Open AccessArticle

Production of Strange and Charm Hadrons in Pb+Pb Collisions at $\sqrt{s_{N N}}$ = 5.02 TeV^†

Wen-Bin Chang

^1,2,

Rui-Qin Wang

Jun Song

Feng-Lan Shao

^1,*,

Qun Wang

^4,*

and

Zuo-Tang Liang

^5,*

School of Physics and Physical Engineering, Qufu Normal University, Qufu 273165, China

Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOS), Central China Normal University, Wuhan 430079, China

School of Physical Science and Intelligent Engineering, Jining University, Qufu 273155, China

⁴

Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China

⁵

Key Laboratory of Particle Physics and Particle Irradiation (MOE), Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao 266237, China

Authors to whom correspondence should be addressed.

^†

This article is dedicated to the memory of Qu-bing Xie.

Symmetry 2023, 15(2), 400; https://doi.org/10.3390/sym15020400

Submission received: 23 December 2022 / Revised: 21 January 2023 / Accepted: 30 January 2023 / Published: 2 February 2023

(This article belongs to the Special Issue Heavy-Ion Collisions and Multiparticle Production)

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Abstract

Using a quark combination model with the equal-velocity combination approximation, we study the production of hadrons with strangeness and charm flavor quantum numbers in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. We present analytical expressions and numerical results for these hadrons’ transverse momentum spectra and yield ratios. Our numerical results agree well with the experimental data available. The features of strange and charm hadron production in the quark–gluon plasma at the early stage of heavy ion collisions are also discussed.

Keywords:

heavy-ion collisions; heavy-flavor production; hadronization

PACS:

27.75.-q; 25.75.Ag; 25.75.Dw; 25.75.Gz

1. Introduction

It is well-known that the hadronic matter is expected to undergo a transition to the quark–gluon plasma (QGP), a strongly coupled state of matter, at high temperatures or baryon densities [1,2,3,4,5,6]. The search for the QGP and the study of its properties have long been the goals of high-energy heavy ion collisions [7,8,9,10]. In heavy ion collisions, strange and heavy-flavor quarks are newly produced (or excited from the vacuum) and most of them are present in the whole stage of the QGP evolution. They interact strongly with the constituents of the QGP medium or they are a part of the QGP. Therefore, strange and heavy-flavor hadrons are usually regarded as special probes to the hadronization mechanism and properties of the QGP [11,12,13,14,15,16,17,18,19,20,21,22].

The relativistic heavy ion collider (RHIC) and the large hadron collider (LHC) have accumulated abundant experimental data on strange and charm hadrons [23,24,25,26,27,28,29,30,31,32,33,34,35,36]. These data show a number of features in the production of strangeness [33,34,37,38,39,40] and baryons [32,35,41,42,43,44]. Many efforts have been made to understand hadron production mechanisms in theory and phenomenology [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]. Hydrodynamic and thermal models [45,46,47,48,49] are commonly used to describe the production of strange hadrons. For the production of heavy-flavor hadrons, some transport models are popular (see, e.g., reference [63] and references therein). In particular, the coalescence or recombination models also provide good descriptions of hadron production especially at low and intermediate transverse momenta [50,51,52,53,54,55,56,64].

Based on Qu-Bing Xie’s works in

e^{+} e^{-}

and pp collisions in early years [65,66,67,68,69,70], we developed a quark combination model (QCM) for hadronization and it works well in explaining yields, rapidity distributions, and transverse momentum spectra for the identified hadrons in high-energy heavy ion collisions at various energies ranging from RHIC to LHC [53,62,71,72]. Recently, inspired by the property of constituent quark number scaling for transverse momentum spectra of strange hadrons in p+Pb collisions at LHC energy [73], we proposed a simplified version of the quark combination model by incorporating the equal-velocity combination (EVC) to replace the near rapidity combination in the original model. Many properties of hadron production can be analytically derived and some of them have been tested by experimental data in high energy pp, pA, and AA collisions [74,75,76,77,78]. Furthermore, our studies show that the EVC of charm and light quarks can explain the transverse momentum spectra of single-charm hadrons at low and intermediate transverse momenta [75,77,78,79,80]. In particular, the model prediction of the

Λ_{c}^{+} / D^{0}

ratio was verified by the latest measurements of the ALICE collaboration [35,81,82].

Recently, the ALICE collaboration published precise measurements of strange and charm hadrons, especially D mesons and

Λ_{c}^{+}

baryons, in Pb+Pb collisions at LHC [31,83,84,85,86]. In this paper, we apply the QCM with EVC to study the production of strange and charm hadrons simultaneously at low and intermediate transverse momenta in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. We will present analytical and numerical results for the

p_{T}

dependence of production ratios between different strange and charm hadrons. We will compare our results with the experimental data available and make predictions for other types of hadrons.

The rest of the paper is organized as follows. In Section 2, we introduce a general phase-space structure of the QCM in heavy ion collisions as well as the idea and formula of the QCM in momentum space based on EVC. In Section 3 and Section 4, we apply the QCM to calculate spectra of various strange and charm hadrons in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV and compare them with data. The final section is a summary of the main results and conclusions.

2. The Quark Combination Model

The QCM developed by the Shandong group led by Qu-Bing Xie [65,66,67,68,69,70] is a kind of exclusive or statistical hadronization model with constituent quarks as building blocks. A quark combination rule (QCR) can be derived for quarks and antiquarks in the neighborhood of the longitudinal phase space (momentum rapidity) to combine into baryons and mesons [65,66,67,68,87]. The QCM based on the QCR has successfully explained experimental data on hadron production in

e^{+} e^{-}

and pp collisions [65,66,67,68,69,70,88] as well as in heavy ion collisions [53,71,89]. A modern version of QCM with spin degrees of freedom in terms of Wigner functions has been developed by some of us and applied to spin polarization of hadrons in heavy ion collisions [90,91,92,93].

In this section, we introduce the general phase-space structure of the QCM in heavy ion collisions as well as its simplified version in momentum space to describe momentum spectra of strange and charm hadrons.

