Quark-Gluon Plasma 4
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Quark-Gluon Plasma 4 edited by
Rudolph C Hwa University of Oregon, USA
Xin-Nian Wang Lawrence Berkeley National Laboratory, USA
World Scientific NEW JERSEY
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QUARK-GLUON PLASMA 4 Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-4293-28-0 ISBN-10 981-4293-28-8
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Preface
The beginning of this decade was also the beginning of a new era when RHIC opened up a new vista in the physics of dense matter. The major theoretical insights gained soon after were reviewed in the previous volume of this series. Now as the decade draws to its end and as RHIC moves on to its mature phase, it is remarkable that new discoveries are still being made and the theoretical understanding shows no sign of reaching a saturation point. The new ideas that continue to spring up are to be tested by future experiments in the next energy domain offered at LHC. A notable claim that has been made is that RHIC has produced the most perfect fluid achievable, which the cover picture of this volume is intended to symbolize. As a springboard to LHC, it is an opportune moment to make an objective assessment of what has been learnt and what to expect next. We are happy to have assembled a group of experts who have all made significant contributions to the various aspects of the subject on many fronts, although regretfully we were not able to cover some fields that are also important. To all the contributors we are very grateful for their time and energy spent in giving the community comprehensive descriptions of the present status of QGP at what will be an auspicious beginning of the next era.
Rudolph C. Hwa Eugene, Oregon Xin-Nian Wang Berkeley, California November 2009
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Contents Preface
v
Energy Loss in a Strongly Coupled Thermal Medium and the Gauge-String Duality
1
S. S. Gubser, S. S. Pufu, F. D. Rocha and A. Yarom 1. 2. 3. 4. 5.
6.
7. 8. 9.
Introduction . . . . . . . . . . . . . . . . . . . . . . The Thermal Medium as a Black Hole . . . . . . . . The Trailing String . . . . . . . . . . . . . . . . . . The Magnitude of the Drag Force . . . . . . . . . . The Perturbed Einstein Equations . . . . . . . . . . 5.1. Metric perturbations in axial gauge . . . . . . 5.2. Boundary conditions . . . . . . . . . . . . . . . 5.3. The boundary stress-energy tensor . . . . . . . Asymptotics . . . . . . . . . . . . . . . . . . . . . . 6.1. Long distance asymptotics . . . . . . . . . . . 6.2. Short distance asymptotics . . . . . . . . . . . Numerical Results for the Holographic Stress Tensor Hadronization, Jet-Broadening, and Jet-Splitting . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Notation . . . . . . . . . . . . . . . . .
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Quarkonium at Finite Temperature
1 2 4 8 13 16 22 24 28 29 33 38 42 51 52 61
A. Bazavov, P. Petreczky and A. Velytsky 1. 2. 3. 4.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pNRQCD at Finite Temperature . . . . . . . . . . . . . . . . . . . . . . Basics of Lattice Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . Correlation Functions of Static Quarks in Lattice Gauge Theory . . . . 4.1. Static meson correlators . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Lattice results on static meson correlators . . . . . . . . . . . . . . 4.3. Color singlet correlator in SU(2) gauge theory at low temperatures 4.4. Color singlet free energy in the deconfined phase . . . . . . . . . . 4.5. Color adjoint free energy . . . . . . . . . . . . . . . . . . . . . . . 5. Quarkonium Spectral Functions . . . . . . . . . . . . . . . . . . . . . . 5.1. Meson correlators and spectral functions . . . . . . . . . . . . . . vii
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61 62 64 66 66 69 71 74 76 78 78
Quark-Gluon Plasma 4
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5.2. Lattice formulations for charmonium physics . . . . . . . . . . . . 5.3. Bayesian analysis of meson correlators . . . . . . . . . . . . . . . . 5.4. Charmonium spectral functions at zero temperature . . . . . . . . 5.5. Charmonium correlators at finite temperature . . . . . . . . . . . 5.6. Charmonium spectral functions at finite temperature . . . . . . . 5.7. Charmonium correlators and spectral functions at finite momenta 5.8. Bottomonium spectral functions at zero temperature . . . . . . . . 5.9. Bottomonium at finite temperature . . . . . . . . . . . . . . . . . 5.10. Zero modes contribution . . . . . . . . . . . . . . . . . . . . . . . 6. Potential Models at Finite Temperature . . . . . . . . . . . . . . . . . . 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 80 . 82 . 83 . 86 . 91 . 94 . 95 . 97 . 99 . 103 . 105
Heavy Quarks in the Quark-Gluon Plasma
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R. Rapp and H. van Hees 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Heavy-Quark Interactions in QCD Matter . . . . . . . . . . . . 2.1. Heavy-quark diffusion in the quark-gluon plasma . . . . . 2.2. Perturbative QCD approaches . . . . . . . . . . . . . . . 2.3. Non-perturbative interactions . . . . . . . . . . . . . . . . 2.4. String theoretical evaluations of heavy-quark diffusion . . 2.5. Comparison of elastic diffusion approaches . . . . . . . . 2.6. Collisional versus radiative energy loss . . . . . . . . . . . 2.7. D mesons in the hadronic phase . . . . . . . . . . . . . . 3. Heavy-Quark Observables in Relativistic Heavy-Ion Collisions 3.1. Relativistic Langevin simulations . . . . . . . . . . . . . . 3.2. Background medium in heavy-ion collisions . . . . . . . . 3.3. Initial conditions and hadronization . . . . . . . . . . . . 3.4 Model comparisons of heavy-quark spectra at RHIC . . . 3.5. Heavy-meson and electron observables . . . . . . . . . . . 3.6. Viscosity? . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Heavy Quarkonia in Medium . . . . . . . . . . . . . . . . . . . 4.1. Spectral properties of quarkonia in the QGP . . . . . . . 4.2. Quarkonium production in heavy-ion collisions . . . . . . 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Viscous Hydrodynamics and the Quark Gluon Plasma
111 117 119 122 128 139 141 144 147 149 150 153 156 159 166 172 174 176 185 197 207
D. A. Teaney 1. Introduction . . . . . . . . . . . . . . . . . . . 1.1. Experimental overview . . . . . . . . . . . 1.2. An interpretation of elliptic flow . . . . . 2. Elliptic Flow — Measurements and Definitions
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Contents
3. 4.
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2.1. Measurements and definitions . . . . . . . . . . . . . . . . . 2.2. Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The eccentricity and fluctuations . . . . . . . . . . . . . . . 2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Shear Viscosity in QCD . . . . . . . . . . . . . . . . . . . . Hydrodynamic Description of Heavy Ion Collisions . . . . . . . . 4.1. Ideal hydrodynamics . . . . . . . . . . . . . . . . . . . . . . 4.2. Ideal Bjorken evolutions and three dimensional estimates . 4.3. Viscous Bjorken evolution and three dimensional estimates 4.4. The applicability of hydrodynamics and η/s . . . . . . . . . 4.5. Time evolution . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Second order hydrodynamics . . . . . . . . . . . . . . . . . 4.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinetic Theory Description . . . . . . . . . . . . . . . . . . . . . Viscous Hydrodynamic Models of Heavy Ion Collisions . . . . . 6.1. Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Corrections to the hydrodynamic flow . . . . . . . . . . . . 6.3. Convergence of the gradient expansion . . . . . . . . . . . . 6.4. Kinetic theory and hydrodynamic simulations . . . . . . . . 6.5. Particle spectra . . . . . . . . . . . . . . . . . . . . . . . . . Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . .
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Hadron Correlations in Jets and Ridges Through Parton Recombination R. C. Hwa 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2. Hadronization by Recombination . . . . . . . . . . . . . 2.1. A historical perspective . . . . . . . . . . . . . . . 2.2. Shower partons . . . . . . . . . . . . . . . . . . . . 2.3. Parton distributions before recombination . . . . . 3. Large Baryon/Meson Ratios . . . . . . . . . . . . . . . 3.1. Intermediate pT in heavy-ion collisions . . . . . . . 3.2. Cronin effect . . . . . . . . . . . . . . . . . . . . . 3.3. Forward production in dAu collisions . . . . . . . 3.4. Forward production in AuAu collisions . . . . . . 3.5. Recombination of adjacent jets at LHC . . . . . . 4. Ridgeology Phenomenology of Ridges . . . . . . . . . . 4.1. Experimental features of ridges . . . . . . . . . . . 4.2. Recombination of enchanced thermal partons . . . 4.3. Trigger from the ridge . . . . . . . . . . . . . . . . 4.4. Dependence of ridge formation on trigger azimuth 5. Azimuthal Anisotropy . . . . . . . . . . . . . . . . . . .
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210 214 216 218 218 223 224 224 230 231 232 234 239 240 244 246 247 249 250 255 260
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267 269 269 272 274 275 275 278 280 282 283 284 285 288 290 292 296
Quark-Gluon Plasma 4
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5.1. Effects of ridge formation at low pT . . . . . 5.2. Effects of shower partons at intermediate pT 5.3. Breaking of quark number scaling . . . . . . 6. Hadron Correlation in Dijet Production . . . . . . 6.1. Distribution of dynamical path length . . . . 6.2. Near-side and away-side yields per trigger . . 6.3. Medium effects on dijets . . . . . . . . . . . . 6.4. Symmetric dijets and tangential jets . . . . . 6.5. Unsymmetric dijets and tomography . . . . . 7. Conclusion . . . . . . . . . . . . . . . . . . . . . .
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Elliptic Flow: A Study of Space-Momentum Correlations in Relativistic Nuclear Collisions
296 299 301 302 302 305 307 311 312 314
323
P. Sorenson 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Two decades in time and five decades in beam energy 1.2. Initial geometry: the reaction plane and eccentricity . 2. Review of Recent Data . . . . . . . . . . . . . . . . . . . . 2.1. Differential elliptic flow . . . . . . . . . . . . . . . . . 2.2. High pT . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Multiply strange hadrons and heavy flavor . . . . . . 2.4. Fluctuations and correlations . . . . . . . . . . . . . . 2.5. Scaling observations . . . . . . . . . . . . . . . . . . . 3. Confronting the Hydrodynamic Paradigm with RHIC Data 3.1. Transport model fits . . . . . . . . . . . . . . . . . . . 3.2. Viscous hydrodynamics . . . . . . . . . . . . . . . . . 3.3. Fluctuating initial conditions . . . . . . . . . . . . . . 3.4. Addressing uncertainties . . . . . . . . . . . . . . . . . 4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Predictions for the Heavy-Ion Programme at the Large Hadron Collider
323 326 327 332 335 339 343 346 351 359 362 364 366 368 368
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N. Armesto 1. Introduction . . . . . . . . . . . . . . . . . . . . . 2. Qualitative Expectations . . . . . . . . . . . . . . 3. Bulk Observables . . . . . . . . . . . . . . . . . . . 3.1. Multiplicities . . . . . . . . . . . . . . . . . . 3.2. Collective flow . . . . . . . . . . . . . . . . . 3.3. Hadrochemistry at low transverse momentum 3.4. Correlations . . . . . . . . . . . . . . . . . . 3.5. Fluctuations . . . . . . . . . . . . . . . . . .
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Contents
4. Hard and Electromagnetic Probes . . . . . 4.1. Particle production at large transverse 4.2. Heavy quarks and quarkonia . . . . . 4.3. Photons and dileptons . . . . . . . . . 5. pA Collisions . . . . . . . . . . . . . . . . . 6. Summary and Discussion . . . . . . . . . .
