EXPANSION VERSUS INTERACTION:

Transcription

1 EXPANSION VERSUS INTERACTION: BOOST-INVARIANT DYNAMICS FROM THE 2PI EFFECTIVE ACTION Gert Aarts Physics Department Swansea University INT, September 6 p.1

2 QUESTIONS EXPANDING SYSTEMS interactions: equilibration and thermalization expansion: dilution, scattering rates decrease what happens: is there enough time? to thermalize or not to thermalize? that s the question INT, September 6 p.2

3 QUESTIONS EXPANDING VS NON-EXPANDING case study in quantum field theory (no kinetic-type theory) prerequisite: non-expanding dynamics should thermalize employ a well-studied approach to non-equilibrium quantum fields 2PI effective action beyond mean field/leading order INT, September 6 p.2

4 OUTLINE 2PI effective action: loop and 1/N expansions real-time dynamics in non-expanding backgrounds: thermalization, precision tests, not kinetic theory boost invariant 2PI dynamics: mean field approximation, non-thermal fixed points memory loss physical versus comoving cutoff (work in progress) summary INT, September 6 p.3

5 OUTLINE 2PI effective action: loop and 1/N expansions real-time dynamics in non-expanding backgrounds: thermalization, precision tests, not kinetic theory boost invariant 2PI dynamics: WITH JÜRGEN BERGES WITH NATHAN LAURIE mean field approximation, non-thermal fixed points memory loss physical versus comoving cutoff (work in progress) summary INT, September 6 p.3

6 NONEQUILIBRIUM QUANTUM FIELDS? WISH LIST scattering beyond mean field approximation essential stable time evolution nontrivial due to secularity: many schemes break down when t 1/expansion parameter connection with well-established approaches, e.g. kinetic theory dynamics at very late times: conservation laws and hydrodynamics, transport... most of these issues are solved by using the two-particle irreducible action INT, September 6 p.4

7 TWO-PARTICLE IRREDUCIBLE EFFECTIVE ACTION CORNWALL, JACKIW & TOMBOULIS (1974), LUTTINGER & WARD (196), BAYM (1962) generating functional with local and bilocal sources Z[J,K] = e iw[j,k] = Dφe i (S[φ]+J i φ i φi K ij φ j ) Legendre transform: δw δj i = φ i, δw δk ij = φ i φ j + G ij Γ[φ,G] = W[J,K] J i φ i 1 2 K ij effective action can be written as ( G ij + φ i φ j) Γ[φ,G] = S[φ] + i 2 Tr ln G 1 + i Tr G 1 (G G ) + Γ 2 [φ,g] 2 variational principe (in absence of sources) δγ δφ =, δγ δg = G 1 = G 1 Σ[G], Σ = 2i δγ 2 δg INT, September 6 p.5

8 2PI TRUNCATIONS LOOP AND 1/N EXPANSIONS TO NEXT-TO-LEADING ORDER truncate the 2PI part of the effective action Γ 2 example: O(N) model, vertex 1/N (with φ = for simplicity) N 1 1/N INT, September 6 p.6

9 2PI TRUNCATIONS LOOP AND 1/N EXPANSIONS TO NEXT-TO-LEADING ORDER truncate the 2PI part of the effective action Γ 2 example: O(N) model, vertex 1/N (with φ = for simplicity) with chain of bubbles = + NNLO contribution ( 1/N): G.A., TRANBERG (PRD 26) INT, September 6 p.6

10 2PI TRUNCATIONS LOOP AND 1/N EXPANSIONS TO NEXT-TO-LEADING ORDER truncate the 2PI part of the effective action Γ 2 example: O(N) model, vertex 1/N (with φ = for simplicity) dressed propagators: G 1 = G 1 Σ D 1 = D 1 Π INT, September 6 p.6

11 NONEQUILIBRIUM DYNAMICS INITIAL VALUE PROBLEM t solve equations in real time: use Schwinger-Keldysh contour for initial value problems i ( x + m 2) G(x,y) = dz Σ(x,z)G(z,y) + δ C (x y) C decompose contour propagator in real and imaginary parts: G(x,y) = F(x,y) i 2 sign(x y )ρ(x,y) statistical function even, anti-commutator spectral function odd, commutator ρ(x,y) x =y =, x ρ(x,y) x =y = δ(x y) INT, September 6 p.7

12 NONEQUILIBRIUM DYNAMICS INITIAL VALUE PROBLEM manifestly real and causal equations [ x + m 2] x F(x,y) = dz dz Σ ρ (x,z)f(z,y) + [ x + m 2] ρ(x,y) = y x dz dz y with Σ F,ρ given in terms of F and ρ dz Σ F (x,z)ρ(z,y), dz Σ ρ (x,z)ρ(z,y) solve on a spacetime lattice, straightforward discretization expensive numerically due to memory kernel predicting the future = remembering the past INT, September 6 p.7

