In hadron colliders, like the LHC, one of the most important tools used when looking for new particles, specifically “dark matter” candidates, is conservation of momentum. Indeed, this simple rule which Newton made rigorous when developing his version of mechanics holds even down to the level of individual fundamental particles so far as we can tell.
Most modern colliders at the energy frontier have focused on smashing together protons with protons or protons with antiprotons. The alternatives are of course smashing together heavier things, like gold, or smashing together lighter things, like electrons. Protons are a great candidate for a few reasons. They have a long, probably infinite, lifetime, they are heavier than electrons, which means it is easier to ramp them up to higher energies, but they are also not made of too many pieces, like heavier atoms, so that the total energy of the particle is split between too many constituents (which means lower energy collisions).
However, protons are still composite objects. When pieces of the protons smash into each other, no matter how well we tune the energies of the individual protons, there is vanishingly small probability that the pieces will have precisely the same the momentum. A Newtonian picture is sufficient here. If we imagine the proton to be a ball filled with other balls called quarks and gluons, then fixing the net momentum of larger ball only tells us about the sum of the momenta of all the individual pieces, and nothing can be said about the momentum of an individual quark or gluon.
When we throw in relativity, the mass-energy relation tells us that no individual quark or gluon can have so much momentum that its energy exceed the mass of the proton. When we throw in quantum mechanics, we have to give up on the idea of balls which are well defined objects that have well defined momenta at every instant of time all together. The result of this is that we end up with what’s called a parton distribution function, and the upshot is that two subatomic particles that collide will generally have some momentum imbalance in the lab frame. This means that any particle made in a proton-proton collision will likely have some nonzero momentum in the lab frame. In electron-positron colliders, like LEP, this was not the case, the beam energies could be tuned so precisely that particles produced would normally be at rest in the lab frame.
However, one thing we can say about the newly produced particles is that they will have almost no transverse momentum, that is momentum in the 2 dimension plane tangent to the beam motion. The most obvious reason for this is that the protons are in a tight beam, which means as a whole, they can not be moving up and down all that much or else they would escape that beam. But this is not enough (it is enough to tell us there will be overall transverse momentum conservation, but not enough to say any new particle made alone will have 0 transverse momentum), remember the protons are composite objects, fixing their net transverse momentum does not tell us, from a Newtonian standpoint, that the individual pieces don’t have high transverse momentum, but thankfully, conservation of energy does.
If we were in the rest frame of one of the traveling protons, the momentum of any individual piece of the proton is bounded by the mass of the proton, about 1 GeV. Since you can get from the lab frame to the rest frame of a proton in the beam by a boost along the direction of the beam, the transverse momentum of a parton in the rest frame is equal to the transverse momentum in the lab frame, the boost is along an orthogonal axis. This means that we can say with absolute certainty that no constituent of the proton can have transverse momentum larger than the rest mass of the proton. This fact, that any object produced in a collision should not have almost any transverse momentum is a huge tool in searching for new ‘dark’ particles.
Dark matter is matter which does not interact electromagnetically, or at least does so extremely weakly, which is why we can’t see it with telescopes. If we make dark matter in a collider, there’s a good chance that we won’t be able to see it in our detectors either, electromagnetically neutral particles are notoriously hard to catch. The main tool we have to search for dark particles is the absence of detection. If the dark particle made from LHC collisions is stable and it is produced alone, then it will be very tough to see in the detector since it probably just escape down the beam line unnoticed.
However, if the particle is produced with a charged particle that we detect (because it had some transverse momentum), or is unstable so that it decays to a standard model particle and another dark particle(which also have transverse momentum), then we can find it by looking for imbalance in transverse momentum in our collision. If the transverse momentum in any collision does not add up to nearly zero, then there’s a good chance we’ve made a particle that escaped the detector unnoticed. Of course, sometimes we ‘mismeasure’ the momentum of things, and the production of neutrinos also leads to momentum imbalance as well, these are the challenges we have to work around. But nevertheless, a lot of the game in looking for dark particles comes from looking for momentum imbalance in collisions where there should not be any.
That’s all we can say about the initial momenta of new particles produced at the hadron colliders. Next time I will talk about what can be said about the momentum of the daughter particles in a two body decay.