# Homework Help: What is the cross section?

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1. May 11, 2015

### unscientific

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

Part (a)

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?

2. May 11, 2015

### thierrykauf

3. May 12, 2015

### unscientific

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?

4. May 14, 2015

### unscientific

bumpp

5. May 14, 2015

### 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.

6. May 16, 2015

### unscientific

bumpp

7. May 17, 2015

### thierrykauf

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

8. May 17, 2015

### unscientific

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

9. May 17, 2015

### 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

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