1.3.3 Non-Abelian Gauge Theory Correlation functions can be calculated perturbatively once the complete Lagrangian for a theory is known. In the SM, interactions are introduced through gauge symmetries. For instance, we have seen that the Lagrangian for free quark fields is given by where we have introduced the index j ∈ { 1 , 2 , 3 } to denote the colour degree of freedom of the quark fields. The Lagrangian (1.22) is invariant under the global gauge transformation where T a is a generator of the non-Abelian group SU (3). The generators satisfy the Lie algebra $^{[}$T $^{a}$, T b ] = if $^{abc}$T c where f abc are the structure constants of SU (3). The generators are related to the Gell-Mann matrices λ a via T a = λ a 2 . Now, suppose that we alter the gauge transformation (1.23) so that θ a → θ $^{a}$( x ). Clearly the Lagrangian (1.22) is not invariant under this local gauge transformation. However, it can be made so by introducing a gauge field. This can be done by replacing the derivative in (1.22) with a covariant derivative where A a µ is the gauge field. In fact, this is the gluon field, which has its own Lagrangian where G a µν is the gluon field strength tensor and the gluon field is massless. This is called the Yang-Mills Lagrangian. It can be shown that the following Lagrangian is invariant under local SU (3) gauge transformations [111]: 18 L = ¯ Q $^{[}$i / D − m $^{]}$Q − 1 4 G a $_{µν}$G µν a . (1.26) L$_{YM}$ = − 1 4 G a $_{µν}$G µν a , G c µν = ∂$_{µ}$A c ν − ∂$_{ν}$A c µ + gf $^{abc}$Aa $_{µ}$A b ν , (1.25) D$_{µ}$ = ∂$_{µ}$ − igT $^{a}$Aa $_{µ}$, (1.24) Q$_{i}$ → Q ′ i = exp $^{[}$− igT a $_{ij}$θ a $^{]}$Q$_{j}$ , ¯ Q$_{i}$ → ¯ Q ′ i = exp $^{[}$igT a $_{ij}$θ a ] ¯ Q$_{j}$ , (1.23) L$_{quarks}$ = ¯ Q$_{j}$ $^{[}$i / ∂ − m $^{]}$Q$_{j}$ , (1.22)