Model investigations of the correlation between the mean transverse momentum and anisotropic flow in shape-engineered events

The correlation between the event mean-transverse momentum $[p_{\mathrm{T}}]$, and the anisotropic flow magnitude $v_n$, $\rho(v^{2}_{n},[p_{T}])$, has been argued to be sensitive to the initial conditions in heavy-ion collisions. We use simulated events generated with the AMPT and EPOS models for Au+Au at $\sqrt{\textit{s}_{NN}}$ = 200 GeV, to investigate the model dependence and the response and sensitivity of the $\rho(v^{2}_{2},[p_{T}])$ correlator to collision-system size and shape, and the viscosity of the matter produced in the collisions. We find good qualitative agreement between the correlators for the string melting version of the AMPT model and the EPOS model. The model investigations for shape-engineered events as well as events with different viscosity ($\eta/s$), indicate that $\rho(v^{2}_{2},[p_{T}])$ is sensitive to the initial-state geometry of the collision system but is insensitive to sizable changes in $\eta/s$ for the medium produced in the collisions. These findings suggest that precise differential measurements of $\rho(v^{2}_{2},[p_{T}])$ as a function of system size, shape, and beam-energy could provide more stringent constraints to discern between initial-state models and hence, more reliable extractions of $\eta/s$.

A central objective of the current heavy-ion programs at the Large Hadron Collider (LHC) and the Relativistic Heavy-Ion Collider (RHIC) is to understand the transport properties of the quarkgluon plasma (QGP) [1,2,3] formed in high-energy heavy-ion collisions. In recent years, particular attention has been given to precision extraction of the specific shear viscosity of the QGP -the ratio of shear viscosity η, to the entropy density s, (η/s). The specific shear viscosity encodes the ability of the QGP to transport momentum. In general observables that characterize the azimuthal anisotropy of particles emitted in the transverse plane, are among the key measurements that have been used to constrain the viscous hydrodynamic response to the initial spatial distribution in energy density, produced in the early stages of the collision [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].
These studies indicated that a significant uncertainty in the η/s extractions stems from the uncertainty in the estimates for the initial-state eccen-1 niseemm@gmail.com tricities employed in the model calculations. Subsequently, several works have sought to construct and investigate new observables insensitive to η/s and more sensitive to initial-state effects leading to new constraints for the initial-state models [22,23].
One such observable, that leverages the correlation between the n th -order flow harmonics v n , and the average transverse momentum of particles in an event [p T ], is the correlation coefficient ρ(v 2 n , [p T ]) [22,24,25,26,27,28,29]; Here, v n is eccentricity-driven and the [p T ] is related to the transverse size of the overlap region, so events that have similar energy-density but smaller initial-state transverse size should have a larger radial expansion and consequently larger mean transverse momentum [30]. It has also been proposed that the ρ(v 2 n , [p T ]) correlator is sensitive to the correlations between the initial size and the initialstate deformation of colliding nuclei [29,31,32].
Initial measurements of the ρ(v 2 n , [p T ]) correlator have been reported for p+Pb and Pb+Pbcollisions at √ s N N = 5.02 TeV by the ATLAS Collaboration [33]. In Pb+Pb collisions the leading order experimental trend for ρ(v 2 2 , [p T ]) reflected negative values in peripheral or very low multiplicity events, but increased with centrality to positive values for mid to central collisions. These measurements provided important insights for initial-state models at LHC energies. The ρ(v 2 2 , [p T ]) correlator has also been studied in hydrodynamic and transport models [25,27,28,34]. Nonetheless, further detailed studies of ρ(v 2 n , [p T ]) are required to optimize its utility to discern between different inital-state models.
In this work, we use detailed simulations with both the AMPT [35], and EPOS [36,37,38] models for Au+Au at √ s N N = 200 GeV, to study the ρ(v 2 2 , [p T ]) correlators model dependence and its response and sensitivity to the magnitude of η/s and initial-state geometry. We exploit the technique of event-shape engineering to obtain a more detailed influence of the initial-state geometry. Our study emphasizes investigations for Au+Au collisions at √ s N N = 200 GeV in anticipation of the need for model predictions to compare to upcoming experimental measurements.
