Post on 18-Mar-2018
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
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)]
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