LLattice B → D ( ∗ ) form factors, R ( D ( ∗ ) ) , and | V cb | Andrew Lytle ∗ INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma RM, ItalyE-mail: [email protected]
I discuss recent progress in lattice calculations of B → D ( ∗ ) (cid:96) ν form factors, important for theprecision determination of | V cb | in the Standard Model (SM), and for testing SM expectations oflepton flavor universality in observables R ( D ( ∗ ) ) . I also discuss progress in calculations of therelated b → c semileptonic decays B s → D ( ∗ ) s (cid:96) ν and B c → J / ψ (cid:96) ν now experimentally accessibleat the LHC. The 37th Annual International Symposium on Lattice Field Theory - LATTICE201916-22 June, 2019Wuhan, China. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] A p r → D ( ∗ ) form factors Andrew Lytle
1. Introduction
The B -meson semileptonic decays B → D ( ∗ ) l ν provide a precise way to determine the CKMmatrix element | V cb | . In addition to experimental data, these determinations require the precisioncalculation of nonperturbative form factors using lattice QCD. There is a long-standing discrepancybetween the values obtained from these exclusive determinations | V cb | excl , and those obtained frominclusive determinations | V cb | incl – this is known as the V cb puzzle [1]. A recent comparison ofinclusive and exclusive determinations of | V cb | from the Flavor Lattice Averaging Group (FLAG)is presented in Fig. 1. There are also long-standing few-sigma discrepancies with Standard Model (SM) predictionsin the measured ‘ R -ratios’ for these decays. The R -ratio for a semileptonic decay is defined as thebranching fraction for that decay into the tau channel divided by that for the muon or electron, R ( D ( ∗ ) ) = B ( B → D ( ∗ ) τν τ ) B ( B → D ( ∗ ) l ν l ) l = µ , e . (1.1)These ratios are independent of | V cb | , but depend on the nonperturbative form factors over theentire kinematic range. Recently a measurement from LHCb found that R ( B c → J / ψ ) also differssignificantly from its SM expectation [2]. A recent synopsis of the situation for R ( D ( ∗ ) ) from theHeavy Flavor Averaging Group (HFLAV) is reproduced in Fig. 2.At the same time, a great deal of new experimental information is expected to become availablein the near future, both at Belle II [3] and from the LHC [4]. This will lead to increasingly preciseexperimental information for B → D ( ∗ ) , and information about newly accessible decays at LHC inchannels B s → D ( ∗ ) s [5] and B c → J / ψ [2], as well as in baryonic channels [6, 7], and increasinglyprecise R -ratio determinations (see Fig. 2). Keeping pace with these advances is an importantchallenge for the lattice community. Therefore now is a good time to take stock of lattice efforts inthese directions, and this forms the main goal of the present article.In the next section I briefly review the theory of semileptonic meson decays relevant for the di-rect determination of | V cb | . The main component of this article is Sec. 3 that attempts to summarisethe current status and works in progress on the lattice. Much of this material is new/preliminaryand was first presented at this conference. Finally in Sec. 4 I will conclude with a short summaryand some considerations for the future.
