††thanks:
Kai-Jia Sun and Rui Wang contributed equally to this work.

# Relativistic kinetic approach to light nuclei production in high-energy
nuclear collisions

Kai-Jia Sun<EMAIL_ADDRESS>Cyclotron Institute and Department of Physics and
Astronomy, Texas A&M University, College Station, Texas 77843, USA  Rui Wang
<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Institute of Modern Physics, Fudan University, Shanghai $200433$, China
Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai
$201800$, 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 Institute of Applied Physics,
Chinese Academy of Sciences, Shanghai $201800$, China  Chun Shen
<EMAIL_ADDRESS>Department of Physics and Astronomy, Wayne State
University, Detroit, MI 48201, USA  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, NY 11973, USA

###### Abstract

Understanding the production mechanism of light (anti-)nuclei in high-energy
nuclear collisions and cosmic rays has been a long-standing problem in nuclear
physics. In the present study, we develop a stochastic method to solve the
relativistic kinetic equations for light nuclei production from many-body
reactions with the inclusion of their finite sizes. The present approach gives
an excellent description of the deuteron and helium-3 data from central Au+Au
(Pb+Pb) collisions at $\sqrt{s_{\rm NN}}$ $=$ $200~{}\rm GeV$ ($2.76~{}\rm
TeV$). It can also naturally explain their suppressed production in $pp$
collisions at 7 TeV as a result of their finite sizes.

###### pacs:

12.38.Mh, 5.75.Ld, 25.75.-q, 24.10.Lx

##  I Introduction

Besides ordinary hadrons in the Standard Model, light nuclei, such as deuteron
($d$), helium-3 (${}^{3}\text{He}$), triton ($t$), helium-4
(${}^{4}\text{He}$), hypertriton (${}^{3}_{\Lambda}\text{H}$) and their
antiparticles, have also been observed in high energy nucleus-nucleus ($AA$),
proton-nucleus ($pA$), and $pp$ collisions at the Relativistic Heavy Ion
Collider (RHIC) and the Large Hadron Collider (LHC) [1, 2, 3]. Although rarely
produced, these weakly bound nuclei carry important information on the space-
time geometry [4], entropy production [5], and the QCD phase transitions [6,
7, 8, 9] in these collisions. They can also provide indirect information on
the dark matter interaction in cosmic rays due to their ultra-low
astrophysical background [10, 11, 12].

Different mechanisms have been proposed to describe light nuclei production in
nuclear reactions. For heavy-ion collisions at high energies, the created
fireball evolves through stages of pre-equilibrium, the Quark-Gluon Plasma
(QGP) or partonic expansion, the hadronization, and subsequent hadronic
scatterings and decays until kinetic freeze-out. In the statistical
hadronization model [13], light nuclei are thermally produced during the
hadronization of QGP and their yields are assumed to remain unchanged during
the hadronic evolution. In the coalescence model [14, 15], light nuclei are
produced from nucleons close in phase space at the kinetic freeze-out. Since
the cross section for the deuteron dissociation by pion reaction $\pi
d\rightarrow\pi NN$ is about $100~{}\rm{mb}$ [16], the rate for the inverse
process $\pi NN\rightarrow\pi d$ in hadronic matter is also large due to the
time-reversal symmetry. Therefore, to understand light nuclei production in
high-energy nuclear collisions, it is essential to include their formation and
dissociation dynamics in these collisions. The process $\pi
d\leftrightarrow\pi NN$ has recently been studied in Ref. [17] by
approximating it as two-step processes of $NN\leftrightarrow d^{\prime}$ and
$\pi d^{\prime}\leftrightarrow\pi d$ with $d^{\prime}$ being a fictitious
dibaryon resonance. For nuclear reactions at low energies, the process
$NNN\leftrightarrow Nd$ is more important because the produced matter consists
mainly of baryons [18]. In all previous transport model studies [18, 19, 20,
17], light nuclei have been considered as point particles. This is a valid
assumption in large collision systems, but no longer holds in small systems
like $pp$ collisions, in which the size of produced matter is comparable to or
even smaller than those of light nuclei such as the $d$ and
${}^{3}_{\Lambda}\text{H}$ [21, 22]. Other proposed mechanisms for light
nuclei production in high-energy nuclear collisions include the decay of a
compact quark droplet [23] or pre-formed nuclear clusters [24]. In spite of
these efforts, the production mechanism of light (anti-)(hyper-)nuclei in
nuclear collisions has not yet been unambiguously understood [25, 26, 27].

