QCD物質の音速のピーク構造と高密度極限
quark meson モデルによる解析
Sound velocity peak structure and conformality in QCD matter
quark meson model analysis
千葉 隆司（東北大学 原子核理論研究室 修士2年）
指導教員：古城 徹
Ref.) Sound velocity peak and conformality in isospin QCD
Ryuji Chiba, Toru Kojo arXiv:2304.13920 [hep-ph]

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

1

Background: T~0 QCD Equation of State
EoS: input of Neutron Star M-R Relationpressure

8⇡G
Einstein eq. Gµ⌫ + ⇤gµ⌫ = 4 Tµ⌫ energy-momentumctensor<latexit sha1_base64="m1V03v4fxZ2vlq6bOrhXK8lTrII=">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</latexit>

P

Equation of State P(ε)

gravity (ε)

M/M⊙

M-R Relation

2.0
Einstein eq.

stiff

stiff

soft-to-stiff
soft

soft
ε(1~2n0)
n₀ ~ 0.16/fm³

ε(~5n0)ε

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

11 - 14 km

R
2

Background: T~0 QCD Equation of StateSound velocity cs²
@P
2
cs =
< 1 (c = 1)
<latexit sha1_base64="gND8Xo1/UGfAtydkgUvie9+lC1Q=">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</latexit>@"→ indicator of stiﬀening of EoS

causality constraint

cs2

P
constraint

stiff

differentiate

not allowed1

integrate with B.C.
constraintsoftε
KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

n03

Closer look of cs²

cs2

artifact1cs2 peakcutoffartifact

Not Observed

NJL

1/3

Conformal (ideal gas) limit

pQCD

nuclear theory

∼ 2n0

∼ 5n0

NS-region

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

∼ 40n0n

4

Closer look of cs²

cs2

artifact1cs2 peakcutoffartifact

Not Observed

NJL

1/3

Conformal (ideal gas) limit

pQCD

nuclear theory

∼ 2n0

∼ 5n0

NS-region

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

∼ 40n0n

5Table of Contents- Why and What is “Isospin QCD”

- p7

- Quark-Meson Model- P8-9- EOS and Sound Velocity- P10-13- QM Model vs pQCD

- P14-15

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

6

Isospin QCD: μI = μu − μd ≠ 0 and μB = μu + μd = 0
Isospin QCD

Real QCD- LQCD: w/o sign problem- LQCD: w/ Sign problem

comparison ✅

comparison ❌

- Model analysis:- Model analysis:- Confinement- BCS-like pairing etc.

→ PredictionTu− quark¯ quarkd−onset of quark matterin dilute regime !

Td ∼ 150 MeV

0.3 n0deconf.

Tc ∼ Δ ∼ 160 MeVBCS

BCS gap of Pion Condensate Δ

BEC
mπ /2

μI

overlap
r
=
0.66
fm
!
n
' 5 n0⇡I
KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.
<latexit sha1_base64="HofpoKXIwinievzHs7NFn65IajA=">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</latexit>

7

Quark Meson Model (at T = 0)meson int.
<latexit sha1_base64="mqSBkFpN1trzmbzdewvdkln9hbI=">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</latexit>

1
1
2
Ldense = (@µ ) + (@µ~⇡ )22
2

1 2 2 1 2
m0
(m0
2 ⇣
2
⌧3
¯
+ i@/ + µI
24µ2I )~⇡ 2( ~ 2 )2 + h
24
Yukawa int.
⌘
0g ¯( + i 5~⌧ · ~⇡ )<latexit sha1_base64="aQS4GcyFPNiV6uASgQJOS94RSBU=">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</latexit>

meson:

