ating functional for the gluon field is where L$_{YM}$ is the Yang-Mills Lagrangian (1.25), L$_{gf}$ is the gauge-fixing Lagrangian (1.37) and L$_{gh}$ is the ghost Lagrangian (1.35). Using (1.38), it can be shown that the gluon propagator is given by Now that the quark and gluon fields have been quantized, the complete QCD Lagrangian is given by The terms in Eq. (1.40) can be interpreted as follows: the first term can be used to derive the quark propagator, the second and third terms can be used to derive the gluon propagator, the fourth term represents an interaction between quark and gluon fields, the fifth term represents an interaction between three gluon fields, the sixth term represents an interaction between four gluon fields, the seventh term can be used to derive the ghost propagator, while the eighth term represents an interaction between ghost and gluon fields. It is important to note that the gauge-fixing Lagrangian in Eq. (1.37) is not gauge invariant. Although the QCD Lagrangian (1.40) is not gauge invariant, it is invariant under a generalized form of gauge symmetry known as BRST symmetry [18, 64]. This can be used to prove the Slavnov-Taylor identities which relate various correlation functions in QCD [109, 120]. Any QCD correlation function can be calculated to any order in g using the perturbative Z [ J ] = ∫ DA Dc D ¯ c exp$^{[}$ i$^{∫}$ d $^{4}$x $^{[}$L$_{YM}$ + J µ $_{a}$A a µ + L$_{gf}$ + L$_{gh}$ ] ] , (1.38) 〈 0 | T [ A a $_{µ}$( x ) A b ν ( y ) ] | 0 〉 = D ab µν ( x − y ) = − iδ ab ∫ d $^{4}$p (2 π ) 4 [ g$_{µν}$ − (1 − a ) p$_{µ}$p$_{ν}$ p 2 + iη$^{]}$ e − ip · ( x − y ) p 2 + iη . (1.39) L$_{QCD}$ = ¯ Q $^{[}$i / ∂ − m $^{]}$Q − 1 4 $^{[}$∂$_{µ}$ A b ν − ∂$_{ν}$A b µ $^{]}$[ ∂ $^{µ}$Aν b − ∂ $^{ν}$A µ $_{b}$] − 1 2 a ∂ $^{µ}$Ab $_{µ}$∂$^{ν}$A b ν + g 2 ¯ Qλ $^{a}$γ$^{µ}$Aa $_{µ}$Q − g 4 $^{[}$∂$_{µ}$ A a ν − ∂$_{ν}$A a µ $^{]}$f $^{abc}$A µ $_{b}$A ν c − g 2 4 f $^{abc}$f $^{ade}$Aa $_{µ}$A b $_{ν}$A µ $_{d}$A ν e − ∂ $^{µ}$¯ c $^{a}$∂$_{µ}$ c a + gf $^{abc}$Ac $_{µ}$∂$^{µ}$¯ c $^{a}$c$^{b}$. (1.40) 1.3.5 Regularization and Renormalization 22