Constraints for the semileptonic $B \to D^{(*)}$ form factors from lattice QCD simulations of two-point correlation functions

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 $\chi_{0^-}^{sub}(m_b^{phys}) = 2.19\,(19) \cdot 10^{-2}$.