Investigating the cluster production mechanism with isospin triggering Thermal models versus coalescence models Apiwit Kittiratpattana13 Tom Reichert12 Pengcheng Li45 Ayut

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Investigating the cluster production mechanism with isospin triggering:

Thermal models versus coalescence models

Apiwit Kittiratpattana1,3, Tom Reichert1,2, Pengcheng Li4,5, Ayut

Limphirat3, Christoph Herold3,∗, Jan Steinheimer6, Marcus Bleicher1,2

1Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt,

Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany

2Helmholtz Research Academy Hesse for FAIR (HFHF),

GSI Helmholtz Center for Heavy Ion Physics, Campus Frankfurt,

Max-von-Laue-Str. 12, 60438 Frankfurt, Germany

3Center of Excellence in High Energy Physics & Astrophysics,

School of Physics, Suranaree University of Technology,

University Avenue 111, Nakhon Ratchasima 30000, Thailand

4School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China

5School of Science, Huzhou University, Huzhou 313000, China and

6Frankfurt Institute for Advanced Studies (FIAS),

Ruth-Moufang-Str.1, D-60438 Frankfurt am Main, Germany∗

Isospin triggering allows to distinguish coalescence from thermal production of light clusters in

heavy ion collisions. Triggering on Y(π−)−Y(π+) allows to select very neutron or proton rich

ﬁnal states. The deuteron (cluster) production with coalescence (d∝n·p) leads then to an inverse

parabolic dependence of the deuteron yield on ∆Yπ. In contrast, in a thermal model, cluster

production is independent on ∆Yπ. The observation of a maximum deuteron (cluster) yield as

function of ∆Yπprovides conﬁrmation of the coalescence mechanism.

I. INTRODUCTION

The exploration of the properties of matter governed

by the theory of strong interaction (Quantum-Chromo-

Dynamics, QCD) is a topic of highest interest. Ab-initio

calculations based on lattice QCD methods have shown

that such matter undergoes a transition at suﬃciently

high temperatures and/or baryonic densities [1, 2]. Such

temperatures are e.g. reached in accelerator facilities like

the CERN-LHC, BNL-RHIC or CERN-SPS. At the high

density frontier, laboratory experiments are performed at

RHIC in the beam energy scan program, at GSI’s SIS18

accelerator or at the future FAIR facility. In nature the

high temperature transition from the deconﬁned Quark-

Gluon-Plasma state to a hadronic system happened ap-

proximately a few microseconds after the Big Bang, while

the high density regime is probed by neutron stars and

neutron star mergers. Especially neutron star merg-

ers have renewed the interest in the equation-of-state of

nuclear matter at highest densities [3] because gravita-

tional wave measurements might allow to pin down the

equation-of-state of QCD matter with very high precision

[4].

A central tool that is often used to infer the properties

of the created matter are light clusters, e.g. deuterons,

tritons and helium. For the production of such states

one uses generally two complementary approaches: The

statistical (thermal) model [5–13] or coalescence [14–31].

While both models provide similar results [25, 32] over a

wide range of collision energies they are very diﬀerent in

their physics assumptions:

∗herold@g.sut.ac.th

(I) The thermal model assumes the creation of a fully

thermalized (mostly assumed grand canonical) ﬁre-

ball, which means that the clusters are produced

at the chemical freeze-out at a temperature of

60-150 MeV (depending on the collision energies

probed in large systems like Au+Au or Pb+Pb)

from √sNN = 2.4−13000 GeV. An often discussed

problem with this model is the fact that lightly

bound clusters may not form or survive in such a

hot environment. This is also known from Big Bang

nucleosynthesis under the term of deuteron bottle-

neck, which means that deuterons and higher mass

light elements can only be formed if the tempera-

ture is on the order or below the binding energy of

a few MeV.

Even within the thermal model itself such a ten-

sion is visible in certain energy ranges [33], and the

clusters are often removed from the thermal ﬁtting

as they worsen the quality of the thermal ﬁt signif-

icantly [34].

(II) In contrast, the coalescence model assumes that

light clusters are formed at kinetic freeze-out, i.e.

after the last collisions/decays have ceased and the

system reaches the free-streaming regime. Here the

formation is possible due to lower temperatures and

due to the fact that no further collisions will destroy

the formed cluster. It is clear that at the earlier

chemical freeze-out, the temperature is higher and

the volume of the source is smaller than at the later

kinetic freeze-out where the temperature is lower

and the volume is larger [35].

