Gauged <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>L</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mi>τ</mml:mi></mml:msub></mml:math> at a muon collider

We investigate the sensitivity of the projected TeV muon collider to the gauged $L^{}_{\mu}$-$L^{}_{\tau}$ model. Two processes are considered: $Z'$-mediated two-body scatterings $\mu^+ \mu^- \to \ell^+ \ell^-$ with $\ell = \mu$ or $\tau$, and scattering with initial state photon emission, $\mu^+ \mu^- \to \gamma Z',~Z' \to \ell \overline{\ell}$, 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 $\sqrt{s} = 3~{\rm TeV}$ and a luminosity $L=1~{\rm ab^{-1}}$. For two-body scattering one can exclude $Z'$ masses $M^{}_{Z'} \lesssim 100~{\rm TeV}$ with $\mathcal{O}(1)$ gauge couplings. When $M^{}_{Z'} \lesssim 1~{\rm TeV} <\sqrt{s}$, one can exclude $g' \gtrsim 2\times 10^{-2}$. The process with photon emission is more powerful than the two-body scattering if $M^{}_{Z'} < \sqrt{s}$. For instance, a sensitivity of $g' \simeq 4 \times 10^{-3}$ can be achieved at $M^{}_{Z'} = 1~{\rm TeV}$. The parameter spaces favored by the $(g-2)^{}_{\mu}$ and $B$ anomalies with $M^{}_{Z'} > 100~{\rm GeV}$ are entirely covered by a muon collider.