Exploiting Perpendicular Momentum Distributions of Semileptonic Decays: $ar{B}_s^0 o D_s^+μ^-arν$ as a Case Study

We derive the differential distribution of semileptonic decays with respect to the perpendicular momentum component of the final state hadron. The benefits and shortfalls arising from measurements of these distributions are discussed. Our approach is illustrated on the LHCb measurement of the $\bar{B}s^0\to D_s^+μ^-\barν$ decay distribution where the publicly available data by the LHCb experiment is used in an independent phenomenological analysis for the first time. We extract the CKM element $|V{cb}|$ and information on the shape of the relevant hadronic form factors from the measurement of the binned rate in the perpendicular momentum component of the hadron.

💡 Research Summary

This paper presents a novel methodological framework for analyzing semileptonic decays by utilizing the distribution of the final-state hadron’s momentum perpendicular to the parent B meson’s flight direction, denoted as k⊥. Traditional analyses rely on the squared momentum transfer to the lepton-neutrino system, q², which is challenging to reconstruct unambiguously in hadron collider experiments like LHCb due to the missing neutrino. In contrast, k⊥ can be determined solely from the measurable kinematics of the B meson flight direction and the daughter hadron momentum, offering a complementary and experimentally cleaner observable.

The authors first establish the theoretical foundation. Using the decay B_s^0 → D_s^+ μ^- ν_bar as a case study, they derive the differential decay rate in k⊥ starting from the standard q²-dependent rate formula. A key insight is that in the B meson rest frame, k⊥ is related to q² and the angle θ_B between the B flight direction and the D_s momentum. By exploiting the isotropic distribution of θ_B for unpolarized initial states, they show that the k⊥ distribution can be obtained through a numerical integration over the theoretically well-known q² distribution and cosθ_B. This forward transformation is essential, as an analytical inversion from k⊥ back to q² is not possible.

The core of the paper is the first independent phenomenological analysis of the LHCb collaboration’s published measurement of the k⊥ distribution for B_s^0 → D_s^+ μ^- ν_bar. To connect theory with experiment, the authors perform a detailed forward modeling of detector effects. They approximate the detector efficiency and k⊥ resolution functions using a third-order Legendre polynomial and a Double-Sided Crystal Ball function, respectively, based on the Monte Carlo information released by LHCb. This allows them to convolve their theoretical prediction with the detector response, enabling a direct comparison with the binned experimental data.

Applying this framework, the authors perform a fit to the LHCb data to simultaneously extract the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |V_cb| and constrain parameters describing the shape of the relevant B_s → D_s hadronic form factors (parameterized via the Boyd-Grinstein-Lebed expansion). Their analysis demonstrates that the k⊥ distribution is sensitive to both the overall normalization governed by |V_cb| and the form factor shapes. The study successfully validates the feasibility and utility of this alternative kinematic variable. It concludes that analyses based on k⊥ (and similar variables) can circumvent some experimental challenges associated with q² reconstruction, providing a valuable cross-check and additional constraints in global determinations of CKM elements and form factors, thereby enhancing the robustness of flavor physics tests in the Standard Model and beyond.