# Dalitz Plot Analysis and Kinematics

I have trouble understanding the fundamental properties of Dalitz plots. For example the Dalitz plotz for the J/Psi (which is treated as unknown) to 3pion decay: I.e. we now have the invariant masses $$m_{12}$$ and $$m_{23}$$: How exactly do I determine the third invariant mass $$m_{13}$$? This should be possible with the "trivial" kinematics, but I don't seem to grasp this, even though I expect this to be very easy. Additionally, I know this relation:

$$m_{12}^{2}+m_{23}^{2}+m_{13}^{2}=M^{2}+m_{1}^{2}+m_{2}^{2}+m_{3}^{2}$$

But here are two unknown, if one does not know the mass of the decay/initial particle (which is the case for our hypothetical problem).

The $$J/\psi$$ cannot be a real unknown here. In a decay $$A \to \pi^+\pi^0\pi^-$$ (where $$A$$ is an unknown particle), the maximal allowed value of the two-body invariant mass $$m(\pi^+\pi^0)$$ is simply $$m(A) - m(\pi)$$. This $$m(\pi^+\pi^0)$$ is the square root of the max value of the X axis of your Dalitz plot, ~8.6 GeV$$^2$$, which gives you directly that $$m(A) = 0.14+2.93 = 3.07$$ GeV, which, within the resolution of whichever detector produced that plot, is compatible with the mass of the $$J/\psi$$ meson. There is no other mesons known at this mass value, while baryons cannot decay to three pions.

I attach below a useful sketch from the PDG review on kinematics. As one can see, the min/max values at X/Y axes of the Dalitz plot are fully determined by the masses of the parent and children particles.

If there were no intermediate resonances, the Dalitz plot would have events distributed uniformly. In your example, there are enhanced populations which are signatures of intermediate resonances: the vertical population is due to a resonance decaying to $$\pi^+\pi^0$$ and $$m^2 \approx 0.5$$ GeV$$^2$$ (which is compatible with the $$\rho^+$$ resonance); the horizontal population is due to $$\rho^- \to \pi^-\pi^0$$ while the diagonal population is due to some resonance decaying to $$\pi^+\pi^-$$ (whose mass would be easier to deduce if they provided a plot with $$m(\pi^+\pi^-)$$ on one axis). This means, the three-body decay $$J/\psi \to \pi^+\pi^0\pi^-$$ is in fact dominated by a cascade of two-body decays of the kind $$J/\psi \to \rho \pi$$ with $$\rho \to \pi\pi$$.

Some useful references:

• This is a very nice answer and made it more clear to me. Funnily enough, I even saw and used this picture. But somehow I completely missed the right part with $(M-m_3)^2$. Commented Jul 4, 2021 at 14:27

Actually you do know the mass of the $$J/\psi$$ as it's a large spike on the distribution of the combined mass $$M(\pi^++\pi^-+\pi^0)$$ or whatever decay channel you are looking at. That's what you see first: now you're asking questions about whether this decay occurs through other 2-body resonances.

OK but suppose you do ignore that: what you actually do is take the measured energies and momenta of the 3 particles and combine them pairwise. $$(E_1,\vec p_1),(E_2,\vec p_2)$$ and $$(E_3,\vec p_3)$$ are what you start with. From them you get $$m_{13}^2=(E_1+E_3)^2 - (\vec p_1+\vec p_3)^2$$ etc.

• That does make sense and I knew you can theoretically calculate this via Energy and momentum (4-momentum vectors so to speak). But there is no way - if you only have the sketch and no additional information about $E_i$ or $p_i$ - to get to $m^2_{13}$? Commented Jul 4, 2021 at 14:26
• Only if $M$ is known - which in an actual case it would be. Commented Jul 4, 2021 at 16:35