2. Theory
The Standard Model parameter | V cb | can be extracted precisely using the semileptonic decayprocesses B → D ( ∗ ) l ν l . In these transitions the initial state b quark is converted to a c quark bythe weak interaction current, with an accompanying factor of V cb in the amplitude. In the StandardModel then the differential partial widths for these decays are represented as follows: d Γ dw ( B → D ) = ( known ) | V cb | ( w − ) / | G ( w ) | (2.1) d Γ dw ( B → D ∗ ) = ( known ) | V cb | ( w − ) / χ ( w ) | F ( w ) | (2.2) A review talk summarising the status of the full CKM matrix was given at this conference by Steve Gottlieb [8]. → D ( ∗ ) form factors Andrew Lytle
36 38 40 42 44|V cb | × 10 | V u b | × B→τν Λ b → p ℓ ν Λ b → Λ c ℓ ν B→πℓν ℓ→D(ℓ→D * ) BGL inclusive 36 38 40 42 44|V cb | × 10 | V u b | × B→τν Λ b → p ℓ ν Λ b → Λ c ℓ ν B→πℓν ℓ→D(ℓ→D * ) CLN inclusive
Figure 1: Figures from the 2019 FLAG review [9] showing the present status of | V cb | using variousexclusive channels compared with inclusive determinations. The vertical yellow band is the resultfrom B → D ∗ l ν decays using either the BGL (left) or CLN (right) parameterisations to fit the data.expressed here in terms of the kinematic variable w , w = v B · v D ( ∗ ) = M B + M D ( ∗ ) − q M B M D ( ∗ ) (2.3)Alternatively the kinematic variable q is often used, where q is the four-momentum transfer be-tween initial and final state mesons. In terms of these variables q = D ( ∗ ) meson in the B rest frame, while w = D ( ∗ ) at rest in the B rest frame, or q = q = ( M B − M ( ∗ ) D ) .In these expressions the non-perturbative QCD dynamics are contained in the functions F ( w ) and G ( w ) . In order to determine | V cb | from the experimental data involving these decays, thesefunctions need to be computed. The functions F ( w ) and G ( w ) can in turn be expressed in termsof a number of form factors, which are related to the following QCD matrix elements: (cid:104) D | V µ | B (cid:105)√ m B m D = ( v B + v D ) µ h + ( w ) + ( v B − v D ) µ h − ( w ) (2.4) (cid:104) D ∗ α | V µ | B (cid:105)√ m B m D ∗ = ε µνρσ v ν B v ρ D ∗ ε ∗ σα h V ( w ) (2.5) (cid:104) D ∗ α | A µ | B (cid:105)√ m B m D ∗ = i ε ∗ να (cid:2) h A ( w )( + w ) g µν − ( h A ( w ) v µ B + h A ( w ) v µ D ∗ ) v ν B (cid:3) (2.6)These matrix elements can be computed from first principles using the methods of latticeQCD, and from them the form factors determined. In Sec. 3 I will review the state-of-the-art inthese calculations, as well as for the related decays involving a b → c transition but with a strange2 → D ( ∗ ) form factors Andrew Lytle − ]0.0010.010.1 A b s o l u t e σ R ( X ) LHCb X = D ∗ , τ − → µ − ¯ ν µ ν τ X = D ∗ , τ − → π − π + π − ν τ X = J/ψ , τ − → µ − ¯ ν µ ν τ R(D) R ( D * ) Average of SM predictions = 1.0 contours cD – R(D) = 0.299 0.005 – R(D*) = 0.258
World Average – – R(D) = 0.340 0.008 – – R(D*) = 0.295 = -0.38 r ) = 27% c P( HFLAV
Spring 2019 s LHCb15LHCb18Belle17 Belle19 Belle15BaBar12
Average
HFLAV
Spring 2019
Figure 2: (Left) Projections for the expected uncertainty achievable at LHCb for ratios R ( D ( ∗ ) ) and R ( J / ψ ) , reproduced from [4]. (Right) Summmary of experimental status of R ( D ( ∗ ) ) measurementscompiled by HFLAV [10].or charm spectator chark. The formalism described above is analogous for these decays, but theform factors will differ.In the expressions (2.1), there is a kinematic suppression factor of ( w − ) raised to either the3/2 or 1/2 power. This results in the experimental rates being damped near w =
1, however, as willbe discussed in Sec. 3, most available lattice QCD results are limited to the region w ≈
