QCD Colour Charges: SU(3) Symmetry & Noether's Theorem

[malawi_glenn]

Hi, I was wondering what the connection between the colour charges;

$$Y^C$$, colour hypercharge and $$I^C_3$$, colour isospin charge

and the $$SU(3)$$ symmetry of the colour interaction?

Has it anything to do with Noethers theorem to do?

Does anyone have good source?

Thanx in advance!

 

[blechman]

strong hypercharge and strong isospin have nothing to do with color at all. color was introduced as an additional symmetry in order to solve the spin-statistics paradox of the $\Delta^{++}$. Years later, it was gauged to make QCD.

Strong isospin/hypercharge are part of the flavor structure of the hadrons, the eight-fold way and whatnot. They are approximate symmetries that emerge from the chiral lagrangian in certain limits ($m_u=m_d\neq m_s$) have nothing to do with QCD, except that QCD is presumably involved in chiral symmetry breaking.

 

[malawi_glenn]

So there is no such things that 'red' has [tex]Y^C = 1/3 [tex] and
[tex] I^C_3 = 1/2[tex]? etc.

What are the conserved quantities related to the symmetry of rotations in colour space?

Do you have a good source?

 

[malawi_glenn]

In SU(3) there are two conserved quantities (two diagonal generators).
For SU(3)color these quantities are:
Yc = “color hypercharge”
I3c = the 3rd component of “color isospin”

From: http://www.physics.ohio-state.edu/~kass/teaching.html - Lecture 13, All about QCD (ppt)

That is all I know, I have no books on this except lecuture notes from my teachers I had in CERN summer School this summer.

Observe again that I am not talking about the SU(3)_flavour symmetry of the 3 lighest quarks (u,d,s) that one construct the meson octet and singlet, which are broken due to the larger mass of the s-quark... etc. This I already are familiar with and I am stressing again that I have read on several places about 'Colour charges' which are colour hypercharge and colour isospin charge, which are according to one source related to the SU(3)c symmetry of QCD... I can give you all the other sources if you want:

- Particle physics, by Martin & Shaw, page 142-143
- http://mesonpi.cat.cbpf.br/e2008/pg2-aula3.pdf , page 5

 

[blechman]

OH, wait a second: SU(3) is a rank-2 group, and so there are two Cartan generators, $T^3$ and $T^8$. I assume this is what you are referring to. It is confusing to talk about these as "color hypercharge" or "color isospin" - I've never heard of that before. It only makes sense to name these generators like that if you are to spontaneously break SU(3) to SU(2)xU(1).

With SU(3) as the full group, ALL EIGHT of the generators correspond to conserved charges, not just those of the Cartan subalgebra. These eight charges are the objects that participate in Noether's theorem.

The reason the Cartan generators are so special gets into representation theory of the Lie algebra: that is, all components of a field in an irrep of SU(N) can be labeled by the eigenvalues of the Cartan generators. But they are not special in any other sense.

 

[malawi_glenn]

Ok, I see. Always when I have studied QCD, colour hypercharge and colour isospin charge are mentioned. But the connection to the local-SU(3) symmetry of the QCD lagrangian has always been vauge (so have no idea if it is confusing since it is mentioned in every book and article I have read about QCD so far..).. that was the reason for why I raised the question: "what does: In SU(3) there are two conserved quantities (two diagonal generators).
For SU(3)color these quantities are:
Yc = “color hypercharge”
I3c = the 3rd component of “color isospin”
" mean... :-)

 

[blechman]

It's kinda weird talking about it like that, although it's not wrong. It is true that there are two quantities that are "conserved" in any QCD interaction, just like $J_z$ is conserved in quantum mechanics, where by "conserved" I mean that the Wigner-Ekhart theorem (or its SU(3) extension) applies. This is probably what those books were trying to say. But all eight SU(3) generators actually correspond to dynamically conserved charges in the sense of Noether's theorem.

It seems VERY misleading to me to call them "isospin" and "hyercharge" though, since "isospin" only makes sense in the SU(2) context, and calling it that suggests you've spontaneously broken QCD, which is course is not what's happening! But I guess historical names get pushed on these objects whether it's sensible to do so or not.

 

[malawi_glenn]

Ok, and those two conserved quantities are not due to Noether? What would you call these two quantities? (maybe I recognize their name..)

