  We study the N\'eel-paramagnetic quantum phase transition in two-dimensional
dimerized $S=1/2$ Heisenberg antiferromagnets using finite-size scaling of
quantum Monte Carlo data. We resolve the long standing issue of the role of
cubic interactions arising in the bond-operator representation when the dimer
pattern lacks a certain symmetry. We find non-monotonic (monotonic) size
dependence in the staggered (columnar) dimerized model, where cubic
interactions are (are not) present. We conclude that there is an irrelevant
field in the staggered model that is not present in the columnar case, but, at
variance with previous claims, it is not the leading irrelevant field. The new
exponent is $\omega_2 \approx 1.25$ and the prefactor of the correction
$L^{-\omega_2}$ is large and comes with a different sign from that of the
formally leading conventional correction with exponent $\omega_1 \approx 0.78$.
Our study highlights the possibility of competing scaling corrections at
quantum critical points.
