QCD is theory of quarks(fermion) and gluons(boson)(\(A_{\mu}=A^a_{\mu }T^a\)). The gluons emerges in the theory from requirement of local gauge invariance in color space. Quarks apart from electric charge also carries color charge and was postulated to save Pauli exclusion principle to explain bound states of sss(\(\Omega^{-}\)) and uuu(\(\delta^{++}\)).

Deepseek_color_charge_requirement

The theory of QED can't be applied to quarks, since the fermion is now 12 component instead of 4 component. The free quark Lagrangian / Lagrangian density can be written as

\[ \mathcal{L}=\overline{\psi}_{\substack{\alpha \\a}}(i\gamma ^{\mu }_{\alpha\beta}\delta_{ab} \partial_{\mu } -m\delta _{ab}\delta_{\alpha\beta})\psi _{\substack{\beta \\b}} \]

, here \(\alpha,\beta=1,2,3,4\) are Dirac indices, and a,b=1,2,3 are color indices.

The aim is to write down a Poincare invariant Lagrangian which is locally gauge invariant in color space.
This gave rise to the idea of covariant derivative which transform with same transformation rule as \(\psi(x)\) itself under local gauge transformation.

\(\psi(x)\) is 3 component object in color space, so its free Lagrangian can be written as

\[ \mathcal{L}=(\overline{\psi_{r}}^f(i\gamma ^{\mu }\partial _{\mu }\psi^f _{r}-m \overline{\psi_{r}}^f \psi^f_{r} ))+\overline{\psi _{g}}^f(i\gamma ^{\mu }\partial _{\mu }\psi^f _{g}-m \overline{\psi_{g}}^f \psi^f_{g} )+\overline{\psi _{b}}^f(i\gamma ^{\mu }\partial _{\mu }\psi^f _{b}-m \overline{\psi_{b}}^f \psi^f_{b} ) \]

here, f is flavour index (f=1,2,3,4,5,6 ; since there are six flavours of quarks).
We can also write \(\psi\) as

\[ \psi = \begin{pmatrix} \psi_{r}\\ \psi_{g}\\ \psi_{b} \end{pmatrix} \] \[ \begin{align*} \psi\to e^{i\alpha(x)^a T^a}\psi \\ \overline{\psi}\to\overline{\psi } e^{-i\alpha^*(x)^a T^{\dagger a}} \\ D_{\mu}\psi\to e^{i\alpha(x)^a T^a}D_{\mu}\psi \\ \end{align*} \]

Here, \(T^a\) (a=1,2...8) are \(3\times3\) matrices which acts upon 3 component \(\psi(x)\) in color space.
The \(\psi(x)_r,\psi(x)_g,\psi(x)_b\) are 4 component Dirac spinors.
In above equations, it is clear that the product \(\overline{\psi} D_{\mu}\psi \to \overline{\psi } e^{-i\alpha^{*}(x)^a T^{\dagger a}}e^{i\alpha(x)^a T^a} D_{\mu} \psi \) is locally gauge invariant if \(\alpha^{* a}(x)=\alpha^a(x), T^{\dagger a}=T^a\) which is indeed true since any element of SU(3) can be written as \(e^{i\alpha^a T^a}\) provided \(\alpha^a \in \mathbb{R}\) and \(T^a\) are hermitian matrices.

• QCD is \(SU(3)_c\) non-Abelian, gauge invariant model for strong interactions written in framework of Quantum field theory.
• The construction of Lagrangian density of non-abelian gauge theories is very similar to that of abelian U(1) case.

