EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-164 LHCb-PAPER-2017-016

Measurement of the shape of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ differential decay
rate

The LHCb collaboration†††Authors are listed at the end of this paper.

A measurement of the shape of the differential decay rate and the associated
Isgur-Wise function for the decay ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ is reported, using
data corresponding to $3\mbox{\,fb}^{-1}$ collected with the LHCb detector in
proton-proton collisions. The ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$(+ anything) final
states are reconstructed through the detection of a muon and a ${{\mathchar
28931\relax}^{+}_{c}}$ baryon decaying into $p{{K}^{-}}{{\pi}^{+}}$, and the
decays ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$
are used to determine contributions from ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}^{\ast+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ decays. The measured
dependence of the differential decay rate upon the squared four-momentum
transfer between the heavy baryons, ${q^{2}}$, is compared with expectations
from heavy-quark effective theory and from unquenched lattice QCD predictions.

Published in Phys. Rev. D

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

In the Standard Model (SM) of particle physics, quarks participate in a rich
pattern of flavor-changing transitions. The relevant couplings form a complex
3$\times$3 matrix, known as the Cabibbo-Kobayashi-Maskawa (CKM) matrix,
characterized by just four independent parameters [1]. A vast body of
measurements of individual CKM elements exists, and thus the overall
consistency of the SM picture of charged current interactions is highly over-
constrained. Decades of experimental work have demonstrated the impressive
consistency of experimental data with the CKM paradigm [2, 3]; nonetheless,
the motivation to probe the CKM matrix remains strong. Effects of physics
beyond the SM may be subtle, thus more precise measurements are necessary to
unveil them. Semileptonic decays of heavy-flavored hadrons are commonly used
to measure CKM parameters, as they involve only one hadronic current,
parameterized in terms of scalar functions known as form factors. The number
of form factors needed to describe a particular decay depends upon the spin of
the initial- and final-state hadrons [4, 5]. A precise calculation of these
form factors has been elusive for many years as it is not possible in
perturbative QCD. Heavy-Quark Effective Theory (HQET) provides the framework
to systematically include nonperturbative corrections in computations
involving hadrons containing heavy quarks. In particular, in the limit of
infinite heavy quark mass, all the form factors describing the semileptonic
decay of a heavy-flavored hadron are proportional to a universal function,
known as the Isgur-Wise (IW) function[6]. Lattice QCD, namely the use of
lattice formulations of QCD in large scale numerical simulations, has emerged
in recent years as a technique with well defined and systematically improvable
uncertainties which can be applied to a wide range of processes and physical
quantities[7]. Predictions from the infinite heavy-quark mass limit are useful
as a check of several Lattice QCD calculations [8].

The decay ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ is described by six
form factors corresponding to the vector and axial-vector components of the
flavor-changing charged current [9]. In HQET, ${{\mathchar
28931\relax}^{0}_{b}}$ decays are particularly simple, as the light ${u}{d}$
quark pair has total spin $j=0$, and thus the chromomagnetic corrections,
which are of the order of a few percent for $B$ mesons, are not present [10].
In the static approximation of infinite heavy quark masses, the six form
factors characterizing the baryonic semileptonic decay111The inclusion of
charge-conjugate modes is implied throughout this paper. ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ can be expressed in
terms of the elastic heavy-baryon Isgur-Wise function $\xi_{B}(w)$ [11]. The
scalar invariant $w\equiv v_{{{\mathchar 28931\relax}^{0}_{b}}}\cdot
v_{{{\mathchar 28931\relax}^{+}_{c}}}$ is related to the squared four-momentum
transfer between the heavy baryons, ${q^{2}}$, by

$w=(m_{{{\mathchar 28931\relax}^{0}_{b}}}^{2}+m_{{{\mathchar
28931\relax}^{+}_{c}}}^{2}-q^{2})/(2m_{{{\mathchar
28931\relax}^{0}_{b}}}m_{{{\mathchar 28931\relax}^{+}_{c}}}),$ (1)

where $v_{{{\mathchar 28931\relax}^{0}_{b}}}$ and $v_{{{\mathchar
28931\relax}^{+}_{c}}}$ are the four-velocities of the ${{\mathchar
28931\relax}^{0}_{b}}$ and ${{\mathchar 28931\relax}^{+}_{c}}$ baryons,
respectively, and $m_{{{\mathchar 28931\relax}^{0}_{b}}}$ and $m_{{{\mathchar
28931\relax}^{+}_{c}}}$ are the corresponding invariant masses.
Nonperturbative corrections to the static limit can be expressed in terms of
an expansion in powers of 1/$m_{c}$ and 1$/m_{b}$, where $m_{c}$ and $m_{b}$
represent the $c$\- and $b$-quark masses respectively. It has been shown in
Ref. [12] that the 1/$m_{c}$ term can be expressed in terms of $\xi_{B}(w)$
and one dimensionful constant. Moreover, partial cancellations lead to small
first-order corrections near $w=1$ [13].

In the static approximation the differential decay width of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ decay is given by

$\frac{d\Gamma}{dw}=GK(w)\xi^{2}_{B}(w),$ (2)

where the constant factor $G$ is given by

$G=\frac{2}{3}\frac{G_{F}^{2}}{(2\pi)^{3}}|V_{cb}|^{2}(m_{{{\mathchar
28931\relax}^{0}_{b}}})^{4}r^{2}~{}~{}\text{with}~{}~{}r=m_{{{\mathchar
28931\relax}^{+}_{c}}}/m_{{{\mathchar 28931\relax}^{0}_{b}}},$ (3)

where $G_{F}$ represents the Fermi coupling constant [14], $|{V_{{c}{b}}}|$ is
the magnitude of the matrix element describing the coupling of the ${c}$ quark
to the ${b}$ quark, and the kinematic factor $K(w)$ is given by

$K(w)=m_{{{\mathchar 28931\relax}^{+}_{c}}}\sqrt{w^{2}-1}\
[3w(1-2rw+r^{2})+2r(w^{2}-1)].$ (4)

The function $\xi_{B}(w)$ cannot be determined from first principles in HQET,
but calculations based on a variety of approaches exist. The kinematic limit
$w=1$ is special in HQET, as only modest corrections in the ($1/m_{b}$,
$1/m_{c})$ expansion are expected, due to the absence of hyperfine corrections
[15]. Thus it is interesting to express $\xi_{B}$ as a Taylor series expansion

$\xi_{B}(w)=1-\rho^{2}(w-1)+\frac{1}{2}\sigma^{2}(w-1)^{2}+\dots,$ (5)

where $\rho^{2}$ is the magnitude of the slope of $\xi_{B}$ and $\sigma^{2}$
is its curvature at $w=1$. Sum rules provide constraints on $\rho^{2}$ and
$\sigma^{2}$. In particular they require the slope at the zero recoil point to
be negative, and give bounds on the curvature and higher-order derivatives
[16, 17]. In addition they predict $\sigma^{2}\geq
3/5[\rho^{2}+(\rho^{2})^{2}]$ [18] and $\rho^{2}\geq 3/4$. Table 1 summarizes
theoretical predictions for $\rho^{2}$ from quenched Lattice QCD, QCD sum
rules, and a relativistic quark model.