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ENERGY LOSS IN A STRONGLY COUPLED THERMAL MEDIUM AND THE GAUGE-STRING DUALITY
STEVEN S. GUBSER∗ , SILVIU S. PUFU† , FABIO D. ROCHA‡ and AMOS YAROM§ Joseph Henry Laboratories, Princeton University Princeton, NJ 08540, USA ∗
[email protected] †
[email protected] ‡
[email protected] §
[email protected]
We review methods developed in the gauge-string duality to treat energy loss by energetic probes of a strongly coupled thermal medium. After introducing the black hole description of the thermal medium, we discuss the trailing string behind a heavy quark and the drag force that it implies. We then explain how to solve the linearized Einstein equations in the presence of the trailing string and extract from the solutions the energy density and the Poynting vector of the dual gauge theory. We summarize some efforts to compare these calculations to heavy ion phenomenology.
1. Introduction A long-standing hope, as yet unrealized, is that quantum chromodynamics (QCD) will be reformulated as a string theory. The gauge-string duality1–3 provides the closest approach to that goal so far attained. It provides useful computational methods for studying strongly coupled gauge theories. The theory that is most accessible via these methods is N = 4 super-Yang-Mills theory (SYM) in the limit of a large number of colors and large ’t Hooft coupling.a Aspects of the progress in using the gauge-string duality to understand QCD have recently been reviewed at a pedestrian level.4 The aim of the current article is a more focused review of efforts to understand energy loss by energetic probes of a thermal medium in a strongly coupled gauge theory, such as SYM, which has a string theory dual; and to review how energy loss in SYM can be compared to energy loss in QCD. At least in simple cases like SYM, the response of an infinite, static, strongly coupled thermal medium to an energetic probe can be presumed to be hydrodynamical far from the probe, because hydrodynamic perturbations are the only longwavelength modes available. In an infinite, interacting thermal medium, there is no aN
= 4 super-Yang-Mills theory is a gauge theory whose matter content consists of gluons, four Majorana fermions in the adjoint representation of the gauge group, and six real scalars, also in the adjoint representation. All the fields are related to one another by the N = 4 supersymmetry, which completely fixes the Lagrangian once the gauge group and gauge coupling are chosen. Our interest is in the gauge group SU (N ). 1
2
S. S. Gubser et al.
radiation, because there are no asymptotic states. In a strongly coupled medium, it is not clear that there is a gauge-invariant distinction between collisional energy loss and radiative energy loss. So the main questions are: (1) What is the rate of energy loss from an energetic probe? (2) What is the hydrodynamical response far from the energetic probe? (3) What gauge-invariant information can be extracted using the gauge-string duality about the non-hydrodynamic region near the probe? (4) Do the rate and pattern of energy loss have some meaningful connection to heavy ion phenomenology? In Sec. 2 we briefly review the dual description of the thermal state of SYM as an AdS5 -Schwarzschild black hole. The reader interested in a more extensive discussion of the AdS/CFT duality is referred to various reviews in the literature.5–7 In Sec. 3 we explain how to describe heavy quarks in N = 4 SYM using strings in AdS5 , and in Sec. 4 we extract the drag force acting on the quark via an “obvious” and “alternative” identification of parameters between SYM and QCD. Also in Sec. 4, we consider how the string theory estimates of drag force relate to the measured nuclear modification factor for heavy quarks. The response of the stress tensor to the motion of the quark is dual to the metric perturbations around an AdS black hole. These are studied in Sec. 5, where we also discuss how the metric perturbations map into the stress tensor of the plasma. In Sec. 6 we provide analytic approximations to the stress tensor both near to the moving quark and far from it. The full numerical solution is described in Sec. 7, and its application to heavy-ion phenomenology can be found in Sec. 8. In Appendix A we provide a glossary of mathematical notations used in the main text. 2. The Thermal Medium as a Black Hole N = 4 SU (N ) super-Yang-Mills theory at finite temperature can be described in terms of N D3-branes near extremality.8 D3-branes are 3 + 1 dimensional objects on which strings may end.9, 10 Each string can end on one of the N D3-branes, which eventually gives rise to the SU (N ) gauge symmetry of SYM.11 Having a non-vanishing tension, D-branes themselves warp the spacetime around them. The near horizon geometry produced by the D3-branes is AdS5 -Schwarzschild times a five-sphere, threaded by N units of flux of the self-dual Ramond-Ramond five-form of type IIB supergravity.12 For most calculations of interest to us in this review, one can ignore the five-sphere and work just with the five non-compact dimensions. Our starting point is the Einstein equations of type IIB supergravity in the non-compact directions, which can be recovered from the action Z √ 1 12 5 Sbulk = 2 d x −G R + 2 . (1) 2κ L
Here κ is the five-dimensional gravitational coupling, G = det Gµν , and L is the radius of the S 5 . Standard relations based ultimately on the quantized charge of
Energy Loss and the Gauge-String Duality
3
the D3-brane lead to L3 = κ2
N 2π
2
.
(2)
AdS5 -Schwarzschild is a solution of the Einstein equations following from (1). Its line element is L2 dz 2 2 µ ν 2 2 ds = Gµν dx dx = 2 −h(z)dt + d~x + , (3) z h(z) where the “blackening function” h(z) is given by h(z) = 1 −
z4 4 . zH
(4)
In this coordinate system, the conformal boundary of AdS5 is located at z = 0. The AdS5 -Schwarzschild solution has a horizon at z = zH , whose temperature is 1 T = . (5) πzH According to the gauge-string duality, the temperature of the horizon (5) is also the temperature of the thermal medium in the dual gauge theory.3 This medium is infinite and static. Its energy density equals the mass per unit coordinate volume, d3 x, of the black hole, and its pressure p can also be straightforwardly computed as minus the free energy of the black hole. To leading order in the number of colors N and the ’t Hooft coupling λ, one finds π2 2 4 =p= N T . 3 8
(6)
These relations reflect the conformal invariance of N = 4 super-Yang-Mills theory, which is exact even at finite N and λ. If we send zH → ∞, or equivalently T → 0, we end up with a pure AdS5 geometry and no black hole. Perturbations of the AdS5 -Schwarzschild black hole with wavelengths much longer than 1/T are described by relativistic fluid dynamics,13, 14 and the viscosity is known to be remarkably small15 : η 1 = , s 4π
(7)
again to leading order in the limit of large N and λ. In summary: A thermal medium of SU (N ) N = 4 super-Yang-Mills theory can be represented as a black hole in AdS5 in the limit of large N and large ’t Hooft coupling. Relations between the radius of AdS5 and the five-dimensional gravitational coupling, between the temperature and the position of the horizon, between the energy density and the temperature, and between the viscosity and the entropy density, can all be derived starting from ten-dimensional type IIB string theory.
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3. The Trailing String N = 4 super-Yang-Mills theory has no fundamentally charged quarks. Instead, its field content is the gluon and its superpartners under N = 4 supersymmetry: namely, four Majorana fermions and six real scalars, all in the adjoint representation of SU (N ). The AdS5 description of SYM makes no direct reference to these colored dynamical fields. The magic of AdS/CFT is to replace the strong coupling dynamics of open strings, whose low-energy quanta participate in the gauge theory, with the gravitational dynamics of closed strings (gravitons, for example) in a weakly curved background. For example, as we saw in Sec. 2, a finite temperature bath in the gauge theory is replaced by a black hole in AdS5 . The absence of colored degrees of freedom in the gravitational theory makes it less than obvious how to discuss energetic colored probes of the medium. A clue comes from the treatment of Wilson loops in the gauge-string duality.1, 16 A static, infinitely massive, fundamentally charged quark can be represented as a string hanging straight down from the boundary of AdS5 , at z = 0, into the horizon. There is no contradiction with the previous statement that N = 4 super-Yang-Mills has no fundamentally charged quarks, because the quark is an external probe of the theory, not part of the theory. An anti-fundamentally charged quark is represented in the same way as a quark, except that the string runs the other way. (In type IIB string theory, strings are oriented, so this statement makes sense.) If a quark and an anti-quark are both present, it is possible for the strings running down into AdS5 to lower their total energy by connecting into a shape like a catenary. From the point of view of the boundary theory, the string configuration gives rise to an attractive potential with a 1/r dependence, as conformal invariance says it must. Let’s go back to a single isolated quark — which makes sense in N = 4 even at zero temperature because there is no confinement. The string dual to the quark can be described by its embedding in AdS5 . The two-dimensional spacetime manifold swept out by the string is called a worldsheet and can be parameterized by two coordinates. If we use t and z to parameterize the string worldsheet, then the string in AdS5 , or in AdS5 -Schwarzschild, dual to a static quark at ~x = 0, is described by the equation static quark:
~x(t, z) = 0 .
(8)
This is a solution to the string equations of motion with boundary conditions such that the string endpoint on the boundary is stationary. By symmetry, (8) is the only possible solution with these boundary conditions. In the absence of a medium, it must be that the description of a moving quark can be obtained by performing a Lorentz boost on the description of a static quark. If the boost is in the x1 direction, and we continue to use t and z to parameterize the worldsheet, then the equations describing the string dual to a quark moving in the absence of a medium are quark moving in vacuum:
x1 (t, z) = vt ,
~x⊥ (t, z) = 0 ,
(9)
Energy Loss and the Gauge-String Duality
5
where v is the velocity of the quark and ~x⊥ = (x2 , x3 ).b Given that (8) solves the equations of motion of classical string theory, it’s guaranteed that (9) does too, because the equations are invariant under the boost in the x1 direction, and the line element (3) with h = 1 is invariant under any boost acting on the (t, ~x) coordinates. When h 6= 1, signaling the presence of a medium, a moving quark is not the same as a static one. Naively, one might nevertheless try to represent the moving quark in terms of the string shape described in (9). Let’s see why this is problematic. String dynamics is defined in terms of the metric on the worldsheet. We chose to parameterize the worldsheet by t and z, but any coordinates σ α = (σ 1 , σ 2 ) could have been chosen. For a general embedding xµ = xµ (σ 1 , σ 2 ) of a string worldsheet into spacetime, the worldsheet metric is gαβ =
∂xµ ∂xν Gµν . ∂σ α ∂σ β
(10)
What (10) says is that times and distances along the string are measured the same way as in the ambient spacetime. Using σ α = (t, z) and the ansatz (9), one immediately finds L2 −h + v 2 0 gαβ = 2 . (11) 1 0 z h This shows that the z direction on the worldsheet is always spacelike and that the t direction is timelike only when h > v 2 . Using (4), this condition is equivalent to p 4 z < z∗ ≡ zH 1 − v 2 . (12)
The part of the string worldsheet with z > z∗ is purely spacelike, which means that the string is locally moving faster than the speed of light. This does not make sense if we are aiming to describe the classical dynamics of a string moving in real Minkowski time. There is a simpler way to arrive at (12): any trajectory of a point particle in the bulk of AdS5 -Schwarzschild with x1 = vt must be spacelike — that is, the speed of the particle will exceed the local speed of light — if the inequality z < z∗ is violated. This conclusion holds even when there are other components of the velocity, but the inequality is sharp only in the case where there aren’t. Physically, what we learn from (12) is that it is harder and harder to move forward as one approaches the horizon. This evokes the idea that there must be some drag force from the medium. But how does one get at that drag force? An answer was provided by two groups,17, 18 and closely related work on fluctuations appeared at the same time.19 The string does not hang straight down from the quark: rather, it trails out behind it. If the shape is assumed not to change as b It
would be more proper to use (x1 , x2 , x3 ) in place of (x1 , x2 , x3 ), and reserve the notation xµ for Gµν xν . However, it simplifies notation to set xi = xi for i = 1, 2, 3, and we will do this consistently.
6
S. S. Gubser et al.
the quark moves forward, and if it respects the SO(2) symmetry rotating the ~x⊥ coordinates, then it must be specified by a small variant of (9): x1 (t, z) = vt + ξ(z)
(13)
for some function ξ(z). If we insist that the quark’s location on the boundary is x1 = vt, then we must have ξ → 0 as z → 0. To determine ξ(z), one must resort to the classical equations of motion for the string. These follow from the action Z √ 1 d2 σ −g , (14) Sstring = − 0 2πα
where g = det gαβ . The parameter α0 is related to L and the ’t Hooft coupling by L4 = λα02 .
