Transcript Diapositiva 1 - Istituto Nazionale di Fisica Nucleare
Color Superconductivity in High Density QCD
Roberto Casalbuoni Dept. of Physics, University of Florence INFN - Florence Budapest, August 1-3, 2005 JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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Outline of the talk
Introduction
Pairing fermions with different Fermi momenta The gapless phases g2SC and gCFL The LOFF phase Preliminary results about the LOFF phase with 3 flavors Conclusions and outlook JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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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 moderate
m
Asymptotic region in
m
fairly well understood : existence of a CS phase
.
Real question : does this type of phase persist at relevant densities (~5-6
r
0 )?
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Pairing fermions with different Fermi momenta
●
M s not zero
●
Neutrality with respect to em and color no free energy cost in
!
( Amore et al. 2003 )
●
Weak equilibrium
All these effects make Fermi momenta of different fermions unequal causing problems to the BCS pairing mechanism
JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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Consider 2 fermions with m 1 = M, m 2 chemical potential = 0 at the same
m
. The Fermi momenta are
p F q ¹ ¡ M p F ¹
Effective chemical potential for the massive quark
¹ q ¹ ¡ M ¼ ¹ ¡ M ¹ Mismatch: ±¹ ¼ M ¹
M 2 /2
m
effective chemical potential JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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If electrons are present, weak equilibrium makes chemical potentials of quarks of different charges unequal:
d !
ue º ) ¹ ¹ ¹ e ¹ d ¡ ¹ Q i ¹ u ¡ ¹ Q ¹ e
N.B.
m
e is not a free parameter: neutrality requires:
Q ¡ @V @¹ e
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Example 2SC: normal BCS pairing when
¹ u ¹
But neutral matter for
d ) n u n d n d ¼ n u ) ¹ d ¼ ¹ u ) ¹ e ¹ d ¡ ¹ u ¼ ¹ u
Mismatch
±¹ p d F ¡ p u F ¹ d ¡ ¹ u ¹ e ¼ ¹ u
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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
Notice that there is also a color neutrality condition
@V @¹
T
@V @¹
T
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In a simple model without supplementary conditions, with two fermions at chemical potentials
m 1
and
m
2 , the system becomes normal at the Chandrasekhar - Clogston point.
Another unstable phase exists.
1.2
1 BCS CC ±¹ B CS 0.8
0.6
0.4
0.2
unstable phase 0.2
0.4
0.6
0.8
1 1.2
dm r 0.4
Normal unstable phase 0.2
0 -0.2
0.2
0.4
BCS 0.6
CC 0.8
1 1.2
dm
±¹
p
B CS -0.4
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The point
|dm| =
is special. In the presence of a mismatch new features are present. The spectrum of quasiparticles is
E j±¹ § E q blocking region j dm = |dm| = |dm| For |dm| < , the gaps are and + dm - dm For |dm| = , an unpairing (blocking) region opens up and gapless modes are present p gapless modes 2 dm Energy cost for pairing E , begins to unpair p ¹ § q ±¹ ¡
±¹ >
2 D Energy gained in pairing
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g2SC
Same structure of condensates as in 2SC 4x3 fermions:
2 quarks ungapped q ub , q db
(
Huang & Shovkovy, 2003
) h ® aL à ¯ bL j i ² ®¯ ² ab
4 quarks gapped q ur , q ug , q dr , q dg General strategy (NJL model):
Write the free energy Solve
V
; ¹
; ¹
e
;
Neutrality Gap equation
@V @¹ e @V @¹ @V @¹ @V
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●
For
|dm| ( dm = m
e
/ )
2 gapped quarks become gapless. The gapped quarks begin to unpair destroying the BCS solution. But a new stable phase exists, the gapless 2SC (g2SC) phase.
●
It is the unstable phase (Sarma phase) which becomes stable in this case (and in gCFL, see later) when charge neutrality is required.
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Q = 0 g2SC V m e = const.
(
Huang & Shovkovy, 2003
) Normal state (MeV) 200 solutions of the gap equation 150 neutrality line m e 2 = V ( GeV /fm 3 ) - 0.081
- 0.082
Normal state 0 gap equation neutrality line 100 - 0.083
- 0.084
- 0.085
50 - 0.086
- 0.087
m e =148 MeV 0 0 100 200 300
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400 m e (MeV) 20 40 60 80
R. Casalbuoni:Superconductivity in high density QCD
m e 100 120 ( MeV ) 13
However, 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 ). Connected with gapless modes? ( Alford & Wang, 2005 ).
