Hot and Magnetized Pions

Transcription

1 .. Hot and Magnetized Pions Neda Sadooghi Department of Physics, Sharif University of Technology Tehran - Iran 3rd IPM School and Workshop on Applied AdS/CFT February 2014 Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 1 / 40

2 Introduction Introduction QCD Phases Under Extreme Conditions Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 2 / 40

3 Introduction QCD under Extreme Conditions Quantum Chromodynamics QCD is the fundamental theory of strong interaction Main Properties: - Asymptotic freedom - Confinement - Chiral symmetry At different scales of energies Different degrees of freedom - High energy scales: Quarks and gluons - Low energy scales: Weakly interacting mesons and baryons Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 3 / 40

4 Introduction QCD under Extreme Conditions Question: How to explore QCD at different energy scales? Experimentally: - Back to Early Universe Big Bang - Relativistic Heavy Ion Experiments Little Bang Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 4 / 40

5 Introduction QCD under Extreme Conditions Question: How to explore QCD at different energy scales? Experimentally: - Back to Early Universe Big Bang - Relativistic Heavy Ion Experiments Little Bang Theoretically: - High energy scales: Perturbative QCD at T = 0 and T 0 - Low energy scales: Nonperturbative QCD at T = 0 and T 0 Lattice QCD Effective models describing the dynamics of mesons (Chiral perturbation theory, NJL like models, Relativistic Hydrodynamics, AdS/CFT, etc.) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 4 / 40

6 Introduction Quark Matter Quark Matter: QCD Phase Diagram Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 5 / 40

7 Introduction Quark Matter in External B Fields Strong Magnetic Fields in Nature and Lab The Early Universe: G - Primordial magnetic fields from superconducting cosmic strings [e.g. T. Kibble, E. Witten] Compact Stars: G (s) and G (c) - Magnetized quark matter in neutron stars [e.g. A. Shabad et al., E. Ferrer et al., K. Rajagopal et al. ] Heavy Ion Experiments: G - Charge asymmetry fluctuations in HIC [e.g. D. Kharzeev et al.] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 6 / 40

8 Introduction Quark Matter in External B Fields Questions - The effect of constant B fields on QCD phases? χsb and CS [-] Sh. Fayazbakhsh and N.S., PRD 82 (2010) [-] Sh. Fayazbakhsh and N.S., PRD 83 (2011) - The effect of constant B fields on meson properties at finite T? [-] Sh. Fayazbakhsh, S. Sadeghian and N.S., PRD 86 (2012) [-] Sh. Fayazbakhsh and N.S., PRD 88 (2013) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 7 / 40

9 Introduction Quark Matter in External B Fields Effects of Constant B Field on Quark Matter Magnetic Catalysis [Klimenko 1992, Miransky 1995] - Bound state formation even at the weakest attractive interaction - Dynamical mass generation Dynamical chiral SB m dyn ) eb exp ( 2π2 GeB ψψ = 0 Chiral Magnetic Effect [Kharzeev et al. 2008] - Exploring the strong CP violation in HIC Magneto-Sono-Luminescence in HIC [Kharzeev et al and 2014] - Photon and dilepton production in HIC Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 8 / 40

10 Introduction Quark Matter in External B Fields Magnetic Catalysis (MC): Applications Cosmology Condensed Matter Physics Particle Physics - Spontaneous creation of axial currents [Son et al 2005] - Formation of π 0 domain walls [Stephanov et al 2008] - Anisotropy in thermodynamic quantities [Rischke et al 2009, N.S. 2009] - Paraelectricity in magnetized (massless) QED [Ferrer et al 2011] - Effects of MC on various phases of QCD [Klimenko 1992, Miransky 1995] Chiral and Color-Superconductivity phase transitions [Shovkovy et al 2007, Fukushima et al 2008, Fayazbakhsh and N.S and 2011] - Effects of MC on meson dynamics [Andersen 2012, Fayazbakhsh and N.S. 2012, 2013] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 9 / 40

