Two-loop Renormalization Group Equations in the $\nu$SMEFT

We calculate two-loop renormalization group equations (RGEs) in the Standard Model Effective Field Theory (SMEFT) with right-handed neutrinos, i.e., the so-called $\nu$SMEFT, up to dimension five. Besides the two-loop RGEs of dimension-five (dim-5) operators, we also present those of the renormalizable couplings, including contributions from dim-5 operators. We check consistency relations among the first and second poles of $\varepsilon \equiv (4-d)/2$ with $d$ being the space-time dimension for all renormalization constants and find that those for lepton doublet and right-handed neutrino wave-function renormalization constants, as well as for renormalization constants of charged-lepton and neutrino Yukawa coupling matrices, do not hold. This leads to divergent RG functions for these fields and Yuwawa coupling matrices. We figure out that such infinite RG functions arise from the non-invariance of fields and Yukawa coupling matrices under field redefinitions, considering that flavor transformations are a kind of linear field redefinitions. Those infinite RG functions will disappear once one restores contributions from the derivative of renormalization constants with respect to the Wilson coefficients of redundant operators or, alternatively, considers the RGEs of flavor invariants, which are physical quantities and remain invariant under field redefinitions.