Directed flow in ultrarelativistic heavy-ion collisions
Abstract
We study the generation of the directed flow in the hydrodynamic expansion of hot matter formed in ultrarelativistic heavy-ion collisions at GeV. The experimentally observed negative directed flow in a wide range of central pseudorapidities is reproduced, assuming that the fireball is tilted away from the collision axis. The tilt of the source is consistent with a preferential emission in the forward/backward hemisphere from forward/backward participating nucleons. The model reproduces the experimentally observed scaling of the directed flow when going from Au-Au to Cu-Cu systems.
pacs:
25.75.Ld, 24.10Nz, 24.10PaI Introduction
The appearance of the transverse, azimuthally asymmetric flow is one of the key observations in the physics of relativistic heavy-ions Arsene:2004fa ; *Back:2004je; *Adams:2005dq; *Adcox:2004mh. It proves that a collectively expanding, strongly interacting medium is formed in the course of the reaction. A number of observables have been studied, both in experiments, and in model calculations, in order to unravel the properties of the dense hot matter created in the collisions. The production of particles with soft momenta can be interpreted as a thermal emission of particles from fluid elements moving with some collective velocity field Schnedermann:1993ws ; *Kolb:2000sd. Relativistic hydrodynamics describes quantitatively the development of the collective velocity from pressure gradients in the fireball Teaney:2000cw ; *Kolb:2003dz; *Hama:2005dz; *Huovinen:2006jp; *Hirano:2005xf; *Broniowski:2008vp; Hirano:2002ds ; Bozek:2009ty .
For non-central collisions the interaction region is azimuthally asymmetric and, as a result of the collective expansion of matter, azimuthally asymmetric emission of particles takes place. The effect can be quantified in terms of Fourier coefficients in the expansion of the measured particle spectra
(1) | |||||
The elliptic flow coefficient is known to be a very sensitive probe of the pressure in the system at early stages Ollitrault:1992bk .
Directed flow, quantified by the coefficient is also measured at energies available at the BNL Relativistic Heavy Ion Collider (RHIC). The coefficient of the directed flow is zero at zero rapidity for collisions of symmetric nuclei, but it increases when moving to forward or backward pseudorapidities . Its size and sign has been the subject of many studies at lower energies, where it is dominated by nucleon flow Herrmann:1999wu ; *Wetzler:2002fi; *Voloshin:2002wa; *Lisa:2000ip. At RHIC energies the spectator nucleons have positive directed flow ( for ), resulting from the deflection of the spectators during the collision. On the other hand, matter in the fireball shows a significant negative (anti-flow) component for pseudorapidities and for centralities , both for Cu-Cu and Au-Au collisions Back:2005pc ; Adams:2005ca ; Abelev:2008jga . A striking characteristic of the measured directed flow is the large negative value of even at the highest energy GeV. Another important observation is the scaling of the measured directed flow with the size of the system. The coefficient is the same for both systems (Au-Au or Cu-Cu), for the same centrality. Whereas, a scaling of with the density of the fireball has been observed Voloshin:2007af .
Transport models of nuclear reactions, describe the directed flow at lower energies, but generally underpredict the amount of anti-flow at RHIC energies at central rapidities Bleicher:2000sx ; *Chen:2009xc. On the other hand, these calculations predict a large negative flow at large rapidities. Some calculations Burau:2004ev predict significant anti-flow around central rapidities, larger than observed, but yield a positive flow for , unlike observed in experiments. It has been noticed that the appearance of negative directed flow around central rapidities could be an effect of the softening of the equation of state Csernai:1999nf . This effect is called the third flow component. Hydrodynamic calculations incorporating such effects yield a negative elliptic flow at central rapidities and a positive directed flow at larger pseudorapidities, unlike the experimental data. A hydrodynamic calculation with initial conditions from a microscopic model gives the correct sign, but a too small magnitude of the directed flow for central rapidities Andrade:2008fa .
