$B_c\to B_{s(d)}$ form factors from lattice QCD

We present results of the first lattice QCD calculations of $B_c\to B_s$ and $B_c\to B_d$ weak matrix elements. Form factors across the entire physical $q^2$ range are then extracted and extrapolated to the physical-continuum limit before combining with Cabbibo-Kobayashi-Maskawa (CKM) matrix elements to predict the semileptonic decay rates $\mathrm{\Gamma }(B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell })=52.4(2.5)\times {10}^9{\mathrm{s}}^{-1}$ and $\mathrm{\Gamma }(B_c^{+}\to B^0\overline{\ell }{\nu }_{\ell })=3.10(21)\times {10}^9{\mathrm{s}}^{-1}$. The lattice QCD uncertainty is comparable to the CKM uncertainty here. Results are derived from correlation functions computed on MILC Collaboration gauge configurations with a range of lattice spacings including $2+1+1$ flavors of dynamical sea quarks in the highly improved staggered quark (HISQ) formalism. HISQ is also used for the propagators of the valence light, strange, and charm quarks. Two different formalisms are employed for the bottom quark: nonrelativistic QCD (NRQCD) and heavy-HISQ. Checking agreement between these two approaches is an important test of our strategies for heavy quarks on the lattice. From chained fits of NRQCD and heavy-HISQ data, we obtain the differential decay rates $d\mathrm{\Gamma }/d{q}^2$ as well as integrated values for comparison to future experimental results.

Figure 1

Three-point correlator $C_{3\mathrm{pt}}(t,T)$. The flavor-changing operator insertion is denoted by a cross at time slice $t$ and the total time length of the three-point correlator is $T$. The random wall source for the $b$ and $s/d$ propagators is at the time slice of the $B_{s(d)}$ interpolator.

Figure 2

The mass $M_{H_c}$ of the heavy-charm meson is plotted against lattice spacing for each of the values of $a{m}_h$ used in the heavy-HISQ calculation. Obtained from fitting the correlators as described in Eq. (7), $M_{H_c}$ is a proxy for the bare lattice heavy-quark mass $a{m}_h$. The continuum-physical point is denoted by a cross at $a=0$ fm and $M_{H_c}=M_{B_c^{+}}$. Note that the $y$-axis scale begins near $M_{{\eta }_c}$.

Figure 3

Effective simulation energies [Eq. (15)] of two-point correlators with an NRQCD spectator quark on set 5 for $|a\mathbf{q}|=0.427$. The black lines with gray error bands show the energies extracted from fitting the correlators in the simultaneous fit of fine lattice data with all the three-point correlators and all the momenta. The $B$-meson energies shown here are offset from their physical values as a consequence applying the NRQCD formalism to the constituent $b$ quark.

Figure 4

Effective simulation energies [Eq. (15)] of two-point correlators with a HISQ spectator quark on set 6 at zero twist with $a{m}_h=0.8$. The horizontal bands show the energies extracted from the full simultaneous correlator fit.

Figure 5

$Z_V$ for the $c\to s$ vector current evaluated at different $q^2$ from the calculation with an NRQCD spectator quark using Eq. (10).

Figure 6

$Z_V$ of the $c\to s$ vector current from both the NRQCD and heavy-HISQ calculations are plotted alongside values from $D\to K$ [3]. The NRQCD data are marked with circles, the heavy-HISQ data are marked with crosses, and finally the $D\to K$ values are given by diamonds. As expected, no significant dependence on the spectator mass is observed.

Figure 7

$Z_V$ from the $s\to s$ vector current from [36] (red crosses) and the $c\to s$ vector current from the heavy-HISQ calculation given here (green squares). The curve is the fitted perturbative expansion, including discretization effects, detailed in [36]. The red circle is an extrapolated value at the lattice spacing associated with the superfine lattice.

Figure 8

$f_{+}$ form factor data for $B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell }$ from both the NRQCD and heavy-HISQ approaches. The NRQCD form factor data is given by filled circles; the heavy-HISQ data, by open circles. Data points on a given set and for a given heavy-quark mass are joined by lines to guide the eye.

Figure 9

$z_p$ expansion coefficients, for the calculation with an NRQCD spectator quark, computed using the variations of correlator fit parameters listed in Table 5 for the $B_c\to B_s$ form factors. The integer $x$ coordinate of each result is given by $n_{\mathrm{var}}^{(1)}+2n_{\mathrm{var}}^{(2)}+4n_{\mathrm{var}}^{(3)}+8n_{\mathrm{var}}^{(4)}+16n_{\mathrm{var}}^{(5)}$ where $n_{\mathrm{var}}^{(i)}=0$, 1 corresponding to the first and second fits, respectively, listed in Table 5 of set $i$.

