# The X(3872) Files: Scanning for new mesons

New mesons! Let’s begin with a picture (from encarta) showing how new particles (states) were discovered back in the good old days:

Work on the big meson paper is continuing. I’m almost done with the isoscalar mesons. So in addition to describing just what these things are, I’m including a few lines on the new J/psi states, the X(3872), X(3940), X(3945), X(4260), and X(4360).

How to Find a New Hadron

Hadrons (mesons and baryons) are found by analyzing large numbers of particle interactions. When you plot the data, the graphs have little, uh, well the technical term is “bump” . Yes, hadrons, like children, begin as bumps. A hadron bump looks like this:

The above is from hep-ph/0510365. This bump happens to be one of the states, the X(3940), with which this post is concerned.

The above plot is experimental estimate of the cross section or probability of the reaction $e^+ \;\;e^- \to J/\psi\;\;X$, as a function of energy. In words, they are colliding positrons with electrons, and looking for results that have a $J/\psi$ or “J/psi” particle detected, along with anything else “X”, and plotting the results according to an energy measurement.

In this sort of business, the J/psi is detected by looking at its decay products. The sum of the charges of the decay products give the charge of the particle that decayed. The sum of the energies give the energy of that particle. And one uses the difference between the sums of the momenta and the sums of the energy to compute the mass of the product, using the equation familiar to all physics (grad) students, $E^2 = m^2c^4 + p^2c^2$. This formula is the generalization of the one everybody else knows, $E = mc^2$. If one finds that the charge is zero and mass m is about 3096 MeV, and certain things about the angular distribution, then one concludes that the particle is a J/psi. Or something very like this; I haven’t been in the business for years and likely have got something wrong.

All the analysis is done by computer program nowadays. Back when I was a kid, data was taken as photos of the contents of cloud chambers or bubble chambers. These beautiful photos were analyzed by “scanners” who examined hundreds of thousands of photos for unusual events. My understanding was that scanners generally did not have degrees in physics, they were looking for patterns. And as per the photograph at the top of the post, most scanners were women.

In 1981, a horror science-fiction movie came out with the title “Scanners“:

The mark of an attack by a scanner was the victim’s head exploding. The movie poster I remember at Brookhaven National Lab (BNL) was better than the above in that it featured this gruesome motif in action (video linked). It was once considered valuable wall art at physics departments doing elementary particles. If any grad students run into one, now you know what it is about (professors already know everything). So send me a photo of your scanner wall art. And enough of discussing times past.

So you get a lot of “events” which have two bunches of decay products observed. One of the bunches of decay products has characteristics that lead you (or more accurately, your scanners or the computer programs that replaced them) to conclude that it is a J/psi. But you can’t identify the other bunch so it is called an “X”. To get the above plot, you take all these sorts of things you can find, and plot the number of them according to the recoil energy (or kinetic energy) of the J/psi. You find that all kinds of energies are measured but that the number has bumps. And there you have it, you’ve discovered a new particle!

Now that you’ve discovered it, what do you call the new particle? Unfortunately, you can’t just name it after your children. Hadrons (i.e. mesons and baryons) have a complicated naming convention described on the Particle Data Group website. The name describes the quantum numbers of the hadron. And finding the quantum numbers is not so easy. To describe the process used to deduce quantum numbers would require a lot more of a blog post than I’m willing to type up. For example, scalar and pseudoscalar particles have angular distributions that do not depend on angle. If you want to see the process in action, try hep-ex/0612053 a paper by the CDF Collaboration on the X(3872). Don’t try this at home. Finding new particles takes amazing amounts of effort. The hep-ex/0612053 paper has more than 500 authors from 61 institutions.

But usually, long before you know the quantum numbers, you have the plot showing the bump. This plot gives the energy (rest mass) of the particle. And until the quantum numbers are defined, the particle is traditionally called an “X”, with different Xs distinguished by adding the approximate mass. So the 3943 bump is called an X(3940). This scheme for the unknown states has come under some pressure recently so Y and Z are also used for unknown states now. The approximate energy measurement is kept in the particle’s name even if the energy is more closely determined later. That’s so guys like me can use google to find information about the particle. An example of this is the $\rho(770)$ whose mass is now estimated at 775.49.

The X(3940)/X(3945), X(4260), X(4360)

The quantum numbers of the X(4260) and X(4360) are both known to have quantum numbers $I^G(J^{PC}) = ?^?(1^{--})$. It’s natural to suppose that these are elements of a Koide triplet. For a third element, none of the J/psi mesons with known quantum numbers are consistent with these two. The X(3872) is known to be inconsistent but the X(3940) and X(3945) as far as quantum numbers go, are blank slates, i.e. they are $?^?(?^{??})$, and so are consistent with anything.

In addition, the X(3940) and X(3945) have consistent mass measurements, $3943\pm 6 \pm 6$ and $3943\pm 11\pm 13$, respectively. So to test to see if the X(3940)/X(3945) is an element of a Koide triplet with the X(4260) and X(4360) we only have one set of numbers to run through the computer program. Sure enough, they are dead on. The formula and (perfect) fit for the triplet are:

where $\mu_e = 25.054309435$ comes from the Koide formula for the charged leptons.

So our prediction is that these states will form a perfect Koide triplet and their quantum numbers will all found to be identical, X(3940)/X(3945) = X(4260) = X(4360).

The X(3872)

The quantum numbers of the remaining unknown J/psi state, the X(3872) are said to be $I^G(J^{PC}) =$ $0^?(1^{++})$ or $0^?(2^{-+})$. The $\chi_{c1}(1P)$ has compatible quantum numbers of $0^+(1^{++})$. If these are two elements of a Koide triplet, then we can predict the mass of the third element, provided we know which of the two likely Koide formulas to choose from.

To remind the reader of the form of the Koide formulas, we reproduce the formulas for the charged leptons and neutral leptons (i.e. neutrinos):

From these, the generalization appropriate for the scalar mesons is:

where s and v are real constants. For instance, the J/psi, psi(3770) and psi(4415) follow the second formula (like that of the neutrinos), while the other $I^G(J^{PC}) = 0^-(1^{--})$ J/psi mesons, the psi(2S), psi(4040), and psi(4160), follow the first formula (like that of the charged leptons, the electron, muon, and tau). This was discussed in more detail a few days ago.

To choose which of these formulas is appropriate, we need to have a Koide fit for another $0^+(1^{++})$ meson. Of course it hasn’t been discussed publicly, but I’ve got more Koide fits than Carter has little pills. So, from our secret stash, we pull out the formula for the $f_1$ = f_1 resonances. The formula and its (as is customary for Koide formulas) perfect fit are:

and we see that the appropriate formula to use is “electron-like.”

Applying this formula to the first two elements of the purported Koide triplet, the $\chi_{1c}(1S)$ and the X(3872), the prediction for the third (so far missing) element of the triplet is 5205.0(1.5) MeV.