𝐶𝑃-violation studies are not the only field in which the B-factories made im-portant discoveries. Another area that held particularly surprising results was charmonium spectroscopy. The detailed investigation ofccbound states was possible thanks to the copious production of charmed mesons inBdecays.
Since the center-of-mass energy at B-factories is mostly fixed to theΥ(4S) mass, resonant production of charmonium, as in dedicated charm factories like the BES III experiment at the BEPC II electron-positron collider in Beijing, is not feasible. States with lower energies and𝐽𝑃𝐶 = 1−−can be produced if either the electron or the positron emits a photon before the collision—a process known as initial state radiation. Another possible production channel is the two-photon processe+e− → e+e−(γ∗γ∗) → e+e−cc, which allows the quantum numbers 𝐽𝑃𝐶 = 0±+, 2±+, 4±+, …and3++, 5++, …[35].
The most important cc-production channel for B-factories, however, is through decays ofBmesons: Theirbquarks must eventually decay weakly into an up-type quark. Their coupling to the charm quark is much stronger than to the up quark, so the processb → cW− → ccsis abundant. This mechanism can, in principle, produce any quantum number. It led to the discovery of the exotic charmonium states described below. Lastly, charmonium states can be produced via double-ccproduction, for examplee+e− → J/ψ + cc. This channel is particularly interesting because its cross section is much larger than predicted by theory [36].
In contrast to the light (u,d, ands) quarks, the mass of the charm quark is in the same order as that of its bound states, with2𝑚c ≈ 2550 MeVand2900 MeV <
𝑚cc < 4700 MeV. The constituent quarks can therefore be associated with a small velocity, and the system can be approximately treated as non-relativistic.
Similarly to the hydrogen and positronium systems in electrodynamics, a simple potential can then be used to model the force between the two quarks, and the energy levels of the system—the masses of the charmonium states—can be obtained by solving the Schrödinger equation.
An example for such a potential is [37]
𝑉0(cc)(𝑟) = −4 3
αs
𝑟 + 𝑏𝑟 + 32παs 9𝑚2c (
𝜎
√π)
3
e−𝜎2𝑟2𝑆⃗c⋅ ⃗𝑆c,
where αs, 𝑏,𝑚c, and 𝜎 are parameters that are determined from fits using known charmonium masses as input. The first term is a Coulomb-like potential that models the binding force at short distances. The difference to the Coulomb potential from electrodynamics lies mainly in the much larger coupling constant αs. The second term, which becomes dominant at larger distances, introduces a
2.4. CHARMONIUM SPECTROSCOPY AND EXOTIC STATES 23
linearly rising potential, resulting in a constant attractive force between the two quarks. I can be seen as a model for color confinement since an ever-increasing amount of energy must be expended to pull the quarks farther apart. The third term models the spin-spin hyperfine interactions between the two quarks.
The referenced model treats additional spin-dependent terms, like spin-orbit coupling, as perturbations that lead to mass shifts of the determined states.
Figure 2.8 shows the masses of the charmonium states predicted by this model in comparison with experimental values. The correspondence between theoretical predictions and measurements are almost perfect in the mass region below the open-charm threshold—the energy above which decays into two charmed mesons are possible. Some of these states, like thehc, have long evaded experimental discovery, but their masses have been correctly predicted for decades.
Above the open-charm threshold, the predictive power of potential models diminishes. While some of the predicted states have not yet been discovered, others miss the experimental values by tens of MeV. The decays into charmed mesons that become possible at these energies complicate the situation. What’s more, other theoretical models predict the existence of exotic states at higher energies in the charmonium system. Potential models cannot predict such states, since they only describe two-quark systems, so more fundamental methods must be used.
Lattice QCD is such a method. It is a non-perturbative approach that calcu-lates QCD on a discrete, four-dimensional spacetime grid using computer simu-lations. Since lattice QCD calculations are based on first principles of QCD, they allow the determination of all bound states that are possible in QCD, including so-called exotic states. Predictions based on lattice QCD exist for the masses of hybrid mesons that have gluonic degrees of freedom (valence gluons) [38] and even glueballs [39]. These calculations require large computational efforts and still suffer from uncertainties, including systematic errors from the discretiza-tion process and statistical errors from Monte Carlo calculadiscretiza-tions. They also require various input parameters like the strong coupling constant and quark masses.
