With the LHC running and experimentalists busy taking real data, one of the things left for theory grad students like me is to learn how to interpret the plots that we hope will be ready next summer.

The most basic feature we can hope for is a resonance, as we described when we met the *Z* boson. The second most basic signal of a new particle is a little more subtle, it’s a bump in the **transverse mass** distribution. I was reminded of this last night because of a new conference proceeding paper on the arXiv last night (1009.2903) presenting the most recent fit to the *W* boson mass from the CDF and D0 collaborations at the Tevatron.

The result isn’t “earth shattering” by any extent. We’ve known that the *W* mass is around 80 GeV for quite some time. The combined results with the most recent data is really an update about the precision with which we measure the value because it is so important for determining other Standard Model relations.

Here’s the plot:

It’s not the prettiest looking plot, but let’s see what we can learn before going into any physics. In fact, we won’t go into very much physics in this post. The art of understanding plots can be subtle and is worth a discussion in itself.

- On the
*x*-axis is some quantity called mT. The plot tells us that it is measured in GeV, which is a unit of energy (and hence mass). So somehow the*x*axis is telling us about the mass of the*W*. - On the
*y*-axis is “Events per 0.5 GeV.” This tells us how many events they measured with a given mT value. - What is the significance of the “per 0.5 GeV” on the
*y*-axis? This means, for example, that they count the number of events between 70 GeV and 70.5 GeV and plot that on the graph. This is called “binning” because it sets how “bins” you divide your data set into. If you have very small bins then you end up with more data points on the*x*axis, but you have much fewer data points per bin (worse statistics per bin). On the other hand, if you have very large bins you end up with lots of data per bin, but less ability to determine the overall shape of the plot. - The label for the plot tells us that we are looking at events where a
*W*boson is produced and decays into a muon and a neutrino. This means (I assume?) that the experimentalists have already subtracted off the “background” events that mimic the signature of a muon and a neutrino in the detector. (In general this is a highly non-trivial step.) - The blue crosses are data: the center of the cross is the measured value and the length of the bars gives the error.
- The values under the plot give us the summary of the statistical fit: it tells us that the
*W*mass is 80.35 GeV and that the χ2/dof is reasonable. This latter value is a measure of how consistent the data is with your theory. Any value near 1 is pretty good, so this is indeed a good fit. - The red line is the expected simulated data using the statistical fit parameters. We can see visually that the fit is very good. You might wonder why it is necessary to simulate data—can’t the clever theorists just do the calculation and give an explicit plot? In general it is necessary to simulate data because of QCD which leads to effects that are intractable to calculate from first principles, but this is a [very interesting] story for another time.

Now for the relevant question: what exactly are we plotting? In order to answer this, we should start by thinking about the big picture. We smash together some particles, somehow produce a *W* boson, which decays into a muon and a neutrino. We would like to measure the mass of the *W* boson from the “final states” of the decay. The primary quantities we need to reconstruct the *W* mass are the energies and momenta of the muon and neutrino. Then we can use energy and momentum conservation to figure out the *W*‘s rest mass. (There’s some special relativity involved in here which I won’t get into.)

**Homework**: for those of you with some background in high school or college physics, think about how you would solve for the

*W*mass if you had a measurement for the muon energy and momentum. For this “toy calculation” you don’t need special relativity, just use E = (rest mass energy) + (kinetic energy) and assume that the neutrino is massless. [The discussion below isn’t too technical, but it will help if you think about this problem a little before reading on.]

The first point is that we cannot measure the neutrino: it’s so weakly interacting that it just shoots out of our detector without any direct signals… like a ninja. That’s okay! Conservation of energy and momentum tells us that it is sufficient to determine the energy and momentum of the muon. We know that the neutrino momentum has to be ‘equal and opposite’ and from this we can reconstruct its energy (knowing that it has negligible mass).

… except that this too is a little simplified. This would be absolutely true if the *W* boson were produced at rest, such as at electron-positron colliders like LEP or SLAC. However, at the Tevatron we’re colliding protons and antiprotons…. which means we’re accelerating protons and antiprotons to equal and opposite energies, but the actual *stuff* that’s colliding are quarks, which each carry an unknown fraction of the proton energy and momentum! Thus the *W* boson could end up having some nonzero momentum along the axis of the beam and this spoils our ability to use a simple calculation based on energy/momentum conservation to determine the *W* mass.

This is where things get slick—but I’ll have to be heuristic because the kinematics involved would be more trouble than they’re worth. The idea is to *ignore* the momentum along the beam direction: it’s worthless information because we don’t know what the initial momentum in that direction was. We only look at the **transverse** momenta, which we know should be conserved and was initially zero.

If we use conservation of energy/momentum on *only* the transverse information, we can extract a “fake” mass. Let us call this the **transverse mass**, mT. (Technically this is not yet the “transverse mass,” but since we’re not giving rigorous mathematical definitions, it won’t matter.) This fake mass is exactly equal to the real mass when the *W* has no initial longitudinal momentum. This is a problem: we have no way to know the initial longitudinal momentum for any particular event… we just know that *sometimes *it is close to zero and other times its not.

The trick, then, is to take a *bunch *of events. Up to this point, in principle you didn’t need more than one event to determine the *W* mass as long as you knew that the one event had zero longitudinal momentum. Now that we don’t know this, we can plot a *distribution* of events. For the events where the longitudinal momentum of the *W* is zero, we expect that our transverse mass measurements are close to the true *W* mass. For the events with a non-negligible longitudinal momentum, part of the “energy” of the *W* goes into the longitudinal direction which we’re ignoring, and thus we end up measuring a transverse mass which is * less* (never greater!) than the true

*W*mass.

Thus we have a strategy: if we can measure a *bunch* of events, we can look at the distribution and the largest possible value that we measure should represent those events with the smallest longitudinal momentum, and hence should give the correct *W* mass.

This is almost right. It turns out that there are a few quantum effects that spoil this. During the production of the *W*, nature can conspire to pollute even the transverse momentum data: the *W* might emit a photon that shifts its transverse momentum a little, or the quarks and gluons might produce some hadrons that also give the *W* some transverse momentum kick. This ends up *smearing out* the distribution. It turns out that these can be taken into account in a very clever—but essentially mathematical—way, and the result is the plot above. You can see that the distribution is still smeared out a little bit towards the tail, but that there is a sharp-ish edge at the true *W* boson mass. This is what experimentalists look for to fit their data to get extract the *W* mass. (For more discussion on the *W* mass and a CMS perspective, see this post by Tommaso a few months ago.)

I really like this story—there’s a lot of intuition and physics that goes into the actual calculations. It turns out, however, that for the LHC things can get a lot more complicated. Instead of single *W* bosons, we hope to produce pairs of exotic particles. These can each end up decaying into things that are visible and invisible, just like the muon–neutrino system that the *W* decays into. However, now that there are *two *such decays, the kinematics ends up becoming *much* trickier.

Recently some very clever theorists from Cambridge, Korea, and Florida have made lots of progress on this problem and have developed an industry for so-called “transverse mass” variables. For those interested in the technical details, there’s now an excellent review article (arXiv:1004.2732). [These sorts of analyses will probably not be very important until after the LHC 2012 shutdown when more data can be collected, but they offer a lot of promise for how we can connect models of new physics to data from experiments.]

Cheers,

Flip