B c → B s(d) form factors from lattice QCD
Laurence J. Cooper, Christine T. H. Davies, Judd Harrison, Javad Komijani, Matthew Wingate
BB c → B s ( d ) form factors from lattice QCD Laurence J. Cooper,
1, a
Christine T. H. Davies, Judd Harrison,
2, b
Javad Komijani,
2, 3 and Matthew Wingate (HPQCD Collaboration) , c Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK Department of Physics, University of Tehran, Tehran 1439955961, Iran
We present results of the first lattice QCD calculations of B c → B s and B c → B d weak matrixelements. Form factors across the entire physical q range are then extracted and extrapolated to thephysical-continuum limit before combining with CKM matrix elements to predict the semileptonicdecay rates Γ( B + c → B s (cid:96)ν (cid:96) ) = 52 . . × s − and Γ( B + c → B (cid:96)ν (cid:96) ) = 3 . × s − .The lattice QCD uncertainty is comparable to the CKM uncertainty here. Results are derivedfrom correlation functions computed on MILC Collaboration gauge configurations with a range oflattice spacings including 2+1+1 flavours of dynamical sea quarks in the Highly Improved StaggeredQuark (HISQ) formalism. HISQ is also used for the propagators of the valence light, strange, andcharm quarks. 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 testof our strategies for heavy quarks on the lattice. From chained fits of NRQCD and heavy-HISQdata, we obtain the differential decay rates d Γ /dq as well as integrated values for comparison tofuture experimental results. I. INTRODUCTION
The semileptonic weak decays B + c → B s (cid:96)ν (cid:96) and B + c → B (cid:96)ν (cid:96) proceed via tree-level flavour changing pro-cesses c → sW + and c → dW + parametrised by theCabbibo-Kobayashi-Maskawa (CKM) matrix of the Stan-dard Model. Associated weak matrix elements can beexpressed in terms of form factors which capture thenon-perturbative QCD physics. Precise determination ofthe normalisation and the q dependence of these formfactors from lattice QCD will allow a novel comparisonwith future experiment to deduce the CKM parameters V cs and V cd . Lattice studies of other semileptonic me-son decays that involve tree-level weak decays of a con-stituent charm quark include [1–6]. Precise determina-tion of these CKM matrix elements is critical for exam-ining the second row unitary constraint | V cd | + | V cs | + | V cb | = 1 . (1)This will complement other unitarity tests of the CKMmatrix. It is possible LHCb could measure B + c → B 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 µν µ is likely to require many more B + c decays.A lattice study of the B + c → B s (cid:96)ν (cid:96) and B + c → B (cid:96)ν (cid:96) decays involves the practical complication of a heavyspectator quark. Care must be taken in placing such aparticle on the lattice to avoid large discretisation effects.We consider two formalisms for the b quark. A valence a [email protected] b [email protected] c NRQCD [7, 8] b quark, a formalism constructed from anon-relativistic effective theory, is used to simulate withphysically massive b quarks. A complementary calcula-tion uses HPQCD’s heavy-HISQ method [9–11]. Here,all flavours of quark are implemented with the HISQ [12]formalism. This is a fully relativistic approach whichinvolves calculations for a set of quark masses on en-sembles 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. Themethod with an NRQCD bottom quark also uses HISQfor the charm, strange and down flavours. This study willdemonstrate the consistency of the NRQCD and heavy-HISQ approaches by comparing the form factors extrap-olated to the physical-continuum limit.In the limit of massless leptons, the differential decayrates for B + c → B s (cid:96)ν (cid:96) and B + c → B (cid:96)ν (cid:96) are given by d Γ dq = G F | V | π | p | | f + ( q ) | , (2)where V is the relevant associated CKM matrix ele-ment V cs or V cd and f + is one of two form factors thatparametrise the continuum weak matrix element (cid:104) B s ( d ) ( p ) | V µ | B c ( p ) (cid:105) = f ( q ) (cid:34) M B c − M B s ( d ) q q µ (cid:35) + f + ( q ) (cid:34) p µ + p µ − M B c − M B s ( d ) q q µ (cid:35) . (3)The 4-momentum transfer is q = p − p , and only thevector part of the V − A weak current contributes sinceQCD conserves parity. The contribution of f to thedecay rate is suppressed by the lepton mass and henceirrelevant for the decays to eν e and µν µ . The phase spaceis sufficiently small to disallow decays to τ ν τ . Form fac-tors are constructed from the matrix elements that are a r X i v : . [ h e p - l a t ] M a r TABLE I: Parameters for the MILC ensembles of gluon fieldconfigurations. The lattice spacing a is determined from theWilson flow parameter w [13] given in lattice units for eachset in column 2 where values were obtained from [14] on sets1 to 5 and [15] on set 6. The physical value w = 0 . f π in [16]. Sets 1 and 2 have a ≈ .
15 [fm], andsets 3 and 4 have a ≈ .
12 [fm]. Sets 5 and 6 have a ≈ . a ≈ .
06 [fm] respectively. Sets 1, 3, 5 and 6 haveunphysically massive light quarks such that m l /m s = 0 . w /a N x × N t n cfg am sea l am sea s am sea c ×
48 1000 0 .
013 0 .
065 0 . ×
48 500 0 . . . ×
64 1053 0 . . . ×
64 1000 0 . . . ×
96 504 0 . .
037 0 . ×
144 250 0 . .
024 0 . w (Table I) to fix the lattice spacing.The fourth and fifth columns give the valence charm quarkmasses for the calculations with NRQCD and HISQ spectatorquarks respectively. In the calculation with NRQCD specta-tor quarks slightly different am val c values were used for histor-ical reasons. Our fits allow for mistuning of the charm quarkmass. am val c set am val l am val s NRQCD spectator HISQ spectator1 0.013 0.0705 0.826 -2 0.00235 0.0677 0.827 -3 0.0102 0.0541 0.645 0.6634 0.00184 0.0507 0.631 -5 0.0074 0.0376 0.434 0.4506 0.0048 0.0234 - 0.274 obtained by fitting the appropriate lattice QCD 3-pointcorrelator data. By calculating correlators at a range oftransfer momenta on lattices with different spacings andquark masses, continuum form factors at physical quarkmasses are obtained and then appropriately integratedto offer a direct comparison with decay rates that couldbe measured in experiment.In this study, we begin with Sec. II in which details ofthe lattice calculations are described. Sec. II A reportson the parameters and gauge configurations used to gen-erate the propagators. Next, Sec. II B explains how thecorrelators are subsequently constructed for the two dif-ferent treatments of the heavy spectator quark, as wellas how the correlator data is fit to extract the matrixelements. Our non-perturbative renormalisation method
TABLE III: The bottom quark masses, NRQCD action pa-rameters c j , and values for the tadpole improvement u wereobtained from [8]. The final columns gives the different mo-menta for the strange and light quarks considered in theNRQCD calculation implemented with twisted boundary con-ditions.set am val b c , c c c u | a q | .
