## Calibrating the muon momentum in CMS October 22, 2007

Posted by dorigo in personal, physics, science, travel.

I arrived at CERN last night after a long trip. I usually travel by car to CERN, because rooms at the Foyer (an internal hostel) are always hard to find, and transportation to the closest cheap hotels in France, past the border, is difficult. The drive lasts about six hours from Venice – it’s roughly 630 km – if traffic is low as usually happens on Sunday afternoons. However, yesterday things conspired to make my drive quite long – at one point I had to take a deviation of 30km to avoid a 5km long queue…

As if my morale wasn’t low enough by being at CERN, I caught a horrible cold last Saturday. We had invited our friends Ivan and Barbara for the evening, and my son Filippo got injured while playing with Sebastiano, Ivan’s son, just minutes before we’d call the evening off. He got a small but deep three-spoke wound on the tip of the chin, resembling the symbol of a german make of cars: it obviously required a couple of stitches. I brought him to the hospital at 11pm, his chin was taken care of, and everything went well – except that on our way back Filippo was feeling cold, as sometimes happens after surgery. So I gave him my leather jacket (in Venice you do not drive, you walk! And the hospital is 25 minutes away…), and as a result, I am now shivering and sneezing as I try to put together a reasonable set of slides for the talk I am going to give this Wednesday at the “Physics days”.

So, about muon momentum calibration: for those of you kind enough to get to read till here, I will describe what it is about.  Together with a few colleagues from Torino (Sara and Chiara), Marco and I have put together an algorithm that uses known resonance decays of J/Psi, Y, and Z particles to muon pairs to correct the scale of the muon momentum measurement.

[Post-scriptum: I sometimes forget -and invariably regret later- to give the right amount of recognition when it is deserved. The current version of our code has been written by Sara Bolognesi, to whom I owe a lot since she is helping me out with the material for the talk!]

The invariant mass of a body (say, a Z) decaying to two particles is a simple quantity that can be derived from the four-momentum of the latter. In case of muons, their mass is known with extremely good accuracy, so you just need to determine the three components of their momentum – i.e. their velocity vector- at the Z decay vertex. We measure with high precision the trajectories of muons as they fly through the inner tracker and then through the calorimeters, finally hitting the outer muon detectors. CMS is certainly well-suited to reconstruct dimuon resonances.

The CMS silicon tracker is immersed in a strong axial magnetic field of 4 Tesla (40,000 gauss), and muon chambers are also sandwiched between 2-Tesla magnetized iron: the result for charged particles subjected to the Lorentz force $q \vec v \times vec B$ (q is the charge, v is the speed, and B is the magnetic field) is a funny trajectory which, in the plane transverse to the beam, sees tracks turning one way in the tracker and then the other way outside it. In the picture on the right you see a slice of the detector in the transverse plane. The magnetized iron is in red, and the track trajectory is measured in the inner tracker (hatched circles inside the blue-yellow calorimeter) and in the pink (blue those hit by the track) muon chambers.

From the precise position of the hits in the detector components, muon trajectories are known with micrometric accuracy within the local coordinates of the sensing elements: however, to go from the local trajectory to the momentum of the particle, we need a very precise model of the position of each sensing element in the detector, and a correct map of the magnetic field!

Now, our particle reconstruction software knows the “nominal” geometry of the detector and the B field, so any difference in terms of alignment and B field map causes a bias in the position of hits and ultimately in the momentum measurement. We can detect a bias by checking whether the measured Z mass is shifted from the real value, as a function of the kinematical characteristics of the measured muons.

The plot on the left shows a tiny but nagging effect found and corrected by CDF in Run I: before scale corrections (plots on the left), the mass of the J/Psi resonance was demonstrated to depend on the azimuthal angle of the muons, if binned properly according to the polar angle. After corrections, the effect was gone (right plots). It is these kinds of effects that our algorithm will address.

We use a Monte Carlo simulation of $Z \to \mu \mu$ decays, where we know the exact value of the generated mass for each particle (not necessarily equal to $M_Z=91.187 GeV$, since the Z has a 2.5 GeV width). We then modify the geometry of the simulated detector such that we obtain a misalignment of the same size of the alignment uncertainty we will face in the analysis of real data. Then, we check whether our algorithm is capable of spotting and correcting the effects of the misalignment, by looking at the reconstructed Z mass values as a function of muon kinematics. It does!

In two words, our algorithm finds a momentum scale correction by minimizing a likelihood function constructed on a sample of measured resonance decays. One hypothesizes that the momentum is biased as a function of the muon kinematical quantities $x_i$ (such as momentum itself, or the particular area of the detector traversed by the muon) via a functional form $p_T = F(p_T; x_i, \alpha_j)$, where $\alpha_j$ are unknown parameters, and $p_T$ is the transverse momentum of the muon. One then computes the mass $M(F(p_T(1), F(p_T(2)))$ for each event, and then the likelihood

$L = \Sigma_k log P[M_k(F_1, F_2)]$

by summing k on the N available events. P is the probability of the mass M, and it just depends on the difference between the true peak of the Z and the value obtained for each event. By minimizing L, one finds the best value of the parameters $\alpha_j$, and thus obtains a correction for the muons as given by the functional form F.

There are tons of subtleties connected to setting the scale of the track momentum measurement, and this is really no place to discuss them… Let me just mention a few, to give you the flavor of the work ahead of us:

• Z simulation is affected by several uncertainties connected to the modeling of their generation, photon radiation effects, and the like;
• The peak of the mass distribution for a Z boson at a proton-proton collider is not sitting at the real Z mass value, because the quarks colliding to produce the Z boson have a much higher probability of having less than the required energy: so the peak sits about 0.3 GeV below the nominal value. Also, that shift depends on the Z kinematics, and if we study the Z mass as a function of its kinematics we have better take the effect into account.
• CMS needs a proper calibration for muons reconstructed by fitting together the signal of both tracker and muon chambers (global muons), or just muon chambers alone (standalone muons), or just tracks. Our algorithm needs to take care of that properly.

What is the goal of such an accurate calibration, anyway? If we are going to discover a Higgs boson in the decay $latex H \to ZZ \to 4 \mu$ we certainly like to measure its mass with “good” precision, but the painstaking calibration of track momentum is not going to be really needed there. One might argue it is still needed for a precise W mass measurement, but the W boson mass will be very hard to measure better at the LHC than what CDF and D0 are doing now, at least for quite a while -the Tevatron experiments might achieve a better than 20 MeV precision on that quantity.

Nevertheless, calibrating one’s detector and perfecting its ability to measure without bias the particle momenta is something that will be of some benefit to all analyses, even if not critical to any of them singularly. We need precise track measurements for b-tagging. We need it to reconstruct exclusive final states. We need it to understand the detector as a whole. In that sense, our task can be considered service work for the experiment, no more and no less than turning a screwdriver…

1. carlbrannen - October 22, 2007

This is a wonderful post. It gives a much better view into the details of how particle experiments are done.

2. riqie arneberg - October 22, 2007

“arrived to CERN”

One “travels to”, and “arrives at”

3. dorigo - October 22, 2007

Hi Rique,

thank you! I will correct the text above because it bothers me when I do these mistakes. I am however always grateful when I get a lesson in English.

Cheers,
T.

4. dorigo - October 22, 2007

Hi Carl,

thank you very much. I will try to give more insight in the tools we use in our complex interpretation of particle collisions…

Cheers,
T.

5. More on the Z lineshape at LHC « A Quantum Diaries Survivor - December 19, 2008

[…] A description of the algorithm, I am afraid, belong to a different post. You can check a rough description I gave in this blog here; and a more general discussion is here. […]

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