Lets talk the standard model. The protons become hydrogen and while they make particle detection possible they are not used up in the process. Because the particles themselves cause radiation the surrounding material in the particle accelerator can become radioactive and is the recognized waste product at the end of the run life of the machine. Standard nuclear waste.

The protons used in the life of a particle accelerator is a very small quantity and I was attempting to find out if there are processes to capture and contain them.

(Alright it may not be standard model here but the mass of the proton is about 100 times more than the quarks that go into making a proton. It is just me but if there is space within the atom that can store that energy introduced when being accelerated then it is such a small quantity that space required means that high level storage is not an unreasonable option).