EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2018-189 LHCb-PAPER-2018-025 July 17, 2018

Search for $C\\!P$ violation in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays

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

A search for $C\\!P$ violation in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays is presented using a
sample of $pp$ collisions collected with the LHCb detector and corresponding
to an integrated luminosity of 3.0$\mbox{\,fb}^{-1}$. The $C\\!P$-violating
asymmetries are measured to be $A_{C\\!P}^{{p}{{K}^{-}}}=-0.020\pm 0.013\pm
0.019$ and $A_{C\\!P}^{{p}{{\pi}^{-}}}=-0.035\pm 0.017\pm 0.020$, and their
difference $A_{C\\!P}^{{p}{{K}^{-}}}-A_{C\\!P}^{{p}{{\pi}^{-}}}=0.014\pm
0.022\pm 0.010$, where the first uncertainties are statistical and the second
systematic. These are the most precise measurements of such asymmetries to
date.

Published in Phys. Lett. B 787 (2018) 124-133

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

##  1 Introduction

The non-invariance of weak interactions under the combined application of
charge conjugation ($C$) and parity ($P$) transformations is accommodated
within the Standard Model by the Cabibbo-Kobayashi-Maskawa mechanism [1, 2].
The violation of the $C\\!P$ symmetry was discovered in neutral-kaon decays
[3], and later observed with ${B}^{0}$ [4, 5, 6, 7, 8, 9, 10, 11, 12],
${{B}^{+}}$ [13] and ${B}^{0}_{s}$ mesons [14, 12]. First evidence for $C\\!P$
violation in the $b$-baryon sector was found more recently [15]. The decays
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ are mediated by the same
quark-level transitions contributing to charmless two-body ${B}^{0}$ and
${B}^{0}_{s}$ decays to charged pions and kaons, where nonzero values of the
$C\\!P$ asymmetries are well established [14]. The inclusion of charge-
conjugate processes is implied throughout.

Predictions for the $C\\!P$ asymmetries in the decays of the ${\mathchar
28931\relax}^{0}_{b}$ baryon to two-body charmless final states ${p}{{K}^{-}}$
or ${p}{{\pi}^{-}}$ range from a few percent in the generalised factorisation
approach [16, 17] up to approximately $30\%$ within the perturbative quantum-
chromodynamics formalism [18]. The only measurements of these quantities
available to date were performed by the CDF collaboration [12]. The
asymmetries were found to be compatible with zero within an uncertainty of $8$
to $9\%$.

This Letter reports on a search for $C\\!P$ violation in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays, using
$pp$-collision data collected with the LHCb detector at centre-of-mass
energies of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$ and corresponding to
3.0$\mbox{\,fb}^{-1}$ of integrated luminosity. The $C\\!P$ asymmetry is
defined as

$A_{C\\!P}^{f}\equiv\frac{\Gamma({{\mathchar 28931\relax}^{0}_{b}}\rightarrow
f)-\Gamma({{\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow\overline{f})}{\Gamma({{\mathchar
28931\relax}^{0}_{b}}\rightarrow f)+\Gamma({{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow\overline{f})},$ (1)

where $\Gamma$ is the partial width of the given decay, with
$f\equiv{p}{{K}^{-}}$ $({p}{{\pi}^{-}})$ and
$\overline{f}\equiv{\overline{p}}{{K}^{+}}$ $({\overline{p}}{{\pi}^{+}})$. In
addition, the difference of the two $C\\!P$ asymmetries, $\Delta
A_{{C\\!P}}\equiv A_{C\\!P}^{{p}{{K}^{-}}}-A_{C\\!P}^{{p}{{\pi}^{-}}}$, is
also reported. As the main systematic uncertainties cancel in the difference,
this quantity will become useful with the increasing size of the data sample.

The Letter is organised as follows. After a brief introduction on the
detector, trigger and simulation in Sec. 2, the formalism needed to relate the
physical $C\\!P$ asymmetries to the experimental measurements is presented in
Sec. 3. Then, the event selection and the invariant-mass fit are described in
Secs. 4 and 5, respectively. The determination of instrumental asymmetries and
systematic uncertainties is discussed in Sec. 6. Finally, results are given
and conclusions are drawn in Sec. 7.

##  2 Detector, trigger and simulation

The LHCb detector [19, 20] 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 [21], 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 [22] placed
downstream of the magnet. The tracking system provides a measurement of the
momentum, $p$, of charged particles with a relative uncertainty that varies
from 0.5% at low momentum to 1.0% at
200${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track to
a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/p_{\mathrm{T}}){\,\upmu\mathrm{m}}$, 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
[23]. 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
[24]. The online event selection is performed by a trigger [25], which
consists of a hardware stage, based on information from the calorimeter and
muon systems, followed by a software stage, which applies a full event
reconstruction.

Simulated events are used to study the modelling of various mass line shapes.
In the simulation, proton-proton collisions are generated using Pythia [26]
with a specific LHCb configuration [27]. Decays of hadronic particles are
described by EvtGen [28], in which final-state radiation is generated using
Photos [29]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [30,
*Agostinelli:2002hh] as described in Ref. [32].

##  3 Formalism

The $C\\!P$ asymmetries of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays are approximated as
the sums of various experimental quantities

$\displaystyle A_{C\\!P}^{{p}{{K}^{-}}}$ $\displaystyle=$ $\displaystyle
A_{\mathrm{raw}}^{{p}{{K}^{-}}}-A_{\mathrm{D}}^{p}-A_{\mathrm{D}}^{{{K}^{-}}}-A_{\mathrm{PID}}^{{p}{{K}^{-}}}-A_{\mathrm{P}}^{{\mathchar
28931\relax}^{0}_{b}}-A_{\mathrm{trigger}}^{{p}{{K}^{-}}},$ (2) $\displaystyle
A_{C\\!P}^{{p}{{\pi}^{-}}}$ $\displaystyle=$ $\displaystyle
A_{\mathrm{raw}}^{{p}{{\pi}^{-}}}-A_{\mathrm{D}}^{p}-A_{\mathrm{D}}^{{{\pi}^{-}}}-A_{\mathrm{PID}}^{{p}{{\pi}^{-}}}-A_{\mathrm{P}}^{{\mathchar
28931\relax}^{0}_{b}}-A_{\mathrm{trigger}}^{{p}{{\pi}^{-}}},$ (3)

where $A_{\mathrm{raw}}^{f}$ is the measured raw asymmetry between the yields
of the decays ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow f$ and ${{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow\overline{f}$, with $f={p}{{K}^{-}}$
$({p}{{\pi}^{-}})$ and $\overline{f}$ its charge conjugate;
$A_{\mathrm{D}}^{h}$ is the asymmetry between the detection efficiencies for
particle $h$ and its charge conjugate, with $h=p,\ {{K}^{-}}$ or
${{\pi}^{-}}$; the symbol $A_{\mathrm{PID}}^{f}$ stands for the asymmetry
between the particle-identification (PID) efficiencies for the final states
$f$ and $\overline{f}$; $A_{\mathrm{P}}^{{\mathchar 28931\relax}^{0}_{b}}$ is
the asymmetry between the production cross-sections of ${\mathchar
28931\relax}^{0}_{b}$ and ${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}$ baryons; and $A^{f}_{\mathrm{trigger}}$ is the
asymmetry between the trigger efficiencies for the particles in the final
states $f$ and $\overline{f}$. This linear approximation is valid to a good
enough accuracy due to the smallness of the terms involved.

The raw asymmetry is defined as

$A_{\mathrm{raw}}^{f}\equiv\frac{N({{\mathchar
28931\relax}^{0}_{b}}\rightarrow f)-N({{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow\overline{f})}{N({{\mathchar
28931\relax}^{0}_{b}}\rightarrow f)+N({{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}}\rightarrow\overline{f})},$ (4)

where $N$ denotes the observed signal yield for the given decay, obtained in
this analysis by means of extended binned maximum-likelihood fits to the
${p}{{K}^{-}}$ and ${p}{{\pi}^{-}}$ invariant-mass spectra.

The proton, kaon and pion detection asymmetries are defined as

$A_{\mathrm{D}}^{p}\equiv\frac{\varepsilon_{\mathrm{rec}}^{p}-\varepsilon_{\mathrm{rec}}^{{\overline{p}}}}{\varepsilon_{\mathrm{rec}}^{p}+\varepsilon_{\mathrm{rec}}^{{\overline{p}}}},\
\ \ \
A_{\mathrm{D}}^{{{K}^{-}}}\equiv\frac{\varepsilon_{\mathrm{rec}}^{{{K}^{-}}}-\varepsilon_{\mathrm{rec}}^{{{K}^{+}}}}{\varepsilon_{\mathrm{rec}}^{{{K}^{-}}}+\varepsilon_{\mathrm{rec}}^{{{K}^{+}}}},\
\ \ \ \
A_{\mathrm{D}}^{{{\pi}^{-}}}\equiv\frac{\varepsilon_{\mathrm{rec}}^{{{\pi}^{-}}}-\varepsilon_{\mathrm{rec}}^{{{\pi}^{+}}}}{\varepsilon_{\mathrm{rec}}^{{{\pi}^{-}}}+\varepsilon_{\mathrm{rec}}^{{{\pi}^{+}}}},$
(5)

where $\varepsilon_{\mathrm{rec}}$ is the total efficiency to reconstruct the
given particle, excluding PID. Such asymmetries are mostly due to the
different interaction cross-sections of particles and antiparticles with the
detector material. The kaon and pion detection asymmetries are measured using
charm-meson control samples employing the procedures described in Refs. [33,
34]. The kaon detection asymmetry is obtained by subtracting the raw
asymmetries of the ${{D}^{+}}\rightarrow{{K}^{0}_{\mathrm{\scriptscriptstyle
S}}}{{\pi}^{+}}$ and ${{D}^{+}}\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}$
decay modes and correcting for the ${{K}^{0}}$ ($A_{\mathrm{D}}^{{{K}^{0}}}$)
[33] and pion detection asymmetries. The latter is measured from the ratio of
partially to fully reconstructed
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{+}}$
decays. The proton detection asymmetry is obtained from simulated events.

The PID asymmetries are measured from large calibration samples and are
defined as

$A_{\mathrm{PID}}^{f}\equiv\frac{\varepsilon_{\mathrm{PID}}^{f}-\varepsilon_{\mathrm{PID}}^{\overline{f}}}{\varepsilon_{\mathrm{PID}}^{f}+\varepsilon_{\mathrm{PID}}^{\overline{f}}},$
(6)

where $\varepsilon_{\mathrm{PID}}^{f\,(\overline{f})}$ is the PID efficiency
for a final state $f\,(\overline{f})$ given a set of PID requirements.

