overview of finite density qcd for string...
TRANSCRIPT
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 1
Roberto CasalbuoniRoberto CasalbuoniDepartment of Physics, INFN and GGI Department of Physics, INFN and GGI --
Florence Florence
Overview of Finite Density QCD for String Theorists
Overview of Finite Overview of Finite Density QCD for Density QCD for String TheoristsString Theorists
[email protected]@fi.infn.it
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 2
An overview for string theorists but not with string techniques.
Problems with lattice.
Calculations from first principles (real QCD) only at very high densities (baryon chemical potential ~ 105 GeV).
But regions of interest as in CSO’s is for μ ~ 400-500 MeV. Most of the calculations made within NJL models.
What can be obtained from these model calculations? The results from QCD calculations extrapolated at low μ
agree with models. The
main thrust is in the qualitative properties like phase transitions and their nature depending mainly on the symmetry properties, not very much in the quantitative results.
The low μ region is strongly interacting, it would be nice to be able to investigate it within the ADS/QCD approach (some attempt already present in the literature)
Main discussion about the ground state.
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 3
SummarySummarySummary
Introduction
Color Superconductivity: CFL and 2SC phases
The case of pairing at different Fermi surfaces
LOFF (or crystalline) phase
Conclusions
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 4
Motivations for the study of high-density QCD
•
Study of the QCD phase in a region difficult in lattice calculations
• Understanding the interior of CSO’s
Asymptotic region in μ
fairly well understood: existence of a CS phase. .
Real question: does this type of phase persist at relevant densities
(~5-6 ρ0)?
Introduction
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 5
CFL and S2C phasesCFL and S2C CFL and S2C phasesphases
Ideas about CS back in 1975 (Collins & PerryIdeas about CS back in 1975 (Collins & Perry--1975, 1975,
BarroisBarrois--1977, Frautschi1977, Frautschi--1978).1978).
Only in 1998 (Alford, Only in 1998 (Alford, RajagopalRajagopal & & WilczekWilczek; Rapp, Schafer, ; Rapp, Schafer, SchuryakSchuryak & & VelkovskyVelkovsky) a real progress. ) a real progress.
Why CS? Why CS? For an arbitrary attractive interaction it is energetically convenient the formation of Cooper pairs (difermions)
Due to asymptotic freedom quarks are almost free at high density and we expect diquark condensation in the color attractive channel 3* (the channel 6 is repulsive).
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 6
In matter In matter SC SC only under particular conditionsonly under particular conditions ((phonon interaction should overcome the phonon interaction should overcome the
Coulomb forceCoulomb force))
430 54
0c 1010
K1010
K 101E(electr.)
(electr.)T −− ÷≈÷
÷≈
SC
much more efficient in QCDSCSC
much more efficient imuch more efficient in QCDn QCD
In QCDIn QCD attractive attractive interactioninteraction ((antitripletantitriplet
channel)channel)≈ ≈cT (quarks) 50 MeV 1
E(quarks) 100 MeV
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 7
AntisymmetryAntisymmetry in spin (in spin (a,ba,b) for better use of ) for better use of the Fermi surfacethe Fermi surface
AntisymmetryAntisymmetry in color (a, b) for attractionin color (a, b) for attraction
AntisymmetryAntisymmetry in flavor (in flavor (i,ji,j) for ) for PauliPauli principleprinciple
α βia jb0 ψ ψ 0
The symmetry breaking pattern of QCD at highThe symmetry breaking pattern of QCD at high-- density depends on the number of flavors with density depends on the number of flavors with mass mass m < m < μμ. Two most interesting cases:. Two most interesting cases: NNff = 2, 3= 2, 3. . Consider the possible pairings at very high densityConsider the possible pairings at very high density
α, β color; i, j flavor;
a,b spin
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 8
pp−
ss− Only possible pairings
LL and RR
Only possible pairings Only possible pairings
LL and RRLL and RR
Favorite state for Nf = 3, CFL (color-flavor locking) (Alford, Rajagopal & Wilczek 1999)
α β α β αβCiL jL iR jR ijC0 ψ ψ 0 = - 0 ψ ψ 0 Δε ε∝
Symmetry breaking patternSymmetry breaking pattern
c L R c+L+RSU(3) SU(3) SU(3) SU(3)⊗ ⊗ ⇒
For For μ >> μ >> mmuu
, , mmdd
, m, mss
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 9
α β αβCiL jL ijC0 ψ ψ 0 Δε ε∝
α β αβCiR jR ijC0 ψ ψ 0 Δε ε∝ −
Why CFL?Why CFL?
