Send two identical particles into each other head-on at high speed. What comes out need not be two particles; many more can scatter in every direction.
Add up the masses of everything that comes out and the total exceeds the two masses that went in. The sum of the masses is not conserved in the collision.
The of the extra mass comes from the kinetic energy before the collision. What is conserved is the total energy, rest energy included.
Not all the energy brought in can go into the mass of new particles. If the whole system is still moving after the collision, kinetic energy remains to carry its momentum.
The frame in which the total momentum is is called the centre of mass frame. There the whole system is at rest, so all of its energy can go into the rest energy of new particles.
The upper row of the figure shows a particle striking a target at rest, where the whole keeps moving forward afterwards. The lower row shows a head-on collision, where the whole is at rest.
Send a particle of energy into an identical particle at rest. The centre of mass energy is , about when is much larger than .
Increase a hundredfold and the usable part grows only tenfold. Collide two particles of energy head-on, and the centre of mass energy is , growing in proportion to .
To reach GeV in the centre of mass with protons, a head-on collision needs GeV each. Against a proton at rest it would take GeV, about GeV.
When an electron meets its antiparticle, the positron, both disappear and two photons remain. Photons have no mass, so the sum of the masses becomes .
For an electron and positron at rest, each has a rest energy of MeV, and the two photons fly off in opposite directions with MeV each. Total energy and total momentum are the same before and after.
The reverse happens too: photons of enough energy colliding with each other produce an electron and a positron. The sum of the masses changes, but the total energy does not.
The rest energy of one proton is GeV. The LHC accelerates protons to GeV and collides them head-on, for a centre of mass energy of GeV, about times the proton's rest energy.
The Higgs boson has a mass of GeV, about protons' worth. It was found in in data from collisions at to GeV in the centre of mass.
A Higgs boson decays into other particles in about seconds, so it cannot be stored. Each time it is studied, it has to be made anew from the energy of a collision.