B_{s}\rightarrow D_{s}\;\ell\,\nu Form Factors and the Fragmentation Fraction Ratio f_{s}/f_{d}

Form Factors and the Fragmentation Fraction Ratio


We present a lattice quantum chromodynamics determination of the scalar and vector form factors for the decay over the full physical range of momentum transfer. In conjunction with future experimental data, our results will provide a new method to extract , which may elucidate the current tension between exclusive and inclusive determinations of this parameter. Combining the form factor results at non-zero recoil with recent HPQCD results for the form factors, we determine the ratios and . These results give the fragmentation fraction ratios and , respectively. The fragmentation fraction ratio is an important ingredient in experimental determinations of meson branching fractions at hadron colliders, in particular for the rare decay . In addition to the form factor results, we make the first prediction of the branching fraction ratio , where is an electron or muon. Current experimental measurements of the corresponding ratio for the semileptonic decays of mesons disagree with Standard Model expectations at the level of nearly four standard deviations. Future experimental measurements of may help understand this discrepancy.

HPQCD Collaboration

I Introduction

Studies of and meson decays at the Large Hadron Collider provide precision tests of the Standard Model of particle physics and are an important tool in the search for new physics. For example, the first observation of the rare decay , through a combined analysis by the LHCb and CMS collaborations Khachatryan et al. (2015); Aaij et al. (2017), tested the Standard Model prediction of the branching fraction. This decay is doubly-suppressed in the Standard Model, but may have large contributions from physics beyond the Standard Model (see, for example, Ge et al. (2016)). Although the observed branching fraction is currently consistent with Standard Model expectations, there is still considerable room for new physics, given the experimental and theoretical uncertainties. Both LHCb and CMS are expected to reduce their errors significantly in Run II and tightening constraints on possible new physics requires a corresponding improvement in the theoretical determination of the Standard Model branching fraction.

Extraction of the meson branching fraction relies on the normalization channels and Adeva et al. (2009). The branching fraction can then be expressed as Khachatryan et al. (2015)


where the are the fragmentation fractions, which give the probability that a -quark hadronizes into a meson. The factors in this equation represent detector efficiencies and the factors denote the observed numbers of events.

The analysis of Khachatryan et al. (2015) used the value of , determined from LHCb experimental data Aaij et al. (2012, 2013); Storaci et al. (2013). The ratio depends on the kinematic range of the experiment, leading to the introduction of an additional systematic uncertainty in the value of to account for the extrapolation of the LHCb result to the CMS acceptance. Reducing sources of systematic uncertainties in the value of this ratio will improve the precision of the determination of the branching fraction. Indeed, an accurate value for the fragmentation fraction ratio is necessary for improved measurements of other meson decay branching fractions at the LHC Adeva et al. (2009).

The ratio of the fragmentation fractions, , can be expressed in terms of the ratios of form factors Fleischer et al. (2010, 2011),


where is the scalar form factor of the semileptonic decay at . The first lattice calculations of the form factor ratios in Equation (2) using heavy clover bottom and charm quarks were published in Bailey et al. (2012). In addition, the form factors, and , for the semileptonic decay were determined with twisted mass fermions for the region near zero recoil in Atoui et al. (2014).

In this article we calculate the form factors, and , for the semileptonic decay . We present a determination of these form factors over the full physical range of momentum transfer, using the modified -expansion for the chiral-continuum-kinematic extrapolation. We combine these form factor results with recent HPQCD results for the decay Na et al. (2015) to determine the ratios of and form factors relevant to the ratio of fragmentation fractions, .

We use the non-relativistic (NRQCD) action for the bottom quarks and the Highly Improved Staggered Quark (HISQ) action for the charm quarks. Our form factors for have appeared already in Na et al. (2015). Here we first present form factor results and then proceed to the form factor ratios. We find


This leads to




respectively. The uncertainties in these results are: the experimental statistical and systematic uncertainties; theoretical uncertainties (predominantly arising from a factor that captures deviations from naive factorization and, in Equation (5), an electroweak correction factor); and the uncertainties in our lattice input. In quoting these results, we have assumed that there are no correlations between the lattice results and the other sources of uncertainty.

