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Non-Standard Interactions (NSI) of neutrinos with matter has received renewed interest in recent years. In particular, it has been shown that NSI can reconcile the $3+1$ solution with IceCube atmospheric data with $E_\nu >500$ GeV, provided that the effective coupling of NSI is large, e.g. $\sim 6 G_F$. The main goal of the present paper is to show that contrary to intuition, it is possible to build viable models with large NSI by invoking a new $U(1)$ gauge symmetry with gauge boson of mass $\sim 10$ eV. We refer to these new constructions as $3+1+ U(1)$ models. In the framework of a $3+1$ solution to LSND and MiniBooNE anomalies, we show that this novel NSI can help to solve the tension with cosmological bounds and constraints from IceCube atmospheric data with $E_\nu>500$ GeV.
We then discuss the implications of the MINOS and MINOS+ results for the 3+1+$U(1)$ scenario.
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