$B_s\to K\ell \nu$ decay from lattice QCD

We use lattice QCD to calculate the form factors $f_{+}(q^2)$ and $f_0(q^2)$ for the semileptonic decay $B_s\to K\ell \nu$. Our calculation uses six MILC asqtad $2+1$ flavor gauge-field ensembles with three lattice spacings. At the smallest and largest lattice spacing the light-quark sea mass is set to $1/10$ the strange-quark mass. At the intermediate lattice spacing, we use four values for the light-quark sea mass ranging from $1/5$ to $1/20$ of the strange-quark mass. We use the asqtad improved staggered action for the light valence quarks, and the clover action with the Fermilab interpolation for the heavy valence bottom quark. We use SU(2) hard-kaon heavy-meson rooted staggered chiral perturbation theory to take the chiral-continuum limit. A functional $z$ expansion is used to extend the form factors to the full kinematic range. We present predictions for the differential decay rate for both $B_s\to K\mu \nu$ and $B_s\to K\tau \nu$. We also present results for the forward-backward asymmetry, the lepton polarization asymmetry, ratios of the scalar and vector form factors for the decays $B_s\to K\ell \nu$ and $B_s\to D_s\ell \nu$. Our results, together with future experimental measurements, can be used to determine the magnitude of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$.

Figure 1

Lowest order Standard Model Feynman diagram shown here for example of semileptonic $B_s^0\to K^{-}{\ell }^{+}{\nu }_{\ell }$ decay.

Figure 2

$B_s$ and kaon meson two-point correlation function for the $a\approx 0.12\mathrm{fm}$, $N_s^3\times N_t=24^3\times 64$ ensemble. The blue points are the effective mass constructed via Eq. (4.3). The prior is shown as a green band. The fitted meson mass is shown as a thin gray horizontal band. Some of the blue-point error bars are too small to be visible. The error of the fitted kaon meson mass is magnified 20 times to make it visible in the plot.

Figure 3

Fitted $B_s$-meson and kaon masses, $M_{B_s}$ and $M_K$ in lattice units at different $t_{\min }$ for the $a\approx 0.12\mathrm{fm}$, $N_s^3\times N_t=24^3\times 64$ ensemble. The left vertical axes show the fitted masses and the right vertical axes show the corresponding $p$ value of the fit. The chosen fit results are shown in wide gray bands. The masses are shown in circles with error bars. The selected value of $t_{\min }$ is plotted using a solid circle and its error band is extended across the plot in gray. Red diamonds denote the $p$ values.

Figure 4

Test of Eq. (4.4) for the $a\approx 0.12\mathrm{fm}$, $N_s^3\times N_t=24^3\times 64$ ensemble. Left: energy-momentum dispersion relation where $E_K$ and $M_K$ come from the kaon 2-point correlators. Right: test of wave function overlap momentum dependence. The dashed lines on both plots show the power-counting estimate of the size of the momentum-dependent discretization error, $\mathcal{O}({\alpha }_s|{\mathit{p}}_K{|}^2{a}^2)$.

Figure 5

The lattice form factor from a combined two-point and three-point correlation function fit for the $a\approx 0.12\mathrm{fm}$, $N_s^3\times N_t=24^3\times 64$ ensemble. The green band is the combined fit result for ${f_{\parallel }}^{\mathrm{lat}}$. The blue points with errors are obtained from the ratio defined in Refs. [16, 18]. The black curve is the ratio constructed directly from combined fit results. The small difference between the green band and the ratio comes from excited state contributions still present in the ratio but accounted for in the fit method used here.

Figure 6

Fit results of $⟨K^{(0)}|V^0|B_s^{(0)}⟩$ from different fit ranges for the $a\approx 0.12\mathrm{fm}$, $N_s^3\times N_t=24^3\times 64$ ensemble with lattice kaon momentum ${\mathit{p}}_K=(2\pi /N_s)(0,0,0)$. Left: The three-point correlator fit maximum are fixed to be $t_{\max }^{3\mathrm{pt}}=15$ and 16 and the minimum are varied between 1 and 10. Right: The three-point correlator fit minimum are fixed to be $t_{\min }^{3\mathrm{pt}}=3$ and the maximum is varied between 12 and 17. The preferred fit ranges are shown with filled points.

Figure 7

Chiral-continuum extrapolated form factors $f_{\parallel }$ and $f_{\perp }$ in $r_1$ units as functions of the recoil energy $r_1{E}_K$. The color denotes the lattice spacings and the symbols denote the ratio of the sea-quark masses $m_l^{\prime }/m_h^{\prime }$. The colored fit lines correspond to the fit results evaluated at the parameters of the ensembles. The cyan band with the black curve shows the chiral-continuum extrapolated results.

Figure 8

Comparison among chiral-continuum extrapolated results for $f_{+}$ with different analytic terms. The gray band shows the preferred fitting result with NNLO SU(2) $\mathrm{HMrS}\chi \mathrm{PT}$. The red (dashed) and blue (solid) curves show the error ranges resulting from the fits with only NLO analytic terms and with all terms up to NNNLO, respectively.

