EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-212 LHCb-PAPER-2016-030 October 27, 2017

Measurement of matter-antimatter differences in beauty baryon decays

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

Differences in the behaviour of matter and antimatter have been observed in
$K$ and $B$ meson decays, but not yet in any baryon decay. Such differences
are associated with the non-invariance of fundamental interactions under the
combined charge-conjugation and parity transformations, known as $C\\!P$
violation. Using data from the LHCb experiment at the Large Hadron Collider, a
search is made for $C\\!P$-violating asymmetries in the decay angle
distributions of ${\mathchar 28931\relax}^{0}_{b}$ baryons decaying to
${p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and ${p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$
final states. These four-body hadronic decays are a promising place to search
for sources of $C\\!P$ violation both within and beyond the Standard Model of
particle physics. We find evidence for $C\\!P$ violation in ${\mathchar
28931\relax}^{0}_{b}$ to ${p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays with a
statistical significance corresponding to 3.3 standard deviations including
systematic uncertainties. This represents the first evidence for $C\\!P$
violation in the baryon sector.

Published in Nature Physics 13, 391–396 (2017)

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

The asymmetry between matter and antimatter is related to the violation of the
$C\\!P$ symmetry (CPV), where $C$ and $P$ are the charge-conjugation and
parity operators. $C\\!P$ violation is accommodated in the Standard Model (SM)
of particle physics by the Cabibbo-Kobayashi-Maskawa (CKM) mechanism that
describes the transitions between up- and down-type quarks [1, 2], in which
quark decays proceed by the emission of a virtual $W$ boson and where the
phases of the couplings change sign between quarks and antiquarks. However,
the amount of CPV predicted by the CKM mechanism is not sufficient to explain
our matter-dominated Universe [3, 4] and other sources of CPV are expected to
exist. The initial discovery of CPV was in neutral $K$ meson decays [5], and
more recently it has been observed in ${B}^{0}$ [6, 7], ${{B}^{+}}$ [8, 9, 10,
11], and ${B}^{0}_{s}$ [12] meson decays, but it has never been observed in
the decays of any baryon. Decays of the ${\mathchar 28931\relax}^{0}_{b}$
($bud$) baryon to final states consisting of hadrons with no charm quarks are
predicted to have non-negligible $C\\!P$ asymmetries in the SM, as large as
20% for certain three-body decay modes [13]. It is important to measure the
size and nature of these $C\\!P$ asymmetries in as many decay modes as
possible, to determine whether they are consistent with the CKM mechanism or,
if not, what extensions to the SM would be required to explain them [14, 15,
16].

The decay processes studied in this article, ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$, are
mediated by the weak interaction and governed mainly by two amplitudes,
expected to be of similar magnitude, from different diagrams describing quark-
level ${b}\rightarrow{u}{\overline{u}}{d}$ transitions, as shown in Fig. 1.
Throughout this paper the inclusion of charge-conjugate reactions is implied,
unless otherwise indicated. CPV could arise from the interference of two
amplitudes with relative phases that differ between particle and antiparticle
decays, leading to differences in the ${\mathchar 28931\relax}^{0}_{b}$ and
${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}$
decay rates. The main source of this effect in the SM would be the large
relative phase (referred to as $\alpha$ in the literature) between the product
of the CKM matrix elements $V_{ub}V_{ud}^{*}$ and $V_{tb}V_{td}^{*}$, which
are present in the different diagrams depicted in Fig. 1. Parity violation
(PV) is also expected in weak interactions, but has never been observed in
${\mathchar 28931\relax}^{0}_{b}$ decays.

Figure 1:  Dominant Feynman diagrams for ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$
transitions. The two diagrams show the transitions that contribute most
strongly to ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays.
In both cases, a pair of ${{\pi}^{+}}{{\pi}^{-}}$ (${{K}^{+}}{{K}^{-}}$) is
produced by gluon emission from the light quarks ($u$,$d$). The difference is
in the $b$ quark decay that happens on the left through a virtual $W^{-}$
boson emission (“tree diagram”) and on the right as a virtual $W^{-}$ boson
emission and absorption together with a gluon emission (“loop diagram”). The
magnitudes of the two amplitudes are expected to be comparable, and each is
proportional to the product of the CKM matrix elements involved, which are
shown in the figure.

To search for $C\\!P$-violating effects one needs to measure $C\\!P$-odd
observables, which can be done by studying asymmetries in the $\widehat{T}$
operator. This is a unitary operator that reverses both the momentum and spin
three-vectors [17, 18], and is different from the antiunitary time-reversal
operator $T$ [19, 20] that also exchanges initial and final states. A non-zero
$C\\!P$-odd observable implies $C\\!P$ violation and similar considerations
apply to $P$-odd observables and parity violation [21]. Furthermore, different
values of $P$-odd observables for a decay and its charge conjugate would imply
CPV. In this paper, scalar triple products of final-state particle momenta in
the ${\mathchar 28931\relax}^{0}_{b}$ centre-of-mass frame are studied to
search for $P$\- and $C\\!P$-violating effects in four-body decays. These are
defined as
${C_{\widehat{T}}}=\vec{p}_{p}\cdot(\vec{p}_{h_{1}^{-}}\times\vec{p}_{h_{2}^{+}})$
for ${\mathchar 28931\relax}^{0}_{b}$ and ${\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}=\vec{p}_{\overline{p}}\cdot(\vec{p}_{h_{1}^{+}}\times\vec{p}_{h_{2}^{-}})$
for ${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}$, where $h_{1}$ and $h_{2}$ are final state hadrons:
$h_{1}=\pi$ and $h_{2}=K$ for ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ and
$h_{1}=h_{2}=\pi$ for ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. In
the latter case there is an inherent ambiguity in the choice of the pion for
$h_{1}$ that is resolved by taking that with the larger momentum in the
${\mathchar 28931\relax}^{0}_{b}$ rest frame, referred to as
$\pi_{\textrm{fast}}$. The following asymmetries may then be defined [22, 23]:

$\displaystyle{A_{\widehat{T}}}({C_{\widehat{T}}})$
$\displaystyle=\frac{N({C_{\widehat{T}}}>0)-N({C_{\widehat{T}}}<0)}{N({C_{\widehat{T}}}>0)+N({C_{\widehat{T}}}<0)},$
(1) $\displaystyle{\kern
1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}}({\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}})$
$\displaystyle=\frac{\overline{N}(-{\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}>0)-\overline{N}(-{\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}<0)}{\overline{N}(-{\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}>0)+\overline{N}(-{\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}<0)},$ (2)

where $N$ and $\overline{N}$ are the numbers of ${\mathchar
28931\relax}^{0}_{b}$ and ${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}$ decays. These asymmetries are $P$-odd and
$\widehat{T}$-odd and so change sign under $P$ or $\widehat{T}$
transformations, i.e.
${A_{\widehat{T}}}({C_{\widehat{T}}})=-{A_{\widehat{T}}}(-{C_{\widehat{T}}})$
or ${\kern 1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}}({\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}})=-{\kern
1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}}(-{\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}})$. The $P$\- and
$C\\!P$-violating observables are defined as

$\displaystyle{a_{P}^{\widehat{T}\text{-odd}}}=\frac{1}{2}\left({A_{\widehat{T}}}+{\kern
1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}}\right),\hskip
85.35826pt{a_{C\\!P}^{\widehat{T}\text{-odd}}}=\frac{1}{2}\left({A_{\widehat{T}}}-{\kern
1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}}\right),$ (3)

and a significant deviation from zero would signal PV or CPV, respectively.

Searches for CPV with triple-product asymmetries are particularly suited to
${\mathchar 28931\relax}^{0}_{b}$ four-body decays to hadrons with no charm
quark [24] thanks to the rich resonant substructure, dominated by ${\mathchar
28929\relax}(1232)^{++}\rightarrow{p}{{\pi}^{+}}$ and
$\rho(770)^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ resonances in the
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ final
state. The observable $a_{C\\!P}^{\widehat{T}\text{-odd}}$ is sensitive to the
interference of $\widehat{T}$-even and $\widehat{T}$-odd amplitudes with
different $C\\!P$-odd (“weak”) phases. Unlike the overall asymmetry in the
decay rate that is sensitive to the interference of $\widehat{T}$-even
amplitudes, $a_{C\\!P}^{\widehat{T}\text{-odd}}$ does not require a non-
vanishing difference in the $C\\!P$-invariant (“strong”) phase between the
contributing amplitudes [25, 19]. The observables $A_{\widehat{T}}$, $\kern
1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}$,
$a_{P}^{\widehat{T}\text{-odd}}$ and $a_{C\\!P}^{\widehat{T}\text{-odd}}$ are,
by construction, largely insensitive to particle-antiparticle production
asymmetries and detector-induced charge asymmetries [26].

