EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2020-026 LHCb-PAPER-2019-046

Measurement of the shape of the $B_{s}^{0}\rightarrow
D_{s}^{*-}\mu^{+}\nu_{\mu}$ differential decay rate

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

The shape of the $B_{s}^{0}\rightarrow D_{s}^{*-}\mu^{+}\nu_{\mu}$
differential decay rate is obtained as a function of the hadron recoil
parameter using proton-proton collision data at a centre-of-mass energy of 13
TeV, corresponding to an integrated luminosity of 1.7$\text{\,fb}^{-1}$
collected by the LHCb detector. The $B_{s}^{0}\rightarrow
D_{s}^{*-}\mu^{+}\nu_{\mu}$ decay is reconstructed through the decays
$D_{s}^{*-}\rightarrow D_{s}^{-}\gamma$ and $D_{s}^{-}\rightarrow
K^{-}K^{+}\pi^{-}$. The differential decay rate is fitted with the Caprini-
Lellouch-Neubert (CLN) and Boyd-Grinstein-Lebed (BGL) parametrisations of the
form factors, and the relevant quantities for both are extracted.

Published in JHEP 12 (2020) 144

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

##  1 Introduction

Semileptonic decays of heavy hadrons are commonly used to measure the
parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [1, 2], as they
involve only one hadronic current that can be parametrised 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 [3, 4, 5]. For the decay of a pseudoscalar $B$ meson to a vector
${D}^{*}$ meson, four form factors are required. The determination of the CKM
matrix element $|V_{cb}|$ using $B\rightarrow{D}^{(*)}\ell{{\nu}_{\ell}}$
decays or via the inclusive sum of all hadronic $B\rightarrow
X_{c}\ell\nu_{\ell}$ decay channels has been giving inconsistent results
during the last thirty years [6]. The exclusive determination relies heavily
on the parametrisation of the form factors, as it requires an extrapolation of
the differential decay rate to the zero recoil point, where the momentum
transfer to the lepton system is maximum.

Recently, the LHCb collaboration has measured $|V_{cb}|$ using
${{B}^{0}_{s}}\rightarrow{D}_{s}^{(*)-}{\mu^{+}}\nu_{\mu}$ decays111The
inclusion of charge-conjugate processes is implied throughout this paper. with
two form-factor parametrisations, giving consistent results [7]. The
determination of the form factors in
${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}{\ell^{+}}{{\nu}_{\ell}}$ decays
obtained using different parametrisations can help to clarify the $|V_{cb}|$
inconsistency between the exclusive and inclusive approaches. It can also be
used to improve the Standard Model (SM) predictions of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\tau^{+}}{{\nu}_{\tau}}$ branching
fraction and the ratio
$\mathcal{R}(D_{s}^{*})={\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\tau^{+}}{{\nu}_{\tau}})/{\mathcal{B}}({{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}})$.
A measurement and precise prediction of the latter could increase the
understanding of the current tension between experimental and theoretical
values of the equivalent ratio
$\mathcal{R}({D}^{(*)})={\mathcal{B}}({B}\rightarrow{D}^{(*)}{\tau^{+}}{{\nu}_{\tau}})/{\mathcal{B}}({B}\rightarrow{D}^{(*)}{\mu^{+}}{{\nu}_{\mu}})$
[6]. Theoretical predictions on ${B}^{0}_{s}$ semileptonic decays are expected
to be more precise than those on ${B}^{0}$ or ${{B}^{+}}$ decays. For example,
the Lattice QCD calculations of the form factors are computationally easier
due to the larger mass of the spectator $s$ quark compared to that of $u$ or
$d$ quarks [8, 9]. Despite these advantages, the study of semileptonic
${B}^{0}_{s}$ decays has received less theoretical attention than the
equivalent ${B}^{0}$ and ${B}^{+}$ decays due to the lack of experimental
results.

This paper reports the first measurement of the shape of the differential
decay rate of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay as a
function of the hadronic recoil parameter $w=v_{{{B}^{0}_{s}}}\cdot
v_{{{D}^{*-}_{s}}}$, where $v_{{{B}^{0}_{s}}}$ and $v_{{{D}^{*-}_{s}}}$ are
the four-vector velocities of the ${B}^{0}_{s}$ and ${D}^{*-}_{s}$ mesons,
respectively. The spectrum of $w$ is unfolded accounting for the detector
resolution on $w$ and corrected for the reconstruction and selection
efficiency. The ${D}^{*-}_{s}$ meson is reconstructed in the
$D_{s}^{*-}\rightarrow D_{s}^{-}\gamma$ mode, where the $D_{s}^{-}$ meson
subsequently decays via the
$D_{s}^{-}\rightarrow\phi(\rightarrow{{K}^{+}}{{K}^{-}})\pi^{-}$ or
$D_{s}^{-}\rightarrow{{K}^{*0}}(\rightarrow{{\pi}^{-}}{{K}^{+}}){{K}^{-}}$
mode. The data used correspond to an integrated luminosity of
1.7$\text{\,fb}^{-1}$ collected by the LHCb experiment in 2016 at a centre-of-
mass energy of 13 TeV.

The ${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay is
described by four form factors. The most commonly used parametrisations to
model these form factors are by Caprini-Lellouch-Neubert (CLN) [10] and by
Boyd-Grinstein-Lebed (BGL) [11, 12, 13]. This paper also describes how the
relevant parameters of these parametrisations can be extracted by fitting the
measured differential decay rate.

##  2 Formalism of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay

The ${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay,
with the subsequent ${{D}^{*-}_{s}}\\!\rightarrow{{D}^{-}_{s}}{\gamma}$ decay,
can be described by three angular variables and the squared momentum transfer
to the lepton system, defined as
${q^{2}}=(p_{{{B}^{0}_{s}}}-p_{{{D}^{*-}_{s}}})^{2}$, where
$p_{{{B}^{0}_{s}}}$ and $p_{{{D}^{*-}_{s}}}$ are the four-momenta of the
${B}^{0}_{s}$ and ${D}^{*-}_{s}$ mesons, respectively. The three angular
variables, indicated in Fig. 1, are two helicity angles $\theta_{\mu}$ and
$\theta_{D_{s}}$, and the angle $\chi$. In this figure the direction of the
$z$-axis is defined in the ${B}^{0}_{s}$ rest frame as
$\hat{z}=\vec{p}_{{{D}^{*-}_{s}}}/|\vec{p}_{{{D}^{*-}_{s}}}|$. The angle
between the muon direction in the virtual $W$ rest frame and the $z$ direction
is called $\theta_{\mu}$, while the angle between the ${D}^{-}_{s}$ meson
direction in the ${D}^{*-}_{s}$ rest frame and the $z$ direction is called
$\theta_{D_{s}}$. Finally, $\chi$ is the angle between the plane formed by the
${D}^{*-}_{s}$ decay products and that formed by the two leptons in the
${B}^{0}_{s}$ rest frame [14]. The angular basis is designed such that the
angular definition for the ${\kern
1.79993pt\overline{\kern-1.79993ptB}}{}^{0}_{s}$ decay is a $C\\!P$
transformation of that of the ${B}^{0}_{s}$ decay.

Figure 1: Schematic overview of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay,
introducing the angles $\theta_{D_{s}}$, $\theta_{\mu}$ and $\chi$.