(15)
2 Here we define λ = gYM N , so that a quantity analogous to the coupling αs in QCD is αYM = λ/4πN . The classical equations of motion following from the action (13) take the form
∇α pα µ = 0 ,
(16)
where 1 αβ g Gµν ∂β xν (17) 2πα0 is the momentum current on the worldsheet conjugate to the position xµ . Plugging (13) into (17), one straightforwardly finds that s πξ dξ h − v2 = , (18) 4 L 2 dz h z 4 h − πξ pα µ ≡ −
where πξ is a constant of integration. In order for ξ(z) to be real, the right hand side of (18) must be real. There are three ways this can happen in a manner consistent with the assumption of steady-state behaviorc : • One can choose πξ = 0. This leads to ξ = 0, which shows that (9) is formally a solution of the equations of motion. But it is not a physical solution — at least, not in the context of classical motions of a string — because of the problem with the signature of the worldsheet metric. The action for this solution is complex. • One can arrange for the worldsheet never to go below z = z∗ , and choose πξ small enough that the denominator inside the square root in (18) is always positive. There is indeed a one-parameter family of such solutions, and they describe a heavy quark and anti-quark in a color singlet state propagating c Technically,
there is a fourth way, but its significance is obscure to us. A string can lead down to z = z∗ in the shape of the trailing string, (20), and then turn around and retrace its path back up again. The energy localized at the kink must grow linearly with time, which means that this configuration is not quite a steady-state solution.
Energy Loss and the Gauge-String Duality
7
without drag (at the level of the current treatment), one behind the other. Similar states were studied by other groups,20–22 but they do not capture the dynamics of a single quark propagating through the plasma, so we do not consider them further here. • One can choose v L2 p L2 (19) πξ = ± 2 h(z∗ ) = ± √ 2 , z∗ 1 − v 2 zH so that the denominator inside the square root in (18) changes sign at the same value of z as the numerator, namely z = z∗ , rendering the ratio inside the square root everywhere positive and finite. Choosing the sign that makes πξ positive means that the string trails out in front of the quark instead of behind it. Although this is technically a solution, it does not describe energy loss and should be discarded. Choosing πξ negative leads to the trailing string solution that we are interested in. Equation (18) then straightforwardly leads to 1 − iy 1+y zH v ξ=− log + i log , (20) 4i 1 + iy 1−y where we have introduced a rescaled depth variable, z y= . zH
(21)
There is another way to justify the choice of the minus sign, not the plus sign, in (19)23 : The solution (20) is non-singular at the future event horizon, whereas the solution one would get with the opposite sign choice is singular. To understand this point, it is convenient to pass to Kruskal coordinates, defined implicitly by the equations UV = −
1 − y −2 tan−1 y e 1+y
V = −e4t/zH . U
(22)
In the region outside the horizon, U < 0 and V > 0, while in the region inside the future horizon, U > 0 and V > 0. The trailing string solution (13), with ξ given by (20), can be extended to a non-singular solution over the union of these two regions: v x1 = log V + v tan−1 y . (23) 2 Thus, the logarithmic singularity in ξ at y = 1 (meaning z = zH ) is a singularity not at the future horizon, which is at U = 0, but at the past horizon, which is at V = 0. Reversing the sign choice in (19) would lead to a solution that is singular at the future horizon but not the past horizon. Causal dynamics in the presence of a black hole horizon can generally be described in terms of functions which are smooth at the future horizon. At the level of our presentation, it has been assumed rather than demonstrated that the trailing string is a stable, steady-state configuration representing the late
8
S. S. Gubser et al.
time behavior of a string attached to a moving quark on the boundary of AdS5 Schwarzschild. In fact, this has been fairly well checked.17, 24 The description we have given of the trailing string is not limited to Schwarzschild black holes in AdS5 . One may extend this analysis to various other black hole geometries which asymptote to AdS5 near their boundary.25–30 These geometries describe theories which are deformations of SYM. The literature also includes a discussion of the distribution of energy along the string31 and an interpretation of the shape of the string in terms of a rapid cascade of strongly coupled partons.32 4. The Magnitude of the Drag Force As we explained in the previous section, an infinitely massive, fundamentally charged quark moving at speed v in the x1 direction through an infinite, static, thermal medium of N = 4 super-Yang-Mills theory can be described at strong coupling in terms of the trailing string solution (20). The quark cannot slow down because it has infinite mass. However, it does lose energy and momentum at a finite, calculable rate. In five-dimensional terms, this energy can be thought of as flowing down the string toward the black hole horizon. In four-dimensional terms, energy and momentum emanates from the quark and eventually thermalizes. To calculate the four-momentum ∆pm delivered from the quark to the bath over a time ∆t, one can integrate the conserved worldsheet current pα m of spacetime energy-momentum over an appropriate line-segment I on the worldsheet. I should cover a time interval ∆t, and it can be chosen to lie at a definite depth z0 in AdS5 . The four-momentum ∆pm isd Z √ ∆pm = − dt −g pz m . (24) I
Because the trailing string is a steady-state configuration, four-momentum is lost at a constant rate: √ dpm = − −g pz m . (25) dt In particular, the drag force can be defined as Fdrag =
√ L2 v dp1 √ = − −gpz 1 = − . 2 0 dt 2πzH α 1 − v 2
(26)
Using (5) and (15), one obtains Fdrag d There
√ π λ 2 v =− T √ . 2 1 − v2
(27)
is an explicit minus sign in (24) which doesn’t appear in the analogous equation of one of the original works.18 This is due to use of the z variable, which increases as one goes deeper into AdS5 , instead of the r = L/z variable, which increases as one goes out toward the boundary.
Energy Loss and the Gauge-String Duality
9
If, instead of an infinitely massive quark, we consider a quark with finite but large mass m, then using the standard relativistic expression mv p= √ (28) 1 − v2
leads to
Fdrag = −
√ π λ 2p T . 2 m
(29)
It has been explained17 that (28) receives corrections when a finite mass quark is described as a string ending at a definite depth z = z∗ on a D7-brane. While these corrections are interesting, it would take us too far from the main purpose of this review to give a proper explanation of how the D7-branes modify the physics. Our discussion is formal because we derived the result (27) in the strict m = ∞ limit and then applied it to finite mass quarks. From (29) it is clear that the drag force causes the momentum of a quark to fall off exponentially: p(t) = p(0)e−t/tquark
where
2 m tquark = √ . π λ T2
(30)
In order to make a physical prediction for QCD, we must plug in sensible values for m, λ, and T . The effective quark mass in the thermal medium is already non-trivial to specify precisely, but mc = 1.5 GeV for charm and mb = 4.8 GeV for bottom are reasonably representative values which were used in a recent phenomenological study.33 We will review here two approaches34 to specifying λ and T . The first approach, used earlier in a calculation of the jet-quenching parameter qˆ from a lightlike Wilson loop in N = 4 super-Yang-Mills theory,35 is to identify the temperature TSYM with the temperature TQCD , and then identifying the gauge coupling gYM of super-Yang-Mills with the gauge coupling gs of QCD evaluated at temperatures typical of RHIC. Because of the proximity of the confinement transition, gs has substantial uncertainty. A standard choice is αs = 0.5, corresponding to gs ≈ 2.5. With the number of colors N set equal to 3, one finds λ ≈ 6π. We will refer to this as the “obvious scheme.” The second approach, called the “alternative scheme,” is based on two ideas. The first idea is that it may make more sense to compare SYM to QCD at fixed energy density than fixed temperature. SYM has ∝ T 4 , and so does QCD, approximately: this approximation is surprisingly good for T ≥ 1.2Tc , according to lattice data.36 But the constant of proportionality is about 2.7 times bigger for SYM than for QCD.e That is, SYM has about 2.7 times as many degrees of freedom as QCD above the confinement transition. So SYM = QCD implies TSYM ≈√TQCD /(2.7)1/4 . This identification leads to a suppression of Fdrag by a factor of 2.7 relative to e This mismatch has previously been stated34 as a factor of 3 rather than 2.7. Some uncertainty exists on both the SYM and the QCD sides, because of finite coupling effects and time discretization, respectively; but 2.7 is probably closer to the true figure.
10
S. S. Gubser et al. Table 1. The obvious and alternative schemes for comparing SYM and QCD.
Scheme
TQCD MeV
TSYM MeV
QCD GeV/fm3
SYM GeV/fm3
λ GeV
mc GeV
mb fm
tc fm
tb
obvious alternative
250a 250a
250 195
5.6b 5.6b
15 5.6
6π 5.5
1.5 1.5
4.8 4.8
0.69 2.1
2.2 6.8
a
We set TQCD = 250 MeV because this is a typical temperature scale for heavy ion collisions at √ sNN = 200 GeV.
b
We use /T 4 ≈ 11 for QCD.
the obvious scheme. The second idea behind the alternative scheme is that the ’t Hooft parameter λ in string theory can be determined by comparing the static force between a quark and an anti-quark, as calculated in string theory, to the same force calculated in lattice gauge theory. The string theory calculation is based on a U-shaped string connecting the quark and the anti-quark. This string pulls on the static quarks in a fashion that is similar to how the trailing string pulls on a moving quark. The lattice calculation is based on computing the excess free energy due to the presence of an external quark and anti-quark in a thermal bath. There is a significant difficulty: in the simplest string theory calculation, based only on the Ushaped string, the force between the quark and anti-quark vanishes for separations larger than some limiting distance r∗ , and this distance is quite small: r∗ ≈ 0.24 fm when TSYM ≈ 195 MeV (corresponding to TQCD = 250 MeV). It has been pointed out37 that exchange of closed strings between two long strings describing the quark and anti-quark at separations r > r∗ contribute to the quark-anti-quark force at the same order in N as the U-shaped string. Unfortunately, it is hard to compute the contribution of closed string exchange. The approach34 is therefore to match the U-shaped string computation to lattice data38 near the limiting distance r∗ . The result of this matching is λ ≈ 5.5. As we show in Table 1, heavy quark relaxation times tc and tb are remarkably short when one uses the obvious scheme, and somewhat larger in the alternative scheme. The uncertainties in tc and tb are substantial: even if one accepts the ideas behind the alternative scheme, one should probably regard the resulting relaxation times as uncertain by a factor of 1.5. An experimental study39 favors a model40 in which tc is roughly 4.5 fm at TQCD = 250 MeV, as estimated from plots from a detailed exposition of that model.41 This seems to indicate that the string theory estimates of tc and tb , even in the alternative scheme, are too short. However, the results of a recent phenomenological study33 favor a range of parameters that is consistent with the string theory predictions translated to QCD using the alternative scheme. Let us briefly review the recent study.33 The starting point is the Langevin equation, which in the Itˆ o discretization scheme takes the form ∆~x(t) =
p ~ ∆t E
~ . ∆~ p(t) = −Γ~ p∆t + ξ(t)
(31)
Energy Loss and the Gauge-String Duality
11
~ Here ξ(t) is a stochastic force, assumed to be Gaussian and uncorrelated from one time-step to the next. The strength of the stochastic force is related to the drag coefficient Γ by demanding that the relativistic Maxwell-Boltzmann distribution p is preserved by the time evolution (31). Ordinary relativistic kinematics, E = p~ 2 + m2 , are assumed. It is also assumed that Γ=γ
T2 , m
(32)
where γ is a dimensionless quantity with no p dependence. Evidently, tquark = 1/Γ. The temperature in (32) is TQCD , whereas the temperature in (30) is TSYM . Comparing these two equations, one finds that the string theory prediction is √ 2 ( 6.8 obvious scheme π λ TSYM γ= = (33) 2 TQCD 2.2 alternative scheme. The alternative scheme value in (33) is fractionally larger than the one quoted in the study under discussion,33 due to the use here of the factor 2.7 for the ratio of degrees of freedom between SYM and QCD, as compared to 3 in previous work.34 (It is a numerical coincidence that the dimensionless factor γ is the same, in the alternative scheme, as tc in femtometers when TQCD = 250 MeV.) The next step of the study33 is to compare Langevin dynamics of heavy quarks in a hydrodynamically expanding plasma to PHENIX39 and STAR42 data on the nuclear modification factor RAA for non-photonic electrons — meaning electrons and positrons coming from decays of heavy-quark mesons. Because of the treatment of hadronization, the theoretical results are deemed trustworthy only when the transverse momentum pT of the non-photonic electron is at least 3 GeV.f For fairly central collisions (impact parameter b = 3.1 fm), agreement between theory (using the alternative scheme) and experiment is best for γ between 1 and 3: see Fig. 1. Thus the prediction (33) of string theory in the alternative scheme can reasonably be said to agree with data to within the uncertainties of the calculations. These uncertainties stem in large part from the difficulty of comparing SYM to QCD; however, it is also clear that the treatment of hadronization is a significant hurdle.33 Fluctuations of the trailing string19, 23, 24, 43 provide direct access to the stochastic forces in (31). In the non-relativistic limit, the size of these forces, relative to the drag force, is exactly what is needed to equilibrate to a thermal distribution. Indeed, the original calculations of drag force17, 18 and stochastic forces19 were done independently. For relativistic quarks, the stochastic forces are enhanced by powers of 1/(1 − v 2 ), including enhancement of longitudinal stochastic forces by the startlingly large factor 1/(1 − v 2 )5/4 . This is larger by 1/(1 − v 2 )3/4 than what is needed to equilibrate to a thermal distribution. Another issue is that the correlation f The
electron carries only a fraction of the pT of the charm quark that led to its production. This fraction varies, but a reasonable rule-of-thumb value is 1/2.