13 REFERENCES Berges, Cox (2) G.A., Berges (21) Berges (21-26) Cooper, Dawson, Mihaila (21-26) G.A., Ahrensmeier, Baier, Berges, Serreau (22) Berges, Borsyani, Serreau, Wetterich (22-25) Juchem, Cassing, Greiner (23-24) Bedingham (23) Arrizabalaga, Smit, Tranberg (24-25) G.A., Martinez Resco (23-25) G.A., Tranberg (26) Rajantie, Tranberg (26)... scalar and fermion fields, d + 1 dimensions with d =, 1, 2, 3 INT, September 6 p.8

14 LOSS OF MEMORY THERMALIZATION first results by Berges and Cox (2): take different initial conditions (or density matrices) with the total energy density identical independence of initial conditions at late times 3-loop expansion in λφ 4 in dimensions time evolution of different momentum modes F(t,t;p) INT, September 6 p.9

15 PRECISION TESTS CLASSICAL 2PI APPROXIMATION 2PI approach in classical statistical field theory possibility to compare with exact solution sampling of initial conditions + numerical integration of classical equation of motion example of classical limit: three-loop approximation classically: Σ ρ (x,z) = λ2 2 ρ(x,z)[ F 2 (x,z) 1 12 ρ2 (x,z) ], Σ F (x,z) = λ2 6 F(x,z)[ F 2 (x,z) 3 4 ρ2 (x,z) ] Σ cl ρ (x,z) = λ2 2 ρ(x,z)f 2 (x,z) Σ cl F (x,z) = λ2 6 F 3 (x,z) INT, September 6 p.1

16 NONEQUILIBRIUM INITIAL CONDITIONS TSUNAMI Gaussian initial conditions far from equilibrium specify F(t,t ;p), t F(t,t ;p), t t F(t,t ;p) at t = t = in terms of initial particle number n(p) tsunami thermal.75 n(p) p/m easily implemented in exact and 2PI dynamics INT, September 6 p.11

17 PRECISION TESTS G.A. & BERGES (PRL 22) G φφ(t,t;p) PI 1/N classical MC 2PI 1/N quantum mt N=1 p/m= p/m=1.9 p/m=4.2 p/m=4.6 p/m=4.9 tsunami initial conditions equal-time correlation function: particle number high energy density: compare quantum and classical evolution evolution from 2PI-1/N expansion in agreement with exact evolution, also for late times. reliable description of both early and late times capable of describing equilibration INT, September 6 p.12

18 PRECISION TESTS G.A. & BERGES (PRL 22).6 N=2 N=1 2PI 1/N classical MC.3 G φφ(p=,t).3 N=2 2PI-1/N expansion unequal-time correlation function mt Monte Carlo: sample of 8. initial conditions 2PI-1/N: one (expensive) numerical solution quantitative agreement for larger N INT, September 6 p.12

19 PRECISION TESTS G.A. & BERGES (PRL 22) γ PI 1/N classical MC 2PI 1/N quantum 2PI-1/N expansion assume ansatz G(t, ;p) e γt cos mt /N fit γ and m compare classical 2PI with classical exact quantitative agreement for larger N compare classical 2PI with quantum 2PI quantum classical! INT, September 6 p.12

20 (NOT) KINETIC THEORY G.A. & BERGES (PRD 22) self-consistent evolution of the spectral function ρ(t,t ;p) no quasiparticle approximation Wigner transform: ρ(t,t ;p) ρ(ω,p;x ) 4 2 E p (X )/m X = (t + t )/ m 2 ρ(x ;ω,p) mx =25. mx =35.4 mx =68.2 mx quasiparticle peak non-zero width slowly evolving ω/m INT, September 6 p.13

21 ET CETERA much more work has been done: quick establishment of equation of state (prethermalization) detailed comparisons with transport theory transport coefficients momentum anisotropy (tachyonic) preheating... but on to: expanding systems INT, September 6 p.14

22 HEAVY ION COLLISIONS EXPANDING GEOMETRY constant η freeze out t hadron gas constant τ longitudinal expansion: use natural coordinates mid rapidity region thermal QGP pre equilibrium z proper time τ = t 2 z 2 rapidity η = 1 2 ln t+z t z transverse x = (x,y) incoming nuclei approximately boost invariant at midrapidity (Bjorken) or ds 2 = dτ 2 dx 2 dy 2 τ 2 dη 2 INT, September 6 p.15

