Determination of $B$-meson distribution amplitudes from $B\to π,K,D$ transition form factors

TL;DR

The paper tackles constraining the inverse moment $\lambda_B$ of the leading-twist $B$-meson LCDA by performing a global analysis of $B\to \pi, K, D$ transition form factors. It combines lattice-QCD data for $f_{B\pi}^{+,0}$, $f_{BK}^{+,0,T}$, and $f_{BD}^{+,0}$ with experimental $B\to \pi\ell\nu$ measurements, computing form factors with light-cone sum rules using a three-parameter $B$-meson LCDA and a Bourrely-Caprini-Lellouch (BCL) $q^2$ expansion. The analysis yields $\lambda_B = 217(19)_{-17}^{+82}$ MeV and $|V_{ub}| = 3.68(13)_{-1}^{+0} \times 10^{-3}$, with a joint 68.3% region $\{\lambda_B, \hat{\sigma}_1\} = [208,324]$ MeV and $\hat{\sigma}_1 = [-0.7,0.27]$; $\hat{\sigma}_2$ is weakly constrained. The NLP corrections in the LCSR reduce form factors by about 30%, lowering $\lambda_B$ relative to some sum-rule estimates, and the work highlights correlations between $\lambda_B$ and $\hat{\sigma}_1$, suggesting future refinements via higher-twist inputs and expanded data sets.

Abstract

Recent work on $B \to π$, $K$ and $B\to D$ form factors from lattice QCD and light-cone sum rules has made it possible to constrain the inverse moment $λ_B$ of the $B$-meson light-cone distribution amplitudes by performing a global fit of $B\to π,K,D$ form factors. We have compiled the $B\to π,K,D$ form factors calculated by the HPQCD, MILC, and RBC/UKQCD collaborations in the large $q^2$ region. By employing an three-parameter ansatz of the $B$-meson light-cone distribution amplitudes, we express the $B\to π,K,D$ form factors at $q^2=0$ that are calculated from light-cone sum rules, in terms of the inverse moment $λ_B$ of the leading-twist $B$-meson light-cone distribution amplitude. In the $B \to π\ell ν$ channel, we also include the available $q^2$-binned experimental data from the BaBar, Belle, and Belle~II collaborations. Using the Bourrely-Caprini-Lellouch parametrization, we perform a global fit and obtain $λ_B=217(19)_{-17}^{+82}$~MeV and $|V_{\text{ub}}|=3.68(13)_{-1}^{+0}\times10^{-3}$. The second uncertainty is obtained by constraining $λ_B>200$ MeV and varying the inverse logarithmic moments $\hatσ_1\in[-0.7,0.7]$ and $\hatσ_2\in[-6,6]$, which represents the model-dependent uncertainty from the $B$-meson light-cone distribution amplitudes. When taking into account $λ_B$ and $\hatσ_1$ as fitting parameters simultaneously, the intervals of our preditions are $λ_B=[208, 324]$~MeV and $\hatσ_1=[-0.7, 0.27]$.

Figures (5)

• Figure 1: The dependence of the $B \to P$ form factors on $\lambda_B$ for three different sets of $\{\hat{\sigma}_1, \hat{\sigma}_2\}$ . For the $B \to \pi,\,K$ cases, the Borel parameter is $M^2 = 1.25$ GeV$^2$, and the effective threshold parameters are $s_0^\pi = 0.7$ GeV$^2$ and $s_0^K = 1.05$ GeV$^2$. For the $B \to D$ case, the Borel parameter $M^2 = 4.5$ GeV$^2$, and the effective threshold parameter is $s_0^D = 6.0$ GeV$^2$.
• Figure 2: Results of the global fit to the $B \to P$ form factors versus $z$ (left pannel) and versus $q^2$ (right pannel) with the parameter set $\{\hat{\sigma}_1, \hat{\sigma}_2\} = \{0, \pi^2/6\}$. The gray points correspond to the LCSR form factor at $q^2=0$ for $\lambda_B = 350~\text{MeV}$, with the upper and lower limits of the gray error bars representing for $\lambda_B = 200~\text{MeV}$ and $\lambda_B = 500~\text{MeV}$, respectively. We also display the results by performing a BCL fit to input data only from lattice QCD as dashed curves for a comparison.
• Figure 3: Theoretical predictions for the CKM-independent differential $q^2$ distributions of the $B \to \pi \ell \bar{\nu}_\ell$ process, using the $z$-series parameters obtained from the global fit.
• Figure 4: Dependence of the form factor $f_{BK}^+(0)$ (left panel) and the best-fit value of $\lambda_B$ (right panel) on the parameters $\hat{\sigma}_1$ and $\hat{\sigma}_2$.
• Figure 5: The joint $68.3\%$ confidence region (red contour) for the parameters $\{\lambda_B, \hat{\sigma}_1\}$ (left panel) and $\{\lambda_B, \hat{\sigma}_2\}$ (right panel). The projections of the blue region onto the $\lambda_B$, $\hat{\sigma}_1$ and $\hat{\sigma}_2$ axes yield the individual $68.3\%$ confidence intervals for each parameter.