Recently, state-of-the-art calculations of the six form factors describing the
decay ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ have been obtained
using Lattice QCD with 2 + 1 flavors of dynamical domain-wall fermions [19].
These form factors are calculated in terms of ${q^{2}}$. More details on this
formalism are given in Appendix I. The resulting theoretical uncertainty
attached to a measurement of $|{V_{{c}{b}}}|$ using this form factor
prediction is about 3.2%. The precision of this calculation makes this
approach an appealing alternative to the ones currently used, all based on
$B$-meson semileptonic decays such as ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}{}^{0}}\rightarrow{{D}^{*+}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$.
Thus it is important to examine the model’s agreement with measured quantities
such as the shape of the $d\Gamma/d{q^{2}}$ spectrum.

The experimental knowledge of ${{\mathchar 28931\relax}^{0}_{b}}$ semileptonic
decays is quite sparse, as this baryon is too heavy to be produced at the
${e^{+}}{e^{-}}\ B$-factories. The only previous experimental study of
$\xi_{B}(w)$ was performed by the DELPHI experiment at LEP, which obtained
$\rho^{2}=2.03\pm 0.46\ {\rm(stat)}^{+0.72}_{-1.00}\ {\rm(syst)}$, with an
overall uncertainty of the order of 50%[20].

In this paper we describe a determination of the shape of the $w$ or ${q^{2}}$
spectrum of the decay ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ and compare it with
functional forms related to a single form factor, inspired by HQET, and the
Lattice QCD prediction of Ref. [19]. Section 2 presents the experimental
procedure and simulated samples, while Sect. 3 describes the method employed
to reconstruct ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ candidates and to
estimate the corresponding kinematic variables $w$ and ${q^{2}}$. Section 4
describes the method adopted to isolate the signal, the unfolding procedure
used to account for experimental resolution effects, and the efficiency
corrections. The fit results for $\xi_{B}(w)$ corresponding to different
functional forms are summarized in Sect. 5. The same analysis procedure is
used in Sect. 6 to derive the shape of the differential decay width
$d\Gamma/d{q^{2}}({{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}})$ and compare with the
predictions of Ref. [19]. These data are also fitted with a single form-factor
parameterization that corresponds to the HQET infinite heavy-quark mass limit.

Table 1: Predictions for the slope at zero recoil of the baryonic Isgur-Wise function $\xi_{B}$. The evaluation from Ref. [21] includes first-order corrections in HQET. $\rho^{2}$ | Approach | Reference
---|---|---
1.35$\pm$0.13 | QCD sum rules | [22]
1.2${{}^{+0.8}_{-1.1}}$ | Lattice QCD (static approximation) | [23]
1.51 | HQET + Relativistic wave function | [21]

##  2 Experimental method

The data used in this analysis were collected with the LHCb detector at the
Large Hadron Collider at CERN and correspond to $1\mbox{\,fb}^{-1}$ of
integrated luminosity collected at a center-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$ in 2011 and $2\mbox{\,fb}^{-1}$ collected at
a center-of-mass energy of $8\mathrm{\,Te\kern-1.00006ptV}$ in 2012.

The LHCb detector [24, 25] is a single-arm forward spectrometer designed for
the study of particles containing ${b}$ or ${c}$ quarks. The detector covers
the pseudorapidity range $2<\eta<5$, where $\eta$ is defined in terms of the
polar angle $\theta$ with respect to the beam direction as
$-\ln(\tan{\theta/2})$. The detector includes a high-precision tracking system
consisting of a silicon-strip vertex detector surrounding the $pp$ interaction
region [26], a large-area silicon-strip detector located upstream of a dipole
magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three stations of
silicon-strip detectors and straw drift tubes [27] placed downstream of the
magnet. The tracking system provides a measurement of the momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at $200\mathrm{\,Ge\kern-1.00006ptV}$.222Natural units with
$c$=$\hbar$=1 are used throughout. The minimum distance of a track to a
primary vertex, the impact parameter (IP), is measured with a resolution of
$(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where $p_{\mathrm{T}}$
is the component of the momentum transverse to the beam, in
$\mathrm{\,Ge\kern-1.00006ptV}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
(RICH) [28]. Photons, electrons and hadrons are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [29]. The online event selection is performed by a trigger [30],
which consists of a hardware stage, based on information from the calorimeter
and muon systems, followed by a software stage, which applies a full event
reconstruction.

Muon candidates are first required to pass the hardware trigger that selects
muons with a transverse momentum $\mbox{$p_{\mathrm{T}}$}>1.6$
($1.8$)$\mathrm{\,Ge\kern-1.00006ptV}$ for the 2011 (2012) data taking period.
In the subsequent software trigger, events with one particle identified as a
muon are selected if at least one of the final-state particles has both
$\mbox{$p_{\mathrm{T}}$}>0.8\mathrm{\,Ge\kern-1.00006ptV}$ and IP larger than
$100{\,\upmu\mathrm{m}}$ with respect to all of the primary $pp$ interaction
vertices (PVs) in the event. In the offline selection, trigger signals are
associated with reconstructed particles. Selection requirements can therefore
be made on the trigger selection itself and on whether the decision was due to
the signal candidate, other particles produced in the pp collision, or a
combination of both. This classification of trigger selections can be used for
data-driven efficiency determination. Finally, the tracks of two or more of
the final-state particles are required to form a vertex that is significantly
displaced from the PVs.

Our study makes use of simulated semileptonic decays, where $pp$ collisions
are generated using Pythia [31, *Sjostrand:2007gs] with a specific LHCb
configuration [33]. Decays of hadronic particles are described by EvtGen [34],
in which final-state radiation is generated using Photos [35]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [36, *Agostinelli:2002hh] as described in
Ref. [38].

