How to Calculate Cross Section for Scalar-Neutrino Interaction?

[ryanwilk]

Homework Statement

Hi. I'm trying to calculate the cross section for this process:

where \phi is a massive neutral scalar, N is a massive Majorana neutrino and \nu_L is the normal SM neutrino.

Homework Equations

N/A

The Attempt at a Solution

Apparently the answer should be something like: \frac{g^4}{4 \pi} \frac{m_N^2}{ {(m_\phi^2+m_N^2)}^2 } \>,
where g is the coupling constant.

There’s a similar case in http://zippy.physics.niu.edu/spinors/DHMspinors.pdf, page 45. However, I’m not sure which part of Eq. (4.5.13) corresponds to the diagram above and I don’t know how to deal with the \sigma \cdot k term (particularly when calculating the invariant amplitude squared)?

Any help would be appreciated,
Thanks!

 

[ryanwilk]

So, I've attempted to do the calculation. Does this look correct? (there are lots of steps I'm unsure about):