The measurement is performed by integrating the full decay rate over the decay
angles. Thus, the expression of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay rate
is given by

$\begin{split}\frac{\mathrm{d}\Gamma({{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}})}{\mathrm{d}{q^{2}}}&=\frac{G^{2}_{\rm
F}\,|{V_{{c}{b}}}|^{2}\,|\eta_{\rm
EW}|^{2}\,|\vec{p}\,|{q^{2}}}{96\,\pi^{3}\,m_{{{B}^{0}_{s}}}^{2}}\left(1-\frac{m_{\mu}^{2}}{{q^{2}}}\right)^{2}\\\
&\quad\times\left[(|H_{+}|^{2}+|H_{-}|^{2}+|H_{0}|^{2})\left(1+\frac{m_{\mu}^{2}}{2\,{q^{2}}}\right)+\frac{3}{2}\frac{m_{\mu}^{2}}{{q^{2}}}|H_{t}|^{2}\right]\,.\end{split}$
(1)

In this equation, $G_{\rm F}$ is the Fermi constant, $V_{{c}{b}}$ is the CKM
matrix element describing the $b$ to $c$ transition, $\eta_{\rm EW}=1.0066$ is
the electroweak correction to $V_{{c}{b}}$ [15], $m_{\mu}$ is the muon mass
[16], and $H_{0}$, $H_{+}$, $H_{-}$, $H_{t}$ are the helicity amplitudes. The
magnitude of the ${D}^{*-}_{s}$ momentum in the ${B}^{0}_{s}$ rest frame is
given by $|\vec{p}|$. The hadronic recoil, $w$, is related to the squared
momentum transfer to the lepton pair, $q^{2}$, by

$w=\frac{p_{{{B}^{0}_{s}}}}{m_{{{B}^{0}_{s}}}}\cdot\frac{p_{{{D}^{*-}_{s}}}}{m_{{{D}^{*-}_{s}}}}=\frac{m^{2}_{{{B}^{0}_{s}}}+m^{2}_{{{D}^{*-}_{s}}}-{q^{2}}}{2\;m_{{{B}^{0}_{s}}}\;m_{{{D}^{*-}_{s}}}},$
(2)

where $m_{{{B}^{0}_{s}}}$ and $m_{{{D}^{*-}_{s}}}$ are the masses of the
${B}^{0}_{s}$ and ${D}^{*-}_{s}$ mesons, respectively. The minimal value,
$w=1$, corresponds to the situation in which the ${D}^{*-}_{s}$ meson has zero
recoil in the ${B}^{0}_{s}$ rest frame. It is also the value for which $q^{2}$
is maximal.

The dependence of the helicity amplitudes on $w$ can be expressed in different
ways, most commonly described in either the CLN or BGL parametrisations, as
discussed further in Sec. 2.1 and Sec. 2.2. This analysis is only sensitive to
a single form-factor contribution while the other form factors are fixed to
existing measurements from ${{B}^{+}}$ and ${B}^{0}$ semileptonic decays [6,
17]. This is supported by Ref. [18], where when imposing unitarity and
analyticity the differences in form factors for semileptonic $B\rightarrow D$
and ${{B}^{0}_{s}}\rightarrow{{D}^{+}_{s}}$ decays are found to be within
$\mathcal{O}(1\%)$ over the entire kinematic range. Also, a simultaneous
analysis of the $B_{q}\rightarrow D_{q}^{(*)}$ form factors for both light
($q=u,d$) and strange ($q=s$) spectator quarks within the Heavy-Quark-
Expansion framework to order $\mathcal{O}(\alpha_{s},1/m_{b},1/m_{c}^{2})$
[19] does not show any significant SU(3) symmetry breaking. Moreover, Lattice
QCD calculations indicate that there is also good agreement of the form
factors at zero recoil [9, 20].

###  2.1 CLN form-factor parametrisation

For the CLN parametrisation [10], the helicity amplitudes $H_{0}$, $H_{+}$,
$H_{-}$ and $H_{t}$ can be written in terms of the form factors $A_{1}(w)$,
$V(w)$, $A_{2}(w)$ and $A_{0}(w)$ as

$\displaystyle H_{\pm}(w)$
$\displaystyle={m_{{{B}^{0}_{s}}}}\,(1+r)\,A_{1}(w)\mp\frac{2}{1+r}\,|\vec{p}\,|\,V(w),$
$\displaystyle H_{0}(w)$
$\displaystyle=\frac{{m_{{{B}^{0}_{s}}}}\,m_{{{D}^{*-}_{s}}}\,(w-r)\,(1+r)^{2}\,A_{1}(w)-2\,|\vec{p}\,|^{2}\,A_{2}(w)}{m_{{{D}^{*-}_{s}}}\,(1+r)\,\sqrt{1+r^{2}-2wr}}\,,$
(3) $\displaystyle H_{t}(w)$
$\displaystyle=\frac{2\,|\vec{p}\,|}{\sqrt{1+r^{2}-2wr}}\,A_{0}(w)\,,$

where $r=m_{{{D}^{*-}_{s}}}/{m_{{{B}^{0}_{s}}}}$. The form factors are
rewritten in terms of a single leading form factor

$h_{A_{1}}(w)=A_{1}(w)\frac{1}{R_{{{D}^{*-}_{s}}}}\frac{2}{w+1}\,,\\\ $ (4)

and three ratios of form factors

$R_{0}(w)=\frac{A_{0}(w)}{h_{A_{1}}(w)}R_{{{D}^{*-}_{s}}}\,,\qquad
R_{1}(w)=\frac{V(w)}{h_{A_{1}}(w)}R_{{{D}^{*-}_{s}}}\,,\qquad
R_{2}(w)=\frac{A_{2}(w)}{h_{A_{1}}(w)}R_{{{D}^{*-}_{s}}}\,,$ (5)

where

$R_{{{D}^{*-}_{s}}}=\frac{2\sqrt{r}}{1+r}.$ (6)

In the CLN parametrisation, the leading form factor and the three ratios are
parametrised in terms of $w$ as

$\displaystyle h_{A_{1}}(w)$
$\displaystyle=h_{A_{1}}(1)[1-8\rho^{2}z(w)+(53\rho^{2}-15)z^{2}(w)-(231\rho^{2}-91)z^{3}(w)]\
,$ (7) $\displaystyle R_{0}(w)$
$\displaystyle=R_{0}(1)-0.11(w-1)+0.01(w-1)^{2}\ ,$ $\displaystyle R_{1}(w)$
$\displaystyle=R_{1}(1)-0.12(w-1)+0.05(w-1)^{2}\ ,$ $\displaystyle R_{2}(w)$
$\displaystyle=R_{2}(1)+0.11(w-1)-0.06(w-1)^{2}\ ,$

where the coefficients, originally calculated for $B$ decays, are assumed to
be the same for ${B}^{0}_{s}$ decays. The function $z(w)$ is defined as

$z(w)=\frac{\sqrt{w+1}-\sqrt{2}}{\sqrt{w+1}+\sqrt{2}}.$ (8)

As this analysis is only sensitive to the shape of the form-factor
parametrisation the term $h_{A_{1}}(1)$ is absorbed in the normalisation. The
values of $R_{1}(1)$ and $R_{2}(1)$ are taken from the HFLAV average of the
corresponding parameters, obtained from $B\rightarrow D^{*}\ell\nu_{\ell}$
decays [6]. The $R_{0}(1)$ parameter is suppressed by $m_{\ell}^{2}/q^{2}$ in
the helicity amplitude and its contribution to the total rate is negligible.
The value predicted by the exact heavy quark limit of $R_{0}(1)=1$ [21] is
used, as no measurement of $R_{0}(1)$ has been performed. The slope,
$\rho^{2}$, of $h_{A_{1}}(w)$ is the only parameter fitted in this
parametrisation.

###  2.2 BGL form-factor parametrisation

In the BGL parametrisation [11, 12, 13], the helicity amplitudes are
parametrised as

$\displaystyle H_{0}(w)$
$\displaystyle=\frac{\mathcal{F}_{1}(w)}{{m_{{{B}^{0}_{s}}}}\,\sqrt{1+r^{2}+2wr}}\
,$ $\displaystyle H_{\pm}(w)$
$\displaystyle=f(w)\mp{m_{{{B}^{0}_{s}}}}\,m_{{{D}^{*-}_{s}}}\sqrt{w^{2}-1}g(w)\
,$ (9) $\displaystyle H_{t}(w)$
$\displaystyle=m_{{{B}^{0}_{s}}}\frac{\sqrt{r}(1+r)\sqrt{w^{2}-1}}{\sqrt{1+r^{2}-2wr}}\mathcal{F}_{2}(w)\
,$

where the form factors are defined as

$\displaystyle f(z)$
$\displaystyle=\frac{1}{P_{1^{+}}(z)\phi_{f}(z)}\sum_{n=0}^{\infty}a_{n}^{f}z^{n}\
,\qquad$ $\displaystyle\mathcal{F}_{1}(z)$
$\displaystyle=\frac{1}{P_{1^{+}}(z)\phi_{\mathcal{F}_{1}}(z)}\sum_{n=0}^{\infty}a_{n}^{\mathcal{F}_{1}}z^{n}\
,$ (10) $\displaystyle g(z)$
$\displaystyle=\frac{1}{P_{1^{-}}(z)\phi_{g}(z)}\sum_{n=0}^{\infty}a_{n}^{g}z^{n}\
,$ $\displaystyle\mathcal{F}_{2}(z)$
$\displaystyle=\frac{\sqrt{r}}{(1+r)P_{0^{-}}(z)\phi_{\mathcal{F}_{2}}(z)}\sum_{n=0}^{\infty}a_{n}^{\mathcal{F}_{2}}z^{n}\
.$

The functions $\phi_{i}$ are the so-called outer functions,
$P_{1^{\pm},0^{-}}$ are Blaschke factors, and the coefficients $a_{n}^{i}$,
where $i=\\{f,\,g,\,\mathcal{F}_{1},\,\mathcal{F}_{2}\\}$, are parameters to
be fit from data.

