We present a novel mode of neutrinoless double-$\beta$ decay with emission of a light Majoron-like scalar particle $\varphi$. We assume it couples via an effective seven-dimensional operator with a ($V+A$) lepton current and ($V\pm A$) quark currents leading to a long-range contribution that is unsuppressed by the light neutrino mass. We calculate the total double-$\beta$ decay rate and determine the fully differential shape for this mode. We find that future double-$\beta$ decay searches are sensitive to scales of the order ${\mathrm{\Lambda }}_{\mathrm{NP}}\approx 1\mathrm{TeV}$ for the effective operator and a light scalar $m_{\varphi }<0.2\mathrm{MeV}$, based on ordinary double-$\beta$ decay Majoron searches. The angular and energy distributions can deviate considerably from that of two-neutrino double-$\beta$ decay, which is the main background. We point out possible ultraviolet completions where such an effective operator can emerge.

• Received 13 November 2018
• Revised 1 February 2019

DOI:https://doi.org/10.1103/PhysRevLett.122.181801

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Published by the American Physical Society