Form factor ratios for B s →Kℓν and B s → D s ℓν semileptonic decays and | V ub / V cb |
Christopher J. Monahan, Chris M. Bouchard, G. Peter Lepage, Heechang Na, Junko Shigemitsu
IINT-PUB-18-046
Form factor ratios for B s → K (cid:96) ν and B s → D s (cid:96) ν semileptonic decays and | V ub /V cb | Christopher J. Monahan, ∗ Chris M. Bouchard, G. Peter Lepage, Heechang Na, and Junko Shigemitsu (HPQCD Collaboration) Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom Laboratory of Elementary Particle Physics, Cornell University, Ithaca, New York 14853, USA Ohio Supercomputer Center, 1224 Kinnear Road, Columbus, Ohio 43212, USA Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA (Dated: December 21, 2018)We present a lattice quantum chromodynamics determination of the ratio of the scalar and vectorform factors for two semileptonic decays of the B s meson: B s → K(cid:96)ν and B s → D s (cid:96)ν . In conjunc-tion with future experimental data, our results for these correlated form factors will provide a newmethod to extract | V ub /V cb | , which may elucidate the current tension between exclusive and inclu-sive determinations of these Cabibbo-Kobayashi-Maskawa mixing matrix parameters. In addition tothe form factor results, we determine the ratio of the differential decay rates, and forward-backwardand polarization asymmetries, for the two decays. I. INTRODUCTION
Semileptonic decays of heavy mesons provide stringenttests of the standard model of particle physics and op-portunities to observe signals of new physics. In partic-ular, experimental measurements of B decays have high-lighted a number of deviations from standard model ex-pectations. These discrepancies include R ( D ( ∗ ) ), the ra-tio of the branching fraction of the B → D ( ∗ ) τ ν and B → D ( ∗ ) e/µν decays, R K ( ∗ ) , the ratio of the branch-ing fraction of the B → K ( ∗ ) µ + µ − and B → K ( ∗ ) e + e − decays, and the long-standing tension between inclusiveand exclusive determinations of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix elements | V ub | and | V cb | .Although none of these differences are conclusive evi-dence of new physics effects, the cumulative weight ofthese tensions suggest a hint of new physics.The ratio | V ub /V cb | , which enters into the length of theside of the CKM unitarity triangle opposite the precisely-determined angle β , is a central input into tests of CKMunitarity. Both | V ub | and | V cb | have been determinedthrough measurements of multiple exclusive mesonic de-cay channels [1, 2], primarily B → π(cid:96)ν (cid:96) [3–11] and B → D ( ∗ ) (cid:96)ν (cid:96) respectively [12–20], although other chan-nels are also used [21–25]. The B s → K decay has gen-erally received less theoretical attention than the cor-responding B decay, largely due to the absence of ex-perimental data, although this channel has been studiedon the lattice in [29 ? ], and using other theoretical ap-proaches [ ? ], including light cone sum rules [ ? ? ],perturbative QCD [ ? ? ] and QCD-inspired models[ ? ? ? ? ? ]. Form factors for both B → π(cid:96)ν (cid:96) and B → D ( ∗ ) (cid:96)ν (cid:96) decays have been calculated by sev-eral lattice groups [26–33] and using light cone sum rules[34–42], which provide complementary coverage of dif-ferent kinematic regions. The leptonic decay B → τ ν ∗ e-mail: [email protected] provides an alternative method to extract | V ub | , but thisapproach is limited by current experimental uncertainties[1]. Most recently, the ratio | V ub /V cb | was determined bythe LHCb collaboration through the ratio of the bary-onic decays Λ b → Λ + c µν and Λ b → pµν [43, 44], usingform factors determined with lattice QCD [45]. Inclusivedeterminations of | V ub | differ from the value extractedfrom exclusive decays at the level of approximately threestandard deviations.Here we undertake a correlated study of the form fac-tors for the B s → K(cid:96)ν (cid:96) and B s → D s (cid:96)ν (cid:96) decays, which,in conjunction with anticipated experimental results fromthe LHCb Collaboration, will provide a new methodto determine the ratio | V ub /V cb | . We perform a chiral-continuum-kinematic fit to the scalar and vector formfactors for both the B s → K(cid:96)ν (cid:96) [46] and B s → D s (cid:96)ν (cid:96) de-cays [47], to determine the correlated form factors overthe full range of momentum transfer. Using the ratioof the form factors significantly reduces the largest sys-tematic uncertainty at large values of the momentumtransfer, which stems from the perturbative matching oflattice nonrelativistic QCD (NRQCD) currents to con-tinuum QCD. We use our form factor results to predictseveral phenomenological ratios, including the differen-tial branching fractions, and the forward-backward andpolarization asymmetries.We briefly summarize the details of the lattice calcu-lations used in the analyses of [46, 47] in Sec. II and thecorresponding form factor results in Sec. III. We thenpresent our new chiral-continuum-kinematic extrapola-tion in Sec. IV, and our phenomenological predictions inSec. V, before summarizing in Sec. VI. We provide fur-ther details of the input two-point correlator data in Ap-pendix A and details required to reconstruct our chiral-continuum-kinematic fit in Appendix B. a r X i v : . [ h e p - l a t ] D ec TABLE I. Details of three “coarse” and two “fine” n f = 2+1MILC ensembles used in the determination of the scalar andvector form factors.Set r /a m l /m s (sea) N conf ( K/D s ) N tsrc L × N t C1 2.647 0.005/0.050 1200/2096 2/4 24 × × × × × II. ENSEMBLES, CURRENTS ANDCORRELATORS
Our determination of the ratio of the form factors forthe exclusive B s → X s (cid:96)ν semileptonic decays closely par-allels the analyses presented in [31, 46, 47]. Throughoutthis work, we use X s to represent a K or D s meson. Weuse the two- and three-point correlator data presented in[46, 47] to perform a simultaneous, correlated fit of theform factors for both B s → K(cid:96)ν and B s → D s (cid:96)ν decays.In this section we outline the details of the ensembles,reproduce the form factor results for convenience, andrefer the reader to [31, 46, 47] for details of the correla-tor analysis.We use five gauge ensembles with n f = 2 + 1 flavorsof AsqTad sea quarks generated by the MILC Collabora-tion [48], including three “coarse” (with lattice spacing a ≈ .
