Nambu-Goldstone Fermion Mode in Quark-Gluon Plasma and Bose-Fermi Cold Atom System

Transcription

1 Nambu-Goldstone Fermion Mode in Quark-Gluon Plasma and Bose-Fermi Cold Atom System Daisuke Satow (ECT*,!)! Collaborators: Jean-Paul Blaizot (Saclay CEA, ") Yoshimasa Hidaka (RIKEN, #)

2 Supersymmetry (SUSY) Symmetry related to interchanging a boson and a fermion = b f b f 2

3 Supersymmetry (SUSY) Q: supercharge b f b f Supercharge operator: annihilate one fermion and create one boson (and its inverse process) SUSY if [Q, H]=0 3

4 SUSY breaking (medium effect) SUSY SUSY breaking (Ground state is doubly degenerated) nf (T=0, ) nb µf E Q E nf nb d d k εk E εk E Nambu-Goldstone (NG) mode? 4

5 V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989) NG fermion which is related to SUSY breaking Nambu-Goldstone theorem: ik µ (fermion ver.) Broken symmetry d 4 xe ik (x y) TJ µ (x)o(y) = {Q, O} J µ : supercurrent Q=J 0 : supercharge NG mode Order parameter When the order parameter is finite, the propagator in the left-hand side has a pole at k 0. 5

6 V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989) NG fermion which is related to SUSY breaking ik µ Broken symmetry d 4 xe ik (x y) TJ µ (x)o(y) = {Q, O} NG mode Order parameter If we set O=Q, NG mode appears in <QQ>. Order parameter is energy-momentum tensor (T µν ) in the Wess-Zumino model. In the presence of medium, SUSY is always broken. (µ=0) SUSY is fermionic symmetry, so the NG mode is fermion (Goldstino). Rare fermionic zero mode { } T µν =diag(ρ,p,p,p), 6

7 Quasi-goldstino in hot QED/QCD V. V. Lebedev and A. V. Smilga, Annals Phys. 202, 229 (1990) Both of the quark and the gluon are regarded as massless at high T There is SUSY approximately if we neglect the interaction. = q g 7

8 Quasi-goldstino in hot QED/QCD Actually, in weak coupling regime, we established the existence of the (quasi) goldstino in QED/QCD. Y. Hidaka, D. S., and T. Kunihiro, Nucl. Phys. A 876, 93 (2012) D. S., PRD 87, (2013). J. P. Blaizot and D. S., Phys. Rev. D 89, (2014). dispersion relation Reω p/3 Consistent with the goldstino picture. Damping rate Im Residue g 2 144π 2 g 2 (4 + N f ) 2 48π 2 QED QCD 8

9 Why Cold Atom System? Lattice Structure (optical lattice) Hubbard model Tunableness of interaction strength (laser intensity, magnetic field strength: Feshbach resonance) Cold atom system can be used as experiment station of many-body system whose experiment is difficult. Wess-Zumino model: Y. Yu, and K. Yang, PRL 105, (2010) Dense QCD: K. Maeda, G. Baym and T. Hatsuda, PRL 103, (2009) Relativistic QED: Kapit and Mueller, PRA 83, (2011) 9

10 Motivation of this research If we prepare a cold atom system with SUSY, we can experimentally observe the goldstino!! 10

11 T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) SUSY in Cold Atom System Trap two kinds of fermion (f, F) and their bound state (boson: b) on optical lattice. b f F 11

12 T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) SUSY in Cold Atom System H α = t α a α i aj α µ α a α i ai α ij i ( ) f b = (tf =tb) (µf =µb) How to tune t: M. Snoek, S. Vandoren, and H. T. C. Stoof, PRA 74, (2006) 12

13 SUSY in Cold Atom System T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) U bb 2 i n b i ( n b i 1 ) + U bf i n b i nf i, = (Ubb =Ubf) When tf =tb, Ubb =Ubf, µf =µb, Q =bf commutes with the Hamiltonian. F decouples. 13

14 NG fermion which is related to SUSY breaking ik µ Broken symmetry d 4 xe ik (x y) TJ µ (x)o(y) = {Q, O} Q =bf Q =b f NG mode Order parameter If we set O=Q, NG mode appears in <QQ >. Order parameter is density (<{Q, Q }>=ρ) in this case. SUSY is always broken when ρ is finite. 14

15 Setup d=2 No BEC. Continuum limit a << (kf) -1, (T) -1 (a=1 unit, for simplicity.) Δµ=µf -µb 0 Explicit SUSY breaking 15

16 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) Calculate the propagator (spectrum) of the Goldstino. Since Q =bf, the propagator at the one-loop order is (<Q (x)q(0)>) <Q (x)q(0)>= b f The Hamiltonian has SUSY, so the bare dispersion relations of the fermion and the boson are the same, and thus the loop integral diverges at ω-δµ, p 0. (pinch singularity) (fermion term)= d 2 k n F (ϵ f k ) (2π) 2 ω µ t(2k p + p 2 ), 2 16

