An electroweak basis for neutrinoless double $\beta$ decay

A discovery of neutrinoless double-$\beta$ decay would be profound, providing the first direct experimental evidence of $\Delta L=2$ lepton number violating processes. While a natural explanation is provided by an effective Majorana neutrino mass, other new physics interpretations should be carefully evaluated. At low--energies such new physics could manifest itself in the form of color and $SU(2)_L \times U(1)_{Y}$ invariant higher dimension operators. Here we determine a complete set of electroweak invariant dimension--9 operators, and our analysis supersedes those that only impose $U(1)_{em}$ invariance. Imposing electroweak invariance implies: 1) a significantly reduced set of leading order operators compared to only imposing $U(1)_{em}$ invariance; and 2) other collider signatures. Prior to imposing electroweak invariance we find a minimal basis of 24 dimension-9 operators, which is reduced to 11 electroweak invariant operators at leading order in the expansion in the Higgs vacuum expectation value. We set up a systematic analysis of the hadronic realization of the 4-quark operators using chiral perturbation theory, and apply it to determine which of these operators have long-distance pion enhancements at leading order in the chiral expansion. We also find at dimension--11 and dimension--13 the electroweak invariant operators that after electroweak symmetry breaking produce the remaining $\Delta L=2$ operators that would appear at dimension--9 if only $U(1)_{em}$ is imposed.