1. This isdue to the fact that the lattice results are more precise here, their signal decays at larger recoil. Inaddition, the expressions for the rates (2.1) simplify at the zero-recoil point, so that only a singleform factor, h A ( ) contributes. Therefore the most precise determinations of | V cb | to date havefocused on precison lattice calculations of h A ( ) , combined with experimental data in B → D ∗ ,over a range of w which is then extrapolated to w = | V cb | using these extrapolations of the experimental data. In this regard, differentmethods are now being utilised by both experimental and lattice collaborations. While the Caprini-Lellouch-Neubert (CLN) [11] uses an expansion based on heavy quark effective theory valid to O ( / m b , c ) , the Boyd-Grinstein-Lebed (BGL) [12] is a model independent parameterisation basedon analyticity and unitarity. The CLN approach has the advantage of relying on few parameters,but this restrictiveness may introduce model dependence particularly once a precision beyond thelevel of approximation is reached [13, 14]. The BGL approach is model independent and as a resultrelies on more parameters, and must be truncated at some order.There have been several studies examining the model dependence from different parameterisa-tions in B → D [13] and B → D ∗ [15, 16, 17] decays [18, 19]. This progress was largely facilitatedby experimental datasets with q and angular distributions being made publicly available, includingfull error budgets and correlations [20, 23]. The situation was summarised recently by FLAG [9],reproduced in Fig. 1, showing their best-fits for | V cb | utilising CLN and BGL parameterisations andcompared with the inclusive determination.In order to match experimental data with theory, it is also extremely important for the latticecommunity to make predictions away from the zero recoil point [1, 18]. Interestingly, in the caseof B → D , the picture appears somewhat more congruent than for B → D ∗ , and here form factors3 → D ( ∗ ) form factors Andrew Lytle
36 38 40 42 44 46 𝖭 𝖿 = + 𝖭 𝖿 = 𝗇 𝗈 𝗇 − 𝗅 𝖺 𝗍𝗍 . HFLAV inclusive𝘉→𝘋ℓ𝘉→(𝘋,𝘋 * )ℓ (CLN)𝘉→(𝘋,𝘋 * )ℓ (BGL)𝘉→𝘋ℓ𝘉→𝘋 * ℓ (CLN)𝘉→𝘋 * ℓ (BGL) |𝖵 𝖼→ |𝗑𝟣𝟢 Figure 3: (Left) Comparison of B → D semileptonic form factors as extracted from experiment(blue, purple) and lattice QCD calculations (green), as summarised in the 2019 FLAG review [9].(Right) Summary of different | V cb | extractions from FLAG varying the included decay channelsand parameterisations used.are available over a large kinematic range both from experiment [22, 23] and lattice [24, 25].This is summarised in Fig. 3. Although the final extraction of | V cb | excl from this mode is lessprecise, it is also in reasonably good agreement with the inclusive determination. With the expectedimprovements from experiment and the lattice community, as more information becomes available,one imagines that the picture from B → D ∗ will become more clear.It is also interesting to determine the related B s → D ( ∗ ) s form factors, which differ only in thesubstitution of the light spectator quark for strange. These decays were recently used by LHCb tomeasure | V cb | [5]. On the lattice, these calculations should be considerably less computationallyexpensive due to the reduced cost of strange inversions as compared to light, and also more sta-tistically precise. Therefore it provides both an interesting laboratory in which to test the effectof different parameterizations, as well as make more precision checks between competing latticedeterminations. If there are systematic effects impacting a particular calculation of B → D ∗ , theseshould show up even more clearly in B s → D ∗ s . Therefore the channels B s → D ( ∗ ) s should be theo-retically pursued.As will be discussed in Sec. 3, there are now several efforts being undertaken by differentgroups to extend these calculations beyond zero recoil, as well as explore other b → c decay modesso that the lattice can be ready for the Belle II and new LHC eras.