##  3 Event reconstruction

To isolate a sample of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ semileptonic decays,
where $X$ represents the undetected particles produced with the ${{\mathchar
28931\relax}^{+}_{c}}$ in the $c$-quark hadronization, we combine ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ candidates with tracks
identified as muons. We consider candidates where a well-identified muon
passing the hardware and software trigger algorithms with momentum greater
than $3\mathrm{\,Ge\kern-1.00006ptV}$ is found. Charmed baryon candidates are
formed from hadrons with momenta greater than $2\mathrm{\,Ge\kern-1.00006ptV}$
and transverse momenta greater than $0.3\mathrm{\,Ge\kern-1.00006ptV}$. In
addition we require that the average of the magnitudes of the transverse
momenta of the hadrons forming the ${{\mathchar 28931\relax}^{+}_{c}}$
candidate be greater than $0.7\mathrm{\,Ge\kern-1.00006ptV}$. Kaons, pions,
and protons are identified using the RICH system. Each track’s IP significance
with respect to the associated primary vertex is required to be greater than
9.333The associated primary vertex to a ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ candidate is selected
as the primary vertex which minimizes the IP significance of the ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}$ system. Moreover, the selected tracks must be
consistent with coming from a common vertex: the $\chi^{2}$ per degree of
freedom ($\chi^{2}$/DOF) of the vertex fit must be smaller than 6. In order to
ensure that the direction of the parent ${{\mathchar 28931\relax}^{0}_{b}}$ is
well measured, the ${{\mathchar 28931\relax}^{+}_{c}}$ vertex must be distinct
from the primary $pp$ interaction vertex. To this end, we require that the
flight-distance significance of the ${{\mathchar 28931\relax}^{+}_{c}}$
candidate (defined as the measured flight distance divided by its uncertainty)
with respect to the associated PV be greater than 100.

Partially reconstructed ${{\mathchar 28931\relax}^{0}_{b}}$ baryon candidates
are formed combining ${\mu^{-}}$ and ${{\mathchar 28931\relax}^{+}_{c}}$
candidates which are consistent with coming from a common vertex, and we
require that the angle between the direction of the momentum of the
${{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}$ candidate and the line from the
associated PV to the ${{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}$ vertex be
less than 45 mrad. As the ${{\mathchar 28931\relax}^{+}_{c}}$ baryon is a
${{\mathchar 28931\relax}^{0}_{b}}$ decay product with a small but significant
lifetime, we require that the difference in the component of the decay vertex
position of the charmed hadron candidate along the beam axis and that of the
beauty candidate be positive. We explicitly require that the ${{\mathchar
28931\relax}^{0}_{b}}$ hadron candidate pseudorapidity be between 2 and 5. We
measure $\eta$ using the line defined by connecting the associated PV and the
vertex formed by the ${{\mathchar 28931\relax}^{+}_{c}}$ and the ${\mu^{-}}$
lepton. Finally, the invariant mass of the ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}$ system must be between $3.3$ and
$5.3\mathrm{\,Ge\kern-1.00006ptV}$. These selection criteria ensure that the
${{\mathchar 28931\relax}^{+}_{c}}$ candidates are decay products of
${{\mathchar 28931\relax}^{0}_{b}}$ semileptonic decays. In particular the
background from directly produced ${{\mathchar 28931\relax}^{+}_{c}}$ (prompt
${{\mathchar 28931\relax}^{+}_{c}}$) is highly suppressed. This is quantified
by an unbinned extended maximum likelihood fit to the two-dimensional
$pK^{-}\pi^{+}$ invariant mass and ln(IP/mm) distributions of the ${{\mathchar
28931\relax}^{+}_{c}}$ candidates, where “/mm” refers to the length unit used
to measure the IP. The ln(IP/mm) component allows us to determine the small
prompt ${{\mathchar 28931\relax}^{+}_{c}}$ background. The parameters of the
IP distribution of the prompt sample are found by examining directly-produced
charm hadrons, as described in Ref. [39]. An empirical probability density
function (PDF) derived from simulation is used for the ${{\mathchar
28931\relax}^{+}_{c}}$ from ${{\mathchar 28931\relax}^{0}_{b}}$ component. We
find $(2.74\pm 0.02)\times 10^{6}$ ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow p{{K}^{-}}{{\pi}^{+}}$ candidates, which can
be interpreted as ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ decays, and we
determine the prompt ${{\mathchar 28931\relax}^{+}_{c}}\rightarrow
p{{K}^{-}}{{\pi}^{+}}$ fraction to be 1.5%, which can be neglected. The
corresponding fit is shown in Fig. 1.

Figure 1: (a) The ln(IP/mm) distribution and (b) $p{{K}^{-}}{{\pi}^{+}}$
invariant mass for ${{\mathchar 28931\relax}^{+}_{c}}$ candidate combinations
with a muon. The red (dashed-dotted) curves show the combinatorial
${{\mathchar 28931\relax}^{+}_{c}}$ background, the green (dashed) curves the
${{\mathchar 28931\relax}^{+}_{c}}$ from ${{\mathchar 28931\relax}^{0}_{b}}$
and the blue-solid curves the total yields.

Our goal is the study of the ground-state semileptonic decay ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$, thus we need to
estimate the contributions from $\mathchar 28931\relax_{c}^{\ast+}$ decaying
into ${{\mathchar 28931\relax}^{+}_{c}}\pi\pi$ states. Theoretical predictions
suggest that the inclusive rate ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ is dominated by the
exclusive channel ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ [40, 41]. The residual
contribution is expected to be accounted for by the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}(2595)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}(2625)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ channels. Other
modes, such as ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28934\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$, are suppressed in the
static limit and to order $1/m_{Q}$, where $m_{Q}$ represents the heavy quark
mass ($m_{c}$ or $m_{b}$) [42], with an additional stronger suppression factor
of the order $(m_{d}-m_{u})/m_{c}$ rather than $(m_{d}-m_{u})/m_{\Lambda_{\rm
QCD}}$ [9].

We use ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$
decays to infer contributions from the excited ${{\mathchar
28931\relax}^{+}_{c}}$ modes, where the ${{\mathchar 28931\relax}^{+}_{c}}$
candidates are selected as $p{{K}^{-}}{{\pi}^{+}}$ combinations whose
invariant mass is within $\pm 20\mathrm{\,Me\kern-1.00006ptV}$ of the nominal
${{\mathchar 28931\relax}^{+}_{c}}$ mass. The ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ candidates are
combined with pairs of opposite-charge pions that satisfy criteria similar to
those used to select the pions from the ${{\mathchar 28931\relax}^{+}_{c}}$
decay. The minimum transverse momentum of these pions is required to be
$0.2\mathrm{\,Ge\kern-1.00006ptV}$ and the transverse momentum of the
${{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}$ system is required
to be greater than $1.5\mathrm{\,Ge\kern-1.00006ptV}$. Lastly, the $\chi^{2}$
per degree of freedom of the vertex fit for the ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}$ system must be smaller than 6.