As the form-factor parametrisation is given through analytic functions, they
must satisfy the unitarity condition in the $z$ expansion

$\sum_{n=0}^{\infty}(a_{n}^{g})^{2}\leq
1\,,\qquad\sum_{n=0}^{\infty}(a_{n}^{f})^{2}+\sum_{n=0}^{\infty}(a_{n}^{\mathcal{F}_{1}})^{2}\leq
1\,,\qquad\sum_{n=0}^{\infty}(a_{n}^{\mathcal{F}_{2}})^{2}\leq 1.$ (11)

This analysis is only sensitive to the form factor $f(z)$, and its series is
truncated at $n=2$, following Refs. [22, 23, 24, 17, 25]. The shapes for
$\mathcal{F}_{1}(z)$ and $g(z)$ are constrained using the results in Ref.
[17], where the $a_{n}^{i}$ coefficients are fit using recent Belle
measurements with ${{B}^{0}}\rightarrow{{D}^{*-}}\ell^{+}\nu_{\ell}$ decays
[26, 27]. The value of $a_{0}^{f}$ in Ref. [17] is determined from the
combination of lattice calculations in Ref. [28]. The parameters
$a_{n}^{\mathcal{F}_{2}}$ for $\mathcal{F}_{2}(z)$ are fixed from predictions
in Ref. [24], where they are called $P_{1}$. As this analysis represents the
first measurement of form factors in ${B}^{0}_{s}$ $\rightarrow$
${D}^{*-}_{s}$ transitions, the choice of the input parameters is driven by
having as much experimental input as possible. An overview of the fit inputs
is given in Tab. 8 in App. C.

##  3 Detector and simulation

The LHCb detector [29, 30] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region [31], 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 [32] 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$\text{\,Ge\kern-1.00006ptV\\!/}c$.
The minimum distance of a track to a primary vertex (PV), the impact parameter
(IP), is measured with a resolution of
$(15+29/p_{\mathrm{T}})\,\upmu\text{m}$, where $p_{\mathrm{T}}$ is the
component of the momentum transverse to the beam, in
$\text{\,Ge\kern-1.00006ptV\\!/}c$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[33]. Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad and preshower detectors, an electromagnetic
and a hadronic calorimeter. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers [34]. The
online event selection is performed by a trigger [35], 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. The
hardware muon trigger selects events containing a high-$p_{\mathrm{T}}$ muon
candidate. The software trigger requires three tracks with a significant
displacement from any primary $pp$ interaction vertex.

Simulation is required to model the effects of the detector acceptance and the
imposed selection requirements. In the simulation, $pp$ collisions are
generated using Pythia [36, *Sjostrand:2007gs] with a specific LHCb
configuration [38]. Decays of unstable particles are described by EvtGen [39],
in which final-state radiation is generated using Photos [40]. The interaction
of the generated particles with the detector, and its response, are
implemented using the Geant4 toolkit [41, *Agostinelli:2002hh] as described in
Ref. [43].

The simulation is corrected for mismodeling of the kinematic properties of the
generated ${B}^{0}_{s}$ mesons and of the photons from the ${D}^{*-}_{s}$
decays, as well as for data-simulation differences in the muon trigger
efficiency and tracking efficiencies of the final-state particles. Corrections
to the ${B}^{0}_{s}$ and $\gamma$ kinematic distributions are determined by
comparing data and simulated samples of
${{{B}^{+}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{{\pi}^{+}}$ decays, respectively.
Kinematic differences between ${B}^{0}_{s}$ and $B^{+}$ mesons due to their
production mechanisms are small and considered to be negligible [44, 45].
Corrections to the trigger and tracking efficiencies are evaluated using data
and simulated samples of
${{{B}^{+}}}\\!\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi}}{{K}^{+}}$ decays
[46]. In the simulated signal sample, the form factors are described following
the CLN parametrisation with numerical values $\rho^{2}=1.205$,
$R_{1}(1)=1.404$ and $R_{2}(1)=0.854$.

##  4 Data selection

Candidate ${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$
decays are selected by pairing ${D}^{*-}_{s}$ and $\mu^{+}$ candidates, where
the ${D}^{*-}_{s}$ candidate is reconstructed through the ${D}^{-}_{s}$
$\gamma$ decay. The ${D}^{-}_{s}$ mesons are reconstructed requiring two
opposite-sign kaons and a pion inconsistent with coming from a PV, and forming
a common vertex that is displaced from every PV. The final-state hadrons and
muon must satisfy strict particle identification (PID) criteria, consistent
with the assigned particle hypothesis.

To suppress the combinatorial background in the ${D}^{-}_{s}$ mass spectrum,
only the regions of the
${{D}^{-}_{s}}\rightarrow{{K}^{+}}{{K}^{-}}{{\pi}^{-}}$ Dalitz plot compatible
with originating from the $\phi{{\pi}^{-}}$ and ${{K}^{*0}}{{K}^{-}}$ decay
modes are retained by requiring the ${K}^{+}$ ${K}^{-}$ mass to be within
$20\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the known $\phi$ mass, or the
reconstructed ${K}^{+}$ ${\pi}^{-}$ mass to be within
$90\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ of the average $K^{*}(892)^{0}$ mass
[16]. Possible backgrounds arising from the misidentification of one of the
${D}^{-}_{s}$ decay products are removed with explicit vetoes which apply more
stringent PID requirements in a small window of invariant mass of the
corresponding particle combination. The main contributions that are removed
come from ${{\kern 1.79993pt\overline{\kern-1.79993pt\mathchar
28931\relax}}{}^{-}_{c}}\rightarrow{{K}^{+}}{\overline{p}}{{\pi}^{-}}$,
${{D}^{-}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}$,
${{D}^{-}_{s}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, and misidentified
or partially reconstructed multibody $D$ decays, all originating from
semileptonic $b$-hadron decays.

Due to the small mass difference between the ${D}^{*-}_{s}$ and ${D}^{-}_{s}$
mesons, the photon must be emitted close to the ${D}^{-}_{s}$ flight
direction. Photons are selected inside a narrow cone surrounding the
${D}^{-}_{s}$ candidate, defined in pseudorapidity and azimuthal angle. Only
the highest $p_{\mathrm{T}}$ photon inside the cone is combined with the
${D}^{-}_{s}$ candidate. Potential contamination from neutral pions
reconstructed as a single merged cluster in the electromagnetic calorimeter is
suppressed by employing a neural network classifier trained to separate
${\pi}^{0}$ mesons from photons [47].

A fit to the ${{D}^{-}_{s}}{\gamma}$ invariant-mass distribution, with the
reconstructed ${D}^{-}_{s}$ mass constrained to the known value [16], is
performed as shown in Fig. 2. The signal is described by a Gaussian function
with a power-law tail on the right hand side of the distribution and the
background by an exponential distribution. The power-law tail accounts for
cases where additional activity in the calorimeter is mistakenly included in
the photon cluster. The sPlot technique [48] is employed to subtract the
combinatorial background from random photons. Weighted signal is used to
create the templates described in Sec. 5. The correlation between the weights
and $w$ is below 4%.

Figure 2: Distribution of the reconstructed ${{D}^{-}_{s}}{\gamma}$ mass,
$m({{D}^{-}_{s}}{\gamma})$, with the fit overlaid. The fit is performed
constraining the ${D}^{-}_{s}$ mass to the world-average value [16]. The
signal and background components are shown separately with dashed red and
dotted green lines, respectively.

The muon candidate is required to have $p_{\mathrm{T}}$ in excess of
1.2$\text{\,Ge\kern-1.00006ptV\\!/}c$. Background arising from $b$-hadrons
decaying into final states containing two charmed hadrons,
$H_{b}\\!\rightarrow{{D}^{*-}_{s}}H_{c}$, followed by a semileptonic decay of
the charmed hadron $H_{c}\\!\rightarrow{\mu^{+}}{{\nu}_{\mu}}X$, where $X$ is
one or more hadrons, are suppressed by using a multivariate algorithm based on
the isolation of the muon [49]. Finally, the ${B}^{0}_{s}$ meson candidates
are formed by combining $\mu^{+}$ and ${D}^{*-}_{s}$ candidates which are
consistent with coming from a common vertex.

