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I am trying to calculate the $g g\to q \bar q$ amplitude but am finding it intractably complex. I have determined that there are only two independent polarizations that contribute non-zero contributions. They are $+-+-$ and $+--+$. Since there are 3 diagrams at tree level, I have 6 separate calculations to do. Even with the spinor helicity formalism, this problem seems very messy and I have no confidence in my final answer. Does anyone know a way to simplify this or point me in the direction of some good notes about how to solve such a problem?

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I cannot give you an advice on how to do this computation by hand in the most efficient way, but if you need a way to cross-check what you have obtained so far and you have access to Mathematica, you could use FeynCalc for that.

The calculation of the $gg \to q \bar{q}$ matrix element squared (unpolarized) is shipped with FeynCalc as an example (see here)

The full result is quite big

enter image description here

But in the limit $m \to 0$ it simplifies to

$\frac{g_s^4 (t^2+u^2)(4t^2-t u + 4 u^2)}{24 s^2 t u}$

Which agrees with Ellis, Stirling and Weber, "QCD and Collider Physics", Table 7.1.

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