One-loop renormalization of VBS with the electroweak chiral Lagrangian in covariant gauges

This work presents a first full one-loop computation of vector boson scattering (VBS) within the non-linear effective field theory given by the bosonic sector of the usually called electroweak chiral Lagrangian (EChL). The computation is performed in the most general case of covariant $R_\xi$ gauges and is compared through all this work with the Standard Model case, whose computation in these covariant gauges is also novel and is presented also here. The calculation of the one-loop VBS amplitude is performed using the diagrammatic method by means of the one-particle-irreducible (1PI) Green functions that are involved in these scattering processes. The central part of this work is then devoted to the renormalization of all the n-legs one-loop 1PI Green functions involved. This renormalization is performed in the most general off-shell case with arbitrary external legs momenta. We then describe in full detail the renormalization program, which within this context of the EChL, implies to derive all the counterterms for both the electroweak parameters, like boson masses and gauge couplings, and those for the EChL coefficients. These later are crucial for the renormalization of the new divergences typically appearing when computing loops with the lowest chiral dimension Lagrangian. We present here the full list of involved divergences and counterterms in the $R_\xi$ gauges and derive the complete set of renormalization group equations for the EChL coefficients. In the last part of this work, we present the EChL numerical results for the one-loop cross section in the WZ channel and compare them with the SM results.