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Page 1: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� � � � � � � � � Ladder � � �� � � �

QCD � Chiral � � � � � ������ �

( � � � "! )#%$'&)(+* ,.- /10

2 31. 4 576 82. 9 : ; < =?> @ A B C D3. E F G HJI K B L4. M N O P Q R S5. T H 6

Page 2: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

1 4 576 8������� ����� �

QCD��������������������

• ���� "! #%$�& �(' ! #%$�& � )+*• �+,-�/.10+2�3 �+4�5• ���� 7698�:• · · ·

;7<1= > '@? ACBED�F��HG�I⇓'@? ACBCJLKM"N+OP�RQ�I

'@? ACB�K M"N+O• Lattice Monte Carlo Simulation

• Schwinger-Dyson equation, Bethe-Salpeter equation, Pagels-Stoker

• Non-Perturbative Renormalization Group

NPRG�TS�U

VHW �EX+YBEZ@� [��H\H]

Page 3: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

2 9 : ; < =?> @ A B C Deffective average action (Wilsonian E.A.

�1PI ��� )

Γ̃Λ[Φ] = ΓΛ[Φ] − Scut[Φ]

ΓΛ[Φ] ≡ −WΛ[J ] + J�Φ

Γ̃Λ[Φ]��� ;�� �

evolution equation

Λd

dΛΓ̃Λ = −

1

2str

Λ

d

dΛ∆−1f (∆−1

f + Γ̃(2)Λ )

∆−1f (q,Λ) = C−1(q/Λ)γ(q) →

0 for q >> Λ

∞ for q << Λ

γ(q) � Γ ���U� �Γ̃(2) � Γ̃ ��� ��� ;�� �������� ��� �C(q/Λ) � cutoff

��C−1(q/Λ) = 1/θε(|q|,Λ) − 1

Sharp cutoff limit

ε→ 0�θε(|q|,Λ) → θ(|q| − Λ)

Sharp cutoff flow equation

d

dΛΓ = −

1

2str

δ(|q| − Λ)

γ(q)̂Γ(2)

(1 +G ̂Γ(2)

) ,

G(q) = θ(|q| − Λ)/γ(q)

Page 4: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

�Sharpcutoff

� � � 4 � ; � � � Q � 4 9 O P � � � $���������������� � K �→V7W �EX+YBEZ@� [�� � & �"! # �%$'&(*) �RD�F �,+.- ��/0� ;H= �+; �132 15476 A98 ��� ;�� ��: ,-�� ;$'&�<>= \

Momentum Scale Expansion

θ(|p− q| − Λ) = θ(p2 + 2p · q) = θ(p̂ · q) +∑ pn

n!δ(n)(p̂ · q)

Sharp cutoff flow equation�LS�U

� � $���������� ��?�@��BA CCB7D�EF�G H I F JLKLM9N �>O�P

Page 5: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

3 E F G H I K B L� � � �

QCD

���1� K���� K ��L =

1

4trFµνF

µν+q̄iD/ q+1

(∂µA

)2+∂µC̄

A[∂µδAB + gfABCA

]CB

� ��� < Landau� � � � #

�+,-�/.10+2�3 � U(Nf )L × U(Nf )R

) ��� �%JComposite Operator ����� �

S = q̄ΛAUXqΛAUX

, P = iq̄ΛAUXγ5qΛAUX

ΛAUX

· · · ���E �� � /������ � ��� ! # � � �"!$# E.qΛAUX

(x) =∫ d4p

(2π)4qi(p)eipxθ(p− Λ

AUX).

0&%�' �(����� σ, π� : �

1 = N −1∫Dσ exp

1

2M2

∫d4x

(σ − yM−2q̄ΛAUX

qΛAUX

)2 ,

1 = N −1∫Dπ exp

1

2M2

∫d4x

(π − yM−2iq̄ΛAUX

γ5qΛAUX

)2 ,

Page 6: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

ERG������ � ����� � � � [6 A98C�

cutoff � � � � � � =�H3 �� = �· · · �� � 2 2 �LN N ��� � J ��� <�� & J ;�

) Modified Slavnov-Taylor ��� � (MSTI)

Λ = 0� � � � =�H3 � � I��

� � ��� ! # � >�� /�������� �non-local 4fermi

��� ! # � �E�1-gluon exchange

G1 ≡∫

q

p

kg2(p, k)q̄(k)γµT

Aq(p)GABµν (p−k)q̄(q)γνT

Bq(−q−p−k),

g(p, k): running gauge coupling constant

GABµν : �T.C "! K 8�# �

�%$ ��&('�)�*�+�,.-�) #"/

g2(Λ) =

1b ln(Λ2/Λ2

QCD)if Λ > Λ1

1b ln(Λ2

1/Λ2QCD)

+((ln(Λ2/Λ))2−(ln(Λ2/Λ1))2)b ln(Λ2/Λ1)(ln(Λ2

1/Λ2QCD))2

if Λ1 ≥ Λ > Λ2

1b ln(Λ2

1/Λ2QCD)

− ln(Λ2/Λ1)b(ln(Λ2

1/Λ2QCD))2

if Λ ≤ Λ2

K-I.Aoki et.al. Prog. Theor. Phys. Vol.84 No.4 (1990), 6830 � 1 )Λ1 �32�4' 576"8�9 :�;"<�= >7? @BADCFE"G.H

Ladder I%J 2�K.?.LNM ) O�P $ !RQ�S T )�U�V :�WX?GABµν 2�4 Y7L Hard Thermal Loop I%J Z\[.]

Page 7: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

����� G�� ) O�P $ !RQ ) ������P $

ΠABµν =

(FPL

µν +GP Tµν

)δAB

��� ���P Tµν = δ̄µν −

p̄µp̄νp̄2

, P Lµν = δµν −

pµpνp2

− P Tµν,

HTL I%J )�&��

F = Z3p2 +m2

HTL(η2 − 1)Φ(η),

G = Z3p2 +

1

2m2

HTL

[1 − (η2 − 1)Φ(η)

],

��� ��� η = ip0/|p̄| ���

m2HTL =

1

3g2T 2

Nc +

1

2Nf

,

Φ(η) =η

2ln

η + 1

η − 1

− 1

Page 8: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

�effective action

ΓΛ =∫ 1/T

0dτ

∫d3x

q̄i∂/q +

1

2

(ZπL (∂0π)2 + ZπT (∂iπ)2

)

+V (q̄, q, σ, π) +G1}

����� G�� : q0 � < → ��������A ωn = (2n + 1)πT)

∫ d4q

(2π)4Λδ(q − Λ) =⇒ T

n

′∫ d3q̄

(2π)3

√Λ2 − ω2

nδ(q̄ −√Λ2 − ω2

n)

∑n′ · · ·Λ2 ≥ ω2

n Z��� F] n 2��.?.L )

� � � � � � � ��� � � � � �! "$# % & ' ( )

Λ∂

∂ΛV =

Λ2T

π2NfNc

n

′√Λ2 − ω2

n ln(Λ2 + V̄ 2

S

)

V̄ ≡ V − g2S2M0/16N2cNf

V̄S = ∂V̄ /∂S, S = q̄q

?+*�� M0 :�, )�- . 20/ >21 543M0(p0/|p̄|) =

8

Λ2 +m2L(p0/|p̄|)

+16

Λ2 +m2T (p0/|p̄|)

.

� 5 6 � � 7 8 9 � � � � � "$# % & ' ( )

Λd

dΛZπL = −2y2

φ

(Λ2 +m2)ζ0 − 2ζ1(Λ2 +m2)3

,

Λd

dΛZπT = −

2

3y2φ

(Λ2 + 3m2)ζ0 − 2ζ1(Λ2 +m2)3

,

ζn ::���

ζn =Λ2T

π2NcNf

n

′√Λ2 − ω2

n

(gσ + g2M0/8N

2cNf

)n.

Ladder� ; < = ' ( ) �

full�?> @ � � A

Page 9: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

4 � � � � � � �� � � � � � �� � � G�� ) π ������A Z���� Y �%I%J )�� � 9 Z�� � 5

fπ =√2/3Z1/2

π σ0.� � � G�� )�! "$# %'&)( E *Λ∂

∂ΛV =

1

4π2NfNc ln

(Λ2 + V̄ 2

S

),

Λ∂

∂ΛZπ = −

1

2π2NcNfy

1

1 +m2/Λ2

3 1

2+m2/Λ2

+', Q.- / P V (S) Z1032)*�I�JV (S) =

N∑

n=1vn−1S

n−1,

Page 10: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

�π � � � � � Truncation � �

2 4 6 8 10order of truncation

70

80

90

100

110

93.4MeV

�fπ : input ��� 5��� 1+', Q.- / P V (S)

) 032)*�I�J : ��� Y7L�?35chiral condensate, quark dynamical mass 2��.?.L· · · ��� 2 ,%A Z�����5�� � �

Truncation 3 4 5 6 7 8 9 10

< ψ̄ψ >1/3 (MeV) [] 181.9 226.9 214.4 216.6 215.5 216.0 215.8 215.8

meff (MeV) [] 931.4 890.4 900.7 883.0 891.0 887.3 889.0 888.3

fπ (MeV) 73.0 101.7 92.3 93.9 93.1 93.5 93.4 93.4

Page 11: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

�< ψ̄ψ > � fπ

�Λ

AUX� �

0.4 0.6 0.8 1.0 −0

40

80

120

160

200

240

280

dt

[Mev]

∆t 0.4 0.5 0.6 0.7 0.8 0.9 1.0

< ψ̄ψ >1/3 (MeV) 215.98 215.97 216.07 216.12 215.76 215.82 216.02

fπ (MeV) 108.31 109.55 108.11 103.53 97.61 92.77 87.47

dt ≡ ln(ΛAUX/Λ

QCD)