11 The Model: Two Flavor NJL Model The Model Quantum Effective Action of a Magnetized Two-Flavor NJL Model Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 10 / 40

12 The Model: Two Flavor NJL Model Quantum Effective Action of Mesons Two-Flavor NJL Model in a Constant B Field - Lagrangian density of two-flavor (gauged) NJL model L NJL = q(iγ D m 0 )q + G [ ( qq) 2 + ( qiγ 5 τq) 2] + F: qf c, f = 1, 2 and c = r, g, b G: D µ µ + iea ext. µ with A ext µ = (0, 0, xb, 0) leading to B = Be 3 - Introduce bosonic fields σ(x) 2G qq, π(x) 2G qiγ 5 τq - Semi-bosonized Lagrangian density L s.b. = σ2 + π 2 4G + q[ iγ D m 0 σ(x) iγ 5 τ π(x) ] q Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 11 / 40

13 The Model: Two Flavor NJL Model Quantum Effective Action of Mesons Quantum Effective Action of Mesons - Integrate out the fermions: Quantum effective action for mesons Γ eff [σ, π] = Tr ln[iγ D m 0 σ(x) iγ 5 τ π(x)] - Choose a SSB direction for Φ(x) = (σ, π) - Expand around mean fields Φ 0 = Φ(x) : Φ(x) = Φ 0 + Φ(x) - Mean fields: σ(x) σ 0, π(x) = 0 In a second order derivative expansion, we obtain: Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 12 / 40

14 The Model: Two Flavor NJL Model Quantum Effective Action of Mesons Quantum Effective Action of Mesons in a Derivative Expansion Γ eff [ Φ] = 1 2 d 4 x Φ(x) ( + m 2 Φ) Φ(x) Ω[m] - In the effective kinetic term [Φ = (σ, π)]: with MΦ 2 Φ(x) FΦ 00 1/2 Φ(x) and 0 2 u (i)2 mφ 2 MΦ 2 FΦ 00 and u (i)2 Φ FΦ ii Φ F 00 Φ d 4 δ 2 Γ eff z δφ(0)δφ(z) and F µν Φ d 4 z z µ z ν δ 2 Γ eff Φ0 δφ(0)δφ(z) 2 i Φ0 Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 13 / 40

15 The Model: Two Flavor NJL Model Quantum Effective Action of Mesons Dispersion Law of σ and π mesons For σ meson q 2 0 = u(1)2 σ q1 2 + u(2)2 σ q2 2 + u(3)2 σ q3 2 + m2 σ For π = (π 1, π 2, π 3 ) mesons q 2 0 = u(1)2 π l q1 2 + u(2)2 q2 2 + u(3)2 π l q3 2 + m2 π l, l = 1, 2, 3 π l π 0 π 3 and π ± (π 1 ± iπ 2 ) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 14 / 40

16 The Model: Two Flavor NJL Model Quantum Effective Action of Mesons Dispersion Law of σ and π mesons For σ meson q 2 0 = u(1)2 σ q1 2 + u(2)2 σ q2 2 + u(3)2 σ q3 2 + m2 σ For π = (π 1, π 2, π 3 ) mesons q 2 0 = u(1)2 π l q1 2 + u(2)2 q2 2 + u(3)2 π l q3 2 + m2 π l, l = 1, 2, 3 π l π 0 π 3 and π ± (π 1 ± iπ 2 ) Later we will show { u (1) π = u (2) π = u (3) π < 1 at T 0, eb = 0 u (1) π = u (2) π > u (3) π = 1 at T 0, eb 0 Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 14 / 40

17 The Model: Two Flavor NJL Model Quantum Effective Action of Mesons Effective Potential of Mesons Γ eff [ Φ] = 1 2 d 4 x Φ(x) ( + m 2 Φ) Φ(x) Ω[m] - In the effective potential Ω(m), m is the constituent quark mass m = m 0 + σ 0 - At finite (T, µ, eb): Ω(m) = Ω(m; T, µ, eb) is the thermodynamic potential leading to the phase diagram of our two-flavor NJL model Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 15 / 40

