Phenomenologies of QCD equa3on of statein high-energy nuclear collisions
Akihiko Monnai (Osaka Institute of Technology)
AM, J. Phys. G 50, 095103 (2023)

Thermal Quantum Field Theory and Their Applica>ons
29th August 2023, KEK, Japan

Introductionn Phase transition of quarks

Protons and neutrons “melt” into quark-gluon plasma (QGP)
above around 2 trillion degrees Kelvin
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Introduc3onn Where is the quark-gluon plasma?

The QGP had filled the Universe around 10-5 seconds after the Big Bang

Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Introduc3onn How to make the quark-gluon plasma on earth

Smash two big nuclei (such as gold or lead) almost at the speed of light
BNL RelaNvisNc Heavy Ion Collider (2000-)CERN Large Hadron Collider (2010-)
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Introductionn What can be observed in nuclear collisions?

Hadrons- bulk observables- informa3ve of the medium “surface”(QGP is color opaque)T ~ 200-600 MeV(primarily u, d, s, g)Direct photons- photons w/o decay photons- informa3ve of the medium “interior”(QGP is electroweak transparent)

Charged leptons, weak bosons, etc.
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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1 GeV ≃ 1.6×10!"# J
1 GeV/𝑐 ≃ 5.3×10!"$kg 1 m/s

Introductionn Momentum distribuNon of charged hadrons
Eskola et al., PRC 72, 044904 (2005)

310-102

1/2 pT dN/dpTd [1/GeV ]Tdec = 150 MeVpQCD + E-loss+(h +h )/22
1105 % most central

010-110-210-4RHIC Au+Au
1/2s = 200 AGeVSTAR 0-5 %PHENIX 0-5 %PHOBOS 0-6 %BRAHMS 0-10 %-510-610-7

10012345+Mid-high momentum region (> 4-5 GeV)
perturbative QCD (weakly-coupled)-3

1010Low momentum region (< 2-4 GeV)
Hydrodynamic model (strongly-coupled)

678910 11pT [GeV]
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

6 / 19Mo3va3onn Relativistic hydrodynamic model

Equationof state

Initialcondition

Hydroevolution*A rela9on among state variables
such as pressure, temperature, etc.

Hadronicdecay

A caveat of the convenNonal model:
Equation of state & particle production include the contributionof high-momentum particles
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

Observables?

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This workn “Red” hydrodynamic model

par>cles*low momentum = long wavelength = “red”

particlesppcp

We introduce an upper limit (pc) for the momenta ofstrongly-coupled components

Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

8 / 19Equation of state

0.1 GeV ≃ 1.16×10"% Kn Results
654P/T4#!

𝑔! 𝑑 $ 𝑝𝐸!
Pressure: 𝑃 = ±𝑇 % &$ ln 1 ± exp − 𝑇2𝜋"pc=∞
pc=5 GeV
pc=4 GeV
pc=3 GeV
pc=2 GeV!

3i: index for par>cle species

2Hadron resonance gas

1(2+1)-flavor parton gas

00

0.1

0.2

0.3 0.4
T (GeV)

0.5

0.6

Smoothly connect thepressures of the 2 phases

0.7

Effects of momentum cutoff is larger at higher temperatures;
mostly negligible in the hadronic phase
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Hydrodynamic evolu3onn (2+1)-dimensional inviscid model

AM, PRC 90, 021901(R) (2014)

IniNal condiNon: Monte-Carlo Glauber model (event-averaged)
2.76 TeV Pb+Pb collisions at b = 4.6 fm (~ 0-20%)
IniNal Nme: τini = 0.4 fm/c
Freezeout temperature: Tf = 140 MeV
Thermal photon esNmated down to: Tph = 110 MeV
Hadronic decay: Sollfrank, Koch, and Heinz, Phys. Lett. B 252, 256 (1990)
SimulaNons of nuclear collisions at the Large HadronCollider
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Hydrodynamic evolution

1 fm/𝑐 ≃ 3.3×10!%& sn Numerical results
0.6

1.3
1.25
1.2
<uτ>

0.5T (GeV)

1.35

pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

0.4

1.15
1.1

0.3

1.05

0.2
0.1

pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV11

τ (fm)

100.951τ (fm)10

Initial temperature is higher with lower cutoff momentum; the flow
development is mostly unaffected
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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PhotonsSee e.g. AM, Int. J. Mod. Phys. A 37(11n12), 2230006 (2022)
for a review

n Sources of direct photons (convenNonal)

Prompt photons
produced at the collision
Turbide, Rapp and Gale, PRC 69, 014903
Heffernan, Hohler, and Rapp, PRC 91, 027902
Holt, Hohler, and Rapp, NPA 945, 1Berges et. al., PRC 95, 054904 (2017)
Tanji and Venugopalan PRD 95, 094009 (2017)
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

Thermal photons

produced from the mediump12 / 19

Photons

See e.g. AM, Int. J. Mod. Phys. A 37(11n12), 2230006 (2022)
for a review

n Sources of direct photons (this work)

Prompt photons
produced at the collision
Turbide, Rapp and Gale, PRC 69, 014903
Heﬀernan, Hohler, and Rapp, PRC 91, 027902
Holt, Hohler, and Rapp, NPA 945, 1Berges et. al., PRC 95, 054904 (2017)
Tanji and Venugopalan PRD 95, 094009 (2017)
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

Thermal photons

produced from the medium(low momentum only)

pcp

High pT photons

produced from the nonthermal components

pcp13 / 19

Par3cle spectran Numerical results
pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

