TwoParticle Correlations in Relativistic HeavyIon Collisions
Abstract
Twoparticle momentum correlations between pairs of identical particles produced in relativistic heavyion reactions can be analyzed to extract the spacetime structure of the collision fireball. We review recent progress in the application of this method, based on newly developed theoretical tools and new highquality data from heavyion collision experiments. Implications for our understanding of the collision dynamics and for the search for the quarkgluon plasma are discussed.
anbury Brown  Twiss (HBT) interferometry, BoseEinstein correlations, twoparticle correlations, threeparticle correlations, quarkgluon plasma, collective expansion flow, source sizes and lifetimes, freezeout
1 Introduction and Overview
1.1 Intensity Interferometry
The method of twoparticle intensity interferometry was discovered in the early 1950’s by Hanbury Brown and Twiss (HBT) [1] who applied it to the measurement of the angular diameter of stars and other astronomical objects. These first measurements used twophoton correlations. An independent development occurred in the field of particle physics in 1960 by Goldhaber, Goldhaber, Lee and Pais [2] who extracted from twopion correlations the spatial extent of the annihilation fireball in protonantiproton reactions. The method exploits the fact that identical particles which sit nearby in phasespace experience quantum statistical effects resulting from the (anti)symmetrization of the multiparticle wave function. For bosons, therefore, the twoparticle coincidence rate shows an enhancement at small momentum difference between the particles. The momentum range of this enhancement can be related to the size of the particle source in coordinate space.
HBT interferometry differs from ordinary amplitude interferometry that it compares intensities rather than amplitudes at different points. It shows the effects of Bose or Fermi statistics even if the phase of the (light or matter) wave is disturbed by uncontrollable random fluctuations (as is, for example, the case for starlight propagating through the earth’s atmosphere) or if the counting rate is very low. To illustrate how the method works and how its applications in astronomy and in particle physics differ let us consider the following simple model [3, 4]: two random point sources and on a distant emitter, separated by the distance , emit identical particles with identical energies which, after travelling a distance , are measured by two detectors 1 and 2, separated by the distance
(see Figure 1a). should to be much larger than or . The total amplitude measured at detector 1 is then
(1) 
where are the amplitudes emitted by points and , are their random phases, are their distances to detector 1. The total intensity in 1 is
(2) 
with a similar result for . The last term in (2) contains information on the distance between the sources and , but it vanishes after averaging the signal over some time, i.e. over the random phases :
(3) 
The product of the averaged intensities is thus independent of both and .
The same is not true for the timeaveraged coincidence rate which is obtained by multiplying the two intensities before averaging:
(4) 
The twoparticle intensity correlation function is thus given by
(5) 
For large , the argument of the second, oscillating term becomes
(6) 
Note the symmetry of this expression in and , the separations of the detectors and of the emitters; this symmetry is lost in the two practically relevant limits:

In astronomical applications the emission points and are part of a star’s surface or of even larger objects, while the detectors on earth are only a few meters or kilometers apart: . In this limit (see Figure 1b) the cosineterm in (5) reduces to , with , the unit vectors giving the directions from the detectors to the two emission points . Experimentally one varies the distance between the detectors, and from the resulting oscillation of the signal one extracts the angular separation of the two emitters via . Absolute determination of the source separation is possible only if their distance is known from elsewhere.
In real life one has, instead of two discrete ones, a continuum of sources described by a distribution of their relative distances. For the case , after averaging (5) over this relative distance distribution, the measured correlation function is then given by the Fourier transform of :
(7) 
As we will see, this expression is only applicable to static sources. The assumption of a static source is adequate for stars. The particles emitted in high energy hadron or nuclear reactions, however, come from sources which feature strong dynamical evolution both along the beam direction and transverse to it. As a result, twoparticle correlation measurements in heavyion physics exhibit a much richer structure, but their interpretation is also considerably more involved. The present review covers the technical tools required to access this richness of information and their practical application in heavyion reactions.
1.2 HBT Interferometry for HeavyIon Collisions
High energy heavyion collisions produce hadronic matter under intense conditions of temperature and density. While the highest densities are reached in the early stages of the collision, most of the observed particles are hadrons, which are emitted rather late in the evolution of the collision. For this reason, the measured momentum spectra and correlations contain direct information only about the size, shape, and dynamics of the source at “freezeout”, i.e. when the hadrons cease to interact.
