The IceCube collaboration reported a $sim 3.5sigma$ excess of $13pm5$ neutrino events in the direction of the blazar TXS 0506+56 during a $sim$6 month period in 2014-2015, as well as the ($sim3sigma$) detection of a high-energy muon neutrino during an electromagnetic flare in 2017. We explore the possibility that the 2014-2015 neutrino excess and the 2017 multi-messenger flare are both explained in a common physical framework that relies on the emergence of a relativistic neutral beam in the blazar jet due to interactions of accelerated cosmic rays (CRs) with photons. We demonstrate that the neutral beam model provides an explanation for the 2014-2015 neutrino excess without violating X-ray and $gamma$-ray constraints, and also yields results consistent with the detection of one high-energy neutrino during the 2017 flare. If both neutrino associations with TXS 05065+056 are real, our model requires that (i) the composition of accelerated CRs is light, with a ratio of helium nuclei to protons $gtrsim5$, (ii) a luminous external photon field ($sim 10^{46}$ erg s$^{-1}$) variable (on year-long timescales) is present, and (iii) the CR injection luminosity as well as the properties of the dissipation region (i.e., Lorentz factor, magnetic field, and size) vary on year-long timescales.
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