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From Peskin and Schroeder's Introduction to Quantum Field theory, in order to calculate the scattering cross section of, for example, $e^-(p)e^+(p')\rightarrow \mu^-(k)\mu^+(k')$, we first want to average the amplitude $\mathcal{M}$ over all spin states: $\frac{1}{4}\sum_{\text{spins}}|\mathcal{M}|^2$.

The book gives a pretty straightforward explanation of why (in the limit $m_e\rightarrow0$) this turns out to be:

$$ \frac{1}{4}\sum_{\text{spins}}|\mathcal{M}|^2= \frac{4e^4}{q^4}\biggl(p'^\mu p^\nu+p'^\nu p^\mu-g^{\mu\nu}(p\cdot p')\biggr)\cdot $$ $$ \cdot\biggl(k_\mu k_\nu'+k_\nu k_\mu'-g_{\mu\nu}(k\cdot k'+m_\mu^2)\biggr) $$

But then it claims that this comes out to: $$ \frac{8e^4}{q^4}\left[(p\cdot k)(p'\cdot k')+(p\cdot k')(p'\cdot k)+m_\mu^2(p\cdot p')\right] $$ With no explanation, but I can't figure out how.

My own calculation gives:

$$ \frac{16e^4}{q^4}\biggl(p'^\mu p^\nu+p'^\nu p^\mu-g^{\mu\nu}(p\cdot p')\biggr)\biggl(k_\mu k_\nu'+k_\nu k_\mu'-g_{\mu\nu}(k\cdot k'+m_\mu^2)\biggr)= $$

$$ \frac{4e^4}{q^4}\biggl((p'\cdot k) (p\cdot k')+(p'\cdot k')(p\cdot k)-(p'\cdot p)(k\cdot k'+m_\mu^2)+ $$$$ (p'\cdot k')(p\cdot k)+(p'\cdot k)(p\cdot k')-(p'\cdot p)(k\cdot k'+m_\mu^2)-(p\cdot p')(k\cdot k')-(p\cdot p')(k\cdot k')+(p\cdot p')(k\cdot k'+m_\mu^2)\biggr) $$ Which gives: $$ \frac{4e^4}{q^4}\biggl(2(p\cdot k')(p'\cdot k)+2(p\cdot k)(p'\cdot k')-3(p\cdot p')(k\cdot k') -p\cdot p' m_\mu^2\biggr) $$

I probably have some identity related to a minus sign that I missed, but even so, I don't see how the term containing $(p\cdot p')(k\cdot k')$ can vanish, seeing how there is an odd number of such terms, meaning that no change of signs can make it vanish.

What is it I'm missing here?

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2 Answers 2

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Okay, my problem was not with the sign but rather because $g^{\mu\nu}g_{\mu\nu}=4$ in 4 dimensions, which makes it work.

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The mistake is in the fact that $g_{\mu \nu}g^{\mu\nu} = 4$ not $1$. Therefore the terms of the form $(p\cdot p')(k\cdot k')$ cancel.

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