A test of heavy quark effective theory using a four-dimensional angular analysis of $\overline{B} \rightarrow D^\ast \ell^- \overline{\nu}_\ell$

An angular analysis of the decay ${\overline{B} \rightarrow D^\ast \ell^- \overline{\nu}_\ell}$, $\ell\in\{e,\mu\}$, is reported using the full $e^+e^-$ collision dataset collected by the BaBar experiment at the $\Upsilon(4S)$ resonance. One $B$ meson from the ${\Upsilon(4S)\to B\overline{B}}$ decay is fully reconstructed in a hadronic decay mode, which constrains the kinematics and provides a determination of the neutrino momentum vector. The kinematics of the semi-leptonic decay is described by the di-lepton mass squared, $q^2$, and three angles. The first unbinned fit to the full four-dimensional decay rate in the Standard Model is performed in the so-called BGL approach, which employs a generic $q^2$ parameterization of the underlying form factors based on crossing symmetry, analyticity and QCD dispersion relations for the amplitudes. The form factor shapes show deviations from previous fits based on the widely used CLN parameterisation, signaling possible deviations from heavy quark effective theory expectations. The latest form factors also provide an updated prediction for the branching fraction ratio $\mathcal{R}(D^\ast)\equiv \mathcal{B}(\overline{B}\to D^\ast \tau^- \bar{\nu}_\tau)/\mathcal{B}({\overline{B} \rightarrow D^\ast \ell^- \overline{\nu}_\ell})$ as $0.253 \pm 0.005$. Finally, using the well measured branching fraction for the ${\overline{B} \rightarrow D^\ast \ell^- \overline{\nu}_\ell}$ decay, a value of $|V_{cb}|=(38.36\pm 0.90)\times10^{-3}$ is obtained that is consistent with the current world average for exclusive ${\overline{B}\to D^{(\ast)}\ell^- \overline{\nu}_\ell}$ decays and remains in tension with the determination from inclusive semi-leptonic $B$ decays to final states with charm.

An angular analysis of the decay B → D * − ν , ∈ {e, µ}, is reported using the full e + e − collision dataset collected by the BABAR experiment at the Υ(4S) resonance.One B meson from the Υ(4S) → BB decay is fully reconstructed in a hadronic decay mode, which constrains the kinematics and provides a determination of the neutrino momentum vector.The kinematics of the semi-leptonic decay is described by the di-lepton mass squared, q 2 , and three angles.The first unbinned fit to the full four-dimensional decay rate in the Standard Model is performed in the so-called BGL approach, which employs a generic q 2 parameterization of the underlying form factors based on crossing symmetry, analyticity and QCD dispersion relations for the amplitudes.The form factor shapes show deviations from previous fits based on the widely used CLN parameterisation, signaling possible deviations from heavy quark effective theory expectations.The latest form factors also provide an updated prediction for the branching fraction ratio R(D variables to fully parametrize the final state.For the analysis presented in this Letter, we adopt the customary choice [8] of the di-lepton invariant mass squared, q 2 , the helicity angles of the D and − , θ V and θ , respectively, and the angle χ between the hadronic and leptonic 2-body decay planes.Denoting dΩ = dcos θ dcos θ V dχ, the four-dimensional differential rate assuming massless leptons in the SM is [8] dΓ where k = (m momentum in the B rest frame and η EW = 1.0066 [6,9] denotes leading electro-weak corrections.In the SM, the helicity amplitudes H ±,0 are the real functions expressed here in terms of the conventional axial-vector and vector form factors, {A 1 , A 2 , V }, as in Caprini et al. (CLN) [10].In the Boyd et al. (BGL) [11] approach, the form factors are written as f = (m B + m D * )A 1 , The BGL formalism parameterizes the i th form factor, F i , in the most generic form, based on crossing symmetry, analyticity and QCD dispersion relations, as The expansion parameter z is given by and is small in the physical region.Here t ≡ q 2 , t ± ≡ (m B ± m D * ) 2 and t 0 = t + − t + (t + − t − ).We adopt the Blaschke factors, P i (z), corresponding to removal of the B c poles of the BD * system, and the outer functions, φ i (z), from Refs.[3,12].The BGL coefficients in Eq. 4 satisfy the relations n |a i n | 2 ≤ 1, known as unitarity constraints.The CLN [10] formalism makes similar expansions up to cubic terms, but imposes heavy-quark symmetry relations and QCD sum rules to relate the expansion parameters.The theoretical uncertainties in the CLN relations have typically been ignored in the form used to report measurements, leading to internal inconsistencies [13].
