Maximum Angle for Neutrino-Electron Scattering

[benjammin]

Hi all,

I'm working on a problem about neutrino-electron scattering $$\overline\nu_ee^-\to e^-\overline\nu_e.$$ The problem says to prove that the maximum angle of emission for the recoil electron relative to the neutrino beam is $$\sqrt{{2m\over E}},$$ but I have worked on this for hours and I don't understand how to proceed. The problem doesn't say that this is an elastic scattering case, but I've been assuming so...any suggestions?

Btw, m and E are the electron mass and energy in the formula for the maximum angle.

Thanks

 

[mfb]

It is elastic, as the particles after the reaction are the same as before, and they do not have any interal degrees of freedom which could get energy.

In which frame is the angle measured, and the angle between what and what? Which energy do the particles have in the system where the angle is measured?

 

[benjammin]

The problem doesn't specify a frame in which the angle is being measured, nor goes it give any initial conditions like the energy of the neutrino beam. I'm actually writing a solutions manual for a textbook that is in progress, so it could be the case that the problem is just not workable as written. I was just wondering if anyone here had any ideas.

The angle is the angle between the outgoing electron (which was stationary before the interaction), and the original neutrino beam direction. Basically just the angle at which the electron recoils with respect to the original neutrino beam.

 

[jtbell]

The kinematics should be the same as for Compton scattering, right?

Set up equations for conservation of energy and conservation of momentum (parallel and transverse to the beam direction), and see if you can come up with a formula for the electron scattering angle in terms of the incoming neutrino energy and either the outgoing neutrino energy or the neutrino scattering angle. I don't know which one would be easier to work with.

 

[AdrianTheRock]

One thing that struck me when I looked at the problem was that it seemed odd that E was the electron energy. On the face of it, it would seem to make more sense if E were the neutrino energy, and the electron was initially at rest.

My other thought was it's unusual to see the constraint applying to the actual angle itself. I'd have thought it more likely it would have applied to sin/cos/tan θ.

All of that said, I did have a quick go with it but couldn't get anything quite like the answer suggested, though it's decades since I really did much maths in anger! What I was trying to do was express the electron recoil angle in terms of one of the other kinematic variables, so I could then differentiate that expression with respect to that 'independent' variable and thus find the maximum by equating the derivative to zero, but couldn't get to a clean expression on which to do this.

 

[mfb]

Let's see where we can get some interesting "maximum angle" at all:


In the center of mass frame, both particles can have any outgoing direction. No interesting angle here.
Rest frame of the electron, incoming neutrino: Similar to compton scattering, the neutrino can have any outgoing angle and the electron should be restricted to 1/2 of the full solid angle (this should be checked with a calculation). No interesting angle here.
Rest frame of the neutrino, incoming electron: The electron has a small maximum deflection angle (shooting the neutrino in an angle of ~90° relative to the electron flight direction). But who works in neutrino rest frames?