The ${\mathchar 28931\relax}^{0}_{b}$ production asymmetry is defined as

$A_{\mathrm{P}}^{{\mathchar 28931\relax}^{0}_{b}}=\frac{\sigma({{\mathchar
28931\relax}^{0}_{b}})-\sigma({{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}})}{\sigma({{\mathchar
28931\relax}^{0}_{b}})+\sigma({{\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}})},$ (7)

where $\sigma$ denotes the inclusive production cross-section in the LHCb
acceptance. The production asymmetry is taken as an external input, following
Ref. [35].

Finally, asymmetries may arise if the hardware and software trigger used to
collect data do not have the same efficiencies on oppositely charged
particles. These effects are estimated through various data-driven techniques,
as described in Sec. 6.

##  4 Event selection

The event selection starts with the reconstruction of $b$ hadrons formed by
two oppositely charged tracks with
$p_{\mathrm{T}}>1{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, inconsistent with
originating from any PV and required to form a common vertex. Each $b$-hadron
candidate needs to have a transverse momentum greater than
$1.2{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an invariant mass, computed
assigning the pion mass to both daughter tracks, in the range between 4.8 and
5.8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. Finally, each $b$-hadron
candidate is required to be consistent with originating from a PV.

Particle-identification selection criteria are applied to divide the data
sample into mutually exclusive subsamples corresponding to the final-state
hypotheses ${p}{{K}^{-}}$, ${\overline{p}}{{K}^{+}}$, ${p}{{\pi}^{-}}$,
${\overline{p}}{{\pi}^{+}}$, ${{K}^{+}}{{\pi}^{-}}$, ${{K}^{-}}{{\pi}^{+}}$,
${{K}^{+}}{{K}^{-}}$ and ${{\pi}^{+}}{{\pi}^{-}}$. The latter four
combinations are selected to study the background due to two-body $B$ decays,
where one or both final-state particles are misidentified.

The event selection is further refined using a boosted decision tree (BDT)
classifier [36, 37] to reject combinatorial background. This algorithm
combines the information from several input quantities to obtain a
discriminant variable used to classify the $b$-hadron candidates as signal or
background. The following properties of the final-state particles are used as
input variables: the transverse momentum of the $b$-hadron decay products, the
logarithms of their $\chi^{2}_{\text{IP}}$ values, 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 candidate under
consideration, the quality of the common vertex fit of the two tracks and the
distance of closest approach between the two tracks. The BDT also exploits the
following properties of the $b$-hadron candidate: the transverse momentum, the
$\chi^{2}_{\text{IP}}$ quantity, and the logarithm of the flight distance with
respect to the associated PV, defined as that with the smallest
$\chi^{2}_{\text{IP}}$ with respect to the $b$-hadron candidate. The BDT is
trained using simulated signal decays and combinatorial background events from
data in the high-mass sideband.

The selection criteria on the BDT classifier and the PID variables are
optimised separately for the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays. Two different
selections, denoted hereafter as $S_{{p}{{K}^{-}}}$ and $S_{{p}{{\pi}^{-}}}$,
are aimed at obtaining the best statistical sensitivity on each of the two
$C\\!P$ asymmetries. Common PID requirements are used for the final states
containing only kaons and pions. Multiple candidates are present in less than
0.05% of the events in the final sample. Only one candidate is accepted for
each event on the basis of a reproducible pseudorandom sequence.

##  5 Invariant-mass fit

For each final-state hypothesis, namely ${p}{{K}^{-}}$,
${\overline{p}}{{K}^{+}}$, ${p}{{\pi}^{-}}$, ${\overline{p}}{{\pi}^{+}}$,
${{K}^{+}}{{\pi}^{-}}$, ${{K}^{-}}{{\pi}^{+}}$, ${{K}^{+}}{{K}^{-}}$ and
${{\pi}^{+}}{{\pi}^{-}}$, the invariant-mass distribution of selected
candidates is modelled by an appropriate probability density function. These
models are used to perform a simultaneous fit to the eight invariant-mass
spectra and determine at once the yields of all two-body $b$-hadron decays
contributing to the spectra. Three categories are considered for the
background: combinatorial, due to random association of tracks; partially
reconstructed, due to multibody $b$-hadron decays with one or more particles
not reconstructed; and cross-feed, arising from other two-body $b$-hadron
decays where one or both final-state particles are misidentified.

The model used to describe each signal is obtained by convolving the sum of
two Gaussian functions with common mean, accounting for mass-resolution
effects, with a power-law function that accounts for final-state photon
radiation effects. The power-law distribution is taken from analytical
quantum-electrodynamics calculations [38] and the correctness of the model is
checked against simulated events generated with Photos [29]. The parameter
governing the tail due to final-state photon radiation effects is different
for each decay mode. This model describes well the invariant-mass
distributions predicted by the simulation.

The combinatorial background is modelled using exponential functions. The
partially reconstructed background is parameterised using ARGUS functions [39]
convolved with the sum of two Gaussian functions with zero mean values, whose
relative fraction and widths are in common with the signal model. Finally, the
cross-feed background is modelled using simulated two-body $b$-hadron decays
and a kernel estimation method [40]. The cross-feed background yields are set
to the corresponding two-body $b$-hadron decay yields, determined by the
simultaneous fit, multiplied by appropriate PID efficiency ratios. The
efficiencies for a given PID requirement are obtained from large calibration
samples of
${{D}^{*+}}\rightarrow{{D}^{0}}(\rightarrow{{K}^{-}}{{\pi}^{+}}){{\pi}^{+}}$,
${\mathchar 28931\relax}\rightarrow{p}{{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays, with the aid
of simulated events in the case of protons to account for phase-space regions
not covered by the calibration samples (about 20% of the protons from signal
decays). The efficiencies are determined in bins of particle momentum,
pseudorapidity and track multiplicity, as the performances of the RICH
detectors depend on such variables. They are then averaged over the momentum
and pseudorapidity distributions of the final-state particles and over the
distribution of track multiplicity in selected events.

After the application of the optimal BDT and PID requirements, an extended
binned maximum-likelihood fit with a bin width of
5${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ is performed simultaneously to the
eight two-body invariant-mass spectra for each of the two selections,
$S_{{p}{{K}^{-}}}$ and $S_{{p}{{\pi}^{-}}}$. The $m_{{p}{{K}^{-}}}$ and
$m_{{p}{{\pi}^{-}}}$ invariant-mass distributions are shown in Fig. 1, with
the results of the fits superimposed.

Figure 1: Invariant-mass distributions: (top left) $m_{{p}{{K}^{-}}}$, (top
right) $m_{{\overline{p}}{{K}^{+}}}$, (bottom left) $m_{{p}{{\pi}^{-}}}$ and
(bottom right) $m_{{\overline{p}}{{\pi}^{+}}}$ for candidates passing the
(top) $S_{{p}{{K}^{-}}}$ and (bottom) $S_{{p}{{\pi}^{-}}}$ selections. The
results of the fits are superimposed.

The values of the raw asymmetries and of the signal yields obtained from the
fits to the candidates passing the respective $S_{{p}{{K}^{-}}}$ or
$S_{{p}{{\pi}^{-}}}$ selection are $A_{\mathrm{raw}}^{{p}{{K}^{-}}}=(1.0\pm
1.3)\%$, $A_{\mathrm{raw}}^{{p}{{\pi}^{-}}}=(0.5\pm 1.7)\%$,
$N^{{p}{{K}^{-}}}_{\mathrm{sig}}+N^{{\overline{p}}{{K}^{+}}}_{\mathrm{sig}}=8\,847\pm
125$ and
$N^{{p}{{\pi}^{-}}}_{\mathrm{sig}}+N^{{\overline{p}}{{\pi}^{+}}}_{\mathrm{sig}}=6\,026\pm
105$.

The fit is validated by generating a large number of pseudoexperimental data
samples according to the total probability density function of the model and
performing an extended binned maximum-likelihood fit to each sample. The
resulting pull distributions for $A_{\mathrm{raw}}^{{p}{{K}^{-}}}$ and
$A_{\mathrm{raw}}^{{p}{{\pi}^{-}}}$ are found to be Gaussian with zero means
and unitary widths.

##  6 Instrumental asymmetries and systematic uncertainties

The determination of the instrumental asymmetries introduced in Eqs. (2) and
(3) is crucial to obtain the $C\\!P$ asymmetries, as described in Sec. 3.

The kaon detection asymmetry is determined as a function of the kaon momentum,
following the approach developed in Ref. [33] and subtracting
$A_{\mathrm{D}}^{{{K}^{0}}}=(0.054\pm 0.014)\%$ [33] and the pion detection
asymmetry. The momentum-dependent values are then weighted with the
background-subtracted [41] momentum distribution of kaons from ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ decays to obtain
$A_{\mathrm{D}}^{{{K}^{-}}}=(-0.76\pm 0.23)\%$, where the dominant uncertainty
is due to the finite size of the samples used. The pion detection asymmetry is
obtained in an analogous way, adopting the approach of Ref. [34], and is
determined to be $A_{\mathrm{D}}^{{{\pi}^{-}}}=(0.13\pm 0.11)\%$. A different
approach is followed for the proton detection asymmetry, since no measurement
of this quantity is available to date. Simulated events are used to obtain the
reconstruction efficiency defined as the number of reconstructed over
generated decays, in bins of proton momentum. Then, according to Eq. (5), an
asymmetry is defined and weights are computed from the background-subtracted
[41] proton-momentum distributions of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays. The proton
detection asymmetries for both decays are found to be equal, consistent with
the fact that the kinematics of the protons for the two decays do not exhibit
significant differences, as shown in Fig. 2. The common value is
$A_{\mathrm{D}}^{p}=(1.30\pm 0.03\pm 0.16\pm 0.65)\%$, where the first
uncertainty is due to the finite amount of simulated events and the second is
associated to the knowledge of the material budget of the LHCb detector. The
third uncertainty is due to the assumptions made on the proton and antiproton
cross-sections used in the computation.

Figure 2: Distributions of (top) momentum, (middle) transverse momentum and
(bottom) pseudorapidity for (left) protons from ${\mathchar
28931\relax}^{0}_{b}$ decays and (right) ${\mathchar 28931\relax}^{0}_{b}$
baryons. The distributions are background-subtracted and normalised to unit
area. Below each plot the ratio between the two distributions corresponding to
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ decays is also shown.

The PID asymmetries are calculated using calibration samples with the aid of
simulation to account for the limited phase-space coverage of the protons from
${\mathchar 28931\relax}\rightarrow{p}{{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{+}_{c}}\rightarrow{p}{{K}^{-}}{{\pi}^{+}}$ decays. The dominant
uncertainty comes from different PID performances in data and simulation in
the phase-space region where simulated events are used. This discrepancy has
been studied using ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays, for
which the phase-space coverage of calibration data is larger. The values of
the PID asymmetries are found to be $A_{\mathrm{PID}}^{{p}{{K}^{-}}}=(-0.30\pm
0.74)\%$ and $A_{\mathrm{PID}}^{{p}{{\pi}^{-}}}=(-0.18\pm 0.73)\%$.