18 The Model: Two Flavor NJL Model Results (I) Results (I) Phases of Magnetized Quark Matter and Meson Properties in an External B Field [1] Sh. Fayazbakhsh and N.S., PRD 82 (2010) [2] Sh. Fayazbakhsh and N.S., PRD 83 (2011) [3] Sh. Fayazbakhsh, S. Sadeghian and N.S., PRD 86 (2012) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 16 / 40

19 The Model: Two Flavor NJL Model Results (I) Two-Flavor NJL Model: Phase Diagram T MeV eb 0.2 GeV 2 eb 0 GeV 2 a eb 0.6 GeV 2 C 3 C 2 C Μ MeV Figure: Phase portrait of a two-flavor magnetized NJL model [2,3] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 17 / 40

20 The Model: Two Flavor NJL Model Results (I) Two-Flavor NJL Model: Phase Diagrams b 200 T MeV 150 Μ 240 MeV C Μ 280 MeV 50 C 1 Μ 320 MeV Μ 320 MeV 0 Μ 340 MeV eb GeV 2 Figure: Phase portrait of a two-flavor magnetized NJL model [2,3] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 18 / 40

21 The Model: Two Flavor NJL Model Results (I) Two-Flavor NJL Model: Phase Diagrams c 350 T 30 MeV Μ MeV T 60 MeV T 100 MeV C 1 C 2 T 150 MeV eb GeV 2 Figure: Phase portrait of a two-flavor magnetized NJL model [2,3] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 19 / 40

22 The Model: Two Flavor NJL Model Conclusion (I) Conclusion (I) Effects of Constant B Field on QCD Phase Diagram.1 Shift of critical T and µ - NJL model [Shovkovy et al., and Ebert et al. (2006)] - (Inverse) Magnetic catalysis T c = m dyn (T = 0) eb exp ) ( 2π2 GeB.2 Changing the type of the phase transition from 2nd to 1st order - Ordinary U(1) superconductivity [Well-known result] - Electroweak phase transition [Elmfors et al. 1998, Bordag et al. 2000, N.S. et al 2008] - Chiral phase transition [Sh. Fayazbakhsh and N.S and 2011] - Color-superconductivity [Sh. Fayazbakhsh and N.S and 2011] In a 2-flavor magnetized NJL model: CS to normal transition always 2nd order[perhaps a model dependent result] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 20 / 40

23 The Model: Two Flavor NJL Model Conclusion (I) Two-Flavor NJL Model: Meson Masses a Μ 0MeV b Μ 0MeV 1.5 eb 0.03 GeV 2 eb 0.2 GeV 2 eb 0.3 GeV eb 0.03 GeV 2 eb 0.2 GeV 2 eb 0.3 GeV 2 m Σ GeV 1.0 m Π 0 GeV T MeV T MeV Figure: T dependence of m σ and m π 0 [3] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 21 / 40

24 The Model: Two Flavor NJL Model Conclusion (I) Two-Flavor NJL Model: Directional Refraction Indices Refraction Index ofσ a Μ 0MeV u trans for eb 0.03 GeV 2 u trans for eb 0.2 GeV 2 u trans for eb 0.3 GeV 2 u long for eb T MeV Figure: T dependence of directional refraction indices of σ meson [3] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 22 / 40

25 The Model: Two Flavor NJL Model Conclusion (I) Two-Flavor NJL Model: Directional Refraction Indices Refraction Index ofπ b Μ 0MeV u trans for eb 0.03 GeV 2 u trans for eb 0.2 GeV 2 u trans for eb 0.3 GeV 2 u long for eb T MeV Figure: T dependence of refraction indices of π 0 meson [3] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 23 / 40

26 The Model: Two Flavor NJL Model Conclusion (I) Conclusion (I) Effects on Meson Dynamics T dependence of meson masses for various B T dependence of directional refraction indices for various B - Anisotropy in u (i), i = 1, 2, 3: u (1) = u (2) u (3) π 0 π 0 π 0 π 0 - It turns out: u (3) = 1 but u (1) = u (2) > 1 π 0 π 0 π 0 (Superluminal velocities...???) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 24 / 40

