Standard Model and Beyond · Problem 3.36

Gauge Theories

Physics — Particle Physics · chiral-fermion-complex-scalar-symmetry-breaking

Problem

This problem involves a chiral fermion and complex scalar, with an internal global symmetry that may be both spontaneously and explicitly broken. Consider the Lagrangian density L=ψˉi∂̸ψ+∂μϕ†∂μϕ−h[ψˉLϕψR+ψˉRϕ†ψL]−μ2ϕ†ϕ−λ(ϕ†ϕ)2+a(ϕ+ϕ†),\mathcal{L} = \bar{\psi}i\not\partial\psi + \partial_\mu\phi^\dagger\partial^\mu\phi - h[\bar{\psi}_L\phi\psi_R + \bar{\psi}_R\phi^\dagger\psi_L] - \mu^2\phi^\dagger\phi - \lambda(\phi^\dagger\phi)^2 + a(\phi + \phi^\dagger), where ψ=ψL+ψR\psi = \psi_L + \psi_R is a fermion, ϕ\phi is a complex scalar, λ>0\lambda > 0, h>0h > 0, and a≥0a \geq 0. (a) Show that L\mathcal{L} has a global chiral symmetry U(1)×U(1)U(1) \times U(1) for the case a=0a = 0. (b) Calculate the spectrum of the model (i.e., the masses) for the cases (i) μ2>0\mu^2 > 0, a=0a = 0 and (ii) μ2<0\mu^2 < 0, a=0a = 0. (c) Calculate the spectrum for the cases (i) μ2>0\mu^2 > 0, a>0a > 0 and (ii) μ2<0\mu^2 < 0, a>0a > 0. In each case assume that λa/∣μ∣3≪1\sqrt{\lambda}a/|\mu|^3 \ll 1, and keep only the leading nonzero term in that parameter. (d) Interpret the spectrum in each of the above cases in terms of the symmetries and symmetry breaking. (e) Now add the Majorana mass term −mL2(ψˉLψRc+ψˉRcψL)-\frac{m_L}{2}(\bar{\psi}_L\psi^c_R + \bar{\psi}^c_R\psi_L) to L\mathcal{L}, where ψRc\psi^c_R defined in (2.301) is essentially the CPCP conjugate of ψL\psi_L. (An analogous term could be added for ψR\psi_R.) Show that this violates the U(1)U(1) fermion number symmetry. Calculate the divergence of the corresponding Noether current.

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