[Source: Kay Königsmann:

*Radiative decays in the Ψ family*, Physics Reports

**139**, Issue 5, June 1986, Pages 243-291, Fig. 5.]

One may wonder, is there a similar phenomenon for bound systems between other particles, say, between quarks, which carry a so-called colour charge and interact via the strong force described by quantum chromodynamics, QCD? The answer is an emphatic yes - and one can learn a lot from it. The figure shows the spectrum of excited states of the J/

*Ψ*meson, a bound charm quark-antiquark pair. Like the electron-proton pair in the hydrogen atom, the quark-antiquark pair can have only specific energies, it can be excited to a series of states with higher energies, and it can emit and absorb light when transiting between these states. The corresponding photon spectrum - the numbers of photons within a small range of energy counted in a detector - is shown in the plot, with a specific pattern of lines superimposed on a smooth background. The numbers help to identify the lines with the transitions between different states, which are shown schematically in the lower part of the figure. Data have been measured at the Crystal Ball, a spherical detector that completely encloses the quark-antiquark pair.

The photons counted in the spectrum do not correspond to visible light, however, but are short-wavelength gamma rays with an energy in the range between 100 and 500 MeV - that's much more than the 1.9 eV corresponding to the red Hα line in the hydrogen spectrum. This should not come as a surprise: The energy scale of the hydrogen atom is set by the Rydberg constant,

*R*= 13.6 eV, which is proportional to

*α²m*, where

*α*≈ 1/137 is the fine structure constant - the coupling constant of QED - and

*m*is the mass of the electron. Now, a charm quark has about 3000 times more mass than an electron, and the coupling constant of QCD, the strong interaction, is about 100 times bigger than that of QED. Hence, one could expect spectral lines with energies by a factor 30 million times bigger for the charm-anticharm pair than for the hydrogen atom - as you can see, that's not that bad an estimate.

However, there is an essential difference between the hydrogen atom and charmonium, as the bound charm-anticharm pair is usually called, and it can be spotted in the known spectrum of charmonium states, which is shown in this plot.

[Source: Ted Barnes:

*The XYZs of charmonium at BES*, Int. J. Mod. Phys.

**A21**(2006) 5583-5591 (arXiv: hep-ph/0608103v1), Fig. 1.]

Here, the known charmonium states are shown as black lines according to their energy, or mass, on the vertical axis, and grouped as per orbital angular momentum of the quark-antiquark pair, labelled by S, P, D, F, along the horizontal axis. In contrast to the hydrogen spectrum, there is no series limit at high energies, which in the hydrogen atom corresponds to ionisation, the separation of the electron and the proton. Instead, energy levels increase rather uniformly, but cross a line, labelled "DD". Above that energy, the charmonium system can decay in a D meson, made up of charm quark and a light antiquark, and the respective antiparticle, the anti-D. This is the manifestation of very characteristic feature of QCD, called colour confinement: No isolated quarks, or other colour charges, can be observed. Instead, if one tries to separate, say, the charm quark and the anticharm quark of the J/

*Psi;*by adding energy, a new quark-antiquark pair will be created, and two mesons will be formed, the D/anti-D pair.

Because of confinement, it is clear that the interaction energy between quarks can not be described by a simple analogy to the Coulomb law for electrical charges. To model confinement, it is stipulated that a Coulomb-type interaction has to be amended by some energy which increases linear with charge separation. In fact, one has tired to reverse-engineer an interaction potential between quarks starting from the charmonium spectrum: Making an ansatz for the interaction energy, one can calculate the corresponding spectrum, and fit the parameters to match the observed spectrum. The most popular ansatz is often called Cornell potential and looks like this:

_{s}instead of that of electrodynamics, the second term is the energy linear in distance, which enforces confinement, and the third term contains spin-dependent terms to model the fine structure of the spectrum. The constant κ is the the so-called string tension. In the spectrum shown in the figure, the best fit to the experimental data with this spectrum is shown in red. It corresponds to a mass of the charm quark of

*m*

_{c}= 1.46 GeV/c², a coupling constant α

_{s}= 0.55, and a string tension (called "b" in the plot) of κ = 0.72 GeV/fm - meaning that an energy of 0.72 GeV is needed to separate the quark-antiquark pair by 10

^{-15}meters.

Understanding confinement of colour charge is an open problem in physics, and the details of the interaction between quarks forming a hadron still contain many riddles. The analysis of the spectrum of quark-antiquark pairs as in charmonium can help to a better understanding of these issues - that's why the spectrum of charmonium is still an active area of research.

The best electrodynamcial analogue to charmonium is not the hydrogen atom, but the bound state of an electron and a positron, called positronium. The nomenclature of charmonia, and the name charmonium itself, derives from positronium physics.

The Cornell potential got his name after the group of physicists at Cornell who had used it within weeks of the discovery of the J/

*Ψ*to analyse the spectrum of its excited states - see E. Eichten et al.:

*Spectrum of Charmed Quark-Antiquark Bound States*, Phys. Rev. Lett.