The resulting spectrum, measured as the mass difference
$m(p{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})-m(p{{K}^{-}}{{\pi}^{+}})$
added to the known ${{\mathchar 28931\relax}^{+}_{c}}$ mass[14], is shown in
Fig. 2. We see peaks corresponding to the $\mathchar
28931\relax_{c}(2595)^{+}$, $\mathchar 28931\relax_{c}(2625)^{+}$, $\mathchar
28931\relax_{c}(2765)^{+}$, and $\mathchar 28931\relax_{c}(2880)^{+}$
resonances. The $\mathchar 28931\relax_{c}(2595)^{+}$ is only a few MeV above
the kinematic threshold and thus it is not well described by a Breit-Wigner
function. The baseline fit for this resonance uses a PDF consisting of the sum
of two bifurcated Gaussian functions. As a check, we use an S-wave
relativistic Breit-Wigner convolved with a Gaussian function with standard
deviation $\sigma=2$ MeV that accounts for the detector resolution. While the
second parameterization is more accurate, the fits to the invariant mass
spectra in different kinematic bins are more stable with the baseline
parameterization. We fit the $\mathchar 28931\relax_{c}(2625)^{+}$ signal with
a double Gaussian PDF with shared mean, as the natural width is expected to be
well below the measured detector resolution. The shape of the combinatoric
background PDF is inferred from wrong-sign (WS) candidates, where a
${{\pi}^{+}}{{\pi}^{+}}$ or ${{\pi}^{-}}{{\pi}^{-}}$ pair is combined with
$\mathchar 28931\relax_{c}^{+}$ instead of ${{\pi}^{+}}{{\pi}^{-}}$. In
addition, we observe peaks corresponding to two higher mass resonances, with
masses and widths consistent with the $\mathchar 28931\relax_{c}(2765)^{+}$
and $\mathchar 28931\relax_{c}(2880)^{+}$ baryons[14]. In order to determine
their yields, we fit the two signal peaks with single Gaussian PDFs with
unconstrained masses and widths. The measured yields for the four ${{\mathchar
28931\relax}^{+}_{c}}$ final states, as well as the ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ final state, are
presented in Table 2.

Figure 2: The mass difference $m(p{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}})-m(p{{K}^{-}}{{\pi}^{+}})$ added to the known ${{\mathchar 28931\relax}^{+}_{c}}$ mass, $m_{\rm PDG}({{\mathchar 28931\relax}^{+}_{c}})$ [14], for candidates with $p{{K}^{-}}{{\pi}^{+}}$ invariant mass within $\pm$20 MeV of the known ${{\mathchar 28931\relax}^{+}_{c}}$ mass in candidate semileptonic decays for the entire $w$ range: data are shown as black dots, the combinatoric background is shown as a blue solid line, and the gray histogram shows the WS spectrum, obtained by combining a ${{\pi}^{+}}{{\pi}^{+}}$ or ${{\pi}^{-}}{{\pi}^{-}}$ pair with ${{\mathchar 28931\relax}^{+}_{c}}$ instead of ${{\pi}^{+}}{{\pi}^{-}}$. The signal fits are identified as follows: (a) for $m<2700$ MeV, the $\mathchar 28931\relax_{c}(2595)^{+}$ as magenta dashed line, and the $\mathchar 28931\relax_{c}(2625)^{+}$ as green long-dashed line, (b) for $m>2700$ MeV, the $\mathchar 28931\relax_{c}(2765)^{+}$ as magenta dashed line, and the $\mathchar 28931\relax_{c}(2880)^{+}$ as green long-dashed line. Table 2: Measured raw yields for the four $\mathchar 28931\relax_{c}^{\ast+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ final states and the inclusive ${{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$.  Final state | Yield
---|---
$\mathchar 28931\relax_{c}(2595)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ |  08569 $\pm$ 144
$\mathchar 28931\relax_{c}(2625)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ |  22965 $\pm$ 266
$\mathchar 28931\relax_{c}(2765)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ |  02975 $\pm$ 225
$\mathchar 28931\relax_{c}(2880)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ |  1602 $\pm$ 95
${{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ |  0(2.74$\pm$0.02)$\times 10^{6}$

The measured contributions from the two heavier $\mathchar
28931\relax_{c}^{\ast+}$ final states, shown in Table 2, are smaller than
those from ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}(2595)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}(2625)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ decays. No
theoretical prediction for nonresonant ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
exists, but we estimate systematic uncertainties due to the subtraction of
this component with an alternative fit of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
spectrum from candidate ${{\mathchar 28931\relax}^{0}_{b}}$ semileptonic
decays, where we derive both the yield and shape of the combinatoric
background from the WS sample.

The kinematical quantities ${q^{2}}$ and $w$ in the decay ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ can be calculated if
the magnitude of the ${{\mathchar 28931\relax}^{0}_{b}}$ momentum is known.
The ${{\mathchar 28931\relax}^{0}_{b}}$ flight direction can be inferred from
the primary and secondary vertex locations, and this input, combined with the
constraints from energy and momentum conservation, implies the following
relationship for $p_{{{\mathchar 28931\relax}^{0}_{b}}}$

$\displaystyle\left[\left(\frac{\hat{p}_{{{\mathchar
28931\relax}^{0}_{b}}}\cdot\vec{p}_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}}{E_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}}\right)^{2}-1\right]p_{{{\mathchar
28931\relax}^{0}_{b}}}^{2}+\left[(m_{{{\mathchar
28931\relax}^{0}_{b}}}^{2}+m_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}^{2})\frac{\hat{p}_{{{\mathchar
28931\relax}^{0}_{b}}}\cdot\vec{p}_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}}{E^{2}_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}}\right]p_{{{\mathchar 28931\relax}^{0}_{b}}}$
(6) $\displaystyle+\left[\left(\frac{(m_{{{\mathchar
28931\relax}^{0}_{b}}}^{2}+m_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}^{2})}{2E_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}}\right)^{2}-m_{{{\mathchar
28931\relax}^{0}_{b}}}^{2}\right]=0,$

