Transcript China-Norway Workshop, May 2011 中国—挪威 研讨会， 2011年 …

The DHBT method to detect rotation
in high-energy heavy-ion collisions
Laszlo P. Csernai,
University of Bergen, Norway
L.P. Csernai
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2014 CBCOS Workshop for Phenomenological
Research on Heavy-Ion Collisions
(CBCOS-2014)
C- CCNU, B-UIB, C-CIAE, O-UIO, S-SUT
(华中师范大学-卑尔根大学-原子能科学研究院-奥斯
陆大学-苏兰拉里理工大学)
研讨会，CBCOS-2014
(May 9 – May 14, CCNU, Wuhan, China)
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)
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String rope --- Flux tube --- Coherent
YM field
Initial
State
This shape is confirmed
by
M.Lisa &al. HBT: PLB496
(2000) 1; & PLB 489
(2000) 287.
3rd flow component
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Initial state – reaching 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
A ATeV
TeV
PIChydro
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)
024914 ]
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Detecting initial rotation
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in preparation
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PIC method !!!
KHI 
ROTATION
KHI
2.4 fm
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2.1 fm
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The Kelvin – Helmholtz instability (KHI)
lz
•
•
•
•
•
V
L
•
•
•
V
Our resolution is (0.35fm)3 and
83 markers/fluid-cell 
~ 10k cells & 10Mill m.p.-s
•
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 V
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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
• Furthermore
Re = 0.3 – 1 for
Re = 3 – 10 for
 we need b > 0.5 bmax
and
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Onset of turbulence around the Bjorken flow
S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: 1108.5535v1
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
P
Green and blue have the same
longitudinal speed (!) in this model.
Longitudinal shear flow is omitted.
T
x
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Onset of turbulence around the Bjorken flow
S. Floerchinger & U. A. Wiedemann, JHEP 1111:100, 2011; arXiv: 1108.5535v1
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
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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!
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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.]
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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:

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Detecting rotation:
Lambda polarization
 From hydro
[ F. Becattini, L.P. Csernai, D.J. Wang,
Phys. Rev. C 88, 034905 (2013)]
LHC
3.56fm/c
RHIC
4.75fm/c
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L.P. Csernai
LHC
RHIC
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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 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) 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
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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).
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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
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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) the
DCF is minimal.
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Results
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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.
L.P. Csernai
Thank you
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