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Differential HBT Method to Analyze Rotation. L.P. Csernai, S. Velle , D.J. Wang University of Bergen, Norway. #395 QM’2014. Peripheral Collisions (A+A). Global Symmetries Symmetry axes in the global CM-frame: ( y  -y) ( x,z  -x,-z) Azimuthal symmetry: φ- even ( cos n φ)

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Differential HBT Method to Analyze Rotation

L.P. Csernai, S. Velle, D.J. Wang

University of Bergen, Norway

#395 QM’2014

L.P. Csernai


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

Global Collective flow and fluctuations canbe separated using symmetries:[Csernai & Stöcker, J. Phys. G (2014) to be p.]for this the participant c.m. and the symmetryaxes should be determined as described in:[ Csernai, Eyyubova & Magas, PRC 86(12)024912, & PRC 88 (13) 019902]

L.P. Csernai

initial state reaching equilibrium
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.

[See Poster # 608 V.K. Magas & L.P. Csernai]





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


in preparation

L.P. Csernai



PICR method !!!



From Rotation and Kelvin-Helmholtz Instabilitywe get high angular momentum and vorticity:

[See Poster # 424 D.J. Wang & L.P. Csernai]

L.P. Csernai


Detecting rotation:

Lambda polarization

From hydro

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





L.P. Csernai




[See Poster # 394 L.P. Csernai, F. Becattini & D.J. Wang]

L.P. Csernai

detection of global collective flow
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

the differential hbt method
The Differential HBT method

The method uses two particle correlations:

with k= (p1+p2)/2 and q=p1-p2 :


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

the space time source function s k q
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

the space time source function s k q1
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.: -


L.P. Csernai

the space time source function s k q2
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


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



We can rotate the frame of reference:

L.P. Csernai



For lower, RHIC energy:

One can evaluate the DCF in these tilted reference frames where (without rotation) the

DCF is minimal.

L.P. Csernai



[20] M.A. Lisa, et al.,

Phys. Lett. B 496, 1 (2000);

Phys. Lett. B 489, 287 (2000);

Phys. Rev. C 84, 014908 (2011);

Phys. Rev. C 89, 014903 (2014).

[21] L.P. Csernai, G. Eyyubova

& V.K. Magas, Phys. Rev. C 86,

024912 (2012).

L.P. Csernai

  • 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