This article describes measurements of the $C\\!P$\- and $P$-violating
asymmetries introduced in equation (3) in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays.
The asymmetries are measured first for the entire phase space of the decay,
integrating over all possible final state configurations, and then in
different regions of phase space so as to enhance sensitivity to localised
CPV. The analysis is performed using proton-proton collision data collected by
the LHCb detector, corresponding to $3.0\mbox{\,fb}^{-1}$ of integrated
luminosity at centre-of-mass energies of 7 and
8$\mathrm{\,Te\kern-1.00006ptV}$, and exploits the copious production of
${\mathchar 28931\relax}^{0}_{b}$ baryons at the LHC, which constitutes around
20% of all $b$ hadrons produced [27]. Control samples of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ decays, with ${{\mathchar
28931\relax}^{+}_{c}}$ decaying to ${p}{{K}^{-}}{{\pi}^{+}}$,
${p}{{\pi}^{-}}{{\pi}^{+}}$, and ${p}{{K}^{-}}{{K}^{+}}$ final states, are
used to optimise the event selection and study systematic effects; the most
abundant control sample consists of ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}$ decays
mediated by ${b}\\!\rightarrow{c}$ quark transitions in which no CPV is
expected [28]. To avoid introducing biases in the results, all aspects of the
analysis, including the selection, phase space regions, and procedure used to
determine the statistical significance of the results, were fixed before the
data were examined.

The LHCb detector [29, 30] is designed to collect data of $b$-hadron decays
produced from proton-proton collisions at the Large Hadron Collider. It
instruments a region around the proton beam axis, covering the polar angles
between 10 and 250$\rm\,mrad$, where approximately 24% of the $b$-hadron
decays occur [31]. The detector includes a high-precision tracking system with
a dipole magnet, providing measurements of the momentum and decay vertex
position of particle decays. Different types of charged particles are
distinguished using information from two ring-imaging Cherenkov detectors, a
calorimeter and a muon system. Simulated samples of ${\mathchar
28931\relax}^{0}_{b}$ signal modes and control samples are used in this
analysis to verify the experimental method and to study certain systematic
effects. These simulated events model the experimental conditions in detail,
including the proton-proton collision, the decays of the particles, and the
response of the detector. The software used is described in Refs. [32,
*Sjostrand:2007gs, 34, 35, 36, *Agostinelli:2002hh, 38]. The online event
selection is performed by a trigger system that takes fast decisions about
which events to record. It consists of a hardware stage, based on information
from the calorimeter and muon systems, followed by a software stage, which
applies a full event reconstruction. The software trigger requires ${\mathchar
28931\relax}^{0}_{b}$ candidates to be consistent with a $b$-hadron decay
topology, with tracks originating from a secondary vertex detached from the
primary $pp$ collision point. The mean ${\mathchar 28931\relax}^{0}_{b}$
lifetime is 1.5${\rm\,ps}$ [39], which corresponds to a typical flight
distance of a few millimetres in LHCb.

The ${{\mathchar 28931\relax}^{0}_{b}}\rightarrow p{{\pi}^{-}}h^{+}h^{-}$
candidates are formed by combining tracks identified as protons, pions, or
kaons that originate from a common vertex. The proton or antiproton identifies
the candidate as a ${\mathchar 28931\relax}^{0}_{b}$ or ${\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}$. There
are backgrounds from $b$-hadron decays to charm hadrons that are suppressed by
reconstructing the appropriate two- or three-body invariant masses, and
requiring them to differ from the known charm hadron masses by at least three
times the experimental resolution. For the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}}$ control mode, only the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}h^{+}h^{-}{{\pi}^{-}}$ events with
reconstructed ${p}h^{+}h^{-}$ invariant mass between $2.23$ and $2.31$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are retained.

A boosted decision tree (BDT) classifier [40] is constructed from a set of
kinematic variables that discriminate between signal and background. The
signal and background training samples used for the BDT are derived from the
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ control
sample, since its kinematics and topology are similar to the decays under
study; background in this sample is subtracted with the $sPlot$ technique
[41], a statistical technique to disentangle signal and background
contributions. The background training sample consists of candidates that lie
far from the signal mass peak, between 5.85 and 6.40
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The control modes ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}(\rightarrow
p{{\pi}^{+}}{{\pi}^{-}}){{\pi}^{-}}$ and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{\mathchar 28931\relax}^{+}_{c}}(\rightarrow
p{{K}^{-}}{{K}^{+}}){{\pi}^{-}}$ are used to optimise the particle
identification criteria for the signal mode with the same final state. For
events in which multiple candidates pass all selection criteria for a given
mode, one candidate is retained at random and the rest discarded.

Unbinned extended maximum likelihood fits to the
${p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and the
${p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ invariant mass distributions are shown in
Fig. 2. The invariant mass distribution of the ${\mathchar
28931\relax}^{0}_{b}$ signal is modelled by a Gaussian core with power law
tails [42] with the mean and the width of the Gaussian determined from the fit
to data. The combinatorial background is modelled by an exponential
distribution with the rate parameter extracted from data. All other parameters
of the fit model are taken from simulations except the yields. Partially
reconstructed ${\mathchar 28931\relax}^{0}_{b}$ decays are described by an
empirical function [43] convolved with a Gaussian function to account for
resolution effects. The shapes of backgrounds from other $b$-hadron decays due
to incorrectly identified particles, e.g. kaons identified as pions or protons
identified as kaons, are modelled using simulated events. These consist mainly
of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{B}^{0}}\rightarrow{{K}^{+}}{{\pi}^{-}}{{\pi}^{-}}{{\pi}^{+}}$ decays for
the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
sample and of similar final states for the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ sample,
as shown in Fig. 2. The yields of these contributions are obtained from fits
to data reconstructed under the appropriate mass hypotheses for the final-
state particles. The signal yields of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ are
$6646\pm 105$ and $1030\pm 56$, respectively. This is the first observation of
these decay modes.

Figure 2:  Reconstructed invariant mass fits used to extract the signal
yields. The invariant mass distributions for (a) ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
(b) ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays
are shown. A fit is overlaid on top of the data points, with solid and dotted
lines describing the projections of the fit results for each of the components
described in the text and listed in the legend. Uncertainties on the data
points are statistical only and are calculated assuming Poisson distributed
entries.

Signal candidates are split into four categories according to ${\mathchar
28931\relax}^{0}_{b}$ or ${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}$ flavour and the sign of $C_{\widehat{T}}$ or ${\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}$ in order to calculate the
asymmetries defined in equations (1) and (2). The reconstruction efficiency
for signal candidates with ${C_{\widehat{T}}}>0$ is identical to that with
${C_{\widehat{T}}}<0$ within the statistical uncertainties of the control
sample, and likewise for $\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}$, which indicates that the
detector and the reconstruction program do not bias this measurement. This
check is performed both on the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}({p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}$ data control sample
and on large samples of simulated events, using yields about 30 times those
found in data, which are generated with no $C\\!P$ asymmetry. The $C\\!P$
asymmetry measured in the control sample is
${a_{C\\!P}^{\widehat{T}\text{-odd}}}({{\mathchar
28931\relax}^{+}_{c}}{{\pi}^{-}})=(0.15\pm 0.31)\%$, compatible with $C\\!P$
symmetry. The asymmetries $A_{\widehat{T}}$ and $\kern
1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}$ in the signal samples are
measured with a simultaneous unbinned maximum likelihood fit to the invariant
mass distributions of the different signal categories, and are found to be
uncorrelated. Corresponding asymmetries for each of the background components
are also measured in the fit; they are found to be consistent with zero, and
do not lead to significant systematic uncertainties in the signal asymmetries.
The values of $a_{C\\!P}^{\widehat{T}\text{-odd}}$ and
$a_{P}^{\widehat{T}\text{-odd}}$ are then calculated from $A_{\widehat{T}}$
and $\kern 1.00006pt\overline{\kern-1.00006ptA}_{\widehat{T}}$.

