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New exact solutions of hydrodynamcsand search for the QCD Critical Point
T. Csörgő 1,2 with I.Barna1 and M. I. Nagy3
1 Wigner Research Center for Physics, Budapest, Hungary2 KRF, Gyöngyös, Hungary2 ELTE, Budapest, Hungary
Introduction
Perfect fluid solutionsNon-relativistic hydro equations
Rotating fluidsTheoretical predictions, the QCD Critical End Point
Recent experimental results
Rotating fluids for QCD CEP
Summary
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Motivation: initial angular momentum
Conserved quantities: important in heavy ion collisions also.Example from L. Cifarelli, L.P. Csernai, H. Stöcker,
EPN 43/22 (2012) p. 91
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Hydrodynamics: basic equations
Basic equations of non-rel hydrodynamics: Euler equation needs to be modified for lattice QCD EoS: no „n”.
Use basic thermodynamical relations for lack of conserved charge (baryon free region)
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Rewrite for v, T and (n, or s)
lattice QCD EoS: modification of the dynamical equations:
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Ansatz for rotation and scaling
The case without rotation:known self-similar solutionsT. Cs, hep-ph/00111139S. V. Akkelin, T. Cs et al, hep-ph/0012127 …
Let’s add ω ≠0, let’s rotate!
First good news: scaling variable remains good→ a hope to find ellipsoidal rotating solutions!
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See Cs. T and M. I. Nagy, arXiv:1309.4390 , PRC89 (2014) 4, 044901
From self-similarity
Ellipsoidal ->spheroidal
time dependence ofR and w coupled
Conservation law!
Common properties of solutions
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Solutions for conserved particle n
T. Cs. and M.I. Nagy, arXiv:1309.4390 , PRC89 (2014) 4, 044901
Similar to irrotational case:
Family of self-similar solutions, T profile free
T. Cs, hep-ph/00111139
but: rotation leads to increased transverse acceleration!
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Role of temperature profiles
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1B: conserved n, T dependent e/p
Rotation: increases the transverse acceleration.
Only if T is T(t):Gaussian density profiles!
But, a more general EoS,similar to T. Cs. et al,
hep-ph/0108067
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n=0, lattice QCD type EoS
Two different class of solutions: p/e = const arbitrary s profile or p/e = f(T) Gaussian s profile only
Notes: increased acceleration (no n vs conserved n)
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First integrals: Hamiltonian motion
1A: n is conserved2A: n is not conseerved
Angular momentum conservedEnergy in rotation → 0
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Summary so far: rotating solutions
New and rotatingexact solutions of fireball hydro
Also for lattice QCD family of EoSNow analyzed in detail
(after 53 years)
Important observation:
Rotation leads to stronger radial expansionlQCD EoS leads to stronger radial expansion
Perhaps connected to large radial flowsin RHIC and LHC data
Observables:Next slides
arXiv:1309.4390 (v2)
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Observables from rotating solutions
Note: (r’, k’) in fireball frame rotated wrt lab frame (r,k)
Note: in this talk, rotation in the (X,Z) impact parameter plane !
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Role of EoS on acceleration
Lattice QCD type Eos is explosive
Tilt angle q integrates rotation in (X,Z) plane: sensitive to EoS!
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Single particle spectra
In the rest frame of the fireball:
Rotation increases effective temperatures both in the longitudinal and impact parameter direction
in addition to Hubble flows
lQCD EoS: observables calculable if s ~ n (Landau freeze-out)
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Directed, elliptic and other flows
Note: model is fully analytic
As of now, only X = Z = R(t)spheroidal solutions are found
→ vanishing odd order flows at y=0
For fluctuations, see:M. Csanád and A. Szabó, Phys.Rev. C90 (2014) 5, 054911
From particle spectra→ flow coefficents vn
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Reminder: Universal w scaling of v2
Rotation does not change v2 scaling, but it modifies radial flow
Black line: Buda-Lund prediction from 2003nucl-th/0310040
Comparision with data:nucl-th/0512078v2 data depend on: particle mass m, centrality %, energy √s, rapidity y, pt
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Details of universal w scaling of v2
nucl-th/0512078
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HBT radii for rotating spheroids
New terms with blue → Rotation decreases HBT radii similarly to Hubble flow.
Diagonal Gaussians in natural frame
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HBT radii in the lab frame
HBT radii without flow
But spheriodal symmetry of the solutionMakes several cross terms vanish → need for ellipsoidal solutions
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HBT radii in the lab frame
HBT radii without ω:
spheriodal symmetry of the hydro solution several cross terms vanish need for ellipsoidal solutions
HBT for ω, X=Y=R:
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Rotating 3d ellipsoid X ≠ Y ≠ Z
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Scaling variables for X ≠ Y ≠ Z
First good news: scaling variable remains good, Ds = 0:→ a hope to find rotating 3d ellipsoidal solutions, too!
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Solution for X ≠ Y ≠ Z
First 3d ellipsoidal exact hydro solution: lQCD EoS can be co-variad with the initial condition
→ hadronic final state may be the same
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Single particle spectraExpectation for tilted sources: tilted elliptical specra ~ OK, but:
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Directed, elliptic and other flows
From the single particle spectra→ flow coefficents vn
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HBT radii in the lab frame
HBT radii without ω:
spheriodal symmetry of the hydro solution several cross terms vanish need for ellipsoidal solutions
HBT for ω, X=Y=R:
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Qualitatively rotation and flow similar
Need for more time dependent calculations
and less academic studies(relativistic solutions)
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Critical End Point from lQCD
Z. Fodor, S.D. Katz,TE = 160 ± 3.5 MeVµB= 725 ± 35 MeVJHEP 0203 (2002) 014
Z. Fodor, S.D. Katz,TE = 162 ± 2 MeVµB= 360 ± 40 MeVJHEP 0404 (2004) 050
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Critical End Point from Au+Au@RHIC
PHENIX:A. Adare et al,
arXiv:1410.2559Non-monotonic behaviour
CEP is close?
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Critical End Point from Au+Au@RHIC
R.LaceyNon-monotonic behaviour
CEP is at
TE ~ 165 MeVµB,E~ 95 MeV
Errors =?
Phys.Rev.Lett. 114 (2015) 14, 142301
Rotation will help…
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Observables calculated
Effects of rotation and flowCombine and have same mass dependence
Spectra: slope increasesv2: universal w scaling remains valid
HBT radii:Decrease with mass intensifies
Even for spherical expansions:v2 from rotation.
Picture: vulcanoHow to detect the rotation?
Next step: penetrating probes(with Dubravko Klabucar & Zagreb)
Summary for hydro solutions
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Summary for hydro solutions
Observables calculated
Effects of rotation and flowCombine and have same mass dependence
Spectra: slope increasesv2: universal w scaling remains valid
HBT radii:Decrease with mass intensifies
Even for spherical expansions:v2 from rotation.
Picture: vulcanoHow to detect the rotation?
Next step: penetrating probes(with Dubravko Klabucar & Zagreb)