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How to order Feynman Rules for Fermions

  1. Dec 8, 2014 #1
    1. The problem statement, all variables and given/known data

    This is more of a general question-- as the title suggests I'm not too sure how to place the terms given by the Feynman rules for fermions (since they involve operators and spinors, the order does of course matter).

    I've been reading Peskin & Schroeder and the rules are simply stated, and we are prompted to e.g., place a spinor at each external leg contraction, but in precisely what order we are supposed to do so isn't clear to me.

    2. Relevant equations


    3. The attempt at a solution

    So for example I have a question on my homework; we must calculate something like [itex] e^{+}e^{-} \rightarrow e^{+}e^{-} [/itex] in the Yukawa potential ([itex] H_{int} = g\Psi \bar{\Psi} \phi [/itex]). To lowest order there are two diagrams, an S and a T channel, and when I write down the corresponding S-matrix element, what I would of course like to happen is for the order of the spinors to be identical for the two diagrams, so that they factor and I get a c-number in the middle (the sum of the two scalar propagators). This seems however too simple, so I have read Shredinki's QFT, and he motivates us to "follow the fermion line," starting from that line which points away from the vertex. What I interpreted this to mean is that, if e.g., one fermion line connects [itex] p_1 [/itex] and [itex] p_2 [/itex], then the spinors corresponding to these momenta will lie the same number of terms away from the propagator connecting them (if [itex] p_1 [/itex] lies on the far left, [itex] p_2 [/itex] lies on the far right. So in the end I end up getting something like

    [tex] iM = g^2 \left[ \bar{v}(p_2)u(p_1)P_1\bar{u}(k_1)v(k_2) + \bar{v}(p_2)v(k_2)P_2\bar{u}(k_1)u(p_1) \right], [/tex]

    where [itex] P_1 [/itex] and [itex] P_2 [/itex] are propagators. The important point is that the order of the spinors isn't identical in the two terms, and this is all I'd really like to be sure of since the resulting calculation of the amplitude is a bit more complicated.

    Thanks for any help!
     
  2. jcsd
  3. Dec 8, 2014 #2
    . Nvm

    I think I got it. The point is that you're supposed to contract the spinors along a single fermion line without ever crossing over the propagator. Then the contraction will be a direct contraction of spinor indices (the spinors will lie next to each other, not "the same number of terms away from the propagator"). The matrix element written above is correct, but I got it in a roundabout and ultimately erroneous way.

    Sorry and thanks
     
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