Editing Quantum chromodynamics (section)

=== Lagrangian ===
The dynamics of the quarks and gluons are defined by the quantum chromodynamics [[Lagrangian (field theory)|Lagrangian]].  The [[gauge invariant]] QCD Lagrangian is

{{Equation box 1
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|equation = :<math>\mathcal{L}_\mathrm{QCD} = \bar{\psi}_i  \left( i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a </math>|border
|border colour = #50C878
|background colour = #ECFCF4}}

where <math>\psi_i(x) \,</math> is the quark field, a dynamical function of spacetime, in the [[fundamental representation]] of the [[SU(3)]] gauge [[Group (mathematics)|group]], indexed by <math>i</math> and <math>j</math> running from <math>1</math> to <math>3</math>; <math>\bar \psi_i \,</math> is the [[Dirac adjoint]] of <math>\psi_i \,</math>; <math>D_\mu</math> is the [[gauge covariant derivative]]; the γ<sup>μ</sup> are [[Gamma matrices]] connecting the spinor representation to the vector representation of the [[Lorentz group]].

Herein, the [[gauge covariant derivative]] <math>\left( D_\mu \right)_{ij} = \partial_\mu \delta_{ij} - i g \left( T_a \right)_{ij} \mathcal{A}^a_\mu \,</math>couples the quark field with a coupling strength <math>g \,</math>to the gluon fields via the infinitesimal SU(3) generators <math>T_a \,</math>in the fundamental representation. An explicit representation of these generators is given by <math>T_a = \lambda_a / 2 \,</math>, wherein the <math>\lambda_a \, (a = 1 \ldots 8)\,</math>are the [[Gell-Mann matrices]].

The symbol <math>G^a_{\mu \nu} \,</math> represents the gauge invariant [[gluon field strength tensor]], analogous to the [[electromagnetic tensor|electromagnetic field strength tensor]], ''F''<sup>μν</sup>, in [[quantum electrodynamics]]. It is given by:<ref>{{cite journal|title=The field strength correlator from QCD sum rules
|author1=M. Eidemüller |author2=H.G. Dosch |author3=M. Jamin
|location=Heidelberg, Germany
|journal=Nucl. Phys. B Proc. Suppl. |volume=86 |pages=421–425 |year=2000
|issue=1–3 |arxiv=hep-ph/9908318|bibcode=2000NuPhS..86..421E|doi=10.1016/S0920-5632(00)00598-3|s2cid=18237543 }}</ref>

:<math>G^a_{\mu \nu} = \partial_\mu \mathcal{A}^a_\nu - \partial_\nu \mathcal{A}^a_\mu + g f^{abc} \mathcal{A}^b_\mu \mathcal{A}^c_\nu \,,</math>

where <math>\mathcal{A}^a_\mu(x) \,</math> are the [[gluon field]]s, dynamical functions of spacetime, in the [[adjoint representation]] of the SU(3) gauge group, indexed by ''a'', ''b'' and ''c'' running from <math>1</math> to <math>8</math>; and ''f<sub>abc</sub>'' are the [[structure constants]] of SU(3) (the generators of the adjoint representation). Note that the rules to move-up or pull-down the ''a'', ''b'', or ''c'' indices are ''trivial'', (+, ..., +), so that ''f<sup>abc</sup>'' = ''f<sub>abc</sub>'' = ''f''<sup>''a''</sup><sub>''bc''</sub> whereas for the ''μ'' or ''ν'' indices one has the non-trivial ''relativistic'' rules corresponding to the [[metric signature]] (+ − − −).

The variables ''m'' and ''g'' correspond to the quark mass and coupling of the theory, respectively, which are subject to renormalization.

An important theoretical concept is the ''[[Wilson loop]]'' (named after [[Kenneth G. Wilson]]).  In lattice QCD, the final term of the above Lagrangian is discretized via Wilson loops, and more generally the behavior of Wilson loops can distinguish [[Color confinement|confined]] and deconfined phases.