Isgur-Wise functions and unitary representations of the Lorentz group: The meson case with $j=\frac{1}{2}$ light cloud

We pursue the group-theoretical method to study Isgur-Wise (IW) functions. We extend the general formalism, formerly applied to the baryon case $j^P=0^{+}$ (for ${\mathrm{\Lambda }}_b\to {\mathrm{\Lambda }}_c\ell {\overline{\nu }}_{\ell }$), to mesons with $j^P={\frac{1}{2}}^{-}$, i.e. $\overline{B}\to D(D^{(*)})\ell \nu$. In this case, which is more involved from the angular momentum point of view, only the principal series of unitary representations of the Lorentz group contribute. We obtain an integral representation for the IW function $\xi (w)$ with a positive measure, recover the bounds for the slope and the curvature of $\xi (w)$ obtained from the Bjorken-Uraltsev sum-rule method, and get new bounds for higher derivatives. We demonstrate also that if the lower bound for the slope is saturated, the measure is a $\delta$ function, and $\xi (w)$ is given by an explicit elementary function. Inverting the integral formula, we obtain the measure in terms of the IW function, allowing us to formulate criteria to decide if a given Ansatz for the Isgur-Wise function is compatible or not with the sum-rule constraints. Moreover, we obtain an upper bound on the IW function valid for any value of $w$. We compare these theoretical constraints to a number of forms for $\xi (w)$ proposed in the literature. The “dipole” function $\xi (w)={(\frac{2}{w+1})}^{2c}$ satisfies all constraints for $c\ge \frac{3}{4}$, while the QCD sum rule result including condensates does not satisfy them. Special care is devoted to the Bakamjian-Thomas relativistic quark model in the heavy-quark limit and to the description of the Lorentz group representation that underlies this model. Consistently, the IW function satisfies all Lorentz group criteria for any explicit form of the meson Hamiltonian at rest.

Figure 1

$\eta (\tau )$ [Eq. (83)] for the exponential Ansatz $c=3/4,1,2$ (higher to lower curves).

Figure 2

$\frac{d\nu (\rho )}{d\rho }$ [Eq. (90)] for the exponential Ansatz, for $c=3/4,1,2$ (higher to lower curves).

Figure 3

$\eta (\tau )$ [Eq. (83)] for the “dipole” Ansatz $c=1.,1.5,2.$ (from higher to lower curves).

Figure 4

$\frac{d\nu (\rho )}{d\rho }$ for the “dipole” Ansatz for $c=1.,1.5,2.$ (from higher to lower curves).

Figure 5

The function ${\eta }_{\mathrm{BSW}}(\tau )$ [Eq. (83)] for $c=0,1,2$ (from higher to lower curves).

Figure 6

The function ${\eta }_{\mathrm{BSW}}^{(0)}(\tau )$ [Eq. (114)].

Figure 7

Fourier transform ${\tilde{\eta }}_{\mathrm{BSW}}^{(0)}(\rho )$ of the function ${\eta }_{\mathrm{BSW}}^{(0)}(\tau )$.

Figure 8

The function ${\eta }_{\mathrm{QCDSR}}(\tau )$ [Eq. (83)] for the QCDSR formula (119) and (120) for the IW function in the cases $\sigma (x)=1$ and $\sigma (x)=\frac{1}{2}(x+1-\sqrt{x^2-1})$ (upper and lower curves, respectively).

Figure 9

$d{\nu }_{\mathrm{QCDSR}}(\rho )/d\rho$ in the cases $\sigma (x)=1$ and $\sigma (x)=\frac{1}{2}(x+1-\sqrt{x^2-1})$ (upper and lower curves at low $\rho$, respectively).