From initial-state fluctuations to final-state observables

Available online at www.sciencedirect.com

Nuclear Physics A 904–905 (2013) 59c–66c www.elsevier.com/locate/nuclphysa

From initial-state fluctuations to final-state observables Kevin Dusling Physics Department North Carolina State University Raleigh, NC 27695, USA

Abstract This talk begins with the interpretation of the azimuthally collimated di-hadron correlation with large rapidity separations observed in heavy-ion and high multiplicity proton-proton collisions known as the “ridge”. The physics of the ridge is intimately related to quantum fluctuations responsible for the decoherence of the strongly occupied initial classical fields generated in heavyion collisions. Progress made in the implementation of a program to resum these instabilities and the connection with the final-state observables is discussed. 1. Introduction It is well known by the heavy-ion community that hydrodynamic modeling has been successful in describing many bulk observables [1, 2, 3, 4]. The agreement of viscous hydrodynamic simulations with measured spectra and flow harmonics is strongly suggestive of an early hydrodynamic starting time and a system having a suitably small shear and bulk viscosity [5, 6]. The fact that hydrodynamics works so well may or may not be surprising depending on the conditions necessary for this framework to be applicable. The necessary conditions for hydrodynamics [7] are twofold: • Isotropy: The system must be isotropic or nearly isotropic for ideal or respectively viscous hydrodynamics to apply. More formally, the stress-energy tensor must be of the form T i j ≈ pδi j in the local rest frame of the medium. • Equation of state: The system must have a well-defined equation of state. In other words, the pressure must be a monotonic function of the energy density, p ≈ p(), where small deviations from this relation can be accounted for by bulk viscosity. While thermalization is a sufficient condition for hydrodynamics it surely is not a necessary one. The temperature never enters into the hydrodynamic equations and in principle observables can be computed without introducing the notion of temperature1 . Even if hydrodynamics is successfully able to describe the system it hides a lot of physics. For example, thermalization 1 In practice one must convert from the hydrodynamic variables to the physically observed hadrons. It is at this point that the assumption of local thermal equilibrium (or near-equilibrium in viscous hydrodynamics) is invoked. However, this only requires thermal equilibrium towards the later stages of the collision where the medium decouples. There are observables, such as the pT integrated elliptic flow, which can be related directly to the stress energy tensor, and are therefore independent of the particle content of the theory [8].

0375-9474/$ – see front matter, Published by Elsevier B.V. http://dx.doi.org/10.1016/j.nuclphysa.2013.01.045

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Figure 1: (Left) NLO parton distribution function[10]. (Right) rcBK unintegrated gluon distribution.

may be taking place in space-time regions where hydrodynamics serves as a suitable effective theory. There are still many open questions; how does a system, which we presuppose is initially comprised of classical gluon fields, decohere, isotropize and finally thermalize? The goal of this talk is to demonstrate that we finally have a theoretical framework available that can address these questions. 2. The nucleus pre-collision 2.1. Wavefunction of the proton Before one can begin to address the problem of thermalization one must have a reasonable description of the initial state. There is a surmounting of evidence that the earliest stages of a heavy-ion collision consist of strongly coherent classical gluon fields [9]. Indeed, fits to hardscattering data within the framework of collinear factorization [10] demonstrate the rise in the gluon’s parton distribution function (see left plot of fig. 1) at smaller values of x for a fixed Q2 . At small x and large center of mass energies the nuclear wavefunction can be described within the color glass condensate framework. For example, the right plot of fig. 1 shows the unintegrated gluon distribution function from the solution of the running coupling Balitsky-Kovchegov [11, 12] equation; a large Nc approximation to the full JIMWLK renormalization group equation containing all of the underlying gluon dynamics in the high energy limit. There is a rich phenomenology supporting this picture as demonstrated by the detailed fits to DIS data [13]. One particularly interesting piece of phenomenology is the near side ridge in high-multiplicity proton and heavy-ion collisions to which we now turn. 2.2. Multiparticle production in QCD In order to understand the mechanism behind the near side ridge let us consider the leading order diagrams contributing to the double inclusive cross-section. There are two possibilities: the first is two gluons produced from the same ladder. The kinematics require the two gluons to be produced primarily back-to-back in azimuthal angle. The resulting di-hadron correlation