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 10
What happens going down with What happens going down with μμ? If ? If μμ
<< m<< mss , we , we get 3 colors and 2 flavors (2SC)get 3 colors and 2 flavors (2SC)
α β αβ3iL jL ij0 ψ ψ 0 = Δε ε
c L R c L RSU(3) SU(2) SU(2) SU(2) SU(2) SU(2)⊗ ⊗ ⇒ ⊗ ⊗
However, if However, if μμ
is in the intermediate region we is in the intermediate region we face a situation with quarks belonging to face a situation with quarks belonging to different Fermi surfaces (see later). Then other different Fermi surfaces (see later). Then other phases could be important (LOFF, etc.)phases could be important (LOFF, etc.)
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 11
Pairing fermions with different Fermi momenta
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
All these effects make Fermi momenta of different fermions unequal causing
problems to the BCS pairing mechanism
no free energy cost in neutral singlet, (Amore et al. 2003)→
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 12
Consider 2 fermions with m1 = M, m2 = 0 at the same chemical potential μ. The Fermi momenta are
Effective chemical potential for the massive quark
Mismatch:Mismatch:
pF1 =qμ2 −M2 pF2 = μ
μeff =
qμ2 −M2 ≈ μ −M
2
2μ
δμ ≈ M2
2μM2/2μ
effective
chemical potential
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 13
•
Weak equilibrium makes chemical potentials of quarks of different charges unequal:
• From this: and
•
N.B. μe is not a free parameter, it is fixed by the neutrality condition:
μi = μ+QiμQ
μe= −μQμe= −μQQ = − ∂V
∂μe= 0
d → ueν ⇒ μd − μu = μe
Neutrality
and β
equilibriumNeutralityNeutrality and and ββ equilibriumequilibrium
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 14
If the strange quark is massless this equation has solutionNu = Nd = Ns , Ne = 0; quark matter electrically neutral with no electrons
μd,s= μu+ μe
The case of non interacting quarksThe case of non interacting quarks
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 15
Fermi surfaces for neutral and color singlet unpaired quark matter at the β
equilibrium and Ms
not zero.
In the normal phase μ3 = μ8 = 0.
By taking into account Ms
μe ≈ pdF − puF ≈ puF − psF ≈M2s /4μ
pdF − psF ≈ 2μe, μe =M2s4μ
The biggest δμ is ds
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 16
• Example 2SC: normal BCS pairing when
• But neutral matter for
Mismatch
μu = μd ⇒ nu = nd
nd ≈ 2nu ⇒ μd ≈ 21/3μu ⇒ μe = μd−μu ≈1
4μu 6= 0
δμ =pdF − puF2
=μd − μu
2=
μe
2≈ μu
8
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 17
• Notice that there are also color neutrality conditions
As long as δμ
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
∂V
∂μ3= T3 = 0,
∂V
∂μ8= T8 = 0
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 18
The problem of two fermions with different chemical potentials:
u d
u d u d
,
,2 2
μ = μ + δμ μ = μ − δμμ + μ μ − μ
μ = δμ =
can be described by an interaction hamiltonian
†I 3H = −δμψ σ ψ
In normal SC:
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 19
u,d (ε(p,Δ)±δμ)/T
1n =e +1
ForFor
T T TT
00
3
3
g d p 11 = (1- θ(-ε - δμ) - θ(-ε + δμ))2 (2π) ε(p,Δ)∫
ε <| δμ |blocking regionblocking region
The blocking region reduces the gap:
BCSΔ << Δ
3
u d3g d p 11 = (1- n - n )2 (2π) ε(p,Δ)∫Gap equation:
( ) ( )2 2ε p,Δ = p -μ +Δ
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 20
In a simple model (no supplementary conditions), with two fermions at chemical potentials μ1
and μ2 , the system goes normal through a 1st
order transition at the Chandrasekhar - Clogston point. Another unstable phase exists.
