Cooper-Frye negative contributions at FAIR energies Dmytro Oliinychenko Frankfurt Institute of Advanced Studies oliiny@fias.uni-frankfurt.de In collaboration with Pasi Huovinen and Hannah Petersen HGS-HIRe Helmholtz Graduate School for Hadron and Ion Research September 22, 2014
Description of heavy ion collision: hybrid models Initial State Relativistic Fluid Pre-equilibrium Hadronization Transport/Freeze-out Dynamics Hydro: local thermal equilibrium, mean free path ≪ system size ∂ µ T µν = 0, ∂ µ j µ = 0, EoS, boundary conditions Transport: Monte-Carlo solution of Boltzmann equation Hydrodynamics and transport are solved independently Transition - on a predefined hypersurface D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 2 / 19
Transition in hybrid models H y d r o m a n n B o l t z Criterion for transition surface: ”hydro equivalent to transport” ◮ Constant energy density surface ǫ ( t , x , y , z ) = ǫ 0 = 0 . 3 − 0 . 6 GeV/fm 3 H. Petersen, Phys.Rev. C78 (2008) ◮ Constant temperature surface T = 150 − 170 MeV D. Teaney et al., 2001, nucl-th/0110037; T. Hirano Phys.Lett.B636, 2006 D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 3 / 19
Particlization and negative contributions Particlization ◮ know ǫ , p , u µ on the surface ◮ from EoS - T , µ "positive" ◮ want particles ”Cooper-Frye formula” "negative" d 3 p p µ d 3 N ( p ) = f ( p ) v p 0 d σ µ (2 π � ) 3 p µ p 0 · d σ µ - analog of n · V � − 1 � p µ u µ − µ e.g. ideal hydro f ( p ) = ± 1 e T Negative contribution ◮ p µ d σ µ > 0: positive contribution, particles fly out ◮ p µ d σ µ < 0: negative contribution, particles fly in d σ µ - normal 4-vector u µ = ( γ, γ − → v ) - 4-velocity T - temperature µ - chemical potential D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 4 / 19
Negative contributions: options Account feedback to hydro - great increase in complexity K. Bugaev, Phys Rev Lett. 2003; L. Czernai, Acta Phys. Hung., 2005 Account effectively - artificial constructions S. Pratt, 2014, nucl-th1401.0136 Neglect - violate conservation laws How large are negative contributions? What changes if we neglect them and how much? How much does the choice of transition surface influence results? Is hydro equivalent to cascade in the transition region? D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 5 / 19
Negative contributions: possible solution Hybrid model Assume: transport equivalent to hydrodynamics Neglect negative Cooper-Frye contributions + remove particles from cascade if they fly to hydrodynamical region Cooper-Frye Is it possible to compensate negative Cooper-Frye contributions? That would solve problem with conservation laws. D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 6 / 19
Coarse-grained microscopic transport approach Hypersurface of constant Landau rest frame energy density: mimic hybrid model transition surface Generate many UrQMD events � � On a (t,x,y,z) grid calculate T µν = p µ i p ν 1 � i p 0 V cell i i ∈ cell event average In each cell go to Landau frame: T 0 ν = ( ǫ L , 0 , 0 , 0) L Construct surface ǫ L ( t , x , y , z ) = ǫ 0 Example: E = 160 AGeV, Au+Au central collision, ǫ 0 = 0.3 GeV/fm 3 t = 4 fm/c t = 11 fm/c t = 13 fm/c t = 18 fm/c D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 7 / 19
Definitions for negative contributions Hypersurface Σ: ǫ L ( t , x , y , z ) = const ◮ A) Cooper-Frye formula on Σ ◮ B) count UrQMD particles crossing Σ A ≡ B if particle distribution from UrQMD is exactly equilibrated A) Cooper-Frye B) ”by particles” p 0 d 3 N + p µ d σ µ = exp ( p ν u ν / T ) ± 1 θ ( p ν d σ ν ) dp 3 p 0 d 3 N − p µ d σ µ = exp ( p ν u ν / T ) ± 1 θ ( − p ν d σ ν ) dp 3 D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 8 / 19
Gaussian smearing and statistics Results are very sensitive to surface lumpiness dN π /dy| y=0 CF positive/10 outward crossings/10 CF negative inward crossings E = 10 A GeV, b =0, σ = 1 fm 25 0 10 4 1 10 1000 N Saturation of results against statistics To get a smooth surface gaussian smearing with σ = 1 fm was used. D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 9 / 19
