Standard Model and Beyond · Problem 3.32

Gauge Theories

Physics — Particle Physics · general-dirac-lagrangian-field-redefinition

Problem

The most general fermion number conserving Lagrangian density for a free Dirac field is L=ψˉi∂̸[a+bγ5]ψ−ψˉ[c+iγ5d]ψ,\mathcal{L} = \bar{\psi}i\not\partial[a + b\gamma^5]\psi - \bar{\psi}[c + i\gamma^5 d]\psi, where Hermiticity requires aa, bb, cc, and dd to be real. The canonical form in (2.152), which corresponds to a=1a = 1, c=mc = m, and b=d=0b = d = 0, is needed to maintain the classical relation E2=p⃗2+m2E^2 = \vec{p}^2 + m^2. However, the more general form is acceptable (and may emerge from an underlying theory) provided that L\mathcal{L} can be put into canonical form by a field redefinition prior to quantization. (a) Show that for a>∣b∣≥0a > |b| \geq 0 (and all of the parameters real), one can find a multiplicative redefinition, ψ′≡Fψ\psi' \equiv \mathcal{F}\psi, with F(a,b,c,d)=FI(a,b,c,d)+F5(a,b,c,d)γ5,\mathcal{F}(a, b, c, d) = \mathcal{F}_I(a, b, c, d) + \mathcal{F}_5(a, b, c, d)\gamma^5, such that L(ψ′)\mathcal{L}(\psi') is canonical, i.e., L=ψˉ′(i∂̸−m)ψ′\mathcal{L} = \bar{\psi}'(i\not\partial - m)\psi', with m≥0m \geq 0. Construct F\mathcal{F} explicitly, and find m(a,b,c,d)m(a, b, c, d). Hint: write ψ\psi in terms of its chiral components, ψ=ψL+ψR=PLψ+PRψ\psi = \psi_L + \psi_R = P_L\psi + P_R\psi, where it will be seen that dd corresponds to a complex mass. (b) Specialize your results to the ``wrong sign mass'' case, a=1a = 1, c=−∣m∣c = -|m|, b=d=0b = d = 0, and interpret the result.

Related theory

Browse the problem library