##  5 Signal yield

The signal yield is determined using a template fit to the distribution of the
corrected mass [50],

$m_{\rm
corr}=\sqrt{m^{2}_{{{D}^{*-}_{s}}{\mu^{+}}}+|p_{\perp}|^{2}}+|p_{\perp}|,$
(12)

where $m_{{{D}^{*-}_{s}}{\mu^{+}}}$ is the measured mass of the ${D}^{*-}_{s}$
$\mu^{+}$ candidate, and $p_{\perp}$ is the momentum of the candidate
transverse to the ${B}^{0}_{s}$ flight direction. When only one massless
final-state particle is missing from the decay, $m_{\rm corr}$ peaks at the
${B}^{0}_{s}$ mass. Only candidates in the range $3500<m_{\rm
corr}<5367\text{\,Me\kern-1.00006ptV\\!/}c^{2}$ are considered.

Extended binned maximum-likelihood fits to the $m_{\rm corr}$ distribution are
performed independently in seven bins of the reconstructed hadronic recoil,
$w$, to obtain the raw yields $N_{\mathrm{meas}}$ per bin. The binning scheme,
detailed in Tab. 1, is chosen such that each $w$ bin has roughly the same
signal yield, based on simulation. Obtaining the value of $w$ requires the
knowledge of the momentum of the ${B}^{0}_{s}$ meson, which in the decays
under study can be solved up to a quadratic ambiguity. By imposing momentum
balance against the visible system with respect to the flight direction, and
assuming the mass of the ${B}^{0}_{s}$ meson, the momentum of the
${B}^{0}_{s}$ meson can be estimated. To resolve the ambiguity in the
solutions, a multivariate regression algorithm based on the flight direction
is used [51] yielding a purity on the solutions of around 70%. The $m_{\rm
corr}$ distribution is fitted using shapes (templates) of signal and of
background distributions mostly obtained from simulation. These simulated
decays are selected as described in Sec. 4, and are corrected for the
simulation mismodeling as described in Sec. 3.

Table 1: Binning scheme used for this measurement. Only the upper bound for
each bin is presented. The lower bound on the first bin corresponds to $w=1$.

bin 1 2 3 4 5 6 7 $w$ 1.1087 1.1688 1.2212 1.2717 1.3226 1.3814 1.4667

The largest contribution to the background is due to
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\tau^{+}}{{\nu}_{\tau}}$ decays,
with
${\tau^{-}}\\!\rightarrow{\mu^{-}}{{\overline{\nu}}_{\mu}}{{\nu}_{\tau}}$. A
small source of background is formed by excited ${D}^{-}_{s}$ mesons decaying
into a ${D}^{*-}_{s}$ resonance. The only such excited state is the
${D}_{{s}1}(2460)^{-}$ meson, and hence templates for
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\mu^{+}}{{\nu}_{\mu}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\tau^{+}}{{\nu}_{\tau}}$
decays are included in the fit. The background arising from $b$ hadrons
decaying into final states containing two charmed hadrons,
$H_{b}\\!\rightarrow{{D}^{*-}_{s}}H_{c}$, is also addressed. The template for
this process is generated using simulated events of ${B}^{0}_{s}$, ${B}^{0}$,
${B}^{+}$ and ${\mathchar 28931\relax}^{0}_{b}$ decays, with an appropriate
admixture of final states, based on their production rates, branching ratios
and relative reconstruction efficiencies taken from simulation. The last
background considered in the fit is the combinatorial background, arising from
random combinations of tracks. This template is obtained from a data sample
where the ${D}^{-}_{s}$ meson and the muon have the same charge.

The free parameters in the fit are the signal yield, the relative abundances
of ${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\tau^{+}}{{\nu}_{\tau}}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\mu^{+}}{{\nu}_{\mu}}$
candidates with respect to that of the signal, and the fraction of
combinatorial background. The total fraction of backgrounds from
$H_{c}\\!\rightarrow{\mu^{+}}{{\nu}_{\mu}}X$ decays is fixed to the expected
value using the measured branching fractions and selection efficiencies
obtained from simulation. A 40% uncertainty is assigned to this component to
account for the uncertainties on the branching fractions [16]. The
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\tau^{+}}{{\nu}_{\tau}}$
contribution is also fixed assuming a value of its ratio with respect to the
muonic mode equal to the SM prediction for
${\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*0}}{\tau^{+}}{{\nu}_{\tau}})/{\mathcal{B}}({{B}^{+}}\\!\rightarrow{{D}^{*0}}{\mu^{+}}{{\nu}_{\mu}})$
[21] under the assumption that this ratio is identical for ${B}^{0}_{s}$ meson
decays. The contribution of this decay to the fit is negligible. The Barlow-
Beeston “lite” technique [52, 53] is applied to account for the limited size
of the simulation samples. The distributions of $m_{\rm corr}$ with the fit
overlaid are shown in Fig. 3.

Figure 3: Distribution of the corrected mass, $m_{\rm corr}$, for the seven
bins of $w$, overlaid with the fit results. The
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\tau^{+}}{{\nu}_{\tau}}$
and the
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\mu^{+}}{{\nu}_{\mu}}$
components are combined in
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\ell^{+}}{{\nu}_{\ell}}$.
Below each plot, differences between the data and fit are shown, normalised by
the uncertainty in the data.

Using the fractions obtained from the fit, data and simulated distributions of
the angular variables $\cos(\theta_{\mu})$, $\cos(\theta_{D_{s}})$, and
$\chi$, as defined in Sec. 2, are shown in Fig. 4. All distributions show good
agreement between data and simulation, indicating that integrating over the
angles does not introduce biases.

Figure 4: Distribution of (top right) $\chi$, (bottom left)
$\cos(\theta_{D_{s}})$ and (bottom right) $\cos(\theta_{\mu})$ integrating
over $w$ and the other decay angles from data (black points) compared to the
distribution from simulation with their relative size extracted from the fit
to the corrected mass. The
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\tau^{+}}{{\nu}_{\tau}}$
and the
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\mu^{+}}{{\nu}_{\mu}}$
components are combined in
${{B}^{0}_{s}}\\!\rightarrow{{D}_{{s}1}(2460)^{-}}{\ell^{+}}{{\nu}_{\ell}}$.
The uncertainties on the templates, indicated by the hashed areas in the
figures, are a combination from all templates.

##  6 Efficiency correction

This analysis requires a precise measurement of all contributions to the
efficiency as a function of the true value of the hadronic recoil $w_{\rm
true}$ extracted from simulation. However, the overall normalisation of the
efficiency is not determined as only its dependency with $w_{\rm true}$ is
relevant.

The total efficiency is the product of the geometrical acceptance of the
detector, the efficiency of reconstructing all tracks, the trigger
requirements, and the full set of kinematic, PID and background rejection
requirements. Most of the contributions to the total efficiency are obtained
using simulation. Only the particle identification and the ${D}^{-}_{s}$
selection efficiencies are derived from data using control samples. The muon
and hadron PID efficiencies are taken from large data samples of
${{J\mskip-3.0mu/\mskip-2.0mu\psi}}\\!\rightarrow{\mu^{+}}{\mu^{-}}$ and
${{D}^{*+}}\\!\rightarrow{{D}^{0}}{{\pi}^{+}}$ decays, respectively [54].
These samples are then used to determine the PID efficiencies in bins of $p$,
$p_{\mathrm{T}}$ and number of tracks in the event. The ${D}^{-}_{s}$
selection efficiency accounts for selecting the regions in the Dalitz plot, as
well as the vetoes described in Sec. 4. This efficiency is determined from a
sample of fully reconstructed
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ decays as a function of
the ${D}^{-}_{s}$ $p_{\mathrm{T}}$. The efficiencies extracted from data are
convolved with the simulation to obtain their dependency on $w_{\rm true}$.

The efficiencies derived from simulation are extracted by comparing the
generator-level simulation, based on Pythia [36, *Sjostrand:2007gs] and EvtGen
[39], to the final reconstructed and selected simulation sample used for the
template fit, omitting the particle identification and the ${D}^{-}_{s}$
selection criteria.

##  7 Unfolded yields

The measured ${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$
spectrum from Sec. 5 must be unfolded to account for the resolution on the $w$
variable, which is 0.07. The unfolding procedure uses a migration matrix
determined from simulation, defined as the probability that a candidate
generated in bin $j$ of the $w_{\rm true}$ distribution appears in bin $i$ of
the $w$ distribution. The unfolded spectrum is then corrected bin-by-bin using
the efficiency described in Sec. 6. The combination of the migration matrix
and the total efficiency, called the response matrix, is shown in App. B.

