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We showed in a previous publication that there are six independentdimension-seven operators violating both lepton and baryon numbers ($L=-B=1$)and twelve ones violating lepton but preserving baryon number ($L=2,~B=0$) instandard model effective field theory, and we calculated one-looprenormalization for the former six operators. In this work we continue ourefforts on renormalization of the operators. It turns out this could becomesubtle because the operators are connected by nontrivial relations when fermionflavors are counted. This kind of relations does not appear in lowerdimensional operators. We show how we can extract anomalous dimension matrixfor a flavor-specified basis of operators from counterterms computed for theabove flavor-blind operators without introducing singular inverse Yukawacoupling matrices. As a phenomenological application, we investigaterenormalization group effects on nuclear neutrinoless double $\beta$ decay. Wealso discuss very briefly its analog in the meson sector,$K^\pm\to\pi^\mp\mu^\pm\mu^\pm$, and indicate potential difficulties to computeits decay width.

Renormalization Group Evolution of Dimension-seven Operators in Standard Model Effective Field Theory and Relevant Phenomenology