What changes if we neglect negative contributions Example: Conservation laws will be dN π /dy violated only positive total Cooper-Frye Spectra will change depending π on 50 ◮ collision energy ◮ centrality 40 ◮ particle sort ◮ transition surface 30 We further investigate 20 [ dN − / dy ] / [ dN + / dy ] in % 10 E = 10 AGeV, b=0 fm 0 -3 -2 -1 1 2 3 y D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 10 / 19
Negative contributions: particle mass dependence E = 40 AGeV, b = 0, ǫ 0 = 0 . 3 GeV/fm 3 , dN / dy distributions K + ( m K = 495 MeV) π ( m π = 139 MeV) N ( m N = 938 MeV) π /dy], % 8 K + /dy], % 8 N /dy], % 8 by particles by particles K + by particles π N Cooper-Frye Cooper-Frye Cooper-Frye π /dy]/[dN + E = 40 AGeV, b=0 fm N /dy]/[dN + K + /dy]/[dN + E = 40 AGeV, b=0 fm E = 40 AGeV, b=0 fm 6 6 6 [dN - [dN - 4 [dN - 4 4 2 2 2 0 0 0 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 y y y Smaller mass - larger negative contribution D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 11 / 19
Negative contributions: energy dependence Pions, ǫ 0 = 0 . 3 GeV/fm 3 , dN / dy distributions 10 AGeV 40 AGeV 160 AGeV π /dy], % 15 π /dy], % 15 π /dy], % 15 by particles by particles by particles 12.5 hydro-style 12.5 Cooper-Frye 12.5 Cooper-Frye π /dy]/[dN tot π /dy]/[dN + π /dy]/[dN + π π E = 40 AGeV, b=0 fm π E = 10 AGeV, b=0 fm E = 160 AGeV, b=0 fm 10 10 10 [dN - [dN - [dN - 7.5 7.5 7.5 5 5 5 2.5 2.5 2.5 0 0 0 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 y y y Lower collision energy - slower expansion - larger negative contributions D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 12 / 19
Negative contributions: dependence on surface ǫ 0 E = 40 AGeV, b = 0 15 15 8 π Cooper-Frye, ε 0 = 0.3 GeV/fm 3 ε 0 = 0.3 GeV/fm 3 t, fm/c Cooper-Frye, ε 0 = 0.6 GeV/fm 3 ε 0 = 0.6 GeV/fm 3 ε 0 = 0.3 GeV/fm 3 by particles, ε 0 = 0.3 GeV/fm 3 by particles, ε 0 = 0.6 GeV/fm 3 E = 40 AGeV, b=0 fm π /dy], % 6 E = 40 AGeV, b=0 fm 10 10 ε 0 = 0.6 GeV/fm 3 dN/dγ ⋅ 10 3 π /dy]/[dN + 5 5 [dN - 2 E = 40 AGeV, b=0 fm 0 0 0 -10 -5 5 10 -3 -2 -1 1 2 3 0.5 1.0 2.0 2.5 z, fm y γ Larger ǫ 0 - slower surface expansion - larger negative contributions D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 13 / 19
Negative contributions: dependence on centrality E = 40 AGeV, b = 0 10 π Cooper-Frye, b = 0 fm Cooper-Frye, b = 6 fm Cooper-Frye, b = 12 fm π /dy], % E = 40 AGeV, ε 0 = 0.3 GeV/fm 3 π /dy]/[dN + 5 [dN - 0 -3 -2 -1 1 2 3 y More peripheral collision - smaller negative contributions D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 14 / 19
Summary Hydro-transport transition (”particlization”) was studied ◮ Coarse-grained UrQMD was used to construct ǫ ( t , x , y , z ) = ǫ 0 isosurface ◮ Negative contributions on this surface are calculated in two ways: from Cooper-Frye formula and explicitly counting particles Negative contributions are larger ◮ for smaller collision energies ◮ for central collisions than for peripheral ◮ for smaller particle masses ◮ for larger ǫ 0 ◮ for lumpy transition surface Negative contributions by particles are smaller than Cooper-Frye ones ◮ no compensation by accident D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 15 / 19
Backup: surface quality check Surface - no holes or double counting: Cornelius routine Conservation laws on hypersurface: accuracy better than 1% Energy conservation GeV 50 GeV GeV E = 10 AGeV, b=0 fm E = 40 AGeV, b=0 fm E = 160 AGeV, b=0 fm E in (t-dt) - E in (t) ∫ dσ μ T μ 0 25 discrep., % % % 5 −5 0 5 −5 0 5 −5 5 10 15 5 15 0 5 10 15 20 t, fm/c D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 16 / 19
Backup: statistics matters E = 160 AGeV, b = 0 Smooth surface Lumpy surface π /dy], % 30 π /dy], % 30 by particles by particles π Cooper-Frye Cooper-Frye 25 25 π /dy]/[dN + π /dy]/[dN + π E = 160 AGeV, b=0 fm E = 160 AGeV, b=0 fm 20 20 [dN - [dN - 15 15 10 10 5 5 0 0 -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 y y Lumpier surface - larger negative contributions D. Oliinychenko (FIAS) Cooper-Frye negative contributions Sep. 2014 17 / 19
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