Constraints for the semileptonic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>*</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> form factors from lattice QCD simulations of two-point correlation functions
G. Martinelli, Silvano Simula, Ludovico Vittorio
Abstract
In this work we present the first nonperturbative determination of the hadronic susceptibilities that constrain the form factors entering the semileptonic $B\ensuremath{\rightarrow}{D}^{(*)}\ensuremath{\ell}{\ensuremath{\nu}}_{\ensuremath{\ell}}$ 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 (ETMC) with ${N}_{f}=2+1+1$ dynamical quarks at three values of the lattice spacing ($a\ensuremath{\simeq}0.062$, 0.082, 0.089 fm) and with pion masses in the range $\ensuremath{\simeq}210--450\text{ }\text{ }\mathrm{MeV}$, 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 ${m}_{b}^{\mathrm{phys}}$. At zero momentum transfer for the $b\ensuremath{\rightarrow}c$ transition we get ${\ensuremath{\chi}}_{{0}^{+}}({m}_{b}^{\mathrm{phys}})=7.58(59)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, ${\ensuremath{\chi}}_{{1}^{\ensuremath{-}}}({m}_{b}^{\mathrm{phys}})=6.72(41)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$, ${\ensuremath{\chi}}_{{0}^{\ensuremath{-}}}({m}_{b}^{\mathrm{phys}})=2.58(17)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ and ${\ensuremath{\chi}}_{{1}^{+}}({m}_{b}^{\mathrm{phys}})=4.69(30)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ 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 ${\ensuremath{\chi}}_{{1}^{\ensuremath{-}}}^{\mathrm{sub}}({m}_{b}^{\mathrm{phys}})=5.84(44)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$ and ${\ensuremath{\chi}}_{{0}^{\ensuremath{-}}}^{\mathrm{sub}}({m}_{b}^{\mathrm{phys}})=2.19(19)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, respectively.