Are there plasminos in superconductors?

Transcription

1 Are there plasminos in superconductors? Barbara Betz and Dirk-H. Rischke Institut für Theoretische Physik Johann Wolfgang Goethe-Universität Frankfurt am Main VI Workshop Rathen 2006 nucl-th/ Are there plasminos in superconductors? p.1/20

2 Contents Introduction/Motivation Are there plasminos in superconductors? p.2/20

3 Contents Introduction/Motivation Normal-conducting fermions Are there plasminos in superconductors? p.2/20

4 Contents Introduction/Motivation Normal-conducting fermions Self-energy Are there plasminos in superconductors? p.2/20

5 Contents Introduction/Motivation Normal-conducting fermions Self-energy Dispersion relation Are there plasminos in superconductors? p.2/20

6 Contents Introduction/Motivation Normal-conducting fermions Self-energy Dispersion relation Spectral density Are there plasminos in superconductors? p.2/20

7 Contents Introduction/Motivation Normal-conducting fermions Self-energy Dispersion relation Spectral density Superconducting fermions Are there plasminos in superconductors? p.2/20

8 Contents Introduction/Motivation Normal-conducting fermions Self-energy Dispersion relation Spectral density Superconducting fermions Self-energy Dispersion relation Spectral density Are there plasminos in superconductors? p.2/20

9 Contents Introduction/Motivation Normal-conducting fermions Self-energy Dispersion relation Spectral density Superconducting fermions Self-energy Dispersion relation Spectral density Conclusions Are there plasminos in superconductors? p.2/20

10 Contents Introduction/Motivation Normal-conducting fermions Self-energy Dispersion relation Spectral density Superconducting fermions Self-energy Dispersion relation Spectral density Conclusions J.-P. Blaizot and J.-Y. Ollitrault, Phys.Rev.D 48,1390 (1993) R.D.Pisarski, Nucl.Phys. A 498,423c (1989) R.D.Pisarski, D.H. Rischke, Phys.Rev.D 60, (1999) Are there plasminos in superconductors? p.2/20

11 Introduction relativistic fermionic systems Are there plasminos in superconductors? p.3/20

12 Introduction relativistic fermionic systems high termperature/ density Are there plasminos in superconductors? p.3/20

13 Introduction relativistic fermionic systems high termperature/ density 2 types of excitations gt 4 p G. Baym, J.-P. Blaizot, B. Svetitsky, Phys. Rev. D 46 (1992) 4043 Are there plasminos in superconductors? p.3/20

14 Introduction relativistic fermionic systems high termperature/ density 2 types of excitations particles and antiparticles gt 4 p G. Baym, J.-P. Blaizot, B. Svetitsky, Phys. Rev. D 46 (1992) 4043 Are there plasminos in superconductors? p.3/20

15 Introduction relativistic fermionic systems high termperature/ density 2 types of excitations particles and antiparticles additional collective excitations: gt 4 G. Baym, J.-P. Blaizot, B. Svetitsky, Phys. Rev. D 46 (1992) 4043 p Are there plasminos in superconductors? p.3/20

16 Introduction relativistic fermionic systems high termperature/ density 2 types of excitations gt 4 particles and antiparticles p additional collective excitations: plasmino and anti-plasmino G. Baym, J.-P. Blaizot, B. Svetitsky, Phys. Rev. D 46 (1992) 4043 Are there plasminos in superconductors? p.3/20

17 Motivation realtivistic description T 160 MeV QGP hadrons 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Are there plasminos in superconductors? p.4/20

18 Motivation realtivistic description T 160 MeV heavy-ion collisions hadrons QGP 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Are there plasminos in superconductors? p.4/20

19 Motivation realtivistic description T 160 MeV heavy-ion collisions compact stellar objects hadrons QGP 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Are there plasminos in superconductors? p.4/20

20 Motivation realtivistic description T 160 MeV heavy-ion collisions compact stellar objects hadrons QGP color superconductor 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Are there plasminos in superconductors? p.4/20

21 Motivation realtivistic description T 160 MeV heavy-ion collisions compact stellar objects hadrons QGP color superconductor 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Do plasminos occur in cold and superconducting matter? Are there plasminos in superconductors? p.4/20

22 Motivation realtivistic description T 160 MeV heavy-ion collisions compact stellar objects hadrons QGP color superconductor 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Do plasminos occur in cold and superconducting matter? consider ultrarelativistic fermions Are there plasminos in superconductors? p.4/20

