Differential HBT Method to Analyze Rotation

1. Differential HBT Method to Analyze Rotation L.P. Csernai, S. Velle, D.J. Wang University of Bergen, Norway #395 QM’2014 L.P. Csernai

2. Peripheral Collisions (A+A) • Global Symmetries • Symmetry axes in the global CM-frame: • ( y  -y) • ( x,z  -x,-z) • Azimuthal symmetry: φ-even (cos nφ) • Longitudinal z-odd, (rap.-odd) for v_odd • Spherical or ellipsoidal flow, expansion Global Collective flow and fluctuations canbe separated using symmetries:[Csernai & Stöcker, J. Phys. G (2014) to be p.]for this the participant c.m. and the symmetryaxes should be determined as described in:[ Csernai, Eyyubova & Magas, PRC 86(12)024912, & PRC 88 (13) 019902] L.P. Csernai

3. Initial state – reaching equilibrium Relativistic, 1D Riemann expansion is added to each stopped streak Initial state by V. Magas, L.P. Csernai and D. Strottman Phys. Rev. C 64 (2001) 014901 & Nucl. Phys. A 712 (2002) 167. [See Poster # 608 V.K. Magas & L.P. Csernai]

4. PIC-hydro A TeV ATeV Pb+Pb 1.38+1.38 A TeV, b= 70 % of b_max Lagrangian fluid cells, moving, ~ 5 mill. MIT Bag m. EoS FO at T ~ 200 MeV, but calculated much longer, until pressure is zero for 90% of the cells. Structure and asymmetries of init. state are maintained in nearly perfect expansion. [ Csernai L P, Magas V K, Stoecker H, Strottman D D, Phys. Rev. C 84 (2011) 024914 ] L.P. Csernai

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6. Detecting initial rotation L.P. Csernai

7. in preparation L.P. Csernai

8. KHI PICR method !!! ROTATION KHI From Rotation and Kelvin-Helmholtz Instabilitywe get high angular momentum and vorticity: [See Poster # 424 D.J. Wang & L.P. Csernai] L.P. Csernai

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10. Detecting rotation: Lambda polarization From hydro [ F. Becattini, L.P. Csernai, D.J. Wang, Phys. Rev. C 88, 034905] LHC RHIC 3.56fm/c 4.75fm/c L.P. Csernai

11. LHC RHIC [See Poster # 394 L.P. Csernai, F. Becattini & D.J. Wang] L.P. Csernai

12. Detection of Global Collective Flow • We are will now discuss rotation (eventually enhanced by KHI). For these, the separation of Global flow and Fluctuating flow is important. (See ALICE v1 PRL (2013) Dec.) • One method is polarization of emitted particles • This is based equilibrium between local thermal vorticity (orbital motion) andparticle polarization (spin). • Turned out to be more sensitive at RHIC than at LHC (although L is larger at LHC)[Becattini F, Csernai L P and Wang D J, Phys. Rev. C 88 (2013) 034905.] • At FAIR and NICA the thermalvorticity is still significant (!) so it might be measurable. • The other method is the Differential HBT method to analyze rotation: • [LP. Csernai, S. Velle, DJ. Wang, Phys. Rev. C 89 (2014) 034916] • We are going to present this method now L.P. Csernai

13. The Differential HBT method The method uses two particle correlations: with k= (p1+p2)/2 and q=p1-p2 : where and S(k,q) is the space-time source or emission function, while R(k,q) can be calculated with, & the help of a function J(k,q): which leads to: This is one of the standard method used for many years. The crucial is the function S(k,q). L.P. Csernai

14. The space-time source function, S(k,q) • Let us start from the pion phase space distribution function in the Jüttnerapproximation,with • Then • and L.P. Csernai

15. The space-time source function, S(k,q) • Let us now consider the emission probability in the direction of k, for sources s : • In this case the J-function becomes: • We perform summations through pairs reflected across the c.m.: - where L.P. Csernai

16. The space-time source function, S(k,q) • The weight factors depend on the Freeze out layer (or surface) orientation: Thus the weight factor is: and for the mirror image source: • Then let us calculate the standard correlation function, and construct a new method L.P. Csernai

17. Results The correlation function depends on the direction and size of k , and on rotation.  we introduce two vectors k+ , k- symmetrically and define the Differential c.f. (DCF): The DCF would vanish for symmetric sources (e.g. spherical and non-rotation sources) L.P. Csernai

18. Results We can rotate the frame of reference:  L.P. Csernai

19. Results For lower, RHIC energy: One can evaluate the DCF in these tilted reference frames where (without rotation) the DCF is minimal. L.P. Csernai

20. Results [20] M.A. Lisa, et al., Phys. Lett. B 496, 1 (2000); Phys. Lett. B 489, 287 (2000); Phys. Rev. C 84, 014908 (2011); Phys. Rev. C 89, 014903 (2014). [21] L.P. Csernai, G. Eyyubova & V.K. Magas, Phys. Rev. C 86, 024912 (2012). L.P. Csernai

21. Summary • FD model: Initial State + EoS + Freeze out & Hadronization • In p+p I.S. is problematic, but Ǝ collective flow • In A+A the I.S. is causing global collective flow • Consistent I.S. is needed based on a dynamical picture, satisfying causality, etc. • Several I.S. models exist, some of these are oversimplified beyond physical principles. • Experimental outcome strongly depends on the I.S. Thank you L.P. Csernai

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