# Quantum Mechanical Softening of the Hypertriton Transverse Momentum Spectrum
in Heavy-Ion Collisions

Dai-Neng Liu  Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Institute of Modern Physics, Fudan University, Shanghai $200433$, China
Shanghai Research Center for Theoretical Nuclear Physics, NSFC and Fudan
University, Shanghai 200438, China  Che Ming Ko<EMAIL_ADDRESS>Cyclotron
Institute and Department of Physics and Astronomy, Texas A&M University,
College Station, Texas 77843, USA  Yu-Gang Ma<EMAIL_ADDRESS>Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai $200433$, China  Shanghai Research
Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai
200438, China  Francesco Mazzaschi  Dipartimento di Fisica dell’Università and
Sezione INFN, Turin, Italy  Maximiliano Puccio  Experimental Physics
Department, CERN, CH-1211 Geneve 23, Switzerland  Qi-Ye Shou  Key Laboratory
of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern
Physics, Fudan University, Shanghai $200433$, China  Shanghai Research Center
for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai 200438,
China  Kai-Jia Sun<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and
Ion-beam Application (MOE), Institute of Modern Physics, Fudan University,
Shanghai $200433$, China  Shanghai Research Center for Theoretical Nuclear
Physics, NSFC and Fudan University, Shanghai 200438, China  Yuan-Zhe Wang  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of
Modern Physics, Fudan University, Shanghai $200433$, China  Shanghai Research
Center for Theoretical Nuclear Physics, NSFC and Fudan University, Shanghai
200438, China

###### Abstract

Understanding the properties of hypernuclei helps to constrain the interaction
between hyperon and nucleon, which is known to play an essential role in
determining the properties of neutron stars. Experimental measurements have
suggested that the hypertriton (${}^{3}_{\Lambda}\text{H}$), the lightest
hypernucleus, exhibits a halo structure with a deuteron core encircled by a
$\Lambda$ hyperon at a distance of about 10 fm. This large $\Lambda-d$
distance in ${}^{3}_{\Lambda}\text{H}$ wave function is found to cause a
suppressed ${}^{3}_{\Lambda}\text{H}$ yield and a softening of its transverse
momentum ($p_{T}$) spectrum in relativistic heavy-ion collisions. Within the
coalescence model based on nucleons and $\Lambda$ hyperons from a microscopic
hybrid hydro model with a hadronic afterburner for nuclear cluster production
in Pb-Pb collisions at $\sqrt{s_{NN}}$= 5.02 TeV, we show how this softening
of the hypertriton $p_{T}$ spectrum appears and leads to a significantly
smaller mean $p_{T}$ for ${}^{3}_{\Lambda}\text{H}$ than for helium-3 (3He).
The latter is opposite to the predictions from the blast-wave model which
assumes that ${}^{3}_{\Lambda}\text{H}$ and 3He are thermally produced at the
kinetic freeze-out of heavy ion collisions. The discovered quantum mechanical
softening of the (anti-)hypertriton spectrum can be experimentally tested in
relativistic heavy-ion collisions at different collision energies and
centralities and used to obtain valuable insights to the mechanisms for light
(hyper-)nuclei production in these collisions.