δμ1 =ΔBCS√2
δμ = ΔBCS
0.2 0.4 0.6 0.8 1 1.2
0.2
0.4
0.6
0.8
1
1.2ΔΔ0
__
δμ___Δ0
ΩρΔ0
2_____
BCS
BCS
Normal unstable phase
unstable phase
0.2 0.4 0.8 1 1.2
-0.4
-0.2
0.2
0.4
δμ___Δ0
CC
CC0
0.6
Sarma phase, tangential to the BCS phase at δμ
= Δ
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 21
The point |δμ| = Δ
is special. In the presence of a mismatch new features are present. The spectrum of quasiparticles is
For For |δμ| = Δ|δμ| = Δ, an , an unpairingunpairing (blocking) region opens up and(blocking) region opens up and
gapless modesgapless modes
are presentare present
For For |δμ| < Δ|δμ| < Δ, the gaps are, the gaps are
Δ Δ −−
δμδμ andand
Δ + δμΔ + δμ
2δμ Energy cost for
pairing
2Δ Energy gained in pairing
begins to unpair
δμ = 0
E
p
|δμ| = Δ
|δμ| > Δ
gapless m odes
blocking region
Δ
E(p) = 0 ⇔ p = μ ±qδμ2 − Δ2
2δμ > 2Δ
E(p) = | ± δμ+q(p− μ)2+Δ2|
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 22
If neutrality constraints are present the situation is different.
For |δμ| > Δ
(δμ = μe /2) 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.
g2SCg2SC ((Huang & Shovkovy, 2003))
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 23
0
e
Δμ
Q = 0
eμ = const.
solutions of thegap equation
V
Normal state
Δ200
150
100
50
00 100 200 300 400
(MeV)
μ e (MeV)
gap equation
neutrality line
μ e2 =Δ
20 40 60 80 100 120Δ (MeV)
- 0.087
- 0.086
- 0.085
- 0.084
- 0.083
- 0.082
- 0.081V ( GeV/fm 3 )
μ =148 MeVe
neutrality line
Normal state
δμ =μe
2
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 24
The case of 3 flavors gCFLgCFL
3
3
3 2
2
1
1
1
2
g2SC : 0,g
CFL :
CF0
L :Δ ≠ Δ =Δ =
Δ >Δ
Δ =Δ =Δ Δ
>Δ
=
Different phases are characterized by different values for the gaps. For instance (but many other possibilities exist)
h0|ψαaLψβbL|0i = Δ1²
αβ1²ab1+Δ2²αβ2²ab2+Δ3²
αβ3²ab3
(Alford, Kouvaris & Rajagopal, 2005)
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 25
0 0 0 -1 +1 -1 +1 0 0ru gd bs rd gu rs bu gs bd
rugdbsrdgursbugsbd
Q
1Δ2Δ3Δ
3Δ
2Δ 1Δ
3−Δ
3−Δ
2−Δ
2−Δ
1−Δ
1−Δ
Gaps in
gCFL
2
1
3
: us pairing: ds pairin
: ud pairing
g
Δ −ΔΔ
−−
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 26
Strange quark mass effects:
Shift of the chemical potential for the strange quarks:
Color and electric neutrality in CFL requires
The transition CFL to gCFL starts with the unpairing of the pair ds with (close to the transition)
μαs ⇒ μαs −M 2s
2μ
μ8 = −M2s2μ
, μ3 = μe = 0
δμds =M2s2μ
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 27
Energy cost for pairing
Energy gained in pairing
begins to unpair
It follows:
Calculations within a NJL model (modelled on one- gluon exchange):
Write the free energy:
Solve:
Neutrality
Gap equations
M 2s
μ
2Δ
M2sμ
> 2Δ
∂V
∂μe=
∂V
∂μ3=
∂V
∂μ8= 0
∂V
∂Δi= 0
V (μ,μ3, μ8,μe,Δi)
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 28
CFL # gCFL 2nd order transition at Ms
2/μ
∼ 2Δ, when the pairing ds starts breaking
0 25 50 75 100 125 150M S
2/μ [MeV]0
5
10
15
20
25
30
Gap
Par
amet
ers
[MeV
]
Δ3
Δ2
Δ1∼ 2Δ
0 25 50 75 100 125 150M2/μ [MeV]
-50
-40
-30
-20
-10
0
10
Ene
rgy
Diff
eren
ce [
106 M
eV4 ]
gCFL
CFL
unpaired
2SC
g2SC
∼ 2Δ
∼ 4Δ
s
(Alford, Kouvaris & Rajagopal, 2005)
(Δ0 = 25 MeV, μ
= 500 MeV)
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 29
4,5,6,78
0 20 40 60 80 100 120
-4-3-2-101
m (M )m (0)
M
M
s
Ms2
μ
2
2
1,23
8
gCFL has gapless quasiparticles, and there are gluon imaginary masses (RC et al. 2004, Fukushima 2005).
