A classification of UV models for higher-dimensional SMEFT operators for neutrinoless double beta decay

The observation of lepton number violation (LNV) would be clear evidence for physics beyond the Standard Model. Non-zero neutrino masses motivate a study of processes that violate lepton number by two units, like e.g.\ neutrinoless double beta decay. In the Standard Model Effective Field Theory (SMEFT), a ubiquitous framework used for indirect new physics searches, $\Delta L =2$ operators appear at dimension-5 and higher odd dimensions. The dimension-5 Weinberg operator, that can explain neutrino masses, does not have to be realised at tree level but could arise at higher loop order in the UV models. These models, however, could produce higher dimensional operators at tree level, leaving the question which contribution dominates the neutrinoless double beta decay. We use a diagrammatic approach to systematically list all possible tree-level models for dimension-9 operators and perform a scan over which lower dimensional operators they produce. Then, we present the matching for one specific model, featuring a scalar leptoquark and a fermionic colour octet, that produces the Weinberg operator at 2-loop. We find that for all BSM couplings of order one the dim-9 operator could only dominate over the loop-suppressed operators at lower dimensions in a small mass range. We conclude that a thorough analysis including a fit to the neutrino sector will be crucial to address the question which operator class dominates in a specific model.