The Higgs field and resulting Higgs boson are a vital part of the Electroweak Interaction and the Standard Model of Particle Physics. In the absence of the Higgs field, when a Local Gauge is applied to the Lagrangian of the Electroweak Interaction we are left with force-carrying bosons that are massive, the W and Z Bosons with masses of ~80GeV and ~90GeV respectively. This would be okay for the Photon as it has no mass. The Higgs mechanism was the most favoured explanation for solving this problem.
In brief, the Higgs field is introduced to ‘break’ the symmetry of the Electroweak theory, which allows particles to have mass.
This Higgs mechanism is important as it not only explains how the heavy bosons become massive but also provides an explanation as to how the fermions come to have mass.
The Mechanism of the interaction is simple to understand. Where the Electroweak Interaction couples to electric and weak (or flavour) charges and the Strong Interaction couples to colour charge, the Higgs interaction couples to mass. The process by which the Higgs gives fermions mass is via the Yukawa potential. This potential gives the coupling strength of the Higgs to all types of fermions, the stronger the coupling, the more mass the particle will have. Hence the Higgs Boson couples more strongly to more massive particles, hence the energies of the LHC were necessary to create the most massive particles for the Higgs to couple with.
Why we needed this boson is a bit more complicated, which corresponds to Peter Higgs, Yoichiro Nambu and Jeffrey Goldstone’s theoretical research.
In the Electroweak interaction you can examine the Lagrangian in a similar way to those for Quantum Electrodynamics (QED) and also Quantum Chromodynamics (QCD). Starting with the Dirac Lagrangian, when a Local Gauge is applied the resulting Lagrangrian is not invariant under the transformation. The local gauge transformation applied to the Langrangian is dependent on the symmetry, for example for the weak force we use SU(2) symmetry as we want physics invariant under swapping up-like and down-like fermions.
When a Local Gauge Symmetry is applied to the Electroweak Lagrangian it does not remain invariant under the gauge transformation. This can be rectified by the introduction of appropriate fields, which have associated mass-less bosons W1, W2, W3 and B. The SU(2)xU(1) symmetry of the electroweak theory is non-abelian which means that the bosons interact with each other as well as with fermions.
The Electroweak theory needs to end up with three massive bosons (2 charged and 1 neutral) and also a mass-less boson. The Goldstone Theorem provides a mechanism by which the 4 mass-less bosons from the original symmetry can become the four Electroweak bosons described above. The Goldstone theorem states “that for any continuous symmetry broken, there exists a mass-less particle, the Goldstone boson.” The result is that for each broken generator, there is a resulting mass-less scalar boson.
The Higgs mechanism is the process applied to Electroweak theory. A complex doublet Higgs field can be included in the theory and this Higgs field breaks the symmetry of the problem while retaining local gauge invariance. This Higgs field (two complex scalar fields which transform under the SU(2) symmetry) will, via the Goldstone Theorem, result in a scalar Higgs boson and 3 Goldstone bosons which will provide mass. The three Goldstone bosons interact with the original fields to provide mass for the W+, W- and Z bosons while leaving the fourth boson mass-less. This can be seen mathematically by looking at the changed form of the Electroweak Lagrangian due to the introduction of the Higgs fields.
There is a reason to believe that the Higgs Boson discovered is not the garden-variety Higgs that physicists were expecting. It’s relatively low mass may place it in the Supersymmetric regime, and may be humanity’s first probe into Supersymmetry. If the Boson was discovered to be a singlet it would also be the first fundamental singlet ever discovered, sparking new interest in finding the last piece of the singlet, vector, tensor boson puzzle: The Graviton, the force carrier for the gravitational force and the key ingredient in the Theory of Everything, “The Promised Land” of Physics that will explain how General Relativity works with Quantum Theory and will explain how The Universe works in completion.