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# Production of light (anti)nuclei in pp collisions at $$\sqrt{s}$$ = 13 TeV

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A bstract Understanding the production mechanism of light (anti)nuclei is one of the key challenges of nuclear physics and has important consequences for astrophysics, since it provides an input for indirect dark-matter searches in space. In this paper, the latest results about the production of light (anti)nuclei in pp collisions at $$\sqrt{s}$$ s = 13 TeV are presented, focusing on the comparison with the predictions of coalescence and thermal models. For the first time, the coalescence parameters B 2 for deuterons and B 3 for helions are compared with parameter-free theoretical predictions that are directly constrained by the femtoscopic measurement of the source radius in the same event class. A fair description of the data with a Gaussian wave function is observed for both deuteron and helion, supporting the coalescence mechanism for the production of light (anti)nuclei in pp collisions. This method paves the way for future investigations of the internal structure of more complex nuclear clusters, including the hypertriton.
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JHEP01(2022)106
Published for SISSA by Springer
Accepted:December 13, 2021
Published:January 20, 2022
Production of light (anti)nuclei in pp collisions at
s= 13 TeV
The ALICE collaboration
E-mail: ALICE-publications@cern.ch
Abstract: Understanding the production mechanism of light (anti)nuclei is one of the
key challenges of nuclear physics and has important consequences for astrophysics, since it
provides an input for indirect dark-matter searches in space. In this paper, the latest results
about the production of light (anti)nuclei in pp collisions at s= 13 TeV are presented,
focusing on the comparison with the predictions of coalescence and thermal models. For
the ﬁrst time, the coalescence parameters B2for deuterons and B3for helions are compared
with parameter-free theoretical predictions that are directly constrained by the femtoscopic
measurement of the source radius in the same event class. A fair description of the data
with a Gaussian wave function is observed for both deuteron and helion, supporting the
coalescence mechanism for the production of light (anti)nuclei in pp collisions. This method
paves the way for future investigations of the internal structure of more complex nuclear
clusters, including the hypertriton.
ArXiv ePrint: 2109.13026
for the beneﬁt of the ALICE Collaboration.
Article funded by SCOAP3.
https://doi.org/10.1007/JHEP01(2022)106
JHEP01(2022)106
Contents
1 Introduction 1
2 Detector and data sample 3
3 Data analysis 4
3.1 Track selection 4
3.2 Eﬃciency and acceptance correction 5
3.3 Fraction of primary nuclei 5
3.4 Systematic uncertainties 6
4 Results and discussion 7
4.1 Coalescence parameter as a function of transverse momentum 9
4.2 Ratio between triton and helion yields 11
4.3 Coalescence parameter as a function of multiplicity 12
4.4 Ratio between integrated yields of nuclei and protons as a function of mul-
tiplicity 13
5 Summary 14
A Theoretical prediction for the coalescence parameter BA16
A.1 Gaussian wave function 17
A.2 Hulthen wave function 17
A.3 Chiral Eﬀective Field Theory wave function 18
A.4 Combination of two Gaussians 18
The ALICE collaboration 24
1 Introduction
In high-energy hadronic collisions at the LHC, the production of light (anti)nuclei and
more complex multi-baryon bound states, such as (anti)hypertriton [1], is observed. An
unexpectedly large yield of light nuclei was observed for the ﬁrst time in proton-nucleus
collisions at the CERN PS accelerator [2]. Twenty-ﬁve years later, the study of nuclear
production was carried out at Brookhaven AGS and at CERN SPS, with the beginning
of the program of relativistic nuclear collisions [3]. Extensive studies of the production of
light (anti)nuclei were later performed at the Relativistic Heavy-Ion Collider (RHIC) [4
7], including the ﬁrst observation of an antihypernucleus [8] and of anti(alpha) [9]. In
this paper, the focus will be on results obtained at LHC. The production yields of light
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JHEP01(2022)106
(anti)nuclei have been measured as a function of transverse momentum (pT) and charged-
particle multiplicity in diﬀerent collision systems and at diﬀerent center-of-mass energies by
ALICE [1017]. One of the most interesting observations obtained from such a large variety
of experimental data is that the production of light (anti)nuclei seems to depend solely
on the charged-particle multiplicity (hereinafter denoted multiplicity). This observation
manifests itself in the continuous evolution of the deuteron-to-proton (d/p) and 3He-to-
proton (3He/p) ratios with the event multiplicity across diﬀerent collision systems and
energies [16,17]. The results presented in this paper complement the existing picture,
providing measurements in yet unexplored multiplicity regions.
These measurements have an important astrophysical value as they provide input for
the background estimates in indirect dark matter searches in space. Indeed, only small
systems like pp collisions are relevant for such searches because the interstellar medium
consists mostly of hydrogen (protons) and helium (alpha particles). In this context, the
observation of a signiﬁcant antimatter excess with respect to the expected background of
antimatter produced in ordinary cosmic ray pp or p-alpha interactions would represent a
signal for dark matter annihilation in the galactic halo or for the existence of antimatter
islands in our universe [1821].
The theoretical description of the production mechanism of (anti)nuclei is still an
open problem and under intense debate in the scientiﬁc community. Two phenomenologi-
cal models are typically used to describe the production of multi-baryon bound states: the
statistical hadronisation model (SHM) [2228] and the coalescence model [2934]. In the
former, light nuclei are assumed to be emitted by a source in local thermal and hadro-
chemical equilibrium and their abundances are ﬁxed at chemical freeze-out. The version
of this model using the grand-canonical ensemble reproduces the light-ﬂavoured hadron
yields measured in central nucleus-nucleus collisions, including those of (anti)nuclei and
(anti)hypernuclei [22]. In pp and p-Pb collisions, the production of light nuclei can be
described using a diﬀerent implementation of this model based on the canonical ensemble,
where exact conservation of the electric charge, strangeness, and baryon quantum numbers
is applied within a pre-deﬁned correlation volume [25,28]. In the coalescence model, light
nuclei are assumed to be formed by the coalescence of protons and neutrons which are
close in phase space at kinetic freeze-out [30]. In the most simple version of this model,
nucleons are treated as point-like particles and only correlations in momentum space are
considered, i.e. the bound state is assumed to be formed if the diﬀerence between the mo-
menta of nucleons is smaller than a given threshold p0, a free parameter of the model which
is typically of the order of 100 MeV/c. This simple version of the coalescence model can
approximately reproduce deuteron production data in low-multiplicity collisions and was
recently used to describe the jet-associated deuteron pT-diﬀerential yields in pp collisions
at s=13 TeV [35]. In recent developments [31,36], the quantum-mechanical properties
of nucleons and nuclei are taken into account and the coalescence probability is calculated
from the overlap between the source function of the emitted protons and neutrons, which
are mapped on the Wigner density of the nucleus. This state-of-the-art coalescence model
describes the d/p and 3He/p ratios measured in diﬀerent collision systems as a function of
multiplicity [33]. On the contrary, the simple coalescence approach provides a description
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JHEP01(2022)106
of pTspectra of light (anti)nuclei measured in high-energy hadronic collisions only in the
low-multiplicity regime [15].
