*Measurement of the Spin of the Omega-Minus Hyperon*, by the BaBar collaboration at SLAC (B. Aubert et al.), Phys. Rev. Lett. 97, 112001 (2006), and hep-ex/0606039.

My first thought was: How comes that such a paper is published as a PRL? I mean, every child knows that the Omega-Minus has spin 3/2 - just read the textbooks! After all, the Omega-Minus is that famous closing particle of the baryon decuplet!

But then, when reading the paper, I was surprised: Although this hyperon was discovered more than 40 years ago, there has not been a really conclusive measurement of its spin so far! And since the story of the Omega-Minus, its discovery, and its spin is, I think, a quite remarkable one, with connections to lots of interesting physics and some twists maybe not known to every child, I had the idea I should post something about it. I started reading more about the history of the Omega-Minus, came across many interesting details which I thought I could mention, and so, eventually, this has become a somewhat longer post... in fact, so long, that I decided to split it in two. So here is something about

- The baryon zoo of the 1960's and flavour SU(3)
- The prediction of the Omega-Minus by Ne'eman and Gell-Mann

What will follow next will cover

- From the Eightfold Way to the quark model and static SU(6)
- Quark colour
- The spin of the Omega-Minus

#### The baryon zoo of the early 1960's and flavour SU(3)

After the discovery of the neutron, it became clear that atomic nuclei are built up of two types of particles, protons and neutrons, bound together by the so-called strong force. The number of protons and neutrons does not change in strong interactions, and the corresponding conserved charge was called the baryon number,

*B*. Moreover, proton and neutron are so similar under the strong interaction that they were considered as two different projections of one particle, the nucleon, in an abstract space called isospin space. As a spin-1/2 particle can come in two projections of its spin on an axis,

*s*=+1/2 and

_{3}*s*-1/2, the nucleon was considered a isospin

_{3}*T*=1/2 particle, with the proton having isospin projection

*T*=+1/2, and the neutron having

_{3}*T*=-1/2. The formalism of isospin is, indeed, completely identical to the formalism of spin.

_{3}During the 1950s, more particles with the same baryon number

*B=1*as the nucleon were discovered: the Λ, named for the V-shaped tracks in a cloud chamber when it decays into a proton and a negative pion, the Σ's, and the "Cascades", Ξ, which got their name because of their cascading decay pattern Ξ → Σ + ... → nucleons + ... The concept of isospin could be applied also to these new baryons: the Λ is a singlet with

*T*=0, the cascades Ξ

^{-}and Ξ

^{0}are a doublet, as the neutron and the proton, and the Σ's form a triplet with

*T*=1 - in fact, Murray Gell-Mann predicted the neutral Σ

^{0}based on the assumption of the triplet, once the Σ

^{+}and Σ

^{-}were known.

In order to classify these baryons, a new quantum number introduced by Gell-Mann, and called "strangeness",

*S*, was useful. Strangeness (or "hypercharge"

*Y*, which is related to strangeness by

*Y*=

*B*+

*S*) can change in weak decays - similar to the decay of the neutron into proton - but not in strong interactions. Then, the Λ and the Σ's have strangeness

*S*= -1, or hypercharge"

*Y*= 0, while the Ξ's have strangeness

*S*= -2, or hypercharge

*Y*= -1. When trying to cast the description of baryons by isospin and strangeness in a unified, symmetric framework, Gell-Mann discovered that the eight baryons can be identified with the eight-dimensional, adjoint representation of the Lie group SU(3). This group, the group of special unitary transformations of a complex three-dimensional vector space, is an extension of the isospin group SU(2) to include strangeness as one further degree of freedom. It has eight generators, instead of the three generators of SU(2), which span the adjoint representation. Gell-Mann called the classification of baryons as an octet of SU(3) the "eightfold way". The very same classification scheme was discovered, independently of Gell-Mann, by the Israeli physicist Yuval Ne'eman. Ne'eman, who was then an army officer on leave to do his Ph.D. with Abdus Salam in London, died earlier this year. The SU(3) scheme discovered by Gell-Mann and Ne'eman is known today as flavour-SU(3).

The "eightfold way": the baryon octet, corresponding to the weight diagram of the eight-dimensional, adjoint representation of the group SU(3).