In this Letter, employing a data sample of 471 × 10

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BB pairs [14] produced at the Υ(4S) resonance and collected by the BABAR detector [15,16], a full 4-dimensional analysis of the B → D * − ν decay rate corresponding to Eq. 1 is reported.One of the B mesons, referred to as the tag-side B, is fully reconstructed via hadronic decays, allowing for the missing neutrino 4-momentum, p miss , to be explicitly reconstructed on the signal-side B, since the initial e ± 4-momenta are known.The hadronic tagging algorithm uses charm-meson seeds (D ( * ) , J/ψ ) combined with ancillary charmless light hadrons (π/K), and is the same as in several previous BABAR analyses [15,17,18].From the remaining particles in the event after the tag-B reconstruction, a D 0 meson reconstructed via one its three cleanest decay modes,  * ) and the missing neutrino.The χ 2 -probability from this highly constrained fit is used as the main discriminant against background.To reject candidates with additional neutral energy deposits, E extra is defined as the sum of the energies of the good quality photons not utilized in the event reconstruction.
The variable E extra is required to be less than 0.4 GeV to 0.6 GeV, depending on the D ( * ) modes.In contrast to analyses of charmless semileptonic decays, backgrounds from continuum q q annihilation events that are typically more jetty compared to BB production, are found to be negligible.Therefore, no additional requirement is placed on event shape variables.Only candidates satisfying q 2 ∈ [0.2, 10.2] GeV 2 are retained.In events with multiple selected candidates, only the candidate with the highest χ 2 -probability from the kinematic fit is retained.
After all selections, the overall background level is estimated to be ∼ 2%, using a simulation of generic Υ(4S) → BB events, where both B-mesons decay to any allowed final state.All selected events enter the 4dimensional angular fit; the small remnant background is treated as a source of systematic uncertainty.Figure 1a shows the comparison between data and simulation in the variable U = E miss − | p miss |, where the resolution in the neutrino reconstruction has been weighted in the signal part of this simulation to match that in the data.Here E miss and p miss correspond to the missing neutrino energy and momentum, respectively.Figure 1b shows the comparison in the discriminating variable E extra .The efficiency in E extra in the E extra → 0 signal region does not affect the angular analysis, so that an exact agreement is not required.The generic BB simulation agrees with the data in all kinematic-variable distributions in the sideband regions, validating its use to estimate the background in the signal region.The final requirement is |U | ≤ 90 MeV.The total number of selected candidates at this stage is 6112, with the estimated signal yield being around 5932.
In addition to the generic BB simulation sample used for the data analysis where both B-mesons are decayed generically, a separate category of BB simulation is employed where the B tag is decayed generically, but B sig → D * (→ Dπ) − ν is decayed uniformly in dq 2 dΩ at the generator level.This latter sample is used to correct for detector acceptance effects in the fit to Eq. 1.The simulation undergoes the same reconstruction and selection steps as the data sample.Unbinned maximum-likelihood fits to the the BABAR data in the four-dimensional decay rate given by Eq. 1 are performed in two variants, both employing BGL expansions of the form factors.For the nominal BABAR-only variant, the negative log likelihood (NLL) is of the nonextended type, implying that the overall normalization factor is not imposed.This fit is used to extract the three form factors in a fashion insulated from systematic uncertainties related to the normalization, in particular with the estimation of the B tag yield.To extract |V cb |, a second version of the fit is performed, where the integrated rate Γ is converted to a branching fraction, B, as Γ = B/τ B , where τ B is the B-meson lifetime.The latest HFLAV [19] values of B and τ B , for B 0 and B − mesons, are employed as additional Gaussian constraints to the BABAR-only NLL, and the entire fit is repeated.Second, at the zero-recoil point, the relation is used to express a F 1 0 in terms of the remaining BGL coefficients in f and F 1 .Therefore, a F 1 0 is not a free parameter in the fit, but is derived from the remaining parameters.The small isospin dependence of these constraints, arising from the differences m B + − m B 0 and m D * 0 − m D * + , is ignored in the calculation.