mean field approximation<latexit sha1_base64="5Q6jW+JYLKlrf3px9UvpbZns5oo=">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</latexit><latexit sha1_base64="J/H1tP2R7KUKZ0dnQrBHwIIK9lY=">AAACj3ichVG7SgNBFD2urxhfURvBJhgUFQkTUSO+CGihheArUTCy7K4TXdyXu5OABn/Ays7CSkFE7G0VbPwBCz9BLBVsLLzZLPhCvcvOnHvmnjtnZlTH0D3B2EOFVFlVXVMbqgvXNzQ2NUdaWjOenXc1ntZsw3ZXVcXjhm7xtNCFwVcdlyumavAVdXuqtL5S4K6n29ay2HX4uqlsWnpO1xRBlBzpy8jFrGtGeS63H534lPTMyTv92WluCGUsa+bl2V45EmNx5kf0J0gEIIYg5u3IObLYgA0NeZjgsCAIG1Dg0beGBBgc4tZRJM4lpPvrHPsIkzZPVZwqFGK3adykbC1gLcpLPT1frdEuBv0uKaPoYvfsgj2zO3bJHtnbr72Kfo+Sl12a1bKWO3LzQfvS678qk2aBrQ/Vn54Fchjxverk3fGZ0im0sr6wd/S8NLrYVexmp+yJ/J+wB3ZLJ7AKL9rZAl88RpgeIPH9un+CzEA8MRwfWhiMpcaDpwihA53ooftOIoUZzCNN+x7iCte4kVqkpDQppcqlUkWgacOXkGbfAUTfmSA=</latexit><latexit sha1_base64="Q1v8yFdaULDeRZGj4G/pKUp5HSk=">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</latexit>= ˜ + h i, ⇡1 = ⇡˜1 + h⇡1 i → Ve↵ = Ve↵ (Mq ,gh i = Mq , gh⇡1 i =( Ve↵ = V0 + O(~) ); µI )

quark:<latexit sha1_base64="Vb7R/3H2F3u5WH03HXc31m64g+A=">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</latexit><latexit sha1_base64="FVnguOXqmViWIfLof5OORboPVhc=">AAACnnichVHLahRBFD3pqInjI6NuhGwKh8gEYaiRJBqJEAgEN5KXMxlIh6G6rZ4pUv2gumYgacYP8AdcuBCFIOJniOBKVy7yCSHLCG5ceKenQTSot+mqc0/dc+tUlZdolVrOj8ac8XPnL0xMXixdunzl6lT52vVmGveMLxt+rGPT8kQqtYpkwyqrZSsxUoSeltve3spwfbsvTari6IndT+RuKDqRCpQvLFHt8qKrZWCr7BlrtjPXhEwGwYA9pIyzO8wNhe36Qmdrg6rb9YSZpULXqE7XzrbLFV7jebCzoF6ACopYj8tv4eIpYvjoIYREBEtYQyClbwd1cCTE7SIjzhBS+brEACXS9qhKUoUgdo/GDmU7BRtRPuyZ5mqfdtH0G1IyzPCv/B0/5Z/4e37Mf/y1V5b3GHrZp9kbaWXSnnp+c+v7f1UhzRbdX6p/erYIcD/3qsh7kjPDU/gjff/gxenWg82Z7DZ/w0/I/2t+xD/QCaL+N/9wQ26+RIkeoP7ndZ8Fzbu1+kJtfmOusrxUPMUkpnELVbrve1jGI6yjQfu+wkd8xheHOavOY2dtVOqMFZob+C2c1k/xjJ4B</latexit>~ = ( , ~⇡ )
✓ ◆
qu
=
qd

<latexit sha1_base64="tdwWMeYQQkO2Y5O6FzzGD7cawTo=">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</latexit>@Ve↵
gap equation:
@

@Ve↵
=
@Mq
⇤ ,M ⇤q=0⇤ ,M ⇤q

VEOS (µI ) = Ve↵ (Mq⇤ ,rewrite potential:<latexit sha1_base64="xLjo2q/48FeYvA7tHu3WpgAOE2E=">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</latexit><latexit sha1_base64="ICP/ZsH6IMw5Wg8tCw0t/H3lzc8=">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</latexit>

thermodynamic quantities:" = nI µ I⇤; µI )@P
P = VEOS , nI =
@µI
@P
nI
2
P, cs =
=
,
@"
µI I

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

@2P
I =
@µ2I
8

Quark Meson Model (at T = 0)Eﬀective potential

Diverge<latexit sha1_base64="S/+8wxn/xVHJmChY8rgySPpojW8=">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</latexit>MS
finiteVe↵ = V0 + Vq + Vc.t.
+ Vc.t.

