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The IceCube report of a $\sim 3.5\sigma$ excess of $13\pm5$ neutrino eventsin the direction of the blazar TXS 05056+056 in 2014-2015 and the 2017detection of a high-energy neutrino, IceCube-170922A, during a gamma-ray flarefrom the same blazar, have revived the interest in scenarios for neutrinoproduction in blazars. We perform comprehensive analyses on the long-termelectromagnetic emission of TXS 05056+056 using optical, X-ray, and gamma-raydata from the All-Sky Automated Survey for Supernovae (ASAS-SN), the NeilGehrels Swift Observatory (Swift), the Monitor of All-sky X-ray Image (MAXI),and the Fermi Large Area Telescope (Fermi-LAT). We also perform numericalmodeling of the spectral energy distributions (SEDs) in four epochs prior to2017 with contemporaneous gamma-ray and lower energy (optical and/or X-ray)data. We find that the multi-epoch SEDs are consistent with a hybrid leptonicscenario, where the gamma-rays are produced in the blazar zone via externalinverse Compton scattering of accelerated electrons, and high-energy neutrinosare produced via the photomeson production process of co-accelerated protons.The multi-epoch SEDs can be satisfactorily explained with the same jetparameters and variable external photon density and electron luminosity. Usingthe maximal neutrino flux derived for each epoch, we put an upper limit of$\sim0.4-2$ on the muon neutrino number in ten years of IceCube observations.Our results are consistent with the IceCube-170922A detection, which can beexplained as an upper fluctuation from the average neutrino rate expected fromthe source, but in strong tension with the 2014-2015 neutrino flare.

Multi-Epoch Modeling of TXS 0506+056 and Implications for Long-Term High-Energy Neutrino Emission