The integrated ${\mathchar 28931\relax}^{0}_{b}$ production asymmetries are
calculated convolving the background-subtracted [41] two-dimensional
transverse-momentum and rapidity distributions of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}$ candidates with the
production asymmetries measured as a function of the same variables reported
in Ref. [35]. Since ${\mathchar 28931\relax}^{0}_{b}$ baryons selected in the
${p}{{K}^{-}}$ or ${p}{{\pi}^{-}}$ final states have very similar kinematics,
as shown in Fig. 2, the value $A_{\mathrm{P}}^{{\mathchar
28931\relax}^{0}_{b}}=(2.7\pm 1.4)\%$, averaged for 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$ data, is obtained for the production
asymmetry of both decays.

Asymmetries related to different trigger efficiencies for the charge-
conjugated final states may arise. The efficiency for a charged hadron to be
responsible for the affirmative decision of the hardware trigger is determined
as a function of transverse momentum, separately for positively and negatively
charged particles, using a sample of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}$ decays.
These efficiencies are used to determine the charge asymmetry introduced by
the hardware trigger for the signal candidates that fire it. The charge
asymmetry introduced by the hardware trigger for candidates that are retained
independently of whether or not they are responsible for an affirmative
hardware-trigger decision is determined studying a sample of
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays [25]. The asymmetry of
the software trigger is also studied using
${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$ decays, determining the charge
asymmetry of the fraction of ${{B}^{0}}\\!\rightarrow{{K}^{+}}{{\pi}^{-}}$
decays for which both final-state hadrons fire the software trigger with
respect to those for which only one hadron fires. The total trigger
asymmetries are measured to be $A_{\mathrm{trigger}}^{{p}{{K}^{-}}}=(0.18\pm
0.53)\%$ and $A_{\mathrm{trigger}}^{{p}{{\pi}^{-}}}=(-0.08\pm 0.55)\%$. The
uncertainties are mainly due to the limited size of the samples used in their
determination.

Several sources of systematic uncertainties associated with the fit model are
investigated. The alternative models used to determine systematic
uncertainties associated with the choices of the invariant-mass shapes consist
in turn of: adding a Gaussian function to the invariant-mass resolution model
used for signals and cross-feed backgrounds to account for long tails due to
candidates with a poor determination of the final-state particles momenta;
changing the value of the parameter governing the final-state photon radiation
effects according to its uncertainty; substituting the exponential function
used to model the combinatorial background with a linear function and removing
the partially reconstructed background component by rejecting candidates with
$m_{{p}{{K}^{-}}}$ $(m_{{p}{{\pi}^{-}}})$ lower than
$5.5$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. When testing alternative
models, 250 pseudoexperiments are generated according to the baseline fit
model and using as input the central values of the baseline results. Fits are
performed to each of the generated samples using the baseline model and then
the alternative models. The mean and the root mean square of the distribution
of the difference between the raw asymmetries determined by the two sets of
fits are added in quadrature and the resulting value is taken as a systematic
uncertainty.

A different approach is adopted to assess systematic uncertainties related to
the knowledge of the PID efficiencies. Samples are generated using the
baseline fit model and results. The baseline fit model is then fitted 250
times to the generated samples, varying the PID efficiencies according to
their uncertainties, which are mainly driven by the choice of the binning
scheme used to divide the phase-space. These uncertainties are assessed by
changing the baseline binning scheme with alternative schemes and computing
again the efficiencies. The largest root mean square of the raw asymmetry
distributions is taken as a systematic uncertainty.

The systematic uncertainties due to the fit model choice, PID efficiencies
determination and instrumental asymmetries measurement, along with the total
uncertainty obtained as the quadratic sum of the individual contributions, are
reported in Table 1.

Table 1: Systematic uncertainties on $A_{C\\!P}^{{p}{{K}^{-}}}$ and $A_{C\\!P}^{{p}{{\pi}^{-}}}$. Systematic uncertainty |  $A_{C\\!P}^{{p}{{K}^{-}}}$ [%] |  $A_{C\\!P}^{{p}{{\pi}^{-}}}$ [%]
---|---|---
Kaon or pion detection asymmetry | $0.23$ | $0.11$
Proton detection asymmetry | $0.67$ | $0.67$
PID asymmetry | $0.74$ | $0.73$
${\mathchar 28931\relax}^{0}_{b}$ production asymmetry | $1.40$ | $1.40$
Trigger asymmetry | $0.53$ | $0.55$
Signal model | $0.02$ | $0.02$
Background model | $0.23$ | $0.47$
PID efficiencies | $0.57$ | $0.74$
Total | $1.91$ | $2.00$

##  7 Results and conclusions

Using in Eqs. (2) and (3) the values of the raw asymmetries reported in Sec. 5
and those of the instrumental and production asymmetries reported in Sec. 6,
the following $C\\!P$ asymmetries are obtained

$\displaystyle A_{C\\!P}^{{p}{{K}^{-}}}$ $\displaystyle=$
$\displaystyle-0.020\pm 0.013\pm 0.019,$ $\displaystyle
A_{C\\!P}^{{p}{{\pi}^{-}}}$ $\displaystyle=$ $\displaystyle-0.035\pm 0.017\pm
0.020,$

where the first uncertainties are statistical and the second systematic. The
correlation between $A_{C\\!P}^{{p}{{K}^{-}}}$ and
$A_{C\\!P}^{{p}{{\pi}^{-}}}$ is found to be 0.5. No evidence for $C\\!P$
violation is observed.

A quantity that is independent from the proton detection and ${\mathchar
28931\relax}^{0}_{b}$ production asymmetries is obtained by taking the
difference

$\displaystyle\Delta A_{{C\\!P}}\equiv
A_{C\\!P}^{{p}{{K}^{-}}}-A_{C\\!P}^{{p}{{\pi}^{-}}}$ $\displaystyle=$
$\displaystyle
A_{\mathrm{raw}}^{{p}{{K}^{-}}}-A_{\mathrm{D}}^{{{K}^{-}}}-A_{\mathrm{PID}}^{{p}{{K}^{-}}}-A_{\mathrm{trigger}}^{{p}{{K}^{-}}}$
(8) $\displaystyle-$ $\displaystyle
A_{\mathrm{raw}}^{{p}{{\pi}^{-}}}+A_{\mathrm{D}}^{{{\pi}^{-}}}+A_{\mathrm{PID}}^{{p}{{\pi}^{-}}}+A_{\mathrm{trigger}}^{{p}{{\pi}^{-}}}.$

The statistical and systematic correlations between the raw asymmetries, the
PID asymmetries and the detection asymmetries are taken into account when
propagating the uncertainty to $\Delta A_{{C\\!P}}$, obtaining

$\Delta A_{{C\\!P}}=0.014\pm 0.022\pm 0.010,$

where the first uncertainty is statistical and the second systematic. These
results represent the world’s best measurements to date, with much improved
precision with respect to previous CDF determinations [12].

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); MOST
and NSFC (China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN
(Italy); NWO (Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); 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 (Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex LLC
(Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open-source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany); EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union); ANR, Labex P2IO and OCEVU,
and Région Auvergne-Rhône-Alpes (France); Key Research Program of Frontier
Sciences of CAS, CAS PIFI, and the Thousand Talents Program (China); RFBR, RSF
and Yandex LLC (Russia); GVA, XuntaGal and GENCAT (Spain); the Royal Society
and the Leverhulme Trust (United Kingdom); Laboratory Directed Research and
Development program of LANL (USA).