*S*is strangeness,

*Y*=

*B*+

*S*= 1 +

*S*is hypercharge, and

*T*is the isospin projection. Representations of SU(3) can be labelled by two numbers,

_{3}*p*and

*q*, which also determine the shape of the multiplet. There are two states in the centre of the multiplet, corresponding to the Λ and the Σ

^{0}. All baryons in the octet have spin 1/2.

#### The prediction of the Ω^{-} by Ne'eman and Gell-Mann

Tables of elementary particles in the early 1960s were quite crowded, and the eightfold way only a first step on the road to a systematic understanding. There were man more baryonic particles known besides the octet baryons. Most of these are very short lived. Typically, they show up in scattering experiments of pions or kaons, the strange mesons, on octet baryons. There, they are visible as bumps in the scattering cross section as a function of energy. They are called resonances for this reason. The first such particles, the Δ resonances, had been discovered in 1952 by Fermi's team using pion-proton scattering experiments. Δ resonances have spin 3/2, and come in four different electric charges (-, 0, +, and ++) at the same mass of 1232 MeV, so they must belong to a state with isospin

*T*=3/2.

In July 1962, when elementary particle physicists from all over the world met at the 11th International Conference on High-Energy Physics at CERN, there was news about resonances in scattering experiments on strange baryons. Two years before, the Σ

^{*}resonances (with isospin 1, then still called Y

^{*}) had been discovered when scattering negative kaons on protons (Margaret Alston et al., PRL 5 (1960) 520-524), which had spin 3/2 as the Δ's (Robert P. Ely et al., PRL 7 (1961) 461-464). At CERN, the detection and properties of Ξ

^{*}resonances was reported. These resonances with strangeness

*S*=-2 seemed to form an isospin doublet, thus to have isospin 1/2 (G. M. Pjerrou et al., PRL 9 (1962) 114-117), and there were strong hints that their spin was 3/2, as for the Δ's and the Σ

^{*}s. Was there a way to make sense of these resonances, or to fit them in a classification scheme?

Two physicists (Gell-Mann on the right?) discussing the table of known hadronic particles and resonances at the CERN conference in July 1962. The second column indicates strangeness, the third column isospin. Further columns give mass and width, and the last one, with the many question marks, spin and parity. (Does anyone know what SNOW stands for?) The then recently discovered Σ

^{*}resonance is noted as Y

_{1}

^{*}in the third to last row, and the brand-new Ξ

^{*}shows up in the last row. (Credits: CERN, via

*Rochester Roundabout: The Story of High Energy Physics*, by J. C. Polkinghorne)

Both Ne'eman and Gell-Mann attended the CERN conference. Ne'eman had submitted an abstract about his work on the SU(3) classification scheme of baryons, but he wasn't given a slot to talk about it. It seems that SU(3) wasn't taken that serious yet. But this didn't stop him from thinking hard about possible ways to integrate the new resonances into his scheme. Several multiplets could, in principle, accommodate for the new resonances: a decuplet, corresponding to (

*p*=3,

*q*=0), a 15-plet, with (

*p*=2,

*q*=1), and a 27-plet, with (

*p*=2,

*q*=2), whose weight diagram would have the same sixfold symmetry as the octet diagram. Ne'eman had no clue about which one to chose, when he met a husband-wive pair of experimentalists originating from Israel on the bus trip from the conference hotel to CERN, Sulamith and Gerson Goldhaber. (Gerson is now involved in the supernova Ia measurements of the Perlmutter group that established cosmic acceleration - that's amazing!). They started talking about physics, the Goldhabers asked him about SU(3), and they told him that they had, without success, tried to repeat the Alston et al. scattering experiments, but using positive kaons on neutrons instead of negative kaons on protons. That was just the piece of information that was missing! Both the 15-plet and the 27-plet, if they were the correct multiplets to classify the baryon resonances, would have required resonances of positive kaons on neutrons! Thus, they were excluded by the negative results of the Goldhaber experiment!