12
S. S. Gubser et al.
1.8 1.6
(a) b=3.1fm, c+b->e±
RAA
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
1
2
3
4 5 6 p [GeV]
γ = 0.3 γ = 1.0 γ = 3.0 PHENIX(0-10%) STAR(0-5%)
7
8
9 10
T
Fig. 1. (Color online) Comparison33 between experimental data for gold-gold collisions with impact parameter b = 3.1 fm and theoretical predictions based on Langevin dynamics of heavy quarks in a hydrodynamically expanding plasma. The open circles correspond to central collisions in a PHENIX experiment,39 while the solid circles come from a STAR experiment.42 Theoretical predictions are plotted for γ = 0.3 (red), 1 (blue), and 3 (green), with γ is as defined in (32). Reprinted with permission from the authors from Y. Akamatsu, T. Hatsuda, and T. Hirano, “Heavy Quark Diffusion with Relativistic Langevin Dynamics in the Quark-Gluon Fluid,” 0809.1499.
time tcor for these stochastic forces grows with velocity: based on results for the relevant Green’s function24 one may estimate tcor ≈
1 . πT (1 − v 2 )1/4
(34)
If we use tcor < tquark as a criterion of validity for the Langevin dynamics, then we get a limit on the Lorentz boost factor: 1 4 m2 √ < . λ T2 1 − v2
(35)
Approximately the same inequality arises from demanding that when the string ends on a D7-brane, its endpoint should not move superluminally. The same inequality, but with m replaced by some fixed scale µ, also arises from demanding that the worldsheet horizon23, 24 should be at a depth in AdS5 corresponding to a scale µ where the dynamics of QCD is far from weakly coupled. Plugging in numbers in the √ alternative scheme, with µ = 1.2 GeV and TQCD = 250 GeV, one finds that 1/ 1 − v 2 < ∼ 30. This inequality should be understood as quite a rough estimate, because of the quadratic dependence on the quantity µ which is only qualitatively defined. For a charm quark, the corresponding limit on the transverse
Energy Loss and the Gauge-String Duality
13
momentum of the non-photonic electron is pT < ∼ 20 GeV. This is well in excess of the highest momentum for which there are statistically significant data; moreover, for√pT more than a few GeV, bottom quarks dominate, and for them the bound < 1/ 1 − v 2 < ∼ 30 translates to pT ∼ 70 GeV for non-photonic electrons. The upshot is that heavy quarks at RHIC do not obviously fall outside the regime of validity of a self-consistent Langevin treatment based on the trailing string — except for the troublesome scaling of longitudinal stochastic forces as 1/(1 − v 2 )5/4 , whose consequences, we feel, are ill-understood. In another study,44 it is argued that energy loss in the perturbative and strongly coupled regimes have experimentally distinguishable signatures for large enough transverse momenta, pT mb , which will be attained at the LHC. A convenient observable that distinguishes between predictions of string theory and perturbative QCD is the ratio of the nuclear modification factors for b and c quarks, Rcb =
c RAA (pT ) . b RAA (pT )
(36)
In particular, the drag force formula (29) implies that at large enough pT , one has mc cb RAdS ≈ ≈ 0.3 , (37) mb where we used the bottom and charm masses quoted right after equation (30). In contrast, perturbative QCD predicts that pcb cb RpQCD ≈1− (38) pT at large pT , where pcb is a relevant momentum scale. So according to perturbative QCD, Rcb should approach unity at large pT . At sufficiently large pT , the trailing string treatment presumably fails, and perturbative QCD presumably is correct. But as discussed following (35), it is difficult to give a good estimate of the characteristic value of pT where the trailing string fails. Absent a reliable estimate of the characteristic pT , the upshot is that if the measured Rcb is significantly below the perturbative prediction, the trailing string should be considered as a candidate explanation. 5. The Perturbed Einstein Equations Given that an external quark dual to the trailing string described in Sec. 3 experiences drag, one might ask what happens to the energy that the quark deposits in the medium. At scales much larger than the inverse temperature, one expects the excitations present in the medium due to interactions with the moving quark to be well-described by linearized hydrodynamics. Earlier investigations of linearized hydrodynamics revealed the presence, for generic sources, of both a sonic boom and a diffusion wake.45 The sonic boom is a directional structure, which, in an ideal fluid, is concentrated on the Mach cone, but in a real fluid there is broadening because of viscous effects. It appears only when the probe is moving faster than the
14
S. S. Gubser et al.
T
mn
q
v
R3,1 AdS 5 −Schwarzschild
h mn
fundamental string
black hole horizon Fig. 2. (Color online) A visual summary46 of the calculation of the response hTmn i of a strongly coupled thermal medium to a heavy quark moving with a speed v via AdS/CFT. hmn is a perturbation of the metric caused by the trailing string.
speed of sound in the medium, and it comes from constructive interference among spherical waves sourced by the quark along its trajectory. The diffusion wake is a flow of the medium behind the quark in the direction of the quark’s motion. It too is broadened by viscous effects. We will discuss the hydrodynamic limit in more detail in Sec. 6.1.2. Apart from a qualitative understanding of energy loss at large distances, little is known a priori about what happens, for example, at small distances close to the quark. An all-scales description can be achieved using the gauge-string duality, where one computes the disturbances in the stress-energy tensor due to the presence of the quark. This was done in a series of papers.46–53 The linearized response of the lagrangian density in the dual field theory was also computed in Fourier space.54, 55 The purpose of this section and the next two is to present a reasonably self-contained summary of how the gauge theory stress-energy tensor is computed starting from the trailing string. A visual summary of the main elements of the computation is shown in Fig. 2. In the context of the AdS/CFT duality, the stress-energy tensor hTmn i of the boundary theory is dual to fluctuations of the metric in the bulk. So in order to compute the expectation value of hTmn i, one first needs to compute to linear order the backreaction of the string describing the quark on√the metric. Non-linear corrections to the Einstein equations will be suppressed by λ/N . The total action describing both the string and the metric is S = Sbulk + Sstring , which one can write as √ Z Z −G(R + 12/L2) 1 2 √ 5 µ µ S = d5 x − d σ −g δ (x − x (σ)) , (39) ∗ 2κ2 2πα0 where xµ∗ (σ) is the embedding function of the string in AdS5 -Schwarzschild. In a gauge where we parameterize the string worldsheet by σ α = (t, z), xµ∗ (σ) is
Energy Loss and the Gauge-String Duality
15
given by xµ∗ = t vt + ξ(z) 0 0 z ,
(40)
with ξ(z) as given in (20). The equations of motion following from (39) are just Einstein’s equations: 6 1 Rµν − Gµν R − 2 Gµν = τ µν , 2 L
(41)
where τ µν = −
3 p κ2 z H y 3 1 − v 2 δ(x1 − vt − ξ(z)) δ(x2 )δ(x3 )∂α xµ∗ ∂ α xν∗ 0 3 2πα L
(42)
is the bulk stress-energy tensor of the trailing string. To compute the backreaction of the string on the metric, we write Gµν = G(0) µν + hµν
(43)
(0)
where Gµν is the unperturbed AdS5 -Schwarzschild metric given in (3), and plug this into (41) to obtain the linearized equations of motion for the metric perturbations hµν . The resulting equations take the form Lhµν = τ µν ,
(44)
νσ 56 where hµν = Gµρ (0) G(0) hρσ and L is the differential operator given by ρ Lhµν = −hµν − 2Rµρνσ hρσ + 2R(µ hν)ρ − ∇µ ∇ν h + 2∇(µ ∇ρ hν)ρ 6 ρσ ρσ + Gµν (−∇ ∇ h + h + R h ) − − R hµν . ρ σ ρσ (0) L2
(45)
The covariant derivatives, the Riemann and Ricci tensors, and the Ricci scalar (0) appearing in (45) are computed using the background metric Gµν , and we have (0) µν denoted h = Gµν h . For a steady-state solution of (44), hµν depends on x1 and t only through the combination x1 − vt. We therefore pass to co-moving Fourier space variables by writing τ
µν
1
2
3
(t, x , x , x , z) =
Z
1 2 3 d3 K µν τ (z)ei[K1 (x −vt)+K2 x +K3 x ]/zH , (2π)3 K
(46)
~ ≡ ~kzH = ~k/πT . We make where we have defined the dimensionless wavevector K a similar expansion for hµν . In Fourier space, (44) can then be written as µν LK hµν K = τK ,
(47)
16
S. S. Gubser et al.
µν with τK being given by
v2 y2 h + v2 y4 v 0 0 h2 h h v 2 2 v 0 0 vy 2 −iK1 ξ(z)/zH 2 5 z y κ e h µν H , √ (48) τK = 0 0 0 0 0 2πα0 L5 1 − v2 0 0 0 0 0 2 2 v y vy 2 0 0 v 2 − h h where y is the rescaled depth coordinate defined in (21), but the tensor components are given in the (t, x1 , x2 , x3 , z) coordinate system. The explicit form of LK is too complicated to be reproduced here. We will decouple equations (47) by passing to a gauge where hµz = 0, which we will refer to as “axial gauge.” Note that hµz = 0 leaves some residual gauge freedom. We will discuss this shortly. The rest of this section is organized as follows. In Sec. 5.1 we explain how to decouple equations (47). In Sec. 5.2 we explain the boundary conditions needed to solve these equations. Lastly, in Sec. 5.3 we explain how the one-point function of the SYM stress-energy tensor is related to the asymptotic behavior of the metric perturbations near y = 0.
