In Gauge Theories of the Strong, Weak, and Electromagnetic Interactions, C. Quigg explains how the process $\nu \overline{\nu} \rightarrow W^+ W^-$ violates unitarity at high energy unless we add the $Z^0$ boson to the EW theory (see sections 6.2 and 6.3).

I have some difficulty with Quigg's proof that the t-channel diagram (with a leptonic mediator) cancels the s-channel amplitude (mediated by $Z^0$) at high energy. The problem comes from equations 6.3.37 and 6.3.39 (see below) where the sign of the s-channel amplitude changes, apparently for no reason. I've been trying the proof myself and I obtain equation 6.3.37, but I cannot get to 6.3.39 (although I understand why the term $S^\mu S^\nu$ vanishes). enter image description here Here $q_1,q_2$ are the 4-momenta of the incoming neutrinos, $S$ is the momentum of the $Z$ boson, $k_+,k_-$ are the 4-momenta of the outgoing $W$ bosons and $\epsilon_\pm^*$ are their polarization 4-vectors (in the high energy limit the longitudinal mode dominates: $\epsilon_\pm^*\simeq k_\pm/M_z$).

Could it be a typo? Obviously, the sign makes all the difference here since we want $\mathcal{M}_s$ to cancel $\mathcal{M}_t$. Also, I noticed that different references use different signs for the Feynman rules of the EW vertices, could that be part of the problem?


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