Particle production at angle relative to beam axis

Question

Why is it that, in general, the particles produced/emitted during a given particle collision tend to be produced/emitted at angles relative to the beam axis?

I can kind of see why this is true for collisions between composite particles, since then, using a classical analogy, as they collide the constituent particles will not necessarily experience a "head-on" collision - if they are not inline with the incoming particle then the force acting on them will do so along the normal to the contact point, and so the force of the collision will cause them to "fly off" at different angles (in this case I'm analogising the classical example of when a billiard ball hits another at an angle, causing the target ball to move off at an angle relative to the incoming ball), however I appreciate that this much this might be taking the classical analogy too far.

What I find more confusing is the case in which a lepton and an antilepton collide, annihilating to produce a particle-antiparticle pair. For example, suppose an incoming positron collides with a stationary electron, annihilating to produce a muon-antimuon. Why is it the case, in general, that the muon and antimuon will propagate outwards at angles relative to the beam axis?

Answer 1

I think you are saying this by looking at the schematic of the process given in various books. They take direction of final particle at arbitrary angle so that they can look at the distribution as a function of $\theta$. If they fix the angle in starting itself, say $\pi/3$, that means they are always looking at a particular angle. But by keeping $\theta$ as a variable, they have cross-section in terms of $\theta$. Now this quantity is more useful as you have theoretical value for all the angles instead of just one. Also, you generally want to calculate total number of final particles. For that you integrate over the whole range of $\theta$ i.e. (0,$\pi$).