# What do the points on a Dalitz plot represent

I have read that a Dalitz plot is nothing but the plot of $$m_{12}^2$$ vs $$m_{23}^2$$, and that the dots correspond to "events". However, this doesn't really tell me anything. My question can boil down to exactly what phase space variable the dots represent. I am currently trying to make a Dalitz plot for the three body decay $$K_{long}\to 3\pi^0$$. Now, I am able to get the boundary for the Dalitz plot by plotting the constraint

$$\cos(\theta_{12})=\frac{\frac{\left({m_\ell}^2+{m_\pi}^2-{m_{23}^2}\right) \left(-2 {m_\pi}^2+{m_{12}^2}+{m_{23}^2}\right)}{2 {m_\ell}^2}+2 {m_\pi}^2-{m_{12}^2}}{\sqrt{\frac{\left({m_\ell}^2+{m_\pi}^2-{m_{23}^2}\right)^2}{4 {m_\ell}^2}-{m_\pi}^2} \sqrt{\frac{\left(-2 {m_\pi}^2+{m_{12}^2}+{m_{23}^2}\right)^2}{{m_\ell}^2}-4 {m_\pi}^2}}$$

in the $$m_{12}^2$$ $$m_{23}^2$$ plane. However, I'm in the dark as to how to populate the inner region. My goal is to be able to answer the following question. Given the constraint above, how do I, for a given ($$m_{12}^2$$ ,$$m_{23}^2$$) know whether or not that corresponds to placing a dot there or not.

• you need data (or an MC). Then you plot the data. – JEB Sep 21 '18 at 2:08
• So you need to run a simulation measuring all that? You're plotting some other phase space variable like $m_{13}^2$ or something? – InertialObserver Sep 21 '18 at 2:14
• try this cds.cern.ch/record/2014715/plots for data – anna v Sep 21 '18 at 17:36
• Hi @InertialObserver, I had a go at answering the question, but I suspect it's an XY question and you might actually be trying to find out how to model decay amplitudes. – dukwon Sep 22 '18 at 19:44