  We demonstrate optical cycling and sub-Doppler laser cooling of a cryogenic
buffer-gas beam of calcium monohydride (CaH) molecules. We measure vibrational
branching ratios for laser cooling transitions for both excited electronic
states A and B. We measure further that repeated photon scattering via the
$A\leftarrow X$ transition is achievable at a rate of $\sim$ $1.6\times10^6$
photons/s and demonstrate the interaction-time limited scattering of $\sim$
$200$ photons by repumping the largest vibrational decay channel. We also
demonstrate the ability to sub-Doppler cool a molecular beam of CaH through the
magnetically assisted Sisyphus effect. Using a standing wave of light, we lower
the molecular beam's transverse temperature from $12.2(1.2)$ mK to $5.7(1.1)$
mK. We compare these results to sub-Doppler forces modeled using optical Bloch
equations and Monte Carlo simulations of the molecular beam trajectories. This
work establishes a clear pathway for creating a magneto-optical trap (MOT) of
CaH molecules. Such a MOT could serve as a starting point for production of
ultracold hydrogen gas via dissociation of a trapped CaH cloud.
