  We investigated the Paschen-Back effect in the (0,0) band of the
A6{\Sigma}+-X6{\Sigma}+ system of the CrH molecule, and we examined its
potential for estimating magnetic fields on stars and substellar objects, such
as brown dwarfs and hot exoplanets. We carried out quantum mechanical
calculations to obtain the energy level structure of the
electronic-vibrational-rotational states considered both in the absence and in
the presence of a magnetic field. Level mixing due to magnetic field
perturbation (the Paschen-Back effect) was consistently taken into account.
Then, we calculated frequencies and strengths of transitions between magnetic
sublevels. Employing these results and solving numerically a set of the
radiative transfer equations for polarized radiation, we calculated Stokes
parameters for both the individual lines and the (0,0) band depending on the
strength and orientation of the magnetic field. We demonstrate that magnetic
splitting of the individual CrH lines shows a significant asymmetry due to the
Paschen-Back effect already at 1 G field. This leads to a considerable signal
in both circular and linear polarization, up to 30 percent at the magnetic
field strength of more than 3 kG in early L dwarfs. The polarization does not
cancel out completely even at very low spectral resolution and is seen as
broad-band polarization of a few percent. Since the line asymmetry depends only
on the magnetic field strength and not on the filling factor, CrH lines provide
a very sensitive tool for direct measurement of the stellar magnetic fields on
faint cool objects, such as brown dwarfs and hot Jupiters, observed with low
spectral resolution.
