The ridge effect and three-particle correlations

Pseudorapidity and azimuthal three-particle correlations are studied based on a correlated-cluster model of multiparticle production. The model provides a common framework for correlations in proton-proton and heavy-ion collisions allowing easy comparison with the measurements. It is shown that azimuthal cluster correlations are definitely required in order to understand three-particle correlations in the near-side ridge effect. This is similar to the explanation of the ridge phenomenon found in our previous analysis of two-particle correlations and generalizes the model to higher-order correlations.


Introduction
Correlation measurements have proven to be extremely useful to understand multiparticle production providing stringent tests for theoretical models of soft hadronic dynamics [1][2][3][4]. Being sensitive to the last stage of particle interactions (freeze-out) in hadronic collisions, correlations become of special interest in high-energy heavy-ion collisions for the study of new matter formation, such as quarkgluon plasma and its properties [5], as well as in the search for hidden sectors beyond the Standard Model [6,7].
Moreover, somewhat unexpected collective effects have been observed both in heavy-ion and proton-proton collisions at RHIC and LHC experiments [5]. Striking ridge-like and dip structures show up in the two-particle correlation spectrum. In particular, the near-side ridge phenomenon corresponds to azimuthal collimated particle production extending over a large pseudorapidity interval, whose physical origin is still unclear especially for proton-proton collisions where no collective flow is expected.
In a previous paper [8] we studied two-particle correlations for the analysis of the near-side ridge in hadron-hadron collisions. The effect was shown to have a natural explanation provided that clusters are produced in a correlated way in the collision transverse plane. On the other hand, if particles are emitted correlated to each other, this effect should also hold for higher-rank particle correlations. Therefore, the analysis of three-particle correlations attracts as well high interest providing further information on hadroproduction mechanisms [1][2][3][4].
In this work we extend our study [8] to three-particle (pseudo)rapidity and azimuthal correlations in the framework of a correlated-cluster model (CCM), providing compact final expressions with predictions under some simple physical assumptions, which can be directly tested by experiments.

Definitions and notations
As usual, two-particle correlations can be studied by means of where indices 1 and 2 stand for the set of kinematic variables relative to particles 1 and 2, respectively. In terms of the rapidity (y) and the azimuthal angle (φ), the one-particle density ρ and the two-particle density ρ 2 are defined through ρ(y, φ) = 1 σ in d 2 σ dydφ ; ρ(y 1 , y 2 , φ 1 , φ 2 ) = 1 σ in where σ in denotes the inelastic cross section and the dependence on the transverse momentum (p T ) has been integrated out. Still a potential dependence on the p T integration range should remain, depending on the selected kinematic cuts. The three-particle rapidity correlation function is defined as: where the three-particle density is defined as As is well known, three-particle correlations can provide a more sophisticated test than twoparticle correlations for the study of partonic dynamics (e.g. jets) in proton-proton collisions [9], or the emergence of a new state of matter in heavy-ion collisions [10].

