The parsec-scale jet of the neutrino-emitting blazar TXS~0506+056

Recently the IceCube collaboration detected very high energy (VHE) neutrinos and associated them with the blazar \txs{}, raising a possible association of VHE neutrinos with this and other individual blazars. Very Long Baseline Interferometry (VLBI) is so far the only technique enabling the imaging of the innermost jet at milli-arcsec resolution (parsec scale), where the high energy emission possibly originates from. Here, we report on the radio properties of the parsec scale jet in \txs{} derived from the analysis of multi-epoch multi-frequency archive VLBI data. The half opening angle of the jet beam is about 3.8\degr, and the jet inclination angle is about 20\degr. The overall jet structure shows a helical trajectory with a precessing period of 5--6 years, likely originating from instabilities operating at parsec scales. The calculated beaming parameters (Doppler boosting factor, bulk Lorentz factor) suggest a moderately relativistic jet. The pc-scale magnetic field strength is estimated in the contexts of core-shift and variability, and is in general agreement in the range of 0.2 - 0.7 G. And it is found to decrease from a relatively larger value during the quiescent period before the ongoing flare. This suggests a conversion of magnetic field energy density to particle energy density that help accelerate injected particles at the jet base and result in variable shocked emission. The neutrino event could be associated with the onset of energetic particle injection into the jet. This scenario then supports the lepto-hadronic origin of the VHE neutrinos and $\gamma$-ray emission owing to a co-spatial origin.