2023 268 20 November

Measurements of chemical potentials in Pb–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV

ALICE Collaboration††thanks: See Appendix D for the list of collaboration
members ALICE Collaboration

This Letter presents the most precise measurement to date of the
matter/antimatter imbalance at midrapidity in Pb–Pb collisions at a center-of-
mass energy per nucleon pair $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV. Using the
Statistical Hadronization framework, it is possible to obtain the value of the
electric charge and baryon chemical potentials, $\mu_{Q}=-0.18\pm 0.90$ MeV
and $\mu_{B}=0.71\pm 0.45$ MeV, with unprecedented precision. A centrality-
differential study of the antiparticle-to-particle yield ratios of charged
pions, protons, $\Omega$-baryons, and light (hyper)nuclei is performed. These
results indicate that the system created in Pb–Pb collisions at the LHC is on
average baryon–free and electrically neutral at midrapidity.

##  1 Introduction

Nuclear matter at extremely high energy densities can be generated in the
laboratory through relativistic heavy-ion collisions [1, 2, 3]. At the LHC,
the beam remnants from the collision are located at rapidities $y\approx\pm 6$
and a fraction of the collision energy is deposited at midrapidity [4]. In
this region, particles are formed from a nearly baryon number and electric
charge free medium. This process can be described in the Color Glass
Condensate model via gluon radiation by static quarks, frozen by time dilation
[5]. Conversely, string-fragmentation models explain it through the breaking
of color flux tubes. Part of the initial baryon number can be transported to
midrapidity via either baryon junction formation [6] or diquark breaking [7].
This phenomenon, known as nuclear stopping, influences the net-baryon density
of the system formed at midrapidity [8, 9, 10]. The baryon number transport is
minimal at the LHC, and the nuclear transparency regime [11] is reached. In
this regime, conditions akin to those of the early Universe are replicated,
where nearly equal abundances of matter and antimatter were present, as
described by the standard cosmological model [12]. Experimentally, one can
gauge the extent to which heavy-ion collisions approach the early Universe
conditions by measuring the antimatter-to-matter yield ratios across various
hadron species.

A comprehensive framework for interpreting these ratios is provided by the
Statistical Hadronization Model (SHM)[13, 14, 15, 16, 17, 18]. Among the
several models that can be used to describe a heavy-ion collision, the SHM is
the most successful in describing the yields of all light-flavor hadronic
species, which are determined starting from the partition function of the
fireball at the freeze-out of inelastic scatterings. This fireball is an
equilibrated gas composed of hadrons and resonances (HRG). Due to the
substantial particle multiplicity and the finite kinematical acceptance, a
Grand Canonical (GC) ensemble description is employed for heavy-ion
collisions. In this approach, the conservation of charges, namely the baryon
number ($B$), the electric charge ($Q$), and strangeness ($S$), is regulated
by the corresponding chemical potentials $\mu_{B}$, $\mu_{Q}$, and $\mu_{S}$,
respectively [19, 20]. The baryon chemical potential $\mu_{B}$ represents the
net-baryon density of the system, with $\mu_{B}=0$ corresponding to an HRG
with same amount of baryons and antibaryons. The electric charge potential
$\mu_{Q}$ encodes the positive-negative charge imbalance of the gas; it is
connected to $\mu_{B}$ by the atomic-to-mass-number ratio $Z/A$ of the
colliding ions [21, 22]. The requirement of strangeness neutrality constrains
$\mu_{S}$ throughout the entire volume of the fireball [21, 22]. Chemical
potentials determine the abundance of hadrons through the fugacity,
$\lambda_{i}=\exp[(B_{i}\mu_{B}+Q_{i}\mu_{Q}+S_{i}\mu_{S})/T_{\mathrm{ch}}]$,
where $B_{i}$, $Q_{i}$, and $S_{i}$ denote the quantum numbers of the
considered species $i$, and $T_{\mathrm{ch}}$ is the chemical freeze-out
temperature, at which hadron yields are determined.

Over the last three decades, the asymmetry between antimatter and matter of
the fireball has been systematically studied at different experimental
facilities [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].
The decreasing trend of $\mu_{B}$, from about $400\ \mathrm{MeV}$ at the SPS
to $20\ \mathrm{MeV}$ at the top RHIC energy of 200 GeV, and $\mu_{B}=0.7\pm
3.8\ \mathrm{MeV}$ at the LHC is consistent with the decrease of baryon number
transport to midrapidity with increasing beam rapidity [39, 40, 41, 36, 37].
The formation of baryon number free matter at midrapidity was first reported
in pp collisions by ALICE, which observed that the
$\bar{\mathrm{p}}/\mathrm{p}$ yield ratio is compatible with unity [42]. At
fixed collision energy, it is also possible to explore nuclear transparency as
a function of centrality, i.e., the transverse displacement between the
centers of the colliding nuclei, as it affects the dynamics of the colliding
nucleons. In particular, a slight increase in $\mu_{B}$ from peripheral to
central (head-on) collisions was observed at low energies by STAR at the RHIC
beam energy scan [43]. These results were obtained by either comparing the SHM
predictions with the measured yields of hadrons and their antimatter
counterparts [44] or by directly fitting antiparticle-to-particle yield ratios
[39, 43]. The latter method yields more precise results, owing to the
cancelation of particle-antiparticle correlated uncertainties and the reduced
dependence on model parameters, such as the system volume, $V$, which is
eliminated in the ratios.

In this Letter, we report the most precise estimation to date of $\mu_{B}$ and
$\mu_{Q}$ obtained from a set of antiparticle-to-particle yield ratios. The
analyzed species are charged pions, protons, $\Omega^{-}$ baryons, and light
(hyper)nuclei. (Anti)protons are the most abundantly produced (anti)baryons at
midrapidity ($\approx$ 35 and $\approx$ 2 protons on average in central and
peripheral Pb–Pb collisions, respectively [45]). Consequently, the antiproton-
to-proton yield ratio can probe the antibaryon-to-baryon imbalance [42, 46]
with high precision. On the other hand, the sensitivity to baryon asymmetry is
enhanced when light (hyper)nuclei are included because of their larger baryon
content. In this work, ${}^{3}\mathrm{He}$, its isobar ${}^{3}\mathrm{H}$, and
hypertriton ${}^{3}_{\Lambda}\mathrm{H}$, which is a bound state of a proton,
a neutron, and a $\Lambda$, along with their antimatter counterparts, are
considered111(Anti)deuterons, $\mathrm{d}$($\bar{\mathrm{d}}$) are not
considered in this Letter since the efficiency correction for
$\bar{\mathrm{d}}$ is based on the $\bar{\mathrm{d}}$ absorption cross section
extracted by the ALICE Collaboration from the measured
$\bar{\mathrm{d}}/\mathrm{d}$ yield ratio itself [47].. The ratio of
oppositely charged pions provides a precise constraint on the imbalance of
electric charge, as the yield ratio depends predominantly on $\mu_{Q}$.
Finally, the dependence of antimatter-to-matter ratios on strangeness is
probed with (anti)$\Omega^{-}$ baryons, which, unlike (anti)$\Lambda$ and
(anti)$\Xi^{-}$, have negligible contamination coming from heavier hadron
decays.

##  2 The ALICE detector and data analysis

The results reported in this analysis are obtained from a sample of Pb–Pb
collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV collected in 2018 by ALICE at
the LHC. The ALICE apparatus and its performance are described in detail in
Refs. [48, 49]. The minimum-bias collision and centrality triggers are
provided by the V0 system [50], which is composed of two arrays of plastic
scintillators covering the forward ($2.8<\eta<5.1$) and backward
($-3.7<\eta<-1.7$) regions of pseudorapidity. The coincidence of signals in
both detectors determines the minimum bias trigger. The amplitude of the V0
signal is proportional to the charge deposited in the detectors, which is
related to the produced charged-particle multiplicity that, in turn, is
controlled by the collision centrality. The V0 amplitude is then used to
trigger specific categories of central and semicentral events, and to estimate
centrality [51]. Five centrality intervals are considered in this Letter,
namely 0-5%, 5-10%, 10-30%, 30-50%, and 50-90%, expressed as percentiles of
the total hadronic cross section for Pb–Pb collisions. The position of the
primary interaction vertex is required to be within a 10 cm wide region
centered at the nominal interaction point to profit from the full acceptance
of the ALICE central barrel detectors. Events with multiple interaction
vertices are rejected to ensure the correct association of reconstructed
tracks and primary vertices. The number of events passing these selections is
approximately 300 million.

