# Exercise 4: Visualizing mathematics

This exercise is a bit different from those we’ve done already. Rather than using data in an NTUPLE, we will visualize mathematical equations. Specifically, we will look at decay rates predicted by isobar models of three-body decays as function of position in the Dalitz plot (two-dimensional phase space). To be specific, we will consider the decay The basic formalism for Dalitz plot amplitudes is discussed in Section 2.1 of Rolf’s thesis, and a discussion of the isobar model in Sections 5.1 and 5.2. Reading these first will help you do the work and understand the underlying physics. There are three major steps:

Create a TH2D object and fill the area within the Dalitz plot boundary with a uniform (non-zero) value. This corresponds to a uniform matrix element.

Now, make a Dalitz plot for

*one*of the amplitudes listed in Table 5.1 of Rolf’s thesis. Note that the formulas in Eqns. 2.13 - 2.17 tell you how to construct the quantum mechanical*amplitude*. To get the rate (which is what is plotted in the Dalitz plot), you need to calculate . To make life a little easier, approximate and as unity. While not precisely true, it is good enough to understand the most important aspects of Dalitz plots.An especially interesting feature of the Dalizt plot is that it allows you to visualize macroscopic quantum interference. Choose two of the amplitudes in Table 5.1 of Rolf’s thesis and make the corresponding Dalitz plot. Then, change the phase of

`*one*`

(multiplying by , for example), and make the Dalitz plot again.

When you have completed one example of each of these three steps, there are several variations on a theme.

Make a Dalitz plot with all the Cabibbo-favored amplitudes in Table 5.1 (when the resonance is neutral or a of some sort, or non-resonant). Then, add the doubly Cabibbo-suppressed (DCS) amplitudes (when the resonance is a of some sort). Where do you see the DCS amplitudes, and what are the effects?

With two amplitudes, systematically change the phase between the two in realtively small increments (for you to determine) and describe how the interference pattern changes. If you are feeling very energetic, learn how to make a gif animation on a web page, and use one to illustrate the variations. I don’t know how to do this, but I should be able to help you find help if you are so inclined.

There are a number of methods for visualizing a two-dimensional plot in ROOT. These include scatter plots, contour plots, lego plots, color plots, and box plots. For each exercise, make plots using at least three of these methods and decide which one is best, or which ones are best for seeing various features.