2.1. General Phase Space Structure of QCM in Heavy Ion Collisions

In the quantum kinetic theory, the formation of a composite particle through the coalescence or combination process of its constituent particles

q_{1} q_{2} \dots q_{n} \to H

can be described by the collision term incorporating the matrix element squared of the process and momentum integrals. In the case we are considering, the composite particle H can be a meson or a baryon, so the constituent particles

q_{1} q_{2} \dots q_{n}

are a quark and an antiquark for the meson, and are three quarks or three antiquarks for the baryon or antibaryon, respectively. In heavy ion collisions, the coalescence process takes place in a space-time region, i.e., the freeze-out hypersurface defined by the proper time

τ_{0}

. The momentum distribution of the hadron (meson or baryon) reads

\begin{matrix} f_{H} (p) & \sim & \int d σ^{μ} p_{μ} \int \prod_{i = 1}^{n} \frac{d^{3} p_{i}}{{(2 π)}^{3} 2 E_{i}} {(2 π)}^{4} δ (p_{1} + p_{2} + \dots + p_{n} - p) \\ \times {|M (q_{1} q_{2} \dots q_{n} \to H)|}^{2} f_{1} (x, p_{1}) f_{2} (x, p_{2}) \dots f_{n} (x, p_{n}), \end{matrix}

(1)

where

p = (E_{p}, p)

is the hadron’s on-shell momentum,

p_{i} = (E_{i}, p_{i})

is the on-shell momentum of the constituent particle

q_{i}

with its momentum distribution

f_{i} (x, p_{i})

at the space-time point x on the freeze-out hypersurface, M is the invariant amplitude of the coalescence process containing the hadron’s wave function, and

d σ^{μ} (x)

is the surface element pointing to the normal direction of the freeze-out hypersurface at x. The momentum distribution can be decomposed into the thermal part and non-thermal part,

f_{i} (x, p_{i}) = f_{i}^{th} (β u \cdot p_{i}) + f_{i}^{nth} (p_{i}),

(2)

where the thermal part

f_{i}^{th}

depends on

β u \cdot p_{i}

with

β (x) = 1 / T (x)

being the inverse temperature and

u^{μ} (x)

being the flow velocity both of which are functions of x on the freeze-out hypersurface, and the non-thermal part

f_{i}^{nth}

depends only on momentum and is independent of the space-time coordinate. We can express the space-time point on the freeze-out hypersurface in terms of the proper time

τ

and space-time rapidity

η

x^{μ} = (τ cosh η, x_{T}, τ sinh η),

(3)

and also the hadron’s on-shell momentum in terms of transverse momentum

p_{T}

and rapidity Y as

p^{μ} = (m_{T} cosh Y, p_{T}, m_{T} sinh Y),

(4)

where

m_{T} = \sqrt{m^{2} + p_{T}^{2}}

is the transverse mass. Then the freeze-out hypersurface element can be expressed as

d σ^{μ} = τ d η d^{2} x_{T} \frac{\partial x^{μ}}{\partial τ} = τ d η d^{2} x_{T} (cosh η, 0, 0, sinh η),

(5)

so its contraction with the hadron’s momentum reads

d σ^{μ} p_{μ} = τ d η d^{2} x_{T} m_{T} cosh (η - Y) .

(6)

We can express the flow velocity with Bjorken’s boost invariance in the longitudinal direction with

η = η_{flow}

\begin{matrix} u^{μ} (x) = & [cosh η cosh ρ (x_{T}, ϕ_{s}), sinh ρ (x_{T}, ϕ_{s}) cos ϕ_{b}, \\ sinh ρ (x_{T}, ϕ_{s}) sin ϕ_{b}, sinh η cosh ρ (x_{T}, ϕ_{s})], \end{matrix}

(7)

where

ρ (x_{T}, ϕ_{s})

is the transverse flow rapidity [94,95] as a function of cylindrical coordinates in the transverse plane

x_{T} = | x_{T} |

and

ϕ_{s}

, and

ϕ_{b}

is the boost angle in the transverse plane which can simply be taken as

ϕ_{s}

in approximation. The elliptic flow can be implemented by [95]

ρ (x_{T}, ϕ_{s}) = \frac{x_{T}}{R} [ρ_{0} + ρ_{2} cos (2 ϕ_{s})],

(8)

where R is the transverse size of the fireball, and

ρ_{2}

is linked to the elliptic flow coefficient

v_{2}

2.2. QCM in Momentum Space with Equal-Velocity Combination

For the purpose of this paper, we will introduce a simplified version of the QCM in momentum space with an equal-velocity combination for hadron production. This corresponds to (a) the quark distributions are homogeneous in space-time and depend only on momentum and (b) the role of the matrix element squared is taken by the EVC. This version of QCM is an approximation to the rigorous one in Section 2.1.

We consider a color-neutral system of

N_{q} = \sum_{i} N_{q_{i}}

quarks and

N_{\bar{q}} = \sum_{i} N_{{\bar{q}}_{i}}

antiquarks where

q_{i} = u, d, s, c

and

{\bar{q}}_{i} = \bar{u}, \bar{d}, \bar{s}, \bar{c}

denote the quark and antiquark flavors, respectively. The momentum distributions

f_{M_{j}} (p) \equiv f_{M_{j}} (p; N_{q}, N_{\bar{q}})

and

f_{B_{j}} (p) \equiv f_{B_{j}} (p; N_{q}, N_{\bar{q}})

for the directly produced meson

M_{j}

and baryon

B_{j}

by combining a pair of quark–antiquarks and three quarks, respectively, can be schematically expressed as

\begin{matrix} f_{M_{j}} (p) = \sum_{{\bar{q}}_{1} q_{2}} \int d p_{1} d p_{2} N_{{\bar{q}}_{1} q_{2}} f_{{\bar{q}}_{1} q_{2}}^{(n)} (p_{1}, p_{2}) R_{M_{j}, {\bar{q}}_{1} q_{2}} (p; p_{1}, p_{2}), \end{matrix}