Charged pions, protons, ${}^{3}\mathrm{He}$, and tritons produced at
midrapidity, $|y|<0.5$, are tracked in the ALICE central barrel: hereafter,
charge conjugates are implied unless stated otherwise. The tracks are
reconstructed within $|\eta|<0.8$ and in the full azimuth using the Inner
Tracking System (ITS) [52] and the Time Projection Chamber (TPC) [53]. These
detectors are placed in a solenoid that provides a uniform magnetic field of
0.5 T parallel to the beam axis. The antiparticle-to-particle yield ratios are
measured as a function of the transverse momentum $p_{\mathrm{T}}$ in the
ranges $0.7\leq p_{\mathrm{T}}<1.6\ \mathrm{GeV}/c$ for $\pi^{-}/\pi^{+}$,
$0.5\leq p_{\mathrm{T}}<3\ \mathrm{GeV}/c$ for $\bar{\mathrm{p}}/\mathrm{p}$,
$1.6\leq p_{\mathrm{T}}<3\ \mathrm{GeV}/c$ for
${}^{3}\overline{\mathrm{H}}/^{3}\mathrm{H}$, and $2\leq p_{\mathrm{T}}<8\
\mathrm{GeV}/c$ for ${}^{3}\overline{\mathrm{He}}/^{3}\mathrm{He}$ to select
the bulk of the production and ensure good identification performance.

The analysis procedure for extracting particle yields is similar to the one
adopted in previous analyses [54, 45, 55]. Standard selections on the
$\chi^{2}$ of the track fit, on the number of reconstructed track points in
the ITS and the TPC, and on the distance of closest approach ($\mathrm{DCA}$)
of the extrapolation of the track to the primary interaction vertex ensure a
good reconstruction of tracks originating from the collisions. Particle
identification (PID) is performed on a statistical basis by measuring the
specific energy loss ($\mathrm{d}E/\mathrm{d}x$) in both the TPC and the ITS,
and particle velocity depending on the transverse momentum of the measured
particles with the Time-Of-Flight detector (TOF). Further details about the
PID are provided in the appendix A.1.

The residual contamination due to hyperon weak decays and spallation reactions
of primary particles in the apparatus is evaluated by fitting the measured DCA
distribution in the plane transverse to the beam axis with templates computed
via Monte Carlo (MC) simulations for the various processes involved [54, 45,
55]. The extracted yields are corrected for the detector acceptance and
candidate selection efficiency, computed using MC simulations, as the fraction
of particles reconstructed out of all MC-generated primary particles. The
Pb–Pb event is generated with HIJING [56], while the particles are transported
through a realistic model of the ALICE apparatus with GEANT4 [57]. To increase
the simulated sample size protons, ${}^{3}\mathrm{He}$ nuclei, and tritons are
injected on top of each HIJING event. The available measurements of hadron
inelastic cross sections are used to correct the GEANT4 parameterizations of
the corresponding reactions [58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69,
70, 71, 72].

The ${}^{3}_{\Lambda}\mathrm{H}$ candidates are reconstructed from their two-
body charged mesonic decay
${}^{3}_{\Lambda}\mathrm{H}\to{}^{3}\mathrm{He}+\pi^{-}$. The reconstruction
algorithm is the same as the one applied in previous measurements [73, 74, 75,
76]. The $\Omega^{-}$ is reconstructed with a similar procedure from the decay
into a charged kaon and a $\Lambda$ baryon, that, in turn, is reconstructed
from its charged two-body decay,
$\Omega^{-}\to\mathrm{K^{-}+\Lambda(\to\pi^{-}+p)}$ [77, 78, 79]. The ratios
are extracted in intervals of proper decay length $ct=cML/p$, with $M$, $L$,
and $p$ being the mass, trajectory length, and candidate momentum,
respectively. In particular, $2\leq ct<35\ \mathrm{cm}$ for
${}^{3}_{\Lambda}\mathrm{H}$ and $1\leq ct<10\ \mathrm{cm}$ for $\Omega^{-}$
are used. The ${}^{3}_{\Lambda}\mathrm{H}$ and $\Omega^{-}$ candidates are
selected with Boosted Decision Tree (BDT) algorithms [80], which are applied
on top of preliminary kinematic and topological selections to enhance the
background rejection. The BDT internal parameters and selections are optimized
using samples of correctly classified signal and background candidates, as
explained in detail in the appendix A.2.

The invariant mass distribution of the selected candidates is fitted with a
probability density function built with a Kernel Density Estimation (KDE) [81,
82] in the MC for ${}^{3}_{\Lambda}\mathrm{H}$, whereas an extended Crystal-
Ball function is used for the $\Omega^{-}$ signal [83]. An exponential
function is used to model the residual background in both cases. The yields
extracted as the integral of the signal functions obtained from the fits are
corrected by the overall selection efficiency and acceptance computed in the
MC simulations. As in previous ${}^{3}_{\Lambda}\mathrm{H}$ analyses [76], an
absorption correction factor is included to account for undetected candidates
absorbed in the detector material before their decay.

The following systematic uncertainty contributions are estimated for the
antiparticle-to-particle yield ratios: candidate selection and signal
extraction, MC data sample size, material budget uncertainty, absorption cross
section uncertainties, and magnetic field polarity. The details about the
estimation and values of such contributions are reported in the appendix A.3.

##  3 Results

The fully corrected antiparticle-to-particle yield ratios do not exhibit any
significant dependence on $p_{\mathrm{T}}$ and $ct$ (see Fig. 3 appendix A.4).
This observation, which is consistent across particle species and centrality
intervals, implies that the production spectra of charge-conjugate species
only differ by normalization factors proportional to their yields. The
antiparticle-to-particle yield ratios of each species are obtained as the
averages weighted with the total uncorrelated uncertainties of the
$p_{\mathrm{T}}$\- and $ct$-differential ratios in each centrality interval.
For ${}^{3}_{\Lambda}\mathrm{H}$, no statistically significant signal is
observed in the 50-90% centrality range.

The chemical potentials $\mu_{B}$ and $\mu_{Q}$ are extracted by fitting the
antiparticle-to-particle yield ratios with the predictions of the GC
statistical hadronization model using the Thermal-FIST code [22]. The measured
ratios and the SHM fit results are reported in Fig. 1. The chemical freeze-out
temperature is set to $T_{\mathrm{ch}}=155\pm 2\ \mathrm{MeV}$, as obtained
from a fit to the ALICE data [84, 85]: its value is compatible with the
pseudo-critical temperature extracted with lattice QCD calculations [86]. This
value is fixed for all centralities, since in heavy-ion collisions only a mild
dependence of $T_{\mathrm{ch}}$ on centrality is observed (less than 3% [43,
84]); additionally, antiparticle-to-particle yield ratios show a negligible
dependence on $T_{\mathrm{ch}}$ for $\mu_{B}\approx 1\ \mathrm{MeV}$ [44]. The
uncertainty on $T_{\mathrm{ch}}$, which is compatible with the range of
variations of $T_{\mathrm{ch}}$ observed as a function of centrality, is
considered as a centrality-correlated source of systematic uncertainty. The
strangeness chemical potential $\mu_{S}$ is constrained in the fit from
strangeness conservation. The contribution of strongly-decaying resonances is
accounted in the model predictions as it cannot be directly disentangled in
the data. For the $\chi^{2}$ minimization, the quadratic sum of statistical
and uncorrelated systematic uncertainty is considered. The effect of the
centrality-correlated sources is evaluated by repeating the fit to ratios
coherently increased or decreased by their uncertainties. The uncertainty
assigned to $\mu_{B}$ and $\mu_{Q}$ is half of the deviation between the
results obtained in the two cases.