where the unit vector $\hat{p}_{{{\mathchar 28931\relax}^{0}_{b}}}$ is the
direction of the ${{\mathchar 28931\relax}^{0}_{b}}$ baryon,
$\vec{p}_{{{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}}$ is the momentum of the
${{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}$ pair, $E_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}$ is the energy of the ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}$ pair, $m_{{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}}$ is the invariant mass of the ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}$ pair, $m_{{{\mathchar 28931\relax}^{0}_{b}}}$
is the nominal mass of the ${{\mathchar 28931\relax}^{0}_{b}}$ baryon, and
${{\mathchar 28931\relax}^{+}_{c}}$ identifies the $p{{K}^{-}}{{\pi}^{+}}$
combination. This is a quadratic equation, reflecting the lack of knowledge of
the neutrino orientation in the ${{\mathchar 28931\relax}^{0}_{b}}$ rest frame
with respect to the ${{\mathchar 28931\relax}^{0}_{b}}$ direction in the
laboratory. The solution corresponding to the lower value of $p_{{{\mathchar
28931\relax}^{0}_{b}}}$, which is correct between 50% and 60% of the time
depending upon the kinematics of the final state, is chosen in the ${q^{2}}$
and $w$ determination as simulation studies have shown that this choice
introduces the smallest bias. The $w$ resolution is estimated from simulated
data in different $w$ intervals. The distributions of differences between
reconstructed and generated $w$ are fitted with double-Gaussian functions and
the effective standard deviations are found to be between 0.01 and 0.05. The
overall $w$ resolution is estimated with a fit with a triple-Gaussian
function, and has an effective standard deviation $\sigma$ equal to $0.028$.

##  4 The spectral distribution ${dN_{\rm corr}/dw({{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}})}$

The ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ candidates are
separated into 14 equal-size bins of reconstructed $w$ in the full kinematic
range $1\leq w\leq 1.43$. The parameters of the PDFs describing the signal and
background components are determined from the fit to the overall
$pK^{-}{{\pi}^{+}}$ mass spectrum. The contributions from semileptonic decays
including higher-mass baryons in the final state is evaluated by fitting the
${{\mathchar 28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}$ mass spectra with
two different methods. In the first, we fit for the four resonances shown in
Fig. 2 using a PDF derived from the WS sample to model the background, and
then use the simulation to correct for efficiency. In the second, we determine
the signal yields of the $\mathchar 28931\relax_{c}^{\ast+}$ states by
subtracting the WS background and treating the residual smooth component of
the spectrum as originating from a semileptonic decay ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$. The second method
provides an estimate of the systematic uncertainty introduced by the
contribution from nonresonant ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}$ components of the hadron
spectrum, as the smooth component of this spectrum is likely to comprise also
combinatoric background.

Next, we correct the raw ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ and ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
signal yields for the corresponding software trigger efficiencies, which are
derived with a data-driven method [30], based on the determination of
${{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ events
where a positive trigger decision is provided by the signal candidates, and
events where the trigger decision is independent of the signal. Then we
subtract the raw yields reported in Table 2, scaled by the corresponding
efficiency ratios $\varepsilon({{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X)\over\varepsilon({{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X)$,
from the corrected ${{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ yields. These ratios
are derived from ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}(2595)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}(2625)^{+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ simulations. The
higher mass yields are scaled by an average of these two corrections, as no
model for these semileptonic decays is available. These corrections account
for the efficiency loss due to the reconstruction of the additional pion
pairs, as well as for the unseen ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{0}}{{\pi}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
decay and are only mildly dependent upon the invariant mass of the final
state. The expectation is that ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$
accounts for two-thirds of the inclusive dipion final state. We have checked
this prediction by studying the inclusive final states ${{\mathchar
28934\relax}^{++}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$, ${{\mathchar
28934\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$, and ${{\mathchar
28934\relax}^{0}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$. Taking into account
the difference in the $\mathchar
28931\relax_{c}^{+}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$
and $\mathchar
28931\relax_{c}^{+}{{\pi}^{0}}{\mu^{-}}{{\overline{\nu}}_{\mu}}X$ detection
efficiencies, estimated with simulations, we measure the ratio $R=N(\mathchar
28931\relax_{c}^{+}{{\pi}^{+}}{{\pi}^{-}})/N(\mathchar
28931\relax_{c}^{+}{{\pi}^{+}}{{\pi}^{-}}+\mathchar
28931\relax_{c}^{+}{{\pi}^{0}}{{\pi}^{0}})$ with

$R=\frac{N({{\mathchar 28934\relax}^{++}_{c}})+N({{\mathchar
28934\relax}^{0}_{c}})}{N({{\mathchar 28934\relax}^{++}_{c}})+N({{\mathchar
28934\relax}^{0}_{c}})+N({{\mathchar
28934\relax}^{+}_{c}})\cdot[\varepsilon(\mathchar
28931\relax_{c}^{+}{{\pi}^{+}}{{\pi}^{-}}\mu)/\varepsilon(\mathchar
28931\relax_{c}^{+}{{\pi}^{0}}\mu)]},$ (7)

where $N({{\mathchar 28934\relax}^{++}_{c}})$ and $N({{\mathchar
28934\relax}^{0}_{c}})$ are the detected yields for the final states
${{\mathchar 28934\relax}^{++}_{c}}{{\pi}^{-}}\mu\nu$ and ${{\mathchar
28934\relax}^{0}_{c}}{{\pi}^{+}}\mu\nu$, $N({{\mathchar
28934\relax}^{+}_{c}})$ is the detected yield for the final state ${{\mathchar
28934\relax}^{+}_{c}}\mu\nu X$ and $\varepsilon(\mathchar
28931\relax_{c}^{+}{{\pi}^{+}}{{\pi}^{-}}\mu)/\varepsilon(\mathchar
28931\relax_{c}^{+}{{\pi}^{0}}\mu)$ is the ratio between the reconstruction
efficiencies of these final states calculated with simulation. A simulation
study gives $\varepsilon(\mathchar
28931\relax_{c}^{+}{{\pi}^{+}}{{\pi}^{-}}\mu)/\varepsilon(\mathchar
28931\relax_{c}^{+}{{\pi}^{0}}\mu)=25.9\pm 2.7$, where the uncertainty
reflects the limited sample size of the simulation. We obtain $R=0.63\pm
0.14$, where the statistical uncertainty is due to limited ${{\pi}^{0}}$
reconstruction efficiency, consistent with the expectation $R=2/3$, and a
negligible ${{\mathchar
28934\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ component in the
denominator of Eq. 7.

The ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ spectrum $dN_{\rm
meas}/dw$ is then unfolded to account for the detector resolution and other
$w$ smearing effects such as the possible choice of the wrong solution of Eq.
6. The procedure adopted is based on the single value decomposition (SVD)
method [43] using the RooUnfold package[44]. We choose to divide the unfolded
spectrum $dN_{\rm u}/dw$ into seven $w$ bins, to be consistent with the
recommendation of Ref. [45] to divide the measured spectrum into a number of
bins at least twice as many as the ones in the corrected spectrum. The SVD
method includes a regularization procedure that depends upon a parameter
$k$[43], ranging between unity and the number of degrees of freedom, in our
case 14. Simulation studies demonstrate that $k=4$ is optimal in our case.
Variations associated with different choices of $k$ have been studied and are
included in the systematic uncertainties. We have performed closure tests with
different simulation models of the ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ dynamics, and verified
that this unfolding procedure does not bias the reconstructed distribution.
The spectra before and after unfolding are shown in Fig. 3. Finally, using
simulated samples of signal events, we correct the unfolded spectrum for
$w$-dependent acceptance and selection efficiency to obtain the distribution
$dN_{\rm corr}/dw$. Various kinematic distributions have been studied in
simulation and data and we find that they are all in good agreement.

