Determining the Octant of $\theta_{23}$ with PINGU, T2K, NOvA and Reactor Data

We explore the prospects of determining the octant of $\theta_{23}$ with atmospheric neutrinos at PINGU. We study in detail the impact of energy and angle resolutions of the neutrino on the octant sensitivity. We show that the systematic uncertainties on the atmospheric neutrino flux predictions, especially the ones which affect the energy and zenith angle spectrum of the neutrinos, make a rather drastic reduction of the sensitivity of PINGU. We also study the prospects of measuring the octant of $\theta_{23}$ in the long baseline experiments T2K and NOvA in conjunction with the reactor experiments. We study this for two configurations of NOvA and T2K and make a comparative analysis of them. From just 3 years of PINGU data, the octant could be determined at more than $3\sigma$ C.L. for $sin^2\theta_{23}<0.419$ and $sin^2\theta_{23}>0.586$ if we add the reactor data and if normal hierarchy is true. On addition of the data from T2K and NOvA, the sensitivity improves so that the octant could be determined at the $4\sigma$ C.L. for $sin^2\theta_{23}<0.426$ and $sin^2\theta_{23}>0.586$ if normal hierarchy is true. Even a $5\sigma$ significance for the right octant can be achieved if $sin^2\theta_{23}<0.413$ for the true normal hierarchy. The sensitivity for the true inverted hierarchy is lower and we expect a $3\sigma$ determination of octant for $sin^2\theta_{23}<0.43$ and $>0.585$ from the combined data set for this case.