In the present article, we study the dynamics of light nuclei in nuclear
collisions by the relativistic kinetic equations that were derived from the
time evolution of the system’s Green’s functions defined on a closed path in
time [28, 29, 18, 30]. In this approach, the effect of finite sizes of light
nuclei, which is usually neglected, can be included by retaining the relative
coordinates between nucleons in the gradient expansion of the Green’s function
[28, 29]. A central quantity in the relativistic kinetic equations is the
many-body transition matrix for light nuclei production and dissociation,
which will be evaluated here by using the impulse approximation. To illustrate
the validity of the impulse approximation, we take $\pi d\to\pi NN$ as an
example. Since the typical temperature of the hadronic matter in high-energy
nuclear collisions is 100-150 MeV, the pion thermal wavelength is then around
0.4-0.5 fm and is much smaller than the deuteron size of about 3 fm. A pion
thus has a sufficiently large momentum to resolve the two constituent nucleons
in the deuteron, resulting its scattering by one nucleon with the other
nucleon being quasifree, as shown in Fig. 1. Therefore, the transition
amplitude $\mathcal{M}_{\pi d\rightarrow\pi NN}$ for this reaction can be
approximately factorized and is proportional to
$\langle\tilde{p}_{N}|\phi_{d}\rangle\mathcal{M}_{\pi N\rightarrow\pi N}$,
with $\langle\tilde{p}_{N}|\phi_{d}\rangle$ and $\mathcal{M}_{\pi
N\rightarrow\pi N}$ being the deuteron wave function and the $\pi$-$N$ elastic
scattering amplitude, respectively. The impulse or quasifree approximation was
previously used in studying deuteron dissociation in low-energy heavy-ion
collisions [31, 18] and also $J/\Psi$ dissociation by partons in relativistic
heavy-ion collisions [32]. Under this approximation, the inverse reaction $\pi
NN\to\pi d$ can be viewed as a superposition of two subprocesses, the
scattering between a pion and a nucleon with the final-state nucleon sightly
off mass shell, and followed by the fusion or coalescence of the off-shell
nucleon and another nucleon to form a deuteron. The impulse approximation can
be generalized straightforwardly to study the production and dissociation of
heavier nuclei that involve more than two nucleons.

Figure 1:  Diagrams for the reaction $\pi^{+}d\leftrightarrow\pi^{+}np$ in the
impulse approximation. Inside the bubble is a $\Delta$ resonance as an
intermediate state.

##  II Relativistic kinetic equation

We start by considering one specific channel
$\pi^{+}d\leftrightarrow\pi^{+}np$. Its effect on deuteron production and
dissociation in nuclear collisions can be described by the relativistic
kinetic equation [29, 18],

$\displaystyle\frac{\partial f_{d}}{\partial t_{d}}+\frac{{\bf P}}{\
{E_{d}}}\cdot\frac{\partial f_{d}}{\partial{\bf
R}}=-\mathcal{K}^{>}f_{d}+\mathcal{K}^{<}(1+f_{d}).$ (1)

where the deuteron distribution function $f_{d}({\bf R},{\bf P})$ is
normalized as $g_{d}(2\pi)^{-3}\int\text{d}^{3}{\bf R}\text{d}^{3}{\bf
P}f_{d}=N_{d}$ with $g_{d}=3$ and $N_{d}$ being the deuteron statistical spin
factor and total yield, respectively.