BGL expansion coefficients beyond the linear terms are essentially unconstrained by our data and allowing them to vary in the fit produces no statistically significant effect on the form factor shapes, but results in violations of the unitarity constraints.Therefore, the BGL expansion fit is performed with N = 1.The background subtraction is performed using a background component estimated from the generic BB simulation sample.To ensure that a global minimum for the NLL is reached, 1000 instances of the fits are executed, with uniform sampling on [-1,+1] for the starting values of the a n coefficients.Among convergent fits, a unique minimum NLL is always found, up to small variations in the least significant digits in the fit parameters.
Many sources of systematic uncertainties cancel in this analysis, since no normalization is required from the BABAR data sample.
Tracking efficiences in simulation show no significant dependence on q 2 or {cos θ , cos θ V , χ}.To account for the resolutions in the reconstructed kinematic variables, the normalization of the probablity density function in the fit is performed using reconstructed variables from the simulation.The dominant systematic uncertainty comes from the remnant background that can pollute the angular distributions.To estimate its effect on the fit results, the fit procedure is repeated excluding the background subtraction and the difference in the results is taken as the systematic uncertainty.For the fit using the HFLAV branching fractions, the uncertainties in those branching fractions are taken from HFLAV [19].I.
Figure 2 shows the comparisons with the CLN world average (CLN-WA) [19] as well as light cone sum rules (LCSR) at the maximum recoil from Ref. [21].Phenomenologically, the most important feature in Fig. 2 is the discrepancy between CLN-WA and BGL at the zerorecoil limit, where HQET is expected to hold.Numerically, the p-value of the consistency between the CLN-WA and BABAR BGL results, computed near the zerorecoil point, is 0.0013.The BGL formalism explicitly avoids placing any HQET-based connections between the form factors.The difference could point to non-negligible corrections that are of higher order in {α s , Λ/m b,c } [3].While experimental tests of the validity of HQET-based form factors have been carried out elsewhere [22], the ratio among the helicity amplitudes obtainable from tagged B → D * − ν is a more unambiguous and clean way to probe HQET.For |V cb |, the result obtained here is well below the value determined from inclusive decays.This is in contrast with those from several recent analyses using the BGL parameterization based on unpublished Belle data [3,[5][6][7]23], where larger values, close to the inclusive result, were typically obtained.None of these latter analyses involved a full four-dimensional fit.Instead, they employed fits to products of the four onedimensional distributions.For the three differential angular rates, information on the q 2 shapes of the form factors is integrated over and thus diluted.On the other hand, dΓ/dq 2 ) does not contain the information to separate out the three helicity amplitudes and the underlying form factors, being a mixture of them.The problem is strictly four-dimensional in nature with the correlation terms being crucial for extracting the form factors in a unique fashion.Additionally, Ref. [23] was an untagged analysis such that the final-state kinematics was not fully reconstructable.
Figure 3 shows the 2-dimensional scatter plots in cos θ V and χ in three bins of cos θ and integrated over the q 2 spectrum, between the data (top row) and simulation (bottom row) after acceptance and reconstruction effects, weighted by the results of the BGL fit.Within uncertainties, the weighted simulation consistently matches the data.The differential rate in Eq. 1 holds under the assumption that the outgoing charged lepton is massless, a valid approxmimation for ∈ {e, µ}.For the τ lepton, additional terms appear in the differential rate, Γ(q 2 , m ), depending on the lepton mass [3].The BGL form factors reported in this Letter lead to an updated prediction for where = {e, µ}.An N = 1 BGL expansion for the additional scalar form factor is performed following Gambino et al.
[3], using the HQET prediction at zero recoil, with a conservative estimate for the uncertainty.At maximum recoil, instead of employing the LCSR form factors [21] with large uncertainties that were adopted in Ref.
