Ground-State Extraction of Heavy-Light Meson Semileptonic Decay Form Factors

Abstract

We discuss the extraction of heavy-light pseudo-scalar to light pseudo-scalar decay form factors from finite time correlation functions. We place particular emphasis on the contamination from excited states employing summed ratios and input from chiral perturbation theory. The analysis is performed on four CLS ensembles with $N_f = 2+1$ flavours of $\mbox{O}(a)$-improved Wilson fermions (presently) at the $\mathrm{SU}(3)$-symmetric point with relativistic heavy-quark masses in the charm region and above. The study presented here is part of the analysis aimed at the computation of the $B \to π\ell ν$ and $B_s \to K \ell ν$ semileptonic form factors, combining the continuum-limit relativistic results with static-limit calculations.

Figures (5)

• Figure 1: Schematic representation of a three-point function on the lattice. The ensembles under consideration have open boundary condition in time and the boundaries are located at $x_0=0$ and $x_0 = T$. The source position, $x_0^{\rm src}$, is kept fixed in the two- and three-point functions of a given ensemble. For the three-point functions, five distinct values of the sink position $x_0^{\rm snk}$ are considered. The injection time of the vector current varies starting from the $B$-meson interpolating operator, at $t=0$, to the $\pi$ interpolating operator at $t = t_s = x_0^{\rm snk}- x_0^{\rm src}$.
• Figure 2: Top row: Effective form factor $h_\perp(t,t_s,E_\pi)$ corresponding to eq. \ref{['Eq:h_perp']} computed using the ratios in eqs. (\ref{['Eq:RI']}-\ref{['Eq:RIII']}). The $B-$meson interpolating operator sits at $t=0$. Bottom row: Fit to the summed ratio $S_\perp(t_s,\mathbf{p}_\pi)$ in eq. \ref{['Eq:Summed ratio']}. The points are shifted in such a way that $S_\perp$ at $t_s \approx 1.6\text{ fm}$ is at $0$. The coloured bands are the result of a correlated linear fit to $S_\perp$ to extract the ground-state matrix element. The bands in the bottom row are drawn over the points that enter the fit. The horizontal bands in the top row correspond to the form factors obtained from the fit shown below. Bands of the same colour share the same $t_s$ fit range. Only fits with $\text{p-value}\ge 0.05$ are shown. The data correspond to the ensemble N202 ($a = 0.063 \text{ fm}$) with heavy quark mass $m_h^{\rm RGI} = 0.93 m_c^{\rm RGI}$, at $|\mathbf{p}_\pi| = 479 \text{ MeV}$.
• Figure 3: $B^*\pi$ contributions to $C_k$ based on HMChPT Broll-thesisBar:2023sef, see eq. \ref{['Eq:Bs_pi_to_Cmu']}. The left panel shows $\delta C_k$ for the $\mathrm{SU}(3)$ symmetric point, the right panel correspond to the physical pion mass. In both cases, $m_\pi L = 4$. The error band comes from $\beta_1 = 0.20(4) \text{ GeV}^{-1}$Gérardin_LAT24 and $\hat{g}=0.49(3)$Bernardoni:2014kla. Comparing the two plots, we clearly see how the contributions coming from the $B^*\pi$ states increase as we approach the physical pion mass.
• Figure 4: Effective form factor $h_\perp(t,t_s,E_\pi)$ corresponding to eq. \ref{['Eq:h_perp']} computed using $\mathcal{R}_k^I$ in eq. \ref{['Eq:RI']}, with $B^*\pi$ state subtraction using HMChPT as in eq. \ref{['Eq:Bs_pi_to_Cmu']}. At LO the LEC $\beta_1$ vanishes, while at NLO $\beta_1 = 0.20(4) \text{ GeV}^{-1}$Gérardin_LAT24 is employed. The data points are slightly displaced horizontally for better visibility. Points with higher opacity in the region $t\gtrsim 1.0\text{ fm}$ identify the region where HMChPT is expected to be better behaved. The data correspond to the ensemble N202 ($a = 0.063 \text{ fm}$) with heavy quark mass $m_h^{\rm RGI} = 0.93 m_c^{\rm RGI}$, at $|\mathbf{p}_\pi| = 479 \text{ MeV}$ and $t_s = 2.7 \text{ fm}$.
• Figure 5: Effective form factor $h_\perp(t,t_s,E_\pi)$ corresponding to eq. \ref{['Eq:h_perp']} computed using $\mathcal{R}^I_k$ in eq. \ref{['Eq:RI']}. Left: unsubtracted $h_\perp$. The green band is the result of the summed ratio method shown in Fig \ref{['Fig:summed_ratios_and_hperp']}. Middle: $h_\perp$ with the $B^*\pi$ excited states subtracted based on HMChPT at LO, i.e eq. \ref{['Eq:Bs_pi_to_Cmu']} with vanishing $\beta_1$. Right: $h_\perp$ with $B^*\pi$ excited states subtracted at NLO, i.e with $\beta_1 = 0.20(4)$ GeV$^{-1}$Gérardin_LAT24. Points with higher opacity in the region $t\gtrsim 1.0\text{ fm}$ identify the region where HMChPT is expected to be better behaved. The data correspond to the ensemble N202 ($a = 0.063 \text{ fm}$) with heavy quark mass $m_h^{\rm RGI} = 0.93 m_c^{\rm RGI}$, at $|\mathbf{p}_\pi| = 479 \text{ MeV}$.