Lecture 5

Heavy Meson States:

We have seen that our attempts at making predictions and characterising the low mass mesons and baryons have met with limited success. The claim is that we are dealing with the strong nuclear force. It is a force so strong that we cannot actually calculate its properties using our most powerful tool - Perturbation theory.

Part of the problem seems to be that, by necessity, the quarks are in the hadronic bound states are relativistic and the one equation that we have which is good at predicting the properties of bound state systems, the Schroedinger equation, is inherently a non-relativistic equation.

One tactic that has been persued both historically and in some of your course work is to try to come up with relativistic versions of the Shroedinger equation so that we can calculate the properties of relavistic bound states. (Don't feel bad if you haven't seen this yet, you will, believe me.) This is how particle physicists eventually came up with the idea of quantum field theories and the Standard Model of particle interactions.

But it never did do what we'd initially hoped we'd accomplish. Though quantum field theories are very good at telling us how particles scatter, how to create other particles, and how fundamental particles like the quarks and leptons decay; relativistic-type Shroedinger equations never did give us a full picture of the properties and spectroscopy of hadronic bound states.

So the last thing we are going to look at before I turn you loose on the world during the long vactation are hadronic bound states with heavy quarks. Our goal is to attempt to learn more about this strong force. We use the heavy quarks because they will not be relativistic in bound states with each other. This holds out hope that we can use a Shroedinger equation to describe them. And also get some idea of what sort of potential is needed for the bound states. We might also hope to discover why we never have seen a quark outside of a meson or baryon. It is possible to strip away electrons from atoms, why can't we strip away the 'up' quark from the neutron?

There are only 2 quarks of the 6 that we can use for this exercise, Charm and Bottom. The reasons for this are many. First both are long lived in some sense. They survive for about 10^-12 seconds. Short by our standards perhaps, but long enough for mesons containing these quarks to be seen in very precise detectors since they will travel a few milimeters away from their creation point before decaying.

Both are pretty heavy. The Charm quark has a mass of 1.4 GeV/c² while the Bottom quark has a mass of 5.0 GeV/c². This means that mesons made of charm-anticharm, charm-antibottom, or bottom-antibottom have both quarks close to, if not in, the non-relativistic regime.

Current accelerators can produce loads of these things. If you want to just make something and say "I've seen it!" then you only need just enough energy to make it. But if you want to study it you have to get very high energies and/or high densities of particle collisions. Top quarks have a mass of 176 GeV/c², about as heavy as an atom of gold. You can't make a lot of them. (They also don't live long enough to form mesons, t=10^-23 seconds.) We can produce bottom and charm much more easily.

We can compare a q q-bar pair to the physics of an e+e- pair in a bound state (called positronium). The only difference being the nature of the force that is binding them.

Finally and most importantly, your lecturer discovered one of these mesons, the B_C meson. Which is the bound state of the Bottom and anti-charm quark. It was the last ground-state heavy meson left to be found. Myself and two graduate students found it in the data from the CDF experiment at the Fermilab accelerator in Batavia Illinois. The data was taken in 1994-1996 but we had to spend 3 more years convincing the collaboration we'd found it before publishing the result. Fermilab was also the site where the J/y (c cbar) and the U (b bbar) were discovered as well in 1974 and 1977 respectively.

The B_S (b sbar) was also found during that same data run at CDF but the ALEPH experiment at CERN published first with 4 dodgy B_S events just before my friend, William Wester, could convince CDF he had seen it (he had over 30 events at the time). But I digress so we should get back to the heavier bound mesons.

Positronium, Charmonium, and Bottomonium

We can calculate nearly every aspect of the 'atom' made up of an electron-positron pair. This is entirely due to the fact that we know a great deal about the electromagnetic interactions. Also the fact that the fine structure constant, a, is equal to 1/137. So all perturbation theory works well in that case.

Positronium ends up with the same sort of structure as the hydrogen atom with a couple of exceptions. First there is the main states of differing energy represented by the quantum number 'n'. These are the states one gets by solving the simple 1/r potential.

E_n = 6.8eV for the n=1 state of positrunium. I've also included the factor for the reduced mass here by taking m=m_e/2

There are many corrections to this that end up breaking the degeneracy in the orbital angular momentum quantum number 'l'. The following are all of the same approximate strength as the case in the hydrogen atom:

Spin-orbit coupling - where we begin to take into account the fact that the intrinsic spin of the electron has a little magnet that can interact with the magnet produced by the electron's orbit.

Relativistic corrections - the Shroedinger equation can be modified to make the first order approximation to the case where relativistic speeds are being approached.

Both of these have the effect on the energy levels of the size:

Which you can see is going to be about 10,000 times smaller than the energy levels themselves. When relativistic and spin-orbit effects are combined the size does not change, but the good quantum number is now the total momentum, J, rather than S and L separately.

There is one exception to positronium being like hydrogen. In hydrogen there is a hyperfine splitting which is due to the magnetic interaction between the spin of the proton and the spin of the electron. This interaction takes the form in hydrogen of:

In hydrogen this is about 2000 times smaller because of the electron/proton mass ratio. Positronium makes this term of the same size as the Spin-orbit and relativistic corrections. But it's still quite tiny.

The hyperfine splitting is also proportional to the dot product of the two spins (S_e·S_p)/(m_em_p). This is a similar situation we encountered with the mesons. We assumed that there was some sort of strong force spin-spin interaction and fit to an arbitrary strength constant of some kind. We were thus able, by fitting the constant, to determine the equivalent hyperfine splitting between the Kaon and K*, and the pion and the r particles.

DE_r-p=630 MeV

Let's be a bit more quantitative regarding this 'hyperfine' splitting in the meson mass states. The difference between the rho and the pion is that the pions have their spins anti-aligned whereas the quarks in the rho have their spins aligned. So this 630 MeV represents the difference between E(1³S₁) and E(1¹S₀) states. This also happens between the D* and D mesons (which are charm quarks paired with up and down) and also with the B* and B mesons. I'm assuming here that the mass of the up and down quarks are equal.

DE(D*-D) = 630(m_u/m_c) = 126 MeV (data = 140 MeV)

DE(B*-B) = 630(m_u/m_b) = 38 MeV (data = 50 MeV)

DE(B*_c-B_c) = 630[m_u²/(m_bm_c)] = 7.5 MeV (data = ??? don't know yet)

We can also do this for the J/y(1³S₁) and the h_c(1¹S₀) (the c-cbar pair). But here we can be a bit more precice since, using a computer, the potential can be better approximated and a wave function found.

The more correct expression is then:

Where 'r' is the equivalent of the Bohr radius and is calculated to be 0.5fm for the pion and 0.3fm for the J/y. Putting in the numbers we get:

The data gives 120 MeV. Not bad. A similar result can be obtained with a computer for the Upsilon (b-bbar pair) but the h_b hasn't been seen yet.