Asymmetry observables and the origin of $R_{D^{(*)}}$ anomalies

The $R_{D^{(*)}}$ anomalies are among the longest-standing and most statistically significant hints of physics beyond the standard model. Many models have been proposed to explain these anomalies, including the interesting possibility that right-handed neutrinos could be involved in the $B$ decays. In this paper, we investigate future measurements at Belle II that can be used to tell apart the various new physics scenarios. Focusing on a number of $\tau$ asymmetry observables (forward-backward asymmetry and polarization asymmetries) which can be reconstructed at Belle II, we calculate the contribution of the most general dimension 6 effective Hamiltonian (including right-handed neutrinos) to all of these asymmetries. We show that Belle II can use these asymmetries to distinguish between new-physics scenarios that use right- and left-handed neutrinos, and in most cases can likely distinguish the specific model itself.

Figure 1

The diagrams that modify the $b\to c\tau \nu$ rate, and subsequently $R_D$ and $R_{D^{*}}$, with a new BSM mediator. The mediator can be one of the three types indicated in the text: (a) uncolored mediators: charged scalar or $W^{\prime }$; or (b) colored mediators: leptoquarks.

Figure 2

Ranges of $R_{D^{(*)}}$ spanned by single operators with complex Wilson coefficients. The SM prediction is denoted by a cyan dot. No other experimental constraints are imposed in this figure. The 1, 2, and $5\sigma$ contours around the current global average are shown as gray-dashed lines. We also show these contours with the projected Belle II precision [51] around the current global average (red ellipses) and a hypothetical average after Belle II that still barely allows a $5\sigma$ discovery (magenta ellipses), assuming the current correlation ${\rho }_{\mathrm{corr}}=-0.2$. (See Sec. 2d for details.)

Figure 3

The range of $R_{D^{(*)}}$ spanned by the simplified models from Table 2 with complex Wilson coefficients. The superscript on $S_1$ and $U_1$ LQ refers to the neutrino chirality which they are coupled to in each figure. No other experimental constraints are imposed in this figure. The other features are as in Fig. 2.

Figure 4

The kinematics of $\overline{B}\to D^{(*)}\tau \nu$ and subsequent $\tau \to d{\nu }^{\prime }$ decay processes, in the center-of-mass frame of the leptonic system (the “$q^2$ frame”). The black plane indicates the original decay plane, defined by the $B$ momentum ${\overrightarrow{p}}_B$ (or the $D^{(*)}$ momentum ${\overrightarrow{p}}_{D^{(*)}}$) and the leptonic pair. The red plane is the decay plane of the $\tau$, defined by the visible daughter meson $d$ and invisible daughter neutrino ${\nu }^{\prime }$ of the $\tau$. The three directions in which we will project the $\tau$ polarization asymmetries are indicated in green.

Figure 5

A schematic showing the Lorentz boost that relates the angles ${\theta }_{\tau d}$ in the $q^2$ frame on the left and ${\theta }_{\mathrm{hel}}$ in the $\tau$ rest frame on the right. The former angle is reconstructible at the $B$-factories, while the latter is used to extract ${\mathcal{P}}_{\tau }^{(*)}$. Although the $\tau$ momentum vector cannot be fully reconstructed at the $B$ factories, its magnitude is measurable, and this is sufficient to relate the two frames.

Figure 6

Two-dimensional plots of asymmetry observables for the $10\sigma$ scenario. We scan over Wilson coefficients that result in $R_{D^{(*)}}$ values within the $2\sigma$ Belle II error ellipse centered on the present-day world averages. We also impose the $\mathrm{Br}(B_c\to \tau \nu )\le 10\%$ bound [56]. The projected Belle II precision for each observable, centered on the SM prediction, is indicated by the dashed gray lines, see the text. Regions which can be realized by models with LH SM neutrinos (shown in green) are from $U_1$ LQ and single operators ${\mathcal{O}}_{LL}^T$ and ${\mathcal{O}}_{RL}^V$, while the one requiring new RH neutrinos (shown in red) corresponds to $U_1$ LQ. We can distinguish all the models from one another by measuring these asymmetry observables.

Figure 7

Two-dimensional plots of asymmetry observables in the $5\sigma$ scenario. We scan over Wilson coefficients which result in $R_{D^{(*)}}$ values within the $2\sigma$ Belle II error ellipse centered at $R_D=0.34$ and $R_{D^{*}}=0.275$. We also impose the $b\to s\nu \nu$ bound [59, 70] and the $\mathrm{Br}(B_c\to \tau \nu )\le 10\%$ bound [56]. The projected Belle II precision for each observable, centered on the SM prediction, is indicated by the dashed gray lines, see the text. All the currently viable models and single operators remain viable in this scenario. Regions which can be realized by models with LH SM neutrinos are shown in green, while those requiring new RH neutrinos are in red.

Figure 8

The ${\mathcal{P}}_T$ and ${\mathcal{P}}_T^{*}$ observables for the points from Fig. 7 that are less than $1\sigma$ apart in our estimation. These figures indicate that the $CP$-odd asymmetries ${\mathcal{P}}_T$ and ${\mathcal{P}}_T^{(*)}$ may be useful for further distinguishing the $R_2$, $U_1$, and $S_1$ leptoquark models coupled to LH neutrinos; however, the fact that they cross at the origin in the left figure also indicates that these asymmetries cannot resolve the difference in all cases.

Figure 9

The $q^2$ distribution for benchmark Wilson coefficients for models interacting with LH neutrinos (green curves) and those interacting with RH neutrinos (red curves). The decay rate for $\overline{B}\to D\tau \nu$ ($\overline{B}\to D^{*}\tau \nu$) is shown on left (right). We show the viable LQ models whose effective operators from Table 2 are related through the symmetry in (3.1)–(3.2). The dashed gray line is the SM prediction. The area under each curve is proportional to its prediction for $R_{D^{(*)}}$. Up to a rescaling factor the plots for different types of neutrinos have indistinguishable shapes.