Srednicki — Quantum Field Theory · Problem 9.2

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · phi4-theory-feynman-diagrams

Problem

Consider a real scalar field with L=L0+L1\mathcal{L} = \mathcal{L}_0 + \mathcal{L}_1, where L0=−12∂μφ∂μφ−12m2φ2,L1=−124Zλλφ4+Lct,Lct=−12(Zφ−1)∂μφ∂μφ−12(Zm−1)m2φ2.\begin{aligned} \mathcal{L}_0 &= -\tfrac{1}{2}\partial^\mu\varphi\partial_\mu\varphi - \tfrac{1}{2}m^2\varphi^2 , \\ \mathcal{L}_1 &= -\tfrac{1}{24}Z_\lambda\lambda\varphi^4 + \mathcal{L}_{\mathrm{ct}} , \\ \mathcal{L}_{\mathrm{ct}} &= -\tfrac{1}{2}(Z_\varphi-1)\partial^\mu\varphi\partial_\mu\varphi - \tfrac{1}{2}(Z_m-1)m^2\varphi^2 . \end{aligned} a) What kind of vertex appears in the diagrams for this theory (that is, how many line segments does it join?), and what is the associated vertex factor? b) Ignoring the counterterms, draw all the connected diagrams with 1≤E≤41 \leq E \leq 4 and 0≤V≤20 \leq V \leq 2, and find their symmetry factors. c) Explain why we did not have to include a counterterm linear in φ\varphi to cancel tadpoles.
Intuitive explanation

The core concept here is renormalization of a φ4\varphi^4 theory: you need to understand how the interaction vertex structure emerges from the Lagrangian, how Feynman diagrams encode the perturbative expansion, and why symmetries constrain which counterterms are necessary. The key insight is that the ZZ-factors are designed to cancel divergences order-by-order, and the absence of certain counterterms reflects underlying symmetries—in this case, the theory's invariance under φ→−φ\varphi \to -\varphi forbids odd powers of the field.

Intuitive explanation

The core concept here is renormalization of a φ4\varphi^4 theory: you need to understand how the interaction vertex structure emerges from the Lagrangian, how Feynman diagrams encode the perturbative expansion, and why symmetries constrain which counterterms are necessary. The key insight is that the ZZ-factors are designed to cancel divergences order-by-order, and the absence of certain counterterms reflects underlying symmetries—in this case, the theory's invariance under φ→−φ\varphi \to -\varphi forbids odd powers of the field.

Related theory

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