27 The Model: Two Flavor NJL Model Conclusion (I) Conclusion (I) Effects on Meson Dynamics T dependence of meson masses for various B T dependence of directional refraction indices for various B - Anisotropy in u (i), i = 1, 2, 3: u (1) = u (2) u (3) π 0 π 0 π 0 π 0 - It turns out: u (3) = 1 but u (1) = u (2) > 1 π 0 π 0 π 0 (Superluminal velocities...???) Note: - External B fields Superluminal velocities [J. Alexandre, J. Ellis and N. E. Mavromatos (2012)] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 24 / 40

28 The Model: Two Flavor NJL Model PCAC Relation and Pion Decay Constant at (T, eb) 0 Questions u (i) π, i = 1, 2, 3 f π Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 25 / 40

29 The Model: Two Flavor NJL Model PCAC Relation and Pion Decay Constant at (T, eb) 0 Questions u (i) π, i = 1, 2, 3 f π Modify: Low energy QCD theorems at finite T and eb PCAC relation at finite T and eb Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 25 / 40

30 Decay Constant of Hot and Magnetized π 0 Decay Constant of Hot and Magnetized π 0 [4] Sh. Fayazbakhsh and N.S., PRD 88 (2013) - Zero T: NJL Model [Buballa et al. (2004)] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 26 / 40

31 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 Reminder: PCAC Relation Global SU A (2) transformation q(x) e iγ 5 α 5 τ q(x) and q(x) e iγ 5 α 5 τ q(x) In the chiral limit m 0 0 δ α5 L QCD = 0 Conserved Noether current J5,µ a = qγ µγ 5 τ a q with µ J5,µ a = 0, a = 1, 2, 3 Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 27 / 40

32 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 J a µ,5 q π b PCAC relation 0 J a 5,µ (0) πb (q) = f π q µ δ ab This ansatz is compatible with Lorentz invariance Pions are Goldstone bosons of SB of chiral invariance (m π = 0) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 28 / 40

33 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 J a µ,5 q π b PCAC relation 0 J a 5,µ (0) πb (q) = f π q µ δ ab This ansatz is compatible with Lorentz invariance Pions are Goldstone bosons of SB of chiral invariance (m π = 0) Check: Using µ J a 5,µ = 0 and q2 0 = q2 + m 2 π, we obtain: =0 {}}{ 0 µ J5,µ a (0) πb (q) = f π q 2 }{{} =mπ 2 δ ab and therefore, m 2 π = 0 Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 28 / 40

34 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 Low Energy Theorems of QCD at (T, eb) = 0 Combinig the PCAC relation: 0 J a 5,µ (0) πb (q) = f π q µ δ ab and the BS relation: g2 πqq q 2 m 2 π = 2G 1 2GΠ π (q 2 ) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 29 / 40

35 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) = 0 Low Energy Theorems of QCD at (T, eb) = 0 We obtain low enegy theorems of QCD.1 The Goldberger-Treiman (GT) relation g πqq f π = m + O(m 2 0 ) with the quark-meson coupling πqq = dπ π dq 2 g 2 q 2 =m 2 π with constituent quark mass m = m 0 + σ 0.2 The Gell-Mann-Oakes-Renner (GOR) relation m 2 πf 2 π = m 0σ 0 2G + O(m2 0 ) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 30 / 40

36 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) 0 Questions Modifications: PCAC and low energy GT and GOR theorems at finite T and eb, that explicitly break the Lorentz invariance - Zero T: NJL Model [Buballa et al. (2004)] - Finite T: Massless QCD [Pisarski (1996)] - Finite B: Chiral perturbation theory [Agasian (2001)] - Finite T and B: NJL Model [Fayazbakhsh and N.S. (2013)] Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 31 / 40