There are two effects leading to a negative directed flow in the models. The first one is the shadowing of the fireball matter by the spectators, which can give a substantial negative for . The second one is the build up of the flow away from the collision axis due to a tilt of the source. The description of the magnitude of the directed flow at different centralities for central rapidities requires, a tilt of the source of the right magnitude as function of the impact parameter, and a sufficient amount of collectivity to generate the flow. We consider two initial conditions for the hot source in the hydrodynamic evolution. The first one is the most commonly used initial conditions in -dimensional (D) calculations incorporating a Bjorken flow in the longitudinal direction and a shift in space-time rapidity due to the local imbalance of the momentum Hirano:2002ds . The second one assumes that the initial density results from a superposition of the energy density radiated by the color sources in the target and the projectile. The preferred emission in the pseudorapidity hemisphere of the emitting charge results in a tilt of the source for non-central collisions. We show that the second choice leads to a satisfactory description of the directed flow generated in heavy ion collisions at GeV for central rapidities.
Ii Initial conditions and early flow
Hydrodynamic evolution in D at RHIC energies is performed in the proper time . The densities are defined as functions of the transverse plane (-) coordinates and the space-time rapidity . The hydrodynamic model requires some initial density and initial flow profile to be chosen at the initial time . Although some guidance from microscopic models of elementary collisions is possible in the choice of the initial conditions, there is still a vast choice of initial energy density profiles that are used in the simulations. Also, most of the calculations assume a Bjorken initial flow in the longitudinal direction and no initial transverse flow
(2) |
In the hydrodynamic evolution the generation of the left-right asymmetry of the flow in the two (forward/backward) halves of the reaction plane requires the presence of an asymmetry in the initial distributions. For non-central collisions the azimuthal asymmetry of the interaction region results in a non-zero initial eccentricity in the transverse plane that gives rise to the collective elliptic flow. For collisions of symmetric nuclei and neglecting the fluctuations, the odd components of the decomposition in Fourier coefficients vanish at space-time rapidity zero. At forward and backward rapidities an imbalance between the contributions from the target and the projectile to the initial source can result in a left-right deformation of the source in the transverse plane and/or in an asymmetric initial flow, that could generate collective directed flow.
The observation of non-zero directed flow implies, that the symmetry in the reaction plane is indeed broken, either in the initial flow or in the initial density, or both. We tried to reproduce the observed directed flow assuming an asymmetric initial flow, different from the Bjorken one (2), but without success. Therefore in the following we assume a Bjorken initial flow and study the effect of asymmetric initial densities on the directed flow.
Starting with a factorized initial energy density in the transverse plane and in the space-time rapidity
(3) |
the subsequent evolution remains symmetric with respect to the axis in the reaction plane (-), with the consequence that the directed flow is exactly zero. Factorized initial conditions in the Glauber Model imply that all the participant nucleons or binary collisions contribute in a similar way to the total density, with a longitudinal profile proportional to .
Modifications of the symmetric distribution (3) could happen during the formation of the initial thermalized state. Momentum imbalance between left and right going participants at a given point in the transverse plane results in a nonzero total momentum of the matter. The longitudinal distribution is shifted in space-time rapidity Hirano:2001yi ; Hirano:2002ds by the value of the center of mass rapidity of the fluid
(4) |
where and are the densities of participants from the two nuclei and is the velocity of the incident nuclei.
(5) |
where is the cross section,
(6) |
is the thickness function calculated from the Woods-Saxon density of colliding nuclei
(7) |
We use the same parameters as in Ref. Bozek:2009ty , where a satisfactory description of spectra and femtoscopy data for Au-Au collisions at GeV has been obtained. The initial energy density takes the form Hirano:2001yi ; Hirano:2002ds
(8) | |||||
The relative contribution from binary collisions is , with . In the following we call these initial conditions shifted initial conditions (Fig. 1).
The form of the initial energy density profile is adjusted to reproduce the measured charged particle distribution in pseudorapidity. The resulting width of the initial distribution depends on the chosen equation of state, initial time and shear viscosity Satarov:2006iw ; *Bozek:2007qt; Bozek:2009ty . In the following we use ideal fluid hydrodynamics with fm/c and a realistic, hard equation of state Chojnacki:2007jc , which requires
(9) |
with a plateau of width and .
