1. Aug 26, 2015

### Safinaz

Hi all,

"Color factors of color -- octet scalars",
and

"The double line notation and the adjoint representation"

* 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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Last edited by a moderator: May 7, 2017
2. Aug 31, 2015