Bounds on the derivatives of the Isgur–Wise function from sum rules in the heavy quark limit of QCD

Abstract

Using the OPE and the trace formalism, we have obtained a number of sum rules in the heavy quark limit of QCD that include the sum over all excited states for any value j P of the light cloud. We show that these sum rules imply that the elastic Isgur-Wise function ξ(w) is an alternate series in powers of (w − 1). Moreover, one gets that any n-th derivative of ξ(w) at w = 1 can be bounded by the (n − 1)-th one. For the curvature σ 2 = ξ ′′ (1), this proves the already proposed bound σ 2 ≥ 5 4 ρ 2 . Moreover, we obtain the absolute lower bound for the n-th derivative (−1) n ξ (n) (1) ≥ (2n+1)!! 2 2n , that generalizes the results ρ 2 ≥ 3 4 and σ 2 ≥ 15 16 .

$$B_s\rightarrow D_s\ell \nu _\ell $$ B s → D s ℓ ν ℓ near zero recoil in and beyond the Standard Model

“…In our case we only obtain ρ 2 = 1.2 (8), or in the case where we dismiss the dependence on the sea quark mass (when the errors on G(w) are smaller) we get ρ 2 = 1.1(3), both being consistent with the experimentally established ρ 2 = 1.19(4)(4) [32]. The same quality of the result for ρ 2 is obtained if the data are fit to [33][34][35] …”

Explicit Form of the Isgur-Wise function in the BPS limit

“…In Sec. IV we demonstrate, using the results of [5,6], that 2 3 4 implies that the curvature attains its lower limit (2), i.e. 00 1 2 15 16 .…”

“…obtained in[5], differentiating relatively to w if and to w and going to the border of the domain (9) w if w 1 (formulas (33), (34),…”

Bound on the curvature of the Isgur-Wise function of the baryon semileptonic decayΛb→Λcℓν¯ℓ

“…In Refs. [4][5][6] we have developed, in the heavy quark limit of QCD, a manifestly covariant formalism within the operator product expansion (OPE), using the whole tower of heavy meson states [7]. We did recover Uraltsev SR plus a general class of SR that allow us to bound also higher derivatives of the IW function.…”