The dynamical information is of particular importance as it allows us to connect the observed final state with the hot and dense early stages. Therefore much of the effort in the last few years has gone into the extraction of this dynamics. It turns out that both the singleparticle spectra and twoparticle correlations are sensitive to certain combinations of thermal and collective motion in the source. A simultaneous analysis of particles with different masses allows for a separation of the two effects: while the thermal motion generates a common momentum distribution in the source, the collective motion creates a flow velocity which combines with the thermal momentum proportionally to the particle’s mass. Further discrimination is achieved by combining the spectra with the twoparticle correlations which reflect the collective source dynamics through a characteristic momentum dependence of their width parameters.
The aim of HBT (or twoparticle BoseEinstein) interferometry is therefore to provide, in conjunction with a simultaneous analysis of singleparticle spectra, a complete and quantitative characterization (both geometrically and dynamically) of the emitting source at “freezeout”. Such a characterization can be used for backward extrapolations into the interesting dense earlier stages of the collision, and it provides stringent constraints on dynamical models aiming to describe the kinetic evolution of these stages.
1.3 New Developments During the Last Decade
The last decade has brought great strides in measurement, theory, and interpretation of twoparticle correlations. Dedicated experiments, optimized for momentum resolution, allow measurement of correlations to very small momentum difference. Much increased statistics of particle pairs has opened the possibility of multidimensional correlation analysis. Correlation functions of identified kaons have become available; as many fewer kaons arise from resonance decays, they provide a more direct picture of the emitting source than the more prevalent pions. Correlations of proton pairs, which will not be discussed here, are also measured in heavyion collisions. The experimental programs at the CERN SPS and Brookhaven AGS have allowed comparison between small and large systems at different energies, using S and Pb beams at CERN and lower energy Si and Au beams at the AGS. The comparisons are greatly aided by the development of a commonly accepted analysis formalism among the experiments. Furthermore, the pair momentum dependence of the correlations is now being used to provide dynamical information about the spacetime structure of the particle source.
Intensive modeling with event generators, combined with methods to extract correlation functions from them, has been used to study experimental effects such as acceptance and to verify the usefulness of simple model parameterizations in theory. New parametrizations of the twoparticle correlation function have been developed which are particularly well adapted for the sources created in high energy collisions. Recently, the necessity of separating geometrical, temporal and dynamical aspects of the correlations has been recognized, and methods to do so have been developed. Intensity interferometry has thus developed into a quantitative tool which at the present stage yields important source parameters at the level of 20% accuracy.
2 Theoretical Tools for Analyzing TwoParticle Correlations
2.1 1 and 2Particle Spectra and Their Relation to the Emitting Source
The 2particle correlation function is defined as the ratio of the Lorentzinvariant 2particle coincidence cross section and the product of the two single particle distributions:
(8) 
The single and twoparticle cross sections are normalized to the average number of particles per event and the average number of particles in pairs , respectively. Different normalizations of the correlation function are sometimes used in the literature [5, 6]. Careful consideration of the normalization is required when analyzing multiparticle symmetrization effects [7, 8, 9, 10, 11].
2.1.1 Pure quantum statistical correlations
The most direct connection between the measured twoparticle correlations in momentum space and the source distribution in coordinate space can be established if the particles are emitted independently (“chaotic source”) and propagate freely from source to detector. Several approaches to derive this connection [12] are worked out in the literature, parametrizing the source as a covariant superposition of classical currents [5, 13, 14, 15, 16, 17, 18] or using a superposition of nonrelativistic wave packets [9, 10]. One finds the simple relations (with the upper (lower) sign for bosons (fermions))
(9)  
(10) 
where the emission function is an effective singleparticle Wigner phasespace density of the particles in the source. (Wigner densities are real but not necessarily everywhere positive.) For the singleparticle spectrum (9) this Wigner function must be evaluated onshell, i.e. at . The correlation function (10) was expressed in terms of the relative momentum , , and average (pair) momentum , . As the two measured particles are onshell, , the 4momenta and are offshell. They satisfy the orthogonality relation
(11) 
That on the rhs of Eq. (10) one needs the emission function for offshell momenta may at first seem troublesome. In practice, however, the onshell approximation is very accurate in heavyion collisions: the corrections of order are small in the interesting domain of small relative momenta , as a result of the large source sizes and the rest masses of the measured particles. A further simplification is achieved by making in the denominator of (10) the smoothness approximation [19, 20], taking the product of singleparticle spectra at their average momentum :
(12) 
It is exact for exponential singleparticle spectra, with corrections proportional to their curvature in logarithmic representation. Both approximations can a posteriori be corrected for in the correlation radii (“HBT radii”, see below), using information from the measured singleparticle spectra [21]. For heavyion collisions, such corrections are usually negligible [20, 21].
