New methods for $B$ meson decay constants and form factors from lattice NRQCD

We determine the normalization of scalar and pseudoscalar current operators made from nonrelativistic $b$ quarks and highly improved staggered light quarks in lattice quantum chromodynamics (QCD) through $\mathcal{O}({\alpha }_s)$ and ${\mathrm{\Lambda }}_{\mathrm{QCD}}/m_b$. We use matrix elements of these operators to extract $B$ meson decay constants and form factors, and then compare to those obtained using the standard vector and axial-vector operators. This provides a test of systematic errors in the lattice QCD determination of the $B$ meson decay constants and form factors. We provide a new value for the $B$ and $B_s$ meson decay constants from lattice QCD calculations on ensembles that include $u$, $d$, $s$, and $c$ quarks in the sea and those that have the $u/d$ quark mass going down to its physical value. Our results are $f_B=0.196(6)\mathrm{GeV}$, $f_{B_s}=0.236(7)\mathrm{GeV}$, and $f_{B_s}/f_B=1.207(7)$, agreeing well with earlier results using the temporal axial current. By combining with these previous results, we provide updated values of $f_B=0.190(4)\mathrm{GeV}$, $f_{B_s}=0.229(5)\mathrm{GeV}$, and $f_{B_s}/f_B=1.206(5)$.

Figure 1

The Feynman diagram for the vertex renormalization in continuum QCD perturbation theory. The heavy quark is denoted by a double line, the light quark by a single line, and the exchange of a gluon by a curly line. The current is denoted by a cross inside a circle.

Figure 2

The different contributions that make up the renormalization coefficients $z_0^{A_0}$ and $z_0^P$ as a function of the bare heavy quark mass in the lattice NRQCD Hamiltonian, as given in Eqs. (21) and (24). The infrared-finite pieces are plotted for infrared-divergent contributions, and $Z_q$ is not included since it does not vary with $a{m}_b$.

Figure 3

The $z_0$ and $z_1$ factors for the $\mathcal{O}({\alpha }_s)$ matching of the temporal axial and pseudoscalar NRQCD-HISQ currents to continuum QCD [Eqs. (19) and (23)] plotted against the bare lattice $b$-quark mass.

Figure 4

The one-loop mixing coefficient ${\zeta }_{02}^{A0}$ for the temporal axial vector NRQCD-HISQ current for massless HISQ quarks divided by the bare heavy quark mass, $a{m}_b$, and plotted against $a{m}_b$. This shows that ${\zeta }_{02}$ grows linearly with $a{m}_b$ as $a{m}_b\to \infty$.

Figure 5

The ratio of the decay constant obtained using the temporal axial current to that obtained using the pseudoscalar density. Results are from lattice QCD calculations of the matrix element from a nonrelativistic expansion of the appropriate current operator between the vacuum and a $B$ meson (upper plot) or $B_s$ meson (lower plot). Results from each of the ensembles of Table 4 are shown plotted against the square of the lattice spacing. Purple open squares denote the lowest (zeroth) order result, while blue open circles include only $J_{A_0,\text{lat}}^{(0)}$ but with $\mathcal{O}({\alpha }_s)$ matching for that current. Green pluses include $J_{A_0,\text{lat}}^{(1)}$ in a matching through $\mathcal{O}({\alpha }_s)$ and red crosses include the full matching of Eqs. (19) and (23). Green pluses have been offset for clarity. The dashed lines show the relative uncertainty on $f_B$ values quoted in [9].

Figure 6

Results for the decay constants of $B$ and $B_s$ mesons (multiplied by the $3/2$ power of the meson mass) for the ensembles in Table 4, obtained from the pseudoscalar current and plotted against the light quark mass in units of the strange quark mass (given as $M_{\pi }^2/M_{{\eta }_s}^2$). The grey bands show the results of the fit described in the text. Errors on the data points include statistics/fitting only; the grey band includes the full error from the fit to lattice spacing and quark mass effects along with the perturbative matching uncertainty.

Figure 7

Results for the ratio of $B_s$ to $B$ decay constants (multiplied by the $3/2$ power of the ratio of meson mass) obtained from the pseudoscalar current. The data points are as shown in Fig. 6, and the grey band is the result of the fit described in the text, including uncertainties from lattice spacing and quark mass effects along with uncertainties from higher-order relativistic corrections to the current.

Figure 8

The ratio of the scalar form factor at zero recoil for the $B$ to $\pi$ decay obtained from the temporal vector current to that from using the scalar current. Results are from lattice QCD calculations of the matrix element of a nonrelativistic expansion of the appropriate current operator between a $\pi$ meson and a $B$ meson. Results from each of the ensembles of Table 4 are shown plotted against the square of the lattice spacing. Symbols are as in Fig. 5, with the green pluses offset for clarity. The dashed line gives the relative error on $f_0(q_{\max }^2)$ quoted in [27].

Figure 9

The ratio of $R_{B\pi }$ values (see text for definition) obtained using a combination of temporal vector and temporal axial vector currents to that from a combination of scalar and pseudoscalar currents. Results from each of the ensembles of Table 4 are shown plotted against the square of the lattice spacing. Symbols are as in Fig. 5, with the green pluses offset for clarity. The dashed line gives the relative error on $R_{B\pi }$ from using the temporal vector current for $f_0$ and temporal axial current for $f_B$ quoted in [27].

Figure 10

A summary of lattice QCD calculations for $f_B$, $f_{B_s}$, and their ratio. The new results reported here are those for the pseudoscalar current given by the red open circles, showing good agreement with earlier results (given by red open squares) using the temporal axial current [9]. The average of these results is given by blue diamonds. Other results in this summary are taken from [36, 37, 38, 39, 40, 41, 42] and use a variety of different quark formalisms for heavy and light quarks as well as working with gluon field configurations that include different numbers of flavors of sea quarks. The results for $f_B$ correspond to those for a light valence quark of mass equal to the average of $u$ and $d$ quark masses except for “RBC/UKQCD15” which correspond to the neutral $b\overline{d}$ meson and “FNAL/MILC11” which correspond to the charged $b\overline{u}$ meson. The experimental result for the charged $B$ meson is an average from the Particle Data Group [1], as are the grey bands.