EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-256 LHCb-PAPER-2017-027 26 April 2018

Test of Lepton Flavor Universality by the measurement of the $B^{0}\rightarrow
D^{*-}\tau^{+}\nu_{\tau}$ branching fraction using three-prong $\tau$ decays

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

The ratio of branching fractions
${\cal{R}}(D^{*-})\equiv{\cal{B}}(B^{0}\rightarrow
D^{*-}\tau^{+}\nu_{\tau})/{\cal{B}}(B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu})$
is measured using a data sample of proton-proton collisions collected with the
LHCb detector at center-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity of
3$\mbox{\,fb}^{-1}$. The $\tau$ lepton is reconstructed with three charged
pions in the final state. A novel method is used that exploits the different
vertex topologies of signal and backgrounds to isolate samples of semitauonic
decays of $b$ hadrons with high purity. Using the ${B}^{0}$ $\rightarrow$
${D}^{*-}$ ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$ decay as the normalization
channel, the ratio ${\cal{B}}(B^{0}\rightarrow
D^{*-}\tau^{+}\nu_{\tau})/{\cal{B}}(B^{0}\rightarrow
D^{*-}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})$ is measured to be $1.97\pm 0.13\pm
0.18$, where the first uncertainty is statistical and the second systematic.
An average of branching fraction measurements for the normalization channel is
used to derive ${\cal{B}}(B^{0}\rightarrow D^{*-}\tau^{+}\nu_{\tau})=(1.42\pm
0.094\pm 0.129\pm 0.054)\%$, where the third uncertainty is due to the limited
knowledge of ${\cal{B}}(B^{0}\rightarrow D^{*-}\pi^{+}\pi^{-}\pi^{+})$. A test
of lepton flavor universality is performed using the well-measured branching
fraction ${\cal{B}}(B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu})$ to compute
${\cal{R}}(D^{*-})=0.291\pm 0.019\pm 0.026\pm 0.013$, where the third
uncertainty originates from the uncertainties on
${\cal{B}}({{B}^{0}}\rightarrow{{D}^{*-}}{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}})$
and ${\cal{B}}(B^{0}\rightarrow D^{*-}\mu^{+}\nu_{\mu})$. This measurement is
in agreement with the Standard Model prediction and with previous
measurements.

Published in Phys. Rev. D 97 (2018) 072013

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

##  1 Introduction

In the Standard Model (SM) of particle physics, lepton flavor universality
(LFU) is an accidental symmetry broken only by the Yukawa interactions.
Differences between the expected branching fraction of semileptonic decays
into the three lepton families originate from the different masses of the
charged leptons. Further deviations from LFU would be a signature of physics
processes beyond the SM.

Measurements of the couplings of $Z$ and $W$ bosons to light leptons, mainly
constrained by LEP and SLC experiments, are compatible with LFU. Nevertheless,
a 2.8 standard deviation difference exists between the measurement of the
branching fraction of the $W^{+}\rightarrow{\tau}^{+}{{\nu}_{\tau}}$ decay
with respect to those of the branching fractions of
$W^{+}\rightarrow{\mu}^{+}{{\nu}_{\mu}}$ and $W^{+}\rightarrow
e^{+}{{\nu}_{e}}$ decays [1].

Since uncertainties due to hadronic effects cancel to a large extent, the SM
prediction for the ratios between branching fractions of semitauonic decays of
$B$ mesons relative to decays involving lighter lepton families, such as

$\displaystyle{\cal{R}}(D^{(*)-})$ $\displaystyle\equiv$
$\displaystyle{\cal{B}}({{B}^{0}}\rightarrow
D^{(*)-}{\tau^{+}}{{\nu}_{\tau}})/{\cal{B}}({{B}^{0}}\rightarrow
D^{(*)-}{\mu^{+}}{{\nu}_{\mu}}),$ (1) $\displaystyle{\cal{R}}(D^{(*)0})$
$\displaystyle\equiv$ $\displaystyle{\cal{B}}({{{B}^{-}}}\rightarrow
D^{(*)0}{\tau^{-}}{{\overline{\nu}}_{\tau}})/{\cal{B}}({{{B}^{-}}}\rightarrow
D^{(*)0}{\mu^{-}}{{\overline{\nu}}_{\mu}}),$ (2)

is known with an uncertainty at the percent level [2, 3, 4, 5]. For ${D}^{*}$
decays, recent papers [6, 5] argue for larger uncertainties, up to 4%. These
decays therefore provide a sensitive probe of SM extensions with flavor-
dependent couplings, such as models with an extended Higgs sector [7], with
leptoquarks [8, 9], or with an extended gauge sector [10, 11, 12].

The $B\\!\rightarrow{D}^{(*)}{\tau^{+}}{{\nu}_{\tau}}$ decays have recently
been subject to intense experimental scrutiny. Measurements of
${\cal{R}}(D^{0,-})$ and ${\cal{R}}(D^{*-,0})$ and their averages
${\cal{R}}(D)$ and ${\cal{R}}(D^{*})$ have been reported by the BaBar [13, 14]
and Belle [15, 16] collaborations in final states involving electrons or muons
from the $\tau$ decay. The LHCb collaboration measured ${\cal{R}}({D}^{*})$
[17] with results compatible with those from BaBar, while the result from the
Belle collaboration is compatible with the SM within 1 standard deviation. The
measurements from both the BaBar and Belle collaborations were performed with
events that were “tagged” by fully reconstructing the decay of one of the two
$B$ mesons from the $\mathchar 28935\relax{(4S)}$ decay to a fully hadronic
final state (hadronic tag); the other $B$ meson was used to search for the
signal. In all of the above measurements, the decay of the $\tau$ lepton into
a muon, or an electron, and two neutrinos was exploited. More recently, the
Belle collaboration published a measurement [16] with events tagged using
semileptonic decays, compatible with the SM within $1.6$ standard deviations.
A simultaneous measurement of ${\cal{R}}({{D}^{*}})$ and of the $\tau$
polarization, using hadronic tagging and reconstruction of the
$\tau^{-}\rightarrow\pi^{-}{{\nu}_{\tau}}$ and
$\tau^{-}\rightarrow\rho^{-}{{\nu}_{\tau}}$ decays, was published by the Belle
collaboration [18, 19]. The average of all these ${\cal{R}}({{D}^{*}})$
measurements is in tension with the SM expectation at $3.3$ standard
deviations. All these ${\cal{R}}(D^{(*)-,0})$ measurements yield values that
are above the SM predictions with a combined significance of $3.9$ standard
deviations [20].

This paper presents a measurement of
${\cal{B}}$(${{B}^{0}}\\!\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}}$),
using for the first time the $\tau$ decay with three charged particles (three-
prong) in the final state, i.e.
${\tau^{+}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\overline{\nu}}_{\tau}}$
and
${\tau^{+}}\\!\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$,
denoted as signal in this paper. The ${D}^{*-}$ meson is reconstructed through
the ${{D}^{*-}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}(\rightarrow
K^{+}{{\pi}^{-}}){{\pi}^{-}}$ decay chain.111The inclusion of charge-conjugate
decay modes is implied in this paper. The visible final state consists of six
charged tracks; neutral pions are not reconstructed in this analysis. A data
sample of proton-proton collisions, corresponding to an integrated luminosity
of $3$$\mbox{\,fb}^{-1}$, collected with the LHCb detector at center-of-mass
energies of $\sqrt{s}$ = 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$ is used. A
shorter version of this paper can be found in Ref. [21].

The three-prong $\tau$ decay modes have different features with respect to
leptonic $\tau$ decays, leading to measurements with a better signal-to-
background ratio and statistical significance. The absence of charged leptons
in the final state avoids backgrounds originating from semileptonic decays of
$b$ or $c$ hadrons. The three-prong topology enables the precise
reconstruction of a $\tau$ decay vertex detached from the ${B}^{0}$ decay
vertex due to the non zero $\tau$ lifetime, thereby allowing the
discrimination between signal decays and the most abundant background due to
$B\\!\rightarrow{{D}^{*-}}3{\pi}X$ decays, where $X$ represents
unreconstructed particles and 3$\pi$
$\equiv{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{+}}$.222The notation $X$ is used when
unreconstructed particles are known to be present in the decay chain and $(X)$
when unreconstructed particles may be present in the decay chain. The
requirement of a 3$\pi$ decay vertex detached from the $B$ vertex suppresses
the ${{D}^{*-}}3{\pi}X$ background by three orders of magnitude, while
retaining about 40% of the signal. Moreover, because only one neutrino is
produced in the $\tau$ decay, the measurements of the ${B}^{0}$ and $\tau$
lines of flight allow the determination of the complete kinematics of the
decay, up to two quadratic ambiguities, leading to four solutions.

After applying the $3\pi$ detached-vertex requirement, the dominant background
consists of $B$ decays with a ${D}^{*-}$ and another charm hadron in the final
state, called double-charm hereafter. The largest component is due to
${B}\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ decays. These decays have the
same topology as the signal, as the second charm hadron has a measurable
lifetime and its decay vertex is detached from the $B$ vertex. The double-
charm background is suppressed by applying vetoes on the presence of
additional particles around the direction of the $\tau$ and $B$ candidates,
and exploiting the different resonant structure of the 3$\pi$ system in
$\tau^{+}$ and ${D}^{+}_{s}$ decays.

The signal yield, $N_{\mathrm{sig}}$, is normalized to that of the exclusive
${{B}^{0}}\rightarrow{{D}^{*-}}3\pi$ decay, $N_{\mathrm{norm}}$, which has the
same charged particles in the final state. This choice minimizes experimental
systematic uncertainties. The measured quantity is

${\cal{K}}({{D}^{*}}^{-})\equiv\frac{{\cal{B}}(B^{0}\rightarrow
D^{*-}\tau^{+}\nu_{\tau})}{{\cal{B}}(B^{0}\rightarrow
D^{*-}3{\pi})}=\frac{N_{\mathrm{sig}}}{N_{\mathrm{norm}}}\frac{\varepsilon_{\mathrm{norm}}}{\varepsilon_{\mathrm{sig}}}\frac{1}{{\cal{B}}(\tau^{+}\rightarrow
3\pi{{\overline{\nu}}_{\tau}})+{\cal{B}}(\tau^{+}\rightarrow
3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}})},$ (3)

where $\varepsilon_{\mathrm{sig}}$ and $\varepsilon_{\mathrm{norm}}$ are the
efficiencies for the signal and normalization decay modes, respectively. More
precisely, $\varepsilon_{\mathrm{sig}}$ is the weighted average efficiency for
the 3$\pi$ and the 3$\pi$ ${\pi}^{0}$ modes, given their respective branching
fractions. The absolute branching fraction is obtained as
${\cal{B}}({{B}^{0}}\\!\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}})={\cal{K}}({{D}^{*}}^{-})\times{\cal{B}}(B^{0}\rightarrow
D^{*-}3{\pi})$, where the branching fraction of the
${{B}^{0}}\rightarrow{{D}^{*-}}{3\pi}$ decay is taken by averaging the
measurements of Refs. [22, 23, 24]. A value for ${\cal{R}}({{D}^{*}}^{-})$ is
then derived by using the branching fraction of the
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{\mu^{+}}{{\nu}_{\mu}}$ decay from Ref.
[20].

This paper is structured as follows. Descriptions of the LHCb detector, the
data and simulation samples and the trigger selection criteria are given in
Sec. 2. Signal selection and background suppression strategies are summarized
in Sec. 3. Section 4 presents the study performed to characterize double-charm
backgrounds due to $B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$,
$B\\!\rightarrow{{D}^{*-}}{{D}^{+}}(X)$ and
$B\\!\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ decays. The strategy used to fit the
signal yield and the corresponding results are presented in Sec. 5. The
determination of the yield of the normalization mode is discussed in Sec. 6.
The determination of ${\cal{K}}({{D}^{*-}})$ is presented in Sec. 7 and
systematic uncertainties are discussed in Sec. 8. Finally, overall results and
conclusions are given in Sec. 9.

##  2 Detector and simulation

The LHCb detector [25, 26] 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 [27], 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 [28] placed
downstream of the magnet. The tracking system provides a measurement of
momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), 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
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors
[29]. 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
[30].

Simulated samples of $pp$ 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 Tauola package [36] is used to simulate the
decays of the $\tau$ lepton into the 3$\pi$ ${\overline{\nu}}_{\tau}$ and
3$\pi$ ${\pi}^{0}$ ${\overline{\nu}}_{\tau}$ final states according to the
resonance chiral Lagrangian model [37] with a tuning based on the results from
the BaBar collaboration [38]. The interaction of the generated particles with
the detector, and its response, are implemented using the Geant4 toolkit [39,
*Agostinelli:2002hh] as described in Ref. [41]. The signal decays are
simulated using form factors that are derived from heavy-quark effective
theory [42]. The experimental values of the corresponding parameters are taken
from Ref. [20], except for an unmeasured helicity-suppressed amplitude, which
is taken from Ref. [43].