297 1 .
36 1 .
21 1 .
22 0 . .
25 1 .
36 1 .
21 1 .
22 0 . .
62 1 .
31 1 .
16 1 .
20 0 . .
91 1 .
21 1 .
12 1 .
16 0 . required to obtain the form factors is set out in Sec. II C.Sec. III presents results of the lattice calculations. Cor-relator fits are examined in Sec. III A, whilst Sec. III Bdiscusses results for the renormalisation of the local lat-tice vector current. In Sec. III C, the form factor datafor the cases of an NRQCD spectator and a HISQ spec-tator are plotted alongside. Sec. IV is concerned withthe methodology and results from fitting the form factordata. An extrapolation of the form factors to physical-continuum point is presented in Sec. IV D and Sec. IV Eshows how the form factors depend on the mass of thespectator quark. Finally, in Sec. V we give our conclu-sions. II. LATTICE CALCULATIONA. Parameters and Set-up
We use ensembles with 2 + 1 + 1 flavours of HISQ seaquark generated by the MILC Collaboration [17–19] anddescribed in Table I. The Symanzik-improved gluon ac-tion used is that from [20], where the gluon action isimproved perturbatively through O ( α s ) including the ef-fect of dynamical HISQ sea quarks. The lattice spacingis identified by comparing the physical value for the Wil-son flow parameter ω = 0 . ω /a from [14] and [15]. Our calculations fea-ture physically massive strange quarks and equal massup and down quarks, with a mass denoted by m l , with m l /m s = 0 . m l /m s = 1 / . B + c is at rest, and TABLE IV: Heavy quark masses and momenta used for theheavy-HISQ calculation. The momenta are in the (1 1 1) di-rection.set am val h | a q | momentum is inserted into the strange or down valencequark through twisted boundary conditions [24, 25] inthe (1 1 1) direction. The values of the momenta usedare given in Tables III and IV. The periodic boundaryconditions of the fermion fields are modified by phases θ i ψ ( n + N x ˆ i ) = e iπθ i ψ ( n ) (4)so that the usual lattice momenta q i = 2 πk i /aN x , forintegers k i , are shifted by πθ i /aN x . The corresponding q is then constructed by taking q to be the differencein energies of the lowest lying initial and final states.The coefficients of operators corresponding to rela-tivistic correction terms in the NRQCD action are givenin Table III. The valence b quark masses used for theNRQCD propagators are also given there. The valueswere taken from [8], where the b quark mass was foundby matching the experimental value for the spin-averagedkinetic mass of the Υ and the η b to lattice data. For thecalculation with an NRQCD spectator bottom quark, weuse sets 1 to 5 in Table I.Bare heavy quark masses am h used for the heavy-HISQmethod are shown in Table IV. The selection of heavyquark masses follows [11]. As well as sets 3 and 5, theheavy-HISQ calculation makes use of a lattice finer thanthe five sets featuring in the calculation with an NRQCDspectator, set 6 in Table I. This is motivated by the ne-cessity to avoid large discretisation effects that grow with( am h ) (as ( am h ) at tree-level) whilst gathering data atlarge masses that will reliably inform the limit m h → m b . B. Correlators
1. NRQCD spectator case
For the case of an NRQCD spectator quark, randomwall source [26] HISQ propagators with the mass of thecharm quark are calculated and combined with randomwall source NRQCD b propagators to generate B + c B s ( d ) aregenerated similarly. The strategy of combining NRQCDrandom wall propagators and HISQ random wall propa-gators to yield 2-point correlators was first developed in[27]. NRQCD propagators are generated by solving aninitial value problem. This is computationally very fastcompared to calculating rows of the inverse of the quarkmatrix. B c B s ( d ) t Tbc q s ( d ) FIG. 1: 3-point correlator C ( t, T ). The flavour-changingoperator insertion is denoted by a cross at timeslice t and thetotal time length of the 3-point correlator is T . The randomwall source for the b and s/d propagators is at the timesliceof the B s ( d ) interpolator. The 3-point correlator needed here is represented dia-grammatically in Fig. 1. A HISQ charm quark propaga-tor is generated by using the random wall bottom quarkpropagator as a sequential source. Following AppendixB in [28], and excluding a spacetime-dependent sign, thesequential source is given by the spin-traceTr spin (cid:110) ΓΩ † ( x , S RW b ( x , (cid:111) , (5)where Γ is the gamma matrix structure at the operatorinsertion, S RW b is the random wall NRQCD propagator,and Ω( x ) := (cid:89) µ =1 ( γ µ ) xµa (6)is the space-spin matrix which transforms the naive quarkfield to diagonalise the HISQ action in spin-space.
2. HISQ spectator case
The case of a HISQ spectator quark proceeds similarlywith the only difference being the use of a HISQ propa-gator instead of an NRQCD propagator for the bottomquark. Again, the charm propagator uses the spectatorbottom quark propagator as a sequential source. Multi-ple masses are used for the spectator quark, each requir-ing a different charm propagator for the 3-point correla-tor. Fig. 2 shows the heavy-charm pseudoscalar mesonmasses that arise from calculations with the am h valuesin Table IV. On set 6, the finest lattice considered, wereach a value for M H c that is 80% of the physical B c mass.The same strange and light random wall HISQ propa-gators on sets 3 and 5 are used in both the NRQCD andthe heavy-HISQ calculations, thus the data on these lat-tices in the two approaches will be correlated. However,the effect of these correlations is small in the physical-continuum limit since the heavy-HISQ data on sets 3 and5 are the furthest away from the physical b quark masspoint, and hence these correlations are safely ignored. .
000 0 .
025 0 .
050 0 .
075 0 .
100 0 . a [fm]3456 M H c [ G e V ] M η c M B + c set 3set 5set 6 FIG. 2: The mass M H c of the heavy-charm meson is plottedagainst lattice spacing for each of the values of am h usedin the heavy-HISQ calculation. Obtained from fitting thecorrelators as described in Eq. (7), M H c is a proxy for thebare lattice heavy quark mass am h . The continuum-physicalpoint is denoted by a cross at a = 0 [fm] and M H c = M B + c .Note that the y -axis scale begins near M η c .
