# Calculating unpolarized scattering cross section (Peskin &Schroeder)

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?

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.
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.