First Lattice Calculation of The B-meson Binding and Kinetic Energies

Abstract

We present the first lattice calculation of the B-meson binding energy $\labar$ and of the kinetic energy $-λ_1/2 m_Q$ of the heavy-quark inside the pseudoscalar B-meson. This calculation has required the non-perturbative subtraction of the power divergences present in matrix elements of the Lagrangian operator $\bar h D_4 h$ and of the kinetic energy operator $\bar h \vec D^2 h$. The non-perturbative renormalisation of the relevant operators has been implemented by imposing suitable renormalisation conditions on quark matrix elements, in the Landau gauge. Our numerical results have been obtained from several independent numerical simulations at $β=6.0$ and $6.2$, and using, for the meson correlators, the results obtained by the APE group at the same values of $β$. Our best estimate, obtained by combining results at different values of $β$, is $\labar =190 \err{50}{30}$ MeV. For the $\overline{MS}$ running mass, we obtain $\overline {m}_b(\overline {m}_b) =4.17 \pm 0.06$ GeV, in reasonable agreement with previous determinations. From a subset of 36 configurations, we were only able to establish a loose upper bound on the $b$-quark kinetic energy in a $B$-meson, $λ_1=\langle B \vert \bar h \vec{D}^{2} h \vert B \rangle /(2 M_B )<$~1\, GeV$^2$. This shows that a much larger statistical sample is needed to determine this important parameter.