5.1. Metric perturbations in axial gauge As mentioned above, we choose a gauge where hµz H00 H01 H10 H11 1 κ2 L √ hK = H20 H21 µν 2πα0 1 − v 2 z 2 H30 H31 0 0
= 0, and let H02 H12 H22 H32 0
H03 H13 H23 H33 0
0 0 0 . 0
(49)
0
Rotational symmetry around the direction of motion of the quark allows us to set ~ = K1 K⊥ 0 with K⊥ > 0. Defining K q K⊥ 2 K = K12 + K⊥ ϑ = tan−1 , (50) K1 one can form the following linear combinations of metric perturbations: −H11 + 2 cot ϑH12 − cot2 ϑH22 + csc2 ϑH33 2v 2 H03 H13 + tan ϑH23 B1 = B2 = − v v2 A=
C = − sin ϑH13 + cos ϑH23
(51a) (51b) (51c)
Energy Loss and the Gauge-String Duality
H01 − cot ϑH02 −H11 + 2 cot 2ϑH12 + H22 D2 = 2 2v 2v 2 1 3 H01 + tan ϑH02 E1 = − H00 + H11 + H22 + H33 E2 = 2 h 2v
D1 =
H11 + H22 + H33 2 −H11 − H22 + 3 cos 2ϑ(−H11 + H22 ) + 2H33 − 6 sin 2ϑH12 . E4 = 4 E3 =
17
(51d) (51e) (51f) (51g)
Using these new variables, the Einstein equations (47) decouple into five sets46 : K2 3 h0 y ∂y + 2 v 2 cos2 ϑ − h A = e−iK1 ξ/zH (52a) ∂y2 + − + y h h h "
∂y2
+
− y3 0
0 − y3 +
h0 h
!
K2 + 2 h
# −h v 2 h cos2 ϑ B1 0 = −1 v 2 cos2 ϑ B2 0
B10 − hB20 = 0 "
∂y2
+
− y3
∂y2
K2 2 3 h0 2 + − + ∂y + 2 v cos ϑ − h C = 0 y h h 0
− y3 +
0
h0 h
!
K2 + 2 h
2 ∂y +
− y3 +
3h0 2h
0
0 0
0
0
0
− y3
0
0
0
0
(52c) (52d)
# y −iK1 ξ/zH 1 −h v 2 h cos2 ϑ D1 = e −1 v 2 cos2 ϑ D2 1 h (52e) D10 − hD20 = 0
(52b)
− y3 + 0
h0 2h
0 −3 y
+
(52f)
h0 h
∂y
−2h 12v 2 cos2 ϑ 6v 2 cos2 ϑ + 2h 0 E1 2 K 0 0 2h h E2 + 2 0 0 −2h −h E3 3h 2 2 2 2 2h −12v cos ϑ 0 3v cos ϑ + h E4
=
y −iK1 ξ/zH e h
2
1 + vh 1 2 −1 + v −
v2 h 1+3 cos 2ϑ 2 v 2
(52g)
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0 1 1 0 −h 0 −3v 2 cos2 ϑ − h −h ∂y h 0 2 0 +
=
0
0
0 −6h −3h 0 1 3(3v 2 cos2 ϑ + h)h0 0 −3hh0 18v 2 cos2 ϑh0 6h 2K 2 yh −2K 2 v 2 y cos2 ϑ −2K 2 y(3v 2 cos2 ϑ − h) 2K 2 yh
−ivy sec ϑ h0 −iK1 ξ/zH e 3ivy cos ϑ(v 2 + h) . 4Kyh K(v 2 − h)
E1 E2 E3
E4
(52h)
In (52), A, Bi , C, Di , and Ei are all functions of the rescaled AdS depth y ≡ z/zH , and primes denote derivatives with respect to y. A few comments are in order. Let’s start by counting the equations. Einstein’s equations (47) consist of fifteen linearly independent equations that split between the A, B, C, D, and E sets as follows: the A and C sets each consist of one second order equation for one unknown function; the B and D sets each consist of two second order equations and one first order constraint for two unknown functions; lastly, the E set consists of four second order equations and three first order constraints for four unknown functions. At first glance, the B, D, and E systems of equations might seem overdetermined. A more careful analysis shows that the constraints are consistent with the second order equations in the sense that if they hold at a particular value of y, they continue to hold at all y. We can therefore think of the constraint equations as reducing by five the number of integration constants in the second order equations. There are fifteen remaining integration constants that are fixed by the boundary conditions which we discuss in Sec. 5.2. Because the Bi and C equations, (52b)–(52d), have no source terms, one can consistently set B1 = B2 = C = 0. This identification is in fact enforced if we insist that the response of the medium should respect the same axial symmetry around the direction of motion of the quark that the trailing string does. In order to keep the discussion of perturbations, boundary conditions, and integration constants general, let us not discard the Bi and C fields just yet. A more general source would force them to be non-zero. Equations (52) can be reduced to just five equations using the residual gauge symmetry.53, 57 The action (39) is reparameterization invariant, and infinitesimal gauge transformations act by sending hµν → hµν + ∇µ ζν + ∇ν ζµ
(53)
for any one-form ζµ dxµ . This allowed us to pass to axial gauge in the first place: if we had started with some arbitrary metric perturbations hµν we could solve, at least locally, ∇z ζµ + ∇µ ζz = −hµz ,
(54)
Energy Loss and the Gauge-String Duality
19
so applying the infinitesimal gauge transformation (53) gives hµz = 0. Residual gauge symmetry arises from the fact that (54) specifies ζµ only up to five integration constants, which in general are functions of t and ~x. Put differently, there are five linearly independent gauge transformations that preserve the axial gauge condition hµz = 0, and the corresponding ζµ are given by the linearly independent solutions to (54) with hµz = 0. For the steady-state solution of (47) the allowed gauge transformations are the m ones where the t and ~x dependence of ζµ is of the form eiKm x /zH , where we have defined Km = −K1 v K1 K2 K3 . (55) As in previous sections, our convention is that lower case Roman indices m, n, . . . take values in (t, x1 , x2 , x3 ). The first four of these gauge transformations are parameterized by m
µ ζ(a) =
κ2 zH eiKm x /zH µ √ δ , 2πα0 L 1 − v 2 a
where a = 0, 1, 2, or 3, while the fifth is given by m 2 √ µ κ2 zH eiKm x /zH µ µ 2 µ 0y j √ ζ(5) = 2y hδ5 − iK δ − iK arcsin y δj . 2πα0 L 1 − v 2 h 0
(56)
(57)
The corresponding pure gauge solutions are given by 2 (0) z Hmn(a) = −2iK(mGn)a 2 a = 0, 1, 2, 3 L √ √ 0 0 Hmn(5) = 4 h ηmn − 2Km Kn arcsin y 2 − 4y 4 h δm δn h √ i 2 2 0 + 2K K(m ηn)0 arcsin y − y h ,
(58)
(59)
where ηmn is the Minkowski metric. One can straightforwardly check that (58)–(59) satisfy equations (47). Using the definitions (51), we can work out the pure gauge solutions in the ABCDE variables. We find that A and C are invariant, which was to be expected since their equations of motion are already fully decoupled. The other variables, however, do transform non-trivially under (53) with (56) and (57): for example the Bi and Di variables vary by iK1 (λ1 − λ2 cot ϑ) 2v
δB1 = iλ3 K1
δD1 =
iλ3 K1 δB2 = 2 v cos2 ϑ
iK1 (λ1 − λ2 cot ϑ) , δD2 = 2v 2 cos2 ϑ sin2 ϑ
(60)
where λa are arbitrary constants multiplying the pure gauge solutions parameterµ ized by ζ(a) .
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From these transformation laws it is easy to see that there is a linear combination of B1 and B2 that is gauge-invariant (and the same is true for D1 and D2 ). A choice of gauge invariants is given by B = B1 − cos2 ϑ v 2 B2
(61)
D = D1 − cos2 ϑ v 2 D2 .
The transformation law for the Ei variables is simple to derive but its exact form is not very enlightening. The corresponding gauge-invariant can be taken to be E = 4E3 + E4 + (12E2 − 3E4 )v 2 cos2 ϑ − (2E1 + E4 )h(z) .
(62)
From (52) we obtain the equations of motion for the B, D, and E invariants: 2 K 2 (v 2 cos2 ϑ − h) 3h + (h − 4)v 2 cos2 ϑ ∂ + B=0 (63) ∂y2 + y yh(v 2 cos2 ϑ − h) h2
∂y2
3h2 + (h − 4)v 2 cos2 ϑ K 2 (h − v 2 cos2 ϑ) ∂ + D y yh(v 2 cos2 ϑ − h) h2 4ivy 5 2 2 −iK1 ξ/zH y 1 − v cos ϑ + =e h K(v 2 cos2 ϑ − h)
∂y2 +
+
1 4 16y 4 − + y yh 4 − 6v 2 sin2 ϑ + 2h
−
∂y
K12 v 2 32y 8 K1 + − E h2 y 2 h(4 − 6v 2 sin2 ϑ + 2h) h cos2 ϑ
y 3v 2 cos2 ϑ − 2 − v 2 h 2 + h − 3v 2 cos2 ϑ 8iy 5 4 4 2 2 8 × 9v cos ϑ − 18v cos ϑ 1 + −y +9 . 3vK1
=e
(64)
−iK1 ξ zH
(65)
The gauge invariants we just described are unique up to an overall z-dependent factor, provided we only consider linear combinations of the Bi , Di , and Ei variables. If we also allow derivatives of the latter there are many other choices of gauge invariants that might prove useful. For instance, 1 2 K E 3 2 → 2v sin ϑzH D10 Z1 = vzH B10 Z0 =
← → Z2 =
−A sin2 ϑv 2 −C
(66) −C
A sin2 ϑv 2
!
.
Energy Loss and the Gauge-String Duality
21
is another set of gauge invariants which has been used in the literature.53 Yet another set, consisting of “master fields,”47 is ψTodd = − ψTeven = − ψVodd = ψVeven
3 zH αv C L6 3 2 zH v αv A sin2 ϑ L6
2 zH vαv hB20 2L4
(67)
z 2 vαv = H 4 hD20 sin ϑ L
2 00 K 2 zH αv 3h yE4 + 6h2 E20 − 3h(3 + y 4 )E40 6L2 hy(K 2 + 6y 2 ) + 2hyK 2 (E1 − E3 ) + 3y 2hy 2 − K12 v 2 (4E2 − E4 ) , √ where αv = 1/2πα0 1 − v 2 . The gauge invariants (67) will prove useful for the asymptotic analysis of Sec. 6.2. Below, we list their equations of motion: 3 h0 K 2 (v 2 cos2 ϑ − h) 00 ψT + − + ψT0 + ψT = −JT (68a) y h h2 3 h0 K 2 v 2 y 2 cos2 ϑ + h(3 − K 2 y 2 + 9y 4 ) 00 ψV + − + ψV0 + ψV = −JV (68b) y h y 2 h2 0 2 2 h 3 K 2 (4 − K 2 y 2 ) + 12y 2 (6 − y 4 ) K1 v ψS00 + − ψS0 + + ψS = −JS (68c) h y h2 y 2 h(K 2 + 6y 2 )2 ψS = −
where JTodd = JVodd = 0 JTeven =
3 zH αv v 2 y sin2 ϑ −iK1 ξ/zH e L6 h
2 zH αv e−iK1 ξ/zH v sin ϑ(5y 4 − 1 − iKvy 3 cos ϑ) + iKy 3 tan ϑ L4 h zH αv K 2 ye−iK1 ξ/zH h JS = (2 + v 2 ) 2y 2 (K 4 − 45) − 3K 2 4 2 2 2 6L (K + 6y )
JVeven =
(69) (70) (71)
+ 3K 2 (2 − 5v 2 )y 4 + 18(2 + 3v 2 )y 6 − 3iKv 3 y 3 (K 2 − 12y 2 ) cos ϑ + 3v 2 90y 2 − K 4 (4 + v 2 )y 2 − 54y 6 + 3K 2 (1 + y 4 − 2v 2 y 4 ) cos2 ϑ + 9iKv 3 y 3 (K 2 − 18y 2 ) cos3 ϑ + 9K 2 v 4 y 2 (K 2 + 6y 2 ) cos4 ϑ i − 6ivy 3 K(K 2 − 2y 2 ) sec ϑ .