In addition to determining the fragmentation fraction ratio relevant to the measurement of the branching fraction for the rare decay, , the semileptonic decay provides a new method to determine the CKM matrix element . There is a long-standing tension between determinations of from exclusive and inclusive measurements of the semileptonic meson decays (see, for example, Amhis et al. (2014, 2016a) and the review in Patrigiani et al. (2016)), although recent analyses suggest the tension has eased Gambino et al. (2016); Bigi and Gambino (2016). The decay has yet to be observed experimentally and consequently has received less theoretical attention than semileptonic decays of the meson. The studies that have been undertaken for the decay include calculations based on relativistic quark models Zhao et al. (2007); Bhol (2014), light-cone sum rules Li et al. (2009), perturbative factorization Fan et al. (2014) and estimates using the Bethe-Salpeter method Li et al. (2010); Chen et al. (2012). At present, there is one unquenched lattice calculation of the form factor at zero recoil Atoui et al. (2014). The FNAL/MILC collaboration has previously studied the ratio of the form factors of the and decays Bailey et al. (2012).

We determine the form factor for the semileptonic decay at zero momentum transfer to be and at zero recoil to be . Although experimental data is frequently presented in the form , the additional information provided by our calculation of the shape of the form factors throughout the kinematic range will, when combined with future experimental data, provide a new method to extract and may elucidate the puzzle of the tension between inclusive and exclusive determinations of this CKM matrix element.

In the next section we briefly outline the details of the calculation, including the gauge ensembles, bottom-charm currents and two- and three-point correlator construction. Our calculation closely parallels that presented in Na et al. (2015) for the semileptonic decay and we refer the reader to that work for further details. In Section III we discuss correlator fits to our lattice data and Section IV covers the chiral-continuum-kinematic extrapolations, which follows closely the methodology of Na et al. (2015). We explain how some of the correlations between the new data and the data are incorporated into the chiral-continuum-kinematic expansion. Section V presents our final results for the form factors, for and , and for and . We summarize in Section VI and in Appendix A we give the information necessary to reconstruct the form factors. The analogous details for form factors were summarized in Appendix A of Na et al. (2015).

Ii Ensembles, currents and correlators

Our determination of the form factors for the semileptonic decay closely parallels the analysis presented in Na et al. (2015). Here we simply sketch the key ingredients of the analysis and refer the reader to Sections II and III of Na et al. (2015) for more details of the lattice calculation.

We use five gauge ensembles, summarized in Table 1, generated by the MILC collaboration Bazavov et al. (2010). These ensembles include three “coarse” (with lattice spacing ) and two “fine” (with ) ensembles and incorporate flavors of AsqTad sea quarks.

Set (sea)
C1 2.647 0.005/0.050 2096 4
C2 2.618 0.010/0.050 2256 2
C3 2.644 0.020/0.050 1200 2
F1 3.699 0.0062/0.031 1896 4
F2 3.712 0.0124/0.031 1200 4
Table 1: Simulation details on three “coarse” and two “fine” MILC ensembles.

In addition, we tabulate the light pseudoscalar masses on these ensembles, for both AsqTad and HISQ valence quarks, in Table 2. The difference in these masses captures discretization effects arising from partial quenching. We account for these effects in the chiral-continuum-kinematic expansion, which we discuss in more detail in Section IV.

C1 0.15971(20) 0.15990(20) 0.36530(29) 0.31217(20) 0.41111(12)
C2 0.22447(17) 0.21110(20) 0.38331(24) 0.32851(48) 0.41445(17)
C3 0.31125(16) 0.29310(20) 0.40984(21) 0.35720(22) 0.41180(23)
F1 0.14789(18) 0.13460(10) 0.25318(19) 0.22855(17) 0.294109(93)
F2 0.20635(18) 0.18730(10) 0.27217(21) 0.24596(14) 0.29315(12)
Table 2: Meson masses on MILC ensembles for both AsqTad Bazavov et al. (2010) and HISQ valence quarks Bouchard et al. (2014). The values are determined with HISQ valence quarks in Bouchard et al. (2014).