A different type of initial conditions studied in this work assumes a preferred emission from participating nucleons in the same hemisphere. Instead of a symmetric distribution of matter in space-time rapidity given by the function in Eq. (3), we assume that the deposited energy depends on the rapidity of the emitting participating nucleon. Such a distribution depending on the rapidity difference between the emitting charge and the emitted gluon is assumed in some phenomenological models Bialas:2004kt ; *Fialkowski:2004wh; *Brodsky:1977de; *Adil:2005qn; *Adil:2005bb. However, there is no direct measurement of the contribution to soft particle production from a single forward or backward moving charge. A phenomenological analysis is possible, by comparing multiplicity distributions in pseudorapidity for different asymmetric systems or by studying multiplicity correlations in different pseudorapidity intervals. These studies indicate that a preferred emission for rapidities close to the rapidity of the participating charge occurs Bialas:2004su ; Bzdak:2009xq ; Gazdzicki:2005rr ; Bzdak:2009dr . In the wounded nucleon model of nuclear collisions, such correlations can be understood as due to a specific distribution of soft particles produced by each participant nucleon. Nucleons from the projectile (with positive rapidity ) emit more particles in the forward () than in the backward hemisphere. The form of the extracted charged particle distribution can be approximated by the function
(10) |
in the interval , where defines the range of rapidity correlations, at GeV it is . The origin of the shift in rapidity is not understood Bialas:2004su ; Bzdak:2009xq . For practical purposes we can treat it as a phenomenological parameter. Particle production in the remaining pseudorapidity intervals close to the fragmentation regions and cannot be reliably described in a hydrodynamic model anyway. Within the framework of relativistic hydrodynamics, we are interested in describing the main characteristics of the soft part of particle spectra in the central region and, in particular, the directed flow. There is another reason why the phenomenological estimates of the emission of particles from participant nucleons Bialas:2004su ; Bzdak:2009xq cannot be directly translated into the initial conditions for hydrodynamics, that we are interested in. Refs. Bialas:2004su ; Bzdak:2009xq study particle distributions and correlations in the final state, whereas, we know that in realistic hydrodynamic simulations the matter distribution in space-time rapidity evolves during the expansion of the fireball Satarov:2006iw ; Bozek:2009ty , also statistical emission broadens the distribution in pseudorapidity. It means that the initial profile is significantly narrower than the final charged particle distribution . The correlation functions in pseudorapidity Bzdak:2009xq can be modified due to the longitudinal transport and the generation of transverse flow as well. We propose as a phenomenological ansatz, inspired by the observations in Refs. Bialas:2004su ; Bzdak:2009xq ; Gazdzicki:2005rr ; Bzdak:2009dr , that the initial energy density of matter produced by a single participant nucleon of rapidity is proportional to
(11) |
where is the initial longitudinal profile (9) fitted to reproduce and
(12) |
The initial energy density of the fireball is constructed as a sum of three terms originating from the forward or backward moving participant nucleons and from the binary collisions that are assumed to contribute in a symmetric way
(13) | |||||
The net result of the difference between forward and backward emission is a tilt of the source in the - plane (Fig. 3). This breaks the symmetry in the longitudinal direction and generates nonzero directed flow in the expansion.
Hydrodynamic equations in D
(14) |
constitute four independent equations, which, together with the equation of state, determine the evolution of the energy density , of the pressure , and of three independent components on the fluid velocity. The fluid four velocity can be written in the form
(15) |
and are the components of the transverse velocity, and
(16) |
In D hydrodynamic evolution, the lack of Bjorken invariance results in a nonzero longitudinal acceleration. The fluid rapidity becomes larger than the space-time rapidity . In Figs. 1 and 3 are shown the vector fields of the initial pressure gradients (). For the shifted initial conditions the gradient in the central region of the fireball is in the transverse direction, and mainly transverse flow is generated in the early stage. This is due to the existence of an approximate Bjorken plateau for central rapidities in the initial stage. The situation is different for tilted initial conditions (Fig. 3); the acceleration in the tilted source is anti-correlated in the transverse and longitudinal directions. The matter that is accelerated to positive rapidities is preferably accelerated in the negative direction. In the same Figures are shown the lines of constant pressure for the initial fireball created in Cu-Cu collisions for the same centrality. The deformation for the shifted fireball (Fig. 1), or the tilt for the tilted fireball (Fig. 3), in the smaller system is very similar as in the larger system. This generates a similar directed flow in the two systems irrespective of their sizes.