The trigger [44] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage, in which all
charged particles with
$\mbox{$p_{\mathrm{T}}$}>500\,(300){\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ are
reconstructed for 7 TeV (8 TeV) data. At the hardware trigger stage,
candidates are required to have a muon with high $p_{\mathrm{T}}$ or a hadron,
photon or electron with high transverse energy. The software trigger requires
a two-, three-, or four-track secondary vertex with significant displacement
from any PV consistent with the decay of a $b$ hadron, or a two-track vertex
with a significant displacement from any PV consistent with a ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\rightarrow K^{+}\pi^{-}$ decay.
In both cases, at least one charged particle must have a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and must be
inconsistent with originating from any PV. A multivariate algorithm [45] is
used for the identification of secondary vertices consistent with the decay of
a $b$ hadron. Secondary vertices consistent with the decay of a ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ meson must satisfy additional
selection criteria, based on the momenta and transverse momenta of the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ decay products
($p>5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $p_{\mathrm{T}}$
$>800$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$), and on the consistency, as a
loose requirement, of the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ momentum vector with the
direction formed by joining the PV and the ${B}^{0}$ vertex.

##  3 Selection criteria and multivariate analysis

The signal selection proceeds in two main steps. First, the dominant
background, consisting of candidates where the 3$\pi$ system originates from
the ${B}^{0}$ vertex, called prompt hereafter, is suppressed by applying a
$3\pi$ detached-vertex requirement. Second, the double-charm background is
suppressed using a multivariate analysis (MVA). This is the only background
with the same vertex topology as the signal.

This section is organized as follows. After a summary of the principles of the
signal selection in Sec. 3.1, the categorization of the remaining background
processes is given in Secs. 3.1.1 and 3.1.2. This categorization motivates
(Sec. 3.1.3) the additional selection criteria that have to be applied to the
tracks and vertices of the candidates in order to exploit the requirement of
vertex detachment in its full power. Section 3.2 describes the isolation tools
used to take advantage of the fact that, for the $\tau^{+}\rightarrow
3{\pi}{{\overline{\nu}}_{\tau}}$ channel, there is no other charged or neutral
particle at the ${B}^{0}$ vertex beside the reconstructed particles in the
final state. Particle identification requirements are presented in Sec. 3.3.
The selection used for the normalization channel is described in Sec. 3.4.
Section 3.5 details the kinematic techniques used to reconstruct the decay
chains in the signal and background hypotheses. Finally, the MVA that is used
to reduce the double-charm backgrounds is presented in Sec. 3.6 and, in Sec.
3.7, the background composition at various stages of the selection process is
illustrated.

###  3.1 The detached-vertex topology

The signal final state consists of a ${D}^{*-}$ meson, reconstructed in the
${{D}^{*-}}\\!\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$, ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
decay chain, associated with a 3$\pi$ system. The selection of ${D}^{*-}$
candidates starts by requiring ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ candidates with masses between
$1845$ and $1885$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, $p_{\mathrm{T}}$
larger than 1.6${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, combined with pions of
$p_{\mathrm{T}}$ larger than $0.11$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ such
that the difference between the ${D}^{*-}$ and the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ masses lies between $143$ and
$148$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The ${{D}^{*-}}3{\pi}$
combination is very common in $B$ meson decays, with a signal-to-background
ratio smaller than $1$%. The dominant background is prompt, i.e. consisting of
candidates where the 3$\pi$ system is produced at the ${B}^{0}$ vertex.
However, in the signal case, because of the significant $\tau$ lifetime and
boost along the forward direction, the 3$\pi$ system is detached from the
${B}^{0}$ vertex, as shown in Fig. 1.

Figure 1: Topology of the signal decay. A requirement on the distance between
the 3$\pi$ and the ${B}^{0}$ vertices along the beam direction to be greater
than four times its uncertainty is applied.

The requirement for the detached vertex is that the distance between the
3$\pi$ and the ${B}^{0}$ vertices along the beam direction, $\Delta z\equiv
z(3{\pi})-z({{B}^{0}})$, is greater than four times its uncertainty,
$\sigma_{\Delta z}$. This leads to an improvement in the signal to noise ratio
by a factor 160, as shown in Fig. 2. The remaining background consists of two
main categories: candidates with a true detached-vertex topology, and
candidates that appear to have such a detached-vertex topology.

####  3.1.1 Background with detached-vertex topology

The double-charm $B\\!\rightarrow{{D}^{*-}}{D}(X)$ decays are the only other
$B$ decays with the same vertex topology as the signal. Figure 2 shows, on
simulated events, the dominance of the double-charm background over the signal
after the detached-vertex requirement. Figure 3 shows the $3\pi$ mass data
distribution after the detached-vertex requirement, where peaking structures
corresponding to the ${{D}^{+}}\\!\rightarrow 3\pi$ decay and
${{D}^{+}_{s}}\\!\rightarrow 3\pi$ decay – a very important control channel
for this analysis – are clearly visible.

Figure 2: Distribution of the distance between the ${B}^{0}$ vertex and the
3$\pi$ vertex along the beam direction, divided by its uncertainty, obtained
using simulation. The vertical line shows the 4$\sigma$ requirement used in
the analysis to reject the prompt background component. Figure 3: Distribution
of the $3\pi$ mass for candidates after the detached-vertex requirement. The
${D}^{+}$ and ${D}^{+}_{s}$ mass peaks are indicated.

####  3.1.2 Background from other sources

Requirements additional to the detached vertex are needed to reject spurious
background sources with vertex topologies similar to the signal. The various
background sources are classified to distinguish candidates where the 3$\pi$
system originates from a common vertex and those where one of the three pions
originates from a different vertex.

The background category, where the 3$\pi$ system stems from a common vertex,
is further divided into two different classes depending on whether or not the
${D}^{*-}$ and 3$\pi$ system originate from the same $b$ hadron. In the first
case, the $3\pi$ system either comes from the decay of a $\tau$ lepton or a
${D}^{0}$, ${D}^{+}$, $D_{s}^{+}$ or ${\mathchar 28931\relax}_{c}^{+}$ hadron.
Candidates originating from $b$ baryons form only 2% of this double-charm
category. In this case, the candidate has the correct signal-like vertex
topology. Alternatively, it comes from a misreconstructed prompt background
candidate containing a ${B}^{0}$, ${{B}^{+}}$, ${B}^{0}_{s}$ or ${\mathchar
28931\relax}_{b}^{0}$ hadron. The detailed composition of these different
categories at the initial and at the final stage of the analysis is described
in Sec. 3.7. In the second case, the ${D}^{*-}$ and the 3$\pi$ systems are not
daughters of the same $b$ hadron. The 3$\pi$ system originates from one of the
following sources: the other $b$ hadron present in the event (B1B2 category);
the decay of charm hadrons produced at the PV (charm category); another PV; or
an interaction in the beam pipe or in the detector material.

The 3$\pi$ background not originating from the same vertex is dominated by
candidates where two pions originate from the same vertex whilst the third may
come directly from the PV, from a different vertex in the decay chain of the
same $b$ hadron, from the other $b$ hadron produced at the PV, or from another
PV. Due to the combinatorial origin of this background, there is no strong
correlation between the charge of the 3$\pi$ system and the $D^{*-}$ charge.
This enables the normalization of the combinatorial background with the wrong-
sign data sample.

####  3.1.3 Summary of the topological selection requirements

The requirements applied to suppress combinatorial and charm backgrounds, in
addition to the detached-vertex criterion, are reported in Table 1. These
include a good track quality and a minimum transverse momentum of
250${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ for each pion, a good vertex
reconstruction quality for the 3$\pi$ system and large $\chi^{2}_{\text{IP}}$
with respect to any PV for each pion of the 3$\pi$ system and for the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ candidate, where
$\chi^{2}_{\text{IP}}$ is defined as the difference in the vertex-fit
$\chi^{2}$ of a given PV reconstructed with and without the particle under
consideration. In addition, the 3$\pi$ vertex must be detached from its
primary vertex along the beam axis by at least 10 times the corresponding
uncertainty. The distance from the 3$\pi$ vertex position to the beam center
in the plane transverse to the beam direction, $r_{3{\pi}}$, must be outside
the beam envelope and inside the beam pipe to avoid 3$\pi$ vertices coming
from proton interactions or secondary interactions with the beam-pipe
material. The attached primary vertex to the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ and 3$\pi$ candidates must be
the same. The number of candidates per event must be equal to one; this cut is
the first rejection step against nonisolated candidates. Finally, the
difference between the reconstructed ${D}^{*-}$ and ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ masses must lie between 143 and
148 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$.

Table 1: List of the selection cuts. See text for further explanation. Variable | Requirement | Targeted background
---|---|---
$[z(3{\pi})-z({{B}^{0}})]/\sigma_{(z(3{\pi})-z({{B}^{0}}))}$ | $>4$ | prompt
$p_{\mathrm{T}}$($\pi$), $\pi$ from 3$\pi$ |  $>250$${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ | all
$3\pi$ vertex $\chi^{2}$ | $<10$ | combinatorial
$\chi^{2}_{\text{IP}}(\pi)$, $\pi$ from 3$\pi$ | $>15$ | combinatorial
$\chi^{2}_{\text{IP}}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})$ | $>10$ | charm
$[z(3{\pi})-z({\text{PV}})]/\sigma_{(z(3{\pi})-z({\text{PV}}))}$ | $>10$ | charm
$r_{3{\pi}}$ | $\in[0.2,5.0]\mathrm{\,mm}$ | spurious 3$\pi$
$\text{PV}({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})$ | $=\text{PV}(3{\pi})$ | charm/combinatorial
number of ${B}^{0}$ candidates | $=1$ | all
$\Delta m\equiv m({{D}^{*-}})-m({{\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}})$ | $\in[143,148]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ | combinatorial

###  3.2 Isolation requirements

####  3.2.1 Charged isolation

A charged-isolation algorithm ensures that no extra tracks are compatible with
either the ${B}^{0}$ or 3$\pi$ decay vertices. It is implemented by counting
the number of charged tracks having $p_{\mathrm{T}}$ larger than
250${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, $\chi^{2}_{\text{IP}}$ with respect
to the PV larger than 4, and $\chi^{2}_{\text{IP}}(3{\pi})$ and
$\chi^{2}_{\text{IP}}({{B}^{0}})$, with respect to the vertex of the 3$\pi$
and ${B}^{0}$ candidates, respectively, smaller than 25. The
${{D}^{*-}}3{\pi}$ candidate is rejected if any such track is found. As an
example, the performance of the charged-isolation algorithm is determined on a
simulated sample of double-charm decays with a ${D}^{0}$ meson in the final
state. In cases where ${{B}^{0}}\rightarrow{{D}^{*-}}{{D}^{0}}{{K}^{+}}(X)$,
with ${{D}^{0}}\rightarrow{{K}^{-}}3\pi(X)$, two charged kaons are present in
the decay chain, one originating from the ${B}^{0}$ vertex and the other from
the ${D}^{0}$ vertex. For these candidates, the rejection rate is 95%. The
charged-isolation algorithm has a selection efficiency of 80% on a data sample
of exclusive ${{B}^{0}}\rightarrow{{D}^{*-}}3\pi$ decays. This sample has no
additional charged tracks from the ${B}^{0}$ vertex and has thus similar
charged-isolation properties as the signal. This value is in good agreement
with the efficiency determined from simulation.

Reversing the isolation requirement provides a sample of candidates from the
inclusive ${D}^{0}$ decay chain mentioned above, where a ${D}^{0}$ meson
decays into ${{K}^{-}}3\pi$ and the charged kaon has been found as a nearby
track. Figure 4 shows the ${{K}^{-}}3{\pi}$ mass distribution featuring a
prominent ${D}^{0}$ peak. This control sample is used to determine the
properties of the ${B}\\!\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ background in the
signal fit.

Figure 4: Distribution of the ${{K}^{-}}3\pi$ mass for $D^{0}$ candidates
where a charged kaon has been associated to the 3$\pi$ vertex.

####  3.2.2 Neutral isolation

Background candidates from decays with additional neutral particles are
suppressed by using the energy deposited in the electromagnetic calorimeter in
a cone of $0.3$ units in $\Delta\eta-\Delta\phi$ around the direction of the
3$\pi$ system, where $\phi$ is the azimuthal angle in the plane perpendicular
to the beam axis. For this rejection method to be effective, the amount of
collected energy in the region of interest must be small when no neutral
particles are produced in the ${B}^{0}$ meson decay. Candidates where the
${B}^{0}$ meson decays to ${{D}^{*-}}3\pi$, with ${{D}^{*-}}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}{{\pi}^{-}}$, are used as a
check. Figure 5 compares the distributions of the ${{D}^{*-}}{3\pi}$ mass with
and without the requirement of an energy deposition of at least
8$\mathrm{\,Ge\kern-1.00006ptV}$ in the electromagnetic calorimeter around the
3$\pi$ direction. Since it is known that no neutral particle is emitted in
this decay, the inefficiency of this rejection method is estimated by the
ratio of the yields of the two spectra within $\pm
30$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the ${B}^{0}$ mass, and
it is found to be small enough to allow the use of this method. The energy
deposited in the electromagnetic calorimeter around the 3$\pi$ direction is
one of the input quantities to the MVA described below, used to suppress
inclusive ${D}^{+}_{s}$ decays to $3{\pi}X$, which contain photons and
${\pi}^{0}$ mesons in addition to the three pions. Photons are also produced
when ${D}^{+}_{s}$ excited states decay to the ${D}^{+}_{s}$ ground state. The
use of this variable has an impact on signal, since it vetoes the
${\tau^{+}}\\!\rightarrow 3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ decay,
whose efficiency is roughly one half with respect to that of the 3$\pi$ mode,
as can be seen later in Table 2.