12 fm) and two “fine” (with a ≈ .
09 fm) ensem-bles. We summarize these ensembles in Table I and tab-ulate the corresponding light pseudoscalar masses, forboth AsqTad and HISQ valence quarks, in Table II.In Table III we list the valence quark masses for theNRQCD bottom quarks and HISQ charm quarks [46, 50].For completeness and ease of reference, we include boththe tree-level wave function renormalization for the mas-sive HISQ quarks [51] and the spin-averaged Υ mass,corrected for electroweak effects, determined in [50].The scalar, f ( X s )0 ( q ), and vector, f ( X s )+ ( q ), form fac-tors that characterize the B s → X s semileptonic decaysare defined by the matrix element (cid:104) X s ( p X s ) | V µ | B s ( p B s ) (cid:105) = f ( X s )0 ( q ) M B s − M X s q q µ + f X s + ( q ) (cid:20) p µB s + p µX s − M B s − M X s q q µ (cid:21) , (1)where V µ is a flavor-changing vector current and the mo-mentum transfer is q µ = p µB s − p µX s . On the lattice it ismore convenient to work with the form factors f ( X s ) (cid:107) and f ( X s ) ⊥ , which are given in terms of the scalar and vector form factors by f ( X s )+ ( q ) = 1 (cid:112) M B s (cid:104) f ( X s ) (cid:107) ( q )+ ( M B s − E X s ) f ( X s ) ⊥ ( q ) (cid:105) , (2) f ( X s )0 ( q ) = (cid:112) M B s M B s − M X s (cid:20) ( M B s − E X s ) f ( X s ) (cid:107) ( q )+ ( E X s − M X s ) f ( X s ) ⊥ ( q ) (cid:21) . (3)Here E X s is the energy of the X s meson in the rest frameof the B s meson. We work in the rest frame of the B s meson and throughout the rest of this work the spatialmomentum, (cid:126)p , denotes the momentum of the X s meson.NRQCD is an effective theory for heavy quarks and re-sults determined using lattice NRQCD must be matchedto full QCD to make contact with experimental data. Wematch the bottom-charm currents, J µ , at one loop in per-turbation theory through O ( α s , Λ QCD /m b , α s / ( am b )),where am b is the bare lattice mass [51]. We rescale allcurrents by the nontrivial massive wave function renor-malization for the HISQ charm quarks, tabulated in Ta-ble III, and taken from [31, 51].The B s and X s meson two-point correlators and three-point correlators of the NRQCD-HISQ currents, J µ , werecalculated in [46, 47]. In those calculations, we usedsmeared heavy-strange bilinears to represent the B s me-son and incorporated both delta-function and Gaussiansmearing, with a smearing radius of r /a = 5 and r /a = 7 on the coarse and fine ensembles, respectively.The three-point correlators were determined with thesetup illustrated in Fig. 1. The B s meson is created attime t and a current J µ inserted at time t , between t and t + T . The X s meson is then annihilated at time t + T . We used four values of T : 12, 13, 14, and 15on the coarse lattices; and 21, 22, 23, and 24 on thefine lattices. We implemented spatial sums at the sourcethrough the U (1) random wall sources ξ ( x ) and ξ ( x (cid:48) ) [52]and generated data for four different values of the X s me-son momenta, (cid:126)p = 2 π/ ( aL )(0 , , (cid:126)p = 2 π/ ( aL )(1 , , (cid:126)p = 2 π/ ( aL )(1 , , (cid:126)p = 2 π/ ( aL )(1 , , L is the spatial lattice extent. III. CORRELATOR AND FORM FACTORRESULTS
The results for the two- and three-point correlatorswere determined with a Bayesian multiexponential fittingprocedure, based on the
PYTHON packages
LSQFIT [53]and
CORRFITTER [54]. The results are summarized forconvenience in Appendix A.We summarize the final results for the form factors, f ( (cid:126)p ) and f + ( (cid:126)p ), for each ensemble and X s momentumin Tables IV and V. For more details, see [31, 46, 47]. TABLE II. Light meson masses on MILC ensembles for both AsqTad [48] and HISQ valence quarks [46]. In the final columnwe list the finite volume corrections to chiral logarithms from staggered perturbation theory [49], for each ensemble.Set aM AsqTad π aM HISQ π aM AsqTad K aM HISQ K aM HISQ η s aM HISQ D s δ FV C1 0.15971(20) 0.15990(20) 0.36530(29) 0.31217(20) 0.41111(12) 1.18755(22) 0.053647C2 0.22447(17) 0.21110(20) 0.38331(24) 0.32851(48) 0.41445(17) 1.20090(30) 0.030760C3 0.31125(16) 0.29310(20) 0.40984(21) 0.35720(22) 0.41180(23) 1.19010(33) 0.003375F1 0.14789(18) 0.13460(10) 0.25318(19) 0.22855(17) 0.294109(93) 0.84674(12) 0.059389F2 0.20635(18) 0.18730(10) 0.27217(21) 0.24596(14) 0.29315(12) 0.84415(14) 0.007567TABLE III. Valence quark masses am b for NRQCD bottomquarks and am s and am c for HISQ strange and charm quarks.The fifth column gives Z (0)2 ( am c ), the tree-level wave functionrenormalization constant for massive (charm) HISQ quarks.The sixth column lists the values of the spin-averaged Υ mass,corrected for electroweak effects.Set am b am s am c Z (0)2 ( am c ) aE sim bb 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)FIG. 1. Lattice setup for the three-point correlators. Seeaccompanying text for details.
