Double beta decay and neutrino mass models

39 pages.- 18 figures.- Minor comments added.- Matches published versionNeutrinoless double beta decay allows to constrain lepton number violating extensions of the standard model. If neutrinos are Majorana particles, the mass mechanism will always contribute to the decay rate, however, it is not a priori guaranteed to be the dominant contribution in all models. Here, we discuss whether the mass mechanism dominates or not from the theory point of view. We classify all possible (scalar-mediated) short-range contributions to the decay rate according to the loop level, at which the corresponding models will generate Majorana neutrino masses, and discuss the expected relative size of the different contributions to the decay rate in each class. Our discussion is general for models based on the SM group but does not cover models with an extended gauge. We also work out the phenomenology of one concrete 2-loop model in which both, mass mechanism and short-range diagram, might lead to competitive contributions, in some detail.T.O. is grateful to Prof. Junji Hisano for insightful comments on flavour structure of the effective interactions. J.C.H. thanks the IFIC for hospitality during his stay. Work supported by the Spanish grants FPA2014-58183-P and Multidark CSD2009-00064 (MINECO), and PROMETEOII/2014/084 (Generalitat Valenciana), by Fondecyt (Chile) under grants 11121557 and by CONICYT (Chile) project 791100017. The research of T.O. is supported by JSPS Grants-in-Aid for Scientific Research on Innovative Areas Unification and Development of the Neutrino Science Frontier Number 2610 5503. The research of F.A.P.S. is supported by the Brazilian Research Council CNPq.Peer reviewe