Figure 5: Distribution of the ${{D}^{*-}}{3\pi}$ mass (blue) before and (red)
after a requirement of finding an energy of at least
8$\mathrm{\,Ge\kern-1.00006ptV}$ in the electromagnetic calorimeter around the
3$\pi$ direction.

###  3.3 Particle identification requirements

In order to ensure that the tracks forming the 3$\pi$ candidate are real
pions, a positive pion identification is required and optimized taking into
account the efficiency and rejection performance of particle identification
(PID) algorithms, and the observed kaon to pion ratio in the 3$\pi$
candidates, as measured through the ${D}^{-}$ peak when giving a kaon mass to
the negatively charged pion. As a result, the kaon identification probability
is required to be less than 17%. To keep the ${D}^{*-}$ reconstruction
efficiency as high as possible, the requirement on the kaon identification
probability for the soft-momentum pion originating from the ${D}^{*-}$ decay
is set to be less than $50$%. The
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$ and
${{D}^{+}}\\!\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{0}}$ decays
have large branching fractions and contribute to the
$B\rightarrow{{D}^{*-}}{{D}^{+}}(X)$ background, that is significant when the
kaon is misidentified as a pion. A remaining kaon contamination of about 5% in
the final sample is estimated by studying the ${K}^{-}$ ${\pi}^{+}$
${\pi}^{+}$ mass when assigning the kaon mass to the negative pion. Figure 6
shows the ${K}^{-}$ ${\pi}^{+}$ ${\pi}^{+}$ mass distribution for candidates
that have passed all analysis requirements, except that the ${\pi}^{-}$
candidate must have a high kaon identification probability. A clear ${D}^{+}$
signal of $740\pm 30$ candidates is visible, with little combinatorial
background. Therefore, an additional requirement on the kaon identification
probability of the ${\pi}^{-}$ candidate is applied. All of these PID
requirements are chosen in order to get the best discrimination between signal
and background. They form, together with the topology selection and the
isolation requirement defined above, the final selection.

Figure 6: Distribution of the ${K}^{-}$ ${\pi}^{+}$ ${\pi}^{+}$ mass for
$D^{+}$ candidates passing the signal selection, where the negative pion has
been identified as a kaon and assigned the kaon mass.

###  3.4 Selection of the normalization channel

The ${{B}^{0}}\rightarrow{{D}^{*-}}3\pi$ normalization channel is selected by
requiring the ${\kern 1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$ vertex to
be located at least 4$\sigma$ downstream of the 3$\pi$ vertex along the beam
direction, where $\sigma$ is the distance between the $B^{0}$ and ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ vertices divided by their
uncertainties added in quadrature. All other selection criteria are identical
to that of the signal case, except for the fact that no MVA requirement is
applied to the normalization channel. Figure 7 shows the ${{D}^{*-}}3\pi$ mass
spectrum after all these requirements. Moreover, the high purity of this
sample of exclusive ${B}^{0}$ decays allows the validation of the selection
efficiencies derived using simulation.

Figure 7: Distribution of the ${{D}^{*-}}3\pi$ mass for candidates passing the
selection.

###  3.5 Reconstruction of the decay kinematics

Due to the precise knowledge of the ${\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}$, 3$\pi$ and ${B}^{0}$ decay
vertices, it is possible to reconstruct the decay chains of both signal and
background processes, even in the presence of unreconstructed particles, such
as two neutrinos in the case of the signal, or neutral particles originating
at the 3$\pi$ vertex in the case of double-charm background. The relevant
reconstruction techniques are detailed in the following.

####  3.5.1 Reconstruction in the signal hypothesis

The missing information due to the two neutrinos emitted in the signal decay
chain can be recovered with the measurements of the ${B}^{0}$ and $\tau$ line
of flight (unit vectors joining the ${B}^{0}$ vertex to the PV and the 3$\pi$
vertex to the ${B}^{0}$ vertex, respectively) together with the known
${B}^{0}$ and $\tau$ masses. The reconstruction of the complete decay
kinematics of both the ${B}^{0}$ and $\tau$ decays is thus possible, up to two
two-fold ambiguities.

The $\tau$ momentum in the laboratory frame is obtained as (in units where
$c=1$)

$|\vec{p}_{\tau}|=\frac{(m_{3\pi}^{2}+m_{\tau}^{2})|\vec{p}_{3\pi}|\cos{\theta_{\tau,3\pi}}\pm
E_{3\pi}\sqrt{(m_{\tau}^{2}-m_{3\pi}^{2})^{2}-4m_{\tau}^{2}|\vec{p}_{3\pi}|^{2}\sin^{2}\theta_{\tau,3\pi}}}{2(E_{3\pi}^{2}-|\vec{p}_{3\pi}|^{2}\cos^{2}\theta_{\tau,3\pi})},$
(4)

where $\theta_{\tau,3\pi}$ is the angle between the $3\pi$ system three-
momentum and the $\tau$ line of flight; $m_{3\pi}$, $|\vec{p}_{3\pi}|$ and
$E_{3\pi}$ are the mass, three-momentum and energy of the $3\pi$ system,
respectively; and $m_{\tau}$ is the known $\tau$ mass. Equation 4 yields a
single solution, in the limit where the opening angle between the $3\pi$ and
the $\tau$ directions takes the maximum allowed value

$\theta^{\max}_{\tau,3\pi}=\arcsin\left(\frac{m_{\tau}^{2}-m_{3\pi}^{2}}{2m_{\tau}|\vec{p}_{3\pi}|}\right).$
(5)

At this value, the argument of the square root in Eq. 4 vanishes, leading to
only one solution, which is used as an estimate of the $\tau$ momentum. The
same procedure is applied to estimate the $B^{0}$ momentum

$\displaystyle|\vec{p}_{B^{0}}|$ $\displaystyle=$
$\displaystyle\frac{(m_{Y}^{2}+m_{B^{0}}^{2})|\vec{p}_{Y}|\cos{\theta_{B^{0},Y}}\pm
E_{Y}\sqrt{(m_{B^{0}}^{2}-m_{Y}^{2})^{2}-4m_{B^{0}}^{2}|\vec{p}_{Y}|^{2}\sin^{2}\theta_{B^{0},Y}}}{2(E_{Y}^{2}-|\vec{p}_{Y}|^{2}\cos^{2}\theta_{B^{0},Y})}$
(6)

by defining

$\displaystyle\theta^{\max}_{B^{0},Y}$ $\displaystyle=$
$\displaystyle\arcsin\left(\frac{m_{B^{0}}^{2}-m_{Y}^{2}}{2m_{B^{0}}|\vec{p}_{Y}|}\right),$
(7)

where $Y$ represents the $D^{*-}\tau$ system. Here, the three-momentum and
mass of the $D^{*-}\tau$ system are calculated using the previously estimated
$\tau$ momentum

$\displaystyle\vec{p}_{Y}={\vec{p}}_{D^{*-}}+{\vec{p}}_{\tau},\,\,\,\,\,\,E_{Y}=E_{D^{*-}}+E_{\tau},$
(8)

where ${\vec{p}}_{D^{*-}}$ and ${\vec{p}}_{\tau}$ are the three-momenta of the
$D^{*-}$ and the $\tau$ candidates, and $E_{D^{\ast-}}$ and $E_{\tau}$ their
energies. Using this method, the rest frame variables
${q^{2}}\equiv(p_{{{B}^{0}}}-p_{{D}^{\ast-}})^{2}=(p_{\tau}+p_{{{\nu}_{\tau}}})^{2}$
and the $\tau$ decay time, $t_{\tau}$, are determined with sufficient accuracy
to retain their discriminating power against double-charm backgrounds, as
discussed in Sec. 5. Figure 8 shows the difference between the reconstructed
and the true value of ${q^{2}}$ divided by the true $q^{2}$ on simulated
events. No significant bias is observed and an average resolution of
1.2${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ is obtained. The relative
$q^{2}$ resolution is $18\%$ full-width half-maximum. The slight asymmetry is
due to the presence at low $q^{2}$ of a tail of reconstructed $q^{2}$ below
the kinematical limit for true $q^{2}$.

Figure 8:  Difference between the reconstructed and true $q^{2}$ variables
divided by the true $q^{2}$, observed in the
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}}$ simulated signal
sample after partial reconstruction.

####  3.5.2 Reconstruction assuming a double-charm origin for the candidate

A full kinematic reconstruction of the $B$ decay chain specifically adapted to
two-body double-charm $B$ decays provides additional discrimination. After the
detached-vertex requirement, the main source of background candidates is
attributed to decays of the form $B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$,
with ${{D}^{+}_{s}}\\!\rightarrow 3{\pi}N$, $N$ being a system of
unreconstructed neutral particles. For these decays, the missing information
is due to a neutral system of unknown mass originating from the ${D}^{+}_{s}$
decay vertex, i.e. four unknowns. The measurements of the ${B}^{0}$ and
${D}^{+}_{s}$ lines of flight, providing four constraints, together with the
known ${B}^{0}$ mass, are sufficient to reconstruct the full decay kinematics

$|\vec{p}_{B}|\hat{u}_{B}=|\vec{p}_{{D}^{+}_{s}}|\hat{u}_{{D}^{+}_{s}}+\vec{p}_{{D}^{*-}}.$
(9)

This equation assumes the absence of any other particles in the $B$ decay. It
is however also valid when an additional particle is aligned with the
${D}^{+}_{s}$ momentum direction, as in the case of ${B}^{0}$ $\rightarrow$
${D}^{*-}$ ${D}^{*+}_{s}$ where the soft photon emitted in the ${D}^{*+}_{s}$
decay has a very low momentum in the direction transverse to that of the
${D}^{+}_{s}$ momentum. It is also a good approximation for quasi-two-body
${B}^{0}$ decays to ${D}^{*-}$ and higher excitations of the ${D}^{+}_{s}$
meson. This equation can be solved with two mathematically equivalent ways,
through a vectorial or scalar product methods, noted $v$ and $s$ respectively.
This equivalence does not hold in the presence of extra particles. This
difference is used to provide some further discrimination between signal and
nonisolated backgrounds. The magnitudes of the momenta obtained for each
method are:

$\displaystyle
P_{B,v}=\frac{|\vec{p}_{{D}^{*-}}\times\hat{u}_{{D}^{+}_{s}}|}{|\hat{u}_{B}\times\hat{u}_{{D}^{+}_{s}}|},$
(10a) $\displaystyle
P_{B,s}=\frac{\vec{p}_{{D}^{*-}}\cdot\hat{u}_{B}-(\vec{p}_{{D}^{*-}}\cdot\hat{u}_{{D}^{+}_{s}})(\hat{u}_{B}\cdot\hat{u}_{{D}^{+}_{s}})}{1-(\hat{u}_{B}\cdot\hat{u}_{{D}^{+}_{s}})^{2}},$
(10b)

for the ${B}^{0}$ momentum, and

$\displaystyle
P_{D_{s},v}=\frac{|\vec{p}_{{D}^{*-}}\times\hat{u}_{B}|}{|\hat{u}_{{D}^{+}_{s}}\times\hat{u}_{B}|},$
(11a) $\displaystyle
P_{D_{s},s}=\frac{(\vec{p}_{{D}^{*-}}\cdot\hat{u}_{B})(\hat{u}_{B}\cdot\hat{u}_{{D}^{+}_{s}})-\vec{p}_{{D}^{*-}}\cdot\hat{u}_{{D}^{+}_{s}}}{1-(\hat{u}_{B}\cdot\hat{u}_{{D}^{+}_{s}})^{2}},$
(11b)

for the ${D}^{+}_{s}$ momentum.

Since this partial reconstruction works without imposing a mass to the
$3{\pi}N$ system, the reconstructed $3{\pi}N$ mass can be used as a
discriminating variable. Figure 9 shows the $3{\pi}N$ mass distribution
obtained on a sample enriched in $B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$
decays, with ${{D}^{+}_{s}}\\!\rightarrow 3\pi N$, by means of the output of
the MVA (see Sec. 3.6). A peaking structure originating from ${D}^{+}_{s}$ and
${D}^{*+}_{s}$ decays is also present around $2000$ MeV/$c^{2}$. Due to the
presence of two neutrinos at different vertices, signal decays are not handled
as well by this partial reconstruction method, which therefore provides a
useful discrimination between signal and background due to
$B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ decays. However, this method
cannot discriminate the signal from double-charm backgrounds due to
$B\\!\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ and
$B\\!\rightarrow{{D}^{*-}}{{D}^{+}}(X)$ decays, where two kaons are missing at
the ${B}^{0}$ and 3$\pi$ vertices.

Figure 9:  Distribution of the reconstructed $3\pi N$ mass observed in a data
sample enriched by ${B}\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ candidates.