4,5,6,7 8 m g ¹ g ¼
But this instability occurs also in the 2SC phase (no gapless modes)
p < ±¹ <
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Generalization to 3 flavors (Alford, Kouvaris & Rajagopal, 2005)
gCFL h ® aL à ¯ bL j i ®¯ ab ² ®¯ ² ab ² ®¯ ² ab
Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist)
CFL : g2SC :
g CF L :
D = D = D
1 2 3
= D D № D = D =
3
0,
1 2
0 D > D
3 2
> D
1
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0 0 0 -1 +1 -1 +1 0 0 ru gd bs rd ru gd bs rd gu rs bu gs bd
D D 2 3 D 3 D 1 D D 1 2 - D 3
gu
- D 3
rs bu
- D 2 - D 2
gs
- D 1
bd
- D 1
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Gaps in gCFL
1 2 3 : ds : us : ud pairin g pairing pairing 16
Strange quark mass effects:
●
Shift of the chemical potential for the strange quarks:
¹ ®s ) ¹ ®s ¡ M ¹ s ●
Color and electric neutrality in CFL requires
¹ ¡ M ¹ s ; ¹ ¹ e ●
The transition CFL to gCFL starts with the unpairing of the pair gs-bd having (close to the transition)
±¹ ds M ¹ s
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M
It follows:
s
Energy cost for pairing
¹
Energy gained in pairing
begins to unpair M s ¹ >
Again, by using NJL model (modelled on one-gluon exchange):
●
Write the free energy:
V
; ¹
; ¹
e
;
●
Solve: Neutrality
@V @¹ e @V @¹ @V @¹
Gap equations JHW Budapest, August 1-3, 2005
@V i
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10 g2SC unpaired ~ 0 ●
CFL
#
gCFL 2 nd order transition at M s 2
/m ~
, when the pairing gs bd starts breaking
-10 -20 -30 2SC gCFL -40 CFL ~ 30 3 -50 0 25 50 M 2 s / 75 m [MeV] 100 125 25 20 2 15
(Alford, Kouvaris & Rajagopal, 2005)
10
(
0 = 25 MeV,
m
= 500 MeV)
5 ~ 1 0 0 25 50 M S 2 / m 75 [MeV] 100
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125 150
R. Casalbuoni:Superconductivity in high density QCD
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●
gCFL has gapless quasiparticles, and there are gluon imaginary masses also in this phase ( RC et al. 2004, Fukushima 2005 ).
m (M ) M s m (0) M 1 0 -1 -2 -3 -4 0 m (M ) M s m (0) M 1.25
1 0.75
0.5
0.25
0 -0.25
0 3 1,2 8 20 40 60 80 100 120 M s 2 m 4,5 6,7 20 40 60 80 100 120 M s 2 m
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Instability cured by gluon condensation? Assuming artificially
m
m
8 > not zero (of order 10 MeV) this can be done. See also a very recent paper ( Gorbar, Hashimoto & Miransky, 2005) about a gluonic phase curing the chromomagnetic instability in 2SC.
Three recent results obtained by Giannakis & Ren:
Chromomagnetic instability of g2SC makes the crystalline phase (LOFF) with two flavors energetically favored ( Giannakis & Ren 2004 )
LOFF with two flavors without requiring electrical neutrality has no magnetic instability although it has gapless modes ( Giannakis & Ren 2005 )
Last week the same result obtained requiring color and electric neutrality in the weak coupling limit ( Giannakis, Hou & Ren 2005 ) JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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LOFF phase
●
LOFF
(
Larkin, Ovchinnikov, Fulde & Ferrel, 1964
):
ferromagnetic alloy with paramagnetic impurities
.
●
The impurities produce a constant exchange field to an acting upon the electron spins giving rise effective difference in the chemical potentials of the opposite spins producing a
mismatch
of the Fermi momenta
●
Studied also in the QCD context (
Alford, Bowers & Rajagopal, 2000
)
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According to LOFF, close to first order point (CC point), possible condensation with non zero total momentum
~ ~ p
More generally
~ hà i X hà i m e i ~ m e i ~ m ¢~
~
p
~
fixed variationally chosen spontaneously JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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E ¡ ¹ )
Single plane wave:
q E p ~ ¡ ¹ ¨ ±¹ ¼ ¨ ¹ ¹ ±¹ ¡ ~ F
Also in this case, for
¹
±¹ ¡ ~
F
an unpairing (blocking) region opens up and gapless modes are present More general possibilities include a crystalline structure
hà (
Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002
) X Ã x i e i ~ i ¢~ ~ i
The q i ’s define the crystal pointing at its vertices.
JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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BCS normal = r 4 ( 2 BCS 2 dm 2 ) LOFF normal = - .
