# Higgs Boson and its relation to the vacuum

The Higgs boson and the electroweak theory used symmetry breaking from condensed matter physics as its inspiration. The BCS theory of superconductivity is one such condensed matter symmetry breaking theory. I believe this theory uses some kind of model for the underlying atomic structure of the metal. In a similar manner the vacuum works in the same way with particles popping in and out with charges all the time. Is there a relation between the vacuum and the atomic structure of condensed matter. If so, could one define the vacuum with particular properties to match the Higgs potential? I know the Higgs potential is a vaccuum, but can properties be assigned to space itself and the fluctuating charged particles that pop in and out that are analogous to the charged atoms in a condensed matter metal and produce the Higgs fields this way? Maybe the vacuum is discrete like a metal lattice instead of continuous.

• It's not entirely clear what you are asking here, primarly because your question title seems to ask about the Higgs field, as in Electro-Weak theory, yet your question paragraph is asking about condensed matter theory. You say regarding BCS "I believe this theory uses some kind of...", as if to suggest you have not yet studied it yet? The vacuum manifold of the Brout-Englert-Higgs mechanism is, well by definition, continuous. I do not know anything about lattice groups, but the Lie group theoretic methods that the mechanism relys upon require continuous groups and a continuous set of vacuua. – Flint72 May 18 '14 at 9:53
• The Higgs field itself is simply a complex-valued scalar field which transforms as a doublet under the $SU(2)$ transformations. The potential energy function that is taken in the Brout-Englert-Higgs mechanism is just the Mexican Hat Potential $$V(\phi^{\dagger} \phi) = \frac{1}{2} m^2 \phi^{\dagger} \phi + \frac{1}{4} \lambda (\phi^{\dagger} \phi)^2 \rightarrow \frac{1}{4} \lambda \left[ (\phi^{\dagger} \phi)^2 - v^2 \right]$$ This potential may be used for other theories to obtain Spontaneous Symmetry Breaking. That is, the field $\phi$ need not necessiarly be the Higgs scalar field. – Flint72 May 18 '14 at 9:58