Melting Pattern of Diquark Condensates in Quark Matter

Home , Diquark

ーGinzburg-Landau approach in Color Superconductivityー

Based on hep-ph/0312363

Motoi Tachibana (RIKEN) Kei Iida（RIKEN-BNL） Taeko Matsuura (Univ. Tokyo) Tetsuo Hatsuda (Univ. Tokyo) 1. Introduction

150MeV Quark-Gluon Plasma

Hadron Color Superconductivity μ 400MeV Neutron Star Core?

(Conjectured) Phase Diagram of Hot and Dense Quark Matter Color Superconductivity

Characterized by diquark condensate

Attractive force via color 3-bar gluon exchange

Lots of internal d.o.f such as spin, charge, color & flavor ‡Complicated phase structures depending on T & μ

Taking into account for the nonzero strange quark mass and the charge neutralities →Much closer situation for the real systems (Already) lots of phases in color superconductivity

Originally 2SC CFL 1997

Then Crystalline Kaon condensed CFL 2000

More recently

Gapless 2SC Gapless CFL 2003

Much more recently CFL-η Last week Purpose of this study

Investigation of thermal phase transition in color superconducting quark matter with 3 flavors and 3 colors, particularly emphasizing on the interplay between effects of the strange quark mass ( m s ) & electric and color charge neutrality near the transition temperatures.

Ginzburg-Landau approach near Tc ＋ † weak coupling analysis

Result Single Phase transition Multiple phase transitions (m 0) ms ≠ 0 s = dSC as a new phase

† † 2. Ginzburg-Landau a la Iida-Baym PRD63(’01)074018 GL potential (mu,d ,s = 0)

r 2 r 2 2 r * r 2 S = a | d a | + b1( | d a | ) + b2 | d a ⋅ d b | Â Â Â i, j,k = u,d,s a a a,b † a,b,c = r,g,b i P + fbcjk = eabceijk (da ) : Paring gap J = 0 d u ~ (ds) † 3x3 matrix † d d ~ (us) In the weak coupling, Density of State † d s ~ (ud) † 7z (3) b ≡ b1 = b2 = 2 N(m) a = 4N(m)t ≡ a 0t 8(pTc ) 2 2 N(m) = m /2p t = (T - Tc )†/T c † †

† † GL free energy Ω

Δ Δ W = Δ Δ ＋＋・・・ Δ Δ

where 1 = 0 0 † g ⋅ q + M + mg + dmg

Quark mass Charge chemical † potential Quark chemical potential Weak coupling analysis (Ms=0) T

a (di = 0) Normal(QGP) i ・All d have the same Tc Tc † (single phase transition) Super(CFL) ・CFL is energetically favorable (d a µd a ) i i † at weak coupling

†

How m s ≠ 0 , charge neutrality affect?

† 3. Effects of m s , charge neutrality and instanton

I. Strange quark mass

u 2 d 2 r 2 s 2 eÂ(| da | + | da | ) = eÂ(| d a | - | da | ) (e > 0) †a a II. Electric charge neutrality d s (ud) Ê ˆ 1 r 2 u 2 † † hÁ Â| d a | -Â| da | ˜ Ë 3 a a ¯ † III. Instanton d d (us) s 2 u † z | da | d Â (ds) a †

† † 2 In the weak coupling, up to O ( m s ) and leading order in g

Strange quark mass † e @ 2a 0s Electric charge neutrality h @ a 0s

Color charge neutrality 2 dmCN /b = (Tc /gm) Instanton † Ê ˆ 9Ê ˆ14 ms LQCD 1 z ~ -a 0 Á ˜ Á ˜ † m Ë m ¯ Ë g¯ † 3p 2 Ê m2 ˆ where s s ≡ Á 2 ˜ negligible! † 8 2 Ë gm ¯

† 4.Hierarchical color-flavor unlockings Parametrization of the paring gap

D1,2,3 ≠ 0 : mCFL Ê D1 0 0 ˆ i Á ˜ D1 = 0,D 2,3 ≠ 0 : uSC d = 0 D 0 a Á 2 ˜ dSC Á ˜ D 2 = 0,D1,3 ≠ 0 : Ë 0 0 D 3 ¯ D1,2 = 0,D 3 ≠ 0 : 2SC

2 D 3 GL action up to O ( m s ) † 2SC mCFL 2 2 2 2 2 † dSC S = a'(D1 + D 2 + D 3 ) -eD 3 -hD1 2 2 2 2 4 4 4 +b (D + D + D ) + b (D + D + D ) D 2 1 1† 2 3 2 1 2 †3 with a'= a + e + h /3 D1 † †

† † Minimization conditions: ∂S /∂D1,2,3 = 0 Ê ˆ 2 a 0 Tc - T 8 1) mCFL phase: D 3 = Á + s˜ 8b Ë Tc 3 ¯

2 a Ê T - T 4 ˆ Ê 16 ˆ D = 0 c - s I † 1 Á ˜ Tc = Á1 - s˜T c 8b Ë Tc 3 ¯ Ë 3 ¯ Ê ˆ 2 a 0 Tc - T 16 D 2 = Á - s˜ 8b Ë Tc 3 ¯

Ê †ˆ 2 a 0 Tc - T 2 2) dSC phase D 3 = Á + s˜ † 6b Ë Tc 3 ¯ Ê 7 ˆ T II = 1- s T Ê ˆ c Á ˜ c 2 a 0 Tc - T 7 Ë 3 ¯ D1 = Á - s˜ 6b Ë Tc 3 ¯

3) 2SC phase Ê ˆ Ê ˆ 2 a 0 Tc - T 1 † III 1 D 3 = Á - s˜ T = Á1 - s˜T 4 T 3 c c † b Ë c ¯ Ë 3 ¯

† † Transition temperatures of the 3-flavor color superconductor 1 sTc normal Tc Tc 3

III Tc 2SC 2sTc 4sTc † † † II 5 s Tc sT T 3 c c dSC 4 † † I T † s c Tc 3 † † (a)† (b) (c) mCFL † (a) All the quarks are †massless (degenerate） (b) Finite Ms is considered (c) Electric charge neutrality is further imposed Fig2. Schematic illustration of the gap as a function of T

mCFL dSC

2SC D2 3 normal

2 D1

2 † D 2 T T I T II T III † c c c

† † † † † Realized symmetries, Gapless modes and # of massive gluons

symmetry gapless quark # of massive modes transv. gluons mCFL [U(1)]2 none 8 dSC [U(1)]4 bu, rs, (ru, bs) 8 2SC [SU(2)]2 ¥[U(1)]2 bu, bd, bs, rs, gs 5 † † More gapless quarks may appear in the close vicinity I II III 2 † of T c , T c and T c where the gaps are less than m s /m

† † † 5. Summary and Discussion

Hierarchical color-flavor unlockings at finite temperature in the Ginzburg-Landau approach

mCFL -> d S C -> 2SC -> normal

O(sTc )

mCFL ? †

CFL gCFL μ Fluctuation of the gauge fields (Matsuura et al., PRD69(‘04)074012)

mCFL -> dSC remains second order dSC -> 2SC and 2SC -> normal become weak 1st order

Future issues

Connection the results at Tc with those at T=0.

Getting to lower densities.