Torqued fireballs in relativistic heavy-ion collisions
aa r X i v : . [ nu c l - t h ] M a r Torqued fireballs in relativistic heavy-ion collisions
Piotr Bo˙zek,
1, 2, ∗ Wojciech Broniowski, † and Jo˜ao Moreira ‡ The H. Niewodnicza´nski Institute of Nuclear Physics,Polish Academy of Sciences, PL-31342 Krak´ow, Poland Institute of Physics, Rzesz´ow University, PL-35959 Rzesz ow, Poland Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland Centro de F´ısica Computacional, Department of Physics,University of Coimbra, 3004-516 Coimbra, Portugal (Dated: 15 November 2010)We show that the fluctuations in the wounded-nucleon model of the initial stage of relativisticheavy-ion collisions, together with the natural assumption that the forward (backward) movingwounded nucleons emit particles preferably in the forward (backward) direction, lead to an event-by-event torqued fireball. The principal axes associated with the transverse shape are rotatedin the forward region in the opposite direction than in the backward region. On the average,the standard deviation of the relative torque angle between the forward and backward rapidityregions is about 20 ◦ for the central and 10 ◦ for the mid-peripheral collisions. The hydrodynamicexpansion of a torqued fireball leads to a torqued collective flow, yielding, in turn, torqued principalaxes of the transverse-momentum distributions at different rapidities. We propose experimentalmeasures, based on cumulants involving particles in different rapidity regions, which should allowfor a quantitative determination of the effect from the data. To estimate the non-flow contributionsfrom resonance decays we run Monte Carlo simulations with THERMINATOR . If the event-by-eventtorque effect is found in the data, it will support the assumptions concerning the fluctuations in theearly stage of the fireball formation, as well as the hypothesis of the asymmetric rapidity shape ofthe emission functions of the moving sources in the nucleus-nucleus collisions.
PACS numbers: 25.75.-q, 25.75Gz, 25.75.LdKeywords: relativistic heavy-ion collisions, RHIC, LHC, forward-backward correlations, wounded nucleons
I. INTRODUCTION
It is believed that the forward-backward (FB) rapid-ity correlations may reveal important information onthe mechanism of particle production in high-energyhadronic and nuclear collisions. Long-range rapidity cor-relations uncover properties of the dynamical system ata very early stage. For that reason the FB multiplic-ity correlations have been studied experimentally [1, 2]and theoretically [3–8]. In this paper we show that thewounded-nucleon approach [9] to the initial stage of thecollisions leads to a new manifestation of the FB fluc-tuations: the torqued fireball . Specifically, parts of thefireball are rotated in the transverse plane in one direc-tion in the forward rapidity region, and in the oppositedirection in the backward rapidity region. The magni-tude and sign of the torque angle fluctuate from event toevent.The following ingredients are responsible for the ap-pearance of the torque effect:1. statistical fluctuations of the transverse density ofthe sources (wounded nucleons) [10], and2. the asymmetric shape [4–8] of the particle emissionfunction, peaked in the forward (backward) rapid- ∗ [email protected] † [email protected] ‡ [email protected] ity for the forward (backward) moving wounded nu-cleons.On the average, for the studied Au+Au collisions atthe highest energies available at the BNL RelativisticHeavy Ion Collider (RHIC), the relative torque angle be-tween the principal axes in the forward and backwardrapidity regions is about 20 ◦ for the central, and 10 ◦ forthe mid-peripheral collisions. On the other hand, an ex-perimental observation of nonzero torque angles of theexpanding fireball could shed light on the mechanism ofthe formation of dense matter. A finite torque betweenreaction planes at different rapidities could influence el-liptic and directed flow studies using reaction planes de-termined in different pseudorapidity intervals [11].The paper has three parts, referring subsequentlyto the early wounded-nucleon phase of the colli-sion (Sect. II), the intermediate hydrodynamic stage(Sect. III), and, finally, the statistical hadronizationphase (Sect. IV), where hadrons are produced. InSect. II A we introduce the necessary elements of thewounded-nucleon approach, in particular, the asymmet-ric rapidity-dependent emission functions [4–8] of theforward and backward moving wounded nucleons. InSect. II B we explain how statistical fluctuations of thedensity of the forward and backward moving woundednucleons in the transverse plane lead to the event-by-event torque effect. Simulations are carried out with GLISSANDO [12] in the so-called mixed model [13, 14],incorporating an admixture of binary collisions into thewounded-nucleon model. We introduce quantitativemeasures of the torque in Sec. II C. Specifically, we usethe difference of the forward and backward angles of theprincipal axes, as well as three angles, forward, back-ward, and central, to construct useful statistical quanti-ties. Next, in Sect. III we present the results of runningthe 3+1 dimensional (perfect fluid) hydrodynamics, asdescribed in [15]. The calculations show that the torquesurvives the hydrodynamic stage of the evolution. Then,in Sect. IV A we pass to discussing the final stage, namely,the statistical hadronization (for a review, see, e.g. [16]).This stage would wash out the torque effect, unless care-ful experimental measures are used. This is because evenat a fixed spatial geometry of the fireball the finitenessof the number of produced particles causes large fluctu-ations of the event-plane angle, covering up the torqueangle. We thus propose to investigate measures based oncumulants [17], introduced in Sect. IV B. They are con-structed in such a way that the relative FB torque anglecan be extracted. We examine the non-flow contribu-tions to these measures via simulations in
THERMINATOR [18], proving that it is possible to find the torque effectwith the proposed methods in the large-statistics RHICdata. In Sect. IV D we repeat this analysis for the par-ticles taken from three rapidity bins, forward, backward,and central, with similar conclusions.
