The impact of solar and atmospheric parameter uncertainties on the measurement of $\theta_{13}$ and $\delta$

We present in this paper the analysis of the measurement of the unknown PMNS parameters $\theta_{13}$ and $\delta$ at future LBL facilities performing complete three parameters fits, each time fully including in the fit one of the atmospheric and solar oscillation parameters within its present (future) error. We show that, due to the presence of degeneracies, present uncertainties on $\theta_{23}$ and $\Delta m^2_{23}$ worsen significantly the precision on ($\theta_{13}$,$\delta$) at future LBL experiments. Only if a precision on the atmospheric parameters at least similar to what expected at T2K-I is reached, then the sensitivities to $\theta_{13}$ and $\delta$ that have been presented in the literature for many facilities (where $\theta_{23}$ and $\Delta m^2_{23}$ are generally considered as fixed external inputs) can indeed be almost recovered.On the other hand, the impact on this measurement of the uncertainties on the solar parameters, $\theta_{12}$ and $\Delta m^2_{12}$ is already negligible. Our analysis has been performed using three reference setups: the SPL Super-Beam and the standard low-$\gamma$ $\beta$-Beam, both aiming toward a Mton Water \v{C}erenkov detector located at L=130 km; the 50 GeV Neutrino Factory with a 40 kton Magnetized Iron Detector to look for the ``golden channel'' $\nu_e \to \nu_\mu$ with baseline L=3000 km and a 4 kton Emulsion Cloud Chamber to look for the ``silver channel'' $\nu_e \to \nu_\tau$ with baseline L=732 km.