Spontaneous Symmetry Breaking of SU(3)

[jazznaz]

Homework Statement

The generators of SU(3) are the Gell Mann matrices, $$\lambda_a$$. Consider symmetry breaking of an SU(3) theory generated by a triplet of complex scalar fields $$\Phi = \left(\phi_1, \phi_2, \phi_3\right)$$. Assuming the corresponding potential has a minimum at $$\Phi_0 = \left(0,0,v\right)$$, write down the kinetic term of the scalar fields and extract the mass term of the gauge bosons.

Homework Equations

The covariant derivative is,

$$D_\mu \phi = \left(\partial_\mu - ig\frac{\lambda_a}{2} G^{a\nu}_{\mu} \right) \phi$$

(I think)

The Attempt at a Solution

Started by writing the kinetic term as $$\|D_\mu \phi\|^2$$, but I'm having trouble getting to anything that looks vaguely like a mass term. :(

Any suggestions would be fantastic!

 

[fzero]

The covariant derivative is similar to

$$D_\mu \phi = \left(\partial_\mu - ig\frac{\lambda_a}{2} G^{a\nu}_{\mu} \right) \phi$$

but you might want to put in the rest of the indices to understand the structure. Also, you want to expand around the VEV, so let

$$\phi_i = \langle \phi_i \rangle + \varphi_i$$

and expand the kinetic term, keeping track of all gauge index structure.