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

We present results of the first lattice QCD calculations of $B_c \to B_s$ and $B_c \to B_d$ weak matrix elements. Results are derived from correlation functions computed on MILC Collaboration gauge configurations with lattice spacings between $0.12$ [fm] and $0.06$ [fm] including 2+1+1 flavours of dynamical sea quarks in the Highly Improved Staggered Quark (HISQ) formalism. Form factors across the entire physical $q^2$ range are then extracted and extrapolated to the physical-continuum limit. Two different formalisms are employed for the bottom quark: non-relativistic QCD (NRQCD) and heavy-HISQ. Checking agreement between these two approaches is an important test of our strategies for heavy quarks on the lattice.


I. INTRODUCTION
The semileptonic weak decays B + c → B 0 s ν and B + c → B 0 ν proceed via tree-level flavour changing processes c → sW + and c → dW + parametrised by the Cabbibo-Kobayashi-Maskawa (CKM) matrix of the Standard Model. Associated weak matrix elements can be expressed in terms of form factors which capture the non-perturbative QCD physics. Precise determination of the normalisation and the q 2 dependence of these form factors from lattice QCD will allow a novel comparison with future experiment to deduce the CKM parameters V cs and V cd . Lattice studies of other semileptonic meson decays that involve tree-level weak decays of a constituent charm quark include [1][2][3][4][5][6]. Precise determination of these CKM matrix elements is critical for examining the second row unitary constraint |V cd | 2 + |V cs | 2 + |V cb | 2 = 1. ( This will complement other unitarity tests of the CKM matrix. It is possible LHCb could measure B + c → B 0 s µν µ using Run 1 and 2 data. For example, normalising by B + c → J/ψµν µ would yield a constraint on the ratio V cs /V cb . Due to CKM suppression, a measurement of B + c → B 0 µν µ is likely to require many more B + c decays. A lattice study of the B + c → B 0 s ν and B + c → B 0 ν decays involves the practical complication of a heavy spectator quark. Care must be taken in placing such a particle on the lattice to avoid large discretisation effects. We consider two formalisms for the b quark. A valence a L.J.Cooper@damtp.cam.ac.uk b Judd.Harrison@glasgow.ac.uk c URL: http://www.physics.gla.ac.uk/HPQCD NRQCD [7,8] b quark, a formalism constructed from a non-relativistic effective theory, is used to simulate with physically massive b quarks. A complementary calculation uses HPQCD's heavy-HISQ method [9][10][11]. Here, all flavours of quark are implemented with the HISQ [12] formalism. This is a fully relativistic approach which involves calculations for a set of quark masses on ensembles of lattices with a range of fine lattice spacings, enabling a fit from which the physical result at the b quark mass in the continuum can be determined. The method with an NRQCD bottom quark also uses HISQ for the charm, strange and down flavours. This study will demonstrate the consistency of the NRQCD and heavy-HISQ approaches by comparing the form factors extrapolated to the physical-continuum limit.
In the limit of massless leptons, the differential decay rates for B + c → B 0 s ν and B + c → B 0 ν are given by dΓ where V is the relevant associated CKM matrix element V cs or V cd and f + is one of two form factors that parametrise the continuum weak matrix element The 4-momentum transfer is q = p 1 − p 2 , and only the vector part of the V − A weak current contributes since QCD conserves parity. The contribution of f 0 to the decay rate is suppressed by the lepton mass and hence irrelevant for the decays to eν e and µν µ . The phase space is sufficiently small to disallow decays to τ ν τ . Form factors are constructed from the matrix elements that are arXiv:2003.00914v1 [hep-lat] 2 Mar 2020  [13] given in lattice units for each set in column 2 where values were obtained from [14] on sets 1 to 5 and [15] on set 6. The physical value w0 = 0.1715 (9) was fixed from fπ in [16]. Sets Table I. For the light quarks, the values for the valence quarks are identical to the masses of the light sea quarks. The masses for the valence strange quarks and the valence charm quarks were tuned in [15] using w0 (Table I)  obtained by fitting the appropriate lattice QCD 3-point correlator data. By calculating correlators at a range of transfer momenta on lattices with different spacings and quark masses, continuum form factors at physical quark masses are obtained and then appropriately integrated to offer a direct comparison with decay rates that could be measured in experiment.
In this study, we begin with Sec. II in which details of the lattice calculations are described. Sec. II A reports on the parameters and gauge configurations used to generate the propagators. Next, Sec. II B explains how the correlators are subsequently constructed for the two different treatments of the heavy spectator quark, as well as how the correlator data is fit to extract the matrix elements. Our non-perturbative renormalisation method required to obtain the form factors is set out in Sec. II C. Sec. III presents results of the lattice calculations. Correlator fits are examined in Sec. III A, whilst Sec. III B discusses results for the renormalisation of the local lattice vector current. In Sec. III C, the form factor data for the cases of an NRQCD spectator and a HISQ spectator are plotted alongside. Sec. IV is concerned with the methodology and results from fitting the form factor data. An extrapolation of the form factors to physicalcontinuum point is presented in Sec. IV D and Sec. IV E shows how the form factors depend on the mass of the spectator quark. Finally, in Sec. V we give our conclusions.

