Equilibration in ϕ 4 theory in 3+1 dimensions

Transcription

1 Equilibration in ϕ 4 theory in 3+1 dimensions Alejandro Arrizabalaga Work in collaboration with Anders Tranberg (Sussex) and Jan Smit (Amsterdam) Physical Review D (2005) NIKHEF (Amsterdam) Summer School on Heavy Ion Phenomenology, Bielefeld, Sep Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

2 Outline 1 Introduction and Motivation 2 2PI-Effective Action 3 Evolution Equations 4 Symmetric Phase 5 Broken Phase 6 Conclusions Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

3 Introduction Motivation Main Motivation: Heavy-Ion Collisions Is an equilibrated Quark-Gluon Plasma achieved during the collisions? Traditional QCD estimates (using transport equations) give a large thermalization time BUT Experimental evidence (large elliptic flow) suggests short thermalization (τ 1 fm/c) Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

4 Introduction Motivation QGP Equilibration: Theoretical proposals Improve Calculations or use effective theories Refined transport equations [Xu,Greiner 04;Xu,Ru,Weber 05;... ] Effective Hadronic theories (J. Bleibel s talk) Look for other mechanisms for thermalization Bottom-Up Thermalization [Baier,Mueller,Schiff,Son 01] Plasma Instabilities [Mrowczynski 93;Arnold,Lenaghan,Moore 03;Romatschke,Strickland 04;... ] Perhaps the system only looks thermalized! Prethermalization [Berges,Borsányi,Wetterich 04] Unruh Effect [Kharzeev,Tuchin 05] Ultimately we would like to understand QGP equilibration from first principles, i.e. from the microscopic quantum theory of QCD. Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

5 Introduction Motivation QGP Equilibration: Theoretical proposals Improve Calculations or use effective theories Refined transport equations [Xu,Greiner 04;Xu,Ru,Weber 05;... ] Effective Hadronic theories (J. Bleibel s talk) Look for other mechanisms for thermalization Bottom-Up Thermalization [Baier,Mueller,Schiff,Son 01] Plasma Instabilities [Mrowczynski 93;Arnold,Lenaghan,Moore 03;Romatschke,Strickland 04;... ] Perhaps the system only looks thermalized! Prethermalization [Berges,Borsányi,Wetterich 04] Unruh Effect [Kharzeev,Tuchin 05] Ultimately we would like to understand QGP equilibration from first principles, i.e. from the microscopic quantum theory of QCD. Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

6 Introduction Motivation QGP Equilibration: Theoretical proposals Improve Calculations or use effective theories Refined transport equations [Xu,Greiner 04;Xu,Ru,Weber 05;... ] Effective Hadronic theories (J. Bleibel s talk) Look for other mechanisms for thermalization Bottom-Up Thermalization [Baier,Mueller,Schiff,Son 01] Plasma Instabilities [Mrowczynski 93;Arnold,Lenaghan,Moore 03;Romatschke,Strickland 04;... ] Perhaps the system only looks thermalized! Prethermalization [Berges,Borsányi,Wetterich 04] Unruh Effect [Kharzeev,Tuchin 05] Ultimately we would like to understand QGP equilibration from first principles, i.e. from the microscopic quantum theory of QCD. Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

7 Introduction Motivation QGP Equilibration: Theoretical proposals Improve Calculations or use effective theories Refined transport equations [Xu,Greiner 04;Xu,Ru,Weber 05;... ] Effective Hadronic theories (J. Bleibel s talk) Look for other mechanisms for thermalization Bottom-Up Thermalization [Baier,Mueller,Schiff,Son 01] Plasma Instabilities [Mrowczynski 93;Arnold,Lenaghan,Moore 03;Romatschke,Strickland 04;... ] Perhaps the system only looks thermalized! Prethermalization [Berges,Borsányi,Wetterich 04] Unruh Effect [Kharzeev,Tuchin 05] Ultimately we would like to understand QGP equilibration from first principles, i.e. from the microscopic quantum theory of QCD. Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

8 First Principle Methods Far from Equilibrium Schwinger-Dyson Equations Schwinger-Dyson Equations Exact equations relating the various full n-point Green functions Truncation to 2-point functions Kadanoff-Baym transport equations on-shell limit, distribution ansatz Boltzmann transport equations Gauge invariance? Possible (ad hoc) Renormalization? Possible (ad hoc) Energy conservation?most likely not out-of-equilibrium Not suited for But very useful in equilibrium at finite T and µ and to study vacuum IR-properties (confinement) Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

9 First Principle Methods Far from Equilibrium 2PI Effective Action 2PI Effective Action Exact representation of the theory in terms of a functional depending on the connected 1- and 2-point functions Example: Pure QCD (with no external fields) Exact Evolution equations derived from the stationary point of the functional Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

