# Ridges in heavy ion collisions and pp scatterings at the LHC

Klaus Werner, Iu Karpenko, T. Pierog

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Klaus Werner, Iu Karpenko, T. Pierog. Ridges in heavy ion collisions and pp scatterings at the LHC. 27th Winter Workshop on Nuclear Dynamics (WWND 2011), Feb 2011, Winter Park, United States. pp.012012, 10.1088/1742-6596/316/1/012012. in2p3-00691537

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# Ridges in heavy ion collisions and pp scatterings at the LHC

K. Werner <sup>(a)</sup>, Iu. Karpenko <sup>(b)</sup>, T. Pierog <sup>(c)</sup>

<sup>(a)</sup> SUBATECH, University of Nantes – IN2P3/CNRS– EMN, Nantes, France

<sup>(b)</sup> Bogolyubov Institute for Theoretical Physics, Kiev 143, 03680, Ukraine

<sup>(c)</sup> Karlsruhe Institute of Technology (KIT) - Campus North, Inst. f. Kernphysik, Germany

**Abstract.** We discuss recent developments in the theoretical description of ultrarelativistic collisions, concerning an event-by-event treatment of the hydrodynamical evolution in heavy ion collisions and also in high multiplicity pp scatterings at the LHC. The initial conditions are flux-tubes, which are formed following elementary (multiple) scatterings. We show that this picture leads in a natural way to the so-called ridge structures, observed in heavy ion and proton-proton collisions

Starting point of our consideration is a multiple scattering model in the spirit of the Gribov-Regge approach[1]. Here, one does not mean simply multiple hard scatterings, the elementary processes correspond to complete parton ladders, which means hard scatterings plus initial state radiation. In this case, this elementary process carries an important fraction of the available energy, and therefore we treat very carefully the question of energy sharing in the multiple scattering process. Open and closed ladders have to be considered in order to have a consistent quantum mechanical treatment. The corresponding graphs are squared, and we employ cutting rule techniques and Markov chains to obtain finally partial cross sections. The cut parton ladders are identified with longitudinal color fields or flux tubes, treated via relativistic string theory.

In case of very high energy pp collisions (at the LHC) or heavy ion scatterings already at RHIC, many flux tubes overlap and produce high energy densities. Let us consider the energy density at an early time in a Au-Au scattering at RHIC, as obtained from an EPOS simulation [1]. In fig. 1, we plot the energy density at different values of space-time rapidity $η_s$, as a function of the transverse coordinates $x$ and $y$. We observe a very bumpy structure concerning the $x-y$-dependence, whereas the variation with $η_s$ is small. There are in particular peaks in the $x-y$-plane, which show up at the same position at different values of $η_s$. So we have sub-flux-tubes which exhibit a long range structure in the longitudinal variable $η_s$. In fig. 1, we clearly identify several sub-flux-tubes, with a typical width of the order of a fermi. This is exactly the width we obtain if we compute the initial energy density in proton-proton scattering at the LHC. This means, if a hydrodynamic treatment is justified for Au-Au collisions at RHIC, it is equally justified for pp scattering at the LHC, provided the energy densities are high enough, which seems to be the case.

We therefore employ a sophisticated hydrodynamical scenario (for details see [1]), for both heavy ions and proton-proton scatterings at the LHC [2, 3, 4], with initial conditions obtained from a flux tube approach (EPOS), compatible with the string model, used since many years for elementary collisions (electron-positron, proton-proton), and the color glass condensate picture
<figure><img src="image_2.png" /><figcaption>Figure 1. Energy density in central Au-Au scattering at 200 GeV.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Radial flow velocity at a proper time $\tau = 4.6$ fm/c, at a space-time rapidities $\eta_s = 0$ and $\eta_s = 1.5$.</figcaption></figure>

[5]. The equation-of-state is compatible with lattice gauge results of ref. [6]. We use a hadronic cascade procedure after hadronization from the thermal system at an early stage [7, 8].

<figure><img src="image_4.png" /><figcaption>Figure 3. Dihadron $\Delta\eta - \Delta\phi$ correlation in a central Au-Au collision at 200 GeV.</figcaption></figure>
In ref. [1], we test the approach by investigating all soft observables of heavy ion physics, in case of AuAu scattering at 200 GeV. Here, we are going to discuss some selected (and interesting) topics. In fig. 1, we see a complicated structure of the initial energy density in the transverse plane, but this structure is longitudinally translational invariant (same structure at different values of η_s). The equations of hydrodynamics preserve this translational invariance, and transport it to different quantities, as the radial flow, see fig. 2. As a consequence, particles emitted from different longitudinal positions get the same transverse boost, when their emission points correspond to the azimuthal angle of a common flow peak position. And since longitudinal coordinate and (pseudo)rapidity are correlated, one obtains finally a strong Δη – Δφ correlation, as seen in fig. 3, where we plot the dihadron correlation $dN/d\Delta\eta d\Delta\phi$, with Δη and Δφ being respectively the difference in pseudorapidity and azimuthal angle of a pair of particles. Here, we consider trigger particles with transverse momenta between 3 and 4 GeV/c, and associated particles with transverse momenta between 2 GeV/c and the $p_t$ of the trigger, in central Au-Au collisions at 200 GeV. Our ridge is very similar to the structure observed by the STAR collaboration [9].