Instability present also in g2SC (Huang & Shovkovy 2004; Alford & Wang 2005)
m2g =μ2g2
3π2
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 30
Gluon condensation. Assuming artificially <Aμ
3 > or <Aμ
8 > not zero (of order 10 MeV) this can be done (RC et
al. 2004) . In g2SC the chromomagnetic instability can be cured by a chromo-magnetic condensate (Gorbar, Hashimoto, Miransky, 2005 & 2006; Kiriyama, Rischke, Shovkovy, 2006). Rotational symmetry is broken and this makes a connection with the inhomogeneous LOFF phase (see later). At the moment no extension to the three flavor case.
Proposals for solving the chromomagnetic
instability
Proposals for solving the chromomagnetic
instability
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 31
CFL-K0 phase. When the stress is not too large (high density) the CFL pattern might be modified by a flavor rotation of the condensate equivalent to a condensate of K0 mesons (Bedaque, Schafer 2002). This occurs for ms > m1/3 Δ2/3. Also in this phase gapless modes are present and the gluonic instability arises (Kryjevski, Schafer 2005, Kryjevski, Yamada 2005). With a space dependent condensate a current can be generated which resolves the instability. Again some relations with the LOFF phase.
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 32
Single flavor pairing. If the stress is too big single flavor pairing could occur but the gap is generally too small. It could be important at low μ
before the
nuclear phase (see for instance Alford 2006)
Secondary pairing. The gapless modes could pair forming a secondary gap, but the gap is far too small (Huang, Shovkovy, 2003; Hong 2005; Alford, Wang, 2005)
Mixed phases of nuclear and quark matter (Alford, Rajagopal, Reddy, Wilczek, 2001) as well as mixed phases between different CS phases, have been found either unstable or energetically disfavored (Neumann, Buballa, Oertel, 2002; Alford, Kouvaris, Rajagopal, 2004).
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 33
Chromomagnetic instability of g2SC makes the crystalline phase (LOFF) with two flavors energetically favored (Giannakis & Ren 2004), also there are no chromomagnetic instability although it has gapless modes (Giannakis & Ren 2005).
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 34
LOFF ((Larkin, Ovchinnikov; Fulde & Ferrel, 1964 ):): ferromagnetic alloy with paramagnetic
impurities.
The impurities produce a constant exchange field acting upon the electron spin giving rise to an effective difference in the chemical potentials of the electrons producing a mismatch of the Fermi momenta
Studied also in the QCD context (Alford, Bowers &
Rajagopal, 2000, for a review R.C. & Nardulli, 2003)
LOFF phaseLOFF phase
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 35
According to LOFF, close to first order point (CC point), possible condensation with non zero total momentum
More generally
fixed variationally
chosen spontaneously
~p1 =~k + ~q, ~p2 = −~k + ~q hψ(x)ψ(x)i=Δe2i~q·~x
hψ(x)ψ(x)i=XmΔme2i~qm·~x
~p1 + ~p2 = 2~q
|~q |
~q /|~q |
rings
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 36
Single plane wave:
Also in this case, foran unpairing (blocking) region opens up and gapless modes are present
More general possibilities include a crystalline structure ((Larkin & Ovchinnikov 1964, Bowers & Rajagopal 2002))
The qi ’s define the crystal pointing at its vertices.
