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.