3. Fitting the correlators
The correlator fits minimise an augmented χ functionas described in [29–31]. The functional forms for the 2-point and 3-point correlators C B s ( d ) ( t ) = (cid:88) i a [ i ] e − E a [ i ] t − (cid:88) i a o [ i ] ( − t e − E ao [ i ] t C B c ( t ) = (cid:88) j b [ j ] e − E b [ j ] t − (cid:88) j b o [ j ] ( − t e − E bo [ j ] t C ( t, T ) = (cid:88) i,j a [ i ] e − E a [ i ] t V nn [ i, j ] b [ j ] e − E b [ j ]( T − t ) − (cid:88) i,j ( − T − t a [ i ] e − E a [ i ] t V no [ i, j ] b o [ j ] e − E bo [ j ]( T − t ) − (cid:88) i,j ( − t a o [ i ] e − E ao [ i ] t V on [ i, j ] b [ j ] e − E b [ j ]( T − t ) + (cid:88) i,j ( − T a o [ i ] e − E ao [ i ] t V oo [ i, j ] b o [ j ] e − E bo [ j ]( T − t ) (7)follow from their spectral decomposition and include os-cillatory contributions from the staggered quark time-doubler. The matrix elements are related to the fit pa-rameters V nn [ i, j ] through V nn [0 ,
0] = (cid:104) B s ( d ) | J | B c (cid:105) (cid:112) E B s ( d ) E B c , (8)where J is the relevant operator that facilitates the c → s ( d ) flavour transition. The pseudoscalar mesons ofinterest are the lowest lying states consistent with their quark content, so we are only concerned with the matrixelements for i = j = 0 since we restrict E [ k ] ≤ E [ k + 1]by using log-normal prior distributions for the energy dif-ferences. The presence of i, j > V nn [0 , c → s and c → d at allmomenta are fit simultaneously to account for all possi-ble correlations. The matrix elements and energies areextracted and form factor values determined, along withthe 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 renor-malisation of the lattice vector current. Since the sameHISQ action is used for the c and s ( d ) quarks that cou-ple to the W + in both the NRQCD and heavy-HISQapproaches, we have the PCVC identity ∂ µ V µ cons = ( m c − m s ( d ) ) S, (9)relating the conserved (point-split) c → s ( d ) lattice vec-tor current and the local lattice scalar density S . Wechoose a local lattice operator V µ loc , thus Eq. (9) must beadjusted by a single renormalisation factor Z V associatedwith that operator, giving q µ (cid:104) B s ( d ) | V µ loc | B c (cid:105) Z V = ( m c − m s ( d ) ) (cid:104) B s ( d ) | S | B c (cid:105) . (10)Since Z V is q independent, in principle Z V need onlybe found at zero-recoil where q µ has only a temporalcomponent [3]. This avoids the need to calculate 3-pointcorrelators associated with the spatial components of thevector current matrix element that appear in Eq. (10)for q (cid:54) = . However, in practice, it is preferable to deter-mine f + near zero-recoil through the spatial componentsof the vector current matrix element, albeit with the ad-ditional cost in computing 3-point correlators with thecorresponding insertion.As in [2], we combine Eqs. (3) and (10) to give a de-termination f (cid:0) q (cid:1) = (cid:104) B s ( d ) | S | B c (cid:105) m c − m s ( d ) M B c − M B s ( d ) (11)of f solely in terms of the scalar density matrix element.We use Eq. (11) and calculation of the vector currentmatrix element to determine f + and f for the full q range following [3, 32]. Thus, we will calculate matrixelements of both the local scalar density J = S and thelocal vector current J = V .Once f is determined, f + is obtained using Eq. (3) for µ = 0 to yield f + ( q ) = Z V V − q f ( q ) M B + c − M B s q p + p − q M B + c − M B s q , (12)where V µ is the vector current matrix element, exceptat zero-recoil where the denominator vanishes and f + cannot be extracted. We find that using Eq. (12) nearzero-recoil is problematic since both the numerator anddenominator grow from 0 as q is decreased from themaximum value at zero-recoil. For the case where thespectator is an NRQCD b quark, we instead use Eq. (3)with µ = i (cid:54) = 0 f + ( q ) = − Z V V i q i + f ( q ) M B + c − M B s q M B + c − M B s q . (13)This method gives much smaller errors near to zero-recoil. Although mathematically equivalent to Eq. (12),extracting f + through Eq. (13) does not suffer an infla-tion of error near zero-recoil since both the numeratorand denominator are non-zero for all physical q . How-ever, since V i appears explicitly in Eq. (13), 3-point cor-relators with an insertion of V i need to be calculated.For the case of the spectator NRQCD b quark, the useof Eq. (13) is straightforward except that it requires in-versions of the charm quark propagator from a differentsequential source (see Eq. (5)) to allow for insertion of thecurrent V i = γ i ⊗ γ i in the mixed NRQCD-HISQ 3-pointfunction. Collecting V i at non-zero 3-momentum trans-fer in the NRQCD calculation will also test for any q dependence of Z V that would appear as a discretisationeffect.Using Eqs. (11) and (12) or (13), form factor data ata variety of lattice spacings, light quark masses and mo-menta are obtained from the energies and matrix ele-ments.
1. NRQCD spectator case
For the case of an NRQCD spectator quark, the formfactor extraction is complicated by the energy offset as aconsequence of the subtraction of the b quark rest massinherent in the NRQCD formalism. Whilst physical en-ergy differences are preserved with NRQCD quarks, en-ergy sums are not. Consequently, Particle Data Group(PDG) [33] values are used where necessary. For exam-ple, we take M B c − M B s ( d ) = (cid:16) E sim B c ( | a q | = 0) − E sim B s ( d ) ( | a q | = 0) (cid:17) × (cid:16) M PDG B c + M PDG B s ( d ) (cid:17) (14)when extracting the form factors. We use interpolatingoperators cγ b and sγ b ( dγ b ) for J P = 0 − pseu-doscalars B + c and B s ( d ) respectively.