17 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) (1) Resum density correction to the dispersion relations. fermion: Uρb Uρf + 2Uρb boson: the pinch singularity is regularized. d 2 k n F (ϵ f k ) (2π) 2 ω ( µ + t(2k p + p 2 )+Uρ), (fermion term)= U -1 17

18 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) All ring diagrams contributes at the same order. U -1 U -1 U U -1 =U -1 We need to sum up infinite ring diagrams. 18

19 Analysis with Resummed Perturbation T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) (2) Random Phase Approximation U The self-consistent equation can be rewritten in explicit form. 19

20 Expansion in terms of energy, momentum Calculate the propagator of the Goldstino by using (1), (2). We are interested in low energy-momentum region. Expand in terms of energy, momentum p <<kf, Uρ/(kf t) 20

21 Spectral properties of Goldstino Dispersion Relation ω Strength 1 (maximum value allowed by sum rule) Type-II NG mode (while it is Type-I in relativistic model). α 1 ρ ( 4πt 2 ρ 2 f Uρ (T=0 case) ) t(ρ f ρ b ) The damping rate is expected to be zero at p=0 because of NG theorem. Damping rate when p is finite can not be evaluated since we neglected collision effect. 21

22 Results at T 0 17 U/t=0.1, f =0.5, b = /t T/t The coefficient α increases when T increases. 22

23 Analogy with Magnon in Ferromagnet Goldstino Conserved Charge: Q, Q, ρ Magnon in ferromagnet Conserved Charge: m +, m -, m z Broken order parameter Broken order parameter Q m + Q m - b Q 2 =Q 2 =0 f up m +2 =m -2 =0 down m ± =m x ±im y 23

24 Analogy with Magnon in Ferromagnet Magnon external magnetic field h Dispersion relation of magnon:! = h + p 2 Type-II 24

25 Analogy with Magnon in Ferromagnet Generally, expectation values of commutators among conserved charges are essentially important to predict whether the NG mode is type-i or II. NII=rank<[Qa, Qb]>/2 NI=NBS-2NII H. Watanabe and H. Murayama, PRL 108, (2012) Y. Hidaka, PRL 110, (2013) ( <[m ±, m z ]>=0 ) <[m +, m - ]>=2m0 rank<[qa, Qb]>/2=1 NII =1 NI =0 NBS =2 25

26 Analogy with Magnon in Ferromagnet The Case of Goldstino Expectation values of (anti-) commutators have the same structure as those in ferromagnet. <[Q, ρ]>=0 <{Q, Q }>=ρ ( <[m ±, m z ]>=0 ) <[m +, m - ]>=2m0 Q, Q m+, m- ρ mz Therefore we understand the reason why the both spectrums have the same form. 26

27 Analogy with Magnon in Ferromagnet! = µ p 2 Δµ h As that of magnon, the damping rate has the form of Dp 4. (D can be calculated by using Kubo formula) T. Hayata, Y. Hidaka, arxiv: [hep-th] The momentum dependence of the energy and the damping rate is modelindependent! 27

28 Summary We obtained the expression of dispersion relation and the strength of the goldstino (at weak coupling, continuum limit). We understand the similarity between the goldstino and the magnon in ferromagnet, by using the fact that the (anti-)commutation relations have the same structure. We obtained the momentum dependence of the excitation energy and the damping rate in modelindependent way. 28

29 Back Up 29

30 Effect on Observable Quantity T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) Photoassociation (PA) process two lasers b f F Binding energy: Eb δ 0 = ω 2 ω 1 E b process is 30

31 T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) Effect on Observable Quantity F couples with goldstino through PA process. H ex = g 0 i ( ) b i f if i e iδ0t + H.c. goldstino Production rate of F should contain information of spectrum of goldstino. 31

32 Effect on Observable Quantity T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) Linear Response Theory R = 2g 2 0 ρ k ImD R (k, δ 0 ) (ε F K-δ0, k-k0) photon propagator One-loop Approximation (δ0, k0) Cut ImDR= 32

33 Effect on Observable Quantity T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) ImDR= (ε F K-δ0, k-k0) (δ0, k0) R = R b Ff R Ff b ( R b Ff = k = 2πg0 2 ρa( k k 0, εk F δ ( 0) θ δ0 εk F ), (ε F K=tFk 2 -µf) ( ) ( ) goldstino spectral function = ( ) ( Fermi distribution R Ff b = k 2πg0 2 ρa( k k 0, εk F δ ) ( ) 0 θ ε F k 33

34 Effect on Observable Quantity T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) F is dilute: µ F 0 k 0 0 Momentum of laser is small: δ0<0 34

35 Effect on Observable Quantity (Result at T=0) R g2 0 N 2t F + α ( δ 0 = µ, ) ( N: total number ) tf: hopping parameter of F α 1 ρ (T=0 case) ( ) t 2 4πρ 2 f t(ρ f ρ b ) Uρ By tuning ρf/b, U, t, and T, we can decrease 2tF +α so that R becomes large. 35

