Calculate the Top Quark Mass

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Introduction: Einstein's celebrated equation is verified daily in particle accelerators around the world. Physicists go about the business of converting energy into mass almost as commonly as high school students flip through channels on the television. Still, this revolutionary idea is not often treated as a classroom activity simply because it seems to be so difficult to convey in a "hands-on" manner. We are able to do just that using a special event recorded by Fermilab's D0 detector. Most events analyzed by the physicists are more complex than this event nature serendipitiously supplied.

The idea that mass and energy are interchangeable is essential to those interested in understanding how two top quarks (actually a top and an antitop) are created from the collision of two protons (actually a proton and an antiproton). The experiment might be thought of as the collision of two ping-pong balls resulting in the production of two ball bearings of the same size, but considerably greater mass. The highly energetic protons collide to create top quarks of about 180 times the mass of the protons. The energy of the less massive protons is converted into the huge mass of the resultant top quarks.

Perhaps the conversion between energy and mass makes more sense when one considers that the proton and antiproton pair were traveling so close to the speed of light that together they had about 1.8 x 1012 eV worth of energy to work with. This energy then becomes the mass of the newly discovered quarks as shown in this diagram of E=mc2 Example: Mass of Quarks coming from Energy of Protons.

Scientists measure the energy of these subatomic particles in units of electron volts. Electron volts are units of energy just as Joules are units of energy. They measure their mass in units of electron volts/c2, energy divided by c2 where c= the speed of light. Here, we are talking about mass and using the fact that E = mc2 to write mass in units of eV/c2. To simplify the units, physicists use a unit called a GeV (a giga electron volt, pronounced two ways, "gee ee vv" or "jev"). 1 GeV equals 109 eV. Physicists use these units because they make it easier to talk about the mass and energy of the particles, just as astronomers use light years to express distances instead of using kilometers or miles.

Scientists at Fermilab first discovered the top quark in 1995 when they collided protons and antiprotons with energies of 900 GeV/c2. You can read Background Material -"Discovery of the Top Quark". The masses of these particles scientists measured are shown below.

Mass of Proton9.38 x 108 eV/c2.938 GeV/c2
Mass of Top Quark1.75 x 1011 eV/c2175 GeV/c2

To help the students understand the idea that mass and energy are interchangeable, this activity examines the fingerprint of a top/antitop production that took place in the D-Zero Detector at Fermilab on July 9, 1995 and is shown in an artist's rendition of top/antitop production.

Make an overhead of the artist's rendition to show the students what the production of a top and an antitop quark might look like if one were to see it in the Fermilab accelerator. It is important to point out that the top and antitop quarks are actually very short-lived particles. They quickly decay into daughter particles and then in turn into "granddaughter" and "great granddaughter" particles. It is these offspring that are actually detected by the scientists at Fermilab. Another diagram shows particles from the collision the D-Zero Detector actually sees. This event shows that the top and antitop quarks (shown as t and t-bar respectively) are never actually directly detected because they decay so rapidly into four "jets" (large blasts of particles), a muon (green) and a neutrino (magenta) in the upper right of the artist's rendition).

This may appear first to be complex, but the mass of the top is easy to calculate from a computer-generated plot of the same event taken from the D-Zero detector.

Teacher Primer: This activity will build on your classes understanding of vector addition and depend upon only a small amount of particle physics explanation from the instructor. The goal of this activity is simple. Your students will determine the mass of the top quark.

Here are three views of one event called Run 92704 Event 14022.

These are computer generated pictures that represent the event discussed previously. To help visualize the event, you may look more carefully at the first color plot labeled CAL+TKS R-Z VIEW which gives a perspective from the side of the detector. Next, look at the second plot called CAL+TKS END VIEW. Here the event is viewed in only two dimensions as seen from the end of the detector. Finally, you may view the third color plot called DST LEGO which shows how the debris could be mapped if the END VIEW were unwrapped starting at the "X" axis.

Once you are familiar with the three views, you should focus attention on the CAL+TKS END VIEW. This view is most similar to the artist's drawing and will be the working document for this activity. Notice the four blue and red "jets" which are the four jets shown on the artist's picture of the event above. You will also notice a solid green line which represents the muon's track. If you look closely at the jet near the bottom of the picture, you will also see a green dotted line. This is a "soft muon" that is hidden in a jet. Please also notice that all of the momenta have been measured and are written on the plot.

You will also see that the computer has calculated the energy of the neutrino and drawn it on the diagram in magenta. Neutrinos are not detectable in most cases, so their existence is found by looking at the total momentum of the system in a collision. The total momentum should remain at zero before and after the collision though the momenta of the particles are far from zero. The process of finding the missing momentum is what your students will do to determine the mass of the top quark. Notice that the momentum for this particle is not included on the picture.

Teacher Instructions for Classroom Presentation: This activity will build on your class's understanding of vector addition and depend upon only a small amount of particle physics explanation from the instructor.

