Uraltsev Sum Rule in Bakamjian-Thomas Quark Models

We show that the sum rule recently proved by Uraltsev in the heavy quark limit of QCD holds in relativistic quark models \`a la Bakamjian and Thomas, that were already shown to satisfy Isgur-Wise scaling and Bjorken sum rule. This new sum rule provides a {\it rationale} for the lower bound of the slope of the elastic IW function $\rho^2 \geq {3 \over 4}$ obtained within the BT formalism some years ago. Uraltsev sum rule suggests an inequality $|\tau_{3/2}(1)|>|\tau_{1/2}(1)|$. This difference is interpreted in the BT formalism as due to the Wigner rotation of the light quark spin, independently of a possible LS force. In BT models, the sum rule convergence is very fast, the $n = 0$ state giving the essential contribution in most of the phenomenological potential models. We underline that there is a serious problem, in the heavy quark limit of QCD, between theory and experiment for the decays $B \to D^*_{0,1}(broad) \ell \nu$, independently of any model calculation.

One of the aims of this note is to show that the SR (3) follows within quark modelsà la Bakamjian and Thomas. Quark models of hadrons with a fixed number of constituents, based on the Bakamjian-Thomas (BT) formalism [6,7], yield form factors that are covariant and satisfy Isgur-Wise (IW) scaling [8] in the heavy mass limit. In this class of models, the lower bound (4) was predicted some years ago [6].
Moreover, this approach satisfies the Bjorken SR that relates the slope of the IW function to the P -wave IW functions τ 1/2 (w), τ 3/2 (w) at zero recoil [9]. In this approach were also computed the P -wave meson wave functions and the corresponding inelastic IW functions [10], and a numerical study of ρ 2 in a wide class of models of the meson spectrum was performed (each of them characterized by an Ansatz for the mass operator M, i.e. the dynamics of the system at rest) [11], together with a phenomenological study of the elastic and inelastic IW functions and the corresponding rates for B → D, D * , D * * ℓν. Moreover, the calculation of decay constants of heavy mesons within the same approach was also performed [12].
The first demonstration of Uraltsev SR within the BT quark models is rather short, relying on formulas established in ref. [10]. Two other demonstrations will follow that will exhibit the underlying physics. The starting point is [10] : where with the radial part of the L = 1 wave functions normalized according to 1 6π 2 p 2 dp pϕ and m, p = |p| and p 0 = √ p 2 + m 2 are the mass, momentum and energy of the spectator quark.
From (13), using closure in the sectors of definite j = 1 2 , 3 2 one finds (page 325 of ref. [10]) : From (14)-(16), the expression for the difference in the left-hand-side of (3) can be integrated by parts, yielding, after some algebra : where the last equality follows from the ground state wave function normalization [6].
Therefore, the SR (17) within the BT quark models provides a rationale for the lower bound ρ 2 ≥ 3 4 that was found within this class of models [6]. The sum rule also establishes that the sum over the j = 3/2 states dominates over the one over the j = 1/2.
The second demonstration, that follows more closely Uraltsev proof, will illustrate quark-hadron duality. Let us first remind the proof of Bjorken SR that was given in [9]. It was shown that the spin averaged hadronic tensor in the BT formalism is, in the heavy quark limit for the active quark, identical to the free quark hadronic tensor :h From this relation, Bjorken SR follows. In equation (18), the free quark tensor is and the hadronic tensor writes where J, λ are the spin and spin projection of the hadron of momentum P.
In BT models, the hadronic tensor can be written [9] : where f λλ s 1f s 1i is the hadronic overlap : and (18) follows from (21) and (22). The wave function ψ λ s 1 ,s 2 (P − p 2 , p 2 ) is the internal moving ground state wave function, with the active quark labelled 1 and λ being the spin projection along some axis. It is defined by deleting the momentum conserving δ-function from the total wave function. In the BT model, it is obtained from a P -depending transformation on the rest internal wave function.
To proceed like Uraltsev, one must generalize the hadronic tensor, allowing for different velocities and angular momentum projections. Let us consider the polarized hadronic tensor : In the BT formalism, this tensor writes, using closure and heavy mass limit [9] † : with the hadronic overlap is the internal moving ground state meson wave function, and the active quark is labelled 1.
Let us choose, like Uraltsev, the vector meson B * as initial and final state, with P i = 0, λ i = 0, λ f = +1, and the vector current with µ = ν = 0. We are thus considering the object to first order in v ′ , v f . There are, in principle, two kinds of terms contributing to this quantity : 1) Spin-flip term coming from the active quark, i.e., from the quark current At the desired order, one can also take the hadronic overlap at P i = P f = 0 : The states |n, P ′ > form a complete set of eigenfunctions at fixed P ′ : n |n, P ′ >< n, P ′ | = 1.

