The origin of high-energy emission in blazars jets ( i.e. , leptonic versus hadronic ) has been a long-standing matter of debate .
Here , we focus on one variant of hadronic models where proton synchrotron radiation accounts for the observed steady \gamma -ray blazar emission .
Using analytical methods , we derive the minimum jet power ( P _ { j, \min } ) for the largest blazar sample analyzed to date ( 145 sources ) , taking into account uncertainties of observables and jet ’ s physical parameters .
We compare P _ { j, \min } against three characteristic energy estimators for accreting systems , i.e. , the Eddington luminosity , the accretion disk luminosity , and the power of the Blandford-Znajek process , and find that P _ { j, \min } is about 2 orders of magnitude higher than all energetic estimators for the majority of our sample .
The derived magnetic field strengths in the emission region require either large amplification of the jet ’ s magnetic field ( factor of 30 ) or place the \gamma -ray production site at sub-pc scales .
The expected neutrino emission peaks at \sim 0.1 - 10 EeV , with typical peak neutrino fluxes \sim 10 ^ { -4 } times lower than the peak \gamma -ray fluxes .
We conclude that if relativistic hadrons are present in blazar jets , they can only produce a radiatively subdominant component of the overall spectral energy distribution of the blazar ’ s steady emission .