Unsterile-Active Neutrino Mixing: Consequences on Radiative Decay and Bounds from the X-ray Background

We consider a sterile neutrino to be an unparticle, namely an \emph{unsterile neutrino}, with anomalous dimension $\eta$ and study its mixing with a canonical active neutrino via a see-saw mass matrix. We show that there is \emph{no unitary} transformation that diagonalizes the mixed propagator and a field redefinition is required. The propagating or ``mass'' states correspond to an unsterile-like and active-like mode. The unsterile mode features a complex pole or resonance for $0 \leq \eta < 1/3$ with an ``invisible width'' which is the result of the decay of the unsterile mode into the active mode and the massless particles of the hidden conformal sector. For $\eta \geq 1/3$, the complex pole disappears, merging with the unparticle threshold. The active mode is described by a stable pole, but ``inherits'' a non-vanishing spectral density above the unparticle threshold as a consequence of the mixing. We find that the \emph{radiative} decay width of the unsterile neutrino into the active neutrino (and a photon) via charged current loops, is \emph{suppressed} by a factor $\sim \Big[2 \sin^2(\theta_0) \frac{M^2}{\Lambda^2}\Big]^\frac{\eta}{1-\eta}$, where $\theta_0$ is the mixing angle for $\eta=0$, $M$ is approximately the mass of the unsterile neutrino and $\Lambda \gg M$ is the unparticle-scale. The suppression of the radiative (visible) decay width of the sterile neutrino weakens the bound on the mass and mixing angle from the X-ray or soft gamma-ray background.