On the l.h.s. of Eq. (1), which denotes the drift term, we have treated
deuteron as a free particle by neglecting its interaction with medium. The two
terms on the r.h.s. of Eq. (1) describes the deuteron dissociation and
production rates, respectively. Under the impulse approximation, the collision
integral on r.h.s. is given by [18, 30]

$\displaystyle\frac{1}{2g_{d}E_{d}}\int\prod_{i=1^{\prime},3^{\prime}}\frac{\text{d}^{3}{\bf
p}_{i}}{(2\pi)^{3}2E_{i}}\frac{\text{d}^{3}{\bf
p}_{\pi}}{(2\pi)^{3}2E_{\pi}}\frac{E_{d}\text{d}^{3}{\bf r}}{m_{d}}$
$\displaystyle\times{2m_{d}W_{d}(\tilde{\bf r},\tilde{\bf
p})\big{(}\overline{|\mathcal{M}_{\pi^{+}n\rightarrow\pi^{+}n}|^{2}}+n\leftrightarrow
p\big{)}}$
$\displaystyle\times\Big{[}-\big{(}\prod_{i=1^{\prime}}^{3^{\prime}}(1\pm
f_{i})\big{)}g_{\pi}f_{\pi}g_{d}f_{d}+\frac{3}{4}\big{(}\prod_{i=1^{\prime}}^{3^{\prime}}g_{i}f_{i}\big{)}$
$\displaystyle~{}~{}~{}~{}\times(1+f_{\pi})(1+f_{d})\Big{]}\times(2\pi)^{4}\delta^{4}(p_{\text{in}}-p_{\text{out}}),$
(2)

from which ${\mathcal{K}}^{>}$ and ${\mathcal{K}}^{<}$ in Eq. (1) can be
identified. In the above equation, the $1\pm f_{i}$ in the square bracket come
from the quantum statistics of fermions ($-$) and bosons ($+$), and the
$\delta$-function denotes the conservation of energy and momentum with
$p_{\text{in}}$ $=$ $\sum_{i=1^{\prime}}^{3^{\prime}}p_{i}$ and $p_{\rm out}$
$=$ $p_{\pi}+p_{d}$. The factor $3/4$ in the third line comes from the spin
factors of initial and final states. The second line of Eq. (2) denotes the
spin-averaged squared amplitude at a relative distance ${\bf r}$ between the
two nucleons. The $W_{d}$ denotes the deuteron Wigner function, with
$\tilde{\bf r}$ $=$ $\tilde{\bf r}_{n}-\tilde{\bf r}_{p}$ and $\tilde{\bf p}$
$=$ $(\tilde{\bf p}_{n}-\tilde{\bf p}_{p})/2$ being the relative coordinate
and momentum in deuteron center-of-mass frame, respectively. For simplicity,
we take $W_{d}$ $=$ $8e^{-\tilde{\bf r}^{2}/\sigma_{d}^{2}-\tilde{\bf
p}^{2}\sigma_{d}^{2}}$ with $\sigma_{d}$ $=$ $3.2~{}\rm fm$ to reproduce the
deuteron root-mean-square radius of $1.96~{}\rm fm$ [14].

For an approximately uniform system, the spatial part of $W_{d}$ in Eq. (2)
can be integrated out, i.e., $|\phi_{d}(\tilde{\bf p})|^{2}$ $=$
$\int\text{d}^{3}{\bf r}\gamma_{d}W_{d}$ $=$
$(4\pi\sigma_{d}^{2})^{3/2}e^{-\tilde{\bf p}^{2}\sigma_{d}^{2}}$ with
$\gamma_{d}$ $=$ $E_{d}/m_{d}$, simplifying the second line of Eq. (2) to
$2m_{d}|\phi_{d}|^{2}\big{(}\overline{|\mathcal{M}_{\pi^{+}n\rightarrow\pi^{+}n}|^{2}}+n\rightarrow
p\big{)}$, which is the usual impulse approximation for
$\overline{|\mathcal{M}_{\pi^{+}d\rightarrow\pi^{+}np}|^{2}}$, e.g. adopted in
Ref. [18] by treating deuteron as a point particle.

