Untangling the spin of a dark boson in $Z$ decays

We analyze the $Z$-boson decay $Z\to \gamma\, X$ into a photon ($\gamma$) plus a hypothetical light boson ($X$) belonging to a dark or secluded sector. Due to its feeble interactions with Standard Model fields, this dark boson is behaving as missing energy in the detector. We consider for $X$ the cases of spin-1 (massless dark-photon), spin-0 (axion-like), and spin-2 (graviton-like) particles and explore the way to untangle its spin origin. All these scenarios predict a universal signature for this decay, characterized by a single mono-chromatic photon in the $Z$ center of mass, with energy about half of the $Z$ mass, plus a neutrino-like missing energy associated to the $X$ boson. We show that if the $Z\to \gamma\, X$ signal should be discovered at $e^+e^-$ colliders, the angular distribution of the mono-chromatic photon in $e^+e^-\to Z\to \gamma\, X$ can provide a clean probe to discriminate between the $J=1$ and alternative $J=0/2$ spin nature of the $X$ dark boson.