FRG approach to QCD phase diagram with isospin chemical potential

Kazuhiko Kamikado (Yukawa Institute for Theoretical Physics, Kyoto Univ)

working with Nils Strodthoff, Lorenz von Smekal and Jochen Wambach (TU Darmstadt)

1

12年1月18日水曜日

QCD phase diagram (2-flavor)

•Three-dimensional phase diagramTemperature [T]Quark chemical potential [µ]Isospin chemical potential [µf]

• Isospin chemical potential is associated by the charge of subgroup of flavor rotation (τ3).

•charged pion condensation occur for large µf.

µu = µ + µf

µd = µ� µf

2

12年1月18日水曜日

Sign problem

•Quark determinant is real (µ=0,µf≠0).

•Lattice Monte Carlo simulation is available.•We can check the reliability of effective models.

3

D = U�1D†U U =�

0 �5

�5 0

�

det[D] = det[D†]

M. Alford, A. Kapustin and F. Wilczek, Phys. Rev. D59, 054502 (1999).

12年1月18日水曜日

Property at T = 0

•Silver blazeCritical isospin chemical potential corresponds to 1/ 2 of lowest meson mass (pion).

•Any effective model must realize this property.

4

T. D. Cohen, Phys. Rev. Lett. 91, 222001 (2003); arXiv:hep-ph/0405043.

“Silver blaze” Arthur Conan Doyle

µcf =1

2M⇡

12年1月18日水曜日

Results of another models

•Pion condensation occur.•Another models satisfy SB relation. 5

!!" = " ,

!#+" = !#!" =$#2

, (80)

we obtain the e!ective Lagrangian at tree level,

Leff = $U(", $) +1

2

!

(%µ!)2 + (%µ#3)2 + (%t#1 $ µI#2)

2

+ (%t#2 + µI#1)2 $ (%#1)

2 $ (%#2)2"

$ g2!

!

#

3"2 + $2 $ f2! +

m2!

2g2!

$

!2

+

#

"2 + 3$2 $ f2! +

m2!

2g2!

$

#21

+

#

"2 + $2 $ f2! +

m2!

2g2!

$

(#22 + #2

3) + 4"$!#1

"

$ 2g2! ("! + $#1)

%

!2 + #2&

$g2

!

2

%

!2 + #2&2

, (81)

with the classical potential

U(", $) = $1

2

%

2g2!f2

! $ m2!

& %

"2 + $2&

+g2

!

2

%

"2 + $2&2

$ f!m2!" $

1

2µ2

I$2 , (82)

where the two condensates are determined by minimizingthe potential,

%U

%"= 0,

%U

%$= 0 . (83)

We have chosen pion condensate to be real as in the NJLmodel. In this section we use " and $ standing for thechiral and pion condensates in order to avoid confusionwith the definitions in the NJL model.

In the vacuum the constraints (83) on the potentialgive " = f! and $ = 0, and the sigma mass can be readout from the quadratic term in the sigma field in thee!ective Lagrangian (81), m2

" = 4g2!f2

! + m2!.

At finite isospin density the two condensates in theground state satisfy " = 0, $ = f!

'

1 + 2µ2I/m2

" in thechiral limit. This is consistent with the NJL model: Anysmall isospin density can force the chiral symmetry to berestored in the chiral limit. In the real world, we obtain" = f!, $ = 0 for µI < m! which is the same as in thevacuum, and [5]

" = f!m2

!

µ2I

,

$ = f!

(

1 $#

m2!

µ2I

$2

+ 2µ2

I $ m2!

m2" $ m2

!, (84)

for µI > m!. This result is qualitatively consistentwith the NJL model and the chiral perturbation theory.Especially the critical behaviors of the two condensates

around the phase transition point µcI = m! in the three

e!ective models are almost the same.The thermodynamic functions can be evaluated from

the potential U . The isospin density nI , pressure p andenergy density & are analytically written as

nI = $%U

%µI= µI$

2

= µIf2!

#

1 $m4

!

µ4I

+ 2µ2

I $ m2!

m2" $ m2

!

$

,

p = $U(", $) + U(f!, 0)

=1

2f2

!(µ2I $ m2

!)2#

1

µ2I

+1

m2" $ m2

!

$

,

& = $p + µInI

=1

2f2

!(µ2I $ m2

!)

#

µ2I + 3m3

!

µ2I

+3µ2

I + m2!

m2" $ m2

!

$

. (85)

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

µI (GeV)

p/ε

LSM

CPT

NJL

mπ

0 0.1 0.2 0.3 0.40

0.5

1

1.5

µI(GeV)

σ/σ0

π/σ0

mπ

LSM NJL

CPT

FIG. 16. The ratio p/! (upper panel) and the chiral andpion condensates (lower panel) as functions of isospin chem-ical potential µI at T = µB = 0 in the NJL model (NJL),linear sigma model (LSM), and chiral perturbation theory(CPT).

If we take the limit m" & ', we reproduce the resultin the chiral perturbation theory [8,12]. However, thefinite sigma mass leads to a big di!erence at large µI .At large µI , the ratio p/& approaches 1 in the chiral per-turbation theory, but only about 1/3 in the linear sigmamodel and 0.7 in the NJL model, as shown in Fig.(16).While the three e!ective models are very di!erent in high

15

L. He, M. Jin, and P. Zhuang, Phys. Rev. D 71 (2005) 116001.

h�i

h⇡i

12年1月18日水曜日

The aim

•Describing QCD phase diagram on T and µf plane by using functional renormalization group equation on Quark Meson (QM) model•Reproducing Silver blaze property

6

12年1月18日水曜日

How to find mass

•Masses are determined by zero of 2 point function instead of curvature of potential.•We need to calculate 2 point function.

