Momentum spectra, anisotropic flow, and ideal fluids
Abstract
If the matter produced in ultrarelativistic heavyion collisions reaches thermal equilibrium, its subsequent evolution follows the laws of ideal fluid dynamics. We show that general predictions can be made on this basis alone, irrespective of the details of the hydrodynamical model. We derive several scaling rules for momentum spectra and anisotropic flow (in particular the elliptic flow, , and the hexadecupole flow, ) of identified particles. Comparison with existing data is briefly discussed, and qualitative predictions are made for LHC.
pacs:
25.75.Ld, 24.10.NzAn ultrarelativistic AuAu collision at RHIC produces a dense system of interacting particles and fields, which then expands into the vacuum. Idealfluid models have been succesful in describing this expansion Kolb:2003dz : they are able, to a certain extent, to reproduce the magnitude and the transverse momentum () dependence of elliptic flow of identified particles, together with their spectra, for momenta GeV/c.
In this Letter, we derive general properties of momentum spectra of identified particles emitted by an ideal fluid, which do not depend on the specific model used. We shall introduce an important distinction between slow particles, whose velocity equals the fluid velocity at some point, and fast particles, whose velocity exceeds the maximum fluid velocity. We discuss in detail the implications of idealfluid behaviour for both slow and fast particles.
The expansion of a plasma into a vacuum goes through successive steps. If the particle mean free path inside the plasma is small enough, the plasma thermalizes, leading to a collisiondominated, isentropic expansion, which is described by ideal (i.e., inviscid) fluid dynamics. On the contrary, the late stage of the expansion is collisionfree. In between, a transition regime takes place. This transition regime must in principle be modelled by transport theory, which encompasses both isentropic and collisionfree limits Bass:2000ib ; Teaney:2001av .
In the context of heavyion collisions, however, the transition regime is most often modelled by a simple Ansatz, the sudden freezeout approximation, which is a sharp transition between the two extremes: one first defines a spacetime hypersurface along the history of the ideal fluid, on which the transition is expected to take place. At each point of , freestreaming particles are emitted according to thermal distributions in the rest frame of the fluid. Integrating over , one obtains for a given particle the following momentum spectrum Cooper:1974mv :
(1) 
where is the fluid 4velocity at point on , is a normalization constant, and we have neglected the effects of quantum statistics (in practice, the latter may only be significant for pions at low ). For simplicity, we also assume that the fluid temperature is everywhere the same on , but the results derived in this paper do not rely on this assumption. The possibility has also been raised that particles of different types Grassi:1994nf or with different transverse momenta Tomasik:2002qt have different freezeout temperatures.
Let us briefly discuss the validity of the CooperFrye Ansatz, Eq. (1). This approximation is expected to be poor when applied to observables which are sensitive to the detailed physics at freezeout, such as HBT radii Wiedemann:1999qn . The reason is that Eq. (1) assumes an isotropic momentum distribution in the rest frame of the fluid, while nonrelativistic studies have shown that freezeout precisely occurs when the relative difference between parallel and transverse components of the kinetic temperature becomes large hamel . On the other hand, the simple freezeout Ansatz may be a reasonable one for computing singleparticle spectra, provided that collective expansion dominates over random, thermal motion. This is known as the hypersonic approximation in the context of nonrelativistic gas dynamics hamel ; cercignani . In the language of heavyion collisions, it can be rephrased as follows: consistency of the idealfluid picture requires that the freezeout temperature be much smaller than the inverse slope parameters obtained by exponential fits to transversemass spectra Bearden:1996dd . The bestfit value of the freezeout temperature at RHIC is MeV Retiere:2003kf ; Hirano:2004rs , while the inverse slope parameter for pions is MeV Adler:2003cb . The value of is large enough to expect significant deviations from idealfluid behaviour, i.e., viscous effects Teaney:2003pb . It is however interesting to study the small limit in view of upcoming heavyion experiments at LHC, and also to have a better grasp on viscous effects, which are easily seen as deviations from this limit.
We therefore investigate systematically the properties of momentum spectra in the limit of small . The general idea is that the integral over in Eq. (1) can then be performed by means of a saddlepoint integration. ^{1}^{1}1The same method was used earlier to predict the behaviour of longitudinal HBT radii Makhlin:1987gm . In physical terms, it means that the dominant contribution comes from the points where the energy of the particle in the fluid frame, , is minimum. For a given , is a function of the space components of the fluid 4velocity, (the fourth one being related to them through ), which themselves depend on the point on . Since is the energy of the particle in the fluid rest frame, its absolute minimum is the particle mass . This minimum is reached when the particle is at rest with respect to the fluid, i.e., when its velocity equals the fluid velocity (or, equivalently, if ).