## References

  * [1] N. Cabibbo, _Unitary symmetry and leptonic decays_ , Phys. Rev. Lett. 10 (1963) 531
  * [2] M. Kobayashi and T. Maskawa, _CP violation in the renormalizable theory of weak interaction_ , Prog. Theor. Phys. 49 (1973) 652
  * [3] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, _Evidence for the $2\pi$ decay of the $K_{2}^{0}$ meson_, Phys. Rev. Lett. 13 (1964) 138
  * [4] BaBar collaboration, B. Aubert et al., _Observation of CP violation in the $B^{0}$ meson system_, Phys. Rev. Lett. 87 (2001) 091801, arXiv:hep-ex/0107013
  * [5] Belle collaboration, K. Abe et al., _Observation of large CP violation in the neutral $B$ meson system_, Phys. Rev. Lett. 87 (2001) 091802, arXiv:hep-ex/0107061
  * [6] BaBar collaboration, B. Aubert et al., _Observation of direct CP violation in $B^{0}\rightarrow K^{+}\pi^{-}$ decays_, Phys. Rev. Lett. 93 (2004) 131801, arXiv:hep-ex/0407057
  * [7] Belle collaboration, K. Abe et al., _Observation of large CP violation and evidence for direct CP violation in $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays_, Phys. Rev. Lett. 93 (2004) 021601, arXiv:hep-ex/0401029
  * [8] BaBar collaboration, B. Aubert et al., _Observation of CP violation in $B^{0}\rightarrow K^{+}\pi^{-}$ and $B^{0}\rightarrow\pi^{+}\pi^{-}$_, Phys. Rev. Lett. 99 (2007) 021603, arXiv:hep-ex/0703016
  * [9] BaBar collaboration, J. P. Lees et al., _Measurement of CP asymmetries and branching fractions in charmless two-body $B$-meson decays to pions and kaons_, Phys. Rev. D87 (2013) 052009, arXiv:1206.3525
  * [10] Belle collaboration, J. Dalseno et al., _Measurement of the CP violation parameters in $B^{0}\rightarrow\pi^{+}\pi^{-}$ decays_, Phys. Rev. D88 (2013) 092003, arXiv:1302.0551
  * [11] Belle collaboration, Y.-T. Duh et al., _Measurements of branching fractions and direct CP asymmetries for $B\rightarrow K\pi$, $B\rightarrow\pi\pi$ and $B\rightarrow KK$ decays_, Phys. Rev. D87 (2013) 031103, arXiv:1210.1348
  * [12] CDF collaboration, T. A. Aaltonen et al., _Measurements of direct ${C\\!P}$-violating asymmetries in charmless decays of bottom baryons_, Phys. Rev. Lett. 113 (2014) 242001, arXiv:1403.5586
  * [13] LHCb collaboration, R. Aaij et al., _Observation of ${C\\!P}$ violation in ${{B}^{\pm}}\rightarrow{D}{{K}^{\pm}}$ decays_, Phys. Lett. B712 (2012) 203, Erratum ibid. B713 (2012) 351, arXiv:1203.3662
  * [14] LHCb collaboration, R. Aaij et al., _First observation of ${C\\!P}$ violation in the decays of ${{B}^{0}_{s}}$ mesons_, Phys. Rev. Lett. 110 (2013) 221601, arXiv:1304.6173
  * [15] LHCb collaboration, R. Aaij et al., _Measurement of matter-antimatter differences in beauty baryon decays_ , Nature Physics 13 (2017) 391, arXiv:1609.05216
  * [16] Y. K. Hsiao and C. Q. Geng, _Direct CP violation in ${\mathchar 28931\relax}^{0}_{b}$ decays_, Phys. Rev. D91 (2015) 116007, arXiv:1412.1899
  * [17] A. Ali, G. Kramer, and C.-D. Lü, _Experimental tests of factorization in charmless nonleptonic two-body B decays_ , Phys. Rev. D58 (1998) 094009, arXiv:hep-ph/9804363
  * [18] C.-D. Lü et al., _Anatomy of the perturbative QCD approach to the baryonic decays ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p\pi,\ pK$_, Phys. Rev. D80 (2009) 034011, arXiv:0906.1479
  * [19] LHCb collaboration, A. A. Alves Jr. et al., _The LHCb detector at the LHC_, JINST 3 (2008) S08005
  * [20] LHCb collaboration, R. Aaij et al., _LHCb detector performance_ , Int. J. Mod. Phys. A30 (2015) 1530022, arXiv:1412.6352
  * [21] R. Aaij et al., _Performance of the LHCb Vertex Locator_ , JINST 9 (2014) P09007, arXiv:1405.7808
  * [22] R. Arink et al., _Performance of the LHCb Outer Tracker_ , JINST 9 (2014) P01002, arXiv:1311.3893
  * [23] M. Adinolfi et al., _Performance of the LHCb RICH detector at the LHC_, Eur. Phys. J. C73 (2013) 2431, arXiv:1211.6759
  * [24] A. A. Alves Jr. et al., _Performance of the LHCb muon system_ , JINST 8 (2013) P02022, arXiv:1211.1346
  * [25] R. Aaij et al., _The LHCb trigger and its performance in 2011_, JINST 8 (2013) P04022, arXiv:1211.3055
  * [26] T. Sjöstrand, S. Mrenna, and P. Skands, _A brief introduction to PYTHIA 8.1_ , Comput. Phys. Commun. 178 (2008) 852, arXiv:0710.3820
  * [27] I. Belyaev et al., _Handling of the generation of primary events in Gauss, the LHCb simulation framework_ , J. Phys. Conf. Ser. 331 (2011) 032047
  * [28] D. J. Lange, _The EvtGen particle decay simulation package_ , Nucl. Instrum. Meth. A462 (2001) 152
  * [29] P. Golonka and Z. Was, _PHOTOS Monte Carlo: A precision tool for QED corrections in $Z$ and $W$ decays_, Eur. Phys. J. C45 (2006) 97, arXiv:hep-ph/0506026
  * [30] Geant4 collaboration, J. Allison et al., _Geant4 developments and applications_ , IEEE Trans. Nucl. Sci. 53 (2006) 270
  * [31] Geant4 collaboration, S. Agostinelli et al., _Geant4: A simulation toolkit_ , Nucl. Instrum. Meth. A506 (2003) 250
  * [32] M. Clemencic et al., _The LHCb simulation application, Gauss: Design, evolution and experience_, J. Phys. Conf. Ser. 331 (2011) 032023
  * [33] LHCb collaboration, R. Aaij et al., _Measurement of ${C\\!P}$ asymmetry in ${{D}^{0}}\rightarrow{{K}^{-}}{{K}^{+}}$ and ${{D}^{0}}\rightarrow{{\pi}^{-}}{{\pi}^{+}}$ decays_, JHEP 07 (2014) 041, arXiv:1405.2797
  * [34] LHCb collaboration, R. Aaij et al., _Measurement of the ${{D}^{+}_{s}}$–${{D}^{-}_{s}}$ production asymmetry in $7$$~{}T\kern-0.50003pteV$ ${p}{p}$ collisions_, Phys. Lett. B713 (2012) 186, arXiv:1205.0897
  * [35] LHCb collaboration, R. Aaij et al., _Measurement of ${B}^{0}$, ${B}^{0}_{s}$, ${{B}^{+}}$ and ${\mathchar 28931\relax}^{0}_{b}$ production asymmetries in $7$ and $8$$~{}T\kern-0.50003pteV$ $p$ $p$ collisions_, Phys. Lett. B774 (2017) 139, arXiv:1703.08464
  * [36] B. P. Roe et al., _Boosted decision trees as an alternative to artificial neural networks for particle identification_ , Nucl. Instrum. Meth. A543 (2005) 577, arXiv:physics/0408124
  * [37] L. Breiman, J. H. Friedman, R. A. Olshen, and C. J. Stone, Classification and regression trees, Wadsworth international group, Belmont, California, USA, 1984
  * [38] E. Baracchini and G. Isidori, _Electromagnetic corrections to non-leptonic two-body B and D decays_ , Phys. Lett. B633 (2006) 309, arXiv:hep-ph/0508071
  * [39] ARGUS collaboration, H. Albrecht et al., _Measurement of the polarization in the decay $B\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{K}^{*}}$_, Phys. Lett. B340 (1994) 217
  * [40] K. S. Cranmer, _Kernel estimation in high-energy physics_ , Comput. Phys. Commun. 136 (2001) 198, arXiv:hep-ex/0011057
  * [41] M. Pivk and F. R. Le Diberder, _sPlot: A statistical tool to unfold data distributions_ , Nucl. Instrum. Meth. A555 (2005) 356, arXiv:physics/0402083

