Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 3.5

Classical field theory — Schwartz M.D. z-lib.org

Physics — QFT · spontaneous-symmetry-breaking

Problem

Spontaneous symmetry breaking is an important subject, to be discussed in depth in Chapter 28. A simple classical example that demonstrates spontaneous symmetry breaking is described by the Lagrangian for a scalar with a negative mass term: L=−12ϕ□ϕ+12m2ϕ2−λ4!ϕ4.(3.86)\mathcal{L} = -\frac{1}{2}\phi\square\phi + \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4. \tag{3.86} (a) How many constants cc can you find for which ϕ(x)=c\phi(x) = c is a solution to the equations of motion? Which solution has the lowest energy (the ground state)? (b) The Lagrangian has a symmetry under ϕ→−ϕ\phi \to -\phi. Show that this symmetry is not respected by the ground state. We say the vacuum expectation value of ϕ\phi is cc, and write ⟨ϕ⟩=c\langle\phi\rangle = c. In this vacuum, the Z2\mathbb{Z}_2 symmetry ϕ→−ϕ\phi \to -\phi is spontaneously broken. (c) Write ϕ(x)=c+π(x)\phi(x) = c + \pi(x) and substitute back into the Lagrangian. Show that now π=0\pi = 0 is a solution to the equations of motion. How does π\pi transform under the Z2\mathbb{Z}_2 symmetry ϕ→−ϕ\phi \to -\phi? Show that this is a symmetry of π\pi's Lagrangian.

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