Note on de Sitter vacua from perturbative and non-perturbative dynamics in type IIB/F-theory compactifications

The properties of the effective scalar potential are studied in the framework of type IIB string theory, taking into account perturbative and non-perturbative corrections. The former modify the K\"ahler potential and include $\alpha'$ and logarithmic corrections generated when intersecting D7 branes are part of the internal geometric configuration. The latter add exponentially suppressed K\"ahler moduli dependent terms to the fluxed superpotential. The possibility of partial elimination of such terms which may happen for particular choices of world fluxes is also taken into account. That being the case, a simple set up of three K\"ahler moduli is considered in the large volume regime, where only one of them is assumed to induce non-perturbative corrections. It is found that the shape of the F-term potential crucially depends on the parametric space associated with the perturbative sector and the volume modulus. De Sitter vacua can be obtained by implementing one of the standard mechanisms, i.e., either relying on D-terms related to $U(1)$ symmetries associated with the $D7$ branes, or introducing $\overline{D3}$ branes. In general it is observed that the combined effects of non-perturbative dynamics and the recently introduced logarithmic corrections lead to an effective scalar potential displaying interesting cosmological and phenomenological properties.