カイラル対称性の超回復

Super restoration of chiral symmetry
稲垣知宏
広島⼤学

T.I., D. Kimura and H. Shimoji, arXiv:2306.00470

Phase Structure of QCD
Are symmetries restored or lost in extreme conditions?
• Hadron• High Temperature• High Density
•…

u

d
u

u

u
u
d
d
• Confinement

Phase Structure of QCD
Heavy Ion Collision,
Early Universe

Thomas Schafer, hep-ph/0304281

Compact StarsPhenomenological Approach• Because of the strong coupling, it is difficult to evaluate theground state from the first principle, i. e. QCD.
• Phenomenological approach• Effective models• SD equationEffective Model of QCD• NJL model (2 flavors)<latexit sha1_base64="N8BG/XYcnzfsA/cI9Bn5uSvG/6g=">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</latexit>S=Z

d x ¯(x)(i µ @µDGm0 ) +( ¯ )2 + ( ¯i 5 ⌧ a )22N

Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1960) 345; 124 (1961) 246.

• GN model<latexit sha1_base64="kWdfxPOJxUnrq2dOhLFv1hRV+eQ=">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</latexit>

S=

Z



dD x ¯(x) (i µ @µm0 ) (x) +0

2N

D. J. Gross and A. Neveu, Phys. Rev. D10, 3235 (1974).

¯(x) (x) 2

Chiral symmetry breaking• Chiral symmetry! exp(i 5 ✓a ⌧ a )<latexit sha1_base64="5+iPXFpa2QDxzCSuaDeO9rp6+hg=">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</latexit>!

5: NJL model: GN model• Auxiliary field method (GN model)<latexit sha1_base64="ZO4PLU+WSakvyVh1pGe4B/6+b24=">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</latexit>

S=

Z



d x ¯(x) (i µ @µD

m0

Explicit breaking(x)) (x)

N
(x)2
2 0

Spontaneous breaking

Effective potential analysis
We assume• Homogeneous chiralcondensationand calculated the effective
potential at the• Leading order of 1/N
expansion

Renormalization at T=µ=0• Effective potential
˜ 2 2m0 ˜VD (˜ ) =2 0<latexit sha1_base64="AURB6qu7zro1kV9VXOQAB7ISbXU=">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</latexit>

CD 2 D/2
(˜ )
,
D

˜=

+ m0

+ CD (D2),<latexit sha1_base64="KIYEWQcr3/zhZKhDKES9ChqUKAs=">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</latexit>• Renormalization condition<latexit sha1_base64="EKYJyt5P9goXhRr/Uxl+Z9y9OmM=">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</latexit>

2@ 2 VD (˜ )
µD
r
=
@ ˜2
r
˜ =µr

<latexit sha1_base64="Kj4+xohxk8S/x4BH4JCL+bOHCuw=">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</latexit>

1@VD (˜ )
1
=
D 1
@˜
µr
r
˜ =µr

✓

1

mr
µr

◆

Effective potential analysis• Following the imaginary time formalism, we introduce thetemperature and the chemical potential and obtain

Super restoration• Order parameter of symmetry breaking (constituent mass)M = h˜Mi = m0r + h i<latexit sha1_base64="ARow7UZsM519ZghEFwqxBo8/azU=">AAAChHichVG7SgQxFD2O7/W1aiNYKC6KhSxZ3wiKaGMj+FoVHFlmZuMazDyYyS7oYmnjD1hYKYiIrX6BjT9g4SeIpYKNhXdmB0RFvSHJyck9NyeJ6UkRKMYeq7Tqmtq6+obGRFNzS2tbsr1jPXCLvsWzlitdf9M0Ai6Fw7NKKMk3PZ8btin5hrk3H+5vlLgfCNdZU/se37aNgiN2hGUoonLJnsVpXRpOQXJdCZnnZT0QBds41P2ITOSSKZZmUfT+BJkYpBDHkpu8hI48XFgowgaHA0VYwkBAbQsZMHjEbaNMnE9IRPsch0iQtkhZnDIMYvdoLNBqK2YdWoc1g0ht0SmSuk/KXvSzB3bFXtg9u2ZP7P3XWuWoRuhln2azouVeru24a/XtX5VNs8Lup+pPzwo7mIy8CvLuRUx4C6uiLx2cvKxOrfSXB9g5eyb/Z+yR3dENnNKrdbHMV04RfkDm+3P/BOvD6cx4enR5NDU7F39FA7rRh0F67wnMYgFLyNK5R7jGDW61Om1IG9HGKqlaVazpxJfQZj4AnJuV8g==</latexit><latexit sha1_base64="Ofc5IIf//Bzeosc6rWoR+844kpU=">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</latexit>• It has been expected that the spontaneously broken chiralsymmetry is restored at high T and µ.

M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit>• Super restoration: If s develops a negative expectation value,
the explicitly broken chiral symmetry can be also restored.