Figure 3:  The spectra (a) $dN_{\rm meas}/dw$ before unfolding and (b)
$dN_{\rm u}/dw$ after unfolding, for the decay ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$. The latter spectrum
is then corrected for acceptance and reconstruction efficiency and fitted to
the IW function $\xi_{B}(w)$ with the procedure discussed in the text.

##  5 The shape of $\xi_{B}(w)$ for ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ decays

In order to determine the shape of the Isgur-Wise function $\xi_{B}(w)$, we
use the square root of $dN_{\rm corr}/dw$ divided by the kinematic factor
$K$($\langle w\rangle$), defined in Eq. 4, evaluated at the midpoint in the
seven unfolded $w$ bins. We derive the IW shape with a $\chi^{2}$ fit, where
the $\chi^{2}$ is formed using the full covariance matrix of $dN_{\rm
corr}/dw$.

We use various functional forms to extract the slope, $\rho^{2}$, and
curvature, $\sigma^{2}$, of $\xi_{B}(w)$. The first functional form is
motivated by the 1/$N_{c}$ expansion [46], where $N_{c}$ represents the number
of colors, and has an exponential shape parameterized as

$\xi_{B}(w)=\exp[-\rho^{2}(w-1)].$ (8)

The second functional form, the so called “dipole” IW function, which is more
consistent with sum-rule bounds [17], is given by

$\xi_{B}(w)=\left(\frac{2}{w+1}\right)^{2\rho^{2}}.$ (9)

Finally, we can use a simple Taylor series expansion of the Isgur-Wise
function and fit for the slope and curvature parameters using the Taylor
series expansion introduced in Eq. 5. Figure 4 shows the measured $\xi_{B}(w)$
and the fit results with this parameterization. Table 3 summarizes the slope
and curvature at zero recoil obtained with the three fit models. Note that the
curvature is an independent parameter only in the last fit, while in the first
two models it is related to the second derivative of the IW function.

Figure 4: (a) The Isgur-Wise function fit for the decay ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}\nu$
with a Taylor series expansion in $(w-1)$ up to second order. The black dots
show the data and the solid (blue) line shows the fitted function with the
second-order Taylor series expansion model. The vertical scale is in arbitrary
units. (b) The correlation between slope $\rho^{2}$ and curvature
$\sigma^{2}/2$: the three ellipses correspond to the 1$\sigma$, 2$\sigma$, and
4$\sigma$ contours.

Table 3: Summary of the values for the slope and curvature of the Isgur-Wise function with different parameterizations. The quoted uncertainties are statistical only. The models marked with “*” have only the slope at zero recoil as a free parameter, thus the curvature is derived from the fitted $\rho^{2}$. Shape | $\rho^{2}$ | $\sigma^{2}$ | correlation coefficient |  $\chi^{2}$/ DOF
---|---|---|---|---
Exponential* | 1.65 $\pm$ 0.03 | 2.72 $\pm$ 0.10 | 100% | 5.3/5
Dipole* | 1.82 $\pm$ 0.03 | 4.22 $\pm$ 0.12 | 100% | 5.3/5
Taylor series | 1.63 $\pm$ 0.07 | 2.16 $\pm$ 0.34 | 97% | 4.5/4

As the slope of the IW function is the most relevant quantity to determine
$|{V_{{c}{b}}}|$ in the framework of HQET [13], we focus our studies on the
systematic uncertainties on this parameter. We consider several sources of
systematic uncertainties, which are listed in Table 4. The first two are
determined by changing the fit models for ${{\mathchar 28931\relax}^{+}_{c}}$
and $\mathchar 28931\relax_{c}(2595)^{+}$ and $\mathchar
28931\relax_{c}(2625)^{+}$ signal shapes in the corresponding candidate mass
spectra. The software trigger efficiency uncertainty is estimated by using an
alternative procedure to evaluate this efficiency using the trigger emulation
in the LHCb simulation. In order to assess systematics associated with the bin
size, we perform the analysis with different binning choices. The sensitivity
to the ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ form factor modeling
is assessed by reweighting the simulated $w$ spectra to correspond to
different $\xi_{B}$ functions with slopes ranging from 1.5 to 1.7. The “phase
space averaging” sensitivity is estimated by comparing the fit to the
expression for $dN_{\rm corr}/dw$ with the fit to $1/K(\langle
w\rangle)\sqrt{dN_{\rm corr}/dw}$. The uncertainty associated with the
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\mathchar
28931\relax_{c}^{\ast+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ modeling is
evaluated by changing the relative fraction of ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}$ versus ${{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{0}}{{\pi}^{0}}$ of the $\mathchar
28931\relax_{c}^{\ast+}$ spectrum by $\pm$ 20%. Finally, we use the
alternative evaluation of the fraction of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{+}}{{\pi}^{-}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$
which includes the maximum possible nonresonant component to assess the
sensitivity to residual $\mathchar 28931\relax_{c}^{\ast+}$ components in the
subtracted spectrum. The total systematic uncertainty in $\rho^{2}$ is 0.08.

The value of $\rho^{2}$ obtained from the Taylor series expansion is

$\rho^{2}=1.63\pm 0.07\pm 0.08,$

which is consistent with Lattice calculations [23], QCD sum rules [22], and
relativistic quark model [21] expectations. The measured slope is compatible
with theoretical predictions and with the bound $\rho^{2}\geq$ 3/4[16]. The
measured curvature $\sigma^{2}$ is compatible within uncertainties with the
lower bound $\sigma^{2}\geq 3/5[\rho^{2}+(\rho^{2})^{2}]$ [18].

