Transcript Diapositiva 1 - Florence Theory Group

Color
Superconductivity in
High Density QCD
Roberto Casalbuoni
Department of Physics and INFN - Florence
Bari, September 29 – October 1, 2004
1
Introduction
Motivations for the study of high-density QCD:
● Understanding the interior of CSO’s
● Study of the QCD phase diagram at
T~0 and high m
Asymptotic region in m fairly well
understood: existence of a CS
phase. Real question: does this
type of phase persists at relevant
densities ( ~5-6 r0)?
2
Summary
● Mini review of CFL and 2SC phases
● Pairing of fermions with different Fermi momenta
● The gapless phases g2SC and gCFL
● The LOFF phase
3
CFL and 2SC
Study of CS back to 1977 (Barrois 1977, Frautschi 1978,
Bailin and Love 1984) based on Cooper instability:
At T ~ 0 a degenerate fermion gas is unstable
Any weak attractive interaction leads to
Cooper pair formation
 Hard for electrons (Coulomb vs. phonons)
 Easy in QCD for di-quark formation (attractive
channel 3 )
(3 Ä 3 = 3 Å 6)
4
In QCD, CS easy for large m due to asymptotic
freedom
At high m, ms, md, mu ~ 0, 3 colors and 3 flavors
Possible pairings:
0 ψαia ψβjb 0
 Antisymmetry in color (a, b) for attraction
 Antisymmetry in spin (a,b) for better use of the
Fermi surface
 Antisymmetry in flavor (i, j) for Pauli principle
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 
s s

