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

Abstract

In this work we present the first non-perturbative determination of the hadronic susceptibilities that constrain the form factors entering the semileptonic $B\to D^{(\ast )}\ell {\nu }_{\ell }$ 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 $B_c^{\ast }$- 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 ${\chi }_{0^{-}}^{sub}(m_b^{phys})=2.19(19)\cdot 10^{-2}$.

Comments: 48 pages, 20 figures, 9 tables. Version to appear in PRD. arXiv admin note: text overlap with arXiv:2105.02497