$Ξ_b \to Ξ$ form factors from lattice QCD and Standard-Model predictions for $Ξ_b \to Ξμ^+μ^-$ and $Ξ_b \to Ξγ$ decays

Abstract

We present the first lattice QCD determination of the $Ξ_b \to Ξ$ vector, axial-vector, and tensor form factors, which are relevant for the theory of rare decays including $Ξ_b \to Ξ\ell^+\ell^-$ and $Ξ_b \to Ξγ$. The calculation is performed with 2+1 flavors of domain-wall fermions at three different lattice spacings and pion masses in the range from approximately 430 to 230 MeV. The bottom quark is implemented using an anisotropic clover action. Three-point functions with a wide range of source-sink separations and model averaging are used to extract the ground-state contributions. We fit the dependence of the form factors on the momentum transfer, the pion mass, and the lattice spacing using modified $z$ expansions that account for subthreshold branch cuts, and apply dispersive bounds and asymptotic-behavior constraints to achieve controlled uncertainties in the full semileptonic kinematic region. Using our form factor results, we present Standard-Model predictions for the $Ξ_b^- \to Ξ^- γ$ and $Ξ_b^- \to Ξ^- μ^+μ^-$ branching fractions and two angular observables.

Figures (9)

• Figure 1: The $t'$ dependence of the different ratios with vector, axial vector and tensor current insertions for three different values of the source-sink separation, $t$. The data from the C005, C01, and F004 ensembles are shown at $|\mathbf{p}^\prime|^2=1 \cdot (\tfrac{2\pi}{L})^2$, and the data from the F1M ensemble are shown at $|\mathbf{p}^\prime|^2=2\cdot (\tfrac{2\pi}{L})^2$. The plots are in units of $\rm{GeV}^{-2}$ for the dimensionful ratios $(\mathscr{R}^V_\perp, \mathscr{R}^V_0, \mathscr{R}^A_\perp, \mathscr{R}^A_0,\mathscr{R}^{TV}_+,\mathscr{R}^{TA}_+)$ and of $\rm{GeV}^{-4}$ for $( \mathscr{R}^{TV}_\perp,\mathscr{R}^{TA}_\perp)$; the uncertainty from the lattice spacing is not shown.
• Figure 2: Evolution of the AIC average and uncertainty of the form factor $g_+$ at $|\mathbf{p}^\prime|^2=1\cdot (\tfrac{2\pi}{L})^2$ as a function of the number of sample fits, for widths of the $t_{\rm min}$ random distributions equal to $\delta=1,...,7$ (in lattice units). As in Ref. Farrell:2025gis, we found that values of $\delta > 4$ could require larger numbers of sample fits for the AIC average to converge, but still tended toward consistent central values. As before, we use $\delta=4$ and $\mathcal{O}(10,000)$ sample fits to obtain the final estimates.
• Figure 3: AIC fit results for $R_f(|\mathbf{p}^\prime|,t)$ on the $\rm{C01}$ ensemble at $|\mathbf{p}^\prime|^2=1(2\pi/L)^2$. The curves going through the data points belong to the sample fit with the highest model weight, with the bands showing only the statistical uncertainty, whereas the horizontal bands depict to the AIC average values of the extracted ground-state form factors and their total uncertainty. Data points plotted with open symbols were omitted in the highest-model-weight fit.
• Figure 4: Chiral and continuum extrapolations of the $\Xi_b \to \Xi$ vector form factors. The solid blue lines show the form factor curves in the physical limit, while the dashed lines show the modified $z$-expansion fits evaluated with the individual lattice spacings and pion masses for each ensemble. The bands include the combined statistical and systematic uncertainties.
• Figure 5: Like Fig. \ref{['fig:vector_extrap']}, but for the axial-vector form factors.