Real Time Evolution of Non-Gaussian Cumulants in the QCD Critical Regime

Transcription

1 Real Time Evolution of Non-Gaussian Cumulants in the QCD Critical Regime Swagato Mukherjee based on: arxiv: with: Raju Venugopalan and Yi Yin September 015, CCNU, Wuhan, China

2 Why non-uilibrium? kurtosis (μ μ c )/ Δ μ freeze-out? unless one accidentally freeze-out almost at the critical point critical signatures will me washed out non-uilibrium memory effects are ruired to preserve remnant of critical signatures

3 Simple non-uilibrium scenario Ansatz for evolution of correlation length: τ ξ = τ eff [ ξ ξ ] with dynamical universality: τ eff ξ z Berdnikov, Rajagopal: arxiv:hep-ph/99174 κ3 ξ 9/ κ 4 ξ 7 is this sufficient? what about signs of higher cumulants? 3

4 Effective action & scope free energy density: 1 λ3 3 λ4 4 Ω0 (σ ) = mσ (σ σ 0 ) + (σ σ 0 ) + (σ σ 0 ) 3 4 uilibrium distribution: P0 (σ) exp ( V Ω0 (σ)/t ) σ m ξ σ : critical mode (zero-momentum) linear combination of chiral condensate and baryon current limit to: scaling regime, but not at the critical point Lmicr < ξ < L ϵ = ξ / V 1 3 mass term ~ σ / ξ kinetic term (momentum dependence) ~ σ /L neglected 4

5 Cumulants M(τ)= σ κ3 ( τ)= ( δ σ ) 3 κ (τ)= ( δ σ ) = dσ ( )P(σ ; τ) / δ σ=σ M( τ) M = σ0 ξ κ = V4 mσ =ξ σ=~ σ 0 T (T ξ )/ M ξ/ 3 κ = λ3 6 ξ d σ P(σ ; τ) κ 4 = V 4 ~ 3/ λ3 = λ 3 T (T ξ ) κ ξ power counting in ϵ: κ 4 (τ)= ( δ σ ) 4 3 κ ( τ) 6 λ ξ ( ) 3 λ 4 ] 3[ V4 ~ λ 4= λ 4 (T ξ ) 9/ κ 3 ξ κ κ κ ϵm,, 3 3, 4 4 O(1) b b ϵb ϵ b assume this power counting holds for non-uilibrium evolution of cumulants in the critical regime and make systematic expansions in ϵ 7 κ 4 ξ ϵ = ξ3 / V b= κ V 4=V / T 5

6 Real time evolution of cumulants Langevin dynamics: soft critical mode receives small, random kicks from a bath of hard modes Fokker-Planck uation: τ P(σ ; τ) = 1 Ω (σ )+ V σ[ σ 0 4 σ ] P(σ ; τ) mσ τ eff [ τ eff : effective relaxation time of the critical modes τ eff ξz z : dynamical critical exponent τ f (σ) = 1 f (σ)ω (σ ) V f (σ) ] [ 0 4 mσ τ eff time evolution of the cumulatns and systematic expansion in ϵ = ξ3 / V 6 ]

7 Real time evolution of cumulants closed set of coupled time evolution uations: evolution of the higher cumulants couples to lower ones pn polynomials τ M = τeff p1 (M) [ 1+O(ϵ) ] τ κn = n τ eff pn (M, κ,, κn ) [ 1+O(ϵ) ] 1 Ω0 (σ ) = mσ (σ σ 0 ) Gaussian limit: τ κn = n τ eff [ κn κ ] n for n= reduces to the old Berdnikov-Rajagopal Ansatz: κ3 =κ 4 =0 evolutions decouple, non-gaussian cumulants are damped, higher cumulants damps faster τ ξ Berdnikov, Rajagopal: arxiv:hep-ph/99174 = τ eff [ ξ ξ ] 7

8 Real time evolution of cumulants near-uilibrium limit: δ κn =κn κ n ~ small, linearize τ κ = τ eff a δ κ τ κ3 = 3 τ eff [ a δ κ +a3 δ κ 3 ] τ κ 4 = 4 τ eff [ a δ κ +a3 δ κ 3 +a 4 δ κ 4 ] lower cumulatns relaxes back to uilibrium first important feature of the complete set of coupled uation 8

9 Time evolutions: emulating heavy-ion collisions universal parameters: known from 3-d Ising universality: σ 0 (r,h), mσ (r,h),λ 3 (r,h), λ 4 (r,h) choose critical region in r-h plane through: ξmax / ξmin =3 r: reduced temperature h: magnetic field dynamical universality class: model H z τ eff =τ rel ξ, z=3 ξmin ( ) τ rel : relaxation time at outside edge of critical region 9

10 Time evolutions: emulating heavy-ion collisions mapping to T μ plane: unknown (major uncertainly) naïve choice: T Tc h = ΔT Δh μ μ c r = Δμ Δr 10

11 Time evolutions: emulating heavy-ion collisions trajectories: T (τ) = ( ττ ) I TI entropy ~ constant, 3-d Hubble-like expansion 3 cs 3 V( τ) τ =(τ ) I VI constant speed of sound: c s =0.15 free parameter: τ rel : relaxation time for the critical mode τ rel / τ I at the boundary of critical region τ I : time when system enters critical region ballpark guess: τrel =1 fm, τi=10 fm 11

12 Cumulants: as time goes by... uil. τ rel / τ I= τ rel / τ I=0.0 critical slow down limits growth of correlation length similar to Berdnikov-Rajagopal kurtosis do not follow growth of correlation length unlike uilibrium expectation sign of kurtosis can be different from uilibrium one τ rel / τ I=0.05 ( Tc T) / Δ T τ rel / τ I=0. K=κ 4 / κ 1

13 Signs of skewness & kurtosis kurtosis skewness uil. uil. τ rel / τ I= 0.05 τ rel / τ I=0. τ rel / τ I= 0.05 τ rel / τ I=0. 13

14 Cumulants at the freeze-out chose 3 different location of the freeze-out line w.r.t the critical region 14

15 Cumulants at the freeze-out F.C.I, τ rel / τ I=0.0 F.C.II, τ rel / τ I=0.05 F.C.III, τ rel / τ I=0. depending on location of the freeze-out line and value of the relaxation time similar non-monotonic features as a function of s 15