_Introduction._ — Observations of anti-nuclei such as anti-deuteron
($\bar{d}$), anti-helium (${}^{3}\overline{\text{He}}$ and
${}^{4}\overline{\text{He}}$), and anti-hypertriton
(${}^{3}\overline{{}_{\bar{\Lambda}}\text{H}}$) have been reported in many
heavy-ion collision experiments Abelev  _et al._ (2010); Agakishiev  _et al._
(2011); Adam  _et al._ (2015a, 2020); Chen  _et al._ (2018); Braun-Munzinger
and Dönigus (2019); Acharya  _et al._ (2023a). Recent experiments at the
Relativistic Heavy Ion Collider (RHIC) Aboona  _et al._ (2023a) and the Large
Hadron Collider (LHC) ALI (2023) have even suggested possible detection of
heavier anti-hypernuclei, such as $\overline{{}^{4}_{\bar{\Lambda}}\text{H}}$
and $\overline{{}^{4}_{\bar{\Lambda}}\text{He}}$. The detection of these bound
nuclei and the study of their properties Gal  _et al._ (2016); Adamczyk  _et
al._ (2015); Acharya  _et al._ (2023b, 2019); Aboona  _et al._ (2023b) are
important in the search for the signals of phase transitions in the strongly
interacting matter created in relativistic heavy-ion collisions Sun  _et al._
(2017, 2018, 2021); Shuryak and Torres-Rincon (2019, 2020); Abdulhamid  _et
al._ (2023); Ko (2023) and in the dark matter detection in space Blum  _et
al._ (2017); von Doetinchem  _et al._ (2020); Acharya  _et al._ (2023c) as
well as for understanding the fundamental CPT theorem in quantum field theory
Adam  _et al._ (2015a, 2020). Also, understanding the properties of
hypernuclei helps to constrain the hyperon-nucleon ($Y\text{-}N$) interaction
Chen  _et al._ (2023); Ma (2023); Zhang  _et al._ (2022), which is essential
for determining the structure of compact astrophysical objects like the
neutron stars Gal  _et al._ (2016).

Various theoretical models based on different assumptions have been used to
describe (anti-)nuclei production in nuclear reactions. These include the
statistical hadronization model (SHM) Andronic  _et al._ (2018), the
coalescence model Scheibl and Heinz (1999); Sun  _et al._ (2019); Zhang and Ko
(2018); Bellini  _et al._ (2021), and the kinetic or transport approach
Danielewicz and Bertsch (1991); Oh and Ko (2007); Oh  _et al._ (2009);
Oliinychenko  _et al._ (2019); Sun  _et al._ (2024). While both (anti-)nuclei
disintegration and regeneration during the evolution of the hadronic matter
are included in the kinetic approach, only (anti-)nuclei production is
considered in the statistical model and the coalescence model by assuming,
respectively, that they are thermally produced from the hadronization of
produced quark-gluon plasma (QGP) and formed from the recombination of
nucleons at the kinetic freeze-out. Another difference between the statistical
and the coalescence model is the role of light cluster wave function in its
production. While the wave function is essential for forming light clusters in
the coalescence model, it is not relevant in the SHM. With the hypertriton
characterized by a halo structure with a deuteron core surrounded by a
$\Lambda$ hyperon, it provides an excellent tool for differentiating between
these two models. If the coalescence model proves to be the production
mechanism of light nuclei and hypernuclei in nuclear collisions, this will
open a new way for investigating their structures and internal wave functions.

According to recent STAR and ALICE measurements, the
${}^{3}_{\Lambda}\text{H}$ has a lifetime close to that of a free $\Lambda$
Abdallah  _et al._ (2022); Acharya  _et al._ (2023b, 2019) and a $\Lambda$
separation energy ($B_{\Lambda}$) of merely a few hundreds keV. While the
$B_{\Lambda}$ reported by the ALICE Collaboration is $0.102\pm
0.063\text{(stat.)}\pm 0.067\text{(syst.)}$ MeV Acharya  _et al._ (2023b), it
is $0.41\pm 0.12\text{(stat.)}\pm 0.11\text{(syst.)}$ MeV according to the
measurement of the STAR Collaboration Adam  _et al._ (2020); Liu  _et al._
(2019). The world average $B_{\Lambda}$ is, however, $0.164\pm 0.043$ MeV
Juric  _et al._ (1973); Keyes  _et al._ (1970); Chaudhari  _et al._ (1968);
Mayeur  _et al._ (1966); Ammar  _et al._ (1962); Cobis  _et al._ (1997); Adam
_et al._ (2020); Acharya  _et al._ (2023b); Eckert  _et al._ (2023),
corresponding to a very large $\Lambda-d$ distance of $\sqrt{\langle
r^{2}_{\Lambda d}\rangle}\approx 9.5$ fm, which suggests that the
${}^{3}_{\Lambda}\text{H}$ has a halo structure with a deuteron core
surrounded by a $\Lambda$ halo. The halo structure is an intriguing quantum
tunneling phenomenon observed in nuclei far from the $\beta$-stability line
Tanihata  _et al._ (1985). Within the coalescence model Sun  _et al._ (2019),
it has been demonstrated that such a large $\Lambda-d$ distance lead to a
pronounced suppression of ${}^{3}_{\Lambda}\text{H}$ production in collisions
of small systems, a phenomenon that has been consistently observed in various
ALICE measurements at the LHC Acharya  _et al._ (2018, 2020a, 2020b, 2022a,
2022b, 2022c). While the grand-canonical ensemble version of the statistical
hadronization model fails to account for the suppression of light clusters in
small systems, the canonical statistical model, which takes into account exact
charge conservation, does predict a similar suppression seen in the
experimental data Vovchenko  _et al._ (2018).

