Exact Hydro Solutions Kasza, Csörgő Exact Solutions for a I. Introduction Rehadronizing, Expanding II. Hydro eqs Basic eqs Fireball Temperature eqs EoS - with lattice QCD Equation of State - III. MC solution Triaxial Spheroidal Boundary cond. Dynamical eqs Gábor Kasza 1 & Tamás Csörgő 2 , 3 Comparison IV. Obser v ables Inverse slope HBT-radii 1 ELTE, Budapest, Hungary V. Time evol. 2 Wigner RCP, Budapest, Hungary Second explosion T , κ, R , ˙ R vs t vs T 0 θ f 3 EKU KRC, Gyöngyös, Hungary VI. Summary Zimányi School 2016 Budapest, 9 th December Talk at Low-X 2016 meeting: arXiv:1610.02197 [nucl-th]
Exact Hydro I. Introduction Solutions Kasza, Csörgő I. Introduction II. Hydro eqs Basic eqs Motivation Temperature eqs EoS ◮ Deeper understanding of rehadronization III. MC solution Triaxial ◮ More precise description of the fireball evolution Spheroidal Boundary cond. ◮ Mass dependence of inverse slope Dynamical eqs Comparison New solution IV. Obser v ables Inverse slope ◮ Non-relativistic, expanding fireball HBT-radii V. Time evol. ◮ Hadro-chemical and kinetic freeze-out stage Second explosion T , κ, R , ˙ R vs t ◮ Multi-component hadronic matter vs T 0 θ f ◮ Equation of state is from lattice QCD VI. Summary
Exact Hydro I. Introduction Solutions Kasza, Csörgő I. Introduction II. Hydro eqs Basic eqs Temperature eqs EoS III. MC solution Triaxial Spheroidal Boundary cond. Dynamical eqs Comparison IV. Obser v ables Inverse slope HBT-radii V. Time evol. Second explosion T , κ, R , ˙ R vs t vs T 0 θ f VI. Summary T = T f + m � u t � 2 = ⇒ T i = T f + m i � u t � 2 PHENIX Collaboration: arXiv:nucl-ex/0307022
Exact Hydro II. Hydro equations Solutions Kasza, Csörgő Non-relativistic, perfect fluid hydrodynamics I. Introduction ◮ Strongly coupled quark matter - QM ( T > T chem ) II. Hydro eqs Basic eqs ∂σ Temperature eqs ∂ t + ∇ ( σ� v ) = 0 EoS III. MC solution ∂ε Triaxial ∂ t + ∇ ( ε� v ) = − p ∇ � v Spheroidal Boundary cond. Dynamical eqs T σ ( ∂ t + � v ∇ ) � v = −∇ p Comparison IV. Obser v ables Inverse slope ◮ Chemically frozen, mc. hadronic matter - HM ( T < T chem ) HBT-radii V. Time evol. ∂ n i Second explosion ∂ t + ∇ ( n i � v ) = 0 T , κ, R , ˙ R vs t vs T 0 θ f ∂ε VI. Summary ∂ t + ∇ ( ε� v ) = − p ∇ � v � ∂ � � ∂ t + � v ∇ � v = −∇ p m i n i i
Exact Hydro II. Hydro equations Solutions Kasza, Csörgő Temperature equations I. Introduction ◮ Strongly coupled quark matter - QM ( T > T chem ) II. Hydro eqs Basic eqs ε = κ QM ( T ) p Temperature eqs EoS σ T III. MC solution p = Triaxial 1 + κ Spheroidal � d Boundary cond. � κ QM T Dynamical eqs ( 1 + κ QM ) ( ∂ t + � v ∇ ) T + T ∇ � v = 0 Comparison dT 1 + κ QM IV. Obser v ables Inverse slope ◮ Chemically frozen hadronic matter - HM ( T < T chem ) HBT-radii V. Time evol. ε = κ HM ( T ) p Second explosion T , κ, R , ˙ R vs t vs T 0 θ f � � p = p i = T n i VI. Summary i i � d � ( ∂ t + � v ∇ ) T + T ∇ � v = 0 dT κ HM T T. Csörgő, M.I. Nagy: arXiv:1309.4390
