Friday, 21 August 2020

quantum field theory - Peskin Schroeder $phi^4$ mass renormalization



In Peskin Schroeder, section 10.2, the contributions to $M^2(p^2)$ of order $\lambda^2$ are calculated. They respective Feynman diagrams are given: enter image description here


why is this diagram NOT included?


enter image description here



Answer



Just before their statement that the first non-zero corrections to the $\phi^4$-propagator are those you indicated, they compute the one-loop corrections, using on-shell renormalization conditions, and find out that they are zero:


The one-loop result


It seems that for the $\mathcal{O}(\lambda^2)$ (two-loop) corrections, one should indeed consider not only the diagram you pointed out but another one with a counterterm insertion, which cancels it due to the above one-loop result in the considered scheme. Schematically:


The full two-loop sum


Where the fourth diagram corresponds to the one you drew and the fifth term is the $\mathcal{O}(\lambda^2)$ contribution with the counterterm insertion.


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