A test of the hadronic origin of $\gamma$-rays from blazars with up to month-later follow-up of IceCube Alerts with Imaging Air Cherenkov Telescopes

The sources of IceCube neutrinos are as yet unknown. The multi-messenger observation of their emission in $\gamma$-rays can be a guide to their identification, as exemplified by the case of TXS 0506+056. We suggest a new method of searching for $\gamma$-rays with Imaging Air Cherenkov Telescopes from sources in coincidence with possible astrophysical neutrinos. We propose that searches of $\gamma$-rays are extended, from the current practice of only a few days, to up to one month from a neutrino alert. We test this strategy on simulated sources modeled after the blazar \emph{TXS 0506+056-like}, emitting neutrinos and $\gamma$-rays via photohadronic interactions: the $\gamma$-rays are subsequently reprocessed in the VHE range. Using MAGIC as a benchmark example, we show that current Cherenkov Telescopes should be able to detect$\gamma$-ray counterparts to neutrino alerts with a rate of approximately one per year. It has been proposed that the high-energy diffuse neutrino flux can be explained by $\sim$ 5% of all blazars flaring in neutrinos once every 10 years, with a neutrino luminosity similar to that of TXS 0506+056 during the 2014-2015 neutrino flare. The implementation of our strategy could lead, over a timescale of one or few years, either to the detection of this subclass of blazars contributing to the diffuse neutrino flux, or to a constraint on this model