2. HISQ spectator case
For the case of a HISQ spectator quark, we work onlywith local scalar and vector currents. Expressed in thespin-taste basis, we use γ ⊗ γ for the H s ( d ) interpolatingoperator and two different operators, γ ⊗ γ and γ γ ⊗ γ γ , for the H c interpolator. The first of these, γ ⊗ γ ,makes a tasteless 3-point correlation function when thescalar density operator ⊗ is used. The second, γ γ ⊗ γ γ , allows for a tasteless 3-point correlation functionwhen we use the local temporal vector current operator γ ⊗ γ [3]. This requires the calculation of two H c H c meson is tinyand, although consistently taken care of, it has no impacton the calculation. III. RESULTSA. Correlators
Figs. 3 and 4 provide samples of the correlator datafrom the NRQCD and heavy-HISQ calculations respec-tively. The quantity plotted is the effective simulationenergies, which we define by the two-step log-ratio aE sim,eff = 12 log (cid:16) C ( t ) C ( t + 2) (cid:17) . (15)This ratio is preferable to an effective energy defined us-ing C ( t ) /C ( t + 1) since the ratio in Eq. (15) better sup-presses the oscillatory contributions in Eq. (7). Errorbars are present in the figure but mostly too small to ob-serve. We exclude t min /a data points from the beginningand end points of the correlators in our fits to reduce thecontributions from excited states.For each of the cases of an NRQCD and HISQ spec-tator quark, we fit all of the correlator data to Eq. (7)on each set simultaneously to obtain the correlations be-tween the fitted parameters. Consequently, the correlatorfits involve a large covariance matrix. Without extremelylarge statistical samples of results small eigenvalues of thecovariance matrix are underestimated [34, 35] and thiscauses problems when carrying out the inversion to find χ . We overcome this by using an SVD (singular-valuedecomposition) cut; any eigenvalue of the covariance ma-trix smaller than some proportion c of the biggest eigen-value λ max is replaced by cλ max . By carrying out thisprocedure, the covariance matrix becomes less singular.These eigenvalue replacements will only inflate our finalerrors, hence this strategy is conservative. The SVD cutreduces the χ / d.o.f. reported by the fit because it low-ers the contribution to χ of the modes with eigenvaluesbelow the SVD cut. In order to check the suitability ofthe SVD cut, we must test the goodness-of-fit from a fitwhere noise (SVD-noise) is added to the data to rein-state the size of fluctuations expected from the modes t/a . . . . aE sim , eff B c B s B d FIG. 3: Effective simulation energies (Eq. (15)) of 2-pointcorrelators with an NRQCD spectator quark on set 5 for | a q | = 0 . B meson energies shown here areoffset from their physical values as a consequence applyingthe NRQCD formalism to the constituent b quark.
20 40 t/a . . . . . . aE sim , eff H c H s H d FIG. 4: Effective simulation energies (Eq. (15)) of 2-pointcorrelators with a HISQ spectator quark on set 6 at zero twistwith am h = 0 .
8. The horizontal bands show the energiesextracted from the full simultaneous correlator fit. below SVD cut, as described in Appendix D of [35]. The χ / d.o.f. is used to check the goodness of fit forboth cases of spectator quark.Many fits were carried out with different SVD cuts,number of exponentials N , and positions t and t of the first timeslice where the correlators are fit. Weselected the fit of the correlators on each lattice for formfactor extraction based on the χ / d.o.f. and Q -value.The parameters used in the fits of correlators with anNRQCD spectator quark are presented in Table V. Theparameters given in bold are those used for our final fits. TABLE V: Input parameters (see text for definition) to thecorrelator fits for the calculation with NRQCD spectatorquarks together with fits including variations of t min /a , N and SVD cut. Bold entries indicate those fits used to obtainour final result. Other values are used in tests of the stabilityof our form factor fits to be discussed in Sec. IV B.set SVD cut t /a t /a N χ / dof1 t min /a , N and SVD cut. Bold entries indicate thosefits used to obtain our final result. Other values will be usedin tests of the stability of our form factor fits in Sec. IV C.set SVD cut t /a t /a N χ / dof3 Other values are used in tests of the stability of our formfactor fits to be discussed in Sec. IV B.We fit the heavy-HISQ correlator data to Eq. (7) oneach set simultaneously, including correlations betweendata with different values of twist, heavy quark mass,and H s/d final state. Values for t min /a , the chosen SVDcut, the number of exponentials used in Eq. (7) and theresultant value of χ / dof including SVD noise are givenin Table VI. We also include in Table VI fits using varia-tions of these parameters. Form factor fit coefficients ob-tained using combinations of these variations are shownin Figs. 19 and 20 in Sec. IV C and demonstrate that ourresults are insensitive to such choices. B. Vector Current Renormalisation Z V In this section we give our results for the renormalisa-tion factor Z V for the vector current (Eq. (10)) and test − . . . . . . q [GeV] . . . . . Z V set 1set 2set 3set 4set 5 FIG. 5: Z V for the c → s vector current evaluated at differ-ent q from the calculation with an NRQCD spectator quarkusing Eq. (10).TABLE VII: Z V obtained at zero-recoil using an NRQCDspectator b quark .set c → s c → d for dependence of Z V on q (for the case of an NRQCDspectator) and on the spectator quark mass (for the caseof a HISQ spectator).The vector current renormalisation factor Z V com-puted at different momentum transfer with NRQCD b quarks shows no significant dependence on q on eachset, demonstrated by Fig. 5. Mild lattice spacing depen-dence is observed, however. For each momenta, we usethe Z V found at the corresponding q from Eq. (10).The Z V factor in Eq. (10) is associated only with thelocal vector current operator and should be indepen-dent of the spectator quark. Z V values obtained in thedifferent calculations are tabulated in Tables VII, VIIIand IX. Good agreement is seen on set 5 at zero-recoilbetween the results with NRQCD and heavy-HISQ spec-tator quarks. Dependence on the mass of the spectator TABLE VIII: Z V for c → s obtained at zero-recoil using aHISQ spectator quark with different values of the heavy quarkmass m h .set/ am h .
274 0 .
450 0 . .
663 0 .
83 - - - 1.026(32) 1.029(36)5 - 1.006(17) 1.003(19) - 1.000(20)6 0.997(14) 0.994(17) 0.995(19) - 0.995(22) TABLE IX: Z V for c → d obtained at zero-recoil using a HISQspectator quark with different values of the heavy quark mass m h .set/ am h .
274 0 .
450 0 . .
663 0 .