3. Lattice QCD results
Here I will briefly summarise the present status of lattice QCD calculations for the group ofsemileptonic decays B ( s ) → D ( ∗ )( s ) l ν and B c → J / ψ l ν , as well as planned efforts in these directionsfocusing on preliminary results presented in this conference.One of the main features that distinguishes amongst these calculations is the choice for thetreatment of the b quark in the simulation. Because am b is not small for the lattice spacings usedin many modern simulations, including b in the simulation on the same footing as the other quarkswould lead to uncontrollable lattice discretisation errors ∼ ( am b ) n . Note that the same considera-4 → D ( ∗ ) form factors Andrew Lytle m π (GeV ) h A ( ) a ≅ ≅ ≅ ≅ ≅ M π / GeV h A ( ) Very CoarseCoarseFine Fit result D ∗ s Phys
Figure 4: Comparison of chiral-continuum extrapolations for the B → D ∗ form factor h A at thezero-recoil point, computed by the FNAL-MILC [27] (left) and HPQCD [31] (right) collaborations.The cusp near the physical m π comes from expectations of chiral perturbation theory.tions also hold for the charm quark, though it is increasingly common to include it relativisticallyas lattice spacings decrease. Therefore a strategy such as an effective theory framework must beadopted to incorporate the b quark.Alternatively, at sufficiently small lattice spacings, it is possible to work at masses m h ap-proaching m b such that am h (cid:46)
1. By working at several mass values approaching m b , the latticedata can be extrapolated to a physical prediction at m b . In what follows I will refer to this as a‘relativistic- b ’ approach to distinguish it from the effective treatments of b . This method benefitsfrom the use of improved actions which formally improve the form of leading heavy mass discreti-sation effects, as used in the ‘heavy-HISQ’ [26] results shown later. A feature of the relativistic- b approach is that it gives not only predictions for the physical decay of a B meson, but also resultsfor masses of the heavy quark between m c and m b . B → D ∗ There are two recent published lattice calculations of the B → D ∗ l ν decay, both at the zero-recoil point, where only the single form factor h A contributes. One is from the Fermilab/MILCcollaboration, calculated on n f = + (cid:104) D ∗ | ¯ c γ j γ b | B (cid:105)(cid:104) B | ¯ b γ j γ c | D ∗ (cid:105)(cid:104) D ∗ | ¯ c γ c | D ∗ (cid:105)(cid:104) B | ¯ b γ b | B (cid:105) = | h A ( ) | , (3.1)to cancel systematic and statistical errors.The other is from HPQCD collaboration on n f = + + b -quark [31]. The chiral extrapolations for the quantity h A ( ) arecompared in Fig. 4. The two groups find compatible results, quoting h A ( ) = . ( )( ) and5 → D ( ∗ ) form factors Andrew Lytle w h A Extrapolation a =0 . fm a =0 . fm a =0 . fm a =0 . fm P r e li m i n a r y Blinded w h V Extrapolation a =0 . fm a =0 . fm a =0 . fm a =0 . fm P r e li m i n a r y Blinded w −1.2−1.0−0.8−0.6−0.4−0.20.00.20.4 h A Extrapolation a =0 . fm a =0 . fm a =0 . fm a =0 . fm P r e li m i n a r y Blinded w h A Extrapolation a =0 . fm a =0 . fm a =0 . fm a =0 . fm P r e li m i n a r y Blinded w d Γ / d w × G e V P r e li m i n a r y Blinded
Best fitLattice × V cb Belle untagged e − Belle untagged µ − Belle taggedBaBar synthetic −1.0 −0.5 0.0 0.5 1.0 cosθ l d Γ / d c o s θ l × G e V P r e li m i n a r y Blinded −1.0 −0.5 0.0 0.5 1.0 cosθ v d Γ / d c o s θ v × G e V P r e li m i n a r y Blinded χ d Γ / d χ × G e V P r e li m i n a r y Blinded
Figure 5: B → D ∗ form factors from the MILC collaboration. The figures on the left show the latticeform factors at different values of the lattice spacing and the extrapolated results, the figures on theright show the kinematic distributions derived from a combination fit of lattice and experimentaldata compared with lattice data and data from Belle and Babar. See also results in [32, 34, 35] Figs.courtesy A. Vaquero.0.895(10)(24) for FNAL/MILC and HPQCD respectively. The dominant error in [31] arises frommissing O ( α s ) matching of NRQCD currents to QCD.The MILC collaboration is extending their calculation to the the full set of form factors, awayfrom zero recoil. Preliminary results in the range w ∈ [ , . ] were presented in [32], and an updatewas presented at this conference [33] showing global fits and comparison with available experi-mental data, as shown in Fig. 5.The JLQCD collaboration have presented their preliminary results for B → D ( ∗ ) form factorsin [36] for a range w ∈ [ , . ] and at two lattice spacings, and an update of these results werepresented at this conference by Kaneko [37], with extended range in w ∈ [ , . ] and includingresults at a finer lattice spacing of a − = . b ’ approachon Möbius domain wall fermions, for which observables are calculated over a range of heavy quarkmasses keeping am h < .
