J.- C. Raynal

The Bakamjian-Thomas relativistic quark model, describing hadrons with a fixed number of constituents, yields in the heavy quark limit of QCD covariant Isgur-Wise functions and satisfies the whole tower of lowest moment sum rules (Bjorken-Uraltsev type sum rules). We first recall, as well as earlier results, the new formalism presented in our recent papers on Lorentz representations, which provides an elegant framework for the analysis of this model in the heavy quark limit and stress the results which have been already obtained in this direction. Then, we give some very explicit demonstrations of the fact that the Bakamjian-Thomas framework satisfies the sum rules by considering simple cases of Isgur-Wise functions. In addition to the specific Bjorken and Uraltsev sum rules, an important sum rule that involves only heavy mesons with light cloud $j^P={\frac{1}{2}}^{-}$ and their radial excitations is demonstrated. This latter sum rule is phenomenologically interesting because it constrains the derivatives of the radially excited Isgur-Wise functions at zero recoil. On the other hand, we recall the limitations of the Bakamjian-Thomas scheme. At finite mass, current matrix elements with the current coupled to the heavy quark are no longer covariant, and higher moment sum rules that hold in the heavy quark limit of QCD are not satisfied.

The Bakamjian-Thomas relativistic quark model provides a Poincaré representation of bound states with a fixed number of constituents and, in the heavy quark limit, form factors of currents satisfy covariance and Isgur-Wise scaling. We compute the light cone distribution amplitudes (LCDA) of $B$ mesons ${\phi }_{\pm }^B(\omega )$ as well as the shape function $S(\omega )$, that enters in the decay $B\to X_s\gamma$, that are also covariant in this class of models. The LCDA and the shape function are related through the quark model wave function. The former satisfy, in the limit of vanishing constituent light quark mass, the integral relation given by QCD in the valence sector of Fock space. Using a Gaussian wave function, the obtained $S(\omega )$ is identical to the so-called roman shape function. From the parameters for the latter that fit the $B\to X_s\gamma$ spectrum we predict the behavior of ${\phi }_{\pm }^B(\omega )$. We discuss the important role played by the constituent light quark mass. In particular, although ${\phi }_{-}^B(0)\ne 0$ for vanishing light quark mass, a nonvanishing mass implies the unfamiliar result ${\phi }_{-}^B(0)=0$. Moreover, we incorporate the short distance behavior of QCD to ${\phi }_{+}^B(\omega )$, which has sizeable effects at large $\omega$. We obtain the values for the parameters $\overline{\Lambda }\cong 0.35\mathrm{GeV}$ and ${\lambda }_B^{-1}\cong 1.43{\mathrm{GeV}}^{-1}$. We compare with other theoretical approaches and illustrate the great variety of models found in the literature for the functions ${\phi }_{\pm }^B(\omega )$; hence the necessity of imposing further constraints as in the present paper. We briefly review also the different phenomena that are sensitive to the LCDA. The value that we find for ${\lambda }_B^{-1}$ fulfills the upper bound recently measured by BABAR.

Using previously formulated sum rules in the heavy quark limit of QCD, we demonstrate that if the slope ${\rho }^2=-{\xi }^{\prime }(1)$ of the Isgur-Wise (IW) function $\xi (w)$ attains its lower bound $\frac{3}{4}$, then all the derivatives $(-1{)}^L{\xi }^{(L)}(1)$ attain their lower bounds $\frac{(2L+1)!!}{2^{2L}}$, obtained by Le Yaouanc et al. This implies that the IW function is completely determined, given by the function $\xi (w)=(\frac{2}{w+1}{)}^{3/2}$. Since the so-called BPS condition proposed by Uraltsev implies ${\rho }^2=\frac{3}{4}$, it implies also that the IW function is given by the preceding expression.