In Table 3 we list the valence quark masses for the NRQCD bottom quarks and HISQ charm quarks Na et al. (2012); Bouchard et al. (2014). For completeness and ease of reference, we include both the tree-level wave function renormalization for the massive HISQ quarks Monahan et al. (2013) and the spin-averaged mass, corrected for electroweak effects, determined in Na et al. (2012).

C1 2.650 0.0489 0.6207 1.00495618 0.28356(15)
C2 2.688 0.0492 0.6300 1.00524023 0.28323(18)
C3 2.650 0.0491 0.6235 1.00504054 0.27897(20)
F1 1.832 0.0337 0.4130 1.00103879 0.25653(14)
F2 1.826 0.0336 0.4120 1.00102902 0.25558(28)
Table 3: Valence quark masses for NRQCD bottom quarks and and for HISQ strange and charm quarks. The fifth column gives , the tree-level wave function renormalization constant for massive (charm) HISQ quarks. The sixth column lists the values of the spin-averaged mass, corrected for electroweak effects.

To study semileptonic decays, we evaluate the matrix element of the bottom-charm vector current, , between and states. We express this matrix element in terms of the form factors and as


where the momentum transfer is . In practice it is simpler to work with the form factors and , which are related to and via


Here is the energy of the daughter meson in the rest frame of the meson. In the following, we work in the rest frame of the meson and when we refer to the spatial momentum, , we mean the momentum of the meson.

NRQCD is an effective theory for heavy quarks and results determined using lattice NRQCD must be matched to full QCD to make contact with experimental data. We match the bottom-charm currents, , at one loop in perturbation theory through , where is the bare lattice mass Monahan et al. (2013). We re-scale all currents by the nontrivial massive wave function renormalization for the HISQ charm quarks, tabulated in Table 3, Na et al. (2015).

We calculate and meson two-point correlators and three-point correlators of the bottom-charm currents, . We use smeared heavy-strange bilinears to represent the meson and incorporate both delta-function and Gaussian smearing, with a smearing radius of and on the coarse and fine ensembles, respectively. Three-point correlators are computed with the setup illustrated in Figure 1. The meson is created at time and a current inserted at timeslice , between and . The daughter meson is then annihilated at timeslice . We use four values of : 12, 13, 14, and 15 on the coarse lattices; and 21, 22, 23, and 24 on the fine lattices. We implement spatial sums at the source through the random wall sources and Na et al. (2010). We generate data for four different values of the meson momenta, , , , and , where is the spatial lattice extent.

Figure 1: Lattice setup for the three-point correlators. See accompanying text for details.

We fit meson two-point functions to a sum of decaying exponentials in Euclidean time, ,


Here the superscripts and indicate the smearing associated with the meson source (delta function or Gaussian); the and are amplitudes associated with the ordinary non-oscillatory states and the oscillatory states that arise in the staggered quark formalism; the meson energies are and for the non-oscillatory and oscillatory states, respectively; and is the number of exponentials included in the fit.

The ground state energy in NRQCD, , is related to the true energy in full QCD, , by


because the -quark rest mass has been integrated out in NRQCD. Here is the spin-averaged mass used to tune the -quark mass and was determined in Na et al. (2012). We tabulate the values for in Table 3.

We fit the meson two-point functions to the form


For the three-point correlator we use the fit ansatz


The amplitudes for energy levels depend on the current , the daughter meson momentum , and the smearing of the meson source, .

The hadronic matrix element between and meson states is then given in terms of the ground state energies and amplitudes extracted from two- and three-point correlator fits by the relation


For more details on this relation, see Section III of Na et al. (2015).

Iii Correlator fit and form factor results

We employ a Bayesian multi-exponential fitting procedure, based on the python packages lsqfit Lepage, G.P. (a) and corrfitter Lepage, G.P. (b), that has been used by the HPQCD collaboration for a wide range of lattice calculations. Statistical correlations between data points, and correlations between data and priors, are automatically captured with the gvar class Lepage, G.P. (c), which facilitates the straightforward manipulation of Gaussian-distributed random variables.