V1R loop

Determine to reproduce
physical quantities<latexit sha1_base64="trG9GWhHE8CeZ/FQTGNFe9DoZII=">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</latexit><latexit sha1_base64="Jg7eqonROWoLJ038AcKYw7jRBo0=">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</latexit>m⇡ = 140MeV,

m = 600MeV,

f⇡ = 90MeV,

Mq = 300MeV

tree-level (meson contribution)<latexit sha1_base64="t88IHnO1JLBN1cr9bIuPipJNh8g=">AAAESnicjVO9b9NQEP+lDlDCR1JYkFgsqqAiIeu5gYJYqISQGPtB2kpNFdnOS/JUf+H3ElEisrEwsDIwgcSAWNkQLCwMrAz9Eyo2ioQQDJztRG2JnPAs+927+/3u7p3v7NAVUjG2m5vS8seOn5g+WTh1+szZYmnm3JoMOpHDq07gBtGGbUnuCp9XlVAu3wgjbnm2y9ft7Tuxfb3LIykC/77aCfmWZ7V80RSOpUhVL8219JoUHn+gV4xKpXJV7/f7es0lBw1raDHnFwyzXpplBkuWPiqYA2F2sbh3+0rh/d2lYCb3BDU0EMBBBx44fCiSXViQ9GzCBENIui30SBeRJBI7x2MUiNshFCeERdpt+rbotDnQ+nSOfcqE7VAUl96ImDrK7Ct7w/bZZ/aW7bE/mb56iY84lx3a7ZTLw3rx6YXVnxNZHu0K7QPW2JwVmriZ5Coo9zDRxLdwUn730fP91Vsr5d5l9op9o/xfsl32iW7gd384r5f5ygvyXkCZmG3KViaVkmNyPIqLY2Rj5eCftOnUGFP70Zv/f2WzkUNvEVmbSa6TvMZobyzOnmAfxlTUM3HnpFHj+h50Ujb7cLdlox4extH0mP/OyqiwNm+YC8b1ZRqja0jXNC7iEuZoVm5gEfewhCr5fYZ3+ICP2hftu/ZL+51Cp3IDznkcWfn8X/hI81M=</latexit>

m20 2 m20 4µ2I
V 0 = 2 Mq +2g
2g 2<latexit sha1_base64="0j+cSgs9hhxCunDuMEoQ/dR0eIA=">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</latexit>

2

2
+
M
q +
4
24g

2 2

Vq =

Nc

counter term
<latexit sha1_base64="O3ZW9sJKy0IiZfxlt0QHHHQaClI=">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</latexit>

Vc.t. =

1 2
m
2 0
"
1
24

✓

Z

g ' 3.333,' 126.1

quasiparticle energy

1-loop (quark single-particle energy)<latexit sha1_base64="YLO4YLW8mAxsci4FMdlG4EgNjo4=">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</latexit>hMqg<latexit sha1_base64="fWcYBp4cpGfZAwuJia9rSpCYFvY=">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</latexit>p(Eu + Ed + Eū + Ed¯)

Eu,d = E(µI ) , Ed,u = E( µI )qq
2
E(µI ) = (ED µI ) + 2 , ED = p2 + Mq2

◆2
✓ ◆2
Mq
1 2
Mq
m0⇡
+ h
g
2
g
g
✓
◆4
✓ ◆4
✓
◆2 ✓ ◆ 2 #
Mq
Mq
+ 4⇡
+2 2 ⇡
4g
g
g
g

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

9

Importance of quark substructure<latexit sha1_base64="tdwWMeYQQkO2Y5O6FzzGD7cawTo=">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</latexit>@Ve↵
Gap equation
@
1200

@Ve↵
=
@Mq
⇤ ,M ⇤q

Mq* [MeV]∆*
Mq* (Tree)∆* (Tree)1000=0⇤ ,M ⇤
q

Diﬀerence on physical picture of meson

artifact

Tree: mesons as elementary particles<latexit sha1_base64="oj1RkAs+E0T26PPgU7UtJJNJKUI=">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</latexit>