We call an effective singleparticle Wigner density since different derivations of the relation (10) yield different microscopic interpretations for . For a detailed discussion of this point we refer to [22]. The differences can become conceptually important in sources with high phasespace density [11]. So far, in heavyion collisions the phasespace densities at freezeout appear to be low enough to neglect them [23, 24].
2.1.2 The invertibility problem
The massshell constraint (11) eliminates one of the four components of the relative momentum ; for example, it can be resolved as
(13) 
which gives the energy difference in terms of and the velocity of the pair. With only three independent components, the Fourier transform in (12) cannot be inverted, i.e. the spacetime structure of cannot be fully recovered from the measured correlator:
(14) 
Separation of the spatial and temporal structure of the source thus requires additional model assumptions about .
We can connect (14) with (7) by introducing the normalized relative distance distribution which is a folding of the singleparticle emission function with itself:
(15) 
is an even function of . It allows to rewrite [22] the correlator in the form (7):
(16) 
In the second equation we used (13) and introduced the relative source function
(17) 
In the pair rest frame where , is the time integral of the relative distance distribution , and the time structure of the source is completely integrated out. On the other hand, is, for each pair momentum , fully reconstructible from the measured correlator by inverting the last Fourier transform in (16). This “imaging method” was recently exploited in [25]. As we will see, interesting information about the source dynamics and time structure is then hidden in the dependence of ; the latter can, however, not be unfolded without additional model assumptions about the source.
2.1.3 Final state interactions and unlike particle correlations
HBT measurements in high energy physics are mostly performed with charged particles. These suffer longrange Coulomb repulsion effects on the way from the source to the detector which, even for boson pairs, cause a suppression of the measured correlator at . Moreover, the charged particle pair feels the total electric charge of the source from which it is emitted. Final state effects from strong interactions play a dominant role in protonproton correlations [26], due to the existence of a strong wave resonance in the twonucleon channel just above threshold. Such final state interaction (FSI) effects are sensitive to the average separation of the two particles at their emission points and thus also contain relevant information about the source size [26, 27, 4].
This has recently led to an increased effort to understand and exploit FSIinduced twoparticle correlations which also exist between pairs of unlike particles [28, 29, 30, 31, 32]. The particular interest in such correlations arises from the fact that, for particles with unequal masses, they allow under certain circumstances to determine the sign of the time difference between the emission points of the two particles or the direction of their separation at emission [28, 29, 30, 31, 32]; this is not possible with correlations between identical particles. In most practical cases, however, the FSIinduced correlations are considerably weaker than the BoseEinstein correlations between pairs of identical particles.
At the level of accuracy of Eqs. (12,16) which use the smoothness approximation, the correlator can be easily corrected for 2body final state interactions by replacing with a suitable distorted wave. Instead of (16) one thus obtains [26]
(18) 
A slightly more general result which avoids the smoothness approximation was derived in [33]. For simplicity the integral in (18) is written in coordinates of the pair rest frame where . is an FSI distorted scattering wave for the relative motion of the two particles with asymptotic relative momentum ; for Coulomb FSI it is given by a confluent hypergeometric function:
(19)  
(20) 
where . It describes the propagation of the particle pair from an initial separation in the pair rest frame, at the time when the second particle was emitted [33], under the influence of the mutual FSI. For identical particle pairs it must be properly symmetrized: . For a pointlike source the correlator (18) with (19) reduces to the Gamov factor (to for identical particles):
(21) 
For Coulomb FSI it was recently shown [34] that a very good approximation for the Coulomb correction can be taken from measured unlikesign particle pairs in the following form:
(22) 
The denominator (which deviates from unity for small ) corrects for the fact that even for a pointlike source the likesign and unlikesign Coulomb correlations are not exactly each other’s inverse. The important observation in [34] is that this correction is essentially independent of the source size.
The effects of the central charge of the remaining fireball on the charged particle pair were studied in [27, 35]. For a static source it was found that at large pair momenta the FSI reduces (increases) the apparent size (HBT radius) for positively (negatively) charged pairs [27, 35] whereas for small pair momenta the apparent radius increases for both charge states [35]. Expanding sources have not yet been studied in this context, nor has this effect been quantitatively confirmed by experiment. Also, combining the central interaction with twobody FSI remains an unsolved theoretical challenge.
2.2 Source Sizes and Particle Emission Times from HBT Correlations
The twoparticle correlation function is usually parametrized by a Gaussian in the relative momentum components. We now discuss different Gaussian parametrizations and establish the relationship of the corresponding width parameters (HBT radii) with the spacetime structure of the source.