(9)

\begin{matrix} f_{B_{j}} (p) = \sum_{q_{1} q_{2} q_{3}} \int d p_{1} d p_{2} d p_{3} N_{q_{1} q_{2} q_{3}} f_{q_{1} q_{2} q_{3}}^{(n)} (p_{1}, p_{2}, p_{3}) R_{B_{j}, q_{1} q_{2} q_{3}} (p; p_{1}, p_{2}, p_{3}), \end{matrix}

(10)

where

f_{{\bar{q}}_{1} q_{2}}^{(n)}

and

f_{q_{1} q_{2} q_{3}}^{(n)}

are normalized joint momentum distributions;

N_{{\bar{q}}_{1} q_{2}}

and

N_{q_{1} q_{2} q_{3}}

are the number of

{\bar{q}}_{1} q_{2}

pairs and that of

q_{1} q_{2} q_{3}

clusters in the system;

R_{M_{j}, {\bar{q}}_{1} q_{2}}

and

R_{B_{j}, q_{1} q_{2} q_{3}}

are combination kernel functions that stand for the probability density for a

{\bar{q}}_{1} q_{2}

pair with momenta

p_{1}

and

p_{2}

to combine into a meson

M_{j}

of momentum p and that for a

q_{1} q_{2} q_{3}

cluster with

p_{1}

p_{2}

, and

p_{3}

to combine into a baryon

B_{j}

of momentum p, respectively.

Just as derived in Refs. [75,76], the combination kernel functions in the EVC can be written as

\begin{matrix} R_{M_{j}, {\bar{q}}_{1} q_{2}} (p; p_{1}, p_{2}) & = & C_{M_{j}} R_{{\bar{q}}_{1} q_{2}}^{(f)} A_{M, {\bar{q}}_{1} q_{2}} δ (p_{1} - x_{q_{1} q_{2}}^{q_{1}} p) δ (p_{2} - x_{q_{1} q_{2}}^{q_{2}} p), \end{matrix}

(11)

\begin{matrix} R_{B_{j}, q_{1} q_{2} q_{3}} (p; p_{1}, p_{2}, p_{3}) & = & C_{B_{j}} R_{q_{1} q_{2} q_{3}}^{(f)} A_{B, q_{1} q_{2} q_{3}} δ (p_{1} - x_{q_{1} q_{2} q_{3}}^{q_{1}} p) \\ \times δ (p_{2} - x_{q_{1} q_{2} q_{3}}^{q_{2}} p) δ (p_{3} - x_{q_{1} q_{2} q_{3}}^{q_{3}} p), \end{matrix}

(12)

where

δ

-functions guarantee the momentum conservation in the EVC and

x_{q_{1} q_{2}}^{q_{i}} = m_{q_{i}} / (m_{q_{1}} + m_{q_{2}})

and

x_{q_{1} q_{2} q_{3}}^{q_{i}} = m_{q_{i}} / (m_{q_{1}} + m_{q_{2}} + m_{q_{3}})

are the momentum fraction of the produced hadron for

q_{i}

. We note that the mass fraction is the same as the momentum fraction in the EVC. Masses of up, down, strange, and charm quarks are taken to be

m_{u} = m_{d} = 0.3

GeV,

m_{s} = 0.5

GeV and

m_{c} = 1.5

GeV, respectively.

The factor

C_{M_{j}}

is the probability for M to be

M_{j}

if the quark content of M is the same as

M_{j}

and similar for

C_{B_{j}}

. In this paper, we only consider hadrons in the ground state, namely mesons with

J^{P} = 0^{-}

and

1^{-}

and baryons with

J^{P} = {(1 / 2)}^{+}

and

{(3 / 2)}^{+}

. In this case,

C_{M_{j}}

is the same for all hadrons in the same multiplet (with the same

J^{P}

) and determined by the production ratio of vector to pseudo-scalar mesons

R_{V / P}

, so is it for

C_{B_{j}}

which is determined by the production ratio of

J^{P} = {(1 / 2)}^{+}

J^{P} = {(3 / 2)}^{+}

baryons

R_{O / D}

with the same flavor content.

The factors

R_{{\bar{q}}_{1} q_{2}}^{(f)}

and

R_{q_{1} q_{2} q_{3}}^{(f)}

contain Kronecker

δ

’s to guarantee the quark flavor conservation, e.g., if

M_{j}

is a D-meson with constituent quark content

\bar{q} c

R_{{\bar{q}}_{1} q_{2}}^{(f)} = δ_{q_{1}, q} δ_{q_{2}, c}

. If

B_{j}

is a single-charm baryon with the quark content

u d c

R_{q_{1} q_{2} q_{3}}^{(f)} = N_{sym} δ_{q_{1}, u} δ_{q_{2}, d} δ_{q_{3}, c}

, where

N_{sym} = 1, 3, 6

is a symmetry factor to account for the number of different permutations of three quarks for (a) three identical flavors, (b) two identical flavors, and (c) all three distinct flavors, respectively.

The factor

A_{M, {\bar{q}}_{1} q_{2}}

is the probability for a quark

q_{2}

to capture a specific antiquark

{\bar{q}}_{1}

to form a meson in the quark–antiquark system; it should be inversely proportional to

N_{q} + N_{\bar{q}}

. Similarly,

A_{B, q_{1} q_{2} q_{3}}

should be inversely proportional to

{(N_{q} + N_{\bar{q}})}^{2}

. Both

A_{M, {\bar{q}}_{1} q_{2}}

and

A_{B, q_{1} q_{2} q_{3}}

are determined by the unitarity and the competition mechanism of meson-baryon production. Note that for light-quark systems produced in

e^{+} e^{-}

and pp collisions,

A_{M, {\bar{q}}_{1} q_{2}}

and

A_{B, q_{1} q_{2} q_{3}}

correspond to combination weights of mesons and baryons that follow the QCR [65,66,87].