A. Parameters and Set-up
We use ensembles with 2 + 1 + 1 flavours of HISQ sea quark generated by the MILC Collaboration [17][18][19] and described in Table I. The Symanzik-improved gluon action used is that from [20], where the gluon action is improved perturbatively through O(α s ) including the effect of dynamical HISQ sea quarks. The lattice spacing is identified by comparing the physical value for the Wilson flow parameter ω 0 = 0.1715(9) fm [16] with lattice values for ω 0 /a from [14] and [15]. Our calculations feature physically massive strange quarks and equal mass up and down quarks, with a mass denoted by m l , with m l /m s = 0.2 and also the physical value m l /m s = 1/27.4 [5]. For sets 1 to 5 in Table I, strange propagators were re-used from [21], a study of the pseudoscalar meson electromagnetic form factor. Light propagators were re-used from [22], an extension of [21] to the pion. The valence quark masses used for the HISQ propagators on these gluon configurations are given in Table II. The valence strange and charm quark masses used here were tuned in [15,21], slightly away from the sea quark masses to yield results that more closely correspond to physical values. The propagators were calculated using the MILC code [23].
We work in the frame where the B + c is at rest, and so that the usual lattice momenta q i = 2πk i /aN x , for integers k i , are shifted by πθ i /aN x . The corresponding q 2 is then constructed by taking q 0 to be the difference in energies of the lowest lying initial and final states. The coefficients of operators corresponding to relativistic correction terms in the NRQCD action are given in Table III. The valence b quark masses used for the NRQCD propagators are also given there. The values were taken from [8], where the b quark mass was found by matching the experimental value for the spin-averaged kinetic mass of the Υ and the η b to lattice data. For the calculation with an NRQCD spectator bottom quark, we use sets 1 to 5 in Table I.
Bare heavy quark masses am h used for the heavy-HISQ method are shown in Table IV. The selection of heavy quark masses follows [11]. As well as sets 3 and 5, the heavy-HISQ calculation makes use of a lattice finer than the five sets featuring in the calculation with an NRQCD spectator, set 6 in Table I. This is motivated by the necessity to avoid large discretisation effects that grow with (am h ) (as (am h ) 4 at tree-level) whilst gathering data at large masses that will reliably inform the limit m h → m b .

NRQCD spectator case
For the case of an NRQCD spectator quark, random wall source [26] HISQ propagators with the mass of the charm quark are calculated and combined with random wall source NRQCD b propagators to generate B + c 2point correlator data. 2-point correlators for B 0 s(d) are generated similarly. The strategy of combining NRQCD random wall propagators and HISQ random wall propagators to yield 2-point correlators was first developed in [27]. NRQCD propagators are generated by solving an initial value problem. This is computationally very fast compared to calculating rows of the inverse of the quark matrix. The 3-point correlator needed here is represented diagrammatically in Fig. 1. A HISQ charm quark propagator is generated by using the random wall bottom quark propagator as a sequential source. Following Appendix B in [28], and excluding a spacetime-dependent sign, the sequential source is given by the spin-trace where Γ is the gamma matrix structure at the operator insertion, S RW b is the random wall NRQCD propagator, and is the space-spin matrix which transforms the naive quark field to diagonalise the HISQ action in spin-space.