100

10-1

10-310-4

11.522.5pT (GeV)

33.5

410-50.5

100

10-2

10-310-4pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

101

10-1

10-2

10-3

10-50.5

100

10-1

10-2

thermal + prompt + *high pT

pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

101
dNγth+pr/2πpTdpTdη

101
dNγth/2πpTdpTdη

thermal + prompt
dNγth+pr+high/2πpTdpTdηthermal

10-4

1

1.5

2
2.5
pT (GeV)33.5

410-50.5

1

1.5

2
2.5
pT (GeV)

33.5

4

Thermal photon distribuNon is reduced above pc
Total distribution is mostly unaffected by the choice of pc
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Elliptic flow

Ollitrault, PRD 46, 229 (1992)
Poskanzer and Voloshin, PRC 58, 1671 (1998)

n From spaNal anisotropy to momentum anisotropy

yPressure gradients drive particles faster in thedirection of the minor axisxThe momentum distribution of hadrons hasanisotropy (elliptic flow v2)

Direct photons are also anisotropic

Prompt photons: isotropic
Thermal photons: anisotropic
High pT photons: isotropic*High pT photons are conjectured as thermal photons with no anisotropy in this study
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Elliptic flown Numerical results
thermal + prompt + *high pT

pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

0.05
0.04

0.04

0.03
0.02
0.01

22.5pT (GeV)

33.5

400.5

pc=∞pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

0.05

v2γ,th+pr

pc=∞
0.18
pc=4 GeV
0.16
pc=3 GeV
0.14 pc=2.5 GeV
pc=2 GeV
0.12
0.1
0.08
0.06
0.04
0.02
00.5
1
1.5

thermal + prompt

v2γ,th+prv2γ,ththermal

0.03
0.02
0.01

1

1.5

22.5
pT (GeV)

33.5

40
0.5

1

1.5

2
2.5
pT (GeV)33.5

4Photon v2 is sensitive to the choice of pc; whether it isenhanced or not depends on the high pT contributions
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Summaryn We developed a “red” hydrodynamic model of the QGP
654P/T4Equation of state is constructed only with low
momentum components
High pT photon emission is assumed to be
non-thermal

32
10

IniNal temperature can be higher

00.2

0.3 0.4
T (GeV)

0.5

0.6

0.7

3.5

4

pc=∞
pc=4 GeV
pc=3 GeV
pc=2.5 GeV
pc=2 GeV

0.04
0.03
0.02
0.01
00.5

Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

0.1

0.05v2γ,th+prDirect photon v2 can be sensitive to thecutoff momentum pc

pc=∞
pc=5 GeV
pc=4 GeV
pc=3 GeV
pc=2 GeV

11.5

2
2.5pT (GeV)

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Outlookn Future direcNons
E d N3 (GeV -2)
dp

Introduction of viscous corrections (including
estimation of transport(b) v
100.2coefficients)
0.15
1
Event-by-event esNmaNon for quanNtaNve 10analyses of ellipNc0.1ﬂow(a) Invariant yield

2

3

PRL114, 072301
and priv. comm.

QGP w/o viscous
QGP w/ viscoussemi-QGPw/o viscous(All three include HG)-1Comparison with the experimental data for understanding
10the photon puzzle*
1010-2E d N3 (GeV -2)
dp-310-4
1-5
1010-1(a) Invariant yield

PRL114, 072301
and priv. comm.

Au+Au 20-40%
QGP w/o viscous
sNN=200GeV QGP w/ viscous

Data
Solid line

3*The discrepancy between the theore>cal es>ma>ons andexperimental data of direct photon v2
41

0.15
0.1

0.05

23

4p (GeV/c)
T

00.2-4

10
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

-510Calorimeter
Conversion(All three include HG)

310-300064901 (2016)(b) v 2(c)PHENIX
30.2

semi-QGP

w/o064904
viscous
Yield is from PRC 91,

102-2
100.05PHENIX, Phys. Rev. C 94,

Au+Au 20-40%
sNN=200GeV

0.15

01(c) v3234PHENIX
p (GeV/c)
TCalorimeter
Conversion

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Backup slides

Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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Equa3on of staten Hadron resonance gas (HRG) + parton gas

Particle Data Group: PRD 98, 030001 (2018)

𝑃678 : hadron resonances < 2 GeV with u,d,s components
𝑃9:; : u,d,s, and gConnecNon of the hadronic and QGP pressures
𝑃 𝑇 = 𝑃!"# 𝑇(𝑇 < 𝑇$ )𝑃 𝑇 = 𝑃!"# 𝑇$ + 𝑃%&' 𝑇 − 𝑃!"# 𝑇$ ×{1 − exp[−𝑐(𝑇 − 𝑇$ )]}(𝑇 ≥ 𝑇$ )𝑃′!"# 𝑇$
where 𝑐 =𝑃%&' 𝑇 − 𝑃!"# (𝑇$ )Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

𝑑𝑃 𝑇$𝑑𝑃!"# 𝑇$
so that=𝑑𝑇
𝑑𝑇

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Equation of staten Beneﬁts of the connecNon method

parton gas

P/T4

exponen>al damping to PQGP
continuous and smooth

HRGTcT1. It puts emphasis on HRG, which is more reliable than parton gas accoding tolaJce QCD
2. Kinetic freezeout is safe below Tc
3. The difference from the lattice QCD pressure is wihtin 10% for 0 < T < 1 TeV
Akihiko Monnai (OIT), TQFT 2023, 29th August 2023

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