23 Motivation realtivistic description T 160 MeV heavy-ion collisions compact stellar objects hadrons QGP color superconductor 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Do plasminos occur in cold and superconducting matter? consider ultrarelativistic fermions interacting via scalar boson exchange Are there plasminos in superconductors? p.4/20

24 Motivation realtivistic description T 160 MeV heavy-ion collisions compact stellar objects hadrons QGP color superconductor 10 MeV color superconductivity 308 MeV DHR, Prog. Part. Nucl. Phys. 52 (2004) 197 Do plasminos occur in cold and superconducting matter? consider ultrarelativistic fermions interacting via scalar boson exchange in the limit of T 0 Are there plasminos in superconductors? p.4/20

25 Normal-conducting fermions Self-energy fermions interacting with scalar bosons Are there plasminos in superconductors? p.5/20

26 Normal-conducting fermions Self-energy fermions interacting with scalar bosons L I = g ψψφ Are there plasminos in superconductors? p.5/20

27 Normal-conducting fermions Self-energy fermions interacting with scalar bosons q=k-p L I = g ψψφ one-loop self-energy Σ(P ) = g 2 T n Σ(p 0,p) = - d 3 k (2π) 3 D 0 (K P )G 0 (K) k (p 0,p) k Are there plasminos in superconductors? p.5/20

28 Normal-conducting fermions Self-energy fermions interacting with scalar bosons q=k-p L I = g ψψφ one-loop self-energy Σ(P ) = g 2 T n Σ(p 0,p) = - d 3 k (2π) 3 D 0 (K P )G 0 (K) k (p 0,p) k projection onto positive/negative-energy solution ] Σ ± (P ) 1 2 [Λ Tr ± p γ 0 Σ(P ) Are there plasminos in superconductors? p.5/20

29 Normal-conducting fermions Self-energy fermions interacting with scalar bosons q=k-p L I = g ψψφ one-loop self-energy Σ(P ) = g 2 T n Σ(p 0,p) = - d 3 k (2π) 3 D 0 (K P )G 0 (K) k (p 0,p) k projection onto positive/negative-energy solution ] Σ ± (P ) 1 2 [Λ Tr ± p γ 0 Σ(P ) imaginary part Im Σ ± (ω, p) = Im a(ω, p) ± p Im b(ω, p) Are there plasminos in superconductors? p.5/20

30 Normal-conducting fermions Self-energy fermions interacting with scalar bosons q=k-p L I = g ψψφ one-loop self-energy Σ(P ) = g 2 T n Σ(p 0,p) = - d 3 k (2π) 3 D 0 (K P )G 0 (K) k (p 0,p) k projection onto positive/negative-energy solution ] Σ ± (P ) 1 2 [Λ Tr ± p γ 0 Σ(P ) imaginary part Im Σ ± (ω, p) = Im a(ω, p) ± p Im b(ω, p) real part Re Σ ± (ω, p) = Re a(ω, p) ± p Re b(ω, p) Are there plasminos in superconductors? p.5/20

31 Normal-conducting fermions Imaginary part of the self-energy Im(a+b*p)/µ Im(a-b*p)/µ ω/µ p/µ ω/µ p/µ massless fermions (left panel) and antifermions (right panel) for g 2 /4π = 1, at T = 0 Are there plasminos in superconductors? p.6/20

32 Normal-conducting fermions Real part of the self-energy Re(a+b*p)/µ Re(a-b*p)/µ ω/µ p/µ ω/µ p/µ massless fermions (left panel) and antifermions (right panel) for g 2 /4π = 1, at T = 0 Are there plasminos in superconductors? p.7/20

33 Normal-conducting fermions Dispersion relation is given by the roots of the full inverse propagator G 1 ± (P ) G 1 0,± (P ) + Σ ±(P ) Are there plasminos in superconductors? p.8/20

34 Normal-conducting fermions Dispersion relation is given by the roots of the full inverse propagator G 1 ± (P ) G 1 0,± (P ) + Σ ±(P ) ω/µ particle plasmino anti-plasmino antiparticle add. coll. exc p/µ Are there plasminos in superconductors? p.8/20

35 Normal-conducting fermions Spectral density is determined from ρ ± (ω, p) = 1 π Im G ±(ω, p) Are there plasminos in superconductors? p.9/20