The spin averaged squared pion-nucleon scattering matrix element
$\overline{|\mathcal{M}_{\pi^{+}n\rightarrow\pi^{+}n}|^{2}}$ can be related to
the $\pi N$ scattering cross section. Under the impulse approximation, the
deuteron dissociation cross section by a pion of momentum $p_{\rm lab}$ is
approximately given by $\sigma_{\pi^{+}d\rightarrow\pi^{+}np}$ $\approx$
$\sigma_{\pi^{+}n\rightarrow\pi^{+}n}$ $+$
$\sigma_{\pi^{+}p\rightarrow\pi^{+}p}$. This relation becomes exact only for
extremely large $p_{\text{lab}}$ [18, 16]. At low $p_{\text{lab}}$, e.g.
$0.3~{}\rm GeV$, one can introduce a renormaliztion factor $F_{d}$ [18] such
that $\sigma_{\pi^{+}d\rightarrow\pi^{+}np}$ $=$
$F_{d}(\sigma_{\pi^{+}n\rightarrow\pi^{+}n}+\sigma_{\pi^{+}p\rightarrow\pi^{+}p})$.
As shown in Fig. 2, using the constant values $F_{d}$ $\approx$ $0.72$ and
$F_{\isotope[3]{He}}$ $\approx$ $0.51$ leads to an excellent description of
the data for the $\pi$ $+$ $d$ and $\pi$ $+$ $\isotope[3]{He}$ cross sections
in the energy region relevant for present study.

Given the above deuteron dissociation and production cross sections with a
pion, the probability for the reaction $\pi^{+}d$ $\rightarrow$ $\pi^{+}np$
between a pion and a deuteron inside a volume $\Delta V$ to take place within
a time interval $\Delta t$ can be obtained as

$\displaystyle P_{23}\big{|}_{\text{IA}}\approx
F_{d}{v}_{12}\sigma_{\pi^{+}p\rightarrow\pi^{+}p}\frac{\Delta
t}{{\Delta}V}+(2\leftrightarrow 3),$ (3)

where ${{v_{12}}}$ is the relative velocity between the pion and one of the
two nucleons inside deuteron, and the two terms on r.h.s correspond to the two
diagrams in Fig. 1. Similarly, the probability for the reaction
$\pi^{+}np\rightarrow\pi^{+}d$ is

$\displaystyle
P_{32}\big{|}_{\text{IA}}\approx\frac{3}{4}F_{d}{v}_{1^{\prime}2^{\prime}}\sigma_{\pi^{+}p\rightarrow\pi^{+}p}\frac{\Delta
t}{\Delta V}W_{d}+(2^{\prime}\leftrightarrow 3^{\prime}).$ (4)

In obtaining Eqs. (3) and (4), we have neglected the factors $1\pm f_{i}$ in
Eq. (2), which is a good approximation in high-energy nuclear collisions. Note
that the deuteron Wigner function $W_{d}$ in Eq. (4) depends on both the
coordinates and momenta of the constituent nucleons. It should be replaced by
$|\phi_{d}|^{2}/(\gamma_{d}\Delta V)$ if the deuteron is treated as a point
particle.

Figure 2:  Comparison of the pion incident momentum ($p_{\text{lab}}$)
dependence of the cross sections for the reactions $\pi+d$ (upper panel) and
$\pi+^{3}$He (lower panel) between experimental data [33, 34, 35, 36, 37, 38,
16] and our theoretical fit based on the impulsive approximation (colored
lines).

In the limit that the production and dissociation rates are equal in Eq. (2),
one obtains the equilibrated deuteron abundance as

$\displaystyle N_{d}\approx\frac{3}{4}\int d\Gamma_{np}g_{n}f_{n}({\bf
r}_{n},{\bf p^{*}}_{n})g_{p}f_{p}({\bf r}_{p},{\bf p}_{p})W_{d}(\tilde{\bf
r},\tilde{\bf p}),$ (5)