In four-body particle decays, the $C\\!P$ asymmetries may vary over the phase
space due to resonant contributions or their interference effects, possibly
cancelling when integrated over the whole phase space. Therefore, the
asymmetries are measured in different regions of phase space for the
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decay
using two binning schemes, defined before examing the data. Scheme A, defined
in Table 1, is designed to isolate regions of phase space according to their
dominant resonant contributions. Scheme B exploits in more detail the
interference of contributions which could be visible as a function of the
angle $\Phi$ between the decay planes formed by the
${p}\pi^{-}_{\textrm{fast}}$ and the $\pi^{-}_{\textrm{slow}}{{\pi}^{+}}$
systems, as illustrated in Fig. 3. Scheme B has 10 non-overlapping bins of
width $\pi/10$ in $|\Phi|$. For every bin in each of the schemes, the
${\mathchar 28931\relax}^{0}_{b}$ efficiencies for ${C_{\widehat{T}}}>0$ and
${C_{\widehat{T}}}<0$ are compared and found to be equal within uncertainties,
and likewise the ${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}$ efficiencies for ${\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}>0$ and ${\kern
1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}}<0$. The analysis technique
is validated on the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}({p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}$ control sample, for
which the angle $\Phi$ is defined by the decay planes of the ${p}{{K}^{-}}$
and ${{\pi}^{+}}{{\pi}^{-}}$ pairs, and on simulated signal events.

Figure 3:  Definition of the $\Phi$ angle. The decay planes formed by the ${p}\pi^{-}_{\textrm{fast}}$ and the $\pi^{-}_{\textrm{slow}}{{\pi}^{+}}$ systems in the ${\mathchar 28931\relax}^{0}_{b}$ rest frame. The momenta of the particles, represented by vectors, determine the two decay planes and the angle $\Phi\in[-\pi,\pi]$ [19] measures their relative orientation. Table 1: Definition of binning scheme A for the decay mode ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$. Binning scheme A is defined to exploit interference patterns arising from the resonant structure of the decay. Bins 1-4 focus on the region dominated by the ${\mathchar 28929\relax}(1232)^{++}\rightarrow{p}{{\pi}^{+}}$ resonance. The other eight bins are defined to study regions where ${p}{{\pi}^{-}}$ resonances are present (5–8) on either side of the $\rho(770)^{0}\rightarrow{{\pi}^{+}}{{\pi}^{-}}$ resonances (5–12). Further splitting for $|\Phi|$ lower or greater than $\pi/2$ is done to reduce potential dilution of asymmetries, as suggested in Ref. [19]. Masses are in units of ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c^{2}}$. Phase space bin | $m(p\pi^{+})$ | $m(p\pi^{-}_{\text{slow}})$ | $m(\pi^{+}\pi^{-}_{\text{slow}})$, $m(\pi^{+}\pi^{-}_{\text{fast}})$ | $|\Phi|$
---|---|---|---|---
1 | $(1.07,1.23)$ |  |  | $(0,\frac{\pi}{2})$
2 | $(1.07,1.23)$ |  |  | $(\frac{\pi}{2},\pi)$
3 | $(1.23,1.35)$ |  |  | $(0,\frac{\pi}{2})$
4 | $(1.23,1.35)$ |  |  | $(\frac{\pi}{2},\pi)$
5 | $(1.35,5.34)$ | $(1.07,2.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})<0.78\textrm{ or\phantom{d}}m(\pi^{+}\pi^{-}_{\text{fast}})<0.78$ | $(0,\frac{\pi}{2})$
6 | $(1.35,5.34)$ | $(1.07,2.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})<0.78\textrm{ or\phantom{d}}m(\pi^{+}\pi^{-}_{\text{fast}})<0.78$ | $(\frac{\pi}{2},\pi)$
7 | $(1.35,5.34)$ | $(1.07,2.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})>0.78\textrm{ and }m(\pi^{+}\pi^{-}_{\text{fast}})>0.78$ | $(0,\frac{\pi}{2})$
8 | $(1.35,5.34)$ | $(1.07,2.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})>0.78\textrm{ and }m(\pi^{+}\pi^{-}_{\text{fast}})>0.78$ | $(\frac{\pi}{2},\pi)$
9 | $(1.35,5.34)$ | $(2.00,4.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})<0.78\textrm{ or\phantom{d}}m(\pi^{+}\pi^{-}_{\text{fast}})<0.78$ | $(0,\frac{\pi}{2})$
10 | $(1.35,5.34)$ | $(2.00,4.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})<0.78\textrm{ or\phantom{d}}m(\pi^{+}\pi^{-}_{\text{fast}})<0.78$ | $(\frac{\pi}{2},\pi)$
11 | $(1.35,5.34)$ | $(2.00,4.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})>0.78\textrm{ and }m(\pi^{+}\pi^{-}_{\text{fast}})>0.78$ | $(0,\frac{\pi}{2})$
12 | $(1.35,5.34)$ | $(2.00,4.00)$ | $m(\pi^{+}\pi^{-}_{\text{slow}})>0.78\textrm{ and }m(\pi^{+}\pi^{-}_{\text{fast}})>0.78$ | $(\frac{\pi}{2},\pi)$

The asymmetries measured in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays with these two binning schemes are shown in Fig. 4 and reported in
Table 2, together with the integrated measurements. For each scheme
individually, the compatibility with the $C\\!P$-symmetry hypothesis is
evaluated by means of a $\chi^{2}$ test, with $\chi^{2}=R^{T}V^{-1}R$, where
$R$ is the array of $a_{C\\!P}^{\widehat{T}\text{-odd}}$ measurements and $V$
is the covariance matrix, which is the sum of the statistical and systematic
covariance matrices. An average systematic uncertainty, whose evaluation is
discussed below, is assigned for all bins. The systematic uncertainties are
assumed to be fully correlated; their contribution is small compared to the
statistical uncertainties. The $p$-values of the $C\\!P$-symmetry hypothesis
are $4.9\times 10^{-2}$ and $7.1\times 10^{-4}$ for schemes A and B,
respectively, corresponding to statistical significances of $2.0$ and $3.4$
Gaussian standard deviations ($\sigma$). A similar $\chi^{2}$ test is
performed on $a_{P}^{\widehat{T}\text{-odd}}$ measurements with $p$-values for
the $P$-symmetry hypothesis of $5.8\times 10^{-3}$ ($2.8\sigma$) and
$2.4\times 10^{-2}$ (2.3$\sigma$), for scheme A and B, respectively.

