An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 11.2

Renormalization and Symmetry — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · spontaneous-symmetry-breaking-linear-sigma-model

Problem

11.2 A zeroth-order natural relation. This problem studies an N=2N = 2 linear sigma model coupled to fermions: L=12(∂μϕi)2+12μ2(ϕi)2−λ4((ϕi)2)2+ψˉ(i∂ ⁣ ⁣ ⁣/)ψ−gψˉ(ϕ1+iγ5ϕ2)ψ(1)\mathcal{L} = \frac{1}{2}(\partial_\mu \phi^i)^2 + \frac{1}{2}\mu^2(\phi^i)^2 - \frac{\lambda}{4}((\phi^i)^2)^2 + \bar{\psi}(i\partial\!\!\!/)\psi - g\bar{\psi}(\phi^1 + i\gamma^5\phi^2)\psi \qquad (1) where ϕi\phi^i is a two-component field, i=1,2i = 1, 2. (a) Show that this theory has the following global symmetry: ϕ1→cos⁡α ϕ1−sin⁡α ϕ2,\phi^1 \to \cos\alpha\,\phi^1 - \sin\alpha\,\phi^2, ϕ2→sin⁡α ϕ1+cos⁡α ϕ2,(2)\phi^2 \to \sin\alpha\,\phi^1 + \cos\alpha\,\phi^2, \qquad (2) ψ→e−iαγ5/2ψ.\psi \to e^{-i\alpha\gamma^5/2}\psi. Show also that the solution to the classical equations of motion with the minimum energy breaks this symmetry spontaneously. (b) Denote the vacuum expectation value of the field ϕi\phi^i by vv and make the change of variables ϕi(x)=(v+σ(x),π(x)).(3)\phi^i(x) = (v + \sigma(x), \pi(x)). \qquad (3) Write out the Lagrangian in these new variables, and show that the fermion acquires a mass given by mf=g⋅v.(4)m_f = g \cdot v. \qquad (4) (c) Compute the one-loop radiative correction to mfm_f, choosing renormalization conditions so that vv and gg (defined as the ψψπ\psi\psi\pi vertex at zero momentum transfer) receive no radiative corrections. Show that relation (4) receives nonzero corrections but that these corrections are *finite*. This is in accord with our general discussion in Section 11.6.

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