We present the first detailed phenomenological analysis of a radiative Majorana neutrino mass model constructed from opening up a $Delta L = 2$ mass-dimension-11 effective operator constructed out of standard model fields. While three such operators are generated, only one dominates neutrino mass generation, namely $O_{47} = overline{L^C} L overline{Q^C} Q overline{Q} Q^C H H$, where $L$ denotes lepton doublet, $Q$ quark doublet and $H$ Higgs doublet. The underlying renormalisable theory contains the scalars $S_1 sim (bar{3},1,1/3)$ coupling as a diquark, $S_3 sim (bar{3},3,1/3)$ coupling as a leptoquark, and $Phi_3 sim (3,3,2/3)$, which has no Yukawa couplings but does couple to $S_1$ and $S_3$ in addition to the gauge fields. Neutrino masses and mixings are generated at two-loop order. A feature of this model that is different from many other radiative models is the lack of proportionality to any quark and charged-lepton masses of the neutrino mass matrix. One consequence is that the scale of new physics can be as high as $10^7$ TeV, despite the operator having a high mass dimension. This raises the prospect that $Delta L = 2$ effective operators at even higher mass dimensions may, when opened up, produce phenomenologically-viable radiative neutrino mass models. The parameter space of the model is explored through benchmark slices that are subject to experimental constraints from charged lepton flavour-violating decays, rare meson decays and neutral-meson mixing. The acceptable parameter space can accommodate the anomalies in $R_{K^{(*)}}$ and the anomalous magnetic moment of the muon.
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