Harmonic Oscillator Based Effective Theory and Effective Operators for Neutrinoless Double Beta Decay

Effective theory has been used to connect BSM physics of 0νββ decay to nucleon-level operators. Such nucleon-level operators must be transformed to the effective operators needed in nuclear structure calculations to quantitatively connect decay rates to BSM physics. Transitions of interest, e.g. ⁷⁶Ge -> ⁷⁶Se and ¹³⁰Te -> ¹³⁰Xe, can be treated in the canonical shell model spaces. As no experimental data exist, a more formal approach to operator renormalization is needed. We implemented HOBET (Harmonic Oscillator Based Effective Theory) operator renormalization Ô^eff_ji=P (E_j/(E_j-HQ)) Ô (E_i/(E_i-QH)) P where indices i and j correspond to eigenstates of the isotopes of interest with P representing a projector for a two-body model space and P+Q=1. The Green's functions with suitable boundary conditions reconstruct the full wave function from the projection. The choice of two-body space will depend on spectator excitation in the A-body wave function. These components enable a calculational path for renormalized operator evaluation.