In this paper, the measurement of the production yields of light (anti)nuclei in pp
collisions at s=13 TeV are presented. In particular, part of the results is obtained from
data collected with a high-multiplicity trigger (see section 2), accessing a multiplicity typ-
ically obtained in p-Pb and peripheral Pb-Pb collisions. For the ﬁrst time, the yields of
(anti)nuclei are measured in a multiplicity region in which high-precision femtostopic mea-
surements of the source size [37] are available. This allows for a parameter-free comparison
of the coalescence measurements with theoretical calculations, showing the potential of this
technique to set constraints on the wave function of (anti)nuclei.
2 Detector and data sample
A detailed description of the ALICE apparatus and its performance can be found in refs. [38]
and [39]. The trajectories of charged particles are reconstructed in the ALICE central barrel
with the Inner Tracking System (ITS) [40] and the Time Projection Chamber (TPC) [41].
The ITS consists of six cylindrical layers of silicon detectors and the two innermost layers
form the Silicon Pixel Detector (SPD). The ITS is used for the reconstruction of primary
and secondary vertices and of charged-particle trajectories. The TPC is used for track
reconstruction, charged-particle momentum measurement and for charged-particle identi-
ﬁcation via the measurement of their speciﬁc energy loss (dE/dx) in the TPC gas [39].
Particle identiﬁcation at high momentum is complemented by the time-of-ﬂight measure-
ment provided by the TOF detector [42]. The aforementioned detectors are located inside
a large solenoid magnet, which provides a homogeneous magnetic ﬁeld of 0.5 T parallel to
the beam line, and cover the pseudorapidity interval |η|<0.9. Collision events are triggered
by two plastic scintillator arrays, V0C and V0A [43], located along the beam axis of the
interaction point, covering the pseudorapidity regions 3.7 < η < 1.7 and 2.8 < η < 5.1,
respectively. The signals from V0A and V0C are summed to form the V0M signal, which
is used to deﬁne event classes to which the measured multiplicity is associated [44]. More-
over, the timing information of the V0 detectors is used for the oﬄine rejection of events
triggered by interactions of the beam with the residual gas in the LHC vacuum pipe.
The results presented in this paper are obtained from data collected in 2016, 2017
and 2018, both with minimum bias (MB) and high multiplicity (HM) triggers. For the
minimum-bias event trigger, coincident signals in both V0 scintillators are required to be
synchronous with the beam crossing time deﬁned by the LHC clock. Events with high
charged-particle multiplicities are triggered on by additionally requiring the total signal
amplitude measured in the V0 detector to exceed a threshold. At the analysis level, the
0–0.1% percentile of inelastic events with the highest V0 multiplicity (V0M) is selected
to deﬁne the high-multiplicity event class. Events with multiple vertices identiﬁed with
the SPD are tagged as pile-up in the same bunch crossing (in-bunch pile-up) and removed
from the analysis [39]. Assuming that all the in-buch pile-up is in the 0–0.01% percentile
of inelastic events, which is the worst-case scenario, only 3% of the selected events (in
the 0–0.01% percentile) would be pile-up events. Therefore, the eﬀect of in-bunch pile-up
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JHEP01(2022)106
on the production spectra is negligible. Pile-up in diﬀerent bunch crossings, instead, is
rejected by requiring track hits in the SPD and its contribution is negligible. The data
sample consists of approximately 2.6 billion MB events and 650 million HM events.
For the measurements of (anti)protons and (anti)deuterons, the high-multiplicity data
sample is divided into three multiplicity classes: HM I, HM II and HM III. The multiplicity
classes are determined from the sum of the signal amplitudes measured by the V0 detectors
and deﬁned in terms of the percentiles of the INEL >0 pp cross section, where an INEL >0
event is a collision with at least a charged particle in the pseudorapidity region |η|<1[45].
For this purpose, charged particles are measured with SPD tracklets, obtained from a pair
of hits in the ﬁrst and second layer of the SPD, respectively. In the case of (anti)triton
(3H) and (anti)helion (3He), due to their lower production rate, it is not possible to divide
the HM sample into smaller classes, but for (anti)helion the MB sample is divided into
two multiplicity classes, MB I and MB II, deﬁned from the percentiles of the INEL >0 pp
cross section. The average charged particle multiplicity hdNch/dηifor all the multiplicity
classes will be reported in table 2. It is deﬁned as the number of primary charged particles
produced in the pseudorapidity interval |η|<0.5. A detailed description of the hdNch/dηi
estimation can be found in ref. [46].
3 Data analysis
3.1 Track selection
(Anti)nuclei candidates are selected from the charged-particle tracks reconstructed in the
ITS and TPC in the pseudorapidity interval |η|<0.8. The criteria used for track selection
are the same as reported in ref. [17]. Particle identiﬁcation is performed using the dE/dx
measured by the TPC and the time-of-ﬂight measured by the TOF. For the TPC analysis,
the signal is obtained from the TPC distribution, where TPC is the diﬀerence between
the measured and expected signals for a given particle hypothesis, divided by the resolution.
For the TOF analysis, the yield in each pTinterval is extracted from the distribution of
the TOF squared-mass, deﬁned as m2=p2t2
TOF/L21/c2, where tTOF is the measured
time-of-ﬂight, Lis the length of the track and pis the momentum of the particle. Similarly
to the TPC case, one deﬁnes TOF as the diﬀerence between the measured and expected
time of ﬂight for a given particle hypothesis, divided by the resolution. For the TOF
analysis, a pre-selection based on the measured TPC dE/dx(|TPC|<3) is performed to
reduce the background originating from other particle species. More details about particle
identiﬁcation with TPC and with TOF can be found in ref. [17].
