  The dielectric loss of silicon nitride (Si$_3$N$_4$) limits the performance
of microwave-frequency devices that rely on this material for sensing, signal
processing, and quantum communication. Using superconducting resonant circuits,
we measure the cryogenic loss tangent of either as-deposited or
high-temperature annealed stoichiometric Si$_3$N$_4$ as a function of drive
strength and temperature. The internal loss behavior of the electrical
resonators is largely consistent with the standard tunneling model of two-level
systems (TLS), including damping caused by resonant energy exchange with TLS
and by the relaxation of non-resonant TLS. We further supplement the TLS model
with a self-heating effect to explain an increase in the loss observed in
as-deposited films at large drive powers. Critically, we demonstrate that
annealing remedies this anomalous power-induced loss, reduces the
relaxation-type damping by more than two orders of magnitude, and reduces the
resonant-type damping by a factor of three. Employing infrared absorption
spectroscopy, we find that annealing reduces the concentration of hydrogen in
the Si$_3$N$_4$, suggesting that hydrogen impurities cause substantial
dissipation.
