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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)