Iii Results
The hydrodynamic equations are solved numerically for the two sets of initial conditions, the shifted initial distributions (8) and the tilted initial conditions (13). The parameter is chosen to reproduce particle spectra and multiplicities in central collisions. We use GeV/fm and GeV/fm for Au-Au and Cu-Cu collision respectively, and a freeze-out temperature of MeV. This gives a satisfactory description of the spectra in collisions up to centralities of Bozek:2009ty .
The distribution of charged particles in pseudorapidity is shown in Fig. 4. The results obtained from the two initial conditions are almost indistinguishable on the plot. Both initial conditions lead to similar results for transverse momentum spectra of particles, interferometry radii, and elliptic flow, as well. The comparison with experimental data can be found in Ref. Bozek:2009ty , giving satisfactory results. The calculated elliptic flow overshoots the experimental data since we do not take viscosity effects into account Bozek:2009mz . The parameters of the initial profile in space-time rapidity are adjusted to reproduce, as closely as possible, the experimental results on pseudorapidity distributions, and transverse momentum spectra at non-zero rapidities Bozek:2009ty . Changing the width of the plateau in the initial profile , within a range compatible with the observed pseudorapidity distributions, does not change the results for the directed flow. Only taking the unrealistic value for shifted initial conditions, causes the wiggle of positive disappear at central pseudorapidities.
By breaking the symmetry in the longitudinal direction, some directed flow can be generated. In Figs. 5 and 6 is shown the development of the asymmetric flow in the direction at different times. The average velocity in the direction is calculated for a given space-time rapidity and time
(17) |
As mentioned above, during the evolution the Bjorken flow is modified, but the final flow conserves a strong correlation between and Bozek:2009ty . Therefore the directed flow as functions of space-time rapidity as shown in Figs. 5 and 6 reflects qualitatively the final flow of particles.
The first observation is that the directed flow is built up for an extended time during the expansion, in the first fm/c. There is a noticeable difference between the evolution from the shifted and the tilted initial conditions. For the shifted initial conditions (Fig. 5) in the central space-time rapidity region a positive directed flow develops gradually in the first fm/c. For large rapidities a negative flow is generated in the first fm/c. The directed flow changes sign around , a behavior very different from the one observed in the experiment. The appearance of the wiggle in the dependence of on pseudorapidity is a consequence of the form of the initial profile of the pressure (Fig. 1). The gradient of the pressure has a significant deflection from the transverse direction only in the forward/backward rapidity regions. On the other hand, the tilted initial condition has a smooth tilted gradient of the pressure that gives a negative directed flow in a broad range of space-time rapidities (Fig. 6). The anti-flow increase for the first fm/c. The negative directed flow of the fluid, increasing with rapidity, leads to a similar pattern in the directed flow of the final particles.
The hydrodynamic evolution is continued until freeze-out, where particle emission from the freeze-out hypersurface takes place. Statistical emission and resonance decays are performed using the event generator THERMINATOR Kisiel:2005hn . In Figs. 7 and 8 are shown the results for three representative centralities, for which experimental data for Au-Au collisions have been published. Other experimental data show that in Cu-Cu interactions almost the same directed flow is generated as in the larger system if the centrality is chosen to be the same Abelev:2008jga . The directed flow for the shifted initial conditions shown in Fig. 7 has an incorrect dependence on pseudorapidity. For all the three centrality classes the flow is positive in the central rapidity region, and switches to anti-flow in the very forward and backward regions. On the same plots are shown the results for Cu-Cu collisions in similar centrality classes. The flow is similar as for the larger system, and hence different than observed experimentally. The similarity between the flow in Au-Au and Cu-Cu systems reflects the similarity in the initial density profiles (Fig. 1).
