Anomalous thresholds and the Isgur-Wise function

Abstract

The original de Rafael-Taron bound on the slope of the Isgur-Wise function at zero recoil is known to be violated in QCD by singularities appearing in an unphysical region. To be consistent, quark models must have corresponding singularity structures. In an existing relativistic quark-loop model, the mesonquark-antiquark vertex is such that the required singularity is an anomalous threshold. We also discuss the implications of another anomalous threshold, whose location is determined by quark masses alone. a holdom@utcc.utoronto.ca b marks@medb.physics.utoronto.caThe properties of QCD as the quark masses m b , m c → ∞ [1] imply that the spectrum of the semileptonic decay B → D * ℓν at the zero-recoil point ω ≡ v B ·v D * = 1 is absolutely normalized at leading order and receives no corrections at order 1/m Q [2]. This gives rise to the possibility of a precise measurement of V cb [3], provided a) the corrections beginning at order 1/m 2 Q are proven to be negligible, and b) a way can be found to extrapolate from the data at nonzero recoil ω > 1 back to the zero-recoil point, where the rate vanishes kinematically. The latter requires knowledge of the shape of the Isgur-Wise function ξ(ω) which determines the spectrum at leading order and satisfies ξ(1) = 1. [In practice, it also requires knowledge of the magnitude and shape of the higher-order corrections, which are nonvanishing away from zero recoil even at order 1/m Q .]In this letter we wish to look at the leading-order problem, namely the shape of the Isgur-Wise function. Some time ago, de Rafael and Taron derived a lower bound on its slope at zero recoil [4], the value of which is ξ ′ (1) ≥ −0.89 [5]. This bound has since been shown to depend on assumptions which are not true in QCD. The bound also seems to be at odds with the data which favors a slope somewhat more negative than −1. We view any significant violation of the bound as providing an interesting clue to the underlying physics.In particular it was pointed out by several groups [6] that the original derivation assumed that the b-number form factor F (q 2 ) defined byis analytic in the region below the BB threshold at q 2 = 4M 2 B . The singularities corresponding to the three Υ's which lie just below threshold were ignored. These singularities can cause ∂F/∂q 2 to become more positive at q 2 = 0. A quantitative estimate of the contribution of these singularities to the slope requires knowledge of the couplings of the Υ's to the vacuum and the BB pair, and involves large uncertainties [5,6].In the heavy-quark limit m b → ∞, the Isgur-Wise function is related to F byso the effect of the singularities is to allow ξ ′ (ω) to become more negative at ω = 1. In the following, we will use ω as the variable. The BB threshold occurs at ω = −1, and the original derivation of the bound assumed that ξ(ω) is analytic for all ω > −1.The authors of the first three papers of Ref.[6] also pointed out that it is possible to construct an example of a meson for which the slope would be large and negative even w...

What can a relativistic quark model tell us about charmed mesons?

“…This prescription preserves the symmetry structure of the theory, and our hypothesis is that it may give reasonable results as long as external momenta of a diagram are such as to be far removed from whatever unphysical singularites the diagram possesses. (A discussion of the singularity structure of our three-point functions, including anomalous thresholds, may be found in [8].) We are finding that this simple, most naive model of confinement is yielding some very encouraging results.…”

“…This introduces singularities into our quark loop amplitudes for momenta outside the physical region [8]. Viewing our amplitudes as functions of the quark masses while keeping the meson masses fixed, there are also cusp-like singularities for special (presumably unphysical) values of the quark masses [3].…”