CMS e-Lab Momentum-Energy Study
From QNFellows
[edit] Introduction to Energy, Momentum and Mass in CMS
E2 = P2 + M2. (Here we should add some introductory motivational material to this: a derivation, perhaps?)
This exercise will highlight two important points for students:
1. In high-energy physics (HEP), the rest mass term is often negligible. Certainly for the lighter, more stable particles that last for very long, since the energies and momenta are so relatively large. (Without awareness of this fact, students will be unable to think their way through the application of conservation laws to the interpretation of CMS data.)
2. Transverse momentum is an especially important quantity in HEP. CMS is a proton-proton collider. But because the energies involved are sufficient in a direct collision to overcome the electromagnetic repulsion of these particles, it is the partons (quarks and gluons) that are colliding in the highest-energy collisions. Since the portion of the energy of the proton present in any parton is unknown, the initial energies and momenta of primary collisions are not known in advance. (In this respect, the LHC as well as its proton-antiproton rival--the Tevatron at Fermilab--are unlike the LEP, which collided fundamental particles--electrons and positrons--at collision energies known and, in fact, carefully tuned in advance.) So while total energy and momenta can be calculated for LHC collisions in a variety of ways, they are not known in advance. What is known from measurement is the quantity of energy deposited in the calorimeters. For small particles, that energy is equivalent to the initial momentum. What is also known at the LHC is that in the primary collisions, particles are carefully steered along the z axis (the beam line), and thus have very small components of transverse momentum (momentum that is radial, orthogonal to the beam line). So initial transverse momentum for primary collisions is treated as zero. Transverse momentum (Pt) for secondary collisions can be calculated through procedures that this exercise will help to motivate (from tracker, calorimeter and timing data.) From that calculated Pt, missing transverse momentum can be used to identify missing energy, neutrinos (not directly observed in CMS) can be identified, and coherent event reconstruction becomes possible.
This exercise will begin by presuming that students know that electron mass (at roughly ½ KeV) is negligible for LHC collisions and thus that momentum and energy are equivalent for electron pair production (one electron, one positron, hereafter, ee), as well as the definition of transverse momentum. By looking at Z --> ee data, students will use transverse momentum to verify for themselves the equivalence of energy and momentum for that subset of particles, all of whose momentum is in a transverse direction.
1. Isolate Z --> ee events in the Data Selection window:
2. Compare E to Pt, and see that not all P is Pt:
3. Return (by clicking on the green ogre 
in the Selection History window) to the plot selection page and plot electron eta (where "eta" is a function of the angle between the y axis and the beam line); then cut the "low eta" events by dragging across the plot from (say) -.4 to .4:
4. Then choose "append" from the "Selection:" choice menu (
) and return again to the plot selection menu via the ogre in the selection history box. Now find the small "my cuts" box, drag it into the plot options area, choose the "electron Pt" and "electron energy" plots, and plot the results:
5. Discuss with students the difference between the Pt vs. E plots (presented again below, for comparison) from both before and after the "low-eta" cut was applied to the data. Students should leave the discussion understanding that our selection of low-eta events was in effect a selection of events, all of whose momentum was in the transverse (radial) direction. (We looked at Pt, rather than P, simply because P was not available in OGRE, but also because Pt of the primary colliding particles is known in advance to be essentially zero. Particle physicists thus use Pt to reconstruct events from this zero-Pt initial state. That's why Pt, not P, is present in OGRE at the beginner's level--to get across its importance in HEP.) Since all of P is Pt for the low-eta selected set of events, then energy should be equivalent to Pt, as we see that it is. This exercise presumed and then verifies the presumption that electron mass is negligible: if it weren't, then energy and momentum would not be equivalent, as we can see through this exercise that they plainly are in the simulated data.
Some students may want to look at the same plot after applying a high-eta cut, as well, where (as in the uncut version, only to a more magnified extent) the Pt and E values will diverge. Students may also be interested in repeating the study for Z-zero to dimuon events.
