  We report the first spectropolarimetric observations and modeling of CaH
transitions in sunspots. We have detected strong polarization signals in many
CaH lines from the A-X system, and we provide the first successful fit to the
observed Stokes profiles using the previously developed theory of the
Paschen-Back effect in arbitrary electronic states of diatomic molecules and
polarized radiative transfer in molecular lines in stellar atmospheres. We
analyze the CaH Stokes profiles together with quasi-simultaneous observations
in TiO bands and conclude that CaH provides a valuable diagnostic of magnetic
fields in sunspots, starspots, cool stars, and brown dwarfs.