IV. CHIRAL, CONTINUUM AND KINEMATICEXTRAPOLATIONS
Form factors determined from experimental data arefunctions of a single kinematic variable, which is typicallythe momentum transfer, q , or the energy of the mesonicdecay product, E X s . Alternatively, the form factors canbe expressed in terms of the z -variable, z ( q ) = (cid:112) t + − q − √ t + − t (cid:112) t + − q + √ t + − t . (4)Here t + = ( M B s + M X s ) and t is a free parame-ter, which we take to be t = ( M B s + M X s )( (cid:112) M B s + (cid:112) M X s ) , as in [46]. This choice minimizes the magni-tude of z over the physical range of momentum transfer.Note that in [47] the choice t = q = ( M B s − M X s ) was used to ensure consistency with the analysis of [31].We have confirmed that our extrapolation results are in-dependent of our choice of t , within fit uncertainties.Lattice calculations of form factors are necessarily de-termined at finite lattice spacing, generally with lightquark masses that are heavier than their physical values,and are thus functions of the lattice spacing and the lightquark mass in addition to the momentum transfer. Weremove the lattice spacing and light quark mass depen-dence of the lattice results by performing a combinedcontinuum-chiral-kinematic extrapolation, through themodified z -expansion, which was introduced in [52, 55]and applied to B s semileptonic decays in [46, 47, 56, 57].Our chiral-continuum-kinematic extrapolation for the B s → X s (cid:96)ν decays closely parallels those studied in [31,46, 47], so here we outline the main components and referthe reader to those references for details.The dependence of the form factors on the z -variableis expressed through a modification of the Bourrely-Caprini-Lellouch (BCL) parametrization [58] P ( X s )0 f ( X s )0 ( q ( z )) = (cid:104) L ( X s ) (cid:105) × J − (cid:88) j =0 a (0 ,X s ) j ( m l , m sea l , a ) z j , (5) P ( X s )+ f ( X s )+ ( q ( z )) = (cid:104) L ( X s ) (cid:105) × J − (cid:88) j =0 a (+ ,X s ) j ( m l , m sea l , a ) (cid:20) z j − ( − j − J jJ z J (cid:21) . (6)Here the P , + are Blaschke factors that take into accountthe effects of expected poles above the physical region, P ( X s )0 , + ( q ) = − q (cid:16) M ( X s )0 , + (cid:17) , (7)where we take [46, 47, 59] M ( K )+ = 5 .
325 20(48) GeV , (8) M ( K )0 = 5 . , (9) M ( D s )+ = M B ∗ c = 6 . , (10) M ( D s )0 = 6 . . (11) TABLE IV. Final results for the form factors f ( K )0 ( (cid:126)p ) and f ( K )+ ( (cid:126)p ). Data reproduced from Table II of [46].Set f ( K )0 (0 , , f ( K )0 (1 , , f ( K )0 (1 , , f ( K )0 (1 , , f ( K )+ (1 , , f ( K )+ (1 , , f ( K )+ (1 , , f ( D s )0 ( (cid:126)p ) and f ( D s )+ ( (cid:126)p ). Data reproduced from Tables VI and VII of [47].Set f ( D s )0 (0 , , f ( D s )0 (1 , , f ( D s )0 (1 , , f ( D s )0 (1 , , f ( D s )+ (1 , , f ( D s )+ (1 , , f ( D s )+ (1 , , In line with [46], we convert these values to lattice unitsin the chiral-continuum-kinematic extrapolation, so thatthe difference between the ground state meson massesand these pole masses is fixed in physical units.The functions L ( X s ) incorporate the chiral logarithmiccorrections, which are fixed by hard pion chiral pertur-bation theory [60, 61] for the B s → K decay L K = − x π (log x π + δ FV ) − g x K log x K − g x η log x η . (12)Here g = 0 . δ FV are finite volume correctionsgiven in Table II, we define x π,K,η,η s = M π,K,η,η s (4 πf π ) , (13) δx π,K = ( M AsqTad π,K ) − ( M HISQ π,K ) (4 πf π ) , (14) δx η s = ( M HISQ η s ) − ( M phys .η s ) (4 πf π ) , (15)and M η = ( M π + 2 M η s ) /
3. We tabulate the mesonmasses required to calculate δx π,K,η s in Table II. For the B s → D s decay, the chiral logarithmic corrections cannotbe factored out in the z -expansion [61] and therefore wefollow [31, 47] and fit the logarithmic dependence by in-troducing corresponding fit parameters in the expansioncoefficients a (0 , + ,D s ) j . In other words, we take L ( D s ) = 0 , (16)and introduce an appropriate fit parameter, c (2) j , in thecorresponding fit function, Eq. (19).The expansion coefficients a (0 , + ,X s ) j include latticespacing and quark mass dependence and can be written as a (0 , + ,X s ) j ( m l , m sea l , a ) = (cid:101) a (0 , + ,X s ) j (cid:101) D (0 , + ,X s ) j ( m l , m sea l , a ) , (17)where the (cid:101) D (0 , + ,X s ) j include all lattice artifacts. Sup-pressing the 0 , + superscripts for clarity, these coefficientsare given by [46] (cid:101) D ( K ) j = 1 + c (1) j x π + d (1) j (cid:18) δx π δx K (cid:19) + d (2) j δx η s + e (1) j (cid:18) aE K π (cid:19) + e (2) j (cid:18) aE K π (cid:19) + f (1) j (cid:18) ar (cid:19) + f (2) j (cid:18) ar (cid:19) , (18)and [47] (cid:101) D ( D s ) j = 1 + c (1) j x π + c (2) j x π log( x π )+ d (1) j (cid:18) δx π δx K (cid:19) + d (2) j δx η s + e (1) j (cid:18) aE D s π (cid:19) + e (2) j (cid:18) aE D s π (cid:19) + m (1) j ( am c ) + m (2) j ( am c ) . (19)Here the c ( i ) j , d ( i ) j , e ( i ) j , f ( i ) j , and m ( i ) j are fit parameters,along with the (cid:101) a (0 , +) j . We incorporate light- and heavy-quark mass dependence in the discretization coefficients f ( i ) j by replacing f ( i ) j → f ( i ) j (1 + l (1 ,i ) j x π + l (2 ,i ) j x π ) × (1 + h (1 ,i ) j δm b + h (2 ,i ) j ( δm b ) ) , (20)where δm b = am b − .
26 [46] and is chosen to minimizethe magnitude of δm b , such that − . < δm b < .
4. Here x π captures sea pion mass dependence and is determinedfrom the AsqTad pion mass [48].The actions we use are highly improved and O ( a ) tree-level lattice artifacts have been removed. The O ( α s a )and O ( a ) corrections are dominated by powers of ( am c )and ( aE X s ), rather than those of the spatial momenta( ap i ). Thus, we do not incorporate terms involving hy-percubic invariants constructed from the spatial momen-tum ap i [62].We follow [46, 47] and impose the kinematic constraint f (0) = f + (0) analytically for the B s → K decay, andas a data point for the B s → D s channel. To incorpo-rate the systematic uncertainty associated with trunca-tion of the perturbative current-matching procedure at O ( α s , Λ QCD /m b , α s / ( am b )), we introduce fit parameters m (cid:107) and m ⊥ , with central value zero and width δm (cid:107) , ⊥ and re-scale the form factors, f (cid:107) and f ⊥ according to f (cid:107) , ⊥ → (1 + m (cid:107) , ⊥ ) f (cid:107) , ⊥ . (21)We take δm (cid:107) , ⊥ = 0 .
04. We refer to this fit Ansatz, in-cluding terms up to z in the modified z -expansion, asthe “standard extrapolation.”To test the convergence of our fit Ansatz and ensure wehave included a sufficient number of terms in the modified z -expansion, we modify the fit Ansatz in the followingways:1. include terms up to z in the z -expansion;2. include terms up to z in the z -expansion;3. include discretization terms up to ( am c ) ;4. include discretization terms up to ( am c ) ;5. include discretization terms up to ( a/r ) ;6. include discretization terms up to ( a/r ) ;7. include discretization terms up to ( aE K /π ) ;8. include discretization terms up to ( aE K /π ) ;9. include discretization terms up to ( aE D s /π ) ;10. include discretization terms up to ( aE D s /π ) ;We show the results of these modifications in Fig. 2,where we label the standard fit Ansatz as “Test 0”. Thesetests demonstrate the stability of the standard fit Ansatz;adding higher order terms does not alter the fit resultsor improve the goodness-of-fit.We also study the stability of the fit with respect tothe following variations:i. omit the x π log( x π ) term;ii. omit the light quark mass-dependent discretizationterms from the f ( i ) j coefficients;iii. add strange quark mass-dependent discretizationterms to the f ( i ) j coefficients; FIG. 2. Comparison of the convergence tests of the “standardextrapolation” fit Ansatz. The top panel shows the χ / doffor each test, normalized by the χ / dof for the standard ex-trapolation. The lower panel shows the fit results for the formfactor ratio f ( K )0 /f ( D s )0 at q = 0. The test numbers label-ing the horizontal axis correspond to the modifications listedin the text. The first data point, the purple square, is the“standard extrapolation” fit result, which is also representedby the purple shaded band. iv. omit the am b -dependent discretization terms fromthe f ( i ) j coefficients;v. omit sea- and valence-quark mass difference, d (1) j ;vi. omit the strange quark mass mistuning, d (2) j ;vii. omit finite volume effects;viii. add light-quark mass dependence to the m ( i ) j fitparameters;ix. add strange-quark mass dependence to the m ( i ) j fitparameters;x. add bottom-quark mass dependence to the m ( i ) j fitparameters; FIG. 3. Analogous to Fig. 2, but for stability tests labeled by“i.” to “xii.” in the text. Details provided in the caption ofFig. 2. xi. incorporate a 2% uncertainty for higher-ordermatching contributions;xii. incorporate a 5% uncertainty for higher-ordermatching contributions;We show the results of these stability tests in Fig. 3.Test 0 represents the standard fit Ansatz. Taken to-gether, these plots demonstrate that the fit has convergedwith respect to a variety of modifications of the chiral-continuum-kinematic extrapolation Ansatz.
