The Hadron-Muon Branching Ratio

Today's plot is my favourite plot from high-energy physics - it's a compilation of data measured at diverse particle colliders that shows what happens when an electron annihilates with its antiparticle, the positron, at very high energies.

Source: Particle Data Group, Plots of cross sections and related quantities, Fig. 6 (PDF file).

As the production of the J/Ψ resonance has already shown us, different things can happen in such a collision: The electron and the positron can just bounce off each other, a process called Bhabha scattering, or they can annihilate, and their energy, if high enough, can materialise in new particles. The most abundant products of this materialisation processes are muon-antimuon pairs (the muon, μ^-, is a massive cousin of the electron), or one or more hadrons, such as pions, which are made of quark-antiquark pairs.

The probability of these different possible events is measured by the so-called cross-section σ: one imagines the particles rushing onto each other as small disks - the larger the area of the disk, the higher the probability that two particles will hit each other and something will happen. The area of these imaginary disks is the cross section.

The figure shows the ratio R of the cross-sections for the creation of hadrons to the creation of muon-antimuon pairs in electron-positron collisions as a function of the centre-of-mass energy of the electron-positron pair, called "square root of s" for historical reasons. Both energy and ratio R are plotted on a logarithmic scale to allow the representation of a larger range of values. Energy is measured in Gigaelectronvolt (GeV, that's roughly the energy equivalent of the mass of a hydrogen atom), and the ratio is given more precisely by


The diagrams on the right-hand side show symbolically what happens in the collision: electron and positron meet and annihilate into a so-called virtual photon γ*, which then materialises as either a muon-antimuon pair or a quark-antiquark pair. But they are not just symbolic: there are very precise rules to convert these Feynman diagrams into actual numbers for the cross-sections σ. Of course, no free quarks have ever been seen in particle detectors, so the creation of a quark-antiquark pairs will be followed by some process that converts them into hadrons. The details of this process are not completely clear yet, but fortunately, the cross-sections entering the ratio R can be calculated without detailed knowledge about the hadronisation process.

Now, there are a few very interesting features about this plot.

First, there are the quite broad peaks shown in blue at relative low energies. These peaks correspond to the creation of mesons made up of light quark-antiquark pairs: the ρ, ρ' and ω of up-antiup and down-antidown pairs, the φ of strange-antistrange pairs. Then follow, as as shown in red, some very sharp spikes: Here, charm-anticharm pairs are created, which hadronise as J/Ψ and Ψ' particles, and at higher even energies, bottom-antibottom pairs materialise, which form the so-called Upsilon Υ and its excited states. From these particles, one can learn a lot about the bound states of quarks and antiquarks and the forces acting between them.

Second, if one takes a closer look, one can see that the flat parts of the curve between the ρ' and J/Ψ spikes, between the J/Ψ and the Υ, and following the Υ are increasing in small steps. This stepwise increase is not difficult to understand: When the energy of the collision becomes high enough that, say, charm-anticharm pairs can be produced, there is a new channel opening for the production of hadrons, while the production of muon-antimuon pairs is not affected. But there is more to learn from these steps: They show that quarks come in three different colours!

In fact, a quite precise first approximation to the ratio R (from a so-called "tree-level" calculation) shows that R is given by the sum of the squares of the charges of the different quarks that can be produced. This estimate is by a factor of three below the experimental data... unless, of course, one takes into account that each quark can come in three different colours!

Third, at the upper end of the energy scale at about 90 GeV, there is a further peak, the so-called Z pole. No new quark-antiquark pairs are created at this peak, but the Z boson, the massive partner of the photon in the electroweak theory. At this pole, the annihilation of the electron-positron pair can happen not only via a virtual photon, but also via a virtual Z boson, and both possibilities have to be added:

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The Z pole contains a very interesting piece of information about the standard model of particle physics, but that will be the story of another plot.

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