HISQ spectator case
The case of a HISQ spectator quark proceeds similarly with the only difference being the use of a HISQ propagator instead of an NRQCD propagator for the bottom quark. Again, the charm propagator uses the spectator bottom quark propagator as a sequential source. Multiple masses are used for the spectator quark, each requiring a different charm propagator for the 3-point correlator. Fig. 2 shows the heavy-charm pseudoscalar meson masses that arise from calculations with the am h values in Table IV. On set 6, the finest lattice considered, we reach a value for M Hc that is 80% of the physical B c mass.
The same strange and light random wall HISQ propagators on sets 3 and 5 are used in both the NRQCD and the heavy-HISQ calculations, thus the data on these lattices in the two approaches will be correlated. However, the effect of these correlations is small in the physicalcontinuum limit since the heavy-HISQ data on sets 3 and 5 are the furthest away from the physical b quark mass point, and hence these correlations are safely ignored.

Fitting the correlators
The correlator fits minimise an augmented χ 2 function as described in [29][30][31]. The functional forms for the 2point and 3-point correlators follow from their spectral decomposition and include oscillatory contributions from the staggered quark timedoubler. The matrix elements are related to the fit pa- where J is the relevant operator that facilitates the c → s(d) flavour transition. The pseudoscalar mesons of interest are the lowest lying states consistent with their quark content, so we are only concerned with the matrix elements for i = j = 0 since we restrict E[k] ≤ E[k + 1] by using log-normal prior distributions for the energy differences. The presence of i, j > 0 terms are necessary to give a good fit and to allow for the full systematic uncertainty from the presence of excited states to be included in the extracted V nn [0, 0]. On each set, the 2-point and 3-point correlator data for both c → s and c → d at all momenta are fit simultaneously to account for all possible correlations. The matrix elements and energies are extracted and form factor values determined, along with the correlations between results at different momenta.

C. Extracting The Form Factors
The Partially Conserved Vector Current (PCVC) Ward identity allows for a fully non-perturbative renormalisation of the lattice vector current. Since the same HISQ action is used for the c and s(d) quarks that couple to the W + in both the NRQCD and heavy-HISQ approaches, we have the PCVC identity relating the conserved (point-split) c → s(d) lattice vector current and the local lattice scalar density S. We choose a local lattice operator V µ loc , thus Eq. (9) must be adjusted by a single renormalisation factor Z V associated with that operator, giving in principle Z V need only be found at zero-recoil where q µ has only a temporal component [3]. This avoids the need to calculate 3-point correlators associated with the spatial components of the vector current matrix element that appear in Eq. (10) for q = 0. However, in practice, it is preferable to determine f + near zero-recoil through the spatial components of the vector current matrix element, albeit with the additional cost in computing 3-point correlators with the corresponding insertion.
As in [2], we combine Eqs. (3) and (10) to give a determination of f 0 solely in terms of the scalar density matrix element. We use Eq. (11) and calculation of the vector current matrix element to determine f + and f 0 for the full q 2 range following [3,32]. Thus, we will calculate matrix elements of both the local scalar density J = S and the local vector current J = V . Once f 0 is determined, f + is obtained using Eq. (3) for µ = 0 to yield where V µ is the vector current matrix element, except at zero-recoil where the denominator vanishes and f + cannot be extracted. We find that using Eq. (12) near zero-recoil is problematic since both the numerator and denominator grow from 0 as q 2 is decreased from the maximum value at zero-recoil. For the case where the spectator is an NRQCD b quark, we instead use Eq.
This method gives much smaller errors near to zerorecoil. Although mathematically equivalent to Eq. (12), extracting f + through Eq. (13) does not suffer an inflation of error near zero-recoil since both the numerator and denominator are non-zero for all physical q 2 . However, since V i appears explicitly in Eq. (13), 3-point correlators with an insertion of V i need to be calculated. For the case of the spectator NRQCD b quark, the use of Eq. (13) is straightforward except that it requires inversions of the charm quark propagator from a different sequential source (see Eq. (5)) to allow for insertion of the Collecting V i at non-zero 3-momentum transfer in the NRQCD calculation will also test for any q 2 dependence of Z V that would appear as a discretisation effect. Using Eqs. (11) and (12) or (13), form factor data at a variety of lattice spacings, light quark masses and momenta are obtained from the energies and matrix elements.