LHCb collaboration

R. Aaij27, B. Adeva41, M. Adinolfi48, C.A. Aidala73, Z. Ajaltouni5, S. Akar59,
P. Albicocco18, J. Albrecht10, F. Alessio42, M. Alexander53, A. Alfonso
Albero40, S. Ali27, G. Alkhazov33, P. Alvarez Cartelle55, A.A. Alves Jr41, S.
Amato2, S. Amerio23, Y. Amhis7, L. An3, L. Anderlini17, G. Andreassi43, M.
Andreotti16,g, J.E. Andrews60, R.B. Appleby56, F. Archilli27, P. d’Argent12,
J. Arnau Romeu6, A. Artamonov39, M. Artuso61, K. Arzymatov37, E. Aslanides6,
M. Atzeni44, B. Audurier22, S. Bachmann12, J.J. Back50, S. Baker55, V.
Balagura7,b, W. Baldini16, A. Baranov37, R.J. Barlow56, S. Barsuk7, W.
Barter56, F. Baryshnikov70, V. Batozskaya31, B. Batsukh61, V. Battista43, A.
Bay43, J. Beddow53, F. Bedeschi24, I. Bediaga1, A. Beiter61, L.J. Bel27, S.
Belin22, N. Beliy63, V. Bellee43, N. Belloli20,i, K. Belous39, I.
Belyaev34,42, E. Ben-Haim8, G. Bencivenni18, S. Benson27, S. Beranek9, A.
Berezhnoy35, R. Bernet44, D. Berninghoff12, E. Bertholet8, A. Bertolin23, C.
Betancourt44, F. Betti15,42, M.O. Bettler49, M. van Beuzekom27, Ia.
Bezshyiko44, S. Bhasin48, J. Bhom29, S. Bifani47, P. Billoir8, A. Birnkraut10,
A. Bizzeti17,u, M. Bjørn57, M.P. Blago42, T. Blake50, F. Blanc43, S. Blusk61,
D. Bobulska53, V. Bocci26, O. Boente Garcia41, T. Boettcher58, A. Bondar38,w,
N. Bondar33, S. Borghi56,42, M. Borisyak37, M. Borsato41, F. Bossu7, M.
Boubdir9, T.J.V. Bowcock54, C. Bozzi16,42, S. Braun12, M. Brodski42, J.
Brodzicka29, A. Brossa Gonzalo50, D. Brundu22, E. Buchanan48, A. Buonaura44,
C. Burr56, A. Bursche22, J. Buytaert42, W. Byczynski42, S. Cadeddu22, H.
Cai64, R. Calabrese16,g, R. Calladine47, M. Calvi20,i, M. Calvo Gomez40,m, A.
Camboni40,m, P. Campana18, D.H. Campora Perez42, L. Capriotti56, A.
Carbone15,e, G. Carboni25, R. Cardinale19,h, A. Cardini22, P. Carniti20,i, L.
Carson52, K. Carvalho Akiba2, G. Casse54, L. Cassina20, M. Cattaneo42, G.
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Chen3, S. Chen22, S.-G. Chitic42, V. Chobanova41, M. Chrzaszcz42, A.
Chubykin33, P. Ciambrone18, X. Cid Vidal41, G. Ciezarek42, P.E.L. Clarke52, M.
Clemencic42, H.V. Cliff49, J. Closier42, V. Coco42, J.A.B. Coelho7, J. Cogan6,
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Montells12, A. Contu22, G. Coombs42, S. Coquereau40, G. Corti42, M. Corvo16,g,
C.M. Costa Sobral50, B. Couturier42, G.A. Cowan52, D.C. Craik58, A.
Crocombe50, M. Cruz Torres1, R. Currie52, C. D’Ambrosio42, F. Da Cunha
Marinho2, C.L. Da Silva74, E. Dall’Occo27, J. Dalseno48, A. Danilina34, A.
Davis3, O. De Aguiar Francisco42, K. De Bruyn42, S. De Capua56, M. De Cian43,
J.M. De Miranda1, L. De Paula2, M. De Serio14,d, P. De Simone18, C.T. Dean53,
D. Decamp4, L. Del Buono8, B. Delaney49, H.-P. Dembinski11, M. Demmer10, A.
Dendek30, D. Derkach37, O. Deschamps5, F. Desse7, F. Dettori54, B. Dey65, A.
Di Canto42, P. Di Nezza18, S. Didenko70, H. Dijkstra42, F. Dordei42, M.
Dorigo42,y, A. Dosil Suárez41, L. Douglas53, A. Dovbnya45, K. Dreimanis54, L.
Dufour27, G. Dujany8, P. Durante42, J.M. Durham74, D. Dutta56, R.
Dzhelyadin39, M. Dziewiecki12, A. Dziurda29, A. Dzyuba33, S. Easo51, U.
Egede55, V. Egorychev34, S. Eidelman38,w, S. Eisenhardt52, U. Eitschberger10,
R. Ekelhof10, L. Eklund53, S. Ely61, A. Ene32, S. Escher9, S. Esen27, T.
Evans59, A. Falabella15, N. Farley47, S. Farry54, D. Fazzini20,42,i, L.
Federici25, P. Fernandez Declara42, A. Fernandez Prieto41, F. Ferrari15, L.
Ferreira Lopes43, F. Ferreira Rodrigues2, M. Ferro-Luzzi42, S. Filippov36,
R.A. Fini14, M. Fiorini16,g, M. Firlej30, C. Fitzpatrick43, T. Fiutowski30, F.
Fleuret7,b, M. Fontana22,42, F. Fontanelli19,h, R. Forty42, V. Franco Lima54,
M. Frank42, C. Frei42, J. Fu21,q, W. Funk42, C. Färber42, M. Féo Pereira
Rivello Carvalho27, E. Gabriel52, A. Gallas Torreira41, D. Galli15,e, S.
Gallorini23, S. Gambetta52, Y. Gan3, M. Gandelman2, P. Gandini21, Y. Gao3,
L.M. Garcia Martin72, B. Garcia Plana41, J. García Pardiñas44, J. Garra
Tico49, L. Garrido40, D. Gascon40, C. Gaspar42, L. Gavardi10, G. Gazzoni5, D.
Gerick12, E. Gersabeck56, M. Gersabeck56, T. Gershon50, D. Gerstel6, Ph.
Ghez4, S. Gianì43, V. Gibson49, O.G. Girard43, L. Giubega32, K. Gizdov52, V.V.
Gligorov8, D. Golubkov34, A. Golutvin55,70, A. Gomes1,a, I.V. Gorelov35, C.
Gotti20,i, E. Govorkova27, J.P. Grabowski12, R. Graciani Diaz40, L.A. Granado
Cardoso42, E. Graugés40, E. Graverini44, G. Graziani17, A. Grecu32, R.
Greim27, P. Griffith22, L. Grillo56, L. Gruber42, B.R. Gruberg Cazon57, O.
Grünberg67, C. Gu3, E. Gushchin36, Yu. Guz39,42, T. Gys42, C. Göbel62, T.
Hadavizadeh57, C. Hadjivasiliou5, G. Haefeli43, C. Haen42, S.C. Haines49, B.
Hamilton60, X. Han12, T.H. Hancock57, S. Hansmann-Menzemer12, N. Harnew57,
S.T. Harnew48, T. Harrison54, C. Hasse42, M. Hatch42, J. He63, M. Hecker55, K.
Heinicke10, A. Heister10, K. Hennessy54, L. Henry72, E. van Herwijnen42, M.
Heß67, A. Hicheur2, R. Hidalgo Charman56, D. Hill57, M. Hilton56, P.H.
Hopchev43, W. Hu65, W. Huang63, Z.C. Huard59, W. Hulsbergen27, T. Humair55, M.
Hushchyn37, D. Hutchcroft54, D. Hynds27, P. Ibis10, M. Idzik30, P. Ilten47, K.
Ivshin33, R. Jacobsson42, J. Jalocha57, E. Jans27, A. Jawahery60, F. Jiang3,
M. John57, D. Johnson42, C.R. Jones49, C. Joram42, B. Jost42, N. Jurik57, S.
Kandybei45, M. Karacson42, J.M. Kariuki48, S. Karodia53, N. Kazeev37, M.
Kecke12, F. Keizer49, M. Kelsey61, M. Kenzie49, T. Ketel28, E. Khairullin37,
B. Khanji12, C. Khurewathanakul43, K.E. Kim61, T. Kirn9, S. Klaver18, K.
Klimaszewski31, T. Klimkovich11, S. Koliiev46, M. Kolpin12, R. Kopecna12, P.
Koppenburg27, I. Kostiuk27, S. Kotriakhova33, M. Kozeiha5, L. Kravchuk36, M.
Kreps50, F. Kress55, P. Krokovny38,w, W. Krupa30, W. Krzemien31, W.
Kucewicz29,l, M. Kucharczyk29, V. Kudryavtsev38,w, A.K. Kuonen43, T.
Kvaratskheliya34,42, D. Lacarrere42, G. Lafferty56, A. Lai22, D. Lancierini44,
G. Lanfranchi18, C. Langenbruch9, T. Latham50, C. Lazzeroni47, R. Le Gac6, A.
Leflat35, J. Lefrançois7, R. Lefèvre5, F. Lemaitre42, O. Leroy6, T. Lesiak29,
B. Leverington12, P.-R. Li63, T. Li3, Z. Li61, X. Liang61, T. Likhomanenko69,
R. Lindner42, F. Lionetto44, V. Lisovskyi7, X. Liu3, D. Loh50, A. Loi22, I.
Longstaff53, J.H. Lopes2, G.H. Lovell49, D. Lucchesi23,o, M. Lucio Martinez41,
A. Lupato23, E. Luppi16,g, O. Lupton42, A. Lusiani24, X. Lyu63, F. Machefert7,
F. Maciuc32, V. Macko43, P. Mackowiak10, S. Maddrell-Mander48, O. Maev33,42,
K. Maguire56, D. Maisuzenko33, M.W. Majewski30, S. Malde57, B. Malecki29, A.
Malinin69, T. Maltsev38,w, G. Manca22,f, G. Mancinelli6, D. Marangotto21,q, J.
Maratas5,v, J.F. Marchand4, U. Marconi15, C. Marin Benito7, M. Marinangeli43,
P. Marino43, J. Marks12, P.J. Marshall54, G. Martellotti26, M. Martin6, M.
Martinelli42, D. Martinez Santos41, F. Martinez Vidal72, A. Massafferri1, M.
Materok9, R. Matev42, A. Mathad50, Z. Mathe42, C. Matteuzzi20, A. Mauri44, E.
Maurice7,b, B. Maurin43, A. Mazurov47, M. McCann55,42, A. McNab56, R.
McNulty13, J.V. Mead54, B. Meadows59, C. Meaux6, F. Meier10, N. Meinert67, D.
Melnychuk31, M. Merk27, A. Merli21,q, E. Michielin23, D.A. Milanes66, E.
Millard50, M.-N. Minard4, L. Minzoni16,g, D.S. Mitzel12, A. Mogini8, J. Molina
Rodriguez1,z, T. Mombächer10, I.A. Monroy66, S. Monteil5, M. Morandin23, G.
Morello18, M.J. Morello24,t, O. Morgunova69, J. Moron30, A.B. Morris6, R.
Mountain61, F. Muheim52, M. Mulder27, C.H. Murphy57, D. Murray56, A. Mödden
10, D. Müller42, J. Müller10, K. Müller44, V. Müller10, P. Naik48, T.
Nakada43, R. Nandakumar51, A. Nandi57, T. Nanut43, I. Nasteva2, M. Needham52,
N. Neri21, S. Neubert12, N. Neufeld42, M. Neuner12, T.D. Nguyen43, C. Nguyen-
Mau43,n, S. Nieswand9, R. Niet10, N. Nikitin35, A. Nogay69, N.S. Nolte42, D.P.
O’Hanlon15, A. Oblakowska-Mucha30, V. Obraztsov39, S. Ogilvy18, R.
Oldeman22,f, C.J.G. Onderwater68, A. Ossowska29, J.M. Otalora Goicochea2, P.
Owen44, A. Oyanguren72, P.R. Pais43, T. Pajero24,t, A. Palano14, M.
Palutan18,42, G. Panshin71, A. Papanestis51, M. Pappagallo52, L.L.
Pappalardo16,g, W. Parker60, C. Parkes56, G. Passaleva17,42, A. Pastore14, M.
Patel55, C. Patrignani15,e, A. Pearce42, A. Pellegrino27, G. Penso26, M. Pepe
Altarelli42, S. Perazzini42, D. Pereima34, P. Perret5, L. Pescatore43, K.
Petridis48, A. Petrolini19,h, A. Petrov69, S. Petrucci52, M. Petruzzo21,q, B.
Pietrzyk4, G. Pietrzyk43, M. Pikies29, M. Pili57, D. Pinci26, J. Pinzino42, F.
Pisani42, A. Piucci12, V. Placinta32, S. Playfer52, J. Plews47, M. Plo
Casasus41, F. Polci8, M. Poli Lener18, A. Poluektov50, N. Polukhina70,c, I.
Polyakov61, E. Polycarpo2, G.J. Pomery48, S. Ponce42, A. Popov39, D.
Popov47,11, S. Poslavskii39, C. Potterat2, E. Price48, J. Prisciandaro41, C.
Prouve48, V. Pugatch46, A. Puig Navarro44, H. Pullen57, G. Punzi24,p, W.
Qian63, J. Qin63, R. Quagliani8, B. Quintana5, B. Rachwal30, J.H.
Rademacker48, M. Rama24, M. Ramos Pernas41, M.S. Rangel2, F. Ratnikov37,x, G.
Raven28, M. Ravonel Salzgeber42, M. Reboud4, F. Redi43, S. Reichert10, A.C.
dos Reis1, F. Reiss8, C. Remon Alepuz72, Z. Ren3, V. Renaudin7, S.
Ricciardi51, S. Richards48, K. Rinnert54, P. Robbe7, A. Robert8, A.B.
Rodrigues43, E. Rodrigues59, J.A. Rodriguez Lopez66, M. Roehrken42, A.
Rogozhnikov37, S. Roiser42, A. Rollings57, V. Romanovskiy39, A. Romero
Vidal41, M. Rotondo18, M.S. Rudolph61, T. Ruf42, J. Ruiz Vidal72, J.J.
Saborido Silva41, N. Sagidova33, B. Saitta22,f, V. Salustino Guimaraes62, C.
Sanchez Gras27, C. Sanchez Mayordomo72, B. Sanmartin Sedes41, R.
Santacesaria26, C. Santamarina Rios41, M. Santimaria18, E. Santovetti25,j, G.
Sarpis56, A. Sarti18,k, C. Satriano26,s, A. Satta25, M. Saur63, D.
Savrina34,35, S. Schael9, M. Schellenberg10, M. Schiller53, H. Schindler42, M.
Schmelling11, T. Schmelzer10, B. Schmidt42, O. Schneider43, A. Schopper42,
H.F. Schreiner59, M. Schubiger43, M.H. Schune7, R. Schwemmer42, B. Sciascia18,
A. Sciubba26,k, A. Semennikov34, E.S. Sepulveda8, A. Sergi47,42, N. Serra44,
J. Serrano6, L. Sestini23, A. Seuthe10, P. Seyfert42, M. Shapkin39, Y.
Shcheglov33,†, T. Shears54, L. Shekhtman38,w, V. Shevchenko69, E. Shmanin70,
B.G. Siddi16, R. Silva Coutinho44, L. Silva de Oliveira2, G. Simi23,o, S.
Simone14,d, N. Skidmore12, T. Skwarnicki61, J.G. Smeaton49, E. Smith9, I.T.
Smith52, M. Smith55, M. Soares15, l. Soares Lavra1, M.D. Sokoloff59, F.J.P.
Soler53, B. Souza De Paula2, B. Spaan10, P. Spradlin53, F. Stagni42, M.
Stahl12, S. Stahl42, P. Stefko43, S. Stefkova55, O. Steinkamp44, S. Stemmle12,
O. Stenyakin39, M. Stepanova33, H. Stevens10, A. Stocchi7, S. Stone61, B.
Storaci44, S. Stracka24,p, M.E. Stramaglia43, M. Straticiuc32, U. Straumann44,
S. Strokov71, J. Sun3, L. Sun64, K. Swientek30, V. Syropoulos28, T. Szumlak30,
M. Szymanski63, S. T’Jampens4, Z. Tang3, A. Tayduganov6, T. Tekampe10, G.
Tellarini16, F. Teubert42, E. Thomas42, J. van Tilburg27, M.J. Tilley55, V.
Tisserand5, M. Tobin30, S. Tolk42, L. Tomassetti16,g, D. Tonelli24, D.Y. Tou8,
R. Tourinho Jadallah Aoude1, E. Tournefier4, M. Traill53, M.T. Tran43, A.
Trisovic49, A. Tsaregorodtsev6, G. Tuci24, A. Tully49, N. Tuning27,42, A.
Ukleja31, A. Usachov7, A. Ustyuzhanin37, U. Uwer12, A. Vagner71, V. Vagnoni15,
A. Valassi42, S. Valat42, G. Valenti15, R. Vazquez Gomez42, P. Vazquez
Regueiro41, S. Vecchi16, M. van Veghel27, J.J. Velthuis48, M. Veltri17,r, G.
Veneziano57, A. Venkateswaran61, T.A. Verlage9, M. Vernet5, M. Veronesi27,
N.V. Veronika13, M. Vesterinen57, J.V. Viana Barbosa42, D. Vieira63, M.
Vieites Diaz41, H. Viemann67, X. Vilasis-Cardona40,m, A. Vitkovskiy27, M.
Vitti49, V. Volkov35, A. Vollhardt44, B. Voneki42, A. Vorobyev33, V.
Vorobyev38,w, J.A. de Vries27, C. Vázquez Sierra27, R. Waldi67, J. Walsh24, J.
Wang61, M. Wang3, Y. Wang65, Z. Wang44, D.R. Ward49, H.M. Wark54, N.K.
Watson47, D. Websdale55, A. Weiden44, C. Weisser58, M. Whitehead9, J. Wicht50,
G. Wilkinson57, M. Wilkinson61, I. Williams49, M.R.J. Williams56, M.
Williams58, T. Williams47, F.F. Wilson51,42, J. Wimberley60, M. Winn7, J.
Wishahi10, W. Wislicki31, M. Witek29, G. Wormser7, S.A. Wotton49, K. Wyllie42,
D. Xiao65, Y. Xie65, A. Xu3, M. Xu65, Q. Xu63, Z. Xu3, Z. Xu4, Z. Yang3, Z.
Yang60, Y. Yao61, L.E. Yeomans54, H. Yin65, J. Yu65,ab, X. Yuan61, O.
Yushchenko39, K.A. Zarebski47, M. Zavertyaev11,c, D. Zhang65, L. Zhang3, W.C.
Zhang3,aa, Y. Zhang7, A. Zhelezov12, Y. Zheng63, X. Zhu3, V. Zhukov9,35, J.B.
Zonneveld52, S. 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
4Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, IN2P3-LAPP, Annecy,
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, Sorbonne Université, Paris Diderot Sorbonne Paris Cité, 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
14INFN Sezione di Bari, Bari, Italy
15INFN Sezione di Bologna, Bologna, Italy
16INFN Sezione di Ferrara, Ferrara, Italy
17INFN Sezione di Firenze, Firenze, Italy
18INFN Laboratori Nazionali di Frascati, Frascati, Italy
19INFN Sezione di Genova, Genova, Italy
20INFN Sezione di Milano-Bicocca, Milano, Italy
21INFN Sezione di Milano, Milano, Italy
22INFN Sezione di Cagliari, Monserrato, Italy
23INFN Sezione di Padova, Padova, Italy
24INFN Sezione di Pisa, Pisa, Italy
25INFN Sezione di Roma Tor Vergata, Roma, Italy
26INFN Sezione di Roma La Sapienza, Roma, Italy
27Nikhef National Institute for Subatomic Physics, Amsterdam, Netherlands
28Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, Netherlands
29Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
30AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
31National Center for Nuclear Research (NCBJ), Warsaw, Poland
32Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
33Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
34Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
35Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
36Institute for Nuclear Research of the Russian Academy of Sciences (INR RAS),
Moscow, Russia
37Yandex School of Data Analysis, Moscow, Russia
38Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
39Institute for High Energy Physics (IHEP), Protvino, Russia
40ICCUB, Universitat de Barcelona, Barcelona, Spain
41Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de
Santiago de Compostela, Santiago de Compostela, Spain
42European Organization for Nuclear Research (CERN), Geneva, Switzerland
43Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
44Physik-Institut, Universität Zürich, Zürich, Switzerland
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
68Van Swinderen Institute, University of Groningen, Groningen, Netherlands,
associated to 27
69National Research Centre Kurchatov Institute, Moscow, Russia, associated to
34
70National University of Science and Technology ”MISIS”, Moscow, Russia,
associated to 34
71National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
34
72Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 40
73University of Michigan, Ann Arbor, United States, associated to 61
74Los Alamos National Laboratory (LANL), Los Alamos, United States, associated
to 61