800

Tree

1-loop: mesons as composite particles

600

q

q

<latexit sha1_base64="oj1RkAs+E0T26PPgU7UtJJNJKUI=">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</latexit><latexit sha1_base64="oj1RkAs+E0T26PPgU7UtJJNJKUI=">AAAD1nicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEJxRRkZMYLKBvoGYCBAibDEMpQdhDsFfgqdXJDQL4gYxxDDEMKQz5DMkMpQy5DKkMeQwmQncOQyFAMhNEMhgwGDAVAsViGaqBYEZCVCZZPZahl4ALqLQWqSgWqSASKZgPJdCAvGiqaB+SDzCwG604G2pIDxEVAnQoMqgZXDVYafDY4YbDa4KXBH5xmVYPNALmlEkgnQfSmFsTzd0kEfyeoKxdIlzBkIHThdXMJQxqDBditmUC3F4BFQL5Ihugvq5r+OdgqSLVazWCRwWug+xca3DQ4DPRBXtmX5KWBqUGzgaZzMagCdWYAXVsMDqliPG5EVQeyA7faYmicZAB5KXjCHtPnxIcsbpUw04qAsmlgtxIyFaQ6F6wOFCKIuMetDzl94FZVgawOmN4N0VM3JiPMSM/QTM80EJjwzRgggINBmkGJQQOYus0ZHBg8GAIYQoHmZjD0MUxnmMEcwVzH3MrcDlHKxAjVI8yAApgnAACVK9dt</latexit>

400<latexit sha1_base64="2zZALmjov9MRIc3A4wN5H5uuyGA=">AAAD7nicjVK7bhNBFD3OAgnmEQcaBM0Kyyg01jUoD9EQCQqaSHlgJ1I2WLubcbzKvrQztghWEDUNHRQIRSCBQBTwAVTQ8AMU+QREGRANBXfHRkmw1smsdufOnXPOnrlzndj3pCLayQ0Zx46fGB45mT91+szZ0cLYuZqMWokrqm7kR8myY0vhe6GoKk/5YjlOhB04vlhyNm6l+0ttkUgvCu+qzVisBvZ66DU811acqhcuWreFr2zTUpF5nch8aHasJDBnRW2rXihSmfQw+4NKLyje/DT+6/MT6+pcNJa7BwtriOCihQACIRTHPmxIflZQASHm3Co6nEs48vS+wBbyzG0xSjDC5uwGf9d5tdLLhrxONaVmu/wXn9+EmSZK9I3e0S59pff0nf5kanW0Ruplk2enyxVxffTxhcXfh7ICnhWae6yBnhUamNZePfYe60x6CrfLbz94trt4Y6HUuUKv6Af7f0k79IVPELZ/uq/nxcJzVs+jxMwmu5W6UnKAx4O49B/ZWNm7kyav1gbUvv/kR69sNvKfWsK7De31MNUUHWhcWpG9u8/m7e+PbNT9/Tju98r/3d0f1K6VK5PliXkqzkyiO0ZwCZcxzt09hRncwRyqrPsIb/ABH43YeGq8MLa70KFcj3MeB4bx9i8HU9/G</latexit>

vac.

200
0

BCS

1-loop! 300 MeVq̄<latexit sha1_base64="qPQHPY3MIk02mR2vGhDTxMCYmYk=">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</latexit>

1<latexit sha1_base64="rQJShIOSqbgGbrV1w2FuYy953vg=">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</latexit>

m⇡ /2

q̄
<latexit sha1_base64="qPQHPY3MIk02mR2vGhDTxMCYmYk=">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</latexit>→ quark Pauli blocking

BEC
0<latexit sha1_base64="oj1RkAs+E0T26PPgU7UtJJNJKUI=">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</latexit><latexit sha1_base64="7cK/cF87s8J3rSYJqNhR1FPIRew=">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</latexit><latexit sha1_base64="7cK/cF87s8J3rSYJqNhR1FPIRew=">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</latexit>

2
3
Chemical Potential [mπ]45Tc ~ 0.57∆ → 170 MeV
agreement with lattice
KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

10

Equation of State๏ ChPT should work in dilute regime
QM model respects Chiral symmetry → reproduce ChPT at low energy
๏ Good agreements with the LQCD result even at higher density ✅
( Also trace anomaly )800
QM: mπ = 140MeV
170MeV
Brandt et al.
Abbott et al. 48x96
64x128700
600
Pressure [mπ4]

0.8

500
0.6

400
0.4

300
0.2

๏ Brandt+ (2022) mπ = 135MeV

200
0
0.6

100
0

0.8

arXiv 2212.14016

1

๏ Abbott+ (2023) mπ = 170MeV
arXiv 2307.15014

0

2

4
6
Chemical Potential [mπ]810KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