###  3.6 Multivariate analysis

Three features are used to reject the double-charm background: the different
resonant structures of ${\tau^{+}}\rightarrow 3{\pi}{{\overline{\nu}}_{\tau}}$
and ${{D}^{+}_{s}}\rightarrow 3{\pi}X$ decays, the neutral isolation and the
different kinematic properties of signal and background candidates. The latter
feature is exploited by using the reconstruction techniques described in Sec.
3.5.

To suppress double-charm background, a set of 18 variables is used as input to
a MVA based upon a boosted decision tree (BDT) [46, 47]. This set is as
follows: the output variables of the neutral isolation algorithm; momenta,
masses and quality of the reconstruction of the decay chain under the signal
and background hypotheses; the masses of oppositely charged pion pairs, the
energy and the flight distance in the transverse plane of the 3$\pi$ system;
the mass of the six-charged-tracks system. The BDT is trained using simulated
samples of signal and double-charm background decays. Figure 10 shows the
normalized distributions of the four input variables having the largest
discriminating power for signal and background: the minimum and maximum of the
masses of oppositely charged pions, $\min[m(\pi^{+}\pi^{-})]$ and
$\max[m(\pi^{+}\pi^{-})]$; the neutrino momentum, approximated as the
difference of the modulus of the momentum of the ${B}^{0}$ and the sum of the
moduli of the momenta of $D^{*-}$ and $\tau$ reconstructed in the signal
hypothesis; and the $D^{*-}3\pi$ mass. The BDT response for signal and
background is illustrated in Fig. 11.

Figure 10:  Normalized distributions of (a) $\min[m(\pi^{+}\pi^{-})]$, (b)
$\max[m(\pi^{+}\pi^{-})]$, (c) approximated neutrino momentum reconstructed in
the signal hypothesis, and (d) the $D^{*-}3\pi$ mass in simulated samples.
Figure 11:  Distribution of the BDT response on the signal and background
simulated samples.

The $B\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$,
$B\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ and $B\rightarrow{{D}^{*-}}{{D}^{+}}(X)$
control samples, described in Sec. 4, are used to validate the BDT. Good
agreement between simulation and control samples is observed both for the BDT
response and the distributions of the input variables.

The signal yield is determined from candidates in the region where the BDT
output is greater than $-0.075$. According to simulation, this value gives the
best statistical power in the determination of the signal yield. Candidates
with the BDT output less than $-0.075$ are highly enriched in ${D}^{+}_{s}$
decays and contain very little signal, as shown in Fig. 11, and represent
about half of the total data sample. They are used to validate the simulation
of the various components in ${{D}^{+}_{s}}\rightarrow 3\pi X$ decays used in
the parametrization of the templates entering in the fit that determines the
signal yield, as explained in Sec. 4.1. No BDT cut is applied in the selection
for the normalization channel.

###  3.7 Composition of the selected sample and selection efficiencies

Figure 12 shows the composition of an inclusive sample of simulated events,
generated by requiring that a ${D}^{*-}$ meson and a 3$\pi$ system are both
part of the decay chain of a $b$ $\overline{b}$ pair produced in a proton-
proton collision before the detached-vertex requirement, at the level of the
signal fit, and with a tighter cut corresponding to the last three BDT bins of
Fig. 16. In the histograms, the first bin corresponds to the signal,
representing only 1% of the candidates at the initial stage, and the second
bin to prompt candidates, where the 3$\pi$ system originates from the
$b$-hadron decay. It constitutes by far the largest initial background source.
The following three bins correspond to cases where the 3$\pi$ system
originates from the decay of a ${D}^{+}_{s}$, ${D}^{0}$ or ${D}^{+}$ meson,
respectively. The plot in the middle corresponds to the BDT output greater
than $-0.075$ used in the analysis to define the sample in which the signal
determination takes place. One can see the suppression of the prompt
background due to the detached-vertex requirement, and the dominance of the
${D}^{+}_{s}$ background. The bottom plot shows for illustration the sample
composition with the harder BDT output cut. The ${D}^{+}_{s}$ contribution is
now suppressed as well. The signal fraction represents about $25\%$ at this
stage. Figure 12 also allows contributions due to decays of other $b$ hadrons
to be compared with those of ${B}^{0}$ mesons.

Figure 12: Composition of an inclusive simulated sample where a ${D}^{*-}$ and
a 3$\pi$ system have been produced in the decay chain of a $b$ $\overline{b}$
pair from a $pp$ collision. Each bin shows the fractional contribution of the
different possible parents of the 3$\pi$ system (blue from a ${B}^{0}$, yellow
for other $b$ hadrons): from signal; directly from the $b$ hadron (prompt);
from a charm parent ${D}^{+}_{s}$, ${D}^{0}$, or ${D}^{+}$ meson; 3$\pi$ from
a $B$ and the ${D}^{0}$ from the other $B$ ($B1B2$); from $\tau$ lepton
following a ${D}^{+}_{s}$ decay; from a $\tau$ lepton following a
$D^{**}{\tau^{+}}{{\nu}_{\tau}}$ decay ($D^{**}$ denotes here any higher
excitation of $D$ mesons). (Top) After the initial selection and the removal
of spurious $3\pi$ candidates. (Middle) For candidates entering the signal
fit. (Bottom) For candidates populating the last 3 bins of the BDT
distribution (cf. Fig. 16).

Table 2 presents the efficiency of the various selection steps, both for
signal and normalization channels. The signal efficiency is computed from the
efficiencies and abundances of the 3$\pi$ and 3$\pi$ ${\pi}^{0}$ channels.

Table 2: Summary of the efficiencies (in %) measured at the various steps of the analysis for simulated samples of the ${{B}^{0}}\rightarrow{{D}^{*-}}3\pi$ channel and the ${B}^{0}$ $\rightarrow$ ${D}^{*-}$ $\tau^{+}$ ${\nu}_{\tau}$ signal channel for both $\tau$ decays to 3$\pi$ ${\overline{\nu}}_{\tau}$ and 3$\pi$ ${\pi}^{0}$ ${\overline{\nu}}_{\tau}$ modes. No requirement on the BDT output is applied for ${{D}^{*-}}3\pi$ candidates. The relative efficiency designates the individual efficiency of each requirement. Requirement |  Absolute efficiencies ($\%$) |  Relative efficiencies ($\%$)
---|---|---
| ${{D}^{*-}}3\pi$ |  ${D}^{*-}$ $\tau^{+}$ ${\nu}_{\tau}$ | ${{D}^{*-}}3\pi$ |  ${D}^{*-}$ $\tau^{+}$ ${\nu}_{\tau}$
|  |  3$\pi$ ${\overline{\nu}}_{\tau}$ |  3$\pi$ ${\pi}^{0}$ ${\overline{\nu}}_{\tau}$ |  |  3$\pi$ ${\overline{\nu}}_{\tau}$ |  3$\pi$ ${\pi}^{0}$ ${\overline{\nu}}_{\tau}$
Geometrical acceptance | 14.65 | 15.47 | 14.64 |  |  |
After: |  |  |  |  |  |
initial selection | 1.382 | 0.826 | 0.729 |  |  |
spurious 3$\pi$ removal | 0.561 | 0.308 | 0.238 | 40.6 | 37.3 | 32.6
trigger requirements | 0.484 | 0.200 | 0.143 | 86.3 | 65.1 | 59.9
vertex selection | 0.270 | 0.0796 | 0.0539 | 55.8 | 39.8 | 37.8
charged isolation | 0.219 | 0.0613 | 0.0412 | 81.2 | 77.0 | 76.3
BDT requirement | - | 0.0541 | 0.0292 | - | 94.1 | 74.8
PID requirements | 0.136 | 0.0392 | 0.0216 | 65.8 | 72.4 | 74.1

##  4 Study of double-charm candidates

The fit that determines the signal yield uses templates that are taken from
simulation. It is therefore of paramount importance to verify the agreement
between data and simulation for the remaining background processes. Control
samples from data are used wherever possible for this purpose. The relative
contributions of double-charm backgrounds and their $q^{2}$ distributions from
simulation are validated, and corrected where appropriate, by using data
control samples enriched in such processes. Inclusive decays of ${D}^{0}$,
${D}^{+}$ and ${D}^{+}_{s}$ mesons to 3$\pi$ are also studied in this way.

###  4.1 The ${{D}^{+}_{s}}$ decay model

The branching fraction of ${D}^{+}_{s}$ meson decays with a 3$\pi$ system in
the final state, denoted as ${{D}^{+}_{s}}\\!\rightarrow 3{\pi}X$ is about 15
times larger than that of the exclusive ${D}^{+}_{s}$ $\rightarrow$ 3$\pi$
decay. This is due to the large contributions from decays involving
intermediate states such as ${K}^{0}_{\mathrm{\scriptscriptstyle S}}$, $\eta$,
$\eta^{\prime}$, $\phi$, and $\omega$, which are generically denoted with the
symbol $R$ in the following. The branching fractions of processes of the type
${D}^{+}_{s}$ $\rightarrow$ $R$${\pi}^{+}$ are well known, but large
uncertainties exist for several decays, such as ${{D}^{+}_{s}}\rightarrow
R(\rightarrow{{\pi}^{+}}{{\pi}^{-}}X)\pi^{+}{{\pi}^{0}}$ and ${D}^{+}_{s}$
$\rightarrow$ $R$3$\pi$.

The $\tau$ lepton decays through the $a_{1}(1260)^{+}$ resonance, which leads
to the $\rho^{0}{\pi}^{+}$ final state [37]. The dominant source of $\rho^{0}$
resonances in ${D}^{+}_{s}$ decays is due to
$\eta^{\prime}\\!\rightarrow{{\rho}^{0}}\gamma$ decays. It is therefore
crucial to control the $\eta^{\prime}$ contribution in ${D}^{+}_{s}$ decays
very accurately. The $\eta^{\prime}$ contribution in the
$\min[m(\pi^{+}\pi^{-})]$ distribution, obtained from simulation, is shown in
Fig. 13. It exhibits a double peaking structure: at low mass, due to the
endpoint of phase space for the charged pion pair in the
$\eta\rightarrow{{\pi}^{+}}{{\pi}^{-}}{{\pi}^{0}}$ and
$\eta^{\prime}\rightarrow\eta{{\pi}^{+}}{{\pi}^{-}}$ decays and, at higher
mass, a ${{\rho}^{0}}$ peak. The shape of this contribution is precisely known
since the $\eta^{\prime}$ branching fractions are known to better than 2%. The
precise measurement on data of the low-mass excess, which consists only of
$\eta^{\prime}$ and $\eta$ candidates, therefore enables the control of the
$\eta^{\prime}$ contribution in the sensitive $\rho$ region. The
${{D}^{+}_{s}}\\!\rightarrow 3{\pi}X$ decay model is determined from a data
sample enriched in $B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ decays by
requiring a low value of the BDT output. The distributions of
$\min[m(\pi^{+}\pi^{-})]$ and $\max[m(\pi^{+}\pi^{-})]$, of the mass of the
same-charge pions, $m(\pi^{+}\pi^{+})$, and of the mass of the 3$\pi$ system,
$m(3{\pi})$, are simultaneously fit with a model obtained from simulation. The
fit model is constructed from the following components:

  * •

${D}^{+}_{s}$ decays where at least one pion originates from the decay of an
$\eta$ meson; the ${{D}^{+}_{s}}\rightarrow\eta{{\pi}^{+}}$ and
${{D}^{+}_{s}}\rightarrow\eta{{\rho}^{+}}$ components are in this category.

  * •

${D}^{+}_{s}$ decays where, in analogy with the previous category, an
$\eta^{\prime}$ meson is involved.

  * •

${D}^{+}_{s}$ decays where at least one pion originates from an intermediate
resonance other than $\eta$ or $\eta^{\prime}$; these are then subdivided into
$R{{\pi}^{+}}$ and $R{{\rho}^{+}}$ final states; these decays are dominated by
$R=\omega$, $\phi$ resonances.

  * •

Other ${D}^{+}_{s}$ decays, where none of the three pions originates from an
intermediate state; these are then subdivided into $K^{0}3\pi$, $\eta 3\pi$,
$\eta^{\prime}3\pi$, $\omega 3\pi$, $\phi 3\pi$, $\tau^{+}(\rightarrow
3\pi(N)\overline{\nu}_{\tau})\nu_{\tau}$, and $3\pi$ nonresonant final states,
$X_{\rm nr}$. Regarding the tauonic
${{D}^{+}_{s}}\rightarrow\tau^{+}\nu_{\tau}$ decay, the label $N$ stands for
any potential extra neutral particle.

