Neutrino oscillation experiments present anomalous results across a vast range of baselines and energies. Here we show that a $3+1$ scenario in which sterile neutrinos feel a novel matter potential $V_s$ proportional to background density of ordinary or (asymmetric) dark matter is able to explain several anomalies. In the limit of low-energies $E\lesssim 1\mathrm{TeV}$ the model gives rise to effective nonstandard interactions (NSI)-like couplings among active flavors arranged in a distinctive nongeneric rank-1 Hamiltonian. Remarkably, unlike the ordinary NSI, the induced NSI-like couplings, being generated via mixing with a gauge-singlet state, are immune to the stringent constraints from $SU(2)$ gauge invariance involving the charged leptons. The NSI-like couplings eliminate the tension between the two long-baseline (LBL) experiments NOvA and T2K provided that the potential is negative and the two sterile mixing angles ${\theta }_{14}$ and ${\theta }_{24}$ are nonzero. Solar neutrino data further constrain the strength of the potential and impose the bound $|U_{e4}{|}^2\simeq {\sin }^2{\theta }_{14}\lesssim 0.04$, while LBL and low-energy atmospheric data establish the hierarchical pattern $|U_{\mu 4}{|}^2\ll |U_{e4}{|}^2$ and $|U_{\tau 4}{|}^2\ll |U_{e4}{|}^2$. A further indication in favor of a nonzero negative potential comes from the anomalous excess of ${\nu }_e$-like events observed in Super-Kamiokande multi-GeV atmospheric neutrinos, which, in the new scenario can be explained by a modification of the 3-flavor resonance at energies of a few GeV. In the high-energy regime ($E\gtrsim 1\mathrm{TeV}$) the new framework reveals its genuine 4-flavor nature and can produce a resonant behavior at $E\simeq 10\mathrm{TeV}$ as hinted at by IceCube which, for the preferred negative values of the potential, occurs in the neutrino channel. The resonance presents unconventional features, reflecting the complex structure of the amplification phenomenon. Specifically, we identify an irreducible three-level dynamics generating a new resonance in the $({\nu }_e,{\nu }_{\mu })$ sector intertwined with two conventional resonances in the $({\nu }_e,{\nu }_s$) and $({\nu }_{\mu },{\nu }_s)$ subsystems. The novel amplification mechanism manifests with the emergence of effective mixing angles in matter (${\theta }_{12}^m$ or ${\theta }_{13}^m$) involving active neutrinos unrelated to their vacuum counterparts and can be operative even in the absence of conventional resonances. In order to have maximal phenomenological relevance the scenario requires values of $f=V_s/|V_{NC}|\sim -20$, $\mathrm{\Delta }{m}_{41}^2\sim 60{\mathrm{eV}}^2$ (both roughly uncertain by a factor of two), $|U_{e4}{|}^2\simeq {\sin }^2{\theta }_{14}\simeq 0.01–0.03$ and $|U_{\mu 4}{|}^2\simeq {\sin }^2{\theta }_{24}\simeq {10}^{-4}–{10}^{-3}$. Such a very small size of $|U_{\mu 4}{|}^2$ eliminates the tension between IceCube and the other (all negative) ${\nu }_{\mu }$ disappearance searches. The third mixing angle can have a sizable impact even for tiny values as small as $|U_{\tau 4}{|}^2\simeq {\sin }^2{\theta }_{34}\sim 3\times {10}^{-4}$. Concerning the nature (Dirac vs Majorana) of the pseudosterile neutrino, the null results from $0\nu 2\beta$ decay strongly favor the Dirac case. However, this conclusion can be evaded in $3+2$ and $3+3$ frameworks if an underlying symmetry forces the effective Majorana mass close to zero. The model can play a substantial role in the Reactor anomaly and can partially explain the Gallium one. More crucially, it can be directly probed by KATRIN, which is very sensitive to the electron-sterile neutrino admixture in the region of high $\mathrm{\Delta }{m}_{41}^2$.

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• Particle interactions
• Particle mixing & oscillations
• Phenomenology
• Neutrinos
• Sterile neutrinos