83 - - - 1.016(47) 1.019(50)5 - 1.009(23) 1.004(25) - 1.000(27)6 0.996(22) 0.993(25) 0.994(28) - 0.995(32) M H c [GeV]0 . . . . . Z V M D M η c M B c set 1set 2set 3set 4set 5set 6 FIG. 6: Z V of the c → s vector current from both theNRQCD and heavy-HISQ calculations are plotted alongsidevalues from D → K [3]. The NRQCD data is marked withcircles, the heavy-HISQ data is marked with crosses, and fi-nally 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 valuesfrom 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 lightquark, much less massive than the heavy spectator quarkin B c → B s . The Z V from B c → B s and D → K in Fig. 6are nevertheless in good agreement, demonstrating neg-ligible dependence on the mass of the quark spectatingthe c → s transition.It is also of interest to compare vector current renor-malisation factors for different masses of quark featuringin the current. For example, [36] calculates the local sγ µ s vector current renormalisation factor from an η s → η s q = 0 on the 2+1+1 MILCensembles. This gave very precise values and it was possi-ble to fit Z V to a perturbative expansion in α s (includingthe known first-order term) along with discretisation ef-fects. This fit is plotted in Fig. 7 alongside Z V for c → s values determined in this study. This plot reveals lat-tice spacing dependence that varies with mass m q of thequark q in the local vector current sγ µ q . In the limit ofvanishing lattice spacing, the renormalisation factors arein agreement.One might worry that the large errors appearing inFig. 7 for the sγ µ c renormalisation factors determined .
000 0 .
005 0 .
010 0 .
015 0 .
020 0 . a (fm ) . . . . . Z V B c → B s ssF (0) FIG. 7: Z V from the s → s vector current from [36] (redcrosses) and the c → s vector current from the heavy-HISQcalculation given here (green squares). The curve is the fit-ted perturbative expansion, including discretisation effects,detailed in [36]. The red circle is an extrapolated value at thelattice spacing associated with the superfine lattice. here would carry forward into our determination of theform factor f + . However, the vector current matrix ele-ment at zero recoil, which contributes the dominant er-ror in Z V , is highly correlated with the vector matrixelements at non-zero recoil. These correlations cancel inthe ratio V / V ( q ) appearing when using Eq. (10) toconstruct the renormalised current Z V V appearing inEqs. (12) and (13). Hence, the uncertainty in the renor-malisation factor is not a large contribution to our finaluncertainty in the form factors. C. Form Factors
Fig. 8 provides an example of the extracted values forthe form factor f + , comparing results from the NRQCDand heavy-HISQ spectator calculations. The lines on thefigures connect data on the same set at a given am h valueand are present as a guide only. The spread of the heavy-HISQ data for different heavy quark masses is small, andthe NRQCD and heavy-HISQ results are in good agree-ment on the fine lattice. Discretisation effects are morenoticeable for the case of an NRQCD spectator quark,especially on the coarsest lattices, sets 1 and 2. We be-lieve that they result from the B c meson in the calcula-tion since the effects are comparable to those seen in the B c meson decay constant study with NRQCD b quarksin [37]. Data points outside the physical region of mo-mentum transfer are unphysical but nevertheless aid thefit. − . . . q [GeV] . . . . . . . f s + heavy-HISQ set 3, am h = 0 . am h = 0 . am h = 0 . am h = 0 . am h = 0 . am h = 0 . am h = 0 . am h = 0 . am h = 0 . NRQCD set 1set 2set 3set 4set 5
FIG. 8: f + form factor data for B + c → B s (cid:96)ν (cid:96) from both theNRQCD and heavy-HISQ approaches. The NRQCD formfactor data is given by filled circles; the heavy-HISQ data, byopen circles. Data points on a given set and for a given heavyquark mass are joined by lines to guide eye. IV. DISCUSSIONA. z Expansion
The four form factors, f and f + for each of the B c → B s and B c → B d processes, at all momenta on allthe lattices, are fit simultaneously to a functional formwhich allows for dependence on the lattice spacing a andbare quark masses. The fit is carried out using the lsq-fit package [38] that implements a least-squares fittingprocedure. As is now standard, we map the semileptonicregion 0 < q < ( M B c − M B s ( d ) ) to a region on the realaxis within the unit circle through z ( q ) = (cid:112) t + − q − √ t + − t (cid:112) t + − q + √ t + − t , (16)so that the form factors can be approximated by a trun-cated power series in z . Here we choose the parameter t to be 0 so that the points q = 0 and z = 0 coincide. Theparameter t + is in principle the threshold for productionof mesons, the lightest being D + K , from the cs currentin the t -channel. It is convenient here, however, to workwith t + = ( M B c + M B s ( d ) ) , but this gives a very smallrange for z because then t + (cid:29) t − . To correct for this werescale z .The rescaling factor that we use is | z ( M p ) | − , where M p is the mass of the nearest cs or cd meson pole (we usethe same mass for both vector and scalar form factors forconvenience). For B c → B s we take M p as the mass ofthe vector meson D ∗ s and for B c → B d , the mass of D ∗ .Thus, we define z p ( q ) = z ( q ) | z ( M p ) | . (17) z p then has a range more commensurate to that for thecorresponding D decay and the polynomial coefficients in z p are O (1). Coefficients of the conventional expansionin terms of z can easily be obtained from the expansionin z p . Using z p also avoids introducing large heavy massdependence through the z transform in the heavy-HISQcase, which otherwise would require large Λ QCD /M H c co-efficients in the heavy-HISQ fit. Note that in the case ofthe heavy-HISQ spectator, the B -meson masses above in t + are replaced by the appropriate heavy meson massesat each value of am h (see Sec. IV C). B. NRQCD Form Factor Fits
The form factor results from the calculation withNRQCD spectator quark are fit to f ( q ) = P ( q ) N (cid:88) n =0 b ( n ) z np . (18)Here, the dominant pole structure is represented by a fac-tor P ( q ) given by (1 − q /M ) − with M res the mass ofthe relevant cs or cd meson (the vector meson for f + andthe scalar for f ). We take the values of M res from cur-rent experiment [33]: M D ∗ s = 2 .