Templates for each category and for the non-${D}^{+}_{s}$ candidates are
determined from $B\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ and
$B\rightarrow{{D}^{*-}}3{\pi}X$ simulation samples, respectively. Figure 13
shows the fit results for the four variables. The fit measures the $\eta$ and
$\eta^{\prime}$ inclusive fractions very precisely because, in the
$\min[m({{\pi}^{+}}{{\pi}^{-}})]$ histogram, the low-mass peak is the sum of
the $\eta$ and $\eta^{\prime}$ contributions, while only the $\eta^{\prime}$
meson contributes to the $\rho^{0}$ region. The ratio between decays with a
${\pi}^{+}$ and a ${\rho}^{+}$ meson in the final state is not precisely
determined because of the limited sensitivity of the fit variables to the
presence of the extra ${\pi}^{0}$. The sensitivity only comes from the low-
yield high-mass tail of the 3$\pi$ mass distribution which exhibits different
endpoints for these two types of decays. Finally, the kinematical endpoints of
the 3$\pi$ mass for each $R3{\pi}$ final state enable the fit to determine
their individual contributions, which are presently either poorly measured or
not measured at all. The ${D}^{+}_{s}$ $\rightarrow$ $\phi$3$\pi$ and
${D}^{+}_{s}$ $\rightarrow$ $\tau^{+}(\rightarrow
3\pi(N)\overline{\nu}_{\tau})\nu_{\tau}$ branching fractions, known with a 10%
precision, are fixed to their measured values [48].

Figure 13: Distributions of (a) ${\mathrm{min}}[m(\pi^{+}\pi^{-})]$, (b)
${\mathrm{max}}[m(\pi^{+}\pi^{-})]$, (c) $m(\pi^{+}\pi^{+})$, (d)
$m(\pi^{+}\pi^{-}\pi^{+})$ for a sample enriched in
$B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ decays, obtained by requiring the
BDT output below a certain threshold. The different fit components correspond
to ${D}^{+}_{s}$ decays with (red) $\eta$ or (green) $\eta^{\prime}$ in the
final state, (yellow) all the other considered ${D}^{+}_{s}$ decays, and
(blue) backgrounds originating from decays not involving the ${D}^{+}_{s}$
meson.

The fit is in good agreement with the data, especially in the critical
$\min[m({{\pi}^{+}}{{\pi}^{-}})]$ distribution. The $\chi^{2}$ per degree of
freedom of each fit is 0.91, 1.25, 1.1 and 1.45 for each histogram,
respectively, when taking into account the simulation sample size. The fit
parameters and their ratios, with values from simulation, are reported in
Table 3. These are used to correct the corresponding contributions from
simulation. In the final fit performed in the high BDT output region, the
shape of each contribution is scaled according to the ratio of candidates in
the two BDT regions, which is taken from simulation.

The fit determines that ($47.3\pm 2.5$)% of the ${D}^{+}_{s}$ decays in this
sample contain $\eta$ and $\eta^{\prime}$ mesons with an additional charged
pion, ($20.6\pm 4.0$)% contain $\phi$ and $\omega$ mesons with an additional
charged pion and ($32.1\pm 4.0$)% are due to $R$3$\pi$ modes. This last
contribution is dominated by the $\eta 3\pi$ and $\eta^{\prime}3\pi$ modes.
The large weighting factors observed in this ${D}^{+}_{s}$ decay-model fit
correspond to channels whose branching fractions are not precisely known.

Table 3: Results of the fit to the ${D}^{+}_{s}$ decay model. The relative contribution of each decay and the correction to be applied to the simulation are reported in the second and third columns, respectively. ${D}^{+}_{s}$ decay | Relative | Correction
---|---|---
| contribution | to simulation
$\eta{{\pi}^{+}}(X)$ | 0.156 $\pm$ 0.010 |
$\eta{{\rho}^{+}}$ |  0.109 $\pm$ 0.016 | 0.88 $\pm$ 0.13
$\eta{{\pi}^{+}}$ | 0.047 $\pm$ 0.014 | 0.75 $\pm$ 0.23
$\eta^{\prime}{{\pi}^{+}}(X)$ | 0.317 $\pm$ 0.015 |
$\eta^{\prime}{{\rho}^{+}}$ | 0.179 $\pm$ 0.016 | 0.710 $\pm$ 0.063
$\eta^{\prime}{{\pi}^{+}}$ | 0.138 $\pm$ 0.015 | 0.808 $\pm$ 0.088
$\phi{{\pi}^{+}}(X),\,\omega{{\pi}^{+}}(X)$ | 0.206 $\pm$ 0.02 |
$\phi{{\rho}^{+}},\,\omega{{\rho}^{+}}$ | 0.043 $\pm$ 0.022 | 0.28 $\pm$ 0.14
$\phi{{\pi}^{+}},\,\omega{{\pi}^{+}}$ | 0.163 $\pm$ 0.021 | 1.588 $\pm$ 0.208
$\eta 3\pi$ | 0.104 $\pm$ 0.021 | 1.81 $\pm$ 0.36
$\eta^{\prime}3\pi$ | 0.0835 $\pm$ 0.0102 | 5.39 $\pm$ 0.66
$\omega 3\pi$ | 0.0415 $\pm$ 0.0122 | 5.19 $\pm$ 1.53
$K^{0}3\pi$ | 0.0204 $\pm$ 0.0139 | 1.0 $\pm$ 0.7
$\phi 3\pi$ | 0.0141 | 0.97
${\tau^{+}}(\rightarrow 3\pi(N){{\overline{\nu}}_{\tau}}){{\nu}_{\tau}}$ | 0.0135 | 0.97
$X_{\rm nr}3\pi$ | 0.038 $\pm$ 0.005 | 6.69 $\pm$ 0.94

###  4.2 The $B\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ control sample

Candidates where the ${D}^{+}_{s}$ meson decays exclusively to the
$\pi^{+}\pi^{-}\pi^{+}$ final state give a pure sample of
$B\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ decays. This sample includes three
types of processes:333In this Section, $D^{**}$ and $D_{s}^{**}$ are used to
refer to any higher-mass excitations of ${D}^{*-}$ or ${D}^{+}_{s}$ mesons
decaying to $D^{*-}$ and $D_{s}^{+}$ ground states.

  * •

${{B}^{0}}\rightarrow{{D}^{*-}}D_{s}^{(*,**)+}$ decays, where a neutral
particle is emitted in the decay of the excited states of the $D_{s}^{+}$
meson. The corresponding $q^{2}$ distribution peaks at the squared mass,
$(p_{B^{0}}-p_{D^{*-}})^{2}$, of the given states.

  * •

${{B}^{0}_{s}}\rightarrow{{D}^{*-}}{{D}^{+}_{s}}X$ decays, where at least one
additional particle is missing. This category contains feed-down from excited
states, both for ${D}^{*-}$ or ${D}^{+}_{s}$ mesons. The $q^{2}$ distribution
is shifted to higher values.

  * •

$B^{0,-}\rightarrow{{D}^{*-}}{{D}^{+}_{s}}X^{0,-}$ decays, where at least one
additional particle originates from either the $B^{0,-}$ decay, or the
deexcitation of charm-meson resonances of higher mass, that results in a
${D}^{*-}$ meson in the final state. These additional missing particles shift
the $q^{2}$ distribution to even higher values.

The $B\\!\rightarrow{{D}^{*-}}{{D}^{+}_{s}}(X)$ control sample is used to
evaluate the agreement between data and simulation, by performing a fit to the
distribution of the mass of the ${{D}^{*-}}3{\pi}$ system,
$m({{D}^{*-}}3{\pi})$. The fitting probability density function $\cal{P}$ is
parametrized as

$\displaystyle{\cal{P}}=f_{\rm c.b.}~{}{\cal{P}}_{\rm c.b.}+\frac{(1-f_{\rm
c.b.})}{k}\sum_{j}f_{j}{\cal{P}}_{j}$ (12)

where
$i,j=\\{D_{s}^{*+};~{}D_{s}^{+};~{}D_{s0}^{*+};~{}D_{s1}^{+};~{}D_{s}^{+}X;~{}(D_{s}^{+}X)_{s}\\}$
and $k=\sum_{i}f_{i}$. The fraction of combinatorial background, $f_{\rm
c.b.}$, is fixed in the fit. Its shape is taken from a sample where the
$D^{*-}$ meson and the $3\pi$ system have the same charge. Each component $i$
is described by the probability density function ${\cal{P}}_{i}$, whose shapes
are taken from simulation. The parameters $f_{i}$ are the relative yields of
$B^{0}\rightarrow D^{*-}D_{s}^{+}$, $B^{0}\rightarrow
D^{*-}D^{*}_{s0}(2317)^{+}$, $B^{0}\rightarrow D^{*-}D_{s1}(2460)^{+}$,
$B^{0,+}\rightarrow D^{*-}D_{s}^{+}X$ and $B_{s}^{0}\rightarrow
D^{*-}D_{s}^{+}X$ decays with respect to the number of $B^{0}\rightarrow
D^{*-}D_{s}^{*+}$ candidates. They are floating in the fit, and
$f_{D_{s}^{*+}}=1$ by definition.

The fit results are shown in Fig. 14 and reported in Table 4, where a
comparison with the corresponding values in the simulation is also given,
along with their ratios. The measured ratios, including the uncertainties and
correlations, are used to constrain these contributions in the final fit. The
large weighting factors observed in this fit correspond to channels whose
branching fractions are not precisely known.

Table 4: Relative fractions of the various components obtained from the fit to the $B\rightarrow D^{*-}D_{s}^{+}(X)$ control sample. The values used in the simulation and the ratio of the two are also shown. Parameter | Simulation | Fit | Ratio
---|---|---|---
$f_{\rm c.b.}$ | — | $0.014$ | —
$f_{D_{s}^{+}}$ | $0.54$ | $0.594\pm 0.041$ | $1.10\pm 0.08$
$f_{D_{s0}^{*+}}$ | $0.08$ | $0.000^{+0.040}_{-0.000}~{}~{}$ | $0.00^{+0.50}_{-0.00}~{}$
$f_{D_{s1}^{+}}$ | $0.39$ | $0.365\pm 0.053$ | $0.94\pm 0.14$
$f_{D_{s}^{+}X}$ | $0.22$ | $0.416\pm 0.069$ | $1.89\pm 0.31$
$f_{(D_{s}^{+}X)_{s}}$ | $0.23$ | $0.093\pm 0.027$ | $0.40\pm 0.12$

Figure 14:  Results from the fit to data for candidates containing a
${D}^{*-}$ ${D}^{+}_{s}$ pair, where ${{D}^{+}_{s}}\rightarrow 3\pi$. The fit
components are described in the legend. The figures correspond to the fit
projection on (a) $m(D^{*-}3\pi)$, (b) $q^{2}$, (c) 3$\pi$ decay time
$t_{\tau}$ and (d) BDT output distributions.  Figure 15:  Distribution of
$q^{2}$ for candidates in the $B\rightarrow D^{*-}D^{0}(X)$ control sample,
after correcting for the disagreement between data and simulation.

###  4.3 The $B\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ and
$B\rightarrow{{D}^{*-}}{{D}^{+}}(X)$ control samples

The decays of ${D}^{0}$ and ${D}^{+}$ mesons into final states with three
pions are dominated by the $D^{0,+}\\!\rightarrow K^{-,0}3\pi({{\pi}^{0}})$
modes, whose subresonant structure is known. The agreement between data and
simulation is validated in the ${D}^{0}$ case by using a control sample. The
isolation algorithm identifies a kaon with charge opposite to the total charge
of the 3$\pi$ system, and compatible with originating from the 3$\pi$ vertex.
The mass of the ${{K}^{-}}3{\pi}$ system must be compatible with the known
${D}^{0}$ mass. Disagreement between data and simulation is found in the
$q^{2}$ and ${D}^{*-}$ ${D}^{0}$ mass distributions, and corrected for. Figure
15 shows the $q^{2}$ distribution after this correction.

A pure sample of $B\\!\rightarrow{{D}^{*-}}{{D}^{+}}(X)$ decays is obtained by
inverting the PID requirements on the negative pion of the 3$\pi$ system,
assigning to this particle the kaon mass and selecting 3$\pi$ candidates with
mass compatible with the known ${D}^{+}$ mass. As in the
$B\\!\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ control sample, disagreement between
data and simulation is found. The limited size of this sample does not allow
the determination of a specific correction. The same correction found in the
$B\\!\rightarrow{{D}^{*-}}{{D}^{0}}(X)$ case is therefore applied, since the
dominant decay ${B}\rightarrow{{D}^{*-}}{D}K$ is identical for both cases.

