Factorization Formula Connecting the Shape Functions of Heavy Meson in QCD and Heavy Quark Effective Theory

The shape function of $B$-meson defined in heavy quark effective theory (HQET) plays a crucial role in the analysis of inclusive $B$ decays, and constitutes one of the dominant uncertainties in the determination of CKM matrix element $\mathrm{\mid }{V}_{ub}\mathrm{\mid }$. On the other hand, the conventional heavy meson shape function defined in QCD is also phenomenologically important and includes shortdistance physics at energy scales of the heavy quark mass. In this work, we derived a factorization formula relating these two kinds of shape functions, which can be invoked to fully disentangle the effects from disparate scales $m_b$ and ${\mathrm{\Lambda }}_{\text{QCD}}$, particularly to facilitate the resummation of logarithms $\ln m_b\mathrm{/}{\mathrm{\Lambda }}_{\text{QCD}}$. In addition, this factorization constitutes an essential component of the recently developed two-step factorization scheme, enabling lattice QCD calculations of lightcone quantities of heavy meson. The results presented here pave the way for first-principles nonperturbative predictions of shape function in the near future.