Geometry of the effective Majorana neutrino mass in the neutrinoless double-beta decay

Abstract

The neutrinoless double-beta ($0νββ$) decay is a unique process to identify the Majorana nature of massive neutrinos, and its rate depends on the size of the effective Majorana neutrino mass $\langle m\rangle_{ee}$. We put forward a novel "coupling-rod" diagram to describe $\langle m\rangle_{ee}$ in the complex plane, by which the effects of the neutrino mass ordering and CP-violating phases on $\langle m\rangle_{ee}$ are intuitively understood. We show that this geometric language allows us to easily obtain the maximum and minimum of $|\langle m\rangle_{ee}|$. It remains usable even if there is a kind of new physics contributing to $\langle m\rangle_{ee}$, and it can also be extended to describe the effective Majorana masses $\langle m\rangle_{eμ}$, $\langle m\rangle_{eτ}$, $\langle m\rangle_{μμ}$, $\langle m\rangle_{μτ}$ and $\langle m\rangle_{ττ}$ which may appear in some other lepton-number-violating processes.