In the last years it has been revealed that if measuring relative to higher order event planes Ψn, higher order flow coefficients vn for n > 2 can be measured. It also turned out that Bose-Einstein (HBT) correlation radii also show 3rd order oscillations if measured versus the third order event plane Ψ3. In this paper we investigate how these observables can be described via analytic hydro solutions and hydro parameterizations. We also investigate the time evolution of asymmetry coefficients and the mixing of velocity field asymmetries and density asymmetries

Bagoly, Attila

Csanád, Máté

Lökös, Sándor

Szabó, András

English

ELTE Digital Institutional Repository (EDIT)

Higher order anisotropies in hydrodynamicsM. Csanád, A. Szabó, S. Lökös, A. Bagoly,Eötvös University, Pázmány P. s. 1/a, 1117 Budapest, HungaryNovember 6, 2018AbstractIn the last years it has been revealed that if measuring relative to higherorder event planes Ψn, higher order flow coefficients vn for n > 2 can bemeasured. It also turned out that Bose-Einstein (HBT) correlation radii alsoshow 3rd order oscillations if measured versus the third order event planeΨ3. In this paper we investigate how these observables can be described viaanalytic hydro solutions and hydro parameterizations. We also investigatethe time evolution of asymmetry coefficients and the mixing of velocity fieldasymmetries and density asymmetries.1 IntroductionIn relativistic heavy ion collisions, an expanding and cooling medium is created,usually referred to as the strongly interacting quark gluon plasma. Hydrodynamicsprovides a tool to investigate the time evolution of this medium, and exact analyticmodels are particularly useful in this regards. Usually spherical, axial or ellipsoidalsymmetry is assumed in these solutions, as these are simple to handle and rep-resent geometries that yield realistic results for many soft observables. However,event-by-event fluctuating nuclear distributions yield event-by-event fluctuating ini-tial conditions, and thus higher order azimuthal asymmetries arise. In order to1arXiv:1504.07932v1  [hep-ph]  29 Apr 2015describe these one has to utilize higher order asymmetries in hydro as well. Thiswas successfully done in numerical calculations, see for example [1, 2].In this paper we discuss the first exact analytic solutions [3] of relativistic hydro-dynamics that assume higher order asymmetries and thus give realistic higher orderflow coefficients. We also discuss possible extensions of this approach, by analyzingthe time evolution of the asymmetries in a numerical approach, and by investigatingthe effect of spatial versus momentum space anisotropies.2 Multipole solutions and higher order anisotropiesThe first 1+3D relativistic solution with ellipsoidal geometry was discovered inRef. [4]. In this solution the thermodynamic quantities at a given proper-time τ areconstant on the surfaces of an expanding ellipsoid, defined by the s scale variables =r2xX2+r2yY 2+r2zZ2, (1)where rx,ry,rz are the spatial coordinates, while X, Y , Z are the time-dependentaxes of the ellipsoid. The velocity profile as a function of space-time coordinates xµis given in form of a 3D Hubble flow, i.e. uµ = xµ/τ . With these, uµ∂µs = 0 holds(if the expansion of the axes is linear in time). In Ref. [4] it was already indicated,that more complicated scale variables can also be written up (with uµ∂µs = 0still holding). In Ref. [3] we showed that this solution can indeed be extended tomultipole symmetries with a generalized scale variables =rNRN(1 + ε cos(Nφ)) (2)With the s given in Eq. (2), the new solutions can be given in cylindrical coor-dinates (r, φ, z) as:s =∑NrNRN(1 + εN cos(N(φ− ψN ))) +zNRN(3)n = nf(τfτ)3ν(s) (4)T = Tf(τfτ)3/κ 1ν(s)(5)p = pf(τfτ)3+3/κ(6)and uµ still representing a Hubble-flow, as in the original paper of Ref. [4]. Inthe formula for s, ψN being the N th order reaction planes (which cancel from theobservables). This way we get new solutions with almost arbitrary shaped initialdistributions, see Fig. 1. It is important to note here that however, the initialstate fluctuation in the observed collision is present through the orientation of theN th order reaction planes and the strength of higher order asymmetries, the event2Figure 1: Heat map of s values in the transverse plane, with multiple superim-posed symmetries. The more εN components are included, the more asymmetricthe shape gets.plane orientation itself does not affect the measured quantities. Thus if every vN ismeasured relatively to the N th order reaction plane, then the (event-through-event)averaged value of vN will correspond to an average n-pole anisotropy εN .We also calculated hadronic observables from the above solution (see detailsof the freeze-out scenario in Ref [3] or Ref. [5]. A comparison to PHENIX dataon higher order harmonics measured in 200 GeV Au+Au collisions [6] is shown inFig. 