L.P. Csernai 1 Laszlo P. Csernai, University of Bergen, Norway Differential HBT method to detect rotation Wigner - CCNU mini Workshop, Budapest, Apr. 7-8, 2014

L.P. Csernai 3  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) Peripheral Collisions (A+A)

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

3 rd flow component This shape is confirmed by M.Lisa &al. HBT: PLB496 (2000) 1; & PLB 489 (2000) 287. Initial State L.P. Csernai 5

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

L.P. Csernai 7 Pb+Pb 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. PIC- hydro..\zz-Movies\LHC-Ec-1h-b7-A.mov ATeV [ Csernai L P, Magas V K, Stoecker H, Strottman D D, Phys. Rev. C 84 (2011) ]

L.P. Csernai 8 Detecting initial rotation

L.P. Csernai 9 KHI  ROTATION KHI 2.4 fm PIC method !!!

L.P. Csernai fm

The Kelvin – Helmholtz instability (KHI) Shear Flow: L=(2R-b) ~ 4 – 7 fm, init. profile height l z =10–13 fm, init. length (b=.5-.7b max ) V ~ ±0.4 c upper/lower speed  Minimal wave number is k = 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-.7b max ) without expansion, and with favorable viscosity/Reynolds no. Re=LV/ν.  this favors large L and large V L.P. Csernai 11 L V V Our resolution is (0.35fm) 3 and 8 3 markers/fluid-cell  ~ 10k cells & 10Mill m.p.-s lzlz

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 b max Furthermore Re = 0.3 – 1 for and Re = 3 – 10 for L.P. Csernai 12

Onset of turbulence around the Bjorken flow Transverse plane [x,y] of a Pb+Pb HI collision at √s NN =2.76TeV at b=6fm impact parameter Longitudinally [z]: uniform Bjorken flow, (expansion to infinity), depending on τ only. L.P. Csernai 13 S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: v1 nucleons energy density [fm] x y P T Green and blue have the same longitudinal speed (!) in this model. Longitudinal shear flow is omitted. y xx

Onset of turbulence around the Bjorken flow 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 L.P. Csernai 14 S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: v1 y Max = 0.2 c/fm

L.P. Csernai 15 Typical I.S. model – scaling flow X t The same longitudinal expansion velocity profile in the whole [x,y]-plane ! No shear flow. No string tension! Usually angular momentum is vanishing! Such a re-arrangement of the matter density is dynamically not possible in a short time! Zero vorticity & Zero shear! P Z T

L.P. Csernai 16 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.] PT c.m. Δy = 2.5 Bjorken scaling flow assumption: Also [Gyulassy & Csernai NPA (1986)] also [Adil & Gyulassy (2005)]

L.P. Csernai 17 Adil & Gyulassy (2005) initial state Considering a longitudinal “local relative rapidity slope”, based on observations in D+Au collisions:  x, y, η, τ coordinates  Bjorken scaling flow

L.P. Csernai 18 Detecting rotation: Lambda polarization  From hydro [ F. Becattini, L.P. Csernai, D.J. Wang, Phys. Rev. C 88, (2013)] RHICLHC 4.75fm/c 3.56fm/c

L.P. Csernai 19 LHC RHIC

Global Collective Flow vs. Fluctuations L.P. Csernai 20

L.P. Csernai 21 Global Collective Flow vs. Fluctuations [Csernai L P, Eyyubova G and Magas V K, Phys. Rev. C 86 (2012) ] [Csernai L P and Stoecker H, (2014) in preparation.]

L.P. Csernai 22 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) and particle 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) ] At FAIR and NICA the thermal vorticity 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) ] We are going to present this method now

The Differential HBT method L.P. Csernai 23 The method uses two particle correlations: with k= (p 1 +p 2 )/2 and q=p 1 -p 2 : 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).

The space-time source function, S(k,q) L.P. Csernai 24 Let us start from the pion phase space distribution function in the Jüttner approximation, with Then and

L.P. Csernai 25 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 26 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 27 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 28 Results We can rotate the frame of reference: 

L.P. Csernai 29 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 30 Results

L.P. Csernai 31 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

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