where $d\Gamma_{np}$ $=$ $(2\pi)^{-6}\text{d}^{3}{\bf r}_{n}\text{d}^{3}{\bf
p}_{n}\text{d}^{3}{\bf r}_{p}\text{d}^{3}{\bf p}_{p}$ and the neutron is
chosen to be off mass shell to conserve energy [14]. The equilibrium solution
given by Eq. (5) is similar to the result from the phase-space coalescence
model based on the sudden approximation [39]. Note that the two nucleons in
Eq. (5) can be $nn$, $np$, or $pp$ pair, while only the $np$ pair is
considered in the coalescence model. For a uniform and thermalized system, Eq.
(5) can be further simplified to $N_{d}$ $\approx$
$g_{d}V(2\pi)^{-3}\int\text{d}^{3}{\bf p}_{d}f_{n}(\frac{{\bf
p}_{d}}{2})f_{p}(\frac{{\bf p}_{d}}{2})$, which is the same as the deuteron
abundance in chemical equilibrium with nucleons, indicating that our approach
has the correct thermal limit.

##  III Numerical implementation and validation

To solve Eq. (1) with the collision integral given by Eq. (2), we adopt the
test particle ansatz [40] (or the particle in cell method), i.e., mimicking
the distribution function $f_{\alpha}$ of a certain particle species of number
$N_{\alpha}$ by a large number of delta functions, $f_{\alpha}({\bf r},{\bf
p})$ $\approx$ $\sum^{N_{\alpha}N_{test}}_{i=1}\delta({\bf r}_{i}-{\bf
r})\delta({\bf p}_{i}-{\bf p})$. Using test particles modifies Eq. (3) and Eq.
(4) by the multiplication of factors $1/N_{\text{test}}$ and
$1/N_{\text{test}}^{2}$, respectively. The number of test particles
$N_{\text{test}}$ should be sufficiently large to ensure the convergence of
the numerical results. To illustrate the numerical algorithm used in the
present study, we take the reaction $\pi d\rightarrow\pi NN$ as an example. In
each time step $\Delta t$, a (test) pion and a (test) deuteron are randomly
chosen from a cell of volume ${\Delta}V$, and the momenta of two nucleons
inside the deuteron is then sampled according to its wave function with one
nucleon slightly off mass shell to conserve energy and momentum. After the
scattering of the pion with the off-shell nucleon, both final state nucleon
and pion are taken to be on mass shell. A similar consideration applies to the
inverse reaction $\pi NN\rightarrow\pi d$. One first randomly chooses a (test)
pion and a (test) nucleon during a time step $\Delta t$ from a cell of volume
${\Delta}V$. After letting the pion to scatter with the nucleon, their final
momenta are then sampled according to the differential cross section but with
the nucleon to be off-shell, which subsequently coalesces with the other
nearby nucleon to form a deuteron. Since we employ the stochastic method [18,
41, 42], the probabilities of these two processes are directly evaluated
according to Eq. (3) and Eq. (4), respectively.

As for triton, it can be formed from the two reactions $\pi
NNN\leftrightarrow\pi t$ and $\pi dN\leftrightarrow\pi t$. The
$3\leftrightarrow 2$ process can be similarly treated as for deuteron
production. The probability of the $4\leftrightarrow 2$ process to occur in a
volume $\Delta V$ during a time step $\Delta t$ is given by

$\displaystyle P_{42}\big{|}_{\text{IA}}$ $\displaystyle\approx$
$\displaystyle\frac{1}{4}F_{t}\frac{{v}_{\pi N}\sigma_{\pi N\to\pi N}\Delta
t}{N_{\text{test}}^{3}\Delta V}W_{t},$ (6)

where $F_{t}\approx F_{\isotope[3]{He}}\approx 0.51$ and the $W_{t}$ is the
triton Wigner function, and its expression can be found in Refs. [14, 43]. The
branching ratio for the dissociation of triton via the two inverse reactions
can be estimated by calculating the deuteron component in the triton wave
function. If we take the wave function to have the Gaussian form, we have the
branching ratio $\mathcal{B}(\pi t\rightarrow\pi Nd)\approx 0.56$. More
realistic Hulthen wave function for the deuteron leads to a similar branching
ratio $\mathcal{B}(\pi t\rightarrow\pi Nd)\approx 0.58$ Similarly, the
branching ratio for 3He dissociation to a deuteron is
$\mathcal{B}(\pi~{}^{3}\text{He}\rightarrow\pi Nd)\approx 0.6$. We note the
following numerical results are insensitive to the value of the branching
ratios in the production and dissociation of triton and helium-3.

