On P-wave meson decay constants in the heavy quark limit of QCD

In previous work it has been shown that, either from a sum rule for the subleading Isgur–Wise function ξ3(1) or from a combination of Uraltsev and Bjorken SR, one infers for P-wave states |τ1/2(1)|⪡|τ3/2(1)|. This implies, in the heavy quark limit of QCD, a hierarchy for the production rates of P-states Γ(B̄d→D(1/2)ℓν)⪡Γ(B̄d→D(3/2)ℓν) that seems at present to be contradicted by experiment. It was also shown that the decay constants of j=3/2 P-states vanish in the heavy quark limit of QCD, f3/2(n)=0. Assuming the model of factorization in the decays B̄d→ D̄s∗∗D, one expects the opposite hierarchy for the emission rates Γ(B̄d→ D̄s(3/2)D)⪡Γ(B̄d→ D̄s(1/2)D), since j=1/2 P-states are coupled to vacuum. Moreover, using Bjorken SR and previously discovered SR involving heavy–light meson decay constants and IW functions, one can prove that the sums ∑nf(n)/f(0)2, ∑nf1/2(n)/f(0)2 (where f(n) and f1/2(n) are the decay constants of S-states and j=1/2 P-states) are divergent. This situation seems to be realized in the relativistic quark models à la Bakamjian and Thomas, that satisfy HQET and predict decays constants f(n) and f1/2(n) that do not decrease with the radial quantum number n.