CMS e-Lab ee Vertex Study
From QNFellows
[edit] CMS e-Lab Z-->ee Vertex Study
Classroom Notes (teacher) Version
Introduction
This study uses Monte Carol data in an introductory activity for the CMS e-Lab. Because it is introductory, it leans more heavily on the side of guiding the inquiry process; subsequent investigations will be less heavily guided. There are two stages to this study, which should be introduced in order from simplest (mapping of ee vertex locations) to more complex (determination of a rough upper bound for Z0 lifetime). The names of these studies can be imposing, but the basic idea for each is pretty simple: the first study asks where in the detector the Z0 decays to two electrons, and the second study asks whether we can tell from the data how long the Z0 lasts before decaying. Both stages are ideal for introducing OGRE functionality. The simpler stage introduces the basic data selection, plotting and plot interaction capabilities of OGRE, and the more complex stage introduces the idea of making and reapplying meaningful cuts in the data. This study is well suited for a class-wide conversation, drawing together some basics regarding particle physics and detector design that students have grasped during their initial reviews of the first three "Getting Started" milestones.
Stage 1: Vertex Mapping
Study presentation and eliciting of student hypotheses
1. Begin by letting students know that this is a study of the vertex--the common point of origin from which particles emerge--of electrons (e+ and e-,hereafter, ee) produced by Z0 decay. Before looking at OGRE, ask students (perhaps using a projected image of full CMS detector) where in the detector they would expect Z0s to decay.
2. In a "milestone seminar", review what students know about ee production (from Z0 decay); mention Z0 lifetime (treated as unknown, for now)--it is best hear as always if the students raise the lifetime issue--and make sure that the impact of a longer or shorter lifetime on vertex distribution gets considered. Discuss the meaning of x, y, and z axes in CMS; x is parallel to floor, positive toward ring center; y is perpendicular to floor, positive toward above ground; z is along the beam line; the origin right in the center of the detector (in the beamline.)
3. Have students draw a 3-d map of where the expect ee vertexes to be located in CMS, and why.
4. Then introduce the idea of a histogram of ee positions in the x, y and z axes. (See the example from OGRE, but begin to discuss these histograms without OGRE. One option is to review histograms first, exploring together the histogram tool from the Basics milestone set at this time.) Ask students to predict where they would expect the electron vertex to be located in x, y and z. Set up a 1-d histogram for position in (say) the x direction, and ask students to predict its shape. Discuss appropriate scale size for such a histogram (1 km wide?) Depending on the level of mathematical preparation of students, you might introduce the idea of a log plot here (even just defining it as changing the y axis so that it proceeds uniformly by powers of ten.) You could take digital pictures of class predictions, or have students find their own way to generate a histogram for their e-Lab logbooks, either an image or a written description.
Begin investigating with OGRE
1. Select ee events; subtract background using the selection tools as shown in the Data Selection Window image. (1b, optional.) (The "one of" box requires that all selected events be in ONLY one of the sets dragged into this box. This serves as a good intro to the logic of event selection (see the "(ee XOR bkg)" notification) in the data window, and you could use this moment to linger on the logic functions and allow students to explore them--or just pass this by, and instruct students where to put the data this first time through.)
2. Choose "Build plots", and drag the Electron Vertex X (Electron Vtx X) plot button into the "Make a Plot" box. (You could introduce plot shading at this time, or just let the students figure it out on their own.) See the Plot Selection Window image below.
3. Do the very first plot together, and use it to introduce the idea of selection in OGRE plots on the analogy of zoom boxing on a graphics calculator. The first plot they encounter will look like every event occurs at x=0. Attend together to the size of the x axis, and invite students to explore more carefully by zooming in using the drag-to-select and "apply selection" tools. Have students plot (separately) electron vertex in x, y, z, size them in a complete graph (define and put in glossary), and post them to their logbooks.
Post-OGRE discussion
Ask students, what did you expect, what did you get, and what do you notice about the difference? Any explanations for the difference? They might have expected symmetry around the origin (0,0,0) with broadly-spread peaks; they would have found a small but clear asymmetry between the x and y axes, with a 1/3mm offset in the positive x direction (toward the center of the ring.) Why? Poor construction? (Could investigate tolerances in construction.) Beam mixing? (Investigate how +,+ beams are brought into crossing.) But both the x and the y vertex distributions are much tighter than the z direction, where the width of the distribution is in tens of centimeters, rather than tenths of millimeters for x and y.
Ask students how we might explain the much large range in the z direction. Z0 decay? (Help students see the potential confusion between the two different uses of "Z", here: one a direction, the other an exchange particle.) How far down the beam pipe could a Z0 travel before it decays? (Does relativity matter, here? The beam pipe is shorter, in effect, for particles at .99999c.) What is the bunch size, and thus the length of potential collisions? Is there a way we might proceed further with the data to explore these questions?
In the vertex mapping thus far, a model of collisions has been developed. Now, we go further to deploy it.
