Non Fermi liquid effects in dense matter. Kai Schwenzer, Thomas Schäfer INT workshop From lattices to stars Seattle,

Transcription

1 Non Fermi liquid effects in dense matter Kai Schwenzer, Thomas Schäfer INT workshop From lattices to stars Seattle,

2 Introduction Possible phases at high density all involve condensed excitations and are no Fermi liquids Rajagopal, Wilzcek, hep-ph/ Alford, Kouvaris, Rajagopal, hep-ph/ Kryjevski, Kaplan, Schaefer, hep-ph/

3 Motivation (Normal) quark matter phase at high density is basically ruled out However: Important to check the stability of superfluid phases Many considered phases include un-gapped quark excitations (g)2sc, gcfl,... Simpler environment to study gluonic effects than in superfluid phases 3

4 High density effective theory Effective degrees of freedom at high density: excitations around the Fermi surface Integrate out high energy excitations No anti-particles N-point functions in the effective theory from matching procedure Counterterm to ensure gauge invariance D. K. Hong, Phys. Lett. B 473 (2000) 4

5 Effective Lagrangian L HDET = v HDET ψ v (iv D)ψ v 1 4 Ga µνg a µν +... Involves an explicit cutoff Λ Fermi surface covered with patches of size Λ involves only relative momenta and energies Consistent power counting scheme Large plays similar role to large µ N f 5

6 Unscreened gluons Gluonic excitations in the medium Static electric gluons are screened by Debye mass V (r) exp( m Dr) r Magnetic gluons are only dynamically screened (( Π t (k 0, k) = m 2 k 0 k 1 ( ) k 0 2 ) ( ) ) 1 k 2 log k 0 +k k 0 k + k 0 k Effective gluon propagator D ij (k 0, k) = δ ij ˆk iˆkj k 2 0 k 2 + iη k 0 / k m 2 D = N f g 2 µ 2 2π 2 = 2m 2 6

7 1-loop self energy Inclusion of dynamically screened gluons iσ(p) = d 4 k (2π) 4 Γa µs(p + k)γ b νd ab µν(k) Approximate analytic result Σ(ω, l) = γ ω log ( Λ ω ) Strong infrared enhancement γ = g2 C F 12π 2 Independent of p to leading order Breakdown of perturbation theory ω < Λ exp 9π2? 7 ( g 2 )

8 Scales in dense matter 0 liquid effects ω bcs Fermi sea non Fermi ω nfl HDET cutoff Λ Pairing instability ω bcs = µ exp( 3π2 2g ) screening & damping Λ m superfluidity µ chosen at the damping scale Non Fermi liquid effects at very low scales ω nfl = µ exp( 9π2 g 2 ) D. T. Son, Phys. Rev. D 59 (1999) ω m 8

9 Dyson-Schwinger equation Selfconsistent analysis of the self energy iσ(p 0 ) = g 2 C F d 4 k 1 ( v ˆk) 2 (2π) 4 p 0 +k 0 l p+k +Σ(p 0 +k 0 ) Momenta scale differently in the vicinity of the Fermi sea -> integrations decouple k l << k t 1 k 2 0 k 2 +iη k 0 / k Gluon propagator dominated for k (ηk 4 ) 1 3 Pick up the fermion pole in the l k integration Self energy is approximately 1-loop exact 9

10 RG analysis Previous result in: Boyanovsky, de Vega, Phys. Rev. D 63 (2001) S 1 (ω, l) = ω ( ω Λ ) γ l In contradiction with DS-result Broken Lorentz invariance due to Fermi see L = ψ v (ω v F l) ψ v + gv F ψ vˆv Aψ v +... Self energy and coupling Σ(ω, l) = g2 v F 9π 2 Bare fields and couplings ψ 0,v = Z 1/2 ψ v ω log ( ) Λ ω g 0 = Z g ZZ F g α = g2 v F 4π v 0,F = Z F v F G (n) 0 (ω i, v 0,F l i, α 0 ) = Z n/2 G (n) (ω i, v F l i, α) 10

11 RG solution Callan Symanzik equation { Λ Λ + β(α) α γ F (α)l i l i + n 2 γ(α) } G (n) (ω i, l i, α) = 0 One loop results β(α) = γ F (α)α RG equation for the two point function same result: Neglecting the running of the coupling yields scaling with an anomalous dimension Higher order logarithmic terms are absent 11 γ(α) = γ F (α) = 4α 9π { Λ Λ + γ [ α α + l l + 1]} S(w, l, α) = 0 S 1 (ω, l) = k a kα k ω [ log ( Λ ω S 1 (ω, l) = ω ( 1 + γlog ( Λ ω )) vf l v F β(α) = 0 )] k vf l a k = 0, k > 1

12 Fermionic self energy solid: full numeric dashed: leading log dotted: anom. dim. 12

13 Asymptotic behavior Ratio between resummed and one loop result Strong µ -> nonperturbative for small µ 13

14 Fermionic excitations Quark propagator is not given by a simple pole but contains a cut Spectral density has not fully Breit-Wigner form ρ(ω) = γω [ω(1 + γ log(λ/ω)) l] 2 + π 2 γ 2 ω 2 Vanishing wavefunction renormalization and Fermi velocity at the Fermi surface Jump at the Fermi surface vanishes Strongly IR-modfied dispersion relation 14

15 Conclusion and Outlook Dense matter is not a Fermi liquid even in the normal phase Effects result from kinematics at the Fermi surface Anomalous specific heat: A. Ipp et. al., PRD 69 (2004) C v = N f (N 2 c 1) g2 µ 2 72π 2 T log ( Λ T Modified neutrino emissivity Higher order n-point functions nonperturbative? Impact on thermal conductivity,...? ) 15