2.2.1 HBT radii as homogeneity lengths
The spacetime interpretation of the HBT radii is based on a Gaussian approximation to the spacetime dependence of the emission function [36, 21, 37, 38, 39, 40, 41]. Characterizing the effective source of particles of momentum by its spacetime variances (“rms widths”)
(23) 
where denotes the (dependent) spacetime average over the emission function defined in (12) and is the center of the effective source, one obtains from (12) the following generic Gaussian form for the correlator [36, 37, 41]:
(24) 
This involves the smoothness and onshell approximations discussed in section 2.1.1 which permit to write the spacetime variances as functions of only. Eq. (24) expresses the width of the correlation function in terms of the rms widths of the singleparticle Wigner density . Note that the absolute spacetime position of the source center does not appear explicitly and thus cannot be measured.
Instead of the widths of the singleparticle function we can also use the widths of the relative distance distribution (see (15)) to characterize the correlation function. Starting from (15) one finds within the same Gaussian approximation ; here denotes the average with the relative distance distribution . Since is even, . One sees that the rms widths of and are related by a factor 2: . This shows that for a Gaussian parametrization of the correlator according to (24), without a factor in the exponent, the width parameters are directly related to the rms widths of the singleparticle emission function whereas a similar parametrization which includes a factor in the exponent gives as width parameters the rms widths of the relative distance distribution (or of the relative source function ). While the latter interpretation may be mathematically more accurate, the former is more intuitive and has therefore been preferred in the recent literature.
In either case, the twoparticle correlator yields rms widths of the effective source of particles with momentum . In general, these width parameters do not characterize the total extension of the collision region. They rather measure the size of the system through a filter of wavelength . In the language introduced by Sinyukov [42] this size is the “region of homogeneity”, the region from which particle pairs with momentum are most likely emitted. The spacetime variances coincide with total source extensions only in the special case that the emission function shows no positionmomentum correlations and factorizes, .
2.2.2 Gaussian parametrizations and interpretation of HBT radii
Relating (24) to experimental data requires first the elimination of one of the four components via the massshell constraint (11). Depending on the choice of the three independent components different Gaussian parametrizations exist.
A convenient choice of coordinate axes for heavyion collisions is the oslsystem [13, 43], with denoting the longitudinal (or ) direction along the beam, the outward (or ) direction along the transverse pair momentum vector , and the third Cartesian direction, the sideward (or ) direction. In this system the sideward component of the pair velocity in (13) vanishes by definition.
The Cartesian parametrization [44] of the correlator (often referred to, historically somewhat incorrectly, as Pratt[13]  Bertsch[43] parametrization) is based on an elimination of in (24) via (13):
(25) 
The Gaussian width parameters (HBT correlation radii) of the Cartesian parametrization are related to the spacetime variances of the emission function by [45, 36, 46]
(26) 
These are 6 functions of three variables: the pair rapidity , the modulus and the azimuthal angle between the transverse pair momentum and the impact parameter . Only these 6 combinations of the 10 independent spacetime variances can be measured.
For azimuthally symmetric collision ensembles the emission function has a reflection symmetry , eliminating 3 of the 10 spacetime variances, and the correlator is symmetric under [37]. Then , and the correlator is fully characterized by 4 functions of only two variables and :
(27) 
with
(28) 
These “HBT radii” mix spatial and temporal information on the source in a nontrivial way, and their interpretation depends on the frame in which the relative momenta are specified. Extensive discussions of these parameters (including the crossterm which originally appeared in the important, but widely neglected paper [44], was recently rediscovered [36] and then experimentally confirmed [47, 23]) can be found in Refs. [18, 21, 36, 37, 38, 39, 40, 41, 52, 48, 49, 50, 51]. The crossterm vanishes in any longitudinal reference frame in which the source is symmetric under [44] (e.g. for pion pairs with vanishing rapidity in the centerofmomentum system (CMS) of a symmetric collision); in general it does not vanish for pion pairs with nonzero CMS rapidity, not even in the longitudinally comoving system (LCMS [53]) [48].
For azimuthally symmetric collisions no direction is distinguished for pairs with . As long as for the emission function reduces to an azimuthally symmetric expression (an exception is a certain class of opaque source models discussed in section 2.3.4), one has at the identities [37] ; these imply that and the crossterm vanish at . At nonzero these identities for the spacetime variances may be broken by transverse positionmomentum correlations in the source, as e.g. generated by transverse collective flow. If the latter are sufficiently weak the leading dependence of the difference
(29) 
is given by the explicit dependence of the first term on the rhs. This yields the duration of the particle emission process for particles with small [54, 55, 53]. (This is sometimes loosely called the “lifetime” of the effective source, but should not to be confused with the total time duration between nuclear impact and freezeout which is not directly measurable.)