##  5 Determination of the signal yield

The yield of $B^{0}\rightarrow D^{*-}\tau^{+}\nu_{\tau}$ decays is determined
from a three-dimensional binned maximum likelihood fit to the distributions of
$q^{2}$, $3\pi$ decay time, and BDT output. Signal and background templates
are produced with eight bins in $q^{2}$, eight bins in $t_{\tau}$, and four
bins in the BDT output, from the corresponding simulation samples. The model
used to fit the data is summarized in Table 5. In the table,

Table 5: Summary of fit components and their corresponding normalization parameters. The first three components correspond to parameters related to the signal. Fit component | Normalization
---|---
$B^{0}\rightarrow D^{*-}\tau^{+}(\rightarrow 3\pi\overline{\nu}_{\tau})\nu_{\tau}$ | ${N_{\rm{sig}}}\times{f_{\tau\rightarrow 3\pi\nu}}$
$B^{0}\rightarrow D^{*-}\tau^{+}(\rightarrow 3\pi\pi^{0}\overline{\nu}_{\tau})\nu_{\tau}$ | ${N_{\rm{sig}}}\times(1-{f_{\tau\rightarrow 3\pi\nu}})$
$B\rightarrow D^{**}\tau^{+}\nu_{\tau}$ | ${N_{\rm{sig}}}\times{f_{D^{**}\tau\nu}}$
$B\rightarrow D^{*-}D^{+}X$ | ${f_{D^{+}}}\times{N_{D_{s}}}$
$B\rightarrow D^{*-}D^{0}X$ different vertices | ${f_{D^{0}}^{v_{1}v_{2}}}\times{N_{D^{0}}^{\rm{sv}}}$
$B\rightarrow D^{*-}D^{0}X$ same vertex | ${N_{D^{0}}^{\rm{sv}}}$
$B^{0}\rightarrow D^{*-}D_{s}^{+}$ | ${N_{D_{s}}}\times f_{D_{s}^{+}}/k$
$B^{0}\rightarrow D^{*-}D_{s}^{*+}$ | ${N_{D_{s}}}\times 1/k$
$B^{0}\rightarrow D^{*-}D_{s0}^{*}(2317)^{+}$ | ${N_{D_{s}}}\times f_{D_{s0}^{*+}}/k$
$B^{0}\rightarrow D^{*-}D_{s1}(2460)^{+}$ | ${N_{D_{s}}}\times f_{D_{s1}^{+}}/k$
$B^{0,+}\rightarrow D^{**}D_{s}^{+}X$ | ${N_{D_{s}}}\times f_{D_{s}^{+}X}/k$
$B_{s}^{0}\rightarrow D^{*-}D_{s}^{+}X$ | ${N_{D_{s}}}\times f_{(D_{s}^{+}X)_{s}}/k$
$B\rightarrow D^{*-}3\pi X$ | ${N_{B\rightarrow D^{\ast}3\pi X}}$
B1B2 combinatorics | ${N_{B1B2}}$
Combinatoric $D^{*-}$ | ${N_{{\rm{not}}D^{*}}}$

  * •

${N_{\rm{sig}}}$ is a free parameter accounting for the yield of signal
candidates.

  * •

${f_{\tau\rightarrow 3\pi\nu}}$ is the fraction of ${\tau^{+}}\rightarrow
3\pi{{\overline{\nu}}_{\tau}}$ signal candidates with respect to the sum of
the ${\tau^{+}}\rightarrow 3\pi{{\overline{\nu}}_{\tau}}$ and
${\tau^{+}}\rightarrow 3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ components.
This parameter is fixed to $0.78$, according to the different branching
fractions and efficiencies of the two modes.

  * •

${f_{D^{**}\tau\nu}}$, fixed to $0.11$, is the ratio of the yield of
$B\rightarrow D^{**}{\tau^{+}}{{\nu}_{\tau}}$ decay candidates to the signal
decays. This yield is computed assuming that the ratio of the decay rates lies
between the ratio of available phase space ($0.18$) and the predictions of
Ref.[49] ($0.06$), and taking into account the relative efficiencies of the
different channels.

  * •

${N_{D^{0}}^{\rm{sv}}}$ is the yield of $B\rightarrow D^{*-}D^{0}X$ decays
where the three pions originate from the same vertex (SV) as the the $D^{0}$
vertex. The $D^{0}\rightarrow K^{+}\pi^{-}\pi^{+}\pi^{-}(\pi^{0})$ decays are
reconstructed by recovering a charged kaon pointing to the $3\pi$ vertex in
nonisolated events. The exclusive $D^{0}\rightarrow
K^{+}\pi^{-}\pi^{+}\pi^{-}$ peak is used to apply a $5\%$ Gaussian constraint
to this parameter, accounting for the knowledge of the efficiency in finding
the additional kaon.

  * •

${f_{D^{0}}^{v_{1}v_{2}}}$ is the ratio of $B\rightarrow D^{*-}D^{0}X$ decays
where at least one pion originates from the $D^{0}$ vertex and the other
pion(s) from a different vertex, normalized to ${N_{D^{0}}^{\rm{sv}}}$. This
is the case when the soft pion from a $D^{*-}$ decay is reconstructed as it
was produced at the $3\pi$ vertex.

  * •

${f_{D^{+}}}$ is the ratio of $B\rightarrow{{D}^{*-}}D^{+}X$ decays with
respect to those containing a $D_{s}^{+}$ meson.

  * •

$N_{D_{s}}$ is the yield of events involving a $D_{s}^{+}$. The parameters
$f_{D_{s}^{+}}$, $f_{D_{s0}^{*+}}$, $f_{D_{s1}^{+}}$, $f_{D_{s}^{+}X}$,
$f_{(D_{s}^{+}X)_{s}}$ and $k$, defined in Sec. 4.2, are used after correcting
for efficiency.

  * •

${N_{B\rightarrow D^{*}3\pi X}}$ is the yield of $B\rightarrow D^{*-}3\pi X$
events where the three pions come from the $B$ vertex. This value is
constrained by using the observed ratio between $B^{0}\rightarrow D^{*-}3\pi$
exclusive and $B\rightarrow D^{*-}3\pi X$ inclusive decays, corrected for
efficiency.

  * •

${N_{B1B2}}$ is the yield of combinatorial background events where the
$D^{*-}$ and the 3$\pi$ system come from different $B$ decays. Its yield is
fixed by using the yield of wrong-sign events
$D^{*-}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ in the region
$m(D^{*-}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}})>5.1$ GeV/${\it{c}}^{2}$.

  * •

${N_{{\rm{not}}D^{*}}}$ is the combinatorial background yield with a fake
$D^{*-}$. Its value is fixed by using the number of events in the ${{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}$ mass sidebands of the
$D^{*-}\rightarrow{{\kern
1.99997pt\overline{\kern-1.99997ptD}{}}{}^{0}}\pi^{-}$ decay.

###  5.1 Fit results

Figure 16:  Projections of the three-dimensional fit on the (a) $3\pi$ decay
time, (b) $q^{2}$ and (c) BDT output distributions. The fit components are
described in the legend.  Figure 17:  Distributions of (a) $t_{\tau}$ and (b)
$q^{2}$ in four different BDT bins, with increasing values of the BDT response
from top to bottom. The fit components are described in the legend.

The results of the three-dimensional fit are shown in Table 6 and Fig. 16. A
raw number of $1336$ decays translates into a yield of $N_{\rm{sig}}=1296\pm
86$ $B^{0}\rightarrow D^{*-}\tau^{+}\nu_{\tau}$ decays, after a correction of
$-3\%$ due to a fit bias is applied, as detailed below. Figure 17 shows the
results of the fit in bins of the BDT output. The two most discriminant
variables of the BDT response are the variables $\min[m(\pi^{+}\pi^{-})]$ and
$m(D^{*-}3\pi)$. Figure 18 shows the fit results projected onto these
variables. A good agreement with data and the post-fit model is found. The fit
$\chi^{2}$ is 1.15 per degree of freedom, after taking into account the
statistical fluctuation in the simulation templates, and 1.8 without. Due to
the limited size of the simulation samples used to build the templates (the
need to use templates from inclusive $b$-hadron decays requires extremely
large simulation samples), the existence of empty bins in the templates
introduces potential biases in the determination of the signal yield that must
be taken into account. To study this effect, a method based on the use of
kernel density estimators (KDE) [50] is used. For each simulated sample, a
three-dimensional density function is produced. Each KDE is then transformed
in a three-dimensional template, where bins that were previously empty may now
be filled. These new templates are used to build a smoothed fit model. The fit
is repeated with different signal yield hypotheses. The results show that a
bias is observed for low values of the generated signal yield that decreases
when the generated signal yield increases. For the value found by the nominal
fit, a bias of $+40$ decays is found, and is used to correct the fit result.

The statistical contribution to the total uncertainty is determined by
performing a second fit where the parameters governing the templates shapes of
the double-charmed decays, $f_{D_{s}^{+}}$, $f_{D_{s0}^{*+}}$,
$f_{D_{s1}^{+}}$, $f_{D_{s}^{+}X}$, $f_{(D_{s}^{+}X)_{s}}$ and
$f_{D^{0}}^{v_{1}v_{2}}$, are fixed to the values obtained in the first fit.
The quadratic difference between the uncertainties provided by the two fits is
taken as systematic uncertainty due to the knowledge of the $B\rightarrow
D^{*-}D_{s}^{+}X$ and $B\rightarrow D^{*-}D^{0}X$ decay models, and reported
in Table 7.

Table 6: Fit results for the three-dimensional fit. The constraints on the parameters $f_{D_{s}^{+}}$, $f_{D_{s0}^{*+}}$, $f_{D_{s1}^{+}}$, $f_{D_{s}^{+}X}$ and $f_{(D_{s}^{+}X)_{s}}$ are applied taking into account their correlations. Parameter | Fit result | Constraint
---|---|---
$N_{\rm sig}$ | $1296\pm 86$ |
$f_{\tau\rightarrow 3\pi\nu}$ |  1$0.78$ |  11$0.78$ (fixed)
$f_{D^{**}\tau\nu}$ |  1$0.11$ |  11$0.11$ (fixed)
$N_{D^{0}}^{\rm{sv}}$ |  1$445\pm 22$ | $445\pm 22$
$f_{D^{0}}^{v_{1}v_{2}}$ |  11$0.41\pm 0.22$ |
$N_{D_{s}}$ |  1$6835\pm 166$ |
$f_{D^{+}}$ |  11$0.245\pm 0.020$ |
$N_{B\rightarrow D^{*}3\pi X}$ |  1$424\pm 21$ | $443\pm 22$
$f_{D_{s}^{+}}$ |  11$0.494\pm 0.028$ |  1$0.467\pm 0.032$
$f_{D_{s0}^{*+}}$ |  .4711$0^{+0.010}_{-0.000}$ |  .461$0^{+0.042}_{-0.000}$
$f_{D_{s1}^{+}}$ |  11$0.384\pm 0.044$ |  1$0.444\pm 0.064$
$f_{D_{s}^{+}X}$ |  11$0.836\pm 0.077$ |  1$0.647\pm 0.107$
$f_{(D_{s}^{+}X)_{s}}$ |  11$0.159\pm 0.034$ |  1$0.138\pm 0.040$
$N_{B1B2}$ |  1$197$ |  11$197$ (fixed)
$N_{{\rm{not}}D^{*}}$ |  1$243$ |  11$243$ (fixed)

Figure 18:  Projection of the fit results on (a)
${\mathrm{min}}[m(\pi^{+}\pi^{-})]$ and (b) $m(D^{*-}3\pi)$ distributions. The
fit components are described in the legend.

##  6 Determination of normalization yield

Figure 7 shows the $D^{*-}3\pi$ mass after the selection of the normalization
sample. A clear ${B}^{0}$ signal peak is seen. In order to determine the
normalization yield, a fit is performed in the region between 5150 and
5400${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The signal component is
described by the sum of a Gaussian function and a Crystal Ball function [51].
An exponential function is used to describe the background. The result of the
fit is shown in Fig. 19. The yield obtained is $17\ 808\pm 143$.

The fit is also performed with alternative configurations, namely with a
different fit range or requiring the common mean value of the signal functions
to be the same in the $7$ and $8$$\mathrm{\,Te\kern-1.00006ptV}$ data samples.
The maximum differences between signal yields in alternative and nominal
configurations are $14$ and $62$ for the $7$ and
$8$$\mathrm{\,Te\kern-1.00006ptV}$ data samples, respectively, and are used to
assign systematic uncertainties to the normalization yields.

Figure 19:  Fit to the $m(D^{*-}3\pi)$ distribution after the full selection
in the (a) $\sqrt{s}=7$ TeV and (b) $8$ TeV data samples.

Figure 20 shows the $m(3\pi)$ distribution for candidates with $D^{*-}3\pi$
mass between $5200$ and $5350$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
the full data sample. The spectrum is dominated by the $a_{1}(1260)^{+}$
resonance but also a smaller peak due to the ${{D}^{+}_{s}}\rightarrow 3\pi$
decay is visible and is subtracted. A fit with the sum of a Gaussian function
modeling the ${D}^{+}_{s}$ mass peak, and an exponential describing the
combinatorial background, is performed to estimate this ${D}^{+}_{s}$
contribution, giving $151\pm 22$ candidates. As a result, the number of
normalization decays in the full data sample is $N_{\mathrm{norm}}=17\ 660\pm
143\mathrm{\,(stat)}\pm 64\mathrm{\,(syst)}\pm 22\mathrm{\,(sub)}$, where the
third uncertainty is due to the subtraction of the
${{B}^{0}}\rightarrow{{D}^{*-}}{{D}^{+}_{s}}$ component.

Figure 20:  (a) Distribution of $m(3\pi)$ after selection, requiring
$m(D^{*-}3\pi)$ to be between 5200 and
5350${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$; (b) fit in the mass region
around the ${D}^{+}_{s}$.

