Independent kinematic variables in elastic and deep inelastic scattering

Question

It is known that we have one independent kinematic and two independent kinematic variables in elastic and deep inelastic scattering. For example in lab frame, experimentally, we generally take scattering angle or energy of scattered electron as independent kinematic variable in elastic scattering while in DIS, we take both of them. (Other way of saying this is that we have single differential cross section in elastic and double differential cross section in DIS.) So is there any intuitive reason why this happens of is it just a mathematical fact?

Answer 1

The starting place is that in both cases you must conserve 4-momentum (and that mass couples the energy and 3-momentum of a particle through $(mc^2)^2 \equiv E^2 - (pc)^2$).

In elastic scattering there are two outgoing particles which have a total of 8 components of 4-momentum and thus 8 degrees of freedom. The conservation rule constrains 4 of them and the fact that each particle has a definite mass constrains two more. So in reality we have two independent variables, but one of these is the angle that the scattering plane makes with some reference direction (i.e. the polar angle in spherical coordinates where the beam direction sets the zero of azimuth); it doesn't affect the physics because the problem has rotational symmetry around the original beam direction. So only one independent parameter that we care about.

In DIS, we can collect all the varied products together and treat them as a system with a single 4-momentum for the purposes of this kind of analysis. So we start with 8 degrees of freedom and remove 4 for conservation again, but this time the products don't have a fixed mass, so we only get one restriction from the mass of the scattered beam particle, and one degree of freedom disregarded due to symmetry. Which leaves two physically interesting parameters for the event.

So the physical reason is that that mass is not a conserved quantity and different DIS events can have different masses, while all elastic events share the same pair of masses.