  Starting from the one-loop divergences we obtained previously, we work out
the renormalization of the Higgs-Electroweak Chiral Lagrangian explicitly and
in detail. This includes the renormalization of the lowest-order Lagrangian, as
well as the decomposition of the remaining divergences into a complete basis of
next-to-leading-order counterterms. We provide the list of the corresponding
beta functions. We show how our results match the one-loop renormalization of
some of the dimension-6 operators in SMEFT. We further point out differences
with related work in the literature and discuss them. As an application of the
obtained results, we evaluate the divergences of the vacuum expectation value
of the Higgs field at one loop and show that they can be appropriately removed
by the corresponding renormalization. We also work out the finite
renormalization required to keep the no-tadpole condition on the Higgs field at
one loop.
