  The question of whether neutrino masses are Dirac or Majorana is one of the
most important, and most difficult, questions remaining in the neutrino sector.
Searches for neutrinoless double beta-decay may help to resolve this question,
but are also sensitive to new, higher dimension Delta L=2 operators. In this
paper we place two phenomenological constraints on these operators at dimension
d<=11. First, we require that the operators obey the quark flavor symmetries of
the Standard Model, with any violation of the symmetries being due to Yukawa
interactions, a scheme known as Minimal Flavor Violation (MFV). Second, we
require that the operators which generate neutrinoless double beta-decay, and
any operators related by the flavor symmetries, do not induce neutrino masses
above 0.05 eV, the limit implied by the atmospheric neutrino data. We find that
these requirements severely constrain the operators which can violate lepton
number, such that most can no longer contribute to neutrinoless double
beta-decay at observable rates. It is noteworthy that quark flavor symmetries
can play such a strong role in constraining new leptonic physics, even when
that physics is not quark flavor changing. Those few operators that can mimic a
Majorana neutrino mass then appear with cutoffs below a TeV, and represent new
physics which could be directly probed at the LHC or a future linear collider.