p
Only possible pairings

p
LL and RR
Favorite state CFL (color-flavor locking)
(Alford, Rajagopal & Wilczek 1999)
α
0 ψαaLψβbL 0 = - 0 ψaR
ψβbR 0 = ΔεαβCεabC
Symmetry breaking pattern
SU(3)c Ä SU(3)L Ä SU(3)R Þ SU(3)c+L+R
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What happens going down with m? If m << ms we get
3 colors and 2 flavors (2SC)
α
aL
0ψ ψ
β
bL
αβ3
0 = Δε εab
SU(3)c Ä SU(2)L Ä SU(2)R Þ SU(2)c Ä SU(2)L Ä SU(2)R
But what happens in real world ?
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● Ms not zero
● Neutrality with respect to em and color
● Weak equilibrium
All these effects make Fermi momenta of
different fermions unequal causing problems to
the BCS pairing mechanism
8
Consider 2 fermions with m1 = M, m2 = 0 at the same
chemical potential m. The Fermi momenta are
pF2  m
pF1  m2  M2
Effective chemical potential for the massive quark
meff =
Mismatch:
2
M
2
2
m - M » m2m
M2
dm»
2m
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If electrons are present, weak equilibrium makes
chemical potentials of quarks of different charges
unequal:
d ® uen Þ
md - mu = me
In general we have the relation:
(mi = m+ QmQ )
me = - mQ
N.B. me is not a free parameter
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¶V
= - Q= 0
¶ me
Neutrality requires:
Example 2SC: normal BCS pairing when
mu = md Þ n u = nd
But neutral matter for
1
n d » 2n u Þ md » 2 mu Þ me = md - mu » mu ¹ 0
4
1/ 3
Mismatch:
d
F
u
F
p - p
md - mu me mu
dm=
=
=
»
¹ 0
2
2
2
8
11
Also color neutrality requires
¶V
¶V
= T3 = 0,
= T8 = 0
¶ m3
¶ m8
As long as dm is small no effects on BCS pairing, but
when increased the BCS pairing is lost and two
possibilities arise:
● The system goes back to the normal phase
● Other phases can be formed
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In a simple model with two fermions at chemical potentials
m+dm, mdm the system becomes normal at the
Chandrasekhar-Clogston point. Another unstable phase exists.
dm= D BCS
D BCS
dm1 =
2
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The point |dm|  D is special. In the
presence of a mismatch new features are
present. The spectrum of quasiparticles is
E(p) = dm±
(p - m) 2 + D 2
For |dm| < D, the gaps
are D  dm and D + dm
For |dm|  D, an unpairing
(blocking) region opens up
and gapless modes are
present.
E(p) = 0 Û p = m±
2dm
Energy cost for pairing
2D
Energy gained in pairing
dm2 - D 2
begins to unpair
2dm> 2D
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g2SC
Same structure of condensates as in 2SC
(Huang & Shovkovy, 2003)
4x3 fermions:
0
● 2 quarks ungapped qub, qdb
α
aL
ψ ψ
β
bL
αβ3
0 = Δε εab
● 4 quarks gapped qur, qug, qdr, qdg
General strategy (NJL model):
● Write the free energy:
● Solve:
Neutrality
Gap equation
V(mm
, 3 , m8 , me , D )
¶V ¶V ¶V
=
=
=0
¶ me ¶ m3 ¶ m8
¶V
= 0
¶D
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● For |dm| > D (dm=me/2) 2 gapped quarks become
gapless. The gapless quarks begin to unpair destroying
the BCS solution. But a new stable phase exists, the
gapless 2SC (g2SC) phase.
● It is the unstable phase which becomes stable in this
case (and CFL, see later) when charge neutrality is
required.
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g2SC
17
● But evaluation of the gluon masses (5 out of 8 become
massive) shows an instability of the g2SC phase. Some of
the gluon masses are imaginary (Huang and Shovkovy 2004).
● Possible solutions are: gluon condensation, or another
phase takes place as a crystalline phase (see later), or this
phase is unstable against possible mixed phases.
● Potential problem also in gCFL (calculation not yet
done).
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gCFL
Generalization to 3 flavors
α
aL
β
bL
0ψ ψ
αβ1
αβ2
αβ3
0 = Δ1ε εab1 + Δ2ε εab2 + Δ3ε εab3
Different phases are characterized by different values for
the gaps. For instance (but many other possibilities exist)
CFL : D 1 = D 2 = D 3 = D
g2SC : D 3 ¹ 0, D 1 = D 2 = 0
gCFL : D 3 > D 2 > D 1
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% 0
Q
0
0
-1 +1 -1 +1
0
0
Gaps
ru gd bs rd gu rs bu gs bd
in
D3 D2
ru
gCFL gd D
D1
3
bs D 2 D 1
- D3
rd
gu
rs
bu
- D3
- D2
- D2
- D1
gs
bd
- D1
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Strange quark mass effects:
● Shift of the chemical potential for the strange
quarks:
M s2
ma s Þ ma s -
2m
● Color and electric neutrality in CFL requires
2
Ms
m8 = , m3 = me = 0
2m
● gs-bd unpairing catalyzes CFL to gCFL
1
Ms2
dmbd- gs = (mbd - mgs ) = - m8 =
2
2m
Ms2
dmrd- gu = me , dmrs- bu = me 2m
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It follows:
M2
m
Energy cost for pairing
2D
Energy gained in pairing
begins to unpair
M2
> 2D
m
Again, by using NJL model (modelled on one-gluon
exchange):
● Write the free energy: V(mm
, ,m ,m , M ,D
3
8
e
s
● Solve:
Neutrality
¶V ¶V ¶V
=
=
=0
¶ me ¶ m3 ¶ m8
Gap equations
¶V
= 0
¶Di
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i
)
● CFL # gCFL 2nd
order transition at Ms2/m ~
2D, when the pairing gsbd starts breaking
● gCFL has gapless
quasiparticles. Interesting
transport properties
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● LOFF (Larkin, Ovchinnikov, Fulde & Ferrel, 1964):
ferromagnetic alloy with paramagnetic impurities.
● The impurities produce a constant exchange
field acting upon the electron spins giving rise to
an effective difference in the chemical potentials
of the opposite spins producing a mismatch of the
Fermi momenta
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According to LOFF, close to first order point (CC point),
possible condensation with non zero total momentum
   p  k + q
p1  k + q 2
More generally
(x)(x)  De
y (x)y (x) =
å

