Factorization formula connecting the shape functions of heavy meson in QCD and heavy quark effective theory

The shape function of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>B</a:mi></a:math>-meson defined in heavy quark effective theory plays a crucial role in the analysis of inclusive <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>B</c:mi></c:math> decays, and constitutes one of the dominant uncertainties in the determination of Cabbibo-Kobayashi-Maskawa matrix element <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo stretchy="false">|</e:mo><e:msub><e:mi>V</e:mi><e:mrow><e:mi>u</e:mi><e:mi>b</e:mi></e:mrow></e:msub><e:mo stretchy="false">|</e:mo></e:math>. On the other hand, the conventional heavy meson shape function defined in QCD is also phenomenologically important and includes short-distance physics at energy scales of the heavy quark mass. In this work, we derive a factorization formula relating these two kinds of shape functions, which can be invoked to fully disentangle the effects from disparate scales <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>m</i:mi><i:mi>b</i:mi></i:msub></i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi mathvariant="normal">Λ</k:mi><k:mi>QCD</k:mi></k:msub></k:math>, particularly to facilitate the resummation of logarithms <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>ln</n:mi><n:msub><n:mi>m</n:mi><n:mi>b</n:mi></n:msub><n:mo>/</n:mo><n:msub><n:mi mathvariant="normal">Λ</n:mi><n:mi>QCD</n:mi></n:msub></n:math>. In addition, this factorization constitutes an essential component of the recently developed two-step factorization scheme, enabling lattice QCD calculations of light cone quantities of heavy meson. The results presented here pave the way for first-principles nonperturbative predictions of shape function in the near future.