In this Letter, yield ratios are analyzed within the GC statistical model also
in the most peripheral events, where canonical ensemble formulation is needed
for an accurate description of hadron yields by requiring exact conservation
of charges over a finite volume [87, 88]. It is known, however, that effects
connected to the canonical conservation of charges cancel out when considering
antiparticle-to-particle yield ratios, and their values are well described by
the GC ensemble [89, 15]. Indeed, good fit quality is obtained across the
0-90$\%$ centrality range using the GC model to quantify these ratios. In
addition, the yield ratio $\overline{\Omega}^{+}/\Omega^{-}$ is compatible
with unity as expected in the SHM, where it is independent of $\mu_{B}$ and
$\mu_{S}$ for $\mu_{B}\sim 0$ [16].

Figure 1: Upper panels: statistical hadronization model fits to the measured
antiparticle-to-particle yield ratios in different centrality intervals. Error
bars show the sum in quadrature of statistical and centrality-uncorrelated
systematic uncertainties. When not visible, error bars are hidden by the
marker. Lower panels: pull distribution, defined as the difference between
data and fit values, normalized to the uncertainty in the data.

The chemical potentials obtained in different centrality intervals are shown
in the left panel of Fig. 2. The centrality dependence of $\mu_{B}$ and
$\mu_{Q}$ is studied by fitting independently the centrality-differential
$\mu_{B}$ and $\mu_{Q}$ results with a constant function, taking into account
the full correlation matrix of the measurements, which are reported in
appendix B. The $\chi^{2}$ profiles of the fits are reported in appendix C.1.
The fit probability is $P=0.97$ for $\mu_{B}$ and $P=0.64$ for $\mu_{Q}$:
therefore, no evidence of centrality dependence is found, even if a larger
$\mu_{B}$ would be expected in more central collisions due to a potentially
larger baryon stopping [4]. The fit of the centrality-differential values
yields chemical potentials $\mu_{B}=0.71\pm 0.45\ \mathrm{MeV}$ and
$\mu_{Q}=-0.18\pm 0.90\ \mathrm{MeV}$, which are compatible with zero within
$1.6\sigma$ and $0.2\sigma$, respectively. The comparison with the previous
data point of $\mu_{B}$ at the LHC [35, 36, 37, 38] shows a significant
improvement in the precision by a factor larger than eight (no direct value of
$\mu_{Q}$ was provided in that study, see below). These results imply that the
system created at midrapidity in Pb–Pb collisions is baryon- and electrically-
neutral on average. As a consequence, this observation shows that the nuclear
transparency regime is reached, i.e., baryon transport from the colliding ions
to the interaction region is negligible. Due to the absence of any centrality
dependence, it is also concluded that nuclear transparency is achieved even in
central Pb–Pb collisions, where a larger-than-zero $\mu_{B}$ could be expected
from a more significant baryon number transport at midrapidity.

As a cross check, the SHM fits described above are repeated by also
constraining $\mu_{Q}$ from initial conditions via conservation laws, as it
was done also in past measurements [39, 43, 36]. Specifically, the
$\mu_{Q}/\mu_{B}$ ratio is fixed by requiring that the average charge-to-
baryon density ratio of the created hadron system, $\langle
n_{Q}\rangle/\langle n_{B}\rangle$, is equivalent to the $Z/A$ ratio of
colliding nuclei, i.e., $\langle n_{Q}\rangle/\langle n_{B}\rangle=Z/A\approx
0.4$ for ${}^{208}\mathrm{Pb}$ [21]. The $\mu_{B}$ values extracted from the
fits in each centrality interval are successfully fitted with a constant
function (fit probability $P=0.09$). The resulting $\mu_{B}$ value is
compatible with the one reported above within uncertainties. Similar results
are obtained by fitting the antiparticle-to-particle yield ratios using the
GSI-Heidelberg model [15, 39, 37], with $T_{\mathrm{ch}}=156.6\pm 1.7\
\mathrm{MeV}$ [38] and $\mu_{Q}$ is fixed to initial conditions: the average
value across centrality is $\mu_{B}=0.90\pm 0.43\ \mathrm{MeV}$. The
$\chi^{2}$ profile of the fit is reported in appendix C.2.

Figure 2: Left panel: $\mu_{B}$ and $\mu_{Q}$ obtained with Thermal-FIST [22]
in different centrality intervals. The centrality-correlated and -uncorrelated
uncertainties are represented with error bars and ellipses, respectively.
Right panel: $\mu_{B}$ extracted from data collected in Au–Au and Pb–Pb
collisions at the AGS (E802, E866, E877, E895, E896, E917 Collaborations), SPS
(NA44, NA49, NA47 Collaborations), RHIC (BRAHMS, PHENIX, STAR Collaboration),
and LHC (ALICE Collaboration) as a function of the center-of-mass energy per
nucleon–nucleon pair [39, 41, 43], and phenomenological parameterization of
$\mu_{B}(\sqrt{s_{\mathrm{NN}}})$ [36]. The inset shows more in detail the
results obtained at the LHC [36].

Using the values of $\mu_{B}$ and $\mu_{Q}$ extracted in the 5% most central
collisions, the inclusive net-proton density at midrapidity, $2/\langle
N_{\mathrm{part}}\rangle\mathrm{d}N_{\mathrm{p}-\overline{\mathrm{p}}}/\mathrm{d}y$,
can be computed in the SHM framework. The value extracted with Thermal-FIST is
$(3.4\pm 1.4)\times 10^{-3}$, while using the GSI-Heidelberg model, a value of
$5.9^{+2.2}_{-2.8}\times 10^{-3}$ is obtained. In both cases, the obtained
results agree with the exponential trend as a function of beam rapidity
predicted by the baryon-junction mechanism [90].

The right panel of Fig. 2 shows the comparison of the current with past
estimations of $\mu_{B}$ as a function of the center-of-mass energy of the
collision [39, 41, 43, 36]. The comparison with the previous LHC data point is
highlighted in the inset of the figure. The result reported in this Letter is
compatible with the extrapolation of the phenomenological parameterization
based on previous data and reported in Ref. [36].

##  4 Conclusions

In summary, the most precise measurement of the asymmetry between matter and
antimatter at the LHC is reported in this Letter. The asymmetry is quantified
through antiparticle-to-particle yield ratios of different hadrons, which are
analyzed within the statistical hadronization framework to extract the
chemical potentials $\mu_{B}$ and $\mu_{Q}$. The GC version of the model
accurately describes the antiparticle-to-particle yield ratios across
centrality, indicating the elimination of effects from canonical charge
conservation in peripheral events. The cancelation of correlated uncertainties
in these ratios leads to a significant improvement in the $\mu_{B}$ precision:
the uncertainty on the obtained value is about one order of magnitude smaller
than the previously published one [36]. In addition, a direct estimation of
$\mu_{Q}$ is provided. Furthermore, the first centrality-differential study of
chemical potentials at the LHC is reported in this Letter. The obtained
chemical potentials are consistent with zero, i.e., with the nuclear
transparency regime being reached across the full centrality range, thus
indicating that baryon transport to midrapidity is negligible even in the most
central events at the LHC. The results reported in this Letter show with
unprecedented precision that the medium created in heavy-ion collisions at the
LHC approaches the early Universe conditions more than any other experimental
facility.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Bulgarian Ministry of Education and Science, within the National Roadmap for Research Infrastructures 2020-2027 (object CERN), Bulgaria; Ministry of Education of China (MOEC) , Ministry of Science & Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à l’Energie Atomique (CEA) and Institut National de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE), Department of Science and Technology, Government of India (DST), University Grants Commission, Government of India (UGC) and Council of Scientific and Industrial Research (CSIR), India; National Research and Innovation Agency - BRIN, Indonesia; Istituto Nazionale di Fisica Nucleare (INFN), Italy; Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Education and Science, National Science Centre and WUT ID-UB, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics, Ministry of Research and Innovation and Institute of Atomic Physics and Universitatea Nationala de Stiinta si Tehnologie Politehnica Bucuresti, Romania; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSTDA) and National Science, Research and Innovation Fund (NSRF via PMU-B B05F650021), Thailand; National Academy of Sciences of Ukraine, Ukraine; Science and Technology Facilities Council (STFC), United Kingdom; National Science Foundation of the United States of America (NSF) and United States Department of Energy, Office of Nuclear Physics (DOE NP), United States of America. In addition, individual groups or members have received support from: European Research Council, Strong 2020 - Horizon 2020 (grant nos. 950692, 824093), European Union; Academy of Finland (Center of Excellence in Quark Matter) (grant nos. 346327, 346328), Finland.