36 Future Work Work in three dimensions (BEC appears.! Beyond the one-loop approximation (damping rate of the goldstino).! Determine the form of the coupling among the goldstino, and between goldstino and other zero modes by using idea of low-energy effective Lagrangian. (ex: chiral perturbation theory) 36

37 Analogy with Magnon in Ferromagnet Derivation of spectrum of magnon P. M. Chaikin, T. C. Lubensky, Principles of condensed matter physics. Expand the free energy density in terms of spatial inhomogeneity. f = ρ s 2M 2 0 ρ (( m x ) 2 +( m y ) 2 ) M0:spontaneous magnetization 37

38 Analogy with Magnon in Ferromagnet df =-hidmi h effective magnetic field external magnetic field (x, y) h a = ρ s (z) hz =h M m a, Equation of Motion of m i t m i = ih j [m i,m j ] = (h m) i, (H-himi) [mi, mj]=iεijkmk Linearization (mi: small) t m x = ρ s M 0 2 m y + hm y, t m y = ρ s M 0 2 m x hm x, 38

39 Analogy with Magnon in Ferromagnet rewrite using m ± instead of m x, m y. t m ± = i ( ) ρs 2 + h M 0 m ±. ( ) Dispersion relation: ω = ± ( ) ρs p 2 + h M 0 39

40 Goldstino in Wess-Zumino model V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989). In Wess-Zumino model, the goldstino exists. Through the fermion-boson coupling, it appears also in fermion propagator. f Q a 40

41 V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989). Goldstino in Wess-Zumino model (T>>m case, weak coupling) dispersion relation Reω p/3 Residue g The dispersion relation is linear (Type-I NG mode). 41

42 Goldstino as Phonino V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989). Phonon case ) conservation of energy-momentum: small deviation in local equilibrium ( ρ T 2 0 p ) T 2 T 00 (x, t) = µ T µν =0, T (x, t) =0 ( 1+ T (x, t) T ) ) ρ, T 0i (x, t) = v i (x, t)(ρ + p), ( T ij (x, t) = δ ij 1+ T (x, t) T ) p. dispersion relation: gation of sound waves with the v ω=vs p v 2 S = p/ T ρ/ T 42

43 { } Goldstino as Phonino V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989). conservation of supercurrent: µ J µ =0 small deviation in local equilibrium (grassmannodd!) must be con µ δ ξ J µ (x, t) = 2iγ ν µ ξ(x, t) T µν (x, t) T µν =diag(ρ,p,p,p), (ργ p γ i i ) ξ(x, t) =0, e Dirac equation for a mass dispersion relation: v SS = p ρ. ω=vss p etric syste Since ρ=3p in conformal system, the velocity is always 1/3 of the speed of light if the system is conformal. 43

44 Goldstino as Phonino V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989). SUSY transformation Tµν (energy-momentum tensor) Jν supercurrent Phonon goldstino (phonino) Goldstino is a supermultiplet of phonon (phonino)!! 44

45 NG fermion which is related to SUSY breaking Generally, symmetry breaking generates zero energy excitation (Nambu-Goldstone (NG) mode). symmetry breaking NG mode We expect that SUSY breaking order parameter also generates NG mode. 45

46 NG fermion which is related to SUSY breaking Example in condensed matter physics ferromagnet: O(3) O(2) magnon Example in high-energy physics QCD: SU(2)L SU(2)R SU(2)V π meson We expect that SUSY breaking also generates NG mode. 46

47 Supercurrent in Wess-Zumino model J µ = ( A iγ 5 B ) γ µ Ψ + i ( F iγ 5 G ) γ µ Ψ. rcharge {Q, J µ } =2T µν γ ν +S.T. δφ i = iξ[q, φ i ] ± 47

48 Cold atom dispersion α B A = t(ρ b ρ f ) ρ + t2 a 4 πuρ 2 0 (3.43 d k k 3 (n f (ϵ f k )+n b(ϵ b k)), 48

49 Analogy with Magnon in Ferromagnet Expand the free energy density in terms of spatial inhomogeneity. f = β ρ 2 q q, Introduce super chemical potential (η, η ). P. Kovtun and L. G. Yaffe, Phys. Rev. D 68, (2003) df =-µf dnf -µbdnb-η dq-dq η η = β ρ 2 2 q. 49

50 Analogy with Magnon in Ferromagnet Equation of Motion of Q i t q = µ f [q, ρ f ] µ b [q, ρ b ] [q, q η] =(µ f µ b )q {q, q }η = µq + β ρ 2 q, η: Grassmann odd As is expected, the spectrum has the same form as that of magnon in ferromagnet. ω = ± ( β ρ p2 + µ ) 50

51 Effect on Observable Quantity T. Shi, Y. Yu, and C. P. Sun, PRA 81, (R) (2010) 1 ω ω δ 0 E b δ 0 = ω 2 ω 1 E b process When Δ0 is large, is 1> can be neglected. 51