The possibility to extract the emission duration from correlation measurements, pointed out by Bertsch and Pratt [54, 55, 53], provided the main motivation for the construction of second generation experiments to measure high quality, high statistics correlation functions. Subsequent model studies for relativistic heavyion collisions [56, 52, 57, 58, 59] where the emission duration is expected to be relatively short (of the order of the transverse source extension) showed, however, that the extraction of is somewhat model dependent; the relative smallness of the last two terms in (29) cannot always be guaranteed, and their implicit dependence can mix with the explicit one of the interesting first term.
The YanoKooninPodgoretskiĭ (YKP) parametrization is an alternative Gaussian parametrization of the correlator for azimuthally symmetric collisions. It uses the massshell constraint (11) to express (24) in terms of and [60, 44, 37, 41]:
(30) 
Like (27) it has 4 dependent fit parameters: the three radius parameters , , , and a velocity parameter with only a longitudinal spatial component:
(31) 
The advantage of fitting the form (30) to data is that the extracted YKP radii do not depend on the longitudinal velocity of the measurement frame, while the fourth fit parameter is simply boosted by that velocity. The frame in which is called the YanoKoonin (YK) frame; the YKP radii are most easily interpreted in terms of coordinates measured in this frame [37]:
(32)  
(33)  
(34) 
where the approximations in the last two lines are equivalent to dropping the last two terms in (29) (see discussion above). To the extent that these hold, the three YKP radii thus have a straightforward interpretation as the transverse, longitudinal and temporal homogeneity lengths in the YK frame. In particular the time structure of the source only enters in . For sources with strong longitudinal expansion, like those created in relativistic heavyion collisions, it was shown in extensive model studies [41, 52, 57, 58, 59] that the YK velocity very accurately reflects the longitudinal velocity at the center of the “homogeneity region” of particles of momentum . The YK frame can thus be interpreted as the rest frame of the effective source of particles with momentum , and the YKP radii measure the transverse, longitudinal and temporal size of this effective source in its own rest frame.
The parametrizations (27) and (30) use different independent components of but are mathematically equivalent. The YKP parameters can thus be calculated from the Cartesian ones and vice versa [41]. The corresponding relations are
(35)  
(36)  
(37)  
(38) 
whose inversion reads
(39)  
(40) 
These last definitions hold in an arbitrary longitudinal reference frame. According to (39) is zero in the frame where vanishes. However, (39) also shows that the YKP parametrization becomes illdefined if the argument of the square root turns negative. This can indeed happen, in particular for opaque sources [61, 58]; this has motivated the introduction of a modified YKP parametrization in [52, 58, 59] which avoids this problem at the expense of a less intuitive interpretation of the modified YKP radii. These remarks show that these relations provide an essential check for the internal consistency of the Gaussian fit to the measured correlation function and for the physical interpretation of the resulting HBT parameters.
2.3 Collective Expansion and Dependence of the Correlator
If the particle momenta are correlated with their emission points (“correlations”), the spacetime variances in (24) depend on the pair momentum . Various mechanisms can lead to such correlations; the most important one for heavyion collisions is collective expansion of the source. Recently a major effort has been launched to extract the collective flow pattern at freezeout from the dependence of the HBT parameters. However, thermalized sources may exhibit temperature gradients along the freezeout surface which cause additional correlations. Moreover, pion spectra receive sizeable contributions from the decay of unstable resonances some time after freezeout. These decay pions tend to come from a somewhat larger spacetime region than the directly emitted ones and, due to the decay kinematics, they preferentially populate the lowmomentum region. Together these two effects also generate correlations for the emitted pions even if the original source did not have them [62].
The separation of these different effects requires extensive model studies some of which will be reviewed below. For didactical reasons we will discuss them in the context of the YKP parametrization where certain mechanisms can be demonstrated most transparently. A translation for the Cartesian fit parameters via the crosscheck relations (35)(38) is straightforward. Furthermore we will show that for sources with strong longitudinal expansion the YK frame (effective source rest frame) is usually rather close to the LCMS (in which the pairs have vanishing longitudinal momentum), i.e. in the LCMS. This allows, at least qualitatively, for a rather direct extraction of source properties in its own rest frame from the Cartesian HBT radius parameters in the LCMS. This is important since initially most multidimensional HBT analyses were done with the Cartesian parametrization in the LCMS, before the YKP parametrization became popular.