Putting all these factors together, for charm hadrons we are considering, Equations (9) and (10) become

\begin{matrix} f_{M_{j}} (p) & = & \frac{N_{c} N_{{\bar{q}}_{1}}}{N_{q} + N_{\bar{q}}} A_{M} C_{M_{j}} f_{{\bar{q}}_{1} c}^{(n)} (x_{q_{1} c}^{q_{1}} p, x_{q_{1} c}^{c} p), \end{matrix}

(13)

\begin{matrix} f_{B_{j}} (p) & = & \frac{N_{c} N_{q_{1}} N_{q_{2}}}{{(N_{q} + N_{\bar{q}})}^{2}} A_{B} C_{B_{j}} N_{sym} f_{q_{1} q_{2} c}^{(n)} (x_{q_{1} q_{2} c}^{q_{1}} p, x_{q_{1} q_{2} c}^{q_{2}} p, x_{q_{1} q_{2} c}^{c} p), \end{matrix}

(14)

where

A_{M}

and

A_{B}

are two global coefficients that can be determined by quark number conservation in the combination process and the baryon-to-meson production ratio

N_{B} / N_{M}

. We are considering a quark–antiquark system in the mid-rapidity region at very high collision energies, so that net baryon number and net quark flavor are negligible, i.e.,

N_{{\bar{q}}_{i}} \approx N_{q_{i}}

for

i = u, d, s, c

. Moreover, we assume that the number of strange quarks is suppressed by a factor

λ_{s}

(strangeness suppression factor) relative to that of up and down quarks, so we have

N_{u} : N_{d} : N_{s} = 1 : 1 : λ_{s}

If we neglect correlations in the joint momentum distributions among different momenta, we have factorization forms for the joint momentum distributions,

\begin{matrix} f_{{\bar{q}}_{1} q_{2}}^{(n)} (p_{1}, p_{2}) & = & f_{{\bar{q}}_{1}}^{(n)} (p_{1}) f_{q_{2}}^{(n)} (p_{2}), \end{matrix}

(15)

\begin{matrix} f_{q_{1} q_{2} q_{3}}^{(n)} (p_{1}, p_{2}, p_{3}) & = & f_{q_{1}}^{(n)} (p_{1}) f_{q_{2}}^{(n)} (p_{2}) f_{q_{3}}^{(n)} (p_{3}) . \end{matrix}

(16)

We will use the above factorization forms in Equations (13) and (14) in our numerical calculation for single-charm hadrons. By using Equations (13) and (14) with Equations (15) and (16), we are able to calculate momentum spectra and yields for different hadrons.

Including strong and electromagnetic decay contributions from short-lived resonances [96], we can obtain the momentum spectra of final state hadrons and make comparison with experimental data. For charm hadrons, we make an approximation that the momentum of the daughter charm hadron is almost equal to that of the mother charm hadron. With this approximation and the production ratio of the vector to the pseudo-scalar meson being set to 1.5 [28,80], we obtain (for the final state D mesons):

\begin{matrix} f_{D^{0}}^{(fin)} (p) & \approx & 3.516 f_{D^{0}} (p), \end{matrix}

(17)

\begin{matrix} f_{D^{+}}^{(fin)} (p) & \approx & 1.485 f_{D^{+}} (p), \end{matrix}

(18)

\begin{matrix} f_{D_{s}^{+}}^{(fin)} (p) & \approx & 2.5 f_{D_{s}^{+}} (p) . \end{matrix}

(19)

Similarly, we can set the production ratio of

J^{P} = {(1 / 2)}^{+}

to the

J^{P} = {(3 / 2)}^{+}

single-charm baryon to 2 [80] and obtain,

\begin{matrix} f_{Λ_{c}^{+}}^{(fin)} (p) & \approx & 5 f_{Λ_{c}^{+}} (p), \end{matrix}

(20)

\begin{matrix} f_{Σ_{c}^{0}}^{(fin)} (p) & \approx & f_{Σ_{c}^{0}} (p), \end{matrix}

(21)

\begin{matrix} f_{Σ_{c}^{+}}^{(fin)} (p) & \approx & f_{Σ_{c}^{+}} (p), \end{matrix}

(22)

\begin{matrix} f_{Σ_{c}^{+ +}}^{(fin)} (p) & \approx & f_{Σ_{c}^{+ +}} (p), \end{matrix}

(23)

\begin{matrix} f_{Ξ_{c}^{0}}^{(fin)} (p) & \approx & 2.5 f_{Ξ_{c}^{0}} (p), \end{matrix}

(24)

\begin{matrix} f_{Ξ_{c}^{+}}^{(fin)} (p) & \approx & 2.5 f_{Ξ_{c}^{+}} (p), \end{matrix}

(25)

\begin{matrix} f_{Ω_{c}^{0}}^{(fin)} (p) & \approx & 1.5 f_{Ω_{c}^{0}} (p) . \end{matrix}

(26)

These analytical results can be used to obtain the

p_{T}

spectra of charm hadrons. For final state strange hadrons, there are no such analytical results, only numerical ones.

3. Transverse Momentum Spectra and Baryon-to-Meson Ratio for Strange Hadrons

In this section, we apply the QCM introduced in Section 2 to study the production of strange hadrons in Pb+Pb collisions at

\sqrt{s_{N N}} = 5.02

TeV. We first calculate the

p_{T}

spectra of strange mesons and baryons. Then we calculate the baryon-to-meson ratio

Λ / K_{s}^{0}

as a function of

p_{T}

in different types of centralities.

3.1. Transverse Momentum Spectra of Strange Hadrons

The inputs of the model are

p_{T}

spectra of quarks and antiquarks. In this paper, we adopt the isospin symmetry and neglect the net quark numbers (

N_{q_{i}} \approx N_{{\bar{q}}_{i}}

for

i = u, d, s, c

) in the mid-rapidity region at LHC energy, so we have only two inputs

f_{d} (p_{T}) = f_{u} (p_{T})

and

f_{s} (p_{T})

, which can be fixed by fitting the experimental data on the

p_{T}

spectra of

ϕ

mesons and

Λ

baryons [40,97,98]. The extracted results for the normalized

p_{T}

spectra of quarks in central 0–5% to peripheral 70–80% Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV are shown in Figure 1. The rapidity densities of d and s quarks are listed in Table 1.