GN model (D=2)

Phasetransition

M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit>

M ! m0r
M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit><latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">AAACd3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsI+SrEFGWmZ5QkFhXllyvkxhtwxQsoG+gZgIECJsMQylBmgIKAfIHlDDEMKQz5DMkMpQy5DKkMeQwlQHYOQyJDMRBGMxgyGDAUAMViGaqBYkVAViZYPpWhloELqLcUqCoVqCIRKJoNJNOBvGioaB6QDzKzGKw7GWhLDhAXAXUqMKgaXDVYafDZ4ITBaoOXBn9wmlUNNgPklkognQTRm1oQz98lEfydoK5cIF3CkIHQhdfNJQxpDBZgt2YC3V4AFgH5Ihmiv6xq+udgqyDVajWDRQavge5faHDT4DDQB3llX5KXBqYGzWYARYAhenBjMsKM9AzN9EwCTZQd7KBRwcEgzaDEoAEMb3MGBwYPhgCGUKC9FQzzGJYzrGD8ySTPpMakAVHKxAjVI8yAApgMAbQMkJg=</latexit>

M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit>GN model (D=3)

M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit>

M ! m0r
M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">AAACd3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsI+SrEFGWmZ5QkFhXllyvkxhtwxQsoG+gZgIECJsMQylBmgIKAfIHlDDEMKQz5DMkMpQy5DKkMeQwlQHYOQyJDMRBGMxgyGDAUAMViGaqBYkVAViZYPpWhloELqLcUqCoVqCIRKJoNJNOBvGioaB6QDzKzGKw7GWhLDhAXAXUqMKgaXDVYafDZ4ITBaoOXBn9wmlUNNgPklkognQTRm1oQz98lEfydoK5cIF3CkIHQhdfNJQxpDBZgt2YC3V4AFgH5Ihmiv6xq+udgqyDVajWDRQavge5faHDT4DDQB3llX5KXBqYGzWYARYAhenBjMsKM9AzN9EwCTZQd7KBRwcEgzaDEoAEMb3MGBwYPhgCGUKC9FQzzGJYzrGD8ySTPpMakAVHKxAjVI8yAApgMAbQMkJg=</latexit><latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit>

M ! m0r<latexit sha1_base64="93VqckmQY7Z3tAjrep7/AJpnGlc=">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</latexit>Phase structure (GN model)

Gap equation with M=m0
• At the limit m0=0 for the gap equation with M=m0

Specific points• µ=0

Phase boundary at m0=01+

<latexit sha1_base64="RQtPeYhALT21xS2Hus4QqMjwaus=">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</latexit>

T /µr = e/⇡
T /µr = 1/ ln 2<latexit sha1_base64="bi7NPywMcBmNIZQpPqtM/hclVhQ=">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</latexit>

D=2<latexit sha1_base64="i8snNhC9q4ppLYZ8XS3Bzw92mAk=">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</latexit>

T /µr = e /⇡T /µr = 1/ ln 4<latexit sha1_base64="qC7pO+PPq+doGaxPnQv5FGqirTU=">AAACdHichVHLSsNAFD2N7/po1Y2gi2KouKoTKSqCILpxaV8qWAlJHGtoXuRR0OIP+AMuXCmKiJ/hxh9w0U8QlxXduPAmDYiKesNkzpy5586Zuapj6J7PWCshdHX39Pb1DyQHh4ZHUunRsS3PDlyNVzTbsN0dVfG4oVu84uu+wXcclyumavBttb4e7m83uOvptlX2jxy+Zyo1Sz/QNcUnSk6nynNVM5DdFWmualiZvJwWWY5FkfkJpBiIiGPTTt+gin3Y0BDABIcFn7ABBR59u5DA4BC3hyZxLiE92uc4QZK0AWVxylCIrdO/RqvdmLVoHdb0IrVGpxg0XFJmkGWP7Ja12QO7Y0/s/ddazahG6OWIZrWj5Y6cOp0ovf2rMmn2cfip+tOzjwMsRV518u5ETHgLraNvHJ+1S8vFbHOGXbJn8n/BWuyebmA1XrTrAi+eI0kNkL4/90+wNZ+TFnL5Ql5cXYtb0Y9JTGOW3nsRq9jAJipRT85xhevEqzAliEK2kyokYs04voSQ+wAxfI7/</latexit>

D=3

• T=0
<latexit sha1_base64="4LPlEjn+g5XnbmGaT86C9gq3zk8=">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</latexit><latexit sha1_base64="vyre0RkdNlF4sSxthDEogjbYlDI=">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</latexit>µ/µr = e/2µ/µr = 2<latexit sha1_base64="Bnsw0AKHEpp6jfZyc72gIlQK7ZU=">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</latexit>

D=2
D=3

p

µ/µr = 1/ 2µ/µr = 1<latexit sha1_base64="L5ycWWzXuqXAiw9UdfY2mVE/K6Y=">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</latexit>

2-flavor NJL model (D=4)• The results depend on the regularization. Here we employ asimple cut off.
• 3DRT<latexit sha1_base64="uKQ8tbwluc2aQjRmvRu9R300FGI=">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</latexit>

⇥

0

T

0

T

h i = 2G itrS (M, ⇤) + itrS (M, ⇤)• 3D<latexit sha1_base64="oKGWCtmdG8A7W1Cnn+X2OHcSET8=">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</latexit>⇥⇤

h i = 2G itrS (M, ⇤) + itrS (M, 1)⇤

H. Kohyama, D. Kimura and T. I. , Nucl. Phys. B896 (2015) 682.

NJL model (D=4)

T.I., D. Kimura and H. Shimoji, arXiv:2306.00470,
E. B. Pasqualotto, R. L. S. Farias, W. R. Tavares, S. S. Avancini,and G. Krein, Phys. Rev. D 107, 096017 (2023)

3DRT
3DRT
3D

3D

NJL model (D=4)⇥

h i3D= 2G itrS 0 (M, ⇤)
⇥
0
h
i
=
2G
itrS
(M, ⇤) + i
3D
⇥
⇤
0
T
h i = 2G itrS (M, ⇤) + itrS (M, 1)
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3DSummary and concluding remarks
Super restoration in four-fermion interaction models:• The spontaneously broken chiral symmetry can be restored
at extremely high temperature or chemical potential.
• We can not find interesting phenomena around the boundaryof the super restoration except for the chiral susceptibility.
What happens in a gauge theory?
• Preliminary result for SD equation in strong coupling QED:Super restoration has not been observed!?