It’s pretty intuitive in the context of confinement theory. Intuitively we imagine something like a rubber band, as you add energy in the form of tension the material passes a threshold and as you say snaps. For confinement however as you add energy to the system it doesn’t snap, you simply reach the moment when you’ve added enough energy to the system to create a new particle pair that are also confined.

This is the explanation for why when we collide beams of protons at near c they don’t produce a new higher energy particle, but a massive shower of secondary and tertiary particles like pions and kaons as a result of the decay chain from initial pair production.

Forgive me for being a layman, but where do these new gluons "go"? It seems to me that pretty much all the energy in the bond between to particles is between those two particles.

But then when the energy becomes great enough and it snaps into a new pair of gluons, do those new gluons go anywhere or do they stay in between the original pair of particles. Do the original particles shoot off into the distance because the force holding them together got converted into some new particles?

Is there a link where I could read more? At these super small scales all my intuitions break down so I'd like something more mathematical to get a grasp on things :)