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You should work out the minimum energy state of your system (classically) to find the vacuum expectation value. I assume you're working with the standard $\phi^4$-Lagrangian $$\mathcal L=\frac{1}{2}(\partial \phi)^2-\frac{1}{2}m^2\phi^2-\frac{\lambda}{4}\phi^4 $$ which corresponds to the Hamiltonian $$\mathcal ...


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The essence of the Higgs mechanism is that it allows the breaking of the (gauge) symmetry to grow a mass for the gauge (vector) bosons, which are necessarily massless in the unbroken symmetry. The Higgs scalar and the two degrees of freedom of the massless vector boson combine to form the three degrees of freedom of a massive vector boson. Goldstone's ...


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" By combining the ideas of Nambu in his study of superconductivity and of Johnson, Baker, and Willey in their approach to electrodynamics we construct a gauge theory of spontaneous symmetry breaking which is free of elementary spin-zero fields. The theory contains two fermions and two vector mesons, one of which acquires a mass via the Higgs mechanism. A ...


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As I understand the question, it is: what are the possible unbroken subgroups when a symmetry group G is spontaneously broken? If we assume that Lorentz invariance is unbroken, then we can look at the possible vacuum expectation values of a scalar field that transforms under some representation R of the symmetry group G. This can be calculated for specific G ...


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The scalar potential of your theory is $$V(\phi) = \lambda (\phi^a \phi^a - v^2)^2,$$ where I suspect you meant to take the square as I've written here. This potential is minimized when $\sqrt{\phi^a \phi^a} = v$. Think of $\phi=\frac{1}{2} \phi^a \sigma^a$ as a vector with components $\phi^a$ in a 3-dimensional vector space with basis vectors $\sigma_a/2$. ...


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The only possible way Bosons admit anomalies in flat space (analogous to the case for fermions you have mentioned) is when they do not admit a covariant lagrangian. These are called by a special name, "Chiral Bosons". The usual Bose lagrangian can always be regulated to give an action free of anomalies. See section 8 of this paper. ...



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