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My impression is that particle colliders produce particles. Then the detectors measure the amount of the scattered flux of particles in a given direction and compare that with theoretical scattering crosssections.

$\bullet$ Is it possible to measure the particle mass, spin and charge independently from the information of scattering cross-section at a given energy scale? If yes, how is that achieved?

$\bullet$ Is it possible to measure the energy $E$ and momentum vector $|\textbf{p}|$ of a scattered particle (or a particle produced as a decay product) in the detector? If yes, using then the relativistic dispersion relation $E^2=|\textbf{p}|^2c^2+m^2c^4$, the mass of the scattered particles can be inferred.

$\bullet$ Is it possible to measure the spin and charge in a similar independent manner which doesn't require any information of the scattering cross-section?

In particular, I'm interested in the principle of such measurements in modern day particle collider experiments such as LHC. We know one would like to know the Higgs mass, its charge and its spin.

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Particle detectors only measure the scattered flux of particles in a given direction. How do they measure the particle mass, spin or charge independent from the information of scattering cross section at a given energy scale?

Particle detectors measure a lot of parameters of the particle tracks.

Tracking detectors are in a magnetic field and from the curvature measure the momentum.

Calorimeters measure the energy of the track impinging on them, so even if neutral the momentum is measured.

Calorimeters can separate hadronic interactions from electromagnetic ones

Muon detectors are especially designed to measure the momentum of weakly interacting charged particles.

All these put together give a lot more information than just a crossection for scattering at an angle. Energy and momentum conservation are used to determine missing mass and missing energy, etc. etc.

Is it possible to measure the energy and momentum of a scattered particle or decay products in the detector and from that using relativistic dispersion relation E2=p2c2+m2c4 , infer the mass from that?

Sure , in speciaized experiments one can fit for specific interactions. In bubble chambers the ionisation of the tracks gave information on the mass of the particle, etc.etc.

Is the possible to measure the spin and charge in a similar independent manner which doesn't require any information of the scattering cross-section? Can one give example of such a modern day particle experiment?

See my answer here for bubble chamber examples. The charge is measured in the tracking detectors of the LHC, for example in this picture, because strong magnetic fields exist and one uses the mv^2/r=Bqv equation. It takes a lot of programming for the huge number of tracks at the LHC.

Spin determinations in particle physics need statistics of angular distributions, and decay products with known spins to get a definite attribution of spin, for example the spin of the recently found Higgs boson in Atlas (fig 15c) . It is not a simple project after all, but it is doable.

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