Standard Model and Beyond · Problem 3.34

Gauge Theories

Physics — Particle Physics · three-scalar-potential-boundedness-flat-directions

Problem

Consider the potential V(ϕ1,ϕ2,ϕ3)=μ22(ϕ12+ϕ22+ϕ32)+μ122ϕ1ϕ2+μ132ϕ1ϕ3+μ232ϕ2ϕ3+λ12(ϕ12−ϕ22)2+λ3ϕ34.V(\phi_1, \phi_2, \phi_3) = \frac{\mu^2}{2}(\phi_1^2 + \phi_2^2 + \phi_3^2) + \mu^2_{12}\phi_1\phi_2 + \mu^2_{13}\phi_1\phi_3 + \mu^2_{23}\phi_2\phi_3 + \lambda_{12}(\phi_1^2 - \phi_2^2)^2 + \lambda_3\phi_3^4. Find the conditions for boundedness (assume λ12≠0\lambda_{12} \neq 0, λ3≠0\lambda_3 \neq 0), and the special cases for which there are flat directions.

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