (370 points) [ no revision ]
Most voted comments show all comments
Slansky Table 14, the starting point of any and all such questions, as your instructor in your GUT course must have emphasized to you.

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user Cosmas Zachos
but those Table seem to include only Lie algebra, I am not sure they have Lie group consideration included. Thank you a good ref I just learned from you!

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user annie marie heart
btw I have one more basic puzzle learning GUT: physics.stackexchange.com/questions/560676

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user annie marie heart
maybe you also know already physics.stackexchange.com/questions/564976

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user annie marie heart
Don't know much about its advantages. Gives you longer life for the proton...

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user Cosmas Zachos
Most recent comments show all comments
Ask a separate question, but be explicit.

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user Cosmas Zachos
thanks - I will <3

This post imported from StackExchange Physics at 2020-11-28 23:03 (UTC), posted by SE-user annie marie heart

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOver$\varnothing$low
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...