The ridge effect and three-particle correlations
Miguel-Angel Sanchis-Lozano, Edward Sarkisyan-Grinbaum
TL;DR
This paper investigates the three-particle correlations in high-energy collisions using a correlated-cluster model, highlighting the necessity of azimuthal cluster correlations to explain the near-side ridge effect.
Contribution
It extends a cluster model to three-particle correlations, demonstrating the importance of azimuthal cluster correlations in understanding the ridge phenomenon.
Findings
Azimuthal cluster correlations are essential for explaining the near-side ridge.
The model successfully describes three-particle correlations in proton-proton and heavy-ion collisions.
Generalizes previous two-particle correlation analysis to higher-order correlations.
Abstract
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.
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Report: IFIC/17-27, FTUV-17-0609
The ridge effect and three-particle correlations
**Miguel-Angel Sanchis-Lozanoa,111Email address: [email protected], Edward Sarkisyan-Grinbaumb,c,222Email address: [email protected]
**
a* Instituto de Física Corpuscular (IFIC) and Departamento de Física Teórica
Centro Mixto Universitat de València-CSIC, Dr. Moliner 50, E-46100 Burjassot, Valencia, Spain
b Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland
c Department of Physics, The University of Texas at Arlington, Arlington, TX 76019, USA*
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.
1 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 quark-gluon 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.
2 Definitions and notations
As usual, two-particle correlations can be studied by means of
[TABLE]
where indices and stand for the set of kinematic variables relative to particles and , respectively. In terms of the rapidity () and the azimuthal angle (), the one-particle density and the two-particle density are defined through
[TABLE]
where denotes the inelastic cross section and the dependence on the transverse momentum () has been integrated out. Still a potential dependence on the integration range should remain, depending on the selected kinematic cuts.
The three-particle rapidity correlation function is defined as:
[TABLE]
where the three-particle density is defined as
[TABLE]
As is well known, three-particle correlations can provide a more sophisticated test than two-particle 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].
2.1 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 ( and , ), use will be made of Dirac’s -functions as in our earlier studies [8]. Then, the two-particle distribution of uncorrelated pairs reads
[TABLE]
and the distribution of correlated pairs can be identified with
[TABLE]
Three-particle correlations are again expressed as a function of the rapidity and azimuthal differences,333Notice that only two rapidity differences are independent: and are chosen as independent, so that . Similarly for the azimuthal variable: and are independent, so that .
[TABLE]
where the shortened notation has been introduced:
[TABLE]
and for the Dirac’s -functions:
[TABLE]
[TABLE]
Non-correlated three-particle distributions correspond to
[TABLE]
According to Eq.(3), a three-particle normalized correlation function depending on the rapidity and azimuthal differences can be defined as
[TABLE]
where the explicit dependence on the rapidity and azimuthal diferences has been omitted in the terms of the r.h.s., and
[TABLE]
while and 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]
[TABLE]
In the following we make use of the expression (12) although the main conclusions would remain the same had we employed Eq.(10) instead.444Additionaly, correlations among so-called event planes [11] (corresponding to different harmonics) have recently emerged as a powerful tool for the analysis of heavy-ion collisions [12, 13]. In this paper, which can be applied to proton-proton collisions as well, we do not consider this analysis.
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 and the particle density from a single cluster , i.e.
[TABLE]
where stands for the average cluster number per collision and denotes the average particle density for single cluster decays. On the other hand, the function 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 single-particle distributions representing the mixed-event background,
[TABLE]
[TABLE]
which suggests to define
[TABLE]
[TABLE]
Next, the two-particle density can be written as
[TABLE]
[TABLE]
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 . Therefore, we conclude for the two-particle density:
[TABLE]
where and stand for correlations stemming from the corresponding two integrals of Eq.(18).
In its turn, the three-particle density can be written as
[TABLE]
[TABLE]
[TABLE]
where we have introduced and . We will write
[TABLE]
where the functions , , encode the rapidity and angular dependence for three-particle correlations in single-cluster production (), double-cluster production (), and triple-cluster production ().
3.1 Factorization hypothesis
As in [8], we apply factorization of the rapidity (longitudinal) and azimuthal (transverse) directions to the -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 \raisebox{-3.69899pt}{~{}\shortstack{>\sim}}~{}1 GeV, we will factorize the rapidity and azimuthal dependences of the above -functions following [8], as
[TABLE]
where the superscripts and 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 small, Eqs.(3.1) 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
[TABLE]
[TABLE]
where and 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 and 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 on is rather weak, i.e. . 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.
[TABLE]
The parameter (\raisebox{-3.69899pt}{~{}\shortstack{<\sim}}~{}1 rapidity units [17]) is usually referred to as the cluster decay (pseudo)rapidity “width”. Regarding the transverse plane, can be seen as another cluster decay width. For small azimuthal angles with respect to the cluster direction, , where and 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:
[TABLE]
[TABLE]
where detailed expressions for the -functions are given in Appendix B.3. For Poisson distribution of clusters, becomes and becomes .
Therefore, the above expression leads to
[TABLE]
The last expression shows that the contribution dominates for large (hence for high-multiplicity events). This is an important feature concerning the correlated-cluster production as discussed later.
In the limit , and keeping the (, ) and (, ) components, Eqs. (73)–(75) of Appendix B.3 read:
- for one cluster:
[TABLE]
[TABLE]
- for two clusters:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
- for three clusters:
[TABLE]
[TABLE]
[TABLE]
Figures 2–4, based on Eqs. (26)–(29), illustrate the above-described interpretation.