This absolute minimum, however, occurs only if there exists a point on where this value of the fluid velocity is reached. This leads us to a qualitative discussion of the values taken by at freezeout. The longitudinal fluid velocity is expected to span almost the whole range from to in ultrarelativistic collisions, and the simple Bjorken picture Bjorken:1982qr shows, at least qualitatively, how it is related to spacetime coordinates. The radial fluid velocity spans a more limited range. The reason is that transverse collective flow is not initially present in the system but builds up progressively. For a given fluid rapidity , the transverse 4velocity, , extends up to some maximum value , which may depend on , and on the azimuthal angle for noncentral collisions. As shown schematically in Fig. 1, is largest at , along the direction of impact parameter. This is due to larger pressure gradients in this direction Ollitrault:1992bk , which explain the large inplane elliptic flow observed at RHIC Ackermann:2000tr . Typical values of are of order 1 at RHIC ( in the notations of Retiere:2003kf , with the bestfit value ).
From now on, we make a distinction between “slow” and “fast” particles as follows: a particle of mass , with rapidity and transverse momentum , is defined as slow if for all (that is, actually, for , where the minimum occurs). Conversely, a fast particle is defined by for all (that is, for , where the velocity is maximal). Between both regimes, there is a small intermediate region, which will not be considered in this Letter.
For a slow particle, there is a point on such that the fluid velocity equals the particle velocity, and the minimum is reached. (Our results are unchanged if there are two or more such points.) If is small enough, the dominant contribution to the integral (1) comes from the neighbourhood of this point (see Fig. 1). The integral can then be evaluated approximately by expanding the exponent to second order around the minimum of . What remains is a Gaussian integral. For a given velocity, is proportional to the particle mass . Hence, the width of the Gaussian varies with like , and the integral over in Eq. (1) is times a function of the particle velocity. This means that the mass dependence is only a global factor for slow particles:
(2) 
where is the same for all particles. As a result, transverse momentum and rapidity spectra (integrated over ) of identified slow particles coincide, up to a normalization factor, when they are plotted as a function of and . The coefficients quantifying azimuthal anisotropies , which are independent of the total yield, should coincide for different identified slow particles at the same and . This property holds for directed flow as well as for elliptic flow and the higher harmonics. The behaviour is qualitatively correct for at RHIC, which rises more slowly with for heavier particles Adams:2004bi ; Adler:2003kt .
The condition under which the saddlepoint approximation is good for slow particles can be roughly stated as for a relativistic fluid, for dimensional reasons: the larger , the smaller the width of the Gaussian, and the better the approximation. A detailed calculation gives the condition
(3) 
with and ; this condition amounts to assuming that collective motion dominates over thermal motion. At RHIC, we expect the approximation to be poor for pions (furthermore, the pion spectrum is contaminated at low by secondary decays, and may also be sensitive to Bose–Einstein statistics), but it might be a reasonable one for kaons and heavier hadrons with /c.
Let us now discuss fast particles. Roughly speaking, these are the particles that move faster than the fluid: the minimum value of is larger than . In order to locate this minimum, we denote by (resp. ) the particle (resp. fluid) longitudinal rapidity, by the transverse component of parallel to the particle transverse momentum , and by the transverse component of orthogonal to . With these notations,
(4) 
where . Minimization with respect to and gives and , i.e., the fluid velocity is parallel to the particle velocity. The minimum is then attained when is maximum, i.e, when , where is the azimuthal angle of the particle. In other words, fast particles come from regions on where the parallel velocity is close to its maximum value (see Fig. 1). A saddlepoint integration then gives ^{2}^{2}2We assume that the maximum value is reached at an inner point of . If it occurs at the edge of the fluid, there is no square root in the preexponential factor.
(5) 
where the dependence is implicit. This result was already obtained long ago for massless particles in Ref. Blaizot:1986bh (see also Schnedermann:1993ws ).
The saddlepoint approximation is valid for fast particles if is large enough. A more precise criterion is
(6) 
together with the condition . At RHIC, the lefthand side (lhs) of Eq. (6) is smaller than the righthand side (rhs) by at least a factor of 2 as soon as GeV/c for pions, GeV/c for kaons, and GeV/c for (anti)protons. On the other hand, ideal fluid dynamics is expected to break down if is too high, since high particles have been shown to be more sensitive to offequilibrium (viscosity) effects Teaney:2003pb . Deviations from fluidlike behaviour are best seen on elliptic flow, for mesons above 1.5 GeV/c, and for baryons above 2.5 GeV/c. The window in which our approximation works is likely to be narrow, which reflects the importance of viscous effects at RHIC. Better agreement should be reached at LHC.
The spectra of identified particles are directly obtained from Eq. (5), neglecting the dependence of . Radial flow results in flatter spectra for heavier particles. In addition, Eq. (5) implies a breakdown of scaling: the slope of the spectrum decreases with increasing for pions, and increases for protons, in qualitative agreement with experimental findings Adler:2003cb .
