Production of $W^+ W^-$ pairs via $\gamma^*\gamma^* \to W^+ W^-$ subprocess with photon transverse momenta

We discuss production of $W^+ W^-$ pairs in proton-proton collisions induced by two-photon fusion including, for a first time, transverse momenta of incoming photons. The unintegrated inelastic fluxes (related to proton dissociation) of photons are calculated based on modern parametrizations of deep inelastic structure functions in a broad range of their arguments ($x$ and $Q^2$). In our approach we can get separate contributions of different $W$ helicities states. Several one- and two-dimensional differential distributions are shown and discussed. The present results are compared to the results of previous calculations within collinear factorization approach. Similar results are found except of some observables such as e.g. transverse momentum of the pair of $W^+$ and $W^-$. We find large contributions to the cross section from the region of large photon virtualities. We show decomposition of the total cross section as well as invariant mass distribution into polarisation states of both W bosons. The role of the longitudinal $F_L$ structure function is quantified. Its inclusion leads to a 4-5 % decrease of the cross section, almost independent of $M_{WW}$.