(72)
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S. S. Gubser et al.
By omitting the “even” and “odd” superscripts in (68a) and (68b) we mean that these equations take the same form (up to the different source terms as indicated in (69)–(71)). 5.2. Boundary conditions In axial gauge, the system of equations (47) consists of ten second order differential equations in y and five first order constraints, so we need to specify fifteen integration constants. The purpose of this section is to show that five of these are fixed by imposing boundary conditions at the horizon (y = 1) and ten of them are fixed from the boundary conditions at the conformal boundary (y = 0). Consider first the tensor field A, defined in (51a). The near horizon solution to (52a) takes the form ivK1
e− 8 (π−ln 4) (1 − y)1−ivK1 /4 +UA (1−y)−ivK1 /4 +VA (1−y)ivK1 /4 +. . . , (73) A= 1 4 1 − ivK 2
where . . . indicates terms which are subleading to one of the ones shown. The first term in (73) is a particular solution to (52a) characterizing the response of A to the trailing string source. The second and third terms are solutions to the homogeneous equation, and UA and VA are constants of integration. The correct boundary condition at the horizon is VA = 0. This corresponds to requiring that there are no outgoing modes at the horizon, and it can be justified by the fact that classical horizons don’t radiate. The result of choosing purely infalling conditions is that in the dual gauge theory, we describe a causal response of the medium to the probe. To see that the UA term in (73) is infalling, let’s define a new coordinate y∗ = log(1 − y) that ranges from −∞ when y = 1 (the horizon) to zero when y = 0 (the conformal boundary of AdS5 ). Recalling that the time dependence of metric perturbations was assumed to be e−iK1 vt/zH , one sees immediately that the (1 − y)−ivK1 /4 term in (73) corresponds to certain metric components behaving as hmn ∼ e−ivK1 (y∗ +4t/zH )/4
(74)
at large negative y∗ . This behavior describes a wave traveling towards negative y∗ , i.e., falling into the black hole horizon. Similarly, the VA term in (73) corresponds to an outgoing mode. The same story goes through for the C combination of metric components defined in (51c), except that since there is no source term, there will be no analog to the first term in (74). A subtlety arises in the horizon boundary conditions for the B, D, and E sets: for each set, in addition to a single infalling solution and a single outgoing solution, there are the pure gauge solutions discussed around (58) and (59). The pure gauge solutions are neither infalling nor outgoing at the horizon. The correct boundary conditions are to exclude the outgoing solution and to permit both the infalling solution and the pure gauge solutions. If one passes to a description only in terms
Energy Loss and the Gauge-String Duality
23
of gauge-invariants, then this subtlety is avoided: each gauge invariant field has only an infalling and outgoing solution, and the latter is excluded by the horizon boundary conditions. Having fixed five integration constants (one for each of the ABCDE sets) using infalling boundary conditions at the horizon, we now discuss the boundary conditions at the boundary of AdS. Close to y = 0, Einstein’s equations can be solved in a series expansion in y. The two homogeneous solutions are (1) Hmn (y) = Rmn 1 + O(y 2 )
(2) Hmn (y) = Qmn y 4 + O(y 6 )
(75)
where Rmn and Qmn are arbitrary constants. The full solution then takes the form (1) (2) Hmn = Hmn (y) + Hmn (y) +
Pmn 3 y + O(y 5 ) . 3
(76)
The components of Pmn are given by 2
+ v2 ) −2v 0 0
3 (2
Pmn =
−2v 2 (1 + 2v 2 ) 3 0 0
0 0 2 2 3 (1 − v ) 0
0 0 . 0 2 2 3 (1 − v )
(77)
The boundary conditions we impose are that Rmn = 0,
(78)
thus fixing the remaining ten integration constants. Allowing non-zero Rmn would correspond to deformations of the gauge theory lagrangian. With this condition imposed, the Qmn are related to the expectation value of the stress tensor in the gauge theory, as we will see in Sec. 5.3. Equation (78) can be translated into boundary conditions for the ABCDE variables. Once the horizon boundary conditions are also taken into account, it follows that each set of equations in (52) is supplemented by just enough boundary conditions to uniquely fix a solution. For example, (78) implies boundary conditions on A which fix one constant of integration. Another integration constant is fixed by the horizon boundary conditions. Since the underlying equation (51a) is second order and linear, the solution is unique. A more complicated example is the E set, where the boundary conditions from (78) fix four integration constants and the horizon boundary conditions fix one more. Since the four second order equations of motion (52g) are subject to three constraints (52h), the number of integration constants available is five. Thus one again finds a unique solution. If we pass to a description in terms of gauge invariants, then (78) fixes a single constant of integration for each gauge-invariant field. Because the horizon fixes another constant and the underlying equation for the gauge-invariant is always second order, we again recover a unique solution.
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S. S. Gubser et al.
5.3. The boundary stress-energy tensor The AdS/CFT duality offers a prescription2, 3 for computing the stress tensor of the boundary theory from the bulk action58, 59 : 2 δStotal hTmn i = lim √ . →0 −g δg mn
(79)
We now explain what Stotal , gmn and are. gmn is the metric on the conformal boundary of AdS5 . After taking the variational derivative, gmn is set equal to the metric of the boundary theory, i.e., the Minkowski metric ηmn in our case. Stotal is given by Stotal = Sbulk + SGH + Scounter .
(80)
Here Sbulk is the bulk action (1); SGH is the Gibbons-Hawking boundary term Z √ 1 SGH = 2 d4 x GΣ K Σ , (81) κ where Σ is a co-dimension one surface close to the AdS boundary with outward normal nµ , induced metric GΣ µν ≡ Gµν − nµ nν ,
(82)
Σ ρ Kµν ≡ −GΣ µρ ∇ nν ;
(83)
and extrinsic curvature tensor
and Scounter is an additional boundary term which renders the on-shell action finite for geometries which do not induce a trace anomaly in the boundary theory. An explicit expression for this term is Z p 2 L2 Σ 1 4 Σ Scounter = 2 d x −G − R , (84) κ L 4
with RΣ the Ricci scalar constructed from GΣ mn . Usually, there are additional terms in (84) coming from the matter action. Since we are working in the probe limit, we do not need to worry about these extra terms.60 The coordinate in (79) specifies a hypersurface Σ() which coincides with the boundary of AdS5 as we take the → 0 limit. Since we are working with a flat boundary metric, the variation of Scounter will not contribute to hTmn i, and we can choose Σ to be a surface of constant z. The outward normal form is L nµ dxµ = − √ dz , z h
(85)
√ z h ∂Gmn = . 2L ∂z
(86)
and then Σ Kmn
Energy Loss and the Gauge-String Duality
25
Equation (79) now reads hTmn i = lim
z→0
L2 Σ Kmn − K Σ GΣ mn . 2 2 z κ
(87)
Using (87), the expectation value of the stress-energy tensor in the absence of the quark is that of a thermal bath hTmn ibath =
π2 2 4 N T diag{3, 1, 1, 1} . 8
(88)
The presence of the quark generates two additional contributions: writing Z 3 1 2 3 d K K K hTmn i = hTmn ibath + hTmn idiv + hTmn i ei[K1 (x −vt)+K2 x +K3 x ]/zH , (2π)3 (89) and using the definitions (49) and (76), as well as the boundary condition Rmn = 0 and the AdS/CFT identities (2) and (15), one obtains √ 1 π2 T 3 λ K √ (90) hTmn idiv = Pmn − ηmn Pl l 4 1 − v 2 √ π3 T 4 λ K (91) hTmn i = √ Qmn − ηmn Qll . 2 1−v
Let’s first understand the divergent contribution. Plugging (77) into (90), it is not hard to see that in position space hTmn idiv takes the form of a contact term √ p λ (92) hTmn idiv = um un 1 − v 2 δ(x1 − vt)δ(x2 )δ(x3 ) 2π
1 where um = √1−v (1, ~v ) is the four-velocity of the quark. A divergence of this form 2 was to be expected, and it is associated to having an infinitely massive quark. In the dual gravity language, the mass of the quark can be identified with the energy of the trailing string. But this energy is both IR and UV divergent. The UV divergence corresponds to the “bare mass” of the quark and is exactly given by √ λ M= , (93) 2π
where is the UV cutoff. Equation (92) then takes the form of the stress-energy tensor of a particle with mass M moving with velocity v along the x1 -direction. It can be shown that any string configuration whose endpoint lies on the boundary of AdS, whether it is stationary, moving with constant velocity relative to the plasma, or accelerating, will generate a divergent contribution of the form (92).60 The finite contribution to the stress-energy tensor given in (91) can be further simplified by using the 55 Einstein equation. Using the series expansion (76) with Rmn = 0, the 55 Einstein equation imposes a tracelessness relation on the Qmn : −Q00 + Q11 + Q22 + Q33 = 0 .
(94)
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S. S. Gubser et al.
This equation implies further that K hTmn i
√ π3 T 4 λ = √ Qmn . 1 − v2
(95)
K Note that even from (91) one can see that hTmn i is traceless, so (94) does not imply tracelessness of the boundary stress-energy tensor. What (94) does is that it allows us to write the one-point function of the stress tensor in the simplified form (95). The 5m Einstein equations imply four more relations among the Qmn : iv K m Qmn = (96) v −1 0 0 . 2 Using (95), (96) shows that the boundary stress-energy tensor fails to be conserved: √ iv π 3 T 4 λ √ K m hTmn i = (97) v −1 0 0 . 2 1 − v2
This non-conservation comes from the fact that the quark is prescribed to move at constant velocity. The drag force can be interpreted as minus the force exerted by the quark on the medium. With this interpretation in mind, we can check explicitly that the drag force (27) can be recovered from (97), as follows. Given the stressenergy tensor, the external force acting on a region V can be computed from Z I Z d in n 3 0n 2 F = d x hT i + d a ni hT i = d3 x ∂m hT mn i . (98) dt V ∂V V
The region V is assumed not to depend on time in the asymptotic rest frame of the plasma. The first term in (98) gives the rate of change of energy-momentum in this region, while the second term corresponds to the energy-momentum flux through the boundary of V . To obtain the last equality we used the divergence theorem. Using (89) and taking the limit where the volume of V goes to infinity, as appropriate for computing the total force on the system, one can see that √ v πT 2 λ mn n 2 F = izH lim Km hTK i = √ . (99) v 1 0 0 ~ 2 1 − v2 K→0
F 1 is then indeed minus the drag force, as can be easily checked by comparing (99) to (27). The prescription (95) gives the stress tensor in terms of the coefficients Qmn that appear in the near boundary asymptotics of the components of Hmn . We can give similar prescriptions in terms of the near boundary asymptotics of the other linear combinations of metric perturbations that we discussed in Sec. 5.1 — all that we need to do is to relate the Qmn to a near boundary expansion of these variables. For the ABCDE variables defined in (51), the near boundary behavior is
PX 3 y + QX y 4 + O(y 5 ) where X = A, Bi , C, Di , Ei , (100) 3 Here, as before, the QX and RX are arbitrary constants while the PX are set by (52). The boundary condition Rmn = 0 is equivalent to RX = 0. X = RX +
Energy Loss and the Gauge-String Duality
27
It is easy to invert the definition (51) and use the near boundary expansions (100) and (76) to write the Qmn components in terms of the QX . However, it is useful to notice that we need not specify all of the QX . The reason for this is that the constraint equations (52c), (52f), and (52h) imply relations among the QX , namely QD1 − QD2 = −
QB1 − QB2 = 0 QE1 − 2QE2 = −
iv 2K1
i 4vK1
QE1 + 2QE3 = 0
(1 − 3v 2 cos2 ϑ)QE1 + 2QE4 =
3iv(1 + v 2 ) cos ϑ . 2K
(101)
Using (101), we can write Qmn in terms of only five of the QX , one from each set. This will be nothing more than a parameterization of the most general Qmn obeying constraints (94) and (96). Recalling that the response to the trailing string has B1 = B2 = C = 0 everywhere because of symmetry, we conclude that we can write Qmn in terms of QA , QD ≡ QD1 and QE ≡ QE1 g : Qmn = amn QA + dmn QD + emn QE + pmn ,
(102)
where
amn
0 0 0 2 v 2 sin2 ϑ 0 −2 sin ϑ sin 2ϑ = 0 sin 2ϑ −2 cos2 ϑ 2 0 0 0
0 4 sin2 ϑ −2 sin 2ϑ 2 2 v 4 sin ϑ −2 sin 2ϑ sin 4ϑ = sin 4ϑ 2 sin2 2ϑ 2 −2 sin 2ϑ 0 0 0 2
dmn
emn
(103)
2
0 0 0 0
−4 4v cos2 ϑ 2v sin 2ϑ 0 1 4v cos2 ϑ 4e11 4e12 0 = 4e12 4e22 0 4 2v sin 2ϑ 2 2 0 0 0 2v cos ϑ − 2
(104)
1 −1 + (1 + v 2 ) cos2 ϑ − 3v 2 cos4 ϑ 2 1 = sin 2ϑ 1 − 3v 2 cos2 ϑ 4 1 = cos2 ϑ −1 − 2v 2 + 3v 2 cos2 ϑ 2
e11 = e12 e22 g In
0 0 0
(105)
later sections, we will continue to use QD to mean QD1 , and likewise QE = QE1 . If the gauge-invariants D and E defined in (61) and (62) were expanded in powers of y, like in (100), the coefficients of y 4 would be related to QD1 and QE1 , but not identically equal.