Figure 9

Percent deviations of alternative chiral-continuum extrapolations from the preferred central fit of $f_{+}$ and $f_0$. The curves show the deviation from the preferred central fit obtained by either omitting the ${\mathit{p}}_K=2\pi (1,1,1)/N_s$ data points or by using SU(2) soft pion $\mathrm{HMrS}\chi \mathrm{PT}$ formula. The gray band shows the statistical errors from the preferred NNLO SU(2) $\mathrm{HMrS}\chi \mathrm{PT}$ fits. The deviations are smaller than the statistical errors.

Figure 10

Distribution of the errors for $f_{+}$ (left) and $f_0$ (right) as a function of $q^2$. The left $y$ axis shows the square of the errors added in quadrature. The right $y$ axis shows the errors themselves. The different bands show the total error when adding individual sources of error in quadrature one by one. The error bands associated with ${\kappa }_b$ and $m_{\ell }$ are too small to be visible on the plots.

Figure 11

A schematic diagram of the conformal mapping of the form factor regions from the complex $q^2$ plane to the complex $z$ plane.

Figure 12

Preferred $K=4$ $z$-parametrization fit results for the form factors $f_{+}$ (upper curve) and $f_0$ (lower curve) as functions of $z$ and $q^2$. The kinematic constraint Eq. (6.9) is applied. The corresponding bands with larger errors are the results of the chiral-continuum extrapolation, as shown in Sec. 4d. They are used as inputs for the $z$-parametrization fit. The bands with smaller errors are the resultant $z$-parametrization fits. The $q^2=0$ point corresponds to $z=0.205$ as shown in Table 8. The meson poles are listed in Table 9.

Figure 13

Theoretical lattice QCD calculations of the $B_s\to K\ell \nu$ form factors from the HPQCD Collaboration [30], the RBC and UKQCD Collaborations [31], the ALPHA Collaboration [32], and the Fermilab Lattice and MILC Collaborations, marked as “This work” in the figure. Different treatments of the bottom quark on the lattice are listed in parenthesis.

Figure 14

Comparison of the theoretical calculations of the $B_s\to K\ell \nu$ form factors at $q^2=0$. The results shown are from light-cone sum rules (LCSR) [34, 35], NLO perturbative QCD (pQCD) [36], relativistic quark model (RQM) [33], and ($2+1$)-flavor lattice QCD (LQCD) from the HPQCD Collaboration [30], the RBC and UKQCD Collaborations [31], and the Fermilab Lattice and MILC Collaborations.

Figure 15

Standard Model predictions of the differential decay rate divided by $|V_{ub}{|}^2$ for $B_s\to K\mu \nu$ (left) and $B_s\to K\tau \nu$ (right).

Figure 16

Standard Model predictions of the ratio of the differential decay rates $R^{\tau /\mu }(q^2)$.

Figure 17

Standard Model predictions of the forward-backward asymmetry divided by $|V_{ub}{|}^2$ for $B_s\to K\mu \nu$ (left) and $B_s\to K\tau \nu$ (right).

Figure 18

Standard Model predictions of the normalized lepton polarization asymmetry for $B_s\to K\mu \nu$ (left) and $B_s\to K\tau \nu$ (right).

Figure 19

Form factor ratios, $f_{+,0}^{2012}(B_s\to D_s)/f_{+,0}^{2012}(B\to D)$, calculated by Fermilab Lattice and MILC Collaborations in Ref. [13] in 2012 (left) and $B\to D\ell \nu$ form factors, $f_{+,0}^{2015}(B\to D)$, calculated by the same collaborations in Ref. [15] in 2015 (right). These are the ingredients to reconstruct the $B_s\to D_s\ell \nu$ form factors $f_{+,0}^{\mathrm{reco}}(B_s\to D_s)$.

Figure 20

The reconstructed form factors $f_{+,0}^{\mathrm{reco}}(B_s\to D_s)$ obtained from Eq. (7.12).

Figure 21

Form factor ratios, $f_{+,0}(B_s\to K)/f_{+,0}^{\mathrm{reco}}(B_s\to D_s)$, as functions of the momentum transfer $q^2$. The result provided by HPQCD [85] at $q^2=0$ is plotted for comparison.

Figure 22

Form factor ratios, $f_{+,0}(B_s\to K)/f_{+,0}^{\mathrm{reco}}(B_s\to D_s)$, as functions of the recoil parameter $w$.

Figure 23

The kinematically allowed region for $B_s\to K\ell \nu$ (upper solid line) and $B_s\to D_s\ell \nu$ (lower solid line) decays in terms of $q^2$ and $w$. The solid lines are the relation between $q^2$ and $w$ as defined in Eq. (7.13). The green and purple areas are the corresponding $B_s\to K\ell \nu$ regions used to construct the form factor ratios as shown in Fig. 21 and Fig. 22, respectively.