Figure 3: Panel (a) Box calculations for deuteron and triton (helium-3).
Transverse momentum spectra of $p$, $d$, and 3He ($t$) in (b) central Au+Au
collisions at $\sqrt{s_{\rm NN}}=200~{}\rm GeV$; (c) central Pb+Pb collisions
at $\sqrt{s_{\rm NN}}=2.76~{}\rm TeV$; (d) $pp$ collisions at $\sqrt{s_{\rm
NN}}=7~{}\rm TeV$. The experimental data points are taken from Refs. [44, 45,
46, 47, 48, 49, 50]. Theoretical predictions from the relativistic kinetic
equations (RKE) are denoted by colored lines. The spectra in panel (d) are
multiplied by a factor of 103.

To validate the above described algorithm, we consider deuteron and triton
production from a hadronic matter in a (20 fm)3 box with periodic boundary
conditions. The box consists of 24 protons, 24 neutrons, and 480 pions for
each of its three charge states. The initial distribution of these particles
is taken to be uniform in the coordinate space and to have a thermal Boltzmann
distribution with temperature $T=155~{}\rm MeV$ in the momentum space. Fig. 3
(a) shows the time evolution of the yields of proton, deuteron, and triton
(helium-3). It is seen that their final numbers are consistent with their
thermal values at chemical equilibrium over 5 orders of magnitude.

##  IV Light nuclei production in high-energy nuclear collisions

We now apply the above described kinetic approach to different collision
systems, i.e., central Au+Au collisions at $\sqrt{s_{\rm NN}}=200~{}\rm GeV$,
Pb+Pb collisions at $\sqrt{s_{\rm NN}}=2.76~{}\rm TeV$, and $pp$ collisions at
$7~{}\rm TeV$. For the evolution of the QGP produced in central Au+Au and
Pb+Pb collisions, we use the (3+1)-d viscous hydrodynamic model MUSIC [51, 52,
53, 54] with initial conditions taken from a newly developed dynamical Glauber
model [52, 55]. The QGP has a smooth crossover to a hadronic matter along a
phase transition line $T(\mu_{B})$ described in Ref. [56]. The initial hadrons
are sampled on a constant energy density particalization hypersurface
according to the Cooper-Frye formula [57]. For $pp$ collisions, the initial
hadron distribution is generated from a multiphase transport (AMPT) model
[58]. In the present article, we also assume that light nuclei are initially
produced from hadronization of QGP, and their yields are estimated from the
canonical statistical model [59] that conserves baryon, electric and strange
charges.

For the evolution of the hadronic matter and the production of light nuclei,
besides the production and dissociation of deuteron and helium-3 by pions,
most scattering channels in AMPT [58] have been included. These channels
describe properly the dynamics of pions and nucleons that are relevant for
light nuclei production from the reactions discussed above.

Fig. 3 (b) and (c) show the comparison of the transverse momentum ($p_{T})$
spectra of $p$, $d$, and 3He between our predictions and the experiment data
[44, 45, 46, 47, 48] in central Au+Au (Pb+Pb) collisions at $\sqrt{s_{\rm
NN}}=200~{}\rm GeV$ (2.76 TeV). The solid lines denote predictions from
relativistic kinetic equations with the inclusion of the finite sizes of $d$
and $\isotope[3]{He}$, and they describe the data very well. We have also
studied the case when treating light nuclei as point particles, and the
results remain in agreement with experiment data within uncertainties. This is
not surprising as the sizes of the produced matter in these collisions are
over $10~{}\rm fm$ and are much larger than those of $d$ and
$\isotope[3]{He}$. Besides, we find the final yields of $d$ and
$\isotope[3]{He}$ are insensitive to their initial values at QGP
hadronization, similar to that found in Refs. [20, 17] for the deuteron. The
information of initial light nuclei is largely lost during the hadronic
evolution which lasts for more than $15~{}{\rm fm}/c$.