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, Vietnam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy
vMSU - Iligan Institute of Technology (MSU-IIT), Iligan, Philippines
wNovosibirsk State University, Novosibirsk, Russia
xNational Research University Higher School of Economics, Moscow, Russia
ySezione INFN di Trieste, Trieste, Italy
zEscuela Agrícola Panamericana, San Antonio de Oriente, Honduras
aaSchool of Physics and Information Technology, Shaanxi Normal University
(SNNU), Xi’an, China
abPhysics and Micro Electronic College, Hunan University, Changsha City, China
†Deceased


# GLD-Net: Improving Monaural Speech Enhancement by Learning Global and Local
Dependency Features with GLD Block

###### Abstract

For monaural speech enhancement, contextual information is important for
accurate speech estimation. However, commonly used convolution neural networks
(CNNs) are weak in capturing temporal contexts since they only build blocks
that process one local neighborhood at a time. To address this problem, we
learn from human auditory perception to introduce a two-stage trainable
reasoning mechanism, referred as global-local dependency (GLD) block. GLD
blocks capture long-term dependency of time-frequency bins both in global
level and local level from the noisy spectrogram to help detecting
correlations among speech part, noise part, and whole noisy input. What is
more, we conduct a monaural speech enhancement network called GLD-Net, which
adopts encoder-decoder architecture and consists of speech object branch,
interference branch, and global noisy branch. The extracted speech feature at
global-level and local-level are efficiently reasoned and aggregated in each
of the branches. We compare the proposed GLD-Net with existing state-of-art
methods on WSJ0 and DEMAND dataset. The results show that GLD-Net outperforms
the state-of-the-art methods in terms of PESQ and STOI.

Index Terms: monaural speech enhancement, global and local dependency,
encoder-decoder architecture, two-stage trainable reasoning mechanism

##  1 Introduction

It has been a long-standing topic in speech processing applications to reduce
background noise and improve quality and intelligibility of degraded speech.
Speech enhancement is frequently used as the preprocessor of the other
acoustic tasks, such as speech recognition [1], cochlear implants [2], and
hearing aids [3]. Monaural speech enhancement provides a versatile and cost-
effective approach to the problem by utilizing recordings from a single
microphone.

Traditional monaural speech enhancement approaches can include Spectral
Subtraction [4], minimum mean-square error (MMSE) estimation [5], Wiener
filter (linear MMSE) [6], Kalman filter [7], subspace method [8], etc.
However, these kinds of speech enhancement methods generally do not perform
well in adverse noise environments. In recent years, deep neural networks
(DNNs) have shown their promising performance on monaural speech enhancement
even in highly non-stationary noise environments because of their superior
capability in modeling complex non-linearity.

Most of current deep learning architectures for speech enhancement are
implemented in time-frequency (T-F) domain by using short-time Fourier
transform (STFT), those methods usually treat the spectral magnitude as
training target. The phase of noisy speech is utilized along with the enhanced
speech magnitude to reconstruct the time-domain signal with inverse short-time
Fourier transform (iSTFT) [9, 10, 11]. Many researchers have investigated the
convolution neural network (CNN) or recurrent neural network (RNN). For
modeling long-range sequence like speech, CNN requires more convolutional
layers to enlarge receptive field. Additionally, although RNN such as long
short-term memory (LSTM) or gated recurrent units (GRU) are commonly used in
modeling long-term sequence with order information, they cannot perform
parallel processing and lead to high computation complexity. Recently, some
improvement is achieved by adding temporal convolutional network (TCN) [12]
blocks or LSTM layers between encoder and decoder for further extracting high-
level features and enlarging receptive fields [13]. However, the contextual
information of observed noisy speech is often ignored, which restricts the
denoising performance.

To tackle the limitations above, we propose a trainable mechanism to infer
speech enhancement which contains a two-stage reasoning process according to
the human auditory perception. It is noted that human auditory system is able
to identify non-speech sounds and human speech through category-selective
responses by capturing long-term context-dependency from the scene and related
speech regions [14], which was particularly selective for acoustic-phonetic
content of speech [15]. Considering the concert (i.e. instrument sounds and
human talk) as an example, the global-level sound information from the scene
(e.g. instrument sounds and mixed speech) can be first captured as prior
knowledge and the speech from the target person is then learned and selected
for further processes. Following this perception, we propose a two-stage
mechanism to infer speech enhancement where global and local dependency of
time-frequency (T-F) bins of speech spectrogram are captured. Aggregating this
innovation synthetically, we propose GLD-Net. The contributions of this work
are summarized as follows:

  * •

A two-stage reasoning mechanism is proposed to learn long-term dependency of
T-F bins both in global level and local level from the noisy spectrogram to
help detecting interactions between speech part and whole noisy scene.

  * •

An efficient framework for monaural speech enhancement is introduced to
efficiently assemble GLD block, namely GLD-Net. The GLD-Net consists of a
speech object branch, interference noise branch, and a global noisy branch. In
particular, the GLD-Net takes encoder-decoder architecture and the GLD block
is inserted in encoder part. In addition, a local-attention is utilized in
speech object branch, and interference branch of GLD-Net to detect speech
region and filter the noise relevant features, respectively.

  * •

To evaluate the proposed GLD-Net, we perform comprehensive ablation studies on
public available speech data. Results show that GLD-Net outperforms all 3
evaluated models.

