Gluonic contribution to $B\to \eta^{(\prime)}$ form factors

We calculate the flavor-singlet contribution to the $B\to\eta^{(\prime)}$ transition form factors from the gluonic content of the $\eta^{(\prime)}$ meson in the large-recoil region using the perturbative QCD approach. The formulation for the $\eta$-$\eta'$ mixing in the quark-flavor and singlet-octet schemes is compared, and employed to determine the chiral enhancement scales associated with the two-parton twist-3 $\eta^{(\prime)}$ meson distribution amplitudes. It is found that the gluonic contribution is negligible in the $B\to\eta$ form factors, and reaches few percents in the $B\to\eta'$ ones. Its impact on the accommodation of the measured $B\to\eta^{(\prime)} K$ branching ratios in the perturbative QCD and QCD-improved factorization approaches is elaborated.