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Unitarity, analyticity, dispersion relations and resonances in strongly interacting $W_L W_L$, $Z_L Z_L$ and $hh$ scattering

If the Electroweak Symmetry Breaking Sector turns out to be strongly interacting, the actively investigated effective theory for longitudinal gauge bosons plus Higgs can be efficiently extended to cover the regime of saturation of unitarity (where the perturbative expansion breaks down). This is achieved by dispersion relations, whose subtraction constants and left cut contribution can be approximately obtained in different ways giving rise to different unitarization procedures. We illustrate the ideas with the Inverse Amplitude Method, one version of the N/D method and another improved version of the K-matrix. In the three cases we get partial waves which are unitary, analytical with the proper left and right cuts and in some cases poles in the second Riemann sheet that can be understood as dynamically generated resonances. In addition they reproduce at Next to Leading Order (NLO) the perturbative expansion for the five partial waves not vanishing (up to J=2) and they are renormalization scale ($μ$) independent. Also the unitarization formalisms are extended to the coupled channel case. Then we apply the results to the elastic scattering amplitude for the longitudinal components of the gauge bosons $V=W, Z$ at high energy. We also compute $h h \rightarrow h h$ and the inelastic process $VV\rightarrow h h$ which are coupled to the elastic $VV$ channel for custodial isospin $I=0$. We numerically compare the three methods for various values of the low-energy couplings and explain the reasons for the differences found in the $I=J=1$ partial wave. Then we study the resonances appearing in the different elastic and coupled channels in terms of the effective Lagrangian parameters.