**At the Large Hadron Collider**

Many particle physics experiments are carried on one of two types of accelerator facilities, (a) colliders in which equal energy particle collide head on, i.e. their momentum vectors point in opposite direction, (b) “fixed target facilities”, where an energetic beam particle collide with a stationary target particle. The two types of facilities have different advantages and

disadvantages. We’ll consider one important difference in this problem.

Suppose our goal is to collide two particles of same mass m to create a more massive particle of mass $M$ $(M > m)$ – in other words the original particles of mass m annihilate with each other and create a new particle of mass $M$. The key question regarding how much expensive our particle accelerator will be “How much energy does our particle beam(s) need to have?’’

(a)

**(Head on)**What is the minimum relativistic total energy of each particle of mass $m$ must be to create the new particle with mass $M$? (Express your answer in terms of $m$ and $M$)

(b)

**(Fixed Particle)**What is the minimum total energy that our incident particle of mass $m$ must have if it is to collide with another stationary particle of mass $m$ and create the single particle with mass $M$? Express your answer in terms of $m$ and $M$. Hint: Don’t forget that momentum must be conserved!

(c) What is the ratio of the fixed target beam and the sum of the energies of two target beams in head on collisions? Which facility do you think should be less expensive?

What is this ratio when $m$ is the mass of the proton $(~ 1 GeV/c2)$ and $M$ is the mass of the Higgs Boson $(~125 GeV/c2)$