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Laszlo P. Csernai, University of Bergen, Norway

Differential HBT method to detect rotation. Wigner - CCNU mini Workshop, Budapest, Apr. 7-8, 2014. Laszlo P. Csernai, University of Bergen, Norway. Peripheral Collisions (A+A). Global Symmetries Symmetry axes in the global CM-frame: ( y  -y) ( x,z  -x,-z)

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Laszlo P. Csernai, University of Bergen, Norway

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  1. Differential HBT method to detect rotation Wigner - CCNU mini Workshop,Budapest, Apr. 7-8, 2014 Laszlo P. Csernai, University of Bergen, Norway 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 • Fluctuations • Global flow and Fluctuations are simultaneously present  Ǝ interference • Azimuth - Global: even harmonics - Fluctuations : odd & even harmonics • Longitudinal – Global: v1, v3 y-odd - Fluctuations : odd & even harmonics • The separation of Global & Fluctuating flow is a must !! (not done yet) L.P. Csernai

  3. String rope --- Flux tube --- Coherent YM field

  4. InitialState This shape is confirmed by M.Lisa &al. HBT: PLB496 (2000) 1; & PLB 489 (2000) 287. 3rd flow component L.P. Csernai

  5. 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.

  6. PIC-hydro ..\zz-Movies\LHC-Ec-1h-b7-A.mov 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

  7. Detecting initial rotation L.P. Csernai

  8. KHI  PIC method !!! ROTATION KHI 2.4 fm L.P. Csernai

  9. 2.1 fm L.P. Csernai

  10. The Kelvin – Helmholtz instability (KHI) lz • Shear Flow: • L=(2R-b) ~ 4 – 7 fm, init. profile height • lz =10–13 fm, init. length (b=.5-.7bmax) • V ~ ±0.4 c upper/lower speed  • Minimal wave number isk = .6 - .48 fm-1 • KHI grows as where  • Largest k or shortest wave-length will grow the fastest. • The amplitude will double in 2.9 or 3.6 fm/c for (b=.5-.7bmax) without expansion, and with favorable viscosity/Reynolds no. Re=LV/ν. •  this favors large L and large V V L V Our resolution is (0.35fm)3 and 83 markers/fluid-cell  ~ 10k cells & 10Mill m.p.-s L.P. Csernai

  11. The Kelvin – Helmholtz instability (KHI) • Formation of critical length KHI (Kolmogorov length scale) • Ǝ critical minimal wavelength beyond which the KHI is able to grow. Smaller wavelength perturbations tend to decay. (similar to critical bubble size in homogeneous nucleation). • Kolmogorov: • Here is the specific dissipated flow energy. • We estimated: • It is required that  we need b > 0.5 bmax • Furthermore Re = 0.3 – 1 for andRe = 3 – 10 for L.P. Csernai

  12. Onset of turbulence around the Bjorkenflow S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: 1108.5535v1 y • Transverse plane [x,y] of a Pb+Pb HI collision at √sNN=2.76TeV at b=6fm impact parameter • Longitudinally [z]: uniform Bjorken flow, (expansion to infinity), depending onτ only. x [fm] x [fm] energy density nucleons y Green and blue have the same longitudinal speed (!) in this model. Longitudinal shear flow is omitted. P T x L.P. Csernai

  13. Onset of turbulence around the Bjorken flow S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: 1108.5535v1 y • Initial state Event by Event vorticity and divergence fluctuations. • Amplitude of random vorticity and divergence fluctuations are the same • In dynamical development viscous corrections are negligible ( no damping) • Initial transverse expansion in the middle (±3fm) is neglected ( no damping) • High frequency, high wave number fluctuations may feed lower wave numbers Max = 0.2 c/fm L.P. Csernai

  14. Typical I.S. model – scaling flow The same longitudinal expansion velocity profile in the whole [x,y]-plane ! No shear flow. No string tension! Usually angular momentum is vanishing! X Zero vorticity & Zero shear! Z P T t Such a re-arrangement of the matter density is dynamically not possible in a short time! L.P. Csernai

  15. Δy = 2.5 Also [Gyulassy & Csernai NPA (1986)] also [Adil & Gyulassy (2005)] Bjorken scaling flow assumption: c.m. T P The momentum distribution, in arbitrary units normalized to the total c.m. energy and momentum. The momentum is zero. Rapidity constraints at projectile and target rapidities are not taken into account! [Philipe Mota, priv. comm.] L.P. Csernai

  16. Adil & Gyulassy (2005) initial state x, y, η, τ coordinates  Bjorken scaling flow Considering a longitudinal “local relative rapidity slope”, based on observations in D+Au collisions:  L.P. Csernai

  17. Detecting rotation: Lambda polarization  From hydro [ F. Becattini, L.P. Csernai, D.J. Wang, Phys. Rev. C 88, 034905 (2013)] LHC RHIC 3.56fm/c 4.75fm/c L.P. Csernai

  18. LHC RHIC L.P. Csernai

  19. Global Collective Flow vs. Fluctuations L.P. Csernai

  20. Global Collective Flow vs. Fluctuations [Csernai L P, Eyyubova G and Magas V K, Phys. Rev. C 86 (2012) 024912.] [Csernai L P and Stoecker H, (2014) in preparation.] L.P. Csernai

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

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

  28. 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

  29. Results L.P. Csernai

  30. 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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