A multi-messenger study of the blazar PKS 0735+178: a new major neutrino
  source candidate
  PKS 0735+178 is a bright radio and $\gamma$-ray blazar that is possibly
associated with multiple neutrino events observed by the IceCube, Baikal,
Baksan, and KM3NeT neutrino telescopes. The source was found to undergo a major
flaring activity in $\gamma$-ray, X-ray, ultraviolet (UV) and optical bands. We
present a long-term detailed study of this peculiar blazar to investigate the
temporal and spectral changes in the multi-wavelength emission when the
neutrino events were observed. The analysis of Swift-XRT snapshots reveal a
flux variability of more than a factor 2 in about $5\times10^3$ seconds during
the observation on December 17, 2021. In the $\gamma$-ray band, the source was
in its historical highest flux level at the time of the arrival of the
neutrinos. The observational comparison between PKS 0735+178 and other neutrino
source candidates, such as TXS 0506+056, PKS 1424+240, and GB6 J1542+6129,
shows that all these sources share similar spectral energy distributions, very
high radio and $\gamma$-ray powers, and parsec scale jet properties. Moreover,
PKS 0735+178, like all the others, is a masquerading BL Lac. We perform
comprehensive modelling of the multiwavelength emission from PKS 0735+178
within one-zone lepto-hadronic models considering both internal and external
photon fields and estimate the expected accompanying neutrino flux. The most
optimistic scenario invokes a jet with luminosity close to the Eddington value
and the interactions of $\sim$ PeV protons with an external UV photon field.
This scenario predicts $\sim 0.067$ muon and antimuon neutrinos over the
observed 3-week flare. Our results are consistent with the detection of one
very-high-energy neutrino like IceCube-211208A.