NRQCD spectator case
For the case of an NRQCD spectator quark, the form factor extraction is complicated by the energy offset as a consequence of the subtraction of the b quark rest mass inherent in the NRQCD formalism. Whilst physical energy differences are preserved with NRQCD quarks, energy sums are not. Consequently, Particle Data Group (PDG) [33] values are used where necessary. For example, we take when extracting the form factors. We use interpolating operators cγ 5 b and sγ 5 b (dγ 5 b) for J P = 0 − pseudoscalars B + c and B 0 s(d) respectively.

HISQ spectator case
For the case of a HISQ spectator quark, we work only with local scalar and vector currents. Expressed in the spin-taste basis, we use γ 5 ⊗γ 5 for the H s(d) interpolating operator and two different operators, γ 5 ⊗ γ 5 and γ 5 γ 0 ⊗ γ 5 γ 0 , for the H c interpolator. The first of these, γ 5 ⊗ γ 5 , makes a tasteless 3-point correlation function when the scalar density operator 1 ⊗ 1 is used. The second, γ 5 γ 0 ⊗ γ 5 γ 0 , allows for a tasteless 3-point correlation function when we use the local temporal vector current operator γ 0 ⊗ γ 0 [3]. This requires the calculation of two H c 2point functions with the two different choices of operator at both the source and the sink. The difference in masses between these two different tastes of H c meson is tiny and, although consistently taken care of, it has no impact on the calculation.