This study is performed with simulated events for Au+Au collisions at √ s N N = 200 GeV, obtained with the AMPT [35], and EPOS [36,37,38] models. Computations were performed for charged hadrons in the transverse momentum range 0.2 < p T < 2.0 GeV/c and the pseudorapidity acceptance |η| < 1.0. The latter choice mimics the acceptance of the STAR experiment at RHIC.
In the current study, simulations were made with the string melting option both on and off. In such a situation when the string melting mechanism is on, hadrons created from the HIJING model are converted to their valence quarks and anti-quarks, and their evolution in time and space is then shaped by the ZPC parton cascade model [48]. The key elements of AMPT include (i) HIJING model [49,50] initial parton-production stage , (ii) a parton scattering stage, (iii) hadronization through coalescence followed by (iv) a hadronic interaction stage [51]. In stage (ii) the used parton scattering cross-sections are evaluated accord-ing to; where α s is the QCD coupling constant and µ is the screening mass in the partonic matter. They principally establish the expansion dynamics of the A+A collision systems [48]; Within the AMPT model framework, the η/s value can be adjusted via an appropriate choice of µ and/or α s for a particular initial temperature T i [52].
In the current work, Au+Au collisions at √ s NN = 200 GeV, were simulated with model version ampt-v2.26t9b for a fixed value α s = 0.47, but the shear viscosity η/s is varied over the range 0.1-0.3 by adjusting µ from 2.26 -4.2 f m −1 for a temperature T i = 378 MeV [52]. The three AMPT sets which will be presented in this work are summarized in Tab. 1. Table 1: The summary of the three AMPT sets which will be presented in this work.
• EPOS Model: The EPOS model [36,37,38] is based on a 3+1D viscous hydrodynamical description of A+A collisions. The initial state conditions are defined in terms of flux tubes estimated via Gribov-Regge multiple scattering theory [36]. EPOS can be subdivided into three main components, (i) the core-corona division, (ii) the hydrodynamical evolution, and (iii) the hadronic cascades.
(i) The separation of the flux tubes fragmentation into core and corona (hadronize as a hadron jet) is based on the probability to escape from the bulk matter which will depends on the fragment transverse momentum and the local string density.
(ii) The hydrodynamical evolution based on the vHLLE, viscous HLLE-based algorithm, 3D+1viscous hydrodynamics employ a realistic Equation of State constrained with Lattice QCD data [53].
(iii) The hadronic cascade, hadronic afterburner, is based on the UrQMD model [54,55], which has been broadly employed to investigate ultra-relativistic heavy-ion collisions [54,55,56]. UrQMD was designed to investigate hadron-hadron, hadron-nucleus, and heavyion collisions from E Lab = 100 A·MeV to √ s NN = 200 GeV. Therefore, it includes a collision term that accounts for the interactions of more than 50 (40) baryon (meson) species as well as their anti-particles. The URQMD model describes the hadron-hadron interactions as well as the system evolution based on covariant propagation of all hadrons in the model with resonance decay, stochastic binary scattering, and color string formation.
The results reported in this work, were obtained for minimum bias Au+Au collisions at √ s NN = 200 GeV. A total of approximately 4.0, 5.0, 3.0, and 0.3 M events of Au+Au collisions were generated with AMPT Set-1, Set-2, Set-3, and EPOS, respectively.
The ρ(v 2 n , [p T ]) correlator is derived from covariances and variances (cf. Eq. 1) which involve both two-and multi-particle correlations that could also be influenced by non-flow effects due to resonance decays, Bose-Einstein correlations, and the fragments of individual jets [57]. Since non-flow contributions mostly involve particles emitted within a localized region in pseudorapidity, η, they can be mitigated via the sub-event cumulant methods [44,58,59,60]. A major mitigating feature of these methods, is the correlation of particles from two or more sub-events which are separated in η. The efficacy of these methods to reduce non-flow effects have been quantified for many different twoand multi-particle correlators [44,58,59]. It is noteworthy that these methods were used in the initial ρ(v n n , [p T ]) measurements by the ATLAS Collaboration [33].