In the present study, we demonstrate that the large $\Lambda-d$ distance in
${}^{3}_{\Lambda}\text{H}$ wave function further results in a softening of its
transverse momentum ($p_{T}$) spectrum in Pb+Pb collisions at
$\sqrt{s_{NN}}=5.02$ TeV. This softening effect on the hypertriton spectrum
occurs not only in non-central collisions but also in central collisions,
where the canonical effects are expected to be insignificant. The quantum
mechanical softening of the hypertriton spectrum in the coalescence model thus
makes it a promising tool to distinguish it from the statistical hadronization
model.

_Suppression of hypertriton production in peripheral collisions_ — We adopt
the hybrid model Zhao  _et al._ (2021) of MUSIC+UrQMD+COAL to study the
production of (anti-)hypertriton in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$
TeV. After the evolution of the QGP produced in these collisions via the
(3+1)-dimensional viscous hydrodynamic model MUSIC Paquet  _et al._ (2016);
Shen and Schenke (2018); Shen and Yan (2020) with the collision-geometric-
based 3D initial conditions Shen and Alzhrani (2020) and a crossover type of
equation of state at finite density NEOS-BQS Monnai  _et al._ (2019), hadrons
are produced from a constant energy density hypersurface according to the
Cooper-Frye formula Cooper and Frye (1974a). The subsequent hadronic
rescatterings and decays are modeled by the UrQMD model Bass  _et al._ (1998).
The (anti-)hypertriton yield in a collision event is then obtained from the
coalescence (COAL) of (anti-)protons, (anti-)neutrons, and (anti-)$\Lambda$
hyperons at their kinetic freeze-out.

In the coalescence model Scheibl and Heinz (1999); Sun  _et al._ (2019) for
light cluster production, the formation probability of a hypertriton from a
proton, a neutron, and a $\Lambda$ hyperon is given by the Wigner function of
the hypertriton internal wave function, which we take as

$\displaystyle W_{\text{ht}}$ $\displaystyle=$ $\displaystyle
8^{2}g_{\text{ht}}e^{-\frac{\rho^{2}}{\sigma_{1}^{2}}-\sigma_{1}^{2}k_{\rho}^{2}}e^{-\frac{\lambda^{2}}{\sigma_{2}^{2}}-\sigma_{2}^{2}k_{\lambda}^{2}},$
(1)

where $g_{ht}=1/4$ is the statistical factor for spin 1/2 proton, neutron, and
$\Lambda$ hyperon to form a spin 1/2 hypertriton. The relative coordinates and
momenta in the above equation are defined by