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##  Appendix A Antiparticle-to-particle ratios

###  A.1 Particle identification of tracked species

The particle identification (PID) of charged pions, protons,
${}^{3}\mathrm{He}$, and tritons is performed by measuring the specific energy
loss ($\mathrm{d}E/\mathrm{d}x$) in both the TPC and the ITS, and particle
velocity depending on the transverse momentum of the measured particles with
the Time-Of-Flight detector (TOF). Due to its electric charge $Z=2$,
${}^{3}\mathrm{He}$ is identified using the TPC $\mathrm{d}E/\mathrm{d}x$
within $|n\sigma^{\mathrm{TPC}}|<3$, where $n\sigma$ is the deviation of the
measured $\mathrm{d}E/\mathrm{d}x$ from the expected one, normalized to the
experimental resolution. A similar approach is applied to protons for
$p_{\mathrm{T}}<1\ \mathrm{GeV}/c$ after a preliminary
$|n\sigma^{\mathrm{ITS}}|<3$ selection, which is required to reduce the
electron contamination. Higher-$p_{\mathrm{T}}$ protons, pions, and tritons
are identified using TOF after preliminary TPC PID selections. The
$n\sigma^{\mathrm{TPC}}$ or $n\sigma^{\mathrm{TOF}}$ background distributions
are fitted outside of the signal window. Their extrapolation within the signal
region is integrated to evaluate contaminations due to either misidentified
particles or the mismatch of TPC tracks and TOF space points.

###  A.2 Machine Learning analysis of $\Omega$ and
${}^{3}_{\Lambda}\mathrm{H}$

The Machine Learning (ML) selection of $\Omega$ and
${}^{3}_{\Lambda}\mathrm{H}$ candidates is based on Boosted Decision Trees
(BDT). The optimization of the BDT internal parameters is performed using
samples of correctly classified signal and background candidates. The signal
sample is built from simulated candidates injected on top of a HIJING Pb–Pb
event with a Blast-wave $p_{\mathrm{T}}$ distribution [91] derived from the
measured production of light flavor hadrons for $\Omega$ [45] and of
${}^{3}\mathrm{He}$ for ${}^{3}_{\Lambda}\mathrm{H}$ [55]. The background
${}^{3}_{\Lambda}\mathrm{H}$ candidates are obtained in the data from same-
sign combinations of ${}^{3}\mathrm{He}$ and $\pi$ tracks. For $\Omega^{-}$,
all the candidates with an invariant-mass deviating more than $7\sigma$ from
the nominal $\Omega^{-}$ mass are considered as background candidates, with
$\sigma\approx 1.7\ \mathrm{MeV}/c^{2}$ being the invariant-mass resolution in
the data. The same training variables used in previous analyses are employed
for ${}^{3}_{\Lambda}\mathrm{H}$ [75, 76]: the cosine of the pointing angle
$\cos(\theta_{p})$ (i.e., the angle between the reconstructed candidate
momentum and the straight line connecting the production and decay vertices),
the DCA between the decay tracks and the primary vertex (PV), and between the
two tracks themselves, the number of TPC space points for the
${}^{3}\mathrm{He}$ track, and the $n\sigma^{\mathrm{TPC}}$ of the decay
tracks. For the $\Omega^{-}$, the BDT input variables include the DCA of the
$\mathrm{K^{-}}$, $\pi^{-}$ and $\mathrm{p}$ to the PV, the DCA of the
reconstructed $\Lambda$ to the PV, the minimum distance between the $\pi^{-}$
and $\mathrm{p}$, and between the $\mathrm{K^{-}}$ and $\Lambda$. The
$\cos(\theta_{p})$ for both the $\Omega^{-}$ and $\Lambda$, and the
$n\sigma^{\mathrm{TPC}}$ for $\mathrm{p}$, are also used as BDT input
variables. Signal candidates are selected requiring a BDT output score larger
than a preset threshold. For ${}^{3}_{\Lambda}\mathrm{H}$, the threshold is
optimized by maximizing the expected signal significance; for the
$\Omega^{-}$, a BDT signal selection efficiency of 50% is required, as it
ensures a consistent BDT response in data and MC.

###  A.3 Systematic uncertainties

The centrality-uncorrelated systematic uncertainty on the yield ratio is
obtained as the variance of multiple reanalyses done by varying the tracking
and PID selections for $\pi$, $\mathrm{p}$, ${}^{3}\mathrm{He}$, and
${}^{3}\mathrm{H}$, and of the BDT output selections for
${}^{3}_{\Lambda}\mathrm{H}$ and $\Omega^{-}$ around their nominal values used
in the analysis. The background fit function is also changed from exponential
to polynomial in the invariant mass fit of ${}^{3}_{\Lambda}\mathrm{H}$ and
$\Omega^{-}$, while the yields of ${}^{3}\mathrm{He}$ and ${}^{3}\mathrm{H}$
are alternatively extracted as the integral of a Gaussian fit to the
$n\sigma^{\mathrm{TPC}}$ and $n\sigma^{\mathrm{TOF}}$ distributions,
respectively. The variations are applied coherently to antiparticles and
particles to allow for the cancelation of correlated contributions in the
antiparticle-to-particle yield ratios. The MC statistical precision is also
considered as a centrality-uncorrelated source of systematic uncertainty. The
uncertainty on the material-budget description in MC simulations is correlated
with centrality. It is evaluated by varying the amount of material crossed by
simulated particles by its uncertainty, estimated to be $\pm 4.5\%$ [49]. The
uncertainties on the measured absorption cross sections used to correct the
GEANT4 ones are also propagated to the ratios. The consistency of the results
obtained with opposite magnetic field polarities is assessed by repeating the
measurement separately with the two configurations: a statistically
significant discrepancy of about $0.4\%$ and $0.6\%$ due to imperfections in
the MC description is observed in the most central events for the proton and
pion yield ratios, respectively. The maximum half dispersion between the
opposite field polarity results is then assigned as a further centrality-
correlated uncertainty. The values of the various contributions are summarised
in Table 1.

Table 1: Relative systematic uncertainty on the average antiparticle-to-particle ratios due to the different sources considered in the analysis. Only the statistically significant contributions to systematic uncertainties are reported in the table. Source | $\overline{\Omega}^{+}/\Omega^{-}$ | $\pi^{+}/\pi^{-}$ | $\overline{\mathrm{p}}/\mathrm{p}$ | ${}^{3}_{\overline{\Lambda}}\overline{\mathrm{H}}/{}^{3}_{\Lambda}\mathrm{H}$ | ${}^{3}\overline{\mathrm{H}}/{}^{3}\mathrm{H}$ | ${}^{3}\overline{\mathrm{He}}/{}^{3}\mathrm{He}$
---|---|---|---|---|---|---
Candidate selection + signal extract. | 0.5% | 0.05% | 0.05% | 10% | 3% | 0.5%
Monte Carlo precision | 0.5% | 0.1% | 0.1% | 1% | 1% | 1%
Material budget | – | 0.1% | 0.5% | – | – | –
Absorption cross section | – | 0.7% | 0.5% | 1% | 10% | 1%
Magnetic field polarity | – | 0.3% | 0.2% | – | – | –

###  A.4 Results for the antiparticle-to-particle ratios

Figure 3: $p_{\mathrm{T}}$\- and $ct$-differential ratios of the species used
for the chemical potential measurement in the various centrality intervals.
Error bars show statistical uncertainties, while boxes represent centrality-
uncorrelated uncertainties. The value of $R$ represents the averages weighted
with the total uncorrelated uncertainties of the differential measurements.
The correlated uncertainties are not shown in the plots.

##  Appendix B Covariance matrices

Figure 4: Covariance matrices of $\mu_{\mathrm{B}}$ (left) and
$\mu_{\mathrm{Q}}$ (right) obtained with Thermal-FIST [22] in the different
centrality intervals.

