  We discuss the systematic decomposition of the dimension nine neutrinoless
double beta decay operator, focusing on mechanisms with potentially small
contributions to neutrino mass, while being accessible at the LHC. We first
provide a (d=9 tree-level) complete list of diagrams for neutrinoless double
beta decay. From this list one can easily recover all previously discussed
contributions to the neutrinoless double beta decay process, such as the
celebrated mass mechanism or "exotics", such as contributions from left-right
symmetric models, R-parity violating supersymmetry and leptoquarks. More
interestingly, however, we identify a number of new possibilities which have
not been discussed in the literature previously. Contact to earlier works based
on a general Lorentz-invariant parametrisation of the neutrinoless double beta
decay rate is made, which allows, in principle, to derive limits on all
possible contributions. We furthermore discuss possible signals at the LHC for
mediators leading to the short-range part of the amplitude with one specific
example. The study of such contributions would gain particular importance if
there were a tension between different measurements of neutrino mass such as
coming from neutrinoless double beta decay and cosmology or single beta decay.