2iqx
D mc me
r r
2iq m ×x
m
 

p1 + p 2  2q

|q|
fixed variationally
 
q/|q|
chosen
spontaneously
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Simple plane wave:
r
r r
E(p) - m® E(± p + q) - mm dm»
(p - m) + D m m
2
2
 
m  dm  v F  q
Also in this case, for
r r
| m=
| dm- v F ×q < D
a unpairing (blocking) region opens up and gapless
modes are present
Possibility of a crystalline structure (Larkin &
Ovchinnikov 1964, Bowers & Rajagopal 2002)
y (x)y (x) = D
å
e
r r
2iqi ×x
r
|qi |= 1.2 dm
The qi’s define the crystal pointing at its vertices.
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Crystalline
structures in LOFF
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The LOFF phase is studied via a Ginzburg-Landau
expansion of the grand potential
b 4  6
  aD + D + D +   
2
3
2
(for regular crystalline structures all the Dq are equal)
The coefficients can be determined microscopically for
the different structures (Bowers and Rajagopal (2002))
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 Gap equation
 Propagator expansion
 Insert in the gap equation
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We get the equation
aD + bD + D +     0
3
Which is the same as
aD 
bD 
3
D 
5
5

 0 with
D The first coefficient has
universal structure,
independent on the crystal.
From its analysis one draws
the following results
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 BCS   normal
r 2
  (D BCS  2dm2 )
4
 LOFF   normal  0.44r(dm  dm2 ) 2
D LOFF  1.15 (dm2  dm)
dm1  D BCS / 2
dm2  0.754D BCS
Small window. Opens up in QCD?
(Leibovich, Rajagopal & Shuster 2001;
Giannakis, Liu & Ren 2002)
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General
analysis
(Bowers and
Rajagopal (2002))
Preferred
structure:
face-centered
cube 32
Effective gap equation for the LOFF phase
(R.C., M. Ciminale, M. Mannarelli, G. Nardulli, M. Ruggieri & R. Gatto, 2004)
For the single plane wave (P = 1) the pairing region is
defined by
r
for (p, v F ) Î PR
elsewhere
D
D eff = D q(E u )q(Ed ) =
0
r r
E u,d = ± (dm- v F ×q) + (p - m) 2 + D 2
{
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For P plane waves we approximate the pairing region
as thesum of the regions Pk such that
r
r
Pk = {(p, v F ) | D E (p, v F ) = kD }
r
D E (p, v F ) =
P
å
r r
D eff (p,v F ×q m )
m= 1
The result can be interpreted as having P quasi-particles
each of one having a gap kD, k =1, …, P.
The approximation is better far from a second order
transition and it is exact for P = 1 (original FF case).
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Multiple phase transitions from the CC
point (Ms2/m  4 D2SC) up to the cube case
(Ms2/m ~ 7.5 D2SC). Extrapolating to CFL
(D2SC ~ 30 MeV) one gets that LOFF
should be favored from about
Ms2/m ~120 MeV up Ms2/m ~ 225 MeV
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Conclusions
● Under realistic conditions (Ms not zero, color
and electric neutrality) new CS phases might exist
● In these phases gapless modes are present. This
result might be important in relation to the
transport properties inside a CSO.
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● gCFL has me not zero, with charge cancelled by
unpaired u quarks
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g2SC parameters:
NJL with chiral (G S ) and diquark (G D ) couplings:
- 2
G S = 5.0163 GeV ,
L = 0.6533 GeV,
G D = hGS ,
h = 0.75
m= 400 MeV
gCFL parameters:
NJL modelled on one gluon-exchange:
D 0 = 25 MeV, L = 800 MeV, m= 500 MeV
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