The 27-plet of SU(3), with (

*p*=2,

*q*=2), could, in principle, accommodate all the baryonic resonances known in 1962. But then, there should also be resonances with strangeness

*S*=1, which should show up in scatterings of positive kaons on neutrons. The experiments of the Goldhabers and their group excluded the existence of such resonances. This negative result, later called the Goldhaber Gap, eliminated the 27-plet, and the 15-plet, from the possible multiplets to classify the resonances, leaving only the decuplet. Amusingly, the Goldhaber Gap corresponds exactly to the position of the elusive pentaquark Θ

^{+}, which made such a fuss in the last three years, but seems not to exist, after all.

Now, Ne'eman had everything he needed to know to come to a conclusion: The baryon resonances fitted neatly in the decuplet (

*p*=3,

*q*=0) of SU(3). Moreover, most excitingly, there was exactly one resonance still missing in this multiplet, the resonance with strangeness

*S*=-3. This resonance should exist, if the decuplet scheme was right, and he could even say something about its mass, using a formula of Gell-Mann and Okubo. This formula predicted a linear splitting of the masses of the decuplet resonances with strangeness, and, indeed, the mass difference between the the Δ's and the Σ

^{*}s was about 150 MeV, as was the mass difference between the Σ

^{*}s and the Ξ

^{*}s. Thus, Ne'eman was confident that there should be a isosinglet resonance with

*S*=-3, negative electric charge, spin 3/2, and a mass of about 1680 MeV, and he decided to make this point in the discussion following a review talk on the new baryonic resonances.

Sulamith Goldhaber, of the Goldhaber gap, with Yuval Ne'eman, visiting the Goldhabers at Berkeley (Credits: Lawrence Berkeley Lab Magnet, Vol. 8, No. 4, April 1964, p. 2)

He didn't have luck. Gell-Mann had also heard the rumours about the negative results of the Goldhaber experiment, and, consequently, he had come to very the same conclusions as had Ne'eman. So, following the presentation on Strong interactions of strange particles by G. A. Snow, both Ne'eman and Gell-Mann raised their hands to ask for permission to speak. The chairman called Gell-Mann, who was the more eminent physicist of both, and Gell-Mann announced that "[...] we should look for the last particle called, say, &Omega

^{-}, with

*S*=-3,

*I*=0. [Here,

*I*is isospin.] At 1685 MeV it would be metastable and should decay by weak interaction [...]". This was the public prediction of the closing resonance of the baryon decuplet, fittingly named after the last letter of the Greek alphabet, and subsequently published in the proceedings of the conference. It seems that Gell-Mann got to know Ne'eman in person for the first time just on the way back to his chair in the auditorium, when he read Ne'emans name on the name tag. And it seems that Ne'eman was not too bitter of being scooped in the last second - at least, they started a long-lasting collaboration, and published together the Eightfold Way.

The SU(3) decuplet of the baryon resonances. The resonances known in 1962 are shown in black. There was one particle missing, at the lower tip of the triangle. This particle was called &Omega

^{-}by Gell-Mann, who predicted its strangeness, spin, isospin, and mass. Ne'eman had come to the very same prediction at the same time.

In the break following the talk of Snow, Gell-Mann and Ne'eman discussed with two experimentalists from Brookhaven National Lab, Nicholas Samios, who later became director of BNL, and Jack Leitner. They thought about possible ways to detect the &Omega

^{-}in experiment. Indeed, a search program was set up at BNL, which was successful two years later: the &Omega

^{-}was found on a bubble chamber picture (V. E. Barnes et al., PRL 12 (1964) 204), and its properties were exactly as predicted by Gell-Mann and Ne'eman.

At that point, there was no more doubt that the SU(3) classification scheme of particles had some truth about it.

The discovery of the &Omega

^{-}in a bubble chamber picture. The &Omega

^{-}leaves the short, thick track in the lower left corner. (Credits: BNL and V. E. Barnes et al., PRL 12 (1964) 204).

