Renormalization group equation improved analysis of $B \to \pi$ form factors from Light-Cone Sum Rules

Within the framework of $B$-meson light-cone sum rules, we compute the one-loop level QCD corrections to $B\to \pi$ transition form factors at small $q^{2}$ region, in implement of a complete renormalization group equation evolution. To solve the renormalization group equations, we work at the "dual" space where the anomalous dimensions of the jet function and the light-cone distribution amplitudes are diagonal. With the complete renormalization group equation evolution, the form factors are almost independent of the factorization scale, which is shown numerically. We also extrapolate the results of the form factors to the whole $q^2$ region, and compare their behavior with other studies.