Joint short- and long-baseline constraints on light sterile neutrinos

Recent studies have evidenced that long-baseline (LBL) experiments are sensitive to the extra CP-phases involved with light sterile neutrinos, whose existence is suggested by several anomalous short-baseline (SBL) results. We show that, within the 3+1 scheme, the combination of the existing SBL data with the LBL results coming from the two currently running experiments NO$\nu$A and T2K, enables us to simultaneously constrain two active-sterile mixing angles $\theta_{14}$ and $\theta_{24}$ and two CP-phases $\delta_{13} \equiv \delta$ and $\delta_{14}$, albeit the information on the second CP-phase is still weak at the moment. The two mixing angles are basically determined by the SBL data, while the two CP-phases are constrained by the LBL experiments, once the information coming from the SBL setups is taken into account. We also assess the robustness/fragility of the estimates of the standard 3-flavor parameters in the more general 3+1 scheme. To this regard we find that: i) the indication of CP-violation found in the 3-flavor analyses persists also in the 3+1 scheme, with $\delta_{13} \equiv \delta$ having still its best fit value around $-\pi/2$; ii) the 3-flavor weak hint in favor of the normal hierarchy becomes even less significant when sterile neutrinos come into play; iii) the weak indication of non-maximal $\theta_{23}$ (driven by NO$\nu$A disappearance data) persists in the 3+1 scheme, where maximal mixing is disfavored at almost the 90\% C.L. in both normal and inverted mass hierarchy; iv) the preference in favor of one of the two octants of $\theta_{23}$ found in the 3-flavor framework (higher octant for inverted mass hierarchy) is completely washed out in the 3+1 scheme.