The Quark Propagator in Superconducting...

The Quark Propagator in Superconducting Phases

Dominik Nickel1

Reinhard Alkofer2 Jochen Wambach1,3

1TU Darmstadt 2U Graz 3GSI Darmstadt

Pairing in Fermionic Systems: Beyond the BCS TheorySeptember 2005, Seattle

GRAZGRAZUNIUNI

Dominik Nickel (TU Darmstadt) Seattle Sept. 2005 1 / 30

Outline

1 Motivation and introduction

2 The quark propagator in the chirally unbroken phase

3 The quark propagator in the 2SC/CFL phase for massless quarks

4 The quark propagator in the 2SC/CFL phase for massive quarks

5 Summary and outlook


Outline







Motivation

cold and dense quark matter expected to be a color superconductor⇒ investigations in

NJL-type models(M. Alford, K. Rajagopal, F. Wilczek, 1998; R. Rapp, T. Schäfer, E. Shuryak, M. Velkovsky, 1998)

weak coupling at asymptotically large densities( T. Schäfer, F. Wilczek, 1999; R. Pisarski, D. Rischke, 1999; D. Hong, V. Miransky, I. Shovkovy, L. Wijewardhana, 1999 )

quark propagator provides information aboutdynamical symmetry breakingpressure, i.e. thermodynamical potentialexcitation spectrum. . .

⇒ Extend successful truncation scheme of DSE’s in vacuum to finitedensities!


Dyson-Schwinger equation for quark propagator

−1=

−1+ Γ

our approach:approximate gluon propagator and quark-gluonvertex separately

other approaches:weak coupling expansion( T. Schäfer, F. Wilczek, 1999; R. Pisarski, D. Rischke, 1999; D. Hong et al., 1999 )

phenomenological coupling( C. Roberts, S. Schmidt, 2000 )


Approximation of quark-gluon vertex

abelian vertex construction

p q

k = p− q

ΓΓaµ(p, q; k = p − q) −→ i g Γ(k2) γµ

λa

2

Γ(k2) determined byDSE studies including Yang-Mills sector in Landau gauge:

chosen such thatpropagator multiplicative renormalizablecorrect anomalous dimension of mass function

analysis of lattice QCD data for propagators in Landau gauge:invert DSE for Γ(k2) for given quark and gluon propagator


Approximation of gluon propagator

vacuum gluon propagator in Landau gauge

Dab vacµν (k) =Z (k2)

k2Tµν(k) δab

incorporate medium effects of particle-hole excitationsadd medium polarisation of bare quarks to inverse vacuum propagator:

Dab−1µν = Dab vac−1µν + Γ

medium gluon propagator in Landau gauge

Dabµν(k) =(

Z (k2)k2 + G(k)

PTµν(k) +Z (k2)

k2 + F (k)PLµν(k)

)δab


Truncated Dyson-Schwinger equation

⇒ S−1(p) = Z2S−10 (p) +Z2

3π3

Zd4q γµS(q) γν

„αs(k2)

k2 + G(k)PTµν +

αs(k2)k2 + F (k)

PLµν

«

similar to HDLstrong running coupling

αs(k2) =Z1FZ 22

g2 Z (k2) Γ(k2)

screening and damping included through

mg(k2)2 =Nf µ2 αs(k2)

π

F (k) = 2 mg(k2)2 + . . .

G(k) =π

2mg(k2)2

k4|~k |

+ . . .


Strong running coupling αs

0 1 2 3 4 5 6 7 8 9

10

0 1 2 3 4 5

αs(k

2)

k[GeV]

DSE studieslatQCD data analysis

with abelian vertex construction DSE studies underestimate chiralsymmetry breaking in vacuum (Mq ≈ 170MeV)(C.S. Fischer, R. Alkofer, 2003)

lattice studies cannot constrain deep infrared(M.S. Bhagwat, M.A. Pichowsky, C.D. Roberts, P.C. Tandy, 2003)


Outline







The unbroken phase - I

quasiparticle propagator in chiral limit:

S−1 = S+−1γ4Λ+ + S−−1γ4Λ−

S+−1 = −ip4 + µ− |~p|+ Σ+

= −ip4(

1− ImΣ+

p4

)+ µ− |~p|+ ReΣ+

Static magnetic gluon exchange renders unbroken phase a non-Fermiliquid. leading non-analytic contribution:

ImΣ+

p4' − 4

3π

∫k>|p−pF |

dkαs(k2)k

k2 + π2 m(k2)2|p4|

−→p=pF

49

αs(Λ12)