Table 4: Summary of the systematic uncertainties on the slope parameter $\rho^{2}$. The total uncertainty is obtained by adding the individual components in quadrature. Source | $\sigma(\rho^{2})$
---|---
Signal fit for ${{\mathchar 28931\relax}^{+}_{c}}$ | $0.02$
Signal PDF for $\mathchar 28931\relax_{c}^{\ast+}$ | $0.02$
Software trigger efficiency | $0.02$
$w$ binning | $0.03$
SVD unfolding regularization | $0.03$
Phase space averaging | 0.03
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ modeling | $0.03$
${{\mathchar 28931\relax}^{0}_{b}}\rightarrow\mathchar 28931\relax_{c}^{\ast+}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ modeling | $0.03$
Additional components of the semileptonic spectrum | $0.02$
${{\mathchar 28931\relax}^{0}_{b}}$ kinematic dependencies | 0.02
Total | $0.08$

##  6 Comparison with unquenched lattice predictions

The lattice QCD calculation in Ref. [19] uses a helicity-based description of
the six form factors governing ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{\mathchar 28931\relax}$ transitions
introduced in Ref. [47]. The calculation uses state-of-the-art techniques
encompassing the entire ${q^{2}}$ region. The stated uncertainties on the
predicted width are therefore larger than what is expected in a high-${q^{2}}$
region, but remain rather small, namely 6.3%. This corresponds to a 3.2%
theoretical uncertainty on $|{V_{{c}{b}}}|$, thus raising the prospect of an
additional precise independent determination of $|{V_{{c}{b}}}|$.

The simplest check on this theoretical prediction consists of a comparison of
the predicted shape of $d\Gamma/d{q^{2}}$ and the measured data. Thus we
measure the distribution $dN_{\rm corr}/d{q^{2}}$ with the same procedure
adopted to derive $dN_{\rm corr}/dw$, including efficiency corrections and the
unfolding procedure, with the same number of bins used to determine the raw
and unfolded yields. We produce seven corrected yields and their associated
covariance matrix, where the nondiagonal terms are related to the unfolding
procedure. We then perform a $\chi^{2}$ fit to the seven experimental $dN_{\rm
corr}/d{q^{2}}$ data points using the theoretical functional shape given in
Eq. 85 of Ref. [19], which also provides the nominal values of the form factor
parameters, and thus we leave only the relative normalization floating. This
fit uses a covariance matrix that combines experimental and theoretical
uncertainties, which yields a $\chi^{2}$ equal to 1.32 for 6 degrees of
freedom, and a corresponding p-value of 97%. This shows that the predicted
shape is in good agreement with our measurement.

The form factor decomposition in Ref. [19] does not allow a straightforward
extrapolation to the HQET limit of infinite heavy-quark masses. However, we
know that in the static limit all the form factors are proportional to a
single universal function. In order to assess how well our data are consistent
with the static limit, we perform a second $\chi^{2}$ fit assuming that all
the form factors are proportional to a single $z$-expansion function [48].
Fits with different pole masses used in the six form factors determined in
Ref. [19] are performed. The overall shape does not change appreciably; the
pole mass of $6.768\mathrm{\,Ge\kern-1.00006ptV}$ is preferred. The two fit
parameters are the coefficients $a_{0}$ and $a_{1}$, giving the strength of
the first two terms in the $z$-expansion. The resulting fitted shape is shown
in Fig. 5. This fit has a $\chi^{2}$ equal to 1.85 for 5 degrees of freedom,
with a corresponding p-value of 87%. Note that the shape obtained with a
single form-factor is very similar to the one predicted in Ref. [19]. This is
consistent with the HQET prediction [15] that the shape of the differential
distribution is well described by the static approximation, modulo a scale
correction of the order of 10%, reflecting higher-order contributions. Further
details of this fit and the fit using the Lattice QCD calculation can be found
in the Appendix.

Figure 5: Comparison between the fit to the seven experimental data points
using either the Lattice QCD calculation of Ref. [19], shown as grey points
with a shaded area corresponding to the binned 1$\sigma$ theory uncertainty,
or a single form factor fit in the $z$-expansion, shown as the solid blue
curve. The data points, modulo a scale factor, are shown as black points with
error bars.

##  7 Conclusions

A precise measurement of the shape of the Isgur-Wise function describing the
semileptonic decay ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}}$ has been performed.
The measured slope is consistent with theoretical models and the bound
$\rho^{2}\geq 3/4$ [16]. The measured curvature $\sigma^{2}$ is consistent
with the lower-bound constraint $\sigma^{2}\geq 3/5[\rho^{2}+(\rho^{2})^{2}]$
[18]. The shape of $d\Gamma/d{q^{2}}$ is studied and found to be well
described by the unquenched lattice QCD prediction of Ref. [19], as well as by
a single form-factor parameterization. Further studies with a suitable
normalization channel will lead to a precise independent determination of the
CKM parameter $|{V_{{c}{b}}}|$.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania);
MinES and FASO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU
(Ukraine); STFC (United Kingdom); NSF (USA). We acknowledge the computing
resources that are provided by CERN, IN2P3 (France), KIT and DESY (Germany),
INFN (Italy), SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom),
RRCKI and Yandex LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF
(Brazil), PL-GRID (Poland) and OSC (USA). We are indebted to the communities
behind the multiple open source software packages on which we depend.
Individual groups or members have received support from AvH Foundation
(Germany), EPLANET, Marie Skłodowska-Curie Actions and ERC (European Union),
Conseil Général de Haute-Savoie, Labex ENIGMASS and OCEVU, Région Auvergne
(France), RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain),
Herchel Smith Fund, The Royal Society, Royal Commission for the Exhibition of
1851 and the Leverhulme Trust (United Kingdom).

## Appendix I: Analytical expression for $d\Gamma/d{q^{2}}$

This Appendix describes the formalism used in the $d\Gamma/d{q^{2}}$ fits. In
particular, we give the expression of $d\Gamma/d{q^{2}}$ in terms of the form
factor basis chosen in Ref. [19], the so-called “helicity form factors”. In
addition, we show the corresponding expression used to model the static limit.

Table 5: Masses of the relevant form factor poles in the physical limit (in GeV). $f$ |  | $J^{P}$ |  | $m_{\rm pole}^{f}({{\mathchar 28931\relax}^{0}_{b}}\rightarrow\Lambda_{c})~{}[\mathrm{\,Ge\kern-1.00006ptV}]$ |  |
---|---|---|---|---|---|---
$f_{+}$, $f_{\perp}$ |  | $1^{-}$ |  | 6.332 |  |
$f_{0}$ |  | $0^{+}$ |  | 6.725 |  |
$g_{+}$, $g_{\perp}$ |  | $1^{+}$ |  | 6.768 |  |
$g_{0}$ |  | $0^{-}$ |  | 6.276 |  |

The Lattice QCD calculations reported in Ref. [19] predict the differential
decay width $d\Gamma({{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}})/d{q^{2}}$ as follows