In this Bayesian multi-exponential approach, one uses a number of indicators of fit stability, consistency, and goodness-of-fit to check the fit results. For example, we check that, beyond a minimum number of exponentials, the fit results are independent of the number of exponentials included in the fit. Figure 2 illustrates the results of this test for the meson two-point fits on ensemble set F1. The upper panel presents our results for four values of the spatial momentum, plotted as a function of the number of exponentials included in the plot. The lower panel shows the results obtained from three types of fits: a simultaneous fit to correlator data for all four spatial momenta, plotted with blue diamonds; a chained fit (discussed in detail in Appendix A of Bouchard et al. (2014)) to correlator data for all four spatial momenta simultaneously, shown with red squares; and an “individual” fit, plotted with purple circles. These individual fits include the correlator data for just a single daughter meson momentum in each fit.

We take the result for from the chained fit as our final result for each momentum. These results are tabulated in Table 4 and shown in Figure 2 as shaded bands in each plot. All three fit approaches give consistent results, as seen in the lower panel of Figure 2, but the simultaneous fits, with or without chaining, have the advantage that they capture the correlations between momenta, which is then reflected in the uncertainty quoted in the fit results. The chained fits give slightly better values of reduced . For example, for the ground state results plotted in the lower panel, the chained fits give for , while the simultaneous fits give . Both fits include 164 degrees of freedom. In addition, the chained fits are about ten percent faster than the simultaneous fits—14.6s to generate all the data in the lower plot for the chained fit compared to 16.4s for the simultaneous fit. This is not an important consideration for the two-point fits, but becomes relevant for the larger three-point fits, which can take many hours. Choosing to use chained fits for both two- and three-point fits ensures a consistent approach throughout the fitting procedure.

Figure 2: Fit results for the meson two-point correlator as a function of the number of exponentials included in the fit on ensemble F1. The upper plot includes data for all four values of the spatial momentum of the meson. The lower plot compares the values for the ground state energy from the simultaneous fit with two alternative fitting strategies, which are described in the text, at zero spatial momentum. Note the magnified scale on the vertical axis in the lower panel.
C1 1.18755(22) 1.21517(34) 1.24284(33) 1.27013(39)
C2 1.20090(30) 1.24013(56) 1.27822(61) 1.31543(97)
C3 1.19010(33) 1.23026(53) 1.26948(54) 1.30755(79)
F1 0.84674(12) 0.87559(19) 0.90373(20) 0.93096(26)
F2 0.84415(14) 0.87348(25) 0.90145(25) 0.92869(33)
Table 4: Fit results for the ground state energies of the meson at each spatial momentum . We take and fit all two-point correlator data simultaneously.

As a further test of the two-point fits for the meson we determine the ratio on each ensemble. We plot the results in Figure 3. The shaded region corresponds to , where we set . In general, the data lie systematically above the relativistic value of unity, indicating that the statistical uncertainties of the fit results are sufficiently small that we can resolve discretization effects at . These discretization effects are less than in the dispersion relation.

Figure 3: Dispersion relation for each ensemble. The shaded region corresponds to where we take .

Figure 4 shows the corresponding two-point fit results for the ground state of the meson for ensemble sets C2 and F1. These ensemble sets have the same sea quark mass ratios, (see Table 1) and the difference between the results stems almost entirely from the lattice spacing. We take the values with as our final results, highlighted in the figure by the square data points and the shaded bands. We tabulate our final results in Table 5.

Figure 4: Fit results for the meson two-point correlator as a function of the number of exponentials included in the fit on two ensemble sets, C2 and F1. We plot our final results, for which , as a green hexagon for C2 and a purple square for F1, with corresponding shaded bands.
C1 C2 C3 F1 F2
0.53714(60) 0.54332(65) 0.53657(86) 0.40873(53) 0.40819(44)
Table 5: Fit results for the ground state , on each ensemble set, with .

For the three-point correlator fits, we use a fitting procedure that diverges slightly from the approach taken in Na et al. (2015) and do not employ a “mixed” fitting strategy. Instead of combining “individual” and “master” fits (see Na et al. (2015) for full details), we use chained fits to correlators at all spatial momenta. This fitting approach ensures that we keep track of all statistical correlations between data at different momenta while maintaining fit stability, which was an issue for simultaneous fits attempted in Na et al. (2015).