A. Correlators
Figs. 3 and 4 provide samples of the correlator data from the NRQCD and heavy-HISQ calculations respectively. The quantity plotted is the effective simulation energies, which we define by the two-step log-ratio This ratio is preferable to an effective energy defined using C(t)/C(t + 1) since the ratio in Eq. (15) better suppresses the oscillatory contributions in Eq. (7). Error bars are present in the figure but mostly too small to observe. We exclude t min /a data points from the beginning and end points of the correlators in our fits to reduce the contributions from excited states. For each of the cases of an NRQCD and HISQ spectator quark, we fit all of the correlator data to Eq. (7) on each set simultaneously to obtain the correlations between the fitted parameters. Consequently, the correlator fits involve a large covariance matrix. Without extremely large statistical samples of results small eigenvalues of the covariance matrix are underestimated [34,35] and this causes problems when carrying out the inversion to find χ 2 . We overcome this by using an SVD (singular-value decomposition) cut; any eigenvalue of the covariance matrix smaller than some proportion c of the biggest eigenvalue λ max is replaced by cλ max . By carrying out this procedure, the covariance matrix becomes less singular. These eigenvalue replacements will only inflate our final errors, hence this strategy is conservative. The SVD cut reduces the χ 2 /d.o.f. reported by the fit because it lowers the contribution to χ 2 of the modes with eigenvalues below the SVD cut. In order to check the suitability of the SVD cut, we must test the goodness-of-fit from a fit where noise (SVD-noise) is added to the data to reinstate the size of fluctuations expected from the modes  below SVD cut, as described in Appendix D of [35]. The f. is used to check the goodness of fit for both cases of spectator quark.
Many fits were carried out with different SVD cuts, number of exponentials N , and positions t 2pt min and t 3pt min of the first timeslice where the correlators are fit. We selected the fit of the correlators on each lattice for form factor extraction based on the χ 2 /d.o.f. and Q-value. The parameters used in the fits of correlators with an NRQCD spectator quark are presented in Table V. The parameters given in bold are those used for our final fits.  Other values are used in tests of the stability of our form factor fits to be discussed in Sec. IV B.
We fit the heavy-HISQ correlator data to Eq. (7) on each set simultaneously, including correlations between data with different values of twist, heavy quark mass, and H s/d final state. Values for t min /a, the chosen SVD cut, the number of exponentials used in Eq. (7) and the resultant value of χ 2 /dof including SVD noise are given in Table VI. We also include in Table VI fits (20) for dependence of Z V on q 2 (for the case of an NRQCD spectator) and on the spectator quark mass (for the case of a HISQ spectator). The vector current renormalisation factor Z V computed at different momentum transfer with NRQCD b quarks shows no significant dependence on q 2 on each set, demonstrated by Fig. 5. Mild lattice spacing dependence is observed, however. For each momenta, we use the Z V found at the corresponding q 2 from Eq. (10).
The Z V factor in Eq. (10) is associated only with the local vector current operator and should be independent of the spectator quark. Z V values obtained in the different calculations are tabulated in Tables VII, VIII and IX. Good agreement is seen on set 5 at zero-recoil between the results with NRQCD and heavy-HISQ spectator quarks. Dependence on the mass of the spectator   . The NRQCD data is marked with circles, the heavy-HISQ data is marked with crosses, and finally the D → K values are given by diamonds. As expected, no significant dependence on the spectator mass is observed.
quark is displayed in Fig. 6. The plot includes values from the analogous calculation for the D → K case [3]. For D → K, a charm quark decays into a strange quark, as in B c → B s , but here the spectator quark is a light quark, much less massive than the heavy spectator quark in B c → B s . The Z V from B c → B s and D → K in Fig. 6 are nevertheless in good agreement, demonstrating negligible dependence on the mass of the quark spectating the c → s transition. It is also of interest to compare vector current renormalisation factors for different masses of quark featuring in the current. For example, [36] calculates the local sγ µ s vector current renormalisation factor from an η s → η s 3point correlation function at q 2 = 0 on the 2+1+1 MILC ensembles. This gave very precise values and it was possible to fit Z V to a perturbative expansion in α s (including the known first-order term) along with discretisation effects. This fit is plotted in Fig. 7 alongside Z V for c → s values determined in this study. This plot reveals lattice spacing dependence that varies with mass m q of the quark q in the local vector current sγ µ q. In the limit of vanishing lattice spacing, the renormalisation factors are in agreement.
One might worry that the large errors appearing in Fig. 7 for the sγ µ c renormalisation factors determined  [36] (red crosses) and the c → s vector current from the heavy-HISQ calculation given here (green squares). The curve is the fitted perturbative expansion, including discretisation effects, detailed in [36]. The red circle is an extrapolated value at the lattice spacing associated with the superfine lattice.
here would carry forward into our determination of the form factor f + . However, the vector current matrix element at zero recoil, which contributes the dominant error in Z V , is highly correlated with the vector matrix elements at non-zero recoil. These correlations cancel in the ratio V 0 /V 0 (q 2 max ) appearing when using Eq. (10) to construct the renormalised current Z V V 0 appearing in Eqs. (12) and (13). Hence, the uncertainty in the renormalisation factor is not a large contribution to our final uncertainty in the form factors. Fig. 8 provides an example of the extracted values for the form factor f + , comparing results from the NRQCD and heavy-HISQ spectator calculations. The lines on the figures connect data on the same set at a given am h value and are present as a guide only. The spread of the heavy-HISQ data for different heavy quark masses is small, and the NRQCD and heavy-HISQ results are in good agreement on the fine lattice. Discretisation effects are more noticeable for the case of an NRQCD spectator quark, especially on the coarsest lattices, sets 1 and 2. We believe that they result from the B c meson in the calculation since the effects are comparable to those seen in the B c meson decay constant study with NRQCD b quarks in [37]. Data points outside the physical region of momentum transfer are unphysical but nevertheless aid the fit.

A. z Expansion
The four form factors, f 0 and f + for each of the B c → B s and B c → B d processes, at all momenta on all the lattices, are fit simultaneously to a functional form which allows for dependence on the lattice spacing a and bare quark masses. The fit is carried out using the lsqfit package [38] that implements a least-squares fitting procedure. As is now standard, we map the semileptonic region 0 < q 2 < (M Bc − M B s(d) ) 2 to a region on the real axis within the unit circle through so that the form factors can be approximated by a truncated power series in z. Here we choose the parameter t 0 to be 0 so that the points q 2 = 0 and z = 0 coincide. The parameter t + is in principle the threshold for production of mesons, the lightest being D + K, from the cs current in the t-channel. It is convenient here, however, to work with t + = (M Bc + M B s(d) ) 2 , but this gives a very small range for z because then t + t − . To correct for this we rescale z.
The rescaling factor that we use is |z(M 2 p )| −1 , where M p is the mass of the nearest cs or cd meson pole (we use the same mass for both vector and scalar form factors for convenience). For B c → B s we take M p as the mass of the vector meson D * s and for B c → B d , the mass of D * 0 . Thus, we define z p then has a range more commensurate to that for the corresponding D decay and the polynomial coefficients in z p are O(1). Coefficients of the conventional expansion in terms of z can easily be obtained from the expansion in z p . Using z p also avoids introducing large heavy mass dependence through the z transform in the heavy-HISQ case, which otherwise would require large Λ QCD /M Hc coefficients in the heavy-HISQ fit. Note that in the case of the heavy-HISQ spectator, the B-meson masses above in t + are replaced by the appropriate heavy meson masses at each value of am h (see Sec. IV C).

