Overview of Finite Density QCD for String Theoriststheory.fi.infn.it/casalbuoni/lavori/Seoul_2008/High_density_QCD.pdf · e = const. solutions of the gap equation V Normal state Δ

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


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.


SummarySummarySummary

Introduction

Color Superconductivity: CFL and 2SC phases

The case of pairing at different Fermi surfaces

LOFF (or crystalline) phase

Conclusions


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


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).


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


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


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


α β αβCiL jL ijC0 ψ ψ 0 Δε ε∝

α β αβCiR jR ijC0 ψ ψ 0 Δε ε∝ −

Why CFL?Why CFL?


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.)


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)→


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


•

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


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


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


• 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


• 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


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:


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


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


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|


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))


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


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)


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

−−


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


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)


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)


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


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


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.


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).


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).


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


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


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


⋅⋅⋅+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


Gap equationGap equation

Propagator expansion

Insert in the gap equation

General strategy


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


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)


Single plane wave

Critical line from

0q

,0 =∂Ω∂

=Δ∂Ω∂

Along the critical line

)2.1q,0Tat( 2δμ==


Bowers and Bowers and RajagopalRajagopal (2002)(2002)

Preferred structure: face- centered cube


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


•

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


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


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).


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.


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


•

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


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


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



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.


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

⋅


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 =

++


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


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

= =

= < =

⎛ ⎞∂= − ∇⎜ ⎟∂⎝ ⎠

− ∂ − ∂ ∂

∑ ∑

∑ ∑


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!

Documents

Overview of Finite Density QCD for String Theoriststheory.fi.infn.it/casalbuoni/lavori/Seoul_2008/High_density_QCD.pdf · e = const. solutions of the gap equation V Normal state Δ