Standard Model and Beyond · Problem 3.31

Gauge Theories

Physics — Particle Physics · scalar-fermion-parity-symmetry

Problem

In theories involving scalars and fermions it is sometimes the case that each mass eigenstate scalar field couples either as a scalar or a pseudoscalar to fermions, i.e., that there is a conserved parity. The SM and the MSSM at tree level are examples of this. As a simpler example, consider a single Dirac fermion ψ\psi and a single complex scalar ϕ\phi, with L=ψˉi∂̸ψ+(∂μϕ)†(∂μϕ)−mψˉLψR−m∗ψˉRψL+hψˉLϕψR+h∗ψˉRϕ†ψL−μ2ϕ†ϕ−κϕϕ−κ∗ϕ†ϕ†,\mathcal{L} = \bar{\psi}i\not\partial\psi + (\partial^\mu\phi)^\dagger(\partial_\mu\phi) - m\bar{\psi}_L\psi_R - m^*\bar{\psi}_R\psi_L + h\bar{\psi}_L\phi\psi_R + h^*\bar{\psi}_R\phi^\dagger\psi_L - \mu^2\phi^\dagger\phi - \kappa\phi\phi - \kappa^*\phi^\dagger\phi^\dagger, where μ2\mu^2 is real, while mm, hh, and κ\kappa are complex. We ignore ϕ\phi self-interactions for simplicity, and choose μ2>2∣κ∣≥0\mu^2 > 2|\kappa| \geq 0 to ensure that ϕ\phi does not acquire a VEV. Consider the cases (a) m=κ=0m = \kappa = 0, (b) m=0m = 0, κ≠0\kappa \neq 0, (c) and m≠0m \neq 0, κ≠0\kappa \neq 0, always with h≠0h \neq 0, μ2≠0\mu^2 \neq 0. In each case, find the spectrum, interpret it in terms of the symmetries of L\mathcal{L}, and determine whether PP, CC, and CPCP are conserved. Hint: identify which of the phases in ψL→eiβLψL,ψR→eiβRψR,ϕ→eiαϕ\psi_L \to e^{i\beta_L}\psi_L, \quad \psi_R \to e^{i\beta_R}\psi_R, \quad \phi \to e^{i\alpha}\phi are symmetries and which are field redefinitions.

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