36 Normal-conducting fermions Spectral density is determined from ρ ± (ω, p) = 1 π Im G ±(ω, p) ω/µ p/µ Are there plasminos in superconductors? p.9/20

37 Superconducting fermions Self-energy taking the superconducting ground state as a basis Are there plasminos in superconductors? p.10/20

38 Superconducting fermions Self-energy taking the superconducting ground state as a basis self-energy is Σ(P ) = g 2 T n d 3 k (2π) 3 D 0 (K P ) G 0 (K) Are there plasminos in superconductors? p.10/20

39 Superconducting fermions Self-energy taking the superconducting ground state as a basis self-energy is Σ(P ) = g 2 T n d 3 k (2π) 3 D 0 (K P ) G 0 (K) quasifermion propagator G 0 G 0 (P ) = e=± p 0 (µ ep) p 2 0 (µ ep)2 φ e (P ) 2 Λ e p γ 0 Are there plasminos in superconductors? p.10/20

40 Superconducting fermions Self-energy taking the superconducting ground state as a basis self-energy is Σ(P ) = g 2 T n d 3 k (2π) 3 D 0 (K P ) G 0 (K) quasifermion propagator G 0 G 0 (P ) = e=± p 0 (µ ep) p 2 0 (µ ep)2 φ e (P ) 2 Λ e p γ 0 calculate imaginary and real part of the self-energy Are there plasminos in superconductors? p.10/20

41 Superconducting fermions Self-energy taking the superconducting ground state as a basis self-energy is Σ(P ) = g 2 T n d 3 k (2π) 3 D 0 (K P ) G 0 (K) quasifermion propagator G 0 G 0 (P ) = e=± p 0 (µ ep) p 2 0 (µ ep)2 φ e (P ) 2 Λ e p γ 0 calculate imaginary and real part of the self-energy gap function of particles φ + = 0.25 µ and gap function of antiparticles φ = 0 leads to... Are there plasminos in superconductors? p.10/20

42 Superconducting fermions Imaginary part of the self-energy Im(a+b*p)/µ Im(a-b*p)/µ ω/µ p/µ ω/µ p/µ massless superconducting fermions (left panel) and antifermions (right panel) for g 2 /4π = 1, at T = 0 Are there plasminos in superconductors? p.11/20

43 Superconducting fermions Real part of the self-energy Re(a+b*p)/µ Re(a-b*p)/µ ω/µ p/µ ω/µ p/µ massless superconducting fermions (left panel) and antifermions (right panel) for g 2 /4π = 1, at T = 0 Are there plasminos in superconductors? p.12/20

44 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± ) 1 Are there plasminos in superconductors? p.13/20

45 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± Σ + was computed above ) 1 Are there plasminos in superconductors? p.13/20

46 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± Σ + was computed above Σ (P ) C [ Σ + ( P ) ] T C 1 is self-energy for charge-conjugate particles ) 1 Are there plasminos in superconductors? p.13/20

47 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± Σ + was computed above Σ (P ) C [ Σ + ( P ) ] T C 1 is self-energy for charge-conjugate particles Φ + is order parameter for condensation Φ + (P ) = e=± φ e (P ) Λ e p γ 5 ) 1 Are there plasminos in superconductors? p.13/20

48 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± Σ + was computed above Σ (P ) C [ Σ + ( P ) ] T C 1 is self-energy for charge-conjugate particles Φ + is order parameter for condensation Φ + (P ) = e=± φ e (P ) Λ e p γ 5 projection onto positive and negative energies Σ + (P ) = γ 0 Σ e (P ) Λ e p e=± Σ (P ) = γ 0 Σ e ( P ) Λ e p e=± ) 1 Are there plasminos in superconductors? p.13/20

49 Superconducting fermions Dispersion relation To distinguish the solutions, we use G ± e (P ) 1 2 Tr [G ± (P ) γ 0 Λ e p ] Are there plasminos in superconductors? p.14/20

50 Superconducting fermions Dispersion relation To distinguish the solutions, we use G ± e (P ) 1 2 Tr [G ± (P ) γ 0 Λ e p G + + (P ) = p 0 µ+p Σ + ( P ) [p 0 +µ p+σ + (P )][p 0 µ+p Σ + ( P )] φ + (P ) 2 G + (P ) = 1 p 0 +µ+p+σ (P ) G + (P ) = 1 p 0 µ p Σ ( P ) G (P ) = p 0 +µ p+σ + (P ) [p 0 +µ p+σ + (P )][p 0 µ+p Σ + ( P )] φ + (P ) 2 ] Are there plasminos in superconductors? p.14/20