We pursue the program of the study of the nonforward amplitude in heavy quark effective theory (HQET). We obtain new sum rules involving the elastic subleading form factors ${\chi }_i(w)$ $(i=1,2,3)$ at order $1/m_Q$ that originate from the ${\mathcal{L}}_{kin}$ and ${\mathcal{L}}_{mag}$ perturbations of the Lagrangian. To obtain these sum rules we use two methods. On the one hand we start simply from the definition of these subleading form factors, and on the other hand, we use the operator product expansion. To the sum rules contribute only the same intermediate states $(j^P,J^P)=({\frac{1}{2}}^{-},1^{-}),({\frac{3}{2}}^{-},1^{-})$ that enter in the $1/m_Q^2$ corrections of the axial form factor $h_{A_1}(w)$ at zero recoil. This allows to obtain a lower bound on $-{\delta }_{1/m^2}^{(A_1)}$ in terms of the ${\chi }_i(w)$ and the shape of the elastic IW function $\xi (w)$. We find also lower bounds on the $1/m_Q^2$ correction to the form factors $h_{+}(w)$ and $h_1(w)$ at zero recoil. An important theoretical implication is that ${\chi }_1^{\prime }(1)$, ${\chi }_2(1)$ and ${\chi }_3^{\prime }(1)$ (${\chi }_1(1)={\chi }_3(1)=0$ from Luke theorem) would vanish if the slope and the curvature of $\xi (w)$, ${\rho }^2\ge \frac{3}{4}$, ${\sigma }^2\ge \frac{15}{16}$, would attain their lowest bounds. We discuss possible implications on the precise determination of $|V_{cb}|$.

Within the OPE, we formulate new sum rules in Heavy Quark Effective Theory in the heavy quark limit and at order 1/m_Q, using the non‐forward amplitude. In the heavy quark limit, 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 the n‐th derivative of ξ(w) at w = 1 can be bounded by the (n − 1)‐th one, and the absolute lower bound for the n‐th derivative ${\text{(−1)}}^n{\xi }^{\mathrm{(n)}}\text{(1)\hspace{0.17em}⩾\hspace{0.17em}}\frac{\text{(2n+1)!!}}{2^{\text{2n}}}$ . Moreover, for the curvature we find $\text{ξ″(1)\hspace{0.17em}⩾\hspace{0.17em}}\frac{1}{5}{\text{[4ρ}}^2{\text{\hspace{0.17em}+\hspace{0.17em}3(ρ}}^2{)}^2]$ where ρ² = −ξ′(1). These results are consistent with the dispersive bounds, and they strongly reduce the allowed region of the latter for ξ(w). The method is extended to the subleading quantities in 1/m_Q. Concerning the perturbations of the Current, we derive new simple relations between the functions ξ₃(w) and Λ̄ξ(w) and the sums ${\displaystyle \sum _n}{\mathrm{\hspace{0.17em}\Delta E}}_j^{\mathrm{(n)}}{\tau }_j^{\mathrm{(n)}}{\text{(1)τ}}_j^{\mathrm{(n)}}\mathrm{(w)\hspace{0.17em}(j\hspace{0.17em}=\hspace{0.17em}}\frac{1}{2}\mathrm{,\hspace{0.17em}}\frac{3}{2})$ , that involve leading quantities, Isgur‐Wise functions ${\tau }_j^{\mathrm{(n)}}\mathrm{(w)}$ and level spacings ${\mathrm{\Delta E}}_j^{\mathrm{(n)}}$ . Our results follow because the non‐forward amplitude depends on three variables (w_i, w_f, w_if) = (v_i ⋅ v′, v_f ⋅ v′, v_i ⋅ v_f), and we consider the zero recoil frontier (w, 1, w) where only a finite number of j^P states contribute $({\frac{1}{2}}^{+}\mathrm{,\hspace{0.17em}}{\frac{3}{2}}^{+})$ . We also obtain new sum rules involving the elastic subleading form factors χ_i(w) (i = 1, 2, 3) at order 1/m_Q that originate from the L_kin and L_mag perturbations of the Lagrangian. To the sum rules contribute only the same intermediate states ${\mathrm{(j}}^P{\mathrm{,\hspace{0.17em}J}}^P\mathrm{)\hspace{0.17em}=\hspace{0.17em}(}{\frac{1}{2}}^{-}{\text{,\hspace{0.17em}1}}^{-}\mathrm{),(}{\frac{3}{2}}^{-}{\text{,\hspace{0.17em}1}}^{-})$ that enter in the ${\text{1/m}}_Q^2$ corrections of the axial form factor hA₁(w) at zero recoil. This allows to obtain a lower bound on ${\mathrm{-\delta }}_{{\text{1/m}}^2}^{{\mathrm{(A}}_1)}$ in terms of the χ_i(w) and the shape of the elastic IW function ξ(w). An important theoretical implication is that ${\chi }_1^{\prime }\text{(1)}$ , χ₂(1) and ${\chi }_3^{\prime }{\text{(1)\hspace{0.17em}(χ}}_1{\text{(1)\hspace{0.17em}=\hspace{0.17em}χ}}_3\text{(1)\hspace{0.17em}=\hspace{0.17em}0}$ from Luke theorem) must vanish when the slope and the curvature attain their lowest values ${\rho }^2\mathrm{\hspace{0.17em}\to \hspace{0.17em}}\frac{3}{4}{\mathrm{,\hspace{0.17em}\sigma }}^2\mathrm{\hspace{0.17em}\to \hspace{0.17em}}\frac{15}{16}$ . These constraints should be taken into account in the exclusive determination of...

