Does any one know what would be $tr(t^a t^b t^c t^d)$, where $t^a$ etc are Gell-Mann matrices? This came about when analyzing the color factor for the compton effect for QCD. So, must be pretty common, but I could not find a proper reference. In general is there any reference for trace of arbitrary number of Gell Mann matrices?


I take the SU(N) generators in the fundamental representation normalized such that $$ \text{Tr}\left[T^a T^b\right] = \frac{1}{2}\delta^{ab} $$

The commutator of two generators define the structure constants $f^{abc}$

$$ \left[T^a,T^b\right] = if^{abc}T^c $$

The anticommutator of two generators is

$$ \left\{T^a,T^b\right\} = \frac{1}{N}\delta^{ab}1 +d^{abc}T^c $$

where by $1$ I mean the identity matrix and $d^{abc}$ are the "d-symbol" defined as

$$ d^{abc} = 2\text{Tr}\left[ \left\{T^a,T^b\right\}T^c \right] $$

Then, there is a useful identity

$$ \text{Tr}\left[T^aT^bT^cT^d\right] = \frac{1}{4N}\delta^{ab}\delta^{cd} + \frac{1}{8}\left(d^{abe}d^{cde} - f^{abe}f^{cde}+if^{abe}d^{cde}+if^{cde}d^{abe}\right) $$

I suggest you this reference http://scipp.ucsc.edu/~haber/ph218/sunid17.pdf where different trace identitites are collected. For your case, look at Equation 75 in Appendix B, page 9.

Check the normalization of the generators before to use this identity.

  • $\begingroup$ As a rule link only answers are actively discouraged, because if the link goes dead the answer is useless. Please use Mathjax to edit in the appropriate equations so the answer can stand alone. $\endgroup$ – StephenG Aug 9 '18 at 7:49
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    $\begingroup$ @Angela if this answers your question, you should mark it as answered. $\endgroup$ – apt45 Aug 9 '18 at 14:27

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