In Figure 2, we show the results for the

p_{T}

spectra of

K_{s}^{0}

and

Λ

in 0–5%, 5–10%, 10–20%, 30–40%, 50–60%, 70–80% centralities, and those of

ϕ

Ξ^{-}

and

Ω^{-}

in 0–10%, 10–20%, 30–40%, 50–60%, 70–80% centralities. The QCM results are displayed in lines and experimental data [97,98] are displayed in open symbols. From Figure 2, we see that our QCM results for strange mesons and baryons agree with the experimental data very well. Such a good agreement provides a piece of evidence for the EVC mechanism in describing strange hadron production in Pb+Pb collisions at the LHC energy.

3.2. Baryon-To-Meson Ratio $Λ / K_{S}^{0}$

Figure 3 shows the multiplicity ratio

Λ / K_{S}^{0}

as a function of

p_{T}

in five centrality ranges 0–5%, 10–20%, 30–40%, 50–60%, and 70–80%. Filled squares are experimental data [98], and lines are the QCM results. We see that

Λ / K_{S}^{0}

exhibits an increase-peak-decrease behavior as a function of

p_{T}

in all centralities, which is regarded as a natural consequence of quark recombination [50,51,52,99,100]. We see that this feature can be well described by the QCM with EVC. The height of the peak increases from about 0.8 in the peripheral (70–80% centrality) to about 1.5 in the most central (0–5% centrality) collisions. The peak positions in the

p_{T}

slightly move to higher values from peripheral to central collisions due to stronger radial flows in more central collisions. The QCM with EVC gives a good description of the

p_{T}

dependence of

Λ / K_{S}^{0}

4. Transverse Momentum Spectra, Yield Ratios, and Nuclear Modification Factor for Charm Hadrons

In this section, we apply the QCM with EVC to study the production of charm hadrons at midrapidity in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. We first calculate the

p_{T}

spectra of D mesons and single-charm baryons. Then we give the

p_{T}

dependence of yield ratios of different charm hadrons. Finally, we give the nuclear modification factor

R_{A A}

for charm hadrons as a function of

p_{T}

4.1. Transverse Momentum Spectra of Charm Mesons and Baryons

In the QCM, the only additional input is the normalized

p_{T}

distribution of the charm quarks, which we adopt a hybrid form based on the simulation of charm quarks propagating in the QGP medium in a Boltzmann transport approach [100,101]

f_{c}^{(n)} (p_{T}) = \frac{1}{N_{norm}} p_{T} [{(\frac{p_{T}}{p_{T 0}})}^{α_{c}} exp (- \frac{\sqrt{p_{T}^{2} + m_{c}^{2}}}{T_{c}}) + {(1.0 + \frac{\sqrt{p_{T}^{2} + m_{c}^{2}} - m_{c}}{Γ_{c}})}^{- β_{c}}] .

(27)

At small

p_{T}

, this parameterized form is very close to the thermal distribution, while at large

p_{T}

, it follows the power law which is a non-thermal distribution. Both the thermal and non-thermal distributions are smoothly connected through the above parameterization. Here, the normalization constant

N_{norm}

can be determined by the condition

\int_{0}^{\infty} d p_{T} f_{c}^{(n)} (p_{T}) = 1

. The parameters

α_{c}

p_{T 0}

T_{c}

Γ_{c}

, and

β_{c}

are fitted using the data of

D^{0}

’s

p_{T}

spectra [31,84] and are listed in Table 2. The shapes of

f_{c}^{(n)} (p_{T})

at different centralities are shown in Figure 4a. We see that there is a stronger suppression in more central collisions in the

p_{T}

range 4 GeV

< p_{T} <

10 GeV. Figure 4b shows

f_{c}^{(n)} (p_{T})

at different centralities normalized by 60–80% centrality, which has similar behavior to the nuclear modification factor

R_{C P}

of the

D^{0}

meson measured in reference [31]. For the rapidity density of charm quarks

d N_{c} / d y

, we assume that it is proportional to the cross-section per rapidity in pp collisions as

\frac{d N_{c}}{d y} = 〈 T_{A A} 〉 \frac{d σ_{c}^{pp}}{d y} .

(28)

Here,

〈 T_{A A} 〉

is the average nuclear overlap function [31],

d σ_{c}^{p p} / d y

is the

p_{T}

integrated cross-section of charm quarks in pp collisions which is about 1.0 mb at

\sqrt{s} =

5.02 TeV [77]. The values of

d N_{c} / d y

at different centralities are listed in Table 2.

In Figure 5, we present the results for the

p_{T}

spectra of

D^{0}

D^{+}

D_{s}^{+}

, and

D^{* +}

mesons at 0–10%, 30–50%, and 60–80% centralities. Open symbols are the experimental data [31,84,85] and lines are the calculated results. The results agree with the experimental data in the

p_{T}

range from 0.5 GeV/c up to 10 GeV/c.

Figure 6 shows the results for charm baryons at 0–10%, 30–50%, and 60–80% centralities. Open symbols [only in Figure 6a] are the data of

Λ_{c}^{+}

[86] and lines are the results from the QCM, which are in good agreement. Predictions from QCM for other charm baryons

Σ_{c}^{0}

Ξ_{c}^{+}

, and

Ω_{c}^{0}

are presented in Figure 6b–d, which can be tested in future experimental measurements.

The agreement between our results and experimental data for various D mesons and

Λ_{c}^{+}

baryons indicates the validity of the EVC mechanism in the QCM in describing the charm hadron production. In this mechanism, the formation of the charm hadron is through the capture of light quarks in the medium by the charm quark with the same velocity.

We also calculated

p_{T}

-integrated yield density

d N / d y

for charm hadrons at midrapidity and 0–10%, 30–50%, and 60–80% centralities as listed in Table 3. The experimental data are taken from Refs. [84,85]. The QCM results are slightly higher than data. This is because we only include single-charm hadrons in the calculation and assume all charm quarks go to single-charm hadrons. In fact, the hidden-charm

J / Ψ

, double-charm baryons, and even heavy-flavor multiquark states can be produced. If including these particles, our results in Table 3 will decrease slightly and the agreement with experimental data will be improved. No data are available for the yield densities of many charm hadrons; our QCM predictions can be tested by their future experimental measurements.