For noncentral collisions, we can also obtain the anisotropic flow coefficients. We expand in Fourier series, and neglect odd harmonics:
(7) 
The parameter is of the order of 4 % for semicentral AuAu collisions at RHIC. It is related to the parameter of blast wave parameterizations Huovinen:2001cy ; Retiere:2003kf by . The distribution is obtained by inserting Eq. (7) into (5). If is small enough, the dependence in Eq. (5) is dominated by the exponential. Expanding the latter to first order in , one obtains
(8) 
A similar equation was already obtained in Ref. Huovinen:2001cy in the framework of a simplified fluid model, and was shown to fit RHIC data rather well. In particular, Eq. (8) shows that the “mass ordering” which follows from Eq. (2) for slow particles persists at high in hydro: at a given , heavier particles have smaller .
We finally make predictions for the hexadecupole flow, . We expand the exponential of Eq. (5) and look for terms in . To leading order one obtains two terms:
(10)  
For large enough , the first term dominates over the second, which gives the simple, universal relation
(11) 
Let us derive the domain of validity of this approximation. If is a smooth function of , one generally expects to be of order . The condition for Eq. (11) is then
(12) 
At RHIC, the lhs is smaller than the rhs by at least a factor of 2 for pions with GeV/c. For heavier particles, Eq. (6) supersedes Eq. (12).
Our result, Eq. (10), is in contradiction with the statement that is a sensitive probe of initial conditions Kolb:2003zi : on the contrary, we find a universal result, which can be directly used as a probe of ideal fluid behaviour, not of initial conditions. The experimental value found by the STAR Collaboration Adams:2003zg is a factor of 2 to 3 higher than our prediction. Deviations from idealfluid behaviour are generally expected to yield higher values of Bhalerao:2005mm .
Our results for fast particles, Eqs. (5), (8), (11), are compared to results from a numerical 3d hydrodynamical calculation in Fig. 2. The calculation has been pushed to very large times, so that the small limit applies. The value of does not go exactly to 0.5 at large but rather to 0.63. This is due to the fact that the initial eccentricity is large for this value of the impact parameter, and Eq. (11) is obtained through a leading order expansion in the anisotropy. We have checked numerically that agreement is better for lower values of , where the eccentricity is smaller.
Before we come to our conclusions, let us compare our approach with the popular blastwave one. The blastwave parameterization, in its simplest form, assumes a unique radial velocity for the fluid Siemens:1978pb ; this framework has recently been refined to take into account the azimuthal dependence of the fluid velocity Huovinen:2001cy and of the freezeout surface Adler:2001nb in noncentral collisions, and even a distribution of fluid velocities Retiere:2003kf . A few parameters (typically four) are then fitted to experimental data. Some of the results we derived above were already obtained within the blastwave approach, namely the massordering of the of different types. However, our present framework is more general in the sense that we do not assume a given fluidvelocity profile, but also more specific in the sense that we assume that collective motion dominates over thermal (random) motion. In addition, blastwave fits treat slow and fast particles on an equal footing, ignoring the distinction between both types of particles. Although fitting the whole spectrum with a single formula is admittedly more convenient, it misses an important feature of the underlying physics, since slow and fast particles originate from different regions of the expanding fluid. In particular, fits using our formulas for fast particles may yield values of and which differ from blastwave fits. Finally, our formulas are significantly simpler than blastwave parameterizations, which involve special functions.
We have obtained the following results for momentum spectra and anisotropies in the framework of idealfluid models using a saddlepoint approximation of the momentum distribution:

At low , identified particles of different masses have the same momentum spectra and anisotropies (up to a normalization for the spectra), when plotted as a function of velocity variables and . This defines “slow” particles. This scaling is due to the fact that slow particles move with the fluid: they come from the regions where the fluid velocity equals their velocity. The scaling is expected to be poor for pions. It is expected to break down when exceeds , the maximum value of the transverse 4velocity of the fluid. may in general depend on the rapidity , and reflects the underlying equation of state of the expanding matter.

Fast particles, defined by Eq. (6), all originate from the region where the fluid is fastest along the direction of the particle velocity. As a result, their transverse momentum spectra and azimuthal anisotropies at a given rapidity are uniquely determined by three parameters , , and , and given by Eqs. (5), (8), (11). Comparing the of different particles should directly give the precise value of , while transverse momentum spectra yield .
These results can be used as signatures of hydrodynamic evolution in heavyion collisions, and also as consistency checks of numerical idealfluid calculations. Idealfluid evolution leads to different behaviours for slow and fast particles. Some of the results obtained for fast particles (in particular for elliptic flow) are already known from blastwave approaches. We have shown that they are in fact more general. The scaling rules for slow particles, which are evidenced here for the first time, should be further tested on available RHIC data. We expect all our results to be in closer agreement with data at LHC than at RHIC. In particular, we predict that the value of the ratio should be lower at LHC than at RHIC.
Acknowledgments
J.Y. O. thanks F. Becattini, T. Hirano, E. Shuryak and R. Snellings for discussions. We thank J.P. Blaizot for careful reading of the manuscript, and the referee for pointing out a mistake in the derivation of Eq. (11).
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