then extract their shapes from the experimental angular distributions of $B\to D^\ast(\to D\pi)l\bar{\nu}$, by using a convenient parameterization. The parameterization considered by the experimental collaborations is the one proposed by Caprini, Lellouch and Neubert (CLN) [@Caprini:1997mu], which suggests $$\begin{aligned}
\label{eq:hA1}
h_{A_1}(w) &= {h_{A_1}(1)}\left[1+8 {\rho^2} z +(53 {\rho^2}-15)z^2-(231 {\rho^2}-91)z^3\right]\,,\\[0.3em]
\label{eq:R1}
R_1(w) &= {R_1(1)}{-0.12} (w-1){+0.05}(w-1)^2\,,\\[0.3em]
\label{eq:R2}
R_2(w) &= {R_2(1)}{-0.11} (w-1){-0.06}(w-1)^2\,,\end{aligned}$$ where $w=(m_B^2+m_D^2-q^2)/(2 m_B m_D)$ is the relative velocities of the initial and final state mesons, and $z=(\sqrt{w+1}-\sqrt{2})/(\sqrt{w+1}+\sqrt{2})$. This parameterization is based on two approximations:

1.  The shape of the form factor $h_{A_1}(w)$, which corresponds to the Isgur-Wise function in the heavy-quark limit, is truncanted after the quadratic term in $w-1$. Unitarity and analyticity are then used to express the shape parameters in Eq.  in terms of a single parameter $\rho$, the slope of the Isgur-Wise function, cf. Ref. [@Caprini:1995wq; @Caprini:1997mu; @Boyd:1997kz];

2.  The ratios $R_{1,2}(w)$ between other form factors and $h_{A_1}(w)$ are also parameterized by a power expansion around zero-recoil values ($w=1$), with their respective shapes and curvatures fixed to phenomenologically plausible values with no error bars. [^3]

The overall normalization $[h_{A_1}(1)]$ is then determined on the lattice [@Bailey:2014tva; @Harrison:2016gup], while the three remaining parameters in –, namely $\rho$ and $R_{1,2}(1)$, are 