We will concentrate on the discussion of azimuthally symmetric sources (central collisions) for which extensive knowledge, both theoretical and experimental, has been accumulated in the last few years. Many analytical model studies [21, 36, 37, 39, 38, 42, 40, 41, 52, 48, 50, 57, 58, 59, 63, 64] are based on the following parametrization of the emission function (or slight variations thereof):
(41) 
Here , , and parametrize the spacetime coordinates , with
The Boltzmann factor parametrizes the momentumspace structure of the source in terms of a collective, directed component, given by a flow velocity field , and a randomly fluctuating component, characterized by an exponential spectrum with local slope , as suggested by the shape of the measured singleparticle spectra. Although this parametrization is somewhat restrictive because it implies that the random component is locally isotropic, it does not require thermalization of the source at freezeout. But if it turns out that all particle species can be described simultaneously by the emission function (41), with the same temperature and velocity fields , , this would indeed suggest thermalization as the most natural explanation.
It is convenient to decompose in the form
(42) 
with longitudinal and transverse flow rapidities and . A simple boostinvariant longitudinal flow () is commonly assumed. For the transverse flow rapidity profile the simplest choice is
(43) 
with a scale parameter . (In our notation, denotes the transverse collective flow rapidity, whereas is the transverse velocity of the particle pair.) In most studies was (like ) set constant such that was a function of only. This cannot reproduce the observed rapidity dependence of and of the inverse slopes of the spectra [65, 66]. In [63, 64] it was shown that an dependence of , with shrinking in the backward and forward rapidity regions keeping the slope in (43) fixed, is sufficient to fully repair this deficiency. Here we discuss only the simpler case of constant .
With these assumptions the exponent of the Boltzmann factor in (41) becomes
(44) 
For vanishing transverse flow () the source depends only on and remains azimuthally symmetric for all .
Since in the absence of transverse flow the dependent terms in (33) and (34) vanish and the source itself depends only on , all three YKP radius parameters then show perfect scaling: plotted as functions of , they coincide for pion and kaon pairs (see Figure 2, left column). This remains true if varies with ; temperature gradients in the source do not destroy the scaling.
For (right column) this scaling is broken by two effects: (1) The thermal exponent (44) receives an additional contribution proportional to . (2) The terms which were neglected in the approximations (33,34) are nonzero, and they also depend on . Both induce an explicit rest mass dependence and destroy the scaling of the YKP size parameters.
2.3.1 Longitudinal flow: YanoKoonin rapidity and dependence of
At each point in an expanding source the local velocity distribution is centered around the average fluid velocity . Thus two fluid elements moving rapidly relative to each other are unlikely to contribute particles with small relative momenta. Only source regions moving with velocities close to that of the observed particle pair contribute to the correlation function. How close is controlled by the width of the random component in the momentum distribution: the larger the local “thermal smearing”, the more the differences in the fluid velocities can be balanced out, and the larger the “regions of homogeneity” in the source become.
Longitudinal expansion is most clearly reflected in the behaviour of the YanoKoonin (YK) rapidity 3 shows (for pion pairs) its dependence on the pair momentum . Transverse flow is seen to have a negligible influence on the YK rapidity. On the other hand, the linear dependence of on the pair rapidity (Figure 3a) is a direct reflection of the longitudinal expansion flow [41]; for a nonexpanding source would be independent of . The correlation between the velocities of the pair () and of the emission region () strengthens as the thermal smearing decreases. For the Boltzmann form (41) the latter is controlled by , and correspondingly as . For small , thermal smearing weakens the correlation, and the effective source moves somewhat more slowly than the observed pairs, whose longitudinal velocities have an additional thermal component. . Figure
If the ratio of the longitudinal source velocity gradient to the thermal smearing factor, defined in (46) below, is large, becomes small and the longitudinal rapidity of the effective source becomes equal to that of the emitted pairs, . This can be true even for slow longitudinal expansion as long as it is strong enough compared to the thermal smearing. Consequently, observation of a behaviour like the one shown in Figure 3a demonstrates strong, but not necessarily boostinvariant longitudinal flow.
Longitudinal expansion is also reflected in the dependence of (second row of Figure 2). Comparison of the left with the right diagram shows only minor effects from transverse expansion; longitudinal and transverse dynamics are thus cleanly separated. A qualitative understanding of the dependence is provided by the following expression, valid for pairs with , which can be derived by evaluating (33) via saddlepoint integration [67, 21, 38, 39]:
(45)  
(46) 
Eq. (46) shows explicitly the competition between the longitudinal velocity gradient and the thermal smearing factor . For strong longitudinal expansion (large velocity gradient and/or weak thermal smearing) and/or large geometric longitudinal extension of the source the second term in the denominator can be neglected, and drops steeply as [67]. Note that quantitative corrections to (45) are not always small [40].