10 First Principle Methods Far from Equilibrium 2PI Effective Action 2PI Effective Action Exact representation of the theory in terms of a functional depending on the connected 1- and 2-point functions Example: Pure QCD (with no external fields) Approximate Evolution equations derived from the truncated functional Φ(Functional)-derivable approximations Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

11 First Principle Methods Far from Equilibrium 2PI Effective Action 2PI Effective Action Exact representation of the theory in terms of a functional depending on the connected 1- and 2-point functions Example: Pure QCD (with no external fields) Approximate Evolution equations derived from the truncated functional Φ(Functional)-derivable approximations Energy conservation! Respect global symmetries [Baym, Kadanoff 61] Renormalization? Systematic [van Hees, Knoll 02; Blaizot,Rebhan,Reinosa 02;Cooper,Mihaila,Dawson ] Gauge invariance? Not completely [AA, Smit 02; Carrington,Kunstatter,Zaraket 03] Recent Equilibration studies: Scalar fields (1+1 dim) [Berges,Cox 00; Aarts,Berges 01; Berges 02, Cooper,Dawson,Mihaila ], (2+1 dim) [Cassing,Juchem,Greiner 02], Fermions and scalars (3+1 dim) [Berges,Borsányi,Serreau 03], Preheating during inflation [Berges,Serreau 03; AA,Smit,Tranberg 04],... Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

12 2PI Effective Action Scalar Theory 2PI Effective Action in scalar theory Scalar λϕ 4 theory Z S[ϕ] = d 4 x» 1 2 µϕ(x) µ ϕ(x) 1 2 m2 ϕ(x) 2 λ 4! ϕ(x)4 Symmetric phase: v = ϕ T=0 = 0 Broken phase: v 0, v tree = p 6 m 2 /λ The 2PI Effective Action for ϕ 4 theory can be written as [Cornwall,Jackiw,Tomboulis 74] for the 1- and 2-point functions φ (crosses) and G (full lines) and G 1 0 (x, y) = ` 2 m λφ2 δ(x, y). Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

13 Truncations 2PI Effective Action Truncations Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

14 Truncations 2PI Effective Action Truncations Hartree Approximation 2-Loop Approximation (only with non-zero mean field) "Basketball" Approximation Equations for the approximate 1- and 2-point functions φ and G obtained from variational principle = Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

15 Truncations 2PI Effective Action Truncations Hartree Approximation 2-Loop Approximation (only with non-zero mean field) "Basketball" Approximation Equations for the approximate 1- and 2-point functions φ and G obtained from variational principle = Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

16 Truncations 2PI Effective Action Truncations Hartree Approximation 2-Loop Approximation (only with non-zero mean field) "Basketball" Approximation Equations for the approximate 1- and 2-point functions φ and G obtained from variational principle = Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

17 2PI Effective Action Field Theory Out of Equilibrium Real-time evolution equations? Field Theory Out of Equilibrium i Equilibrium: Ô 1... Ô 2 = Tr he βĥ Ô 1... Ô 2 i Out-of-Equilibrium: Ô 1... Ô 2 (t) = Tr hˆρ(t) Ô 1... Ô 2 h Z = Tr e βĥ i h i Z = Tr ρ(t) ˆ h i Z (t) = Tr [ˆρ(t)] = Tr U(t, 0)ˆρ(0)U (t, 0) = Z jz t Z Z (t) = Dϕ + Dϕ ϕ + in ˆρ(0) ϕ in exp dt 0 d 3 xl ˆϕ + Z t Z dt 0 d 3 xl ˆϕ ff Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

18 2PI Effective Action Real-time 1- and 2-Point Functions Real-time 1- and 2-point functions 1-point functions 2-point functions φ + (x) = φ (x) φ(x) G(x, y) = Θ C(x 0 y 0 )G > (x, y)+θ C(y 0 x 0 )G < (x, y) with ( G > (x, y) ϕ(x)ϕ(y) G < (x, y) ϕ(y)ϕ(x) Real scalar theory ˆG > (x, y) = G < (x, y) only 2 independent real functions. G > (x, y) = F (x, y) i ρ(x, y), 2 G < (x, y) = F (x, y) + i ρ(x, y). 2 The functions F /ρ contain statistical/spectral information F (x, y) = 1 {ϕ(x), ϕ(y)}, ρ(x, y) = i [ϕ(x), ϕ(y)] 2 Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