II. TORQUED FIREBALL
In this Section we describe the earliest stage of therelativistic heavy-ion collision in terms of the wounded-nucleon model [9]. All simulations are carried out for theAu+Au collisions at the highest RHIC energy of √ s NN =200 GeV. A. Wounded nucleons with rapidity profiles
The wounded-nucleon [9] approach is commonly usedto describe the early stage of the heavy-ion collisions. Inthe collision process N w nucleons get wounded, i.e. inter-act inelastically at least once, as well as N bin binary colli-sions occur. Roughly speaking, wounded nucleon (partic-ipants) are responsible for the soft emission, while binarycollisions describe hard processes. Both serve as sources for the formation of the density of energy or entropy inthe initial fireball. At RHIC, the mixed model [13], wherethe total number of the produced particles is given by thecombination N prod = A (cid:18) − α N w + αN bin (cid:19) , (1)describes quite successfully the multiplicity data [14].The proportionality constant A depends on the energyof the collision but not on its centrality. Throughout thiswork we use α = 0 . √ s NN = 200 GeV [14]. ff + f - - - Η ° FIG. 1. (Color online) The emission profiles in space-time ra-pidity for the wounded nucleons (dashed lines) and the binarycollisions (solid line). The profile f + ( f − ) corresponds to theforward (backward) moving wounded nucleons. The concept of sources can be extended to account forthe rapidity dependence of the produced particles. Thepseudorapidity distribution is given as a sum of contri-butions from the emission of forward and backward go-ing wounded nucleons. Within such a scheme Bia las andCzy˙z successfully described [4] the distribution of chargedparticles in pseudorapidity in the deuteron-gold collisions[19]. The independent emission from the forward andbackward going nucleons determines specific FB multi-plicity correlations [6, 7]. In particular, this hypothesishas been tested successfully in Ref. [7] with the FB mul-tiplicity correlation data from the PHOBOS collabora-tion [2]. Based on this idea, Ref. [8] postulated thatin nucleus-nucleus collisions the emission profile definingthe initial density (in the space-time rapidity η k and thetransverse plane coordinates x , y ) has the form F ( η k , x, y ) = (1 − α )[ ρ + ( x, y ) f + ( η k ) + ρ − ( x, y ) f − ( η k )]+ αρ bin ( x, y ) f ( η k ) , (2)where ρ ± ( x, y ) is the density of the forward and back-ward going wounded nucleons at a given point in thetransverse plane, ρ bin ( x, y ) is the binary collisions den-sity, while f + ( η k ) and f − ( η k ) describe the correspondingwounded-nucleon emission profiles. Finally, f ( η k ) is theemission profile for the binary collisions. These func-tions are chosen appropriately, taking into account thefollowing features: the profile f + ( f − ) is peaked in theforward (backward) direction, i.e. the wounded nucleonemits mostly in its own forward hemisphere, with a quitebroad distribution, whereas the binary profile f is (forthe collision of identical nuclei) symmetric.We remark that the asymmetric wounded-nucleonemission functions in Eq. (2) give a tilt away from thebeam axis for the initial fireball (in the ( x, η k ) plane).The hydrodynamic expansion [8] of such a tilted fireballgenerates the directed flow and for the first time the ex-perimental observations at √ s NN = 200 GeV [20] couldbe reproduced. B C F FB
FIG. 2. A cartoon visualization of the torque effect. A ran-dom cluster of wounded nucleons, drawn here at the edge ofthe ellipse, with three nucleons moving in the forward (F)direction (open circles) one one moving in the backward (B)direction (filled circle), causes a random torque of the prin-cipal axes. The angle of the torque is higher in the forwarddirection than in the backward direction. In the the centralrapidity region (C) the effect is between the F and B cases.