Correlations as function of (pseudo)rapidity and azimuthal differences
In order to match our theoretical approach to experimental results in terms of (pseudo)rapidity and azimuthal differences (∆y ij = y i − y j and ∆φ ij = φ i − φ j , i, j = 1, 2, 3, i = j), use will be made of Dirac's δ-functions as in our earlier studies [8]. Then, the two-particle distribution of uncorrelated pairs reads and the distribution of correlated pairs can be identified with Three-particle correlations are again expressed as a function of the rapidity and azimuthal differences, 3 where the shortened notation has been introduced: ∆y, ∆φ for ∆y ij , ∆φ ij , y = (y 1 , y 2 , and for the Dirac's δ-functions: Non-correlated three-particle distributions correspond to b 3 ( ∆y, ∆φ) = d y d φ δ(∆y) δ(∆φ) ρ(y 1 , φ 1 ) ρ(y 2 , φ 2 ) ρ(y 3 , φ 3 ) .
According to Eq.(3), a three-particle normalized correlation function depending on the rapidity and azimuthal differences can be defined as where the explicit dependence on the rapidity and azimuthal diferences has been omitted in the terms of the r.h.s., and while s 231 and s 132 terms are obtained straigthforwardly by permutation. On the other hand, sometimes a simplified version of the three-particle correlation function, also of common use in experimental analyses of data, is given by [5] In the following we make use of the expression (12) although the main conclusions would remain the same had we employed Eq.(10) instead. 4 3 Two-and three-particle correlations in the CCM It is generally accepted that particle production in soft hadronic interactions occurs via an intermediate step of decaying strings/clusters/fireballs yielding final-state particles [1,4]. It should be noted that the "cluster" concept has to be understood in a broad sense, i.e. a group of particles with some correlated properties, probably comming from a common ancestor. We keep the same notation as in our paper [8] for two-particle correlations. Hence, the single particle density can be expressed as the convolution of the cluster density ρ (c) (y c , φ c ) and the particle density from a single cluster ρ (1) (y, φ; y c , φ c ), i.e.
where N c stands for the average cluster number per collision andρ (1) denotes the average particle density for single cluster decays. On the other hand, the function E 1 (y, φ) encodes the expected dependence on the rapidity and azimuthal variables of the emitted particles.
For uncorrelated particle pairs and triplets we introduce the product of the two and three singleparticle distributions representing the mixed-event background, which suggests to define Next, the two-particle density can be written as ) . The first term on the r.h.s. corresponds to the emission of secondaries from a single cluster while the second term corresponds to the emission of the two particles from two distinct clusters, whose density is noted as ρ (c) 2 (y c1 , φ c1 , y c2 , φ c2 ). Therefore, we conclude for the two-particle density: where E s (y 1 , φ 1 , y 2 , φ 2 ) and E 2 s (y 1 , φ 1 , y 2 , φ 2 ) stand for correlations stemming from the corresponding two integrals of Eq. (18).

