Extraction of heavy-quark-expansion parameters from unquenched lattice data on pseudoscalar and vector heavy-light meson masses

We present a precise lattice computation of pseudoscalar and vector heavy-light meson masses for heavy-quark masses ranging from the physical charm mass up to $\simeq 4$ times the physical $b$-quark mass. We employ the gauge configurations generated by the European Twisted Mass Collaboration (ETMC) with $N_f=2+1+1$ dynamical quarks at three values of the lattice spacing ($a\simeq 0.062,0.082,0.089\mathrm{fm}$) with pion masses in the range $M_{\pi }\simeq 210–450\mathrm{MeV}$. The heavy-quark mass is simulated directly on the lattice up to $\simeq 3$ times the physical charm mass. The interpolation to the physical $b$-quark mass is performed using the ETMC ratio method, based on ratios of the meson masses computed at nearby heavy-quark masses, and adopting the kinetic mass scheme. The extrapolation to the physical pion mass and to the continuum limit yields $m_b^{\mathrm{kin}}(1\mathrm{GeV})=4.61(20)\mathrm{GeV}$, which corresponds to ${\overline{m}}_b({\overline{m}}_b)=4.26(18)\mathrm{GeV}$ in the $\overline{\mathrm{MS}}$ scheme. The lattice data are analyzed in terms of the heavy-quark expansion (HQE) and the matrix elements of dimension-four and dimension-five operators are extracted with a good precision, namely, $\overline{\mathrm{\Lambda }}=0.552(26)\mathrm{GeV}$, ${\mu }_{\pi }^2=0.321(32){\mathrm{GeV}}^2$, and ${\mu }_G^2(m_b)=0.253(25){\mathrm{GeV}}^2$. The data also allow for a rough estimate of the dimension-six operator matrix elements. As the HQE parameters play a crucial role in the inclusive determination of the Cabibbo-Kobayashi-Maskawa matrix elements $V_{ub}$ and $V_{cb}$, their precise determination on the lattice may eventually validate and improve the analyses based on fits to the semileptonic moments.

Figure 1

Left panel: Effective masses of the four correlators $C_{\mathrm{PS}}^{LL}(t)$, $C_{\mathrm{PS}}^{LS}(t)$, $C_{\mathrm{PS}}^{SL}(t)$, and $C_{\mathrm{PS}}^{SS}(t)$, calculated for a ($c\ell$) meson using Eq. (8) in the case of the ETMC gauge ensemble B55.32 (corresponding to a pion mass $\simeq 380\mathrm{MeV}$). Right panel: The same as in the left panel, but for the vector correlators $C_{\mathrm{V}}^{LL}(t)$, $C_{\mathrm{V}}^{LS}(t)$, $C_{\mathrm{V}}^{SL}(t)$, and $C_{\mathrm{V}}^{SS}(t)$.

Figure 2

Left panel: Effective masses of the correlator $C_{\mathrm{PS}}^{SL}(t)$ calculated for various ($h\ell$) mesons using Eq. (8) in the case of the ETMC gauge ensemble A40.32 (corresponding to a pion mass $\simeq 320\mathrm{MeV}$). Right panel: The same as in the left panel, but for the vector correlator $C_{\mathrm{V}}^{SL}(t)$. The solid lines identify the plateau region $t_{\min }\le t\le t_{\max }$ selected for each value of the heavy-quark mass.

Figure 3

The quantity $M_{av}({\tilde{m}}_h^{(1)})=M_{av}({\tilde{m}}_c)$ versus the (renormalized) light-quark mass ${\overline{m}}_{\ell }={\overline{m}}_{\ell }(2\mathrm{GeV})$ for the various ETMC gauge ensembles. The dashed lines are the results of a linear fit in both ${\overline{m}}_{\ell }$ and $a^2$ at each value of the lattice spacing and in the continuum limit. The diamond is the result at the physical light-quark mass $m_{ud}^{\mathrm{phys}}$ (see Table 2) in the continuum limit.

Figure 4

The ratios $y_M({\tilde{m}}_h^{(4)},\lambda )$ (left panel) and $y_M({\tilde{m}}_h^{(8)},\lambda )$ (right panel) versus the (renormalized) light-quark mass ${\overline{m}}_{\ell }={\overline{m}}_{\ell }(2\mathrm{GeV})$ for the various ETMC gauge ensembles. The dashed lines are the results of a linear fit in both ${\overline{m}}_{\ell }$ and $a^2$. The diamonds correspond to the values ${\overline{y}}_M({\tilde{m}}_h^{(4)},\lambda )$ and ${\overline{y}}_M({\tilde{m}}_h^{(8)},\lambda )$, obtained at the physical light-quark mass $m_{ud}^{\mathrm{phys}}$ (see Table 2) in the continuum limit.