4.2. Yield Ratios for Charm Hadrons

In this subsection, we calculate two kinds of yield ratios as functions of

p_{T}

for charm hadrons: one is

D_{s}^{+} / D^{0}

which is related to strangeness production, and the other is the baryon-to-meson ratio.

We first look at the results for

D_{s}^{+} / D^{0}

in Figure 7 at 0–10%, 30–50% and 60–80% centralities. The symbols in panels (a) and (b) are from the most recent data [85], while those in panel (c) are from previous measurements [31]. Different lines are the QCM results. The agreement between the data and our results with the same value of

λ_{s}

extracted from strange hadrons implies the same ’strangeness’ environment for both strange and charm hadrons and supports the QCM works as the hadronization mechanism for charm quarks in the QGP medium.

We then look at the baryon-to-meson ratio

Λ_{c}^{+} / D^{0}

, which is considered a probe to the charm quark hadronization. Recalling Equations (17) and (20), we have

\frac{Λ_{c}^{+}}{D^{0}} = \frac{4.267}{2 + λ_{s}} \cdot \frac{A_{B}}{A_{M}} \cdot \frac{{[f_{d}^{(n)} (x_{d d c}^{d} p_{T})]}^{2} f_{c}^{(n)} (x_{d d c}^{c} p_{T})}{f_{d}^{(n)} (x_{d c}^{d} p_{T}) f_{c}^{(n)} (x_{d c}^{c} p_{T})} .

(29)

In our calculation, we set

A_{B} / A_{M}

to 0.45 by

R_{B / M} = 0.6

in the charm sector [75]. Following Equation (29), the results for

Λ_{c}^{+} / D^{0}

as a function of

p_{T}

at different centralities are given in Figure 8. We see that

Λ_{c}^{+} / D^{0}

as a function of

p_{T}

shows a similar shape to

Λ / K_{S}^{0}

in Figure 3 except a larger shift in the

p_{T}

at the peak value from the central to peripheral collisions; in the central to peripheral collisions, the peak values decrease from about 1.3 to 0.9 and their locations in the

p_{T}

shift from 5 to 3 GeV. The peak locations shifting lower

p_{T}

from the central to peripheral collisions is due to the stronger collectivity in more central collisions.

4.3. Nuclear Modification Factor $R_{A A}$

We finally investigate the nuclear modification factor

R_{A A}

for charm hadrons, which is defined as

R_{A A} (p_{T}) = \frac{1}{〈 T_{A A} 〉} \frac{d N_{A A} / d p_{T}}{d σ_{p p} / d p_{T}} .

(30)

The results for differential cross-sections of charm hadrons in

p p

collisions at

\sqrt{s} = 5.02

TeV are taken from reference [77] by some of us. Figure 9 shows

R_{A A}

for prompt

D^{0}

D^{+}

and

D^{* +}

mesons as well as

Λ_{c}^{+}

at 0–10% and 30–50% centralities. Symbols are experimental data [84,86], and solid lines are the QCM results. One can see in Figure 9 that both D mesons and

Λ_{c}^{+}

have similar

p_{T}

behaviors. The peaks are located at

p_{T}^{peak}

which shifts towards higher values from peripheral to central collisions. This shift is mainly due to the stronger collectivity in central collisions which can boost thermal quarks to larger transverse momenta that are passed to charm hadrons by the EVC mechanism. We also see that the peak shift for

R_{A A}

Λ_{c}^{+}

is more obvious than that of D mesons, because

Λ_{c}^{+}

contains two light quarks and therefore is more influenced by centrality-dependent collectivity.

5. Summary

The comparative study of the production properties of strange and charm hadrons can provide information on the hadronization mechanism in relativistic heavy ion collisions. We use a quark combination model in momentum space with the approximation of equal-velocity combination to study these properties in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. We used experimental data of

Λ

ϕ

and

D^{0}

to fix the

p_{T}

spectra of up, strange, and charm quarks at hadronization. We computed the

p_{T}

spectra and rapidity densities of

K_{s}^{0}

Ξ^{-}

Ω^{-}

D^{+}

D_{s}^{+}

D^{* +}

Λ_{c}^{+}

Σ_{c}^{0}

Ξ_{c}^{+}

and

Ω_{c}^{0}

from central to peripheral collisions. The QCM results agree with the available experimental data quite well.

We calculated the yield ratios

Λ / K_{S}^{0}

D_{s}^{+} / D^{0}

and

Λ_{c}^{+} / D^{0}

as functions of

p_{T}

. We found that the EVC-based QCM in momentum space can naturally describe their non-trivial behaviors as functions of the

p_{T}

and the centrality. The

p_{T}

locations of the peaks in these ratio curves shift to lower values from central to peripheral collisions, an effect arising from collectivity in heavy ion collisions, absent in pp collisions. The calculated results for the nuclear modification factor

R_{A A}

show similar behaviors for both D and

Λ_{c}^{+}

, which can be tested by more precise experimental measurements, especially at a low

p_{T}

. All our results support the validity of the EVC-based QCM in describing the hadronization mechanism of charm quarks in high energy heavy ion collisions.

Author Contributions

Conceptualization, F.-L.S. and Z.-T.L.; methodology, J.S., F.-L.S. and Z.-T.L.; formal analysis, R.-Q.W. and Q.W.; investigation, J.S., W.-B.C. and R.-Q.W.; data curation, W.-B.C.; writing—original draft preparation, R.-Q.W.; writing—review and editing, Q.W. and F.-L.S.; supervision, F.-L.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China under grant nos. 12175115, 11975011, 12135011, 11890710 and 11890713, and the Natural Science Foundation of Shandong Province, China, under grant nos. ZR2019MA053 and ZR2020MA097.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

We dedicate this work to Qu-Bing Xie (1935–2013) who was the teacher, mentor, and friend of Z.-T.L., F.-L.S. and Q.W.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. The normalized

p_{T}

distributions of (a) d quark and (b) s quark in different centralities in Pb+Pb collisions at

\sqrt{s_{N N}} = 5.02

TeV.