Following Ref. [8], we choose the following parameter-izations: f ( η k ) = exp (cid:18) − ( | η k | − η ) σ η θ ( | η k | − η ) (cid:19) ,f + ( η k ) = f F ( η k ) f ( η k ) ,f − ( η k ) = f F ( − η k ) f ( η k ) , (3)with f F ( η k ) = , η k ≤ − η mη k + η m η m , − η m < η k < η m , η m ≤ η k (4)The values of parameters, describing the RHIC data afterthe hydrodynamic evolution [8], are η = 1 ,η m = 3 . ,σ η = 1 . . (5)The profile functions are shown in Fig. 1. We note that byconstruction f + ( η k ) + f − ( η k ) = f ( η k ). Parametrization(4) is chosen in such a way, that after the hydrodynamicevolution and statistical emission [8, 15] one correctlyreproduces the spectra [21] of particles produced at dif-ferent rapidities in the Au+Au collisions at the highestRHIC energy. B. Fluctuations and the torque
The initial density of the fireball (2) can be obtainedin a Glauber Monte Carlo approach. The densities of theforward and backward going wounded nucleons ρ ± ( x, y )and of the binary collisions ρ bin ( x, y ) are obtained with FIG. 3. (Color online) The schematic figure of the torquedfireball, elongated along the η k axis. The direction of theprincipal axes in the transverse plane rotates as η k increases.The left and right pictures correspond to the rank-2 (elliptic)and rank-3 (triangular) cases, respectively. The effect occursevent-by-event. Monte Carlo simulations by
GLISSANDO [12]. These dis-tributions fluctuate on event-by-event basis. The phe-nomenon has a purely statistical origin, as the positionson nucleons in nuclei fluctuate. The event-by-event fluc-tuations in the wounded-nucleon approach are know tocause important effects, such as the increase of the ellip-tic deformation, resulting in larger elliptic flow [10], orthe recently discussed triangular flow [22].Another effect due to fluctuations, focal to this work,is the event-by-event torque of the fireball. Its appear-ance is simple to understand. For the sake of simplicitylet us consider the situation depicted in Fig. 2. A clus-ter of wounded nucleons is formed, here drawn at theedge of the fireball. It contains three wounded nucleonsmoving forward and one moving backward. The clustercauses the twist of the principal axis. However, due tothe shape of the emission functions of Fig. 1, the shift isdifferent at various values of the space-time rapidity. Atforward η k ∼ η k region all four nucleons contribute,but according to Eq. (4) their relative weight is reducedbe a factor of 2. Thus the relative weight of the effect ofthe cluster in the forward, backward, and central regionsis as 3 : 1 : 2. The result is the schematic arrangement ofthe principal axes as drawn in Fig. 2.In an actual Monte Carlo simulation many clusters oc-cur and the situation is more complicated, but the originis as described above. The effect appears on the event-by-event basis and by symmetry the mean value of thetorque angle vanishes upon averaging over events. Thus,the effect may only be revealed in event-by-event studiesof fluctuations, see Sect. II C.A similar phenomenon occurs for the axes of the trian-gular shape and the axes corresponding to higher Fouriermoments in the azimuthal angle. The shape of the fire-ball is depicted schematically in Fig. 3, where the torqueangle, somewhat exaggerated, is shown for the ellipticand triangular deformations. As mentioned above, thetorque appears on the event-by-event basis, varying inthe direction and value. C. Characteristics of the torqued the fireball
In this subsection we provide quantitative studies ofthe torque effect at the instant of formation. In a givenevent, the angle of the principal axis for the Fourier mo-ment of rank- k for n sources at some η k is given by theformula Ψ ( k ) = 1 k arctan (cid:18) P ni =1 w i r i sin( kφ i ) P ni =1 w i r i cos( kφ i ) (cid:19) , (6)where ( r i , φ i ) are the polar coordinates of the source po-sition with respect to the center of mass of the slice ofthe fireball at some fixed η k , and w i is the weight ofthe source. Explicitly, for the forward-moving woundednucleons w i = (1 − α ) f + ( η k ), for the backward ones w i = (1 − α ) f − ( η k ), while for the binary collisions w i = αf ( η k ). All spatial angles are measured relative tothe axes perpendicular to the reaction plane. The inter-pretation of the angle Ψ ( k ) is that the azimuthal Fouriermoment of rank k is highest along that direction. Theangle is defined modulo 2 π/k . For k = 2 the angle Ψ (2) isthe angle of the principal axes of the moment of inertia.For brevity of notation, we shall skip the superscript (2)from the rank-2 quantities, while retaining superscriptsindicating higher ranks.The simplest measure of the torque effect is the differ-ence of the Ψ ( k ) angles at forward and backward valuesof η k , ∆ ( k ) F B ( η k ) = Ψ ( k ) ( η k ) − Ψ ( k ) ( − η k ) . (7)As argued above, this quantity fluctuates event-by-event.The result of the GLISSANDO simulations is shown inFig. 4. We plot the event-by-event distribution of ∆
F B for the 20-30% centrality class and η k = 0 . η k = 2 . η k is expected. The rms width of the distributionsof Fig. 4 for c = 20 −
30% are 2 ◦ for η k = 0 . ◦ for η k = 2 .
