J/ψ discovery:

Ting's experiment at Brookhaven with p+Be ("J" particle):
How do you know the opening angle needed for your spectrometer, given that you don't know the mass of the resonance?
The reasoning is that the most probable value for the momentum of the resonance (in proton-nucleon collisions) is 0, ie produced at rest.
If you consider e+e- pairs produced at 90 degrees wrt the boost direction, P_x=0, P_y=M/2, E=M/2. Applying the boost, P'_x=βγM/2, so tg θ=1/(βγ)=m_p/P_p, which does not depend on M.
The spectrometer has a vertical angle and a horizontal opening; this is to have independent determinations for p and θ, simplifying the analysis. The vertical magnetic fields measure p.
Target: the several Berillium planes were spaced by ~7cm, and the magnetic fields are so intense that the particle produced by one plane is magnetically deviated before it traverses other planes (otherwise it would be absorbed). This way, it is also easier to identify pairs with a common origin.
Check on accidental combinations: signal is proportional to the beam intensity, while accidental combinations are proportional to the square of the beam intensity; so, variating the intensity one can estimate this background. For the same reason they variated also the target thickness (doubling the thickness, the yield increased by a factor of 2 and not 4).
Variating the target thickness also gave the possibility to rule out the possibility of a two step process: p+N -> π+... followed by π+N -> e+e- + ... (it would increase quadratically with the thickness, since N enters both in the first and second steps, while a one-step process would increase linearly, as it was observed).

quantum numbers
How do you know the J, P, C quantum numbers? From the interference with the photon (Figure 1b and 2).
But how do you know that it is a pure state? (*) It it were not, you would have a mixture of different P's, so the state would not be P-invariant, so you would have a F-B asymmetry in the decay products (Figure 3).

(*) For example, it had to be checked if there were a component from the Z; in that case, it would have a 1^+- component in addition to the standard 1^-- of the photon.