I was doing higher order calculations for purely gluonic system and came across complicated color factors like the product of six structure constants product $f_{a_1a_2a_3}f_{a_4a_2a_7}f_{a_7a_8a_1}f_{a_5a_6a_3}f_{a_9a_4a_5}f_{a_8a_9a_6}$. I checked a few textbooks on perturbative QCD calculations and only found formula like $f_{abc}f_{dbc}=N_c\delta_{ad}$ and $f_{ade}f_{bef}f_{cfd} =\frac{N_c}{2}f_{abc}$. These can be easily derived from expressing the structure constant as trace over group generators like $f_{abc}= -2iTr[T_a,[T_b,T_c]]$. But it will looks more complicated as we have more structure constants in the expression as the one I listed above. I want to ask if anyone know how to systematically simplify color factors with increasing complexity? any references would be appreciated.

  • $\begingroup$ It depends on how much complex is your calculations, at some points you have to use FORM by Vermaseren. $\endgroup$
    – Karozo
    Sep 28, 2015 at 22:17
  • 1
    $\begingroup$ I believe the only way to simplify them is to repeatedly use the trace formula and simplify. There is no magic formula to help you here. What you could do however - is use a computer to solve it. There are packages in Mathematica such as FeynArts or ColorMath that help you do this. $\endgroup$
    – Prahar
    Sep 29, 2015 at 13:58
  • $\begingroup$ @Prahar,Thanks for your comments. I used the above trace expression and two Fierz identities, then I was able to get the result $\frac{N_c^3}{4}(N_c^2-1)$. It just needs more pages of calculations. $\endgroup$
    – Ming Li
    Sep 29, 2015 at 22:17

1 Answer 1


Here is one possible way to evaluate this expression using FeynCalc:

<< FeynCalc`
SUNF[a1, a2, a3] SUNF[a4, a2, a7] SUNF[a7, a8, a1] SUNF[a5, a6, a3] *
SUNF[a9, a4, a5] SUNF[a8, a9, a6] // 
SUNSimplify[#, Explicit -> True, SUNNToCACF -> False] & // Simplify

and the answer is:

1/4 SUNN^3 (-1 + SUNN^2)

i.e. what the OP got from doing the computation by hand.


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