Transcript QCD a densita’ finita

Color Superconductivity
Giuseppe Nardulli
Physics Department and INFN Bari
Varenna, June 29, 2006
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Collaboration Bari - FirenzeGeneva
• R. Casalbuoni, M. Ciminale, R. Gatto,
N. Ippolito, M. Mannarelli, G. Nardulli,
M. Ruggieri
Review papers
• F. Wilczek, K. Rajagopal, hep-ph/0011333
• G. N. Riv. N. Cim.(2002) hep-ph/0202037
• R. Casalbuoni and G.N. Rev. Mod. Phys,
(2004) hep-ph/0305069
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Nuclear matter and QCD
1. At small nuclear density and low temperature the
relevant degrees of freedom of nuclear matter are the
hadrons: nucleons, hyperons, mesons.
2. At high density and/or high temperature the relevant
dof are quarks and gluons. The corresponding theory
is Quantum ChromoDynamics
3. QCD is a gauge theory based on the local gauge
group SU(3), with quarks belonging to the
fundamental and gluons to the adjoint representation
4. QCD enjoys the property of Asymptotic Freedom
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Accelerators (RHIC,LHC, GSI) and compact stars
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Structure of a Pulsar
Continuous emission of e.m. radiation and slowing down
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Phenomenology of pulsar:
glitches
Sudden increases of pulsar rotational frequency
6
(Ω/Ω  10 )
Vela Psr
0833-45
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Further phenomenology
• Effects on EOS
• Cooling by neutrino emission
• Neutrinos from supernovae
• Shear viscosity and other transport properties
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Nuclear matter at T=0, large μ
1. Quarks have S=1/2. They are in three colors
(r,g,b) and, if light, in three flavors (u,d,s).
2. When two quarks scatter (3 x 3 ) they are
either in a sextet (color symmetric) or in
antitriplet (color antisymmetric) state
3. Gluon exchange in the antitriplet is attractive
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Free fermion gas and BCS mechanism
Per
T  0 (β  1/kT  )
f(E)
E
EF  μ
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Free energy
(F  E  μN)
• Adding a particle at the Fermi surface
• Extracting a particle, i.e. forming a hole
Free energy
unchanged
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For an arbitrary attractive interaction it is
energetically convenient to form particle pairs
(Cooper pairs)
E  (2EF  EB )  μ(N  2)  F  EB
In metals SC realized only in special conditions, when the
attractive interactions between electrons, due to phonon
exchange beats the repulsive e.m. forces
0
Tc (elettr.)
1  10 K
3
4
 4

10

10
E(elettr.) 10  105 0 K
QCD:attractive channels with
SC more efficient in QCD
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QCD phases at T=0, large μ
As a consequence of the attractive gluonic
interaction in the color antisymmetric
channel and the Cooper’s theorem:
Formation of Cooper pairs of quarks
(diquarks) and color superconductivity
The condensate is colored, not white as in
the quark-antiquark channel, relevant at
zero density
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Cooper pairs
a) Normal preferred state: Ppair=0
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b) Preferred total spin Spair=0
Two particle w.f.
ψ(r) = χ Σk gk eikr
r = r1 – r2
The spin wf χ with S=0 is antisymmetric in spin
therefore the orbital part has behavior
cos kr (symmetric in r), while S=1 has behavior sin
kr; coupling disfavored at small r
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Color Superconductivity at extreme densities
• QCD with 3 flavors: attractive interaction in color
antisymmetric (antitriplet) channel
• For S=0 state Pauli principle implies antisymmetry in
flavor
• Asymptotically (very large densities):Color Flavor
Locking (CFL)
< 0 | ψi α ψj β |0> = Δ εijγ ε αβγ
• Valid for μ >> mq
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Symmetries of the
CFL (color flavor locking) phase
< 0 | ψ ψ |0>  0
SU(3)c  SU(3)L  SU(3)R
 SU(3)cL R
broken spontaneously. Also U(1) global symmetry
(baryonic number) broken. 8 gluons acquire mass
(Higgs-Anderson). 8+1 Goldstone bosons similar
to π and K of chiral symmetry breaking
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CS at intermediate densities
1. Analysis complicated because the three light
quarks are different: strange quark has mass
comparable with μ for μ of the order of 500
MeV, while up, down are massless
2. These are the densities expected in the core of
neutron stars where CS most likely takes place
3. In stars quark matter must be neutral and in
beta equilibriun due to
d -> u  e
and
u e -> d 
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Neutrality and β equilibrium
Non interacting quarks
μd,s = μu + μe
If the strange quark is massless this equation has solution
Nu = Nd = Ns , Ne = 0; quark matter electrically neutral with no electrons
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Strange quark mass
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Mass effects, neutrality, beta equilibrium:
The result is that there are different gaps for quarks
of different flavor.
Gapless phases: uniform (gCFL) or not uniform
(LOFF). Three gaps
< 0 | ψi α ψj β |0> = γΔγ εijγ ε αβγ
Choosing the true vacuum:
The true ground state has the minimum
value of the free energy (grand potential)
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Gap Parameters
Free energy
M.Alford, P.Jotwani, C.
Kouvaris, J. Kundu,
K.Rajagopal
Gapless CFL
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The importance of being gapless
Gapless modes: relevant for astrophysical effects
(phenomenology of compact stars) µ= 500 MeV
Δ= 2523MeV
M2s/2µ= 80 MeV
Meissner masses in the gCFL phase
Solid line a=1,2;
Gluons 3, 8:
same behavior;
4,5,6,7: no instability
R. Casalbuoni,
M. Mannarelli,
G. N., M. Ruggieri, R. Gatto
Imaginary masses: instability
Symptom that it is not the true vacuum
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Loff Phase
If δμ0 it can be energetically
favourable to have
states where the Cooper pair
has total momentum 2q 0
LOFF= Larkin-Ovchinnikov&
Fulde-Ferrel (1964);
R. Casalbuoni, G.N.
Rev. Mod. Phys.(2004)
Δ ( r) = < 0 | ψ( r)ψ( r)|0> = Δ exp[i(p1 +p2 )r]
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Two flavors
Condensate BCS Condensate
(independent of δμ)
LOFF
Ω(BCS) - Ω (normal)
Ω(LOFF)-Ω (normal)
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Symmetry breaking for
LOFF phase with 3 flavors
SU(3)c  SU(3)L  SU(3)R