The unfolding procedure adopted is based on the singular value decomposition
(SVD) method [55] using the RooUnfold package implemented in the Root package
[56]. The SVD method includes a regularisation procedure that depends upon a
parameter $k$, ranging between unity and the number of degrees of freedom,
seven in this case. Using simulation, the optimal value for $k$ is found to be
$k=5$, which minimises the difference between the yield from the unfolding
procedure and the expected yield in each bin. The final yields, labelled
$N_{\rm corr}^{\rm unf}$, are normalised to unity and presented in Tab. 2.

###  7.1 Systematic uncertainties

Systematic uncertainties on $N_{\rm corr}^{\rm unf}$ originate from the fitted
${D}^{*-}_{s}$ and
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ yields, and
the efficiency corrections. By varying the determination of the unfolded
yields, systematic uncertainties are quantified. Since this analysis is
sensitive only to the shape of the decay distribution and the absolute
normalisation is unknown, every such variation is normalised to unity. After
normalising, the values are compared to those from the default normalised
unfolded yields, and from this the uncertainties are extracted.

The size of the simulated samples, which are very CPU intensive to generate,
is the dominating systematic uncertainty on the unfolded yields. The simulated
sample size is accounted for in the fit by applying the Barlow-Beeston “lite”
technique [52, 53] when determining the signal yield. Its relative
contribution to the systematic uncertainty is assessed by not applying this
technique and comparing the obtained uncertainties. The uncertainties due to
the size of the control samples used to determine the efficiencies and
corrections are obtained by varying each of the efficiency and correction
inputs within their uncertainty, repeating this 1000 times, and taking the
spread as the uncertainty on $N_{\rm corr}^{\rm unf}$.

The uncertainty on the SVD unfolding procedure is determined by repeating the
regularisation procedure with a different regularisation parameter, $k$. The
nominal value used is $k=5$, which is changed to $k=4$ and $k=6$, and the
difference with the nominal value is assigned as the systematic uncertainty.

Two systematic uncertainties are determined to account for assumptions in the
simulation. Radiative corrections simulated by the Photos package are known to
be incomplete [40, 57]. The difference in $N_{\rm corr}^{\rm unf}$ from
simulated samples with and without Photos is evaluated and a third of the
difference is assigned following Ref. [58]. The efficiency due to the detector
acceptance, and thus the shape of the efficiency correction, may be affected
by the form factors in the HQET model used to generate the simulation, which
are based on the 2016 HFLAV averages [59]. This is studied by weighting both
the generator level and fully reconstructed simulated samples to the 2019
HFLAV averages [6]: $\rho^{2}=1.122\pm 0.024$, $R_{1}(1)=1.270\pm 0.026$, and
$R_{2}(1)=0.852\pm 0.018$, with correlations
$\mathrm{corr}[\rho^{2},R_{1}(1)]=-0.824$,
$\mathrm{corr}[\rho^{2},R_{2}(1)]=0.566$, and
$\mathrm{corr}[R_{1}(1),R_{2}(1)]=-0.715$. The values of each pair are varied
within one standard deviation of their mean, taking into account their
correlation. The value of $R_{0}(1)$ is varied by a $20\%$ uncertainty
accounting for finite $b$\- and $c$-quark masses [21]. These variations result
in small changes of the total efficiency and the average difference is taken
as the uncertainty.

The effect of the ${B}^{0}_{s}$ and $\gamma$ kinematic corrections is assessed
by changing the kinematic binning schemes in which the corrections are
evaluated. The large effect induced by this change has been checked for
statistical fluctuations of the calibration samples. The sample is split
randomly into two, after which new corrections and $N_{\rm corr}^{\rm unf}$
yields are calculated. No relevant differences between the $N_{\rm corr}^{\rm
unf}$ values of these two samples are found in any $w$ bin. Hence, the
systematic uncertainty is based on the change of binning schemes alone.

The corrections to the hardware and software trigger efficiencies applied to
the simulated samples depend on the kinematics and PID of the candidates. The
systematic uncertainty is evaluated by changing the binning scheme and the PID
selection of the control sample.

The systematic uncertainty due to the kinematic dependence of the
${D}^{-}_{s}$ selection efficiency is assessed by extracting the efficiency as
a function of $p$ instead of $p_{\mathrm{T}}$ from the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{-}_{s}}{{\pi}^{+}}$ control sample.

The systematic uncertainty due to the photon background subtraction, performed
through the sPlot method with fits to the ${D}^{*-}_{s}$ invariant mass, is
assessed by implementing the fit with a third-order Chebyshev polynomial for
the background description, and repeating the background subtraction process.

Systematic uncertainties induced by the tracking corrections, detector
occupancy and PID efficiencies are found to be negligible as they do not
affect the corrected mass distribution nor the shape of the efficiency
correction.

Table 2: Fraction of the unfolded yields corrected for the global
efficiencies, $N^{\mathrm{unf}}_{\mathrm{corr}}$, for each $w$ bin. Also shown
in this table is the breakdown of the systematic and statistical uncertainties
on $N^{\mathrm{unf}}_{\mathrm{corr}}$. These are shown as a fraction of the
unfolded yield.

$w$ bin 1 2 3 4 5 6 7 Fraction of $N^{\mathrm{unf}}_{\mathrm{corr},i}$ 0.183
0.144 0.148 0.128 0.117 0.122 0.158 Uncertainties (%) Simulation sample size
3.5 3.0 2.8 3.1 3.4 3.0 3.7 Sample sizes for effs and corrections 3.6 3.2 3.0
2.8 2.8 2.7 2.8 SVD unfolding regularisation 0.5 0.5 0.1 0.7 1.2 0.0 0.5
Radiative corrections 0.1 0.2 0.1 0.3 0.4 0.2 0.2 Simulation FF
parametrisation 0.3 0.1 0.1 0.1 0.2 0.4 0.2 Kinematic corrections 2.4 1.0 1.1
0.1 0.2 0.1 0.9 Hardware-trigger efficiency 0.3 0.3 0.0 0.2 0.2 0.3 0.1
Software-trigger efficiency 0.0 0.1 0.0 0.0 0.1 0.0 0.0 ${D}^{-}_{s}$
selection efficiency 0.5 0.2 0.3 0.3 0.2 0.1 0.3 Photon background subtraction
0.0 2.3 0.8 2.9 2.0 0.9 0.4 Total systematic uncertainty 5.6 5.1 4.4 5.2 5.0
4.2 4.8 Statistical uncertainty 3.4 2.9 2.7 3.1 3.2 2.9 3.4

Table 3: Correlation matrix for the unfolded data set in bins of $w$,
including both statistical and systematic uncertainties.

$w$ bin 1 2 3 4 5 6 7 1 1.00 2 0.44 1.00 3 0.13 0.60 1.00 4 0.19 0.32 0.48
1.00 5 0.30 0.30 0.15 0.60 1.00 6 0.34 0.38 0.33 0.22 0.54 1.00 7 0.27 0.34
0.34 0.27 0.07 0.32 1.00

###  7.2 Results

The $N_{\rm corr}^{\rm unf}$ yields and corresponding systematic and
statistical uncertainties per $w$ bin are shown in Tab. 2. The correlations
between the $N_{\rm corr}^{\rm unf}$ yields including statistical and
systematic uncertainties are given in Tab. 3, and the covariance matrix is
presented in Tab. 6 in App. B. The detector response combined with the
reconstruction efficiency is presented in App. B. Together these can be used
to constrain form-factor parametrisations.

##  8 Form factor fits

The yields $N_{\rm corr}^{\rm unf}$ with corresponding correlation matrix
presented in Sec. 7 can be fit using various form-factor parametrisations.
Fits using the commonly used CLN and BGL parametrisations, with the
assumptions described in Sec. 2, are presented in the following.

The values of the form-factor parameters are derived from a $\chi^{2}$ fit
with

$\chi^{2}=\sum_{i,j}\left(N^{\mathrm{unf}}_{\mathrm{corr},i}-N_{\mathrm{exp},i}\right)C_{ij}^{-1}\left(N^{\mathrm{unf}}_{\mathrm{corr},j}-N_{\mathrm{exp},j}\right).$
(13)

In this expression, $N^{\mathrm{unf}}_{\mathrm{corr},i(j)}$ is the normalised,
unfolded and efficiency-corrected yield in bin $i(j)$, $N_{\mathrm{exp},i(j)}$
is the expected yield in bin $i(j)$ obtained from integrating
$\mathrm{d}\Gamma_{i(j)}/\mathrm{d}w$ from the CLN or BGL parametrisation over
the bin, and $C_{ij}$ is the covariance matrix describing the statistical
uncertainties from the yields and efficiency corrections. This $\chi^{2}$
function is minimised for the CLN and BGL parametrisations separately. For the
CLN parametrisation, the fitted value is $\rho^{2}=1.16\pm 0.05$, where the
uncertainty is only statistical in nature.

