Unraveling Soft Components in the Shape Function for Inclusive B Decays

Inclusive B-meson decays, such as semileptonic B → Xulν̄ decays and penguininduced B → Xsγ decays, are of special interest because they are sensitive probes of electroweak parameters as well as new physics. Especially for the precise determination of the CKM matrix element |Vub|, the region near the kinematic endpoint for the lepton energy spectrum, El ∼ E max l = mB/2, in the decay B → Xulν̄ plays an important role,1 avoiding large backgrounds from the decays into charmed particles, B → Xclν̄. In this endpoint region, the hadronic decay products evolving from the u-quark have large energy but small invariant mass, so that the hadronic final state becomes jet-like with collinear interactions of an outgoing light quark. This implies that the corresponding differential decay rates are expressed as the light-cone expansion, analogously with deep inelastic lepton-nucleon scattering (DIS) cross sections: the leading term in the light-cone expansion is described by an analogue of the leading twist in the DIS, i.e., by a factorization formula where a structure function with all nonperturbative, long-distance (∼ 1/ΛQCD) contribution is convoluted with the perturbatively calculable function.2 This factorization formula is valid in the leading power of ΛQCD/mb and to all orders in αs, 2,3,4 and the structure function, called the shape function, is expressed as the B-meson matrix element of a bilocal light-cone operator in the heavy-quark effective theory (HQET):2 〈B̄(v)|h̄v(tn)Pe ig ∫ t