The big paper on the mesons is coming along. Yesterday I was checking the mass formulas for heavy quarkonium, a subject which was discussed here a few months ago. While checking numbers for the paper, I found that the Particle Data Group has changed the values for psi(3770) in our favor! For us it was exciting so Kea says I have to blog it. And in fact I feel that I did an inadquate job the last posting on this.

So first, a little background. Mesons are made of a quark and an anti-quark, plus the color and electric force that binds them together. The usual method of modeling them is to simplify the color force and treat it as if it were just a scalar force like the electric force, and then calculate the binding energies (and therefore the masses of the mesons) by using the methods used to calculate atomic energies.

What I’ve been doing instead is looking at the problem from the point of view of quantum information theory. With this one ignores the spatial and momentum information and looks only at the information content of the particle states. When one does this to a spin-1/2 particle like the electron, one uses qubits. Since I want to model the color force but don’t care about spin, I use qutrits instead with the three states being red, green, and blue.

The usual method of modeling the mesons works best for lowest energy states of the heaviest quarkonium. This is because these states are the least relativistic (because the quarks are so heavy) and the color force isn’t as strong (since these quarks are so heavy, their deepest bound states are smaller than other mesons, and since the quarks are close to each other, the color force is reduced by asymptotic freedom). With my method, the reverse should be true; I should be more accurate at bigger states where color is more important. These states are either higher excitations or have lighter quarks. This is because I treat the color states correctly but don’t work on getting momentum modeled correctly.

There are two heavy quarkonium meson resonance series, the c-cbar (J/psi) and the b-bbar (Upsilon). Each includes 6 masses. We split these into two Koide triplets each. The lowest triplet is a 1Sg color triplet, the higher energy (and greater mass) 2Sg color triplet. The “g” stands for “generation” and runs 1, 2, and 3. The designations, 1S, 2S, 2P, etc., are borrowed from the designations of the orbits of an electron in an atom. Most of my readers have seen them before as they are taught to high school students as the electron shells. It’s part of the theory explaining the periodic table of the elements that is on the wall in every chemistry classroom.

The 1S, 2S, etc., states are common to all radial quantum mechanical systems due to the symmetry of rotations and all that. You derive them by using symmetry to organize the solutions to the differential equations. The subject was discussed around here some time ago in the post Quantum Numbers and Differential Equations.

The mass equations look like this:

There are four of them because each gives the masses of a triplet. The value is taken from the Koide formula for the charged leptons (electron, muon and tau) and is around the square root of 625 MeV. The values 3.99332, 0.12667 are computed to fit the three measured meson masses using least squares. Some of these values look awfully non random but that’s another subject. Anyway, each formula uses 2 of these fitting numbers to approximate 3 measured masses so in using them, there is one degree of freedom removed by each formula. And note that other than the fits, all of which are fairly similar, the four equations look a lot alike.

In removing one degree of freedom from three masses, these equations amount to specifying the middle mass given the other two. So you can think of them as defining the ratio of the mass gaps. The square root makes it a little more messy, but this is about what is going on. The claim is that one has two sorts of mass gaps available, the type used in the first and third of the above 4 equations, and the type used in the second and fourth. The values 3.99332, 0.12667, etc., define how wide the gaps are, and average mass.

According to the approximations described above, the worst fit should be the Upsilon 1S (as these three resonances are the least color dominated). And in fact, it *is* a pretty ugly fit. Here are how well the Upsilon equations fit their masses:

In the above, the left most column is my designation for the state. Where the electron shells are split into two states per entry (spin up and spin down), with color I end up with three states per entry. No, they do not correspond to “red”, “green”, and “blue”. Instead, they correspond to different relative phases in the color states of the gluon sea (or which wikipedia calls the parton sea. The second column is the designation of the meson state as used in the Particle Data Group’s (PDG) website. This website is where particle physicists keep a list of all the particles (and there are a lot of them). The third column is the result of my formula, and the rightmost column is the measured mass from the PDG. The number in parentheses is the error in the measurement. You can see that the formula for the 2Sx Upsilon mesons is dead perfect, but that the Upsilon 1Sx is not.

So in checking the data for the paper against the latest PDG numbers, I took a look at the 2008 PDG’s data for the psi(3770) meson. This is what they have now for the top few inches of its entry:

So they have two values, “our fit” and “our average”. Before, they only had one value. Hmmm. Notice that the two mass values, 3772.92(.35) and 3775.2(1.7) exclude each other? The mass “ideogram” giving the spread in measured masses from several experiments looks like this:

The above is the source of the 3775.2 figure. Where does the “our fit” come from? It comes from the analysis of the difference in energy between the psi(2S) and the psi(3770) states. This is more difficult analysis, but it claims to give more accurate measurements, compared to the direct mass measurements.

Since there are two psi(3770) mass figures, I can fit both and see which fits better. When I did this, I laughed out loud. I had been using the 2006 PDG data which gave the psi(3770) mass as 3771.1:

When I used that data, I ended up higher, 3773.8 (see old post for the numbers). The fit was “okay” but it was excluded. The 2008 “our fit” data gives me a better fit.

Since the PDG has separated out their fit from their average measurements of the psi(3770) I can try both numbers. Turns out that “our average” leaves me with perfect fits. Hey, I’m going with “our average”. Here’s how the psi mass fits work now:

I haven’t typed the numbers into my “significance” computer program to see just how good the new fits are, but the old PDG figures gave a 1% fit to the J/psi resonances. That is, given 6 random numbers between 1 and 2, the probability that they could be split into two groups of 3 with a lower total error than that seen in the measured masses was about 1%. I’m guessing that it’s closer to 0.1% now, but I feel that getting to choose between the “our fit” and “our average” mass numbers is unfair.

So how should I claim the goodness of fit? Use the “our fit” numbers or the “our average” numbers? Or both and explain?

The PDG giveth and the PDG taketh away. This time they giveth.

Excellent! Do you think the PDG will start looking at Koide fits now? It seems a shame that such an important source of data should be allowed to contain errors in the masses.

Well I think it will be some time before the PDG learns about Koide hadron fits.

And it’s really not an “error” it’s just that they were a little too enthusiastic in their modeling.

Hmmm…all of those “bad fits” appear to be within 1% error margin, which is pretty damn good! I remember reading a paper by Feynman sometime ago where he said something along the lines of the error being fairly good, only 40%(!)…so it’s all about how much you want to be off in comparison to the experimental results.

Thanks for the comment, Angry, I respect your opinions. Let me amplify the issue a little.

The “bad fits” in the Upsilons are around 0.1% of their masses, but one of the parameters in their equation (i.e. 3.99332) defines the center of the three masses. (The other parameter defines the spread.) So in addressing the fit you have to use relative errors. I’m guessing you did just that because when you look at the relative errors you get just about exactly 1%.

However that’s not all. There are something like 20 ways to split the 6 Upsilon meson masses into two groups of 3. So the goodness of fit is nowhere near as good as what it would like from just nailing down a 1% error. To pay for that freedom, you want to get really, really, really accurate results.

I’ve got a computer program that takes random mass values and computes best fits. You get a distribution for the error total. This says that the fits are extremely good. But there is also choice in the equations so it’s not really cut and dried. And extraordinary claims need extraordinary evidence.

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