Dalitz Plots

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.

The 2006 edition of that standard repository of all knowledge and wisdom*, the Particle Data Group’s Review of Particle Physics, has gone live. I’ve even helped contribute to a small corner of fit space (in the form of HE 05). There is a new feature, the oddly-named pdgLive, which puts the data in HTML format, with lots of hyperlink goodness to clarify just what numbers go into the fits and point you at the papers’ SPIRES references.

* I’m not joking. Experimental particle physicists hold the PDG, as it’s called, in such high esteem that some people have set up a Flickr photo pool in honor of it.

Collaboration Meeting!

July 17, 2006

Experimental particle physics is a very social endeavor. We love to see each other so much that we schedule lots of meetings, from small person-to-person get-togethers to huge conferences. It’s just so much fun to sit in windowless rooms!

One common form of assembly is the Collaboration Meeting. Every collaboration has a Meeting. It occurs with some frequency, which is set by the difficulty of getting everyone together in the same place at once. Mega-international collaborations have them relatively infrequently, sometimes in pretty places on different continents from the actual experiment.

The Collaboration Meeting, as a platonic form, is a forum where all the people involved in an experiment get together, catch up with the latest status of everything, plan for the future, and have a jolly good time too. Papers are approved, votes are taken, decisions of far-reaching import are made. The Collaboration is, for a shining instant, realized as a single entity. Needless to say it’s usually less exciting.

CLEOns, being a gregarious bunch, want to have collaboration meetings all the time, sometimes twice in one month. Where other experiments may have three a year, we had eleven in 2005. Since many of us are on-site here in Ithaca, and most of the remainder are in the northeast or midwest, it’s relatively easy for lots of us to come and take part in the excitement. Our meetings are so frequent, in fact, that the physics groups (e.g. bottomonium or hadronic D decays) meet on the same schedule to review the progress of analyses; we don’t hold with this weekly physics meeting nonsense that CDF engages in — once a month’s good enough.

A frequent CLEO meeting scenario goes like this. You are a poor grad student/postdoc, working hard on your analysis, but distracted as always by other responsibilities. A few months/weeks before a meeting, someone (your advisor/the Analysis Coordinator/a random faculty member from another institution) starts dropping broad hints/threats that it would be really nice if, you know, you could have some numbers ready for a conference two months from now, which means you have to have them approved for public release during the next meeting.

So you work really hard and succeed/fail in getting everything ready. If the first, you give a plenary talk (in principle to the whole collaboration) on your work, where you will promptly be asked lots of questions which you may or may not be prepared to answer. The questions need not actually relate to your work, in which case members of the audience will proceed to have irrelevant arguments while you stand in front of everybody, hiding your distress, wishing you were eating one of those nice donuts they provide in the mornings instead. If you didn’t quite make it to plenary talk level, you go through the same thing, except it’s now a parallel-session physics group talk instead. This will all most likely happen on a Friday.

On Saturday, you come in early in the morning and sit in a large room on the 7th floor of a different building. The power outlets are pretty much all in the back of the room. A continuous strip of tables is set up there, in front of the windows; a long row of people sit here, their backs to the outside, facing the speakers but looking intently at their laptop monitors the entire morning. You will listen to talks on the status of the accelerator, detector, and software (useful) and then be asked to vote on papers you haven’t read, didn’t know existed, but have your name on anyway (er…) Then there will be a lunch featuring ice cream sandwiches, in all probability.

On the whole, collaboration meetings are fun if you’re not presenting anything. If you are, they can be stressful deadlines, just another rung in the infinite ladder of similar-yet-not-identical talks one seems to always be giving on the same topic. But, when they’re over, you get to pay attention to other things again, like blogs.

I’m a bit of a type afficionado. Not a very good one, and certainly without the time to get really into the business; but I know my Janson from my Garamond, my Bookman from my Bodoni, and as they say, I know what I like. Even for physicists, fonts are important. Here follow some ramblings…


There are two classes of high-energy physicists in this regard: those who write talks in Powerpoint or similar programs, and those who write talks in (La)TeX. (Disclaimer: I do my talks using the LaTeX Beamer class, in Computer Modern Sans, sometimes with Euler for math.) The latter are a small minority, and I suspect are considered certifiable by the former. The LaTeX users pretty much restrict themselves to Computer Modern (Regular or Sans), though Helvetica and the like have been seen from time to time.

The offenses against typography tend to come from the Powerpoint folk. I must admit that some people do great things with the tool. B.L.’s Futura-themed talks are spectacular, and J.N. and students have an inspiring commitment to Gill Sans. There’s nothing wrong, exactly, with Arial or Times New Roman in presentations, although Times’s serifs get in the way, and the insistent overuse of Arial these days makes Univers seem dangerously radical. R.B. uses Textile throughout every talk, giving them the impression of being entirely in bold italics — this counts as a minor infraction.

But no, the evil of which I speak is Comic Sans:

Comic Sans sample

a font supposedly inspired by comic books and which is regarded as so horrific by so many there is a society dedicated to its extermination, where its use is described as “analogous to showing up for a black tie event in a clown costume.” If nothing else, that bent middle stem of the “m” should set anyone’s teeth on edge. I can only hope that the otherwise savvy people who use it frequently are intentionally trying to create an undertone of silliness. (Sure, it’s nonthreatening and legible. So is Tahoma, which doesn’t look like a kindergartener was tracing BIG letters.)

The Horror of Papyrus

What really gets my goat these days is the overuse of Papyrus (especially in movie promotion). I’m sure you’ve seen it around:

Papyrus sample

It’s not actually a bad font, but it is so distinctive that any use echoes all previous ones. This is a typeface whose main role is to put the reader in mind of something, and that something is exotic and non-Western (and wholesome). Nobody would use it for a film about the Scottish fishing industry or mobsters in Japan, but morality tales involving animals on African grasslands or isolated Buddhist monks can expect to get the full fat “y” treatment. The use of Papyrus is usually transparently manipulative, extremely cloying by your fifth exposure, and can’t even claim to be original.

And a Final Note

Speaking of manipulative fonts, there’s a booming business in typefaces intended to suggest the good life. Things like Trajan can be seen easily on any sign advertising Olde Brooke Towne-type subdivision or luxury goods store. But now, Orange Italic have prepared all three fonts you will ever need to sell a $1500 handbag or a McMansion: the Luxury line.