We emphasize that only the first equation in (46) is general. The appearance of the parameter in the second equation is due to the choice of a Bjorken profile for the longitudinal flow for which the longitudinal velocity gradient is given by the total proper time between impact and freezeout. This is not true in general; the interpretation of the length in terms of the total expansion time is therefore a highly modeldependent procedure which should be avoided. As a matter of principle, the absolute temporal position of the freezeout point is not measurable, see section 2.2.1.
Strong longitudinal correlations also occur in string fragmentation. In fact, in the Schwinger model of string breaking [68] the quark pairs created from the chromoelectric field are assumed to have longitudinal momentum , without thermal fluctuations. Thus a similar linear rise of the YKrapidity with the pair rapidity and a strong decrease of would also be expected in jet fragmentation (with the axis oriented along the jet axis). It would be interesting to confirm this prediction [57] in or collisions.
2.3.2 Transverse flow: dependence of
Just as longitudinal expansion affects , transverse flow causes an dependence of . This is seen in the first row of diagrams in Figure 2, which also shows that longitudinal flow does not contribute to this feature: for the transverse radius does not depend on , in spite of strong longitudinal expansion of the source. A qualitative understanding of this behaviour is given by the analogue of (45), again obtained by evaluating (32) via saddle point integration [21, 39, 37]:
(47)  
(48) 
Once again there is competition between flow velocity gradients in the source, this time in the transverse direction, which tend to reduce the homogeneity regions, and thermal smearing by the factor resulting from the random component in the momentum distribution, enlarging the regions of homogeneity. The left equations in (47,48) are generic while the right ones apply to the specific transverse flow profile (43).
Transverse flow must be built up from zero during the collision while longitudinal correlations may contain a sizeable primordial component from incomplete stopping of the two nuclei and/or the particle production process (e.g. string fragmentation, see above). One thus expects generically weaker transverse than longitudinal flow effects at freezeout. Correspondingly, in realistic simulations (e.g. [39, 57, 56]) the longitudinal homogeneity length turns out to be dominated by the expansion (i.e. by ) while in the transverse direction the geometric size dominates at small , with flow effects taking over only at larger values of . Correspondingly, decreases more slowly at small than [40, 57].
2.3.3 The emission duration
Saddlepoint integration of (34) with the source (41) yields, with from (45),
(49) 
The dependence of thus induces an dependence of the temporal YKP parameter. Eq. (49) reflects the proper time freezeout assumed in the model (41): particles emitted at different points are also emitted at different global times , and the total temporal width of the effective source is thus given by the Gaussian width plus the additional variation along the proper time hyperbola, integrated over the longitudinal homogeneity region [40].
Although (49) suggests that the proper emission duration can be measured via in the limit (where ), this is has not been true in systems studied to date. For transversely expanding sources receives additional contributions from the dependent terms in (34), in particular at large (see Figure 2). The most important correction is due to the term which can have either sign and usually grows with [57, 59, 61]. The extraction of the emission duration must thus be considered the most modeldependent aspect of the HBT analysis.
2.3.4 Temperature gradients and opacity effects
A different source of transverse correlations which can compete with transverse flow in generating an dependence of are transverse temperature gradients in the emission function. Since particle densities and mean free paths (which control the freezeout process [69, 70]) depend very strongly on temperature, one would a priori not expect strong temperature variations across the freezeout surface [71]. While transverse temperature gradients and transverse flow affect similarly (except that the latter weakly breaks the scaling), they have quite different effects on the single particle spectra [39, 50]: transverse temperature gradients strongly reduce the flattening effect of transverse flow on the spectra which is needed to reproduce the data [70, 72]. Thus constrained by singleparticle data, their phenomenological usefulness is limited. Temporal temperature gradients only reduce the emission duration, but do not affect the transverse parameter [39, 50].
One possible source feature that parametrization (41) cannot describe is “opacity”, i.e. surface dominated emission. Heiselberg and Vischer generated opaque sources by multiplying (41) (or a similar source with bulk freezeout) with an exponential absorption factor [61] (see also [58])
(50)  
(51) 
The ratio controls the degree of opacity of the source; as , the source becomes an infinitely thin radiating shell. The parametrization (50) together with (41) leads to sources with negative for all values of (including the limit ). According to (29) and (34) this leads to negative values of and ( even diverges as [58]); the data (see below) are consistent with vanishing or positive at small .
