Are the detectors in a typical particle accelerator experiment, either in Fermilab, or now in LHC, sensitive to negative energy particles?
How would a negative energy particle, (say, a negative energy $\gamma$-ray photon) be found in the data collected by those detectors? would it have to be detected by the missing momentum approach (just like neutrinos were found)?
If you wanted to look for traces of negative-energy particles in the on-shell, final, outgoing, asymptotic states in existing collision data, what strategy would you choose, that could be applied on existing HEP datasets?
Regarding Anna's answer about invariant mass measurement, I have a doubt, which is related to a few discussions by experimentalists regarding neutrinos possibly being tachyons due to the fact that the error bars of the squared-mass observable leaked a bit into negative territory, I bring that up because if squared-invariant-mass is the actual observable, instead of invariant-mass, then I'm not sure how that can be sensitive to the sign of the invariant mass.
But moving on to light particles (like an hypothetical negative-energy γ-ray, which I used as an example) that like neutrinos, have energy dispersion curves that are experimentally indistinguishable from $|E|=c|P|$. Does the situation change significantly when the invariant-mass is zero or nearly zero, but the time-like component of the four-momentum could be both very big, and pointing backwards? can we measure or at least infer such signed components?
Note: If someone has experience with Geant4, I'll appreciate some brief remarks about how to modify a gamma particle definition to simulate a negative energy photon hitting a detector volume, and/or how to add a reaction process that would produce it