The actual detection of the &Omega

^{-}was a big success of the SU(3) classification scheme, but it was not the end of the story. Surprisingly, the spin of the &Omega

^{-}could not been measured so easily - even the 2006 particle data book entry on the &Omega

^{-}still states on the baryon summary page:

*J*

^{P}is not yet measured; 3/2^{+}is the quark model prediction.So, stay tuned, and read next time about:

and why, maybe, SLAC could have pushed the headline;

### Quark model prediction finally proven after 40 years!

The story leading to the prediction of the &Omega

^{-}at the 1962 CERN conference is related in several sources. I have used (thanks, in part, goes to Google Search Inside):

- The Second Creation: Makers of the Revolution in Twentieth-Century Physics, by Robert P. Crease, Charles C. Mann
- The Particle Century, edited by G Fraser, with contributions by Ne'eman and Samios
- Strange Beauty: Murray Gell-Mann and the Revolution in Twentieth-Century Physics by George Johnson
- Rochester Roundabout: The Story of High Energy Physics by J. C. Polkinghorne
- The Periodic Table and Genetic Code of the Hadrons by Yuval Ne'eman
- From SU(3) to Gravity: Festschrift in Honor of Yuval Ne'eman edited by Errol Gotsman and Gerald Tauber, with a contribution of Gerson Goldhaber about the encounter in the bus on pages 103-105.
- The Eightfold Way, by Murray Gell-Mann, Yuval Ne'eman. The new edition contains the historically oriented survey
*Hadron Symmetry, Classication and Compositeness*by Yuval Ne'eman, first published in Symmetries in Physics (1600-1980): Proceedings of the 1st International Meeting on the History of Scientific Ideas, held at Sant Feliu de Guíxols, Catalonia, Spain, September 20-26, 1983. edited by M. G. Doncel, A. Hermann, L. Michel and A. Pais (Barcelona, 1987).

Physics Omega-Minus SU(3) Quark Model

The Standard Model begins massless and requires almost two dozen parameter inserts (mostly particle masses) plus the Higgs mechanism. That is disturbing. Standard Model extensions are disasters of empirical absence. If the Higgs remains invisible in the LHC, what originates mass?

ReplyDeleteIs the vacuum isotropic and angular momentum thereby conserved (Noether)? A counterexample with 0.27x10^(-12)/day divergence orbits 240,000 miles overhead. Are two days in a pair of differential scanning calorimeters testing for a chiral pseudoscalar vacuum background so onerous?

Hi Stefan, thanks for that very informative post! What do you think would have happend if they had measured a spin 1? Best, B.

ReplyDeleteHi Stefan, excellent post

ReplyDeleteMost informative, laced and flavoured with historical facts.

Particle Physics is becoming more and more exciting

PS - loved Bee's previous post on microstate bhs too.

ReplyDeleteI read all the links too, Thanks!

Just that it would not be fair for me not to expess differences of point of view, from the macroscale.

Hope you take any differences and contradictory opinions with a pinch of salt - or as constructive criticism and debate.

Hope You Are Having Fun.

Laters ...

Stefan, another great post!

ReplyDeleteThis:

http://physicsweb.org/articles/news/10/9/16

Thus:

http://blogs.nature.com/nature/peerreview/trial/2006/09/boseeinstein_condensation_of_m.html

and this:

http://en.wikipedia.org/wiki/Heat_equation

can have interesting consequences.

1-D, 2-D and 3-D factors ?

Form the first link:"But Deveaud-Plédran happily dismisses both assertions. "BEC is forbidden only in two dimensions for an infinite system without disorder. We have a finite system with disorder, so standard BEC is allowed. And despite the quasiparticle nature and very short lifetime, we have shown that we are able to get a thermal equilibrium."

Radio active decay is a 3-D time ordered function?

Dimensional Spin must also surely be factors?

There are no physical examples of a Solid existing in less than 3-D form?

If one read through the Heat Equation, one can see where the facts deviate away from Special Relativity.

Dear Stefan:

ReplyDeleteThis is an interesting story. What do these multiplet diagrams look like for SU(N) groups with N>3? Are they higher dimensional? Sorry for the dumb question. Yours,

Rob

question:

ReplyDelete> What do these multiplet diagrams look

> like for SU(N) groups with N>3? Are

> they higher dimensional?

--- good question!

It is convenient for us that the largest group (to be more precise, Lie algebra) that is so far compulsory to understand particle physics has rank 2, which means that its representations can be drawn on a two-dimensional surface, i.e. a blackboard.