The (anti)deuteron yield is extracted from the TPC signal for pT<1 GeV/c, while
at higher pTthe yield is extracted from the TOF after the pre-selection using the TPC
signal. For (anti)protons, the TOF is used for the entire pTrange. (Anti)tritons are
identiﬁed through the TPC signal and after a pre-selection with TOF (|TOF|<3) for
pT<2 GeV/c. The (anti)helion identiﬁcation is based only on the TPC dE/dx, which
provides a good separation of its signal from that of other particle species. This is due to
the charge Z=2of this nucleus.
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JHEP01(2022)106
3.2 Eﬃciency and acceptance correction
The estimation of reconstruction eﬃciencies of both nuclei and antinuclei, as well as those of
the contamination to the raw pTspectra of nuclei from spallation and of the signal loss due
to event selection, requires Monte Carlo (MC) simulations. Simulated pp collision events,
generated using Pythia 8 [47] (Monash 2013 tune [48]), are enriched by an injected sample
of (anti-)nuclei generated with a ﬂat pTdistribution in the transverse-momentum range
0< pT<10 GeV/cand a ﬂat rapidity distribution in the range |y|<1. The interactions
of the generated particles with the experimental apparatus are modeled by GEANT4 [49].
The detector conditions during the data taking are reproduced in the simulations.
The raw pTspectra of (anti)nuclei are corrected for the reconstruction eﬃciency and
acceptance, deﬁned as
(pT) = Nrec(|η|<0.8,|y|<0.5, prec
T)
Ngen(|y|<0.5, pT),(3.1)
where Nrec and Ngen are the number of reconstructed and generated (anti)nuclei, respec-
tively. The same criteria for the track selection and particle identiﬁcation used in the
real data analysis are applied to reconstructed tracks in the simulation. Considering that
(anti)nuclei are injected with a ﬂat pTdistribution into the simulated events, their input
distributions are reshaped using pT-dependent weights to match the real shape observed
in data. The latter is parameterised using a Lévy-Tsallis function whose parameters are
determined using an iterative procedure: they are initialized using the values taken from
ref. [17], the corrected spectrum is then ﬁtted with the same function and a new set of
parameters is determined and used for the next iteration. The parameters are found to
converge after two iterations, with diﬀerences from the previous one of less than one per
mille. Protons are abundantly produced by Pythia 8 and their spectral shape is consis-
tent with the one obtained in real data. For this reason, (anti)protons are not injected
additionally into the simulation and their shape is not modiﬁed.
3.3 Fraction of primary nuclei
Secondary nuclei are produced in the interaction of particles with the detector material.
To obtain the yields of primary nuclei produced in a collision, the number of secondaries
must be subtracted from the measured yield. Since the production of secondary antinuclei
is extremely rare, this correction is applied only to nuclei and not to antinuclei. For
(anti)protons, instead, also a contribution from weak decays of heavier unstable particles
(for example Λhyperons) is present and cannot be neglected. The fraction of primary
nuclei is evaluated using diﬀerent techniques according to the analysis.
For deuterons and (anti)protons, the primary fraction is obtained by ﬁtting the distri-
bution of the measured distance of closest approach to the primary vertex in the transverse
plane (DCAxy). For the ﬁt, templates obtained from MC are used, as described in ref. [10].
The DCAxy distribution of anti-deuterons is used as a template for primary deuterons
considering the negligible feed-down from weak decays of hypertriton. The production of
secondary deuterons is more relevant at low pT(at pT=0.7 GeV/cthe fraction of sec-
ondary deuterons is 40%), decreases exponentially with the transverse momentum (<
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JHEP01(2022)106
5% for pT=1.4 GeV/c) and becomes negligible for pT>1.6 GeV/c. For the (anti)proton
analysis, all the templates are taken from MC. In this case, also a template for weak decay
is used. The fraction of secondary protons from material is maximum at low transverse
momentum (5% for pT=0.4 GeV/c) and decreases exponentially, becoming negligible
for pT>1 GeV/c. Also, the fraction of secondary (anti)protons from weak decays is maxi-
mum at low transverse momentum (30% for pT=0.4 GeV/c) and decreases exponentially
(10% for pT=5 GeV/c).
For helion and triton, the primary fraction is obtained by ﬁtting the DCAxy distribution
with two Gaussian functions with diﬀerent widths, one for the primary and one for the
secondary nuclei, respectively. The spallation background is also ﬁtted using a parabola
and a constant function to estimate the systematic uncertainties. For the latter, variations
of the binning and ﬁt range are also considered. A smooth function can be used in this case,
considering that the peak in the DCAxy distribution of secondary nuclei, which typically
appears close to zero and is caused by the wrong association of one SPD cluster to the track
during reconstruction, is negligible for pT>1.5GeV/c. The fraction of secondary helions
and tritons, for both MB and HM pp collisions, are found to be about 15%in the pTinterval
1.5< pT<2GeV/c, about 3%in the pTinterval 2< pT<2.5GeV/cand negligible for
higher pT. For tritons and helions, only antinuclei are used for pT<1.5GeV/c, where the
secondary fraction for nuclei becomes large and it is diﬃcult to constrain the value of the
correction.
3.4 Systematic uncertainties
A summary of the systematic uncertainties for all the measurements is reported in table 1.
Values are provided for low (1.5GeV/c) and high (4 GeV/c) transverse momentum. Where
the systematic uncertainty diﬀers between matter and antimatter, the latter is reported
within brackets. The ﬁrst source of systematic uncertainty is related to track selection.