V. RESULTSA. Form factor ratios
Our final results, from a simultaneous fit to both decaychannels, for the ratio of form factors at zero momentumtransfer are f ( K )0 (0) f ( D s )0 (0) = 0 . , (22) where the uncertainties account for correlations betweenthe form factor results for each decay channel. Thecorresponding results for the individual form factors atzero momentum transfer are f ( K )0 (0) = 0 . f ( D s )0 (0) = 0 . f ( K )0 (0) /f ( D s )0 (0) = 0 . / . . χ of χ / dof = 1 . Q = 0 . Q -value (or p -value) corresponds to the probabil-ity that the χ / dof from the fit could have been larger,by chance, assuming the data are all Gaussian and con-sistent with each other. The simultaneous fit ensuresthat the uncertainties associated with the perturbativematching procedure for the heavy-light currents largelycancel in the form factor ratio. This can be seen bycomparing the error budget contribution from perturba-tive matching in Table VI, with the individual fits, forwhich the perturbative truncation uncertainty was thesecond-largest source of uncertainty. The uncertaintiesin our ratio results are dominated by the B s → K chan-nel, which has fewer statistics and a larger extrapolationuncertainty, because, in the region of momentum transferreported here, 0 − − . , the corresponding formfactors are extrapolated further from the region in whichwe have lattice results.We tabulate our choice of priors and the fit results inthe Appendix, where we provide the corresponding z -expansion coefficients and their correlations. Following[47], based on the earlier work of [31, 52, 55], we splitthe priors into three groups. Broadly speaking, Group Ipriors includes the typical fit parameters, Group II theinput lattice scales and masses, and Group III priors theinputs from experiment, such as physical meson masses.We plot our final results for the ratios of the form factors, f ( K )0 /f ( D s )0 ( q ) and f ( K )+ /f ( D s )+ ( q ), as a function of themomentum transfer, q , in Fig. 4. Details required toreconstruct the fully correlated form factors are given inAppendix B. B. Form factor error budget
We tabulate the errors in the ratios of the form factorsat zero momentum transfer, Eq. (22), in Table VI. Thesources of uncertainty listed in Table VI are: a. Statistical.
Statistical uncertainties include thetwo- and three-point correlator fit errors and those as-sociated with the lattice spacing determination, r and r /a . These effects are the second largest source of un-certainty in our results, and are dominated by the smallerstatistics available in the B s → K analysis. b. Chiral extrapolation. Includes the uncertaintiesarising from extrapolation in both valence and sea quark
FIG. 4. Chiral and continuum extrapolated form factor ra-tios, f ( K )0 /f ( D s )0 ( q ) (upper panel) and f ( K )+ /f ( D s )+ ( q ) (lowerpanel), as a function of the momentum transfer, q . Thedashed lines indicate the central values of the extrapolatedform factors and the uncertainty bands include all sources ofstatistical and systematic uncertainty.TABLE VI. Error budget for the form factor ratios at zeromomentum transfer, Eq. (22). We describe each source ofuncertainty in more detail in the accompanying text.Type Partial uncertainty (%)Statistical 6.63Chiral extrapolation 0.89Quark mass tuning 2.18Discretization 4.16Kinematic 9.31Matching 0.28Total 13.03 masses and from the B s → D s chiral logarithms in thechiral-continuum extrapolation, corresponding to the fitparameters c ( i ) j in Eqs. (18) and (19). c. Quark mass tuning. These uncertainties arisefrom tuning the light and strange quark masses at finitelattice spacing and partial quenching effects. d. Discretization.