Calculations of Momenta of Products of the Collision.

The momentum of each jet or particle was determined by computer and is printed on the color END VIEW plots. See the Data plots for Displaying and Data Plots for Printing. These numbers will be used in creating a vector diagram of the debris that comes from the collision as students attempt to find the momentum of the "undetectable" neutrino.

The teacher should simply explain to the students that this is an exercise in momentum conservation. They are to determine the momentum of the 'undetectable' neutrino by adding up the vectors with directions shown on the diagram and magnitudes indicated by the numbers listed. The result should be a value close to the same number the D0 collaboration determines for each event. These values are listed below for the teacher's reference.

D0 Collaboration Values Using Fermilab Computers
Actual Event14022momentum of neutrino53.9 GeV/c
Computer Simulated Event26momentum of neutrino76.1 GeV/c
Computer Simulated Event153momentum of neutrino43.6 GeV/c
Computer Simulated Event553momentum of neutrino45.3 GeV/c

The direction of each neutrino may be verified by examining the color plots. Please be aware that the students will not all get the exact values given above nor will they get the precise directions shown on the pictures. This is due to their selection of the directions of the debris in the first place as well as effects introduced by problems similar to the one noted below[1]. The value we got using a protractor and ruler was abou 42.

[1] It is important to realize that this only works if the debris has no motion in the direction of the beams which we define as z direction. The event takes place in a plane perpendicular to the axis of the proton and antiproton. The third color plot labeled DST LEGO shows that Mother Nature was indeed kind in giving us this type of event. Notice that all of the tracks happen to lie close to the ETA = 0.0 axis. This allows us to approximate this as a two-dimensional problem. You will also see that there is some "noise" seen on the computer plot. This will adversely affect your vector diagram and represents the uncertainty that is present in any experiment.

Still, after vector diagrams are drawn by various groups of students, a reasonable value for the momentum of each of neutrino may be found.

An example of a possible vector diagram for event 14022 is shown below for the teacher's convenience. If you click on it, you can see a graphic that shows the correspondence between the vectors and the data.

Vector Diagram of the Event

Background Energy, Mass and Momentum Calculations for the Teacher: In order for your students to find the mass of the top quark, they need to understand that their discovery of the missing momentum of the neutrino is crucial. This value gives them all they need to find the mass of the top quark. You will need to supply them with the following information. In all honesty, much of this material is beyond the scope of high school physics (and many college courses as well), but the leap is not so great that it cannot be done. Perhaps a bit of faith is needed here. The teacher is certainly the best judge.

A common relation in high-energy physics is the following.

E2 - p2 = m2

The reason energy, momentum and mass are shown as equal is actually due to the convention of choosing a system where the speed of light, c, is set equal to one. In this case, where particles are traveling with speeds of almost c, E = mc2 becomes E=m and p=mv becomes p=mc or p=m. This does change scale somewhat to be sure, but it allows for a simpler conversion between energy, mass and momentum.

In our particular case, it follows that one should write energy and momentum in terms of the mass of the top quark.

E2 - p2 = (2mt)2

When one observes that the net momentum in a plane perpendicular to the beam direction before the collision is the same as the momentum after and that value is zero, we write:

E2 = (2mt)2

or, taking the square root of both sides,

E = 2mt

Because almost all of the energy of the collision is the result of top and antitop decay, we simply add the energies of the four jets, the soft muon, the muon and the neutrino before dividing by the two tops (actually a top and an antitop quark) to obtain the mass of the most recently discovered quark.

Students will use the values they calculated for momentum (now as energy values) and incorporate their new value for the missing neutrino before adding all the energies as scalars to find 2mt.

If we use the 42 GeV we got with our vectors produced with protractor and straight edge, we would have the equation.

61.2 GeV + 7.3 GeV + 95.5 GeV + 58.6 GeV + 54.8 GeV + 17.0 GeV + 42 GeV = 336.3 GeV

348.2 GeV/2 = 168 GeV

If we use the value of 53.0 calculated by the computers used by the D0 collaboration, we would have the equation:

61.2 GeV + 7.3 GeV + 95.5 GeV + 58.6 GeV + 54.8 GeV + 17.0 GeV + 53.9 GeV = 348.2 GeV

348.2 GeV/2 = 174.2 GeV

which is even closer to the currently accepted value of about 175 GeV.

As was indicated, the "missing momentum" may be found in a more careful analysis of the event as well as a better understanding of the event.

This relatively simple procedure may be repeated by your students in events generated by the Fermilab computers that were directed to simulate various energies that would show this type of event. We have found that when students average their results for all events and for multiple students analyzing all the events that the average comes reasonably close to the accepted value of the mass of the top quark.

Conclusion: The final result of this exercise should be that the students have gained some experience in using actual data to see how scientists analyze collisions. Further, your students will begin to understand that the last pieces in The Standard Model puzzle have been assembled from the energy in the collider. Mass does indeed come from energy as Einstein predicted.