This source takes into account the imprecision in the description of the detector response
in the MC simulation. The uncertainties are evaluated by varying the relevant selection
criteria, as done in ref. [11]. It is worth mentioning that at low pTuncertainties are generally
larger for matter than for antimatter, due to the increasing number of secondary nuclei
selected when loosing the selection on the DCA. It is one of the main sources. The second
source is related to signal extraction. It is evaluated by changing the ﬁt function used to
evaluate the raw yield or, when the direct count is used, by varying the interval in which
the count is performed (see ref. [11] and ref. [17] for further details). Its value increases with
transverse momentum. The eﬀect of the incomplete knowledge of the material budget of the
detector is evaluated by comparing diﬀerent MC simulations in which the material budget
is varied by ±4.5%. This value corresponds to the uncertainty on the determination of
the material budget by measuring photon conversions [39]. Similarly, the limited precision
in the measurements of inelastic cross sections of (anti)nuclei with matter is a source
of systematic uncertainty. It is evaluated by comparing experimental measurements of
the inelastic cross section with the values implemented in GEANT4, following the same
approach used in ref. [50]. For antihelion, the diﬀerence between the momentum-dependent
inelastic cross sections implemented in GEANT3 [51] and GEANT4 is also considered. This
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JHEP01(2022)106
Source pT= 1.5GeV/c pT= 4 GeV/c
p (p) d (d)3He (3He)3H (3H) p (p) d (d)3He (3He)
Track selection 3% 1% 14% (10%) 14% (10%) 3% 2% 10% (7%)
Signal extraction <1% 3% 13% (<1%) <1% 5% 2% <1%
Material budget 1% 1% 2% 2% 1% <1% 2%
Hadronic interaction 1% (2%) 2% (6%) 1% (2%) 9% (6%) 1% (2%) 2% (6%) 1% (1%)
ITS-TPC matching 2% 2% 2% 2% 2% 2% 2%
TPC-TOF matching 3% 2% 3% 3% 2%
Total 6% (7%) 4% (7%) 22% (11%) 20% (17%) 7% (8%) 4% (7%) 8%
Table 1. Summary of the main contributions to the systematic uncertainties for all the particle
species under study at pT= 1.5GeV/cand at pT= 4 GeV/c. Values in brackets refer to antiparti-
cles. If they are not present, the systematic uncertainty is the same for particles and antiparticles.
A dash symbol is used where the uncertainty from the corresponding source is not applicable. More
details about the sources of the uncertainties can be found in the text.
contribution is maximum (3%) for pT<1.5GeV/cand decreases to a negligible level going
to higher pT. Finally, the last sources of systematic uncertainty are related to the matching
of the tracks between ITS and TPC and between TPC and TOF. They are evaluated from
the diﬀerence between the ITS-TPC (TPC-TOF) matching in data and MC.
4 Results and discussion
The production spectra for all the species under study are shown in ﬁgure 1. The multi-
plicity classes used for this measurement are reported in Tab 2, together with the corre-
sponding pT-integrated yields. (Anti)protons, (anti)deuterons and (anti)helions are ﬁtted
with a Lévy-Tsallis function [52], which is used to extrapolate the yields in the unmea-
sured pTregion. For (anti)triton, the ﬁt parameters (except for the mass and the nor-
malisation) are ﬁxed to those of (anti)helion, due to the few data points available. The
extrapolation amounts to about 20% of the total pT-integrated yield for (anti)protons and
ﬁt functions such as a simple exponential depending on mT, a Boltzmann function or a
Blast-wave function [5355], are used to evaluate the systematic uncertainty on the pT-
integrated yield as done in refs. [1517]. This uncertainty varies between 0.5% and 3% for
protons, deuterons and helions in the HM analysis. For tritons it is around 8% due to the
narrower pTcoverage. In the MB analysis, it is around 8% (14%) for helions (tritons).
7
JHEP01(2022)106
Multiplicity hdNch/dηi|ηlab|<0.5
dN/dy
p d 3He 3H
HM 31.5 ±0.3 0.80 ±0.01 ±0.05 (23.3 ±1±3) ×107(25 ±2±4) ×107
HM I 35.8 ±0.5 0.91 ±0.01 ±0.05 (22.1 ±0.1 ±1.4) ×104
HM II 32.2 ±0.4 0.83 ±0.01 ±0.05 (19.8 ±0.1 ±1.3) ×104
HM III 30.1 ±0.4 0.77 ±0.01 ±0.04 (18.4 ±0.1 ±1.1) ×104
MB 6.9 ±0.1 (2.4 ±0.3 ±0.4) ×107(1.7 ±0.3 ±0.4) ×107
MB I 18.7 ±0.3 (11 ±2±2) ×107
MB II 6.0 ±0.2 (1.5 ±0.2 ±0.3) ×107
Table 2. Multiplicity classes for the diﬀerent measurements, with the corresponding multiplicity
hdNch/dηi|ηlab |<0.5, and pT-integrated yields dN/dyfor the diﬀerent species (average for particle and
antiparticle). For hdNch /dηi|ηlab|<0.5, only the systematic uncertainty is reported, because the sta-
tistical one is negligible. For dN/dy, the ﬁrst uncertainty is statistical and the second is systematic.
3
10
2
10
1
10
1
10
-1
)c (GeV/
yd
T
pd
N
2
d
ev
N
1
) / 2p(p +
= 13 TeVspp,
8)×HM ( 4)×HM I (
2)×HM II ( HM III
vy-TsalliseL
ALICE
10
10
9
10
8
10
7
10
6
10
) / 2He
3
He +
3
(
= 13 TeVspp,
HM MB
MB I MB II
vy-TsalliseL
0 1 2 3 4 5
)c (GeV/
T
p
5
10
4
10
3
10
-1
)c (GeV/
yd
T
pd
N
2
d
ev
N
1
) / 2d(d +
= 13 TeVspp,
4)×HM I ( 2)×HM II (
HM III vy-TsalliseL
0 1 2 3 4 5 6
)c (GeV/
T
p
10
10
9
10
8
10
7
10
6
10
) / 2H
3
H +
3
(
= 13 TeVspp,
HM MB
vy-TsalliseL
Figure 1. Transverse-momentum spectra of (anti)protons, (anti)deuterons, (anti)helion, and
(anti)triton, measured in HM and MB pp collisions at s=13 TeV at midrapidity (|y|<0.5). The
results are shown in the multiplicity classes reported in table 2. Vertical bars and boxes represent
statistical and systematic uncertainties, respectively. The dashed lines are individual ﬁts with a
Lévy-Tsallis function.
8
JHEP01(2022)106
0.5 1.0 1.5 2.0
)c (GeV/A/
T
p
2
10
1
10
)
3
c/
2
(GeV
2
B
= 13 TeVspp,
HM I 2)×HM II (
4)×HM III (
ALICE
(a) (Anti)deuterons.
0.5 1.0 1.5 2.0
)c (GeV/A/
T
p
5
10
4
10
3
10
2
10
)
6
c/
4
(GeV
3
B
= 13 TeVspp,
HM 2)×MB (
MB I MB II
ALICE
(b) (Anti)helions.