2. Fit parameters of the model are εN (for N = 2, 3, 4), ut and b (T0 and τ0was fixed to values given from spectra and HBT comparisons of a similar model, asdescribed in Refs. [3, 5]).3 Time evolution of the anisotropiesIn the above described solution, the anisotropies don’t change over time - due tothe lack of pressure gradients and the Hubble-flow. In a numerical framework, weinvestigated how the introduction of pressure gradients, various speeds of soundand viscosity coefficients influence the time evolution of the asymmetries, whenstarting from an initial condition that is very similar to one described by knowanalytic solutions – except in pressure, where we used a pressure profile similarto the density profile given in usual Hubble-expansion models [4, 3]. We used amulti-stage predictor-corrector method outlined in Ref. [7]. This is a finite volumescheme, where the initial flux is a weighted average of the Lax-Friedrichs and Lax-Wendroff fluxes, called GFORCE. This flux is used to make a new prediction onthe grid points, which is used to get a better flux approximation. This procedure isrepeated for a number of times, as described e.g. in Ref. [7]. This multi-stage fluxgives results that are comparable to those of the Godunov method. We tested ourmethod with known analytic solutions given in Refs. [8, 4, 3]. We analyzed bothnon-relativistic and relativistic hydrodynamics, and arrived at similar conclusions.Fig. 3 shows the result of a nonrelativistic calculation of the time evolution ofthe energy density. If we assume a small amount of viscosity, it makes the flow3Figure 2: Fits to PHENIX 200 GeV Au+Au data [6] in 5 centrality bins. Fitparameters are summarized in Ref. [3]4Figure 3: Time evolution of energy density. Top row shows viscosity free case,the bottom row with µ = 10MeV · fm viscosity.Figure 4: The time evolution of the asymmetry coefficients ε1,2,3,4 in the energydensity, as modified by a small amount of viscosity (left plot) and the change inspeed of sound (right plot).5Figure 5: The time evolution of the asymmetry coefficients in the energy densityin a relativistic calculation, as modified by the change in κ = c−2s (right plot).itself and thus the disappearance of asymmetries slower. The time evolution of theasymmetries themselves is shown in Fig. 4. In this figure we also see the effect ofspeed of sound in this nonrelativistic calculation: the reduction of speed of soundmakes the asymmetries disappear slower – due to the reduction of the speed ofsound waves. A similar effect is seen in case of a relativistic calculation, as shownin Fig. 5: the increase of κ = c−2s makes the disappearance of asymmetries slower.It is important to see that this also slows down the speed of cooling, which meansthe system will freeze out latter.4 Anisotropy mixingIt is important to see that there may be asymmetries in both momentum space(i.e. in the velocity field) and in density (i.e. in energy density or pressure), andboth influence the measured anisotropies. To investigate this effect, we created amultipole version of the Buda-Lund model [9], with a scale variable given in Eq. 3,but we introduced a multipole flow field as well. We start from a “flow potential”Φ, which gives us the flow:v = (∂xΦ, ∂yΦ, ∂zΦ). (7)The flow field at a given time is spherically symmetric if Φ = r22H with H beinga Hubble-coefficient at that given time. Elliptical symmetry is obtained with Φ =6r22H (1 + χ2 cos(2ϕ)), whileΦ =r22H(1 + χ2 cos(2ϕ)) +r33H2χ3 cos(3ϕ)) (8)represents a triangular perturbation of the elliptical flow. Of course the variousanisotropies can have various event planes (symmetry planes), but the specific angleof these does not enter into the results. Inspired by Ref. [10], we analyzed how χ2,3and ε2,3 influence flow coefficients v2 and v3 – see results in Fig. 6. Compared tothe spatial anisotropy, velocity field anisotropy has a much larger effect on ellipticand triangular flow coefficients.Figure 6: Dependence of flow anisotropy coefficients v2,3 on asymmetry param-eters χ2,3 and ε2,3. Velocity field asymmetry has a stronger influence on flowcoefficients.5 Summary and acknowledgmentsIn this paper we showed an extension of the scope of analytic relativistic hydro-dynamics to higher order azimuthal asymmetries, compatible with realistic (event-by-event fluctuating) geometries. Higher order flow observables were calculatedfrom this model, and are found to be compatible with data. In the analytic model,anisotropy parameters were independent of time, thus we investigated their timeevolution in a numerical framework, developed for this purpose. We also inves-tigated how velocity- and density-field anisotropies “mix”, in the framework of amultipole Buda-Lund model. We are thankful to Tamás Csörgõ and Márton Nagyfor useful discussions with respect to this project. We thank the WPCF commu-nity and the WPCF 2014 organizers, in particular the local hosts, Tamás Novákand Tamás Csörgõ, for the possibility to present this work. We also thankfullyacknowledge the support of the OTKA grant NK 101438.7References[1] R. Chatterjee, D. K. Srivastava, and T. Renk arXiv:1401.7464 [hep-ph].[2] L. Yan and J.-Y. Ollitrault Phys.Lett. B744 (2015) 82–87,arXiv:1502.02502 [nucl-th].[3] M. Csanád and A. Szabó Phys.Rev. C90 no. 5, (2014) 054911,arXiv:1405.3877 [nucl-th].[4] T. Csörgő, L. P. Csernai, Y. Hama, and T. Kodama Heavy Ion Phys. A21(2004) 73–84, nucl-th/0306004.[5] M. Csanád and M. Vargyas Eur. Phys. J. A44 (2010) 473–478,arXiv:0909.4842 [nucl-th].[6] A. Adare et al. Phys.Rev.Lett. 107 (2011) 252301, arXiv:1105.3928[nucl-ex].[7] E. F. Toro and V. A. Titarev J. Comp. Phys. 216 (2016) 403–429.[8] T. Csörgő, S. V. Akkelin, Y. Hama, B. Lukács, and Y. M. Sinyukov Phys.Rev. C67 (2003) 034904, hep-ph/0108067.[9] M. Csanád, T. Csörgő, and B. Lörstad Nucl. Phys. A742 (2004) 80–94,nucl-th/0310040.[10] M. Csanád, B. Tomásik, and T. Csörgő Eur. Phys. J. A 37 (2008) 111–119,arXiv:0801.4434 [nucl-th].8

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Higher order anisotropies in hydrodynamics

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