  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.