Factorization hypothesis
As in [8], we apply factorization of the rapidity (longitudinal) and azimuthal (transverse) directions to the E-functions. Factorization of production cross sections and decay distributions into transverse and longitudinal momentum parts is a hypothesis widely used in many high-energy physics processes.
Although not yet rigorioulsy proven from first principles, it works very well when contrasted with experimental data, especially for high transverse momentum where such a hypothesis can also be theoretically justified to some extent.
Since the ridge phenomenon shows up for particles with transverse momentum typically of order > ∼ 1 GeV, we will factorize the rapidity and azimuthal dependences of the above E-functions following [8], as where the superscripts L and T denote the longitudinal and transverse parts, respectively. According to different (hydrodynamic, cascade) models, fluctuating initial conditions should lead to decorrelations of the orientation of initial event-planes in heavy ion collisions. In particular, the authors of [14,15] argue that event-to-event early state fluctuations (termed "torque effect" in [16]) should lead to a (pseudo)rapidity-azimuthal factorization breaking for well separated (pseudo)rapidity particles in heavy ion collisions. Notice, however, that keeping the (pseudo)rapidity difference ∆y ij small, Eqs.(22) should remain reliable. Moreover, any observed deviation from our later predictions on rapidity and azimuthal correlations for larger (pseudo)rapidity separations might be interpreted as a hint of the existence of such kind of torque effect.
On the other hand, as usual in cluster models we shall adopt Gaussian distributions in rapidity and azimuthal spaces for both cluster density and particle density from clusters, as developed below. Thus, we shall write the single, two-cluster and three-cluster densities as where δ cy and δ cφ stand for the rapidity and azimuthal cluster correlation lengths, respectively. Let us remark that Eqs. (23) can be regarded as parameterizations especially suitable to determine the near-ridge effect using the CCM. The rapidity Gaussians with arguments y c1 + y c2 and y c1 + y c2 + y c3 can be seen as a consequence of (partial) longitudinal momentum conservation for two-cluster and three-cluster poduction. The azimuthal conditions are implemented in the Gaussians following [8], in order to include collinear emission of particles in the near-side ridge effect. On account of the plateau structure of multiplicity distribution in pseudorapidity phase space, one may assume that the dependence of ρ (c) (y c , φ c ) on y c is rather weak, i.e. δ 2 cy 1. On the other hand, the particle density from single cluster decay, the rapidity and azimuthal dependence can be approximately expressed in terms of Gaussians, i.e.
The parameter δ y ( < ∼ 1 rapidity units [17]) is usually referred to as the cluster decay (pseudo)rapidity "width". Regarding the transverse plane, δ φ can be seen as another cluster decay width. For small Figure 1: Illustrative picture of three clusters produced in a primary hadron collision at the origin of the transverse plane with azimuthal angles φ c1 , φ c2 and φ c3 , decaying into final-state particles.
azimuthal angles with respect to the cluster direction, where v T and γ T denote the cluster velocity in the transverse plane and its associated Lorentz factor, as shown in [8].
In Fig. 1 we illustrate the particle emission from three clusters produced at the same primary hadron collision leading to different elliptic shapes due to different Lorentz boosts. Clusters are assumed to be correlated both in rapidity and azimuth according to Eqs. (23) (for more details see Appendices).
4 Interpretation of the near-side ridge effect according to CCM Rewriting the three-particle correlation function by adding the terms of contributions from one, two and three cluster productions, one gets: 3 ( ∆y, ∆φ) + s 3 ( ∆y, ∆φ) + s . Therefore, the above expression leads to The last expression shows that the h (3) contribution dominates for large N c (hence for high-multiplicity events). This is an important feature concerning the correlated-cluster production as discussed later.
In the right panel of Fig. 2 we show the contour-plot of c 3 (∆y 12 , ∆y 13 , ∆φ 12 , ∆φ 13 ) now as a function of the rapidity differences ∆y 12 and ∆y 13 , having fixed ∆φ 12 = ∆φ 13 = 0. A quite different behaviour can be appreciated as compared to the azimuthal dependence on the left. Aside the central peak corresponding to dominating short-range correlations from single clusters decays, it is now a rather structureless plot, in agreement with some early experimental measurements [19].
In Fig. 3, the projection plots of the three-particle correlation function c 3 (∆y 12 , ∆y 13 , ∆φ 12 , ∆φ 13 ) for the azimuthal-difference dependence along the diagonal (∆φ 12 = ∆φ 13 , left panel) and off the     diagonal (∆φ 12 = −∆φ 13 , right panel) are shown under the ∆y 12 = ∆y 13 = 0 condition. Again a different behaviour can be readly observed in both plots, as the on-diagonal correlation length is appreciably longer than the off-diagonal correlation length. We interpret this difference as an indication that the former is dominated by cluster correlations whose correlation length is larger than for particles emitted from the same cluster that mainly populate the off-diagonal line.

Summary
A study of the ridge phenomenon is presented for three-particle correlations, extending our previous work in the context of the correlated-cluster model (CCM). Gaussians are employed for azimuth and (pseudo)rapidity distributions, encoding short-and long-range correlations for clusters and finalstate hadrons. The CCM provides a common framework to explain the ridge effect in proton-proton, proton-nucleus and heavy-ion collisions. As obtained for two-particle correlations in our earlier study, we conclude again that azimuthal correlations among clusters are definitely needed to explain the ridge phenomenon. A Two-particle correlations A.1 (Pseudo)rapidity dependence We will assume throughout that both clusters and particles stemming from clusters obey Gaussian distributions in rapidity space (for more detais on this Appendix we refer the reader to [8]): Upon integration over the cluster rapidity y c , the E L 1 (y) function, introduced in Eq. (13), reads Hence, for two particles emitted from the two clusters one gets for the longitudinal part of the E b function, introduced in Eq. (14), Upon integration on both rapidities keeping the rapidity interval ∆y = y 1 − y 2 fixed, one gets For two particles stemming from the same cluster with rapidity y c After integration using the Dirac's δ-function, δ(∆y − y 1 + y 2 ), the above expression leads to Notice that δ cy drops off in the last expression so that it can be considered as a short-range correlation contribution.
For two particles with rapidity y 1 and y 2 coming from two (correlated) clusters with rapidities y c1 and y c2 , respectively, we have (see Eqs. (23)) . (36) Using again the Dirac δ-function, one gets e (2) s (∆y) ∼ const., which corresponds to a long-range correlation contribution.
In sum, we get two pieces with different behaviours (short-range and long-range correlations) in rapidity space:

