Constraints on very light sterile neutrinos from $\theta_{13}$-sensitive reactor experiments

Three dedicated reactor experiments, Double Chooz, RENO and Daya Bay, have recently performed a precision measurement of the third standard mixing angle $\theta_{13}$ exploiting a multiple baseline comparison of $\nu_e \to \nu_e$ disappearance driven by the atmospheric mass-squared splitting. In this paper we show how the same technique can be used to put stringent limits on the oscillations of the electron neutrino into a fourth very light sterile species (VLS$\nu$) characterized by a mass-squared difference lying in the range [$10^{-3} - 10^{-1}$] eV$^2$. We present accurate constraints on the admixture $|U_{e4}|^2$ obtained by a 4-flavor analysis of the publicly available reactor data. In addition, we show that the estimate of $\theta_{13}$ obtained by the combination of the three reactor experiments is rather robust and substantially independent of the 4-flavor-induced perturbations provided that the new mass-squared splitting is not too low ($\gtrsim 6 \times 10^{-3}$ eV$^2$). We briefly comment on the possible impact of VLS$\nu$'s on the rest of the neutrino oscillation phenomenology and emphasize their potential role in the cosmological "dark radiation" anomaly.