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.
Categories: hep-ph hep-ex