  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.