A nonvanishing difference in the limit violates the postulated azimuthal symmetry of the source (see discussion before Eq. (29)). It is easy to see that shortlived sources can never be opaque for particles with : the source shrinks to zero before such particles can be reabsorbed. The particular behaviour excluded in [58] is thus anyhow rather unphysical. At larger , on the other hand, the “opacity signal” (leading, if strong enough, to [61]) can be “faked” by other mechanisms: Tomášik found [52, 59] that expanding sources with a boxlike transverse density profile generate exactly such a signature. At the moment it is thus unclear how to uniquely distinguish “opaque” from “transparent” sources.
2.4 NonGaussian Features of the Correlator and Moments
The (dependent) HBT radii provide a full characterization of the twoparticle correlation function only if it is a Gaussian in or, equivalently, if the effective source is a Gaussian in . However, in many physical situations the source is not characterized by just one, but by several distinct length scales. In this case the Gaussian approximation of section 2.2 breaks down.
2.4.1 Resonance decays
The most important physical processes leading to a nonGaussian shape of the correlator are resonance decays [62, 73, 74, 75, 76, 77, 18]. Especially longlived resonances which decay into pions cause a longrange exponential tail in the pion emission function which distorts the twoparticle correlation function at small relative momentum (see example in Figure 4 below). According to [77, 78] the resonances can be classified into three classes:

Shortlived resonances ( MeV) which (especially if heavy) decay very close to their production point. Their most important effect is to add a contribution proportional to their lifetime to the emission duration [74, 77], thereby affecting and in the Cartesian and in the YKP parametrization, but not the transverse radius . They do not spoil the Gaussian parametrization.

Longlived resonances ( MeV), mostly the and hyperons. These resonances travel far outside the original source before decaying. The resulting wide tail in the emission function contributes to the correlator only at very small relative momenta. This region is experimentally inaccessible due to finite twotrack and momentum resolution, and the contribution from this “halo” [76] to the correlator is thus missed in the experiment. The result is an apparent decrease of the correlation strength (i.e. the intercept ). In the measurable range the shape of the correlator is not affected.

The meson ( MeV) is not sufficiently longlived to escape detection in the correlator. Its lifetime is, however, long enough to create a measurable exponential tail in the pion emission function which distorts the shape of the correlator, giving it extra weight at small and destroying its Gaussian form.
In practice the pions from shortlived resonances can thus be simply added to the directly emitted ones into an emission function for the “core” [73, 76]. The “halo” from longlived resonances is accounted for by a reduced intercept parameter
(52) 
where the sum goes over all longlived resonances and is the fraction of pions with momentum stemming from resonance . A correspondingly modified Cartesian parametrization for the correlator reads
(53) 
Pions from decays must, however, be considered explicitly and, if sufficiently abundant, the resulting correlator is no longer well described by the ansatz (53).
In heavyion collisions the resonance fractions are unknown since most resonances cannot be reconstructed in the highmultiplicity environment. Thus in (53) is an additional fit parameter. Its value is very sensitive to nonGaussian distortions in the correlator, and so are the HBT radii extracted from a fit to the function (53). In theoretical studies [77] it was found that differences of more than 1 fm in the fitted HBT radii can occur if the fit is performed with fixed to its theoretical value (52) or if, as done in experiment, is fitted together with the radii. In the latter case resonance contributions (including the ) affect the fitted radii much less than in the former. This difference in procedure may largely explain the consistently larger resonance contributions to the HBT radii found by Schlei et al. [73, 75], compared to the much weaker effects reported in [77]. Whereas Schlei et al. [73, 75] find that resonances, whose decay pions contribute only to the region of small , add considerably to the dependence of and thus contaminate the transverse flow signature, practically no such effect was found in [77].
2.4.2 moments
In view of these systematic uncertainties one may ask for a more quantitative characterization of correlation functions whose shape deviates from a Gaussian. This can be achieved via the socalled moments of the correlator [77]. In this approach the matrix of Cartesian HBT parameters and the correlation strength are calculated from the following integrals:
(54)  
(55) 
Similar expressions exist for the YKP parameters [77]. For a Gaussian correlator this gives the same HBT parameters as a Gaussian fit; for nonGaussian correlators the HBT radius parameters and intercept are defined by (54,55).
Deviations of the correlator from a Gaussian shape are then quantified by higher order moments. Formally they can be obtained as derivatives at the origin of the relative source function which acts as generating function [77]: