Light Nuclei and Hyper Nuclei Collectivity Measurements at High Baryon Density Region

The energy dependence of $v_{1}$ for proton and net-proton at $\sqrt{s_{NN}}=$ 7.7 to 62.4 GeV have been measured by STAR experiment Adamczyk_2014 . The value of $v_{1}$ slopes at midrapidity, $dv_{1}/dy|_{y=0}$, change signs from positive to negative at $\sqrt{s_{NN}}\sim$ 10 GeV with the increased energy and show a minimum at $\sqrt{s_{NN}}\sim$ 15 GeV as shown in Fig. 1 Adam_2020 , which may be a signature of first-order phase transition

according to a hydrodynamical model Stocker_2005 . According to the picture of nucleon coalescence, the light nuclei $v_{1}$ should have a similar but more pronounced energy dependence behavior in the same energy region.

The deuteron $v_{1}$ has been measured by the STAR experiment and the extracted $dv_{1}/dy|_{y=0}$ as a function of collision energy is shown in Fig. 1 Adam_2020 . At $\sqrt{s_{NN}}=7.7$ GeV, the value of $dv_{1}/dy|_{y=0}$ for deuteron is higher than that for proton and follows the $A$ scaling considering the statistical and systematic uncertainties. Above 7.7 GeV, there is a hint that the deuteron $dv_{1}/dy|_{y=0}$ have a positive sign with large uncertainties, which is opposite to the corresponding proton $dv_{1}/dy|_{y=0}$. The $p_{\rm T}$ dependence of deuteron $v_{1}$ show an enhancement towards very low $p_{\rm T}$ at 7.7 GeV Adam_2020 . This measurement may suggest a more complicated mechanism than the simple coalescence picture for light nuclei production, while a stronger conclusion needs more event statistics which can be achieved with the second phase of STAR beam energy scan program.

Moving to lower energies, the light nuclei $v_{1}$ slopes reach maximum and have a weak energy dependence from $\sqrt{s_{NN}}=$ 2 to 3 GeV Adam_2020 ; Abdallah_2022 ; Reisdorf_2012 , as shown in Fig. 2. There is a clear mass ordering and the value of $dv_{1}/dy|_{y=0}$ follow the $A$ scaling approximately. Above 3 GeV, the $v_{1}$ slopes decrease with the increasing of collision energies. While the $dv_{1}/dy|_{y=0}$ has a steep drop with the decreasing energy below 2 GeV Reisdorf_2012 . The $p_{\rm T}$ dependencies of light nuclei $v_{1}$ also show the $A$ scaling at midrapidity Abdallah_2022 . The scaling, however, worsens for higher $p_{\rm T}$ and away from midrapidity; one of possible reasons is the increasing contamination of target/beam rapidity fragments. The proton $v_{1}$ at 3 GeV can be well reproduced by the baryon mean-field configuration of JAM transport model Abdallah_2022 , while the cascade of the model underestimates the proton $v_{1}$. A nucleon coalescence using the proton and neutron phase space distributions from this JAM mean-field calculations can qualitatively describes the $v_{1}$ for all measured light nuclei species, which may suggest that the baryonic interactions dominate the evolution and light nuclei are likely formed via the coalescence of nucleons at energy of several GeV.

The light nuclei $v_{2}$ as a function of $p_{\rm T}$ has been measured by the STAR experiment in $\sqrt{s_{NN}}=$ 7.7 to 200 GeV Au+Au collisions Adamczyk_2016 . The $v_{2}$ value shows a monotonic rise with increasing $p_{\rm T}$ and a mass ordering with increasing mass number. The difference of $v_{2}$ between deuteron and anti-deuteron is qualitatively follow the difference between proton and anti-proton. For all measured collision energies, the $A$ number scaling holds generally for all (anti-)light nuclei at low $p_{\rm T}$ ($p_{\rm T}/A<1.5$ GeV/$c$), while the scaling is broken, especially for low energies, towards high $p_{\rm T}$. The results have been updated with more statistic events by STAR recently qm22 , as shown in Fig. 3. Compared to $v_{2}$ of protons, the $v_{2}/A$ have a overall 20-30% deviations from $A$ scaling for deuteron, triton, and ³He at both $\sqrt{s_{NN}}=$ 27 GeV and $\sqrt{s_{NN}}=$ 54.4 GeV. The facts that both the deuteron $v_{1}$ and light nuclei $v_{2}$ do not have $A$ scaling imply that light nuclei may not formed only via a simple coalescence of nucleons at the energy region $\sqrt{s_{NN}}=7.7-200$ GeV.

With decreasing of collision energies, the $v_{2}$ value at midrapidity for protons as well as light nuclei become negative at energy close to 3 GeV, as in Fig. 4.

This negative $v_{2}$ may be caused by the squeeze-out and/or shadowing of the spectators due to their passage time becoming comparable with the expansion time of the fireball at $\sqrt{s_{NN}}=$ 2 to 3 GeV Reisdorf_2012 . The light nuclei $v_{2}$ at the mid-rapidity have a clear mass hierarchy that increases with increasing collision energy from 1.9 GeV to 2.2 GeV and then decreases from 2.2 GeV to 3 GeV. The location of minimum value of $v_{2}$ varies with the nuclei mass. The mass hierarchy disappears at 3 GeV for all light nuclei. Similar to their $v_{1}$ slope behavior as a function of energy, the heavier nuclei have a stronger energy dependence which may suggest more sensitive to the EoS Reisdorf_2012 . Although the light nuclei $v_{2}$ violate $A$ scaling at 3 GeV, the JAM plus coalescence calculations can qualitatively describe the rapidity dependence Abdallah_2022 . The reason of the $A$ scaling broken may partly be that the Eq. 1 is only valid under the assumption of small $v_{1}$ Abdallah_2022 . At 3 GeV, the $v_{1}$ is however not negligibly small as discussed in Section II.1.

HADES experiment measure the light nuclei flow from $v_{1}$ to $v_{6}$ using the first-order event plane at $\sqrt{s_{NN}}=2.4$ GeV Au+Au collisions. It is expected that high order flow coefficients ($n>3$) relative to the first-order event plane are zero if they originate from an initial state fluctuation. However, all the $v_{3}-v_{6}$ are not zero but have strong dependencies on particle $p_{\rm T}$ and rapidity for all the measured light nuclei species. No mass hierarchy is observed for $v_{3}-v_{6}$ in the midrapidity. Non-zero high order flow may imply these high order flows have more complicated causes at high baryon density and could be more sensitive to the EoS than $v_{1}$ and $v_{2}$ according to transport model calculations Hillmann_2018 .