Renormalization-Group Evolution of the $B$-Meson Light-Cone Distribution Amplitude

An integro-differential equation governing the evolution of the leading-order $B$-meson light-cone distribution amplitude is derived. The anomalous dimension in this equation contains a logarithm of the renormalization scale, whose coefficient is identified with the cusp anomalous dimension of Wilson loops. The exact solution of the evolution equation is obtained, from which the asymptotic behavior of the distribution amplitude is derived. These results can be used to resum Sudakov logarithms entering the hard-scattering kernels in QCD factorization theorems for exclusive $B$ decays.

Figure 1

One-loop diagrams contributing to the calculation of the anomalous dimension. The crossed circle denotes an insertion of the operator $O_{+}^{\mathrm{b}\mathrm{a}\mathrm{r}\mathrm{e}}({\omega }^{\prime })$. The double lines represent heavy-quark fields in HQET.

Figure 2

Results for the LCDA and the kernel (inset) for different values of the renormalization scale such that ${\alpha }_s=1$ (solid), 0.5 (dashed), 0.3 (dash-dotted). We use ${\lambda }_B=0.3\mathrm{G}\mathrm{e}\mathrm{V}$ at $\mu ={\mu }_0$.