QCD Corrections and Long-Range Mechanisms of neutrinoless double beta decay

Recently it has been demonstrated that QCD corrections are numerically important for short-range mechanisms (SRM) of neutrinoless double beta decay ($0\nu\beta\beta$) mediated by heavy particle exchange. This is due to the effect of color mismatch for certain effective operators, which leads to mixing between different operators with vastly different nuclear matrix elements (NMEs). In this note we analyze the QCD corrections for long-range mechanisms (LRM), due to diagrams with light-neutrino exchange between a Standard Model (V-A)$\times$(V-A) and a beyond the SM lepton number violating vertex. We argue that in contrast to the SRM in the LRM case, there is no operator mixing from color-mismatched operators. This is due to a combined effect of the nuclear short-range correlations and color invariance. As a result, the QCD corrections to the LRM amount to an effect no more than 60%, depending on the operator in question. Although less crucial, taken into account QCD running makes theoretical predictions for $0\nu\beta\beta$-decay more robust also for LRM diagrams. We derive the current experimental constraints on the Wilson coefficients for all LRM effective operators.