  We show that for multi-orbital quantum impurity models the non-crossing
approximation and one-crossing approximation versions of the self-consistent
hybridization expansions violate the sum rules relating the coefficients of the
high-frequency expansion of the self energy and the product of the self energy
and Green function to thermodynamic expectation values. Comparison of
non-crossing/one-crossing results to numerically exact quantum Monte-Carlo
calculations shows that the consistency with sum rules provides a useful
estimate of the reliability of the approximations. The sum rule violations are
more pronounced, and therefore the quality of the non-crossing/one-crossing
approximation is poorer, in situations with multiple orbitals and away from
particle-hole symmetry but becomes less severe as the correlation strength
increases. The one crossing approximation is markedly superior to the
non-crossing approximation.