In Fig. 8 are shown results for the directed flow of charged particles emitted after a hydrodynamic evolution from the tilted initial fireball (Eq. 13). The thick solid lines represent the results for Au-Au collisions. The experimental data are reproduced in the central rapidity region in a satisfactory way. For the semi-peripheral and peripheral collisions the large anti-flow at large pseudorapidities is not reproduced by the model. This kinematic region is at the limit of applicability of the hydrodynamic model, assuming the collective expansion of a thermalized fluid. The directed flow in the fragmentation region can have a different origin, like in the transport models Bleicher:2000sx . On the panels in Fig. 8 are shown the results for the Cu-Cu system in the same centrality classes. Again, as expected from the similarity of the tilt of the source in the two systems (Fig. 3), the final directed flows for the two systems almost overlap. To test some of the uncertainty in the choice of the initial conditions for the hydrodynamic evolution, we take as the initial density an ansatz where also the contribution from binary collisions is asymmetric
(18) | |||||
The tilt is stronger in this case and the magnitude of the directed flow bigger (thin solid lines in Fig. 8). The shaded bands in the figure represent the uncertainty of the model related to this assumption. This uncertainty is the largest among those that we tested.
In the following we study the effect of other details of the model on the final directed flow. First, a different initial time for the hydrodynamic evolution is chosen fm/c. The initial time is given by the rate of the initial thermalization processes, that lead to the formation of a dense, almost perfect fluid. The precise mechanisms of thermalization are not known and it is instructive to test the influence of the value of the thermalization time on the build up of the directed flow. In Fig. 9 is shown the result for the centrality class %. The directed flow is reduced when compared to the previous calculation using fm/c. The fluid that starts to expand at fm/c has a smaller pressure. In fact, the experimental points for central rapidities lie between the two lines representing the two calculations starting at different thermalization times.
Another effect worth to be studied is the dependence on the details of the form-factor , describing the emission from a single participant nucleon. The range of forward-backward correlations is set by the parameter . We make another calculation using . With this choice, the emission from a forward going participant nucleon decreases when going from rapidity to zero at the rapidity . The initial tilt of the source is smaller than for the choice , used before. The effect of this change on the directed flow is small and the results are close to the experimental data at central rapidities (dotted line in Fig. 9).
A third effect, that can be tested, is the assumption that the initial energy density is proportional to the density of wounded nucleons (binary collisions) as in Eq. (13). Assuming that this proportionality applies to the entropy density instead, we have
(19) | |||||
The directed flow is very similar to the one obtained using the energy density initial profile (13) (dashed-dotted line in Fig. 9). We notice that some of the details of the initial conditions can influence the final directed flow. This situation resembles the conclusion from the elliptic flow studies, where a crucial ingredient is the initial eccentricity of the fireball.
Iv Summary
We study the formation of the directed flow in the D hydrodynamic expansion of the fireball created in heavy-ion collisions at the highest energies at RHIC. The directed flow of charged particles has been measured as function of pseudorapidity, finding a substantial negative flow Back:2005pc ; Adams:2005ca ; Abelev:2008jga . We use two different initial conditions for the evolution. The first one (Eq. 8) is quite commonly used in hydrodynamic model calculations Hirano:2002ds . It incorporates a shift of the densities in the initial fireball, due to the local imbalance of the longitudinal momentum. Initial densities of the second type are constructed as a sum of contributions from forward and backward going participants (Eq. 13). The asymmetry in the emission from individual participants leads to a tilt of the source. Model calculations, incorporating a hydrodynamic expansion stage, particle emission at freeze-out, and resonance decays, indicate that the second type of initial conditions can reproduce the sign and the magnitude of the observed directed flow at central rapidities.
The large negative flow close to the fragmentation regions is of different origin and cannot be described in our calculation. Deviations from the Bjorken flow in the initial conditions Snellings:1999bt ; *Becattini:2007sr could influence the results. In particular, they could lead to a different flow pattern for baryons and for the bulk of the matter, if baryons do not follow the Bjorken flow. The directed flow is expected to decrease with the collision energy for two reasons. With increasing kinematic range of rapidities the tilt of the source goes down, also the contribution of binary collisions to the Glauber Model density is believed to increase with the energy, with a similar consequence. Finally let us note, that our calculation reproduces the experimentally observed similarity of the flow in Au-Au and Cu-Cu collisions at the same centrality.
Acknowledgements.
The authors are very grateful to Jean-Yves Ollitrault for correspondence and discussions that have been of vital importance in finding errors in early studies and in reaching the correct conclusions. The work is supported by Polish Ministry of Science and Higher Education under grants N202 034 32/0918 and N N202 263438.References
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