These effects include the( aE X s /π ) n , ( a/r ) n , and ( am c ) n terms in the modified z -expansion, corresponding to the fit parameters e ( i ) j , f ( i ) j and m ( i ) j in Eqs. (18) and (19). e. Kinematic. Uncertainties that arise from the z -expansion coefficients, including the Blaschke factors.These effects are the dominant source of uncertaintyin our results, and again predominantly arise from the B s → K channel. f. Matching. The perturbative matching uncertain-ties stemming from the truncation of the expansion ofNRQCD-HISQ effective currents in terms of QCD cur-rents. These are the second largest source of uncertaintyin the results for the individual channels, but the effectslargely cancel in the ratio. This is further demonstratedby tests (xi) and (xii) of the previous section, in whichchanging the matching uncertainty from 2% to 5% haspractically negligible effect on the fit, and in particular,the ratio at zero momentum transfer.We propagate all uncertainties from the largemomentum-transfer region, for which we have lattice re-sults, to zero momentum transfer. We do not include theuncertainties associated with physical meson mass inputerrors and finite volume effects, which are both less than0 . f ( q ) and f + ( q ), as a function of themomentum transfer, q , in Fig. 5. C. Semileptonic decay phenomenology
The experimental measurements of the ratio R ( D ) = B ( B → Dτ ν ) B ( B → D(cid:96)ν ) , (23)which measures the ratio of branching fraction of thesemileptonic decay to the τ lepton to the branching frac-tion to an electron or muon (represented by (cid:96) ), are cur-rently in tension with the standard model result. Theglobal experimental average is [63–66] R ( D ) exp . = 0 . stat . (28) sys . , (24)whereas the standard model expectation, neglecting cor-relations between the calculations [31, 67, 68], is R ( D ) theor . = 0 . . (25)We determine the corresponding ratio of the R -ratiosfor the semileptonic B s → X s (cid:96)ν decays, R ( K ) R ( D s ) = 2 . , (26) FIG. 5. Error budget estimates for the ratios of the form fac-tors, f ( K )0 /f ( D s )0 ( q ) (upper panel) and f ( K )+ /f ( D s )+ ( q ) (lowerpanel), as a function of the momentum transfer, q . which is in agreement with, but with slightlysmaller errors than, the value of R ( K ) /R ( D s ) =0 . / . . B s → X s (cid:96)ν is given in terms of the corresponding scalarand vector form factors byd Γ( B s → X s (cid:96)ν )d q d cos θ (cid:96) = G F | V xb | π M B s (cid:18) − m (cid:96) q (cid:19) | (cid:126)p X s |× (cid:20) M B s (cid:126)p X s (cid:18) sin θ (cid:96) + m (cid:96) q cos θ (cid:96) (cid:19) | f + | + 4 m (cid:96) q (cid:0) M B s − M X s (cid:1) M B s | (cid:126)p X s | cos θ (cid:96) f f + + m (cid:96) q (cid:0) M B s − M X s (cid:1) | f | (cid:21) . (27)Here θ (cid:96) is defined as the angle between the final statelepton and the B s meson, in the frame in which (cid:126)p (cid:96) + (cid:126)p ν = (cid:126)
0. Integrating over the angle θ (cid:96) , one obtains the standard FIG. 6. Ratio of the differential decay rates, γ ( K ) (cid:96) /γ ( D s ) (cid:96) , di-vided by | V ub /V cb | , as a function of the momentum transfer, q . model differential decay rate, γ ( X s ) (cid:96) = d Γ( B s → X s (cid:96)ν )d q = G F | V xb | π M B s (cid:18) − m (cid:96) q (cid:19) | (cid:126)p X s |× (cid:20) (cid:18) m (cid:96) q (cid:19) M B s (cid:126)p X s | f + | + 3 m (cid:96) q (cid:0) M B s − M X s (cid:1) | f | (cid:21) . (28)In Fig. 6 we plot the ratio of the differential decay rates, γ ( K ) (cid:96) /γ ( D s ) (cid:96) , as a function of the momentum transfer, forthe semileptonic decays to muons ( (cid:96) = µ ) and to tauleptons ( (cid:96) = τ ).We combine our results for these decay rate ratios withthe experimental world average results for | V ub /V cb | [1],using both inclusive and exclusive determinations,exclusive | V ub /V cb | = 0 . , (29)inclusive | V ub /V cb | = 0 . , (30)and plot the results in Fig. 7. The LHCb Collabora-tion has measured this ratio to be | V ub /V cb | = 0 . | V ub /V cb | = 0 . b → p + µ − ν andΛ b → Λ + c µ − ν [43]. This result is sufficiently close to theworld average given in Eqs. (29) that we do not includeit in Fig. 7. A correlated average, | V ub /V cb | = 0 . FIG. 7. Ratio of the differential decay rates, γ ( K ) (cid:96) /γ ( D s ) (cid:96) , usinginclusive and exclusive world average results for | V ub /V cb | , asa function of the momentum transfer, q . The upper panelshows the decay rates for (cid:96) = τ , and the lower panel (cid:96) = µ . Defining the partially integrated ratio ζ ( X s ) (cid:96) = 1 | V xb | (cid:90) q m (cid:96) dΓ( B s → X s (cid:96)ν )d q d q , (31)where q = ( M B s − M X s ) , we integrate our resultsnumerically to obtain ζ ( K ) µ ζ ( D s ) µ = 0 . , (32) ζ ( K ) τ ζ ( D s ) τ = 1 . . (33)Asymmetries in the differential decay rate can bedefined from the angular distribution, Eq. (27). The forward-backward asymmetry is given by A ( X s ) (cid:96) ( q ) = (cid:20)(cid:90) − (cid:90) − (cid:21) d cos θ (cid:96) d Γd q d cos θ (cid:96) = G F | V xb | π M B s (cid:18) − m (cid:96) q (cid:19) | (cid:126)p X s | m (cid:96) q × (cid:0) M B s − M X s (cid:1) f f + , (34)and the polarization asymmetry by P ( X s ) (cid:96) ( q ) = dΓ(LH)d q − dΓ(RH)d q , (35)where the differential decay rates to left-handed (LH) andright-handed (RH) final state leptons are given bydΓ(LH)d q = G F | V xb | | (cid:126)p X s | π (cid:18) − m (cid:96) q (cid:19) f , (36)dΓ(RH)d q = G F | V xb | | (cid:126)p X s | π m (cid:96) q (cid:18) − m (cid:96) q (cid:19) × (cid:20)