Figure 2. Coalescence parameters B2(a) and B3(b) as a function of pT/A for the multiplicity
classes reported in table 2. Vertical bars and boxes represent statistical and systematic uncertainties,
respectively.
4.1 Coalescence parameter as a function of transverse momentum
In the coalescence model, the production probability of a nucleus with mass number Ais
proportional to the coalescence parameter BA, deﬁned as
BA(pp
T) = 1
2πpA
T
d2NA
dydpA
T 1
2πpp
T
d2Np
dydpp
T!A
,(4.1)
where the labels pand Arefer to protons and the (anti)nucleus with mass-number A,
respectively. The invariant spectra of the (anti)protons are evaluated at the transverse
momentum of the (anti)nucleus, divided by the mass-number A. For (anti)deuterons, for
example, BA=B2and pp
T=pA
T/A =pd
T/2.
The coalescence parameters for (anti)deuterons (B2) and for (anti)helions (B3) are
shown in ﬁgure 2as a function of pT/A for diﬀerent multiplicity classes. Some of the
measurements are scaled for better visibility (see the legend), but for B2the three curves
are consistent with each other. A clear increase of both B2and B3with increasing pT/A
is observed. Previous measurements of B2in pp [11,17] and p-Pb [15] collisions indicated
an almost ﬂat trend with pT/A. However, in ref. [17] and in ref. [11] it was shown that
even though B2evaluated in multiplicity classes was ﬂat, B2evaluated in the multiplicity-
integrated sample showed a rise with pT/A. The trend shown in ref. [11] is a consequence
of the mathematical deﬁnition of B2and of the hardening of the proton spectra. Given
the narrow multiplicity intervals used in the present measurement, the signiﬁcant rise of
the coalescence parameters with pT/A cannot be attributed to eﬀects originating from a
diﬀerent hardening of the (anti)proton and (anti)nuclei spectra within these multiplicity
intervals [11].
9
JHEP01(2022)106
0.5 1.0 1.5 2.0
)c (GeV/A/
T
p
5
0
5
10
15
20
25
3
10×
)
3
c/
2
(GeV
2
B
= 13 TeV, HM Ispp,
Gaussian Hulthen
EFTχTwo Gaussians
ALICE
(a) (Anti)deuterons.
0.5 1.0 1.5 2.0
)c (GeV/A/
T
p
0.00
0.05
0.10
0.15
0.20
0.25
3
10×
)
6
c/
4
(GeV
3
B
= 13 TeVsHM pp,
Gaussian
ALICE
(b) (Anti)helions.
Figure 3. Comparison between measurements and theoretical predictions for the coalescence pa-
rameters B2for (anti)deuterons (a) and B3for (anti)helions (b) as a function of pT/A. Vertical bars
and boxes represent statistical and systematic uncertainties, respectively. Theoretical predictions
are obtained using diﬀerent wave functions to describe nuclei: Gaussian (yellow), Hulthen (blue),
χEFT (gray) and two Gaussians (green).
The measurement of the coalescence parameter as a function of transverse momentum
is compared with predictions from the coalescence model, using diﬀerent nuclei wave func-
tions [56] and the precise measurement of the source radii for the same data set [37]. In
the case of (anti)deuterons, the following wave functions are used: single and double Gaus-
sian [34], Hulthen [32], and a function obtained from chiral Eﬀective Field Theory (χEFT)
of order N4LO with a cutoﬀ at 500 MeV [57]. For (anti)helion, only calculations using
a Gaussian wave function are currently available, because the general recipe used for B2
cannot be extended to B3but new calculations ab initio are needed. These wave functions
and more details about the adopted theoretical models can be found in appendix A. In the
coalescence model, the coalescence parameter depends on the radial extension of the parti-
cle emitting source [56]. The source radius is measured in HM pp collisions at s=13 TeV
by ALICE using p-p and p-Λcorrelations as a function of the mean transverse mass hmTi
of the pair [37]. In ref. [37] two diﬀerent measurements of the source radius are reported,
reﬀective and rcore, respectively. The diﬀerence between the two is that rcore takes into ac-
count the contributions coming from the strong decay of resonances by subtracting them.
It is shown that rcore is universal, since it could describe simultaneously p-p and p-Λcor-
relations. In this analysis, rcore is used. The diﬀerence between rcore and reﬀective is small:
B2is on average 7% smaller using reﬀective, while B3is on average 20% smaller, due to the
stronger dependence on the system size for A >2 (see eq. (A.7)). For B2, only the HM I
class is considered, because the B2values in the three multiplicity classes are compatible.
The data in ref. [37] are parameterised as rsource(hmTi) = c0+ exp(c1+c2hmTi), with
cifree parameters, to map the transverse-mass to the source radius. The value of pTcor-
10
JHEP01(2022)106
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
)c (GeV/
T
p
0.5
1.0
1.5
2.0
2.5
3.0
He
3
H /
3
= 13 TeV, HMspp,
Coal. - Phys.Rev.C 99 (2019) 004913
Two-body coal. - Phys.Lett.B 792 (2019) 132-137
Three-body coal. - Phys.Lett.B 792 (2019) 132-137
SHM (T = 155 MeV) - Phys.Lett.B 785 (2018) 171-174
ALICE
Figure 4. Ratio between the pTspectra of triton and helion for the HM data sample. Vertical
bars and boxes represent statistical and systematic uncertainties, respectively. The measurements
are compared with the prediction of thermal (red) and coalescence models from ref. [36] (yellow)
and ref. [33] (green and blue).
responding to mTis taken from mT=qm2
p+ (pT/A)2, where mpis the proton mass. The
radius of the deuteron and 3He are taken from ref. [58]. The coalescence predictions are
shown in comparison with the data in ﬁgure 3. Bands represent the uncertainty propa-
gated from the measurement of the source radius. In the case of B2, the Gaussian wave
function provides the best description of the data, even though the Hulthen wave func-
tion is favoured by low-energy scattering experiments [59]. The other wave functions are
signiﬁcantly larger than the measurement. For the B3, the coalescence predictions using
a Gaussian wave function of helion are above the data by almost a factor of 2 except for
the last pTinterval, which is consistent with the measured B3within the uncertainties. In
the future, a systematic investigation of the coalescence parameter B3using diﬀerent wave
functions, in the context of the coalescence model, will gauge the potential of coalescence
measurements to further constrain the wave function of helion. This technique can be used
in a more general context to obtain information on the internal wave function of more
complex (hyper)nuclei, such as alpha (4He) and hypertriton.
