Résumé
Although the annihilation of positrons towards the Galactic centre was established more than 50 years ago through the detection of a 511 keV $\gamma$-ray line, the origin of the positrons remains unknown. The $\gamma$-ray line should be accompanied by continuum emission from positron in-flight annihilation, which, until now, has not been detected. We aim to detect positron in-flight annihilation emission, as it provides information on the kinetic energy of the positrons that is key in determining the origin of Galactic positrons. We analysed archival data obtained by the COMPTEL instrument on the Compton Gamma-RayObservatory satellite to search for positron in-flight annihilation emission in the MeV energy range. Our analysis revealed extended emission in the MeV energy range towards the bulge of the Galaxy, which we attribute to in-flight annihilation of positrons produced with kinetic energies of $\sim$2 MeV. The observed spectrum suggests that positrons are produced quasi-mono-energetically, which could occur by the annihilation of dark matter particles with masses of $\sim$3 MeV, or through bulk motion in the jet of the microquasar 1E 1740.7-2942. We furthermore detected a point-like emission component in the MeV energy range towards the Galactic centre that is the plausible low-energy counterpart of the Fermi-LAT source 4FGLJ1745.6-2859. The broad band spectrum of the source may be explained by the injection of pair plasma from the supermassive black hole Sgr A$^\ast$ into the interstellar medium, which would also explain the point-like 511 keV line emission component that was discovered by INTEGRAL/SPI at the Galactic centre. The observed positron in-flight annihilation spectrum towards the Galactic bulge excludes $\beta^+$ decays from radioactive isotopes, as well as any mechanism producing highly relativistic positrons as the origin of the Galactic bulge positrons.