$\displaystyle{\bf\rho}$ $\displaystyle=$ $\displaystyle\frac{{\bf r}_{1}-{\bf
r}_{2}}{\sqrt{2}},~{}{\bf\lambda}=\sqrt{\frac{2}{3}}\left(\frac{m_{1}{\bf
r}_{1}+m_{2}{\bf r}_{2}}{m_{1}+m_{2}}-{\bf r}_{3}\right),$ (2) $\displaystyle
k_{\rho}$ $\displaystyle=$ $\displaystyle\sqrt{2}\frac{({m_{2}\bf
k}_{1}-m_{1}{\bf k}_{2})}{m_{1}+m_{2}},$ (3) $\displaystyle k_{\lambda}$
$\displaystyle=$ $\displaystyle\sqrt{\frac{3}{2}}\frac{m_{3}({\bf r}_{1}+{\bf
r}_{2})-(m_{1}+m_{2}){\bf r}_{3}}{m_{1}+m_{2}+m_{3}}.$ (4)

Here, ${\bf r}_{1}$, ${\bf r}_{2}$, and ${\bf r}_{3}$ are the spatial
coordinates of neutron, proton, and $\Lambda$ hyperon, respectively at an
equal time in their rest frame, and they are determined by propagating the
constituent particles at earlier freeze-out time to the time of the last
freeze-out one. The size parameter $\sigma_{1}$ in the Wigner function is
related to the deuteron root-mean-squared radius $r_{{d}}$ by
$\sigma_{1}=\sqrt{4/3}~{}r_{d}\approx 2.26$ fm Sun and Chen (2017); Ropke
(2009). The size parameter $\sigma_{2}$ is related to the $\Lambda-d$ distance
by $\sigma_{2}=\frac{2}{3}\sqrt{\langle r_{\Lambda d}^{2}\rangle}$, and it can
be parameterized as
$\sigma_{2}\approx(2.15(B_{\Lambda}/\text{MeV})^{-1/2}+1.23)~{}$fm, with
details provided in the AppendixA. For the binding energies measured by the
STAR and ALICE collaboration, we obtain $\sigma_{2}=5.45$ fm and
$\sigma_{2}=7.96$ fm, respectively, while using the world averaged
$B_{\Lambda}$ value gives $\sigma_{2}=6.52$ fm. For helium-3 production in the
coalescence model, we use $\sigma_{1}=\sigma_{2}\approx 1.76$ fm Ropke (2009)
as it has a spherical shape.

Figure 1: Collision centrality dependence of ${}_{\Lambda}^{3}$H/3He yield
ratio in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV for different
$B_{\Lambda}$ values of 0.42 MeV (solid stars), 0.164 MeV (solid circles), and
0.102 MeV (solid triangles).

Figure 1 displays the centrality dependence of the yield ratio
${}^{3}_{\Lambda}\text{H}/^{3}\text{He}$ for different values of
$B_{\Lambda}$. The yield ratio ${}^{3}_{\Lambda}\text{H}/^{3}\text{He}$ is
seen to be suppressed in more peripheral collisions (30-50%) as compared to
central collisions (0-10%). This suppression is due to the decrease of the
emission source volume with increasing collision centrality. Also, the yield
ratio ${}^{3}_{\Lambda}\text{H}/^{3}\text{He}$ is suppressed for smaller
$\Lambda$-separation energies or larger $\Lambda-d$ distance. The suppression
of hypertriton production in non-central collisions is in accordance with
earlier studies on light cluster production in collisions of small systems
like $pp$ or $p$+Pb collisions at the LHC energies Sun  _et al._ (2019).

Figure 2: The yield ratio of hypertriton to helium-3 as a function of
transverse momentum $p_{T}$ in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV
for 0-10% (a), 10-30% (b), and 30-50% (c) centralities. Results from the
coalescence approach (MUSIC+UrQMD+COAL) and blast-wave model are denoted by
shaded regions and solid lines, respectively.

