The enhancement of $v_4$ in nuclear collisions at the highest densities signals a first order phase transition

The beam energy dependence of $v_4$ (the quadrupole moment of the transverse radial flow) is sensitive to the nuclear equation of state (EoS) in mid-central Au + Au collisions at the energy range of $3<\sqrt{s_{NN}}<30$ GeV, which is investigated within the hadronic transport model JAM. Different equations of state, namely, a free hadron gas, a first-order phase transition and a crossover are compared. An enhancement of $v_4$ at $\sqrt{s_{{NN}}}\approx 6$ GeV is predicted for an EoS with a first-order phase transition. This enhanced $v_4$ flow is driven by both the enhancement of $v_2$ as well as the positive contribution to $v_4$ from the squeeze-out of spectator particles which turn into participants due to the admixture of the strong collective flow in the shocked, compressed nuclear matter.

where φ is the azimuthal angle with respect to the event plane Φ n , which is estimated experimentally in various ways. The harmonic flow coefficients measure the strength of the system response to the initial coordinate space anisotropy and fluctuations in the collision zone.
To investigate the phase structure of QCD, both the beam energy-, centrality-, and system sizedependence are studied to access the different regions of T − µ B phase diagram [27]. In particular, the search for a first-order phase transition and the critical end point at high baryon density is a challenging goal of high energy heavy ion collisions [28].
At lower beam energies ( √ s N N < 10 GeV), the strength of the elliptic flow is determined by the interplay between out-of-plane (squeeze-out) and in-plane emission [4,29]. In a previous work we predicted a first-order phase transition [30,31] will cause an enhancement of the elliptic flow v 2 as function of the beam energy by the suppression of the squeeze-out due to the softening of EoS [32].
Does this enhancement of v 2 suggest that v 4 is also enhanced in the vicinity of a first-order phase transition? This letter presents the beam energy dependence of v 4 as calculated with the microscopic transport model JAM [33], using the modified scattering style method [34,35] and confirms our conjecture. In JAM, particle production is modeled by the excitations of hadronic resonances and strings, and their decays in a similar way as in the RQMD and UrQMD models [36][37][38]. Secondary products are allowed to scatter again, which generates collective effects within our approach. In the standard cascade version of the model, one usually chooses the azimuthal scattering angle randomly for any two-body scattering. (The effects of a preserved two-body reaction plane have been studied in Ref. [39]). Thus, cascade simulations yield the free-hadronic gas EoS in equilibrium, as then two-body scatterings, on average, do not generate additional pressure. In our approach, the pressure of the system is controlled by changing the scattering style in the twobody collision terms. It is well known that an attractive orbit reduces the pressure, while repulsive orbit enhances the pressure [40,41]. Thus, the pressure is controlled by appropriately choosing the azimuthal angle in the two-body scatterings. Specifically, the pressure difference from the free streaming pressure ∆P is obtained by the following constraints: [42]: where ρ is the local particle density and δτ i is the proper time interval of the i-th particle between successive collisions, (p i −p i ) is the momentum change and r i is the coordinate of the i-th particle.
Momenta and coordinates in Eq. (3) refer to the values in the c.m. frame of the respective binary collisions. We had demonstrated that a given EoS can be simulated by choosing the azimuthal angle according to the constraint in Eq.(3) in the two-body scattering process [35]. We note that the total cross section and scattering angle of the two-body scattering are not changed by this method; the only modification is the choice of the azimuthal angle.
In this work, we use the same EoS as developed and used in Ref. [35] to simulate both the conjectured first-order phase transition (1OPT) and also the alternative crossover transition (Xover). The EoS with a first-order phase transition (EOS-Q) [19,43] is constructed by matching a free, massless quark-gluon phase with the bag constant B 1/4 = 220 MeV with the hadron gas EoS. In the hadronic gas phase, hadron resonances with mass up to 2 GeV are included, with a repulsive, baryon density ρ B dependent mean field potential V (ρ B ) = 1 2 Kρ 2 B , with K = 0.45 GeV fm 3 . For the crossover EoS, we use the chiral model EoS from Ref. [44], where the EoS at vanishing and at finite baryon density is consistent with a smooth crossover transition, i.e. this EoS is consistent with recent lattice QCD results.
For all presented results we compute v 4 with respect to the reaction plane: Φ n = Φ RP , where Φ RP is the reaction plane angle of the collision. As usual, the reaction plane anisotropies in the even-order Fourier coefficients are in good agreement with the anisotropies taken with respect to the event plane, while odd-order Fourier coefficients are generated by event-by-event fluctuations.     Predicted v 4 signal can be studied experimentally at future experiments such as RHIC-BESII [49], FAIR [50,51] NICA [52], and J-PARC-HI [53], which offer the best opportunities to explore the compressed baryonic matter, and reveal the phase structure of QCD.