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S. S. Gubser et al.
pmn
0 4v 4v tan ϑ 0 iv cos ϑ ℘11 ℘12 0 4v = 4v tan ϑ ℘12 ℘22 0 8K 0 0 0 2 + 2v 2
(106)
℘11 = (1 − 3v 2 ) cos 2ϑ − 5 − v 2
℘12 = (1 − 3v 2 ) sin 2ϑ − 4 tan ϑ
℘22 = (3v 2 − 1) cos 2ϑ + 3 − v 2 .
For the master fields (67), the near boundary behavior is PT QT ψTeven = 3 y 3 + 8 y 4 + O(y 5 ) L L ψVeven =
PV 2 QV 3 y + 6 y + O(y 4 ) L4 L
(107a) (107b)
QS PS y + 4 y 2 + O(y 3 ) . (107c) 2 L L Here we write asymptotics after having imposed the condition Rmn = 0. Similarly to before, the Q coefficients are arbitrary while the P are set by (68). The odd master fields ψTodd and ψVodd have similar expansions, but since their equations of motion are homogeneous, we can set them to be identically zero and worry about them no longer. Using the definitions (67) and (101) it is easy to relate the Q coefficients for the master fields to the QX . The result is QT QA = − (108a) 3 sin2 ϑ 2 2 αv v L zH ψS =
i QV + (108b) 2 sin ϑ 4vK1 4αv vL2 zH iv QS [5pt]QE = − − . (108c) 2K1 2αv L2 zH Similar formulas relating the QX to the asymptotics of the other gauge-invariants discussed can be easily derived, but we will not need them. QD = −
6. Asymptotics Equations (52) or (68) are difficult to solve exactly and we eventually resort to numerics to obtain a full solution. However, there are various approximations which can be used in order to get a handle on the large and small momentum asymptotics of the metric fluctuations. These may be Fourier transformed to real space, giving us approximations to the near- and far-field behavior of the boundary theory stressenergy tensor. In Sec. 6.1 we focus on the small momentum asymptotics of the solution: in Sec. 6.1.1 we construct a small momentum series expansion of the metric perturbations, while in Sec. 6.1.2 we explain how this maps into hydrodynamic behavior on the boundary theory. In Sec. 6.2 we focus on the near-field of the stress-energy tensor.
Energy Loss and the Gauge-String Duality
29
6.1. Long distance asymptotics 6.1.1. Momentum space analysis The small momentum asymptotics of the solution can be obtained by formally expanding the appropriate fields in power series in the momentum K. In what follows we will go over such an expansion in the A, B, C, D, E variables of (51).46 A similar construction can be carried out for the other parameterizations of the Einstein equations47, 51 given in (66) and (67). We start by formally expanding the A variable in (51a) such that A=
∞ X
αn K n ,
(109)
n=0
~ which are invariant under rescalings K ~ → λK. ~ where the αn are functions of K ~ ~ That is, the αn only depend on the direction of K, not its magnitude K = |K|. Plugging the expansion (109) into (52a) and collecting terms with identical powers of K, we obtain a set of equations for αn of the form y3 h ∂y ∂y αn = Sn , h y3 where Sn can depend on αm with m < n. For example, 2 y y cos ϑξ y cos ϑ v 2 cos2 ϑ − h S0 = S1 = −i S2 = − − α0 . h h hzH h zH h2 The most general solution to (110) is Z y Z y˜ h(y˜˜) y˜3 Sn (y˜˜) . dy˜˜ αn = d˜ y h(˜ y ) y1 y˜˜3 y0
(110)
(111)
(112)
In order to satisfy the boundary conditions (78) at the asymptotically AdS boundary, we require that y0 = 0. The other integration constant in (112) is obtained by matching the near horizon behavior of (112) to the series approximation (73) with VA = 0, expanded at small K. For example, at order n = 0, we find that 1 1−y 1 + y2 −1 α0 = 2 tan y + log + y1 log . (113) 4 1+y 1 − y2 Expanding (113) near the horizon and matching it to a small K expansion of (73), A = (1 − y) + UA + O(K) ,
(114)
we find that we need to set y1 = 1. According to (95) and (102), to obtain the stress tensor we will eventually need QA . Expanding (113) for small y and recalling (100), we can read off that QA = 1/4 + O(K). The next order corrections to QA can be obtained in the same manner. We find that to order O(K), QA =
1 i ln 2 − vK1 + O(K 2 ) . 4 8
(115)
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S. S. Gubser et al.
A similar treatment gives us the asymptotic values of D1 and E1 . The result is QD = − QE =
i sec ϑ sec2 ϑ − 4v 2 − + O(K 2 ) 4vK 16v 2
3iv(1 + v 2 ) cos ϑ 3v 2 cos2 ϑ(2 + v 2 (1 − 3 cos2 ϑ)) − . 2 2 2K(1 − 3v cos ϑ) 2(1 − 3v 2 cos2 ϑ)2
(116) (117)
Note that QD exhibits a pole structure at K = 0, while QE exhibits a pole structure at K 2 = 3K12 v 2 . We will see shortly that these correspond to the diffusion pole and sound pole expected of the hydrodynamic behavior of the plasma far from the moving quark. 6.1.2. Relating the large momentum asymptotics to hydrodynamics At scales much larger than the mean free path, we expect to be able to describe a thermal gauge theory by effective, hydrodynamic, slowly varying degrees of freedom. Let’s first see what such a description entails, and then check how well our asymptotics match it. Consider a conformal theory for which the hydrodynamic energy-momentum tensor (T hydro )mn is traceless: (T hydro )m m = 0.
(118)
A static configuration of the fluid will be given by 0 (T hydro )mn = diag {3, 1, 1, 1} . (119) 3 If we now perturb this fluid slightly, and choose our hydrodynamic variables to be = (T hydro )00 and Si = (T hydro )0i with i = 1, 2, or 3, then we should be able to write the remaining space-space components of the energy-momentum tensor, (T hydro )ij , in terms of gradients of and Si . The only possible combination consistent with the tracelessness condition (118) is 1 3 δij − Γik(i Sj) + O(k 2 ) , (120) 3 2 where the free parameter Γ is the sound attenuation length, related to the shear viscosity, temperature, and entropy density through Γ = 4η/3sT . Using (7) we find that Γ = 1/3πT in the theories we are considering. Brackets denote the symmetric traceless combination, i.e., (T hydro )ij =
1 1 (Mmn + Mnm ) − ηmn η pq Mpq . (121) 2 4 The energy density and the energy flux Si can now be computed from the conservation equations, M(mn) =
hydro ikn (T hydro )nm = fm ,
(122)
hydro where fm is the source which perturbs the hydrodynamic stress-energy tensor. Separating the spatial part of the source, f~ hydro , into longitudinal and transverse
Energy Loss and the Gauge-String Duality
31
ˆ kˆ and f~ hydro = f~ hydro − f~ hydro , with (k) ˆ i = ki /k, components, f~Lhydro = (f~ hydro · k) T L we find from (120) and (122) that −3i f~ hydro · ~k − f0hydro ω − 3f0 k 2 Γ + O(f k) (123a) = k 2 − 3ω 2 − 3iΓk 2 ω hydro 2 k + 3f~ hydro · ~kω ~ · ~k = −f0 S + O(f k) k 2 − 3ω 2 − 3iΓk 2 ω
(123b)
~ hydro ˆ kˆ = −ifT ~ − (S ~ · k) S + O(f k) . ω + 34 Γik 2
(123c)
In real space, the pole at approximately ω ∼ 0 corresponds to a diffusive wake behind the source, and the pole at roughly k 2 ∼ 3ω 2 corresponds to the shock wave which appears along the Mach cone behind the moving probe. The displacement of the two poles from the real axis due to the shear viscosity of the fluid is responsible for viscous broadening of the wake and Mach cone. We can now compare the small momentum results of Sec. 6.1.1 to the hydrodynamic behavior (123). To do so, we use (115)–(117) in (102) and (95) to obtain the small momentum stress-energy tensor. We find −S1 −S⊥ 0 √ (πT )4 λ K + Tmn + O(K) −S1 (124) i= √ hTmn 2 −S⊥ Tij 1−v 0 with
3K12 v 2 −3K12 v 2 + K 2 (2 + v 2 ) 3iK1 v(1 + v 2 ) =− + 2 2π (K 2 − 3K12 v 2 ) 2π (K 2 − 3K12 v 2 ) −6K14 v 5 − K 4 v + K12 K 2 6v 2 + 1 v iK1 v 2 + 1 + S1 = − i 2 2π (K 2 − 3K12 v 2 ) 2π (K 2 − 3K12 v 2 ) +
2 K⊥ i + 2K1 π 32K12 πv 2
(125b)
K1 K2 v K 2 + 3K12 v 4 iK2 (1 + v 2 ) S⊥ = − + 2 2π (K 2 − 3K1 v 2 ) 2π(K 2 − 3K12 v 2 )2 −
(125a)
K2 8πK1 v
(125c)
and Tij = Tmn =
1 1 δij − iK(i Sj) 3 6
(126)
v2 diag {0, −2, 1, 1} . 6π
(127)
32
S. S. Gubser et al.
Note that we may always carry out a resummation K2
O(K 4 ) αi Ki + O(K 2 ) αi Ki + = 2 2 2 2 2 2 2 − 3K1 v (K − 3K1 v ) K − 3K12 v 2 − iK1 K 2 v
(128)
(where αi are constants) or
1 1 + O(K) O(K 2 ) . + 2 2 = vK1 v K1 vK1 + 41 iK 2
(129)
~ in (125) take the form (123) After such a resummation, the expressions for and S with 1 (130) fnhydro = −v 2 v 0 0 n − iK m Tmn 2π and Γ = 1/3πT . Thus, if we identify the first term in the parenthesis on the right hand side of (124) with the hydrodynamic contribution to the stress tensor of the SYM theory, (T hydro,SYM)mn , then we find that (T hydro,SYM)mn satisfies iK m (T hydro,SYM)mn = fnhydro,SYM .
