Abstract
We consider the renormalization-group evolution of the matrix element of 〈0|q¯(z)β[z,0]b(0)α|B¯〉, which can be used to define the distribution amplitudes for B meson and widely applied in studies of B meson decays. The contribution to the renormalization constant of the non-local operator q¯(z)β[z,0]b(0)α is considered up to one-loop order in QCD. Since the quark fields in this operator are not directly coupled fields, momentum can not flow freely through this non-local operator. Momentum involved in this operator can be treated stringently in coordinate space. We find that the ultraviolet divergences regulated by dimensional parameter ϵ cancel with each other, and the evolution effect vanishes. The matrix element 〈0|q¯(z)β[z,0]b(0)α|B¯〉 escapes from the renormalization-group evolution. We then apply the matrix element in calculating B→π transition form factor, where the matrix element is obtained by using the B meson wave function calculated in QCD-inspired potential model. By comparing with experimental data for the semileptonic decay of B→πℓν and light-cone sum rule calculation, we analyze the perturbative and non-perturbative contributions to B→π transition form factor in the frame work of perturbative QCD approach. We find that the effectiveness of the suppression of Sudakov factor to soft contribution depends on the end-point behavior of B meson wave function, and with the B-meson wave function used in this work, soft contribution can not be well suppressed. The hard contribution to the B→π transition form factor is about 59%, and soft contribution can be as large as 41% in the naive calculation. To make the perturbative calculation reliable, a soft momentum cutoff in the calculation and soft form factor have to be introduced.
Highlights
Studies of B meson decays are important for researching the source of CP asymmetry and testing the methods of treating the decay amplitude developed on the basis of QCD
The B meson wave function obtained by solving bound-state equation and the relevant B-meson matrix element are applied in the calculation of B → π transition form factor for semileptonic decay B → πlν within the approach of kT factorization
The numerical calculation shows that the effectiveness of the suppression to the soft contribution depends on the end-point behavior of the Bmeson wave function other than Sudakov factor
Summary
Studies of B meson decays are important for researching the source of CP asymmetry and testing the methods of treating the decay amplitude developed on the basis of QCD. The evolution of the matrix element should lead to the scale-dependence of the distribution amplitudes of B meson. In this work we study the evolution of the matrix element 0|q(z)β[z, 0]b(0)αBby considering QCD correction up to one-loop order. There have been several works on the evolution of the matrix element or the distribution amplitudes of B meson existing in the literature [12,13,14,15],.
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