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.

Figure 1

Effective $d=9$ operator description of the short-range mechanisms (SRM) of $0\nu \beta \beta$ decay. Diagram (a) gives the tree-level description, diagrams (b)–(d) are one-loop QCD corrections to the SRM $0\nu \beta \beta$ decay in the effective theory.

Figure 2

Long-range mechanism (LRM) contribution to $0\nu \beta \beta$ decay. The diagram on the left shows the tree-level effective diagram: Exchange of a light neutrino between a SM charged current vertex and a beyond-SM LNV vertex (indicated by the grey blob). The diagram to the right shows the one-loop QCD correction to this diagram, correcting the right-handed non-SM vertex proportional to $C(\mu )$, see text.

Figure 3

Long-range contributions to $0\nu \beta \beta$ decay: From the high energy point of view, where $SU(2{)}_L\times U(1{)}_Y$ is unbroken, these contributions are generated from $d=7$ operators, always involving one Higgs field. At low energies, the Higgs is replaced by its vacuum expectation value.