From a complex scalar field to the

hydrodynamics of dense quark matter Stephan Stetina

Institute for Theoretical Physics, TU Wien Mark G. Alford, S. Kumar Mallavarapu, Andreas Schmitt arXiv:1212.0670 [hep-ph]

dense matter in the QCD phase diagram

• r mode instability [e.g., N. Andersson, Astrophys. J. 502, 708 (1998)]

• pulsar glitches [e.g., B. Link, MNRAS 422, 1640 (2012)]

Superfluids in dense quark matter

• CFL breaks chiral symmetry and Baryon conservation:

octet of (pseudo) goldstone modes

CFL is a superfluid! - (exact) Goldstone mode due to U(1)B

• Kaon condensation in CFL at high (but not asymptotically high) densities:

going down in density: ms becomes non negligible

CFL reacts on stress on pairing pattern by developing a kaon condensate

[(P. F. Bedaque, T. Schäfer, NPA 697, 802,2002) ]

on top of CFL breaking pattern: (not exact) Goldstone mode due to U(1)S

• how many superfluid components are to expect in CFL with kaon condensation?

U(1)s ! 1

Landau´s model of superfluidity

Comparison: Landau vs microscopic model

Landau model microscopic model

• Phenomenological model based on

hydrodynamic equations such as:

• Fluid formally divided into superfluid and normal density (very successful describing superfluid 4He )

• Variables : , , …

~g = ½s~vs + ½n~vn

² = ²n + ²s +½sv

2s

2+

½nv2n

2

• QFT model based on SSB and the existence of Goldstone modes

• Condensate related to superfluid, elementary excitations to the normal fluid part

• Variables :

- gradients of Goldstone fields:

,…

@0ª~rª@0Ã; ~rÃ; T

½; ~vs; T

SSB in a φ4 model

• from effective theory for CFL mesons:

• ansatz for the condensate:

• assumption:

homogenius superflow

L= @¹'@¹'¤ ¡m2j'j2 ¡¸j'j4

¾2 = @¹Ã@¹Ã

½; @¹Ã = cons

L=¡U +L(1) +L(2) +L(3) +L(4)

U = ¡12@¹½@

¹½¡ ½2

2

¡¾2 ¡m2

¢+¸

4½4

'(x)! ½(x)p2eiÃ(x) + fluctuations

¹K0 =m2s ¡m2

d

2¹¸eff =

4¹2K0 ¡m2

K0

6f2¼

Comparison: Landau vs microscopic model

hydrodynamics microscopic model

• super current

• stress-energy tensor

• superfluid density

• superfluid velocity

• energy density

• pressure

• Noether current

• stress-energy tensor

• superfluid density

• superfluid velocity

• energy density

• pressure:

v¹ = ° (1;~vs)

j¹ = nsv¹

T¹º = (²s +Ps)v¹vº ¡ g¹ºPs

ns =pj¹j¹ = v

¹j¹

²s = v¹vºT¹º

Ps = ¡13(g¹º ¡ v¹vº)T¹º

v¹ = @¹Ã=¾

j¹ = @L=@ (@¹Ã) = @¹Ã(¾2¡m2)=¸

ns = ¾(¾2¡m2)=¸

²s = v¹@¹Ãns ¡L

Ps = L+(v¹@¹Ã¡ ¾)ns

T¹º = @¹Ã@ºÃ(¾2¡m2)=¸¡ g¹ºL

Superfluidity from φ4 model (T=0)

• connection to thermodynamics:

chemical potential and flow velocity of the superfluid are both determined in

terms of the phase of the condensate:

-rotations of the phase around the U(1) circle generate the chem. pot.

-number of rotations/unit length gives rise to the superflow velocity

Lorentz factor in σ:

¹s = ¾ = v¹@¹Ã²s +Ps = ¹sns

¾ =

r(@0Ã)

2 ¡³~rô2= @0Ã

vuut1¡Ã~rÃ@0Ã

!2= ¹

p1¡ v2

finite temperature

• use pressure as effective action:

• Anisotropic dispersions (Goldstone + massive):

• Anisotropic pressure:

• low T approximation: use linear and cubic terms in k in the dispersions.

• to go (numerically) up to Tc : use 2 particle irreducible formalism (CJT) (J.M. Cornwall, R. Jackiw, and E. Tomboulis, Phys. Rev. D10, 2428 (1974))

(M.G. Alford, M. Braby, A. Schmitt: J.Phys.G35:025002 (2008))

²1;k =

q¾2¡m2

3¾2¡m2 ³(cos µ) jkj+O(jkj3)

²2;k =p2p3¾2 ¡m2 +2(rÃ)2 +O(jkj)

¡eff [©; S] =¡U(©)¡ 12Tr lnS¡1 S¡1(k) =

à ¡k2 + 2(¾2 ¡m2) 2ik ¢ @Ã

¡2ik ¢ @Ã ¡k2

!

