Nuclear matrix element ($\mathcal{M}=⟨f|\hat{O}|i⟩$)

A transition-operator matrix element between specified nuclear states. It combines their many-nucleon wavefunctions with the relevant interaction operator. In double beta decay, different assumed mechanisms and nuclear-structure calculations give different matrix elements, so converting a half-life bound to a neutrino-mass bound requires that input. A measured energy endpoint alone does not determine it.