Angular analyses of exclusive $\overline{B}\to X{\ell }_1{\ell }_2$ with complex helicity amplitudes

We present the differential rates for exclusive $\overline{B}\to X{\ell }_1{\ell }_2$, where ${\ell }_1$ is a charged massless lepton and ${\ell }_2$ is a charged or neutral massless lepton and $X$ is a mesonic system up to spin 2. The cases of interest are semileptonic $\overline{B}\to X_{\{c,u\}}{\ell }^{-}{\overline{\nu }}_{\ell }$ decays, and $\overline{B}\to X_s{\ell }^{-}{\ell }^{+}$ where the dilepton can be $c\overline{c}$ resonances or nonresonant electroweak penguins. We consider helicity amplitudes having nonzero relative phases that can be potential new sources for $CP$ violation. Our motivations for these additional phases include a complex right-handed admixture in the hadronic weak charged current for the semileptonic decays and complex Wilson coefficients in the effective Hamiltonians for the electroweak penguin decays. We demonstrate the efficacy of a novel technique of projecting out the individual angular moments in the full rate expression in a model-independent fashion. Our work is geared toward ongoing data analyses by BABAR and LHCb.

Figure 1

Effective RH coupling arising in (a) gluino loop in supersymmetric models and (b) a heavy RH $W$ boson in left-right symmetric models.

Figure 2

The quark level diagram for the SL decay $\overline{B}(b{\overline{q}}^{\prime })\to X(q{\overline{q}}^{\prime }){\ell }^{-}{\overline{\nu }}_{\ell }$ in the SM.

Figure 3

The set of four kinematic variables $\varphi \in \{q^2,\chi ,\cos {\theta }_{\ell },\cos {\theta }_V\}$ for the SL decay chain $\overline{B}\to \rho (\to \pi \pi )W^{*}(\to {\ell }^{-}{\overline{\nu }}_{\ell })$.

Figure 4

The reference frames for calculating ${\theta }_{\ell }$, ${\theta }_V$, and $\chi$ for $\overline{B}\to V{W}^{*}(\to {\ell }^{-}{\overline{\nu }}_{\ell })$ conforming to (a) Fig. 2 in Gilman–Singleton [1] and (b) Fig. 3 in Hagiwara [5]. The relations between the two sets of angles are given in the text.

Figure 5

Pull distributions from a thousand toy samples of fits to Eq. (61) with the efficiency function as set III in Table 5. The upper and lower plots use the MOM and UML techniques, respectively. No background is included.

Figure 6

A toy sample incorporating both efficiency and background effects used in the validation study. A fit to the profile is shown as well.

Figure 7

Pull distributions from a thousand toy samples of fits to Eq. (61) with the efficiency function as set III in Table 5. The upper and lower plots use the MOM and UML techniques, respectively. The samples incorporate a signal to background ratio corresponding to that in Fig. 6.