51 Superconducting fermions Dispersion relation ω/µ 1 ω/µ p/µ p/µ particle antiplasmino add. coll. exc. hole antiplasmino-hole plasmino antiparticle add. coll. exc. plasmino-hole antihole positive energy dispersion relation, g 2 /(4π) = 1, T = 0 negative energy dispersion relation, g 2 /(4π) = 1, T = 0 Are there plasminos in superconductors? p.15/20

52 Superconducting fermions Spectral density ρ ± e (ω, p) = 1 π Im G± e (ω, p) Are there plasminos in superconductors? p.16/20

53 Superconducting fermions Spectral density ρ ± e (ω, p) = 1 π Im G± e (ω, p) ω/µ 1 ω/µ p/µ p/µ spectral density ρ + + for g2 /(4π) = 1, T = 0 spectral density ρ for g2 /(4π) = 1, T = 0 Are there plasminos in superconductors? p.16/20

54 Superconducting fermions Spectral density ω/µ 0 ω/µ p/µ spectral density ρ + for g2 /(4π) = 1, T = p/µ spectral density ρ + for g2 /(4π) = 1, T = 0 Are there plasminos in superconductors? p.17/20

55 Conclusions We studied the fermionic excitation spectrum Are there plasminos in superconductors? p.18/20

56 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, Are there plasminos in superconductors? p.18/20

57 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, for massless fermions interacting with massless bosons, Are there plasminos in superconductors? p.18/20

58 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, for massless fermions interacting with massless bosons, vanishing temperature and finite chemical potential. Are there plasminos in superconductors? p.18/20

59 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, for massless fermions interacting with massless bosons, vanishing temperature and finite chemical potential. Plasminos occur Are there plasminos in superconductors? p.18/20

60 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, for massless fermions interacting with massless bosons, vanishing temperature and finite chemical potential. Plasminos occur in the normal-conducting as well as Are there plasminos in superconductors? p.18/20

61 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, for massless fermions interacting with massless bosons, vanishing temperature and finite chemical potential. Plasminos occur in the normal-conducting as well as in the superconducting systems. Are there plasminos in superconductors? p.18/20

62 Conclusions We studied the fermionic excitation spectrum in normal- and superconducting systems, for massless fermions interacting with massless bosons, vanishing temperature and finite chemical potential. Plasminos occur in the normal-conducting as well as in the superconducting systems. Superconducting excitation spectrum is the extension of the normal-conducting one. Are there plasminos in superconductors? p.18/20

63 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± ) 1 Are there plasminos in superconductors? p.19/20

64 Superconducting fermions Dispersion relation is given by the poles of the propagator G ± = ( [G ± 0 ] 1 + Σ ± Φ { [G 0 ] 1 + Σ } 1 Φ ± for quasiparticles G + = ([G 0 ] 1 + Σ ) {([G +0 ] 1 + Σ + )([G 0 ] 1 + Σ ) ) 1 Φ ( [G 0 ] 1 + Σ ) 1 Φ + ( [G 0 ] 1 + Σ )} 1 Are there plasminos in superconductors? p.19/20

65 Superconducting fermions Dispersion relation condensation occurs in the even-parity channel Φ + (P ) = e=± φ e (P ) Λ e p γ 5 Are there plasminos in superconductors? p.20/20

66 Superconducting fermions Dispersion relation condensation occurs in the even-parity channel Φ + (P ) = e=± φ e (P ) Λ e p γ 5 so we have Φ ( [G 0 ] 1 + Σ ) 1 Φ + ( [G 0 ] 1 + Σ ) = e=± φ e (P ) 2 Λ e p Are there plasminos in superconductors? p.20/20

67 Superconducting fermions Dispersion relation condensation occurs in the even-parity channel Φ + (P ) = e=± φ e (P ) Λ e p γ 5 so we have Φ ( [G 0 ] 1 + Σ ) 1 Φ + ( [G 0 ] 1 + Σ ) 2SC-Phase Φ ( [G 0 ] 1 +Σ ) 1 Φ + ( [G 0 ] 1 +Σ ) = = e=± e=± L p = (J 3 ) 2 (τ 2 ) 2 = (δ ij δ i3 δ j3 )δ fg φ e (P ) 2 Λ e p φ e (P ) 2 L p Λ e p Are there plasminos in superconductors? p.20/20