New heavy quark limit sum rules involving Isgur-Wise functions and decay constants

Abstract We consider the dominant c c contribution to ΔΓ for the B s 0 − B s 0 system in the heavy quark limit for both b and c quarks. In analogy with the Bjorken-Isgur-Wise sum rule in semileptonic heavy hadron decay, we impose duality between the parton model calculation of ΔΓ and its estimation by a sum over heavy mesons. Varying the mass ratio m c m b and assuming factorization and saturation by narrow resonances (Nc → ∞), we obtain new sum rules that involve the Isgur-Wise functions ξ(n)(w) and τ 1 2 (n) (w) and the decay constants f(n), f 1 2 (n) (n stands for any radial excitation). Alternatively, we deduce the sum rules with another method free of the factorization hypothesis, from the saturation of the expectation value of a product of two currents by heavy hadrons and by the corresponding free quarks. The sum rules read ∑ n f (n) f (0) ξ (n) (w)= 2 ∑ n f (n) 1 2 f (0) τ (n) 1 2 (w)=1; valid for all w. Moreover, we obtain, in the heavy quark limit, f 3 2 (n) = 0 . As a consequence, unlike the BIW sum rule, the slope of the elastic function ξ(w) is related to radial excitations alone. These are generalizations, rigorous for QCD in the heavy quark limit, of results that have an easy understanding in the non-relativistic quark model.