Getting the wrong sign in optical theorem
In Peskin&Schroeder chapter 16.3 we are showing how ghosts ensure unitarity in non-Abelian theories. My problem comes when I try to calculate a specific amplitude. 843
In equation 16.37 we have the term $$ \frac{1}{2}(i\mathcal{M}^{\mu\nu})g_{\mu\rho}g_{\nu\sigma}(i\mathcal{M}'^{\rho\sigma}) $$ where the amplitude $i\mathcal{M}^{\mu\nu}$ corresponds to the annihillation of a fermion=anti-fermion pair to a pair of gauge bosons while the amplitude $i\mathcal{M}'^{\rho\sigma}$ corresponds to the creation of a fermion-anti-fermion pair from a gauge boson pair. My problem is in the calculation of the second interation.
Before going there let me explain that the above expression is within a phasespace integral and using the completeness relation for the metrics we get $$ i\mathcal{M}^{\mu\nu}\left(\epsilon^{-}\mu\epsilon^+\rho+\epsilon^{+}\mu\epsilon^-\rho-\sum_{i=1}^2\epsilon^T_{i\mu}\epsilon_{i\rho}^{T}\right)\left(\epsilon^{-}\nu\epsilon\sigma^{+}+\epsilon^{+}\nu\epsilon\sigma^{-}-\sum_{i=1}^2\epsilon_{i\nu}^T\epsilon_{i\sigma}^{T}\right)i\mathcal{M}'^{\rho\sigma} $$ where most of the terms are zero and the only contributions we have are $$ i\mathcal{M}^{\mu\nu}\epsilon^{-}\mu\epsilon\nu^{+}\epsilon^+\rho\epsilon^-\sigma i\mathcal{M}'^{\rho\sigma}+i\mathcal{M}^{\mu\nu}\epsilon_\mu^{+}\epsilon_\nu^{-}\epsilon_\rho^-\epsilon_\sigma^+ i\mathcal{M}'^{\rho\sigma}=\ \left(i\mathcal{M}^{\mu\nu}\epsilon^{-}_\mu(k_1)\epsilon^{+}\nu(k_2)\right)\left(i\mathcal{M}'^{\rho\sigma}\epsilon^+\rho(k_1)\epsilon^-\sigma(k_2)\right)+\left(i\mathcal{M}^{\mu\nu}\epsilon^{+}_\mu(k_1)\epsilon^{-}\nu(k_2)\right)\left(i\mathcal{M}'^{\rho\sigma}\epsilon^-\rho(k_1)\epsilon^+\sigma(k_2)\right) $$ where we omit contributions of the transverse polarization vectors because they will vanish in our calculation of the Ward Identity.
Now I haven't found anywhere a clear calculation of the amplitude $i\mathcal{M}'^{\rho\sigma}$ so I tried to go through it on my own. The calculation is identical to $i\mathcal{M}^{\mu\nu}$ but there is an extra negative sign which I don't understand where it comes from. 818