We find that the evolution equation for the three-particle quark-gluon $B$-meson light-cone distribution amplitude (DA) of subleading twist is completely integrable in the large $N_c$ limit and can be solved exactly. The lowest anomalous dimension is separated from the remaining, continuous spectrum by a finite gap. The corresponding eigenfunction coincides with the contribution of quark-gluon states to the two-particle DA ${\varphi }_{-}(\omega )$ so that the evolution equation for the latter is the same as for the leading-twist DA ${\varphi }_{+}(\omega )$ up to a constant shift in the anomalous dimension. Thus, “genuine” three-particle states that belong to the continuous spectrum effectively decouple from ${\varphi }_{-}(\omega )$ to the leading-order accuracy. In turn, the scale dependence of the full three-particle DA turns out to be nontrivial so that the contribution with the lowest anomalous dimension does not become leading at any scale. The results are illustrated on a simple model that can be used in studies of $1/m_b$ corrections to heavy-meson decays in the framework of QCD factorization or light-cone sum rules.

• Received 14 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.074044