  In this work we present the first non-perturbative determination of the
hadronic susceptibilities that constrain the form factors entering the
semileptonic $B \to D^{(*)} \ell \nu_\ell $ transitions due to unitarity and
analyticity. The susceptibilities are obtained by evaluating moments of
suitable two-point correlation functions obtained on the lattice. Making use of
the gauge ensembles produced by the Extended Twisted Mass Collaboration with
$N_f = 2+1+1$ dynamical quarks at three values of the lattice spacing ($a
\simeq 0.062, 0.082, 0.089$ fm) and with pion masses in the range $\simeq 210 -
450$ MeV, we evaluate the longitudinal and transverse susceptibilities of the
vector and axial-vector polarization functions at the physical pion point and
in the continuum and infinite volume limits. The ETMC ratio method is adopted
to reach the physical $b$-quark mass $m_b^{phys}$. At zero momentum transfer
for the $b \to c$ transition we get $\chi_{0^+}(m_b^{phys}) = 7.58\,(59) \cdot
10^{-3}$, $\chi_{1^-}(m_b^{phys}) = 6.72\,(41) \cdot 10^{-4}$ GeV$^{-2}$,
$\chi_{0^-}(m_b^{phys}) = 2.58\,(17) \cdot 10^{-2}$ and $\chi_{1^+}(m_b^{phys})
= 4.69\,(30) \cdot 10^{-4}$ GeV$^{-2}$ for the scalar, vector, pseudoscalar and
axial susceptibilities, respectively. In the case of the vector and
pseudoscalar channels the one-particle contributions due to $B_c^*$- and
$B_c$-mesons are evaluated and subtracted to improve the bounds, obtaining:
$\chi_{1^-}^{sub}(m_b^{phys}) = 5.84\,(44) \cdot 10^{-4}$ GeV$^{-2}$ and
$\chi_{0^-}^{sub}(m_b^{phys}) = 2.19\,(19) \cdot 10^{-2}$.