23 BOOST INVARIANT 2PI DYNAMICS equilibration in expanding system very different interactions drop as 1/τ: interaction vs expansion decoupling, freeze-out at late enough τ heavy ion physics: effective early thermalization? essential for hydrodynamics first results: scalar quantum fields in dimensions 2PI approximation using three-loop expansion IN PROGRESS, WITH NATHAN LAURIE INT, September 6 p.16

24 BOOST INVARIANT 2PI DYNAMICS 2PI equations easily adapted [ 2 τ + 1/4 + k2 η = 1 τ + 1 τ τ 2 τ τ ] + k 2 + M2 (τ) F(τ,τ ;k) dτ τ dτ τ Σρ (τ,τ ;k) F(τ,τ ;k) ΣF (τ,τ ;k) ρ(τ,τ ;k) time dependent mass parameter M 2 (τ) = m 2 + λ N N τ p F(τ,τ;p) INT, September 6 p.16

25 BOOST INVARIANT 2PI DYNAMICS 2PI equations easily adapted [ 2 τ + 1/4 + k2 η = 1 τ τ 2 τ dτ τ τ ] + k 2 + M2 (τ) ρ(τ,τ ;k) Σρ (τ,τ ;k) ρ(τ,τ ;k) time dependent mass parameter M 2 (τ) = m 2 + λ N N τ p F(τ,τ;p) factor out the analytic time dependence at late times ρ(τ,τ ;k) = 1 (ττ ) 1/2 ρ(τ,τ ;k) Σ(τ,τ ;k) = 1 (ττ ) 3/2 Σ(τ,τ ;k) INT, September 6 p.16

26 BOOST INVARIANT 2PI DYNAMICS QUANTUM SCALAR FIELDS IN DIMENSIONS 2 15 λ= equal-time function F(τ,τ;k η ) F(τ,τ;p) τ/τ free evolution initial tsunami start at τ = τ no interactions λ = increase λ, interaction strength INT, September 6 p.17

27 BOOST INVARIANT 2PI DYNAMICS QUANTUM SCALAR FIELDS IN DIMENSIONS 8 λ=5 F(τ,τ;p) equal-time function F(τ,τ;k η ) initial tsunami include interactions λ = τ/τ attempt to erase memory of initial conditions (unsuccessful) INT, September 6 p.17

28 MEAN FIELD DYNAMICS INTERLUDE dynamics at LO: mean field approximation lots of work in the 9 s, e.g. Cooper, Mottola et. al. Disoriented chiral condensates 12 NLO LO (mean field) λ=3 F(τ,τ;p) mean field dynamics vs dynamics with scattering (NLO) τ/τ INT, September 6 p.18

29 MEAN FIELD DYNAMICS INTERLUDE follow LO evolution to late times 15 LO mean field, λ=3 1 F(τ,τ;p) τ/τ where is mean field evolution heading? INT, September 6 p.18

30 MEAN FIELD DYNAMICS EQUAL-TIME FORMULATION consider Heisenberg equation of motion ( τ ( τ 2 η 2 1 ) + m 2 + λ6τ ) 4 φ 2 φ = φ = τφ, π = τ φ replace interaction term by mean field term φ 3 3 φ 2 φ (take φ = ) use equal-time correlation functions G φ φ(x y,τ) = φ(x,τ) φ(y,τ) G π π (x y,τ) = π(x,τ) π(y,τ) G π φ(x y,τ) = 1 2 π(x,τ) φ(y,τ) + φ(x,τ) π(y,τ) INT, September 6 p.19

31 MEAN FIELD DYNAMICS Gaussian approximation: EQUAL-TIME FORMULATION τ G φ φ(k,τ) = 2G π φ(k,τ) τ G π φ(k,τ) = ω 2 k G φ φ(k,τ) + G π π (k,τ) τ G π π (k,τ) = 2 ω k 2 G π φ (k,τ) with ω 2 k = k2 + k2 η + 1/4 τ 2 + m 2 + λ 2τ φ 2 (τ,x) conserved quantity: C 2 (k) = G φ φ(k,τ)g π π (k,τ) G (k,τ) 2 π φ INT, September 6 p.19

32 MEAN FIELD DYNAMICS non-thermal fixed point: EQUAL-TIME FORMULATION G π π(k) = ω k 2 G φ φ(k) G (k) = π φ ω k = k 2 + m2 C 2 (q) = G φ φ(q)g π π (q) fixed by initial ensemble explicit solution: G π π (k) = C(k) ω k G φ φ(k) = C(k)/ ω k determined by initial ensemble for non-expanding systems: G.A., BONINI, WETTERICH (PRD 21) analysis here easier INT, September 6 p.19

