July 31, 2006
You’ve heard of the Feynman diagram, the amazingly versatile figure used to represent terms of certain theoretical computations and visualize them in terms of physical processes. Experimental particle physicists have another named figure of great fame and utility: the Dalitz Plot.
These are named for the late Richard Dalitz, who seems to have invented the plots while at Cornell in the mid 50s. (He had earlier shown that the neutral pion, which usually decays to two photons, could also decay to one photon and an electron-positron pair, a mode now called the “Dalitz decay.”) His name has become attached to the process of interpreting these plots, the dark art referred to as “Dalitz analysis.”
So what is a Dalitz plot? They are representations of the transition of an initial state to three particles. (You can’t make an interesting plot with two particles, and four particles requires too many dimensions to fit on a page.) They represent the kinematics of the process — how the particles are moving with respect to each other. An example (from the decay Ds → K- K+ π+) is shown below. It is relatively simple as these things go, but still shows some complexity.
Let’s go over the plot in detail. We have three possible pairings of the three particles. In each case, we can compute the mass that the particle pair would have if they came from a single parent. In a Dalitz plot we actually plot the combination’s invariant mass squared, m2. Here, the x and y axes are m2 for the K- K+ and K- π+ combinations, respectively. Because the whole K- K+ π+ system is required to have come from something that looks like the Ds, conservation of energy and momentum sets the boundaries of the plot. The third combination, K+ π+, isn’t as interesting, because it has charge two (unlike the other combinations which have charge zero). It’s actually on the plot — it’s the top right to bottom left diagonal, due to energy-momentum conservation again — but usually you don’t draw the axis for that. Why do we use m2 instead of m? It turns out that, in the absence of any “dynamics,” i.e. if the decays are randomly distributed and not modified by any other processes, they will uniformly populate a plot of m2, but not one of m. Using m2 stops us from being confused by this.
Each observed event in data is represented by a point. You can see immediately that the distribution of events is not uniform: they are concentrated in one horizontal and one vertical band, and within each of those bands there is a gap in the middle. These bands, centered on specific masses, tell us that the final particles produced in the Ds decay are not produced equally in all possible ways: they in fact prefer to have particular mass combinations. These are signatures of particles: The main vertical band on the left, from the K- K+ combination, is called the φ(1020), while the broader, lower horizontal band, in the K- π+ combination, is the K*0(892). (No, we don’t just know that; we look it up in a book, or do a Dalitz analysis.)
Our usual model for understanding Dalitz plots is that all decays happen as a sequence of two-particle decays, for example Ds → φπ+, followed by φ → K- K+. (This goes by the fancy name of the isobar model.) The bands on the Dalitz plot, then, are in fact intermediate particles in the various decay chains that feed the final state we observe.
There’s also some gunk in the rest of the plot. Some of it is probably from some other particles, but some of it is background, random combinations of two kaons and a pion that came from something completely different but happened to have the correct mass to be a candidate for a Ds.
Dalitz plots demonstrate interesting features of quantum mechanics. For example, the bands have widths (more easily seen with the K*0). Although some fuzziness is introduced in our detector — you can’t measure anything to infinite precision — we are much more accurate than that. The width is actually a feature of the particle. Due to the (ahem) “energy-time uncertainty principle”, a particle can only have a precisely defined mass if it has an infinite lifetime. (In other words, a state with a precisely defined energy — mass energy in this case — can never change.) Since, by definition, these intermediate particles have decayed, with a rather short lifetime (or we wouldn’t be seeing their decay products!), the bands have an intrinsic width. We can use Dalitz plots to read off these widths, and hence determine the particles’ lifetimes. (For example, the K*0 average lifetime is about 1.3 × 10-23 seconds; the φ, much narrower, lives ten times as long: 1.5 × 10-22 seconds.)
What about those gaps in the middle of the bands? Another quantum mechanical effect, this time due to angular momentum. Particles can carry an intrinsic (“spin”) angular momentum, independent of the “orbital” angular momentum due to their relative motions. In this case, the initial Ds and the final K and π have zero spin angular momentum, but the φ and K*0 have one unit of spin angular momentum. So we have a spin 0 going to a spin 0 and spin 1, with the spin 1 going to spin 0 and spin 0. For reasons that are a little too involved to discuss in detail, in this chain, the daughters of the spin 1 particle tend to be aligned or antialigned with the first spin 0 particle that’s produced. Alignment makes the m2 of the combination small, while anti-alignment makes it large. Because the “perpendicular” region is depleted, we observe the two lobes.
The final effect of quantum mechanics I’ll mention here is interference. You can’t actually see it in the plot above, so here’s a doozy of a Dalitz plot from the Crystal Barrel experiment:
This is a Dalitz plot for the annihilation of a proton and antiproton, producing three π0
particles. Because the three neutral pions are identical, the plot is symmetric around any axis that exchanges two particles (an overall six-fold symmetry). Because the Crystal Barrel experiment recorded so many events, this Dalitz plot is presented using shading instead of dots (blue for few events, red for a lot).
There are a few areas with very large concentrations of events. The two areas marked f2 are particles which have spin 2. Again, because they involve changes in angular momentum, the contributions of these particles have weird structures and don’t form lines. There is a very nice simple line, due to the particle f0(1500), which has spin 0.
The line labeled f0(980) is different: it is a dip in the plot. How can a lack of events indicate a particle? This is a fun feature of quantum mechanics: the possibility of interference. The thing that determines the probability of an event showing up in a particular part of the plot is the square of the size of a complex number, called an amplitude. Conceptually, you can think of each particle contributing an arrow, with a direction and a length:
The direction and length from each particle depends on where you are in the Dalitz plot: in particular they are long in the m2 region where the particle contributes most. Theorists spend a lot of time trying, essentially, to predict the arrows at each point in the Dalitz plot.
The arrows from each particle are added together to get the overall amplitude. In the Ds → K- K+ π+ Dalitz plot (the first one), the amplitudes for the φ and K*0 are never large at the same time, so we get this sort of situation:
The small arrow, from the particle that doesn’t contribute in the local mass region, doesn’t change the length of the sum much from the length of the big arrow. However, for the f0(980) in the second plot, the situation looks more like this:
so, although the individual particle contributions are large, they mostly cancel, and the overall rate is smaller than what you would get from either individually.
Some random links on Dalitz plots: there’s a lovely review article of the Crystal Barrel data here; David Asner’s summary of the formalisms is here; and a very readable summary, including some historical information on Dalitz plots, is in this thesis.