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}|{V_{{c}{b}}}|^{2}\sqrt{s_{+}s_{-}}}{768\pi^{3}m_{{{\mathchar
28931\relax}^{0}_{b}}}^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}$ (10)
$\displaystyle\times\Bigg{\\{}4\left(m_{\ell}^{2}+2q^{2}\right)\left(s_{+}\left[g_{\perp}\right({q^{2}})]^{2}+s_{-}\left[f_{\perp}({q^{2}})\right]^{2}\right)$
$\displaystyle\hskip
8.61108pt+2\frac{m_{\ell}^{2}+2q^{2}}{q^{2}}\left(s_{+}\left[\left(m_{{{\mathchar
28931\relax}^{0}_{b}}}-m_{X}\right)g_{+}({q^{2}})\right]^{2}+s_{-}\left[\left(m_{{{\mathchar
28931\relax}^{0}_{b}}}+m_{X}\right)f_{+}({q^{2}})\right]^{2}\right)$
$\displaystyle\hskip
8.61108pt+\frac{6m_{\ell}^{2}}{q^{2}}\left(s_{+}\left[\left(m_{{{\mathchar
28931\relax}^{0}_{b}}}-m_{X}\right)f_{0}({q^{2}})\right]^{2}+s_{-}\left[\left(m_{{{\mathchar
28931\relax}^{0}_{b}}}+m_{X}\right)g_{0}({q^{2}})\right]^{2}\right)\Bigg{\\}},$

where $g_{\perp}$, $f_{\perp}$, $g_{+}$, $f_{+}$, $g_{0}$, and $f_{0}$
represent the six form factors necessary to describe this decay,
$X\equiv\Lambda_{c}$ denotes the final-state baryon, $m_{\ell}$ represents the
mass of the muon, ${q^{2}}$ is the squared four-momentum transfer between the
heavy baryons, and

$s_{\pm}=(m_{{{\mathchar 28931\relax}^{0}_{b}}}\pm m_{X})^{2}-q^{2}.$ (11)

The six form factors are cast in terms of the $z$-expansion [48] up to first
order, and have the functional form

$f({q^{2}})=\frac{1}{1-{q^{2}}/(m^{f}_{\rm
pole})^{2}}\times\left[a_{0}^{f}+a_{1}^{f}z^{f}({q^{2}})\right],$ (12)

where $z^{f}({q^{2}})$ is given by

$\displaystyle z^{f}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{t^{f}_{+}-q^{2}}-\sqrt{t_{+}^{f}-t_{0}}}{\sqrt{t_{+}^{f}-q^{2}}+\sqrt{t_{+}^{f}-t_{0}}},$
(13) $\displaystyle t_{0}$ $\displaystyle=$ $\displaystyle(m_{{{\mathchar
28931\relax}^{0}_{b}}}-m_{X})^{2},$ (14)

and $t_{+}^{f}$ is given by

$t_{+}^{f}=(m^{f}_{\rm pole})^{2},$ (15)

and the pole masses used in the calculations are shown in Table 5. The
parameters $a_{0}^{f}$ and $a_{1}^{f}$ for the six form factors describing
this decay are given in Table VIII of Ref. [19].

In the static limit all the helicity form factors are proportional to a single
universal function. Thus, we use a common $z$-expansion parameterization

$\displaystyle\frac{\mathrm{d}\Gamma}{\mathrm{d}q^{2}}$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}|{V_{{c}{b}}}|^{2}\sqrt{s_{+}s_{-}}}{768\pi^{3}m_{{{\mathchar
28931\relax}^{0}_{b}}}^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}g_{\perp}^{2}({q^{2}})$
(16)
$\displaystyle\times\Bigg{\\{}4\left(m_{\ell}^{2}+2q^{2}\right)\left(s_{+}+s_{-}\right)$
$\displaystyle\hskip 8.61108pt+\frac{4}{q^{2}}\left[s_{+}\left(m_{{{\mathchar
28931\relax}^{0}_{b}}}-m_{X}\right)^{2}+s_{-}\left(m_{{{\mathchar
28931\relax}^{0}_{b}}}+m_{X}\right)^{2}\right]\left[2m_{\ell}^{2}+q^{2}\right]\Bigg{\\}},$

where the choice of $g_{\perp}$ reflects the choice of the pole mass used in
the single $z$-expansion fit given in Sect. 6. We have performed the fits with
various choices of pole masses and examined the effects on the shape
$d\Gamma/d{q^{2}}$ and found the shape did not vary significantly, though it
was found that the parameters defining $g_{\perp}$ yielded the optimal fit. In
this case, the fit parameters are the coefficients $a_{0}$ and $a_{1}$ in the
$z$-expansion parameterization of $g_{\perp}({q^{2}})$, which has the form
shown in Eq. 12.

## Appendix II: Measured normalized spectra $dN_{\rm corr}/d{q^{2}}$ and
associated covariance matrix

In this appendix we report the seven measured data points $dN_{\rm
corr}/d{q^{2}}$ and the corresponding covariance matrix, shown in Table 6 and
Table 7 respectively.

Table 6: Measured normalized yields $dN_{\rm corr}({{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}})/d{q^{2}}$ .  ${q^{2}}$ [GeV2] | $dN_{\rm corr}/d{q^{2}}$
---|---
0.80 |  1.50 $\pm$ 0.10
2.38 |  1.80 $\pm$ 0.10
3.97 |  2.04 $\pm$ 0.10
5.56 |  2.23 $\pm$ 0.08
7.14 |  2.35 $\pm$ 0.07
8.73 |  2.28 $\pm$ 0.05
10.32 |  1.50 $\pm$ 0.04
Table 7: Covariance matrix of the measured normalized yields ${\rm Cov}[dN_{\rm corr}({{\mathchar 28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}{\mu^{-}}{{\overline{\nu}}_{\mu}})/d{q^{2}}$] .  ${q^{2}}$ [GeV2] | $dN_{\rm corr}/d{q^{2}}$
---|---
0.80 | 0.0103 | 0.0052 | -0.0032 | -0.0033 | -0.0009 | 0.0004 | 0.0005
2.38 | 0.0052 | 0.0100 | 0.0044 | 0.0011 | -0.0002 | -0.0006 | 0.0002
3.97 | -0.0032 | 0.0044 | 0.0090 | 0.0048 | 0.0004 | -0.0013 | -0.0007
5.56 | -0.0035 | -0.0011 | 0.0048 | 0.0070 | 0.0031 | -0.0006 | -0.0013
7.14 | -0.0009 | -0.0019 | -0.0004 | 0.0031 | 0.0044 | 0.0015 | 0.0006
8.73 | 0.0004 | -0.0006 | -0.0013 | -0.0006 | 0.0015 | 0.0023 | 0.0013
10.32 | 0.0005 | 0.0002 | -0.0007 | -0.0013 | -0.0006 | 0.0013 | 0.0018

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