# SU(3) decomposition

1. Feb 24, 2006

### Uma

Hello,

I have a question with respect to the decomposition in irreducible representations of antiquark - antiquark ( SU(3) color ).

In the case of quark - quark what you have is a triplet with an antitriplet and what you obtain is an antitriplet and a sextet, and from the Young tables correspondent to them you can immediately see that the antitriplet is completly antisymmetric while the sextet is symmetric.

When the same is done for the antiquark - antiquark case, I cannot see the symmetry from the Young tables, in this case the triplet and the antisextet seems to me to not being symmetric or antisymmetric .

My question is with respect to the symmetry or antisymmetry of the triplet and antisextet in this last case, also, I would like to know if I can infer such symmetries from the adjoint representation (the first case).

Thank you

2. Feb 24, 2006

### arivero

Uma, where did you learn of the quark quark case? I mean, I am interested on diquarks, and I havent see a lot of books working straighly from them.

Also, the quark antiquark case should produce an octet, shouldnt it? The eight Gell-Mann matrices. (and a singlet, with the wrong (anti?)symmetry.). I guess it appears not explicitly in most books because they have already done the calculation for SU(3) flavour, producing the pion+kaon+eta in the octect and the eta' in the singlet.

3. Feb 24, 2006

### Uma

Hi arivero,

What you can easily find in most of the books, as you said it is the decomposition of quark-antiquark into the singlet and octet, and the reason for this is that as a physical system you are interested in the color singlet. The case of diquark or antidiquark by themselves do not produce such singlets, but, they can in the so called "exotic" mesons, which would be systems of two quarks two antidiquarks (tetraquarks) or four quarks and one antiquark (pentaquarks).

I actually haven´t found information of diquarks on books, but you can find that information in some papers, I don´t know if there are previous works but the first that I found was from R.L. Jaffe in 1977 and also F. Close has discussed such systems recently (2000).

4. Feb 25, 2006

### arivero

I think that the introductory chapters of Huang build some barions in a double step, first a couple quarks, then the third one. So it could be a good text-based clue. I'll look at it this weekend.

So the origin of your question is because you are interested on diquarks after all? I ask because I am interested myself, as you may knoew: the diquarks should be used as scalar partners of quarks, the supersymmetry generator being the barions (a baryon anhiquilates an antiquark and produces a diquark, as a SUSY generator does) and then being a clearly broken susy.

5. Feb 26, 2006

### arivero

Ok yep, the Huang in table 2.6 does thow (throw+show) the triplet-triplet boxes amd the triplet-antritriplet ones, but as you could expect he does not work out the antitriplet-antitriplet. Perhaps a reference could be the 8x8 example, deleting out all the 2,2' boxes, so that only two different tableaus remain, one with two rows of two squares each, and another of thre rows (so 2,1,1, squares). Before, in section three, it is stressed that you never go beyond three rows, for SU(3).

6. Feb 26, 2006

### Uma

Hi arivero,

Yea, they don´t usually work that case, so I am still with my original question.

The origin of my question is because I am interested in tetraquarks, I actually didn´t know about diquarks in SUSY theories.

7. Feb 27, 2006

### arivero

I see. I still think that the 8x8 example could help, as labeled in Huang's book. I guess you have already perused all the bibliography; Jaffe has a recent one, in addition to the old 1977 you told. Also Wilczek.

I am interested in any reason fo forbid the +4/3 (uu, cc, uc) diquarks but at the same time still allowing dd or ss diquarks.

8. Feb 28, 2006

### Uma

With respect to the question I had, the solution didn´t came to me from Huang´s book, but finally I understood what I needed to, thank you for the help.

So, in what you are working, you wouldn´t have SU(3) flavor symmetry anymore?

9. Mar 1, 2006

### arivero

I am not sure how the diquarks could carry flavour symmetry, in fact was recalling all of these group theory just to understand how to pair each fermionic quark with two scalar diquarks. I do not discard that the mixing could illuminate the need for Cabibbo angles. At the moment, all I have is the double coindence between degrees of freedom: that binding five diferent quarks into diquarks you have 6 scalar degrees of charge 2/3, 6 scalar degrees of charge 1/3 and its antiparticles (plus three spoureous? of charge 4/3), and that by binding five diferent quarks into mesons you have 6 scalar degrees of charge +1 plus their antiparticles plus some bunch of charge 0 things. So I could control the loops making the higgs diverge if the coupling of higgs to diquarks were of the same order that the coupling to quarks.

As for the sextet... well, better a different post, wait a moment

10. Mar 1, 2006

### arivero

As for the colour sextet, I am surprised it is an older object than I had first guessed. There are a PhysRevD of Freund and Hill in 1979 speculating if the b quark was to be adscribed to a color sextet instead of a triplet; the goal was to increase the binding force of Upsilon and to adjust the first excited state. Very intriguing. And there is a preprint from Fukazawa, Muta and others from 1990 suggesting that "color-sextet quark condensates" could be the origin of electroweak symmetry breaking; it seems that the idea goes back to a PhysRevD of Marciano in 1980.

The most persistent player is Lichtenberg, who has published on diquarks in 2004 but that really started the word in 1968, PhysRev 167 p 1535 as a follow-up to PhysRev 155 p 1601, of 1967!