Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

In former threads, namely:

"Color factors of color -- octet scalars",

and

"The double line notation and the adjoint representation"

I were asking about the difference between the interaction vertices among:

* three gluons (GGG), as in SM,

* three colored octet scalars, call ## S = S^A T^A ##, where ## T^A ## are the ## SU_C(3)## generators, and A=1,..,8,

* GSS.

Let me summarize the answer here:

In SM, the interactions between three gluons comes from the Lagrangian term;

$$ \mathcal{L} = F_{\mu \nu} F^{\mu\nu}, $$

where ## F_{\mu \nu} ## is the YM field strength, given by

$$ F_{\mu\nu} = \partial_{[\mu} A_{\nu]} + i g [ A_\mu, A_\nu], $$

So that GGG vertex comes from ## ig \text{Tr}(\partial_\mu A_\nu [ A^\mu, A^\nu]) \sim \text{Tr}(T_A [T_B,T_C]) \sim g f_{ABC} . ##

The interactions between three coloured octet fields (i.e. in the adjoint representation as gluons), is given by the interaction term:

\begin{equation*}

\begin{split}

\textrm{Tr} (S^A S^B S^C) &= \text{Tr} (T^A T^B T^C) S^A S^B S^C

\\& = \frac{1}{4} (d^{ABC} + i f^{ABC} ) S^A S^B S^C \sim d^{ABC}~~ S^A S^B S^C .

\end{split}

\end{equation*}

The term includes ## f^{ABC} ## has vanished because ## f^{ABC} ## is a totally symmetric tensor times a symmetric product.

The interactions between gluons and octet scalars come from the covariant derivative:

$$ \mathcal{L}_S = D^\mu S^\dagger D_\mu S, $$

\begin{equation*}

\begin{split}

( D_\mu S) ^A & = \partial_\mu S^A - i g A_{\mu B} (T_B)^{AC} S^C

\\ &= \partial_\mu S^A + g A_{\mu B} f_{ABC} S^C.

\end{split}

\end{equation*}

So that ## G_A S_B S_C ## vertex ## \sim f_{ABC} ##.

Where in the adjoint representation ## (T_{adj}^a) = -i f^{abc}##

A useful reference for that is of course:

An Introduction To Quantum Field Theory (Frontiers in Physics) (Michael E. Peskin, Dan V. Schroeder), Ch:15,

Hopefully that's useful for you and thanks for the advisors who helped me, fzero & samalkaiat

:)

S

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# An answer about former question about QCD interactions

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