$B\to K$ and $D\to K$ form factors from fully relativistic lattice QCD

We present the result of lattice QCD calculation of the scalar, vector and tensor form factors for the $B\to K{\ell }^{+}{\ell }^{-}$ decay, across the full physical range of momentum transfer. We use the highly improved staggered quark (HISQ) formalism for all valence quarks on eight ensembles of gluon-field configurations generated by the MILC collaboration. These include four flavors of HISQ quarks in the sea, with three ensembles having the light $u/d$ quarks at physical masses. In the first fully relativistic calculation of these form factors, we use the heavy-HISQ method. This allows us to determine the form factors as a function of heavy-quark mass from the $c$ to the $b$, and so we also obtain new results for the $D\to K$ tensor form factor. The advantage of the relativistic formalism is that we can match the lattice weak currents to their continuum counterparts much more accurately than in previous calculations; our scalar and vector currents are renormalized fully nonperturbatively and we use a well-matched intermediate momentum-subtraction scheme for our tensor current. Our scalar and vector $B\to K$ form factors have uncertainties of less than 4% across the entire physical $q^2$ range and the uncertainty in our tensor form factor is less than 7%. Our heavy-HISQ method allows us to map out the dependence on heavy-quark mass of the form factors and we can also see the impact of changing spectator quark mass by comparing to earlier HPQCD results for the same quark weak transition but for heavier mesons.

Figure 1

Schematic of our three-point correlation function.

Figure 2

The range of $a{m}_h$ values used in this work (Table 2), and their corresponding $M_H/M_B$ values. Ensembles with physical light quarks are shown in blue.

Figure 3

Stability plot for different correlator fit choices on set 8, showing the mass of the ground-state non-Goldstone $\widehat{H}$ meson for $a{m}_h=0.6$, the ground-state energy of the $K$ with twist $\theta =4.705$ and $T_{00}^{\mathrm{nn}}$ for $a{m}_h=0.45$, $\theta =2.235$. Test 0 is the final result, corresponding to $N_{\exp }=5$ exponentials. Tests 1 and 2 use one fewer and one more exponential respectively. Tests 3–6 double and halve the prior widths and SVD cut. Test 7 increases $t_{\min }$ by 2 across the whole fit. The final test, 8, is when the fit is done on its own, or in the case of the $T_{00}^{\mathrm{nn}}$, just with the $\widehat{H}$ and $\widehat{K}$ two-point correlators required, as opposed to being part of one big simultaneous fit. The ${\chi }^2$ per degree of freedom and log(GBF) value for each test are shown in the bottom pane in blue and red respectively. For the later tests (5–7), data are removed from the fit, resulting in a lower log(GBF) which is not comparable with the others and not displayed. As discussed in Sec. 3a, ${\chi }^2$ values are artificially lowered by our SVD cut and priors so are only meaningful relatively. ${\chi }^2$ values for tests 3–6, which change prior width and SVD, are thus not directly comparable with other tests. The final fit gives a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ close to 1 with SVD and prior noise.

Figure 4

Stability tests for the $z$-expansion fit evaluated at the physical $B$ mass. Test 0 is the final result, 1 and 2 take different priors for ${\zeta }_0$, test 3 allows ${\zeta }_0$ to vary between the form factors and test 4 (see text) allows for ${\zeta }_{n\ne 0}\ne 0$. Test 5 drops the term containing $\zeta$ entirely. Test 6 increases the number of the terms in the $z$ expansion, $N$, by 1 (to 4) and test 7 does the same for each component of $N_{ijkl}$ in each $a_n$ coefficient. Test 8 doubles the width of ${\zeta }_n$ and all $d$ and $\rho$ priors, and 9 halves them. Test 10 removes the chiral logarithm term by setting $\mathcal{L}=1$, and 11 tightens the prior on the $\rho$ coefficients considerably. Test 12 allows for logarithmic heavy-mass dependence $(a{m}_h{)}^2\log (a{m}_h)$ in the fit. Test 13 removes the $f_0(0)=f_{+}(0)$ constraint; in this case the black point is $f_0(0)$ and the red is $f_{+}(0)$. Tests 14, 15 and 16 remove all the lattices with physical light masses, all of set 8 data, and results with $m_l=m_s$ respectively. Test 17 removes the spatial vector data, and 18 removes the largest mass from all ensembles with multiple masses. The ${\chi }^2$ per degree of freedom and log(GBF) value for each test are shown in the bottom panel in blue and red respectively. For the latter tests, data are removed from the fit, resulting in a lower log(GBF) which is not comparable with others and so not displayed. As in our correlator fits, ${\chi }^2$ values are artificially lowered by our SVD cut and priors so are only meaningful relatively. ${\chi }^2$ values for tests 7 and 8, which change widths on many priors, are thus not directly comparable with other tests. No SVD cut is required, and our final fit has a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ of 0.3 when prior noise is included.