_Softening of hypertriton $p_{T}$ spectrum_— Figure 2 displays by shaded bands
the ratio ${}^{3}_{\Lambda}\text{H}/^{3}\text{He}$ as a function of transverse
momentum ($p_{T}$) from the coalescence model. It is observed that for all
three different values of $\Lambda$ separation energies and all centralities,
there is a pronounced decreasing trend in this ratio with an increase in
$p_{T}$. This trend is in sharp contrast with the behavior predicted by the
blast-wave (BW) model, shown by solid lines, which shows an increase of
${}^{3}_{\Lambda}\text{H}/^{3}\text{He}$ with increasing $p_{T}$. In the BW
model, both hypertriton and helium-3 are assumed in local thermal equilibrium
at their kinetic freeze-out with an invariant momentum spectrum
$f=E\text{d}^{3}N/\text{d}p^{3}$ given by Cooper and Frye (1974b);
Schnedermann  _et al._ (1993):

$\displaystyle f_{\text{BL}}\propto\int
m_{T}I_{0}\left(\frac{p_{T}\sinh(\rho)}{T_{\text{kin}}}\right)K_{1}\left(\frac{m_{T}\cosh(\rho)}{T_{\text{kin}}}\right)r\text{d}r.$
(5)

In the above, $I_{0}$ and $K_{1}$ are the modified Bessel functions with their
arguments depending on the transverse mass $m_{T}=\sqrt{m^{2}+p_{T}^{2}}$, the
kinetic freeze-out temperature $T_{\rm kin}$, and the flow rapidity profile
$\rho=\tanh^{-1}[(r/R_{0})^{n}\beta_{s}]$ with $r$ and $R_{0}$, $\beta_{s}$,
and $n$ being the radial distance in the transverse plane, the transverse
radius, the transverse expansion velocity at the surface, and the exponent of
the velocity profile, respectively. The blast-wave model can be viewed as a
simple approximation to the hydrodynamic model used in describing heavy ion
collisions. The results shown in Fig. 2 for the blast-wave model are obtained
by using the fitted kinetic freeze-out temperature ($T_{\text{kin}}$), flow
velocity ($\beta_{s}$) and exponent ($n$) in Ref. Acharya  _et al._ (2020c),
i.e., 0.090 GeV, 0.907, and 0.735 for 0-10% centrality, 0.094 GeV, 0.90, and
0.739 for 10-30% centrality, and 0.105 GeV, 0.88, and 0.828 for 30-50%
centrality. For large $m_{T}$, $f_{\text{BL}}$ takes the form
$f_{\text{BL}}\propto\text{exp}(-m_{T}/T_{\text{eff}})$ with
$T_{\text{eff}}=T_{\rm
kin}\sqrt{\frac{1+\langle\beta_{T}\rangle}{1-\langle\beta_{T}\rangle}}$ being
the effective (blue-shifted) temperature. The ratio of the hypertriton to the
helium-3 spectrum at large $m_{T}$ is then given by

$\displaystyle\frac{f_{\text{BL}}^{\text{ht}}}{f_{\text{BL}}^{\text{he3}}}\propto\text{exp}\left(-\frac{m^{2}_{\text{ht}}-m^{2}_{\text{he3}}}{(\sqrt{m^{2}_{\text{ht}}+p_{T}^{2}}+\sqrt{m^{2}_{\text{he3}}+p_{T}^{2}})T_{\text{eff}}}\right),$
(6)

which increases as a function of $p_{T}$ because the transverse flow drives
the heavier hypertriton to higher momentum than the lighter helium-3, and this
explains the increasing trend of the blast-wave model results (solid lines)
shown in Fig. 2.

Figure 3: Ratio $\langle p_{T}\rangle_{{}^{3}_{\Lambda}\text{H}}/\langle
p_{T}\rangle_{{}^{3}\text{He}}$ of hypertriton to helium-3 as a function of
$\Lambda-d$ distance in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV for 0-10%
(a), 10-30% (b), and 30-50% (c) centralities. The solid lines denote the
prediction of the blast-wave model, whereas the shaded bands denote the
prediction from the coalescence model. The solid symbols correspond to results
obtained for various values of $\Lambda$ separation $B_{\Lambda}$.