In the current work, the two-subevents method is used to construct the v 2 2 variance. Thus, we use two separate η selections specified as −1.0 < η A < −0.35 and 0.35 < η C < 1.0 to determine the v 2 2 variance as: where v 2 {2} and v 2 {4} are the flow coefficients obtained from the two-and four-particle correlations respectively, with the sub-event method [58] with particles in the region η A and η C ; where φ A(C) is the azimuthal angle of particles in the region A (C). where, The variance of the dynamical p T fluctuations [61], c k ∼ Var([p T ]), defined in region |η B | < 0.35, can be given as: where is an average over all events. The event mean pT , [pT ], is given as, where MB is the event multiplicity in sub-event B. The covariance between v 2 2 and [pT ], cov(v 2 2 , [pT ]), is defined using the three-subevents method [33,62] as, The resulting ρ(v 2 2 , [pT ]) correlator, obtained via Eqs. 4, 10 and 12; is similar to that used in prior studies [22,24,25,26,27,28,29]. The ρ(v 2 n , [pT ]) correlator is derived from the correlations and fluctuations of vn and pT . Therefore, it is instructive to investigate the dependence of these variables on the models and the corresponding parameters tabulated in Table 1. Fig. 1 shows a comparison of the centrality dependence of v2{2} (a), v2{4} (b) the ratios v2{4}/v2{2} (c), and pT (d) for the AMPT and EPOS models. Panels (a) and (b) indicate that the AMPT results are sensitive to both the viscosity and whether or not string melting is turned on. They also indicate similar qualitative patterns between both models and the data reported by the STAR collaboration [63] (hatched bands) over the range of the model parameters summarized in Table    The sensitivity of the ρ(v 2 n , [pT ]) correlator to the shape and size of the collision system was investigated using the Event Shape Engineering (ESE) technique [66]. This technique leverages the observation that selections on the magnitude of the event-by-event fluctuations of the vn coefficients, for a fixed centrality, serve to influence the shape of the collision system [67].
The event-shape selections were performed via a frac- where Q2 is the magnitude of the second-order harmonic flow vector calculated within the sub-event 1.5 < η < 2.5, and M is the charged hadron multiplicity for this sub-event. The sub-event |η| < 1.0 was used to evaluate ρ(v 2 n , [pT ]) to ensure the separation between the sub-event used to evaluate q2 and ρ(v 2 n , [pT ]). Note that there are two caveats to the ESE method. First, the q2 selective power depends on the magnitude of v2 and the event multiplicity, so the benefit of the technique is handicapped by weak flow values and small event multiplicities [69]. Second, the non-flow effects, such as resonance decays, jets, etc. [70], could bias the q2 selections. The latter can be minimized via a ∆η separation between the sub-events used for the q2 selections and ρ(v 2 n , [pT ]) evaluations. Fig. 5 (a) shows a representative q2 distribution for 10-50% central Au+Au collisions at 200 GeV. The v2{2} values which result from the q2% selections indicated in panel (a), are shown in Fig. 5 (b). They indicate an essentially linear increase of v2{2} with q2%. Figure 6 compare the q2% dependence of the values for Var(v and ρ(v 2 2 , [pT ]) (d), computed for tracks with 0.2 < pT < 2.0 GeV/c in Au+Au collisions simulated with the AMPT and EPOS models. A striking feature of these results is the q2%-independence of c k and the essentially quadratic dependence of cov(v 2 2 , [pT ]) and ρ(v 2 2 , [pT ]) on q2%. These dependencies suggests that data-model comparisons of the ρ(v 2 2 , [pT ]) correlators extracted in shape-engineered events, could serve as a sensitive constraint for the initial-state eccentricity, and give important insight on the deformation of colliding systems.
In summary, we have presented extensive model studies to evaluate the model dependence, as well as the response and sensitivity of the ρ(v 2 2 , [pT ]) correlator to collision-system size and shape and the η/s of the matter produced in the collisions. We find that ρ(v 2 2 , [pT ]) is sensitive to the event shape selections of the collision system, but insensitive to sizable changes in the η/s of the medium produced in the collisions. Initial comparisons of the calculated and experimental Var(v 2 2 ), and √ c k / pT values, also indicate good qualitative agreement. These findings strongly suggest that precise differential measurements of ρ(v 2 2 , [pT ]) as a function of system-size, shape, deformation and beam-energy could provide more stringent constraints to discern between initial-state models and hence, more reliable extractions of η/s.