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We consider the hadronic description of the $B^0_d\to \pi^+\pi^-$ decay, withthe aim to investigate the strong phases generated by the final stateinteractions. The derivation of the dispersion relations using theLehmann-Symanzik-Zimmermann formalism and the Goldberger-Treiman method toinclude inelastic effects in the spectral function are presented. We discussthe problem of quark-hadron duality and estimate in the hadronic formalism thecorrections to the factorized amplitude in the heavy quark limit.Comment: 16 pages, revised version, introduction of a scale dependenc

Quark–hadron duality, factorization and strong phases in $B^0_d\to \pi^+\pi^-$ decay