_Suppression of hypertriton mean transverse momentum_ — Figure 3 further shows
the ratio ($\langle p_{T}\rangle_{{}^{3}_{\Lambda}\text{H}}/\langle
p_{T}\rangle_{{}^{3}\text{He}}$) of hypertriton to helium-3 mean-$p_{T}$ as a
function of $\Lambda-d$ distance. The solid lines denote the prediction of the
blast-wave model, whereas the shaded bands denote the prediction from the
coalescence model. The solid symbols correspond to results obtained with
various values of $B_{\Lambda}$. According to the blast-wave model prediction,
the value of the hypertriton to helium-3 mean-$p_{T}$ ratio has a slightly
larger than unity value of about 1.03-1.04. The coalescence model predicts, on
the other hand, a value for this ratio that is substantially smaller than one
for all three values of $B_{\Lambda}$, and it decreases with increasing
$\Lambda-d$ distance or decreasing $B_{\Lambda}$. This trend of a smaller mean
$p_{T}$ for hypertriton than that for helium-3 is seen for all collision
centralities. The centrality dependence of the softening effect shown in Fig.
3 is, however, rather weak, and this may be attributed to the reason discussed
qualitatively below.

Within the coalescence model, the ratio of hypertriton spectrum to helium-3
spectrum is approximately given by Scheibl and Heinz (1999); Bellini and
Kalweit (2019); Blum and Takimoto (2019)

$\displaystyle\frac{f_{\text{COAL}}^{\text{ht}}}{f_{\text{COAL}}^{\text{he3}}}\approx\frac{f_{\text{BL}}^{\text{ht}}}{f_{\text{BL}}^{\text{he3}}}\times\frac{[1+\frac{\sigma_{\text{he3}}^{2}}{2R^{2}(p_{T})}]^{3}}{[1+\frac{\sigma_{1}^{2}}{2R^{2}(p_{T})}]^{3/2}[1+\frac{\sigma_{2}^{2}}{2R^{2}(p_{T})}]^{3/2}},$
(7)

where the last term (denoted by $\mathcal{C}$ hereafter) in the above equation
is due to the quantum mechanical corrections Scheibl and Heinz (1999) from the
hypertriton wave function. Because the inhomogeneity size $R(p_{T})$ decreases
with increasing $p_{T}$ Adam  _et al._ (2015b) and
$\sigma_{2}\gg\sigma_{\text{he3}}\sim\sigma_{1}$, the quantum correction
factor $\mathcal{C}$ leads to a suppression the ${}^{3}_{\Lambda}\text{H}$
yield at larger $p_{T}$. For a large emission source with $R\geq\sigma_{2}$,
the correction factor can be expressed as

$\displaystyle\mathcal{C}\approx
1-\frac{3(\sigma_{2}^{2}+\sigma_{1}^{2}-2\sigma_{\text{he3}}^{2})}{4R(p_{T})^{2}},$
(8)

which makes the ratio in Eq. (7) becoming less than one when
$\sigma_{2}\gtrsim 4$ fm, resulting in a large quantum correction and thus a
decreasing ${}^{3}_{\Lambda}\text{H}/^{3}\text{He}$ ratio with increasing
$p_{T}$ shown in Fig. 2 at all collision centralities. The $p_{T}$ dependence
of the quantum correction factor $\mathcal{C}$ has a different behavior for a
small emission source with $R\ll\sigma_{\text{he3}}$. In this case, it can be
simplified as

$\displaystyle\mathcal{C}\approx\frac{\sigma_{\text{he3}}^{6}}{\sigma_{1}^{6}\sigma_{2}^{6}}\times\left[1+3R^{2}(p_{T})\left(\frac{2}{\sigma_{\text{he3}}^{2}}-\frac{1}{\sigma_{1}^{2}}-\frac{1}{\sigma_{2}^{2}}\right)\right],$
(9)

which shows a weaker $p_{T}$ dependence than in the case of a large emission
source because of $\sigma_{\text{he3}}/R\ll 1$ as assumed. This effect due to
the relative hypertrition and source sizes indicates that the softening of
hypertriton transverse momentum is weaker in collisions of small systems like
$pp$ and $pA$ collisions.