B. NRQCD Form Factor Fits
The form factor results from the calculation with NRQCD spectator quark are fit to Here, the dominant pole structure is represented by a factor P (q 2 ) given by (1 − q 2 /M 2 res ) −1 with M res the mass of the relevant cs or cd meson (the vector meson for f + and the scalar for f 0 ). We take the values of M res from current experiment [33]: M D * s = 2.112 GeV, M D * s0 = 2.317 GeV, M D * = 2.01027 GeV, and M D * 0 = 2.300 GeV. We do not include uncertainties in these values since P (q 2 ) is a purely fixed factor designed to remove much of the q 2 -dependence from the form factors. For our lattice results uncertainties enter P (q 2 ) from the uncertainty in our determination of q 2 in physical units, including that from the determination of the lattice spacing.
P (q 2 ) multiplies a polynomial in z p , and the polyno- The parameters κ For the b quark, we take tuned values 1 of the quark mass from Table XII in [8].
For each of the sea and valence quark flavours, δm sea and δm val are given by giving estimates of the extent that the quark masses deviate from the ideal choices in which appropriate meson masses are exactly reproduced.
For prior values on the parameters in Eq. (19), we use 0(1) for A (n) , B (n) and C (n) , and 0.0(5) for κ (j) . The power series in Eq. (18) is truncated to include up to the z 3 p term. Fits without a pole, i.e. P (q 2 ) = 1, yield no statistically significant discrepancies. This is not surprising since the poles are far away from the physical region of q 2 , and so the pole effect on the form factor can be reasonably absorbed into the polynomial. Finally, the kinematic relation is imposed on the fit as a constraint (we have tested that removing this constraint makes very little difference to the fit in fact and f + (0) − f 0 (0) is zero to well within 1σ.). Constraints on b (n) from unitarity, as in the BCL [40] and BGL [41] expansions, are unnecessary here since the full range of physical momentum transfer can be reached and so extrapolation in q 2 , which may benefit in accuracy from imposing these constraints, is not required. Hence, more complicated fit forms that impose additional physical constraints are not expected to be appreciably advantageous.
In Figs. 9 and 10, we demonstrate that the form factors in the physical-continuum limit are insensitive to the choice of the parameters in the fits of the correlators. As can be seen in the figures, the coefficients in the fits of the form factors are stable, within their uncertainties, as the correlator fits on different sets are varied.  Table V for the Bc → Bs form factors. The integer x coordinate of each result is given by n (1) var + 2n (2) var + 4n (3) var + 8n (4) var + 16n (5) var where n (i) var = 0, 1 corresponding to the first and second fits respectively listed in Table V of set i.  Table V for the Bc → B d form factors. The x coordinate is the same as that in Fig. 9.
TABLE X: A selection of f0 and f+ fit parameters from our fit to Eq. (19) with an NRQCD-b quark, demonstrating the leading order momentum and lattice spacing dependence. Note that the discretisation effects in the f0 and f+ fits are allowed to vary independently of each other with separate B (0) parameters. In practice, as the  The fitted form factors from the NRQCD spectator case exhibit errors no greater than 4% across the entire physical range of q 2 when tuned to the physicalcontinuum limit. Figs. 11, 12, 13 and 14 show the results on all the lattices along with the fitted function for the form factors in the physical-continuum limit.
The z 0 p and z 1 p behaviour of the form factors is well resolved by our fit to Eq. (19), as well as the (am c /π) 2 z 0 p discretisation effect. Table X summarises