5. An analogous study for c = 0 − ◦ (notshown) yields 4 ◦ for η k = 0 . ◦ for η k = 2 .
5. Thusthe spread enhances with increasing η k , moreover, thedistributions are wider for the central collisions.For the rank-3 analysis (triangular flow) we have qual-itatively the same effect. For c = 20 −
30% the rms width One could also overlay a statistical distribution of weights, aswas done in Ref. [12]. [deg] FB D -40 -30 -20 -10 0 10 20 30 40 F B D d d N FIG. 4. Distribution of the difference of the forward andbackward torque angles, Ψ F − Ψ B , for the elliptic deformation.The narrower and wider distributions correspond to the space-time rapidities η k = 0 . .
5, respectively. Centrality 20 − α = 0 . Σ H D F B L Η þ c = - % c = - % c = - % c = - % c = - % c = - % c = - % FIG. 5. The dependence of the rms width of the ∆
F B distri-bution on η k at various centralities c . are 4 ◦ for η k = 0 . ◦ for η k = 2 .
5, which are largerthan for the rank-2 case at the same centrality.In Figs. 5 and 6 we show the dependence of the rmswidth of the distributions of ∆
F B and ∆ (3)
F B as functionsof η k at various centralities. For the rank-2 case we notethat the widths grow gradually from 0 at η k = 0 to 10 − ◦ at η k = 4, with the largest angle for the most centralevents. The rank-3 widths are somewhat larger than inthe rank-2 case, except for the most central case. Wenote that at fixed η k the dependence on centrality is non-monotonic, with the lowest value for c = 20 −
30% for therank-2 case and 40 −
50% for the rank-3 case. We alsonote that for the rank-2 case the most central collisionslead to significantly larger torque fluctuations than forthe less central collisions, cf. Fig. 5. Σ H D H L F B L Η þ c = - % c = - % c = - % c = - % c = - % c = - % c = - % FIG. 6. Same as Fig. 5 for the rank-3 case (triangular flow).
One may introduce other measures, involving three an-gles associated with the forward, backward, and central η k regions. This might be advantageous, as the centralrapidity region is experimentally better covered experi-mentally. We define the two relative angles∆ ( k ) F C = Ψ ( k ) ( η k ) − Ψ ( k ) (0) , ∆ ( k ) BC = Ψ ( k ) ( − η k ) − Ψ ( k ) (0) , (8)and their covariance,cov ( k ) F BC = h ∆ ( k ) F C ∆ ( k ) BC i events . (9)The forward-backward-central correlation coefficient isdefined as ρ ( k ) F BC ≡ cov ( k ) F BC / ( σ (∆ ( k ) F C ) σ (∆ ( k ) BC )) . (10)In Fig. 7 we present the 2-dimensional distribution plotof ∆ F C and ∆ BC for the rank-2 case for η k = ± . Figures 8 and 9 show the rapidity dependence ofcov ( k ) F BC for the rank-2 and rank-3 cases. We note thatthese measures drop monotonically with η k from zero tonegative values in the range − .
005 to − .
02, reaching aplateau near η k ≈ .
5. For a fixed η k the dependence onthe centrality class is non-monotonic. From the data ofFigs. 8 and 5 one may obtain the correlation of Eq. (10).This quantity grows from the value − η k = 0 up toabout − . η k = 4, similarly for the rank-2 and rank-3cases. We note that the points in Fig. 7 occupy all quadrants, i.e. thereare cases where both shifts ∆
F C and ∆ BC have the same sign.This is because for each rapidity we evaluate independently thecenter of mass, which is the origin for the transverse coordinatesystem. If we evaluated the angle in the common system as-sociated with the central rapidity, only the second and fourthquadrants in Fig. 7 would be filled. [deg] FC D -10 -8 -6 -4 -2 0 2 4 6 8 10 [ d e g ] BC D -10-8-6-4-20246810 FIG. 7. (Color online) The 2-dimensional distribution plotof the relative torque angles ∆ FC and ∆ BC , for centrality50 − η k = 2 .
