Question about Higgs fields in SU(5) GUT.

[Max]

[[Mod. note -- I have "repaired" various non-ASCII characters as best
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-- jt]]

In the SU(5) grand unified theory, there are two sets of Higgs fields.
They are the {5} and {24} representations (we don't consider {45}
here). {24} breaks SU(5) down to SU(3)c*SU(2)l*U(1)y, and gives mass
to X gauge bosons via Higgs mechanism. {5} breaks SU(3)c*SU(2)l*U(1)y
further down to SU(3)c*U(1)em, and gives mass to W+- and Z gauge
bosons.

However, only {5} gives mass directly to the fermions via a Yukawa
type coupling. The conventional wisdom is that the "Higgs
representation must appear in the tensor product of the SU(5)
representations in which the fermion and its antifermion
appear." (H.Georgi's Lie Algebras in Particle Physics, Page 234-235).

This reasoning sounds empirical to me, because it's based on the way
fermion and its antifermion are assigned to different representations
of SU(5). Is there any first principle which explains why {24} is not
mass generating for fermions?

 

[BenTheMan]

On Dec 13, 5:07 pm, Max <aws...@yahoo.com> wrote:
> [[Mod. note -- I have "repaired" various non-ASCII characters as best
> as I can, but it's *much* safer if postings are submitted in 7-bit-ASCII
> only.  In particular, Microsoft Windows "smart quotes" tend to get
> quite mangled before they arrive at s.p.r moderstors' mailboxes... :(
> -- jt]]
>
> In the SU(5) grand unified theory, there are two sets of Higgs fields.
> They are the {5} and {24} representations (we don't consider {45}
> here). {24} breaks SU(5) down to SU(3)c*SU(2)l*U(1)y, and gives mass
> to X gauge bosons via Higgs mechanism. {5} breaks SU(3)c*SU(2)l*U(1)y
> further down to SU(3)c*U(1)em, and gives mass to W+- and Z gauge
> bosons.
>
> However, only {5} gives mass directly to the fermions via a Yukawa
> type coupling. The conventional wisdom is that the "Higgs
> representation must appear in the tensor product of the SU(5)
> representations in which the fermion and its antifermion
> appear." (H.Georgi's Lie Algebras in Particle Physics, Page 234-235).
>
> This reasoning sounds empirical to me, because it's based on the way
> fermion and its antifermion are assigned to different representations
> of SU(5). Is there any first principle which explains why {24} is not
> mass generating for fermions?[/color]

Good question!

First, a bit of group theory. Let's ignore U(1) for now, because it
only will complicate things, and I think I can show you qualitatively
how it works without complicating things.

The first thing to know is how the representations of SU(5) break down
under SU(3)xSU(2). I am mostly wrong when I try to remember things,
but I can tell you where to look to check my answers: there is a
wonderful reference by Richard Slansky called "Group Theory for
Unified Model Building", which is available online at spires. There
are extensive tables in the back that show you all of this info.
Anyway, the SU(5) reps break into the following reps under SU(3)xSU
(2):

5 -> (3,1) + (1,2)
10 -> (3,2) + (3,1) + (1,1)
24 -> (8,1) + (1,3) + (3,2) + (3*,2*) + (1,1).

Now hopefully you can start to see the problem---the 24 doesn't have
an appropriate electroweak higgs candidate! The EW higgs HAS to
transform like (1,2), so it is ONLY the 5 that has the appropriate
content.

Now, you mentioned the 45. It COULD be that the 45 can give you a
correction to fermion masses. Indeed, it has to because that's why
people use it. In this case, you can look in Slansky's book yourself
and see how the 45 decomposes. Then, if you're ambitious, you can
work out how they couple to the SU(5) fermions, and compute mass
corrections. There's probably a review paper by Pavel Perez where
this is all worked out---he's the leader (I guess) when it comes to
non-SUSY SU(5) theories.

Anyway, hope this helps.

 

[Max]

On Dec 14, 3:55 am, BenTheMan <BenDun...@gmail.com> wrote:
> there is a
> wonderful reference by Richard Slansky called "Group Theory for
> Unified Model Building",[/color]

Thanks for the reference

> the 24 doesn't have
> an appropriate electroweak higgs candidate!  The EW higgs HAS to
> transform like (1,2), so it is ONLY the 5 that has the appropriate
> content.[/color]

I am fully aware that 24 doesn’t break the electroweak symmetry. The
question is why 24 is not mass generating for fermions including
leptons and quarks. In other words, why are fermion masses tied to
electroweak symmetry breaking, rather than SU(5) -> SU(3)c*SU(2)l*U(1)
y symmetry breaking?

-Max

 

[Hendrik van Hees]

On Dec 14, 3:59 pm, Max <aws...@yahoo.com> wrote:
> On Dec 14, 3:55 am, BenTheMan <BenDun...@gmail.com> wrote:
>
> >  there is a
> > wonderful reference by Richard Slansky called "Group Theory for
> > Unified Model Building",[/color]
>
> Thanks for the reference
>
> > the 24 doesn't have
> > an appropriate electroweak higgs candidate!  The EW higgs HAS to
> > transform like (1,2), so it is ONLY the 5 that has the appropriate
> > content.[/color]
>
> I am fully aware that 24 doesn’t break the electroweak symmetry. The
> question is why 24 is not mass generating for fermions including
> leptons and quarks. In other words, why are fermion masses tied to
> electroweak symmetry breaking, rather than SU(5) -> SU(3)c*SU(2)l*U(1)
> y symmetry breaking?
>
> -Max[/color]

Write out a fermion mass term and look at the quantum numbers. You
should see that, in order to be gauge invariant, the mass term has to
transform like (1,2). The only possible (dimension 4) mass term that
you can write involves the normal, electroweak higgs.

 

[Max]

On Dec 15, 12:01 pm, h...@fb07-hees.theo.physik.uni-giessen.de
(Hendrik van Hees) wrote:
> Write out a fermion mass term and look at the quantum numbers. You
> should see that, in order to be gauge invariant, the mass term has to
> transform like (1,2). The only possible (dimension 4) mass term that
> you can write involves the normal, electroweak higgs.-[/color]

Thanks for the explanation. How about a Majorana mass term, instead of
the Dirac mass term? Would 24 generate Majorana mass for fermions?

-Max