Up to now it has not been possible to distinguish be-

tween both methods for cluster production, because the

arXiv:2210.11699v2 [nucl-th] 7 May 2023

−150 −100 −50 0 50 100 150 200

∆Yπ

hNiAu + Au,b=0.0 fm

3He

FIG. 1. [Color online] Estimates for the deuteron (red full

line), triton (blue dashed line) and 3He (green dotted line)

production in Au+Au reactions as a function of ∆Yπ.

results for the mean values have been shown to be sim-

ilar in the thermal model [36] and the coalescence ap-

proaches [37]. Here, we propose a new method based on

the isospin ﬂuctuations that allows to distinguish thermal

cluster production from the coalescence model.

The idea is the following: In the coalescence model,

the deuteron production is proportional to the prod-

uct of the number (densities) of the protons and neu-

trons at kinetic freeze-out. The number of protons and

neutrons at kinetic freeze-out is related to the amount

of emitted charged pions1due to isospin conservation.

E.g., assuming a ﬁxed volume 2and no production of

(charged) pions, the ratio of neutrons to protons at ki-

netic freeze-out is equal to the initial ratio of neutrons

to protons Nfr/Zfr =α=NAu/ZAu. The deuteron yield

din the coalescence model is then proportional to the

product of the neutron and proton numbers (or densi-

ties) d∝Nfr ·Zfr =αZfr ·Zfr.

1It is clear that also other charged particles, e.g. Kaons can be

produced. However, their yield is small and for the present ar-

gument it is suﬃcient to restrict the discussion to pions.

2To obtain a ﬁrst estimate of the eﬀect, we assume that the num-

ber of participants and the N/Z ratio in the participant nucleons

does not ﬂuctuate. In that sense, we speak of a ﬁxed volume. In

line with Ref. [31] we use these primordial Nand Zvalues for

the estimates of the deuteron (and higher mass cluster) yields.

Of course in a realistic situation this is not exactly true which

is why we contrast our simple model estimates in the following

with a detailed microscopic simulation of the UrQMD model in

which no such assumptions are made and which shows almost

the same behavior as our simpliﬁed model.

0 10 20 30 40 50 60

tfr [fm]

(dN/dt)|fr [1/fm]

Au + Au,3.0 GeV,b=0.0 fm

UrQMD,4π

Nucleon

d×10

t×100

3He ×100

FIG. 2. [Color online] Freeze-out time distribution of nucleons

(full blue line), pions (dashed black line), deuterons (dotted

red line), tritons (dotted green line) and 3He (dotted blue

line).

Let us now consider the case of charged pion produc-

tion. At low energies, pions are produced during the

collision, however travel inside the ∆ resonance until the

∆ escapes from the ﬁreball and decays into a nucleon

and a pion (kinetic freeze-out) [38, 39]. The production

of pions leads on average to an equipartion of isospin

in the nucleon system. However, the decay of the ∆’s is

stochastic (given by the branching ratios) and introduces

isospin ﬂuctuations in the nucleon system at freeze-out.

Let us look at the most extreme case: Initially the sys-

tem in Au+Au reactions consisted of 2·79 = 158 protons,

triggering on an event with 158 emitted π+and no π−

would lead to a nucleonic system at freeze-out consisting

only of neutrons. In this case the production probability

of a deuteron vanishes in the coalescence approach be-

cause there is no proton left to form a deuteron. The

maximal deuteron yield is obviously reached when the

pion emission created a system with Nfr =Zfr. In sum-

mary, the emission of positive and negative pions changes

the N/Z ratio at freeze-out. This modiﬁed N/Z ratio

enters as an input into the coalescence model. Thus,

the deuteron yield in the coalescence model depends on

∆Yπ=Y(π−)−Y(π+) and shows a distinct maximum.

Analog arguments lead to maxima for higher mass clus-

ters.

In case of the thermal model the situation is completely

diﬀerent [5–13]. In a grand canonical model only the av-

erage positive and negative pion numbers are ﬁxed, at

chemical freeze-out, by an isospin chemical potential ob-

tained from the initial N/Z ratio (again we assume a

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Investigatingtheclusterproductionmechanismwithisospintriggering:ThermalmodelsversuscoalescencemodelsApiwitKittiratpattana1;3,TomReichert1;2,PengchengLi4;5,AyutLimphirat3,ChristophHerold3;,JanSteinheimer6,MarcusBleicher1;21InstitutfurTheoretischePhysik,GoetheUniversitatFrankfurt,Max-von-Laue-Stras...

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Investigating the cluster production mechanism with isospin triggering Thermal models versus coalescence models Apiwit Kittiratpattana13 Tom Reichert12 Pengcheng Li45 Ayut

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