ladder qcd chiral - riise.hiroshima-u.ac.jp · 1 4 576 8 qcd "! #%$ & ('! )+* +,-...
TRANSCRIPT
� � � � � � � � � Ladder � � �� � � �
QCD � Chiral � � � � � ������ �
( � � � "! )#%$'&)(+* ,.- /10
2 31. 4 576 82. 9 : ; < =?> @ A B C D3. E F G HJI K B L4. M N O P Q R S5. T H 6
1 4 576 8������� ����� �
QCD��������������������
• ���� "! #%$�& �(' ! #%$�& � )+*• �+,-�/.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]
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)
�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
� � $���������� ��?�@��BA CCB7D�EF�G H I F JLKLM9N �>O�P
3 E F G H I K B L� � � �
QCD
���1� K���� K ��L =
1
4trFµνF
µν+q̄iD/ q+1
2α
(∂µA
Aµ
)2+∂µC̄
A[∂µδAB + gfABCA
Cµ
]CB
� ��� < Landau� � � � #
�+,-�/.10+2�3 � U(Nf )L × U(Nf )R
) ��� �%JComposite Operator ����� �
S = q̄ΛAUXqΛAUX
, P = iq̄ΛAUXγ5qΛAUX
ΛAUX
· · · ���E �� � /������ � ��� ! # � � �"!$# 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 ,
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\[.]
����� 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
�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
4 � � � � � � �� � � � � � �� � � G�� ) π ������A Z���� Y �%I%J )�� � 9 Z�� � 5
fπ =√2/3Z1/2
π σ0.� � � G�� )�! "$# %'&)( E *Λ∂
∂ΛV =
1
4π2NfNc ln
(Λ2 + V̄ 2
S
),
Λ∂
∂ΛZπ = −
1
2π2NcNfy
2φ
1
1 +m2/Λ2
3 1
2+m2/Λ2
+', Q.- / P V (S) Z1032)*�I�JV (S) =
N∑
n=1vn−1S
n−1,
�π � � � � � Truncation � �
2 4 6 8 10order of truncation
70
80
90
100
110
fπ
93.4MeV
�fπ : input ��� 5��� 1+', Q.- / P V (S)
) 032)*�I�J : ��� Y7L�?35chiral condensate, quark dynamical mass 2��.?.L· · · ��� 2 ,%A Z�����5�� � �
Truncation 3 4 5 6 7 8 9 10
< ψ̄ψ >1/3 (MeV) [] 181.9 226.9 214.4 216.6 215.5 216.0 215.8 215.8
meff (MeV) [] 931.4 890.4 900.7 883.0 891.0 887.3 889.0 888.3
fπ (MeV) 73.0 101.7 92.3 93.9 93.1 93.5 93.4 93.4
�< ψ̄ψ > � fπ
�Λ
AUX� �
0.4 0.6 0.8 1.0 −0
40
80
120
160
200
240
280
dt
[Mev]
∆t 0.4 0.5 0.6 0.7 0.8 0.9 1.0
< ψ̄ψ >1/3 (MeV) 215.98 215.97 216.07 216.12 215.76 215.82 216.02
fπ (MeV) 108.31 109.55 108.11 103.53 97.61 92.77 87.47
dt ≡ ln(ΛAUX/Λ
QCD)
< ψ̄ψ > :������ �fπ :�6"8�9�� % T 1 5 ( 032)*�I%J � O(∂2) Z truncate)
full � : ΛAUX2 6"8 Y �%? :��
?+*.: dt = 0.88 ����� 3
� � � � � � �Chiral � � �
��� +', Q.- / P 2��.?.L ! "$# %'&)( 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
dΛ
(
−gσ2
)
=1
Λ2 +m2
ζ2 + 2mG6ζ
0 −2m2ζ2
Λ2 +m2
Λd
dΛ
G6
3
= −
2G6ζ1
Λ2 +m2
1 −
2m2
Λ2 +m2
+
2mζ3
(Λ2 +m2)2
Y �ζn =
Λ2T
π2NcNf
∑
n
′√Λ2 − ω2
n (gσ +GA)n .
� � � � � � � ��� � � � � � (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
� � � � � � � ��� � � � � � (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
� � � � ��� � �
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
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√2/3Z
1/2πL σ0, fπT =
√2/3Z
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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����