8, ( m h up to 3.05 m c ), with an extrapolation required to the physical m b .Their results are shown in Fig. 6 for h A , , and h V . The extrapolated results for h A ( ) agree wellwith FNAL/MILC and HPQCD [27, 31].The LANL/SWME collaboration have also released preliminary results for the h A form factorat zero recoil [38, 39], and at this conference [40]. Their calculation is carried out on the n f = + + a ≈ . , .
09 fm and a single pion mass m π ≈
310 MeV.The OK action is improved at a higher order in λ c , b ∼ Λ QCD m c , b than the Fermilab action being usedby the MILC collaboration – This is important to reduce the charm quark discretisation error, thedominant (1%) error in [27], to below the percent level [43]. Preliminary results for h A ( ) areshown in Fig. 7. 6 → D ( ∗ ) form factors Andrew Lytle w h A ( w ) m Q ~ 1.25 m c m Q ~ 1.56 m c m Q ~ 1.95 m c m Q ~ 2.44 m c m Q ~ 3.05 m c Fermilab/MILC ’14HPQCD ’17 M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV 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a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV M π ~ 300 MeV a -1 ~ 4.5 GeV w -1.00.0 h A ( w ) w h A ( w ) w h V ( w ) m b ~ 1.25 m c m b ~ 1.56 m c m b ~ 1.25 m c m b ~ 1.56 m c m b ~ 1.95 m c m b ~ 2.44 m c a -1 ~ 3.6 GeV, M π ~ 300 MeV a -1 ~ 2.5 GeV, M π ~ 300 MeV Figure 6: B → D ∗ form factors from the JLQCD collaboration, using a ‘relativistic- b ’ approach.New results on a finer a − ≈ h A ( ) the comparison with previousresults from [27, 31] are also shown. Figs. courtesy Takashi Kaneko. B → D The FNAL/MILC and HPQCD collaborations have both computed B → D form factors atzero and non-zero recoil on n f = + a ∼ .
045 fm, the latter using HISQ (rel-ativistic) c and NRQCD b [25] at two lattice spacings of a ∼ .
09, 0.12 fm. A comparison of theseresults reproduced from [9] is shown in Fig. 3 along with experimental data [22, 23]. Their re-sults are in good agreement, although HPQCD has larger errors coming mainly from discretizationeffects and NRQCD matching uncertainties, similar to the situation for B → D ∗ .In contrast to the present situation with B → D ∗ , here the form factors from lattice are availableover an extended range in q . After the new lattice data beyond zero-recoil became available aswell as new experimental data from Belle, a careful analysis [13] of the available data and differentform factor parameterisations found a value for | V cb | = . ( ) − , this value being between,and compatible with, both the inclusive determination and the exclusive value from B → D ∗ . Itis clear from this study the importance of having lattice data away from zero recoil, as well ascarefully assessing parameterisation dependence. The lattice and experimental data for the formfactors are shown together in Fig. 3.Preliminary results for B → D form factors from JLQCD collaboration were presented in [36],7 → D ( ∗ ) form factors Andrew Lytle . . .
95 0 0.2 0.4 0.6 0.8 1.0a12m3100 . . .
95 0 0.2 0.4 0.6 0.8 1.0a12m310 | h A ( ) / ρ A j | m x /m s O ( λ )-improved O ( λ )-improvedUnimproved | h A ( ) / ρ A j | m x /m s . . .
95 0 0 .
03 0 .
06 0 .
09 0 . . . .
95 0 0 .
03 0 .
06 0 .
