An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 10.2

Systematics of Renormalization — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · yukawa-theory-renormalization

Problem

10.2 Renormalization of Yukawa theory. Consider the pseudoscalar Yukawa Lagrangian, L=12(∂μϕ)2−12m2ϕ2+ψˉ(i∂ ⁣ ⁣ ⁣/−M)ψ−igψˉγ5ψϕ,\mathcal{L} = \frac{1}{2}(\partial_\mu \phi)^2 - \frac{1}{2}m^2\phi^2 + \bar{\psi}(i\partial\!\!\!/ - M)\psi - ig\bar{\psi}\gamma^5\psi\phi, where ϕ\phi is a real scalar field and ψ\psi is a Dirac fermion. Notice that this Lagrangian is invariant under the parity transformation ψ(t,x)→γ0ψ(t,−x)\psi(t,\mathbf{x}) \to \gamma^0\psi(t,-\mathbf{x}), ϕ(t,x)→−ϕ(t,−x)\phi(t,\mathbf{x}) \to -\phi(t,-\mathbf{x}), in which the field ϕ\phi carries odd parity. (a) Determine the superficially divergent amplitudes and work out the Feynman rules for renormalized perturbation theory for this Lagrangian. Include all necessary counterterm vertices. Show that the theory contains a superficially divergent 4ϕ4\phi amplitude. This means that the theory cannot be renormalized unless one includes a scalar self-interaction, δL=λ4!ϕ4,\delta\mathcal{L} = \frac{\lambda}{4!}\phi^4, and a counterterm of the same form. It is of course possible to set the renormalized value of this coupling to zero, but that is not a natural choice, since the counterterm will still be nonzero. Are any further interactions required? (b) Compute the divergent part (the pole as d→4d \to 4) of each counterterm, to the one-loop order of perturbation theory, implementing a sufficient set of renormalization conditions. You need not worry about finite parts of the counterterms. Since the divergent parts must have a fixed dependence on the external momenta, you can simplify this calculation by choosing the momenta in the simplest possible way.

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