Form factors for the processes $B_c^{+}\to D^0{\ell }^{+}{\nu }_{\ell }$ and $B_c^{+}\to D_s^{+}{\ell }^{+}{\ell }^{-}(\nu \overline{\nu })$ from lattice QCD

We present results of the first lattice QCD calculations of the weak matrix elements for the decays $B_c^{+}\to D^0{\ell }^{+}{\nu }_{\ell }$, $B_c^{+}\to D_s^{+}{\ell }^{+}{\ell }^{-}$ and $B_c^{+}\to D_s^{+}\nu \overline{\nu }$. Form factors across the entire physical $q^2$ range are then extracted and extrapolated to the continuum limit with physical quark masses. Results are derived from correlation functions computed on MILC Collaboration gauge configurations with three different lattice spacings and including $2+1+1$ flavors of sea quarks in the highly improved staggered quark (HISQ) formalism. HISQ is also used for all of the valence quarks. The uncertainty on the decay widths from our form factors for $B_c^{+}\to D^0{\ell }^{+}{\nu }_{\ell }$ is similar in size to that from the present value for $V_{ub}$. We obtain the ratio $\mathrm{\Gamma }(B_c^{+}\to D^0{\mu }^{+}{\nu }_{\mu })/{|{\eta }_{\mathrm{EW}}{V}_{ub}|}^2=4.43(63)\times {10}^{12}{\mathrm{s}}^{-1}$. Combining our form factors with those found previously by HPQCD for $B_c^{+}\to J/\psi {\mu }^{+}{\nu }_{\mu }$, we find ${|V_{cb}/V_{ub}|}^2\mathrm{\Gamma }(B_c^{+}\to D^0{\mu }^{+}{\nu }_{\mu })/\mathrm{\Gamma }(B_c^{+}\to J/\psi {\mu }^{+}{\nu }_{\mu })=0.257(36{)}_{B_c\to D}(18{)}_{B_c\to J/\psi }$. We calculate the differential decay widths of $B_c^{+}\to D_s^{+}{\ell }^{+}{\ell }^{-}$ across the full $q^2$ range and give integrated results in $q^2$ bins that avoid possible effects from charmonium and $u\overline{u}$ resonances. For example, we find that the ratio of differential branching fractions integrated over the range $q^2=1{\mathrm{GeV}}^2–6{\mathrm{GeV}}^2$ for $B_c^{+}\to D_s^{+}{\mu }^{+}{\mu }^{-}$ and $B_c^{+}\to J/\psi {\mu }^{+}{\nu }_{\mu }$ is $6.31(90{)}_{B_c\to D_s}(65{)}_{B_c\to J/\psi }\times {10}^{-6}$. We also give results for the branching fraction of $B_c^{+}\to D_s^{+}\nu \overline{\nu }$. Prospects for reducing our errors in the future are discussed.

Figure 1

The mass $M_{H_c}$ of the heavy-charm pseudoscalar meson is plotted against the lattice spacing squared for each of the values $a{m}_h=0.5$, 0.65, 0.8 used in the heavy-HISQ calculation. Values for $M_{H_c}$ are obtained from fitting the correlation functions as described in Sec. 2e. The continuum-physical point is denoted by a cross at $a=0\mathrm{fm}$ and $M_{H_c}=M_{B_c}$ from experiment [28]. Data from sets 1–4 are denoted by the colors red, blue, green and magenta, respectively. Data for $a{m}_h=0.5$, 0.65, 0.8 can be identified by the diamond, triangle and circle markers, respectively. These choices will be repeated in all subsequent plots.

Figure 2

The $q^2$ values on each set as a proportion of the maximum value $q_{\max }^2=(M_{H_c}-M_{D_{l(s)}}{)}^2$. From top to bottom, data from sets 1–4 are displayed (see Table 1). For different $a{m}_h$ on a given set, the same twists were used. As described in the caption for Fig. 1, data from sets 1–4 and heavy quark masses $a{m}_h$ are denoted by different colors and marker styles. Values used here for the masses of the initial and final mesons are found from fits of correlation functions (to be discussed in Sec. 2e).

Figure 3

Diagrammatic representations of the three-point functions we calculate on the lattice. The top two diagrams are relevant for extracting matrix elements of the scalar density and temporal vector current, and the bottom diagram is calculated for the case $B_c\to D_s$ and the tensor current. Each operator insertion is shown by a cross and is labeled by its description given in the spin-taste basis, while the lines represent lattice quark propagators. The heavy quark propagator is represented by the line, labeled by the flavor $h$, between the leftmost operator and the insertion. The daughter quark propagator is represented by the line, labeled by the flavor $l(s)$, between the insertion and the rightmost operator. The remaining quark propagator is the spectator quark, labeled by the flavor $c$.

