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Challenging $\bar{B}_{(s)}\to D_{(s)}^{(*)}$ Form Factors with the Heavy Quark Expansion

Marzia Bordone, Nico Gubernari, Martin Jung, Danny van Dyk

Abstract

Recent publications by three lattice QCD collaborations have provided an unprecedented wealth of theoretical predictions for the $\bar{B}_q \to D_q^{(*)}$ form factors, for spectator flavours $q=u/d$ and $q=s$. We analyse these predictions within the framework of the heavy-quark expansion (HQE) to order $α_s$, $1/m_b$, and $1/m_c^2$. For the first time, our analysis imposes unitarity bounds for all of the $\bar{B}_q^{(*)} \to D_q^{(*)}$ form factors; this includes newly identified tensor form factors arising in $\bar{B}_q^*\to D_q^{(*)}$. This enables us to treat all form factors in the same fashion. At the level of our present analysis, the inclusion of the tensor bounds is not yet constraining the HQE parameter space. We find the lattice QCD results to be well compatible with each other in a joint HQE fit as well as with QCD sum rule estimates that were used in previous HQE analyses. This is in contrast to the strong variability of the posterior predictions, in particular of the form factors ratios $R_0$ and $R_2$. Using the posterior distributions of our HQE analysis, we provide predictions for angular observables and LFU ratios in the $\bar{B}_q \to D_q^{(*)}\ell^-\barν$ decays.

Challenging $\bar{B}_{(s)}\to D_{(s)}^{(*)}$ Form Factors with the Heavy Quark Expansion

Abstract

Recent publications by three lattice QCD collaborations have provided an unprecedented wealth of theoretical predictions for the form factors, for spectator flavours and . We analyse these predictions within the framework of the heavy-quark expansion (HQE) to order , , and . For the first time, our analysis imposes unitarity bounds for all of the form factors; this includes newly identified tensor form factors arising in . This enables us to treat all form factors in the same fashion. At the level of our present analysis, the inclusion of the tensor bounds is not yet constraining the HQE parameter space. We find the lattice QCD results to be well compatible with each other in a joint HQE fit as well as with QCD sum rule estimates that were used in previous HQE analyses. This is in contrast to the strong variability of the posterior predictions, in particular of the form factors ratios and . Using the posterior distributions of our HQE analysis, we provide predictions for angular observables and LFU ratios in the decays.

Paper Structure

This paper contains 20 sections, 61 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Form factor definitions
  4. Heavy-quark expansion
  5. Unitarity bounds
  6. Analysis setup
  7. Statistical framework
  8. Theoretical data sets
  9. Fit models and parameters
  10. Implementation of the unitarity bounds
  11. Results
  12. Summary of the fit results for all models
  13. Phenomenological predictions
  14. Form factor ratios
  15. Lepton-flavour universality ratios
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Posterior predictive distributions for the unitarity bounds saturation for each $\bar{c}b$ current. The two plots on the right show the saturation of the tensor currents for the first time.
  • Figure 2: Plot of the $\bar{B}\to D^*$ form factor $A_1$ and the form factor ratios $R_{0,1,2}$, all as functions of $q^2$. Solid curves correspond to the median values and shaded regions correspond to the central $68\%$ credible interval. The nominal results of our previous work involving sum-rule results are shown in green. Our nominal results of this work in the $3/2/1^*$ model with $\text{SU(3)}_F\xspace$ symmetry are shown in blue. Auxiliary results of this work in the $3/2/1$ model are shown in orange.
  • Figure 3: Plot of the angular observable $S_{6s}$ for the $\bar{B} \to D^* \mu^- \bar{\nu}_\ell$ decays in bins of $q^2$.