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Irreducible graphs and srednicki's book

  1. Feb 7, 2014 #1
    Srednicki's QFT book uses scalar [tex]\phi^3 [/tex] as an example of a QFT. To make a calculation to all orders, Srednicki claims (chpt 19) you calculate all 1P1 graphs for 2 external lines, giving the self-energy. He then says you calculate all the 1p1 graphs for 3 external lines, giving the 3-point vertex function. Then you calculate n>3 vertex functions by drawing 1p1 graphs, but using the 3-point vertex for vertices and exact propagators for propagators. Then any amplitude can be calculated by tree level graphs using these n-point vertices and exact propagators.

    I think this is mostly correct, but doesn't Srednicki need to say that you have to calculate the 3-point vertex function using the exact propagator? If not, you'd be missing a lot of diagrams that can contribute to a scattering process.
  2. jcsd
  3. Feb 7, 2014 #2


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    No, because the sum of all 1PI diagrams with 3 external lines includes the loop corrections to the internal propagators.
  4. Feb 8, 2014 #3
    I see. So in ø4, the sum of 1PI with 4 external lines already includes loop corrections to internal propagators. Or more generally, in øn, saying that the n-point vertex is the sum of 1PI diagrams already takes into account loop corrections to internal propagators.

    I'm a little confused about the meaning of the vertex function and 1PI with regards to the 2-point function however. The self-energy is the 1PI diagram, but the 2-point vertex function is actually defined as the full 2-point Green's function, divided by the external lines (each represented by a full 2-point Green's function), so the renormalized 2-point vertex is:

    1/ΔR ΔR 1/ΔR=1/ΔR =p2-mR2-∑R(p2R)

    whereas if you just take


    If you keep careful track of i's, then you get the same result, that the two expressions differ in the sign of the renormalized self energy ∑R.

    It would seem to me that the former is not 1PI, but the latter is.

    So is it safe to say that the 2-point vertex (the former expression) is not 1PI? Only the 3-point vertex you can say is constructed from 1PI? I mean, you took ΔR, which is not 1PI (only ∑R is), and you cut two of the ends, so everything in between remains the same, and everything in between wasn't 1PI.
  5. Feb 8, 2014 #4


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    Your 2nd expression has no meaning.

    The self-energy ∑ is the sum of 1PI diagrams with two external lines (and no propagators for those lines). The exact propagator is then given by a geometric series that sums up to Δ as given by your first expression.
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