Figure 4:  Distributions of the asymmetries. The results of the fit in each region of binning schemes A and B are shown. The asymmetries $a_{P}^{\widehat{T}\text{-odd}}$ and $a_{C\\!P}^{\widehat{T}\text{-odd}}$ for ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays are represented by open boxes and filled circles, respectively. The error bars indicate the total uncertainties, calculated as the sum in quadrature of the statistical uncertainty resulting from the fit to the invariant mass distribution and the systematic uncertainties estimated as described in the main text. The values of the $\chi^{2}$/ndf are quoted for the $P$\- and $C\\!P$-conserving hypotheses for each binning scheme. Table 2:  Measurements of $C\\!P$\- and $P$-violating observables. The $C\\!P$\- and $P$-violating observables, $a_{C\\!P}^{\widehat{T}\text{-odd}}$ and $a_{P}^{\widehat{T}\text{-odd}}$, resulting from the fit to the data are listed with their statistical and systematic uncertainties. Each value is obtained through an independent fit to a region of the phase space as described in the text and Table 1. Results for schemes A and B are outlined for ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ decays, and in two bins of phase space for ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays, as defined in the text. The first column lists the bin number. For both decay modes the measurement integrated over the phase space, performed independently, is also shown.  |  $a_{P}^{\widehat{T}\text{-odd}}$ [%] |  $a^{\widehat{T}-\text{odd}}_{{C\\!P}}$ [%]
---|---|---
Scheme A | ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
1 | $21.64\pm 8.28\pm 0.60$ | $-7.69\pm 8.28\pm 0.60$
2 | $-2.04\pm 3.26\pm 0.60$ | $-0.33\pm 3.26\pm 0.60$
3 | $2.03\pm 6.12\pm 0.60$ | $1.94\pm 6.12\pm 0.60$
4 | $-2.45\pm 4.60\pm 0.60$ | $-3.49\pm 4.60\pm 0.60$
5 | $-10.04\pm 4.13\pm 0.60$ | $10.29\pm 4.13\pm 0.60$
6 | $-6.40\pm 5.23\pm 0.60$ | $6.51\pm 5.23\pm 0.60$
7 | $-11.91\pm 5.00\pm 0.60$ | $8.40\pm 5.00\pm 0.60$
8 | $0.94\pm 5.60\pm 0.60$ | $-1.88\pm 5.60\pm 0.60$
9 | $-5.38\pm 4.67\pm 0.60$ | $7.20\pm 4.67\pm 0.60$
10 | $-4.26\pm 4.98\pm 0.60$ | $-11.24\pm 4.98\pm 0.60$
11 | $13.94\pm 7.19\pm 0.60$ | $-2.90\pm 7.19\pm 0.60$
12 | $-7.64\pm 4.79\pm 0.60$ | $-5.35\pm 4.79\pm 0.60$
Scheme B |  |
1 | $-0.42\pm 4.92\pm 0.60$ | $\phantom{-0}1.81\pm 4.92\pm 0.60$
2 | $-1.63\pm 4.88\pm 0.60$ | $\phantom{-0}2.86\pm 4.88\pm 0.60$
3 | $-14.73\pm 5.13\pm 0.60$ | $\phantom{-}2.87\pm 5.13\pm 0.60$
4 | $-0.32\pm 4.95\pm 0.60$ | $\phantom{-}19.79\pm 4.95\pm 0.60$
5 | $-2.71\pm 5.16\pm 0.60$ | $\phantom{-}4.47\pm 5.16\pm 0.60$
6 | $-3.85\pm 4.79\pm 0.60$ | $-7.23\pm 4.79\pm 0.60$
7 | $-14.40\pm 4.65\pm 0.60$ | $-5.44\pm 4.65\pm 0.60$
8 | $-3.75\pm 4.14\pm 0.60$ | $\phantom{-}0.76\pm 4.14\pm 0.60$
9 | $-4.16\pm 4.01\pm 0.60$ | $\phantom{-}7.74\pm 4.01\pm 0.60$
10 | $4.21\pm 3.84\pm 0.60$ | $-9.16\pm 3.84\pm 0.60$
Integrated | $-3.71\pm 1.45\pm 0.32$ | $1.15\pm 1.45\pm 0.32$
Phase space bin | ${{\mathchar 28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$
1 | $3.27\pm 6.07\pm 0.66$ | $-4.68\pm 6.07\pm 0.66$
2 | $4.43\pm 6.73\pm 0.66$ | $\phantom{-}4.73\pm 6.73\pm 0.66$
Integrated | $3.62\pm 4.54\pm 0.42$ | $-0.93\pm 4.54\pm 0.42$

The overall significance for CPV in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays from the results of schemes A and B is determined by means of a
permutation test [44], taking into account correlations among the results. A
sample of 40,000 pseudoexperiments is generated from the data by assigning
each event a random ${\mathchar 28931\relax}^{0}_{b}$/${\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}}{}^{0}_{b}$ flavour
such that $C\\!P$ symmetry is enforced. The sign of $C_{\widehat{T}}$ is
unchanged if a ${\mathchar 28931\relax}^{0}_{b}$ candidate stays ${\mathchar
28931\relax}^{0}_{b}$ and reversed if the ${\mathchar 28931\relax}^{0}_{b}$
candidate becomes ${\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}}{}^{0}_{b}$. The $p$-value of the $C\\!P$-symmetry hypothesis is
determined as the fraction of pseudoexperiments with $\chi^{2}$ larger than
that measured in data. Applying this method to the $\chi^{2}$ values from
schemes A and B individually, the $p$-values obtained agree with those from
the $\chi^{2}$ test within the uncertainty due to the limited number of
pseudoexperiments. To assess a combined significance from the two schemes, the
product of the two $p$-values measured in data is compared with the
distribution of the product of the $p$-values of the two binning schemes from
the pseudoexperiments. The fraction of pseudoexperiments whose $p$-value
product is smaller than that seen in data determines the overall $p$-value of
the combination of the two schemes [45]. An overall $p$-value of $9.8\times
10^{-4}$ (3.3$\sigma$) is obtained for the $C\\!P$-symmetry hypothesis,
including systematic uncertainties.

For the ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays,
the smaller purity and signal yield of the sample do not permit PV and CPV to
be probed with the same precision as for ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$, and
therefore only two regions of phase space are considered. One spans
$1.43<m({p}{{K}^{-}})<2.00{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (bin 1)
and is dominated by excited $\mathchar 28931\relax$ resonances decaying to
${p}{K}$ and the other covers the remaining phase space,
$2.00<m({p}{{K}^{-}})<4.99{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (bin 2).
The observables measured in these regions are given in Table 2 and are
consistent with $C\\!P$ and $P$ symmetry.

The main sources of systematic uncertainties for both $p$ ${\pi}^{-}$
${\pi}^{+}$ ${\pi}^{-}$ and $p$ ${\pi}^{-}$ ${K}^{+}$ ${K}^{-}$ decays are
experimental effects that could introduce biases in the measured asymmetries.
This is tested by measuring the asymmetry
$a_{C\\!P}^{\widehat{T}\text{-odd}}$, integrated over phase space and in
various phase space regions, using the control sample ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{{\mathchar
28931\relax}^{+}_{c}}(\\!\rightarrow{p}{{K}^{-}}{{\pi}^{+}}){{\pi}^{-}}$,
which is expected to exhibit negligible CPV. The results are in agreement with
the $C\\!P$-symmetry hypothesis; an uncertainty of 0.31% is assigned as a
systematic uncertainty for the $a_{C\\!P}^{\widehat{T}\text{-odd}}$ and
$a_{P}^{\widehat{T}\text{-odd}}$ integrated measurements; an uncertainty of
0.60%, the largest asymmetry from a fit to scheme B measurements using a range
of efficiency and fit models, is assigned for the corresponding phase space
measurements. The systematic uncertainty arising from the experimental
resolution in the measurement of the triple products $C_{\widehat{T}}$ and
$\kern 1.00006pt\overline{\kern-1.00006ptC}_{\widehat{T}}$, which could
introduce a migration of events between the bins, is estimated from simulated
samples of ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays
where neither $P$\- nor $C\\!P$-violating effects are present. The difference
between the reconstructed and generated asymmetry is taken as a systematic
uncertainty due to this effect and is less than 0.06% in all cases. To assess
the uncertainty associated with the fit models, alternative functions are
used; these tests lead only to small changes in the asymmetries, the largest
being 0.05%. For ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays,
this contribution is larger, about 0.28% for the
$a_{C\\!P}^{\widehat{T}\text{-odd}}$ and $a_{P}^{\widehat{T}\text{-odd}}$
asymmetries.

Further cross-checks are made to investigate the stability of the results with
respect to different periods of recording data, different polarities of the
spectrometer magnet, the choice made in the selection of multiple candidates,
and the effect of the trigger and selection criteria. Alternative binning
schemes are studied as a cross-check, such as using 8 or 12 bins in $|\Phi|$
for ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays. For these alternative binning schemes, the significance of the CPV
measurement of the modified scheme B is reduced to below $3\sigma$.
Nonetheless, the overall significance of the combination of these two
additional binnings with schemes A and B remains above three standard
deviations, with a $p$-value of $1.8\times 10^{-3}$ ($3.1\sigma$), consistent
with the $3.3\sigma$ result seen in the baseline analysis. An independent
analysis of the data based on alternative selection criteria confirmed the
results. It used a similar number of events, of which 73.4% are in common with
the baseline analysis, and gave $p$-values for $C\\!P$ symmetry of $3.4\times
10^{-3}$ ($2.9\sigma$) for scheme A and $1.4\times 10^{-4}$ ($3.8\sigma$) for
scheme B.