To improve stability and goodness-of-fit, we thin the three-point correlator data on the fine ensembles by keeping every third timeslice. We illustrate the stability of these fits with the number of exponentials in the fit in Figure 5.

Figure 5: Fit results for the three-point amplitudes as a function of the number of exponentials on two ensemble sets, C2 and F1. We fit to correlator data for all values of the spatial momentum simultaneously and thin by keeping every third timeslice. We plot our final results, for which , as a green hexagon for C2 and a purple square for F1, with corresponding shaded bands. Note that the amplitudes on set C2 are approximately three times larger than the amplitudes on set F1, as indicated by the left (F1) and right (C2) vertical axes.

We test our choice by comparing fit results for the three-point amplitudes with thinning (keeping both every third and every fifth timeslice) and without thinning and plot the results in Figure 6. We do not consider thinning by an even integer, which removes information about the oscillatory states generated by the staggered quark action.

Figure 6: Fit results for the three-point amplitudes as a function of the number of exponentials for different choices of data thinning: no thinning, represented by turquoise triangles; keeping every third timeslice, represented by blue circles and the label “Thinning = 3”; and every fifth timeslice, shown by yellow pentagons and the label “Thinning = 5”. Our final result, for which we use thinning by every third timeslice and , is shown as a purple square and the corresponding purple shaded band.

In Figure 7 we present results for the three-point fits when different combinations of source-sink separations, , are used. For our final results we take the full set, on the coarse ensembles and on the fine ensembles.

Figure 7: Fit results for the three-point amplitude as a function of the number of source-sink separations, , incorporated in the fit on ensemble set F1. We fit to correlator data for all values of the spatial momentum simultaneously and thin by keeping every third timeslice. For our final results we take the full set, on the coarse ensembles and on the fine ensembles, indicated by the first point, the purple square, and the purple shaded band. Fit results from other combinations of source-sink separations are plotted as blue circles.

We fit the three-point correlator data after matching the bottom-charm currents to full QCD, as described briefly in Section II and in more detail in Na et al. (2015). In Na et al. (2015) this approach was compared with fitting the data first and then matching to full QCD and, as expected, the results are in good agreement within errors.

We summarize our final results for the form factors, and , for each ensemble and momentum in Tables 7 and 7. We represent the correlations between form factors at different momenta as a heat map in Figure 8 for ensemble set F2.

Figure 8: Correlations between form factors at different momenta for the ensemble set F2.
C1 0.8885(11) 0.8754(14) 0.8645(13) 0.8568(13)
C2 0.8822(13) 0.8663(15) 0.8524(16) 0.8418(18)
C3 0.8883(13) 0.8723(16) 0.8603(16) 0.8484(21)
F1 0.90632(98) 0.8848(13) 0.8674(13) 0.8506(17)
F2 0.9047(12) 0.8855(16) 0.8667(15) 0.8487(19)
Table 7: Final results for the form factor .
C1 1.1384(35) 1.1081(20) 1.0827(21)
C2 1.1137(29) 1.0795(22) 1.0470(21)
C3 1.1260(34) 1.0912(24) 1.0552(28)
F1 1.1453(29) 1.0955(24) 1.0549(24)
F2 1.1347(42) 1.0905(26) 1.0457(33)
Table 6: Final results for the form factor .

Iv Chiral, continuum and kinematic extrapolations

The form factor results presented in the previous section are determined at finite lattice spacing, with sea quark masses that are heavier than their physical values. These form factors are therefore functions of the momentum transfer, the lattice spacing, and the sea quark masses. The form factors determined from experimental data are functions of a single kinematic variable only. Typically this variable is the momentum transfer, , or the daughter meson energy, , but the form factors can also be expressed in terms of the -variable, defined by


where or the -variable,


Here and is a free parameter, which we take to be to ensure consistency with the analysis of Na et al. (2015). In Figure 9 we compare our results for the form factors, and , with the corresponding form factors for the decay, taken from Na et al. (2015), as a function of the -variable. From the plot, we see that there is little dependence on the light spectator quark species in the form factor results.

