Gauged $L_{\mu }-L_{\tau }$ 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: $Z^{\prime }$-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.

Figure 1

The leading Feynman diagrams that are sensitive to the gauged $L_{\mu }-L_{\tau }$ model in a muon collider. The detectable final states are highlighted in red and purple colors.

Figure 2

The differential cross sections as a function of the polar angle $\cos \theta$. We have taken $g^{\prime }=e/3$ and $M_{Z^{\prime }}=0.5\mathrm{TeV}$ for the left two panels and $g^{\prime }=e$ and $M_{Z^{\prime }}=10\mathrm{TeV}$ for the right ones. The dashed black histogram stands for the SM backgrounds, and the red ones for the contribution of the $Z^{\prime }$ and its interference with SM processes. Their sum is given by the solid black histogram. Note that, for the upper-right panel, the new physics contribution is negative, but for clarity we reflect it to the positive axis. The integrated cross sections over the polar angle range $10°<\theta <170°$ for the SM background and the sum of SM and $Z^{\prime }$ contributions are indicated in the bottom-right corner.

Figure 3

The cut efficiencies and $R_{\mathrm{sig}}^2/R_{\mathrm{bkg}}$ with respect to $\lambda$. We take for illustration channel T1a at $m_{\mu \mu }=1000\mathrm{GeV}$. The efficiency of the cut can be roughly measured by ${\chi }^2\propto R_{\mathrm{sig}}^2/R_{\mathrm{bkg}}$.

Figure 4

The invariant mass spectrum of dimuon tracks without a photon (upper panel) and with an accompanying photon (lower panel). The purple and black curves stand for the cross sections mediated by $Z^{\prime }$ and SM particles, respectively, with the kinematic cut $|t_{{\mu }^{+}}{t}_{{\mu }^{-}}|>0.01s^2$ to remove the large $t$-channel photon-exchange backgrounds. In comparison, the original cross sections without the kinematic cut are given as the pink and gray curves. From left to right, three peaks correspond to $M_{Z^{\prime }}=200,600$, and 1000 GeV, respectively. In the upper panel, the thick black curve represents the background of ${\mu }^{+}+{\mu }^{-}\to {\mu }^{+}+{\mu }^{-}+\gamma$, while the thinner black one is the background ${\mu }^{+}+{\mu }^{-}\to {\mu }^{+}+{\mu }^{-}+{\mu }^{+}+{\mu }^{-}$.

Figure 5

The invariant mass spectrum of ditau (upper panel) and dineutrino (lower panel) processes. The purple and black curves stand for the cross sections mediated by $Z^{\prime }$ and SM particles, respectively. From left to right, three peaks correspond to $M_{Z^{\prime }}=200,600$, and 1000 GeV, respectively. The black curves are the SM backgrounds (see the text for details).

Figure 6

The sensitivity of the muon collider with a center-of-mass energy $\sqrt{s}=3\mathrm{TeV}$ and luminosity $1{\mathrm{ab}}^{-1}$, with SM background observed only (left panel) and with a $Z^{\prime }$ signal (right panel). In the left panel, the orange region is derived by assuming that only the SM background is observed, i.e., no excess of events. The darker orange region is obtained from channel I, ${\mu }^{+}+{\mu }^{-}\to {\ell }^{+}+{\ell }^{-}$, and the lighter orange region is from channels II, ${\mu }^{+}+{\mu }^{-}\to {\mu }^{+}+{\mu }^{-}+\gamma$, III ${\mu }^{+}+{\mu }^{-}\to {\tau }^{+}+{\tau }^{-}+\gamma$, and IV ${\mu }^{+}+{\mu }^{-}\to \nu +\overline{\nu }+\gamma$. The parameter space favored by the ${(g-2)}_{\mu }$ anomaly by $2\sigma$ is shown in the yellow band, and the region supported by the $B$ anomalies is reproduced as the light blue band [31]. The neutrino trident production limit is given as the green curve [57], and the LHC bound recast from $3\ell$ data is shown as the purple curve [50]. The projection of LHC and ILC with different channels is given in three dashed black curves [51]. In the right panel, we choose $M_{Z^{\prime }}=1\mathrm{TeV}$ to generate the experimental events. From outer to inner contours, $g^{\prime }$ is taken to be 0.01, 0.05, and 0.1, respectively. The white cross in the center of the contours is the prediction of the gauged $L_{\mu }-L_{\tau }$ model.