# $e^+e^-\rightarrow \mu^+\mu^-$ polarized scattering cross section

In chapter 5 of Peskin and Schroeder, on bottom of page 142, beginning of equation 5.19 (the numbering of the equation is on the top of page 143), the book writes

$$\tag{5.19} \sum_{\text{spins}}\bigg|\overline{v}(p')\gamma^\mu\bigg(\frac{1+\gamma^5}{2}\bigg)u(p)\bigg|^2=\sum_{\text{spins}}\overline{v}(p')\gamma^\mu\bigg(\frac{1+\gamma^5}{2}\bigg)u(p)\,\overline{u}(p)\gamma^\mu\bigg(\frac{1+\gamma^5}{2}\bigg)v(p')=...$$

Should the left hand side of above be omitted? We want to compute the write hand side of the equation above anyways, and I don't see how the equality holds here. $$\mu,\nu$$ are fixed indices here that come from equation 5.2 (page 132)

$$\tag{5.12} |\mathcal{M}|^2=\frac{e^4}{q^2}\big(\overline{v}(p')\gamma^\mu u(p)\overline{u}(p)\gamma^\nu v(p')\big)\big(\overline{u}(k)\gamma_\mu v(k')\overline{v}(k')\gamma_\nu u(k)\big)$$

I expand everything: $$\frac{1}{4}tr[\gamma^\alpha p_\alpha\prime \gamma^\mu (1+\gamma^5) \gamma^\beta p_\beta \gamma^\nu(1+\gamma^5)]$$ $$=\frac{1}{4}tr[\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu + \gamma^\alpha p_\alpha \prime \gamma^\mu \gamma^5 \gamma^\beta p_\beta\gamma^\nu+\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu\gamma^5+\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^5\gamma^\beta p_\beta\gamma^\nu\gamma^5]$$ swap $$\gamma^5$$ in second term for two times gets third term, swap $$\gamma^5$$ in forth term and using $$\gamma^5\cdot\gamma^5=1$$, it is the first term $$=\frac{1}{4}tr[\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu + \gamma^\alpha p_\alpha \prime \gamma^\mu \gamma^\beta p_\beta\gamma^\nu \gamma^5+\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu\gamma^5+\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu\gamma^5\gamma^5]$$ $$=\frac{1}{4}tr[\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu + \gamma^\alpha p_\alpha \prime \gamma^\mu \gamma^\beta p_\beta\gamma^\nu \gamma^5+\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu\gamma^5+\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu]$$ $$=\frac{1}{2}tr[\gamma^\alpha p_\alpha \prime \gamma^\mu\gamma^\beta p_\beta\gamma^\nu + \gamma^\alpha p_\alpha \prime \gamma^\mu \gamma^\beta p_\beta\gamma^\nu \gamma^5]$$ $$=2(p\prime^\mu p^\nu+p\prime^\nu p^\mu-g^{\mu\nu}p\prime.p-i\epsilon^{\alpha\mu\beta\nu}p\prime_\alpha p_\beta)$$ For $$|M|^2$$, $$2(p\prime^\mu p^\nu+p\prime^\nu p^\mu-g^{\mu\nu}p\prime.p-i\epsilon^{\alpha\mu\beta\nu}p\prime_\alpha p_\beta).2(k\prime_\mu k_\nu+k_\nu\prime k_\mu-g_{\mu\nu}k\prime.k-i\epsilon_{\rho\mu\sigma\nu}k^\rho k\prime ^\sigma)$$
Terms like $$p^\mu p\prime^\nu k_\mu k\prime_\nu = (p.k)(p\prime.k\prime)$$ $$p^\mu p\prime^\nu g_{\mu\nu}k\prime.k = (p.p\prime) (k\prime.k)$$ $$p^\mu p\prime^\nu i\epsilon_{\rho\mu\sigma\nu}k^\rho k\prime ^\sigma = i\epsilon_{\rho\mu\sigma\nu} p^\mu p\prime^\nu k^\rho k\prime ^\sigma$$ You got four terms like these but they cancel pairwise as, $$i\epsilon_{\rho\mu\sigma\nu} p^\mu p\prime^\nu k^\rho k\prime ^\sigma+i\epsilon_{\rho\mu\sigma\nu} p^\nu p\prime^\mu k^\rho k\prime ^\sigma$$ $$=i\epsilon_{\rho\nu\sigma\mu} p^\nu p\prime^\mu k^\rho k\prime ^\sigma+i\epsilon_{\rho\mu\sigma\nu} p^\nu p\prime^\mu k^\rho k\prime ^\sigma$$ interchange dummy variable in first term $$=-i\epsilon_{\rho\mu\sigma\nu} p^\nu p\prime^\mu k^\rho k\prime ^\sigma+i\epsilon_{\rho\mu\sigma\nu} p^\nu p\prime^\mu k^\rho k\prime ^\sigma =0$$ swap index of $$\epsilon$$ in first term
$$g^{\mu\nu}p\prime.p g_{\mu\nu}k\prime.k=4(p\prime .p)(k\prime .k)$$ as summation over index we got a four
$$g^{\mu\nu}p\prime.p i\epsilon_{\rho\mu\sigma\nu}k^\rho k\prime ^\sigma =0$$ as $$g^{\mu\nu}$$ is non-zero only for repeated index but $$\epsilon_{\rho\mu\sigma\nu}$$ is zero for repeated index. $$\epsilon^{\alpha\mu\beta\nu}p\prime_\alpha p_\beta \epsilon_{\rho\mu\sigma\nu}k^\rho k\prime ^\sigma=\epsilon^{\nu\mu\beta\alpha}p\prime_\alpha p_\beta \epsilon_{\nu\mu\sigma\rho}k^\rho k\prime ^\sigma=-2(\delta^\beta_\sigma \delta^\alpha_\rho-\delta^\beta_\rho \delta^\alpha_\sigma) p\prime_\alpha p_\beta k^\rho k\prime^\sigma =-2 (p.k\prime)(p\prime.k)+2(p\prime.k\prime)(p.k)$$