# Mesons and Young Tableaux

I need some help conecting Young Tableaux with actual particles.
I think I have some feel for using Young Tableaux for instance: a baryon in SU(3) where the states are u,d,s can be represented by $\square \bigotimes \square \bigotimes \square$ which gives some tableaux which I can arrive at fine. (I don't know how to latex that in so I'll just leave it at that)

Now my question is what about mesons? Here the tableuaux is $\square \bigotimes$ (A two box column) for the conjugate. What I am confused about is when I consider a box, I think of a particle. How then can an antiparticle have two boxes in SU(3) and the overall tableaux have three boxes. One a mixed state with 8 dimensionality, and a antisymmetric diagram with 3 boxes.

How can one tell then that these diagrams are truly 2 particle diagrams for a meson rather than say a baryon in an anti or mixed symmetric state? How would one assign the 8 mixed states to the actual particle states explicitly might help?

I have a feeling it has to be related to the idea when considering mesons, we have to think of color and not quarks as the degrees of freedom? However, it still seems like that would need two boxes. So I am at a loss...

• I have no idea why you say "When I consider a box, I think of a particle". Young tableaux are a purely abstract graphic notation for representations of $\mathrm{SU}(N)$ and related groups, so I do not understand what you are trying to ask for - it is just a fact of Young tableaux that the anti-fundamental representation of $\mathrm{SU}(3)$ is drawn as a 2-box column. Jan 26, 2016 at 20:24
• What I am trying to ultimately figure out is given SU(3) $\bigotimes$ SU(3)_bar for a meson, how to assign the 3 x 3 (nine) states to the 8 + 1 states and why. I am very confused by group theory and what we actually mean by representations. I have spent a long time trying to grasp it to no great success. So maybe a source for a beginner trying to learn group theory would be helpful to point out as well. I have tried wiki and few other sites, but they have me confused. I'll post a side question regarding that. Jan 27, 2016 at 15:34

## 1 Answer

You need a two box column for the anti fundamental-representation of SU(3) to accommodate the rules for filling in the boxes in the columns of the Young tableau and having the right number of anti-particles or anti-colors. There are three distinct states 1,2,3 in SU(3). A two box column has exactly 3 possible different configurations using these numbers. In a vertical column of boxes, the state numbers must increase from top to bottom by the rules of generating Young tableau. So the possible combinations for this in a two box vertical column are 1 over 2, 1 over 3 and 2 over 3, and these represent the three anti-quarks for SU(3) flavor and the three anti-colors for SU(3) color.

• Ok, so combining the comment from ACuriousMind above with your answer: while a single box lends itself to the interpretation of a single quark state in the fundamental representation, this is not generally true. A two box column is the state for an antiparticle and this is due to the rules of building Tableaux. I guess I was confused because for instance a two box row is the symmetric state say 12 + 21 and a column 12 - 21 where I interpret the 1 and 2 as two quark states. Jan 27, 2016 at 15:42
• I hit enter but wanted to ask one more thing. Say I drew a two box column. If all I gave you was the two boxes, you would not know exactly what I was talking about unless I also gave information on the group (say SU(3)), AND what representation I was in. Are there more than just the fundamental and antifundamental representations? How would I know something like that? Jan 27, 2016 at 15:49
• I agree that this is confusing. Need to know in what context we are using the two box column. For instance, if it appears on the left side of direct product equation of SU(3), that says to me it can only be an anti-3 state of SU(3) and one antiparticle (or anti-color). If it appears by itself on the right side of the equation in a direct sum, then it must be an anti-symmetric combination of two different states of two different particles. A two box column by itself really doesn't mean anything to me unless I know the SU(whatever) it's being used in. Hope I'm not making things worse for you. Jan 27, 2016 at 16:31
• I used this to help jog my memory hepwww.rl.ac.uk/Haywood/Group_Theory_Lectures/Lecture_4.pdf Jan 27, 2016 at 16:59
• Scroll down to page 32 of previous ref for the 3 by 3_bar layout. Fill in the boxes on RHS mixed symmetry tableau (row numbers cannot decrease left to right and column numbers must increase top to bottom). You should get 8 combos for the mixed symm tableau and one for the anti symm one, Jan 27, 2016 at 17:12