##  Appendix C $\chi^{2}$ profiles of chemical potential fits

###  C.1 Thermal-FIST model

Table 2: Confidence intervals (C.I.) at $1\sigma$, $2\sigma$, and $3\sigma$ levels for $\mu_{B}$ and $\mu_{Q}$. Observable |  $1\sigma$ C.I. (MeV) |  $2\sigma$ C.I. (MeV) |  $3\sigma$ C.I. (MeV)
---|---|---|---
$\mu_{B}$ | $[0.26,1.16]$ | $[-0.20,1.61]$ | $[-0.65,2.07]$
$\mu_{Q}$ | $[-1.08,0.73]$ | $[-1.98,1.63]$ | $[-2.89,2.54]$

Figure 5: Profiles of the $\chi^{2}$ variable minimized in the fit of
$\mu_{B}$ (left) and $\mu_{Q}$ (right) obtained with the Thermal-FIST model
[22]. The values obtained from the minimization, as well as the $1\sigma$,
$2\sigma$, and $3\sigma$ confidence intervals, are reported in the figures.

###  C.2 GSI-Heidelberg model

Table 3: Confidence intervals (C.I.) at $1\sigma$, $2\sigma$, and $3\sigma$ levels for $\mu_{B}$ and $\mu_{Q}$. Observable |  $1\sigma$ C.I. (MeV) |  $2\sigma$ C.I. (MeV) |  $3\sigma$ C.I. (MeV)
---|---|---|---
$\mu_{B}$ | $[0.47,1.33]$ | $[0.05,1.75]$ | $[-0.38,2.18]$
Figure 6: Profiles of the $\chi^{2}$ variable minimized in the fit of
$\mu_{B}$ obtained with the GSI-Heidelberg model [15, 39, 37]. The values
obtained from the minimization, as well as the $1\sigma$, $2\sigma$, and
$3\sigma$ confidence intervals, are reported in the figures.

##  Appendix D The ALICE Collaboration

S. Acharya 127, D. Adamová 86, G. Aglieri Rinella 33, M. Agnello 30, N.
Agrawal 26, Z. Ahammed 135, S. Ahmad 16, S.U. Ahn 72, I. Ahuja 38, A.
Akindinov 141, M. Al-Turany 97, D. Aleksandrov 141, B. Alessandro 57, H.M.
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C. Cheshkov 128, V. Chibante Barroso 33, D.D. Chinellato 111, E.S. Chizzali
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P. Dupieux 127, M. Durkac106, N. Dzalaiova13, T.M. Eder 126, R.J. Ehlers 74,
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T. Gahlaut48, C.D. Galvan 109, D.R. Gangadharan 116, P. Ganoti 78, C.
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V. Grigoriev 141, S. Grigoryan 142,1, F. Grosa 33, J.F. Grosse-Oetringhaus 33,
R. Grosso 97, D. Grund 36, N.A. Grunwald94, G.G. Guardiano 111, R. Guernane
73, M. Guilbaud 103, K. Gulbrandsen 83, T. Gündem 65, T. Gunji 124, W. Guo 6,
A. Gupta 91, R. Gupta 91, R. Gupta 49, K. Gwizdziel 136, L. Gyulai 47, C.
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P. Hauer 43, L.B. Havener 138, E. Hellbär 97, H. Helstrup 35, M. Hemmer 65, T.
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C. Jahnke 110, M.J. Jakubowska 136, M.A. Janik 136, T. Janson71, S. Ji 17, S.
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J. Kaewjai105, P. Kalinak 61, A.S. Kalteyer 97, A. Kalweit 33, D. Karatovic
89, O. Karavichev 141, T. Karavicheva 141, P. Karczmarczyk 136, E. Karpechev
141, U. Kebschull 71, R. Keidel 140, D.L.D. Keijdener60, M. Keil 33, B. Ketzer
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## Affiliation Notes

I Deceased
II Also at: Max-Planck-Institut fur Physik, Munich, Germany
III Also at: Italian National Agency for New Technologies, Energy and
Sustainable Economic Development (ENEA), Bologna, Italy
IV Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy
V Also at: Department of Applied Physics, Aligarh Muslim University, Aligarh,
India
VI Also at: Institute of Theoretical Physics, University of Wroclaw, Poland
VII Also at: An institution covered by a cooperation agreement with CERN

## Collaboration Institutes

1 A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
2 AGH University of Krakow, Cracow, Poland
3 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine
4 Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
5 California Polytechnic State University, San Luis Obispo, California, United
States
6 Central China Normal University, Wuhan, China
7 Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
8 Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and
Mérida, Mexico
9 Chicago State University, Chicago, Illinois, United States
10 China Institute of Atomic Energy, Beijing, China
11 China University of Geosciences, Wuhan, China
12 Chungbuk National University, Cheongju, Republic of Korea
13 Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovak Republic
14 COMSATS University Islamabad, Islamabad, Pakistan
15 Creighton University, Omaha, Nebraska, United States
16 Department of Physics, Aligarh Muslim University, Aligarh, India
17 Department of Physics, Pusan National University, Pusan, Republic of Korea
18 Department of Physics, Sejong University, Seoul, Republic of Korea
19 Department of Physics, University of California, Berkeley, California,
United States
20 Department of Physics, University of Oslo, Oslo, Norway
21 Department of Physics and Technology, University of Bergen, Bergen, Norway
22 Dipartimento di Fisica, Università di Pavia, Pavia, Italy
23 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
24 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
25 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
26 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
27 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
28 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
29 Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
30 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
31 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy
32 Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy
33 European Organization for Nuclear Research (CERN), Geneva, Switzerland
34 Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia
35 Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway
36 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
37 Faculty of Physics, Sofia University, Sofia, Bulgaria
38 Faculty of Science, P.J. Šafárik University, Košice, Slovak Republic
39 Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany
40 Fudan University, Shanghai, China
41 Gangneung-Wonju National University, Gangneung, Republic of Korea
42 Gauhati University, Department of Physics, Guwahati, India
43 Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
44 Helsinki Institute of Physics (HIP), Helsinki, Finland
45 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico
46 Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania
47 HUN-REN Wigner Research Centre for Physics, Budapest, Hungary
48 Indian Institute of Technology Bombay (IIT), Mumbai, India
49 Indian Institute of Technology Indore, Indore, India
50 INFN, Laboratori Nazionali di Frascati, Frascati, Italy
51 INFN, Sezione di Bari, Bari, Italy
52 INFN, Sezione di Bologna, Bologna, Italy
53 INFN, Sezione di Cagliari, Cagliari, Italy
54 INFN, Sezione di Catania, Catania, Italy
55 INFN, Sezione di Padova, Padova, Italy
56 INFN, Sezione di Pavia, Pavia, Italy
57 INFN, Sezione di Torino, Turin, Italy
58 INFN, Sezione di Trieste, Trieste, Italy
59 Inha University, Incheon, Republic of Korea
60 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht
University/Nikhef, Utrecht, Netherlands
61 Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovak Republic
62 Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India
63 Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic
64 Institute of Space Science (ISS), Bucharest, Romania
65 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
66 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
67 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil
68 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico
69 iThemba LABS, National Research Foundation, Somerset West, South Africa
70 Jeonbuk National University, Jeonju, Republic of Korea
71 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany
72 Korea Institute of Science and Technology Information, Daejeon, Republic of
Korea
73 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS-IN2P3, Grenoble, France
74 Lawrence Berkeley National Laboratory, Berkeley, California, United States
75 Lund University Department of Physics, Division of Particle Physics, Lund,
Sweden
76 Nagasaki Institute of Applied Science, Nagasaki, Japan
77 Nara Women’s University (NWU), Nara, Japan
78 National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece
79 National Centre for Nuclear Research, Warsaw, Poland
80 National Institute of Science Education and Research, Homi Bhabha National
Institute, Jatni, India
81 National Nuclear Research Center, Baku, Azerbaijan
82 National Research and Innovation Agency - BRIN, Jakarta, Indonesia
83 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
84 Nikhef, National institute for subatomic physics, Amsterdam, Netherlands
85 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom
86 Nuclear Physics Institute of the Czech Academy of Sciences, Husinec-Řež,
Czech Republic
87 Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
88 Ohio State University, Columbus, Ohio, United States
89 Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia
90 Physics Department, Panjab University, Chandigarh, India
91 Physics Department, University of Jammu, Jammu, India
92 Physics Program and International Institute for Sustainability with Knotted
Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan
93 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany
94 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
95 Physik Department, Technische Universität München, Munich, Germany
96 Politecnico di Bari and Sezione INFN, Bari, Italy
97 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Darmstadt, Germany
98 Saga University, Saga, Japan
99 Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata,
India
100 School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
101 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica
del Perú, Lima, Peru
102 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
103 SUBATECH, IMT Atlantique, Nantes Université, CNRS-IN2P3, Nantes, France
104 Sungkyunkwan University, Suwon City, Republic of Korea
105 Suranaree University of Technology, Nakhon Ratchasima, Thailand
106 Technical University of Košice, Košice, Slovak Republic
107 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Cracow, Poland
108 The University of Texas at Austin, Austin, Texas, United States
109 Universidad Autónoma de Sinaloa, Culiacán, Mexico
110 Universidade de São Paulo (USP), São Paulo, Brazil
111 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
112 Universidade Federal do ABC, Santo Andre, Brazil
113 Universitatea Nationala de Stiinta si Tehnologie Politehnica Bucuresti,
Bucharest, Romania
114 University of Cape Town, Cape Town, South Africa
115 University of Derby, Derby, United Kingdom
116 University of Houston, Houston, Texas, United States
117 University of Jyväskylä, Jyväskylä, Finland
118 University of Kansas, Lawrence, Kansas, United States
119 University of Liverpool, Liverpool, United Kingdom
120 University of Science and Technology of China, Hefei, China
121 University of South-Eastern Norway, Kongsberg, Norway
122 University of Tennessee, Knoxville, Tennessee, United States
123 University of the Witwatersrand, Johannesburg, South Africa
124 University of Tokyo, Tokyo, Japan
125 University of Tsukuba, Tsukuba, Japan
126 Universität Münster, Institut für Kernphysik, Münster, Germany
127 Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France
128 Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de
Lyon, Lyon, France
129 Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France,
Strasbourg, France
130 Université Paris-Saclay, Centre d’Etudes de Saclay (CEA), IRFU, Départment
de Physique Nucléaire (DPhN), Saclay, France
131 Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France
132 Università degli Studi di Foggia, Foggia, Italy
133 Università del Piemonte Orientale, Vercelli, Italy
134 Università di Brescia, Brescia, Italy
135 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata,
India
136 Warsaw University of Technology, Warsaw, Poland
137 Wayne State University, Detroit, Michigan, United States
138 Yale University, New Haven, Connecticut, United States
139 Yonsei University, Seoul, Republic of Korea
140 Zentrum für Technologie und Transfer (ZTT), Worms, Germany
141 Affiliated with an institute covered by a cooperation agreement with CERN
142 Affiliated with an international laboratory covered by a cooperation
agreement with CERN.


