Exploiting Perpendicular Momentum Distributions of Semileptonic Decays: $\bar{B}_s^0\to D_s^+μ^-\barν$ as a Case Study

TL;DR

This work develops a theory-to-detector framework to extract CKM information from semileptonic decays using the perpendicular momentum component $k_\bot$ of the final-state hadron. By deriving $d\mathcal{B}/dk_\bot^2$ from the standard $d\mathcal{B}/dq^2$ and implementing forward modelling of detector effects, the authors connect theory with LHCb data on $\bar{B}_s^0\to D_s^+\mu^-\bar{\nu}$. They demonstrate the robustness of the detector-response treatment and perform a Bayesian analysis combining lattice QCD priors for the form factors with LHCb binned data, obtaining $|V_{cb}| = 38.60^{+0.81}_{-0.80} \times 10^{-3}$ and informative constraints on the form-factor ratios; results are in good agreement with LHCb and highlight the potential of $k_\bot$-based measurements to inform global CKM analyses. The study emphasizes forward-modelling as essential for exploiting $k_\bot$ distributions and advocates for more such measurements to cross-check and augment traditional $q^2$-based analyses.

Abstract

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.

Figures (6)

• Figure 1: Sketch of the decay kinematics and illustration of the $k_\perp$ variable in the rest frame of the $\bar{B}_{s}^0$ meson. The momentum of the $D_{s}^+$ meson is labelled $k$, and the momentum transferred to the lepton-neutrino system by means of a virtual $W^*$ boson is labelled $q$. The $\hat{z}_B$ direction corresponds to the direction of flight of the $\bar{B}_{s}^0$ meson in the laboratory frame. The solid angle between the axis $D_{s}^+$--$W^*$ and the $\hat{z}_B$ axis is described by the azimuthal angle $\theta_B$ and the polar angle $\phi_B$.
• Figure 2: The rate of $\bar{B}_s^0\to D_s^+\mu^-\bar{\nu}$ as measured by the LHCb experiment LHCb:2020cyw (black error bars labelled as LHCb 2020), which is still affected by some detector effects, overlaid with our theoretical prediction \ref{['eq:theoretical-predictions:differential-br-kperp2']} in bins of $k_\perp$ (shown as blue bands), which does not account for any detector effects.
• Figure 3: The efficiency (left-hand side) and resolution (right-hand side) as documented in the supplementary material LHCb-2019-041-CDS of the LHCb measurement LHCb:2020cyw (black error bars). We fit the efficiency with a Legendre polynomial of degree three and the resolution with a double-sided Crystal Ball density (blue curves).
• Figure 4: The left-hand plot shows the rate of $\bar{B}_s^0\to D_s^+\mu^-\bar{\nu}$ as obtained from the LHCb experiment LHCb:2020cyw including detector effects (black error bars, labelled LHCb 2020). It is overlaid with our theory-level prediction \ref{['eq:theoretical-predictions:differential-br-kperp2']} in bins of $k_\perp$ (blue bands) and our detector level prediction obtained by multiplication with the response matrix \ref{['eq:detector-effects:response-matrix:def']} obtained from the full acceptance function $A$ (orange bands). The right-hand plot shows the lack of variability of the forward folding by plotting the detector-level effect using two different response matrices $R^\text{flat}$ (grey bands) and $R$ (orange bands), which are visually indistinguishable. The lack of variation of the response matrix indicates little to no dependence on the underlying signal model.
• Figure 5: The one-dimensional marginal posterior distributions the ratios of form factor parameters $\alpha_1^{(+)}/\alpha_0^{(+)}$ and $\alpha_2^{(+)}/\alpha_0^{(+)}$, accompanied by their joint two-dimensional marginal posterior distribution. Curves show the posterior density, and shared areas/contours indicate the central interval/region at $68\%$ probability (theory likelihood: grey curves, areas, and contours; LHCb Run-3 projection: dashed red curves, areas, and contours).