**34**(1975) 369.

Charmonium physics will be one main topic of the PANDA (Proton ANtiproton DArmstadt) experiment at the new "Facility of Antiproton and Ion Research" (FAIR) at GSI, Darmstadt, Germany (see, e.g., Bertram Kopf:

*Physics with Antiprotons at PANDA*, J. Phys.: Conf. Ser.

**69**012026).

The Crystal Ball Detector is now in use at the Mainzer Mikrotron (MAMI), Germany.

This post is part of our 2007 advent calendar A Plottl A Day.

## 7 comments:

One can do this not only towards spectroscopy of smaller structures, but also larger, like e.g. for molecules. It's all about excitations... Best,

B.

Since the study of the single-electron atom ultimately led to quantum mechanics, I am optimistic that the study of charmonium will ultimately lead to a proper theory of strong interactions.

The problem with the 2-body microscopic bound state is that it appears not to be an advantage to use more sophisticated treatments: naive solving for eigenstates of a classical potential actually does much better than any highbrow approach based on quantum field theory. It also means that one does not need to find a way of dealing with the infinities as none occur.

The Cornell potential is - to my mind - only significant in that it shows that ultimately the EM and Strong interactions are manifestations of the same thing. Without actually being the solution itself, the fact that the same bag of tricks can work at the MeV scale as at the eV scale suggests that a proper, relativistic, axiomatic theory of bound states, if possible at all, can be devised in a way that works for both types of interaction.

(reply to Chris Oakley)

I've always been bothered by how quantum field theory doesn't really look at the bound state problem extensively (with the exception of maybe the lattice folks).

On the other hand, attempting to calculate QED results using conventional quantum mechanics (ie. without any field theory) is really messy, such as in the first edition of Heitler's book.

Is there an analogy to the Lamb shift in the Charmonium spectrum, and what does it "tell us" if any? tx

Also, I am curious re:

"I've always been bothered by how quantum field theory doesn't really look at the bound state problem extensively (with the exception of maybe the lattice folks)."

I thought QFT was so complete and effective to nth decimal, so do you mean what people do instead of what the theory in principle can do? Anyone's thoughts are welcome.

tx

(reply to Neil)

> I thought QFT was so complete and

> effective to nth decimal, so do you

> mean what people do instead of what

> the theory in principle can do?

What I meant is what people do.

In the case of charmonium, it's largely due to the fact that there are not many easy ways to do non-perturbative calculations in the QFT formalism. Doing non-perturbative calculations via lattice gauge theory, seems to produce the correct qualitative features. (An expert on lattice QCD can say whether more recent results can be compared directly to the experimental data or not).

Hi Chris,

I think one should keep in mind that using a potential interaction is a nonrelativistic concept, and that that the analogy between charmonium and the hydrogen atom via quantum states in the Cornell and Coulomb potentials works because the charm quark has a very big mass - it's a bit less than half the mass of the J/Ψ (about 1.3 GeV, compared to 3.096 GeV for the J/Ψ). This means that the interaction energy contributes only little to the J/Ψ mass, and the nonrelativistic potential models have a chance to work. Keep in mind also that there is no mass defect in the J/Ψ.

On the other hand, for the up and down quarks making up protons and neutrons, this cannot work - their masses are very small, less than 10 MeV, and interaction contributes to the rest, the 940 MeV making the nucleon. Here, the low quark mass is the so-called "current mass", which enters the QCD Lagrangian. It has to be distiguished from the constitutent mass, about 300 MeV. Using the constituent mass, one can, again, try to work with phenomenological potentials, but then, one has to remember that the constituent mass subsumes a complicated interaction with the gluon fields, and the breaking of chiral symmetry. So, this way of analysing the nucleon is very phenomenologic, and it is not clear a priori what can be learned for QCD on a more fundamental level.

For charmonia, the high charm mass is a Higgs-created mass in the standard model, so at least in this respect, the using the Cornell potential is closer to QCD. But here, again, one has to keep in mind that there is not yet (to my knowledge) a derviation of the Cornell potential from QCD.

On the other hand, as anonymous has pointed out, first-principles lattice QCD calculations with heavy quark flavours are very precise nowadays - for example, for the B_c meson (beauty-anticharm pair), there is a permille-consistency between lattice calculations and experimental data.

As for using QFT for looking at bound states, I guess via the Bethe-Salpeter equation, that's techenically very involved... that's probably the reason that it's not used more often... But it says nothing about the issue that solving Bethe-Salpeter, instead of calculating eigentstates in a nonrelativistc potential, is the thing one should actually do...

Best, Stefan

Hi Neil,

to my knowledge, there is nothing like the Lamb shift involved in the analysis of charmonia with Cornell-type potentials. The reason is, I think, that these potentials are just ansätze, they are definitely not thought to be fundamental in the sense that the Coulomb potential is the fundamental potential between electrical charges. Hence, thinking about QFT induced corrections to this potential probably makes not much sense..

About the successes of QFT you're alluding to - these are so great in the case of QED, where the coupling constant is small and calculations based on perturbation theory work fine and can be carried through to high orders, yielding very high precision. As we have seen last Friday ;-), in QCD the coupling constant in the energy range relevant for hadrons is much bigger, and perturbation theory does not work. Hence, so far the only first-principles high-precision methods are lattice calculations.

Best, Stefan

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