4.2 Ratio between triton and helion yields
The statistical hadronisation and coalescence models predict diﬀerent yields of nuclei with
similar masses but diﬀerent radii. To test the production model, the ratio of triton and
helion is measured as a function of pTfor HM pp collisions (ﬁgure 4) and compared with the
model predictions. Two diﬀerent versions of coalescence are considered, based on ref. [33]
11
JHEP01(2022)106
1 10 2
10 3
10
|<0.5
lab
η|
lab
η/d
ch
Nd
4
10
3
10
2
10
)
3
c/
2
(GeV
2
B
= 13 TeV, HMspp,
= 13 TeVspp,
= 7 TeVspp,
= 5 TeV
NN
sp-Pb,
= 2.76 TeV
NN
sPb-Pb,
(d) = 3.2 fmr coalesc.
2
B
Param. A (fit to source radii)
)
2
BParam. B (constrained to ALICE Pb-Pb
ALICE
c = 0.75 GeV/A/
T
p
(a) (Anti)deuterons.
1 10 2
10 3
10
|<0.5
lab
η|
lab
η/d
ch
Nd
8
10
7
10
6
10
5
10
4
10
3
10
2
10
)
6
c/
4
(GeV
3
B
= 13 TeV, HMspp,
= 13 TeVspp,
= 7 TeVspp,
= 5 TeV
NN
sp-Pb,
= 2.76 TeV
NN
sPb-Pb,
He) = 2.48 fm
3
( r coalesc.
3
B
Param. A (fit to source radii)
)
2
BParam. B (constrained to ALICE Pb-Pb
ALICE
c = 0.73 GeV/A/
T
p
(b) (Anti)helions.
Figure 5. (a): B2at pT/A =0.75 GeV/cas a function of multiplicity in HM pp collisions at
s= 13 TeV, in MB pp collisions at s= 13 TeV [17] and at s= 7 TeV [11], in p-Pb
collisions at sNN = 5.02 TeV [15], and in Pb-Pb collisions at sNN = 2.76 TeV [10]. (b): B3at
pT/A =0.73 GeV/cas a function of multiplicity in HM pp collisions at s= 13 TeV, in MB pp
collisions at s= 13 TeV, in p-Pb collisions at sNN = 5.02 TeV [15], and in Pb-Pb collisions at
sNN = 2.76 TeV [10]. Vertical bars and boxes represent statistical and systematic uncertainties,
respectively. The two lines are theoretical predictions based on two diﬀerent parameterisations of
and ref. [36], respectively. The main diﬀerence between the two models concerns the source
size R: while in ref. [33] the value of Ris constrained from the parameters of a thermal
ﬁt, in ref. [36]Ris an independent variable, for which the aforementioned pTdependence
has been taken into account. In the former approach, Ris about a factor of 2 larger than
in the latter, determining very diﬀerent predictions. The coalescence model predicts a
slightly larger yield of triton as compared to helion due to its smaller nuclear radius. In
the statistical hadronisation model, the yield ratio between these two nuclei is given by
exp(m/Tchem), where mis the mass diﬀerence between triton and helion, taken from
ref. [60], and Tchem the chemical freeze-out temperature. For the latter, Tchem = 155 MeV
is used, as done in the canonical statistical model [61]. Given the small mass diﬀerence, the
statistical hadronisation model predicts a ratio which is very close to unity. The precision
of the present data prevents distinguishing between the models. The 3H/3He ratio will be
measured with higher precision in Run 3 [62] of the LHC. Indeed, the new ITS, which is
characterised by a low material budget, will reduce the systematic uncertainty related to
track reconstruction [63]. Moreover, with a better description of the nuclear absorption
cross section, it will be possible to reduce the corresponding systematic uncertainties.
4.3 Coalescence parameter as a function of multiplicity
The evolution of BAwith multiplicity hdNch/dηiprovides an insight on the dependence
of the production mechanisms of light (anti)nuclei. Figure 5(a) shows B2as a function
of hdNch/dηifor diﬀerent collision systems and energies at pT= 0.75 GeV/c and B3at
pT= 0.73 GeV/c. The measurements are compared with the theoretical predictions from
ref. [58], using r(d) = 3.2fm and r(3He) = 2.48 fm as deuteron and helion radii, respectively.
12
JHEP01(2022)106
Two diﬀerent parameterisations (named A and B in the following) of the system radius
as a function of multiplicity are used. Parameterisation A is based on a ﬁt to the ALICE
measurements of system radii Rfrom femtoscopic measurement as a function of multiplic-
ity [64]. In parameterisation B, the relation between the system radius and the multiplicity
is ﬁxed to reproduce the B2of deuterons in Pb-Pb collisions at sNN = 2.76 TeV in the cen-
trality class 0–10% (see ref. [58] for more details). The B2measurement in HM pp collisions
agrees with observations in 0-1b collisions at sNN =5 TeV [15] at similar multiplicity and
conﬁrms the trend observed in all the previous measurements. This measurement further
strengthen the idea of a production mechanism that depends only on the multiplicity and
not on the collision system nor the centre-of-mass energy. Similarly, ﬁgure 5(b) shows the
evolution of B3as a function of multiplicity. The measurements are compared with the
theoretical prediction from ref. [58], using the same two paremeterisations as for B2. Also
in this case, the coalescence model qualitatively describes the trend but fails in accurately
describing the measurements in the whole multiplicity range. For both B2and B3, one
reason could be that multiplicity is not a perfect proxy for the system size, because for each
multiplicity the source radius depends also on the transverse momentum of the particle of
interest, as shown in ﬁgure 2. In the future, it is important to have more measurements of
the source radius as a function of mTfor the diﬀerent multiplicity classes in order to test the
agreement between the models and the BAmeasurement over the whole multiplicity range.
