We investigate the sensitivity of the projected TeV muon collider to the gauged $L_{\mu }-L_{\tau }$ model. Two processes are considered: $Z^{\prime }$-mediated two-body scatterings ${\mu }^{+}{\mu }^{-}\to {\ell }^{+}{\ell }^{-}$ with $\ell =\mu$ or $\tau$ and scattering with initial state photon emission, ${\mu }^{+}{\mu }^{-}\to \gamma Z^{\prime },Z^{\prime }\to \ell \overline{\ell }$, where $\ell$ can be $\mu$, $\tau$, or ${\nu }_{\mu /\tau }$. We quantitatively study the sensitivities of these two processes by taking into account possible signals and relevant backgrounds in a muon collider experiment with a center-of-mass energy $\sqrt{s}=3\mathrm{TeV}$ and a luminosity $L=1{\mathrm{ab}}^{-1}$. For two-body scattering, one can exclude $Z^{\prime }$ masses $M_{Z^{\prime }}\lesssim 100\mathrm{TeV}$ with $\mathcal{O}(1)$ gauge couplings. When $M_{Z^{\prime }}\lesssim 1\mathrm{TeV}<\sqrt{s}$, one can exclude $g^{\prime }\gtrsim 2\times {10}^{-2}$. The process with photon emission is more powerful than the two-body scattering if $M_{Z^{\prime }}<\sqrt{s}$. For instance, a sensitivity of $g^{\prime }\simeq 4\times {10}^{-3}$ can be achieved at $M_{Z^{\prime }}=1\mathrm{TeV}$. The parameter spaces favored by the $(g-2{)}_{\mu }$ and $B$ anomalies with $M_{Z^{\prime }}>100\mathrm{GeV}$ are entirely covered by a muon collider.

• Received 21 January 2021
• Accepted 6 April 2021

DOI:https://doi.org/10.1103/PhysRevD.103.095005

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Published by the American Physical Society