The situation, however, changes dramatically in $pp$ collisions at
$\sqrt{s_{\rm NN}}=7~{}\rm TeV$ as shown in Fig. 3 (d). For the initial $d$
and $\isotope[3]{He}$ generated from hadronization, there are uncertainties of
the canonical effects [59] for $pp$ collisions, which are absent in large
systems. The initial $d$ and $\isotope[3]{He}$ in central rapidity ranges
approximately from $1.3\times 10^{-4}$ to $5.3\times 10^{-4}$, and from
$7.3\times 10^{-8}$ to $1.3\times 10^{-6}$, respectively. The lower values
correspond to an aggressive estimation of the canonical effects, while the
upper ones are obtained by neglecting the canonical effects. The uncertainty
in our theoretical predictions on the $p_{T}$ spectra of $d$ and [3]He, shown
in Fig. 3 (d) by the shaded bands, reflects the uncertainty in the initial $d$
and $\isotope[3]{He}$ yields. For the case of treating light nuclei as point
particles, we find the spectra of $d$ and [3]He are significantly
overestimated, regardless of the initial yields of $d$ and [3]He. After taking
into account of their sizes, both spectra are reduced, and they now agree with
the experimental data. Unlike in central Au+Au and Pb+Pb collisions, we find
the final yields of $d$ and [3]He in $pp$ collisions to be sensitive to their
initial values after their sizes are taken into account. This is mainly
because the reaction rate is not large enough to achieve chemical equilibrium
among nucleons, deuterons and helium-3 during the very short collision time of
merely a few fm/$c$. As a result, both the hadronization mechanism and the
many-body hadronic reactions play important roles in light (anti-)nuclei
production in high-energy $pp$ collisions. This finding will also be relevant
for the indirect search of dark matter from studying light (anti-)nuclei in
cosmic rays.

##  V Summary

We have developed a stochastic method to solve the relativistic kinetic
equations for light nuclei production from many-body reactions in the hadronic
matter produced in high-energy nuclear collisions. With the initial hadron
distributions generated from the MUSIC hydrodynamic model or the AMPT model,
we have found that the $d$ and 3He $p_{T}$ spectra measured in central Au+Au
(Pb+Pb) collisions at $\sqrt{s_{\rm NN}}=200~{}\rm GeV$ ($2.76~{}\rm TeV$) and
in $pp$ collisions at $7~{}\rm TeV$ are well described by the present
approach. In particular, the inclusion of light nuclei sizes, which has been
neglected in previous transport studies, is essential for describing their
suppressed production in $pp$ collisions. The present kinetic approach has
thus significantly advanced our understanding of light nuclei production in
high-energy nuclear collisions. This is different from the usual statistical
hadronization model and the nucleon phase-space coalescence model, which
correspond to the limits of our dynamic model if the final hadronic matter in
high-energy nuclear collisions is a uniform and thermalized system.

The present study can be extended straightforwardly to investigate the
production of other light nuclei like $\isotope[4]{He}$ ([4]Li),
${}^{3}_{\Lambda}$H, ${}^{4}_{\Lambda}$H, and exotic states like $X$(3872)
[60] and the possible $\Omega$-dibaryon [61, 62, 63] in various collision
systems, such as the $e\bar{e}$, $pp$, $pA$ and $AA$ collisions. With the
upcoming high-quality data of light nuclei from RHIC and LHC, the present
kinetic approach will further help shed light on more fundamental questions in
physics, such as cosmic-ray dark matter detection and the QCD phase structure
in relativistic heavy-ion collisions.

###### Acknowledgements.

_Acknowledgments_ —We thank Lie-Wen Chen, Zi-Wei Lin, Feng Li, Zhen Zhang, and
Xiao-Jian Du for helpful discussions, and Chen Zhong for setting up and
maintaining the GPU server. This work was supported in part by the U.S.
Department of Energy under Award No. DE-SC0015266, No. DE-SC0013460, the Welch
Foundation under Grant No. A-1358, the National Science Foundation (NSF) under
grant number PHY-2012922, National Natural Science Foundation of China under
contract No. 11891070 and No. 11890714, and the Guangdong Major Project of
Basic and Applied Basic Research No. 2020B0301030008.

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