5. The correspondingcorrelation coefficient is ρ F CB = − . - - - - C ov F BC Η þ c = - % c = - % c = - % c = - % c = - % c = - % c = - % FIG. 8. Covariance of the ∆
F C and ∆ BC angles plotted as afunction of the space-time rapidity η k for various centralities. III. HYDRODYNAMICS
The intermediate evolution of the dense system formedin relativistic heavy-ion collisions is believed to be gov-erned by hydrodynamics [15, 23, 24]. The hydrodynamicexpansion of the fireball generates a collective velocityfield of the hot fluid. The direction of the acceleration ofthe fluid element is given by pressure gradients. This waythe azimuthal eccentricity of the fireball is transformedinto the elliptic flow [25], the triangular deformation intothe triangular flow [22], and the source tilt into the di-rected flow of the final hadrons [8]. A similar mechanismgenerates, on event-by-event basis, a torqued transverse - - - - C ov F BC H L Η þ c = - % c = - % c = - % c = - % c = - % c = - % c = - % FIG. 9. Same as in Fig. 8 for the rank-3 angles (triangularflow). velocity field at different rapidities.The discussion of the torque of the fireball in the pre-ceding sections concerned the earliest stage of the col-lision, described within the wounded-nucleon approach.That stage, essentially, prepares the initial condition forthe subsequent phases of the evolution. It is necessaryto check that the signatures of the torque fluctuationssurvive these later stages, such that measurable effectscan be detected in experiment. Here we investigate thebehavior of the rank-2 torque under the hydrodynamicevolution, as the rank-3 case is expected to behave sim-ilarly. A rotation of the density in the transverse planein the torqued fireball scenario, would generate a rotatedfluid velocity field. This rotation of the transverse ve-locity field at each space-time rapidity would lead to arotation of the reconstructed reaction plane for particlesemitted in the corresponding rapidity range.We apply the 3+1 hydrodynamic evolution of the per-fect fluid with a realistic equation of state, implementedfor the first time in [26]. This approach is capable of uni-formly describing the main experimental features, suchas the transverse momentum spectra, v , as well as theHanbury Brown–Twiss correlation radii [24]. Extensionto 3+1 dimensions allows to describe also the spectra atnon-central rapidities and the directed flow v [8, 15].To demonstrate the effects of the torqued fireball onfinal particle spectra, we generate the hydrodynamic evo-lution for the sample value of the impact parameter, b = 6 . c = 20-25% [27]. The initial energy density ǫ is taken in the formgiven in Eq. (2), with the wounded nucleon and and bi-nary collision densities calculated in the Glauber model.The details and the parameters of the distribution can befound in [8, 15]. The initial proper time for the evolutionis τ = 0 .
25 fm/c. To study the torque effect, instead ofdoing tedious event-by-event hydrodynamic simulationswith a whole distribution of torque angles as in Fig. 4,we perform one simulation where the densities of the for-ward and backward going wounded nucleons are rotatedin opposite directions by a fixed angle of 5 ◦ which is avalue corresponding to the rms width of the distribution Τ + (cid:144) c3fm (cid:144) c6fm (cid:144) c - - - - Η ´ Y - Y C @ deg D FIG. 10. (Color online) The dependence of the torque an-gle of the fluid velocity field on space-time rapidity after the3+1-dimensional hydrodynamics of Ref. [8] (solid, dotted anddashed lines). Subsequent curves are for different evolutiontimes. of ∆
F B . Thus the initial energy density (2), which is thestarting point of our hydrodynamics, becomes ǫ ( η k , x, y ) = (1 − α )[ ρ + ( Rx, Ry ) f + ( η k )+ ρ − ( R T x, R T y ) f − ( η k )] + αρ bin ( x, y ) f ( η k ) . (11)The operator R rotates the density of forward goingwounded nucleons by the fixed angle, while operator R T rotates the backward going wounded nucleons in the op-posite direction. The density of binary collisions is notrotated, as its emission component is symmetric in η k .During the hydrodynamic evolution the direction ofthe transverse flow is determined by the orientation ofthe fireball density at a given space-time rapidity. Thetorque angle of the fluid, Ψ( η k ), is determined by therequirement that in the frame defined by the principalaxes of the transverse flow we have h T xy i ( η k ) = 0, where h T µν i ( η k ) = R dxdy T µν ( η k , x, y ) are the components ofthe energy-momentum tensor averaged over the trans-verse plane. In Fig. 10 we show the evolution of thetorque angle of the fluid velocity field in the hydrody-namic calculation. We present the angle as