1. Zagreb, Croatia, 2015/04/20 Csörgő, T. 1 New exact solutions of hydrodynamcs and search for the QCD Critical Point T. Csörgő 1,2 with I.Barna 1 and M. I. Nagy 3 1 Wigner Research Center for Physics, Budapest, Hungary 2 KRF, Gyöngyös, Hungary 2 ELTE, Budapest, Hungary Introduction Introduction Perfect fluid solutions Non-relativistic hydro equations Rotating fluids Theoretical predictions, the QCD Critical End Point Recent experimental results Rotating fluids for QCD CEP Summary

2. Zagreb, Croatia, 2015/04/20 Csörgő, T. 2 Motivation: initial angular momentum 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

3. Zagreb, Croatia, 2015/04/20 Csörgő, T. 3 Hydrodynamics: basic equations 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)

4. Zagreb, Croatia, 2015/04/20 Csörgő, T. 4 Rewrite for v, T and (n, or  ) Rewrite for v, T and (n, or  ) lattice QCD EoS: modification of the dynamical equations:

5. Zagreb, Croatia, 2015/04/20 Csörgő, T. 5 Ansatz for rotation and scaling Ansatz for rotation and scaling The case without rotation: known self-similar solutions T. Cs, hep-ph/00111139 S. 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!

6. Zagreb, Croatia, 2015/04/20 Csörgő, T. 6 See Cs. T and M. I. Nagy, arXiv:1309.4390, PRC89 (2014) 4, 044901 arXiv:1309.4390 From self-similarity Ellipsoidal ->spheroidal time dependence of R and  coupled Conservation law! Common properties of solutions Common properties of solutions

7. Zagreb, Croatia, 2015/04/20 Csörgő, T. 7 Solutions for conserved particle n Solutions for conserved particle n T. Cs. and M.I. Nagy, arXiv:1309.4390, PRC89 (2014) 4, 044901 arXiv:1309.4390 Similar to irrotational case: Family of self-similar solutions, T profile free T. Cs, hep-ph/00111139hep-ph/00111139 but: rotation leads to increased transverse acceleration!

8. Zagreb, Croatia, 2015/04/20 Csörgő, T. 8 Role of temperature profiles Role of temperature profiles

9. Zagreb, Croatia, 2015/04/20 Csörgő, T. 9 1B: conserved n, T dependent e/p 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

10. Zagreb, Croatia, 2015/04/20 Csörgő, T. 10 n=0, lattice QCD type EoS n=0, lattice QCD type EoS Two different class of solutions: p/e = const  arbitrary  profile or p/e = f(T)  Gaussian  profile only Notes: increased acceleration (no n vs conserved n)

11. Zagreb, Croatia, 2015/04/20 Csörgő, T. 11 First integrals: Hamiltonian motion First integrals: Hamiltonian motion 1A: n is conserved 2A: n is not conseerved Angular momentum conserved Energy in rotation → 0

12. Zagreb, Croatia, 2015/04/20 Csörgő, T. 12 Summary so far: rotating solutions Summary so far: rotating solutions New and rotating exact solutions of fireball hydro Also for lattice QCD family of EoS Now analyzed in detail (after 53 years) Important observation: Rotation leads to stronger radial expansion lQCD EoS leads to stronger radial expansion Perhaps connected to large radial flows in RHIC and LHC data Observables: Next slides arXiv:1309.4390arXiv:1309.4390 (v2)

13. Zagreb, Croatia, 2015/04/20 Csörgő, T. 13 Observables from rotating solutions 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 !

14. Zagreb, Croatia, 2015/04/20 Csörgő, T. 14 Role of EoS on acceleration Role of EoS on acceleration Lattice QCD type Eos is explosive Tilt angle  integrates rotation in (X,Z) plane: sensitive to EoS!

