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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:enter image description here 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).

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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.

Extract from the PDG review on kinematics: Dalitz plot

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:

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  • $\begingroup$ 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 $(M-m_3)^2$. $\endgroup$ Commented Jul 4, 2021 at 14:27
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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.

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  • $\begingroup$ 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 $m^2_{13}$? $\endgroup$ Commented Jul 4, 2021 at 14:26
  • $\begingroup$ Only if $M$ is known - which in an actual case it would be. $\endgroup$ Commented Jul 4, 2021 at 16:35

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