One obtains
where the factor 1/ √ 2 comes from the hadronic overlap, and 1 labels the active quark.
2) Terms without spin-flip of the active quark. Then, to have a contribution to (26), one needs to appeal to a Wigner rotation of the spectator quark 2, giving a contribution ∼ p 2 × P f . But, by integration, this term is zero, because there is no other hadron momentum than P f -in the hadronic overlap there is no dependence We are then left with expression (28), that means that we have exact duality, just like in the unpolarized, P i = P f case : We need now to compute the same hadronic tensor (23) in terms of the phenomenological Isgur-Wise functions τ j (w), within the same approximations. After a good deal of algebra, we find, using the definitions of [14], and taking into account that the states are not normalized according to the usual normalization where the different contributions are (a sum over a radial quantum number is implicit) and the ground state does not contribute. One obtains, Identifying the expressions (28) and (32), Uraltsev SR follows.
Some words of caution about the general scope and limitations of Bakamjian-Thomas quark models are in order here. Both zero order moment sum rules, the ones of Bjorken [9] and Uraltsev are satisfied by this class of models. However, higher moment sum rules as Voloshin sum rule [4] are not satisfied. These higher moments sum rules seem to be specific to the gauge nature of QCD. Anyhow, one limitation of BT models is the following, as exposed in [6].
where ϕ 1/2 (p) ∼ = ϕ 3/2 (p) = ϕ (0) L=1 (p) (assuming small LS coupling) are the internal hadron wave functions at rest. We assume, as it is natural, that for the ground state ϕ (0) L=1 (p) is positive. One finds that τ (0) 3/2 (1) is larger than τ (0) 1/2 (1) even in the limit of vanishing LS coupling. The difference (33) has a simple physical interpretation, outlined in ref. [11] : it is essentially due to the relativistic structure of the matrix elements in terms of the wave functions. More precisely, it is due to the light spectator quark Wigner rotations, i.e. a relativistic effect due to the center-of-mass boost, and not due to the difference coming from the spin-orbit force between the 1/2 and 3/2 internal wave functions at rest, which is small and has a rather moderate effect. On the contrary, the difference (33) is quite large, at least for the lowest L = 1 states, since for a constituent quark mass m ∼ = 0.3 GeV, the quantity p p 0 +m is of O(1).
Expression (33), that comes from a specific relativistic effect, is to be contrasted with the equality for any non-relativistic quark model with spin-orbit independent potential [13], also used in ref. [14], that analizes 1/m Q corrections : Let us see how, in terms of internal wave functions at rest, the Wigner rotation of the spectator quark is responsible for the difference between τ In this expression we see the basic ingredients of the model. There is a change of variables of the quark momenta e.g. for the initial state (p 1 , p 2 ) → (P, k 2 ), where P is the center-of mass momentum, and k 2 the internal relative momentum, and likewise for the final state (p ′ 1 , p ′ 2 ) → (P ′ , k ′ 2 ). The first term under the integral comes from the Jacobian of this change of variables. The matrix element u s ′ 1 γ µ u s 1 expresses the fact that the quark 1 is the active heavy quark. The relation between e.g. k 2 and p 2 is given by the boost k 0 where |0(v) > stands for the ground state wave function in motion and likewise |0) for the internal ground state at rest in terms of Pauli spinors. The first operator −i(p 0 2 r 2 + r 2 p 0 2 ), where r 2 is the operator i ∂ ∂p 2 , comes from the variation of the Jacobian factor and the variation of the argument k of the wave function, while the second operator i(σ 2 × p 2 ) p 0 2 + m is the Wigner rotation. Equation (36) becomes, in the non-relativistic limit, the matrix element of the electric dipole operator, and leads to the difference (33) through the latter spin-dependent term. To demonstrate Uraltsev SR, we are interested in the hadronic tensor The ground state does not contribute to the sum rule over intermediate states in (37), in HQET and likewise in BT quark models, that satisfy HQET. We have indeed demonstrated in ref. [6] (formulas (26)-(29)) that BT quark models in the heavy quark limit satisfy HQET relations for all ground state form factors. More specifically, in BT quark models, as follows after some algebra from (35), the contributions of the active quark (28) cancels with the one of the spectator quark for the ground state. We are then left with the L = 1 intermediate states for which we apply formula (36).
Defining the frame v i = (1, 0, 0, 0), v f = (v 0 f , 0, 0, v z f ), the hadronic tensor can then be written, at first order in the velocities v f and v ′ , where the |n) states are L = 1. The spin flip B * (0) → B * (+1) can occur because of the Wigner rotation on the spectator light quark. Using completeness n |n)(n| = 1, two kinds of terms contribute : crossed terms between a Wigner rotation and a spinindependent operator, and products of two Wigner rotations. After some algebra, the final result reads :

