Srednicki — Quantum Field Theory · Problem 32.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spontaneous-breaking-continuous-symmetries

Problem

We define the decay constant ff of the Goldstone boson via ⟨k∣jμ(x)∣0⟩=ifkμe−ikx,(32.18)\langle k|j^\mu(x)|0\rangle = ifk^\mu e^{-ikx} , \tag{32.18} where ∣k⟩|k\rangle is the state of a single Goldstone boson with four-momentum kk, normalized in the usual way, ∣0⟩|0\rangle is the θ=0\theta = 0 vacuum, and jμ(x)j^\mu(x) is the Noether current. a) Compute ff at tree level. (That is, express jμj^\mu in terms of the ρ\rho and χ\chi fields, and then use free field theory to compute the matrix element.) A nonvanishing value of ff indicates that the corresponding current is spontaneously broken. b) Discuss how your result would be modified by higher-order corrections.
Intuitive explanation

The decay constant ff measures how strongly the Noether current couples to the Goldstone boson—it's the amplitude for the current to create a single massless excitation from the vacuum. At tree level, you'll construct the current from the Lagrangian using Noether's theorem, then sandwich it between the vacuum and a one-particle state using free-field mode expansions to extract ff. A nonzero result confirms spontaneous symmetry breaking, since only a broken symmetry can have a current that creates Goldstone modes.

Intuitive explanation

The decay constant ff measures how strongly the Noether current couples to the Goldstone boson—it's the amplitude for the current to create a single massless excitation from the vacuum. At tree level, you'll construct the current from the Lagrangian using Noether's theorem, then sandwich it between the vacuum and a one-particle state using free-field mode expansions to extract ff. A nonzero result confirms spontaneous symmetry breaking, since only a broken symmetry can have a current that creates Goldstone modes.

Related theory

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