Figure 9: Form factor results for the decay, compared to those for the decay from Na et al. (2015), as function of . We plot four sets of results, for and for both and meson decays. We distinguish the data in four ways. First, the shape of each data marker indicates the corresponding ensemble set, as shown in the legend in the upper left corner: squares represent set C1; diamonds set C2; circles C3; left-triangles F1; and triangles F2. Second, the upper set of points are those for and the lower set of points show the data for , as indicated by the annotations. Third, the color of the points distinguishes the data as follows: the turquoise-green points represent ; the light purple points are ; the blue points are ; and the orange-yellow points are . Finally, we distinguish the data by size: the larger markers represent the decay, while the smaller points are from those for the decay.

To relate the form factor results determined at finite lattice spacing and unphysical sea quark masses to experimental data, we must therefore perform continuum and chiral extrapolations, along with a kinematic extrapolation in terms of one of the choices of kinematic variable. We combine these extrapolations through the modified -expansion, introduced in Na et al. (2010, 2011), and applied to heavy-light decays in Bouchard et al. (2013a, b); Bouchard et al. (2014). Our analysis of the chiral-continuum-kinematic extrapolation for decay closely parallels that for the decay in Na et al. (2015), so we only briefly outline the key components and refer the reader to Na et al. (2015) for details.

We express the dependence of the form factors on the -variable through a modification of the BCL parameterization Bourrely et al. (2009)


Here the are Blaschke factors that take into account the effects of expected poles above the physical region,


where we take Gregory et al. (2010), and . We find little dependence on the value of , in line with the results of Na et al. (2015). The expansion coefficients include lattice spacing and light quark mass dependence and can be written as


where the include all lattice artifacts and chiral logarithms. These coefficients are given by




and the , , , and are fit parameters, along with the . We use the fit function form of Na et al. (2015), with a new fit parameter, , to account for the tuning of the valence strange quark mass on each ensemble. We tabulate the meson masses required to calculate in Table 2.

We further modify the -expansion parameterization of the form factors to accommodate the systematic uncertainty associated with the truncation of the matching procedure at . We introduce fit parameters and , with central value zero and width and re-scale the form factors, and according to


We take the systematic uncertainties in these fit parameters as 3% and refer the reader to the detailed discussion of this approach in Na et al. (2015).

In Figure 10 we plot our fit results for , as a function of the -variable. We obtain a reduced of with 36 degrees of freedom (dof), with a quality factor of . The -value (or -value) corresponds to the probability that the from the fit could have been larger, by chance, assuming the data are all Gaussian and consistent with each other. We plot the lattice data and the results of the chiral-continuum-kinematic extrapolation for as the upper, red shaded band and for as the lower, purple shaded band. We use the fit ansatz outlined above, including terms up to in the modified -expansion, and refer to these results as the “standard extrapolation”. We tabulate our choice of priors and the fit results in Appendix A, and provide the corresponding -expansion coefficients and their correlations in Table 11. Following Na et al. (2015) and the earlier work of Na et al. (2010, 2011), we group the priors into Group I and Group II variables, and add a third group. Broadly speaking, Group I priors are the typical fit parameters, Group II includes the input lattice scales and masses, and Group III priors are physical input masses. See the appendix of Na et al. (2015) for more details.

Figure 10: Fit results from the “standard extrapolation” fit ansatz detailed in the text. The purple data points show the fit results at finite lattice spacing and the red and purple shaded bands are the physical extrapolations.

To test the convergence of our fit ansatz, we follow a procedure similar to that outlined in Na et al. (2015). This can be summarized as modifying the fit ansatz in the following ways:

  1. include terms up to in the -expansion;

  2. include terms up to in the -expansion;

  3. add light-quark mass dependence to the fit parameters ;

  4. add strange-quark mass dependence to the fit parameters ;

  5. add bottom-quark mass dependence to the fit parameters ;

  6. include discretization terms up to ;

  7. include discretization terms up to ;

  8. include discretization terms up to ;

  9. include discretization terms up to ;

  10. omit the term;

  11. incorporate a 2% uncertainty for higher-order matching contributions;

  12. incorporate a 4% uncertainty for higher-order matching contributions;

  13. incorporate 4% and 2% uncertainties on coarse and fine ensembles, respectively, for higher-order matching contributions.