Figure 4

Fit functions for the $B_c\to D_l$ and $B_c\to D_s$ form factors $f_{0,+}^l$ and $f_{0,+,T}^s$, respectively, tuned to the continuum limit with physical quark masses. The tensor form factor is at the scale 4.8 GeV.

Figure 5

Fit functions for the four form factors $f_{0,+}^{l,s}$ tuned to the continuum limit with physical quark masses.

Figure 6

Ratio of the tensor and vector form factors of $B_c\to D_s$ across the entire range of physical $q^2$. The behavior is in agreement with LEET [41], which predicts a constant ratio $(M_{B_c}+M_{D_s})/M_{B_c}$.

Figure 7

Errors on the form factors $f_{0,+}^l$. The black curve shows the total error, and the other lines show a particular partition of the error. When added in quadrature, these contributions yield the black curve. The dashed curves show uncertainties from the fit coefficients in Eq. (16). The solid blue curve shows the statistical errors resulting from our fits of correlation functions. The solid red curve represents the contribution to the final error from the determinations of the quark mass mistunings on each lattice [see Eq. (20)].

Figure 8

Errors on the form factors $f_{0,+,T}^s$. The curves are labeled similarly to Fig. 7.

Figure 9

Differential decay rate ${\eta }_{\mathrm{EW}}^{-2}|V^{ub}{|}^{-2}d\mathrm{\Gamma }(B_c^{+}\to D^0{\ell }^{+}{\nu }_{\ell })/d{q}^2$ as a function of $q^2$ for the cases $\ell =\mu$ in blue and $\ell =\tau$ in red.

Figure 10

We plot the ratio of $d\mathrm{\Gamma }/d{q}^2$ for each of the processes $B_c^{+}\to J/\psi {\ell }^{+}{\nu }_{\ell }$ and $B_c^{+}\to D^0{\ell }^{+}{\nu }_{\ell }$ for the $q^2$ range of the $B_c^{+}\to J/\psi {\ell }^{+}{\nu }_{\ell }$ decay. The decay width for the former process is derived from form factors found in [3], and the decay width of the latter process is derived from form factors determined in this study. The case $\ell =\mu$ is shown in blue, and the case $\ell =\tau$ is shown in red.

Figure 11

Plot of the $b_{\ell }(q^2)$, as defined in Eqs. (35) and (34), for $B_c^{+}\to D_s^{+}{\ell }^{+}{\ell }^{-}$. The top plot shows the case $\ell =\mu$ (blue) for the region $m_{\mu }^2<q^2<1{\mathrm{GeV}}^2$. The middle plot shows the case $\ell =\mu$ (blue) for the region $1{\mathrm{GeV}}^2<q^2<q_{\max }^2$. Finally, the lower plot shows the case $\ell =\tau$ (red).

Figure 12

Plot of the $B_c^{+}\to D_s^{+}{\ell }^{+}{\ell }^{-}$ differential branching ratio for $\ell =\mu$ (top) and $\ell =\tau$ (bottom) in the final state. The yellow bands show regions where charmonium resonances (not included in our calculation) could have an impact. The grey band is between the two yellow regions labeling the charmonium resonances. Through the yellow and gray bands, we interpolate the function $d{\mathcal{B}}_{\mu }/d{q}^2$ linearly when integrating to find the branching fraction and related quantities.

Figure 13

From top to bottom, we show plots of the flat terms $F_H^{\ell }$ for each of $\ell =e$, $\mu$, $\tau$, respectively. We use a log scale for the cases $\ell =e$, $\mu$. Error bands are presented, though the errors are small due to the correlations in the construction of the flat term.

Figure 14

Differential branching fraction for $B_c^{+}\to D_s^{+}\nu \overline{\nu }$ as a function of $q^2$.

Figure 15

We show data from the exafine lattice in blue with squares, denoting $a{m}_h=0.625$, and circles, denoting $a{m}_h=0.35$. Alongside the exafine data, we show the fits of the form factors $f_0^s$ (top) and $f_{+}^s$ (bottom) from ultrafine sets to coarser sets as presented in the upper plot of Fig. 4. The lattice data at $a{m}_h=0.625$ ($\approx a{m}_b$) closely follow the fit curves.

