A Comprehensive Analysis of $B_s \to D_s^{**}\ell\nu_\ell$ Decays Within and Beyond the Standard Model

Technical Summary: A Comprehensive Analysis of $B_s \to D^{**}_s \ell\nu_\ell$ Decays Within and Beyond the Standard Model

Problem Statement
The paper addresses the phenomenology of exclusive semileptonic decays of the $B_s$ meson into orbitally excited charmed-strange mesons ($D^{**}_s$), specifically the scalar $D^*_{s0}$, axial-vector $D^*_{s1}$ and $D_{s1}$, and tensor $D^*_{s2}$ states. These decays are critical for two primary reasons:

1. Background to LFU Anomalies: The $D^{**}_s$ states predominantly decay into $D^{(*)}K$, potentially mimicking signals in the measurement of lepton flavor universality (LFU) ratios $R(D^{(*)})$ at colliders like LHCb. Current measurements of $R(D^*)$ require corrections for feed-down from $B_s \to D^{**}_s \ell\nu_\ell$, and the anticipated increase in $B_s$ data at Belle II and LHCb necessitates precise theoretical predictions to reduce associated uncertainties.
2. Nature of Excited States: The internal structure of the $D^*_{s0}$ and $D^*_{s1}$ states remains ambiguous (simple $q\bar{q}$ vs. $DK$ molecular states). Precise analysis of their semileptonic decays offers a pathway to clarify their nature.

A significant theoretical challenge is the lack of Lattice QCD (LQCD) results for the $B_s \to D^{**}_s$ transition form factors across the full kinematic region.

Methodology
The authors employ a multi-stage theoretical framework to predict observables and test for New Physics (NP):

1. Heavy Quark Effective Theory (HQET) Parameter Extraction:

   • Since direct LQCD inputs are unavailable, the study relies on HQET form factors incorporating corrections up to $\mathcal{O}(\alpha_s, \Lambda/m_Q)$.
   • A data-driven approach is used to extract HQET parameters. The authors utilize $SU(3)$ flavor symmetry to relate $B_s \to D^{**}_s$ transitions to the experimentally measured $B \to D^{**}$ decays ($D^{**} = \{D^*_0, D^*_1, D_1, D^*_2\}$).
   • Using updated branching fraction measurements and bin-wise normalized differential decay rates from Belle, a $\chi^2$ analysis extracts the relevant Isgur-Wise (IW) function parameters ($\zeta, \tau$) and subleading parameters.
   • Chromomagnetic contributions are dropped based on mass splitting arguments, reducing the parameter space to $\{\zeta(1), \zeta'(1), \zeta_1\}$ for $s^\pi_\ell = 1/2^+$ states and $\{\tau(1), \tau'(1), \tau_1, \tau_2\}$ for $s^\pi_\ell = 3/2^+$ states.

2. $z$-Parametrization (BSZ):

   • To describe the form factors across the full kinematic region ($q^2$) in a model-independent manner, the authors employ the Bharucha-Straub-Zwicky (BSZ) $z$-expansion.
   • Synthetic data is generated from the central values and correlations of the extracted HQET parameters.
   • A combined fit is performed to determine the expansion coefficients ($a_n$). For the $B_s \to D^*_{s0}$ mode, Light Cone Sum Rule (LCSR) estimates at $q^2=0$ are included as additional constraints. The series is truncated at linear order ($N=1$), with truncation errors estimated as systematic uncertainties.

3. Standard Model (SM) Predictions:

   • Using the resulting form factors, the authors compute the two-fold angular decay distribution and derive various observables: differential decay rates, forward-backward asymmetries ($A_{fb}$), convexity parameters ($C_\ell$), lepton polarization asymmetries ($P_\ell$), longitudinal polarization of the daughter hadron ($F_L$), and LFU ratios ($R_{D^{**}_s}$).

