Hydrodynamics and femtoscopy in heavy-ion physics B alint Kurgyis - PowerPoint PPT Presentation

Hydrodynamics and femtoscopy in heavy-ion physics B´ alint Kurgyis E¨ otv¨ os University, Budapest Bolyai Physics Seminar Budapest, 20 February 2019

Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary Big bang in the laboratory - heavy-ion collisions Timeline of the Universe Galaxies Atoms Nuclei Elementary particles How should we investigate? Reproduce in the laboratory! Create “little bangs” → Heavy-ion collisions Detect the created particles → Study the sQGP B´ alint Kurgyis Hydrodynamics and femtoscopy 3 / 38

Introduction Hydrodynamics in heavy-ion physics Femtoscopy at PHENIX Summary The strongly interacting quark-gluon plasma (sQGP) Discovered at the RHIC, created at LHC Hot, expanding, perfect quark-fluid Hadrons created at the freeze-out Photons and leptons ”shine through” B´ alint Kurgyis Hydrodynamics and femtoscopy 4 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Equations of non-relativistic hydrodynamics Euler-equation ∂ u ∂ t + ( u ∇ ) u = − 1 ρ ∇ p Looking for ( u , p , ρ ) fields Assumptions: Continuity equation zero viscosity ∂ρ ∂ t + ∇ ( ρ u ) = 0 zero heat conductivity Equation of state p − ρ relation B´ alint Kurgyis Hydrodynamics and femtoscopy 6 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Perturbative equations Perturbed fields Perturbed equations u → u + δ u first order perturbation p → p + δ p using another solution ρ → ρ + δρ B´ alint Kurgyis Hydrodynamics and femtoscopy 7 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Wave solution Known solution: Standing fluid u = 0 p = const. Perturbed Euler-equation ρ = const. ∂δ u ∂ t = − 1 ρ ∇ δ p Sound speed from equation of state: ∂ p Perturbed continuity equation ∂ρ = c 2 ∂δρ ∂ t + ρ ∇ δ u = 0 Wave solution for pressure ∂ 2 δ p ∂ t 2 = c 2 ∆ δ p B´ alint Kurgyis Hydrodynamics and femtoscopy 8 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Equations for relativistic hydrodynamics Looking for ( u µ , p , ǫ, n or σ ) fields Assumptions: Energy, momentum conservation zero viscosity ∂ µ T µν = 0 zero heat-conductivity T µν = ( ǫ + p ) u µ u ν − pg µν local energy-momentum conservation Continuity equation Properties: ∂ µ ( nu µ ) = 0 or ∂ µ ( σ u µ ) = 0 u µ u µ = 1 g µν = diag ( 1 , − 1 , − 1 , − 1 ) Equation of state (EoS) Temperature: T = ( ǫ + p ) /σ Locally conserved entropy density ǫ = κ p n → conserved charge density B´ alint Kurgyis Hydrodynamics and femtoscopy 9 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Known solutions for relativistic hydrodynamics Many numerical solutions Exact, analytic solutions important: connect initial/final state Famous 1+1D solutions: Landau-Khalatnikov & Hwa-Bjorken L. D. Landau, Izv. Akad. Nauk Ser. Fiz. 17 , 51 (1953) I.M. Khalatnikov, Zhur. Eksp. Teor. Fiz. 27 , 529 (1954) R. C. Hwa, Phys. Rev. D 10 , 2260 (1974) J. D. Bjorken, Phys. Rev. D 27 , 140 (1983) Discovery of sQGP → Many new solutions First truly 3D relativistic solution: Hubble-flow Cs¨ org˝ o, Csernai, Hama, Kodama, Heavy Ion Phys. A21 , 73 (2004), nucl-th/0306004 Describes well experimental data Csan´ ad, Vargyas, Eur. Phys. J. A 44 , 473 (2010) nucl-th/09094842 B´ alint Kurgyis Hydrodynamics and femtoscopy 10 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Perturbative handling of the relativistic hydrodynamics Perturbed fields: Start from a known Equations for perturbations: solution: ( u µ , p , n ) substitute perturbations into equations u µ → u µ + δ u µ substract 0 th order equations p → p + δ p neglect 2 nd or higher order perturbations n → n + δ n remainder: perturbed equation works similarly for n → σ solution yields perturbations δ u µ , δ n , δ p Orthogonality: u µ δ u µ = 0 (1) B´ alint