For the BGL parametrisation, the unitarity constraint is considered in the
minimisation by adding a Gaussian penalty function [60] to the $\chi^{2}$
defined in Eq. 13. This function is of the form

$\theta(U-1)\Bigg{(}\frac{U-1}{\sigma}\Bigg{)}^{2},$ (14)

where $\theta$ is the Heaviside function, $U$ is the unitarity constraint
$\sum_{n=0}^{2}(a_{n}^{f})^{2}+\sum_{n=0}^{2}(a_{n}^{\mathcal{F}_{1}})^{2}$,
and $\sigma$ is the theoretical uncertainty associated with the bound [61].
The correlation between the external parameter $a_{0}^{f}$ and the fitted
parameters $a_{1}^{f}$ and $a_{2}^{f}$ is not considered due to its small
uncertainty. To assess the impact of this choice, the value of $a_{0}^{f}$ has
been increased (decreased) by $+1(-1)\sigma$. The change in the fitted
parameters is observed to be negligible compared with the overall systematic
uncertainty which covers for it for any value of the correlation between
$a_{0}^{f}$ and the rest of the parameters. This can be explained as
$a_{0}^{f}$ only enters as a nuisance parameter in the unitarity bounds, which
is the only source of correlation between these parameters. As the two scaled
parameters $a_{1}^{f}/a_{0}^{f}$ and $a_{2}^{f}/a_{0}^{f}$ can be much larger
than one (as shown in Fig. 5) the average magnitude of the correlation
diminishes.

The fitted values are $a_{1}^{f}=-0.005\pm 0.034$, and
$a_{2}^{f}=1.00^{+0.00}_{-0.19}$, where the uncertainties are only statistical
in nature.

###  8.1 Systematic uncertainties

The systematic uncertainties on the parameters $\rho^{2}$, $a_{1}^{f}$ and
$a_{2}^{f}$ originate from the same sources as those described in Sec. 7.1.
Additional systematic uncertainties originate from the external parameters
used in the form-factor fits. A summary of all systematic uncertainties for
$\rho^{2}$, $a_{1}^{f}$ and $a_{2}^{f}$ is shown in Tab. 4.

The impact of changes in signal yields or efficiencies has been assessed by
repeating the fit with different conditions and comparing the obtained values
to the nominal ones. In the $\chi^{2}$ fit, the parameters $R_{1}(1)$ and
$R_{2}(1)$ are fixed to the HFLAV averages [6]. The uncertainties on these
values are propagated to the CLN fit outcome by changing $R_{1}(1)$ and
$R_{2}(1)$ within one standard deviation from their average, while accounting
for the correlation between these values. For the BGL fit, the values of the
external parameters of the $f(z)$, $g(z)$ and $\mathcal{F}_{1}(z)$ functions
are varied simultaneously within their uncertainty. When the uncertainties are
asymmetric the largest is chosen. This process is repeated 1000 times applying
the unitarity constraint and the difference between the average of the
variations and the nominal value is assigned as a systematic uncertainty.

Table 4: Summary of the systematic and statistical uncertainties on the
parameters $\rho^{2}$, $a_{1}^{f}$ and $a_{2}^{f}$ from the unfolded CLN and
BGL fits. The total systematic uncertainty is obtained by adding the
individual components in quadrature.

Source $\sigma(\rho^{2})$ $\sigma(a_{1}^{f})$ $\sigma(a_{2}^{f})$ Simulation
sample size 0.053 0.036 ${}^{\,+\,0.00}_{\,-\,0.35}$ Sample sizes for
efficiencies and corrections 0.020 0.016 ${}^{\,+\,0.00}_{\,-\,0.15}$ SVD
unfolding regularisation 0.008 0.004 – Radiative corrections 0.004 – –
Simulation FF parametrisation 0.007 0.005 – Kinematic corrections 0.024 0.012
– Hardware-trigger efficiency 0.001 0.008 – Software-trigger efficiency 0.004
0.002 – ${D}^{-}_{s}$ selection efficiency – 0.008 – Photon background
subtraction 0.002 0.015 – External parameters in fit 0.024 0.002
${}^{\,+\,0.00}_{\,-\,0.04}$ Total systematic uncertainty 0.068 0.046
${}^{\,+\,0.00}_{\,-\,0.38}$ Statistical uncertainty 0.052 0.034
${}^{\,+\,0.00}_{\,-\,0.19}$

###  8.2 Results

An analysis to extract the leading parameters of the form factor describing
the semileptonic transition
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ has been
performed. Using the CLN parametrisation the result obtained is

$\rho^{2}=1.16\pm 0.05\text{\,(stat)}\pm 0.07\text{\,(syst)},$

where the mass of the muon has not been neglected. To compare with other
published results, the fit is repeated assuming a massless muon, resulting in
a small shift of the central value of the $\rho^{2}$ parameter of about 1.5%,
as shown in Tab. 5. The world-average value of $\rho^{2}$ for the equivalent
${B}^{0}$ $\rightarrow$${D}^{*+}$ $\mu^{-}$ ${\nu}_{\mu}$ decay is
$\rho^{2}=1.122\pm 0.015\text{\,(stat)}\pm 0.019\text{\,(syst)}$ [6]. Both
values of $\rho^{2}$ are consistent within their uncertainties. The
measurement is also in agreement with the value obtained in Ref. [7],
$\rho^{2}=1.23\pm 0.17\text{\,(stat)}\pm 0.05\text{\,(syst)}\pm
0.01{\text{\,(ext)}}$, where the last uncertainty comes from external inputs.
That analysis uses ${{B}^{0}_{s}}\rightarrow{D}_{s}^{*-}{\mu^{+}}\nu_{\mu}$
decays from an independent data set, and where the photon from the
${D}^{*-}_{s}$ decay is not reconstructed. A comparison with the normalised
$\Delta\Gamma/\Delta w$ spectra inferred from the CLN and BGL parametrisations
in Ref. [7] gives consistent results with the measured $w$ spectrum in this
paper, as shown in App. D.

Using the BGL parametrisation, the results obtained are

$\displaystyle a_{1}^{f}$ $\displaystyle=-0.005\pm 0.034\text{\,(stat)}\pm
0.046\text{\,(syst)},$ $\displaystyle a_{2}^{f}$
$\displaystyle=\phantom{-}1.00^{\,+\,0.00}_{\,-\,0.19}\text{\,(stat)}^{\,+\,0.00}_{\,-\,0.38}\text{\,(syst)}.$

In Fig. 5, the $\Delta$$\chi^{2}$ contours for the scaled parameters
$a_{1}^{f}/a_{0}^{f}$ versus $a_{2}^{f}/a_{0}^{f}$ are shown; Fig. 8 in App. C
shows the contours of the unscaled $a_{1}^{f}$ versus $a_{2}^{f}$ parameters.
The unitarity constraint results in a non-gaussian distribution of the
uncertainty on the $a_{2}^{f}/a_{0}^{f}$ parameter. The fits to the
differential decay rate using both parametrisations are shown in Fig. 6. The
$p$-values are $8.2\%$ and $1.3\%$ for the CLN and BGL parametrisations,
respectively. The low $p$-values are found to be caused by the third bin in
$w$, which is higher than expected for both parametrisations. When
artificially decreasing the central value of this bin by one standard
deviation, the $p$-values increase to $69.7\%$ and $8.3\%$ for the CLN and BGL
parametrisations, respectively. The low $p$-value for the latter fit is
explained by the fact that the minimum of the $\chi^{2}$ function without the
unitarity constraint lies in the region excluded by this constraint.

Figure 5: $\Delta$$\chi^{2}$ contours for the scaled parameters
$a_{1}^{f}/a_{0}^{f}$ versus $a_{2}^{f}/a_{0}^{f}$. The black cross marks the
best-fit central value. The solid (dashed) contour encloses the
$\Delta$$\chi^{2}$ = 2.3 (6.17) region. The observed shape is due to the
applied unitarity condition, see Eq. 11.

