The recent availability of high-resolution far-infrared (FIR) polarization observations of galaxies using HAWC+/SOFIA has facilitated studies of extragalactic magnetic fields in the cold and dense molecular disks.We investigate if any significant structural differences are detectable in the kpc-scale magnetic field of the grand design face-on spiral galaxy M51 when traced within the diffuse (radio) and the dense and cold (FIR) interstellar medium (ISM). Our analysis reveals a complex scenario where radio and FIR polarization observations do not necessarily trace the same magnetic field structure. We find that the magnetic field in the arms is wrapped tighter at 154um than at 3 and 6 cm; statistically significant lower values for the magnetic pitch angle are measured at FIR in the outskirts (R > 7 kpc) of the galaxy. This difference is not detected in the interarm region. We find strong correlations of the polarization fraction and total intensity at FIR and radio with the gas column density and 12CO(1-0) velocity dispersion. We conclude that the arms show a relative increase of small-scale turbulent B-fields at regions with increasing column density and dispersion velocities of the molecular gas. No correlations are found with HI neutral gas. The star formation rate shows a clear correlation with the radio polarized intensity, which is not found in FIR, pointing to a small-scale dynamo-driven B-field amplification scenario. This work shows that multi-wavelength polarization observations are key to disentangling the interlocked relation between star formation, magnetic fields, and gas kinematics in the multi-phase ISM.
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