09 0 . M π ≈ | h A ( ) | a [fm] O ( λ )-improved, M π ≈ O ( λ )-improved, M π ≈ M π ≈ | h A ( ) | a [fm]FNAL/MILC ’14HPQCD ’18HPQCD ’18, Figure 7: Results for B → D ∗ at zero recoil from LANL/SWME [39] showing the effect of higherorders of current improvement (left) and compared with prior results in the literature (right) – Notethat ρ A is here set to 1 so the comparison is only indicative. w R ( w ) Belle un+tagged + BGL (Gambino et al. ’19)
Belle tagged + CLN (Bernlochner et al. ’17)
HQET + QCDSR w R ( w ) Belle un+tagged + BGL (Gambino et al. ’19)
Belle tagged + CLN (Bernlochner et al. ’17)
HQET + QCDSR
Figure 8: Comparison of lattice form factor data from JLQCD (symbols) with fits to experimentalBelle data [20, 21] using different parameterisations (colored bands) [19, 1] and predictions ofHQET. For more discussion see [37]. Figs. courtesy Takashi Kaneko.with an update presented at this conference [37] including lighter pion masses, and a third latticespacing with a − ∼ . m h ∼ . m c . RBC/UKQCD also presented [44] preliminary results on n f = + a ≈ . , .
09 fm) and two pion masses ( m π ≈ h + / − ( w ) form factors are shown in Fig. 10. B s → D ∗ s There are two determinations of the B s → D ∗ s zero-recoil form factor h sA ( ) , both from theHPQCD collaboration using n f = + + b -quark. The calculation of [31] uses an NRQCD b -quark on relatively coarser ensembles,while [45] uses the relativistic ‘heavy-HISQ’ approach on fine ensembles down to a ∼ .
45 fm.The main systematic uncertainty in the NRQCD calculation comes from the perturbative currentmatching known to O ( α s ) , this error is absent from the heavy-HISQ calculation where the current8 → D ( ∗ ) form factors Andrew Lytle q [GeV ] f B → D + q C1M1M2M3F1 P R E L I M I N A R Y q [GeV ] f B s → D s f B s → D s + ( K + ,K )=(3 , f f + P R E L I M I N A R Y Figure 9: Preliminary results from the RBC/UKQCD collaboration [44]. (Left) Data for f + ( q ) for B → D decay. The colors correspond to the different gauge ensembles ( a / L / m π ), and differentshapes correspond with different input "charm" masses used to interpolate/extrapolate to physicalcharm. (Right) The f + / ( q ) form factors for B s → D s with fits to a CLN parameterization. . . . . . . . . . . . h + ( w ) wV µ : λ -improved , Z V : blinded a m a m a m a m − . . . . . . . . . . h − ( w ) wV µ : λ -improved , Z V : blinded a m a m a m a m Figure 10: Preliminary results for B → D h + ( w ) (left) and h − ( w ) (right) form factors fromLANL/SWME [40], at two lattice spacings ( a ≈ . , .
09 fm) and two pion masses ( m π ≈ h sA ( ) = . ( ) stat ( ) sys (3.2) h sA ( ) = . ( ) stat ( ) sys (3.3)It is also interesting to note that in [31] the ratio of zero-recoil form factor with light/strangespectator was calculated to be h A ( ) / h sA ( ) = . ( ) stat ( ) sys . In this ratio the main system-atic from the current matching largely cancels. B s → D s There have been a few calculations of the B s → D s form factors, using different methodologies.The MILC collaboration determined f ( q ) and f + ( q ) using n f = + n f = → D ( ∗ ) form factors Andrew Lytle q [GeV ]0 . . . . . . f s ( q ) a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am physl , am h = 0 . a (cid:39) . am physl , am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . a (cid:39) . am h = 0 . q [GeV ]0 . . . . . . f s + ( q ) Figure 11: Figure from [51] showing results for B s → D s f / + form factors using the ‘heavy-HISQ’approach. The colored open symbols show raw data for form factors calculated at unphysicallylight b -quark masses on a range of ensembles with lattice spacings from a ∼ .