37 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) 0 Modified PCAC Relation at (T, eb) 0 Keypoint: Modified pion dispersion relation q 2 0 = u(i)2 π q 2 i + m 2 π arising from Γ eff [ Φ] = 1 2 d 4 x Φ(x) ( + m 2 Φ) Φ(x) Ω(m) with + m 2 Φ 2 0 u(i)2 Φ 2 i + m 2 Φ Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 32 / 40

38 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) 0 Modified PCAC Relation at (T, eb) 0 - Define: u (µ) π = (1, u (i) π ) - Modified PCAC relation at (T, eb) 0 0 J a 5,µ (0) πb (q) = f b u (µ)2 π q µ δ ab, µ This is correct, because using µ J a 5,µ = 0 and q2 0 u(i)2 π q 2 i = m 2 π 0 = 0 µ J a 5,µ (0) πb (q) = f b m 2 πδ ab = m 2 π = 0 Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 33 / 40

39 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) 0 Combining - Modified PCAC relation at (T, eb) 0 0 J5,µ a (0) πb (q) = f b u π (µ)2 q µ δ ab - Modified Bethe-Salpeter relation at (T, eb) 0 Keeping in mind: [ π a = F 00 aa 1/2 π a ] g2 π a qq Faa 00 1 q0 2 = u(i)2 π a qi 2 mπ 2 a 2G 1 2GΠ πa (q) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 34 / 40

40 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) 0 Modified Low Energy Theorems of QCD at (T, eb) 0 we obtain.1 The modified Goldberger-Treiman (GT) relation f (µ) π g (µ) qqπ = m + O(m 2 0 ).2 The modified Gell-Mann-Oakes-Renner (GOR) relation mπf 2 π (µ)2 = u π (µ)2 m 0 σ 0 2G + O(m2 0 ) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 35 / 40

41 Decay Constant of Hot and Magnetized π 0 PCAC Relation and Pion Decay Constant at (T, eb) 0 Definitions: - Directional decay constant of pions: f (µ) π f a F µµ aa 1/2 - Directional quark-pion coupling: (g qqπ) (µ) 2 g µµ dπ π(q) dq 2 µ qπ=(m π,0) = g 2 qqπ F µµ aa Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 36 / 40

42 Decay Constant of Hot and Magnetized π 0 Results (II) Results (II) Sh. Fayazbakhsh and N.S., PRD 88 (2013) Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 37 / 40

43 Decay Constant of Hot and Magnetized π 0 Results (II) Directional Quark-Pion Coupling a Μ 0MeV, eb 0.03 GeV 2 c Μ 0MeV, eb 0.3 GeV g qqπ 0 g qqπ g qqπ 0 g qqπ 0 Effective Coupling toπ g qqπ 0 g qqπ 0 Effective Coupling toπ g qqπ 0 g qqπ T MeV T MeV Figure: T dependence of longitudinal and transverse g (µ) qqπ 0 g qqπ > g qqπ related to u < u Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 38 / 40

44 Decay Constant of Hot and Magnetized π 0 Results (II) Directional weak decay constant b Μ 0MeV c Μ 0MeV 150 eb 0.03 GeV 2 eb 0.2 GeV 2 eb 0.3 GeV eb 0.03 GeV 2 eb 0.2 GeV 2 eb 0.3 GeV 2 MeV 0 3 f Π 0 fπ MeV 1 2 f Π 0 fπ T MeV T MeV Figure: T dependence of longitudinal and transverse f (µ) π 0 f π < f π related to u < u Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 39 / 40

45 Decay Constant of Hot and Magnetized π 0 In Summary In Summary u (i) π, i = 1, 2, 3 f π We have shown, analytically and numerically, that the results for the refraction indices u i π, i = 1, 2, 3 at finite T and/or B are compatible with the modified low energy theorems of QCD [Remember: For T 0 and B = 0, u π (1) = u π (2) = u π (3) < 1 and for T 0 and B 0, u (1) π = u (2) π > u (3) π = 1]. Neda Sadooghi (Dept. of Physics, SUT) Hot and Magnetized Pions 40 / 40