An answer about former question about QCD interactions

[Safinaz]

Hi all,

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


[PLAIN]https://www.physicsforums.com/members/samalkhaiat.35381/[/PLAIN]

 

[AndyW]

Hi S,

Thank you for summarizing the differences between the interaction vertices for three gluons, three colored octet scalars, and gluons with octet scalars. Your explanation is very clear and concise. I also appreciate you mentioning the reference for further reading on the topic.

One thing I would like to add is that the interactions between gluons and octet scalars can also be written in terms of the GGG vertex, as you mentioned, but with an additional term involving the octet scalar field strength. This can be seen by expanding the covariant derivative term:
$$ D^\mu S^\dagger D_\mu S = \partial^\mu S^\dagger \partial_\mu S + g f_{ABC} A_\mu^A S^\dagger T^B \partial^\mu S^C + g^2 f_{ABC} A_\mu^A A^\mu_B S^\dagger T^B S^C $$
The last term in the above equation involves both gluons and octet scalars, and is proportional to the GSS vertex.

Thanks again for your contribution to the discussion!