[3], the present BABAR result is employed.These values at the two ends of the q 2 spectra completely specify the scalar form factor in the linear expansion.The resultant SM prediction is For a different choice of t 0 = t − , a value 0.253 ± 0.005 is found, consistent with the above.The result is consistent with the CLN-based calculation of 0.252 ± 0.003 in Ref. [24], although with a larger uncertainty, mostly driven by the uncertainty in the scalar form factor from HQET.The degree of HQET violation is an important consideration, impacting the uncertainties, although the central value of R(D We are grateful for the excellent luminosity and machine conditions provided by our PEP-II colleagues, and for the substantial dedicated effort from the computing organizations that support BABAR.The collaborating institutions wish to thank SLAC for its support and kind hospitality. * ) ≡ B(B → D * τ − ντ )/B(B → D * − ν ) as 0.253 ± 0.005.Finally, using the well measured branching fraction for the B → D * − ν decay, a value of |V cb | = (38.36± 0.90) × 10 −3 is obtained that is consistent with the current world average for exclusive B → D ( * ) − ν decays and remains in tension with the determination from inclusive semi-leptonic B decays to final states with charm.PACS numbers: 13.20.-v,13.20.He, 12.15.Hh, 12.15.-yThe Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2], V CKM , describing quark flavor mixing due to the charged weak current, is one of pillars of the Standard Model (SM) of particle physics.It contains the only source of charge-parity (CP ) violation in the SM.Validating this picture requires precise determinations of the CKM matrix elements |V ub | and |V cb |.These are measured by the tree-level semi-leptonic decays, b → {u, c} − ν , where refers to an electron or muon.The hadronization of the final-state {u, c} quark can be probed via inclusive or exclusive final states, the theoretical treatment being quite different for the two processes.For the heavy-to-heavy b → c transition, the inclusive and exclusive procedures use an operator product expansion and form factors based on heavy quark effective theory (HQET), respectively [3].The theoretical and experimental uncertainties are different in the two cases, and a long-standing tension of about 3σ exists between them, with the inclusive results systematically higher than the exclusive ones, for both |V ub | and |V cb |.The different results from inclusive and exclusive measurements could arise from non-SM physics.This motivates better quantification of uncertainties in the measurements and underlying theoretical treatment of strong interaction effects.For exclusive |V cb | from B → D * − ν [4], based on unpublished Belle results [5], several authors [3, 6, 7] have recently pointed out that removal of input from HQET in the theoretical parameterization of the underlying B → D * form factors can play a role in reducing the tension.The measurement described here is a test of this suggestion.The B → D * − ν process, with the subsequent D * → Dπ decay, requires four independent kinematic is combined with a π 0 or π + , to form a D * 0 or D * + , respectively.For each D * candidate, the reconstructed invariant mass of the D 0 and the difference of the reconstructed masses, ∆m ≡ (m D * − m D ), are required to be within four standard deviations of the expected resolution from their nominal values, at this stage.The D * is combined with a charged lepton ∈ {e, µ}, with the laboratory momentum of the lepton required to be greater than 0.2 GeV and 0.3 GeV for e and µ, respectively.The six D * decay modes along with the two charged lepton species comprise twelve signal channels that are processed as independent data samples.No additional tracks are allowed in the event.The entire event topology, e + e − → Υ(4S) → B tag B sig (→ D * − ν ) is considered in a kinematic fit including constraints on the beam-spot, relevant secondary decay vertices and masses of the reconstructed B tag , B sig , D FIG. 1. Comparisons between data and generic BB simulation in the discriminating variables (a) U and (b) E extra .For each plot, selections in all other variables have been applied.

FIG. 3 .
FIG. 3. Comparisons as binned scatter plots between the BABAR data (top row) and simulation weighted by the BGL fit result (bottom row) in (a) backward, (b) mid and (c) forward angles in cos θ .The multi-dimensional features in the data are well-represented by the model.The z-axes indicate the number of events in each bin and the simulation is normalized to the number of data events.The binned χ 2 differences between the data and weighted simulation are (a) 103, (b) 89 and (c) 96, evaluated over 100 bins.

TABLE I .
Table I summarizes the main results from the BGL The N = 1 BGL expansion results of this analysis, including systematic uncertainties.
) might be sensitive to variations in the BGL form factors since the overall efficiency calculation for the measurement is a convolution of the form factor model and the four-dimensional detector acceptance function.In summary, using the BABAR BB data sample with one of the B mesons fully reconstructed in hadronic modes, an unbinned four-dimensional fit to tagged B → D * − ν decays is performed to extract the form factors in the more model-indepedent formalism of BGL.The BGL form factors show differences with those obtained in the CLN formalism, including at zero-recoil, where higher order corrections to HQET are expected to be minimal.The value of |V cb | is found to be lower than those obtained in recent analyses based on unpublished Belle data [3, 5-7, 23] that did not use a four-dimensional fit.The tension with inclusive determinations of |V cb | persists, even with the more model-independent BGL parameterization of the form factors.The central value of the SM R(D * ) prediction obtained in this work is consistent with the CLN based prediction, but with a larger uncertainty due removal of HQET constraints, thereby reducing the overall tension with the latest average of experimental measurements.An extended version of the results presented here, including unfolded four-dimensional angular moments will be presented in a forthcoming publication.
* ) is largely unaffected.It is important to note that the experimental measurement of R(D *