Figure 10

$z_p$ expansion coefficients, for the calculation with an NRQCD spectator quark, computed using the variations of correlator fit parameters listed in Table 5 for the $B_c\to B_d$ form factors. The $x$ coordinate is the same as that in Fig. 9.

Figure 11

Lattice results and fitted $f_0$ form factor data for $B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell }$ with an NRQCD $b$ quark. The gray band shows the fitted form factor tuned to the limit of vanishing lattice spacing and physical quark masses.

Figure 12

Lattice results and fitted $f_{+}$ form factor data for $B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell }$ with an NRQCD $b$ quark. The gray band shows the fitted form factor tuned to the limit of vanishing lattice spacing and physical quark masses.

Figure 13

Lattice results and fitted $f_0$ form factor data for $B_c^{+}\to B^0\overline{\ell }{\nu }_{\ell }$ with an NRQCD $b$ quark. The grey band shows the fitted form factor tuned to the limit of vanishing lattice spacing and physical quark masses.

Figure 14

Fitted $f_{+}$ form factor data for $B_c^{+}\to B^0\overline{\ell }{\nu }_{\ell }$ with an NRQCD $b$ quark. The gray band shows the fitted form factor tuned to the limit of vanishing lattice spacing and physical quark masses.

Figure 15

Heavy-HISQ form factor results for $f_0^s$ together with the fitted curve at the physical point with its error band.

Figure 16

Heavy-HISQ form factor results for $f_{+}^s$ together with the fitted curve at the physical point with its error band.

Figure 17

Heavy-HISQ form factor results for $f_0^d$ together with the fitted curve at the physical point with its error band.

Figure 18

Heavy-HISQ form factor results for $f_{+}^d$ together with the fitted curve at the physical point with its error band.

Figure 19

$z_p$ expansion coefficients, for the calculation with a HISQ spectator quark, computed using the variations of correlator fit parameters listed in Table 6 for the $B_c\to B_s$ form factors. The integer $x$ coordinate of each result is given by $n_{\mathrm{var}}^{(3)}+3n_{\mathrm{var}}^{(5)}+9n_{\mathrm{var}}^{(6)}$ where $n_{\mathrm{var}}^{(i)}=0$, 1, 2 corresponding to original, first, and second variations, respectively, listed in Table 6 of set $i$.

Figure 20

$z_p$ expansion coefficients, for the calculation with a HISQ spectator quark, computed using the variations of correlator fit parameters listed in Table 6 for the $B_c\to B_d$ form factors. The $x$ coordinate is the same as that in Fig. 19.

Figure 21

Fits of $f_0$ for $B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell }$ tuned to the physical-continuum limit. The form factor is plotted against $z_p/|z_p(t_{-})|$.

Figure 22

Fits of $f_{+}$ for $B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell }$ tuned to the physical-continuum limit.

Figure 23

Fits of $f_0$ for $B_c^{+}\to B^0\overline{\ell }{\nu }_{\ell }$ tuned to the physical-continuum limit.

Figure 24

Fits of $f_{+}$ for $B_c^{+}\to B^0\overline{\ell }{\nu }_{\ell }$ tuned to the physical-continuum limit.

Figure 25

Values for the physical-continuum form factors $f_0^s=f_{+}^s$ at $q^2=0$ and $f_0^s$ and $f_{+}^s$ at $q_{\max }^2$ are plotted against the mass of the heavy-charm pseudoscalar meson. The curve is the continuum limit of the heavy-HISQ fit function [Eq. (23)] extrapolated to the physical $B_c$ and $D$ masses. Note that the region in which the heavy-HISQ calculation has results is the region above $M_{{\eta }_c}$. See the text for a description of how the extrapolation down to the $D$ was done. Also plotted are the form factor results for $D\to K$ [3] (green squares) as well as the NRQCD $B_c\to B_s$ result presented in this work (red circles).

Figure 26

Final form factors from the chained fits of $f_0$ (below) and $f_{+}$ (above) for $B_c^{+}\to B_s^0\overline{\ell }{\nu }_{\ell }$ in the physical-continuum limit, plotted against the entire range of physical $q^2$. This fit is described in Sec. 4d.

Figure 27

Final form factors from the chained fits of $f_0$ (below) and $f_{+}$ (above) for $B_c^{+}\to B^0\overline{\ell }{\nu }_{\ell }$ in the physical-continuum limit, plotted against the entire range of physical $q^2$. This fit is described in Sec. 4d.