$\displaystyle\frac{1}{p+1}\int_{0}^{T}\|\overline{\Pi}_{n}u-\underline{\Pi}_{n}u\|_{X}^{p}\,dt$

and a similar computation for $\Lambda_{n}u-\underline{\Pi}_{n}u$. Passing to
the limit in A.1c and A.1d as $p\to\infty$, we conclude that both of them hold
for $p=\infty$ as well.

Let $p\in[1,\infty)$. Estimating the norm of the arising Bochner’s integral
and then using Hölder’s inequality, we obtain

$\displaystyle\frac{1}{n}\sum_{k=1}^{nT}\left\|P_{n}\widetilde{u}\left(\frac{k}{n}\right)\right\|_{X}^{p}=$
$\displaystyle
n^{p-1}\sum_{k=1}^{nT}\left\|\int_{\frac{k-1}{n}}^{\frac{k}{n}}\widetilde{u}(t)\,dt\right\|_{X}^{p}$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\|\widetilde{u}(t)\|_{X}^{p}\,dt$
$\displaystyle=$ $\displaystyle\int_{0}^{T}\|\widetilde{u}(t)\|_{X}^{p}\,dt,$

yielding A.2 for finite $p$. Passing to the limit in A.2 as $p\to\infty$, we
confirm that the inequality holds for $p=\infty$ as well.

For a Hilbert space $(X,(\cdot,\cdot)_{X})$, we compute using the Cauchy-
Schwartz inequality and A.1a and A.1b

$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\|\Lambda_{n}u\|_{X}^{2}\,dt=$
$\displaystyle\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\left\|\frac{\frac{k}{n}-t}{\frac{1}{n}}u\left(\frac{k-1}{n}\right)+\frac{t-\frac{k-1}{n}}{\frac{1}{n}}u\left(\frac{k}{n}\right)\right\|_{X}^{2}\,dt$
$\displaystyle=$ $\displaystyle
n^{2}\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\left(\frac{k}{n}-t\right)^{2}\left\|u\left(\frac{k-1}{n}\right)\right\|_{X}^{2}+\left(t-\frac{k-1}{n}\right)^{2}\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{2}$
$\displaystyle\phantom{n^{2}\sum_{k=1}^{nT}\int_{\frac{k-1}{n}}^{\frac{k}{n}}}+2\left(\frac{k}{n}-t\right)\left(t-\frac{k-1}{n}\right)\left(u\left(\frac{k-1}{n}\right),u\left(\frac{k}{n}\right)\right)_{X}\,dt$
$\displaystyle=$
$\displaystyle\frac{1}{3n}\sum_{k=1}^{nT}\left(\left\|u\left(\frac{k-1}{n}\right)\right\|_{X}^{2}+\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{2}+\left(u\left(\frac{k-1}{n}\right),u\left(\frac{k}{n}\right)\right)_{X}\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{6n}\sum_{k=1}^{nT}\left(\left\|u\left(\frac{k-1}{n}\right)\right\|_{X}^{2}+\left\|u\left(\frac{k}{n}\right)\right\|_{X}^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{6}\left(\|\underline{\Pi}_{n}u\|_{L^{2}((0,T);X)}^{2}+\|\overline{\Pi}_{n}u\|_{L^{2}((0,T);X)}^{2}\right),$

so A.5 follows.

######  Step 2 (A.3).

Let us first assume

$p=\infty\text{ and }\widetilde{u}\in C([0,T];X).$

In this case, estimating the norm of the arising Bochner’s integral, we find
that

$\displaystyle\max\left\\{\underset{t\in\left(\frac{k-1}{n},\frac{k}{n}\right)}{\sup}\left\|P_{n}\widetilde{u}\left(\frac{k}{n}\right)-\widetilde{u}(t)\right\|_{X}:\
k\in\\{1,2,\dots,nT\\}\right\\}$ $\displaystyle=$
$\displaystyle\max\left\\{n\underset{t\in\left(\frac{k-1}{n},\frac{k}{n}\right)}{\sup}\left\|\int_{\frac{k-1}{n}}^{\frac{k}{n}}(\widetilde{u}(s)-\widetilde{u}(t))\,ds\right\|_{X}:\
k\in\\{1,2,\dots,nT\\}\right\\}$ $\displaystyle\leq$
$\displaystyle\max\left\\{n\underset{t\in\left(\frac{k-1}{n},\frac{k}{n}\right)}{\sup}\int_{\frac{k-1}{n}}^{\frac{k}{n}}\|\widetilde{u}(s)-\widetilde{u}(t))\|_{X}\,ds:\
k\in\\{1,2,\dots,nT\\}\right\\}$ $\displaystyle\leq$
$\displaystyle\max\left\\{\|\widetilde{u}(s)-\widetilde{u}(t))\|_{X}:\
s,t\in[0,T]\text{ and }|t-s|\leq\frac{1}{n}\right\\}$
$\displaystyle\underset{n\rightarrow\infty}{\rightarrow}0,$

proving A.3 for ${\cal T}=\overline{\Pi}$. A similar calculation yields A.3
for ${\cal T}=\underline{\Pi}$.

Now let

$p\in[1,\infty)\text{ and }\widetilde{u}\in C([0,T];X).$

Since $[0,T]$ is finite, A.3 for ${\cal
T}\in\\{\underline{\Pi},\overline{\Pi}\\}$ and $p\in[1,\infty)$ directly
follows from the corresponding convergences for $p=\infty$.