In conclusion, a search for $P$ and $C\\!P$ violation in ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$ and
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays is
performed on signal yields of $6646\pm 105$ and $1030\pm 56$ events. This is
the first observation of these decay modes. Measurements of asymmetries in the
entire phase space do not show any evidence of $P$ or $C\\!P$ violation.
Searches for localised $P$ or $C\\!P$ violation are performed by measuring
asymmetries in different regions of the phase space. The results are
consistent with $C\\!P$ symmetry for ${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{K}^{+}}{{K}^{-}}$ decays
but evidence for $C\\!P$ violation at the $3.3\sigma$ level is found in
${{\mathchar
28931\relax}^{0}_{b}}\\!\rightarrow{p}{{\pi}^{-}}{{\pi}^{+}}{{\pi}^{-}}$
decays. No significant $P$ violation is found. This represents the first
evidence of $C\\!P$ violation in the baryon sector and indicates an asymmetry
between baryonic matter and antimatter.

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

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar6, J. Albrecht10,
F. Alessio40, M. Alexander53, S. Ali43, G. Alkhazov31, P. Alvarez Cartelle55,
A.A. Alves Jr59, S. Amato2, S. Amerio23, Y. Amhis7, L. An41, 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, C. Baesso62, S. Baker55, W. Baldini17, R.J. Barlow56, C.
Barschel40, S. Barsuk7, W. Barter40, M. Baszczyk27, V. Batozskaya29, B.
Batsukh61, V. Battista41, A. Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24,
I. Bediaga1, L.J. Bel43, V. Bellee41, N. Belloli21,i, K. Belous37, I.
Belyaev32, E. Ben-Haim8, G. Bencivenni19, S. Benson43, J. Benton48, A.
Berezhnoy33, R. Bernet42, A. Bertolin23, F. Betti15, M.-O. Bettler40, M. van
Beuzekom43, I. Bezshyiko42, S. Bifani47, P. Billoir8, T. Bird56, A.
Birnkraut10, A. Bitadze56, A. Bizzeti18,u, T. Blake50, F. Blanc41, J. Blouw11,
S. Blusk61, V. Bocci26, T. Boettcher58, A. Bondar36, N. Bondar31,40, W.
Bonivento16, A. Borgheresi21,i, S. Borghi56, M. Borisyak35, M. Borsato39, F.
Bossu7, M. Boubdir9, T.J.V. Bowcock54, E. Bowen42, C. Bozzi17,40, S. Braun12,
M. Britsch12, T. Britton61, J. Brodzicka56, E. Buchanan48, C. Burr56, A.
Bursche2, J. Buytaert40, S. Cadeddu16, R. Calabrese17,g, M. Calvi21,i, M.
Calvo Gomez38,m, A. Camboni38, P. Campana19, D. Campora Perez40, 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,i, L. Castillo Garcia41, M. Cattaneo40, Ch. Cauet10, G. Cavallero20,
R. Cenci24,t, M. Charles8, Ph. Charpentier40, G. Chatzikonstantinidis47, M.
Chefdeville4, S. Chen56, S.-F. Cheung57, V. Chobanova39, M. Chrzaszcz42,27, X.
Cid Vidal39, G. Ciezarek43, P.E.L. Clarke52, M. Clemencic40, H.V. Cliff49, J.
Closier40, V. Coco59, J. Cogan6, E. Cogneras5, V. Cogoni16,40,f, L.
Cojocariu30, G. Collazuol23,o, P. Collins40, A. Comerma-Montells12, A.
Contu40, A. Cook48, S. Coquereau38, G. Corti40, M. Corvo17,g, C.M. Costa
Sobral50, B. Couturier40, G.A. Cowan52, D.C. Craik52, A. Crocombe50, M. Cruz
Torres62, S. Cunliffe55, R. Currie55, C. D’Ambrosio40, F. Da Cunha Marinho2,
E. Dall’Occo43, J. Dalseno48, P.N.Y. David43, A. Davis59, O. De Aguiar
Francisco2, K. De Bruyn6, S. De Capua56, M. De Cian12, J.M. De Miranda1, L. De
Paula2, M. De Serio14,d, P. De Simone19, C.-T. Dean53, D. Decamp4, M.
Deckenhoff10, L. Del Buono8, M. Demmer10, D. Derkach35, O. Deschamps5, F.
Dettori40, B. Dey22, A. Di Canto40, H. Dijkstra40, F. Dordei40, M. Dorigo41,
A. Dosil Suárez39, A. Dovbnya45, K. Dreimanis54, L. Dufour43, G. Dujany56, K.
Dungs40, P. Durante40, R. Dzhelyadin37, A. Dziurda40, A. Dzyuba31, N.
Déléage4, S. Easo51, M. Ebert52, U. Egede55, V. Egorychev32, S. Eidelman36, S.
Eisenhardt52, U. Eitschberger10, R. Ekelhof10, L. Eklund53, Ch. Elsasser42, S.
Ely61, S. Esen12, H.M. Evans49, T. Evans57, A. Falabella15, N. Farley47, S.
Farry54, R. Fay54, D. Fazzini21,i, D. Ferguson52, V. Fernandez Albor39, A.
Fernandez Prieto39, F. Ferrari15,40, F. Ferreira Rodrigues1, M. Ferro-Luzzi40,
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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
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, 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
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN 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) and Novosibirsk State
University, 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
41Ecole 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
64Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
65Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
66Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
67National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 38
69Van 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


# Threshold-Independent Fair Matching through Score Calibration

Mohammad Hossein Moslemi  University of Western OntarioLondonCanada
<EMAIL_ADDRESS>0009-0002-0278-4665 and  Mostafa Milani  University of
Western OntarioLondonCanada<EMAIL_ADDRESS>0000-0002-3386-7079

###### Abstract.

Entity Matching (EM) is a critical task in numerous fields, such as
healthcare, finance, and public administration, as it identifies records that
refer to the same entity within or across different databases. EM faces
considerable challenges, particularly with false positives and negatives.
These are typically addressed by generating matching scores and apply
thresholds to balance false positives and negatives in various
contexts.However, adjusting these thresholds can affect the fairness of the
outcomes, a critical factor that remains largely overlooked in current fair EM
research. The existing body of research on fair EM tends to concentrate on
static thresholds, neglecting their critical impact on fairness. To address
this, we introduce a new approach in EM using recent metrics for evaluating
biases in score-based binary classification, particularly through the lens of
distributional parity. This approach enables the application of various bias
metrics—like equalized odds, equal opportunity, and demographic parity—without
depending on threshold settings. Our experiments with leading matching methods
reveal potential biases, and by applying a calibration technique for EM scores
using Wasserstein barycenters, we not only mitigate these biases but also
preserve accuracy across real-world datasets. This paper contributes to the
field of fairness in data cleaning, especially within EM, which is a central
task in data cleaning, by promoting a method for generating matching scores
that reduce biases across different thresholds.

##  1\. Introduction

Entity matching (EM), also known as entity linkage or record matching, plays a
crucial role in various applications across multiple domains, including
healthcare, finance, e-commerce, and public administration. It identifies
records referring to the same real-world entity across different or single
databases, integrating fragmented information, improving data quality, and
providing a unified view of data entities. Despite extensive research and
numerous matching methods, challenges in entity matching (EM) due to data
heterogeneity continue to cause false positives and false negatives across
domains and datasets. These errors can introduce biases and discrimination, as
highlighted in recent research (Shahbazi et al., 2023; Nilforoushan et al.,
2022; Efthymiou et al., 2021; Louis et al., 2023; Makri et al., 2022). This
short paper discusses overlooked biases in state-of-the-art EM methods related
to fairness.

EM methods often generate a matching score and use a matching threshold to
derive a binary decision. This process inherently allows for the tuning of
thresholds across different applications to balance between false positives
and false negatives. However, for varying matching thresholds, biases in
matching must be evaluated independent of a particular threshold. Previous
research on fairness in EM has often overlooked the significant effects of
changing thresholds (Efthymiou et al., 2021; Louis et al., 2023; Makri et al.,
2022), concentrated on specific thresholds (Shahbazi et al., 2023), or used
metrics that can be misleading for evaluating bias in a threshold-independent
manner (Nilforoushan et al., 2022).