SU(n) has rank n-1, so if we wanted to model something with SU(4) we would need to use 3-dimensional diagrams to display all the particles in their proper relationships.

SU(2) is actually used to describe spin itself and, sure enough, when you write down all the possible spin states the Omega^- can be found in they just lie on a one-dimensional line

-3/2 -1/2 +1/2 +3/2 .

Now, I am waiting to find out how the spin was measured! It ought to be some difficult measurements of angular distribution ...?

Dear bee,

ReplyDeleteWhat do you think would have happend if they had measured a spin 1?I really do not know. Probably they would have checked again everything, and again, and again...

I cannot think of any explanation of spin 1 that would be possible within the quark model. Even if you consider possible contributions to spin through angular momentum or glue, it's not possible to obtain spin 1, I think.

So, such a result, if solidly established, would have been a really big challenge and definitly beyond the standard model.

Best, stefan

Hi Thomas,

ReplyDeletethanky you for your answer!

Hi Rob,

yes, as Thomas has explained, the multiplets of SU(N) can be drawn in N-1 dimensions. Physically speaking, this means that if an interaction Hamiltonian has SU(N) symmetry, there are N-1 conserved quantum numbers, which are plotted along the N-1 axis of the multiplet diagrams. For the strong interactions, its isospin for N=2, isospin and strangeness (or hypercharge) for N=3, and isospin, strangeness and charm for N=4.

The multiplets for SU(4) are three-dimensional, and they are used indeed to classify charmed hadrons - see for example page 3 and page 11 of the quark model review of the PDG for the corresponding multiplets.

SU(4) symmetry may be badly broken by the large mass of the charm quark, but as a means of classification of hadrons (or, sometimes, interaction terms) it can be used nonetheless.

If you cut slices through these three-dimensional SU(4) diagrams at constant charm, you find again multiplets of SU(3) (or sums of multiplets of SU(3)). For example, the "lowest planes" of the SU(4) baryon multiplets on page 11 are, again, the SU(3) octet and decuplet, respectively.

Best, stefan

Hi Paul,

ReplyDeletethanks fo the links to the news BECs... All this is a bit off-topic here, but just two comments:

BEC is forbidden only in two dimensions for an infinite system without disorder.I guess they are alluding to the Mermin-Wagner-Hohenberg theorem, which states that there is no spontanous braking of symmetry in two dimensions, hence no perfect solids, which break translational symmetry.

Instead, there is the Kosterlitz-Thouless-Berezinsky transition between quasi-long-range order and "fluid".

If one read through the Heat Equation, one can see where the facts deviate away from Special Relativity.That has been known since long... and it concerns not only the heat equation, but also hydrodynamics if you want to include viscosity. There are also remedies known in order to formulate relativistic theories, but the equations are quite complicated. This has the practical consequence that, for example, in the simulation of relativistic heavy ion collisions using hydrodynamical models, a proper treatment of effects of viscosity has only recently become available.

Best, stefan

Dear Thomas, dear Stefan:

ReplyDeleteThanks so much for your answers. In the danger of appearing even more dumb, I still don't completely understand it. I know that the group SU(N) has N-1 Casimir Operators. In my understanding the quantum numbers belonging to these operators are needed to uniquely specify the multiplet. Thats what is said here..

But what you say is that these quantum numbers uniquely specify the particle IN the multiplet? What then classifies the multiplet? I mean, if I want to know which particle I am talking about do I need the quantum numbers of the Casimir operators + knowing which multiplet, and how many of these are there? Infinitly many?

Thanks,

Rob

Come on, Stefan.

ReplyDeleteIt's great if hundreds of skillful people have well-defined tasks to do so that experimental physics is kept in good shape but I am sure that every reasonable member of the collaboration knows that this is not a new discovery or overly interesting result, and most citations of them will likely be self-citations for some time.

How could a member of the decuplet of simple bound states OmegaMinus have a different spin than others, namely 3/2? The quarks have been observed and their properties are known. Of course that it must work. Any serious doubt about these experiments would mean a complete distrust in theory (and theorists) in physics.

There are thousands of similar experiments one can invent to confirm basic facts about QCD. The overhyping of some results that would really be interesting only in the early 1960s shows how important it is for experimenters to have theorists who tell them what is interesting and what is trivial.