Figure 1. The normalized

p_{T}

distributions of (a) d quark and (b) s quark in different centralities in Pb+Pb collisions at

\sqrt{s_{N N}} = 5.02

TeV.

Figure 2. The

p_{T}

spectra of (a)

K_{s}^{0}

, (b)

Λ

, (c)

Ξ^{-}

, (d)

ϕ

and (e)

Ω^{-}

in different centralities in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. Open symbols are experimental data [97,98]. Lines for

Λ

and

ϕ

are fitting results that are used to fix

p_{T}

spectra of light-flavor quarks at hadronization in QCM. Lines for

K_{S}^{0}

Ξ^{-}

and

Ω^{-}

are predictions from QCM.

Figure 2. The

p_{T}

spectra of (a)

K_{s}^{0}

, (b)

Λ

, (c)

Ξ^{-}

, (d)

ϕ

and (e)

Ω^{-}

in different centralities in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. Open symbols are experimental data [97,98]. Lines for

Λ

and

ϕ

are fitting results that are used to fix

p_{T}

spectra of light-flavor quarks at hadronization in QCM. Lines for

K_{S}^{0}

Ξ^{-}

and

Ω^{-}

are predictions from QCM.

Figure 3. The

p_{T}

dependence of

Λ / K_{S}^{0}

in (a) 0–5%, (b) 10–20%, (c) 30–40%, (d) 50–60%, (e) 70–80% centralities in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. Filled squares are experimental data [98] and lines are the QCM results.

Figure 3. The

p_{T}

dependence of

Λ / K_{S}^{0}

in (a) 0–5%, (b) 10–20%, (c) 30–40%, (d) 50–60%, (e) 70–80% centralities in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV. Filled squares are experimental data [98] and lines are the QCM results.

Figure 4. (a) Normalized

p_{T}

distribution of the charm quarks

f_{c}^{(n)} (p_{T})

at different centralities. (b) The distribution

f_{c}^{(n)} (p_{T})

at different centralities normalized by that at 60–80% centrality.

Figure 4. (a) Normalized

p_{T}

distribution of the charm quarks

f_{c}^{(n)} (p_{T})

at different centralities. (b) The distribution

f_{c}^{(n)} (p_{T})

at different centralities normalized by that at 60–80% centrality.

Figure 5. The

p_{T}

spectra of (a)

D^{0}

, (b)

D^{+}

, (c)

D_{s}^{+}

, and (d)

D^{* +}

mesons at different centralities. Open symbols are the experimental data [31,84,85]. Lines for

D^{0}

are fitting results that are used to fix the shape parameters in the

p_{T}

spectrum of charm quarks at hadronization in Equation (27). Lines for

D^{+}

D_{s}^{+}

and

D^{* +}

are predictions from QCM.

Figure 5. The

p_{T}

spectra of (a)

D^{0}

, (b)

D^{+}

, (c)

D_{s}^{+}

, and (d)

D^{* +}

mesons at different centralities. Open symbols are the experimental data [31,84,85]. Lines for

D^{0}

are fitting results that are used to fix the shape parameters in the

p_{T}

spectrum of charm quarks at hadronization in Equation (27). Lines for

D^{+}

D_{s}^{+}

and

D^{* +}

are predictions from QCM.

Figure 6. The

p_{T}

spectra of (a)

Λ_{c}^{+}

, (b)

Σ_{c}^{0}

, (c)

Ξ_{c}^{+}

, and (d)

Ω_{c}^{0}

baryons at different centralities. Open symbols are the data of

Λ_{c}^{+}

[86], and lines are the QCM results.

Figure 6. The

p_{T}

spectra of (a)

Λ_{c}^{+}

, (b)

Σ_{c}^{0}

, (c)

Ξ_{c}^{+}

, and (d)

Ω_{c}^{0}

baryons at different centralities. Open symbols are the data of

Λ_{c}^{+}

[86], and lines are the QCM results.

Figure 7.

D_{s}^{+} / D^{0}

as a function of

p_{T}

for (a) 0–10%, (b) 30–50%, and (c) 60–80% centralities. The symbols are experimental data [31,85], and solid lines are the QCM results.

Figure 7.

D_{s}^{+} / D^{0}

as a function of

p_{T}

for (a) 0–10%, (b) 30–50%, and (c) 60–80% centralities. The symbols are experimental data [31,85], and solid lines are the QCM results.

Figure 8. The baryon-to-meson ratio

Λ_{c}^{+} / D^{0}

as a function of

p_{T}

for (a) 0–10%, (b) 30–50%, and (c) 60–80% centralities. Symbols are experimental data [86], and solid lines are the QCM results.

Figure 8. The baryon-to-meson ratio

Λ_{c}^{+} / D^{0}

as a function of

p_{T}

for (a) 0–10%, (b) 30–50%, and (c) 60–80% centralities. Symbols are experimental data [86], and solid lines are the QCM results.

Figure 9. The nuclear modification factor

R_{A A}

for (a) prompt

D^{0}

D^{+}

and

D^{* +}

mesons and (b)

Λ_{c}^{+}

baryons as functions of

p_{T}

at different centralities. Symbols are experimental data from Refs. [84,86], and solid lines are the QCM results.

Figure 9. The nuclear modification factor

R_{A A}

for (a) prompt

D^{0}

D^{+}

and

D^{* +}

mesons and (b)

Λ_{c}^{+}

baryons as functions of

p_{T}

at different centralities. Symbols are experimental data from Refs. [84,86], and solid lines are the QCM results.

Table 1. Rapidity densities of d and s quarks in different centralities in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV.

Table 1. Rapidity densities of d and s quarks in different centralities in Pb+Pb collisions at

\sqrt{s_{N N}} =

5.02 TeV.

Centrality	${dN}_{d} / dy$	${dN}_{s} / dy$
0–5%	840	370
5–10%	686	302
10–20%	516	227
30–40%	267	115
50–60%	97	40
70–80%	27	10

Table 2. The parameters in the normalized

p_{T}

distribution of the charm quarks at different centralities.

