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What is the cross section?

  1. May 11, 2015 #1
    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.
    2011_B4_Q8.png

    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
    2011_B4_Q8_2.png

    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. jcsd
  3. May 11, 2015 #2
  4. May 12, 2015 #3
    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?
     
  5. May 14, 2015 #4
  6. May 14, 2015 #5
    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.
     
  7. May 16, 2015 #6
  8. May 17, 2015 #7
    As I remember you integrate over angle but not over momenta at tree level.
     
  9. May 17, 2015 #8
    So How do I find the cross section at tree level feynman diagrams?
     
  10. May 17, 2015 #9
    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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