The Omega-Minus gets a Spin (part 1)

Last Friday, I could not come to any decision which of the many Perimeter photos I could write something sensible about, and browsed through the latest issues of the PRL instead. There, I stumbled across a paper whose title really amazed me: 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

1. The baryon zoo of the 1960's and flavour SU(3)


2. The prediction of the Omega-Minus by Ne'eman and Gell-Mann

What will follow next will cover

1. From the Eightfold Way to the quark model and static SU(6)


2. Quark colour


3. The spin of the Omega-Minus

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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, s₃=+1/2 and s₃-1/2, the nucleon was considered a isospin T=1/2 particle, with the proton having isospin projection T₃=+1/2, and the neutron having T₃=-1/2. The formalism of isospin is, indeed, completely identical to the formalism of spin.

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 Ξ⁰ 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 Σ⁰ 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 T₃ is the isospin projection. Representations of SU(3) can be labelled by two numbers, 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 Σ⁰. 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 Y₁^* 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:

1. From the Eightfold Way to the quark model and static SU(6)


2. Quark colour


3. The spin of the Omega-Minus

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

Quark model prediction finally proven after 40 years!

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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):

1. The Second Creation: Makers of the Revolution in Twentieth-Century Physics, by Robert P. Crease, Charles C. Mann


2. The Particle Century, edited by G Fraser, with contributions by Ne'eman and Samios


3. Strange Beauty: Murray Gell-Mann and the Revolution in Twentieth-Century Physics by George Johnson


4. Rochester Roundabout: The Story of High Energy Physics by J. C. Polkinghorne


5. The Periodic Table and Genetic Code of the Hadrons by Yuval Ne'eman


6. 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.


7. 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).

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Physics Omega-Minus SU(3) Quark Model