μ= δμ − ~vF · ~q > Δ
hψ(x)ψ(x)i= ΔX~qi
e2i~qi·~x
E(~k)−μ⇒ E(±~k+~q)−μ∓δμ ≈q(|~k| − μ)2 +Δ2∓μ
μ = δμ− ~vF · ~q
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 37
⋅⋅⋅+Δγ
+Δβ
+Δα=Ω 642
32(for regular crystalline structures all the Δq are
equal)
The coefficients can be determined microscopically for the different structures (Bowers and Bowers and RajagopalRajagopal (2002)(2002) ))
The LOFF phase has been studied via a Ginzburg-Landau expansion of the grand potential
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 38
Gap equationGap equation
Propagator expansion
Insert in the gap equation
General strategy
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 39
We get the equation
053 =⋅⋅⋅+Δγ+Δβ+Δα
Which is the same as 0=Δ∂Ω∂
with
=Δα=Δβ 3
=Δγ 5
The first coefficient has universal structure,
independent on the crystal. From its analysis one draws
the following results
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 40
22normalLOFF )(44. δμ−δμρ−0=Ω−Ω
)2(4
2BCS
2normalBCS δμ−Δ
ρ−=Ω−Ω
)(15.1 2LOFF δμ−δμ≈Δ
2/BCS1 Δ=δμ BCS2 754.0 Δ≈δμ
Small window. Opens up in QCD (Leibovich, Rajagopal & Shuster 2001)
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 41
Single plane wave
Critical line from
0q
,0 =∂Ω∂
=Δ∂Ω∂
Along the critical line
)2.1q,0Tat( 2δμ==
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 42
Bowers and Bowers and RajagopalRajagopal (2002)(2002)
Preferred structure: face- centered cube
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 43
Results about LOFF with three flavors
Results about LOFF with three flavors
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 μ3 and μ8 taken equal to zero, OK if close to the transition to the normal state).
Ms treated as in gCFL. Pairing similar to gCFL with inhomogeneity in terms of simple plane waves, as for the simplest LOFF phase.
hψαaLψβbLi=3XI=1
ΔI(~x)²αβI²abI, ΔI(~x) = ΔIe
2i~qI ·~x
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 44
•
A further simplifications is to assume only the following geometrical configurations for the vectors qI , i = 1,2,3
• The free energy, in the GL expansion, has the form
•
with coefficients αΙ,
βΙ and βIJ calculable from an effective NJL four-fermi interaction simulating one-gluon exchange
1 2 3 4
Ω−Ωnormal=3XI=1
⎛⎝αI2Δ2I +
βI4Δ4I +
XI 6=J
βIJ4Δ2IΔ
2J
⎞⎠ + O(Δ6)
Ωnormal= −3
12π2(μ4u+ μ4d + μ4s)−
1
12π2μ4e
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 45
βI(qI, δμI) =μ2
π21
q2I − δμ2I
αI(qI , δμI) = −4μ2
π2
Ã1 − δμI
2qIlog
¯¯qI + δμIqI − δμI
¯¯ − 12 log
¯¯4(q2I − δμ2I )
Δ20
¯¯!
β12 = −3μ2
π2
Zdn
4π
1
(2q1 · n + μs − μd) (2q2 · n + μs − μu)
Others by the exchange :
12 → 23, μs ↔ μd
12 → 13,μs ↔ μu
Δ0 ≡ΔBCS, μu = μ−23μe, μd = μ+
1
3μe, μs = μ+
1
3μe−
M2s2μ
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 46
We require:∂Ω
∂ΔI=
∂Ω
∂qI=
∂Ω
∂μe= 0
At the lowest order in ΔI
∂Ω
∂qI= 0⇒ ∂αI
∂qI= 0
since αI depends only on qI and δμi we get the same result as in the
simplest LOFF case:
|~qI | = 1.2δμI
In the GL approximation we expect to be pretty close to the normal phase, therefore we will assume μ3 = μ8 = 0. At the same order we expect Δ2 = Δ3 (equal mismatch) and Δ1 = 0 (ds mismatch is twice the ud and us).
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 47
Once assumed Δ
1 = 0, only two configurations for q2 and q3 , parallel or antiparallel. The antiparallel is disfavored due to the lack of configurations space for the up fermions.
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 48
2
1
3
: us pairing: ds pairin
: ud pairing
g
Δ −ΔΔ
−−
(we have assumed the same parameters as in Alford et al. in gCFL,
Δ0 = 25 MeV, μ
= 500 MeV)
Δ1 = 0, Δ2 =Δ3
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 49
•
LOFF phase takes over gCFL at about 128 MeV and goes over to the normal phase at about 150 MeV
(RC, Gatto, Ippolito, Nardulli, Ruggieri, 2005)
Confirmed by an exact solution of the gap equation (Mannarelli, Rajagopal, Sharma, 2006)
Comparison with other phases
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 50
Longitudinal masses
Transverse masses
No chromo-magnetic instability in the LOFF phase with three flavors (Ciminale, Gatto, Nardulli, Ruggieri, 2006)
M1 =M2 =M4 = M5
M6 =M7
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 51
Extension to a crystalline structure (Rajagopal, Sharma 2006), always within the simplifying assumption Δ1 = 0 and Δ2 = Δ3
hudi ≈Δ3Xaexp(2i~q a3 ·~r), husi ≈ Δ2
Xaexp(2i~q a2 ·~r)
The sum over the index a goes up to 8 qia. Assuming also Δ2 = Δ3
the favored structures (always in the GL approximation up to Δ6) among 11 structures analyzed are
CubeX 2Cube45z
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 52
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 53
Conclusions
Various phases are competing, many of them having gapless modes. However, when such modes are present a chromomagnetic instability arises.