A.2 Azimuthal dependence
In addition to the hypothesis of isotropically decaying clusters in their own rest frame, we will assume axial symmetry for cluster production in the transverse plane, i.e.
Thus, the distribution for two particles, emitted from the same cluster with azimuthal angle φ c should obey for small azimuthal angles. Therefore, regarding the azimuthal dependence we can write As for the above rapidity correlations, the dependence on δ cφ drops off in this expression, so that it can be referred as a short-range correlation term.
On the other hand, we will assume that clusters are produced in a correlated way according to Eqs.(23). Hence for two particles with azimuthal angles φ 1 and φ 2 coming from two (correlated) clusters with azimuthal angles φ c1 and φ c2 , we will write , that directly leads to which corresponds to a long-range correlation contribution.

A.3 Final expressions
In sum, we find that the short-range and long-range pieces of the e s (∆y, ∆φ) function can be written as .
Note that e b (∆y, ∆φ) only retains dependence on the rapidity variable for isotropic cluster production in the transverse plane, .

B Three-particle correlations B.1 (Pseudo)rapidity dependence
Similarly to the above two-particle calculations in Appendix A.1, for three particles emitted from three clusters one gets for the longitudinal part of the E b function, where E L 1 (y) is defined in Eq. (31). Upon integration over all three rapidities and keeping the rapidity intervals ∆y 12 = y 1 − y 2 and ∆y 13 = y 1 − y 3 fixed, one gets .
Notice again that δ cy drops off in the last expression so that it can be referred to as a short-range correlation contribution. Again, using the ∆y interdependence, the above equation can be expressed as For three particles coming from two clusters emitted in a correlated way (see Eqs. (23)), we have three possibilities: cy (y 1 − y 2 ) 2 + 2δ 2 y (y 2 1 + y 2 2 + y 2 3 + y 1 y 3 + y 2 y 3 − y 1 y 2 ) 2δ 2 y (3δ 2 y + 2δ 2 cy ) .
Using again the Dirac's δ-functions (8) and Similar to a single-cluster case, the δ cy drops off in the these expressions which therefore can be referred to as other short-range correlation contributions.
As commented in the main text, the Gaussian depending on the sum of rapidities y c1 + y c2 + y c3 stems from the requirement of partial (longitudinal) momentum conservation. It takes into account different topologies for cluster emission once integrated upon their rapidities.
Applying again the Dirac's δ-functions, one gets which corresponds to a long-range correlations contribution in rapidity phase-space.
In sum, we get several pieces with different behaviours in rapidity space:

B.2 Azimuthal dependence
The distribution for three particles, emitted from the same cluster with azimuthal angle φ c should obey for small azimuthal differences. Therefore, regarding the azimuthal dependence we can write (61) For three particles with azimuthal angles φ 1 and φ 2 coming from two clusters with azimuthal angles φ c1 and φ c2 emitted in a correlated way (see Eqs. (23)), three possible topologies are: and and The above integrals lead to , which can be rewritten as .
Notice that, conversely to the (pseudo)rapidity dependence, we consider pairwise azimuthal correlations among clusters as potential contributions to the ridge effect. Therefore, no two-cluster contributions similar to the three ones just above to appear for pseudorapidities.
For three clusters, all of them independently emitted, one gets leading to e (3) s ( ∆φ) ∼ const.

B.3 Final expressions
Taking into account that in the transverse direction, in addition to the hypothesis of isotropically decaying clusters in their own rest frame, we assume axial symmetry for cluster production i.e.