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## 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 

<img src='content_image/301655.jpg'>

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 

<img src='content_image/301654.jpg'>

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 

<img src='content_image/301653.jpg'>

where A a µ is the gauge field. In fact, this is the gluon field, which has its own Lagrangian 

<img src='content_image/301652.jpg'>

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]: 

<img src='content_image/301651.jpg'>