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Low color charge density (min. bias)

High color charge density (central) g → 1/g

g

g Jet Graph

α4s

Glasma Graph

α6s

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Table 1: Power counting for multiparticle production in QCD in the dilute (left column) and dense (right column) limit of color sources. The top row shows the diagram responsible for mini-jet production (hadrons produced primarily backto-back in azimuthal angle). The bottom row shows the “glasma” diagram responsible for the near-side collimation that is long-range in rapidity known as the ridge.

has a jet-like structure: long range in Δη with collimation on the away side. The second diagram consists of two gluons emitted from different ladders. In this case the gluons are produced nearly isotropically with a slight preference for emission when the two gluons are collinear or back-toback (the signal is symmetric around Δφ = π/2). In the dilute limit (which we associate with min. bias collisions) each vertex in the upper left diagram of figure 1 brings one power of g. The amplitude squared therefore goes as α4s . In the dense limit the occupation number of gluons is of order α−1 s and therefore each connection to the “valence” (larger-x) partons is adjusted from g → 1/g. In the dense limit the cross section for mini-jet production is increased by α−2 s . Performing the same exercise for the so-called “glasma” graph responsible for the ridge, we find that the cross-section in the dilute limit is suppressed as α6s . However, when the sources are highly occupied (as we expect in the most central protonproton or nucleus-nucleus collisions) this graph attains an α−8 s enhancement surpassing all other production mechanisms. The same power counting can be applied to the computation of the triple inclusive cross-section [14] and higher n-particle productions [15].

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p+p Ridge

A+A Ridge

Ridge is generated by an intrinsic correlation present in a single flux tube.

In A+A there are many such flux tubes each with an intrinsic correlation further collimated by transverse flow.

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Including transverse flow in p+p collisions creates a discrepancy with measurements.

Transverse flow is necessary to explain identical measurements in Pb+Pb collisions.

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Table 2: Interpretation of the ridge in p+p (left) and A+A (right) collisions. The experimental data is from the CMS collaboration [16, 17].

2.3. The ridge: from p+p to A+A One may wonder about the relationship between the p+p ridge and the ridge seen in heavyion collisions. As we shall now argue, these are two closely related phenomenon. In both cases the long range rapidity correlation originates as a result of particle production from nearly boost invariant glasma flux tubes. However the mechanism responsible for the collimation of this signal in azimuthal angle is different in p+p and A+A collisions. In p+p collisions the near side collimation is intrinsic to the initial-state particle production mechanism and follows from the evaluation of the Feynman diagrams shown in table 1 with gluon distribution functions strongly peaked around Q s . This intrinsic near-side collimation is indeed present in A+A collisions, however the signal from this mechanism is completely dwarfed by final state hydrodynamic flow. Final state flow further collimates the already present intrinsic correlation, creating a nearside associated yield that is ten times larger in nucleus-nucleus collisions. The close connection between the p+p and A+A ridge is driven home by the PHOBOS data shown in the right plot of figure 2. The correlation clearly persists to rapidity separations of four units. This observation combined with the hypothesis of a strong identification between the longitudinal momentum and position (y ∼ η) dictates that the correlation must have formed early.

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Figure √ 2: (Left) Event display from the CMS collaboration[18] of a high-multiplicity event in proton-proton collisions at s = 7 TeV. (Right) Di-hadron correlation from the PHOBOS experiment[19] in Au+Au collisions showing the ridge extending up to four units in rapidity.

