Consider a $2\rightarrow2$ scattering process in $\phi^4$-theory. On p. 326 in the book of Peskin & Schroeder, they consider the 3 1-loop corrections in the parenthesis:
My question is: Why don't they include below self-loop diagram?
Is this 0? Why?
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asked Feb 28, 2019 at 15:41
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2 Answers
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Peskin & Schroeder (P&S) on p. 326 are discussing a connected, amputated 4-pt amplitude$^1$ ${\cal M}_4$. It is clear from various places in the P&S textbook [see e.g. eq. (10.21)] that ${\cal M}_4$ is an amputated amplitude. Now OP's self-loop diagram can be understood as an amputated diagram, so OP's question is still in principle a valid question.
However, P&S explain on p. 113-114 that such diagrams should be cut away. This is e.g. to avoid over-counting when we attach connected propagators $G_c$ to the external legs of the amputated amplitude ${\cal M}_4$. [Technically, self-loop diagrams in principle contribute to the connected propagator $G_c$, but are typically cancelled via renormalization conditions on the self-energy, see e.g. eqs. (10.28) & (10.29) on p. 328.]
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$^1$ By the way, the last explicit diagram in P&S is not a 4-loop diagram: It denotes an interaction counterterm, cf. fig. 10.3 on p. 325 :-)
answered Jul 6, 2019 at 17:28
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if you separate the new diagram you added to 2 diagram it means:
- diagram of the first order
- diagram with a 1 particle that scatter with itself and continue in its trajectory
the probability for a particle to continue its trajectory when it only "interacts" with itself is 1 due to conservation of momentum.
therefore the probability is the same and the physical meaning is the same as the first order diagram
