Study of Form Factors and Observables in $B_c^- \rightarrow D_{s}^{*-}\ell^+\ell^-$ and $B_c^- \rightarrow D_{s}^{*-}ν\barν$ decays

Technical Summary: Study of Form Factors and Observables in $B_c^- \to D_s^{*-} \ell^+ \ell^-$ and $B_c^- \to D_s^{*-} \nu \bar{\nu}$ Decays

Problem Statement
The $B_c$ meson, a unique heavy-heavy quark system ($\bar{b}c$), offers a distinct laboratory for testing the Standard Model (SM) and probing New Physics (NP), particularly through rare Flavor-Changing Neutral Current (FCNC) transitions like $b \to s \ell^+ \ell^-$ and $b \to s \nu \bar{\nu}$. While the $b \to s \ell^+ \ell^-$ transitions in $B \to K^{(*)}$ decays have been extensively studied, the corresponding rare decays in the $B_c$ sector, specifically $B_c^- \to D_s^{(*)-} \ell^+ \ell^-$ and $B_c^- \to D_s^{(*)-} \nu \bar{\nu}$, remain theoretically less constrained. A primary challenge in predicting these rates is the lack of precise knowledge regarding the $q^2$-dependence of the hadronic form factors, which encode the non-perturbative QCD dynamics of the transition. Furthermore, the $B_c \to D_s^{(*)}$ transitions involve a heavy-heavy to heavy-light transition, complicating the application of standard heavy-quark symmetry relations compared to $B \to D^{(*)}$ decays.

Methodology
The authors employ a multi-stage theoretical framework combining Perturbative QCD (pQCD), Heavy Quark Spin Symmetry (HQSS), and model-independent parametrizations to bridge the gap between theoretical inputs and physical observables.

1. Extraction of LCDA Shape Parameters at $q^2=0$:
   The study utilizes the pQCD framework to calculate form factors at zero momentum transfer ($q^2=0$). Unlike previous works that fixed Light-Cone Distribution Amplitude (LCDA) shape parameters (e.g., $\omega_{B_c}$, $\omega_{D_s}$), this work treats them as free parameters. The authors perform a $\chi^2$ minimization fit using available Lattice QCD (HPQCD) and Light-Cone Sum Rule (LCSR) inputs for $B_s \to D^{(*)}_s$ and $B_c \to D_s$ transitions at $q^2=0$. This simultaneous fit extracts consistent shape parameters for $B_c$, $B_s$, $D_s$, and $D_s^*$ mesons, along with nuisance parameters for quark masses and theoretical uncertainties. A 30% systematic uncertainty is assigned to account for neglected higher-order radiative corrections and higher-twist effects.

2. Connecting $B_c \to D_s$ and $B_c \to D_s^*$ via HQSS:
   To access the form factors for the vector meson final state ($D_s^*$) over the full physical $q^2$ range, the authors utilize Heavy Quark Effective Theory (HQET) trace formalism. They derive relations connecting the $B_c \to D_s$ and $B_c \to D_s^*$ form factors through two universal functions, $\Sigma_1(w)$ and $\Sigma_2(w)$, where $w$ is the recoil parameter. These universal functions are expanded in a Taylor series around $w=1$ (zero recoil) and fitted using lattice data for $B_c \to D_s$ form factors at high $q^2$.

3. Full $q^2$ Parametrization (BGL):
   Recognizing that HQSS relations are reliable only near zero recoil ($w \approx 1$) and pQCD is reliable near maximum recoil ($q^2 \approx 0$), the authors employ the Boyd-Grinstein-Lebed (BGL) $z$-expansion parametrization. They map the physical $q^2$ region to the unit disk using a conformal transformation. The coefficients of the BGL expansion are constrained by:

   • The $q^2=0$ pQCD predictions (converted to the helicity basis).
   • The high-$q^2$ values derived from the HQSS universal functions.
   • Unitarity bounds.
     This allows for a model-independent determination of the form factors across the entire kinematic range.

4. Prediction of Observables:
   Using the extracted form factors, the authors calculate differential decay widths, branching fractions, and a comprehensive set of angular observables for the cascade decay $B_c^- \to D_s^{*-} (\to D_s^- \pi^0) \ell^+ \ell^-$. This includes CP-averaged angular coefficients ($S_i$), forward-backward asymmetry ($A_{FB}$), polarization fractions ($F_L, F_T$), and "clean" angular observables ($P_i, P'_i$) that are less sensitive to hadronic uncertainties.

Key Contributions and Results

• Consistent LCDA Parameters: The work provides a self-consistent set of LCDA shape parameters ($\omega_{B_c} = 1.011(60)$ GeV, $\omega_{Bs} = 0.493(31)$ GeV, etc.) extracted by fitting multiple channels simultaneously, improving upon previous model-dependent estimates.
• Form Factor Determination: The study presents the first complete determination of $B_c \to D_s^*$ form factors over the full physical $q^2$ region, combining pQCD inputs at low $q^2$ and HQSS-derived constraints at high $q^2$ via the BGL formalism.
• Branching Ratios: Predictions for the branching fractions are provided:
  • $\mathcal{B}(B_c^- \to D_s^{*-} \ell^+ \ell^-) \approx 4.74 \times 10^{-7}$
  • $\mathcal{B}(B_c^- \to D_s^{*-} \nu \bar{\nu}) \approx 3.54 \times 10^{-6}$
  • The lepton flavor universality ratio $R_{B_c \to D_s^*} = \mathcal{B}(\tau^+\tau^-)/\mathcal{B}(\ell^+\ell^-)$ is predicted to be $0.0988(40)$.
• Angular Observables: The paper provides detailed $q^2$-binned predictions for angular observables. Notably, the zero-crossing points for the angular coefficients $S_5$ and $S_{6s}$ are identified at $q^2_0 = 1.89(16)$ GeV$^2$ and $3.26(24)$ GeV$^2$, respectively. The clean observables $P_1, P_2, P_3$ and $P'_4, P'_5, P'_6, P'_8$ are calculated, with specific attention to their sensitivity to lepton mass effects and potential NP contributions.
• CP Asymmetry: A non-zero CP asymmetry is predicted, primarily driven by the phase of the CKM element $V_{ub}$ in the effective Wilson coefficient $C_9^{\text{eff}}$, with integrated values of $\langle A_{CP} \rangle \approx 6.57 \times 10^{-4}$ for light leptons.

Significance
The paper establishes a rigorous Standard Model baseline for rare $B_c$ decays involving $D_s^*$ mesons. By extracting shape parameters from lattice data and employing a hybrid approach (pQCD + HQSS + BGL) to constrain form factors, the authors reduce theoretical uncertainties compared to previous pQCD-only studies. The authors emphasize that these predictions serve as essential benchmarks for future experimental measurements at the LHCb experiment, which is expected to produce a large sample of $B_c$ mesons. The precise SM predictions for angular observables, particularly the "clean" ones, are highlighted as critical tools for identifying deviations that could signal New Physics, analogous to the anomalies observed in $B \to K^* \ell^+ \ell^-$ decays. The work underscores the importance of $B_c$ decays in testing SU(3) flavor symmetry breaking and lepton flavor universality in a heavy-heavy quark system.