document: Including the O(αs) corrections, all the operators listed in (9) and (10) have to be included. A convenient framework to carry out these calculations is the QCD factorization framework [14] which allows to express the hadronic matrix elements in the schematic form: (11)〈Vγ|Oi|B〉=FB→VTiI+∫dk+2π∫01duφB,+(k+)TiII(k+,u)φV⊥(u), where FB→V are the transition form factors defined through the matrix elements of the operator O7, φB,+(k+) is the leading-twist B-meson wave-function with k+ being a light-cone component of the spectator quark momentum, φ⊥V(u) is the leading-twist light-cone distribution amplitude (LCDA) of the transversely-polarized vector meson V, and u is the fractional momentum of the vector meson carried by one of the two partons. The quantities TiI and TiII are the hard-perturbative kernels calculated to order αs, with the latter containing the so-called hard-spectator contributions. The factorization formula (11) holds in the heavy quark limit, i.e., to order ΛQCD/MB. This factorization framework has been used to calculate the branching fractions and related quantities for the decays B→K∗γ [15–17] and B→ργ [15,17]. The isospin violation in the B→K∗γ decays in this framework have also been studied [18]. (For applications to B→K∗γ∗, see Refs. [16,19,20].) Very recently, the hard-spectator contribution arising from the chromomagnetic operator O8 have also been calculated in next-to-next-to-leading order (NNLO) in αs showing that the spectator interactions factorize in the heavy quark limit [21]. However, the numerical effect of the resummed NNLO contributions is marginal and we shall not include this in our update.
extractive_keyphrases: fb→v fractional momentum of the vector meson carried by one of the two partons heavy quark isospin violation in the b→k∗γ decays lcda nnlo qcd factorization framework schematic form spectator interactions the factorization formula this factorization framework u
abstractive_keyphrases: express the hadronic matrix elements in the schematic form: hard-spectator contribution arising from the chromomagnetic operator o8 +(k+) leading-twist b-meson wave-function with k+ being a light-cone component of the spectator quark momentum leading-twist light-cone distribution amplitude next-to-next-to-leading orde operators listed in (9) and (10) have to be included transition form factors defined through the matrix elements of the operator o7, φb 〈vγ|oi|b〉=fb→vtii+∫dk+2π∫01duφb,+(k+)tiii(k+,u)φv⊥(u) φ⊥v(u