Probing hidden sectors with a muon beam: implication of spin-0 dark matter mediators for muon $(g-2)$ anomaly and validity of the Weisz\"acker-Williams approach

In addition to vector ($V$) type new particles extensively discussed previously, both CP-even ($S$) and CP-odd ($P$) spin-0 Dark Matter (DM) mediators can couple to muons and be produced in the bremsstrahlung reaction $\mu^- + N \rightarrow \mu^- + N + S(P)$. Their possible subsequent invisible decay into a pair of Dirac DM particles, $S(P) \to \chi \overline{\chi}$, can be detected in fixed target experiments through missing energy signature. In this paper, we focus on the case of experiments using high-energy muon beams. For this reason, we derive the differential cross-sections involved using the phase space Weisz\"acker-Williams approximation and compare them to the exact-tree-level calculations. The formalism derived can be applied in various experiments that could observe muon-spin-0 DM interactions. This can happen in present and future proton beam-dump experiments such as NA62, SHIP, HIKE, and SHADOWS; in muon fixed target experiments as NA64$\mu$, MUoNE and M3; in neutrino experiments using powerful proton beams such as DUNE. In particular, we focus on the NA64$\mu$ experiment case, which uses a 160 GeV muon beam at the CERN Super Proton Synchrotron accelerator. We compute the derived cross-sections, the resulting signal yields and we discuss the experiment projected sensitivity to probe the relic DM parameter space and the $(g-2)_\mu$ anomaly favoured region considering $10^{12}$ and $10^{13}$ muons on target.