19 2PI Effective Action Real-time 1- and 2-Point Functions Real-time 1- and 2-point functions 1-point functions 2-point functions φ + (x) = φ (x) φ(x) G(x, y) = Θ C(x 0 y 0 )G > (x, y)+θ C(y 0 x 0 )G < (x, y) with ( G > (x, y) ϕ(x)ϕ(y) G < (x, y) ϕ(y)ϕ(x) Real scalar theory ˆG > (x, y) = G < (x, y) only 2 independent real functions. G > (x, y) = F (x, y) i ρ(x, y), 2 G < (x, y) = F (x, y) + i ρ(x, y). 2 The functions F /ρ contain statistical/spectral information Example: Free Theory F (k) = 2πδ (d) k 2 m 2» n BE (k 0 ) + 1 2, ρ(k) = 2iπδ (d) k 2 m 2 Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

20 Evolution Equations Evolution Equations Equations of Motion 2-point functions h i 2 x + M2 (x) F (x, y) = h i 2 x + M2 (x) ρ(x, y) = Z x0 Z dz 0 0 Z x0 Z y 0 dz 0 Z y0 Z d 3 z Σ ρ (x, z)f(z, y) dz 0 0 d 3 z Σ ρ (x, z)ρ(z, y), d 3 z Σ F (x, z)ρ(y, z), with M 2 (x) = m 2 + λ 2 φ(x)2 + λ F (x, x) 2 " # " # Σ F (x, y) = λ2 2 φ(x)φ(y) F 2 (x, y) ρ2 (x, y) + λ2 F (x, y) F 2 (x, y) 3ρ2 (x, y) " # Σ ρ (x, y) = λ 2 φ(x)φ(y)ˆf (x, y)ρ(x, y) + λ2 ρ(x, y) 3F 2 (x, y) ρ2 (x, y) point function h i 2 x + M2 (x) φ(x) = λ Z x0 Z 3 φ(x)3 + dz 0 0 d 3 z Σ eρ (x, z)φ(z), " # with Σ e ρ (x, z) = λ2 ρ(x, z) 3F(x, z) 2 ρ(x, z)2 6 4 Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

21 Initial Conditions Evolution Equations Initial Conditions Spatially homogeneous situation j ff F (x, y) = F (t, t, x y), ρ(x, y) = ρ(t, t, x y) j ff = F k (t, t ), ρ k (t, t ) Mean Field φ = 0 Symmetric Phase φ = v tree Broken Phase Spectral Function ρ k (t, t) = 0, t ρ k (t, t ) t=t = 1 F k (t, t ) t=t =0 = {ϕ k (t), ϕ k (t )} t=t =0 = 1» n k + 1 ω k 2 Symmetric Function t F k (t, t ) t=t =0 = {π k (t), ϕ k (t )} t=t =0 = 0» t t F k (t, t ) t=t =0 = {π k (t), π k (t )} t=t =0 = ω k n k Thermal Top-Hat 1 n k = e (ω k /T in ) 1 q with ω k = m in 2 + k2 n k = H Θ(k 2 max k2 )Θ(k 2 k 2 min ) Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

22 Observables Evolution Equations Observables Statistical Information q Dispersion relation ω k (t) = t t F k (t, t ) t=t /F k (t, t) q Quasiparticle distribution function n k (t) = c k t t F k (t, t ) t=t F k (t, t) Close to equilibrium Effective quasiparticle mass m eff ω 2 k (t) = c2 (t) m eff (t) 2 + k 2 1 Effective Temperature T eff and chemical potential µ eff n p(t) = hωp(t) µ i e eff (t) /T eff (t) 1 Energy and Memory Kernels We monitor the memory kernels, i.e. Σ F (t, t ), Σ ρ (t, t ) and Σ eρ (t, t ) Only a finite memory is kept, i.e. Σ(t, t ) for t t > t cut We check that the energy E(t) = R d 3 x T 00 (x, t) is conserved Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

23 Evolution Equations Numerical Implementation Numerical Implementation The system is discretized on a N 3 = 16 3 spatial lattice of spacing a. Time is discretized with spacing a t a Renormalization General method quite involved (solution of Bethe-Salpeter equations) In our discretized case we use an approximate 2-loop renormalization Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

24 Symmetric Phase Equilibration Symmetric Phase: Equilibration Simulation Parameters: φ = 0, am = 0.7, λ = 6, a t = 0.1a, mt cut = 28 T1, T2 and T3: same energy T1 and T2: similar total particle number density Distribution function n k vs. ω k, Hartree and Basketball for T1 T1 appears to equilibrate at mt Dispersion relation ω 2 k vs. k2, Hartree and Basketball for T1 Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

25 Symmetric Phase Kinetic vs. Chemical Equilibration Kinetic vs. Chemical Equilibration Evolution of individual modes Evolution of total particle number n tot Kinetic equilibration occurs relatively fast (mt 1000), dominated by 2 2 processes Chemical equilibration is much slower (caused by 1 3, 2 4,... processes). Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