Figure 1: The proposed GLD block in speech branch (one of the three branches
from our proposed GLD-Net), where Conv-E, Conv-D, BN, and LA denote
convolution encoder, convolution decoder, batch normalization, and local
attention, respectively.

##  2 Model Architecture

###  2.1 GLD Block

While listening to a target speech, a person is able to capture its long-range
context information from the scene and target speech, rather than just orderly
focus on target region of the speech signal like convolution operations do.
Thus, we learn from human auditory perception to build a two-stage reasoning
mechanism which captures long-range dependency in global-level and local-level
to predict target speech. To make the above assumption computationally
feasible, we design the GLD block containing two parts to exploit both global
and local dependency. Taking speech branch (one of three branches from our
proposed GLD-Net) as an example, Figure 1 shows the detailed procedure of GLD
block.

We divide the process of GLD block into two parts, which are global-dependency
part and the local-dependency part. While a speech happens throughout the
whole noisy observation, we firstly capture global dependency from the whole
noisy speech through the global-dependency part. Afterwards, we capture local
dependency from speech region through the local-dependency part.

Figure 2: The schematic diagram of local attention.

####  2.1.1 Global-dependency part

In global-dependency part, given an input feature map $F_{\text{in}}$, two 2D
convolution layers, each of which is followed by batch normalization [16] and
exponential linear unit (ELU) [17], are adopt to generate new feature map K
and V respectively, where $\\{\textbf{K},\textbf{V}\\}\in\mathbb{R}^{T\times
F\times C}$, and $T$, $F$, and $C$ denote the number of time steps, frequency
bands, and convolution channels. Concretely, a matrix multiplication between
K, $\textbf{V}^{\top}$ is operated, and to obtain the attention map X, a
softmax layer is connected as:

$x_{ji}=\frac{exp(K_{i}\times V_{j})}{\sum^{C}_{i=1}exp(K_{i}\times V_{j})},$
(1)

in which $x_{ji}$ means the $i^{th}$ channel’s influence on the $j^{th}$
channel. The closer feature representation of the two bins signifies a
stronger correlation between them.

Finally, the results of matrix multiplication between $\textbf{X}^{\top}$ and
V are weighted by a parameter of scale $\alpha$ to acquire the output of
global-dependency part G:

$G_{j}=\alpha\sum^{C}_{i=1}(x_{ji}V_{j}),$ (2)

where $\alpha$ is initialized as zero and can be learned gradually.

Figure 3: The proposed GLD-Net, where “SB”, “NB”, and “IB” denote speech
branch, noisy branch, and interference branch, respectively.

####  2.1.2 Local-dependency part

In local-dependency part, the intermediate feature map G is computed from the
output of global-dependency part, G, and feature map Q which is computed from
initial input $F_{\text{in}}$ processing by a 2D convolution block and a local
attention (LA).

The LA [18] is described in Figure 2. E is the $F_{\text{in}}$ processing by
the 2D convolution block, and K is the feature map in the global-dependency
part. Two additional 2D convolution block, referring as $\textbf{W}_{g}^{att}$
and $\textbf{W}_{x}^{att}$, are used to mapping E and K to high-dimensional
space feature:

$\textbf{R}=\sigma(\textbf{W}_{g}^{att}\textbf{E}^{\top}+\textbf{W}_{x}^{att}\textbf{K}^{\top}),$
(3)

which is fed into a single 2D deconvolution block, $\textbf{W}_{f}^{att}$, to
give the local attention mask:

$\textbf{P}=\sigma(\textbf{W}_{f}^{att}\textbf{R}^{\top}),$ (4)

where $\sigma$ denotes sigmoid function. Then, the feature map Q is generated
by

$\textbf{Q}=\textbf{R}\times\textbf{P},$ (5)

The local attention is learning a soft voice activity detection (VAD) and is
keeping features focus more on the voice activity region [19].

Next, a multiplication of metrics is executed between Q and G, and a softmax
layer is attached subsequently to calculate the attention maps Y:

$y_{ji}=\frac{exp(Q_{i}\times G_{j})}{\sum^{C}_{i=1}exp(Q_{i}\times G_{j})},$
(6)

where $y_{ji}$ measures the impact of $i^{th}$ bin to the $j^{th}$ bin. Then,
a multiplication of metrics is performed between Y and G, and the result is:

$L_{j}=\beta\sum^{C}_{i=1}(y_{ji}G_{j}),$ (7)

where $\beta$ with a zero initial value can be learned to assign larger weight
gradually. Finally, the result feature map L is fed into a 2D deconvolution
block to form the output of the GLD block, $F_{\text{out}}$.

The goal of the matrix multiplications and softmax operation in GLD block is
to dynamically conduct target-specfic reasoning and capture long-range
dependency for a speech sequence. In addition, different T-F bins are
considered in the spectrogram from global-dependency part and local-dependency
part.

###  2.2 GLD-Net

The proposed GLD-Net uses convolutional encoder-decoder structure based on the
standard U-Net [20], shown in Figure 3. The encoder contains several GLD
layers, which are introduced in Section 2.2.1. The bottleneck consists of two
LSTM layers to improve its generalization capabilities [21]. The decoder
contains several 2D dilated decovolution blocks [22]. Skip connections are
introduced to compensate for information loss during the encoding process.

In addition, the input of GLD-Net is noisy real-imaginary spectrogram computed
by short-time Fourier transform (STFT), denoted by
$\textbf{Y}_{r,i}\in\mathbb{R}^{T\times F\times 2}$ [23]. The output of GLD-
Net, $\textbf{S}_{out}$, is processed by a learnable decoder layer to
transform the feature back to the predicted time-domain signal,
$\textbf{s}_{pred}\in\mathbb{R}^{N\times 1}$. The network structure of the
learnable decoder is designed to be exactly the same as the convolutional
implementation of inverse STFT. Specifically, the learnable decoder is a 1D
transposed convolution layer whose kernel size and stride are the window
length and hop size in the STFT [24, 25].

####  2.2.1 GLD layer

Each GLD layer of GLD-Net contains three branches, speech branch (SB), noisy
branch (NB), and interference branch (IB), as shown in Figure 3. The local-
level feature (SB feature and IB feature), and the global-level feature (NB
feature) are extracted from incoming convolution feature map by using three 2D
convolution blocks. Then, speech local-global feature and noise local-global
feature obtained from GLD block are concatenated to SB and IB features,
respectively. It is noted that the GLD blocks for SB and IB are different, in
which GLD block for SB is as shown in Figure 1 and GLD block for IB which we
change the Eq.(5) of GLD block for SB to
$\textbf{Q}=(1-\textbf{P})\times\textbf{E}$. The speech and interference
branches takes speech T-F features as inputs to encode long-range dependency.

With the feature vectors, two 2D convolution blocks are employed to fuse the
concatenated features, then a matrix addition between the two fused features
followed by a sigmoid function are operated to generate local-global
dependency gate. Next, several convolution blocks are adopted to extract the
feature from original input, obtaining the intermediate feature map. After
that SB and NB features are concatenated to intermediate feature, and fed into
a 2D convolution block to generate the confidence feature. Our motivation for
the concatenation above is that the global context and speech region
information are both important for extracting the target speech. Finally, the
result of multiplication between confidence feature and local-global
dependency gate is the output of a GLD layer, and is passed to the next layer.

Table 1: Evaluation results of proposed model compared with baseline models with unseen speakers. BOLD indicates the best result of each column. Metric | PESQ | STOI (in %)
---|---|---
Test SNR | -5 | 0 | 5 | 10 | Avg. | -5 | 0 | 5 | 10 | Avg.
Unprocessed | 1.37 | 1.88 | 2.12 | 2.49 | 1.97 | 57.06 | 69.33 | 80.59 | 87.63 | 73.40
CRN [13] | 2.14 | 2.49 | 2.79 | 3.09 | 2.63 | 78.30 | 88.61 | 91.11 | 94.74 | 88.19
GRN [26] | 2.20 | 2.58 | 2.84 | 3.12 | 2.69 | 81.45 | 88.09 | 92.43 | 94.96 | 89.23
T-GSA [27] | 2.33 | 2.70 | 2.96 | 3.24 | 2.81 | 83.17 | 89.54 | 93.47 | 95.76 | 90.49
Proposed GLD-Net | 2.41 | 2.86 | 3.03 | 3.28 | 2.90 | 84.06 | 90.12 | 94.28 | 96.19 | 91.16
Table 2: Ablation study of proposed model. BOLD indicates the best result of
each column.

Metric | PESQ | STOI (in %)
---|---|---
Test SNR | -5 | 0 | 5 | 10 | Avg. | -5 | 0 | 5 | 10 | Avg.
Unprocessed | 1.37 | 1.88 | 2.12 | 2.49 | 1.97 | 57.06 | 69.33 | 80.59 | 87.63 | 73.40
GLD-Net w/o speech branch | 2.23 | 2.59 | 2.86 | 3.11 | 2.70 | 81.36 | 88.41 | 92.21 | 94.86 | 89.21
GLD-Net w/o interference branch | 2.38 | 2.84 | 2.97 | 3.20 | 2.85 | 83.45 | 89.79 | 94.06 | 95.42 | 90.68
GLD-Net w/o speech and interference branches | 2.17 | 2.48 | 2.77 | 3.04 | 2.62 | 80.32 | 88.24 | 92.05 | 94.76 | 88.84
Proposed GLD-Net | 2.41 | 2.86 | 3.03 | 3.28 | 2.90 | 84.06 | 90.12 | 94.28 | 96.19 | 91.16

##  3 Experimental Setup

In our experiments, we evaluate the models on the WSJ0 SI-84 training set [28]
including 7138 utterances from 83 speakers (42 males and 41 females). Among
these speakers, 6 speakers (3 males and 3 females) are treated as untrained
speakers. Hence, we train the models with 77 remaining speakers. To obtain
noise-independent models, we select 10 types of noise signals from Demand
dataset [29]. For training, 4.5 hours clean speech randomly mix with noises at
a signal-to-noise ratio (SNR) that is randomly chosen from
$\\{-5,-3,0,3,5,7,10\\}$dB. The validation and test set are constructed in the
same way, using 1.5 hours of clean speech each, where speakers do not overlap
in between the three sets. An additional test sets employing unseen noise
types is constructed by using the clean test set speech files and two unseen
noise types from the NOISE-92 database [30]. For the evaluation, additional
unseen SNR conditions of $\\{-5,0,5,10\\}$dB are employed.

All data are downsampled to 16 kHz and features are extracted by using frames
of length 512 with frame shift of 256, and Hann windowing followed by STFT of
size $K=512$ with respective zero-padding. Real and imaginary parts of the
complex spectrum are divided into separate feature maps, resulting in $C=2$
input channels. The models are trained with the Adam optimizer [31]. The
numbers of output channels for the layers in the encoder are changed to 16,
32, 64, 128 and 256 successively, and those for each layer in the decoder to
128, 64, 32, 16 and 1 successively. We set the learning rate to 0.0002. The
mean squared error (MSE) is used as the objective function. We train the model
with minibatch size of 16.