Figure 5

Upper plot: A comparison of values and their statistical errors for the vector form factor derived from matrix elements for the spatial and temporal vector currents on ensembles where both are available. The filled symbols are the temporal vector results and the open symbols the spatial vector results. We have offset spatial vector results slightly on the $q^2$-axis for clarity. Lower plot: The ratio of the $f_{+}$ values for the spatial and temporal vector cases. We see no evidence of any differences between them (within our uncertainties) that would indicate discretisation effects.

Figure 6

$(1-\frac{q^2}{M_{H_{s0}^{*}}^2})f_0(z)$ data points and final result at the physical point (blue band). Data points are labelled by heavy quark mass, where e.g. m0.8 indicates $a{m}_h=0.8$ on that ensemble. Lines between data points of a given heavy mass are the result of the fit evaluated on this ensemble and mass with all lattice artefacts present. Sets 9 and 10 are the $H_s\to {\eta }_s$ data from sets 1 and 2 in [32], which were fitted simultaneously with sets 6 and 7 respectively.

Figure 7

$(1-\frac{q^2}{M_{H_s^{*}}^2})f_{+}(z)$ data points and final result at the physical point (red band). Data points are labelled by heavy quark mass, where e.g. m0.8 indicates $a{m}_h=0.8$ on that ensemble. Lines between data points of a given heavy mass are the result of the fit evaluated on this ensemble and mass with all lattice artefacts present. Sets 9 and 10 are the $H_s\to {\eta }_s$ data from sets 1 and 2 in [32], which were fitted simultaneously with sets 6 and 7 respectively. At large $|z|$ (large $q^2$), data obtained from both temporal and spatial components of $V^{\mu }$ are shown, the latter with end caps specifying the associated uncertainty. As discussed in Sec. 2, errors for $f_{+}$ at large $q^2$ are significantly smaller when obtained from spatial vector components.

Figure 8

$(1-\frac{q^2}{M_{H_s^{*}}^2})f_T(z)$ data points and final result at the physical point (green band). Data points are labelled by heavy quark mass, where e.g. m0.8 indicates $a{m}_h=0.8$ on that ensemble. Lines between data points of a given heavy mass are the result of the fit evaluated on this ensemble and mass with all lattice artefacts present.

Figure 9

Final $B\to K$ form factor results at the physical point across the full $q^2$ range.

Figure 10

The contributions to the total percentage error (black line) of $B\to K$ form factors $f_0(q^2)$ (top) and $f_{+}(q^2)$ (middle) and $f_T(q^2)$ (bottom) from different sources, shown as an accumulating error. The red dashed line (‘inputs’) includes values for parameters, such as masses, taken from the particle data group (PDG) [57] and used in the fit as described above. The purple dotted line (“$q$ mistunings”) adds, negligibly, to the inputs of the error contribution from the quark mistunings associated with $c$ fit parameters and errors from the light-quark chiral extrapolation, while the solid blue line (“statistics”) further adds the error from our correlator fits. The green dotted-dashed line (“HQET”) includes the contribution from the expansion in the heavy-quark mass, and, finally, the thick black line (“Discretization), the total error on the form factor, also includes the discretization errors. In the case of the tensor form factor, the difference here is so small as to obscure the HQET line. The percentage variance adds linearly and the scale for this is given on the left-hand axis. The percentage standard deviation, the square root of this, can be read from the scale on the right-hand side.

