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Vector bosons, such as Dark Photon $A'$ or $Z'$, can couple to muons and beproduced in the bremsstrahlung reaction $\mu^- + N \rightarrow \mu^- + N +A'(Z')$. Their possible subsequent invisible decay can be detected in fixedtarget experiments through missing energy/momentum signature. In suchexperiments, not only is the energy transfer to $A'(Z')$ important, but alsothe recoil muon angle $\psi_{\mu'}$. In this paper, we derive the total and thedouble differential cross-sections involved in this process using the phasespace Weisz\"acker-Williams and improved Weisz\"acker-Williams approximations,as well as using exact-tree-level calculations. As an example, we compare thederived cross-sections and resulting signal yields in the NA64$\mu$ experimentthat 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 $Z'$ boson considering $10^{12}$muons accumulated on target.

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