We turn to the general case of

$p\in[1,\infty)\text{ and }\widetilde{u}\in L^{p}((0,T);X).$

Consider a sequence $(\widetilde{u}_{m})\subset C([0,T];X)$ such that

$\displaystyle\widetilde{u}_{m}\underset{m\to\infty}{\to}\widetilde{u}\qquad\text{in
}L^{p}((0,T);X).$

Combining A.1b and A.2 with A.3 for (continuous) $\widetilde{u}_{m}$ and
${\cal T}=\overline{\Pi}$, we obtain

$\displaystyle\underset{n\to\infty}{\lim\sup}\|\overline{\Pi}_{n}P_{n}\widetilde{u}-\widetilde{u}\|_{L^{p}((0,T);X)}$
$\displaystyle\leq$
$\displaystyle\underset{m\to\infty}{\lim\sup}\,\underset{n\to\infty}{\lim\sup}\left(\|\overline{\Pi}_{n}P_{n}\widetilde{u}_{m}-\widetilde{u}_{m}\|_{L^{p}((0,T);X)}+\|\widetilde{u}_{m}-\widetilde{u}\|_{L^{p}((0,T);X)}+\|\overline{\Pi}_{n}P_{n}(\widetilde{u}_{m}-\widetilde{u})\|_{L^{p}((0,T);X)}\right)$
$\displaystyle\leq$
$\displaystyle\underset{m\to\infty}{\lim\sup}\,\underset{n\to\infty}{\lim\sup}\left(\|\overline{\Pi}_{n}P_{n}\widetilde{u}_{m}-\widetilde{u}_{m}\|_{L^{p}((0,T);X)}+2\|\widetilde{u}_{m}-\widetilde{u}\|_{L^{p}((0,T);X)}\right)$
$\displaystyle=$ $\displaystyle 0.$

A proof for ${\cal T}=\underline{\Pi}$ is analogous. Since $\Lambda_{n}$ is a
convex combination of $\underline{\Pi}_{n}$ and $\overline{\Pi}_{n}$ on each
interval $((k-1)/n,k/n)$ by 2.5, A.3 holds also for ${\cal T}=\Lambda$.

######  Step 3 (A.4 and A.6).

Thanks to A.3, we have

$\displaystyle\|{\cal
T}_{n}u_{n}-\widetilde{u}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\Leftrightarrow\quad\|{\cal
T}_{n}(u_{n}-P_{n}\widetilde{u})\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\qquad\text{for
}{\cal T}\in\\{\underline{\Pi},\overline{\Pi},\Lambda\\}.$ (A.8)

Further, estimating the norm of the arising Bochner’s integral and using
Hölder’s inequality, we find that

$\displaystyle\frac{1}{n^{\frac{1}{p}}}\|P_{n}\widetilde{u}(t)\|_{X}=$
$\displaystyle
n^{1-\frac{1}{p}}\left\|\int_{t-\frac{1}{n}}^{t}\widetilde{u}(t)\,dt\right\|_{X}$
$\displaystyle\leq$ $\displaystyle
n^{1-\frac{1}{p}}\int_{t-\frac{1}{n}}^{t}\|\widetilde{u}(t)\|_{X}\,dt$
$\displaystyle\leq$
$\displaystyle\int_{t-\frac{1}{n}}^{t}\|\widetilde{u}(t)\|_{X}^{p}\,dt$
$\displaystyle\underset{n\rightarrow\infty}{\rightarrow}$ $\displaystyle
0\qquad\text{for }t\in\\{1,T\\}.$ (A.9)

Due to A.9 and A.8, it suffices to prove A.4 and A.6 for $\widetilde{u}=0$,
i.e. that

$\displaystyle\quad\|\underline{\Pi}_{n}u_{n}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n^{\frac{1}{p}}}\|u_{n}(T)\|_{X}\underset{n\rightarrow\infty}{\rightarrow}0$
$\displaystyle\Leftrightarrow$
$\displaystyle\quad\|\overline{\Pi}_{n}u_{n}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n^{\frac{1}{p}}}\|u_{n}(0)\|_{X}\underset{n\rightarrow\infty}{\rightarrow}0$
(A.10a) $\displaystyle\Rightarrow$
$\displaystyle\quad\|\Lambda_{n}u_{n}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0,$
(A.10b)

and, for $p=2$ and $X$ a Hilbert space, also the reverse implication

$\displaystyle\quad\|\Lambda_{n}u_{n}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0$
$\displaystyle\Rightarrow$
$\displaystyle\quad\|\overline{\Pi}_{n}u_{n}\|_{L^{p}((0,T);X)}\underset{n\rightarrow\infty}{\rightarrow}0\quad\text{and}\quad\frac{1}{n^{\frac{1}{p}}}\|u_{n}(0)\|_{X}\underset{n\rightarrow\infty}{\rightarrow}0$
(A.11)

The equivalence in A.10a is a direct consequence of A.1a and A.1b. Combining
A.10a and A.1d, we obtain implication A.10b. Finally, implication A.11
directly follows with A.1a, A.1b, and A.5.

∎

######  Remark A.2.

Lemma A.1 is motivated by [39, Chapter 11 Lemma 11.4]. There, the implication

$\displaystyle\|\Lambda_{n}u_{n}-\widetilde{u}\|_{L^{2}((0,T);X)}\to
0\qquad\Rightarrow\qquad\|\overline{\Pi}_{n}u_{n}-\widetilde{u}\|_{L^{2}((0,T);X)}\to
0$

was proved with the help of an estimate similar to A.5.

######  Lemma A.3.

Let $n\in\mathbb{N}$, $T\in\frac{1}{n}\mathbb{N}$,
$u:\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\to\mathbb{R}$, and
$g:\mathbb{R}\rightarrow\mathbb{R}$ be an increasing function with a primitive
$G$. Then

$\displaystyle\partial_{t}\Lambda_{n}ug\left(\overline{\Pi}_{n}{u}\right)\geq$
$\displaystyle\partial_{t}\Lambda_{n}{G(u)}\qquad\text{in
}[0,T]\backslash\left(\frac{1}{n}\mathbb{N}_{0}\right).$ (A.12)

The proof is a consequence of the following elementary inequality: under the
assumptions on $g$ from the Lemma it holds that

$\displaystyle(b-a)g(b)\geq G(b)-G(a)\qquad\text{for all }a,b\in\mathbb{R}.$
(A.13)

######  Proof of Lemma A.3..

Thanks to A.13 it holds that

$\displaystyle\partial_{t}\Lambda_{n}ug\left(\overline{\Pi}_{n}{u}\right)=$
$\displaystyle\frac{u\left(\frac{k}{n}\right)-u\left(\frac{k-1}{n}\right)}{\frac{1}{n}}g\left(u\left(\frac{k}{n}\right)\right)$
$\displaystyle\geq$
$\displaystyle\frac{G\left(u\left(\frac{k}{n}\right)\right)-G\left(u\left(\frac{k-1}{n}\right)\right)}{\frac{1}{n}}$
$\displaystyle=$ $\displaystyle\partial_{t}\Lambda_{n}{G(u)}\qquad\text{in
}\left(\frac{k-1}{n},\frac{k}{n}\right)\text{ for }k\in\\{1,2,\dots,nT\\}.$

∎

We conclude this Section with a version of the discrete Gronwall lemma.

######  Lemma A.4.

Let $n\in\mathbb{N}$, $T\in\frac{1}{n}\mathbb{N}$, and $C\in(-\infty,n)$.
Define

$\displaystyle\operatorname{exp}_{n,C}:\frac{1}{n}\mathbb{N}_{0}\to(0,\infty),\qquad\operatorname{exp}_{n,C}(t):=\left(\left(1-\frac{C}{n}\right)^{-n}\right)^{t}.$

Let $u,v:\frac{1}{n}\mathbb{N}_{0}\cap[0,T]\to[0,\infty)$. Then

$\displaystyle\partial_{t}\Lambda_{n}u\leq
C\overline{\Pi}_{n}u+\overline{\Pi}_{n}v\qquad\text{in
}[0,T]\backslash\left(\frac{1}{n}\mathbb{N}_{0}\right)$ (A.14)

implies

$\displaystyle\|\overline{\Pi}_{n}\operatorname{exp}_{n,C}^{-1}u\|_{L^{\infty}(0,T)}\leq
u(0)+\|\underline{\Pi}_{n}\operatorname{exp}_{n,C}^{-1}\overline{\Pi}_{n}v\|_{L^{1}((0,T))}.$
(A.15)

###### Proof.

Set

$\displaystyle a_{n}:=u\left(\frac{k}{n}\right),\quad
g_{n}:=v\left(\frac{k}{n}\right),\quad\Delta t:=\frac{1}{n}.$