• Define a comparator \(U(y,x)\) with gauge transformation property \(U(y,x)\to g(y)U(y,x)g^{\dagger}(x)\) , where \(g(x)=e^{i \alpha(x)^a T^a}\), such that we can define a covariant derivative \(D_{\mu}\) such that,
• \[ \eta^{\mu}D_{\mu}\psi(x)=\lim_{\epsilon \to 0}\frac{\psi(x+n\epsilon)-U(x+n\epsilon,x)\psi(x)}{\epsilon} \]
• Expanding \(U(x+n\epsilon,x)\) around \(U(x,x)=\mathcal{1}\), we get \[ U(x+n\epsilon,x)=\mathcal{1}+ig\epsilon \eta^{\mu}A_{\mu}(x)+\mathcal{O}(\epsilon^2) \] covariant derivative becomes,\(D_{\mu}\psi(x)=(\partial_{\mu}-igA_{\mu}(x))\psi(x)\)
• Under gauge transformation \(\psi(x) \to g(x)\psi(x)\), where g(x) is exponential of 3x3 matrix which implies that \(\psi(x)\) is 3 component object and similarly, \(A_{\mu}(x)=A^a_{\mu}(x)T^a\)., where \(T^a\) are generators of SU(3) satisfying \([T^a,T^b]=if_{abc}T^{c}\) and \(Tr(T^a T^b)=\frac{\delta^{ab}}{2}\).
• Under local SU(3) gauge transformations, these \(3^2-1\) gauge bosons \(A^a_{\mu}(x)\) also transform as \(A_{\mu}(x)\to g(x)(A_{\mu}(x)+\frac{i}{g} \partial_{\mu})g^{\dagger}(x)\), which can be seen from infinitesimal transformation of comparator: \(U(x+n\epsilon,x)\to g(x+n\epsilon)U(x+n\epsilon,x)g^{\dagger}(x)\).
• We can now write kinetic term of local SU(3) invariant Lagrangian density with a trivial mass term. \[ \mathcal{L}_{kinetic}=\overline{\psi}(x)i\gamma^{\mu}D_{\mu}\psi(x)-m\overline{\psi}(x)\psi(x) \]
• Together with gauge invariant kinetic term for these 8 gauge bosons \(A^{a}_{\mu}(x)\), \[ \mathcal{L}_{QCD}=\overline{\psi_{f}}(x)(i\gamma^{\mu}D_{\mu}-m)\psi_f(x)-\frac{1}{2}Tr[F^{\mu\nu}F_{\mu\nu}] \] where, \( F_{\mu\nu}(x)=\frac{-i}{g}[D_{\mu},D_{\nu}]=(\partial_{\mu}A^a_{\nu}-\partial_{\nu}A^a_{\mu}+gf_{abc}A^b_{\mu}A^c_{\nu})T^a=F^a_{\mu\nu}T^a \).

• We would like to calculate the partition function of fields at finite temperature for calculation of equilibrium properties of the fields at equilibrium with heat bath at some temperature T.
• Consider Scalar fields (\(\phi(\vec{x},t)\)) at some temperature (T): Statistical Mechanics \(\implies Z=Tr[e^{-\beta H}]=\int d\phi\bra{\phi}e^{-\beta H}\ket{\phi}\), and from path integral formulation in euclidean time we get: \[ \begin{align*} \bra{\phi_f(\tau_f)}e^{-H(\tau_f-\tau_i)}\ket{\phi_i(\tau_i)} &=\int_{\phi_i(\tau_i)}^{\phi_f(\tau_f)}[D\phi]e^{-S_{E}[\phi]} \\ Z &=Tr[e^{-\beta H}]=\int d\phi\bra{\phi}e^{-\beta H}\ket{\phi}=\int d\phi \int_{\phi(\tau_i)}^{\phi(\tau_i+\beta)=\phi(\tau_i)}[D\phi]e^{-S_E[\phi]} \end{align*} \]
  • Similarly, for QCD: \[ \begin{align*} Z&=\int d\overline{\psi} d\psi dA \int_{A(\tau_i),\psi(\tau_i)}^{A(\tau_i + \beta)=A(\tau_i),\psi(\tau_i+\beta)=-\psi(\tau_i)}[D\overline{\psi}][D\psi][DA]e^{-S_E[\overline{\psi},\psi,A]} \\ \text{where}, S_E[\overline{\psi},\psi,A] &=\int_{\tau_i}^{\tau_i+\beta}d\tau \int_{-\infty}^{\infty}d^3x \mathcal{L}^{E}_{QCD}[\overline{\psi},\psi,A] \end{align*} \] Because of trace we are restricted to have Bosonic and fermionic fields with periodic and anit-periodic boundary conditions with time period \(\tau=\tau_f-\tau_i=\beta\).