Exact Hydro II. Hydro equations Solutions Kasza, Csörgő I. Introduction II. Hydro eqs Basic eqs Temperature eqs EoS III. MC solution Triaxial Spheroidal Boundary cond. Dynamical eqs Comparison IV. Obser v ables Inverse slope HBT-radii V. Time evol. Second explosion T , κ, R , ˙ R vs t vs T 0 θ f VI. Summary
Exact Hydro III. Multi-component solution Solutions Triaxial (X � = Y � = Z) solution Kasza, Csörgő ◮ Velocity field ( ω 0 = 0) I. Introduction ˙ ˙ ˙ X ( t ) Y ( t ) Z ( t ) II. Hydro eqs v x = X ( t ) r x , v y = Y ( t ) r y , v z = Z ( t ) r z Basic eqs Temperature eqs EoS ◮ Entropy and particle density III. MC solution Triaxial � � 2 X 2 − r 2 Spheroidal − r 2 2 Y 2 − r 2 V 0 y Boundary cond. x z σ ( � r , t ) = σ 0 V exp Dynamical eqs 2 Z 2 Comparison IV. Obser v ables � � 2 X 2 − r 2 − r 2 2 Y 2 − r 2 V h Inverse slope y x z n i ( � r , t ) = n i , h V exp HBT-radii 2 Z 2 V. Time evol. Second explosion T , κ, R , ˙ R vs t ◮ Where V h = V ( t h ) , n i , h = n i ( � r = 0 , t h ) vs T 0 θ f VI. Summary Landau’s idea σ ( � r , t h ) n i ( � r , t h ) r = 0 , t h ) = σ ( � n i ( � r = 0 , t h ) S.V. Akkelin, T. Csörgő and others: arXiv:hep-ph/0012127 (for single component)
Exact Hydro III. Multi-component solution Solutions Kasza, Csörgő Spheroidal (X=Y � = Z), rotating solution I. Introduction ◮ Velocity field ( ω 0 � = 0) II. Hydro eqs Basic eqs ˙ ˙ ˙ R ( t ) R ( t ) Z ( t ) Temperature eqs v x = R ( t ) r x − ω r y , v y = R ( t ) r y + ω r x , v z = Z ( t ) r z EoS III. MC solution Triaxial R 2 Spheroidal 0 ω ( t ) = ω 0 Boundary cond. R 2 ( t ) Dynamical eqs Comparison IV. Obser v ables ◮ Entropy and particle density Inverse slope HBT-radii V. Time evol. � � Second explosion 2 R 2 − r 2 − r 2 2 R 2 − r 2 V 0 T , κ, R , ˙ y R vs t x z σ ( � r , t ) = σ 0 V exp vs T 0 θ f 2 Z 2 VI. Summary � � − r 2 2 R 2 − r 2 2 R 2 − r 2 V h y x z n i ( � r , t ) = n i , h V exp 2 Z 2 T. Csörgő, M.I. Nagy: arXiv:1309.4390 (for single component)
Exact Hydro III. Multi-component solution Solutions Kasza, Csörgő ◮ Boundary conditions I. Introduction II. Hydro eqs t h : no more quarks in the medium, only hadrons Basic eqs Temperature eqs EoS T QM ( t h , ✁ r ) = T HM ( t h , ✁ � � r ) ≈ T chem III. MC solution Triaxial � v QM ( t h ) = � v HM ( t h ) Spheroidal Boundary cond. κ QM ( T QM ( t h )) = κ HM ( T HM ( t h )) Dynamical eqs Comparison { X QM ( t h ) , Y QM ( t h ) , Z QM ( t h ) } = { X HM ( t h ) , Y HM ( t h ) , Z HM ( t h ) } IV. Obser v ables Inverse slope HBT-radii V. Time evol. Second explosion Ansatz T , κ, R , ˙ R vs t vs T 0 θ f Even for the hadronic matter phase, the scales are independent of VI. Summary the particle species: { X i , Y i , Z i } = { X , Y , Z } , ∀ i .
