Model-independent analysis of semileptonic $B$ decays to $D^{**}$ for arbitrary new physics

We explore semileptonic $B$ decays to the four lightest excited charm mesons, $D^{**}=\{D_0^{*},D_1^{*},D_1,D_2^{*}\}$, for nonzero charged lepton mass and for all $b\to c\ell \overline{\nu }$ four-Fermi interactions, including calculation of the $\mathcal{O}({\mathrm{\Lambda }}_{\mathrm{QCD}}/m_{c,b})$ and $\mathcal{O}({\alpha }_s)$ corrections to the heavy quark limit for all form factors. In the heavy quark limit, some form factors are suppressed at zero recoil; therefore, the $\mathcal{O}({\mathrm{\Lambda }}_{\mathrm{QCD}}/m_{c,b})$ corrections can be very important. The $D^{**}$ rates exhibit sensitivities to new physics in $b\to c\tau \overline{\nu }$ mediated decays complementary to the $D$ and $D^{*}$ modes. Since they are also important backgrounds to $B\to D^{(*)}\tau \overline{\nu }$, the correct interpretation of future semitauonic $B\to D^{(*)}$ rate measurements requires consistent treatment of both the $D^{**}$ backgrounds and the signals. Our results allow more precise and more reliable calculations of these $B\to D^{**}\ell \overline{\nu }$ decays and are systematically improvable by better data on the $e$ and $\mu$ modes. As an example, we show that the $D^{**}$ rates are more sensitive to a new $\overline{c}{\sigma }_{\mu \nu }b$ tensor interaction than the $D^{(*)}$ rates.

Figure 1

Predictions for $R(D^{**})$ as functions of a NP tensor coupling (left), and for $R(X)/R(X{)}_{\mathrm{SM}}$ for all six $D^{(*,**)}$ states (right).

Figure 2

Predicted $\mathrm{d}\mathrm{\Gamma }/\mathrm{d}w$ distributions normalized to the SM rates for each of the four $D^{(**)}$ states, in the SM (dashed curves) and for ${\tilde{\alpha }}_L^T{\tilde{\beta }}_L^T=0.35$ (solid curves), as determined from the Approximation C best-fit result, including ${\mathrm{\Lambda }}_{\mathrm{QCD}}/m_{c,b}$ and ${\alpha }_s$ corrections.

Figure 3

Allowed ranges of $R(D_1)$ and $R(D_2^{*})$ for the two narrow states, in the presence of any one of the four non-SM currents with arbitrary weak phases, as determined from the Approximation C best-fit result, including ${\mathrm{\Lambda }}_{\mathrm{QCD}}/m_{c,b}$ and ${\alpha }_s$ corrections. The SM prediction is shown by the black dot.

Figure 4

Helicity angle definitions.