##  4 Results and Analysis

Evaluation metrics are short-time objective intelligibility measure (STOI)
[32] and perceptual evaluation of speech quality (PESQ) [33]. All metrics are
better if higher.

###  4.1 Overall Performance

Table 1 presents comprehensive evaluations for three baseline models on
untrained noises and untrained speakers. The best results in each case are
highlighted by boldface. The three selected baseline models are: (1) CRN [13],
a CRN speech enhancement model with complex spectral mapping, (2) GRN [26], a
gated convolution speech enhancement model with complex spectral mapping, and
(3) T-GSA [27], a Gaussian-weighted self-attention based speech enhancement
model in frequency domain.

We first compare CRN and GRN, GRN yields significant better STOI and PESQ than
CRN. At -5 dB SNR, for example, the GRN improves STOI by $3.15\%$, absolutely
and PESQ by $0.06$ over the CRN. It is noted that GRN that takes dilated
convolutions, has larger receptive fields than CRN, and thus can leverage
long-term contexts. In addition, the T-GSA model substantially outperforms the
GRN. Take the -5 SNR case as an example, the T-GSA yields a $1.63\%$ STOI
improvement and a $0.13$ PESQ improvement compared with the GRN model. This is
observed that transformer neural network promise a stronger long-term
dependencies modelling capability than GRN and CRN, which is consistent with
the findings in [27].

The proposed GLD-Net consistently outperforms CRN, GRN, and T-GSA in all
conditions. Take the $0$ dB SNR case as an example, the proposed GLD-Net
improves STOI by $0.58\%$ and PESQ by $0.16$ over T-GSA. These results
demonstrate that the proposed GLD-Net which detects the global and local
dependency, is more competitive than others in speech enhancement.

###  4.2 Ablation Study

The proposed GLD-Net consists of three branches, SB, NB, and IB, for exploring
the the speech local-global dependency and noise local-global dependency. In
this ablation study, we evaluate variants of our method when removing SB or IB
in this subsection. We set the same parameters for the algorithm with previous
section but control the usage of different branches.

As shown in Table 2, taking results in $-5$dB SNR case as examples, removing
the SB from GLD-Net reduces STOI by $3.74\%$ and PESQ by $0.18$, while
removing the IB from GLD-Net reduces STOI by $0.61\%$ and PESQ by $0.04$. That
is to say, while adopting just two branches for speech enhancement, SB shows
greater strength than IB, which can be interpreted as the speech region
contribute that more information than noise region in an observed noisy
signal.

##  5 Conclusion

In this paper, we introduce a two-stage reasoning mechanism for detecting the
local and global dependency of speech observation for speech enhancement,
namely GLD block. Conforming to the human auditory perception, GLD block not
only learns the speech feature from local regions, but also captures both
global and local dependency of T-F bins from whole noisy speech RI
spectrograms. Without employing any additional inputs, this is a significant
try to exploit speech and noise correlation in global-level and local-level in
a single trainable mechanism. On this basis, we construct GLD-Net, a three-
stream speech enhancement framework consisting of SB, NB, and IB. Through
experiments, we show the superiority of the proposed method over other methods
compared in this paper.

## References

  * [1] F. Weninger, H. Erdogan, S. Watanabe, E. Vincent, J. Le Roux, J. R. Hershey, and B. Schuller, “Speech enhancement with LSTM recurrent neural networks and its application to noise-robust ASR,” in _International conference on latent variable analysis and signal separation_. Springer, 2015, pp. 91–99.
  * [2] L.-P. Yang and Q.-J. Fu, “Spectral subtraction-based speech enhancement for cochlear implant patients in background noise,” _The journal of the Acoustical Society of America_ , vol. 117, no. 3, pp. 1001–1004, 2005.
  * [3] C. K. A. Reddy, N. Shankar, G. S. Bhat, R. Charan, and I. Panahi, “An individualized super-Gaussian single microphone speech enhancement for hearing aid users with smartphone as an assistive device,” _IEEE signal processing letters_ , vol. 24, no. 11, pp. 1601–1605, 2017.
  * [4] S. Boll, “Suppression of acoustic noise in speech using spectral subtraction,” _IEEE Transactions on acoustics, speech, and signal processing_ , vol. 27, no. 2, pp. 113–120, 1979.
  * [5] Y. Ephraim and D. Malah, “Speech enhancement using a minimum-mean square error short-time spectral amplitude estimator,” _IEEE Transactions on acoustics, speech, and signal processing_ , vol. 32, no. 6, pp. 1109–1121, 1984\.
  * [6] J. Lim and A. Oppenheim, “All-pole modeling of degraded speech,” _IEEE Transactions on Acoustics, Speech, and Signal Processing_ , vol. 26, no. 3, pp. 197–210, 1978.
  * [7] K. Paliwal and A. Basu, “A speech enhancement method based on Kalman filtering,” in _1987 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)_ , vol. 12. IEEE, 1987, pp. 177–180.
  * [8] Y. Ephraim and H. L. Van Trees, “A signal subspace approach for speech enhancement,” _IEEE Transactions on speech and audio processing_ , vol. 3, no. 4, pp. 251–266, 1995.
  * [9] S.-W. Fu, T.-y. Hu, Y. Tsao, and X. Lu, “Complex spectrogram enhancement by convolutional neural network with multi-metrics learning,” in _2017 IEEE 27th international workshop on machine learning for signal processing (MLSP)_. IEEE, 2017, pp. 1–6.
  * [10] M. H. Soni, N. Shah, and H. A. Patil, “Time-frequency masking-based speech enhancement using generative adversarial network,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 5039–5043.
  * [11] X. Xu, Y. Wang, D. Xu, Y. Peng, C. Zhang, J. Jia, and B. Chen, “Multi-stage progressive speech enhancement network,” _Proc. Interspeech 2021_ , pp. 2691–2695, 2021.
  * [12] A. Pandey and D. Wang, “TCNN: Temporal convolutional neural network for real-time speech enhancement in the time domain,” in _2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 6875–6879.
  * [13] K. Tan and D. Wang, “Complex spectral mapping with a convolutional recurrent network for monaural speech enhancement,” in _2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 6865–6869.
  * [14] B. Chandrasekaran, J. Hornickel, E. Skoe, T. Nicol, and N. Kraus, “Context-dependent encoding in the human auditory brainstem relates to hearing speech in noise: implications for developmental dyslexia,” _Neuron_ , vol. 64, no. 3, pp. 311–319, 2009.
  * [15] A. M. Leaver and J. P. Rauschecker, “Cortical representation of natural complex sounds: effects of acoustic features and auditory object category,” _Journal of Neuroscience_ , vol. 30, no. 22, pp. 7604–7612, 2010.
  * [16] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” in _International conference on machine learning_. PMLR, 2015, pp. 448–456.
  * [17] D.-A. Clevert, T. Unterthiner, and S. Hochreiter, “Fast and accurate deep network learning by exponential linear units (elus),” _arXiv preprint arXiv:1511.07289_ , 2015.
  * [18] R. Giri, U. Isik, and A. Krishnaswamy, “Attention wave-u-net for speech enhancement,” in _2019 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA)_. IEEE, 2019, pp. 249–253.
  * [19] F. Deng, T. Jiang, X. Wang, C. Zhang, and Y. Li, “NAAGN: Noise-aware attention-gated network for speech enhancement.” in _INTERSPEECH_ , 2020, pp. 2457–2461.
  * [20] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in _International Conference on Medical image computing and computer-assisted intervention_. Springer, 2015, pp. 234–241.
  * [21] M. Strake, B. Defraene, K. Fluyt, W. Tirry, and T. Fingscheidt, “Fully convolutional recurrent networks for speech enhancement,” in _2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 6674–6678.
  * [22] Y. Li, X. Li, Y. Dong, M. Li, S. Xu, and S. Xiong, “Densely connected network with time-frequency dilated convolution for speech enhancement,” in _2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2019, pp. 6860–6864.
  * [23] Z.-Q. Wang, P. Wang, and D. Wang, “Complex spectral mapping for single-and multi-channel speech enhancement and robust ASR,” _IEEE/ACM transactions on audio, speech, and language processing_ , vol. 28, pp. 1778–1787, 2020.
  * [24] R. Gu, J. Wu, S.-X. Zhang, L. Chen, Y. Xu, M. Yu, D. Su, Y. Zou, and D. Yu, “End-to-end multi-channel speech separation,” _arXiv preprint arXiv:1905.06286_ , 2019.
  * [25] C. Tang, C. Luo, Z. Zhao, W. Xie, and W. Zeng, “Joint time-frequency and time domain learning for speech enhancement,” in _Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence_ , 2021, pp. 3816–3822.
  * [26] K. Tan and D. Wang, “Learning complex spectral mapping with gated convolutional recurrent networks for monaural speech enhancement,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 28, pp. 380–390, 2019.
  * [27] J. Kim, M. El-Khamy, and J. Lee, “T-gsa: Transformer with gaussian-weighted self-attention for speech enhancement,” in _2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2020, pp. 6649–6653.
  * [28] D. B. Paul and J. Baker, “The design for the wall street journal-based CSR corpus,” in _Speech and Natural Language: Proceedings of a Workshop Held at Harriman, New York, February 23-26, 1992_ , 1992.
  * [29] J. Thiemann, N. Ito, and E. Vincent, “DEMAND: a collection of multi-channel recordings of acoustic noise in diverse environments,” in _Proc. Meetings Acoust_ , 2013, pp. 1–6.
  * [30] A. Varga and H. J. Steeneken, “Assessment for automatic speech recognition: Ii. NOISEX-92: A database and an experiment to study the effect of additive noise on speech recognition systems,” _Speech communication_ , vol. 12, no. 3, pp. 247–251, 1993.
  * [31] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” _arXiv preprint arXiv:1412.6980_ , 2014.
  * [32] C. H. Taal, R. C. Hendriks, R. Heusdens, and J. Jensen, “An algorithm for intelligibility prediction of time–frequency weighted noisy speech,” _IEEE Transactions on Audio, Speech, and Language Processing_ , vol. 19, no. 7, pp. 2125–2136, 2011.
  * [33] A. W. Rix, J. G. Beerends, M. P. Hollier, and A. P. Hekstra, “Perceptual evaluation of speech quality (PESQ)-a new method for speech quality assessment of telephone networks and codecs,” in _2001 IEEE international conference on acoustics, speech, and signal processing. Proceedings (Cat. No. 01CH37221)_ , vol. 2. IEEE, 2001, pp. 749–752.