4. New Physics Sensitivity Analysis:

   • The study probes NP effects assuming they manifest only in the $\tau$ channel.
   • Weak Effective Theory (WET): A global fit to existing $b \to c\tau\nu_\tau$ data ($R(D^{(*)})$, $R(\Lambda_c)$, etc.) determines benchmark Wilson Coefficients (WCs) for scalar, vector, and tensor operators. Sensitivity is tested for real and complex WCs.
   • Standard Model Effective Field Theory (SMEFT): The analysis extends to dimension-6 operators, evolving couplings from the NP scale ($\Lambda = 1$ TeV) to the $b$-quark mass scale via Renormalization Group Evolution (RGE) and matching to WET.
   • General Two Higgs Doublet Model (G2HDM): The authors analyze the Type-III 2HDM, fitting the product couplings $\rho^d_{23}\rho^{*\ell}_{33}$ and $\rho^{*u}_{32}\rho^{*\ell}_{33}$ to the same $b \to c\tau\nu_\tau$ data.

Key Contributions and Results

• Form Factor Determination: The paper provides the first comprehensive determination of $B_s \to D^{**}_s$ form factors across the full kinematic region using a data-driven HQET approach combined with $z$-expansion. This includes specific estimates at $q^2=0$ for all four decay modes, serving as benchmarks for future LQCD or LCSR calculations.
• SM Predictions: Precise predictions for integrated observables and LFU ratios are provided:
  • $R_{D^*_{s0}} = 0.158(20)$
  • $R_{D^*_{s1}} = 0.045(5)$
  • $R_{D_{s1}} = 0.073(4)$
  • $R_{D^*_{s2}} = 0.066(9)$
  • The authors note that errors in LFU ratios are reduced due to high positive correlations between numerator and denominator terms.
• NP Sensitivity and Tensions:
  • WET: The scalar ($C_{S2}$) and tensor ($C_T$) operators induce the largest deviations. Notably, the observable $P_{D^*_{s1}}^\tau$ exhibits a zero-crossing in the SM which disappears in the presence of complex $C_T$. Several observables, such as $A_{fb, \tau}^{D^*_{s0}}$ and $R_{D_{s1}}$, show deviations exceeding $2\sigma$.
  • SMEFT: The analysis identifies that the observable $A_{fb, \tau}^{D^*_{s0}}$ deviates by more than $2\sigma$ in the presence of the complex $C^{(1)}_{\ell equ}$ coupling. The zero-crossing of $P_{D_{s1}}^\tau$ is also absent in the presence of complex $C^{(3)}_{\ell equ}$.
  • 2HDM: The General 2HDM shows significant deviations for $B_s \to D^{(*)}_{s1} \tau\nu_\tau$ modes. Specifically, $A_{fb, \tau}^{D_{s1}}$ deviates by over $5\sigma$, and $P_{D_{s1}}^\tau$ by over $3\sigma$. The absence of zero-crossings for $A_{fb, \tau}^{D^{(*)}_{s1}}$ and $P_{D^{(*)}_{s1}}^\tau$ in the 2HDM is highlighted as a distinct signature.

Significance and Claims
The authors claim that their work establishes a robust theoretical baseline for $B_s \to D^{**}_s \ell\nu_\ell$ decays, which is essential for:

1. Precision LFU Measurements: Reducing theoretical uncertainties in $R(D^{(*)})$ measurements by accurately accounting for $B_s$ feed-down backgrounds.
2. Probing New Physics: Demonstrating that specific angular observables (particularly forward-backward asymmetries and lepton polarization) and LFU ratios are highly sensitive to scalar and tensor NP operators.
3. Experimental Guidance: Identifying specific observables (e.g., the zero-crossing behavior of $P_\ell^\tau$ and large deviations in $A_{fb, \tau}$) that can serve as clear experimental tests to distinguish between the SM and various NP scenarios (WET, SMEFT, 2HDM).

The paper concludes that while current data limits the extraction of all HQET parameters, the combined approach of data-driven fits and $z$-parametrization yields precise predictions that will be crucial for interpreting future high-statistics data from Belle II and LHCb.