Search for 100 MeV to 10 GeV gamma-ray lines in the Fermi-LAT data and implications for gravitino dark matter in the $\mu\nu$SSM

Dark matter decay or annihilation may produce monochromatic signals in the gamma-ray energy range. In this work we argue that there are strong theoretical motivations for studying these signals in the framework of gravitino dark matter decay and we perform a search for gamma-ray spectral lines from 100\,MeV to 10\,GeV with Fermi-LAT data. In contrast to previous line searches at higher energies, the sensitivity of the present search is dominated by systematic uncertainties across most of the energy range considered. We estimate the size of systematic effects by analysing the flux from a number of control regions, and include the systematic uncertainties consistently in our fitting procedure. We have not observed any significant signals and present model-independent limits on gamma-ray line emission from decaying and annihilating dark matter. We apply the former limits to the case of the gravitino, a well-known dark matter candidate in supersymmetric scenarios. In particular, the $R$-parity violating "$\mu$ from $\nu$" Supersymmetric Standard Model ($\mu\nu$SSM) is an attractive scenario in which including right-handed neutrinos solves the $\mu$ problem of the Minimal Supersymmetric Standard Model while simultaneously explaining the origin of neutrino masses. At the same time, the violation of $R$-parity renders the gravitino unstable and subject to decay into a photon and a neutrino. As a consequence of the limits on line emission, $\mu\nu$SSM gravitinos with masses larger than about 5\,GeV, or lifetimes smaller than about $10^{28}$\,s, are excluded at 95% confidence level as dark matter candidates.