# QCD Feynman Diagrams: Understanding Particle Interactions

• Thor Shen
In summary: I don't really know what his process is, as it doesn't seem to have a specific time direction, so I just used generic spinors. Its its a 1 -> 5 process then yes, some of those need to be changed from u to v on the two bottom channels.
Thor Shen

the number stand for the index of particles (quarks and gluons)
$M=\bar{v}(p_2) ig_sT_{12}\gamma^\mu(12)u(p_1)\frac{-i}{p_7^2}\bar{u}(p_5) ig_sT_{56}\gamma_\mu(56)\bar{v}(p_6)\frac{-i}{m-\gamma^\mu p_{9\mu}}\bar{v}(p_3) ig_sT_{34}\gamma^\mu(34)u(p_4)\frac{-i}{p_8^2}\bar{u}(p_5)ig_sT_{56}\gamma_\mu(56)\bar{v}(p_6)$
$u(p_i)$ and $v(p_i)$ stand for the wave function of quark and antiquark, respectively. $p_i$ stand for the four momentum
I am studying the QCD right now. Do I write the amplitude above right?
Someone recommend me a textbook by T.Muta. But I make some confusion when I confront complex diagrams like above. Which book or paper can tell me about this? Thank you!

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I like to structure it more as to not make mistakes:

##\def\lts#1{\kern+0.1em /\kern-0.45em #1}
\bar{u}_6 (-i g_s \gamma_{\mu} T^a) \frac{i}{\lts{p}_9 - m}(-i g_s \gamma_{\nu} T^b) u_5##
##\times \left(\bar{u}_4 (-i g_s \gamma_{\alpha} T^c) u_3\right)##
##\times \left(\bar{u}_2 (-i g_s \gamma_{\beta} T^d) u_1\right)##
##\times \left( -i \frac{g^{\mu \alpha} \delta^{ac}}{p_8^2} \right) \times \left( -i \frac{g^{\nu \beta} \delta^{bd}}{p_7^2} \right) ##

My convention is to always start with outgoing, working AGAINST the arrows (particle flow), and for fermion propagators write the momentum such that it goes WITH the arrow, and use ##\def\lts#1{\kern+0.1em /\kern-0.45em #1} i \frac{\lts{p}+m}{p^2-m^2}## which will be right.

You have a couple small errors I think. Also, I assumed those gauge bosons were gluons, though normally that would be the springy/curly line, not the wavy one. Wavy is reserved for basically all spin-1 bosons (gamma, w z) except the gluon.

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Yes, the gauge bosons were gluons. I will take care next time,thanks!

Thor Shen said:
Do I write the amplitude above right?

First thing I noticed was that you used the index μ twice (two pairs). You should make sure to use different indices for different gluons.

@ Hepth: Shouldn't you have used u's and v's instead of all u's for the external legs? I think Thor Shen had that right.

dauto said:
First thing I noticed was that you used the index μ twice (two pairs). You should make sure to use different indices for different gluons.
Yes. Firstly, I write the same index for omitting the delta functions. But the two pairs will mislead using Einstein's reduction rule, the latter one should be $\nu$. Of course, the complete form should be written by Hepth.

dauto said:
@ Hepth: Shouldn't you have used u's and v's instead of all u's for the external legs? I think Thor Shen had that right.

I don't really know what his process is, as it doesn't seem to have a specific time direction, so I just used generic spinors. Its its a 1 -> 5 process then yes, some of those need to be changed from u to v on the two bottom channels.

I guess it might make the most sense as a strong decay of some meson from the top down now that I am thinking about it deeper than a diagram. I always go from left to right as into out states.

The diagram is from the proton-antiproton annihilation into two mesons.

## 1. What is the purpose of QCD Feynman diagrams?

QCD Feynman diagrams are used to visualize and understand the interactions between particles in quantum chromodynamics (QCD). They allow scientists to interpret complex mathematical equations and predict the behavior of particles in different scenarios.

## 2. How do QCD Feynman diagrams work?

QCD Feynman diagrams use graphical representations to show the exchange of virtual particles between interacting particles. These diagrams follow specific rules and conventions, such as the conservation of energy and momentum, to accurately depict particle interactions.

## 3. What is the significance of color charge in QCD Feynman diagrams?

Color charge is a property of particles in QCD that describes their strong nuclear interaction. In QCD Feynman diagrams, color charge is represented by different lines and vertices, and plays a crucial role in understanding the interactions between particles.

## 4. Can QCD Feynman diagrams be used to study all types of particle interactions?

QCD Feynman diagrams are specifically designed to study the strong nuclear force, which is responsible for interactions between quarks and gluons. They cannot be used to study other fundamental forces, such as electromagnetism or gravity.

## 5. What are the limitations of QCD Feynman diagrams?

QCD Feynman diagrams are a simplified representation of complex particle interactions and have limitations in accurately predicting the behavior of particles in certain scenarios, such as at extremely high energies. They also do not account for the effects of gravity, which is a fundamental force not included in QCD.

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