##  7 Determination of ${\cal{K}}({{D}^{*-}})$

The result

${\cal{K}}({{D}^{*-}})=1.97\pm 0.13\mathrm{\,(stat)}\pm
0.18\mathrm{\,(syst)},$

is obtained using Eq. 3. The ratio of efficiencies between the signal and
normalization modes, shown in Table 2, differs from unity due to the softer
momentum spectrum of the signal particles and the correspondingly lower
trigger efficiency. The effective sum of the branching fractions for the
${\tau^{+}}\rightarrow 3\pi{{\overline{\nu}}_{\tau}}$ and
${\tau^{+}}\rightarrow 3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ decays is
$(13.81\pm 0.07)\%$ [48]. This includes the 3$\pi$ mode (without ${K}^{0}$), a
very small feed-down from $\tau$ five-prong decays, the 3$\pi$ ${\pi}^{0}$
mode (without ${K}^{0}$), and only 50% of the 3$\pi$ ${\pi}^{0}$ ${\pi}^{0}$
mode due to the smaller efficiency of this decay mode. This latter
contribution results in a $1\%$ correction (see Sec. 8.1). Finally, a
correction factor $1.056\pm 0.025$ is applied when computing
${\cal{K}}({{D}^{*-}})$ in order to account for residual efficiency
discrepancies between data and simulation regarding PID and trigger. The event
multiplicity, measured by the scintillating-pad detector, affects the
efficiency for the fraction of the data sample which is triggered at the
hardware trigger level by particles in the event other than those from the
${{D}^{*-}}\tau^{+}{{\nu}_{\tau}}$ candidate. An imperfect description of this
multiplicity in the simulation does not cancel completely in
${{\cal{K}}({{D}^{*-}})}$. The correction factor also includes a small feed-
down contribution from ${{B}^{0}_{s}}\\!\rightarrow
D_{s}^{**-}{\tau^{+}}{{\nu}_{\tau}}$ decays, where
$D_{s}^{**-}\\!\rightarrow{{D}^{*-}}K^{0}$, that is taken into account
according to simulation.

As a further check of the analysis, measurements of ${\cal{K}}({{D}^{*-}})$
are performed in mutually exclusive subsamples, obtained by requiring
different trigger conditions and center-of-mass energies. All of these results
are found to be compatible with the result obtained with the full sample.
Changing the requirement on the minimal BDT output value, as well as the
bounds of the nuisance parameters, does not change the final result.

##  8 Systematic uncertainties

The systematic uncertainties on ${\cal{K}}({{D}^{*-}})$ are subdivided into
four categories: the knowledge of the signal model, including $\tau$ decay
models; the modeling of the various background sources; possible biases in the
fit procedure due to the limited size of the simulated samples; and trigger
selection efficiencies, external inputs and particle identification
efficiency. Table 7 summarizes the results.

###  8.1 Signal model uncertainties

The uncertainty in the relative proportion of signal events in the mode
${\tau^{+}}\rightarrow 3\pi{{\overline{\nu}}_{\tau}}$ and
${\tau^{+}}\rightarrow 3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ affects the
fit results. Taking into account the relative efficiencies, an uncertainty of
$0.01$ is assigned to $f_{\tau\rightarrow 3\pi\nu}$. A fit is performed with
this fraction constrained to $0.78\pm 0.01$ using a Gaussian function. A
second fit is done fixing the fraction to the value found by the first fit.
The squared difference between the uncertainties of the two fits is taken as
the systematic uncertainty due to the signal composition, resulting in a
$0.7\%$ systematic uncertainty.

To estimate the systematic uncertainty due to the knowledge of the
$B^{0}\rightarrow D^{*-}\tau^{+}\nu_{\tau}$ form factors, a study based on
pseudoexperiments is performed. A total of 100 fits to generated samples is
done by varying the values of the parameters $R_{1}(1)$, $R_{2}(1)$ and
$\rho^{2}$ of Ref. [2] which govern the fraction of each spin configuration in
the form-factor templates. The parameters are varied according to a
multivariate Gaussian distribution using their uncertainties and correlations.
The parameter $R_{0}(1)=1.14\pm 0.11$ is varied under the conservative
assumption that it is not correlated with the other parameters. A systematic
uncertainty of $0.7\%$ on the signal yield is obtained by taking the standard
deviation of the distribution of the fitted signal yields.

A value of 1% systematic uncertainty of the efficiency due to the form factor
reweighting is computed by repeating the fit without it.

The effect of the $\tau$ polarization is studied separately for
${\tau^{+}}\rightarrow 3\pi{{\overline{\nu}}_{\tau}}$ and
${\tau^{+}}\rightarrow 3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ decays. Due
to the $a_{1}(1260)^{+}$ dominance observed in the ${\tau^{+}}\rightarrow
3\pi{{\overline{\nu}}_{\tau}}$ decay, the sensitivity of the 3$\pi$ momenta to
the polarization is negligible and therefore no systematic uncertainty is
assigned due to this effect. For the ${\tau^{+}}\rightarrow
3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ decay mode, the signal is simulated
in two configurations: using either the TAUOLA [52] model or a pure phase-
space model. The effect of the $\tau$ polarization is evaluated by multiplying
the efficiency by the ratio of the distributions of the cosine of $\alpha$
(the angle between the 3$\pi$ momentum in the $\tau$ rest frame and the $\tau$
direction in the laboratory frame) generated with the two configurations. This
produces a relative change in the efficiency of $1.5\%$. This value, scaled by
the relative fraction of the ${\tau^{+}}\rightarrow
3\pi{{\pi}^{0}}{{\overline{\nu}}_{\tau}}$ component with respect to the total,
gives a systematic uncertainty of 0.4%.

Other $\tau$ decays could contribute to the signal yield. They are either
decays with three charged tracks in the final states ($K^{+}$${\pi}^{-}$
${\pi}^{+}$, $K^{+}K^{-}$${\pi}^{+}$, ${\pi}^{+}$ ${\pi}^{-}$ ${\pi}^{+}$
${\pi}^{0}$ ${\pi}^{0}$) or five charged tracks, all of them having very small
branching fractions compared to the ${\tau^{+}}\rightarrow
3\pi({{\pi}^{0}}){{\overline{\nu}}_{\tau}}$ decay mode. The study of a
dedicated simulation sample with inclusive $\tau$ decays indicates an effect
of $1\%$ that is taken as a systematic uncertainty.

The $B\rightarrow D^{**}\tau\nu_{\tau}$ fraction used for the nominal fit,
$0.11$, is assigned a 40% uncertainty, based on the results of an auxiliary
study of ${{{B}^{-}}}\\!\rightarrow D_{1}(2420)^{0}{\tau^{+}}{{\nu}_{\tau}}$
decays, where $D_{1}(2420)^{0}\rightarrow{{D}^{*-}}{{\pi}^{+}}$. These results
give a systematic uncertainty on the signal yield of 2.3%.

An additional systematic uncertainty of 1.5% due to the feed-down from
${{B}^{0}_{s}}\rightarrow D_{s}^{**}\tau^{+}{{\nu}_{\tau}}$ decays is
assigned, under the assumption that the yield of these decays in the
simulation has an uncertainty of 50%, determined to be the upper limit from a
study performed on simulated data.

###  8.2 Background-related systematic uncertainties

This section lists the systematic uncertainties due to the modeling of
different background sources, such as the ${D}^{+}_{s}$ decay model, double-
charm and combinatorial contributions.

Candidates in the low BDT output region are used to correct the composition of
${D}^{+}_{s}$ decays in simulation. From the fit to this data sample
corrections are obtained, which are used to generate $1000$ alternative
$D_{s}^{+}$ templates for each $D_{s}^{+}$ component in the nominal three-
dimensional fit. Each alternative template is produced by varying the nominal
template accounting for the uncertainty and correlations between the
$D_{s}^{+}$ subcomponents according to a Gaussian distribution. These
alternative templates are employed to refit the model to the data. The
difference between the signal yield of the alternative and the nominal fits,
divided by the yield of the nominal fit, is fitted with a Gaussian function
and a systematic uncertainty of $2.5\%$ is determined.

The mass variables that are expected to be significantly correlated with the
fit variables $q^{2}$ and BDT, are $m({{D}^{*-}}3\pi)$, $m(3\pi)$,
$\text{min}[m(\pi^{+}\pi^{-})]$, $\text{max}[m(\pi^{+}\pi^{-})]$ and
$m(\pi^{+}\pi^{+})$.444Only $m({{D}^{*-}}3{\pi})$ is considered for the
${{D}^{*-}}{{D}^{+}_{s}}$ case, since the effect of the other three variables
is included in the systematic uncertainty due to the ${D}^{+}_{s}$ decay
model. The corresponding effect on the fit result of these variables is
empirically studied by varying the distributions using a quadratic
interpolation method: for each template, two alternative templates are
produced, with a variation of $\pm 1\sigma$. Then, the fit enables the
interpolation between the nominal and the alternative templates to be made
with a linear weight. Each nuisance parameter is allowed to float in the range
$[-1,+1]$ and a loose Gaussian constraint with $\sigma=1$ is included. This
method is used to compute systematic uncertainties due to the knowledge of the
shape of the templates. The corresponding systematic uncertainty is $2.9\%$. A
systematic uncertainty of 2.6% arises due to the composition of the
$B\rightarrow D^{*-}D_{s}^{+}(X)$ and $B\rightarrow D^{*-}D^{0}(X)$ decays, as
discussed in Sec. 4. The use of the ${D}^{+}_{s}$ exclusive reconstruction in
$3\pi$ helps to limit the size of this uncertainty.

The systematic uncertainty due to the knowledge of the shape of the residual
prompt background component is estimated by applying the same interpolation
technique to the corresponding template. When combined with the knowledge of
the normalization of this background, this gives an overall uncertainty of
2.8%.

The same method is again used to assess the systematic uncertainty due to the
shape of the combinatorial background. The change in the signal yield provides
a systematic uncertainty of $0.7\%$.

Another systematic uncertainty is due to the normalization of this background.
This uncertainty is computed by performing the fit with a $30\%$ Gaussian
constraint around the nominal value. The resulting difference with respect to
the nominal fit is $0.1\%$, which is assigned as systematic uncertainty. This
uncertainty has a negligible effect on the total systematic uncertainty
associated with the shape of the combinatorial background.

###  8.3 Fit-related systematic uncertainties

To assess the systematic uncertainty relative to the bias due to empty bins in
the templates used in the fit the study performed using the KDE method is
repeated implementing different smoothing parameters. A difference in the
signal yield of 1.3% is assigned as the systematic uncertainty due to the bias
observed in the fit.

In order to estimate the systematic uncertainty due to the limited size of the
simulated samples, a bootstrap method is used. Each template from the nominal
model is used to produce new templates sampled from the originals by using a
bootstrap procedure based on random selection with replacement, varied bin-by-
bin according to a Poisson distribution. This procedure is repeated $500$
times. A Gaussian fit to the distributions of signal yields provides a $4.1\%$
effect taken as the systematic uncertainty due to the limited size of the
simulated samples.

###  8.4 Uncertainties related to the selection

In this section systematic uncertainties related to the selection criteria are
discussed. Such uncertainties stem from the choice of the trigger strategy,
the online and offline selection of the candidates, the normalization and
external inputs, and the efficiency of the PID criteria.

The trigger efficiency is studied on data using the fraction of the events
where the trigger was fired by particles other than the six tracks forming the
signal candidate, as a function of the two most important variables in this
analysis, $t_{\tau}$ and $m({{D}^{*-}}3\pi)$, the latter being highly
correlated with $q^{2}$. Corrections on the $t_{\tau}$ and $m({{D}^{*-}}3\pi)$
distributions due to different trigger efficiency between data and simulation
are applied. This gives a change in the number of signal candidates of $1.0\%$
for the $t_{\tau}$ and $0.7\%$ for the $m({{D}^{*-}}3\pi)$ corrections. The
sum in quadrature of these two contributions, taken as systematic uncertainty
related to the trigger efficiency, is $1.2\%$.

An additional $1\%$ systematic uncertainty arises from a mismatch between data
and simulation in the occupancy of the event.

The relative efficiency between the signal and the normalization channels is
precisely determined from simulated samples. Discrepancies between data and
simulation, due to online and offline selection criteria, introduce a 2% of
systematic uncertainty for both.

A 1% systematic uncertainty is assigned on the charged isolation criterion,
due to differences observed between the
$B^{0}\rightarrow{{D}^{*-}}\tau^{+}{{\nu}_{\tau}}$ and the
$B^{0}\rightarrow{{D}^{*-}}3\pi$ simulations.

All selection criteria, except the detached-vertex topology requirement, are
common to the signal and normalization decays. The corresponding efficiencies
are therefore directly determined from data by fitting the number of events in
the ${{B}^{0}}\rightarrow{{D}^{*-}}3{\pi}$ mass peak before and after each
selection, and no systematic uncertainty is assigned. To compute the
systematic uncertainty attributed to the knowledge of the relative
efficiencies corresponding to the different signal and normalisation vertex
topologies, the vertex position uncertainty distribution is split into three
regions: between $-4\sigma$ and $-2\sigma$, between $-2\sigma$ and $2\sigma$
and between $2\sigma$ and $4\sigma$, where $\sigma$ is the reconstructed
uncertainty on the distance along the beam line of the ${B}^{0}$ and 3$\pi$
vertices. Then a ratio between the number of candidates in the outer regions
and the number of candidates in the inner region is computed for the
candidates which have $m({{D}^{*-}}3\pi)$ in the exclusive
$B^{0}\rightarrow{{D}^{*-}}3\pi$ peak. The same procedure is performed for the
candidates outside the $B^{0}\rightarrow D^{*-}3\pi$ peak, which exhibit a
signal-like behavior. The procedure is repeated for data, and the ratio
between data and simulation gives rise to a 2% systematic uncertainty.

