Standard Model and Beyond · Problem 3.33

Gauge Theories

Physics — Particle Physics · o2-breaking-potential-minimum-masses

Problem

The potential corresponding to (3.130) can be written for

\lambda_{11} = \lambda_{22}

V(\phi_1, \phi_2) = \frac{1}{2}\mu^2(\phi_1^2 + \phi_2^2) + \frac{\lambda}{4}\left(\phi_1^2 + \phi_2^2\right)^2 + \frac{\lambda'\phi_1^2\phi_2^2}{2},

which is clearly

O(2)

invariant for

\lambda' = 0

, but only invariant under sign changes and

\phi_{1,2}

interchange for

\lambda' \neq 0

. Assume that

\mu^2 < 0

and

\lambda > 0

. Find the minimum, the physical particle masses, and the unbroken symmetries for the three cases

\lambda' > 0

\lambda' = 0

, and

\lambda' < 0

. In the last case impose the necessary boundedness condition.