πln(|p4|Λ2

)Λ1 ∝ |p4|

13 dominating scale of contributions


The unbroken phase - II

for coupling from DSE studies and µ = 1GeV:

-2

-1.5

-1

-0.5

0 0.02 0.04 0.06 0.08 0.1

ImΣ

+ +(p

4,|

~p|=

pF)/

p4

p4[GeV]

numericalanalytical for α(µ2)analytical for α(0)

-1.5

-1

-0.5

0

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

ImΣ

+ +(p

4,|

~p|)/p4

p4 = 15MeVp4 = 25MeVp4 = 50MeV

|~p|[GeV]

logarithmic divergence constrained to small energies(temperatures)sensitive to infrared behavior of strong couplingdependence on ~p included


Outline







Nambu-Gor’kov formalism

In Nambu-Gor’kov space with bispinors Ψ =(ψψc

), the quark DSE

S−1 = Z2S−10 + Z1F Σ

includes the gap-equation

Σ =

(Σ+ Φ−

Φ+ Σ−

)= −

∫d4q

(2π)4Γµ0 aS(q)Γ

νb(p, q)D

abµν(p − q).

→ room for diquark condensation (through finite Φ’s)

→ apply same truncations as in the unbroken case


Dirac and Color-Flavor structure in chiral limit

Dirac structureT - and χ- symmetric, even-parity and color-flavor symmetric(R. Pisarski, D. Rischke, 1999)

Σi = −i ~p · ~γ ΣA,i − i ωpγ4 ΣC,i = γ4(Σ+i Λ

+ + Σ−i Λ−)

φi =(φC,i + γ4 p̂ · ~γ φA,i

)γ5 = γ5

(φ+i Λ

+ + φ−i Λ−)

color-flavor structure of gap functions (similar for Σ+)

2SC with 3 flavors: Φ+ =Z2Z1F

φ2SC λ2 ⊗ τ2

CFL: Φ+ =Z2Z1F

(φ3̄

∑A

λA ⊗ τA + φ6∑

S

λS ⊗ τS

)


Gap-functions on the Fermi surface - I

analytical result in weak coupling (Q. Wang, D. Rischke, 2001):

φ+weak = 512 π4(

2Nf g2

) 52

e−π2+4

8 µ e− 3π

2√

2g ×{

1 2SC2−1/3CFL

. . . with αs from DSE studies:

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8 9 10

φ(p

4=

0,p

3=

pf)[M

eV]

µ[GeV]

φ+

φ−

φ+weak

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10

φ(p

4=

0,p

3=

pf)[M

eV]

µ[GeV]

φ+weak

φ+3̄

φ−3̄

φ+6φ−6

2SC CFL


Gap-functions on the Fermi surface - II

. . . with αs from lattice QCD studies:

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8 9 10

φ(p

4=

0,p

3=

pf)[M

eV]

µ[GeV]

φ+

φ−

φ+weak

0

10

20

30

40

50

60

70

80

0 1 2 3 4 5 6 7 8 9 10

φ(p

4=

0,p

3=

pf)[M

eV]

µ[GeV]

φ+weak

φ+3̄

φ−3̄

φ+6φ−6

2SC CFL

huge deviations from extrapolated weak coupling result!φ+2SC(µ ≈ 400MeV) > 60MeV!ratio of φ+

3̄to φ+2SC similar to weak coupling result

large values for anti-quasiparticle pairing


Momentum dependence of gap-functions

relative dependence of φ+2SC on |~p| on Fermi surface (p4 = 0):

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

0.95

1

0.99 1 1.01

φ+(p

4=

0,|

~p|)/φ

+(p

4=

0,|

~p|=

pf)

0.5GeV1GeV

10GeV

|~p|/pf

weak coupling regime: gap function concentrated around Fermimomentum (colinear scattering dominates)strong coupling regime: no scale separation (ΛQCD ≈ pF )


Spectral functions via Maximum Entropy Method

dispersion relation for masslessfermions:

S+(|~p|, p4) = −Z ∞−∞

dω2π

ρ(|~p|, ω)−ip4 + µ− ω

Fredholm equation (1st kind):→ need positivity of ρ for inversion→ MEM

for gapped 2SC at µ = 1GeV:

0.6 0.8 1. 1.2p[GeV]

0.6

0.8

1.

1.2

1.4

.