Figure 11

Breakdown of the contributions to the statistical uncertainty of the $B\to K$ form factors at their extremes from data on each ensemble. Uncertainty from each ensemble ${\sigma }_i$ is added in quadrature, normalized by the total uncertainty squared ${\sum }_i{\sigma }_i^2$. Sets 6 and 7 include contributions from $H_s\to {\eta }_s$ data.

Figure 12

The $f_0(q_{\max }^2)$ data points on each ensemble, plotted against $M_H$. Points in blue have physical $m_l$ values, black have $m_l=m_s/5$ and red have $m_l=m_s$. The blue band indicates the continuum result from our full fit [i.e. Eq. (25)]. The green band indicates the continuum results of a fit of just the $f_0(q_{\max }^2)$ data to Eq. (26). Dashed lines between data points indicate the full fit evaluated at that lattice spacing.

Figure 13

The form factors at $q_{\max }^2$ and $q^2=0$ evaluated across the range of physical heavy masses from the $D$ to the $B$. Other lattice studies [25, 28, 71, 72] of both $D\to K$ and $B\to K$ are shown for comparison. We also include some $B\to K$ results at $q^2=0$ from Gubernari et al. [73], a calculation using light-cone sum rules. We do not include HPQCD’s $D\to K$ results that share data with our calculation here [36]; see text for a discussion of that comparison. At the $B$ end, data points are offset from $M_B$ for clarity. Note that we have run $Z_T$ to scale $\mu$ in this plot, where $\mu$ is defined linearly between 2 GeV and $m_b=4.8\mathrm{GeV}$, according to Eq. (27). The full running to 2 GeV from $m_b$ results in a factor of 1.0773(17), applied to $f_T^{D\to K}$.

Figure 14

The leading power of $M_H$ dependence in the form factors at $q^2=0$ and $q_{\max }^2$. The scale associated with the tensor form factor, $\mu$, is varied using Eq. (27).

Figure 15

Combinations of $B\to K$ form factors, $f_0$, $f_{+}$ and $f_T(\mu =4.8\mathrm{GeV})$ in Eqs. (28) and (29) compared with expectations $\frac{M_B}{M_B+M_K}$ from HQET [69] (dashed line). Uncertainties on the HQET expectations are not included.

Figure 16

Comparison of our $B\to K$ scalar and vector form factors with those of $B_s\to {\eta }_s$ [32] and $B_c\to D_s$ [38] to show the impact of changing the spectator quark mass. In the lower panel, we have multiplied the form factors by their common pole factors to reduce the $y$-axis range and highlight the variation between the form factors. We take $P^0(q^2)=1-\frac{q^2}{M_{B_{s0}^{*}}^2}$, $P^{+}(q^2)=1-\frac{q^2}{M_{B_s^{*}}^2}$, using the central values of the masses in Table 5.

Figure 17

Comparison of our $B\to K$ tensor form factor (at $\mu =4.8\mathrm{GeV}$) with those of $B_c\to D_s$ [38] to show the impact of changing the spectator quark mass.

Figure 18

The green band gives our $D\to K$ tensor form factor at $\mu =2\mathrm{GeV}$, across the physical $q^2$ range. Results from [72] are included for comparison.

Figure 19

Breakdown of the contributions to the statistical uncertainty of the $D\to K$ form factors at their extremes from data on each ensemble. Uncertainty from each ensemble ${\sigma }_i$ is added in quadrature, normalized by the total uncertainty squared ${\sum }_i{\sigma }_i^2$. Sets 6 and 7 include contributions from $H_s\to {\eta }_s$ data.

Figure 20

Comparison of our $D\to K$ scalar and vector form factors with those for $D_s\to {\eta }_s$ [32] and $B_c\to B_s$ [34] to show the effect of changing spectator quark mass.

Figure 21

The $M_H$ dependence of our form factors at $q^2=0$ and $q^2=M_D^2$.

Figure 22

The dominant power of the $M_H$ dependence of our form factors at $q^2=0$ and $q^2=M_D^2$, isolated using $(M_H/f)\times df/d{M}_H$.