In related setup, the experiments at RHIC (and perhaps others) about QGP are more nontrivial because in the 1960s, people really didn't have tools to predict. So these experiments became cutting edge science, despite having low energies. But the measurement of the OmegaMinus spin is not cutting edge experimental physics in 2006.

Best

Lubos

Hi Stefan,

ReplyDeleteIt's not that I'd doubt the spin was 3/2, I just found it an interesting sociological speculation what might have happened if. Yes, they would have measured again and again. Can reality be in contradiction to what we thought all the time? I guess it would have severely traumatized a whole generation of particle physicist.

Hi Lubos,

No, it's certainly not cutting edge physics in 2006. But then, if everybody did cutting edge experiments, then they wouldn't be cutting edge any more. I think it's just necessary to measure these things at least once. Otherwise we'll run into a problem drawing a line between what needs to be experimentally tested, and things we then 'know' because we trust our theories.

Best,

B.

Dear Rob,

ReplyDeleteno, that's no dumb question at all, there is, IMHO, quite a lot of confusing nomenclature around...

So, first of all, the multiplets of SU(N) can be uniquely described by N-1 nonnegative integers, but these are in no immeditate relation to the N-1 Casimir operators of the Cartan subalgebra, I think.

For SU(3), I have used the quite standardized notation (p,q) for this purpose. These numbers also fix the shape of the multiplet. There is one way to do this as described in the PDG text you have cited, the other one which I know of, is to first outline the border of the multiplet, and then filling the innner arts. You ouline the multiplet by starting at one state "on the far right" (which is sometimes called state of heighest weight), then going p steps in the direction "left downwards", and q steps in the horizontal direction to the left. You complete the diagram by threefold rotational symmetry and fill up the "inner states". To fill up, when going inwards from the border, multiplicities of the states increase by one until the inner states have a triangluar shape (either p or q is 0 for the inner states)... I hope this is somehow clear... Anyhow, from these two numbers p and q, you get the shape of the multiplet, and the dimension, by the formula dim = (p+1)(q+1)(p+q+2)/2. You can check that this works with the (1,1), (3,0), and (2,2) multiplets (dim=8,10,27).

There are inifitely many multiplets.

Similar rules apply for SU(N) with N>3, where you need N-1 nonnegative integers.

I am not sure how, or if, the numbers (p,q), or more for SU(N) with N>3,relate to the N-1 Casimir operators you mention. However, the states in the multiplet specified by (p,q) are eigenstates under these operators, and the eigenvalues just give the position of the state within the multiplet, or, what is the same, the quantum numbers of the state. For SU(3), these eigenvalues/quantum numbers are the 3-component of isospin, and hypercharge.

Hope this helps?

Better explanations or corrections are welcome!

Best, stefan

Hi bee,

ReplyDeleteI completely agree :-)

Hi Lubos,

I also agree: the experimental result as such is neither cutting edge physics, nor is there the tiniest reason to doubt that the result could be anything else than what was eventually measured.

But I was quite surprised, nevertheless, that such a comparably elementary property as the spin of the Omega-Minus has not been measured before.

I could imagine that this is exactly because no one thought it worthwile to take the effort/means/costs to do so, just because there was no reason to doubt the result. Even now, the paper seems to be merely a by-product of the cutting edge B-factory CP violation physics projects at SLAC. I can also imagine that someone in the BaBar team had the idea, hey, we could easily do this analysis, measure the spin of the omega-Minus, could be a nice exercise project for a student, and perhaps we can make a paper out of it...

I think, as Bee says, it's important to measure things that can be measured at least once. If there is no reason to doubt the outcome, then the experiment better should not cannibalise more important or interesting experiments. This seems to be just the case with this Omega-Minus experiment - so it is beautiful to see it done, IMHO. And I guess the referees for PRL had similar views ;-)..

Best, stefan.

Dear Stefan,

ReplyDeletenow I am also confused. I also thought the number of Casimir invariants (independent elements of the max. abelian subgroup, rank of the group) is the dimension of the diagram. The operators themselves however are not uniquely defined, since you can always redefine the base, or, change the axis in the diagram. There is a way to define the operators such that the diagrams are so nicely symmetric, but the precise prescription I forgot.