ω[G

eV]

access to dispersion relation and width of quasiparticlesparticle/holes vanishing below/above Fermi momentasmall group velocity (≈ .5c)


Occupation number and number densities

density given by

ρ =

∫d4p

(2π)4Tr(Z2γ4S+(p)

)=∑

i

gi(2π)3

∫d3p ni(p) =:

∑i

gi6π2

p3i

0

0.2

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4

|~p|[GeV]

n(|

~p|)

gapped 2SC

ungapped 2SCunbroken

µ = 1GeV

unbroken phase non-Fermi liquidLuttinger’s theorem not valid, i.e. pi 6= pF


Coherence lengths

extract coherence length from diquark correlator

〈〈ψ(x)T CMiψ(y)〉〉 =Z

d4p(2π)4

eip(x−y) MiXe=±

T +i,e(p) Λe~p

by

ξ2i,e =

∫d4p |∇~pT +i,e(p)|2∫

d4p |T +i,e(p)|2

0

5

10

15

20

25

30

0 1 2

µ[GeV]

ξ i,+

/λ

i

2SCCFL octet

CFL singlet ξ+/λ ∼ O(10)→ BEC-BCS transition?even smaller with couplingfrom lattice studies


UV-behavior of gap-functions

For µ, φ,ΛQCD � p the gap-equation takes the formXi

φ+C,i(p)Mi ' −3

16π

Xi

λTa Miλa

αs(p2)

p2

Z p2dq2 φ+C,i(q) +

Zp2

dq2αs(q2)φ+C,i(q)

q2

!,

with regular asymptotic form

φ+C,i(p) ∝1p2

(ln(

p2

Λ2

))γφ−1and

γφ =

{γm2 attractive channel

−γm4 repulsive channel,

where γm = 12/(33− 2Nf ).


Which is the preferred phase?

effective potential, i.e. the negative pressure, via"Cornwall-Jackiw-Tomboulis" formalism:

A[S] = −12

Trp,D,c,f ,NLnS−1 +14

Trp,D,c,f ,N(

1− S−10 S)

pressure difference between 2SC and CFL to unbroken phase:

0

0.0005

0.001

0.0015

0.002

0 0.5 1 1.5 2 2.5 3

CFL2SC

µ[GeV]

∆p[G

eV4]


Outline







Dirac and Color-Flavor structure with massive strangequarks

Dirac strutureself-energy and gap-function in T - symmetric, even-parity andcolor-flavor symmetric phase parametized by(R. Pisarski, D. Rischke, 1999):

Σi = −i~p · ~γ ΣA,i − iγ4 (p4 + iµ) ΣC,i + ΣB,i + γ4 ~p · ~γ ΣD,iφi = (γ4 p̂ · ~γ φA + γ4 φB + φC + p̂ · ~γ φD) γ5

→ selfconsistent, dynamical treatment of mass function!→ is there a simple physical interpretation for this functions?

color-flavor strutureextend ansatz from (M. Alford, J. Berges, K. Rajagopal, 1999) tobecome selfconsistent


Dependence of pressure difference on ms

-2e-05

0

2e-05

4e-05

0 50 100 150 200

ms(ν = 2GeV)[MeV]

∆p[G

eV4]

2SCµ = 300MeVµ = 400MeV

first order phase transition at critical strange quark mass


Critical strange quark mass

0

50

100

150

200

250

300 350 400 450 500

ms(ν

=2G

eV)[M

eV]

µ[MeV]

ms,critical

ms, PDG’04

2SC favored

CFL favored

light quark screen interaction also in strange quark sector→ only small dynamical chiral symmetry breaking (different to NJL)!!!→ 2SC never favored?


Outline







Summary and outlook

Summaryselfconsistent solution of DSE by approximating gluon propagator,incorporating medium effectsaccess to exitation spectrum via MEMhuge deviations from extrapolated weak coupling results2SC phase for physical strange quark mass?

Outlookspin-1 phasestemperatureneutralityincorporate Meissner effect(quasiparticle RPA, fully selfconsistent incl. of Yang-Mills sector)


Color-Flavor structure with massive strange quarks

Extend ansatz from (M. Alford, J. Berges, K. Rajagopal, 1999):

Σ =

0BBBBBBBBBBB@

b + e b c1b b + e c1c2 c2 d

ee

fg

fg

1CCCCCCCCCCCA, Φ =

0BBBBBBBBBBB@

b + e b c1b b + e c1c2 c2 d

ee

f1f2

f1f2

1CCCCCCCCCCCA,

where rows and columns correspond to

(color , flavor) = (r , u), (g, d), (b, s), (r , d), (g, u), (r , s), (b, u), (g, s), (b, d).


OutlineMotivation and introductionThe quark propagator in the chirally unbroken phaseThe quark propagator in the 2SC/CFL phase for massless quarksThe quark propagator in the 2SC/CFL phase for massive quarksSummary and outlook

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