4.4 Ratio between integrated yields of nuclei and protons as a function of
multiplicity
The measurements of the ratios between the pT-integrated yields of nuclei and protons as a
function of multiplicity are shown. Figure 6shows the measurement for diﬀerent collision
systems and energies for deuterons (d/p) and helions (3He/p), on the left and right panels,
respectively. The new measurements complement the existing picture and are consistent
with the global trend obtained from previous measurements [1017], for both d/p and
3He/p: the ratio increases as a function of multiplicity and eventually saturates at high
multiplicities. This trend can be interpreted as a consequence of the interplay between the
evolution of the yields and of the system size with multiplicity. The measurements are com-
pared with the prediction of both Thermal-FIST Canonical Statistical Model (CSM) [61]
and coalescence model [33]. The predictions from CSM suggest correlation volumes VC
between 1 and 3 units of rapidity. However, the measurement of the proton-to-pion (p/π)
ratio is better described by a correlation volume of 6 rapidity units [28]. On the contrary,
the coalescence model provides a better description of the data. For d/p, the agreement is
good for the whole multiplicity range. For 3He/p, instead, there are more tensions between
data and model in the multiplicity region corresponding to p-Pb and high-multiplicity pp
collisions. Remarkably, the two-body coalescence appears to describe the data better than
the three-body coalescence prediction.
13
JHEP01(2022)106
1 10 2
10 3
10
|<0.5
lab
η|
lab
η/d
ch
Nd
0
1
2
3
4
5
6
3
10×
d / p
= 2.76 TeV
NN
sPb-Pb,
= 5 TeV
NN
sp-Pb,
= 7 TeVspp,
= 13 TeVspp,
= 13 TeV, HMspp,
= 155 MeV
ch
TThermal-FIST CSM,
y/dV = 3 d
c
V
y/dV = d
c
V
Coalescence
ALICE
(a) (Anti)deuterons.
10 2
10 3
10
|<0.5
lab
η|
lab
η/d
ch
Nd
8
10
7
10
6
10
5
10
He / p
3
= 2.76 TeV
NN
sPb-Pb,
= 5 TeV
NN
sp-Pb,
= 13 TeVspp,
= 13 TeV, HMspp,
Thermal-FIST CSM
y/dV = 3 d
c
V = 155 MeV,
ch
T
y/dV = d
c
V = 155 MeV,
ch
T
Coalescence
Two-body
Three-body
ALICE
(b) (Anti)helions.
Figure 6. Ratio between the pT-integrated yields of nuclei and protons as a function of multiplicity
for (anti)deuterons (a) and (anti)helions (b). Measurements are performed in HM pp collisions at
s= 13 TeV, in MB pp collisions at s= 13 TeV [17] and at s= 7 TeV [11], in p-Pb collisions
at sNN = 5.02 TeV [15,16], and in Pb-Pb collisions at sNN = 2.76 TeV [10]. Vertical bars and
boxes represent statistical and systematic uncertainties, respectively. The two black lines are the
theoretical predictions of the Thermal-FIST CSM [61] for two sizes of the correlation volume VC.
For (anti)deuterons, the green line represents the expectation from a coalescence model [33]. For
(anti)helion, the blue and green lines represent the expectations from a two-body and three-body
coalescence model, respectively [33].
5 Summary
In this paper, the measurements of the production yields of (anti)nuclei in minimum bias
and high-multiplicity pp collisions at s=13 TeV are reported.
A signiﬁcant increase of the coalescence parameter B2with increasing pT/A is observed
for the ﬁrst time in pp collisions. Indeed, previous measurements in small collision systems
were consistent with a ﬂat trend within uncertainties. Given the very narrow multiplicity
intervals used in the present measurement, this rising trend cannot be attributed to eﬀects
coming from a diﬀerent hardening of the proton and deuteron spectra within the measured
multiplicity intervals and thus points to some other physics eﬀect. Moreover, the coa-
lescence parameters are compared with theoretical calculations based on the coalescence
approach using diﬀerent internal wave functions of nuclei. This comparison was possible
due to the availability of the measurement of the source radii in the same data sample.
While the predictions for B2using a Gaussian approximation for the deuteron wave func-
tion are in very good agreement with the experimental results, for B3they overestimate
the data by up to a factor of 2 at the lowest pT. Updated theoretical calculations including
also more complex wave functions for 3He would help in providing a better description of
this measurement.
The multiplicity evolution of the coalescence parameters for a ﬁxed pT/A and of the
ratios of integrated yields d/p and 3He/p are consistent with the global trend from previous
measurements. Moreover, the d/p ratio is consistent with predictions from the coalescence
model, while signiﬁcant deviations are observed between the 3He/p and coalescence expec-
14
JHEP01(2022)106
tations at intermediate multiplicities. The canonical statistical model predictions provide a
qualitative description of the particle ratios presented in this paper at low and intermediate
multiplicities covered by pp and p-Pb collisions and are consistent with the data only in
the grand-canonical limit (multiplicities covered by Pb-Pb collisions).
Acknowledgments
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 ac-
knowledges the following funding agencies for their support in building and running the
ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Insti-
tute) 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; 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 Tecnológi-
cas 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 Foun-
dation (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 Technol-
ogy, Ministry of Education, Research and Religions, Greece; National Research, Develop-
ment 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; Indonesian Institute of Science, Indonesia; Istituto Nazionale di
Fisica Nucleare (INFN), Italy; Japanese Ministry of Education, Culture, Sports, Science
and Technology (MEXT), Japan Society for the Promotion of Science (JSPS) KAKENHI
and Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT)of
Applied Science (IIST), 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 Or-
15
JHEP01(2022)106
ganisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council
of Norway, Norway; Commission on Science and Technology for Sustainable Development
in the South (COMSATS) and Pakistan Atomic Energy Commission, 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 Edu-
cation and Scientiﬁc Research, Institute of Atomic Physics and Ministry of Research and
Innovation and Institute of Atomic Physics, Romania; Joint Institute for Nuclear Research
(JINR), Ministry of Education and Science of the Russian Federation, National Research
Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Ba-
sic Research, Russia; 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; Euro-
pean Organization for Nuclear Research, Switzerland; Suranaree University of Technology
(SUT), National Science and Technology Development Agency (NSDTA) and Oﬃce of the
Higher Education Commission under NRU project of Thailand, Thailand; Turkish Energy,
Nuclear and Mineral Research Agency (TENMAK), Turkey; 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
Council, European Union.