We consider the nonforward amplitude within the heavy quark effective theory. We show that one can obtain new information on the subleading corrections in $1/m_Q$. We illustrate the method by deriving new simple relations between the functions ${\xi }_3(w)$ and $\overline{\Lambda }\xi (w)$ and the sums $\sum _n\Delta E_j^{(n)}{\tau }_j^{(n)}(1){\tau }_j^{(n)}(w)$ $(j=\frac{1}{2},\frac{3}{2})$, that involve leading quantities, namely, the Isgur-Wise functions ${\tau }_j^{(n)}(w)$ and the level spacings $\Delta E_j^{(n)}$. Our results follow because the nonforward amplitude $B(v_i)\to D^{(n)}(v^{\prime })\to B(v_f)$ depends on three variables $(w_i,w_f,w_{if})=(v_i·v^{\prime },v_f·v^{\prime },v_i·v_f)$ independent in a certain domain, and we consider the zero recoil frontier $(w,1,w)$ where only a finite number of $j^P$ states contribute $({\frac{1}{2}}^{+},{\frac{3}{2}}^{+})$. These sum rules reduce to known results at $w=1$, for $\overline{\Lambda }$ obtained by Voloshin, and for ${\xi }_3(1)$ obtained by Le Yaouanc et al. and by Uraltsev, and generalizes them to all values of $w$. We discuss phenomenological applications of these results, in particular the check of Bakamjian-Thomas quarks models and the comparison with the QCD sum rules approach.

In the leading order of the heavy quark expansion, we propose a method within the operator product expansion and the trace formalism that allows us to obtain, in a systematic way, Bjorken-like sum rules for the derivatives of the elastic Isgur-Wise (IW) function $\xi \mathrm{}\left(w\right)\mathrm{}$ in terms of corresponding Isgur-Wise functions of transitions to excited states. A key element is the consideration of the nonforward amplitude, as introduced by Uraltsev. A simplifying feature of our method is to consider currents aligned along the initial and final four-velocities. As an illustration, we give a very simple derivation of Bjorken and Uraltsev sum rules. On the other hand, we obtain a new class of sum rules that involve the products of IW functions at zero recoil and IW functions at any w. Special care is given to the needed derivation of the projector on the polarization tensors of particles of arbitrary integer spin. The new sum rules give further information on the slope ${\rho }^2=-{\xi }^{\prime }\mathrm{}\left(1\right)\mathrm{}$ and also on the curvature ${\sigma }^2={\xi }^{″}\mathrm{}\left(1\right)\mathrm{}$ and imply, modulo a very natural assumption, the inequality ${\sigma }^2>~\mathrm{}(5/4)\mathrm{}{\rho }^2,$ and therefore the absolute bound ${\sigma }^2>~15/16.$

