Quarkonium Quantum Mechanics
Chris Quigg

The discovery of the J/psi and upsilon families of new particles has had a revolutionary effect upon particle physics. The rich array of states observed with the psions and expected with the upsilons has given rise to a new spectroscopy. This has encouraged the hope that heavy quark–antiquark systems might serve as a "hydrogen atom" of hadron physics.

Grotrian diagram

Heavy quark-antiquark bound states — experiment and theory: the left-hand side shows the observed spectra of psions and upsilons (the psions labelled 13S1 and 23S1 correspond to the J/ψ and ψ’). The right-hand side gives the predictions of a non-relativistic quark-antiquark potential which behaves logarithmically. In this picture, the level spacing between the ground state (1S) and the first excited state (2S) is the same for psions and upsilons.

High-energy physics is concerned with the relativistic domain in which particle creation and annihilation are routine. However it has frequently been possible to find restricted situations in which nonrelativistic quantum mechanics is useful. Some familiar illustrations are the application of the Weisskopf–Wigner formalism to the mixing of short- and long-lived kaons lnd the meson classification scheme lased on the nonrelativistic harmonic oscillator model for quark–antiquark states.

In scattering theory, nonrelativistic examples have been an important source of inspiration for the relativistic domain. Regge poles, on which the phenomenological description of high energy scattering is based, were derived and understood in nonrelativistic theory and applied to the relativistic problem by conjecture. Similarly, the Glauber multiple scattering approach emerged from nonrelativistic theory and optics. With the discovery of the psion and upsilon families, theorists are turning again to nonrelativistic quantum mechanics for inspiration and understanding.

Candidate gauge theories of the strong interactions (see November 1977 issue, page 380) suggest that the coupling between quarks is weak at short distances but becomes very strong at large distances. This 'explains' the paradox where quarks appear to behave as quasi-free particles within hadrons but cannot be liberated from the hadrons. On the basis of these arguments it was conjectured that heavy quarks would move nonrelativistically within hadrons. Thus bound states of a heavy quark and antiquark should be a hadronic analogue of the positronium system of a bound electron and positron.

This so-called quarkonium system might then be interpreted according to the familiar rules of nonrelativistic quantum mechanics using a potential to describe the interquark force. An explicit form for this potential has not yet been derived from the general theory but forms for very short and very long distances can be suggested. At very small distances, the potential is expected to take a form like the Coulomb force, corresponding to the exchange of a single massless gluon. At very large distances, a linear confining potential seems to be appropriate.

The first opportunity to test the applicability of atomic-physics ideas to hadrons came with the discovery in 1974 of the J/ψ and ψ’, which are long-lived by strong interaction standards. These particles could readily be interpreted as the first two levels of a charm quark–antiquark system. A natural starting point is to regard the potential as a combination of the Coulomb and linear forms, and then to vary potential strengths and the charmed quark mass to reproduce the experimental results.

Looked at from the left, this diagram represents the decay of a vector meson, V0, into an electron and positron through an intermediate photon (wavy line). The meson could be a psion, an upsilon, or any other heavy vector (spin one, negative parity) particle. Looked at from right to left, the diagram shows the production of the vector meson in an electron-positron collision, via an intermediate photon.

The masses of the psions are eigenvalues of the Schrödinger equation. Another observable, the square of the charmonium wave function at the origin, is measured by the leptonic decay widths of the spin-one levels where the quark and antiquark have their spins pointing in the same direction. In the quarkonium picture, the decay of a vector (spin one, negative parity) meson into a lepton pair is described by the annihilation of the quarks into a virtual photon, which subsequently decays into the lepton pair. The rate of the process is governed by the probability for the quarks to coincide, i.e., the square of the charmonium wave function at zero separation.

Once adjusted to fit the J/ψ and ψ’ positions and leptonic widths, the potential can predict the positions of other levels and rates for radiative transitions among levels, along with other quantities. Many of the predicted levels have been found with masses remarkably close to the theoretical expectations. For those accustomed to a purely descriptive hadron spectroscopy, this agreement supported the nonrelativistic potential model approach. Inevitably, complications appeared in the form of relativistic corrections, tensor forces, and coupled-channel effects.

New opportunities arose with the discovery of the upsilon family. If these particles are a heavier quarkonium family, the nonrelativistic approximation should be more reliable than for the psions, because the new quark is two or three times more massive than the charmed quark. The comparison of the psion and upsilon families should provide an incisive probe of the potential.

An elementary way to make this comparison is to exploit the variation in behaviour due to different simple potentials. So far, the only data to which these considerations can now be applied is the apparent equality of the three lowest level spacings for the psion and upsilon families. This disagrees with the prediction of the most popular charmonium potential, but is reproduced instead by a logarithmic potential, which has been adopted as a simple form useful for making predictions.

What lies ahead? Theorists are busily refining potential models and calculating the potential from the presumed underlying field theory of the strong interactions. Many useful theorems can be proved and the use of special techniques to reconstruct the potential from the data is an appealing dream. Experiment, of course, will be the final arbiter of the value of the nonrelativistic approach. With work continuing on the upsilon at Fermilab, DESY and the ISR, and with the advent of higher energy electron–positron colliding beams at PETRA, SLAC and CESR, the best is yet to come.

Published in CERN Courier 18, 215 (1978).