###### Example 1.1.

Adjusting matching thresholds in EM applications is key to optimizing matching
performance. For example, using lower thresholds increases the chances of
identifying individuals on a no-fly list, accepting the risk of some wrongful
identifications to ensure criminals are not missed. Conversely, in financial
services, a higher threshold is used to avoid wrongly linking and merging
accounts due to the significant privacy and financial risks involved; here,
it’s acceptable to miss some matches. Likewise, in healthcare, setting a
higher threshold is crucial to prevent the merging of wrong patient records,
which helps avoid treatment mistakes, incorrect diagnoses, and life-
threatening treatment errors. However, changing thresholds can affect
fairness. A method that seems fair at one threshold might show significant
biases at another, as adjusting the threshold can create or increase biases.
To illustrate, the ROC curves in Figure 1(a) display the true positive and
false positive rates (TPR and FPR) across various thresholds for minority and
majority groups in the matching method Magellan (Konda, 2018) using the
Amazon-Google dataset (Mudgal et al., 2018). Additionally, Figure 1(b)
demonstrates the TPR for these groups at different thresholds. These figures
show that although the matching method can perform similarly for both groups
at certain thresholds (where the curves for the two groups intersect), at
other thresholds, significant biases emerge, as shown by the differences in
the curves.

Recent work in (Nilforoushan et al., 2022) explores threshold-independent
fairness in EM, following the AUC-based bias metrics for binary classifiers in
(Kallus and Zhou, 2019; Yang et al., 2023; Vogel et al., 2021). These metrics
use the AUC gap between minority and majority groups to measure bias. However,
these bias measures have been criticized for potential ambiguity or misleading
interpretations, as pointed out by Kwegyir et al. (Kwegyir-Aggrey et al.,
2023b). The critique is based on the fact that AUC, as an aggregate measure,
may not capture specific threshold biases. For instance, Figure LABEL:fig:auc-
intro illustrates that although the AUC values for both groups are almost
identical, the ROC curves significantly differ at certain thresholds,
revealing biases at those thresholds.

(a)

(b)

Figure 1. The ROC of Magellan (Konda, 2018) (Figure 1(a)) shows similar AUC,
0.897 and 0.894 while there is a significant performance difference in certain
thresholds. Figure 1(b) demonstrates a significant difference in TPR at
different thresholds.

To overcome the shortcomings highlighted in previous EM fairness research, we
employ the recent metrics for evaluating biases in score-based binary
classification without depending on a specific threshold (Kwegyir-Aggrey et
al., 2023a; Jiang et al., 2020; Gouic et al., 2020; Chzhen et al., 2020). We
use distributional parity (DSP), which limits the expected value of the
difference in measures such as positive rate (PR), TPR, or FPR between groups.
The gray area in Figure 1(b) illustrates this for TPR. Using PR, TPR, or both
TPR and FPR, DSP extends the principle of demographic parity (DP), equal
opportunity (EO), and equalized odds (EOD) across various thresholds.

Through DSP, we examine state-of-the-art matching techniques to uncover
biases, then formulate fair EM by calibrating and adjusting matching score
distributions across groups. Using Wasserstein barycenters from optimal
transport (OT) theory, this post-processing fairness approach corrects biases
without retraining the initial method. Our experiments show that this
calibration reduces biases measured by DSP while maintaining accuracy.

We provide an overview of the relevant background, formulate the matching
problem, and explain distributional parity within the context of EM in §2. The
calibration method is detailed in §3, §4 presents our experimental results,
and §5 concludes the paper.

##  2\. Background and Problem Definition

Let
$f\mathrel{\mathop{\ordinarycolon}}\mathcal{X}\times\mathcal{A}\mapsto\\{0,1\\}$
be a binary classifier where $\mathcal{X}$ is a feature space, and
$\mathcal{A}=\\{a,b\\}$ is a set of binary protected (sensitive) attribute
values with $a$ and $b$ representing the majority and minority groups,
respectively. We assume $f$ is derived from a score function
$s\mathrel{\mathop{\ordinarycolon}}\mathcal{X}\times\mathcal{A}\mapsto[0,1]$
and a classification threshold $\tau\in[0,1]$, where
$f(r)=\mathbbm{1}_{s(r)\geq\tau}$ for every $r\in\mathcal{X}\times\mathcal{A}$
($\mathbbm{1}$ is the indicator function). We use $f_{a}$ and $f_{b}$ ($s_{a}$
and $s_{b}$) to refer to the classifier (the score function) within majority
and minority groups.

By employing a probability distribution “Pr” with random variables ${\bf X}$,
$A$, and $Y$, which denote the features, the protected attribute, and the true
label, respectively, we define the positive rate (PR) within a specific group
$g\in\mathcal{A}$ as $\text{PR}_{g}(s,\tau)=\text{Pr}(s(X)\geq\tau\mid A=g)$.
Similarly, we define TPR and FPR for each group. From PR, we derive DP as
$|\text{PR}_{a}(s,\tau)-\text{PR}_{b}(s,\tau)|\leq\epsilon$, where $\epsilon$
represents a limit on disparity or bias. TPR and FPR are used to measure
fairness in terms of EO and EOD; for EO, we have
$|\text{TPR}_{a}(s,\tau)-\text{TPR}_{b}(s,\tau)|\leq\epsilon$, and for EOD,
both $|\text{TPR}_{a}(s,\tau)-\text{TPR}_{b}(s,\tau)|\leq\epsilon$ and
$|\text{FPR}_{a}(s,\tau)-\text{FPR}_{b}(s,\tau)|\leq\epsilon$ are required.

The AUC for the scoring function $s$ is expressed as
$\textit{AUC}(s)=\int_{0}^{1}\text{TPR}(s,\text{FPR}^{-1}(s,\tau))\;d\tau$. It
quantifies the likelihood that a positive record is accurately ranked above a
negative one, offering a measure of accuracy that does not depend on any
specific threshold $\tau$. AUC-based fairness introduces a threshold-
independent fairness criterion inspired by the concept of cross-group AUC
(xAUC) between two groups $a$ and $b$, defined as
$\text{xAUC}_{a}^{b}(s)=\int_{0}^{1}\text{TPR}_{a}(s,\text{FPR}_{b}^{-1}(s,\tau))$
$d\tau$ (Kallus and Zhou, 2019). This effectively measures the chance that a
positively labeled record from group $a$ is ranked above a negatively labeled
record from group $b$. Fairness in AUC terms is captured by
$\Delta\text{xAUC}(s)=\text{xAUC}_{a}^{b}(s)-\text{xAUC}_{b}^{a}(s)\leq\epsilon$,
reflecting the bias or disparity in scoring between the groups, $a$ and $b$.

Distributional Parity (Kwegyir-Aggrey et al., 2023a): Distributional parity,
or DSP, extends DP, EO, and EOD to define threshold-independent fairness.
Intuitively, it measures these traditional measures across varying thresholds
by computing their expected value. More precisely, for any measure
$\gamma\in\\{\text{PR},\text{TPR},\text{FPR}\\}$, DSP is achieved when
$\mathbb{E}_{\tau\sim[0,1]}|\gamma_{a}(\tau)-\gamma_{b}(\tau)|=0,$ where
$\mathbb{E}_{\tau\sim[0,1]}$ is the expected value for iid sampled thresholds
$\tau$ from $[0,1]$. When the expected value is nonzero, we denote it by
$\mathcal{U}(s,\gamma)$ as a measure of bias and refer to it as DSP.

The Fair EM Problem: In EM, a matcher acts as a binary classifier within a
feature space $\mathcal{X}=\mathcal{E}\times\mathcal{E}$, where $\mathcal{E}$
is the feature space for individual records. Each record would have its own
protected attribute. A record pair is defined as a minority pair if both
records belong to the minority group and a majority pair if both belong to the
majority group. Combinations of one minority and one majority record are
excluded, as these are identified as non-matches by blocking and removed
before classification. The matching score, denoted as
$s\mathrel{\mathop{\ordinarycolon}}\mathcal{E}^{2}\times\mathcal{A}\rightarrow[0,1]$,
and other performance metrics like PR, TPR, FPR, and DSP are defined as in
prior discussions, using the protected attribute $\mathcal{A}$, which is the
same in both records.