C. Heavy-HISQ Form Factor Fits
We take a similar approach to fitting the form factor results for the case of a heavy-HISQ spectator. Now we have results at multiple heavy-quark masses and the conversion from q 2 to z-space (Eq. (16)) uses the values of M Hc and M Hs or M H d , as appropriate, from our calculation. We then rescale z at each m h as described in Sec. IV A (Eq. (17)). This rescaling gives a similar z-range for each m h and avoids introducing spurious dependence on m h that comes simply from the z-transform.
The heavy-HISQ results are then fit to a form that is a product of P (q 2 ) and a polynomial in z p as for the NRQCD case. We now require a fit form for the polynomial coefficients that accounts for (am h ) 2n discretisation effects as well as physical dependence on m h , however. Motivated by HQET we express this physical heavy mass dependence as a power series in Λ QCD /M Hc . The form factor data from the heavy-HISQ approach is fit to where, for k = 0, ∆ where we only include the term proportional to δm val s for the B c → B s case. P (q 2 ), δm and the tuned masses have the same definitions as in the NRQCD case (Sec. IV B). In the physical continuum limit, this form collapses to P (q 2 ) n z n p A (n) 000 . Again we apply the constraint f 0 (0) = f + (0) in the continuum limit (by fixing A the B c → B d case we take prior values of 0(1) for a n and e n and 0(0.3) for b n and d n . In both cases we take prior values of 0(1) for A n ijk except for when i = 1 or j = 1 where we use a prior values of 0(0.3) to account for the HISQ one loop improvement.
As in the case of an NRQCD spectator quark, we present coefficients of the form factors fits from many different fits of the correlator data. Figs. 19 and 20, show that the coefficients are insensitive to the choice of the parameters in the fits of the correlators.

D. Chained Fit
The form factor functions tuned to the physicalcontinuum limit from NRQCD and heavy-HISQ are compared in Figs ticularly good agreement for the more accurate B c → B s case.
Plotted among the functions from the heavy-HISQ and NRQCD calculations is a function arising from a 'chained' fit where the A (n) 000 from the heavy-HISQ fit were used as prior distributions for the A (n) in the form factor fit forms in the NRQCD study. We label this fit NRQCD from heavy-HISQ in Figs. 21, 22, 23 and 24. As with the separate fits for each case of spectator quark, the form factors for B c → B s and B c → B d are fit simultaneously. This chained fit has χ 2 /d.o.f. = 1.3 and is consistent with both the separate fits. We make our final predictions for the decay rates and values for Γ|V | 2 using the chained fit.
We include the coefficients A  Table VI for the Bc → Bs form factors. The integer x coordinate of each result is given by n (3) var + 3n (5) var + 9n (6) var where n (i) var = 0, 1, 2 corresponding to original, first and second variations respectively listed in Table VI of set i.  Table VI for the Bc → B d form factors. The x coordinate is the same as that in Fig. 19. for c to s/d decay depend on the mass of the spectator quark? We can answer that question with our heavy-HISQ calculation because we have results at a range of spectator quark masses from m c upwards (see Fig. 2). Our form factor fits (Sec. IV C) enable us to extrapolate up to m b . Our most accurate results are for the c to s decay case and we concentrate on that here. Fig. 25 shows the fit curve from the heavy-HISQ results for f s + and f s 0 as a function of the heavy-charm meson mass (as a proxy for the spectator quark mass). The form factor curves that are plotted are those for q 2 = 0 (where f + = f 0 ) and for the zero-recoil point (q 2 max ). At q 2 max the daughter meson is at rest in the rest-frame of the H c meson. The q 2 value at q 2 max falls slowly as the heavy-quark mass increases above m c because the mass difference between H c and H s mesons falls. Examining the region between M ηc and M Bc in Fig. 25 we see almost no dependence on the spectator mass. The form factor value that shows the most dependence is f + (q 2 max ). This  is not surprising because f + shows the biggest slope in q 2 close to q 2 max and hence sensitivity to the value of q 2 max . Note that the curve from the heavy-HISQ analysis agrees with the NRQCD results at a spectator mass equal to that of the b. As discussed in the previous subsection, the form factors obtained from the two calculations agree across the full q 2 range.
We can also investigate the behaviour of the heavy-HISQ fit function as m h is taken below m c to m l where contact is made with results for D → K from [3]. For the form factors at q 2 = 0, we have P (q 2 ) = 1 and our fit form at Eq. (25) depends only on M Hc . This permits a straightforward extrapolation to the point M Hc = M D in the continuum limit. For the form factors at zero-recoil (q 2 max ), constructing the extrapolation curve is complicated by requiring the dependence of q 2 max on the mass of the spectator quark. This requires knowledge of M Hs as a function of M Hc . To achieve this, we fit our values of M Hs taken from set 6, together with physical values from experiment [33] at m h = m l , m b (i.e. M K and M Bs ), using a simple fit form M Hs = M Hc (1 + 4 n=1 ω n (Λ QCD /M Hc ) n + A(aΛ QCD ) 2 + B(aΛ QCD ) 4 ). Here A, B and ω n take prior values 0(2) and we do not include aΛ terms for data from [33]. We find this fit function reproduces our data, as well as the physical values, well. Fig. 25 also shows the result of this downward extrapolation. Whilst this extrapolation below m c is outside the region where HQET is expected to be valid, the curves nevertheless show approximately the correct amount of upward movement necessary to reproduce the D → K results in [3] for f + and f 0 at zero-recoil and q 2 = 0. The form factors at q 2 = 0 continue to show almost no spectator mass dependence, and this is in agreement with the D → K results.

