Addendum to the Proposal P348: Search for dark sector particles weakly coupled to muon with NA64 μ

The NA64 collaboration proposes to carry out further searches for dark sector and other rare processes in missing energy events from high energy muon interactions in the hermetic detector at the CERN SPS starting from the run in 2021. \par The dark sector of particles predominantly weakly-coupled to the second and possibly third generations of the Standard Model is motivated by several theoretically interesting models. Additional to gravity this new very weak interaction between the visible and dark sector could be mediated either by a scalar ($S_\mu$) or $U'(1)$ gauge bosons ($Z_\mu$) interacting with ordinary muons. In a class of $L_\mu -L_\tau$ models the corresponding $Z_\mu$ could be light and have the coupling strength lying in the experimentally accessible region. If such $Z_\mu$ mediator exists it could also explain the muon $(g-2)_\mu$ anomaly - the discrepancy between the predicted and measured values of the muon anomalous magnetic moment. \par We propose an extension of the experiment NA64 called NA64$\mu$ to search for invisible decays of the $Z_\mu$ either to neutrinos or light DM aiming to probe the still unexplored area of the coupling strengths and masses $M_{Z_\mu} \lesssim 200$ MeV with the $\simeq 100-160$ GeV M2 beam. The NA64$\mu$ has the capability of a sensitive search for $S_\mu$s decaying invisibly to dark-sector particles, such as dark matter and millicharged particles and will cover a part of the parameter space complementary to what is intended to be probed by other searches Morevoer, the experiment will probe completely the $Z_\mu$ parameter space (coupling vs mass) that could explain the muon $(g-2)_\mu$ anomaly and could be used to search lepton flavor violation process of the $\mu - \tau$ conversion.