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I'm reading Appendix 1 of Section N2 (Gluon Scattering) in "Quantum Field Theory in a Nutshell" by Anthony Zee. The generators for SU(N) have the usual algebra

[tex][T^a, T^b] = i \epsilon^{a b c}T^c[/tex]

Suppose we adopt the following normalization

[tex]\text{tr}(T^a T^b) = \frac{1}{2}\delta^{a b}[/tex]

Then, we have

[tex]\text{tr}([T^a, T^b]T^c) = \text{tr}(i f^{a b e} T^{e} T^{c}) = i f^{a b e}\text{tr}(T^e T^c) = \frac{i f^{a b e} \delta^{e c}}{2} = \frac{i f ^{a b c}}{2}[/tex]

so that

[tex]f^{a b c} = - 2 i \text{tr}([T^a, T^b]T^c)[/tex]

Also,

[tex]\text{tr}([T^a, T^b][T^c, T^d]) = \text{tr}(i f^{a b \alpha} T^{\alpha} i f ^{c d \beta} T^{\beta}) = -f^{a b \alpha} f^{c d \beta}\text{tr}(T^\alpha T^\beta) = -\frac{1}{2}f^{a b e} f^{c d e}[/tex]

which implies that

[tex]f^{a b e} f^{c d e} = -2 \text{tr}([T^a, T^b][T^c, T^d])[/tex]

However, the author explicitly states that

[tex]f^{a b e} f^{c d e} = -4 \text{tr}([T^a, T^b][T^c, T^d])[/tex]

I don't get this additional factor of 2. What am I missing? Is there an error somewhere?

Thanks in advance!

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# Gluon Scattering - Colored Feynman Rules for Yang Mills Theory

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