In this notation, inequality A.14 can be rewritten as

$\displaystyle\frac{a_{n+1}-a_{n}}{\frac{1}{n}}\leq
Ca_{n+1}+g_{n+1}\qquad\text{for }n\in\\{0,\cdots,nT-1\\}.$

Since $1-C/n>0$, a discrete version of Gronwall’s lemma [30, Proposition 3.1]
can be applied in this setting, followed by division throughout by
$\operatorname{exp}_{n,C}(k/n)$. The result is

$\displaystyle\operatorname{exp}_{n,C}^{-1}u\left(\frac{k}{n}\right)\leq$
$\displaystyle
u(0)+\frac{1}{n}\sum_{j=0}^{k-1}\operatorname{exp}_{n,C}^{-1}\left(\frac{j}{n}\right)v\left(\frac{j+1}{n}\right)\qquad\text{for
}k\in\\{1,\dots,nT\\}.$ (A.16)

Since $v$ is non-negative, A.16 implies

$\displaystyle\operatorname{exp}_{n,C}^{-1}u\left(\frac{k}{n}\right)\leq$
$\displaystyle
u(0)+\frac{1}{n}\sum_{j=0}^{nT-1}\operatorname{exp}_{n,C}^{-1}\left(\frac{j}{n}\right)v\left(\frac{j+1}{n}\right)\qquad\text{for
}k\in\\{1,\dots,nT\\}.$ (A.17)

Re-writing A.17 in terms of continuous-time interpolations gives A.15.

∎

##  Appendix B Auxiliary results about functions and measures with values in
Banach spaces

The two results in this Section help dealing with functions and vector
measures that take values in Banach spaces. Our first Lemma is about
representation of such measures by weakly-$*$ measurable functions. It relies
on some known facts related to vector measures and presents them in the form
that is useful for our analysis in Section 4.4.

######  Lemma B.1.

Let $I$ be a finite interval and $X$ be a Banach space.

  1. (1)

Each $f\in L^{1}_{w-*}(I;X^{*})$ generates a vector measure $\mu_{f}\in{\cal
M}(I;X^{*})$ that is given by the Gelfand integral [28, Chapter II §3
Definition 2] of $f$, i.e.

$\displaystyle\left<\mu_{f}(E),\varphi\right>:=\int_{E}\left<f,\varphi\right>dt\qquad\text{for
all }\varphi\in X\text{ and Borel }E\subset I$ (B.1)

and satisfies

$\displaystyle\|\mu_{f}(E)\|_{X^{*}}\leq\int_{E}\|f\|_{X^{*}}\,dt\qquad\text{for
all Borel }E\subset I.$ (B.2)

  2. (2)

Let $\mu\in{\cal M}(I;X^{*})$ and assume that there exists $0\leq g\in
L^{1}(I)$ such that

$\displaystyle\|\mu(E)\|_{X^{*}}\leq\int_{E}g\,dt\qquad\text{for all Borel
}E\subset I.$ (B.3)

Then there exists some $\zeta\in L^{\infty}_{w-*}(I;X^{*})$ such that

$\displaystyle\|\zeta\|_{L^{\infty}_{w-*}(I;X^{*})}\leq 1$

and

$\displaystyle\mu=\mu_{\zeta g}.$

If $X$ is separable, then any $\zeta_{1}$ and $\zeta_{2}$ that satisfy the
conditions on $\zeta$ coincide a.e. in $\\{g>0\\}$.

###### Proof.

  1. (1)

Since $f\in L^{1}_{w-*}(I;X^{*})$, its Gelfand integral exists. Hence,
$\mu_{f}$ is a well-defined $X^{*}$-valued set function on the set of Borel
subsets of $I$. Estimate B.2 is a trivial consequence of B.1 and the
definition of the $X^{*}$-norm. Countable additivity directly follows with the
corresponding property of the Lebesgue integral of scalar functions, whereas
regularity and variation finiteness are direct consequences of the
corresponding properties of the scalar measure defined by the right-hand side
of the inequality in B.2.

  2. (2)

Thanks to B.3 the given vector measure $\mu$ is continuous with respect to the
finite measure $g\,dt$. By [29, Chapter II §13 4. Theorem 5] (take there
$T:=I$, $E:=X$, $F:=Z:=\mathbb{R}$, $\nu:=g\,dt$, and $m:=\mu$) there exists
some $\zeta\in L^{1}_{w-*,loc}(I;X^{*})$ such that $\mu=\mu_{\zeta g}$ and
(take $\varphi$ in that Theorem to be the characteristic function of $E$)

$\displaystyle\int_{E}\|\zeta\|_{X^{*}}\,g\,dt=|\mu|(E)\qquad\text{for all
Borel }E\subset I,$ (B.4)

where by $|\mu|(E)$ we denote the variation of $\mu$ over $E$. Combining B.4
and B.3, we obtain

$\displaystyle\int_{E}\|\zeta\|_{X^{*}}\,g\,dt\leq\int_{E}g\,dt\qquad\text{for
all Borel }E\subset I,$

and this implies $\|\zeta\|_{X^{*}}\leq 1$ a.e. in $I$, as required.

Now let $X$ be separable, ${\cal C}$ being a countable dense subset of $X$.
Assume that $\zeta_{1}$ and $\zeta_{2}$ satisfy the conditions on $\zeta$.
Then B.1 for $f_{i}:=\zeta_{i}g$ leads to

$\displaystyle\int_{E}\left<\zeta_{1},\varphi\right>gdt=\int_{E}\left<\zeta_{2},\varphi\right>gdt\qquad\text{for
all }\varphi\in{\cal C}\text{ and Borel }E\subset I.$

Consequently,

$\displaystyle\left<(\zeta_{1}-\zeta_{2}),\varphi\right>g=0\qquad\text{ a.e.
in }I\text{ for all }\varphi\in{\cal C}.$

Since ${\cal C}$ is countable, we conclude that

$\displaystyle\left<(\zeta_{1}-\zeta_{2}),\varphi\right>g=0\qquad\text{ for
all }\varphi\in{\cal C}\text{ a.e. in }I.$

Finally, utilising the density of ${\cal C}$ in $X$, we deduce that
$\zeta_{1}-\zeta_{2}=0$ a.e. in $\\{g>0\\}$.

∎

The next Lemma deals specifically with weak measurability of parts of a time-
dependent Radon measure.

######  Lemma B.2.

Let $I$ be a finite interval and $O$ be a bounded domain. Let $f:I\to{\cal
M}(\Omega)$ be weakly measurable. Then: $f^{ac}$ is
$L^{1}(\Omega)$-measurable, $f^{s}$ is weakly ${\cal M}(\Omega)$-measurable,
and $|f^{s}|$ is weakly-$*$ ${\cal M}(\Omega)$-measurable.

###### Proof.

Since operators $(\cdot)^{ac}:{\cal M}(\Omega)\to L^{1}(\Omega)$ and
$(\cdot)^{s}:{\cal M}(\Omega)\to{\cal M}(\Omega)$ are linear and bounded, we
conclude that $f^{ac}:I\to L^{1}(\Omega)$ and $f^{s}:I\to{\cal M}(\Omega)$ are
weakly measurable. Moreover, since $L^{1}(\Omega)$ is separable, Pettis’
theorem applies and yields the strong measurability of $f^{ac}$.

In order to prove that $|f^{s}|$ is weak-$*$ measurable it suffices to verify
that $\left<|f^{s}|,\varphi\right>$ is measurable for any $0\leq\varphi\in
C_{0}(\overline{\Omega})$. Observe that for such $\varphi$

$\displaystyle\left<|f^{s}(t,\cdot)|,\varphi\right>=$
$\displaystyle\left\|\varphi f^{s}(t,\cdot)\right\|_{{\cal M}(\Omega)}$
$\displaystyle=$
$\displaystyle{\sup}\left\\{\left<f^{s}(t,\cdot),\varphi\psi\right>:\ \psi\in
C_{0}(\overline{\Omega})\text{ such that }\|\psi\|=1\right\\}.$ (B.5)

Since $C_{0}(\overline{\Omega})$ is separable, the supremum in B.5 can be
replaced by the supremum over a countable set. Exploiting the weak
measurability of $f^{s}$, we conclude that the function on the right-hand side
of B.5 is measurable. The proof of the Lemma is thus complete. ∎