We study the Neel-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.
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