3. Quantum Decoherence While far from obvious, the dynamics of the ridge discussed in the previous sections is intimately related to our understanding of thermalization in heavy-ion collisions. At leading order in the strong coupling constant the di-hadron correlation discussed previously is produced from boost invariant classical fields [20]. However, quantum fluctuations, of which there are two main types [21] modify this picture. The first are zero modes (pη = 0), which can be resummed into the nuclear wavefunction prior to the collision and are responsible for the decoherence of the ridge at rapidity separations Δy  α−1 s [22]. The second are non-zero modes (pη  0) which give rise to secular divergences. Numerical simulations have shown that the occupation number √ of these rapidity dependent fluctuations are unstable [23, 24, 25], and grow with time as ∼ e QS τ. ln2 g−1 . Any perturbative calculation will therefore be spoiled at times of order τmax  Q−1   √ S n In order to properly include quantum corrections all terms behaving as g exp QS τ must be resummed. We will argue that this resummation can be accomplished in a semi-classical framework. In order to motivate this claim, let us look at the quantum mechanical problem of the inverted harmonic oscillator [26]. In this example, the system is initialized with a particle confined to the ground state of a harmonic oscillator (see left diagram of figure 3). At time t = 0 the potential is inverted and we wish to find the time development of the system. Formally, we want to find the expectation value a quantum operator which is a function of position xˆ and momentum pˆ at time t f . It was shown [27] that this problem can be recast in terms of a classical probability distribution f (x, p, t) such that     dx ψ∗ (x, tF )O( xˆ, p)ψ(x, ˆ tF ) = dxd p f (x, p, t f ) O(x, p) + O e−ωt f . (1) O( xˆ, p) ˆ tF =   Thus, at late times, terms of order e−ωt f (which originated from the commutator xˆ, pˆ ) can be ignored. The solution to the inverted harmonic oscillator at late times can then be solved by finding the evolution of a classical probability distribution with an initial condition determined from the ground state wavefunction. These general arguments for the simple harmonic oscillator can be extended to field theory [28] and have proved useful in other contexts. For example; in understanding the phenomenon of reheating during slow-roll inflation [29, 30, 31] or in possibility of elucidating the dynamics of collapsing and exploding Bose-Einstein condensates (known as Bosenova) [32, 33].

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Figure 3: Quantum mechanical problem of the inverted harmonic oscillator. At late times the quantum expectation value can be found from classical solutions of a particle in the inverted potential.

For a quantum field theory the evaluation of a quantum expectation value resumming all leading instabilities amounts to averaging over classical solutions of the field equations of motion over a Gaussian ensemble of initial conditions. For example, the expectation value of the stress energy tensor which is a functional of a classical field A can be written as      μν μν T (x, t)LInst. = Dα ΓA α T LO [A + α](x, t) . (2) The precise form of the Gaussian ensemble, ΓA , depends on the classical background field and the LO operator is evaluated from the classical evolution of the field equations with the quantum fluctuations, α, overlaid on the initial classical field configurations. A first study of the above resummation scheme has been implemented for a scalar φ4 theory [34, 35, 36]. It was found that such a system, when initialized with a non-perturbatively large classical field configuration, reached thermal equilibrium. The resulting system first forms a well defined equation of state (i.e. the pressure is a monotonic function of the energy density) and then evolves in accordance with near ideal hydrodynamics. These observations are demonstrated in figure 4. The functional form of the initial Gaussian spectrum of fluctuations relevant for heavy-ion collisions was derived in [37] and a numerical implementation of the above resummation scheme in this framework is currently underway. 4. Towards Thermalization On time scales larger than the inverse saturation scale (τ  Q−1 s ) we presume the quantum system has decohered, the classical fields are almost linear (g2 f 1) and quasi-particles have formed. At this point a description of the system in terms of a Boltzmann equation is applicable. One of the seminal works within this framework is bottom-up thermalization [38] which demonstrated that thermalization can happen relatively quickly when inelastic processes are taken into account. This work did not include the effect of instabilities [39], which numerical simulations within the Hard Thermal Loop framework [40, 41] have shown to be important in driving the system toward thermalization. There has also been a revival of analytic studies in weak coupling QCD [42, 43]. The main take away message from these works is that thermalization in a weakly coupled gauge theory does indeed occur and happens on the timescale τ ∼ α−5/2 Q−1 S . S