Previous research (Nilforoushan et al., 2022; Shahbazi et al., 2023) has shown
that current matching techniques are biased. We plan to assess these
techniques using DSP to identify biases without specific thresholds. Our
findings reveal distributional disparity, indicating uneven performance across
groups. To address these biases, we introduce fair EM, which adjusts scores
from current methods to meet a broad fairness criterion.

###### Definition 2.1.

Given a score function $s$ and a fairness constraint
$\sigma\mathrel{\mathop{\ordinarycolon}}\mathcal{U}(s,\gamma)\leq\epsilon$,
where $\gamma$ is any measure in $\\{\text{PR},\text{TPR},\text{FPR}\\}$, the
problem of fair EM is to find a score $s^{*}$ that satisfies $\sigma$ and
minimally differs from $s$, i.e., it minimizes
$\delta(s^{*},s)=\mathbb{E}_{r\sim\mathcal{E}^{2}\times\mathcal{A}}|s^{*}(r)-s(r)|$.

The objective of fair EM is to find a fair score function, $s^{*}$, that
closely resembles the original matching score, $s$, and therefore enjoys high
accuracy while adhering to the fairness constraint, $\sigma$, in terms of DSP
with respect to $\gamma$.

##  3\. Score Calibration using Barycenter

Dataset | Method | DSP (before) | before | DSP (after) | after
---|---|---|---|---|---
DP | EO | EOD | AUC | $\Delta\text{xAUC}$ | DP | EO | EOD | AUC | $\Delta\text{xAUC}$
AMZ-GOG | Ditto | 7.17 | 12.40 | 14.71 | 95.95 | 2.26 | 2.37  4.8 | 5.83  6.57 | 6.46  8.24 | 95.86  0.09 | 2.23  0.03
HierGAT | 11.10 | 17.33 | 21.57 | 96.75 | 0.41 | 6.64  4.45 | 11.60  5.73 | 12.20  9.37 | 94.91  1.84 | 9.13  8.72
DeepMatcher | 8.53 | 11.74 | 15.22 | 84.99 | 5.62 | 3.46  5.06 | 10.34  1.39 | 11.19  4.03 | 84.64  0.35 | 5.68  0.06
HierMatcher | 8.64 | 17.58 | 20.83 | 94.23 | 0.61 | 5.03  3.61 | 8.03  9.56 | 8.35  12.48 | 93.93  0.3 | 0.87  0.26
WAL-AMZ | Ditto | 3.35 | 12.69 | 12.29 | 96.16 | 1.25 | 2.51  0.84 | 9.72  2.97 | 9.76  2.53 | 93.79  2.36 | 3.91  2.66
HierGAT | 2.59 | 14.64 | 13.45 | 96.00 | 3.06 | 2.59  | 14.64  | 13.45  | 96.0  | 3.06
DeepMatcher | 3.57 | 31.80 | 31.22 | 83.35 | 5.86 | 3.44  0.13 | 16.10  15.7 | 16.73  14.49 | 83.34  0.01 | 5.9  0.04
HierMatcher | 2.15 | 11.99 | 12.08 | 94.06 | 5.21 | 2.15  | 11.99  | 12.08  | 94.06  | 5.21
DBLP-GOGS | Ditto | 4.17 | 2.19 | 2.62 | 99.43 | 0.44 | 4.00  0.18 | 1.99  0.2 | 2.62  | 99.43  | 0.44
HierGAT | 4.55 | 3.46 | 2.97 | 99.68 | 0.34 | 3.41  1.14 | 1.17  2.29 | 1.53  1.44 | 99.68  | 0.26  0.09
DeepMatcher | 5.71 | 8.61 | 8.32 | 98.58 | 4.26 | 4.71  1.0 | 8.32  0.29 | 6.96  1.36 | 98.59  0.01 | 4.32  0.05
HierMatcher | 4.76 | 3.64 | 3.66 | 99.59 | 0.64 | 3.36  1.4 | 3.28  0.36 | 3.51  0.15 | 99.59  | 0.65  0.01
DBLP-ACM | Ditto | 1.06 | 0.55 | 0.52 | 99.88 | 0.08 | 1.06  | 0.55  | 0.52  | 99.88  | 0.08
HierGAT | 0.94 | 1.24 | 0.94 | 99.99 | 0.03 | 0.82  0.11 | 0.79  0.45 | 0.64  0.3 | 99.99  | 0.03
DeepMatcher | 0.64 | 0.51 | 0.54 | 99.94 | 0.18 | 0.64  | 0.51  | 0.54  | 99.94  | 0.18
HierMatcher | 0.81 | 1.05 | 1.19 | 99.96 | 0.05 | 0.81  | 1.05  | 1.19  | 99.96  | 0.05

Table 1. Bias (DP, EO, EOD, $\Delta\text{xAUC}$) and accuracy (AUC) for
matching methods (Ditto, HierGAT, DeepMatcher, HierMatcher) over benchmarks
(AMZ-GOG, WAL-AMZ, DBLP-GOGS, DBLP-ACM), evaluated before and after score
calibration. Values are scaled by $\times 100$.  means no change after
calibration. The color green shows improvement, for instance, an increase in
accuracy (  ) or a reduction in bias (  ). Orange denotes deterioration,
such as a decrease in accuracy (  ) or a rise in bias (  ).

In this paper, we introduce a novel method for fair EM by calibrating scores
from $s$ using the Wasserstein barycenter, derived from optimal transport
theory. This method was previously suggested to improve fairness in (Chzhen et
al., 2020). As an initial solution to fair EM in Definition 2.1, our method
doesn’t fully meet all fairness constraints or promise a minimal change from
the original scores defined in the definition. However, our experiments show
it can reduce bias without losing much score accuracy. We postpone developing
a more detailed solution to our future work and discuss. First, we explain the
basics of the Wasserstein distance and barycenter.

In OT theory, the Wasserstein distance between two probability measures
$\mu_{1}$ and $\mu_{2}$ over a metric space is defined as:

(1) $W^{p}_{p}(\mu_{1},\mu_{2})=\inf_{\pi\in\Pi(\mu_{1},\mu_{2})}\int_{M\times
M}d(x,y)^{p}\,d\pi(x,y),$

where $p\geq 1$ is the order of the distance, $d(x,y)$ is the cost of moving
between points $x$ and $y$ in the metric space, and $\Pi(\mu_{1},\mu_{2})$
represents the set of all joint distributions $\pi$ in the metric space with
marginals $\mu_{1}$ and $\mu_{2}$. The Wasserstein barycenter $\mu_{1}$ and
$\mu_{2}$ with respect to weights $\alpha$ is defined as the measure
$\hat{\mu}$ that minimizes the weighted sum of the Wasserstein distances to
$\mu_{1}$ and $\mu_{2}$ in $\hat{\mu}=\operatorname*{argmin}_{\mu}\left(\alpha
W_{p}^{p}(\mu_{1},\mu)+(1-\alpha)W_{p}^{p}(\mu_{2},\mu)\right).$ This
barycenter can be interpreted as a generalized mean of the measures $\mu_{1}$
and $\mu_{2}$ under the Wasserstein distance metric.

Our fair EM method treats the scoring functions $s_{a}$ and $s_{b}$ for groups
$a$ and $b$ as distributions that are moved towards a common central
Wasserstein barycenter, $\hat{s}$. This alignment reduces unfairness by making
score distributions more similar across groups. To ensure fairness in EM with
respect to DP, the score distributions for both minority and majority groups
should be similar. This approach, based on barycenters, was chosen for its
promising results and balance of fairness and accuracy. We plan to conduct an
extensive search for alternative calibration methods in future research.

Moving scores to the barycenter, $\hat{s}$, does not always result in the best
scoring function. For instance, when $\gamma$ is PR, $\hat{s}$ makes PR even
for all groups at every level, which meets the fairness goals for DP and DSP
(meaning $\mathcal{U}(\hat{s},\text{PR})=0$). Yet, this does not ensure the
scores are as close as possible to the original scores $s$. When $\gamma$ is
either TPR or FPR, using $\hat{s}$ may not lead to the best scores. Unlike
with $\gamma=$ PR, $\hat{s}$ does not reduce $\mathcal{U}(\hat{s},\gamma)$ to
its lowest possible value because TPR and FPR depend on the actual labels, not
just on the distribution of scores as PR does.

