Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

Suppose that a particle z is expected to be produced in a reaction x + y → z. (For example, the Z° particle could be produced in the reaction .) (a) By far, the simplest experimental arrangement is to fire the x particle at a stationary y target (or vice versa). Prove that in this case the required incident total energy (E = Ex + myc2) is equal to

[Hint: Use conservation of energy and momentum, and apply the Pythagorean relation (2.23) to particles x and z.] (b) A much more efficient arrangement (although much harder experimentally) is to use colliding beams, in which the x and y collide head-on with equal but opposite momenta. Prove that in this case the required total energy (E = Ex + Ey) is

 E = mzc2 (colliding beams)

(c) Suppose that mx = my (as is the case in pp or collisions). Show that the particle z that can be produced has

for a stationary target, but

 mzc2 = E (colliding beams)

for colliding beams. If E = 400 GeV and mx and my equal the proton mass, what is the mass mz that can be produced in each case? (d) If mz = 90 GeV/c2 (about the Z° mass) and mx and my equal the proton mass, what is the energy, E, needed to produce the z particle in each case?