How Do You Calculate Cross Sections in Electron-Neutrino Scattering?

[unscientific]

Homework Statement

[/B]
(a) Find the ratio of cross sections.
(b) Find the cross section for electron-neutrino scattering by first writing down relevant factors.

Homework Equations

The Attempt at a Solution

Part (a)[/B]
These represent the neutral current scattering for the muon-neutrino and neutral/charged scattering for electron-neutrino. Feynman diagrams are given by

Given that there are 2 possibilities for the electronic case, I say $R = 2$?

Part (b)

Propagator factor is given by $\frac{1}{P \cdot P - m_w^2}$ which in the zero-momentum frame is $\approx \frac{1}{m_w^2}$.
There are two vertices, so another factor of $g_w^2$.
Thus amplitude is $\frac{g_w^2}{m_w^2}$.
By fermi's golden rule, $\Gamma = 2\pi |M_{fi}|^2 \frac{dN}{dE_0}$.
Cross section is $d\sigma = \frac{\Gamma}{v_e}$.

How do I proceed?

 

[thierrykauf]

Correct me if I'm wrong but it seems the course asks you to do things you haven't seen in class. If that's the case, these lecture notes might help http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf
It's been a while since I've computed cross-sections. If I see something better I'll let you know.

 

[unscientific]

Correct me if I'm wrong but it seems the course asks you to do things you haven't seen in class. If that's the case, these lecture notes might help http://isites.harvard.edu/fs/docs/icb.topic521209.files/QFT-Schwartz.pdf
It's been a while since I've computed cross-sections. If I see something better I'll let you know.

I'm looking for a hand-wavy approach in the sense we avoid explicitly calculating the feynman probabilities.

I think the density of states is something like: $\frac{dN}{dE_0} = \frac{dN_e}{dp_e} \frac{dp_e}{dE_0} = \frac{1}{(2\pi)^6} p_e^2 dp_e \frac{dp_e}{dE_0}$. How do I proceed?

 

[unscientific]

bumpp

 

[thierrykauf]

Sorry I've been busy! Didn't find time to reply more. I know better what kind of answer is needed. I'll try to post later today.

 

[unscientific]

bumpp

 

[thierrykauf]

As I remember you integrate over angle but not over momenta at tree level.

 

[unscientific]

As I remember you integrate over angle but not over momenta at tree level.

So How do I find the cross section at tree level feynman diagrams?

 

[thierrykauf]

For each tree diagram you have a coupling constant at each vertex, a delta function that says momentum is conserved so inner momentum, that of the Z or W is fixed, because in and out particles are on-shell. So the integration over d3p, 3d momentum. becomes integral over solid angle omega. Let me know if this helps. http://www.iop.vast.ac.vn/theor/conferences/vsop/18/files/QFT-6.pdf