We show the results of these modifications in Figure 11. This plot demonstrates that the fit has converged with respect to a variety of modifications of the chiral-continuum-kinematic extrapolation ansatz. As part of this process, we also tested the significance of the Blaschke factor in the fit results. In line with the results of Na et al. (2015), we found that, while the results agreed within uncertainties, removing the Blaschke lowered the central value and increased the uncertainty of the result. This test is not strictly a test of convergence and is therefore not included in Figure 11.

Figure 11: Fit results from modifications to the “standard extrapolation” fit ansatz, plotted as blue circles representing the form factor at (the lower set of data points) and at (the upper set of points). The test numbers labeling the horizontal axis correspond to the modifications listed in the text. The first data point, the purple square for and turquoise diamond for , are the “standard extrapolation” fit results, which are also represented by the purple and turquoise shaded bands, respectively.

To determine the ratio of form factors, we simultaneously fit the lattice form factor data for the and decays in a single script. We take the form factor results from Table III of Na et al. (2015) for the decay. Fitting the results simultaneously ensures that statistical correlations between the two data sets, such as those stemming from the lattice spacing determination on each ensemble set, are included in the final result for the ratio at zero momentum transfer. We do not re-analyze the to account for statistical correlations between the correlators themselves, which have negligible effect on the final result, given the current precision. This analysis would require fitting both and two- and three-point correlators simultaneously. To ensure that these statistical correlations are not important, we tested the correlations between the three-point correlators on different ensemble sets. We show an example of the corresponding correlations as a heat map in Figure 12, from which one can see that statistical correlations are less than . We have found that correlations of this size have negligible impact at our current level of precision.

Figure 12: Correlations between and ensemble-averaged, three-point correlators for ensemble set C1. The data correspond to a single meson source with Gaussian smearing , a source-sink separation of and with .

We fit the form factor data using the standard extrapolation ansätze for both the and data. For the decay, we choose the priors for the coefficients in the modified -expansion to be equal to those for the corresponding expression for the -expansion. These priors reflect the close agreement between the values for the and decays, illustrated in Figure 9. We list our choice of priors and the fit results for the ratio of form factors in Appendix A, and provide the corresponding -expansion coefficients and their correlations in Table 12.

V Results

v.1 Form factors

We plot our final results for the form factors, and , as a function of the momentum transfer, , in Figure 13.

Figure 13: Chiral and continuum extrapolated form factors, (lower band) and (upper band), as a function of the momentum transfer.

Our final result for the form factor at zero momentum transfer is


We provide an estimate of the error budget for this result in Table 8. For the ratio of form factors, we find




with corresponding error budgets in Table 9. We show the extrapolation bands as a function of momentum transfer for both (purple hatched band) and (plain turquoise band) semileptonic decays in Figure 14.

Figure 14: Chiral and continuum extrapolated form factors, (lower band) and (upper band), as a function of the momentum transfer, for both (purple hatched band) and (plain turquoise band) semileptonic decays. The lattice data for each decay cannot be distinguished on this plot and are therefore not included. See Figure 10 for a detailed plot of the results for the form factors at finite lattice spacing for both decays.

We find agreement, within errors, with the results of Bailey et al. (2012), which are


Here we have combined the uncertainties quoted in Bailey et al. (2012), which are statistical and systematic, in quadrature.

For the form factor at zero recoil, , which is often quoted as


where , we find


This result is in good agreement with the value of determined in Atoui et al. (2014), with a slightly smaller uncertainty. The corresponding values for the form factors are Na et al. (2015) and Bailey et al. (2012) (where the quoted uncertainty includes only statistical uncertainties).

The slope of the form factor, , is given by


where the derivative is with respect to the -variable of Equation (14). In the CLN parameterization, Caprini et al. (1998), the form factor is then parameterized by


with the -variable of the previous section:


We obtain