Table 2. The parameters in the normalized

p_{T}

distribution of the charm quarks at different centralities.

Centrality	0–10%	30–50%	60–80%
$p_{T 0}$ (GeV/c)	0.0051	0.10	0.63
$α_{c}$	0.5	1.0	1.5
$T_{c}$ (GeV)	0.46	0.38	0.34
$β_{c}$	3.10	3.00	2.95
$Γ_{c}$ (GeV)	0.6	0.6	0.7
$d N_{c} / d y$	23.07	3.90	0.417

Table 3. The yield density

d N / d y

for charm hadrons at different centralities. The experimental data are from Refs. [84,85].

Table 3. The yield density

d N / d y

for charm hadrons at different centralities. The experimental data are from Refs. [84,85].

Hadron	0–10%		30–50%		60–80%
Hadron	Data	QCM	Data	QCM	Data	QCM
$D^{0}$	$6.819 \pm 0 . 457_{- 0.936}^{+ 0.912} \pm 0.054$	8.438	$1.275 \pm 0 . 099_{- 0.173}^{+ 0.167} \pm 0.010$	1.436	—	0.157
$D^{+}$	$3.041 \pm 0 . 073_{- 0.155}^{+ 0.154} \pm 0 . 052_{- 0.618}^{+ 0.352}$	3.563	$0.552 \pm 0 . 008_{- 0.024}^{+ 0.024} \pm 0 . 009_{- 0.114}^{+ 0.068}$	0.606	—	0.0665
$D^{* +}$	$3.803 \pm 0 . 037_{- 0.085}^{+ 0.084} \pm 0 . 041_{- 1.175}^{+ 0.854}$	3.600	$0.663 \pm 0 . 023_{- 0.039}^{+ 0.038} \pm 0 . 007_{- 0.165}^{+ 0.149}$	0.613	—	0.0672
$D_{s}^{+}$	$1.89 \pm 0 . 07_{- 0.16 - 0.55}^{+ 0.13 + 0.36} \pm 0.07$	2.417	$0.34 \pm 0 . 01_{- 0.03 - 0.09}^{+ 0.02 + 0.11} \pm 0.01$	0.395	—	0.0368
$Λ_{c}^{+}$	—	5.983	—	1.026	—	0.115
$Σ_{c}^{0}$	—	0.997	—	0.171	—	0.0192
$Σ_{c}^{+ +}$	—	0.997	—	0.171	—	0.0192
$Ξ_{c}^{0}$	—	1.211	—	0.199	—	0.0189
$Ξ_{c}^{+}$	—	1.211	—	0.199	—	0.0189
$Ω_{c}^{0}$	—	0.246	—	0.0386	—	0.00311

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Chang, W.-B.; Wang, R.-Q.; Song, J.; Shao, F.-L.; Wang, Q.; Liang, Z.-T. Production of Strange and Charm Hadrons in Pb+Pb Collisions at $\sqrt{s_{N N}}$ = 5.02 TeV. Symmetry 2023, 15, 400. https://doi.org/10.3390/sym15020400

AMA Style

Chang W-B, Wang R-Q, Song J, Shao F-L, Wang Q, Liang Z-T. Production of Strange and Charm Hadrons in Pb+Pb Collisions at $\sqrt{s_{N N}}$ = 5.02 TeV. Symmetry. 2023; 15(2):400. https://doi.org/10.3390/sym15020400

Chicago/Turabian Style

Chang, Wen-Bin, Rui-Qin Wang, Jun Song, Feng-Lan Shao, Qun Wang, and Zuo-Tang Liang. 2023. "Production of Strange and Charm Hadrons in Pb+Pb Collisions at $\sqrt{s_{N N}}$ = 5.02 TeV" Symmetry 15, no. 2: 400. https://doi.org/10.3390/sym15020400

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Production of Strange and Charm Hadrons in Pb+Pb Collisions at $\sqrt{s_{N N}}$ = 5.02 TeV^†

Abstract

1. Introduction

2. The Quark Combination Model

2.1. General Phase Space Structure of QCM in Heavy Ion Collisions

2.2. QCM in Momentum Space with Equal-Velocity Combination

3. Transverse Momentum Spectra and Baryon-to-Meson Ratio for Strange Hadrons

3.1. Transverse Momentum Spectra of Strange Hadrons

3.2. Baryon-To-Meson Ratio $Λ / K_{S}^{0}$

4. Transverse Momentum Spectra, Yield Ratios, and Nuclear Modification Factor for Charm Hadrons

4.1. Transverse Momentum Spectra of Charm Mesons and Baryons

4.2. Yield Ratios for Charm Hadrons

4.3. Nuclear Modification Factor $R_{A A}$

5. Summary

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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Production of Strange and Charm Hadrons in Pb+Pb Collisions at sNN = 5.02 TeV †

Abstract

1. Introduction

2. The Quark Combination Model

2.1. General Phase Space Structure of QCM in Heavy Ion Collisions

2.2. QCM in Momentum Space with Equal-Velocity Combination

3. Transverse Momentum Spectra and Baryon-to-Meson Ratio for Strange Hadrons

3.1. Transverse Momentum Spectra of Strange Hadrons

3.2. Baryon-To-Meson Ratio Λ / K S 0

4. Transverse Momentum Spectra, Yield Ratios, and Nuclear Modification Factor for Charm Hadrons

4.1. Transverse Momentum Spectra of Charm Mesons and Baryons

4.2. Yield Ratios for Charm Hadrons

4.3. Nuclear Modification Factor R A A

5. Summary

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

Guidelines

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Production of Strange and Charm Hadrons in Pb+Pb Collisions at $\sqrt{s_{N N}}$ = 5.02 TeV^†

3.2. Baryon-To-Meson Ratio $Λ / K_{S}^{0}$

4.3. Nuclear Modification Factor $R_{A A}$