The simulation is corrected in order to match the performance of PID criteria
measured in data. Correction factors are applied in bins of momentum,
pseudorapidity and global event multiplicity, after having adjusted the
simulated event multiplicity to that observed using real data. To assess the
systematic uncertainty due to the choice of the binning scheme used to correct
simulation, two new schemes are derived from the default with half and twice
the number of bins, the default configuration consisting of fifteen bins in
momentum, seven in pseudorapidity and three in the global event multiplicity.
The correction procedure is repeated with these two alternate schemes, leading
to a systematic uncertainty related to PID of $1.3\%$.

The normalization channel consists of exactly the same final state as the
signal. In this way, differences between data and simulation are minimized.
The systematic uncertainty in the normalization yield is determined to be
equal to 1%. The statistical uncertainty attributed to the normalization yield
is included in the statistical uncertainty quoted for each result in this
paper. Differences between data and simulation in the modeling of the
$B^{0}\rightarrow{{D}^{*-}}3\pi$ decay impact the efficiency of the
normalization channel and result in a $2.0\%$ systematic uncertainty on
${\cal{K}}({{D}^{*-}})$.

The branching fraction for the normalization channel, obtained by averaging
the measurements of Refs. [22, 23, 24], has an uncertainty of 3.9%. A 2.0%
uncertainty arising from the knowledge of the $B^{0}\rightarrow
D^{*-}\mu^{+}\nu_{\mu}$ branching fraction is added in quadrature to obtain a
4.5% total uncertainty on ${\cal{R}}({{D}^{*-}})$ due to external inputs.

###  8.5 Summary of systematic uncertainties

Table 7 summarizes the systematic uncertainties on the measurement of the
ratio ${\mathcal{B}}({{B}^{0}}\rightarrow
D^{*-}\tau^{+}\nu_{\tau})/{\mathcal{B}}({{B}^{0}}\rightarrow D^{*-}3\pi)$. The
total uncertainty is $9.1\%$. For ${\cal{R}}({{D}^{*-}})$, a $4.5\%$
systematic uncertainty due to the knowledge of the external branching
fractions is added.

Table 7: List of the individual systematic uncertainties for the measurement of the ratio ${\mathcal{B}}({{B}^{0}}\rightarrow D^{*-}\tau^{+}\nu_{\tau})/{\mathcal{B}}({{B}^{0}}\rightarrow D^{*-}3\pi)$.  Contribution | Value in %
---|---
${\cal{B}}({\tau^{+}}\\!\rightarrow 3{\pi}{{\overline{\nu}}_{\tau}})/{\cal{B}}({\tau^{+}}\\!\rightarrow 3{\pi}({{\pi}^{0}}){{\overline{\nu}}_{\tau}})$ | $0.7$
Form factors (template shapes) | $0.7$
Form factors (efficiency) | $1.0$
$\tau$ polarization effects | $0.4$
Other $\tau$ decays | $1.0$
$B\rightarrow D^{**}\tau^{+}\nu_{\tau}$ | $2.3$
$B_{s}^{0}\rightarrow D_{s}^{**}\tau^{+}\nu_{\tau}$ feed-down | $1.5$
$D_{s}^{+}\rightarrow 3\pi X$ decay model | $2.5$
$D_{s}^{+}$, $D^{0}$ and $D^{+}$ template shape | $2.9$
$B\rightarrow D^{*-}D_{s}^{+}(X)$ and $B\rightarrow D^{*-}D^{0}(X)$ decay model | $2.6$
${{D}^{*-}}3{\pi}X$ from $B$ decays | $2.8$
Combinatorial background (shape + normalization) | $0.7$
Bias due to empty bins in templates | $1.3$
Size of simulation samples | $4.1$
Trigger acceptance | $1.2$
Trigger efficiency | $1.0$
Online selection | $2.0$
Offline selection | $2.0$
Charged-isolation algorithm | $1.0$
Particle identification | $1.3$
Normalization channel | $1.0$
Signal efficiencies (size of simulation samples) | $1.7$
Normalization channel efficiency (size of simulation samples) | $1.6$
Normalization channel efficiency (modeling of ${{B}^{0}}\rightarrow{{D}^{*-}}3\pi$) | $2.0$
Total uncertainty | $9.1$

##  9 Conclusion

In conclusion, the ratio of branching fractions between the
${{B}^{0}}\\!\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}}$ and the
${{B}^{0}}\\!\rightarrow{{D}^{*-}}3\pi$ decays is measured to be

${\cal{K}}({{D}^{*-}})=1.97\pm 0.13\mathrm{\,(stat)}\pm
0.18\mathrm{\,(syst)},$

where the first uncertainty is statistical and the second systematic. Using
the branching fraction
${\cal{B}}({{B}^{0}}\rightarrow{{D}^{*-}}3\pi)=(7.214\pm 0.28)\times 10^{-3}$
from the weighted average of the measurements by the LHCb [22], BaBar [23],
and Belle [24] collaborations, a value of the absolute branching fraction of
the ${{B}^{0}}\\!\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}}$ decay is
obtained

${\cal{B}}({{B}^{0}}\rightarrow{{D}^{*-}}{\tau^{+}}{{\nu}_{\tau}})=\left(1.42\pm
0.094\mathrm{\,(stat)}\pm 0.129\mathrm{\,(syst)}\pm
0.054\mathrm{\,(ext)}\right)\times 10^{-2},$

where the third uncertainty originates from the limited knowledge of the
branching fraction of the normalization mode. The precision of this
measurement is comparable to that of the current world average of Ref. [48].
The first determination of ${\cal{R}}({{D}^{*-}})$ performed by using three-
prong $\tau$ decays is obtained by using the measured branching fraction of
${\cal{B}}({{B}^{0}}\rightarrow{{D}^{*-}}{\mu^{+}}{{\nu}_{\mu}})=(4.88\pm
0.10)\times 10^{-2}$ from Ref. [20]. The result

${\cal{R}}(D^{*-})=0.291\pm 0.019\mathrm{\,(stat)}\pm
0.026\mathrm{\,(syst)}\pm 0.013\mathrm{\,(ext)}$

is one of the most precise single measurements performed so far. It is $1.1$
standard deviations higher than the SM prediction ($0.252\pm 0.003$) of Ref.
[2], and consistent with previous determinations. This R(${D}^{*}$)
measurement, being proportional to
${\cal{B}}({{B}^{0}}\rightarrow{{D}^{*-}}3\pi)$, and inversely proportional to
${\cal{B}}({{B}^{0}}\rightarrow{{D}^{*-}}{\mu^{+}}{{\nu}_{\mu}})$, will need
to be rescaled accordingly when more precise values of these inputs are made
available in the future. An average of this measurement with the LHCb result
using
${\tau^{+}}\\!\rightarrow{\mu^{+}}{{\nu}_{\mu}}{{\overline{\nu}}_{\tau}}$
decays [17], accounting for small correlations due to form factors, $\tau$
polarization and $D^{**}{\tau^{+}}{{\nu}_{\tau}}$ feed-down, gives a value of
${\cal{R}}({{D}^{*-}})=0.310\pm 0.0155\mathrm{\,(stat)}\pm
0.0219\mathrm{\,(syst)}$ , consistent with the world average and $2.2$
standard deviations above the SM prediction. The overall status of
${\cal{R}}({D})$ and ${\cal{R}}({{D}^{*}})$ measurements is reported in Ref.
[20]. After inclusion of this result, the combined discrepancy of
${\cal{R}}({D})$ and ${\cal{R}}({{D}^{*}})$ determinations with the SM
prediction is 4.1$\sigma$.

The novel technique presented in this paper, allowing the reconstruction and
selection of semitauonic decays with $\tau^{+}\rightarrow\
3{\pi}(\pi^{0}){{\overline{\nu}}_{\tau}}$ transitions, can be applied to all
the other semitauonic decays, such as those of ${{B}^{+}}$, ${B}^{0}_{s}$,
${B}_{c}^{+}$ and ${\mathchar 28931\relax}^{0}_{b}$. This technique also
allows isolation of large signal samples with high purity, which can be used
to measure angular distributions and other observables proposed in the
literature to discriminate between SM and new physics contributions. The
inclusion of further data collected by LHCb at
$\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}$ will result in an overall
uncertainty on ${\cal{R}}({{D}^{*-}})$ using this technique comparable to that
of the current world average.

## 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),
ANR, Labex P2IO, ENIGMASS and OCEVU, and Région Auvergne-Rhône-Alpes (France),
RFBR and Yandex LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith
Fund, the Royal Society, the English-Speaking Union and the Leverhulme Trust
(United Kingdom).

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
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An3, L. Anderlini18, G. Andreassi41, M. Andreotti17,g, J.E. Andrews60, R.B.
Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S.
Bachmann12, J.J. Back50, A. Badalov38,m, C. Baesso62, S. Baker55, V.
Balagura7,b, W. Baldini17, A. Baranov35, R.J. Barlow56, C. Barschel40, S.
Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V. Battista41, A.
Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1, A. Beiter61,
L.J. Bel43, N. Beliy63, V. Bellee41, N. Belloli21,i, K. Belous37, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, S. Beranek9, A.
Berezhnoy33, R. Bernet42, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt42, F. Betti15, M.-O. Bettler40, M. van Beuzekom43, Ia. Bezshyiko42,
S. Bifani47, P. Billoir8, A. Birnkraut10, A. Bitadze56, A. Bizzeti18,u, M.
Bjørn57, T. Blake50, F. Blanc41, J. Blouw11,†, S. Blusk61, V. Bocci26, T.
Boettcher58, A. Bondar36,w, N. Bondar31, W. Bonivento16, I. Bordyuzhin32, A.
Borgheresi21,i, S. Borghi56, M. Borisyak35, M. Borsato39, F. Bossu7, M.
Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12, T.
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Bursche16,f, J. Buytaert40, W. Byczynski40, S. Cadeddu16, H. Cai64, R.
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Camboni38,m, P. Campana19, D.H. Campora Perez40, L. Capriotti56, A.
Carbone15,e, G. Carboni25,j, R. Cardinale20,h, A. Cardini16, P. Carniti21,i,
L. Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina21, L. Castillo
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Chapman48, M. Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M.
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Lesiak27, B. Leverington12, P.-R. Li63, T. Li3, Y. Li7, Z. Li61, T.
Likhomanenko35,68, R. Lindner40, F. Lionetto42, X. Liu3, D. Loh50, A. Loi16,
I. Longstaff53, J.H. Lopes2, D. Lucchesi23,o, M. Lucio Martinez39, H. Luo52,
A. Lupato23, E. Luppi17,g, O. Lupton40, A. Lusiani24, X. Lyu63, F. Machefert7,
F. Maciuc30, V. Macko41, P. Mackowiak10, S. Maddrell-Mander48, O. Maev31,40,
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Martinez Santos39, F. Martinez Vidal70, D. Martins Tostes2, L.M. Massacrier7,
A. Massafferri1, R. Matev40, A. Mathad50, Z. Mathe40, C. Matteuzzi21, A.
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D. Melnychuk29, M. Merk43, A. Merli22,40,q, E. Michielin23, D.A. Milanes66, E.
Millard50, M.-N. Minard4, L. Minzoni17, D.S. Mitzel12, A. Mogini8, J. Molina
Rodriguez1, T. Mombächer10, I.A. Monroy66, S. Monteil5, M. Morandin23, M.J.
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S. Ogilvy19, R. Oldeman16,f, C.J.G. Onderwater71, A. Ossowska27, J.M. Otalora
Goicochea2, P. Owen42, A. Oyanguren70, P.R. Pais41, A. Palano14,d, M.
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S. Playfer52, M. Plo Casasus39, F. Polci8, M. Poli Lener19, A. Poluektov50,36,
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Popov11,40, S. Poslavskii37, C. Potterat2, E. Price48, J. Prisciandaro39, C.
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S. Roiser40, A. Rollings57, V. Romanovskiy37, A. Romero Vidal39, J.W.
Ronayne13, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P. Ruiz Valls70, J. Ruiz
Vidal70, J.J. Saborido Silva39, E. Sadykhov32, N. Sagidova31, B. Saitta16,f,
V. Salustino Guimaraes1, C. Sanchez Mayordomo70, B. Sanmartin Sedes39, R.
Santacesaria26, C. Santamarina Rios39, M. Santimaria19, E. Santovetti25,j, G.
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Zucchelli15.

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
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Univ. Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Universita & INFN, Milano-Bicocca, Milano, Italy
22Sezione di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
71Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

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
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
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
vIligan Institute of Technology (IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
†Deceased