1. Λ Polarization in an Exact rotating and expanding model for peripheral heavy ion reactions Yilong Xie NorSAC-2015 Department of Physics and Technology, University of Bergen

2. 1. Introduction 2. Analytical solution 3. Results 4. Summary

3. 1 Introduction: Angular momentum Department of Physics and Technology, University of Bergen L. P. Csernai, et al. Physical Review. C 87, 034906 (2013) Tilted initial state (Fig. 1) carries the angular momentum from impact. The velocity shear (Fig. 2b) will further rotate this initial state, and even leads to Kelvin Helmholtz Instability(KHI). Fig. 2 (a)Transverse view of collisions. (b) Velocity shear streaks are formed after collision. Fig. 1 3-D view of the collision shortly after the impact. Tilted initial state. Rotation System

4. 1 Introduction: Vorticity and Polarization Department of Physics and Technology, University of Bergen F. Becattini, et al. PRC 88, 034905 (2013) Fig. 3 The vorticity calculated in the reaction (xz) plane at t = 0.17 fm/c after the start of fluid dynamical evolution. The rotation can lead to observable vorticity (Fig. 3), and polarization (Fig. 4). Fig. 4. The dominant y component of the observable polarization, Π 0 (p) in the Λ’s rest frame. L. P. Csernai, et al, PRC 87, 034906 (2013)

5. 1 Introduction: Exact model Department of Physics and Technology, University of Bergen Fig. 4 Space-time evolution in Bjorken model  A color-glass condensate is formed after Lorentz contracted nuclei penetrate each other, then system evolves as a (3+1)D fluid dynamics describes.  Our exact model starts a short time after fluid dynamics stage, prior to the freeze-out, which is featured with a rotating and expanding system.

6. 2 Analytic solution Department of Physics and Technology, University of Bergen F. Becattini, V. Chandra, et al. Annals of Physics 338, 32 (2013) The polarization for a spin ½ particle with mass m and momentum p is defined as: The space-integrated mean polarization vector in participant reference frame is derived as: The polarization 3-vector in the rest frame of particle can be found by Lorentz-boosting the above four-vector:

7. 2 Analytic solution Department of Physics and Technology, University of Bergen F. Becattini, L.P. Csernai, and D.J. Wang, Phys. Rev. C 88, 034905 (2013). The spatial part of polarization vector can be simplified as: Based on our Exact model, this polarization 3-vector can be analytically solved as (in non-relativistic limit):, Rotation Expansion

8. 3 Results Department of Physics and Technology, University of Bergen  The second term arising from expansion is of comparable magnitude to the first term arising from local vorticity Fig. 5 The first term (left) and second term (right) of polarization at time t = 0.5 fm/c after the equilibration of rotation in the Exact model

9. 3 Results Department of Physics and Technology, University of Bergen  The x and z components are both p y asymmetric, with values of 8% and 3% at the corners. Fig. 6 The x (left) and z (right) components of second term arose by expansion at time t = 0.5 fm/c after the equilibration of rotation in the Exact model

10. 3 Results Department of Physics and Technology, University of Bergen Fig. 7 The y component of second term (left) and total polarization (right) at time t = 0.5 fm/c after the equilibration of rotation in the Exact model  The y component of polarization reaches -16% in the corners, much greater than the x and z components.

11. 3 Results Department of Physics and Technology, University of Bergen  The plots are not symmetric any more, due to the Lorentz Boost.  The structure of Π 0y is similar to the one obtained in Ref [2], but it reaches 12% at high p x values, greater than 9% in Ref [2], which shows the effect of expansion dynamics on polarization. Becattini, Csernai, Wang, PRC 88, 034905 (2013); Fig. 7 The radial, and axial components of polarization in Λ ‘s rest frame, at time t = 0.5 fm/c.

12. 4 Summary Department of Physics and Technology, University of Bergen  The polarization arising from the rotation is not too large due to the moderate energy in Exact model.  The polarization arising from the dynamics of radial and spherical expansion is discussed, which shows non-negligible contribution to the overall polarization.  This study indicates that the assumptions regarding the initial state are influencing the predictions on the observed vorticity, and polarization.

14. Fig.2 Due to the azimuthal symmetry the rotation is in the reaction plane. The radial component of momentum is chosen to be p x only. Fig. 1. Sketch of peripheral heavy ion collisions at high energy. The Λ polarization points essentially into the direction of the total angular momentum (-y) of the interaction region, it is orthogonal to the reaction plane. Configuration