Stage 2: Preliminary investigation of Z0 lifetime
Study presentation and eliciting of student hypotheses
Z0 lifetime is on the order of 10^-20 seconds; even at the speed of light (~1ft/10^-9 seconds), that's a very small distance before decay, and thus Z decay is not the explanation for the large spread (in centimeters) of Z0-decay vertexes along the beamline. Some students might think to look up Z0 decay time and eliminate this explanation right from the start; teachers should be prepared for this. But the question remains: can students detect the role which Z0 decay time might play in the large spread of vertexes along the beamline?
Begin by asking students whether and how Z0 decay time would impact the shape of distribution of vertexes along the z axis. Students will come across the possibility that the width of the distribution would begin to increase if Z0 lifetime were long enough. Help students see that the much narrower width of the x and x distributions is a better place to investigate whether Z0 decay time accounts for the variance in vertex location.
Once they see this, they can come to see that Z0s with a high momentum perpendicular to the beamline (i.e., with high transverse momentum, Pt--introduce and translate for students--would travel further in x and y before they decay, and those with low transverse momentum, less far in x and y. This leads up to the question: could difference in transverse momentum be part of the reason for the width of these distributions? You might want to use a projected image of the CMS detector to help them visualize this question.
Applying Cuts to the data in OGRE
Tell students that when we selected Z0-->ee data by applying a background cut to the ee MC data, we made one cut in the data, one selection. We made other cuts when we dragged across the plots to look only at the events near to the origin in x and y. What we're about to make another set of cuts by plotting Pt for this same Z0-->ee set, and then selecting for first high, then low Pt, and then re-run our vertex study on these two different (hight Pt, low Pt) data sets.
Begin by isolating Z0 -->ee data (ee minus background), and then select the Electron Pt plot. (Discuss with students what justification there might be for using decay product--electron--Pt to distinguish between high and low parent particle Pt. Better ways of differentiating between high and low Pt Z0s will be developed in OGRE soon.) In the resulting plot, select high Pt data (say, from 90 to 150 GeV). To see the resulting plot you may choose "apply selection", which will change the plot, but not the data set. What we want to do is eliminate all the non 50-to-100-GeV events from the set. This is done by choosing the "APPEND" button next to the "Selection" indicator.
Once your cut is "saved" (or "appended" to the existing set of criteria for your dataset), choose the "Selection History" button and then choose the small green OGRE image in the upper left corner of the resulting window. Return to the plot selection page, choose Electron Vertex X, and apply the saved cut (by dragging it from the right into the top middle box), and plot. Expand the distribution by manipulating the plot and then save it. Then repeat this whole procedure for low Pt events. (You might let students determine for themselves what energies to accept as low-Pt electrons.)
Post-OGRE discussion
Once you've produced two different histograms of ee Vertex location in the same direction (say, x), compare the resulting plots: do students notice any difference in the distributions--do high Pt events get measurably further away from the point of collision before they decay, producing a wider distribution? How does your answer bear on the lifetime of the Z0 boson? (If it were a long lifetime, what would you expect to see? If short? If not measurable at all? Can we quantify a limit on the Z0 lifetime?) Have students research the Z0 lifetime and compare their result to the theoretical lifetime of the Z0 boson. What they'll see for themselves in the data is that Z0 lifetime does not seem to impact the observable ee vertex location, and the theoretical estimate of Z0 lifetime on the order of 10^-25 seconds will confirm their observation. Students should leave this study with a better feel for CMS detector geometry and some confidence that they could use OGRE's selection and cutting tools as they think their way through questions about collisions in the detector.
Stage 3: Using Calculator Functions in OGRE
(an optional supplemental activity)
For further exploration: invite students to try the same study (at either or both stages) using photon or muon vertex locations. This is a good time to introduce the idea of plotting two data sets on the same histogram; see the pairing of photon and electron vertexes in x. (Students can learn how to use the plot coloring options by trial and error.) This time could also be used to introduce OGRE's calculator functions. If we divide the number of events with a given x vertex position for photons by the number with that same x location for electron vertexes--see the plot selection window and the guide to OGRE's calculator functions, below--we get a histogram with the main distribution centered on 1 (since for the most part the photon and electon vertexes overlap), with the width of the distribution on either side reflected in the difference between the shapes of the photon and electron vertex distributions. Notice also the peak at zero in the plot of the division: there are regions in the comparative distribution plot where there are electron vertexes but no photon vertexes, resulting in a zero value over a non-zero value, for a net zero. The calculation tools in OGRE are powerful, and they could be introduced to students (or advanced students could introduce themselves to these tools) as a supplement to this vertex location study.
Using OGRE's Calculating Functions
To use any of OGRE's calculator functions, just drag them into the plot selection box.
The +, -, *, and / keys perform their standard arithmetic operations in OGRE.
Parentheses also act as expected; drag them in one at a time. (There is no auto-close function.)