112 GeV, M D ∗ s = 2 . M D ∗ = 2 . M D ∗ = 2 .
300 GeV. Wedo not include uncertainties in these values since P ( q )is a purely fixed factor designed to remove much of the q -dependence from the form factors. For our lattice re-sults uncertainties enter P ( q ) from the uncertainty inour determination of q in physical units, including thatfrom the determination of the lattice spacing. P ( q ) multiplies a polynomial in z p , and the polyno-mial coefficients are b ( n ) = A ( n ) (cid:110) B ( n ) ( am c /π ) + C ( n ) ( am c /π ) + κ ( n )1 δm sea l m tuned s + κ ( n )2 δm sea s m tuned s + κ ( n )3 δm sea c m tuned c + κ ( n )4 δm val s m tuned s + κ ( n )5 δm val c m tuned c + κ ( n )6 δm val b m tuned b (cid:111) . (19)The parameters κ ( n ) j allow for errors associated withmistunings of both sea and valence quark masses. Theterm accounting for mistuning of valence strange quarksis included only for the B c → B s transition. The tunedmasses m tuned s and m tuned c are the valence quark massesthat yield physical η s and η c meson masses respectivelyin the sea of 2+1+1 flavours of sea quark. The tuned s sea quark mass is determined through am tuned s = am val s (cid:32) m phys η s m η s (cid:33) (20) and m tuned l is fixed by multiplying m tuned s by the physicalratio m l m s = 127 . c quark mass is deter-mined from am tuned c = am val c m phys η c m η c . (22)For the b quark, we take tuned values of the quark massfrom Table XII in [8].For each of the sea and valence quark flavours, δm sea and δm val are given by δm sea = m sea − m tuned δm val = m val − m tuned , (23)giving estimates of the extent that the quark masses de-viate from the ideal choices in which appropriate mesonmasses are exactly reproduced.For prior values on the parameters in Eq. (19), we use0(1) for A ( n ) , B ( n ) and C ( n ) , and 0 . κ ( j ) . Thepower series in Eq. (18) is truncated to include up to the z p term. Fits without a pole, i.e. P ( q ) = 1, yield nostatistically significant discrepancies. This is not surpris-ing since the poles are far away from the physical regionof q , and so the pole effect on the form factor can bereasonably absorbed into the polynomial. Finally, thekinematic relation f (0) = f + (0) (24)is imposed on the fit as a constraint (we have tested thatremoving this constraint makes very little difference tothe fit in fact and f + (0) − f (0) is zero to well within1 σ .).Constraints on b ( n ) from unitarity, as in the BCL [40]and BGL [41] expansions, are unnecessary here since thefull range of physical momentum transfer can be reachedand so extrapolation in q , which may benefit in accuracyfrom imposing these constraints, is not required. Hence,more complicated fit forms that impose additional phys-ical constraints are not expected to be appreciably ad-vantageous.In Figs. 9 and 10, we demonstrate that the form fac-tors in the physical-continuum limit are insensitive to thechoice of the parameters in the fits of the correlators. Ascan be seen in the figures, the coefficients in the fits ofthe form factors are stable, within their uncertainties, asthe correlator fits on different sets are varied. To ensure consistency, we convert values from [8] in lattice units tophysical units by using the lattice spacing determined in [8] fromthe Υ(2 S − S ) splitting. . . . . . . . A (0) f s − . − . − . − . − . A (1) f s − . − . − . − . − . A (1) f s + FIG. 9: z p expansion coefficients, for the calculation with an NRQCD spectator quark, computed using the variations ofcorrelator fit parameters listed in Table V for the B c → B s form factors. The integer x coordinate of each result is given by n (1)var + 2 n (2)var + 4 n (3)var + 8 n (4)var + 16 n (5)var where n ( i )var = 0 , i . . . . . A (0) f d − . − . − . − . − . − . . A (1) f d − . − . − . − . − . − . A (1) f d + FIG. 10: z p expansion coefficients, for the calculation with an NRQCD spectator quark, computed using the variations ofcorrelator fit parameters listed in Table V for the B c → B d form factors. The x coordinate is the same as that in Fig. 9.TABLE X: A selection of f and f + fit parameters from our fitto Eq. (19) with an NRQCD- b quark, demonstrating the lead-ing order momentum and lattice spacing dependence. Notethat the discretisation effects in the f and f + fits are allowedto vary independently of each other with separate B (0) pa-rameters. In practice, as the Table shows, the fit returns verysimilar values. f s f d f s + f d + A (0) A (1) -0.52(14) -0.19(22) -0.74(14) -0.48(21) A (2) -0.63(63) 0.05(74) -0.29(72) 0.12(77) B (0) The fitted form factors from the NRQCD spectatorcase exhibit errors no greater than 4% across the en-tire physical range of q when tuned to the physical-continuum limit. Figs. 11, 12, 13 and 14 show the resultson all the lattices along with the fitted function for theform factors in the physical-continuum limit.The z p and z p behaviour of the form factors is wellresolved by our fit to Eq. (19), as well as the ( am c /π ) z p discretisation effect. Table X summarises the correspond-ing parameters from the fit. After fitting, other param- − . . . . . . q [GeV] . . . . f s set 1set 2set 3set 4set 5 FIG. 11: Lattice results and fitted f form factor data for B + c → B s (cid:96)ν (cid:96) with an NRQCD b quark. The grey band showsthe fitted form factor tuned to the limit of vanishing latticespacing and physical quark masses. eters show errors comparable to the width of their priorand are consistent with 0. In particular, quark mass mis-tuning coefficients simply return their prior value.1 − . . . . . . q [GeV] . . . . . . f s + set 1set 2set 3set 4set 5 FIG. 12: Lattice results and fitted f + form factor data for B + c → B s (cid:96)ν (cid:96) with an NRQCD b quark. The grey band showsthe fitted form factor tuned to the limit of vanishing latticespacing and physical quark masses. . . . . . . q [GeV] . . . . . f d set 1set 2set 3set 4set 5 FIG. 13: Lattice results and fitted f form factor data for B + c → B (cid:96)ν (cid:96) with an NRQCD b quark. The grey band showsthe fitted form factor tuned to the limit of vanishing latticespacing and physical quark masses. C. Heavy-HISQ Form Factor Fits
We take a similar approach to fitting the form fac-tor results for the case of a heavy-HISQ spectator. Nowwe have results at multiple heavy-quark masses and theconversion from q to z -space (Eq. (16)) uses the val-ues of M H c and M H s or M H d , as appropriate, from ourcalculation. We then rescale z at each m h as describedin Sec. IV A (Eq. (17)). This rescaling gives a similar z -range for each m h and avoids introducing spurious de-pendence on m h that comes simply from the z -transform.The heavy-HISQ results are then fit to a form thatis a product of P ( q ) and a polynomial in z p as for theNRQCD case. We now require a fit form for the polyno-mial coefficients that accounts for ( am h ) n discretisation . . . . . . q [GeV] . . . f d + set 1set 2set 3set 4set 5 FIG. 14: Fitted f + form factor data for B + c → B (cid:96)ν (cid:96) with anNRQCD b quark. The grey band shows the fitted form factortuned to the limit of vanishing lattice spacing and physicalquark masses. effects as well as physical dependence on m h , however.Motivated by HQET we express this physical heavy massdependence as a power series in Λ QCD /M H c . The formfactor data from the heavy-HISQ approach is fit to f ( q ) = P ( q ) (cid:88) n,i,j,k =0 A ( n ) ijk z np × (cid:16) am c π (cid:17) i (cid:16) am h π (cid:17) j ∆ ( k ) H c N ( n )mis , (25)where, for k = 0, ∆ ( k ) M = 1 and, for k (cid:54) = 0,∆ ( k ) H c = (cid:18) Λ QCD M H c (cid:19) k − (cid:18) Λ QCD M B c (cid:19) k (26)where we take Λ QCD = 500MeV. The mistuning termsare given by N ( n )mis = 1 + δm val c m tuned c a n + δm sea c m tuned c b n + δm val s m tuned s c n + δm sea s m tuned s d n + δm l m tuned s e n , (27)where we only include the term proportional to δm val s forthe B c → B s case. P ( q ), δm and the tuned masses havethe same definitions as in the NRQCD case (Sec. IV B).In the physical continuum limit, this form collapses to P ( q ) (cid:80) n z np A ( n )000 . Again we apply the constraint f (0) = f + (0) in the continuum limit (by fixing A (0)000 to be thesame in the two cases).Results for the extrapolated form factors are given inFigs. 15, 16, 17 and 18 together with the correspondinglattice data. For the B c → B s case a n and c n take priorvalues 0(1) and b n , d n and e n take prior values 0(0 .
