Sum rules in the heavy quark limit of QCD

Abstract

In the leading order of the heavy quark expansion, we propose a covariant method, within the OPE and the trace formalism, that allows to obtain, in a systematic way, Bjorken-like sum rules for the derivatives of the elastic Isgur-Wise function ξ(w) in terms of corresponding Isgur-Wise functions of transitions to excited states. To illustrate the method, we give a simultaneous derivation of Bjorken and Uraltsev sum rules, with generalizations of the latter for w = 1. On the other hand, we obtain a new class of sum rules that involve the products of IW functions at zero recoil and of IW functions at any w.These sum rules give new information on the slope ρ 2 = −ξ ′ (1) and also on the curvature σ 2 = ξ ′′ (1), and imply, modulo a very natural assumption, the inequality σ 2 ≥ 5 4 ρ 2 , and therefore the absolute bound σ 2 ≥ 15 16 .

Bounds on the derivatives of the Isgur-Wise function with a nonrelativistic light quark

“…The main general reason is that the shape of the latter is linked to the determination of the CKM matrix element |V cb | through the exclusive processes B → D ( * ) ℓν. As pointed out in [3], a more quantitative reason is that, beyond the first derivative, higher derivatives will play a non-negligible role at the edge of the phase space, at high w, in the region where the data are presently rather precise, and will become more and more precise in the near future.…”

“…Using the OPE in the heavy quark limit of QCD, new Bjorken-like sum rules (SR) have been obtained [1,2,3]. It has been shown that the Isgur-Wise (IW) function ξ(w) is an alternate series in powers of (w − 1), and lower bounds have been found on the absolute magnitude of its derivatives.…”

“…In particular, it has been found that the n-th derivative is bounded by the (n−1)-th one [2] (−1) n ξ (n) (1) ≥ 2n + 1 4 (−1) n−1 ξ (n−1) (1) ≥ (2n + 1)!! 2 2n…”

“…where the absolute lower bound (independent of ρ 2 ) follows from (2). Radiative corrections to the bounds (3) and (4) have been computed by M. Dorsten [6].…”

“…It has been underlined in [3] that the quadratic term in (4) has a clear physical interpretation, as it is leading in a non-relativistic (NR) expansion in the mass of the light quark :…”