The eventful story of charmonium singlet states

The discovery of the J/Ψ particle in November 1974 by the teams led by Burton Richter at SLAC and Sam Ting at Brookhaven came as a great surprise. However, after a period of uncertainty, ended by the discovery of the Ψ′ at SLAC, the J/Ψ was identified as a bound state of a charm quark and an antiquark, c-cbar, which had been explicitly predicted in 1970 by Sheldon Glashow, John Iliopoulos and Luciano Maiani. In the J/Ψ and the Ψ′, the spins of the c and ccbar are parallel and form a triplet state (spin 1) associated with a space wave function of orbital momentum l=0. However, as in positronium (e⁺e^–), there also exist singlet states in which the spins are antiparallel, with orbital angular momentum l=0 or l>=1, as shown in figure 1.

The story of the experimental search for the l=0 singlet states and the efforts of theoreticians to explain the successive and contradictory experimental results, is an interesting one. The table summarises the history of the ground and first excited singlet states, η_c and η_c′ (or η_c(2S)). Δm and Δm′ give the hyperfine splittings, or in other words, the mass differences between these singlet states and the related triplet states.

In the late 1970s, experiments found Δm, the mass difference between the η_c and the J/Ψ, to be about 300 MeV (Braunschweig et al. 1977, Apel et al. 1978). However, this result was difficult to swallow for two reasons. First, naive estimates of the hyperfine splitting give much smaller values, and second, the radiative decay width J/Ψ → η_c + γ is proportional to Δm³, so any theory correctly predicting Δm ~300 MeV would overestimate this width. This is why most theoreticians were extremely sceptical about the result from the DASP experiment (Braunschweig et al. 1977). Fortunately, the Mark II and Crystal Ball groups found in J/Ψ → η_c + γ what we believe is the true η_c, with a splitting of Δm = 119 MeV (Himel et al. 1980, Partridge et al. 1980).

A little later, the Crystal Ball group also found a candidate for the η_c′, again by radiative decay, but from the Ψ′ (Edwards et al. 1982). The Δm′ =~ 90 MeV splitting they found is acceptable – for instance Wilfried Buchmuller, Yee Jack Ng and Henry Tye found 80±10 MeV in a QCD-inspired calculation (Buchmuller, Ng and Tye 1981). However, the ratio Δm′/Δm seems difficult to accept. First, a naive estimate using a Fermi-like hyperfine interaction suggests that Δm′/Δm is related to the ratio of the leptonic widths of the Ψ′ and J/Ψ. This gives Δm′/Δm =~ 0.6±0.1, which is hardly consistent with the Crystal Ball result. In addition, there are effects due to the coupling of the ccbar bound states to the charm-anticharm meson pairs, D^(*)Dbar^(*), as we pointed out in 1981. The coupling to the very close DDbar threshold is allowed for a vector state, so this should make the Ψ′ lower than predicted by naive potential-model calculations. The pseudoscalar η_c, on the other hand, does not couple to DDbar and so is shifted much less. Using the Cornell model (Eichten et al. 1978, 1980), we found that this effect reduces Δm′ by at least 20 MeV.

The puzzling Crystal Ball result on η_c′ was never confirmed. Searches for the η_c′ in formation experiments in proton-antiproton collisions, first at the ISR and then at the Fermilab accumulator, were unsuccessful. This may be because these experiments had too high a resolution in energy, and perhaps because of prejudice that the η_c′ would not be too close to the Ψ′. The coupling of the η_c′ to proton-antiproton might also be less favourable than for η_c. Meanwhile the η_c was seen at LEP, in its γγ decay mode, but no signal was found for the η_c′.

Charmonium can also be investigated through B decay, as proposed by several authors (e.g. Eichten et al. 2002). The Belle experiment at KEK, whose primary purpose is to study the CP violation in B decays, has seen both the η_c and η_c′ in two distinct channels, which we can call Belle I and Belle II. The BaBar experiment at SLAC should also produce similar results.

In Belle I, the decays B → Kη_c(η_c′) → KK_sK^–π⁺ reveal two main peaks, as in figure 2 (Choi et al. 2002). The first is clearly the η_c, while the second is most likely the η_c′, as the background from B → K + J/Ψ or K + Ψ′ is expected to be rather small. This implies that m(η_c′) = 3654±6 MeV, i.e. Δm′ = 32±6 MeV, which is much smaller than the Crystal Ball-obtained value, and even smaller than we expected from the effect of the coupling to charm-anticharm channels.

In Belle II, the reaction studied is e⁺e^– → J/Ψ + ccbar, i.e. double ccbar production with one pair constrained to match the J/Ψ (Abe et al. 2002). The recoil spectrum against the J/Ψ gives a set of ccbar bound states. If the process takes place via e⁺e^– annihilating into one photon, charge conjugation conservation strictly forbids J/Ψ and Ψ′, and three peaks corresponding to η_c, Χ₀ and η_c′ can be seen (figure 2). This time Δm′ is somewhat higher, about 60 MeV, which is more consistent with our 1981 expectation. On the other hand, the η_c is shifted with respect to the standard value of the Particle Data Group. The imperfect agreement between Belle I and Belle II will hopefully disappear in the final analysis and in particular it should be decided whether or not a background B → Ψ′ + K or (unlikely) e⁺e^– → J/Ψ + Ψ′ contributes to the observed spectrum. In any case, we are very close to the complete clarification of the η_c′, with a mass much closer to the Ψ′ than was indicated by the Crystal Ball group.

Theory also predicts a ccbar singlet P-state called h_c. Paradoxically, the corresponding state in positronium has only been observed relatively recently (Conti et al. 1993). First indications for h_c came from the R704 experiment at the ISR, in which a cooled antiproton beam collided with a gas jet target (Baglin et al. 1986). This was at the time when the ISR was to be stopped and dismantled. At the request of one of us (A M), a few extra days running were granted by the director-general, Herwig Schopper, but no firm conclusion could be reached. Years later a similar experiment, E760, was carried out at Fermilab and gave strong indications of the h_c at the same mass that happens to agree with the most naive prediction, i.e. the weighted average of the triplet P-state masses (Armstrong et al. 1992). However, these indications have disappeared in the latest experiment, E835 (Patrignani et al. 2001). Assuming that E760 was right, it is tempting to wonder if the same scenario will not repeat itself with the Higgs search: indication in the last runs of LEP of a Higgs at 115 GeV, which might be right and so be definitely seen years later with the LHC.