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

3. 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

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

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

6. 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) & Nucl. Phys. A 712 (2002) 167.

7. PIC- hydro 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.
..\zz-Movies\LHC-Ec-1h-b7-A.mov
[ 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 
PIC method !!!
ROTATION
KHI
2.4 fm
L.P. Csernai

10. 2.1 fm
L.P. Csernai

11. 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 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-.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

12. 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 and Re = 3 – 10 for
L.P. Csernai

13. Onset of turbulence around the Bjorken flow
S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: v1 y
nucleons x
[fm]
energy density x
[fm]
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 on τ only. y
Green and blue have the same longitudinal speed (!) in this model.
Longitudinal shear flow is omitted. P T x
L.P. Csernai

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

15. 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 T P t
Such a re-arrangement of the
matter density is dynamically
not possible in a short time!
L.P. Csernai

16. Bjorken scaling flow assumption:
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

17. 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

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

19. LHC
RHIC
L.P. Csernai

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

23. 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

24. The space-time source function, S(k,q)
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
L.P. Csernai

32. L.P. Csernai

33. L.P. Csernai