15. Zagreb, Croatia, 2015/04/20 Csörgő, T. 15 Single particle spectra 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 ~ n (Landau freeze-out)

16. Zagreb, Croatia, 2015/04/20 Csörgő, T. 16 Directed, elliptic and other flows 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 v n

17. Zagreb, Croatia, 2015/04/20 Csörgő, T. 17 Reminder: Universal w scaling of v 2 Reminder: Universal w scaling of v 2 Rotation does not change v2 scaling, but it modifies radial flow Black line: Buda-Lund prediction from 2003 nucl-th/0310040 Comparision with data: nucl-th/0512078 v 2 data depend on: particle mass m, centrality %, energy √s, rapidity y, p t

18. Zagreb, Croatia, 2015/04/20 Csörgő, T. 18 Details of universal w scaling of v 2 nucl-th/0512078

19. Zagreb, Croatia, 2015/04/20 Csörgő, T. 19 HBT radii for rotating spheroids HBT radii for rotating spheroids New terms with blue → Rotation decreases HBT radii similarly to Hubble flow. Diagonal Gaussians in natural frame

20. Zagreb, Croatia, 2015/04/20 Csörgő, T. 20 HBT radii in the lab frame HBT radii in the lab frame HBT radii without flow But spheriodal symmetry of the solution Makes several cross terms vanish → need for ellipsoidal solutions

21. Zagreb, Croatia, 2015/04/20 Csörgő, T. 21 HBT radii in the lab frame 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:

22. Zagreb, Croatia, 2015/04/20 Csörgő, T. 22 Rotating 3d ellipsoid X Y Z Rotating 3d ellipsoid X ≠ Y ≠ Z

23. Zagreb, Croatia, 2015/04/20 Csörgő, T. 23 Scaling variables for X Y Z Scaling variables for X ≠ Y ≠ Z First good news: scaling variable remains good, Ds = 0: → a hope to find rotating 3d ellipsoidal solutions, too!

24. Zagreb, Croatia, 2015/04/20 Csörgő, T. 24 Solution for X Y Z 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

25. Zagreb, Croatia, 2015/04/20 Csörgő, T. 25 Single particle spectra Single particle spectra Expectation for tilted sources: tilted elliptical specra ~ OK, but:

26. Zagreb, Croatia, 2015/04/20 Csörgő, T. 26 Directed, elliptic and other flows Directed, elliptic and other flows From the single particle spectra → flow coefficents v n

27. Zagreb, Croatia, 2015/04/20 Csörgő, T. 27 HBT radii in the lab frame 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:

28. Zagreb, Croatia, 2015/04/20 Csörgő, T. 28 Qualitatively rotation and flow similar Qualitatively rotation and flow similar Need for more time dependent calculations and less academic studies (relativistic solutions)

29. Zagreb, Croatia, 2015/04/20 Csörgő, T. 29 Critical End Point from lQCD Critical End Point from lQCD Z. Fodor, S.D. Katz, T E = 160 ± 3.5 MeV µ B = 725 ± 35 MeV JHEP 0203 (2002) 014 Z. Fodor, S.D. Katz, T E = 162 ± 2 MeV µ B = 360 ± 40 MeV JHEP 0404 (2004) 050

30. Zagreb, Croatia, 2015/04/20 Csörgő, T. 30 Critical End Point from Au+Au@RHIC Critical End Point from Au+Au@RHIC PHENIX: A.Adare et al, arXiv:1410.2559 arXiv:1410.2559 Non-monotonic behaviour  CEP is close?

31. Zagreb, Croatia, 2015/04/20 Csörgő, T. 31 Critical End Point from Au+Au@RHIC Critical End Point from Au+Au@RHIC R.Lacey Non-monotonic behaviour  CEP is at T E ~ 165 MeV µ B,E ~ 95 MeV Errors =? Phys.Rev.Lett. 114 (2015) 14, 142301 Rotation will help…

32. Zagreb, Croatia, 2015/04/20 Csörgő, T. 32 Observables calculated Effects of rotation and flow Combine and have same mass dependence Spectra: slope increases v 2 : universal w scaling remains valid HBT radii: Decrease with mass intensifies Even for spherical expansions: v 2 from rotation. Picture: vulcano How to detect the rotation? Next step: penetrating probes (with Dubravko Klabucar & Zagreb) Summary for hydro solutions

33. Zagreb, Croatia, 2015/04/20 Csörgő, T. 33 Summary for hydro solutions Observables calculated Effects of rotation and flow Combine and have same mass dependence Spectra: slope increases v 2 : universal w scaling remains valid HBT radii: Decrease with mass intensifies Even for spherical expansions: v 2 from rotation. Picture: vulcano How to detect the rotation? Next step: penetrating probes (with Dubravko Klabucar & Zagreb)