Probing the $R_{K^{(*)}}$ anomaly at a muon collider

The LHCb measurements of the $\mu /e$ ratio in $B\to K\ell \ell$ decays $(R_K)$ indicate a deficit with respect to the Standard Model prediction, supporting earlier hints of lepton universality violation observed in the $R_{K^{(*)}}$ ratio. Possible explanations of these $B$-physics anomalies include heavy $Z^{\prime }$ bosons or scalar and vector leptoquarks mediating $b\to s{\mu }^{+}{\mu }^{-}$. We note that a muon collider can directly measure this process via ${\mu }^{+}{\mu }^{-}\to b\overline{s}$ and can shed light on the lepton nonuniversality scenario. Investigating currently discussed center-of-mass energies $\sqrt{s}=3$, 6 and 10 TeV, we show that the parameter space of $Z^{\prime }$ and leptoquark solutions to the $R_{K^{(*)}}$ anomalies can be mostly covered. Effective operators explaining the anomalies can be probed with the muon collider setup $\sqrt{s}=6\mathrm{TeV}$ and integrated luminosity $L=4{\mathrm{ab}}^{-1}$.

Figure 1

Tree-level processes at a muon collider directly related to $R_{K^{(*)}}$: $Z^{\prime }$ or leptoquark.

Figure 2

The sensitivity contours for the $Z^{\prime }$ model with ${\lambda }_{22}^{\mathrm{L}}=1$ (upper panel) and ${\lambda }_{22}^{\mathrm{L}}=\sqrt{4\pi }$ (lower panel) via the process ${\mu }^{+}{\mu }^{-}\to b\overline{s}$ at muon colliders with the following setups: $\sqrt{s}=3\mathrm{TeV}$ and $L=1{\mathrm{ab}}^{-1}$ (red curves), $\sqrt{s}=6\mathrm{TeV}$ and $L=4{\mathrm{ab}}^{-1}$ (blue curves), as well as $\sqrt{s}=10\mathrm{TeV}$ and $L=10{\mathrm{ab}}^{-1}$ (green curves). The $2\sigma$ parameter space favored by a fit of $B$ anomalies is shown as the yellow band [32]. Dashed (solid) curves stand for the case with (without) flavor tagging. The $B_s$ mixing bounds are given as gray shaded regions [56]. The limits from neutrino trident production are recast as brown shaded regions [63]. The regions disfavored by LHC dimuon resonance searches are shown as black shaded regions, rescaled from Ref. [64]. This limit is overestimated as all light quarks are assumed to couple identically to $Z^{\prime }$. The projected sensitivity of HL-LHC is given by the vertical dotted lines near 1 TeV [65]. The ${\mu }^{+}{\mu }^{-}\to {\mu }^{+}{\mu }^{-}$ process at the muon collider can probe all $M_{Z^{\prime }}$ values smaller than 100 TeV with order one ${\lambda }_{22}^{\mathrm{L}}$ [20]. These are shown as two vertical lines near 100 TeV.

Figure 3

The sensitivity contours for $S_3$ (upper panel) and $U_1$ (lower panel) leptoquark models via the process ${\mu }^{+}{\mu }^{-}\to b\overline{s}$ at muon colliders with the following setups: $\sqrt{s}=3\mathrm{TeV}$ and $L=1{\mathrm{ab}}^{-1}$ (red curves), $\sqrt{s}=6\mathrm{TeV}$ and $L=4{\mathrm{ab}}^{-1}$ (blue curves), as well as $\sqrt{s}=10\mathrm{TeV}$ and $L=10{\mathrm{ab}}^{-1}$ (green curves). The $2\sigma$ parameter space favored by a fit of $B$ anomalies is shown as the yellow band [32]. Dashed (or solid) curves stand for the case with (or without) the flavor tagging. The $B_s$ mixing bound on leptoquark is given as gray shaded regions [57]. The constraints by LHC searches of leptoquark pair production (or indirect high-energy tails $q\overline{q}\to {\mu }^{+}{\mu }^{-}$) as well as the future projection of high-luminosity LHC [94, 96] are given by the brown (or black) shaded region and the dotted line, respectively. Note that for $U_1$ leptoquark, the assumption $\mathcal{L}\supset \kappa g_{\mathrm{s}}{U}_1^{†\mu }{G}_{\mu \nu }{U}_1^{\nu }$ with $\kappa =1$ has been made in deriving the constraints from leptoquark pair production [96].

Figure 4

The ratio of leptoquark signal to SM background at a muon collider with $\sqrt{s}=6\mathrm{TeV}$ and $L=4\mathrm{a}{\mathrm{b}}^{-1}$. The red curves stand for the scenario $y_{23}^{\mathrm{LQ}}·y_{22}^{\mathrm{LQ}}=0.02$ and $M_{S_3}=5\mathrm{TeV}$, where the leptoquark mass is comparable to the collision energy. The blue curves stand for the scenario $y_{23}^{\mathrm{LQ}}·y_{22}^{\mathrm{LQ}}=0.7$ and $M_{S_3}=30\mathrm{TeV}$. Here the leptoquark can safely be integrated out, and the scattering is described by effective operators. The case with (without) flavor tagging is shown as dashed (solid) curves.

Figure 5

The sensitivity of muon colliders to the square sum of effective operator coefficients $|C_9^{\mu }{|}^2+|C_{10}^{\mu }{|}^2$ as a function of the colliding energy $\sqrt{s}$. The luminosity has been assumed to satisfy the benchmark value $L=4\mathrm{a}{\mathrm{b}}^{-1}·[\sqrt{s}/(6\mathrm{TeV}){]}^2$. The blue region (the dashed blue curve) is the $3\sigma$ exclusion parameter space assuming no excess beyond the Standard Model background has been observed without (with) the flavor tagging, while the yellow band indicates the $2\sigma$ range favored by the global fit of $B$ anomalies assuming $C_9^{\mu }=-C_{10}^{\mu }$ [31, 32]. The LHC limit and the HL-LHC projection by looking for high-energy dimuon tails, assuming only the $bs\mu \mu$ couplings, are given as black shaded region and dotted line, respectively [93].