The interest in charmonium spectroscopy was fueled by a discovery made in 2003: The Belle collaboration found a new state in theπ+π−J/ψinvariant-mass spectrum of the decayB± → K±π+π−J/ψ [40]. This state, known asX(3872), appeared very close to theD∗0D0threshold and could not be accounted for with naive potential models. It was classified ascharmonium-like, since it decayed into final states with charmonium, indicating that the it must contain “hidden charm” (accpair). Its mass, however, did not fit any of the missing charmonium states, and its width was decidedly too narrow for a charmonium state above the open-charm threshold, which should be able to decay into aDDpair quickly. At
1S0 3S1/3D1 1P1 3P0 3P1 3P2
0−+ 1−− 1+− 0++ 1++ 2++
2,800 3,000 3,200 3,400 3,600 3,800 4,000 4,200 4,400 4,600 4,800
2S+1LJ(meson model)
D0D0 D∗0D0 D+sD−s D∗0D∗0 D∗+s D−s
D∗+s D∗−s
ηc(1S)
J/ψ(1S)
χc0(1P)
χc1(1P)
hc(1P) χc2(1P)
ηc(2S) ψ(2S)
ψ(3770) Zc(3900)± χc0(2P) X(3872) χc2(2P) ψ(4040)
ψ(4160) Y(4260) Y(4360)
ψ(4415)
Z(4430)± Y(4660)
JPC
Mass(MeV)
Figure 2.8: Charmonium and charmonium-like states that are listed as con-firmed in the current PDG Review of Particle Physics [17]. The triangle marks are mass predictions from a non-relativistic potential model [37]. The vertical lines are thresholds for the production of charmed meson pairs. States are or-dered in columns according to their quantum numbers𝐽𝑃𝐶. Quark spin (𝑆) and orbital angular momentum (𝐿) are assumptions based on the potential model and do not apply to the exotic statesX,Y, andZ.
2.4. CHARMONIUM SPECTROSCOPY AND EXOTIC STATES 25
the time of its discovery, the quantum numbers of theX(3872)were not known.
They were not completely established until 2013, when the LHCb collaboration reported the value𝐽𝑃𝐶 = 1++[41]. The state is shown in figure 2.8 along with the conventional charmonium states. Its quantum numbers would fit the missing χc1(2P)state, but its mass is too far off. As of today, the situation is still unclear, but the closeness of theX(3872)mass to the combined masses of theD∗0andD0 mesons indicates that it could be a loosely bound molecule of the two mesons.
Since the discovery of theX(3872), a number of new charmonium-like states have been identified. In 2004, the BaBar collaboration found theY(4260)with quantum numbers𝐽𝑃𝐶 = 1−−in initial-state radiation processes [42]10. Once again, potential models could not provide a fitting candidate (see figure 2.8), especially since the predicted 1−−-states in the mass regions of theY(4260) had already been discovered. More states with the same quantum numbers showed up, including theY(4360)andY(4660). As in the case of theX(3872), their nature is still unknown. Possible explanations include tetraquarks, meson molecules, and hybrid mesons.
The clearest evidence to date of an exotic charmonium state was found in 2008 by the Belle collaboration [43]: In the decayB → Kπ±ψ(2S), Belle found a distinct peak in theπ±ψ(2S)invariant-mass spectrum. Once again, the decay intoψ(2S) indicated that the discovered state must contain a ccpair;
in contrast to theXandY states, however, the additionalπ±, meant that the state carries electric charge. Consequently, it must be composed of at least two additional quarks, making it a very strong candidate for a tetraquark or meson molecule. The resonance was labeledZ(4430)±. It was at first not seen by the BaBar collaboration [44], but it was later confirmed with high significance by LHCb [45], and its quantum numbers were determined to be𝐽𝑃𝐶 = 1+−. In the meantime, theZc(3900)±had been discovered by the BES III and Belle collaborations in 2013 [46, 47], making it the first charged charmonium-like state observed by to independent experiments.
More charged charmonium-like states have been observed since then, but there is still no unambiguous explanation for any of the exotic candidates. A similar situation has evolved in thebbsystem, where several “bottomonium-like”
states, both neutral and charged, were discovered. Finding a theoretical model that is able to predict all of these mysterious states would greatly enhance our understanding of QCD. Precision measurements in the charmonium and bottomonium sector have therefore become a hot topic for current and future experiments.
10I use the nomenclature ofX,Y, andZthat is currently prevalent among the physics community when referring to these exotic states. It should be noted that the PDG labels all mesons with unknown quark content withXand the state’s mass [17, p. 120].