a function ofthe space-time rapidity after a hydrodynamic evolutionlasting 1, 3 or 6 fm/c. Due to the longitudinal push,the effect decreases somewhat as the evolution time in-creases, but this quenching is very small. Therefore thetorque effect survives the hydrodynamic phase with analmost unchanged magnitude. A sizable twist of the col-lective velocity field at the freeze-out should give a twistof the distribution of the emitted particles.We note that the space-time rapidity η k is not equal tothe fluid rapidity, as in a boost-invariant model. More-over, the transverse momenta and rapidities of finalhadrons include a thermal component besides the col-lective velocity. These effects are addressed in the nextsection, through the use of a realistic model of statisticalhadronization. - - -
50 0 50 100 1500.00.20.40.60.8 k Θ @ deg D f H k Θ L FIG. 11. (Color online) The event-by-event distribution of kθ for v k = 5% for several values of the event multiplicity n : 600(solid), 100 (dashed), and 20 (dotted). IV. HADRONIZATION AND EXPERIMENTALMEASURES
The final question, of key practical importance for thewhole idea, is how to observe the torque of the fire-ball from the data consisting of momenta of detectedparticles. In our model approach we adopt the statis-tical hadronization picture [28], where hadrons (stableand resonances [29], which subsequently decay) are pro-duced at freeze-out according to the Frye-Cooper for-malism [30], with a freeze-out temperature of 150 MeV[8, 15]. The difficulty lies in the fact that the finitenessof the number of particles produced in each event causessizable fluctuations of the principal axes (or the eventplane) as determined from the transverse momenta. Wedenote the relative angle between the event-plane axisand the fireball spatial principal axis Ψ as θ . Despitethis difficulty, as we shall see, one can propose measuresof even-by-event torque fluctuations that should be pos-sible to be observed in the RHIC data.Throughout this section η denotes the momentumpseudorapidity of produced particles, in distinction of thespace-time rapidity η k of the preceding parts. A. Fluctuations of principal axes from statisticalhadronization
To appreciate the phenomenon of the fluctuation of θ ,let us recall formulas from [10] concerning the fluctua-tion of the eccentricity and the principal axes due to thefinite number of (independent) particles. The angle θ (dependent on the rank k ) is defined in each event astan( kθ ) = P ni =1 sin( kφ i ) P ni =1 cos( kφ i ) , (12)where all angles are measured with respect to the angleof the principal axis of the fireball, Ψ ( k ) , at a selected η window. From Appendix C of Ref. [10], which derives theresults based on the central limit theorem, we find that(at a fixed multiplicity n ) the event-by-event distributionof θ is given by f ( kθ ) = e − nv k /w πs n √ πnv k cos( kθ ) e nv k cos ( kθ ) / ( s w ) × (cid:20) sgn[cos( kθ )]erf (cid:18) √ nv k | cos( kθ ) | sw (cid:19) + 1 (cid:21) + sw (cid:27) , (13)where the short-hand notation is w = q − v k ,s = q − v k sin ( kθ ) . (14)This distribution is a function of the product kθ and v k √ n , therefore we expect universality with respect tothe rank k , as well as scaling with v k √ n . Both largermultiplicity and larger v k reduce the fluctuations of theangle θ .In Fig. 11 we plot the distribution f ( kθ ) for the typicalvalue of the flow coefficient for the elliptic flow ( k = 2)with v k = 5% for three values of the multiplicity n . Wenote a large width of the distributions, of the order of 30 ◦ for k = 2, or even more as n or v decrease. This spreadwill have the tendency of washing out the smaller torqueangle, unless a suitable method, sensitive to differencesof angles in the same pseudorapidity bin, is used. B. Cumulants
To achieve the goal of observing the torque, similarlyto analyses of flow [17, 31], we consider cumulants. Inthe simplest case of the two-particle cumulant we maytake D e in ( φ F − φ B ) E = 1 N events X events n F n B n F X i =1 n B X j =1 e ik ( φ i − φ j ) , (15)with k denoting the Fourier rank and φ i ( φ j ) being the az-imuthal angles of particles emitted in the forward (back-ward) η windows in a selected centrality class. The quan-tities n F and n B are the corresponding multiplicities.The measure is averaged over events, with the numberof events in the sample equal to N events .When no correlations between particles are present,the distribution function of n particles is the product ofone-body distributions. The one-body distribution canthen be written in the form f ( φ ) = v + 2 X k =1 v k cos[ k ( φ − Ψ ( k ) )] . (16)Then D e ik ( φ F − φ B ) E = h v k,F v k,B cos( k ∆ F B ) i events , (17)where h . i events indicates the averaging over events. Non-flow contributions modify the right-hand side at the level1 /n , where n is the effective multiplicity of particles inthe event. These effects, hard to estimate, include res-onance decays, conservation laws, Bose-Einstein correla-tions of identical particles, short-range correlations, etc.The influence of resonance decays will be analyzed viasimulations below.Since we are interested in measuring the averagecos[ n (Ψ F − Ψ B )], we need to divide Eq. (17) by v k,F v k,B .We can do it, for instance, by evaluating the ratio of cu-mulants, defined ascos( k ∆ F B ) { } ≡ (cid:10) e ik ( φ F − φ B ) (cid:11)q(cid:10) e ik ( φ F, − φ F, ) (cid:11) (cid:10) e ik ( φ B, − φ B, ) (cid:11) = h cos( k ∆ F B ) i events + nonflow . (18)One may also use higher-order cumulants to generatemeasures of the torque. For example, the ratio of four-particle cumulants, yieldscos(2 k ∆ F B ) { } ≡ h e ik [( φ F, + φ F, ) − ( φ B, + φ B, )] ih e ik [( φ F, − φ F, ) − ( φ B, − φ B, )] i = h cos(2 k ∆ F B ) i events + nonflow (19)The practical issue in this kind of studies is the influ-ence of the non-flow contributions on the result. C. THERMINATOR simulations
The precise estimate of the non-flow effects in the for-mulas of the previous section is not easy. To obtain arealistic estimate of the influence of resonance decays,we have run
THERMINATOR [18], generating 100000 events(in one centrality class) from a fireball with a torquedhypersurface, resulting from running the 3+1 perfect hy-drodynamics [8] on the torqued initial condition, as de-scribed in Sect. III. In Figs. 12 we present the results forcentrality c = 20 − ≃ ◦ , as in Fig. 10.First, we check if THERMINATOR is capable of reproduc-ing the (fixed) input torque angle. For this purpose wetake into account primordial particles (those created atthe freeze-out hypersurface) only, thus disregarding reso-nance decays. The result, shown in Fig. 12 with squares,shows a nice agreement between the cumulant measures(18,19) and the functions cos(2∆
F B ) and cos(4∆
F B ),shown with lines, evaluated directly from the fireballtorque angle shown in Fig. 10 (we take here the case ofthe evolution time equal of 6 fm/c). For comparison, thetriangles indicate the result of the calculation withoutthe torque, ∆
F B = 0. The agreement shows that with-out the non-flow contribution from resonance decays theaccumulated statistics of 100000
THERMINATOR events percentrality class is sufficient to see the torque effect. The error bars in figures of this section correspondto the statistical errors of our Monte Carlo simulation.These errors increase fast with η , as the number of pro-duced particles decreases rapidly above | η | ∼
3. In exper-imental data samples, which at RHIC have a very largestatistics, these errors should be significantly smaller.Next, we show the realistic case, accounting for allcharged pions, kaons, protons, and antiprotons, includ-ing those coming from resonance decays, in the deter-mination of the principal axes. We set the transversemomentum in the window 0 .
45 GeV < p T < p T leads to alarger elliptic flow coefficient, which reduces the statisti-cal noise (cf. Eq.(13) and its discussion), thus is advan-tageous for our calculation with a limited-size sample.The result of our simulation is shown in Fig. 13, with thesquares corresponding to the calculation with the torqueas in Fig. 10, and the triangles giving the base-line re-sults with no torque. We note that both cos(2∆ F B ) { } and cos(4∆ F B ) { } pick up the non-flow effects, as thetriangles are shifted down from unity.We see in Fig. 13 that the non-flow effect from res-onance decays depends weakly on η . This is to be ex-pected, as the effect comes from correlations of the twoparticles in the forward pseudorapidity window and thetwo particles in the backward pseudorapidity window.Once these windows are sufficiently separated, the effectdoes not depend on the separation η . We also remarkthat the simulation is much more noisy (not shown infigures) for the central events, which is due to the smallvalue of v . Thus the torque effect has the best chanceof being observed for the mid-central or mid-peripheralcentrality classes. Thus, if a decrease of the proposed cu-mulant measures with η is observed in real data, it wouldhint to the torque effect.In a complete simulation, an average should be taken ofthe observables over the torque angle distribution. Withour calculation, taking the torque angle corresponding tothe rms width of the angle distribution, we get a correctestimate for the averages of quantities of the order of thesquare of ∆ F B (Eqs. 18, 19), which is sufficient.