38 ( M B s − M X s ) M B s f + 12 | (cid:126)p X s | f (cid:21) . (37)In the standard model, the production of right-handed fi-nal state leptons is helicity suppressed, and so this asym-metry offers a probe for helicity-violating interactionsgenerated by new physics.In Figs. 8 and 9, we plot the ratios of the forward-backward and polarization asymmetries, respectively, forthe B s → K(cid:96)ν and B s → D s (cid:96)ν decays. We plot theasymmetry ratios using both inclusive and exclusive val-ues of | V ub /V cb | . Integrating over q , and multiplying bythe appropriate combination of CKM matrix elements todefine the QCD contribution, we find | V cb | | V ub | (cid:82) q m (cid:96) A ( K ) µ d q (cid:82) q m µ A ( D s ) µ d q = 0 . , (38) | V cb | | V ub | (cid:82) q m (cid:96) A ( K ) τ d q (cid:82) q m τ A ( D s ) τ d q = 1 . , (39) | V cb | | V ub | (cid:82) q m (cid:96) P ( K ) µ d q (cid:82) q m µ P ( D s ) µ d q = 0 . , (40) | V cb | | V ub | (cid:82) q m (cid:96) P ( K ) τ d q (cid:82) q m τ P ( D s ) τ d q = − . . (41)Normalizing these asymmetry ratios by the corre-sponding differential decay rate ratio removes the am-0 FIG. 8. Ratio of the forward-backward asymmetries, A ( K ) τ /A ( D s ) τ (upper panel) and A ( K ) µ /A ( D s ) µ (lower panel), us-ing inclusive and exclusive world average results for | V ub /V cb | ,as a function of the momentum transfer, q . biguity arising from | V ub /V cb | , A ( X s ) (cid:96) = (cid:82) q m (cid:96) A ( X s ) (cid:96) d q (cid:82) q m (cid:96) (dΓ / d q )d q , (42) P ( X s ) (cid:96) = (cid:82) q m (cid:96) P ( X s ) (cid:96) d q (cid:82) q m (cid:96) (dΓ / d q )d q . (43)We integrate over the momentum transfer numerically tofind A ( K ) µ A ( D s ) µ = 0 . , P ( K ) µ P ( D s ) µ = 1 . A ( K ) τ A ( D s ) τ = 0 . , P ( K ) τ P ( D s ) τ = − . , (45)where the smaller relative uncertainties compared to theasymmetries themselves demonstrates that most of the FIG. 9. Ratio of the polarization asymmetries, P ( K ) τ /P ( D s ) τ (upper panel) and P ( K ) µ /P ( D s ) µ (lower panel), using inclusiveand exclusive world average results for | V ub /V cb | , as a functionof the momentum transfer, q . hadronic uncertainties have canceled in these normalizedresults. VI. SUMMARY
We have presented a study of the ratio of the scalarand vector form factors for the B s → X s (cid:96)ν semileptonicdecays, where X s is a K or D s meson, over the full kine-matic range of momentum transfer. These ratios combinecorrelator data results determined in [46] for the B s → K decay and in [47] for the B s → D s decay. Our simultane-ous, correlated chiral-continuum kinematic extrapolationreduces the uncertainty in the form factor ratio and, inparticular, largely removes the uncertainty arising fromthe perturbative matching procedure.In addition to the form factor ratios, we predict R ( K ) /R ( D s ), where R ( X s ) is the ratio of the branchingfractions of the corresponding semileptonic B s decay to1 TABLE VII. Fit results for the ground state energies of the K meson at each spatial momentum (cid:126)p K . Data reproducedfrom Table V of [46].Set aM K aE K (1 , , aE K (1 , , aE K (1 , , D s meson at each spatial momentum (cid:126)p D s . Data reproducedfrom Table IV of [47].Set aM D s aE D s (1 , , aE D s (1 , , aE D s (1 , , tau and to electrons and muons. We determine the ratioof the differential decay rates for the two decay channels,as well as the ratio of the forward-backward and polar-ization asymmetries.The LHC is scheduled to significantly improve the sta-tistical uncertainties in experimental measurements of B s decays with more data over the next decade. In particu-lar, experimental data on the ratio of the B s → K(cid:96)ν and B s → D s (cid:96)ν decays, when combined with our form factorresults, will provide a new determination of | V ub /V cb | . ACKNOWLEDGMENTS
Numerical simulations were carried out on facilitiesof the USQCD Collaboration funded by the Office ofScience of the Department of Energy and at the OhioSupercomputer Center. Parts of this work were sup-ported by the National Science Foundation. C.J.M. wassupported in part by the U.S. Department of Energythrough Grant No. DE-FG02-00ER41132 and J.S. in partby the U.S. Department of Energy through Grant No. de-sc0011726. We thank the MILC Collaboration for use oftheir gauge configurations.
Appendix A: Two-point fit results
Here we reproduce the two-point fit results of [46] inTable VII for the K meson and for the D s meson [47] inTable VIII. Appendix B: Reconstructing form factors
In this Appendix we provide our fit results for the co-efficients of the z -expansion for the B s → K(cid:96)ν decay inTable IX, for B s → D s (cid:96)ν in Table X, and for the cor-related fit to both decays in Table XI. We also tabulateour choice of priors for the chiral-continuum extrapola-tion for the B s → K(cid:96)ν decay in Tables XII and XIV, forthe B s → D s (cid:96)ν decay in Tables XIII and XV, and forpriors common to both channels in XVI, and XVII.2 TABLE IX. Coefficients of z -expansion and the corresponding Blaschke factors for the B s → K(cid:96)ν decay. a (0)1 a (0)2 a (0)3 P a (+)0 a (+)1 a (+)2 P + z -expansion and the corresponding Blaschke factors, for the B s → D s (cid:96)ν decay. a (0)0 a (0)1 a (0)2 a (0)3 P a (+)0 a (+)1 a (+)2 P + z -expansion and the corresponding Blaschke factors for the simultaneousfit to the B s → K(cid:96)ν and B s → D s (cid:96)ν decays. The rows correspond to the columns, moving from top to bottom and left toright, respectively. a (0) ,K a (0) ,K a (0) ,K P ( K )0 a (+) ,K a (+) ,K × − × − × − -3.95599616 × − × − × − × − × − × − × − × − × − × − × − -1.81816269 × − × − × − -5.88646696 × − × − a (+) ,K P ( K )+ a (0) ,D s a (0) ,D s a (0) ,D s a (0) ,D s × − × − × − × − × − -1.50864482 × − × − × − × − -1.12557927 × − -4.15916006 × − × − × − × − -8.00096682 × − -1.57760368 × − × − × − × − -2.93868039 × − × − × − -2.93053824 × − × − × − × − × − × − -9.57576314 × − × − -1.42002142 × − × − -7.74705909 × − -1.63296714 × − × − × − × − × − × − -1.12948406 × − -1.17310027 × − × − -4.00252884 × − × − × − × − × − -1.32946477 × − -2.95921529 × − -1.18940865 × − × − × − -1.47064962 × − × − P ( D s )0 a (+) ,D s a (+) ,D s a (+) ,D s P ( D s )+ × − × − -1.00202940 × − × − -1.42966100 × − × − × − -8.93653944 × − × − -3.15809640 × − -3.14364934 × − × − × − × − × − × − -1.49607346 × − -1.05105378 × − × − -4.41710197 × − × − × − -9.47821843 × − -7.78344712 × − -1.33434640 × − -6.39749633 × − -1.59677031 × − × − × − × − × − × − -3.87458330 × − × − -8.16004467 × − -1.19874155 × − -3.95882072 × − × − -1.34953976 × − × − × − × − × − × − -6.23834522 × − -1.51873367 × − × − × − × − -2.61624148 × − × − × − × − × − -2.65313186 × − -2.72916676 × − -1.76329307 × − -1.96104068 × − -8.21918389 × − × − × − -5.12869129 × − -5.75838181 × − -9.37738726 × − × − × − -1.31877655 × − -8.10703811 × − × − × − × − -1.70915346 × − × − × − TABLE XII. Group I priors and fit results for the parametersin the modified z -expansion for the B s → K(cid:96)ν decay. Notethat these parameters are fit simultaneously with those ofTable XIII, but displayed separately for clarity.Prior [ f ] Fit result [ f ] Prior [ f + ] Fit result [ f + ] a a a c (1)1 c (1)2 c (1)3 d (1)1 d (1)2 d (1)3 d (2)1 d (2)2 d (2)3 e (1)1 e (1)2 e (1)3 e (2)1 e (2)2 e (2)3 × − (1.0) f (1)1 f (1)2 f (1)3 f (2)1 f (2)2 f (2)3 l (1 , l (1 , l (1 , l (1 , l (1 , l (1 , l (2 , l (2 , l (2 , l (2 , l (2 , l (2 , h (1 , h (1 , h (1 , h (1 , h (1 , h (1 , h (2 , h (2 , h (2 , h (2 , h (2 , h (2 , z -expansion for the B s → D s (cid:96)ν decay. Notethat these parameters are fit simultaneously with those ofTable XII, but displayed separately for clarity.Prior [ f ] Fit result [ f ] Prior [ f + ] Fit result [ f + ] a a a a c (1)0 c (1)1 c (1)2 c (1)3 c (2)0 c (2)1 c (2)2 × − (0.3) c (2)3 × − (0.3) - - d (1)0 d (1)1 d (1)2 × − (0.3) d (2)3 × − (0.3) - - d (2)0 d (2)1 × − (0.3) 0.00(30) 0.01(30) d (2)2 × − (0.3) 0.00(30) -1 × − (0.3) d (3)2 × − (0.3) - - e (1)0 e (1)1 e (1)2 × − (0.3) e (1)3 × − (0.3) - - e (2)0 e (2)1 e (2)2 × − (1.0) e (2)3 × − (1.0) - - m (1)0 m (1)1 m (1)2 × − (0.3) m (1)3 × − (0.3) - - m (2)0 m (2)1 m (2)2 m (2)3 TABLE XIV. Group II priors and fit results for the parame-ters in the modified z -expansion for the B s → K(cid:96)ν decay.Quantity Prior Fit result aE K (0 , ,
0) 0.31195(14) 0.31197(14)0.32870(17) 0.32865(17)0.35744(21) 0.35747(21)0.22861(12) 0.22862(12)0.24566(13) 0.24565(13) aE K (1 , ,
0) 0.40661(49) 0.40662(48)0.45434(73) 0.45432(70)0.47507(71) 0.47566(69)0.32020(61) 0.31986(58)0.33310(50) 0.33293(49) aE K (1 , ,
0) 0.48408(63) 0.48393(62)0.5506(11) 0.5511(11)0.57218(80) 0.57168(78)0.39192(82) 0.39240(79)0.40184(72) 0.40204(70) aE K (1 , ,
1) 0.5513(13) 0.5511(13)0.6273(35) 0.6290(34)0.6539(18) 0.6534(17)0.4528(16) 0.4527(15)0.4624(11) 0.4624(11) M + M z -expansion for the B s → D s (cid:96)ν decay.Quantity Prior Fit result aE D s (0 , ,
0) 1.18750(15) 1.18749(15)1.20126(21) 1.20132(20)1.19031(24) 1.19020(24)0.84674(12) 0.84674(12)0.84419(10) 0.84421(10) aE D s (1 , ,
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