The Need for an Higgs Boson in the SM

In this subsection we review the role of the Higgs boson in preserving unitarity of the scattering amplitudes in the Standard Model (SM). We will look at the processes $\nu_e + \overline{\nu}_e \rightarrow W^-_L +W^+_L$, $W^-_L + W^+_L \rightarrow W^-_L +W^+_L$ and $e^- + e^+ \rightarrow W^-_L +W^+_L$ for longitudinally polarized gauge bosons. Special emphasis will be put in using algebraic methods (package FeynCalc for mathematica) to evaluate the amplitudes and cross sections. A discussion, in the form of an article, can be found here.

The calculations were done using the following programs for FeynCalc:

• For the scattering $\nu_e + \overline{\nu}_e \rightarrow W^+_L +W^-_L$ we use the program nunuWLWL.m
• For the scattering $W^+_L + W^-_L \rightarrow W^+_L +W^-_L$ we use the program WLWLWLWL.m
• For the scattering $e^- + e^+ \rightarrow W^+_L +W^-_L$ we use the program eEWLWL-LinearPol.m for the case of linearly polarized $W$ bosons (for the transverse case) and eEWLWL-Hel.m for the more realistic case of helicity polarizations. If we sum over all cases the results should be equal.