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Дата изменения: Wed Aug 9 20:40:47 2000
Дата индексирования: Mon Oct 1 22:53:54 2012
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QCD Lagrangian

QCD is a gauge theory based on the $SU(3)$ group. The corresponding gauge field $G^{\alpha}_{\mu}(x)$ is called a gluon. The matter fields $q_k(x)$ are quarks. They are triplets and are transformed according to the fundamental representation. Index $k$ enumerates a sort of quarks.

QCD Lagrangian in the Feynman gauge is written down following the rules (1), (2), (3).

$$L_{phys} = -\, \frac{1}{4} {F^{\alpha}}_{\mu \nu}(x) {F_{\a... ..._{\mu}q_k(x) - \nabla_{\mu}\bar{q}_k(x)\gamma^{\mu}q_k(x) )\;.$$ (5)

where

$$\nabla_{\mu}q(x) = \partial_{\mu} q(x) - i\,g\, G^{\alpha}_{\mu}(x) \hat{t}_{\alpha}q(x)\;;$$

$$F^{\alpha}_{\mu \nu}(x) = \partial_{\mu}G^{\alpha}_{\nu}(x) -... ...{\alpha}_{\beta\gamma}G^{\beta}_{\mu}(x)G^{\gamma}_{\nu}(x)\;;$$

$$L_{GF} = -\,\frac{1}{2} \sum_{\alpha}(\partial^{\mu} G^{\alpha}_{\mu})^2 \;;$$ (6)

$$L_{FP}= - \bar{c}_{\alpha}(x) (\Box c^{\alpha}(x) + g\, \p... ...^{\alpha}_{\beta \gamma} G^{\beta}_{\mu}(x) c^{\gamma}(x)))\;,$$ (7)

$c^{\alpha}(x)$ are Faddeev-Popov ghosts, $g$ is coupling constant, $f^{\alpha}_{\beta \gamma}$ are the $SU(3)$ structure constants, $t_{\alpha}$ are the generators in the fundamental representation¹.

Footnote

1  $t_{\alpha}$ are equal to $\lambda_{\alpha}/2$ where $\lambda_{\alpha}$ are the Gell-Mann matrices.