$\mathrm{\Lambda }$ polarization in peripheral collisions at moderately relativistic energies

The polarization of $\mathrm{\Lambda }$ hyperons from relativistic flow vorticity is studied in peripheral heavy ion reactions at FAIR and NICA energies, just above the threshold of the transition to the quark-gluon plasma. Previous calculations at higher energies with larger initial angular momentum, predicted significant $\mathrm{\Lambda }$ polarization based on the classical vorticity term in the polarization, while relativistic modifications decreased the polarization and changed its structure in the momentum space. At the lower energies studied here, we see the same effect namely that the relativistic modifications decrease the polarization arising from the initial shear flow vorticity.

Figure 1

The three-dimensional view of the collisions shortly after the impact. The projectile spectators are going along the $z$ direction, and the target spectators are going along the $z$ axis. The collision region is assumed to be a cylinder with an almond-shaped profile and tilted end surfaces, where the top side is moving to the right and the bottom is moving to the left. The participant cylinder can be divided into streaks, and each streak has its own velocity, as shown in Fig. 2. The velocity differences among the streaks result in rotation, turbulence, and even KHI.

Figure 2

The schematic hydroflow velocity after the collisions shown in Fig. 1. (a) The longitudinal velocity profile along the $x$ direction, and it gives rise to the $v_1$ type of flow in the reaction plane, i.e., (b). (c) The anti-$v_2$ type of flow in the $[y-z]$ plane, and (d) is the $v_2$ type of flow in the $[x-y]$ plane.

Figure 3

The first (top) and second (bottom) term of the dominant $y$ component of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the FAIR U+U reaction at $\sqrt{s_{NN}}=8.0$ GeV.

Figure 4

The first (top) and second (bottom) term of the $x$ component of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the FAIR U+U reaction at $\sqrt{s_{NN}}=8.0$ GeV.

Figure 5

The first (top) and second (bottom) term of the $z$ component of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the FAIR U+U reaction at $\sqrt{s_{NN}}=8.0$ GeV.

Figure 6

The $y$ component of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the FAIR U+U reaction at $\sqrt{s_{NN}}=8.0$ GeV. The top figure is in the calculation frame, while the bottom figure is boosted to the frame of the $\mathrm{\Lambda }$ [18].

Figure 7

The modulus of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the FAIR U+U reaction at $\sqrt{s_{NN}}=8.0$ GeV. The figure is in the frame of the $\mathrm{\Lambda }$.

Figure 8

The $y$ component (top) and the modulus (bottom) of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the NICA Au+Au reaction at $\sqrt{s_{NN}}=9.3$ GeV. The figure is in the frame of the $\mathrm{\Lambda }$.

Figure 9

The $y$ component (top) and the modulus (bottom) of the $\mathrm{\Lambda }$ polarization for momentum vectors in the transverse, $[p_x,p_y]$, plane at $p_z=0$, for the FAIR U+U reaction at $\sqrt{s_{NN}}=8$ GeV at the earlier freeze-out time of $t=4.2$ fm/c. The figure is in the frame of the $\mathrm{\Lambda }$.