The reason Iam asking is also that I am about to do my master thesis abous meson physics and will do some Chiral Perturbation theory, and thus, I want to deepen my knowledges in QCD :-)

Here is one more paper talking about these 'charges' that I am referring to in my questions.
http://www.iop.org/EJ/article/0143-0807/2/4/009/ejv2i4p232.pdf?request-id=ecfb9c33-2298-443d-9c27-8a53b239b50a

 

[blechman]

Ok, and those two conserved quantities are not due to Noether? What would you call these two quantities? (maybe I recognize their name..)

no, of course they are. But so are $T^1,T^2,T^4,T^5,T^6,T^7$! They all come from Noether's theorem.

It's exactly analogous to angular momentum in QM: if AM is conserved, then it's conserved! And yet only $J_z$ is diagonal. Why is that? What happened to angular momentum in the x and y direction? Are they NOT conserved?!

I wouldn't call them anything special. I would call them the two Cartan generators.

The reason Iam asking is also that I am about to do my master thesis abous meson physics and will do some Chiral Perturbation theory, and thus, I want to deepen my knowledges in QCD :-)

Here is one more paper talking about these 'charges' that I am referring to in my questions.

Well, I'm not a nuclear physicist, so maybe that's common usage among that community.

 

[malawi_glenn]

Well, I'm not a nuclear physicist, so maybe that's common usage among that community.

Well first, mesons and baryons are not longer belonging to nuclear physics, but have their own branch called 'Hadron Physics' (but maybe some persons refer it to nuclear physics or, rather, Intermediate energy nuclear physics)

Second, all these concepts I have found in ELEMENTARY PARTICLE physics textbooks and articles.

I guess I am still waiting for a answer :-(

 

[blechman]

Well first, mesons and baryons are not longer belonging to nuclear physics, but have their own branch called 'Hadron Physics' (but maybe some persons refer it to nuclear physics or, rather, Intermediate energy nuclear physics)

whatever. call it what you want.

Second, all these concepts I have found in ELEMENTARY PARTICLE physics textbooks and articles.

I guess I am still waiting for a answer :-(

Hey, man, all I can say is that my old profs and colleagues never called them that, and neither do I! I stand by all my previous comments and justifications. What more do you want from me?!

As I said before, fields in an irrep of SU(N) are given by their eigenvalues under the Cartan generators, so that's what these things are. It's identical if you're doing SU(5) GUTs (except now there are 4 such generators - you run out of names!). But I just never call them "isospin" or "hyercharge". Apparently, others do, but I would claim that it is a misleading name, for the reasons I already used. It's not the first time people have stuck to tradition over sensibility!

Hmmm... have I offended anyone yet?!

 

[malawi_glenn]

Ok, but if you can tell me more about the Cartan generators and why they are so special I would love you til the end of time :-D

As far as I know (now), the $T^3$ and $T^8$ commutes and I know what their eigenvalues are.

One wants to find the colourless states for Mesons and Baryons (well hadrons in general), so one writes the colour charges as a linear combination of $T^3$ and $T^8$ since they commute? Or is this just a choice one has done?

 

[blechman]

A Lie algebra of rank n (SU(n+1), for instance) has n simultaneously diagonalizable generators that form the Cartan subalgebra. The eigenvalues of these operators depend on what representation you are in. It is an old theorem of Lie Group theory that objects that transform under the group are labeled by these eigenvalues.

For the canonical example: consider a "rank-k" operator transforming under SU(2) (k is like integer spin). Such an operator has 2k+1 objects inside it. Each of these operators has a specific eigenvalue of $J_z$ that is {-k,-k+1,...,k-1,k}. For example, the position operator $\vec{x}$ is a rank-1 operator with $x_{\pm 1}=x\pm iy,x_0=z$.

Another result of Lie algebra theory is that you can think of a general Lie group of rank n as n copies of SU(2) groups with nontrivial relations between them (that's kinda fuzzy, but I'm trying to avoid details here). Then each SU(2) has a corresponding Cartan generator $J_z$ and as such, each element of an operator transforming under SU(n+1) is described by the eigenvalues of each of these n eigenvalues.

But the values of these eigenvalues (and the number of them) depend on what rep you're in. For example, as above, an operator in the "spin-k" representation has 2k+1 eigenvalues.

That's a VERY crude introduction to Lie group representation theory. For what little it's worth, I have some notes I typed up to help some of my students, where I describe some of this in more detail, but it's still very low-level. For a more solid understanding you need to get deep into algebra. Georgi has a good book, as does Bob Cahn.

If you would like to see my notes, they're at:

http://www.physics.utoronto.ca/~blechman/papers/mpri.pdf

Chapter 2 is on this stuff.

 

[malawi_glenn]

Thank you for all the help :-)

Will tr

 

[malawi_glenn]

I got it now, I suppose those 'hypercharge' stuff were just historical stuff. Thanx again!