Kurgyis Hydrodynamics and femtoscopy 11 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Known solution: Hubble-flow Hubble-flow: Cs¨ org˝ o, Csernai, Hama, Kodama , Heavy Ion Phys. A21 , 73 (2004), nucl-th/0306004 u µ = x µ τ � τ 0 � 3 N ( S ) n = n 0 τ � τ 0 � 3+ 3 p = p 0 κ τ Scaling variable: u µ ∂ µ S = ∂ τ S = 0 Describes well hadronic data and photons Csan´ ad, Vargyas, Eur. Phys. J. A 44 , 473 (2010) nucl-th/09094842 Csan´ ad, M´ ajer, Central Eur. J. Phys. 10 (2012) Multipole solutions also possible Csan´ ad, Szab´ o , Phys. Rev. C 90 , 054911 (2014) B´ alint Kurgyis Hydrodynamics and femtoscopy 12 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Finding solutions for perturbations The way of solution: Choosing test functions: δ p , δ u µ , δ n Fixing arbitrary functions Choosing scaling variable S Satisfying all the restrictions instead, one could look for sound waves on top of Hubble-flow S. Shi, J. Liao, P. Zhuang Phys.Rev. C90 no.6, 064912 (2014) arXiv:1405.4546 Perturbations: u µ = x µ → δ u µ = δ · F ( τ ) g ( x ν ) ∂ µ S · χ ( S ) (2) τ � τ 0 � 3 � τ 0 � 3 n = n 0 N ( S ) → δ n = δ · n 0 h ( x ν ) ν ( S ) (3) τ τ � 3+ 3 � 3+ 3 � τ 0 � τ 0 κ κ p = p 0 → δ p = δ · p 0 π ( S ) (4) τ τ B´ alint Kurgyis Hydrodynamics and femtoscopy 13 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables General form of solutions Perturbations: δ u µ → defined by g ( x ν ) δ u µ = δ · F ( τ ) g ( x ν ) ∂ µ S · χ ( S ) (5) χ ( S ) and F ( τ ) fixed by g ( x ν ) � 3+ 3 � τ 0 κ δ p → π ( S ) determined by δ u µ δ p = δ · p 0 π ( S ) (6) τ � τ 0 � 3 δ n → ν ( S ) fixed by h ( x ν ) δ n = δ · n 0 h ( x ν ) ν ( S ) (7) g ( x ν ) , h ( x ν ) , S need to fullfill (8)-(10) τ Equations for N ( S ) , χ ( S ) , ν ( S ) , π ( S ) , h ( x µ ) , g ( x µ ) , S functions ∂ µ S ∂ µ S − ∂ µ S ∂ µ ln g ( x ν ) χ ′ ( S ) χ ( S ) = − ∂ µ ∂ µ S (8) ∂ µ S ∂ µ S π ′ ( S ) � � u µ ∂ µ g ( x ν ) − 3 g ( x ν ) � � + F ′ ( τ ) g ( x ν ) χ ( S ) = ( κ + 1) F ( τ ) (9) κτ χ ( S ) N ′ ( S ) = − F ( τ ) g ( x ν ) ∂ µ S ∂ µ S ν ( S ) (10) u µ ∂ µ h ( x ν ) B´ alint Kurgyis Hydrodynamics and femtoscopy 14 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Looking for concrete solution To compute measurables → fix the g ( x ν ), F ( τ ), h ( x ν ) functions: g ( x ν ) = 1 , � τ � 3 κ F ( τ ) = τ + c τ 0 τ 0 � 3 κ − 1  � � � τ κ τ if κ � = 3 ln + c  τ 0 3 − κ τ 0 h ( x ν ) = � � τ (1 + c ) ln if κ = 3  τ 0 Restrictions: u µ ∂ µ S = 0 ∂ µ ∂ µ S ∂ µ S ∂ µ S is a function of the scaling variable τ 2 ∂ µ S ∂ µ S also a function of the scaling variable S = r m S = r m S = τ m Scaling variables found so far: t m , τ m , t m B´ alint Kurgyis Hydrodynamics and femtoscopy 15 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Scaling variable: S = r m / t m Scaling variable Perturbations � 3+ 3 � τ 0 κ π ( S ) δ p = δ · p 0 S = r m τ (11) � 3 � κ � δ u µ = δ · � t m τ + c τ 0 τ ∂ µ S χ ( S ) τ 0 � 3 � κ − 1 � � τ 0 � 3 � � � δ n = δ · n 0 ln + c ν ( S ) τ κ τ The functions of 3 − κ τ τ 0 τ 0 the scaling variable: � r � − m − 1 χ ( S ) = (12) t π ( S ) = − ( κ + 1)( κ − 3) � r � − 1 (13) m κ t � r m � m − 1 �� r � � � − 2 � � ν ( S ) = m 2 � r � 2 � r N ′ − 1 1 − (14) t m t t t B´ alint Kurgyis Hydrodynamics and femtoscopy 16 / 38

Introduction Introduction Hydrodynamics in heavy-ion physics Perturbative handling Femtoscopy at PHENIX The new class of solutions Summary Observables Concrete solution with S = t / r , N ( S ) = exp ( − S − 2 ) Particular case: m = − 1 S = t (15) r Let us choose Gaussian N ( S ): N ( S ) = e − r 2 t 2 = e − S − 2 The functions of the scaling variable: χ ( S ) = 1 (16) π ( S ) = ( κ + 1)( κ − 3) � t � (17) κ r � 2 � 2 � − 3 � � t � t � t � 1 − N ν ( S ) = 2 (18) r r r B´ alint Kurgyis Hydrodynamics and femtoscopy 17 / 38