Beyond $\mathcal{R}\mathbf{(}{D}^{\mathbf{(}*\mathbf{)}}\mathbf{)}$ with the general type-III 2HDM for $b\to c\tau \nu$

We review the parameter regions allowed by measurements of $\mathcal{R}(D^{(*)})$ and by a theoretical limit on $\mathcal{B}(B_c\to \tau \nu )$ in terms of generic scalar and pseudoscalar new physics couplings, $g_s$ and $g_p$. We then use these regions as constraints to predict the ranges for additional observables in $b\to c\tau \nu$ including the differential decay distributions $d\mathrm{\Gamma }/d{q}^2$; the ratios $\mathcal{R}(J/\psi )$ and $\mathcal{R}({\mathrm{\Lambda }}_c)$; and the tau-lepton polarization in $B\to D^{(\star )}\tau \nu$, with emphasis on the $CP$-violating normal polarization. Finally we map the allowed regions in $g_s$ and $g_p$ into the parameters of four versions of the Yukawa couplings of the general type-III 2HDM model. We find that the model is still viable but could be ruled out by a confirmation of a large $\mathcal{R}(J/\psi )$.

Figure 1

Left panel: The blue ring indicates the region allowed by $\mathcal{R}(D)$ at the $2\sigma$ level and the black ring shows how this region is modified following the prescription of Refs. [30, 57] described in the text. Center panel: The red ring indicates the region allowed by $\mathcal{R}(D^{*})$ at the $2\sigma$ level and the green circle results in $\mathcal{B}(B_c\to \tau \nu )\le 30\%$. The small crescent region where these two intersect is the constraint of $g_P$ that we use for our predictions. This region is magnified as the red crescent on the right panel where it is also compared with the larger orange region which uses the CCQM form factors for $\mathcal{R}(D^{*})$ and with the black region which shows the intersection between $\mathcal{R}(D^{*})$ at the $3\sigma$ level and $\mathcal{B}(B_c\to \tau \nu )\le 10\%$.

Figure 2

Normalized distributions $d\mathrm{\Gamma }(B\to D\tau \nu )/(\mathrm{\Gamma }d{q}^2)$ (left) and $d\mathrm{\Gamma }(B\to D^{\star }\tau \nu )/(\mathrm{\Gamma }d{q}^2)$ (right) as measured by BABAR [2] compared to the predictions in the SM with form factors from the CCQM of Ref. [66].

Figure 3

Predictions for $r_{J/\psi }$ compatible with the measured $\mathcal{R}(D^{\star })$ at $2\sigma$ as well as $\mathcal{B}(B_c\to \tau \nu )\le 30\%$.

Figure 4

Average tauonic polarizations $P_{L,N,T}^{\tau }(D)$ for the allowed parameter space.

Figure 5

Average tauonic polarizations $P_{L,N,T}^{\tau }(D^{*})$ for the allowed parameter space.

Figure 6

Normalized distribution $d\mathrm{\Gamma }({\mathrm{\Lambda }}_b\to {\mathrm{\Lambda }}_c\mu {\overline{\nu }}_{\mu })/(\mathrm{\Gamma }d{q}^2)$ with the CCQM form factors. The points are the LHCb data [70]. The dashed black line, green and yellow areas mark the central CCQM results and a 10% and 20% deviation respectively.

Figure 7

Predictions for $r({\mathrm{\Lambda }}_c)$ compatible with the measured $\mathcal{R}(D^{\star })$ at $2\sigma$ as well as $\mathcal{B}(B_c\to \tau \nu )\le 30\%$. The left and right panels show separately the contributions from $g_S$ and $g_P$.

Figure 8

Left panel: Region where $\mathcal{R}(D^{(*)})$ agrees with experiment at $2\sigma$ and $\mathcal{B}(B_c\to \tau \nu )\le 30\%$ in the $\tan \beta -m_{H^{\pm }}$ plane. Center and right panels: We map solutions with ${\tilde{\chi }}_{sb}^d={\tilde{\chi }}_{\tau \tau }^l={\tilde{\chi }}^1{e}^{0.8i}$, ${\tilde{\chi }}_{ct}^u={\tilde{\chi }}^2{e}^{2i}$, $\tan \beta =0.1$ and $m_{H^{\pm }}=400\mathrm{GeV}$ onto the allowed regions of Fig. 1 shown in green.

Figure 9

Left panel: Region where $\mathcal{R}(D^{(*)})$ agrees with experiment at $2\sigma$ and $\mathcal{B}(B_c\to \tau \nu )\le 30\%$ in the $\tan \beta -m_{H^{\pm }}$ plane. Center and right panels: We map solutions with ${\tilde{\chi }}_{sb}^d={\tilde{\chi }}_{\tau \tau }^l={\tilde{\chi }}^1{e}^{-0.8i}$, ${\tilde{\chi }}_{ct}^u={\tilde{\chi }}^2{e}^{-1.2i}$, $\tan \beta =15$ and $m_{H^{\pm }}=500\mathrm{GeV}$ onto the allowed regions of Fig. 1 shown in green.

Figure 10

Left panel: Region where $\mathcal{R}(D^{(*)})$ agrees with experiment at $2\sigma$ and $\mathcal{B}(B_c\to \tau \nu )\le 30\%$ in the $\tan \beta -m_{H^{\pm }}$ plane. Center and right panels: We map solutions with ${\tilde{\chi }}_{sb}^d={\tilde{\chi }}_{\tau \tau }^l={\tilde{\chi }}^1{e}^{-2.4i}$, ${\tilde{\chi }}_{ct}^u={\tilde{\chi }}^2{e}^{-1.2i}$, $\tan \beta =15$ and $m_{H^{\pm }}=400\mathrm{GeV}$ onto the allowed regions of Fig. 1 shown in green.

Figure 11

Left panel: Region where $\mathcal{R}(D^{(*)})$ agrees with experiment at $2\sigma$ and $\mathcal{B}(B_c\to \tau \nu )\le 30\%$ in the $\tan \beta -m_{H^{\pm }}$ plane. Center and right panels: We map solutions with ${\tilde{\chi }}_{sb}^d={\tilde{\chi }}_{\tau \tau }^l={\tilde{\chi }}^1{e}^{0.7i}$, ${\tilde{\chi }}_{ct}^u={\tilde{\chi }}^2{e}^{2i}$, $\tan \beta =20$ and $m_{H^{\pm }}=400\mathrm{GeV}$ onto the allowed regions of Fig. 1 shown in green.