Probing a cosmogenic origin of astrophysical neutrinos and cosmic rays using gamma-ray observations of TXS 0506+056

In September 2017, a high-energy neutrino event detected by the IceCube Neutrino Observatory (IceCube-170922A) was associated, at the $3?$ level, with a gamma-ray flare from the blazar TXS 0506+056. Cosmic rays that are accelerated in astrophysical sources can escape from their jets and interact with background radiation fields. Interactions with the extragalactic background light can produce pions and hence neutrinos, while interactions with the cosmic microwave background predominantly drive inverse Compton scattering, contributing to electromagnetic cascades in intergalactic space. The resulting secondary gamma-ray emission can be detected with high-energy gamma-ray telescopes. Here, we report on a new search for such cosmogenic cascade emission from the blazar TXS 0506+056, using a combined data set from the Fermi-Large Area Telescope and VERITAS. We compare the gamma-ray spectrum and neutrino observations with the predictions of cosmic-ray induced cascades in intergalactic space. The observed gamma-ray spectrum is modeled as a combination of the primary spectrum and the cascade spectrum. We apply a Monte Carlo simulation with a $??^2$-based likelihood analysis to jointly determine the best-fit parameters of a proton emission spectrum describing the data and derive constraints on the proton escape luminosity. Assuming a log-parabola primary photon spectrum, we find consistency with a proton injection spectral index of $?_{p} \simeq 2.0$ and a cutoff energy of $E_{p,\text{max}} \simeq 1.3 \times 10^{16}$ eV, and constrain the isotropic proton escape luminosity to $1 \times 10^{44}$ erg s$^{-1}$ $\lesssim L_{p, esc} \lesssim 3 \times 10^{45}$ erg s$^{-1}$ at the 90 % confidence level.