Quantum Diaries

Perhaps in your wanderings of physics papers, you’ve seen plots which look like this:

\( D^{+}\rightarrow K^{-} \pi^{+} \pi^{+}\)Dalitz Plot. Borrowed from Lectures by Brian Meadows, Cincinnati.

While yes, you may think that Easter has come early, this is actually an honest-to-goodness physics analysis technique. Developed by R.H. Dalitz in 1953, this plot illustrates visually the interference of the quantum mechanical amplitudes of the final state particles. Let’s take a step-by-step walk through of the plot.

The Setup

Dalitz plots were originally used to investigate a three body final state, for instance \( D^{+}\rightarrow K^{-} \pi^{+} \pi^{+}\). Taking this example, let’s imagine we’re in the \( D^{+}\) rest frame (it’s just sitting there), then the \( D^{+}\) decays. The decay products can go a variety of directions, so long as momentum is conserved.

The directions in which the particles fly and with what momentum will determine where we are in the plot. For reference, we can label the daughters as 1, 2 and 3, then assign them masses \( m_1, m_2\) and \( m_3 \), and momenta \( p_1, p_2\) and \(p_3\), respectively. Finally, let the \( D^{+} \) have mass M. It’s momentum is 0 since it’s just sitting there. With a bit of algebraic manipulation, and Einstein’s relation \(E^2=p^2+m^2\) (c=1, for simplicity of calculation), we can define a whole host of new variables, for instance \( m_{12}^2 = (E_1+E_2)^2-(p_1+p_2)^2\).

The Axes

Let’s take  \( m_{12}^2 = (E_1+E_2)^2-(p_1+p_2)^2\) as our guinea pig. Physically, we can think of this as combining particles 1 and 2 into a single particle, and then plot its effective invariant mass spectrum. This is quite similar to looking at the invariant dimuon mass squared of the Higgs searches. In this case, however, we then plot either \( m_{13}^2 \) or \(m_{23}^2\) on the remaining axis. Since all of the momenta and energies are related, picking either \( m_{13}^2 \) or \(m_{23}^2\) fully defines the system. This gives us all the ingredients we need for the plot!

The Boundary

PDG, Review of Kinematics

After setting up the axes above, we need to plot the actual figure. The boundary is completely described by energy and momentum conservation. For example, if you can ask “What is the minimum energy squared that particle 12 could have?” After a bit of consideration, you would say “why the addition of the two masses, then squared!” Likewise, the maximum energy it could have is the mass of the parent minus the mass of the other daughter, then squared. In this case, all of the momentum is then available to the \( m_{12}\) system. Repeating this process for all values of \( m_{12}^2 \) then gives the complete boundary of the Dalitz plot. Some special spots are shown in the PDG plot above. Forming the complete boundary is not necessarily a simple task, especially if the particles are indistinguishable. For the sake of explanation, we will stick to our simple example here.

The Innards

Finally, the bulk of the Dalitz plot is defined by interactions of the final state particles. If these particles did not interact, then we would expect a completely flat distribution along the inside of the plot. The fact that these particles do interfere is due to the quantum mechanical probability of the initial state transforming into the final state given the interaction potential of the system. The result is a vast array of structure and symmetries across the plot. For the example of  \(D^{+}\rightarrow K^{-} \pi^{+} \pi^{+}\), the result is shown above.  Each little dot is one event, and we can clearly see that there are places where the density is high (resonances, the so called “isobar model”), and places where there is almost no density at all (destructive interference).

The structures can be quite different depending on the spin of the resonance as well. For instance, the first plot shown below shows the resonance (where the boxes are bigger). This plot is actually Monte Carlo simulation for the process \( \pi^- p\rightarrow f_0 n\rightarrow\pi^0 \pi^0 n\), produced with a \(f_0\) mass of 0.4 GeV/c². Since the \(f_0\) is a scalar (spin 0), the resonance extends across the entire plot. In the second plot, the \(\rho(770)\) is produced in the decay \(D^{-}\rightarrow K^{-} \pi^{+} \pi^{-}\). This too is Monte Carlo. The fact that the \(\rho(770)\) is a vector (spin 1) is what produces the distinct shape shown below.  This simple example shows how one can identify the spin of a resonance by visually inspecting the Dalitz plot.

MC \(f_0\)Dalitz Plot. From Crystal Ball Collaboration : http://arxiv.org.proxy.libraries.uc.edu/abs/nucl-ex/0202007"

\(\rho(770)\) resonance in \(D^{-}\rightarrow K^{-} \pi^{+} \pi^{-}\) From lectures by Brain Meadows.

Now, there’s a lot more to Dalitz plot analysis that what I’ve presented here. There can be reflections across the plot and different resonances interfering with each other in quite complicated ways. For example, in the decay \(D^{-}\rightarrow K^{-} \pi^{+} \pi^{-}\), if we had a \(K^{*}_{0} (800)\) interfere with the \(f_0\), the Dalitz plot might look something like this:

\(K^{*}_{0} (800)\) interfering with \(f_0\) in decay \(D^{-}\rightarrow K^{-} \pi^{+} \pi^{-}\). From Brian Meadows.

The distinct shape, which looks to my eye a bit like a butterfly, is due to the phase difference between the two resonances.

So now you at least have a bit of an intro to the Dalitz plot, in this all too brief and quite simplified example.