³(µ) =

p1¡ v2s

q1¡ v2s

3(1 + 2cos2µ) +

2jvsjp3cosµ

1¡ v2s3

P±ij !fP?; Pkg

Carter`s canonical two fluid formalism [B. Carter and I. M. Khalatnikov, PRD 45, 4536 (1992)]

• generalization of TD relation (generalized Pressure and Energy):

hydro description either in terms of conserved currents or their conjugated momenta! (this set of variables differs from and )

• definition of coefficients (A,B,C), A “entrainment coefficient”

• stress energy tensor:

@¹T¹º = 0

@¹j¹ = 0

@¹s¹ = 0d¤= @¹Ãdj

¹+£¹ds¹ dª= j¹d(@

¹Ã) + s¹d£¹

@¹Ã = @¤@j¹

= Bj¹ +As¹ ;£¹ = @¤@s¹

=Aj¹ + Cs¹

T¹º =¡g¹ºª+ j¹@ºÃ+ s¹£º

@¹j¹s 6= 0 @¹j

¹n 6= 0

T¹º =¡g¹ºª+Bj¹jº + Cs¹sº +A(j¹sº + s¹jº)

ª=ª[(@Ã)2; µ2; @Ã ¢ µ]¤ = ¤[j2; s2; s ¢ j]

²+P = ¹n+Ts ¤+ª= @Ã ¢ j + µ ¢ s

relativistic two fluid formalism

• relation to formalism of Landau\Khalatnikov: [e.g., D. T. Son, Int. J. Mod. Phys. A 16S1C, 1284 (2001)]

• expressible through coefficients:

• the microscopic calculations are performed in the normal fluid restframe (restframe of the heat bath) , in this frame, we can identify:

T¹º = (²n+Pn)u¹nu¹n+Png

¹º +(²s+Ps)u¹su¹s +Psg

¹º

j¹ = nnu¹ +ns

@¹Ã

¾

A= ¡¾nnsns

; B = ¾ns; C = ¾n2n

s2ns+ ¹nn+sT

s2

ª= 13(g¹º ¡ u¹uº) (j¹@ºÃ¡ T¹º) ª = T

V¡ u¹ = (1;~0)

¤+ª= @Ã ¢ j + µ ¢ s µ0 = T

s¹ = su¹

s0 =@ª

@T

relation to microscopic theory

• coefficients:

• some exemplary results:

A = ¡ @0Ã

s0~rà ¢~j

³~v2sj

0@0à ¡~j ¢ ~rô

C =j0@0Ã

³~v2sj

0@0à ¡~j ¢ ~rô+ s0µ0

³~j ¢ ~rÃ

´

(s0)2³~j ¢ ~rÃ

´

T

V¡eff '

¹4

4¸

¡1¡ v2s

¢2+¼2T 4

10p3

¡1¡ v2s

¢2

(1¡ 3v2s)2¡ 4¼4T6

105p3

¡1¡ v2s

¢2

(1¡ 3v2s)5¡5 + 30v2s + 9v

4s

¢

ns '¹3

¸(1¡ v2s)¡

4¼2T 4

5p3¹

1¡ v2s(1¡ 3v2s)3

+8¼4T6

105p3

1¡ v22(1¡ 3v2s)6

¡95 + 243v2s ¡ 135v4s ¡ 27v6s

¢

nn '4¼2T 4

5p3¹

¡1¡ v2s

¢2

(1¡ 3v2s)3¡ 16¼4T 6

35p3¹3

¡1¡ v2s

¢2

(1¡ 3v2s)6¡15 + 38v2s ¡ 9v4s

¢

B = ¡

³~rô2

~rà ¢~j

first and second sound

• two fluid system allows for two sound modes:

first sound:

normal and super fluid densities

oscillate in phase , pressure wave,

weak temperature dependence

second sound:

normal and super fluid densities

oscillate out of phase , temperature wave

figure: [R.J. Donnelly, Physics Today 62, 10 (2009)]

first and second sound

T = 0

T=¹ = 0:02

T=¹ = 0:04

Outlook

towards a complete hydro description of CFL and kaon

condensation

• use the CJT formalism to numerically go up to Tc

[work in progress; Mark G. Alford, S. Kumar Mallavarapu, Andreas Schmitt , Stephan Stetina]

• include effect of weak interactions (include a U(1)s breaking term into the effective Lagrangian)

[work in prgress; Denis Parganlija, Andreas Schmitt]

• start from a fermionic system of Cooper pairs