Also the LOFF phase is gapless but the gluon instability does not seem to appear.
Recent studies of the LOFF phase with three flavors seem 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.
The problem of the QCD phases at moderate densities and low temperature is still open.
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 54
PhononsPhonons
In the LOFF phase translations and rotations are brokenIn the LOFF phase translations and rotations are broken
phononsphononsPhonon field through the phase of the condensate (R.C., Phonon field through the phase of the condensate (R.C.,
GattoGatto, , MannarelliMannarelli
& & NardulliNardulli
2002):2002):
)x(ixqi2 ee)x()x( Φ⋅ Δ→Δ=ψψ xq2)x( ⋅=Φ
introducingintroducing 1 φ(x) = Φ(x) - 2q xf
⋅
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 55
2 2 22 2 2
2 2 2
12phonon llL v v
x y zϕ ϕ ϕϕ ⊥
⎡ ⎤⎛ ⎞∂ ∂ ∂= − + −⎢ ⎥⎜ ⎟∂ ∂ ∂⎝ ⎠⎣ ⎦
Coupling phonons to fermions (quasiCoupling phonons to fermions (quasi--particles) trough particles) trough the gap termthe gap term
ψψΔ→ψψΔ Φ CeC)x( T)x(iT
It is possible to evaluate the parameters of It is possible to evaluate the parameters of LLphononphonon (R.C., (R.C., GattoGatto, , MannarelliMannarelli
& & NardulliNardulli
2002)2002)
153.0|q|
121v
22 ≈
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ δμ−=⊥ 694.0
|q|v
22|| ≈⎟⎟
⎠
⎞⎜⎜⎝
⎛ δμ=
( )1,0,0q =
++
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 56
Cubic structureCubic structure
∑∑∑±=ε=
Φε
±=ε=
ε
=
⋅ Δ⇒Δ=Δ=Δi
)i(i
i
iik
;3,2,1i
)x(i
;3,2,1i
x|q|i28
1k
xqi2 eee)x(
3 scalar fields 3 scalar fields ΦΦ(i)(i)(x)(x)
i)i( x|q|2)x( =Φ
i)i()i( x|q|2)x()x(
f1
−Φ=ϕ
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 57
Coupling phonons to fermions (quasiCoupling phonons to fermions (quasi--particles) trough particles) trough the gap termthe gap term
∑±=ε=
Φε ψψΔ→ψψΔi
)i(i
;3,2,1i
T)x(iT CeC)x(
ΦΦ(i)(i)(x) transforms under the group O(x) transforms under the group Ohh
of the cube. of the cube.
Its e.v. ~ xIts e.v. ~ xi i breaks O(3)xObreaks O(3)xOhh
-->>
OOhh
diagdiag. Therefore we get. Therefore we get
( ) ( )
2( ) 2( )
1,2,3 1,2,3
2( ) ( ) ( )
1,2,3 1,2,3
1 | |2 2
2
ii
phononi i
i i ji i j
i i j
aLt
b c
ϕ ϕ
ϕ ϕ ϕ
= =
= < =
⎛ ⎞∂= − ∇⎜ ⎟∂⎝ ⎠
− ∂ − ∂ ∂
∑ ∑
∑ ∑
Seoul, xx/05/2008 R. Casalbuoni, On the ground state of… 58
we get for the coefficientswe get for the coefficients
121a = 0b = ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛ δμ= 1
|q|3
121c
2
One can evaluate the effective One can evaluate the effective lagrangianlagrangian
for the gluons in for the gluons in thatha
anisotropic medium. For the cube one findsanisotropic medium. For the cube one finds
Isotropic propagationIsotropic propagation
This because the second order invariant for the cube This because the second order invariant for the cube and for the rotation group are the same!and for the rotation group are the same!