11

Sound velocity
2) n ~ 0.8 n₀: peak structure
1
QM Tree: mπ = 140MeV
QM: mπ = 140MeV
170MeV
Brandt et al.
Abbott et al. 48x96
64x128Sound Velocity cs2

0.8

0.6

0.4
Conformal Limit

0.2

0

0

2

1) n < 0.8 n₀: rapid increase

46
Isospin Debsity [n0]8103) Larger n : relax tothe conformal value from above

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

≠ pQCD

12

Occupation Probability f(p); onset of quark substructure
Momentum distribution of quark wavefunction… two diﬀerent mode

1
0.9
0.8
0.7

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f (p)

0.6
0.5
0.4
0.3
0.2
0.1
0

Vertical Evolution

Horizontal Evolution

0

2

60

<latexit sha1_base64="iVKOCsC1VumunvQAhw/IHQl/KOg=">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</latexit>→ "/nI ⇠ const. small P, soft EoS<latexit sha1_base64="KbyuDlkDIyKZybR2K+j8nkGR17g=">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</latexit>

Horizontal Evolution: increasing large p mode( ∵ Interactions and Pauli blocking )→ stiﬀen EoS

10⇠ f (p; nI )/'vac⇡ (p)Horizontal Evolution

50

c2s > 1/3

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f(p)/f(p)n=0.1n

Vertical Evolution: increasing uniformly
f (p) ⇠ nI 'vac(low density limit)
⇡ (p)

8

<latexit sha1_base64="lWPrS3bmlXvzXkK8rLDFQAys76g=">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</latexit><latexit sha1_base64="cAiCty3xQdHyn30/WTmj6QeQmFI=">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</latexit>

2 @"/nI
P = nI@nI

4
6
Momentum [mπ]40transition n ∼ 0.3n0

30
20
10
0

2

Vertical Evolution cs < 1/3<latexit sha1_base64="FlThamqA/jfe/+/oPvKeiYb7XGc=">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</latexit>

0

2

4
6
Momentum [mπ]810r⇡ = 0.66fm ! noverlap⇠ 5.2n0I<latexit sha1_base64="pRko23GY7EXm2/k31eDoEHB3DqY=">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</latexit>

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

13

QM model vs pQCD; non-perturbative eﬀect
2 0.34cs<latexit sha1_base64="HOL07YntRxzls4L/exgw2Bv6A+g=">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</latexit>

QM model:approaches from above
pQCD:
approaches from below

1/3

0.32
Sound Velocity cs2

cs² asymptotic behavior

larger αs

0.3

pQCD: from below

0.28<latexit sha1_base64="JlgloUGwDro2AEp4+hiX98Ru7bk=">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</latexit>↵s (Q = XµI )0.26

X = 0.50
1.00
2.00
Conformal Limit

0.24
0.22

20

40

60

80

100
120
140
Isospin Density [n0]a2 ⇠ ⇤2QCD

160

180

200

<latexit sha1_base64="ibFn1IFdq1mNKuijINLth9VA/H4=">AAAD8HicjVK7ThtBFD1m8yDOA5M0kaBYxXIUKZI1tpSHUiFIQUGBIQYkDKvd9RiP2Jdm1lbAokmZfECQKBAIpDyq5BfS5AdS8AlRShKlSZG7Y0dArDXManfu3Dnn7Jk714k8oWLGjjJDxqXLV64OX8tev3Hz1khu9PaCClvS5VU39EK55NiKeyLg1VjEHl+KJLd9x+OLzvpUsr/Y5lKJMHgRb0R8xbfXAtEQrh1TysqN21bZrCnhm7UZYtVtq1OTvlmZer61WrZyeVZkepj9QakX5Cce/jp4w/PRbDiaWUUNdYRw0YIPjgAxxR5sKHqWUQJDRLkVdCgnKRJ6n2MLWeK2CMUJYVN2nb5rtFruZQNaJ5pKs136i0evJKaJAvvG3rNj9pV9ZN/Zn1StjtZIvGzQ7HS5PLJGXt+d/30uy6c5RvOENdBzjAaeaq+CvEc6k5zC7fLbm2+P55/NFTr32T77Qf732BH7QicI2j/dwwqf2yH1LArEbJJbpSulBng8i0v+kY5VvTtp0qo+oPb9J794ZdOR/9Qk7Ta01/NUE7SvcUlFTu4+nXe6P9JRL0/jqN9L/3d3f7BQLpYeFx9VqPEn0R3DGMM9PKDufoIJTGMWVdJ9hXf4hM+GNLaNXWOvCx3K9Dh3cGYYH/4C8xngnA==</latexit><latexit sha1_base64="f3zfhltyi59MHTd0z1943/uixqc=">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</latexit>

