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I'm trying to work through a calculation in the Peskin QFT textbook in chapter 5, specifically getting equation 5.10. They take two bracketed terms

$$ 4[p'^{\mu}p^{\nu}+p'^{\nu}p^{\mu}-g^{\mu\nu}(p \cdot p'+m_e^2)] $$

and

$$ 4[k_{\mu}k'_{\nu}+k_{\nu}k'_{\mu}-g_{\mu\nu}(k \cdot k'+m_{\mu}^2)] $$

they set $m_e=0$ and take the dot product of these two to get

$$ {32e^4}[(p \cdot k)(p' \cdot k')+(p \cdot k')(p' \cdot k)+m^2_{\mu}(p \cdot p')] $$

When I do this I get

$$ 16[2(p' \cdot k)(p \cdot k')+2(k \cdot p)(p' \cdot k')-3(p' \cdot p)(k' \cdot k)-(p' \cdot p)m^2_{\mu}] $$

In this scattering problem the two incoming momenta are $p$ and $p'$ and outgoing $k$ and $k'$, so working in the COM frame I suspect there is a reduction you can make but I can't figure out what it is. Any help is appreciated!

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$g^{\mu\nu} g_{\mu\nu}=4$. I have not seen your calculations, but my guess is you have taken this to be one.

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