In the left panel of Fig.2 we show the contour-plot of as a function of the azimuthal differences and , having fixed . A quite asymmetric two-dimensional plot can be seen, resulting from the existence of two correlation scales: a short-range correlation length (set by single-cluster decay) and a long-range correlation length (set by cluster formation).555Let us remark that this plot agrees quite well with the results presented in [18] based on the framework of the glasma interpretation. One can see that, in fact, long-range azimuthal correlations mainly come from the term, i.e. they are originated by correlated-cluster emission. This important point is in agreement with our main conclusion on the near-side ridge effect in hadronic collisions obtained in our study of two-particle correlations [8]. In a way similar to two-particle azimuthal correlations [8], we conclude here that, in the absence of correlated-cluster emission, no long-range azimuthal three-particle correlations would be seen.
In the right panel of Fig. 2 we show the contour-plot of now as a function of the rapidity differences and , having fixed . 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 for the azimuthal-difference dependence along the diagonal (, left panel) and off the diagonal (, right panel) are shown under the 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.
In Fig. 4, the projections of the three-particle correlation function for the rapidity dependence only (having fixed ) along the diagonal (, left plot) and off the diagonal (, right plot) are shown. Now, as expected, the two Gaussian-like plots are quite similar, reflecting that both are determined by short-distance correlations from single-cluster decays.
5 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 final-state 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.
Acknowledgements
This work has been partially supported by MINECO under grant FPA2014-54459-P, and Generalitat Valenciana under grant PROMETEOII/2014/049. One of us (M.A.S.L.) acknowledges support from IFIC under grant SEV-2014-0398 of the “Centro de Excelencia Severo Ochoa” Programme.
Appendix 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]):
[TABLE]
Upon integration over the cluster rapidity , the function, introduced in Eq.(13), reads
[TABLE]
Hence, for two particles emitted from the two clusters one gets for the longitudinal part of the function, introduced in Eq.(14),
[TABLE]
Upon integration on both rapidities keeping the rapidity interval fixed, one gets
[TABLE]
For two particles stemming from the same cluster with rapidity
[TABLE]
[TABLE]
After integration using the Dirac’s -function, , the above expression leads to
[TABLE]
Notice that drops off in the last expression so that it can be considered as a short-range correlation contribution.
For two particles with rapidity and coming from two (correlated) clusters with rapidities and , respectively, we have (see Eqs. (23))
[TABLE]
[TABLE]
Using again the Dirac -function, one gets , 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:
[TABLE]
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.
[TABLE]
Thus, the distribution for two particles, emitted from the same cluster with azimuthal angle should obey
[TABLE]
for small azimuthal angles. Therefore, regarding the azimuthal dependence we can write
[TABLE]
As for the above rapidity correlations, the dependence on 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 and coming from two (correlated) clusters with azimuthal angles and , we will write
[TABLE]
[TABLE]
that directly leads to
[TABLE]
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 function can be written as
[TABLE]
and
[TABLE]
Note that only retains dependence on the rapidity variable for isotropic cluster production in the transverse plane,
[TABLE]
Appendix 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 function,
[TABLE]
where is defined in Eq. (31). Upon integration over all three rapidities and keeping the rapidity intervals and fixed, one gets
[TABLE]
Since , the above equation can be also expressed in terms of all three rapidity intervals
[TABLE]
For three particles stemming from the same cluster with rapidity ,
[TABLE]
[TABLE]
After integration using the Dirac’s -functions (see Eq.(8), the above expression leads to
[TABLE]
Notice again that drops off in the last expression so that it can be referred to as a short-range correlation contribution.
Again, using the interdependence, the above equation can be expressed as
[TABLE]
For three particles coming from two clusters emitted in a correlated way (see Eqs. (23)), we have three possibilities:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Using again the Dirac’s -functions (8), one gets respectively
[TABLE]
[TABLE]
and
[TABLE]
Similar to a single-cluster case, the drops off in the these expressions which therefore can be referred to as other short-range correlation contributions.
For three particles with rapidities , and coming from three (correlated) clusters with rapidities , and , respectively, we have (see Eqs. (23)):
[TABLE]
[TABLE]
As commented in the main text, the Gaussian depending on the sum of rapidities 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 -functions, one gets
[TABLE]
which corresponds to a long-range correlations contribution in rapidity phase-space.
In sum, we get several pieces with different behaviours in rapidity space:
[TABLE]
[TABLE]
[TABLE]
B.2 Azimuthal dependence
The distribution for three particles, emitted from the same cluster with azimuthal angle should obey
[TABLE]
for small azimuthal differences. Therefore, regarding the azimuthal dependence we can write
[TABLE]
For three particles with azimuthal angles and coming from two clusters with azimuthal angles and emitted in a correlated way (see Eqs. (23)), three possible topologies are:
[TABLE]
and
[TABLE]
and
[TABLE]
The above integrals lead to
[TABLE]
[TABLE]
[TABLE]
which can be rewritten as
[TABLE]
[TABLE]
[TABLE]
For three clusters with azimuthal angles , and , all of them emitted in a correlated way one has (see Eqs. (23)):
[TABLE]
[TABLE]
[TABLE]
which can be rewritten as
[TABLE]
For two clusters (out of three) emitted in a correlated way, the three possible topologies are:
[TABLE]
[TABLE]
which can be rewritten as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
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
[TABLE]
leading to
[TABLE]
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.
[TABLE]
similarly to two-particle correlations, Eq. (38), the final experssions are as follows:
- for one cluster:
[TABLE]
[TABLE]
*- for two clusters:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
- for three clusters:
[TABLE]
[TABLE]
[TABLE]
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