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It has recently been proposed [44], on short time scales, elastic scattering may lead to the formation of a transient Bose-Einstein condensate. This overpopulated stage of the collision may behave as a strongly interacting fluid even though the coupling is weak. While there is a general consensus that condensation occurs in a scalar field theory, which has also been confirmed by numerical simulations[45], the role played by inelastic processes in the QCD case is still not entirely clear. Whether a condensate is observed in classical Yang-Mills is subtle to discern [46, 47, 48]. 5. Outlook I hope this talk convinced the audience that our field has a fairly comprehensive theoretical framework for the various stages of heavy-ion collisions. However, these frameworks have yet to be merged into a single framework relevant for phenomenology. In the meantime progress has still been made in understanding how initial-state fluctuations affect final-state observables [49]. The important role played by color-charge fluctuations (in addition to Glauber fluctuations) has been emphasized and shown to improve the description of higher harmonics [50, 51]. The importance played by thermal fluctuations, which may need to be disentangled from initial-state fluctuations is still a work in progress [52]. To summarize, there is clear evidence that the initial state survives into the final state. The compatibility of the high multiplicity proton ridge with the nucleus-nucleus ridge as well as the consistency of the initial spatial n and final momentum vn anisotropies confirm this hypothesis. The significant amount of progress in understanding the early dynamics since Quark Matter 2011 in Annecy, France will have far reaching consequences for phenomenology. 6. Acknowledgements I would like to thank Adrian Dumitru, Yuri Kovchegov, Thomas Schaefer, Shu Lin, Bjorn Schenke, Prithwish Tribedy and Raju Venugopalan for useful discussions throughout the prepa-

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ration of this talk. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]

