Probing the inverse moment of $B_s$-meson distribution amplitude via $B_s \to \eta_s$ form factors

We investigate the inverse moment of the $B_s$-meson light-cone distribution amplitude (LCDA), denoted as $\lambda_{B_s}$ and defined within the heavy quark effective theory, through the calculation of $B_s \to \eta_s$ form factors. The presence of the $s$-quark inside the $B_s$-meson dictates a notable departure of approximately $20\%$ in the $\lambda_{B_s}$ value compared to the non-strange case $\lambda_{B_q}$, as computed within the QCD sum rule approach, albeit with significant uncertainty. First, we compute the decay constant of the $\eta_s$-meson utilizing two-point sum rules while retaining finite $s$-quark mass contributions. Next, we constrain the parameter $\lambda_{B_s}$ by calculating $B_s \to \eta_s$ form factors within the light-cone sum rule approach, using $B_s$-meson LCDAs, and leveraging Lattice QCD estimates at zero momentum transfer from the HPQCD collaboration. Our findings yield $\lambda_{B_s}$ = 480 $\pm$ 92 MeV when expressing the $B_s$-meson LCDAs in the Exponential model, consistent with previous QCD sum rule estimate yet exhibiting a 1.5-fold improvement in uncertainty. Furthermore, we compare the form factor predictions, based on the extracted $\lambda_{B_s}$ value, with earlier analyses for other channels such as $B_s \to D_s$ and $B_s \to K$.