section_path: Theoretical aspects/Construction of the Standard Model Lagrangian/Quantum chromodynamics sector
section_name: Quantum chromodynamics sector
paragraph_text: Quantum chromodynamics sector
  
  The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, which is a Yang–Mills gauge theory with SU(3) symmetry, generated by T a = λ a / 2 {\displaystyle T^{a}=\lambda ^{a}/2}. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by
  
  L QCD = ψ ¯ i γ μ D μ ψ − 1 4 G μ ν a G a μ ν, {\displaystyle {\mathcal {L}}_{\text{QCD}}={\overline {\psi }}i\gamma ^{\mu }D_{\mu }\psi -{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu },}
  
  where ψ {\displaystyle \psi } is a three component column vector of Dirac spinors, each element of which refers to a quark field with a specific color charge (i.e. red, blue, and green) and summation over flavor (i.e. up, down, strange, etc.) is implied.
  
  The gauge covariant derivative of QCD is defined by D μ ≡ ∂ μ − i g s 1 2 λ a G μ a {\displaystyle D_{\mu }\equiv \partial _{\mu }-ig_{s}{\frac {1}{2}}\lambda ^{a}G_{\mu }^{a}}, where
  
  The QCD Lagrangian is invariant under local SU(3) gauge transformations; i.e., transformations of the form ψ → ψ ′ = U ψ {\displaystyle \psi \rightarrow \psi '=U\psi }, where U = e − i g s λ a ϕ a (x) {\displaystyle U=e^{-ig_{s}\lambda ^{a}\phi ^{a}(x)}} is 3 × 3 {\displaystyle 3\times 3} unitary matrix with determinant 1, making it a member of the group SU(3), and ϕ a (x) {\displaystyle \phi ^{a}(x)} is an arbitrary function of spacetime.