3) toreflect the fact that they enter through loop effects. In2 − . . . . q [GeV] . . . . . f s a \ am h . . . .
274 0 .
450 0 .
600 0 .
663 0 . FIG. 15: Heavy-HISQ form factor results for f s together withthe fitted curve at the physical point with its error band. − . . . q [GeV] . . . . . f s + a \ am h . . . .
274 0 .
450 0 .
600 0 .
663 0 . FIG. 16: Heavy-HISQ form factor results for f s + together withthe fitted curve at the physical point with its error band. 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 priorvalues of 0(1) for A nijk except for when i = 1 or j = 1where we use a prior values of 0(0 .
3) to account for theHISQ one loop improvement.As in the case of an NRQCD spectator quark, wepresent coefficients of the form factors fits from many dif-ferent fits of the correlator data. Figs. 19 and 20, showthat the coefficients are insensitive to the choice of theparameters in the fits of the correlators.
D. Chained Fit
The form factor functions tuned to the physical-continuum limit from NRQCD and heavy-HISQ are com-pared in Figs. 21, 22, 23 and 24 in z -space. There is goodagreement across the entire physical range of z , with par- . . . q [GeV] . . . f d a \ am h . . . .
274 0 .
450 0 .
600 0 .
663 0 . FIG. 17: Heavy-HISQ form factor results for f d together withthe fitted curve at the physical point with its error band. .
00 0 .
25 0 .
50 0 .
75 1 . q [GeV] . . . . . f d + a \ am h . . . .
274 0 .
450 0 .
600 0 .
663 0 . FIG. 18: Heavy-HISQ form factor results for f d + together withthe fitted curve at the physical point with its error band. ticularly good agreement for the more accurate B c → B s case.Plotted among the functions from the heavy-HISQand NRQCD calculations is a function arising from a‘chained’ fit where the A ( n )000 from the heavy-HISQ fit wereused as prior distributions for the A ( n ) in the form factorfit forms in the NRQCD study. We label this fit NRQCDfrom heavy-HISQ in Figs. 21, 22, 23 and 24. As with theseparate fits for each case of spectator quark, the formfactors for B c → B s and B c → B d are fit simultaneously.This chained fit has χ / d.o.f. = 1 . | V | using the chainedfit.We include the coefficients A ( n )0 , + from the chained fitin the ancillary json file BcBsd ff.json .3 . . . . . . . A (0)000 ,f s − . − . − . − . − . − . − . − . − . A (1)000 ,f s − . − . − . − . − . − . A (1)000 ,f s + FIG. 19: z p expansion coefficients, for the calculation with a HISQ spectator quark, computed using the variations of correlatorfit parameters listed in Table VI for the B c → B s form factors. The integer x coordinate of each result is given by n (3)var +3 n (5)var + 9 n (6)var where n ( i )var = 0 , , i . . . . . . . . . . A (0)000 ,f d − . − . − . − . − . − . A (1)000 ,f d − . − . − . − . − . − . − . A (1)000 ,f d + FIG. 20: z p expansion coefficients, for the calculation with a HISQ spectator quark, computed using the variations of correlatorfit parameters listed in Table VI for the B c → B d form factors. The x coordinate is the same as that in Fig. 19. − . − . − . − . − . . z p / | z p ( t − ) | . . . . . f s NRQCDheavy-HISQNRQCD from heavy-HISQ
FIG. 21: Fits of f for B + c → B s (cid:96)ν (cid:96) tuned to thephysical-continuum limit. The form factor is plotted against z p / | z p ( t − ) | . − . − . − . − . − . . z p / | z p ( t − ) | . . . . . f s + NRQCDheavy-HISQNRQCD from heavy-HISQ
FIG. 22: Fits of f + for B + c → B s (cid:96)ν (cid:96) tuned to the physical-continuum limit. E. Dependence of the form factors on thespectator quark mass
In order to build up a picture of the behaviour of formfactors it is interesting to ask: how do the form factors4 − . − . − . − . − . . z p / | z p ( t − ) | . . . . . f d NRQCDheavy-HISQNRQCD from heavy-HISQ
FIG. 23: Fits of f for B + c → B (cid:96)ν (cid:96) tuned to the physical-continuum limit. − . − . − . − . − . . z p / | z p ( t − ) | . . . . . f d + NRQCDheavy-HISQNRQCD from heavy-HISQ
FIG. 24: Fits of f + for B + c → B (cid:96)ν (cid:96) tuned to the physical-continuum limit. for c to s/d decay depend on the mass of the spectatorquark? We can answer that question with our heavy-HISQ calculation because we have results at a range ofspectator quark masses from m c upwards (see Fig. 2).Our form factor fits (Sec. IV C) enable us to extrapolateup 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 resultsfor f s + and f s as a function of the heavy-charm mesonmass (as a proxy for the spectator quark mass). Theform factor curves that are plotted are those for q = 0(where f + = f ) and for the zero-recoil point ( q ). At q max the daughter meson is at rest in the rest-frame ofthe H c meson. The q value at q falls slowly as theheavy-quark mass increases above m c because the massdifference between H c and H s mesons falls. Examiningthe region between M η c and M B c in Fig. 25 we see almostno dependence on the spectator mass. The form factorvalue that shows the most dependence is f + ( q max ). This M H c [GeV]0 . . . . . . M D M η c M B + c M D M η c M B + c f s , + (0) f s + ( t − ) f s ( t − ) B c → B s NRQCD D → KB c → B s heavy-HISQ FIG. 25: Values for the physical-continuum form factors f s = f s + at q = 0 and f s and f s + at q are plotted against themass of the heavy-charm pseudoscalar meson. The curve isthe continuum limit of the heavy-HISQ fit function (Eq. (25))extrapolated to the physical B c and D masses. Note that theregion in which the heavy-HISQ calculation has results is theregion above M η c . See the text for a description of how theextrapolation down to the D was done. Also plotted are theform factor results for D → K [3] (green squares) as wellas the NRQCD B c → B s result presented