Exact Hydro III. Multi-component solution Solutions Kasza, Csörgő Dynamical equations ◮ Strongly coupled quark matter ( T > T chem ) I. Introduction 1 II. Hydro eqs X ¨ X = Y ¨ Y = Z ¨ Z = ( ω 0 = 0 ) Basic eqs 1 + κ ( T ) Temperature eqs EoS 1 III. MC solution R − R 2 ω 2 = Z ¨ R ¨ Z = ( ω 0 � = 0 ) Triaxial 1 + κ ( T ) Spheroidal Boundary cond. � d � ˙ Dynamical eqs ˙ κ QM T T V Comparison ( 1 + κ QM ) T + V = 0 dT 1 + κ QM IV. Obser v ables Inverse slope HBT-radii V. Time evol. ◮ Chemically frozen, mc. hadronic matter ( T < T chem ) Second explosion T , κ, R , ˙ R vs t Z = T X ¨ X = Y ¨ Y = Z ¨ vs T 0 ( ω 0 = 0 ) θ f � m � VI. Summary Z = T R − R 2 ω 2 = Z ¨ R ¨ ( ω 0 � = 0 ) � m � ˙ ˙ d ( κ HM T ) T V T + V = 0 dT
Exact Hydro III. Multi-component solution Solutions Compare to the single component hadronic matter Kasza, Csörgő ◮ Ellipsoidal symmetry ( ω 0 = 0) I. Introduction Z = T II. Hydro eqs X ¨ X = Y ¨ Y = Z ¨ Basic eqs m Temperature eqs EoS III. MC solution ◮ Spheroidal symmetry ( ω 0 � = 0) Triaxial Spheroidal Z = T Boundary cond. R − R 2 ω 2 = Z ¨ R ¨ Dynamical eqs m Comparison IV. Obser v ables Inverse slope HBT-radii ◮ Difference: m ⇐ ⇒ � m � V. Time evol. � m i n i , h Second explosion T , κ, R , ˙ R vs t i � m � = ≈ 280 MeV vs T 0 θ f � n i , h VI. Summary i Conclusion The X, Y and Z scales are independent of the particle species. M. Kaneta, N. Xu: arXiv:nucl-th/0405068
Exact Hydro IV. Observables Solutions Kasza, Csörgő Inverse slope parameter I. Introduction ◮ Single particle spectrum of the MC scenario II. Hydro eqs Basic eqs � � Temperature eqs p 2 p 2 p 2 EoS x , i y , i z , i N 1 , i ( p i ) ∝ exp − − − III. MC solution 2 m i T x , i 2 m i T y , i 2 m i T z , i Triaxial Spheroidal Boundary cond. Dynamical eqs Inverse slope Single-component Multi-component Comparison IV. Obser v ables 2 2 T x = T f + m ˙ T x , i = T f + m i ˙ X f X f Inverse slope HBT-radii T y = T f + m ˙ 2 T y , i = T f + m i ˙ 2 ω 0 = 0 Y f Y f V. Time evol. Second explosion T z = T f + m ˙ 2 T z , i = T f + m i ˙ 2 ( ellipsoidal ) Z f Z f T , κ, R , ˙ R vs t vs T 0 θ f 2 + ω 2 2 + ω 2 � � � � ˙ f R 2 ˙ f R 2 T x = T f + m R f T x , i = T f + m i R f VI. Summary f f 2 + ω 2 2 + ω 2 � � � � ˙ f R 2 ˙ f R 2 ω 0 � = 0 T y = T f + m R f T y , i = T f + m i R f f f 2 2 T z = T f + m ˙ T z , i = T f + m i ˙ ( spheroidal ) Z f Z f T. Csörgő, S.V. Akkelin and others: arXiv:hep-ph/0108067v4 T. Csörgő, M.I. Nagy, I.F. Barna: arXiv:1511.02593v1
Exact Hydro IV. Observables Solutions Kasza, Csörgő I. Introduction II. Hydro eqs Basic eqs Temperature eqs EoS III. MC solution Triaxial Spheroidal Boundary cond. Dynamical eqs Comparison IV. Obser v ables Inverse slope HBT-radii V. Time evol. Second explosion T , κ, R , ˙ R vs t vs T 0 θ f VI. Summary T i = k 1 · m i + k 2 PHENIX Collaboration: arXiv:nucl-ex/0307022
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