Complete UV Resonances of SMEFT Dim-9 Operators for Short-range Neutrinoless Double Beta Decay

Abstract

We present a systematic classification of tree-level ultraviolet (UV) completions for dimension-nine SMEFT operators relevant to short-range neutrinoless double beta decay. Using the SMEFT J-basis framework, we categorize distinct UV completions, including both all-boson and boson-fermion-boson topologies. A primary objective is the identification of minimal UV realizations, defined as the smallest set of genuine heavy degrees of freedom required to generate each operator. Out of the 505 unique mediator combinations identified, 440 are found to be minimal, with 12 cases necessitating only two distinct heavy species. While our findings reproduce the scalar- and fermion-mediated results of Ref.[1], we significantly extend the classification by providing the first comprehensive compilation of 324 minimal UV completions featuring vector resonances -- a category previously unexplored in this context.

Figures (3)

• Figure 1: The two generic three-propagator tree topologies relevant for genuine short-range $d=9$$0\nu\beta\beta$ operators. External lines ($f$) are SM fermions, while the internal lines denote heavy bosonic or fermionic mediators. Blue and red dots indicate gauge-invariant interaction vertices in Topology I and Topology II, respectively. In the present paper these topologies are embedded into an analysis performed in the unbroken electroweak phase, so that inequivalent $\mathrm{SU}(2)_L$ and color channels are kept separate before symmetry breaking.
• Figure 2: Schematic workflow of the $J$-basis method. Starting from an independent operator basis, one iterates over all tree partitions, represents the corresponding commuting Casimir operators on the local amplitude space, and finds their common eigenbasis to obtain the $J$-basis operators whose eigenvalues encode the admissible UV mediator quantum numbers. The yellow box indicates the construction of the commuting Casimir family; the orange boxes highlight the core diagonalization step and its output. The final three boxes indicate the additional physical filters used in the strict short-range $0\nu\beta\beta$ classification.
• Figure 3: Illustrative Higgs-like SM channel in a Topology II factorization of the operator type $d_{\mathbb{C}}^{\dagger 2}L^{\dagger 2}Q^{\dagger 2}$. The bilinears $L^\dagger Q^\dagger$ and $d_{\mathbb{C}}^\dagger L^\dagger$ are matched to $S_{15}^\dagger(\bar{\mathbf{3}},\mathbf{1},1/3)$ and $S_{20}(\mathbf{3},\mathbf{2},1/6)$, while the color-singlet part of $d_{\mathbb{C}}^\dagger Q^\dagger$ is carried by the Higgs-like channel $S_4(\mathbf{1},\mathbf{2},1/2)\sim H$. Reinterpreting this line as an off-shell $H^\dagger$ insertion and applying the Higgs equation of motion returns the short-range branch shown in the green box, whose $Y_d^\dagger$ term reproduces $d_{\mathbb{C}}^{\dagger 2}L^{\dagger 2}Q^{\dagger 2}$.