The prediction of the decay rate can also be transformed to a prediction of
the expected normalised event yields taking into account the efficiency and
resolution, which then is fit to the experimental spectrum. Both procedures
provide similar results with small differences induced by slightly different
bin-by-bin correlations shown in Tab. 5.

Figure 6: Unfolded normalised differential decay rate with the fit superimposed for the CLN parametrisation (green), and BGL (red). The band in the fit results includes both the statistical and systematic uncertainty on the data yields. Table 5: Results from different fit configurations, where the first uncertainty is statistical and the second systematic.  CLN fit |
---|---
Unfolded fit | $\rho^{2}=1.16\pm 0.05\pm 0.07$
Unfolded fit with massless leptons | $\rho^{2}=1.17\pm 0.05\pm 0.07$
Folded fit | $\rho^{2}=1.14\pm 0.04\pm 0.07$
BGL fit |
Unfolded fit | $a_{1}^{f}=-0.005\pm 0.034\pm 0.046$
$a_{2}^{f}=1.00\kern-3.00003pt\phantom{.}^{\,+\,0.00}_{\,-\,0.19}\kern-3.00003pt\phantom{.}^{\,+\,0.00}_{\,-\,0.38}$
Folded fit | $a_{1}^{f}=0.039\pm 0.029\pm 0.046$
$a_{2}^{f}=1.00\kern-3.00003pt\phantom{.}^{\,+\,0.00}_{\,-\,0.13}\kern-3.00003pt\phantom{.}^{\,+\,0.00}_{\,-\,0.34}$

##  9 Conclusions

In conclusion, this paper presents for the first time the unfolded normalised
differential decay rate for
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decays as a
function of the recoil parameter $w$. The unfolded spectrum as a function of
$w$ with the systematic uncertainty per bin is given in Tab. 2 and the
correlations between these bins in Tab. 3. This result allows to constrain
${{B}^{0}_{s}}\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ form-factor
parametrisations. The CLN and BGL form-factor parametrisations have been used
to fit the measured spectrum with additional input from
${{B}^{0}}\rightarrow{{D}^{*-}}\ell^{+}\nu_{\ell}$ decays. Both fits give
consistent results when compared to data.

## 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 (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MSHE
(Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC
(United Kingdom); DOE NP and NSF (USA). We acknowledge the computing resources
that are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN
(Italy), SURF (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); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom).

## Appendices

##  Appendix A Fitted yields and efficiency

Figure 7 shows the total efficiency applied to the unfolded signal yields, as
a function of $w_{\rm true}$. It is the combination of the reconstruction and
selection efficiencies, including the acceptance of the LHCb detector.

Figure 7: Total efficiency as a function of $w_{\rm true}$, including the
acceptance of the LHCb detector as well as the reconstruction and selection
efficiencies.

##  Appendix B Covariance and response matrices

This section contains the information needed to reproduce a form-factor fit.
To perform the fit using the unfolded, efficiency-corrected and normalised
yields given in Tab. 2, the corresponding covariance matrix with the combined
statistical uncertainties is given in Tab. 6.

To transform theoretical predictions into expected signal yields, the response
matrix, given in Tab. 7 is needed. This contains the migration matrix (from
the true value of $w$ to the reconstructed one) combined with the
reconstruction efficiency. The migration matrix is normalised such that the
entries within a given bin of $w$ sum up to unity. The absolute efficiencies
have not been measured for this analysis.

Table 6: Covariance matrix for the unfolded data set in bins of $w$, including
both statistical and systematic uncertainties in units of $10^{-5}$.

$w$ bin [$10^{-5}$] 1 2 3 4 5 6 7 1 16.10 2 4.73 7.05 3 1.21 3.81 5.63 4 1.87
2.12 2.81 6.10 5 2.74 1.80 0.78 3.37 5.12 6 2.42 1.82 1.38 0.98 2.17 3.19 7
3.24 2.69 2.43 2.02 0.44 1.69 8.95

Table 7: Response matrix, containing the migration from $w_{\rm true}$ to $w$
bins together with the total efficiency in units of $10^{-4}$.

$[10^{-4}]$ $w_{\rm true}$ $w$ 1 2 3 4 5 6 7 1 132.0 29.9 11.0 6.1 2.7 2.4 1.0
2 22.4 111.0 36.3 11.1 5.0 3.8 1.4 3 6.0 28.7 109.0 35.9 12.3 6.6 4.8 4 4.6
9.8 27.0 102.0 34.6 12.3 5.7 5 1.4 4.4 8.9 30.3 98.0 33.7 10.3 6 0.8 0.7 5.0
8.5 34.5 97.0 30.9 7 $-0.1$ 0.7 2.2 5.7 11.0 33.5 98.5

##  Appendix C Additional information BGL fit

Table 8 gives an overview of the fit inputs for the BGL fit.

Table 8: Fit inputs used for the BGL fit, taken from Ref. [17] and Ref. [24].  BGL parameter | Value
---|---
$a_{0}^{f}$ | $0.01221\pm 0.00016$
$a_{1}^{\mathcal{F}_{1}}$ |  - 0$0.0042\pm 0.0022$
$a_{2}^{\mathcal{F}_{1}}$ | $-0.069^{\,+\,0.041}_{\,-\,0.037}$
$a_{0}^{g}$ | $0.024^{\,+\,0.021}_{\,-\,0.009}$
$a_{1}^{g}$ | $0.05^{\,+\,0.39}_{\,-\,0.72}$
$a_{2}^{g}$ | $1.0^{\,+\,0.0}_{\,-\,2.0}$
$a_{0}^{\mathcal{F}_{2}}$ |  -0$0.0595\pm 0.0093$
$a_{1}^{\mathcal{F}_{2}}$ | $-0.318\pm 0.170$

Figure 8 shows the unscaled $a_{1}^{f}$ versus $a_{2}^{f}$ contours,
equivalent to Fig. 5.

Figure 8: $\Delta$$\chi^{2}$ contours for the scaled parameters $a_{1}^{f}$
versus $a_{2}^{f}$. The black cross marks the best-fit central value. The
solid (dashed) contour encloses the $\Delta$$\chi^{2}$ = 2.3 (6.17) region.
The observed shape is due to the applied unitarity condition, see Eq. (11).

##  Appendix D Comparison with Phys. Rev. D101 (2020) 072004

The $w$ spectrum measured in this analysis can be compared with the results
obtained in Ref. [7] where the form-factor parameters of the
${{B}^{0}_{s}}\\!\rightarrow{{D}^{*-}_{s}}{\mu^{+}}{{\nu}_{\mu}}$ decay are
measured using a version of the CLN and BGL parametrisations. From this, the
normalised $\Delta\Gamma/\Delta w$ spectrum can be inferred, which is shown in
Fig. 9. The spectrum measured in this paper is consistent with the normalised
spectra inferred from both CLN and BGL parametrisations used in Ref. [7].

Figure 9: Comparison between the $w$ spectrum measured in this paper to the
normalised $\Delta\Gamma/\Delta w$ spectra inferred from the CLN and BGL
parametrisations in Ref. [7].

