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In the paper, we apply the $k_T$ factorization approach to deal with the$B\to K$ transition form factor $F^{B\to K}_{+,0}(q^2)$ in the large recoilregions. The B-meson wave functions $\Psi_B$ and $\bar\Psi_B$ that include thethree-particle Fock states' contributions are adopted to give a consistent PQCDanalysis of the form factor up to ${\cal O} (1/m^2_b)$. It has been found thatboth the wave functions $\Psi_B$ and $\bar\Psi_B$ can give sizablecontributions to the form factor and should be kept for a better understandingof the $B$ meson decays. Then the contributions from different twist structuresof the kaon wavefunction are discussed, including the $SU_f(3)$-breakingeffects. A sizable contribution from the twist-3 wave function $\Psi_p$ isfound, whose model dependence is discussed by taking two group of parametersthat are determined by different distribution amplitude moments obtained in theliterature. It is also shown that $F^{B\to K}_{+,0}(0)=0.30\pm0.04$ and$[F^{B\to K}_{+,0}(0)/F^{B\to \pi}_{+,0}(0)]=1.13\pm0.02$, which are morereasonable and consistent with the light-cone sum rule results in the largerecoil regions.

$B\to K$ Transition Form Factor up to ${\cal O}(1/m^2_b)$ within the $k_T$ Factorization Approach