<figure><img src="image_5.png" /><figcaption>**Figure 4.** (Color online) Two particle correlation function $R$ versus $\Delta\eta$ and $\Delta\phi$ for high multiplicity events in $pp$ collisions at 7 TeV.</figcaption></figure>

The CMS collaboration published recently results [10] on two particle correlations in $\Delta\eta$ and $\Delta\phi$, in $pp$ scattering at 7 TeV. Most remarkable is the discovery of a ridge-like structure around $\Delta\phi = 0$, extended over many units in $\Delta\eta$, referred to as “the ridge”, in high multiplicity $pp$ events. A similar structure has been observed in heavy ion collisions at RHIC, as discussed earlier, and there is little doubt that the phenomenon is related to the hydrodynamical evolution of matter. This “fluid dynamical behavior” is actually considered to be the major discovery at RHIC.

So does $pp$ scattering provide as well a liquid, just ten times smaller than a heavy ion collision? It seems so! We showed recently [11] that if we take exactly the same hydrodynamic approach which has been so successful for heavy ion collisions at RHIC [1], and apply it to $pp$ scattering, we obtain already very encouraging results compared to $pp$ data at 0.9 TeV. In this paper, we apply this fluid approach, always the same procedure, to understand the 7 TeV results. In fig. 4, we show that our hydrodynamic picture indeed leads to a near-side ridge, around $\Delta\phi = 0$, extended over many units in $\Delta\eta$. One should note that the correlation functions are defined and normalized as in the CMS publication, so we can say that our “ridge” is quite close in
<figure><img src="image_6.png" /><figcaption>**Figure 5.** (Color online) Initial energy density (upper panel) and radial flow velocity at a later time (lower panel) for a high multiplicity pp collision at 7 TeV at a space-time rapidity $η_s = 0$ (left) and $η_s = 1.5$ (right).</figcaption></figure>

shape and in magnitude compared the experimental result. The experimental high multiplicity bin corresponds to about 7 times average, whereas in our calculation (extremely demanding concerning CPU power) “high multiplicity” refers to 5.3 times average (we actually trigger on events with 10 elementary scatterings). We cannot go beyond at the moment.

It is easy to understand the origin of the ridge, in a hydrodynamical approach based on flux tube initial conditions. Imagine many (say 20) flux tubes of small transverse size (radius ≈ 0.2 fm), but very long (many units of space-time rapidity $η_s$). For a given event, their transverse positions are randomly distributed within the overlap area of the two protons. Even for zero impact parameter (which dominated for high multiplicity events), this randomness produces azimuthal asymmetries, as shown in fig. 5, upper panel. The energy density obtained from the overlapping flux tubes (details will be discussed later) shows an elliptical shape. And since the flux tubes are long, and only the transverse positions are random, we observe the same asymmetry at different longitudinal positions ($η = 0$ and $η = 1.5$ in the figure). So we observe a translational invariant azimuthal asymmetry!

If one takes this asymmetric but translational invariant energy density as initial condition for a hydrodynamical evolution, the translational invariance is conserved, and in particular translated into other quantities, like the flow. In fig. 5, lower panel, we show the radial flow velocity at a later time again at the two space-time rapidities $η_s = 0$ (left) and $η_s = 1.5$ (right). In both cases, the flow is more developed along the direction perpendicular to the principal axis of the initial energy density ellipse. This is a very typical fluid dynamical phenomenon, referred to as elliptical flow.

Finally, particles are produced from the flowing liquid, with a preference in the direction of large flow. This preferred direction is therefore the same at different values of $η_s$. And since $η_s$
and pseudorapidity $\eta$ are highly correlated, one observes a $\Delta\eta\Delta\phi$ correlation, around $\Delta\eta = 0$, extended over many units in $\Delta\eta$: a particle emitted a some pseudorapidity $\eta$ has a large chance to see a second particle at any pseudorapidity to be emitted in the same azimuthal direction.

**Acknowledgments** This research has been carried out within the scope of the ERG (GDRE) "Heavy ions at ultra-relativistic energies. Iu.K. acknowledges partial support by the State Fund for Fundamental Researches of Ukraine (Agreement of 2011) and National Academy of Sciences of Ukraine (Agreement of 2011). T.P. and K.W. acknowledge partial support by a PICS (CNRS) with KIT (Karlsruhe).

[1] K. Werner, Iu. Karpenko, T. Pierog, M. Bleicher, K. Mikhailov, Phys. Rev. C 82, 044904 (2010)

[2] K. Werner, Iu. Karpenko, T. Pierog, Phys. Rev. Lett. 106:122004, 2011

[3] K. Werner, Iu. Karpenko, T. Pierog, M. Bleicher, K. Mikhailov, Phys. Rev. C 83, 044915 (2011)

[4] K. Werner, Y. Karpenko, T. Pierog, Il Nuovo Cimento Vol. 34 C, 85 (2011)

[5] L.D. McLerran, R. Venugopalan, Phys. Rev. D 49, 2233 (1994). ibid. D49, 3352 (1994); D 50, 2225 (1994)

[6] S. Borsanyi et al., arXiv:1007.2580

[7] M. Bleicher et al., J. Phys. G25 (1999) 1859

[8] H. Petersen, J. Steinheimer, G. Burau, M. Bleicher and H. Stocker, Phys. Rev. C78 (2008) 044901

[9] STAR Collaboration, B. Abelev et al., Phys. Rev. C 80 (2009) 64192

[10] CMS Collaboration, JHEP 1009:091, 2010

[11] K. Werner, Iu. Karpenko, T. Pierog, M. Bleicher, K. Mikhailov, arXiv:1010.0400, Phys. Rev. C 83, 044915 (2011)