_Conclusion and outlook._ — In the present study, we have investigated
hypertriton production in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV using
the coalescence model with kinetic freeze-out nucleons and $\Lambda$ hyperons
from a microscopic hybrid approach based on the MUSIC hydrodynamic model and
the UrQMD hadronic transport model. We have found that the halo structure of
hypertriton with a large $\Lambda-d$ distance of approximately $10$ fm leads
to not only a suppression of ${}^{3}_{\Lambda}\text{H}$ yield but also a
softening of its transverse momentum ($p_{T}$) spectrum with a weak centrality
dependence. In particular, the mean $p_{T}$ of ${}^{3}_{\Lambda}\text{H}$ is
found to be significantly smaller than that of helium-3 even in the most
central collisions, which is in sharp contrast with the predictions of the
blast-wave model. Such a quantum mechanical softening on (anti-)hypertriton
spectrum is a general feature and a natural outcome of the hypertriton wave
function used in the final-state coalescence model, which can be readily
tested in high-energy experiments with different beam energies and collision
systems, providing thus a good opportunity to unravel the production mechanism
of (anti-)hypernuclei in high-energy nuclear collisions and also to obtain
constraints on the $\Lambda$-nucleon interaction.

_Acknowledgments._ — We thank Song Zhang and Bo Zhou for fruitful discussions
and Chen Zhong for the maintenance of the CPU and GPU clusters. This work was
supported in part by the National Natural Science Foundation of China (Grants
No. 12375121, No. 11891070, No. 11890714, No. 12147101, No. 12322508), the
National Key Research and Development Project of China under Grant No.
2022YFA1602303, the STCSM (Grant No. 23590780100), the 111 Project, and the
U.S. Department of Energy under Award No. DE-SC0015266.

Figure 4: The $B_{\Lambda}$ dependence of $\Lambda$-d distance ($\sqrt{\langle
r^{2}_{\Lambda d}\rangle}$) of hypertriton. The solid squares denote
theoretical results taken from Ref. Cobis  _et al._ (1997).

##  Appendix A APPENDIX

_$\Lambda$ separation energy and $\Lambda-d$ distance._— Figure 4 depicts the
$\Lambda-d$ distance $\sqrt{\langle r^{2}_{\Lambda d}\rangle}$ of hypertriton
as a function of $\Lambda$ separation energy $B_{\Lambda}$. The symbols in
Fig. 4 are theoretical results taken from Ref. Cobis  _et al._ (1997). Because
the hypertriton is a loosely bound state of a deuteron and a $\Lambda$
hyperon, one has the relation (black dashed line of Fig. 4)

$\sqrt{\langle r_{\Lambda
d}^{2}\rangle}\approx\frac{197.33~{}\text{MeV}\cdot\text{fm}}{\sqrt{4\mu
B_{\Lambda}}},$ (10)

where $\mu=m_{\Lambda}m_{d}/(m_{\Lambda}+m_{d})=700$ MeV is the reduced mass.
Instead of Eq. (10), we use the following parametrization for the $\Lambda-d$
distance (red solid line in Fig. 4),

${\displaystyle\sqrt{\langle r_{\Lambda
d}^{2}\rangle}\approx\frac{a}{\sqrt{B_{\Lambda}/\text{MeV}}}+b,}$ (11)

with parameters $a=3.226\pm 0.062$ fm and $b=1.836\pm 0.173$ fm. For
$B_{\Lambda}$=0.42 MeV (STAR), 0.164 MeV (world average), and 0.102 MeV
(ALICE), the corresponding values of $\sqrt{\langle r_{\Lambda d}^{2}\rangle}$
are 6.8 fm, 9.5 fm, and 11.9 fm, respectively. The size parameter $\sigma_{2}$
in the coalescence model is related to the $\Lambda-d$ distance by
$\sigma_{2}=\frac{2}{3}\sqrt{\langle r_{\Lambda d}^{2}\rangle}$, and the
values of $\sigma_{2}$ are 5.45 fm, 6.52 fm, and 7.96 fm, respectively, for
the above three values of $B_{\Lambda}$.

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