A Theoretical prediction for the coalescence parameter BA
In this appendix, the details about the theoretical prediction used for the coalescence
parameter BAas a function of the source radius are reported. The general recipe is taken
from ref. [36]. B2is deﬁned as
B2(p)3
2mZd3q D(~q)CPRF
2(~p, ~q),(A.1)
where mis the proton mass, pis the momentum of the nucleus, qis the relative momentum
of the nucleons, D(~q)is the deuteron Wigner density and CPRF
2(~p, ~q)is the correlation
between two nucleons in the rest frame of the pair (PRF), assuming a Gaussian source
model. For these calculations, we assume a homogeneous source, i.e. R=Rk=R.
Hence, the correlation function has the form
CPRF
2(~p, ~q) = eR2q2,(A.2)
where Ris the source radius. Finally, the Wigner density is deﬁned as
D(~q) = Zd3r|φd(~r)|2ei~q·~r,(A.3)
where φdis the deuteron wave function. In the following, we will provide diﬀerent predic-
tions for B2as a function of the source radius Rstarting from eq. (A.1) and using diﬀerent
16
JHEP01(2022)106
wave functions φd. The theoretical predictions for B2as a function of the source radius R
is shown in ﬁgure 7(a). At large values of the source radius they all show the same trend.
On the contrary, for small values they diﬀer, with a maximum spread of around a factor
of 10. Eq. (A.1) has not an equivalent for B3and ab initio calculations are needed. For
this reason, it is currently not possible to obtain B3predictions for diﬀerent wave functions
as easily as for B2. However, it is possible to obtain a prediction for the case of a simple
Gaussian wave function (see eq. (A.7)).
A.1 Gaussian wave function
The most simple assumption is a Gaussian wave function
φd(r) = er2
2d2
(πd2)3/4,(A.4)
where dis the nucleus radius. For this calculations, d= 3.2fm, as in ref. [36]. The
corresponding Wigner density is
D(~q) = eq2d2
4.(A.5)
This brings to the expression for B2as a function of the source radius R
B2(R) = 3π2
2mR2+d
223
2
.(A.6)
This function is shown in ﬁgure 7(a), together with the other predictions for B2. As shown
in ref. [36], eq. (A.6) can also be generalised for a nucleus with mass number A
BA(R) = 2JA+ 1
2AA
1
mA1"2π
R2+ (rA/2)2#3
2(A1)
,(A.7)
where JAis the spin of the nucleus. Eq. (A.7) is used to calculate the theoretical prediction
for B3, shown in ﬁgure 7(b).
A.2 Hulthen wave function
The second hypothesis tested here is a Hulthen wave function
φd(r) = sαβ(α+β)
2π(αβ)2
eαr eβr
r,(A.8)
where α= 0.2fm1and β= 1.56 fm1are parameters taken from ref. [32]. The corre-
sponding expression for B2is
B2(R) = 3π2
R2
αβ(α+β)
(αβ)2he4α2R2erfc(2αR)2e(α+β)2R2erfc((α+β)R) + e4β2R2erfc(2βR)i.
(A.9)
This function is shown in ﬁgure 7(a), together with the other predictions for B2.
17
JHEP01(2022)106
100101
(fm)
10 5
10 4
10 3
10 2
10 1
100
(GeV / )
Gaussian
Hulthen
CET
Two Gaussians
(a)
1002 × 1003 × 1004 × 100
(fm)
10 8
10 7
10 6
10 5
10 4
(GeV / )
Gaussian
(b)
Figure 7. Coalescence parameters B2(a) and B3(b) as a function of the source radius Rfor
diﬀerent wave functions (see the text for more details).
A.3 Chiral Eﬀective Field Theory wave function
The third hypothesis for deuteron wave function is obtained from Chiral Eﬀective ﬁeld
theory (χEFT) calculations (N4LO). It is based on ref. [57] and the normalisation is based
on ref. [65]. A cutoﬀ at Λc= 500 MeV is used. The deuteron wave function is
φd(~r) = 1
4π r u(r) + 1
8w(r)S12(ˆr)χ1m,(A.10)
where u(r)and w(r)are radial wave functions, S12 r)is the spin tensor and χ1mis a spinor.
The spin-averaged density of the deuteron can be hence expressed as
|φd(r)|2=1
4π r2hu2(r) + w2(r)i.(A.11)
Using eq. (A.1) and eq. (A.3), one obtains
B2(R) = 6π
mZΛc
0
dqZ
0
dr q hu(r)2+w(r)2isin(qr)
reR2q2.(A.12)
Integrating eq. (A.12) over q, one obtains
B2(R) = 3π
mR2Z
0
dr
rhu2(r) + w2(r)i(eΛ2
cR2sin(Λcr)(A.13)
+πr
4Rer2
4R2"erf ir + 2R2Λc
2R!erf ir 2R2Λc
2R!#).
This function is shown in ﬁgure 7(a), together with the other predictions for B2.
A.4 Combination of two Gaussians
The last considered wave function is a combination of two Gaussians, ﬁtted to the Hulthen
wave function [34]:
φd(r) = π3/4"1/2
d3/2
1
er2/(2d2
1)+e (1 ∆)1/2
d3/2
2
er2/(2d2
2)#(A.14)
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JHEP01(2022)106
where ∆=0.581,d1= 3.979 fm and d1= 0.890 fm. The corresponding density is
|φd(r)|2=π3/2
d3
1
er2/d2
1+1
d3
2
er2/d2
2.(A.15)
B2can be hence written as
B2(R) = 24π5/2
mZ
0
dqZ
0
dr|φd(r)|2sin(qr)r q eR2q2,(A.16)
and after integrating over qand r, one obtains:
B2(R) = 3π3/2
2mR3
1 + d2
1
4R2!3/2
+ (1 ∆) 1 + d2
2
4R2!3/2
.(A.17)
This function is shown in ﬁgure 7(a), together with the other predictions for B2.
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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M. Angeletti35, V. Anguelov107 , F. Antinori58, P. Antonioli55, C. Anuj16, N. Apadula82,
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M. Arslandok149,107, A. Augustinus35, R. Averbeck111 , S. Aziz80, M.D. Azmi16, A. Badalà57 ,
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F. Bellini26, R. Bellwied128 , S. Belokurova116, V. Belyaev96, G. Bencedi148,71 , S. Beole25 ,
A. Bercuci49, Y. Berdnikov101, A. Berdnikova107, L. Bergmann107, M.G. Besoiu69 , L. Betev35 ,
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24
JHEP01(2022)106
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M. Ivanov111 , V. Ivanov101, V. Izucheev