Fluid dynamic propagation of initial baryon number perturbations

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1 Fluid dynamic propagation of initial baryon number perturbations Stefan Flörchinger (Heidelberg U.) Initial Stages 2016, Lisbon,

2 mainly based on S. Floerchinger & M. Martinez: Fluid dynamic propagation of initial baryon number perturbations on a Bjorken flow background [Phys. Rev. C 92 (2015), ] S. Floerchinger & U. A. Wiedemann: Kinetic freeze-out, particle spectra and harmonic flow coefficients from mode-by-mode hydrodynamics [Phys. Rev. C 89 (2014) ]

3 Baryon number & fluctuations Total baryon number B B is conserved. For 208 Pb Pb collisions B B = 416 Determines total integrated baryon number Small compared to total number of baryons B + B and other produced particles at at RHIC or LHC energies. Standard assumption: µ B = 0 What about local and event-by-event fluctuations? Dynamics should be governed by universal fluid dynamics. 1 / 14

4 Evolution of baryon number in fluid dynamics Small perturbation in static medium with u µ = (1, 0, 0, 0) t δn(t, x) = D 2 δn(t, x) Baryon number diffusion constant [ ] 2 ( ) nt (µ/t ) D = κ ɛ + p n Heat capacity κ appears here because ɛ baryon diffusion in Landau frame ˆ= heat conduction in Eckart frame Is D finite for n 0? 2 / 14

5 Heat conductivity Heat conductivity of QCD rather poorly understood theoretically so far. From perturbation theory [Danielewicz & Gyulassy, PRD 31, 53 (1985)] κ T 4 µ 2 α 2 s ln α s (µ T ) From AdS/CFT [Son & Starinets, JHEP 0603 (2006)] κ = 8π 2 T st η = 2π µ 2 µ 2 (µ T ) Baryon diffusion constant D finite for µ 0! 3 / 14

6 Relativistic fluid dynamics Evolution of baryon number density from conservation law u µ µ n + n µ u µ + µ ν µ = 0 Diffusion current ν α determined by heat conductivity κ [ ] 2 nt ( µ ) ν α = κ αβ β ɛ + p T 4 / 14

7 Bjorken expansion Consider Bjorken type expansion τ ɛ + (ɛ + p) 1 τ ( 4 3 η + ζ) 1 τ 2 = 0 τ n + n 1 τ = 0 Heat conductivity κ does not enter by symmetry argument Compare ideal gas to lattice QCD equation of state [Borsanyi et al., JHEP 08 (2012) 053] 0.4 (a) 0.4 (b) T [GeV ] 0.2 μ 0 (GeV) T [GeV ] μ [GeV] μ [GeV] 5 / 14

8 Perturbations around Bjorken expansion Consider situation with n(x) = µ(x) = 0 Local event-by-event fluctuation δn 0 Concentrate now on Bjorken flow profile for u µ Consider perturbation δn τ δn + 1 ( τ δn D(τ) x 2 + y ) τ 2 2 η δn = 0 Structures in transverse and rapidity directions are flattened out by heat conductive dissipation 6 / 14

9 Solution by Bessel-Fourier expansion Expand perturbations like δn(τ, r, φ, η) = 0 dk k m= dq 2π δn(τ, k, m, q) ei(mφ+qη) J m (kr) Leads to ODE τ δn + 1 ( ) τ δn + D(τ) k 2 + q2 τ 2 δn = 0. For q = 0 and different k 1/fm, AdS/CFT value κ = 8π 2 T µ 2 η 1.0 η/s = 1/(4π) 1.0 η/s = 10/(4π) δn(τ,k l (m ) ) /δn τ τ 7 / 14

10 Evolution of perturbations For k = 0 and different q = 1, 3, 5, AdS/CFT value κ = 8π 2 T µ 2 η 1.0 η/s = 1/(4π) 1.0 η/s = 10/(4π) 0.8 δn(τ,q) /δn At τ = 10fm/c δn(τ f,k) /δn η/s = 1/(4π) δn τ δn(τ f,q) /δn0 k[1/fm] η/s = 1/(4π) δn Only long-range fluctuations survive diffusive damping. τ q 8 / 14

11 Initial baryon number fluctuation Initial baryon number density fluctuations must be known to learn about diffusive transport properties... What can be said from first principles? How are baryon number fluctuations generated by QCD processes? What is the dynamics at very early times? Maybe answers at this conference...? 9 / 14

12 Glauber type model Fluctuations due to nucleon positions: used so far for energy density N part ɛ(τ 0, x, η) = ˆɛ w (x x i ) i=1 Can be generalized to baryon number fluctuations n(τ 0, x, η) = N part i=1 ˆn w (x x i ) Would generate baryon number fluctuations on nucleon scale More general origin of fluctuations is initial state physics and early-time, non-equilibrium dynamics 10 / 14

13 Baryon number fluctuations at freeze-out On freeze-out surface dn i d 3 pd 3 x = f i(p µ ; T, u µ, µ, π µν, π bulk, ν µ ) Close-to-equilibrium expansion f i = f i,eq + δf i Equilibrium distribution functions f i,eq = 1 e pν uν µ i T ± 1 Baryons and anti-baryons have opposite baryon chemical potential Non-equilibrium correction δf i =p µ p ν π µν g i (p µ u µ, T, µ i ) + p µ p ν µν π bulk hi (p µ u µ, T, µ i ) + p µ ν µ ki (p µ u µ, T, µ i ) 11 / 14

14 Fluctuations at freeze-out Background-perturbation splitting can also be used at freeze-out Interesting observable is net baryon number n(φ, η) = (B B)(φ, η) Correlation functions and distributions contain information about baryon number fluctuations Two-particle correlation function of baryons minus anti-baryons C Baryon (φ 1 φ 2, η 1 η 2 ) = n(φ 1, η 1 ) n(φ 2, η 2 ) c 12 / 14

15 Baryon number correlation function In Fourier representation C Baryon ( φ, η) = m= dq 2π C Baryon (m, q) e im φ+iq η heat conductivity leads to exponential suppression C Baryon (m, q) = e m2 I 1 q 2 I 2 CBaryon (m, q) κ=0 I 1 and I 2 can be approximated as τf I 1 dτ 2 [ ] nt 2 ( ) (µ/t ) τ 0 R 2 κ ɛ + p n τf I 2 dτ 2 [ ] nt 2 ( ) (µ/t ) τ 0 τ 2 κ ɛ + p n I 2 I 1 would lead to long-range correlations in rapidity direction ( baryon number ridge ) ɛ ɛ 13 / 14

16 Conclusions Baryon number diffusion constant heat conductivity is well defined transport property of the quark-gluon plasma for µ 0 Baryon number fluctuations could allow to constrain it Early time baryon diffusion should lead to long-range rapidity correlations in net baryon number More knowledge about initial state welcome Seems to be interesting topic for further experimental and theoretical studies 14 / 14