Figure 16

Diagrammatic representation of the three-point functions we calculate on set 1 for insertions of the local spatial vector current ${\gamma }_x\otimes {\gamma }_x$ as described in the text in Sec. 4b. Each operator is shown by a cross and is labeled by its description given in the spin-taste basis, while the lines represent lattice quark propagators as in Fig. 3.

Figure 17

Lattice data on set 1 (see Table 1) for $f_{+}^s$ for different methods of extraction which we differentiate by color. The blue points are the $f_{+}^s$ data extracted via Eq. (51) using the local spatial vector current ${\gamma }_x\otimes {\gamma }_x$. The red points are the $f_{+}^s$ data extracted via Eq. (8) using the local temporal vector current. The blue and red points agree very well at all $q^2$. Near zero recoil, the errors on blue points are much smaller than the red points.

Figure 18

Plots of effective energies for the pseudoscalar heavy-charm meson at each $a{m}_h$ value for set 1. The left plot shows the ${\gamma }_5{\gamma }_t\otimes {\gamma }_5{\gamma }_t$ meson. The right plot shows the ${\gamma }_5\otimes {\gamma }_5$ meson.

Figure 19

Plots of effective energies for the $D_s$ meson with taste ${\gamma }_5\otimes {\gamma }_5$ at each twist in Table 2 for set 1.

Figure 20

Plots of effective energies for the $D_s$ meson with taste ${\gamma }_5{\gamma }_0\otimes {\gamma }_5{\gamma }_0$ at each twist in Table 2 for set 1.

Figure 21

Results for the local vector current renormalization factor $Z_V$ obtained from Eq. (6) by the ratio of scalar density and temporal vector current matrix elements at zero recoil. The top and bottom plots show the results from the calculation of $B_c\to D_l$ and $B_c\to D_s$. The different colors and shapes of markers relate to sets and $a{m}_h$ values as described in Fig. 2.

Figure 22

Parameter $V_{\mathrm{nn},00}$ from Eq. (13) corresponding to the $H_c\to D_s$ three-point correlator at zero recoil with $a{m}_h=0.65$ plotted against the fit index $I$ [defined in Eq. (a6)]. From top to bottom, results on sets 1, 2, 3 and 4 (see Table 1) are presented respectively. Red, green and blue points indicate that the fit used $N_{\mathrm{n}}+N_{\mathrm{o}}=4$, 5, 6 exponentials respectively (see Eqs. (10) and (13). The different marker styles reflect the SVD cut chosen: squares, circles, triangles, right and left pointing triangles correspond to SVD cuts of 0.001,0.025,0.05,0.075 and 0.01 respectively. The scale of the y axis is shared by the four plots. We scrutinise the form factors associated with correlator fits detailed in Table 9.

Figure 23

Data and fit for the form factors $f_{0,+}^l$. The scale of the $y$-axis is the same as for Fig. 24. The different colours and shapes of markers relate to sets and $a{m}_h$ values as described in Fig. 2.

Figure 24

Data and fit for the form factors $f_{0,+,T}^s$. The scale of the $y$-axis is shared with Fig. 23.

Figure 25

Data and fit for the form factors $f_{0,+}^l$ multiplied by the pole factor $P(q^2)$ [see Eq. (16)]. The fit band is the polynomial ${\sum }_n{c}^{(n)}(-z{)}^n$ [coefficients $c^{(n)}$ are defined in Eq. (c2)].

Figure 26

Data and fit for the form factors $f_{0,+,T}^s$ multiplied by the pole factor $P(q^2)$] see Eq. (16)]. The fit band is the polynomial ${\sum }_n{c}^{(n)}(-z{)}^n$ [coefficients $c^{(n)}$ are defined in Eq. (c2)].

Figure 27

Data and fit for the form factor $f_0$ multiplied by the pole factor [see Eq. (16)] plotted at zero recoil as a continuous function of $M_{H_c}$. The vertical dotted lines show the masses of the $H_c$ meson for the cases in which the heavy quark coincides with the charm and bottom quarks.

Figure 28

For each of the 16 different correlator fits indexed by $J$ [see Eq. (b1)], we show the fitted values of the physical-continuum form factors for $B_c\to D_l$ (top) and $B_c\to D_s$ (bottom) evaluated at maximum $q^2$ and $q^2=0$. These plots show results from all possible combinations of the correlation function fits described in Table 9 and demonstrate the stability of our results under these changes. The filled black points show the results from our final fit.

Figure 29

For each of the different form factor fits described in Appendix pp2-s3, we show the physical-continuum form factors for $B_c\to D_l$ (top) and $B_c\to D_s$ (bottom) evaluated at maximum $q^2$ and $q^2=0$. The filled black points show the results from our final fit.