Zhao Zhang ( Kyoto University )

Vector-vector interaction, Charge neutrality and the number of QCD critical points

contents Introduction to QCD phase diagram Competition between chiral condensate and diquark

condensate Effect from vector-vector four quark interaction Effect from electrical chemical potential μe under charge-

neutrality & β-equilibrium Multiple critical-point structures for chiral restoration Summary & Outlook

Collaborators: K. Fukushima, T. Kunihiro

1. 1 Schematic Phase Diagram of QCD

?

What’ s the possible phase of non-CFL ?What ‘s the effect of non-CFL on chiral restoration ? New critical points and crossover in dense QCD ?

Low T region :

Two critical points (end of first order transition) Liquid-gas transition confirmed by experiment Chiral critical point existence and location ?

Recent review on CSC: Alford, Rajagopal, Schaefer and Schmitt , RMP, (’08)

2nd order phase transition, massless mode, critical fluctuations

Ruster, Werth, Buballa, Shovkovy and Rischke, PRD (’05)See also, Abuki and Kunihiro, NPA (’06)

1.2 Phase diagram from 3-flavor NJL model

Strong diquark coupling Weak diquark coupling

The possible candidate for non-CFL: 2SC or g2SC

The influence of 2SC (g2SC) on chiral restoration ?

1.3 The number of CP’s in QCD phase diagram ?The results from NJL: 1,2,3,4,0 are all possible

Mechanisms: Vector interaction and charge-neutrality

T

μ

T T

T

μ

μ

μ

The key for the appearance of miltiple critical-point structures

• The competition between chiral condensate and diquark condensate:

I. The emergence of coexisting phase for chiral symmetry breaking and color superconducting phase

II. Abnormal T dependence of the gap of the mismatching diquark condensate in coexisting region

Order parameter Chiral Symmetry Breaking

2. Competition between cSB and CSC

2.1 Chiral symmetry breaking phase (cSB)

In the chiral limit (mq =0):

mq Mq

current quark mass

dynamical quark mass

2.2 Color superconducting phase ( CSC) Cooper instability:

BCS: In sufficiently cold fermionic matter, any attractive interaction leads to the instability to form Cooper pairs.

QCD at asymptotic density: weak interaction: asymptotic freedom quark Fermi surface Attractive channel: one-gluon exchange interaction

QCD at moderate density: Instanton-induced four-quark interaction models support CSC, for example NJL model

Many internal degrees of freedom of quark: spin, color, flavor, patterns of quark-quark cooper paring are more complicated than that of normal SC

• u

• CFL• dSC• uSC• 2SC

ud s

u

d s

u

d s sd

u

CFL : (3) (3) (3) (3)C L R C L RSU SU SU SU 2SC : (3) (2)C CSU SU

Chiral restoration with increasing μ

E

0

E

0

Large μ and small M means large Fermi sphere and strong

Cooper instability

mq

qq

q

Baryon density suppresses the quark-antiquark pairing.

2.3 Competition between at moderate density

Formation of quark Cooper pair with increasing m

and

MM

2.4 .1 Repulsive vector interaction

However, vector interaction channel also naturally appears in the effective theories.

Instanton-anti-instanton molecule model, Schaefer,Shuryak (‘98)

82

52

22

52

2)()(

2

1)()(

2L

Ni

NGL aa

C

aa

C

Renormalization-group analysis, N.Evans et al. (‘99), Schaefer, Wilczek (‘99)

2200 )()( LLLLllLL GL

Dyson-Schwinger equation model of QCD, Cahill, Roberts (‘85)

GV /GS =0.25

GV /GS =0.5

Hadron spectroscopy: Klimt,Luts,&Weise (’90), Roberts, Williams (’94)

In the 4-quark interaction models, scalar and pseudoscalar interaction channels have been extensively used to study cSB and CSC

2.4 Two mechanisms for enhancing the competition between cSB and CSC

2

VG

2

2

2

52C

AC

AD iiG 2

5

2 iGS

Asakawa,Yazaki ’89 /Klimt,Luts,&Weise ’90 / Buballa,Oertel ’96

Chiral restoration is weakened and delayed towards higher μ

With increasing GV:

2

VG density-density correlation

0 Quark number density:

1. Negative dynamical chemical potential: 2 20 0 2

V V VG G G 2. Negative contribution to the free energy

, vector-type quark-antiquark condensate

In NJL:

2SG Sqq GmM 2 dynamical quark mass

VG2

Enhancing the competition between cSB and CSC

Kitazawa, Koide, Kunihiro & Nemoto (’02)

0/ SV GG 35.0/ SV GG 5.0/ SV GG

2 critical points

COE becomes broader with increasing Gv due to the delay of cSB restoration

the competition of two condensates in COE leads to new CP

Coexistence (COE) region is the key for the appearance of two CP’s:

1 critical point no critical point

μu =μ - 2μe/3

μd =μ + μe/3

μe

du 6

1

2

μe 100MeV for μ400MeV

2.4.2 Electrical chemical potential (μe) under charge-neutrality & β-equilibrium

vs VG2

eVdu GQQ ;6

1

Role 1: μe > 0, an effective vector interaction

Role 2: μe > 0, the mismatch between μu and μd .