44 r ( dm dm 2 ) 2 LOFF 1 .
15 ( dm 2 dm ) dm 1 dm 2 = BCS / 2 0 .
754 BCS
±¹
Small window. Opens up in QCD? ( Leibovich, Rajagopal & Shuster 2001; Giannakis, Liu & Ren 2002
)
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Preliminary results about LOFF with three flavors
Recent study of LOFF with 3 flavors within the following simplifying hypothesis ( RC, Gatto, Ippolito, Nardulli & Ruggieri, 2005 )
Study within the Landau-Ginzburg approximation.
Only electrical neutrality imposed (chemical potentials
m
3 taken equal to zero).
and
m
8
M s treated as in gCFL. Pairing similar to gCFL with inhomogeneity in terms of simple plane waves, as for the simplest LOFF phase.
hà ® aL à ¯ bL i I X I ² ®¯I ² abI ; I
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I e i ~ I 26
A further simplifications is to assume only the following geometrical configurations for the vectors q I , I=1,2,3 (a more general angular dependence will be considered in future work) 1 2 3 4
The free energy, in the GL expansion, has the form
¡ norm al I 0 X @ ® I I ¯ I I 1 X I 6 J ¯ I J I J A O
with coefficients
I, I
and
IJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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´ B CS ; ¹ u ¹ ¡ ¹ e ; ¹ d ¹ ¹ e ; ¹ s ¹ ¹ e ¡ M ¹ s ® I I ; ±¹ I ¡ ¹ ¼ Ã ±¹ q I I ¯ ¯ q I q I ±¹ ¡ ±¹ I I ¯ ¯ ¡ ¯ I I ; ±¹ I ¹ ¼ q I ¡ ±¹ I ¯ ¯ I ¡ ±¹ I ¯ ¯ !
¯ ¡ ¹ ¼ Z d n ¼ q 1 ¢ n ¹ s ¡ ¹ d q 2 ¢ n ¹ s ¡ ¹ u !
!
; ¹ s $ ¹ d ; ¹ s $ ¹ u
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¡ norm al I 0 X @ ® I I ¯ I I 1 X I 6 J ¯ I J I J A O
We require:
@ I @q I @ @¹ e
At the lowest order in
I
@ ) @q I @® @q I I
since
I depends only on q I and
dm
i we get the same result as in the usual LOFF case:
j~
I
j
±¹
I
JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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-2 -4 -6 -8 2 0 2 1 0.8
0.6
0.4
0.2
25 1 2 50 1 3 4 75 M s 3 2 / 100 m [MeV] 125 150 175
Structure 4 dominates starting from about 30 MeV (we have assumed the same parameters as in
0 Alford et al. in gCFL, = 25 MeV,
m
= 500 MeV)
3 1 2
: ds : us : ud
-
pairin g pairing pairing
25 50 75 M 2 / s 100 m [MeV] 125
JHW Budapest, August 1-3, 2005
150 175
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Comparison with other phases:
If no chromo-magnetic instability LOFF takes over gCFL at about 120 MeV and goes over to the normal phase at about 150 MeV (both first order transitions)
10 g2SC unpaired ~ 0 -10 LOFF -20 2SC gCFL -30 -40 CFL ~ -50 0 25
JHW Budapest, August 1-3, 2005
50 M s 2 / 75 m [MeV] 100 125 150
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Otherwise it should go over to the CFL phase at 75 MeV
10 0 -10 -20 -30 -40 -50 0 CFL 25 unpaired 2SC ~ 50 ~ gCFL M s 2 / 75 m [MeV] 100 g2SC LOFF 125 150
JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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m e [MeV] 40 30 20 10 50 75 M 2 / s 100 125 m [MeV] 150 175
The behaviour of
m
e in LOFF is pretty similar to the one in gCFL. If the same is true for
m
3 and
m
8 our assumption of their vanishing is not too bad for M s 2 /
m
in the region where LOFF is the favored phase.
50 40 30 20 10 0 -10 -20 -30 0 CFL m e , m 3 m 8 gCFL m e m 8 m 3 25 50 M S 2 / 75 m [MeV] 100 125 150
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Conclusions
●
The problem of the QCD phases at moderate densities and low temperature is still open.
●
Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises (but this happens also under different conditions, see 2SC).
●
Also the LOFF phase is gapless but the gluon instability does not seem to appear.
●
Our recent study of the LOFF phase with three flavors seems to suggest that this should be the favored phase after CFL , although this study is very much simplified and more careful investigations should be performed. JHW Budapest, August 1-3, 2005 R. Casalbuoni:Superconductivity in high density QCD
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