Adding the power correction (non-perturbative)
Pwithpowers = a0 µ4I + a2 µ2I2
2a0 µI + a2
1
2a2
2
cs =
=
1+
2
6a0 µI + a2
3
6a0 µ2I + a2

⇤QCD = Q exp<latexit sha1_base64="hLQdWKSmMm2TN3pFkHnNBIRMacs=">AAAEG3icjVJNaxNRFL3p+FHjR1PdCG4elogghDvBWhGFoqAVLKRN0xaSdJiZvjSPzBczL2nrkD/gH3AhLhRciPgruhFxJ10U9AeICxcV3LjwzptIW8OkvmFm7rvvnPPOu+9agSMiibiXG9NOnDx1evxM/uy58xcmCpMXlyO/G9q8ZvuOH65aZsQd4fGaFNLhq0HITddy+IrVeZCsr/R4GAnfW5LbAW+65oYnWsI2JaWMwqOKETdCl20K2WaBv0nQPrvHTAMbbtd4vHaT3WANybek2qo+tzT/pBnfn9ZnEPuxaZRTVLlvFKawhGqw4UAfBFOzbL76o9N8WPEnc2vQgHXwwYYuuMDBA0mxAyZE9NRBB4SAck2IKRdSJNQ6hz7kidslFCeESdkOfTdoVh9kPZonmpFi27SLQ29ITAZF3MW3uI8f8B1+w9+ZWrHSSLxs099KuTwwJp5drv46luXSX0L7gDXSs4QW3FZeBXkPVCY5hZ3ye0+f71fvLBbja/gav5P/V7iHO3QCr/fTfrPAF1+Qeh6KxGyT20hVKhrh8Sgu2SMbGw3upE2z9RG1Hz75/1c2G/lXLaTVlvJ6nGqCdhUuqcjB3WfzDvdHNmrrMI76Xf+3u4eD5XJJv1WaXqDGvwvpGIcrcBWuU3fPwCzMQQVqpPsePsMX+Kq91Ha0j9qnFDqWG3AuwZGh7f4BRALwvg==</latexit><latexit sha1_base64="clkxdOmU4aogT+gGB9/Cr+L1AHY=">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</latexit>✓

2⇡20 ↵s (Q )◆Conformality from above is possible
for large power correction.

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

14

pQCD + Power Corrections
X = 0.5 ⇠ 2

2<latexit sha1_base64="pTl/uXVCRD+SsNpx5Dik1bABXfE=">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</latexit>

P = PpQCD +⇡<latexit sha1_base64="lGv4O+ih054VNXm4APi8g1WxcN8=">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</latexit>

2
µ
2 I

solid line is for X = 1<latexit sha1_base64="fhZB9MefDHXfQMFgIcgp1RUj5oI=">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</latexit>

0.4
0.38

from above !

Sound Velocity cs2

0.36
0.34

1/3

0.32

From below

0.3
0.28

Δ[MeV] = 0
200
400
Conformal Limit

0.26
0.24
0.22

20

40

60

80

100
120
140
Isospin Density [n0]power correction in pressure … 5-10 %

160

180

200

change the qualitative behavior of cs²= origin of diﬀerence between QM and pQCD
KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

15

Summary๏ Soft-to-stiﬀ EOS and quark dof- quark dof may appear before the hadron overlap- Importance of quark substructure๏ Quark-meson model and low- and high density physics- low density: respects ch. symmetry → reproduce Ch.PT and LQCD
- high density: includes quark dof, matched to LQCD ✅

๏ Non-perturbative eﬀects ( = BSC-like pairing )- low density: attractive power correction → stiffens EOS- high density: change the asymptotic behavior from pQCD

KEK 熱場の量子論とその応用 Day2, Rj.Chiba, Tohoku Uni.

16