F. Decay rate
The hadronic quantity required for determining the decay rate and branching fraction is the integral where V is the CKM element V cs or V cd . Table XI gives values for this quantity for each of the B c → B s and B c → B d processes based on the NRQCD and heavy-HISQ chained form factor fit described in Sec. IV D. Values for different q 2 bins can also be obtained. Proceeding with the total decay rate, combining these results with existing CKM matrix values [33] V cs = 0.997 (17) where the CKM matrix elements are responsible for the first errors and the second errors arise from our lattice calculations. The dominant source of lattice QCD uncertainty is the fitting of 2-point and 3-point correlators described in Sec. II B 3. We can convert these results for the decay width into a branching fraction using the lifetime of the B c meson, 513.49(12.4) fs [42]. This gives B(B + c → B 0 s ν ) = 0.0269(9)(9)(6) B(B + c → B 0 ν ) = 0.00159(6)(9)(7) where now the third uncertainty is from the lifetime. We also present the ratio of the Γ|V | −2 for B c → B s to B c → B d taking correlations into account between the numerator and denominator. From the chained fit of In fact the uncertainty is roughly the same as if we were to treat the numerator and denominator as uncorrelated. and f+ (above) for B + c → B 0 ν in the physical-continuum limit, plotted against the entire range of physical q 2 . This fit is described in Sec. IV D.

V. CONCLUSIONS
We have reported here the first calculations of the decay rates Γ(B + c → B 0 s ν ) and Γ(B + c → B 0 ν ), demonstrating the success of lattice QCD in studying decays of heavy-light mesons. The use of HISQ-HISQ c → s(d) currents allows for a non-perturbative renormalisation using the PCVC. We used two different formulations for the spectator b quark, heavy-HISQ and NRQCD. Results from the heavy-HISQ calculations are in good agreement with the physical-continuum form factors derived from the calculations using NRQCD b quarks, giving us confidence in assessing and controlling the systematic errors in each formulation. Simulating at a variety of spectator masses in the heavy-HISQ calculation has provided a check of the spectator-independence of the renormalisation procedure for the vector current. The NRQCD study also accessed Z V away from zero-recoil to scrutinise momentum independence.
Our final form factors from the chained fit that combines both NRQCD and heavy-HISQ results are plotted against q 2 in Figs. 26 and 27. The decay rates are predicted from our calculation with 4% and 6% uncertainty for Γ(B + c → B 0 s ν ) = 52.4(2.5) × 10 9 s −1 and Γ(B + c → B 0 ν ) = 3.10(21) × 10 9 s −1 respectively. There is scope for significant improvement should future experiment demand more precision from the lattice. Such improvement would be readily achieved by the inclusion of lattices with a finer lattice in the heavy-HISQ calculation. 'Ultrafine' lattices with a ≈ 0.045 [fm] were used in [11] to provide results nearer to the physical-continuum limit with am h ≈ am b . Larger statistical samples could also be obtained on the lattices used here, at the cost of more computational resources.