Figure 5

Lattice data for the ratio ${\overline{y}}_M({\tilde{m}}_h,\lambda )$ versus the inverse heavy-quark mass $1/{\tilde{m}}_h$. The solid line is the result of the HQE-constrained fit (16) with ${\epsilon }_2=0$, taking into account the correlation matrix among the lattice points. The vertical dotted line corresponds to the position of the inverse physical $b$-quark mass $1/{\tilde{m}}_b$.

Figure 6

The quantity $\mathrm{\Delta }M({\tilde{m}}_h^{(1)})=\mathrm{\Delta }M({\tilde{m}}_c)$ versus the (renormalized) light-quark mass ${\overline{m}}_{\ell }={\overline{m}}_{\ell }(2\mathrm{GeV})$ for the various ETMC gauge ensembles. The dashed lines are the results of a linear fit in both ${\overline{m}}_{\ell }$ and $a^2$ at each value of the lattice spacing and in the continuum limit. The black diamond is the result at the physical light-quark mass $m_{ud}^{\mathrm{phys}}$ (see Table 2) in the continuum limit.

Figure 7

The Wilson coefficient $c_G$ evaluated at orders $\mathcal{O}({\alpha }_s)$ (dashed lines) and $\mathcal{O}({\alpha }_s^2)$ (solid lines) in the kinetic scheme (red lines) and in the pole-mass scheme (blue lines), i.e., using Eq. (34) with $\tilde{x}\ne 0$ and $\tilde{x}=0$, respectively. The vertical dotted lines correspond to the locations of the inverse physical $b$-quark and $c$-quark masses.

Figure 8

The ratios $y_{\mathrm{\Delta }M}({\tilde{m}}_h^{(4)},\lambda )$ (left panel) and $y_{\mathrm{\Delta }M}({\tilde{m}}_h^{(8)},\lambda )$ (right panel) versus the (renormalized) light-quark mass ${\overline{m}}_{\ell }={\overline{m}}_{\ell }(2\mathrm{GeV})$ for the various ETMC gauge ensembles. The solid lines are the results of a linear fit in both ${\overline{m}}_{\ell }$ and $a^2$. The black dots correspond to the values ${\overline{y}}_{\mathrm{\Delta }M}({\tilde{m}}_h^{(4)},\lambda )$ and ${\overline{y}}_{\mathrm{\Delta }M}({\tilde{m}}_h^{(8)},\lambda )$, obtained at the physical light-quark mass $m_{ud}^{\mathrm{phys}}$ (see Table 2) in the continuum limit.

Figure 9

Lattice data for the ratio ${\overline{y}}_{\mathrm{\Delta }M}({\tilde{m}}_h,\lambda )$ versus the inverse heavy-quark mass $1/{\tilde{m}}_h$. The solid line is the result of the HQE-constrained fit (36) with $\mathrm{\Delta }{\epsilon }_2=0$, taking into account the correlation matrix among the lattice points. The vertical dotted line corresponds to the position of the inverse physical $b$-quark mass $1/{\tilde{m}}_b$.

Figure 10

Lattice data for the quantity $M_{av}({\tilde{m}}_h)/{\tilde{m}}_h$ [Eq. (40)] versus the inverse heavy-quark mass ${\tilde{m}}_h$. The dashed and solid lines are the results of the HQE fit (41) in which the correlation matrix between the lattice data is taken into account. The dashed line corresponds to the central values of the fits, while the solid lines represent 1 standard deviation. The vertical dotted lines correspond to the positions of the inverse physical $b$-quark and $c$-quark masses, $1/{\tilde{m}}_b$ and $1/{\tilde{m}}_c$.

Figure 11

Lattice data for the quantity ${\tilde{m}}_h\mathrm{\Delta }M({\tilde{m}}_h)$ [see Eq. (38)]. The dashed and solid lines are the results of the HQE fit (42), in which the correlation matrix between the lattice data is taken into account. The dashed line corresponds to the central values of the fit, while the solid lines represent 1 standard deviation. The vertical dotted lines correspond to the positions of the inverse physical $b$-quark and $c$-quark masses, $1/{\tilde{m}}_b$ and $1/{\tilde{m}}_c$, respectively.

Figure 12

Lattice data for the quantity $M_{\mathrm{V}}^2-M_{\mathrm{PS}}^2$ versus the inverse heavy-quark mass ${\tilde{m}}_h$. The dashed and solid lines are the result of the HQE fit (57), in which the correlation matrix between the lattice data is taken into account. The dashed line corresponds to the central values of the fits, while the solid lines represent 1 standard deviation. The vertical dotted lines correspond to the positions of the inverse physical $b$-quark and $c$-quark masses, $1/{\tilde{m}}_b$ and $1/{\tilde{m}}_c$. At the charm mass the experimental value and the error from Ref. [3] are adopted.