J. Adams et al. (STAR Collaboration), Nucl.Phys. A757, 102 (2005), nucl-ex/0501009. K. Adcox et al. (PHENIX Collaboration), Nucl.Phys. A757, 184 (2005), nucl-ex/0410003. I. Arsene et al. (BRAHMS Collaboration), Nucl.Phys. A757, 1 (2005), nucl-ex/0410020. B. Back, M. Baker, M. Ballintijn, D. Barton, B. Becker, et al., Nucl.Phys. A757, 28 (2005), nucl-ex/0410022. P. Romatschke, Int.J.Mod.Phys. E19, 1 (2010), 0902.3663. T. Schafer and D. Teaney, Rept.Prog.Phys. 72, 126001 (2009), 0904.3107. P. B. Arnold, J. Lenaghan, G. D. Moore, and L. G. Yaffe, Phys.Rev.Lett. 94, 072302 (2005), nucl-th/0409068. J.-Y. Ollitrault, Phys.Rev. D46, 229 (1992). F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, Ann.Rev.Nucl.Part.Sci. 60, 463 (2010), 1002.0333. A. Martin, W. Stirling, R. Thorne, and G. Watt, Eur.Phys.J. C63, 189 (2009), 0901.0002. I. Balitsky, Phys.Rev. D75, 014001 (2007), hep-ph/0609105. Y. V. Kovchegov and H. Weigert, Nucl.Phys. A784, 188 (2007), hep-ph/0609090. J. L. Albacete, N. Armesto, J. G. Milhano, and C. A. Salgado, Phys.Rev. D80, 034031 (2009), 0902.1112. K. Dusling, D. Fernandez-Fraile, and R. Venugopalan, Nucl.Phys. A828, 161 (2009), 0902.4435. F. Gelis, T. Lappi, and L. McLerran, Nucl.Phys. A828, 149 (2009), 0905.3234. V. Khachatryan et al. (CMS Collaboration), JHEP 1009, 091 (2010), 1009.4122. CMS-PAS-HIN-11-006 (2011), URL http://cdsweb.cern.ch/record/1353583. W. Li, Mod.Phys.Lett. A27, 1230018 (2012), 1206.0148. B. Alver et al. (PHOBOS Collaboration), Phys.Rev.Lett. 104, 062301 (2010), 0903.2811. A. Dumitru, F. Gelis, L. McLerran, and R. Venugopalan, Nucl.Phys. A810, 91 (2008), 0804.3858. F. Gelis, T. Lappi, and R. Venugopalan, Phys.Rev. D78, 054019 (2008), 0804.2630. K. Dusling, F. Gelis, T. Lappi, and R. Venugopalan, Nucl.Phys. A836, 159 (2010), 0911.2720. P. Romatschke and R. Venugopalan, Phys.Rev.Lett. 96, 062302 (2006), hep-ph/0510121. P. Romatschke and R. Venugopalan, Phys.Rev. D74, 045011 (2006), hep-ph/0605045. K. Fukushima and F. Gelis, Nucl.Phys. A874, 108 (2012), 1106.1396. G. Barton, Annals Phys. 166, 322 (1986). A. H. Guth and S.-Y. Pi, Phys.Rev. D32, 1899 (1985). K. Fukushima, F. Gelis, and L. McLerran, Nucl.Phys. A786, 107 (2007), hep-ph/0610416. D. Son (1996), hep-ph/9601377. S. Y. Khlebnikov and I. Tkachev, Phys.Rev.Lett. 77, 219 (1996), hep-ph/9603378. D. Polarski and A. A. Starobinsky, Class.Quant.Grav. 13, 377 (1996), gr-qc/9504030. E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Nature. 412, 295 (2001). E. A. Calzetta and B. L. Hu, Phys. Rev. A 68, 043625 (2003). K. Dusling, T. Epelbaum, F. Gelis, and R. Venugopalan, Nucl.Phys. A850, 69 (2011), 1009.4363. T. Epelbaum and F. Gelis, Nucl.Phys. A872, 210 (2011), 1107.0668. K. Dusling, T. Epelbaum, F. Gelis, and R. Venugopalan (2012), 1206.3336. K. Dusling, F. Gelis, and R. Venugopalan, Nucl.Phys. A872, 161 (2011), 1106.3927. R. Baier, A. H. Mueller, D. Schiff, and D. Son, Phys.Lett. B502, 51 (2001), hep-ph/0009237. P. B. Arnold, J. Lenaghan, and G. D. Moore, JHEP 0308, 002 (2003), hep-ph/0307325. S. Mrowczynski, A. Rebhan, and M. Strickland, Phys.Rev. D70, 025004 (2004), hep-ph/0403256. A. Rebhan, P. Romatschke, and M. Strickland, Phys.Rev.Lett. 94, 102303 (2005), hep-ph/0412016. A. Kurkela and G. D. Moore, JHEP 1112, 044 (2011), 1107.5050. A. Kurkela and G. D. Moore, JHEP 1111, 120 (2011), 1108.4684. J.-P. Blaizot, F. Gelis, J.-F. Liao, L. McLerran, and R. Venugopalan, Nucl.Phys. A873, 68 (2012), 1107.5296. J. Berges and D. Sexty, Phys.Rev.Lett. 108, 161601 (2012), 1201.0687. A. Kurkela and G. D. Moore, Phys.Rev. D86, 056008 (2012), 1207.1663. S. Schlichting, Phys.Rev. D86, 065008 (2012), 1207.1450. J. Berges, S. Schlichting, and D. Sexty (2012), 1203.4646. G.-Y. Qin, H. Petersen, S. A. Bass, and B. Muller, Phys.Rev. C82, 064903 (2010), 1009.1847. B. Schenke, P. Tribedy, and R. Venugopalan (2012), 1206.6805. B. Schenke, P. Tribedy, and R. Venugopalan, Phys.Rev.Lett. 108, 252301 (2012), 1202.6646. J. Kapusta, B. Muller, and M. Stephanov, Phys.Rev. C85, 054906 (2012), 1112.6405.