  We discuss new diffractive mechanism of central exclusive production of
$W^+W^-$ pairs in proton-proton collisions at the LHC. We include diagrams with
intermediate virtual Higgs boson as well as quark box diagrams. Several
observables related to this process are calculated. Predictions for the total
cross section and differential distributions in $W$-boson rapidity and
transverse momentum as well as $WW$ invariant mass are presented. We also show
results for different polarization states of the final $W^{\pm}$ bosons. We
compare the contribution of the $\gamma \gamma \to W^+ W^-$ mechanism
considered in the literature with the contribution of the diffractive mechanism
through the $gg \to W^+ W^-$ subprocess for the different observables. The
phase space integrated diffractive contribution when separated is only a small
fraction of fb compared to 115.4 fb of the $\gamma\gamma$-contribution without
absorption. The latter contribution dominates at small four-momentum transfers
squared in the proton lines and in a broad range of $W^+ W^-$ invariant masses.
This offers a possibility of efficient searches for anomalous triple-boson
($\gamma W W$) and quartic-boson ($\gamma \gamma WW$) couplings and testing
models beyond the Standard Model. We discuss shortly also the $p p \to p p
\gamma \gamma$ process, where the box contribution is very similar to that for
$W^+ W^-$ and compare our results with recent CDF data. Nice agreement has been
achieved without additional free parameters.