09 – 0.045 fm. Thecontinuum extrapolated results for the form factors at physical b -quark mass are given by the graybands.in which they also determined the ratio of the tensor form factor to f + near zero recoil. TheRBC/UKQCD collaboration presented preliminary results in [49, 50] and these were updated atthis conference [44]. The preliminary results for their form factors are shown in Fig. 9.Recently the f / + form factors were computed over the entire kinematic range using the heavy-HISQ approach by the HPQCD collaboration [51]. The raw data at unphysically light b mass andthe form factors extrapolated to the b mass are shown in Fig. 11. HPQCD also determined bothform factors using NRQCD b in [52]; the results from both calculations are shown in Fig 12.Until recently the lattice QCD results for B s → D ( ∗ ) s form factors could not be compared withexperiment, that changed with the LHCb measurement [5], resulting in a new determination of | V cb | based on B s decays. Their analysis was performed using both BGL and CLN parameterizations,and the extracted | V cb | is compatible between the two within errors. Their result is compatiblewith both inclusive and exclusive determinations from B decays, but with larger errors. Theseencouraging results increase the urgency for B s → D ∗ s results away from zero recoil and increasingprecision in both channels B s → D ( ∗ ) s . B c → J / ψ There are currently only preliminary results available for the B c → J / ψ l ν lattice form fac-tors [55, 56, 57], by HPQCD using the ‘heavy-HISQ’ approach. The R -ratio for this decay wasmeasured by LHCb [2], who found R ( J / ψ ) = . ( ) stat ( ) syst . This value is ∼ σ larger thanwhat is expected in the SM and although this value has large uncertainties it is desirable to have alattice determination, particularly as the experimental precision improves (see Fig. 2). A prelimi-nary value R ( J / ψ ) = . ( ) was given in [55], and Fig. 13 shows the differential decay widthas a function of q . 10 → D ( ∗ ) form factors Andrew Lytle q [GeV ]0 . . . . . . . NRQCD [1703.09728]this work f s + ( q ) f s ( q ) 3 4 5 6 7 8 9 10 M η h [GeV]0 . . . . . . . S (1) /h A (1) HQET S (1) /V (1) HQET h A (1) /V (1) HQET M η c M η b S s (1) /h sA (1) S s (1) /V s (1) h sA (1) /V s (1) M η h [GeV]0 . . . . . . . M η c M η b f s + ( q max ) f s ( q max ) f s , + (0) q [GeV ]0 . . . . . . ( / | V c b | ) d / d q ⇥ [ G e V ] B s ! D s µ⌫ µ B s ! D s ⌧⌫ ⌧ Figure 12: (Top-left) HPQCD’s B s → D s form factors f / + from [51] using the ‘heavy-HISQ’approach compared with results from NRQCD [52]. (Top-right) Results for ratios of zero-recoilform factors as a function of M η h used as a physical proxy for the heavy quark mass m h used insimulation, compared with predictions of HQET [18]. (Bottom-left) Evolution of the form factorendpoints f / + ( , q ) as a function of M η h . (Bottom-right) Partial differential decays widths for B s → D s l ν l for l = µ , τ .
4. Conclusions
I would hazard that the study of b → c transitions is at somewhat of a crossroads. There areseveral long-standing puzzles in this sector where experimental data and theoretical predictionsdo not quite square, and there are a number of welcome developments on the horizon that willbe essential to a precise understanding that can either confirm or rule out these discrepancies.Among these are improved predictions from the lattice community over a larger kinematic rangethan has heretofore been available, and results in new channels B s → D ∗ s and B c → J / ψ , and alsoin the baryon sector, that can match experimental breakthroughs from LHC. With the imminentresults from Belle II, B → D ∗ will surely remain the gold standard for extractions of | V cb | , and itis therefore a challenge for the lattice community to put these calculations on a solid footing, andin particular away from zero recoil. As reviewed briefly above, fortunately there are several latticecollaborations that have embarked upon this endeavour.
5. Acknowledgements
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