22eud

Effect from mismatch quark-quark paring under charge-neutrality & β-equilibrium

ρd > ρu >ρs Mismatch paring or pair breaking

, Fm p

Standard BCS paring

For 2-flavor asymmetric homogenous CSC:

(1)Abnormal thermal behavior of diquark gap

( )n p

pF p

TDouble effects of T :

The competition between two condensates may be enhanced with increasing T

Fermi sphere is smeared by T

(2)Chromomagnetic instability: imaginary Meissner mass

dus

e

MM

diagQ

,

)3

1,

3

1,

3

2(

Fdue PMM ,

2

2

1

Schematic critical-point structure with charge-neutrality and CSC

Possible critical-point structures induced by μe

SD GG /weak strong

“survivor”“remnant”

β-equilibrium condition

Zhang, Fukushima and Kunihiro, PRD, (’09), arXiv: 0808.3371

Local charge-neutrality

Two constraints for bulk matter:

Results from a simple 2-flavor NJL:

3. Multi-critical-point structures of QCD ?

3.1 Multi-critical-point structures induced by μe under charge-neutrality & β-equilibrium

)( miNJL

])()[( 25

2

iGS

)])([( 2525 ATT

AD CCG

Result : Impact on chiral restoration without CSC

Increasing Qμ corresponds to increasing GV

Chiral restoration is delayed and weakened with Increasing Qμ

Same effect as vector interaction

(Qd = -1/3 fixed).

In nature, electric charges : Qu = 2/3 , Qd = -1/3

edu QQ -

Without charge-neutrality With charge-neutrality

First report on 3 critical-point structure

Intermediate coupling: GD / GS = 0.75

COE is significantly broadened by nonzero μe Line E-F becomes crossover due to the abnormal thermal behavior of diquark

condensate D and E are free from chromomagnetic instaibility, the fate of F is unknown

1 CP 3 CP’s !

Result : Impact on chiral restoration with and without charge-neutrality including CSC

A general phenomenon for asymmetric homogenoues superconductivity or superfluidity system

Leading to 3 critical points structure

Large Small

In COE region In CSC region

SD GG /

For stronger diquark coupling cases

2 CP’s no CP

For weaker diquark coupling cases

1 CP 1 CP

The multiple critical point structures are not observed in traditional NJL model if the first order chiral phase transition is relatively strong.

Ruster, Werth, Buballa, Shovkovy and Rischke, PRD (’05)

3.2 Combined effects of vector-interaction and charge-neutrality&β-equilibrium on chiral phase transition

For more real case, both vector interaction and neutral charge-constraint should be taken into account simultaneously Combined effect on chiral transition should be more significant.

Role 1:helping ue to fulfill the multi-critical-point structure

Role 2: effect on chromomagnetic instability

Suppressing the chromomagnetic instability ?

suppressing

0 ud

Results from a nonlocal two flavor NJL with different sets of model parameters

We used 3 sets of the model Parameters to check the model- parameter sensitivity of the main results.

Model parameters fixing (Gs and Λ):

MeV 250 -,4.92,135,53

1

, MeVfMeVmMeVm du

taking Gv as free parameter with GD/Gs =0.75

Two flavor NJL model with vector interaction

Parameters set 1 : M(p=0)= 400 MeV, the stronger first order chiral transition case

Increasing Gv/Gs

Parameters set 2 : M(p=0)= 367.5 MeV , the relatively strong chiral first order trannsition case

Parameters set 3 : M(p=0)= 330 MeV, the weak chiral transition case

In contrast to the case with parameters set 1, the remnant first vanishes with increasing Gv

Effect of Vector interaction on chromomagnetic instability

With increasing Gv/Gs:

The ratio μ/Δ is effectively suppressed Unstable region shrinks to larger μ and lower T

Unstable region

3.3 Extension to 3 flavor caseEffect from μe with charge-neutrality & β-equilibrium • For 3 flavor limit, μe=0. • For 2+1 flavor case, μe still plays important role on chiral restoration

due to the relatively heavy s quark.Effect from vector interaction and UA(1) breaking

interaction?

Mean field thermal potential :

Results from 2+1 flavor NJL under charge-neutrality

vector-vector interaction’t Hooft six-quark interaction

Only considering 2CSCOther possible cubic coupling terms are ignored

Order of critical-point numbers with increasing Gv/Gs: 1,2,4,2,0.

For example:

2 CP’s 4 CP’s 2 CP’s

All of the critical-point structures appearing for 2 flavor case are confirmed in 2+1 flavor case with some range of model parameters.

95.0/ SD GG

Vector interaction can effectively suppress the unstable asymmetric homogenous CSC region related to chromomagnetic instability.

Summary : Vector interaction can delay and weaken chiral restoration, enhance the the competition between chiral condensate and diquark condensate. Under charge-neutrality & β-equilibrium, electric chemical potential can be

regarded as an effective vector interaction. The abnormal thermal behavior of mismatch paring can enhance the

competition in COE with increasing T.

5 types of critical-point structure are found in NJL study, Multi-critical-point structures are possible to happen in QCD

Outlook:Effect of Vector interaction on asymmetric inhomogeneous CSC,

especially the role of space-like vector-type quark-antiquark condensate.

Z.Zhang and T. Kunihiro, in preparation