How Do Techni-Baryons Transform in SU(N_{TF}) and SU(2)_{spin} Symmetries?

In summary, the conversation discusses the behavior of TCb under different symmetries, specifically SU(N_TC) and SU(2)_spin. The fully symmetric representation of SU(2N_TF) is needed, but any representation that reduces to the trivial representation of S_NTC will work. Georgi's chapter 15 is referenced and the results for N_TC = 3 and N_TC = 4 are given. The discussion also mentions the rules for Young Tableaux and how they apply to this problem.
  • #1
Andrea M.
28
1
Consider a model with an exact ##SU(N_{TC})## techni-color symmetry and a ##SU(N_{TF})_L\otimes SU(N_{TF})_R## global techni-flavour symmetry which is spontaneously broken to the diagonal sub-group ##SU(N_{TF})## by condensates producing techni-pions (TC\pi) and techni-baryons(TCb).
What I'm trying to understand is how the various TCb transform under ##SU(N_{TF})## and ##SU(2)_{spin}##.
Because the wave-function is totally antisymmetric in techni-color we expect that TCb must be fully symmetric in spin and techni-flavour. Following the line of thought of Georgi's chapter 15 I have embedded ##SU(N_{TF})## and ##SU(2)_{spin}## in a ##SU(2N_{TF})##. The TCb transform under this ##SU(2N_{TF})## like a ##N_{TC}## completely symmetric combination, or in Young tableaux notation like a tableaux with ##N_{TC}## horizontal box. Now Georgi says that to understand what representation of ##SU(N_{TF})## and ##SU(2)_{spin}## are contained in the fully symmetric representation of ##SU(2N_{TF})## we must take the tensor product of all tableaux with ##N_{TC}## boxes of ##SU(N_{TF})## and ##SU(2)_{spin}## and see if they contain the fully symmetric representation. The results (see for example Chivukula, R.S. & Walker, T.P., 1989. Technicolor cosmology, Boston, MA: Boston Univ.) should be the following:
-For ##N_{TC}=3##
Schermata 2015-09-07 alle 17.55.43.png


-For ##N_{TC}=4##
Schermata 2015-09-07 alle 17.55.28.png

The problem is that, except for the ##N=3## case, I'm not able to obtain the fully symmetric representation by taking the tensor product of this tableaux.
 
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  • #2
I don't think we necessarily need to have the totally symmetric representation of ##SU(2N_{TF})##, just a representation that reduces to the trivial representation of ##S_{N_{TC}}##. Any diagram that is trivial after deleting all rows with ##N_{TC}## or more boxes from the diagram will work. The totally symmetric representation for ##SU(2N_{TF}## will work, but so will others.

Looking at the products of the representations in your reference, we have
young1.JPG

where in the middle line we actually have a few copies of the trivial ##S_4## representation appearing. For the missing representations like
young2.JPG

we can see that we don't have enough boxes in the rows to make a trivial ##S_4## representation.
 
  • #3
fzero said:
I don't think we necessarily need to have the totally symmetric representation of SU(2NTF)SU(2N_{TF}), just a representation that reduces to the trivial representation of SNTCS_{N_{TC}}.
Yes I agree with this.
fzero said:
Any diagram that is trivial after deleting all rows with NTCN_{TC} or more boxes from the diagram will work. The totally symmetric representation for SU(2NTFSU(2N_{TF} will work, but so will others
Why I can delete all the rows with ##N_{TC}## or more boxes?
Am I wrong or the trivial representation of ##S_4## is a Young tableaux with 4 horizontal boxes?
 
  • #4
Andrea M. said:
Yes I agree with this.

Why I can delete all the rows with ##N_{TC}## or more boxes?
Am I wrong or the trivial representation of ##S_4## is a Young tableaux with 4 horizontal boxes?

Yes it is, but a Young tableaux with 4 horizontal columns of any equal size is also the trivial representation. So I mean delete in the same way that you would if you had a column with ##N## boxes for ##SU(N)##.
 
  • #5
Could you suggest some reference where this result is derived?
 
  • #6
Andrea M. said:
Could you suggest some reference where this result is derived?

Georgi discusses the Young tableaux for ##S_n## in sects 1.21-24. The trouble is that we already know that irreps of ##S_4## never have more than 4 boxes so not all of the rules there apply (for instance we can't naively compute the dimension using 8 as the number of boxes). But we can assign ##j##-cycles to the the boxes, so we get 4 2-cycles on 4 objects, which should be easy to show is reducible to 4 1-cycles.
 
  • #7
fzero said:
But we can assign jj-cycles to the the boxes, so we get 4 2-cycles on 4 objects, which should be easy to show is reducible to 4 1-cycles.
Are you sure? For example the four 2-cycles (12)(23)(14)(23) do not seem to be equivalent to four 1-cycles.
 
  • #8
Andrea M. said:
Are you sure? For example the four 2-cycles (12)(23)(14)(23) do not seem to be equivalent to four 1-cycles.

The rules of the Young Tableau are that the columns are the ##j##-cycles and that we can't repeat an index in the same row. If we had a valid diagram, we would never repeat an index in any box, but with more than ##n## boxes we have to relax something.

With the above rules we only get elements like (13)(21)(34)(42) that are always reducible.
 

FAQ: How Do Techni-Baryons Transform in SU(N_{TF}) and SU(2)_{spin} Symmetries?

What is Technibaryon transformation?

Technibaryon transformation is a theoretical process in particle physics that involves the conversion of ordinary matter into a new type of particle called a technibaryon.

Why is Technibaryon transformation important?

Technibaryon transformation is important because it could provide an explanation for the existence of dark matter in the universe. It is also a key concept in understanding the behavior of elementary particles at high energies.

How does Technibaryon transformation occur?

Technibaryon transformation is believed to occur through the strong interaction between quarks, which are the building blocks of protons and neutrons. This interaction creates a new type of force that can transform ordinary matter into technibaryons.

What evidence supports the existence of Technibaryons?

Currently, there is no direct evidence for the existence of technibaryons. However, there are several theoretical models and experiments that suggest their existence, such as the Large Hadron Collider at CERN.

What are the potential implications of Technibaryon transformation?

If technibaryons are proven to exist, it could greatly impact our understanding of the fundamental forces and particles in the universe. It could also lead to advancements in fields such as cosmology and particle physics.

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