User:Maschen/Gluon field strength tensor

In theoretical particle physics, the gluon field strength tensor is a tensor field characterizing the gluon interaction between quarks.

Background
The quantum field theory (QFT) of the strong interaction (one of the fundamental interactions of nature) is called quantum chromodynamics (QCD). Quarks interact with each other by the strong force due to their color charge, mediated by gluons. Gluons themselves possess color charge and can mutually interact.

The gauge covariant derivative
The gauge covariant derivative $$\mathcal{D}_\mu$$ is required to transform quark fields in manifest covariance, the four-gradient $∂_{μ}$ alone is not enough. The components are given by:


 * $$\mathcal{D}_\mu =\partial_\mu \pm ig\mathcal{A}_\mu\,,$$

wherein $i$ is the imaginary unit and $$\mathcal{A}_\alpha$$ is the spin-1 gluon field. Different authors choose different signs.

The field strength
The tensor is denoted $G$, (or $F$, $\overline{F}$, or some variant), and has components defined proportional to the commutator of gauge covariant derivative above;


 * $$ G_{\alpha\beta}\propto[\mathcal{D}_\alpha,\mathcal{D}_\beta]\,,$$

The proportionality constant is $± 1/ig_{s}$, in which


 * $$g_s=\sqrt{4\pi \alpha_s}$$

is the coupling constant for QCD. Again, different authors choose different signs.

Expanding the commutator gives;


 * $$G_{\alpha\beta} =\partial_{\alpha}\mathcal{A}_\beta-\partial_\beta\mathcal{A}_\alpha \pm ig[\mathcal{A}_{\alpha}, \mathcal{A}_{\beta}]$$

This almost parallels the electromagnetic field tensor (also denoted F) in quantum electrodynamics, given by the electromagnetic four-potential $A$ describing a spin-1 photon;


 * $$F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}\,.$$

The gluon field strength has extra terms which lead to self-interactions between the gluons and asymptotic freedom. This is a complication of the strong force making it inherently non-linear, contrary to the linear theory of the electromagnetic force.

QCD Lagrangian densities
Characteristic of field theories, the dynamics of the field strength are summarized by a suitable Lagrangian density and substitution into the Euler-Lagrange equation (for fields) obtains the equation of motion for the field. The Lagrangian for quarks, bound by gluons, is


 * $$\mathcal{L}=-\frac{1}{2}\mathrm{tr}\left(G_{\alpha\beta}G^{\alpha\beta}\right)+ \bar{\psi}\left(P_\mu + g\mathcal{A}_\mu\right)\gamma^\mu\psi $$

where "tr" denotes trace, and the quark fields are in triplet representation,


 * $$\psi=\begin{pmatrix}\psi_{1}\\

\psi_{2}\\ \psi_{3} \end{pmatrix},\overline{\psi}=\begin{pmatrix}\overline{\psi}^*_{1}\\ \overline{\psi}^*_{2}\\ \overline{\psi}^*_{3} \end{pmatrix} $$

The quark field $ψ$ belongs to the fundamental representation (3) and the antiquark field $\overline{ψ}$ belongs to the complex conjugate representation (3*), complex conjugate is denoted by $$ (not overbar).

Gauge transformations
The quark fields are invariant under the gauge transformation;


 * $$\psi_a\rightarrow e^{i\lambda\bar{\theta}(x)}\psi$$

where


 * $$\bar{\theta}(x)=\frac{1}{2}\lambda_n \theta^n\,,$$

in turn $n = 1, 2,... 8$, $λ_{n}$ are the eight matrix representations of the SU(3) group, and $θ^{n}(x)$ are eight global phase factors as a function of spatial position $x$. The matrix is exponentiated. The gauge covariant derivative transforms similarly.

The gluon field strength tensor is not gauge invariant, due to the extra coupled terms $$\mathcal{A}_\alpha\mathcal{A}_\beta$$. The transformation of each $$\mathcal{A}_\alpha$$ alone is;


 * $$\mathcal{A}_\alpha\rightarrow e^{i\lambda\bar{\theta}(x)} \left(\mathcal{A}_\alpha + \frac{i}{g}\partial_\alpha\right)e^{-i\lambda\bar{\theta}(x)}$$