Hadronic production of a Higgs boson in association with two jets at ...

1.1. The Standard Model of Particle Physics 5follows,V (x) = 1 + iα a t a + O(α 2 ). (1.11)Here t a is a matrix and the set of t’s are the basic generators of the symmetry group.Indeed since V (x) is unitarity we find thatV (x)V † (x) = 1 =⇒ t a − (t † ) a = 0 (1.12)so t a are Hermitian. A continuous group with Hermitian operators of this kind isknown as a Lie group and the vector space spanned by the generators defined withthe following commutation relation,[t a , t b ] = if abc t c (1.13)defines a Lie algebra. Here f abc are called the structure constants of the group. Liegroups can be quite diverse but in this discussion we restrict ourselves to the groupof N × N unitary matrices with determinants equal to 1 (SU(N)). These theorieswere first studied by Yang and Mills [14], hence the resulting gauge invariantSU(N) Lagrangian is referred to as the Yang-Mills Lagrangian. The traceless Hermitianmatrices t a define the fundamental representation of the group, and it is thisrepresentation which will govern how fermions will transform given an infinitesimaltransformation. The structure constants f abc define the adjoint representation of thegroup and it is this representation that determines how vector bosons transform. Wealso note that if the structure constants all vanish (an example of which is the U(1)gauge group) then the group is called Abelian. If however, there are non-zero commutationrelations between group generators the group is non-Abelian. We shall seepresently that this has a huge effect on the physics of a gauge theory.Now that we have defined the properties of the groups with which we wantphysics to be invariant under, we must define the infinitesimal field transformationsand gauge invariant combinations of fields that can be used to construct Lagrangians.The generalisation of the covariant derivative eq. (1.6) is straightforward,D µ = ∂ µ − igA a µ ta . (1.14)