Rob, yes, besides the location in the diagram you also need to know which representation you are looking at, and there are infinitely many. The multiplet is somehow defined by the maximal value the quantum numbers can take.

Best,

B.

Dear Quasar,

ReplyDeleteHope you take any differences and contradictory opinions with a pinch of salt - or as constructive criticism and debate.Thanks, I appreciate your contributions, and I find your opinions undoubtedly interesting. I just honestly didn't know what further to say to your comments. But the world is not just black and white, so its always good to have different perspectives on a topic.

Best

B.

Dear Bee, Rob,

ReplyDeletesorry, I didn't want to increase confusion... so

the number of Casimir invariants (independent elements of the max. abelian subgroup, rank of the group) is the dimension of the diagramYes!

For SU(N), this number (= number of Casimir invariants = dimension of the maximal abelian subgroup = rank of the group = dimension of the Cartan subalgebra = dimension of the multiplet diagram) is N-1.

What I wanted to say is that, moreover, there are N-1 nonnegative integers necessary to uniquely define the infinitely many multiplet diagrams, or, the irreducible representations of SU(N).

I think this is a special property of the SU(N) groups - there may well be more than N-1 numbers necessary to uniquely define a multiplet diagram that is drawn in N-1 dimensions?

For SU(3), there are two integers, usually called (p, q). They define the shape of the multiplet diagram and the multiplicities. Furthermore, they determine the allowed range of the quantum numbers of the states of the multiplet. In flavour-SU(3), these quantum numbers are, as mentioned before, isopin 3-component T_3 and hypercharge Y. Now, this range is

Y: -(2p+q)/3, ... , +(p+2q)/3

and

T_3: -(p+q)/2, ... , +(p+q)/2 for Y = (p-q)/3.

At other hypercharges, the range of possible T_3 values is smaller.

Hope not to increase entropy - Best, stefan

So in toto it takes 2*(N-1) numbers to define the particle?

ReplyDeletecum grano salis, yes:

ReplyDeleteN-1 to fix the multiplet, and another N-1 to fix the position within the multiplet.

However, the inner positions in a multiplet can have a multiplicity higher than 1. For these states, more numbers are necessary. For example, the central positition of the baryon octet is doubly occupied, by the Σ_0 and by the Λ. Both have Y=0 and T_3=0, but the Σ_0 has total isospin T=1, while the Λ has T=0.

Unfortunately, I do not know right now how this is expressed in a formal way, and how it generalises for SU(N) with N higher than 3...

The best review of this is Slansky, ``Group Theory for Unified Model Building'', Phys. Rep. 79 (1981)

ReplyDelete... not having read about it for some time I can't remember the formal solution to the problem of different particles sitting at the same place in the diagram.

So far as I remember the representation can be specified by the 'state of highest weight' which is unique within each representation and as Stefan says sits at the edge of the diagram.

Then the other states are obtained by applying 'ladder operators' which change the values of the Casimir operators. Since the ladder operators do not commute with one another there may be more than one inequivalent way to reach a point in the middle from a point at the edge. So although the states in the middle have the same values of Casimir operators, you can tell the difference by hitting them with something made up of the ladder operators.

At least, that's my guess at an explanation.

Dear Bee, Stefan and Thomas:

ReplyDeleteThanks so much for your explanations! I think, I finally get it. Me, I come more from the mathematical side, and its equally fascinating as complicated to understand the physics behind group theory. But isn't this totally awsome how real particles can be classified in abstract weight diagrams?

This is a great blog by the way, and thanks again,

-- Rob

Dear Thomas,

ReplyDeletethanks for the explanations, and the reference to the Slansky review!

However, I have to say I am still looking for a good paper or text about all this representation stuff that combines mathematical exactness with thourough physical understanding...

Dear Rob,

But isn't this totally awsome how real particles can be classified in abstract weight diagrams?Absolutely! To me, it's a big mystery!

Besides, I am also fascinated by the mathematics of the representations of the simple Lie groups, and though I have learned how to use some recipes, I wish I would really understand it...

And thanks that you like the blog :-)