D. Forward-backward-central correlations
On may also use correlation measures based on threerapidity windows, forward, backward, and central. Onepossibility is the following combination: A F BC { } = h e i φ F − φ C, ) − ( φ B − φ C, )] i − h e i φ F − φ C, )+( φ B − φ C, )] i v ,F v ,B v ,C = h F C ) sin(2∆ BC ) i events + nonflow . (20) õ õ õ õ õ õ õ õ á á á á á á á á H a L primordial c o s H D F B L < õ õ õ õ õ õ õ õ á á á á á á á á H b L primordial Η c o s H D F B L < FIG. 12. (Color online) Cumulant measures of the torqueobtained for the Au+Au collisions at c = 20 −
25% with theprimordial particles only (i.e. with no resonance decays), plot-ted as functions of pseudorapidity. Triangles correspond to notorque, squares to the torque of Fig. 10. The solid line repre-sents evaluation directly from the fireball torque angle shownin Fig. 10. The agreement of the line and the squares showsthat the statistics is sufficient to detect the torque effect. The η windows have the width of one unit. The error bars corre-spond to the statistical errors of the THERMINATOR simulation.
For small torque angles this measure reduces to the co-variance (9), namely A F BC { } ∼ F C , ∆ BC ) + nonflow . (21)The results of the THERMINATOR simulations for A F BC are shown in Fig. 14. The conclusions are similar as forthe previously discussed measures. With primordial par-ticles (no non-flow effects, Fig. 14(a)) we reproduce theexpected behavior. The squares are very close to thesolid line, showing 8cov
F BC (cf. Fig. 8). For the realisticcase of all charged pions, kaons, protons, and antiprotons(Fig. 14(b)), we observe the non-flow effect from the res-onance decays. We note that the cases of the torquedand untorqued fireballs are qualitatively different, withthe former giving positive, and the latter negative valuesof A F BC . õ õ õ õ õ õ õ õ á á á á á á á á H a L < p T < c o s H D F B L < õ õ õ õ õ õ õ õ á á á á á á á á H b L < p T < Η c o s H D F B L < FIG. 13. (Color online) Same as Fig. 12 obtained fromall charged pions, kaons, protons, and antiprotons, with450 MeV < p T < V. CONCLUSIONS
In conclusion, we summarize our results:1. The space-time rapidity emission profile, where thewounded nucleons emit predominantly in the direc-tion of their motion, combined with the statisticalfluctuations of the source densities in the transverseplane, lead to event-by event torqued fireballs.2. The standard deviation of the relative torque an-gle between the forward ( η k ∼
3) and backward( η k ∼ −
3) regions varies from 20 ◦ for the most cen-tral collisions to 10 ◦ for the mid-central and mid-peripheral Au+Au collisions at the highest RHICenergies.3. This initial torque is transformed, via hydrodynam-ics, into the torque of the transverse collective flowof the fluid, and subsequently into the torque of theprincipal axes of the transverse-momentum distri-butions of the detected particles.4. Statistical measures based on cumulants containing0 õ õ õ õ õ õ õ á á á á á á á H a L primordial - - A F BC < õ õ õ õ õ õ õ á á á á á á á < p T < H b L - - Η A F BC < FIG. 14. The measure A F BC obtained from for the primor-dial particles (a) and all charged pions, kaons, protons, andantiprotons (b). Triangles correspond to no torque, squaresto the torque of Fig. 10. The solid line shows 8cov
F BC ofEq. (9). The η windows have the width of one unit. The errorbars correspond to the statistical errors of the THERMINATOR simulation. particles in different pseudorapidity bins should beuseful in detecting the torque effect experimentally.5. The non-flow corrections can be sizable, but should not overshadow the effect. In particular this isthe case of the effects of resonance decays, esti-mated realistically with
THERMINATOR
Monte Carlosimulations. We find that there is a clear differ-ence between the behavior of the proposed cumu-lant measures for the untorqued and torqued cases.Therefore the torque fluctuations should possiblybe observed in the high-statistics RHIC data bythe PHOBOS and STAR collaborations.6. Since the statistical noise increases as the productof the particle multiplicity and the v k flow coef-ficient, the best chance for looking for the torqueeffect is in the mid-central or mid-peripheral cen-trality classes, such as c = 20 − ACKNOWLEDGMENTS
Supported by Polish Ministry of Science and HigherEducation, grants N N202 263438 and N N202 249235,and by the Portuguese Funda¸c˜ao para a Ciˆencia e Tec-nologia, FEDER, OE, grant SFRH/BPD/63070/2009,CERN/FP/116334/201. [1] S. Uhlig, I. Derado, R. Meinke, and H. Preiss-ner, Nucl. Phys.
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