in this work (redcircles). is not surprising because f + shows the biggest slope in q close to q and hence sensitivity to the value of q . Note that the curve from the heavy-HISQ analysisagrees with the NRQCD results at a spectator mass equalto that of the b . As discussed in the previous subsection,the form factors obtained from the two calculations agreeacross the full q range.We can also investigate the behaviour of the heavy-HISQ fit function as m h is taken below m c to m l wherecontact is made with results for D → K from [3]. Forthe form factors at q = 0, we have P ( q ) = 1 and our fitform at Eq. (25) depends only on M H c . This permits astraightforward extrapolation to the point M H c = M D inthe continuum limit. For the form factors at zero-recoil( q ), constructing the extrapolation curve is compli-cated by requiring the dependence of q on the massof the spectator quark. This requires knowledge of M H s as a function of M H c . To achieve this, we fit our val-ues of M H s taken from set 6, together with physicalvalues from experiment [33] at m h = m l , m b (i.e. M K and M B s ), using a simple fit form M H s = M H c (1 + (cid:80) n =1 ω n (Λ QCD /M H c ) n + A ( a Λ QCD ) + B ( a Λ QCD ) ).Here A , B and ω n take prior values 0(2) and we donot include a Λ terms for data from [33]. We find thisfit function reproduces our data, as well as the physicalvalues, well. Fig. 25 also shows the result of this down-ward extrapolation. Whilst this extrapolation below m c is outside the region where HQET is expected to be valid,the curves nevertheless show approximately the correctamount of upward movement necessary to reproduce the5 TABLE XI: Final results of the weighted integral of | f + ( q ) | over the physical range of squared 4-momentum transfer.Units are MeV. B + c → B s (cid:96)ν (cid:96) B + c → B (cid:96)ν (cid:96) Γ | V | − . × − . × − D → K results in [3] for f + and f at zero-recoil and q = 0. The form factors at q = 0 continue to show al-most no spectator mass dependence, and this is in agree-ment with the D → K results. F. Decay rate
The hadronic quantity required for determining the de-cay rate and branching fraction is the integralΓ | V | − = G F π (cid:90) t − dq | p | | f + ( q ) | , (28)where V is the CKM element V cs or V cd . Table XI givesvalues 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. Val-ues for different q bins can also be obtained. Proceedingwith the total decay rate, combining these results withexisting CKM matrix values [33] V cs = 0 . V cd = 0 . B + c → B s (cid:96)ν (cid:96) ) = 52 . . . × s − Γ( B + c → B (cid:96)ν (cid:96) ) = 3 . × s − (29)where the CKM matrix elements are responsible for thefirst errors and the second errors arise from our latticecalculations. The dominant source of lattice QCD un-certainty is the fitting of 2-point and 3-point correlatorsdescribed in Sec. II B 3.We can convert these results for the decay width intoa branching fraction using the lifetime of the B c meson,513.49(12.4) fs [42]. This gives B ( B + c → B s (cid:96)ν (cid:96) ) = 0 . B ( B + c → B (cid:96)ν (cid:96) ) = 0 . | V | − for B c → B s to B c → B d taking correlations into account betweenthe numerator and denominator. From the chained fit of B + c → B s (cid:96)ν (cid:96) and B + c → B (cid:96)ν (cid:96) form factors, we obtainΓ( B + c → B s (cid:96)ν (cid:96) ) | V cd | Γ( B + c → B (cid:96)ν (cid:96) ) | V cs | = 0 . . (31)In fact the uncertainty is roughly the same as if we wereto treat the numerator and denominator as uncorrelated. .
00 0 .
25 0 .
50 0 .
75 1 . q [GeV] . . . . . f s , + FIG. 26: Final form factors from the chained fits of f (below)and f + (above) for B + c → B s (cid:96)ν (cid:96) in the physical-continuumlimit, plotted against the entire range of physical q . This fitis described in Sec. IV D. .
00 0 .
25 0 .
50 0 .
75 1 . q [GeV] . . . . . f d , + FIG. 27: Final form factors from the chained fits of f (below)and f + (above) for B + c → B (cid:96)ν (cid:96) in the physical-continuumlimit, plotted against the entire range of physical q . This fitis described in Sec. IV D. V. CONCLUSIONS
We have reported here the first calculations of the de-cay rates Γ( B + c → B s (cid:96)ν (cid:96) ) and Γ( B + c → B (cid:96)ν (cid:96) ), demon-strating the success of lattice QCD in studying decays ofheavy-light mesons. The use of HISQ-HISQ c → s ( d )currents allows for a non-perturbative renormalisationusing the PCVC. We used two different formulations forthe spectator b quark, heavy-HISQ and NRQCD. Resultsfrom the heavy-HISQ calculations are in good agreementwith the physical-continuum form factors derived fromthe calculations using NRQCD b quarks, giving us con-fidence in assessing and controlling the systematic errorsin each formulation. Simulating at a variety of specta-tor masses in the heavy-HISQ calculation has provided6a check of the spectator-independence of the renormal-isation procedure for the vector current. The NRQCDstudy also accessed Z V away from zero-recoil to scruti-nise momentum independence.Our final form factors from the chained fit that com-bines both NRQCD and heavy-HISQ results are plottedagainst q in Figs. 26 and 27.The decay rates are predicted from our calculationwith 4% and 6% uncertainty for Γ( B + c → B s (cid:96)ν (cid:96) ) =52 . . × s − and Γ( B + c → B (cid:96)ν (cid:96) ) = 3 . × s − respectively. There is scope for significant im-provement should future experiment demand more preci-sion from the lattice. Such improvement would be readilyachieved by the inclusion of lattices with a finer latticein the heavy-HISQ calculation. ‘Ultrafine’ lattices with a ≈ .
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