Towards the discovery of new physics with lepton-universality ratios of $b\to s\ell \ell$ decays

Tests of lepton-universality as rate ratios in $b\to s\ell \ell$ transitions can be predicted very accurately in the Standard Model. The deficits with respect to expectations reported by the LHCb experiment in muon-to-electron ratios of the $B\to K^{(*)}\ell \ell$ decay rates thus point to genuine manifestations of lepton nonuniversal new physics. In this paper, we analyze these measurements in the context of effective field theory. First, we discuss the interplay of the different operators in $R_K$ and $R_{K^{*}}$ and provide predictions for $R_{K^{*}}$ in the Standard Model and in new-physics scenarios that can explain $R_K$. We also provide approximate numerical formulas for these observables in bins of interest as functions of the relevant Wilson coefficients. Secondly, we perform frequentist fits to $R_K$ and $R_{K^{*}}$. The Standard Model disagrees with these measurements at $3.7\sigma$ significance. We find excellent fits in scenarios with combinations of ${\mathcal{O}}_{9(10)}^{\ell }=\overline{s}{\gamma }^{\mu }{b}_L\ell {\gamma }_{\mu }({\gamma }_5)\ell$ operators, with pulls relative to the Standard Model in the region of $4\sigma$. An important conclusion of our analysis is that a lepton-specific contribution to ${\mathcal{O}}_{10}$ is important to understand the data. Under the hypothesis that new-physics couples selectively to the muons, we also present fits to other $b\to s\mu \mu$ data with a conservative error assessment and comment on more general scenarios. Finally, we discuss new lepton universality ratios that, if new physics is the origin of the observed discrepancy, should contribute to the statistically significant discovery of new physics in the near future.

Figure 1

$R_K$ and $R_{K^{*}}$ (in the $[1.1,6]{\mathrm{GeV}}^2$ bin) parametric dependence on one Wilson coefficient at a time, for NP affecting only the muonic coefficients, where the nodes indicate steps of $\mathrm{\Delta }(\delta C^{\mu })=+0.5$ from the SM point and in the direction of the arrows. The red solid line shows the dependence on $\delta C_9^{\mu }$, dashed blue line on $\delta C_{10}^{\mu }$, green dot-dashed on $\delta C_9^{\prime \mu }$ and orange dotted on $\delta C_{10}^{\prime \mu }$.

Figure 2

Results for $R_{K^{*}}$ and $R_K$ in the SM and various NP scenarios as a function of the invariant mass of the lepton pair, $q^2$. Solid gray line corresponds to the SM, solid red line to the scenario in which $\delta C_9^{\mu }=-1$, dot-dashed orange line to $\delta C_{10}^{\mu }=1$, dashed green line to $\delta C_9^{\prime \mu }=-1$ and blue dotted line to $\delta C_L^{\mu }=-0.5$. The shadings around each curve indicates our estimate of the hadronic uncertainties (see main text). We overlay the experimental LHCb results shown in black as points with error bars.

Figure 3

The top panel shows the result of a fit to a model that includes muon-specific NP Wilson coefficients $\delta C_9^{\mu }$ and $\delta C_{10}^{\mu }$. The cross indicates the position of the minimum. The first two graphs in the bottom row give the ${\chi }^2$ distribution projected onto each Wilson coefficient, while the third one is projected onto the difference $\delta C_L^{\mu }=\delta C_9^{\mu }-\delta C_{10}^{\mu }$. Ranges in orange and light red correspond to 1 and $3\sigma$ intervals of Wilson coefficients, respectively. [$\mathrm{\Delta }{\chi }^2=1(9)$ for $1\sigma (3\sigma )$ in the 1-parameter cases, $\mathrm{\Delta }{\chi }^2=2.3(11.83)$ for $1\sigma (3\sigma )$ in the 2-parameter fit.]

Figure 4

Contours at $1\sigma$ and $3\sigma$ level in the $(\delta C_9^{\mu },\delta C_{10}^{\mu })$ plane, in solid lines and orange and light-red colors, for the fit to $R_K$, $R_{K^{*}}$ and $\overline{\mathrm{BR}}(B_s\to \mu \mu )$. We also show the $1\sigma$ and $3\sigma$ constraints given individually by $R_K$, $R_{K^{*}}$ in the $[1.1,6]{\mathrm{GeV}}^2$ bin and $\overline{\mathrm{BR}}(B_s\to \mu \mu )$ using blue, green and gray contours, respectively. The cross indicates the position of the minimum.

Figure 5

Contours at $1\sigma$ and $3\sigma$ level in the $(\delta C_9^{\mu },\delta C_{10}^{\mu })$ plane, in solid lines and orange and light-red colors, for the fit to LUV data, $\overline{\mathrm{BR}}(B_s\to \ell \ell )$, $\mathrm{BR}(B\to K^{*}\gamma )$, the $B\to K^{*}\mu \mu$ angular distribution, the $Q_i$ observables, and the $B\to K^{*}{e}^{+}{e}^{-}$ angular observables in the ultralow bin. The cross indicates the position of the minimum. Excluding the latter electronic data produces a very similar fit. Underneath, and for comparison purposes, we show in gray colors and dashed lines the $1\sigma$ and $3\sigma$ regions for the fit to only LUV data and $\overline{\mathrm{BR}}(B_s\to \ell \ell )$ described in the Sec. 4b and plotted in Fig. 4.

Figure 6

Robustness of fit: Plot of the $p$-values of the SM as a function of $x$ and in two different fits: Solid (Blue) line represents the fit to the $R_K$ and $R_{K^{*}}$ ratios and $B_s\to \mu \mu$ and the dashed (red) line represents the global fit including the $B\to K^{*}\mu \mu$ angular observables. The variable $x$ is a factor by which we multiply all the uncertainty ranges of the theoretical parameters in Eq. (14).

Figure 7

Left panel: Two-parameter fit to $C_L^{\mu }$ and $C_L^e$ where the gray regions and dashed contours include only $R_K$, $R_{K^{*}}$ and $B_s\to \mu \mu$. The overlaid orange and light-red colored regions, enclosed by solid lines, represent the $1\sigma$ and $3\sigma$ bounds including the full data set. Right panel: Two-parameter fit to $C_9^e=C_9^{\mu }=C_9^{\text{univ}}$ and $C_L^{\mu }$ where we only show the $1\sigma$ and $3\sigma$ regions for the fit to the full data set. The crosses indicate the positions of the minima.

Figure 8

Studies of sensitivity of the new precision probes $R_6$, Eq. (18), and $R_6^{\prime }$, Eq. (19), to NP scenarios. Left panel: $R_6[0.045,1.1]$ as a function of $\delta C_{10}$ for different values of $\delta C_9$: $\delta C_9=0$ (black solid), $\delta C_9=-\delta C_{10}$ (red dashed), $\delta C_9=-1$ (blue dot-dashed). Right panel: Same for $R_6^{\prime }[0.045,1.1]$.

Figure 9

Constraining power of the hypothetical measurement of $R_6^{\prime }$ in the bin $[0.045,1.1]{\mathrm{GeV}}^2$ discussed in the main text, represented at $1\sigma$ and $3\sigma$ by the horizontal green bands. This is shown overlaid to the $1\sigma$ and $3\sigma$ gray, arc-shaped regions given by the fit to the LUV observables $R_K$ and $R_{K^{*}}$ discussed in Sec. 4a and plotted in Fig. 3. The combination of the two constraints results in the regions plotted with orange and light-red colors.