7

@U(�)

@�

������=�min

= 0

�(0,2)� (p0 = m�,�min) = 0

�(0,2)⇡ (p0 = m⇡0 ,�min) = 0

Our way

@U(�)

@�

������=�min

= 0

m2� = 2U 0 + 4�2U 00

m2⇡ = 2U 0

conventional way

12年1月18日水曜日

Functional Renormalization Group (FRG)

•Γk is effective action at scale k.

8

C. Wetterich, Phys. Lett. B301, 90 (1993)

k�k�k[�] =12Tr

�k�kRkB

RkB + �(0,2)k [�]

�� Tr

�k�kRkF

RkF + �(2,0)k [�]

�

�k=⇤ = S �k=0 = �

classical quantum

12年1月18日水曜日

Diagramatic representation

•Equations never close.•We need some truncation.

9

p + q

p

qq

p

p p

qq

+

q@�k[�]

@k⇠

@�(0,2)k [p,�]

@k⇠

�(0,3)k

�(0,3)k

�(0,4)k

@kRk(q)

@kRk(q) @kRk(q)

12年1月18日水曜日

Local Potential approximation (LPA)

•LPA leads to the flow equation for effective action.•We neglect wave function renormalization.

10

�k[�] = ̄ [i/@ + g(� + i�5~⇡ · ~⌧) + µf�0⌧3]

+1

2@�@� +

1

2@⇡0@⇡0 +

1

2~@⇡+~@⇡+ +

1

2~@⇡�~@⇡�

+ (@0 + 2µf )(⇡+ + i⇡�)(@0 � 2µf )(⇡+ � i⇡�)

+ Uk(X = �2 + ⇡23 , Y = ⇡2

+ + ⇡2�)� c�

12年1月18日水曜日

BMW approximation

•Equations are closed up to two point function.

11

q << k

J. P. Blaizot, R. Mendez-Galain and N. Wschebor, Phys. Rev. E 74, 051116 (2006)

�(0,3)abi [p,�p+ q,�q] ! �(0,3)

abi [p,�p, 0] =@�(0,2)

ab [p]

@�i

�(0,4)abij [p,�p, q,�q] ! �(0,4)

abij [p,�p, 0, 0] =@2�(0,2)

ab [p]

@�i@�j

12年1月18日水曜日

Consistency

•BMW approximation does not have consistency with curvature mass.•We can realize consistency under the another approximation

(RPA- like).

12

�(0,3)abi [p,�p, 0] =

@�(0,2)ab [p]

@�i! @3U

@�a@�b@�i

�(0,4)abij [p,�p, 0, 0] =

@2�(0,2)ab [p]

@�i@�j! @4U

@�a@�b@�j@�k

�(0,2)� [p = 0] = M2

�,curv �(0,2)⇡ [p = 0] = M2

⇡,curv

12年1月18日水曜日

How to calculate real time correlation

•After all analytic calculation along the Matsubara formalism, we make analytic continuation.•We directory solve flow equation for real time correlation

function.

13

@k�(0,2)(!n) = f(!n,�)

@k�(0,2)R (p0) = Re f(!n ! ip0 + ✏,�)

12年1月18日水曜日

pion 2 point function

• In this parameter set, LPA and BMW have good agreement at zero external momentum.

14

Approx � = �p

20 +M

2c

0

50

100

150

200

250

300

350

400

0 50 100 150 200

|K(p0)|1/2

p0

BMW/RPA/approx K/

McMp

At the vacuum

12年1月18日水曜日

phase transiton (T = 0)

•normalized by curvature mass (Mc)•There is the difference between onset of pion condensation

and curvature mass 15

0

10

20

30

40

50

60

70

80

90

100

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Ord

er p

aram

eter

µf/M/

m/

12年1月18日水曜日

phase transition (T = 0)

•normalized by pole mass (Mp)•pole mass almost satisfy silver blaze relation

16

0

10

20

30

40

50

60

70

80

90

100

0 0.1 0.2 0.3 0.4 0.5 0.6

Ord

er p

aram

eter

µf/M/

m/

12年1月18日水曜日

Results

•The value of curvature mass and pole mass are different at 17%.•Numerical result support pole mass.•We must use pole mass as meson masses instead of

curvature mass.

17

12年1月18日水曜日

Phase diagram (mean field)

•Neglect meson contribution for flow equation.•Only quark determinant term remains (mean field).

18

a = 0.1�2[Mev2]b = 3.5

c = 0.08�3[Mev3]� = 600[Mev]g = 3.2

U� = a(X + Y ) + b(X + Y )2 � 2µ2fY � c

�X

@UK

@k= Meson part + Fermion part

12年1月18日水曜日

phase diagram

•phase transition of sigma is cross over.•For large isospin chemical potential sigma became small.

19

</>

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1µf/ M/0

0

50

100

150

200

250

300

T

-10 0 10 20 30 40 50 60 70 80 90 100

<m>

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1µf/ M/0

0

50

100

150

200

250

300

T

20

30

40

50

60

70

80

90

100

12年1月18日水曜日

Conclusion

•We consider phase diagram with isospin chemical potential by using Functional renormalization group equation.•We calculate real time mesonic 2 point function and

redefine meson masses by pole of correlation function.

20

12年1月18日水曜日

21

Thank you for your attention

12年1月18日水曜日