Phenomenological remarks.
From the calculations of ref. [11] in the BT formalism for a wide class of potentials, one can see from Table 1   The Godfrey and Isgur potential [15] is the one that describes the meson spectrum in the most complete way, from light meson spectroscopy to heavy quarkonia.
The agreement of the contribution of lowest n = 0 states with the right-hand-side of the SR (17) is quite striking. Within the BT class of quark models, one gets [11] a value ρ 2 ∼ = 1, not inconsistent with present experimental data on the ξ(w) slope, and also, consistently, with small values for τ (n) 1/2 (1). It is interesting to remark that, among the three potential models quoted in Table 1, only the more complete one by Godfrey and Isgur contains a L.S coupling.
There are indeed in this case L.S splittings (M 1/2 (w) even for these latter potentials. However, even in the case of the Godfrey-Isgur potential, the L.S force is small.
In Table 2 we compare the predictions of the BT quark models for the different semileptonic decays. While the BR for the modes B → D 2 ℓν and B → D 1 ( 3 2 )ℓν have the right order of magnitude, and are consistent with experiment within 1σ, ‡ This fast convergence of the sum rules has also been observed in QCD 2 in the N c → ∞ limit [19]. is opposite to experiment. This moderate disagreement could be explained by 1/m Q corrections [20]. However, in the case of the j = 1 2 the disagreement is very strong. QCD in the heavy quark limit predicts, according to Uraltsev SR, that the j = 3 2 states are dominant over the j = 1 2 . This general trend could be hardly reversed by the small hard QCD corrections to Uraltsev [1] and Bjorken [20] sum rules. As to the 1/m Q corrections [14], their magnitude is poorly known, since the numerical estimate of ref. [14], although the formalism is completely general, relies on a large number of dynamical hypotheses.   ℓν come from ALEPH (a), DELPHI (b) and CLEO (c) data [18], with the errors added in quadrature. The last entry corresponds to DELPHI data for the wide states.
The serious problem for the decays B → D 0,1 ( 1 2 )ℓν goes beyond the specific BT quark models and appears to be, more generally, a problem between experiment and the heavy quark limit of QCD.

Conclusion.
We have shown that the sum rule proved recently by Uraltsev in the heavy quark limit of QCD holds in relativistic quark modelsà la Bakamjian and Thomas. Its physical interpretation is the Wigner rotation of the spectator light quark spin, and not a possible LS perturbation. We have underlined that, since |τ 3/2 (1)| > |τ 1/2 (1)| [22], there is a serious problem between theory and experiment for the decays B → D * 0,1 (broad)ℓν. This problem goes beyond the BT quark models and appears to be a general one, within the heavy quark limit of QCD.