33 NON-THERMAL FIXED POINTS INTERLUDE 15 LO mean field, λ=3 1 F(τ,τ;p) mean field evolution is heading to non-thermal fixed point asymptotic evolution can be predicted from initial ensemble τ/τ INT, September 6 p.2

34 NON-THERMAL FIXED POINTS INTERLUDE NLO LO (mean field) λ=3 NLO LO (mean field) λ=5 9 1 F(τ,τ;p) 6 F(τ,τ;p) τ/τ τ/τ at stronger coupling λ = 3 at weak coupling λ = 5 mean field approximation gives correct initial response but does not work significantly longer at weaker coupling interactions almost immediately relevant INT, September 6 p.2

35 BOOST INVARIANT 2PI DYNAMICS QUANTUM SCALAR FIELDS IN DIMENSIONS signs of effective equilibration in dimensions? 8 λ=5 F(τ,τ;p) equal-time function F(τ,τ;k η ) initial tsunami include interactions λ = τ/τ momentum dependence drops fast as k 2 η/τ 2 fixed k η and late τ all modes equivalent? INT, September 6 p.21

36 BOOST INVARIANT 2PI DYNAMICS QUANTUM SCALAR FIELDS IN DIMENSIONS signs of effective equilibration in dimensions? 7 6 λ=1 F(τ,τ;p) equal-time function F(τ,τ;k η ) initial tsunami stronger interactions λ = τ/τ erase memory of initial conditions before freeze-out competition between interaction and expansion INT, September 6 p.21

37 BOOST INVARIANT 2PI DYNAMICS QUANTUM SCALAR FIELDS IN DIMENSIONS signs of effective equilibration in dimensions? 5 F(τ,τ;p) λ=3 equal-time function F(τ,τ;k η ) initial tsunami strong interactions λ = τ/τ fixed k η : all modes equal memory loss complete interactions beat expansion! INT, September 6 p.21

38 BOOST INVARIANT 2PI DYNAMICS EVOLUTION OF ENERGY-MOMENTUM TENSOR 5 stress tensor covariantly conserved: T µν ;µ = homogeneous: E = T, p η = T 33 energy conservation: τ E = E + p η τ 4 λ=3 τ E total τ E kinetic τ E gradient τ (E,P) 3 2 τ E potential τ P E and p η expressed in terms of F and ρ τ/τ INT, September 6 p.22

39 BOOST INVARIANT 2PI DYNAMICS EVOLUTION OF ENERGY-MOMENTUM TENSOR 3 25 λ=3 τ P τ d/dτ(τ E) 2 energy density: τ E = E + p η τ τ/τ covariant energy conservation (test of numerics): τ (τe) = p η INT, September 6 p.22

40 BOOST INVARIANT 2PI DYNAMICS EVOLUTION OF ENERGY-MOMENTUM TENSOR 1 1 λ=3 τ/τ =1 1 τ/τ =3 equation of state : P.1 τ/τ =2 p(e).1 τ/τ = E read off equation of state during time evolution relation well-established throughout the whole evolution INT, September 6 p.22

41 FIXED PHYSICAL CUTOFF IN PROGRESS technical problem: regularize QFT use a cutoff on physical momenta k η /τ INT, September 6 p.23

42 FIXED PHYSICAL CUTOFF IN PROGRESS technical problem: regularize QFT use a cutoff on physical momenta k η /τ 4 physical momenta Λ 3 k/τ 2 1 see e.g. Cooper, Mottola et. al τ/τ number of momentum modes not constant INT, September 6 p.23

43 FIXED PHYSICAL CUTOFF IN PROGRESS technical problem: regularize QFT use a cutoff on physical momenta k η /τ 4 comoving momenta 3 k 2 comoving momenta 1 Λ τ numerically expensive τ/τ number of modes should increase linearly in time INT, September 6 p.23

44 FIXED PHYSICAL CUTOFF IN PROGRESS technical problem: regularize QFT use a cutoff on physical momenta k η /τ 8 LO mean field, λ=3 F(τ,τ;p) implementation at LO easy minor quantitative effect τ/τ implementation at NLO in progress INT, September 6 p.23

45 SUMMARY goal: QFT dynamics in boost invariant geometry 2PI effective action beyond leading order non-expanding systems: precision tests works (surprisingly) well first results for 1 + 1D boost invariant λφ 4 interaction vs expansion memory loss when coupling is strong enough to do: use physical cutoff add transverse directions... INT, September 6 p.24