Friday, December 9, 2016

quantum field theory - Doubts about spontaneous symmetry breaking


I have been exposed to the usual treatment about spontaneous symmetry breaking in the standard model but it shames me to admit that there are some loose ends I still have to tie up. For simplicity, instead of the standard model let's consider a U(1) gauge theory with a complex scalar ϕ given by the Lagrangian


L=|Dμϕ|214(Fμν)2V(ϕϕ)


The V part is called the scalar potential and we take it to be



V=μ2ϕϕ+λ2(ϕϕ)2


where both μ and λ are positive and whose shape is the logo of this very site. It is straightforward to check that the minimums of the potential occur at the field value


ϕ0=(μ2λ)1/2


or at any other related to this one by the U(1) symmetry ϕ0=


ϕ0=(μ2λ)1/2eiα(x)


Until here I have no problem. In the next step it is assumed that ϕ0=(μ2λ)1/2 is the vacuum expectation value (I will use the letter v henceforth) of the field ϕ. FIRST QUESTION. How does this follow? why does the minimum of the scalar potential give the vacuum expectation value of the field?


Be that as it may, we have that ϕ has a vacuum expectation value. The next step is to expand ϕ around its VEV


ϕ=v+ψ


and by introducing this in the Lagrangian we get a massive gauge boson that eats a degree of freedom from ϕ. My SECOND QUESTION is, why do we have to expand around the VEV of ϕ to get the spectrum of the theory?




No comments:

Post a Comment

classical mechanics - Moment of a force about a given axis (Torque) - Scalar or vectorial?

I am studying Statics and saw that: The moment of a force about a given axis (or Torque) is defined by the equation: $M_X = (\vec r \times \...