Higgs boson
The Higgs boson is an elementary particle in the Standard Model of particle physics. It is the quantum excitation of the Higgs field—a fundamental field of crucial importance to particle physics theory, first suspected to exist in the 1960s, that unlike other known fields such as the electromagnetic field, takes a non-zero constant value almost everywhere. The question of the Higgs field's existence has been the last unverified part of the Standard Model of particle physics and, according to some, "the central problem in particle physics".
The presence of this field, now believed to be confirmed, explains why some fundamental particles have mass when, based on the symmetries controlling their interactions, they should be massless. The existence of the Higgs field would also resolve several other long-standing puzzles, such as the reason for the weak force's extremely short range.
Higgs field (classical)
Spontaneous symmetry breaking, a vacuum Higgs field, and its associated fundamental particle the Higgs boson are quantum phenomena. A vacuum Higgs field is responsible for spontaneous symmetry breaking the gauge symmetries of fundamental interactions and provides the Higgs mechanism of generating mass of elementary particles.
At the same time, classical gauge theory admits comprehensive geometric formulation where gauge fields are represented by connections on principal bundles. In this framework, spontaneous symmetry breaking is characterized as a reduction of the structure group 
of a principal bundle 
to its closed subgroup 
. By the well-known theorem, such a reduction takes place if and only if there exists a global section 
of the quotient bundle 
. This section is treated as a classical Higgs field.
A key point is that there exists a composite bundle 
where 
is a principal bundle with the structure group . Then matter fields, possessing an exact symmetry group , in the presence of classical Higgs fields are described by sections of some composite bundle 
, where 
is some associated bundle to . Herewith, a Lagrangian of these matter fields is gauge invariant only if it factorizes through the vertical covariant differential of some connection on a principal bundle , but not .
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