Higgs Expectation Value with Classical vs Quantum Potential

[jim burns]

I'm having a hard time following the arguments of how the Higgs gives mass in the Standard Model. In particular, the textbook by Srednicki gives the Higgs potential as:

$$V(\phi)=\frac{\lambda}{4}(\phi^\dagger \phi-\frac{1}{2}\nu^2)^2 $$

and states that because of this, $$\langle 0 | \phi(x) |0 \rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}v \\ 0 \end{pmatrix}$$

However, I thought the proper procedure would be to write $$\phi=q+\tilde{\phi}$$, and substitute this

$$\tilde{V}(\tilde{\phi})=\frac{\lambda}{4}((q+\tilde{\phi})^\dagger (q+\tilde{\phi})-\frac{1}{2}\nu^2)^2 $$

Then you would calculate the minimum of the quantum effective potential $$\tilde{V}_{eff}(\tilde{\phi})$$, which will give you an equation that gives you 'q' in terms of 'ν' and your other couplings. You then plug this value into 'q' to get

$$\tilde{V}(\tilde{\phi})=\frac{\lambda}{4}((q(\nu,\lambda)+\tilde{\phi})^\dagger (q(\nu,\lambda)+\tilde{\phi})-\frac{1}{2}\nu^2)^2 $$

Moreover, any field M that multiplies the Higgs in an interaction now has a component

$$q(\nu,\lambda)*M$$

Did Srednicki skip all these steps? How can you just say that the vacuum expectation value of a field is the minimum of the classical potential? Is he renormalizing all his couplings so that the minimum of the classical potential is the minimum of the quantum effective potential?

 

[DelcrossA]

Did Srednicki skip all these steps? How can you just say that the vacuum expectation value of a field is the minimum of the classical potential? Is he renormalizing all his couplings so that the minimum of the classical potential is the minimum of the quantum effective potential?

I think this is an interesting question that may be a bit out of my scope, but I'll try to provide at least a qualitative answer. Basically, It is not exact, but an approximation that can be justified in a regime of really weak couplings. In reality the quantum-mechanical ground state would be some superposition of all the "classical" vacuum states that is invariant under the symmetries of the Lagrangian/Hamiltonian, but for really weak couplings, this is unstable and falls into one of the potential states the way a classical system works. Essentially, my reasoning is similar to that used to explain why macroscopic objects are not in rotationally invariant ground states despite being governed by a rotationally invariant Hamiltonian. The Higgs Field is then a first order perturbation about this semiclassical vacuum state to get a perturbative spectrum (spectrum refers to the masses of particles).

The validity of this assumption seems to be supported due to the couplings to particles in this model are proportional to M/v , where M is the particle's mass and v is the v.e.v, which are all fairly small using experimental values. The top quark being a potential exception as well as some of the Higgs self-couplings

I believe a more rigorous answer could be found in talking about Effective Actions and the tree approximation, which are explained in Weinberg's Theory of Quantum Fields, vol. 2,