L.P. Csernai 1

Laszlo P. Csernai, Laszlo P. Csernai, University of Bergen, Norway

Differential HBT method Differential HBT method to detect rotationto detect rotation

Wigner - CCNU mini Workshop,Wigner - CCNU mini Workshop,Budapest, Apr. 7-8, 2014Budapest, Apr. 7-8, 2014

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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)Peripheral Collisions (A+A)

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

3rd flow component

This shape is confirmed by M.Lisa &al. HBT: PLB496 (2000) 1; & PLB 489 (2000) 287.

InitialInitialStateState

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Initial stateInitial state – – reaching equilibriumreaching equilibrium

Initial state by V. Magas, L.P. Csernai and D. Strottman Phys. Rev. C 64 (2001) 014901 & Nucl. Phys. A 712 (2002) 167.

Relativistic, 1D Riemann expansion is added to each stopped streak

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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.

PIC-PIC-hydrohydro

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

ATeV A TeV

[ Csernai L P, Magas V K, Stoecker H, Strottman D D, Phys. Rev. C 84 (2011) 024914 ]

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Detecting initial rotation Detecting initial rotation

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KHI

ROTATION

KHI2.4 fm

PIC method !!!

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2.1 fm

The Kelvin – Helmholtz instability (KHI)The Kelvin – Helmholtz instability (KHI)

• 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 = .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 VL.P. Csernai 11

L

V

V

Our resolution is (0.35fm)3 and83 markers/fluid-cell ~ 10k cells & 10Mill m.p.-s

lz

The Kelvin – Helmholtz instability (KHI)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

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Onset of turbulence around the Bjorken flowOnset of turbulence around the Bjorken flow

• 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.

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S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: 1108.5535v1

nucleons energy density [fm] [fm]

x

y

P T

Green and blue have the same longitudinal speed (!) in this model.Longitudinal shear flow is omitted.

y

x x

Onset of turbulence around the Bjorken flowOnset 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

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S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: 1108.5535v1

y

Max Max = 0.2 = 0.2 c/fmc/fm

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Typical I.S. model – scaling flowTypical 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 dynamicallynot possible in a short time!

Zero vorticity&Zero shear!

PZ

T

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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)]

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Adil & Gyulassy (2005) initial stateAdil & 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

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Detecting rotation: Detecting rotation: Lambda polarizationLambda polarization

From hydro

[ F. Becattini, L.P. Csernai, D.J. Wang, Phys. Rev. C 88, 034905 (2013)]

RHICLHC

4.75fm/c3.56fm/c

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LHC RHIC

Global Collective Flow vs. Fluctuations

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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.]

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Detection of Global Collective FlowWe 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) 034905.] • 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) 034916]

• We are going to present this method now

The Differential HBT method

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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 calculatedwith, &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)

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• Let us start from the pion phase space distribution function in the Jüttnerapproximation,with

• Then

• and

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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

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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

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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)

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Results

We can rotate the frame of reference:

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Results

For lower, RHIC energy:

One can evaluate the DCF in these tilted reference frames where (without rotation) theDCF is minimal.

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Results

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SummarySummary

• 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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