We propose the determination of the CKM matrix element $|V_{\mathrm{ub}}|$ by the measurement of the spectrum of $\overrightarrow{B}{D}_s^{+}{X}_u,$ dominated by the spectator quark model mechanism $\overline{b}\to D_s^{\left(*\right)+}ū.$ The interest in considering $\overrightarrow{B}{D}_s^{+}{X}_u$ versus the semileptonic decay is that more than 50% of the spectrum for $\overrightarrow{B}{D}_s^{+}{X}_u$ occurs above the kinematical limit for $\overrightarrow{B}{D}_s^{+}{X}_c,$ while most of the spectrum $\overrightarrow{B}l\nu X_u$ occurs below the $\overrightarrow{B}l\nu X_c$ one. Furthermore, the measure of the hadronic mass $M_X$ is easier in the presence of an identified $D_s$ than when a $\nu$ has been produced. As a consistency check, we point out that the rate $\overline{b}\to D_s^{\left(*\right)+}\overline{c}$ (including QCD corrections) is consistent with the measured $\mathrm{BR}\left(\overrightarrow{B}{D}_s^{\pm }X\right)$ by CLEO. Although the hadronic complications may be more severe in the decay mode that we propose than in the semileptonic inclusive decay, the end of the spectrum in $\overrightarrow{B}l\nu X_u$ is not well understood on theoretical grounds. We argue that, in our case, the excited mesons $D_s^{**},$ decaying into $DK,$ do not contribute and, if there is tagging of the B meson, the other mechanisms to produce a $D_s$ of the right sign are presumably small, of ${O(10\mathrm{}}^{-2})$ relative to the spectator amplitude, or can be controlled by kinematical cuts. We discuss in detail the hadronic uncertainties of the method and present an error calculation on $|V_{\mathrm{qb}}|$ $\mathrm{}(q=u,c)\mathrm{}$ and on the ratio

In the quark models of the Bakamjian-Thomas type that yield covariance and Isgur-Wise scaling of form factors in the heavy quark limit we compute the decay constants $f^{\mathrm{}\left(n\right)\mathrm{}}$ and $f_{1/2}^{\mathrm{}\left(n\right)\mathrm{}}$ of S- and P-wave mesons composed of heavy and light quarks. Lorentz invariance of the decay constants and heavy quark limit scaling $\sqrt{M}f=\mathrm{const}$ is obtained, and it is shown that this class of models satisfies the sum rules involving decay constants and Isgur-Wise functions recently formulated by us in the heavy quark limit of QCD. Moreover, the model also satisfies selection rules of the type $f_{3/2}^{\mathrm{}\left(n\right)\mathrm{}}=0,$ which must hold in this limit. We discuss different Ansätze for the dynamics of the mass operator at rest. For nonrelativistic kinetic energies $p^2/2m,$ the decay constants are finite even if the potential $V\left(r\right)\mathrm{}$ has a Coulomb part. For the relativistic form $\sqrt{p^2{+m}^2},$ the S-wave decay constants diverge if there is a Coulomb singularity. Using phenomenological models of the spectrum with relativistic kinetic energy and a regularized short-distance part (Godfrey-Isgur model or Richardson potential of Colangelo et al.) that yield ${\rho }^2\cong 1$ for the elastic Isgur-Wise function, we compute the decay constants in the heavy quark limit and obtain $f_B\cong 300\mathrm{MeV},$ of the same order although slightly smaller than in the static limit of lattice QCD. We find the decay constants of $D^{**}$ with $j=\frac{1}{2}$ of the same order of magnitude. The convergence of the heavy quark limit sum rules is also studied.