An answer about former question about QCD interactions

In summary: SamalkhaiatIn summary, the differences between the interaction vertices for three gluons, three colored octet scalars, and gluons with octet scalars were discussed. In SM, the GGG vertex is derived from the Lagrangian term, while the interactions between three colored octet fields come from the interaction term. The interactions between gluons and octet scalars involve both the GGG vertex and an additional term involving the octet scalar field strength. A useful reference for further reading is also mentioned.
  • #1
Safinaz
259
8
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] [Broken]
 
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  • #2


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!
 

1. What are QCD interactions?

QCD interactions, or Quantum Chromodynamics interactions, are the fundamental forces that govern the behavior of subatomic particles, specifically quarks and gluons. These interactions are responsible for the strong nuclear force, which binds quarks together to form protons and neutrons.

2. How do QCD interactions work?

QCD interactions work through the exchange of gluons, which are particles that carry the strong nuclear force. Gluons bind quarks together by constantly exchanging between them, creating a strong force that holds the quarks together in a nucleus.

3. What is the role of QCD interactions in particle physics?

QCD interactions play a crucial role in particle physics as they are responsible for the formation of protons and neutrons, which make up the nucleus of atoms. They also play a role in the behavior of quarks and gluons in high-energy colliders, allowing scientists to study the fundamental particles of matter.

4. How do scientists study QCD interactions?

Scientists study QCD interactions through experiments using high-energy particle accelerators, such as the Large Hadron Collider (LHC). These experiments involve colliding particles at high speeds, allowing scientists to observe the behavior of quarks and gluons and study the effects of QCD interactions.

5. What are the implications of understanding QCD interactions?

Understanding QCD interactions is crucial for understanding the behavior of matter at the subatomic level. It also has implications for our understanding of the early universe, as QCD interactions played a significant role in the formation of matter after the Big Bang. Additionally, research on QCD interactions can lead to advancements in technology, such as new materials and energy sources.

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