This means that just moving $s_{a}$ and $s_{b}$ directly to the barycenter
$\hat{s}$ might not result in the best score, like $s^{*}$ mentioned in
Definition 2.1. To fix this issue, (Kwegyir-Aggrey et al., 2023a) introduces
geometric repair, adjusting a score function $s$ towards another score
function $\hat{s}$ using a weighted average: $s_{\lambda}=(1-\lambda)\cdot
s+\lambda\cdot\hat{s}$. Here, $\lambda$ determines the extent of adjustment.
As (Kwegyir-Aggrey et al., 2023a) explains, the term “geometric” is used
because the adjustment process is a linear interpolation between the initial
score s and the barycenter $\hat{s}$, which will result in $s_{\lambda}$. In
our fair EM method, we apply geometric repair, adjusting $\lambda$ to minimize
$\mathcal{U}(s_{\lambda},\gamma)$, focusing on $\\{\text{TPR},\text{FPR}\\}$
as $\gamma$ values. AUC is primarily influenced by TPR and FPR, so
concentrating on these values minimizes any adverse impact on AUC while
ensuring fairness. Without a closed-form formula for
$\mathcal{U}(s_{\lambda},\gamma)$, we employ a linear search across $\lambda$
in [0,1].

##  4\. EXPERIMENTS

Our experiments focus on achieving two goals. Firstly, we investigate the
effectiveness of traditional fairness metrics, including DP, EO, EOD, and
$\Delta\text{xAUC}$, for measuring biases in EM. We find these measures often
do not accurately reflect biases in EM, particularly when the matching
thresholds vary. Instead, our metrics based on DSP prove to be more effective.
Secondly, we explore the impact of calibrating matching scores by aligning the
score distributions of minority and majority groups towards their barycenters.
This approach shows promising results in reducing DSP without significantly
affecting the accuracy of the corrected matching scores.

Experimental Setup: Our approach builds upon existing state-of-the-art EM
techniques, selecting four methods HierMatcher (Fu et al., 2021), Ditto (Li et
al., 2021), DeepMatcher (Mudgal et al., 2018), and HierGAT (Yao et al., 2022).
We use datasets from popular EM benchmarks: Amazon-Google (AMZ-GOG), Walmart-
Amazon (WAL-AMZ), DBLP-Google Scholar (DBLP-GOGS), and DBLP-ACM. We classify
entities into minority and majority groups across various datasets, similar to
previous research (e.g., (Nilforoushan et al., 2022; Shahbazi et al., 2023)).
Specifically, in AMZ-GOG, the “manufacturer” attribute’s inclusion of
“Microsoft” identifies a minority group. In WAL-AMZ, the minority group
consists of entities categorized as “printers” within the “category”
attribute. For DBLP-GOGS, the “venue” attribute containing “vldb j” denotes
the minority, while in DBLP-ACM, entities are considered part of the minority
if the “authors” attribute includes a female name. Comprehensive details on
these datasets—including their statistics, the division between training and
test sets, record and attribute counts, and the distribution of matched versus
unmatched records—are available in our GitHub repository (moh, 2024). The
repository also contains additional experiments with other benchmarks and all
the source code needed to reproduce the findings of our study.

DSP in Current Matching Methods: Our first experiment assesses the matching
methods’ biases using DSP. Table 1 shows the results in the “before” columns.
The methods exhibit varying bias levels across the four datasets. AMZ-GOG,
WAL-AMZ, and DBLP-GOGS demonstrate significant biases in all three DSP
measures (DP, EO, and EOD), whereas DBLP-ACM shows minimal biases. The AUC-
based bias measure does not consistently identify these biases in methods with
noticeable DSP biases. For instance, neither HierGAT nor HierMatcher in AMZ-
GOG, nor Ditto, HierGAT, and HierMatcher in DBLP-GOGS, are flagged by
$\Delta\text{xAUC}$. This inconsistency occurs because $\Delta\text{xAUC}$ may
overlook biases when xAUC is nearly identical for different groups despite
variations at specific thresholds. In general, DP experiences more significant
changes than EOD and EO. This is because DP focuses on ensuring that both
groups have identical distributions. By aligning both distributions to their
barycenter, we achieve this objective. We also analyzed biases at two
threshold values, $\tau=0.5$ and $0.9$, to support the theory that biases can
vary significantly across thresholds. For instance, EO and EOD in HierMatcher
with AMZ-GOG are 0.3337 and 0.2995 at $\tau=0.5$, but drop to 0.1074 and
0.0946 at $\tau=0.9$, showing a significant decrease in biases at the higher
threshold. However, biases do not always decrease at higher thresholds. For
example, in AMZ-GOG, DeepMatcher’s EO and EOD scores are 0.1720 and 0.1306 at
$\tau=0.5$, increasing to 0.2084 and 0.1847 at $\tau=0.9$. This demonstrates
that a comprehensive understanding of biases requires an examination across
different thresholds, as enabled by DSP.

The experiment demonstrates that DSP identifies biases in the current matching
methods undetected by existing bias measures.

Score Calibration: We investigated the impact of calibrating matching scores
from existing methods to mitigate biases. We optimized parameter $\lambda$ for
each bias measure, such as DP and EOD, recognizing that these measures can
sometimes yield conflicting outcomes. The calibration results are presented in
Table 1 in columns labeled “after.” These results demonstrate that calibration
effectively reduces DSP biases where there are significant biases, such as
AMZ-GOG, WAL-AMZ, DBLP-GOGS, without increasing biases where the biases were
insignificant, such as in DBLP-ACM. This outcome results from the optimization
that can only reduce biases as $\lambda=0$ gives the same initial biases. An
important point from the results is that, although DSP biases decrease
$\Delta\text{xAUC}$ does not necessarily follow suit, highlighting a distinct
behavior between $\Delta\text{xAUC}$ and DSP.

Figures 2(a) to 2(c) illustrate the DSP biases before and after the
calibration. Areas highlighted in these figures denote biases at various
thresholds (the difference between the curves for the minority and majority
groups), with a reduction in these highlighted areas post-calibration
indicating a bias decrease across thresholds. The figures also detail the DSP
biases, scaled by a factor of $\times 100$. Figure 2(d) compares the ROC of
the matching score before and after calibration, labeled as “initial” and with
tags PR, EOD, and EO, respectively. The legend numbers represent AUC values.
“Initial” is the initial AUC, while “PR”, “EOD”, and “EO” are AUCs after
calibration for each metric. The initial AUC and the AUCs for EOD and EO are
nearly identical, resulting in overlapping curves that appear as a single blue
curve. Despite a slight decrease in accuracy post-calibration, as indicated by
the AUC values, the reduction is minimal. The more significant accuracy drop
observed for PR, which correlates with DP, is because calibration modifies the
scores of both groups to the barycenter. In contrast, EO and EOD adjustments
are less drastic, merely moving scores toward the barycenter based on
$\lambda$. Therefore, DP calibration results in the most substantial deviation
from the initial scores, leading to a larger impact on accuracy.

The experiment indicates score calibration efficiently lowers DSP in modern
matching methods while maintaining high accuracy.

(a)

(b)

(c)

(d)

Figure 2. DSP (Fig 2(a)-2(c)) and AUC (Fig 2(d)), before and after score
calibration, using DeepMatcher and WAL-AMZ

##  5\. Conclusion

In this paper, we have explored the evaluation of bias at various matching
thresholds in EM methods, using DSP to assess biases across these thresholds.
Our experimental findings confirm the utility of DSP in identifying biases
that traditional bias metrics may miss. We also formulated the problem of fair
EM, and we implemented a calibration technique as a post-processing debiasing
approach that adjusts scores within groups towards a common barycenter to
rescue biases in scores. This method has been shown to reduce biases while
effectively maintaining high accuracy. Although our implementation yields
promising results, future work is needed to address the challenge of achieving
fair EM under strict fairness constraints, especially in providing guarantees
regarding the accuracy of the calibrated scores to their original optimal
values. A future approach will be reformulating the fair EM problem as an
optimization problem with an objective that minimizes change on the score and
also incorporates the fairness constraints in the objective function with the
space of possible score functions beyond geometric repairs.

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