Calculate the Top Quark Mass

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Many teachers have tried the Top Quark activity on their own in the way in which they imagine their students working, using a protractor and ruler. And many teachers have initially been disappointed by the results. They find that, when they calculate the neutrino momentum, neither the magnitude nor direction seem to match Fermilab's published values. Fear not! This is an activity that works better with more people than with one.

What follows is a set of observations and ideas that will help make this activity a winner. They have been developed after watching Bob Grimm, Fremd High School in Palatine, IL, run this activity with two of his classes. His students NAILED the top quark mass! Bob revealed to them, after their calculations, that the current value of the top quark mass is 174 GeV. The first class average was 177.1; the second was 177.4! Bob suggested that Fermilab might want to revise its number upward, based on the classroom results!


Bob's students were studying conservation laws, so the activity fit right in. They didn't know about quarks, etc., but they spent some time, before the lab day, looking at overheads (taken right from the Website) that describe the top quark experiment. So, with their particle questions addressed, they were able to focus on the vector addition portion of the activity. This is a good investment; on lab day, the students' questions about vectors and momentum were not clouded by questions about particles. This is not to say they did not have questions about particles. Indeed, the questions they had were fairly sophisticated. But they did not seem distracted by details of the experiment.


The errors that teachers experience in their practice of this activity are due to the fact that there are two ways in which the data gets used. First, they're used to generate a VECTOR DIAGRAM that does NOT demonstrate the conservation of momentum unless you fill in the missing (anti-) neutrino. (By the way, Bob had some nice short discussions about what might be wrong . . . there are great teachable moments lurking inside this activity.) Second, the data are used to create a SCALAR SUM (no direction needed) that, with the information from the neutrino, gives a total energy that is very close to twice the top quark mass.

Now, if teachers evaluate the activity in terms of its ability to get the magnitude and direction of the neutrino momentum vector, it's a bust! This is one place where teachers will stop their own work and question the value of doing the activity with students. But, if you ignore that, and evaluate the activity as a way to determine the mass of the top, it's remarkable - but only if you have enough data! A single data point is pretty inconclusive. This is the second place where teachers will feel ill at ease. Their single data point might be off by 10% or more and the thought of leading students through the activity begins to lose some luster. But, in this activity, more data is better. Bob, for example, uses the four scenarios liberally (the three simulated events and one actual event), and gets as many as ten data points on the board before averaging. The results, as mentioned before, are quite impressive. The real trick is to get the students to use the vector diagram to get the MAGNITUDE of the neutrino momentum while simultaneously ignoring the DIRECTION of that same vector.

Some students noticed that the direction of their calculated neutrino momentum was in grave disagreement with the computer-generated direction (the drawings Bob used included a pink neutrino vector drawn in by the computer). It was good to see kids check themselves like this and Bob and I (I took on a reasonably active role with the students) were able to allay their fears with some discussions about error sources, most notably, the true three-dimensional nature of the collision versus the two-dimensional nature of the data and the difficulties in determining EXACTLY the direction of the jets. But most importantly, we guided them toward the addition of the ENERGIES and away from the MOMENTUM except as much as the momentum measurement can give you a working value for the neutrino momentum/energy.


At the relativistic collision speeds like those achieved by the protons and anti-protons in the top quark experiment, the concepts of mass, energy and momentum intertwine to such an extent that they can all be substituted for one another without changing scale. For example, the energies in this activity are measured in GeV (giga-electron volts), the momenta are measured in GeV/c (where c is the speed of light in a vacuum), and masses are measured in GeV/c2. Now, a favorite trick of particle physicists is to normalize c so that it equals 1. This is a natural way of thinking about c, particularly when the particle velocities are expressed in terms of the percent (typically 99% or higher) of the speed of light. Having done so, the units for energy, momentum and mass can all be expressed as GeV. This is an important point since, right at the end of the activity, you're going to ask your students to use the magnitudes of the momenta (vectors) as addends in an energy sum (a scalar) without changing the values at all. Some students may object; applaud them for their skepticism (and their correctness at much lower speeds), then reassure them that the mathematical differences between these concepts evaporate the closer you get to the speed of light.


Just a thought: Even if a student did not add in the neutrino momentum/energy, they could get a mass for the top quark that was within 15%. Since the neutrino "only" adds about 50 GeV to the 350 GeV total, the worst error possible is less than 15%. That's the hidden fail-safe mechanism for this activity: students will generally be within 10% even if they have serious issues with their neutrino calculation.


Keep the following ideas in mind while leading your students through this very worthy activity: