QCD Feynman Diagrams: Understanding Particle Interactions

[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!

 

[Hepth]

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.

 

[Thor Shen]

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

 

[dauto]

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.

 

[dauto]

@ 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.

 

[Thor Shen]

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.

 

[Hepth]

@ 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.

 

[Thor Shen]

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