Probing Hidden Sectors with a muon beam: Total and differential cross-sections for vector boson production in muon bremsstrahlung

Abstract

Vector bosons, such as Dark Photon $A^{\prime }$ or $Z^{\prime }$, can couple to muons and be produced in the bremsstrahlung reaction ${\mu }^{-}+N\to {\mu }^{-}+N+A^{\prime }(Z^{\prime })$. Their possible subsequent invisible decay can be detected in fixed target experiments through missing energy/momentum signature. In such experiments, not only is the energy transfer to $A^{\prime }(Z^{\prime })$ important, but also the recoil muon angle ${\psi }_{{\mu }^{\prime }}$. In this paper, we derive the total and the double differential cross-sections involved in this process using the phase space Weiszäcker-Williams and improved Weiszäcker-Williams approximations, as well as using exact-tree-level calculations. As an example, we compare the derived cross-sections and resulting signal yields in the NA64$\mu$ experiment that uses a 160 GeV muon beam at the CERN Super Proton Synchrotron accelerator. We also discuss its impact on the NA64$\mu$ expected sensitivity to explore the $(g-2{)}_{\mu }$ anomaly favoured region with a $Z^{\prime }$ boson considering $10^{12}$ muons accumulated on target.