26 Symmetric Phase Kinetic vs. Chemical Equilibration Evolution of effective mass, temperature and chemical potential Very slow evolution towards final equilbrium (mt 10 4 ) Effective mass: Comparing the Basketball mass M with the Hartree mass M H (T eff, µ eff ) indicates that the contribution to the mass from the basketball diagram is not very large. Chemical equilibration seems to be much smaller than in 2+1 dimensions [Juchem,Cassing,Greiner 03] Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

27 Symmetric Phase Damping Symmetric Phase: Damping Close to thermal equilibrium (Initial conditions: Thermal) Mean field slightly displaced from φ = 0.. φ(t) + M 2 (T, t)φ(t) = λ Z t 6 φ(t)3 0 Linearization Z t.. φ(t) + M 2 (T )φ(t) = φ(t) = 2φ i π Z dω 0 0 Solvable dt Σ ρ 0 (t, t ) φ(t ), dt Σρ 0 (t t ) φ(t ) ω Im Σ R 0 (ω) cos(ωt) h ω 2 M 2 Re Σ R 0 (ω) i 2 + Im Σ R 0 (ω)2 Narrow width φ(t) φ i Ze γt cos `M eff t α, γ = Z Im Σ R 0 (M eff ) M eff, M 2 eff = M2 + Re e Σ R 0 (ω) Spectral Function ρ k (t, t ) = 1 ω k e γ k t t sin [ω k (t t )] Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

28 Symmetric Phase Damping Damping: 2-loop vs. Basketball vs. Perturbative Basketball damping slightly larger than 2-loop damping Basketball damping (20-40)% larger than Perturbative Spectral function zero-mode mass and damping closely follow mean field values Effective masses almost identical and close to Hartree Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

29 Broken Phase Equilibration Broken Phase: Equilibration φ 0 allows to compare the 2-loop and basketball approximations for equilibration The 2-loop perturbative approximation contains no on-shell scattering, But the 2-loop Φ-derivable approximation contains on-shell scattering (through resummation of higher orders) We take φ = v tree v so that the time evolution of φ(t) does not affect the dynamics of the 2-point functions Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

30 Broken Phase Equilibration Broken Phase: Equilibration Simulation Parameters: φ = v tree, am = 0.7, λ = 1, a t = 0.1a, mt cut = 84, Distribution function n k vs. ω k, 2-loop and Basketball for T1 Dispersion relation ω 2 k vs. k2, 2-loop and Basketball for T1 Early equilibration in 2-loop almost as fast as in Basketball Further chemical and final equilibration very slow Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

31 Broken Phase Broken Phase: Damping Damping Close to thermal equilibrium (Initial conditions: Thermal) Mean field φ = v tree slightly displaced from true v Linearization around v: φ(t) = v + σ(t) Z t.. σ(t) + M 2 (T, t)σ(t) = dt Σρ 0 (t, t )σ(t ) 0 Vacuum expectation value v M 2 (T, t)v λ Z t 3 v Close enough to equilibrium dt Σρ 0 (t, t )v = 0 σ(t) σ in Ze γt cos `M eff t α Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

32 Broken Phase Damping Damping: 2-loop vs. Basketball Effective masses and v practically identical and close to Hartree Similar damping in both approximations (rough estimates) Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

33 Conclusions Conclusions Equilibration stages Early Stabilization of occupation numbers and dispersion relation Intermediate Kinetic equilibration Very late Chemical and final equilibration Prethermalization? Hartree/2-loop/Basketball Φ-derivable approximations Hartree vs. 2-loop/Basketball: Not large changes in masses and v Enhanced mean field damping (w.r.t. perturbation theory) Equilibration almost as fast in 2-loop as in Basketball (broken phase) Prospects in Heavy-Ion Collisions Study effective hadronic/mesonic models, heavy-quarks,... Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24

34 Conclusions Conclusions Equilibration stages Early Stabilization of occupation numbers and dispersion relation Intermediate Kinetic equilibration Very late Chemical and final equilibration Prethermalization? Hartree/2-loop/Basketball Φ-derivable approximations Hartree vs. 2-loop/Basketball: Not large changes in masses and v Enhanced mean field damping (w.r.t. perturbation theory) Equilibration almost as fast in 2-loop as in Basketball (broken phase) Prospects in Heavy-Ion Collisions Study effective hadronic/mesonic models, heavy-quarks,... Thanks! Alejandro Arrizabalaga (NIKHEF) Equilibration in ϕ 4 theory in 3+1 dimensions Heavy Ion Phenomenology, Sep / 24