SU(3)cL R
T(3) x O(3) Oh
Cube group
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LOFF phase: three flavors
Needed for realistic calculations
• R.Gatto,R.Casalbuoni,N.Ippolito,GN, M.Ruggieri :
Ginzburg Landau expansion of the gap equation and free
energy and M.Mannarelli, K. Rajagopal, R. Sharma: one
plane wave
• K. Rajagopal, R. Sharma: Cristallography
< 0 | ψi α ψj β |0> = γΔγ εijγ ε αβγ q exp iqr
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Ginzburg-Landau expansion
of the gap equation Δ (small)
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Gap parameters: one plane wave
K( r) = K e2i qk r
For each inhomogeneous pairing a Fulde-Ferrell ansatz;
2 qk
represents the momentum of the Cooper pair.
This is the simplest ansatz, other structures should be examined
Three independent functions
1( r),
2( r),
3( r),
describing respectively d-s, u-s and u-d pairing.
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Near the transition point
du,us couplings, equal
strength, no ds pairing
Expected:
μe =m2s/4μ
Δ1 = 0; Δ2 = Δ3
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q2 q3 parallel favored by phase space
q  1.2 δμ
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Ginzburg-Landau expansion
n+ 1/4I=13 (2I I2+ II4 +
JIIJI2J2) +O(6)
n = -3/122(u4+d4+s4) -1/ 122 e4
-k  = 0 minimum in k
R. Casalbuoni, R.Gatto,,
N.Ippolito,GN,
M.Ruggieri
-e = 0 electrical neutrality
Approximation: 3'8' 0 (confirmed by M.Ciminale et al., in
preparation: Valid near the transition point, since this is the result
for the normal phase)
The numerical results in this regime confirm 1=0; 2  3
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Comparison among different CSC phases
Ginzburg Landau Approximation
Calculation at
leading order in μ
M.Mannarelli,
K. Rajagopal,
R. Sharma
R.Gatto,
R.Casalbuoni,
N.Ippolito,
G.Nardulli,
M.Ruggieri
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Free energy expansion for more plane waves
Ω= αΔ2+βΔ4+γΔ6 +δΔ8…
The favored structures for
both 2 and 3 flavors have cubic
symmetry (Bowers, Rajagopal,
Sharma).
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Chromomagnetic instability
In gCFL : imaginary gluon Meissner masses
Studied in LOFF model with 2 flavors (Giannakis&Ren,
Fukushima, Gorbar, Hashimoto&Miransky).
LOFF favored in comparison to gapless uniform
conductive states. At least in the GL region (small gap) no
chromomagnetic instability
For 3 flavors: recent results (Ciminale,Gatto,GN,Ruggieri)
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Feynman Diagrams at
4
O(Δ )
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Gluon Meissner masses:no instability
Transverse mass
Longitudinal mass
M.Ciminale,
R.Gatto,GN,
M.Ruggieri
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Compact stars and CS
Laboratory for the study ogf color superconductivity
Model of neutron
star with quark
core
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Specific heat and neutrino emissivity
• Cv  T for electrons and gapless or free
quarks;
• Cv exponentially suppressed for gapped quarks
• E=neutrino emissivity
• E T6 for gapless or free quarks;
• E T8 for nuclear matter (URCA [beta decay]
processes forbidden, modified URCA allowed
[ n+X->p+X+e+ν]
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Cooling of a compact star
• L dt = heath lost in time dt
• -V Cv dT =change in the energy of star
• Taking into account quark and nuclear
matter for neutrino emission as well as
photon emission, the rate of change in T:
dT
Vnm Enm + Vqm Eqm + Lγ
= -dt
Vnm Cnm + Vqm Cqm
V=volumes of nm (nuclear) or qm (quark) 42
Cooling of neutron star
(preliminary)
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Conclusions
• Color superconductivity is a theoretical prediction
based on the theory of hadronic interactions, QCD
• Best laboratory to study it: future dedicated
experiments with heavy ions or the core of
compact stars
• Compact star lab: remote, but experimental
signatures can be found and should allow to find
the correct QCD vacuum at high, but not very high
nuclear density
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