Consider a Lagrangian:
L=12∂μϕi∂μϕi−V(ϕ);V(ϕ)=−12μ2ϕiϕi+14λϕiϕiϕjϕj.
I understand that the mass spectrum of the particles in the theory can be obtained by:
m2ij=∂2V∂ϕi∂ϕj|ϕ=ϕ0.
The minimum of the potential is given by:
∂V∂ϕk=−μ2ϕk+λϕkϕjϕj=0⟹ϕk=0 or ϕjϕj=μ2λ
Now to obtain the mass spectrum:
∂2Vϕkϕl=∂∂ϕl(−μ2ϕk+λϕkϕjϕj)=−μ2δkl+λϕjϕjδkl+2λϕkϕl
Substituting the minimum obtained above:
∂2Vϕkϕl|ϕ=ϕ0=−μ2δkl+λδklμ2λ+2λμ2λ=2μ2.
I was expecting to obtain a matrix with the masses of the particles, but I just have one mass. What did I do wrong?