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LHCb collaboration

R. Aaij31, C. Abellán Beteta49, T. Ackernley59, B. Adeva45, M. Adinolfi53, H.
Afsharnia9, C.A. Aidala80, S. Aiola25, Z. Ajaltouni9, S. Akar64, P.
Albicocco22, J. Albrecht14, F. Alessio47, M. Alexander58, A. Alfonso Albero44,
Z. Aliouche61, G. Alkhazov37, P. Alvarez Cartelle60, A.A. Alves Jr45, S.
Amato2, Y. Amhis11, L. An21, L. Anderlini21, G. Andreassi48, A. Andreianov37,
M. Andreotti20, F. Archilli16, A. Artamonov43, M. Artuso67, K. Arzymatov41, E.
Aslanides10, M. Atzeni49, B. Audurier11, S. Bachmann16, M. Bachmayer48, J.J.
Back55, S. Baker60, P. Baladron Rodriguez45, V. Balagura11,b, W. Baldini20,47,
J. Baptista Leite1, A. Baranov41, R.J. Barlow61, S. Barsuk11, W. Barter60, M.
Bartolini23,47,h, F. Baryshnikov77, J.M. Basels13, G. Bassi28, V.
Batozskaya35, B. Batsukh67, A. Battig14, A. Bay48, M. Becker14, F. Bedeschi28,
I. Bediaga1, A. Beiter67, L.J. Bel31, V. Belavin41, S. Belin26, V. Bellee48,
K. Belous43, I. Belyaev38, G. Bencivenni22, E. Ben-Haim12, S. Benson31, S.
Beranek13, A. Berezhnoy39, R. Bernet49, D. Berninghoff16, H.C. Bernstein67, C.
Bertella47, E. Bertholet12, A. Bertolin27, C. Betancourt49, F. Betti19,e, M.O.
Bettler54, Ia. Bezshyiko49, S. Bhasin53, J. Bhom33, L. Bian72, M.S. Bieker14,
S. Bifani52, P. Billoir12, F.C.R. Bishop54, A. Bizzeti21,u, M. Bjørn62, M.P.
Blago47, T. Blake55, F. Blanc48, S. Blusk67, D. Bobulska58, V. Bocci30, J.A.
Boelhauve14, O. Boente Garcia45, T. Boettcher63, A. Boldyrev78, A. Bondar42,x,
N. Bondar37, S. Borghi61,47, M. Borisyak41, M. Borsato16, J.T. Borsuk33, S.A.
Bouchiba48, T.J.V. Bowcock59, A. Boyer47, C. Bozzi20, M.J. Bradley60, S.
Braun16, A. Brea Rodriguez45, M. Brodski47, J. Brodzicka33, A. Brossa
Gonzalo55, D. Brundu26, E. Buchanan53, A. Büchler-Germann49, A. Buonaura49, C.
Burr47, A. Bursche26, A. Butkevich40, J.S. Butter31, J. Buytaert47, W.
Byczynski47, S. Cadeddu26, H. Cai72, R. Calabrese20,g, L. Calero Diaz22, S.
Cali22, R. Calladine52, M. Calvi24,i, M. Calvo Gomez44,m, P. Camargo
Magalhaes53, A. Camboni44,m, P. Campana22, D.H. Campora Perez47, A.F.
Campoverde Quezada5, S. Capelli24,i, L. Capriotti19,e, A. Carbone19,e, G.
Carboni29, R. Cardinale23,h, A. Cardini26, I. Carli6, P. Carniti24,i, K.
Carvalho Akiba31, A. Casais Vidal45, G. Casse59, M. Cattaneo47, G.
Cavallero47, S. Celani48, R. Cenci28, J. Cerasoli10, A.J. Chadwick59, M.G.
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Materok13, R. Matev47, A. Mathad49, Z. Mathe47, V. Matiunin38, C. Matteuzzi24,
K.R. Mattioli80, A. Mauri49, E. Maurice11,b, M. Mazurek35, M. McCann60, L.
Mcconnell17, T.H. Mcgrath61, A. McNab61, R. McNulty17, J.V. Mead59, B.
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Meloni24,i, M. Merk31, A. Merli25, L. Meyer Garcia2, M. Mikhasenko47, D.A.
Milanes73, E. Millard55, M.-N. Minard8, O. Mineev38, L. Minzoni20,g, S.E.
Mitchell57, B. Mitreska61, D.S. Mitzel47, A. Mödden14, A. Mogini12, R.A.
Mohammed62, R.D. Moise60, T. Mombächer14, I.A. Monroy73, S. Monteil9, M.
Morandin27, G. Morello22, M.J. Morello28,t, J. Moron34, A.B. Morris10, A.G.
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Müller47, K. Müller49, C.H. Murphy62, D. Murray61, P. Muzzetto26, P. Naik53,
T. Nakada48, R. Nandakumar56, T. Nanut48, I. Nasteva2, M. Needham57, I.
Neri20,g, N. Neri25,q, S. Neubert16, N. Neufeld47, R. Newcombe60, T.D.
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Pappalardo20,g, C. Pappenheimer64, W. Parker65, C. Parkes61, B. Passalacqua20,
G. Passaleva21,47, A. Pastore18, M. Patel60, C. Patrignani19,e, A. Pearce47,
A. Pellegrino31, M. Pepe Altarelli47, S. Perazzini19, D. Pereima38, P.
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Pietrzyk48, M. Pili62, D. Pinci30, J. Pinzino47, F. Pisani19, A. Piucci16,
Resmi P.K10, V. Placinta36, S. Playfer57, J. Plews52, M. Plo Casasus45, F.
Polci12, M. Poli Lener22, M. Poliakova67, A. Poluektov10, N. Polukhina77,c, I.
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S. Popov41, S. Poslavskii43, K. Prasanth33, L. Promberger47, C. Prouve45, V.
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Quagliani12, B. Quintana8, N.V. Raab17, R.I. Rabadan Trejo10, B. Rachwal34,
J.H. Rademacker53, M. Rama28, M. Ramos Pernas45, M.S. Rangel2, F.
Ratnikov41,78, G. Raven32, M. Reboud8, F. Redi48, F. Reiss12, C. Remon
Alepuz46, Z. Ren3, V. Renaudin62, R. Ribatti28, S. Ricciardi56, D.S.
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Paula2, B. Spaan14, E. Spadaro Norella25,q, P. Spradlin58, F. Stagni47, M.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4School of Physics State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing, China
5University of Chinese Academy of Sciences, Beijing, China
6Institute Of High Energy Physics (IHEP), Beijing, China
7Institute of Particle Physics, Central China Normal University, Wuhan, Hubei,
China
8Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
France
9Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
10Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
11Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
12LPNHE, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3,
Paris, France
13I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
14Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
15Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
16Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
17School of Physics, University College Dublin, Dublin, Ireland
18INFN Sezione di Bari, Bari, Italy
19INFN Sezione di Bologna, Bologna, Italy
20INFN Sezione di Ferrara, Ferrara, Italy
21INFN Sezione di Firenze, Firenze, Italy
22INFN Laboratori Nazionali di Frascati, Frascati, Italy
23INFN Sezione di Genova, Genova, Italy
24INFN Sezione di Milano-Bicocca, Milano, Italy
25INFN Sezione di Milano, Milano, Italy
26INFN Sezione di Cagliari, Monserrato, Italy
27Universita degli Studi di Padova, Universita e INFN, Padova, Padova, Italy
28INFN Sezione di Pisa, Pisa, Italy
29INFN Sezione di Roma Tor Vergata, Roma, Italy
30INFN Sezione di Roma La Sapienza, Roma, Italy
31Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
32Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
33Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
34AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
35National Center for Nuclear Research (NCBJ), Warsaw, Poland
36Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
37Petersburg Nuclear Physics Institute NRC Kurchatov Institute (PNPI NRC KI),
Gatchina, Russia
38Institute of Theoretical and Experimental Physics NRC Kurchatov Institute
(ITEP NRC KI), Moscow, Russia
39Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
40Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
41Yandex School of Data Analysis, Moscow, Russia
42Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
43Institute for High Energy Physics NRC Kurchatov Institute (IHEP NRC KI),
Protvino, Russia, Protvino, Russia
44ICCUB, Universitat de Barcelona, Barcelona, Spain
45Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
46Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain
47European Organization for Nuclear Research (CERN), Geneva, Switzerland
48Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
49Physik-Institut, Universität Zürich, Zürich, Switzerland
50NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
51Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
52University of Birmingham, Birmingham, United Kingdom
53H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
54Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
55Department of Physics, University of Warwick, Coventry, United Kingdom
56STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
57School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
58School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
59Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
60Imperial College London, London, United Kingdom
61Department of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
62Department of Physics, University of Oxford, Oxford, United Kingdom
63Massachusetts Institute of Technology, Cambridge, MA, United States
64University of Cincinnati, Cincinnati, OH, United States
65University of Maryland, College Park, MD, United States
66Los Alamos National Laboratory (LANL), Los Alamos, United States
67Syracuse University, Syracuse, NY, United States
68Laboratory of Mathematical and Subatomic Physics , Constantine, Algeria,
associated to 2
69School of Physics and Astronomy, Monash University, Melbourne, Australia,
associated to 55
70Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
71Guangdong Provencial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou, China, associated to 3
72School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
73Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 12
74Institut für Physik, Universität Rostock, Rostock, Germany, associated to 16
75Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 31
76National Research Centre Kurchatov Institute, Moscow, Russia, associated to
38
77National University of Science and Technology “MISIS”, Moscow, Russia,
associated to 38
78National Research University Higher School of Economics, Moscow, Russia,
associated to 41
79National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
38
80University of Michigan, Ann Arbor, United States, associated to 67

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mDS4DS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vUniversità di Siena, Siena, Italy
wMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
xNovosibirsk State University, Novosibirsk, Russia
yINFN Sezione di Trieste, Trieste, Italy
zPhysics and Micro Electronic College, Hunan University, Changsha City, China
aaUniversidad Nacional Autonoma de Honduras, Tegucigalpa, Honduras