< ψ̄ψ > :������ �fπ :�6"8�9�� % T 1 5 ( 032)*�I%J � O(∂2) Z truncate)

full � : ΛAUX2 6"8 Y �%? :��

?+*.: dt = 0.88 ����� 3

Page 12: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� � � � � � �Chiral � � �

��� +', Q.- / P 2��.?.L ! "$# %'&)( E *Λ∂

∂ΛV =

Λ2T

π2NfNc

n

′√Λ2 − ω2

n ln

Λ2 +

VS −

g2σM0

8N2cNf

2

��� ���

M0(p0/|p̄|) =8

Λ2 +m2L(p0/|p̄|)

+16

Λ2 +m2T (p0/|p̄|)

032)*�I%J V = v0 +mS − gσS2/2 +G6S

3/3 + · · ·

Λd

dΛv0 = ζ0 ln(Λ2 +m2)

Λd

dΛm = −

2mζ1

Λ2 +m2

Λd

(

−gσ2

)

=1

Λ2 +m2

ζ2 + 2mG6ζ

0 −2m2ζ2

Λ2 +m2

Λd

G6

3

= −

2G6ζ1

Λ2 +m2

1 −

2m2

Λ2 +m2

+

2mζ3

(Λ2 +m2)2

Y �ζn =

Λ2T

π2NcNf

n

′√Λ2 − ω2

n (gσ +GA)n .

Page 13: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� � � � � � � ��� � � � � � (Thermal mass U�V Z�W�� Y ���� )

0 0.25 0.5 0.75 1 1.25 1.5φ [100 MeV]

−1

0

1

2

3

4

5

effe

ctiv

e po

tent

ial V

(φ)

[100

MeV

]4

T = 135 MeVT = 140 MeVT = 145 MeV

Chiral symmetry is restored

Page 14: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� � � � � � � ��� � � � � � (Thermal mass U�V Z�� "�� 1 ���� )

0 0.25 0.5 0.75 1 1.25 1.5φ [100 MeV]

−2

−1

0

1

2

3

4

5

6

Effe

ctiv

e po

tent

ial V

(φ)

[100

MeV

]4

T = 90 MeVT = 100 MeVT = 110 MeV

� ,���� @��X5�? : �� ��D2��.?� ,����XH� � � � � �

three flavors ��

Thermal mass Z�� "�� 1 ������ � �Tc = 103.62 MeV�

Thermal mass Z W�� Y ������ � �Tc = 147.18 MeV�

Tc = 166 MeV by M.Harada and A.Shibata�Tc = 129 MeV by O.Kiriyama, M.Maruyama and F.Takagi�Tc = 150 ∼ 180 MeV by Lattice Monte-Carlo Method�Tc ≈ 100 MeV by J.Berges, D.U.Jungnickel and C.Wetterich

Page 15: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� � � � ��� � �

Miransky

M2L =

1

3g2T 2

Nc +

1

2Nf

, M 2

T =π

4M2

L

|p0|

|p̄|.

Static limit

M2L =

1

3g2T 2

Nc +

1

2Nf

, M 2

T = 0.

Screening� � Z�W��

M2L = M 2

T = 0.

Scheme Tc (Mev)

HTL approx. 103.62

Miransky et.al. 82.026

Static limit of HTL 131.90

Neglectted 147.18

Page 16: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� � � � � � � 5 � � � � � � �

� � �

� �

� �

� � �

� � �

� � � � � � � � � � � � � � �� � � � � � ! " # $ % & ' ( )

* +,,-/.01

2 3 4 5 6 7 8 9 : ; < = > ? @ AB C DE F G

�����IH � π �����KJfπL =

√2/3Z

1/2πL σ0, fπT =

√2/3Z

1/2πT σ0.� � �IH �MLON YQP ��� �IH Z����SRUT�� 1 ��1 �WV XOR

Page 17: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

� �ZT/ZL

� � � � �

� � � � � �

� � � � � �

� � � � � �

� �

� � � �

� � � � � �

� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � ! " # $ % & ' ( ) *

Page 18: Ladder QCD Chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 QCD "! #%$ & ('! )+* +,- /.10+23 +45 7698 : ;7 '@? ACBEDF HGI + '@? ACBCJLK M"N+OP RQ I '@? ACBK M"N+O Lattice

5 � � �������� ���• Thermal mass �� "�� 1��� ���� H Chiral ���� ��������������� �"!�� #%$'&)(*��+-,/.103254)6)�87

�������:9);��Thermal mass

0=<�>�(@?�A)BDC( E �-F"G�HI�"J�K'L�M-$ON�P ) Q Tc ≈ 100 MeV

• R ST� H � π U�VXWKJZY\[�]�SD ^�_/`�a �cb5deH*f"4)6)g Q fπ = 93.4 MeV

• h�ikj\iel mon\iqprY�� H�sutev Y\[�]wux Y�yez• {o|u}o~ �����o� T��IR Debye ������qY\[�]• ����� H � Y����