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Determining Amplitude queries

  1. Mar 8, 2015 #1
    Hi,
    I have a question regarding pair production, regarding the amplitude that I am trying to understand.

    I have attached a photograph of the feynmann diagram, which I believe to be correct - although I don't like the vertex/propogator combination as shown below: (I have split up integral so it fits onto more than 1 line)



    [itex]-iM = \int u(p_{3},s_{3}) (-ie\gamma_{\mu}) \epsilon(k_{1},\lambda_{1}) [/itex]

    [itex](i\frac{q_{\mu}\gamma^{mu}+m}{q^{2}-m^{2}}) [/itex]

    [itex]\bar{u}(p_{4},s_{4}) (-ie\gamma_{\mu}) \epsilon(k_{2},\lambda_{2}) (2\pi)^{4} \delta^{4}(q-p_{3}-{k1}) (2\pi)^{4} \delta^{4}(q-p_{3}-{k1}) \frac{d^{4}q}{(2\pi)^4}[/itex]

    (p = mmt, s = spin, k = 4-mmt, lambda = polorisation, integrated over all mmt space q.)

    My questions are:
    - The bit I don't like is the fact the \mu and \nu covariants don't have contravariant partners (just \gamma_{\mu} \gamma_{\nu}). If the propagator was a photon, these would nicely have partners. Isn't the idea it should be invariant, so isn't this an issue?

    -Labels, I am doing the 1, 2, 3, 4 based on the order it happens in (this makes sense given time axis goes horizontally) - Are these correct?

    - When I integrate this, will I get two terms, e.g. one for when [itex]q = p_{3} + p_{1}[/itex] - I guess these just add to give an overall amplitude?

    -On notation, should it be u(p_{3}, s_{3}) or [itex]\bar{u}(p_{3}, s_{3})[/itex] - i.e. Am I saying, oh it's a positron, so I should make that known, or do I follow the feynmann digram and say it's an electron going backwards in time. I'm sure the former is true.

    Many thanks for you help
     

    Attached Files:

    Last edited: Mar 8, 2015
  2. jcsd
  3. Mar 10, 2015 #2

    ChrisVer

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    Gold Member

    What are the [itex]\epsilon[/itex]s you have written?

    I don't get this... also are you sure about your delta functions? you have given the same expression for both, while (I think) you wanted to apply the conservation of energy/momentum in each vertex... eg one of the deltas should have k2 and p4 in the argument.

    add up?

    The 4spinors are [itex]u,v[/itex] the [itex]u[/itex] is for particles and [itex]v[/itex] for antiparticles.
     
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