(131)
This should be compared to the full conservation law (99), K iK m hTmn i = fn .
(132)
The extra term Tmn in (124) holds information on the deviation of the stress-energy tensor from its hydrodynamic form. Alternately, Jn − ik m Tmn gives us an effective hydrodynamic four-force which sources the hydrodynamic stress-energy tensor. It is no coincidence that the large distance asymptotics of the stress-energy tensor agree with a hydrodynamic expansion. In fact, it can be shown that generic probesources excite the metric in such a way that the resulting large distance asymptotics of the boundary theory stress tensor will have hydrodynamic behavior.60 There is also mounting evidence that such a connection between hydrodynamics and gravity goes beyond the linearized approximation.14 To see that the pole structures in (128) and in (129) really correspond to a laminar wake and a shock wave, we Fourier transform them to real space. Consider the last two terms in (125b), resummed as in (129). Using n−d/2 Z d ~ ~ d K eiK·X 2 X = Kn−d/2 (µX) , (133) (2π)d (K 2 + µ2 )n (4π)d/2 Γ(n) 2µ
we can Fourier transform the resummed expression to position space: Z 3 1 2 3 d K 2v ei(K1 (x −vt)+K2 x +K3 x )/zH 3 2 (2π) π(K − 4iK1v) √ − 2v (x1 −vt)+ (x1 −vt)2 +x2⊥ e zH vzH p = . 2 2 1 2 2π (x − vt) + x⊥
(134)
As expected of a diffusion wake, we find that the configuration (134) exhibits a directional energy flow, with a parabolic shape far behind the moving quark.
Energy Loss and the Gauge-String Duality
33
Fourier transforming the resummed sound pole (128) is difficult due to the cubic terms in the denominator, and we eventually resort to numerics to convert such expressions to real space. To see that the pole at K 2 ∼ 3K12 v 2 really corresponds to a shock wave, it is sufficient to Fourier transform only the leading order contributions to these poles. Since we are neglecting the viscous contribution to the pole structure, this corresponds to the inviscid limit. Using contour integration and the identities61 Z ∞ 0 0
0 , (136) 2 a + b2 0 where J is a Bessel function of the first kind, we find that, for example, the leading contribution to the energy density (125a) reads Z 3 d K 3iK1 v(1 + v 2 ) i(K1 (x1 −vt)+K2 x2 +K3 x3 )/zH e − (2π)3 2π(K 2 − 3K12 v 2 ) 2 3v v 2 + 1 x1 zH v 2 < 1/3 − 2 ((x1 )2 + (1 − 3v 2 ) x2 )3/2 8π ⊥ 2 √ = (137) 3v v 2 + 1 x1 zH − v 2 > 1/3, −x⊥ 3v 2 − 1 < x1 < 0 3/2 2 2 1 2 2 4π ((x ) + (1 − 3v ) x⊥ ) 0 otherwise . In obtaining (125a) we assumed that the sound poles are slightly below the real K1 axis for v 2 > 1/3. This assumption is motivated by the fact that the viscous corrections that appear at the next order do shift the poles to the lower-half complex K1 plane. As expected, since we have treated viscous contributions as infinitesimal in the leading order result (137), the real space expression for the energy density is singular along the Mach cone. 6.2. Short distance asymptotics As was the case for the large distance asymptotics, the short distance asymptotics of the solution can be obtained by formally expanding all variables in large momenta. Starting from the z coordinate system in (3), we look at momenta which are much larger than the inverse temperature scale, K = kzH 1. In this case, it is more practical to use the dimensionless radial coordinate Z = zk instead of y = z/zH . Setting K 1 means that we’re pushing the black hole horizon off to Z → ∞, implying that we’re nearing the zero temperature limit. To see this explicitly, consider the expansion ψTeven = K −3
∞ X
n=0
tn K −n
(138)
34
S. S. Gubser et al.
of the tensor modes defined in (67), similar to (109). Plugging the expansion (138) into (68a) and collecting terms with similar powers of K, we find that the tn ’s satisfy k12 2 3 2 (139) ∂Z tn − ∂Z tn − 1 − 2 v tn = −(JT )n Z k where, as before, we need to compute (JT )n order by order in a large K expansion. For n = 0, 1, and 2 we find −(JT )0 = −
3 2 zH v αv sin2 ϑZ L6
−(JT )1 = 0 −(JT )2 =
(140) (141)
z 3 v 2 αv − H 6 L
sin2 ϑ
4
iK1 vZ . 3K
(142)
To solve (68a) we use the method of Green’s functions. The homogeneous version of (139) can be easily solved. The solutions are ! ! r r k12 v 2 k12 v 2 (1) 2 2 (2) t (Z) = Z I2 1− 2 Z 1− 2 Z t (Z) = Z K2 (143) k k where I2 and K2 are modified Bessel functions of the first and second kind. In (143) we suppressed the index n because the homogeneous parts of (139) are the same p for all n. To simplify the notation we define α = 1 − k12 v 2 /k 2 . The solution which vanishes near the boundary, and therefore does not correspond to a deformation of the theory, is t(1) . The other solution, t(2) , is the only solution which does not diverge exponentially in the deep interior of AdS. Clearly, t(1) is the solution which captures the boundary asymptotics we have in mind, and it seems physically reasonable to disallow a solution which is exponentially divergent at large Z. A more rigorous approach would be to find a uniform approximation to the two linearly independent solutions of the homogeneous version of (68a) and to show that the solution that behaves like t(2) near the boundary is not purely infalling at the horizon. This can be carried out via a WKB approximation.62 Thus, the solution to 3 k2 (144) ∂Z2 G(Z, Z 0 ) − G(Z, Z 0 ) − 1 − 12 v 2 G(Z, Z 0 ) = δ(Z − Z 0 ) Z k is G=
(
(Z 0 )−3 t(2) (Z 0 )t(1) (Z)
Z < Z0
(Z 0 )−3 t(1) (Z 0 )t(2) (Z)
Z > Z0 ,
(145)
and then tn (Z) =
Z
dZ 0 G(Z, Z 0 )(JT )n (Z 0 ) .
(146)
Energy Loss and the Gauge-String Duality
For n = 0 and n = 1 these integrals may be carried out exactly.61 We find 3 2 zH v αv 2α 3 2 2 2 sin ϑ Z L2 (αZ) − Z I2 (αZ) + Z t0 (Z) = − L6 3π t1 (Z) = 0
z 3 v 2 αv − H 6 L
(147)
4
iK1 vZ , 3Kα2 with L2 a modified Struve function. Recalling (107a) we can read off q ivK1 1 3 2 QT = −zH v αv sin2 ϑL2 π K 2 − K12 v 2 − 16 3(K 2 − K12 v 2 ) t2 (Z) =
35
sin2 ϑ −
(148)
from (147). The equation of motion for t0 coincides with the one that would have been obtained starting from a string hanging straight down from the boundary of AdS space, boosted to a velocity v in the x1 direction. The solution t0 then corresponds to the tensor mode metric perturbation in response to this string and, as we will see shortly, it captures the near-field physics of the stress-energy tensor in response to a massive quark. At scales much smaller than the mean free path, one may effectively ignore the interaction of the quark with the plasma. The function t2 corresponds to the first thermal corrections to the near-field of the quark. The computation of the large K asymptotics for the vector and scalar modes follows in a similar manner. Let ψVeven = K −4
0 X
vn K −n ,
(149)
n=−∞
similar to (138). Expanding (68b) at large K, we find that the vn ’s satisfy 3 ∂Z vn + (3 − Z 2 α2 )vn = −(JV )n . Z The first few terms in −(JV )n are given by ∂Z2 vn −
αv vK⊥ Z 4 K5 −(JV )1 = 0 −(JV )0 =
−(JV )2 = −i
(150)
(151) (152) 2
4K12 v 2 )
αv K⊥ (3K − 3K 2 K1
.
(153)
The homogeneous solutions to (150) are v (1) = Z 2 I1 (αZ)
v (2) = Z 2 K1 (αZ) ,
(154)
and using the Green’s function method we find ψVeven =
αv K⊥ πv 2 Z (L1 (αZ) − I1 (αZ)) 2Kα2 +
iαv K⊥ (3K 2 − 4K12 v 2 )Z 3 (8 + Z 2 α2 ) 3 Z , 3K1 K 2 α4
(155)
S. S. Gubser et al.
36
which, recalling (107b), implies " # p π k 2 − v 2 k12 L2 αv vk⊥ 3k 2 − 4k12 v 2 L4 −4 QV = − − + 2 + O(zH ) . (156) k 4 3ivkk1 (k 2 − k12 v 2 ) zH The details of the computation of the scalar modes can be found elsewhere.47, 49, 50 The final result is " π 2 + v 2 (k 2 − k12 v 2 ) − v 2 (1 − v 2 )k12 p QS = αv − 12 k 2 − k12 v 2 # iv 9 k1 (5 − 11v 2 ) 2v 2 (1 − v 2 )k13 −4 − − + O(zH ) (157) − 2 9zH k1 k 2 − v 2 k12 (k 2 − k12 v 2 )2 where ψS = PS L−2 y + QS L−4 y 2 + O(y 5 ) .
(158)
With QT , QV , and QS at hand we can use (108), (102), and (95) to obtain the leading large momentum asymptotics of the stress-energy tensor. The momentum space expressions for the energy density and Poynting vector are p √ " π3 T 4 λ (2 + v 2 ) K 2 − K12 v 2 K12 v 2 (1 − v 2 ) K − + p hT00 i = − √ 24 1 − v2 24 K 2 − K12 v 2
iK1 v(11v 2 − 5) 14 + 7v 2 iK13 v 3 (1 − v 2 ) + + 18π(K 2 − K12 v 2 ) 24 (K 2 − K 2 v 2 )3/2 9π(K 2 − K12 v 2 )2 1 # K12 v 2 10v 2 − 1 K14 v 4 (1 − v 2 ) + − (159) 24(K 2 − K12 v 2 )5/2 8(K 2 − K12 v 2 )7/2 p √ " v K 2 − K12 v 2 π3 T 4 λ K 2 v(1 − v 2 ) K hT01 i = − √ + p1 − 8 1 − v2 24 K 2 − K12 v 2 −
iK1 v 2 9v iK13 v 2 (1 − v 2 ) + + 2 2 3π(K 2 − K1 v 2 ) 16(K 2 − K1 v 2 )3/2 9(K 2 − K12 v 2 )2 π # K12 v(1 + 17v 2 ) K14 v 3 (1 − v 2 ) + − 48(K 2 − K12 v 2 )5/2 8(K 2 − K12 v 2 )7/2 −
(160)
and
√ " 3 4 π T λ K1 K2 v(1 − v 2 ) iK2 v 2 K p hT02 i = −√ − 2π(K 2 − K12 v 2 ) 1 − v 2 24 K 2 − K12 v 2
# iK12 K2 v 2 (1 − v 2 ) K1 K2 v(1 + 14v 2 ) K13 K2 v 3 (1 − v 2 ) + + − . 9π(K 2 − K12 v 2 )2 48(K 2 − K12 v 2 )5/2 8(K 2 − K12 v 2 )7/2
(161)
Energy Loss and the Gauge-String Duality
37
K The expression for hT03 i can be obtained from (161) by exchanging K2 with K3 . In real space, using the notation in (91), we find that to leading order in xT , K hTmn i = Λ−1 T quark Λ
(162)
mn
where Λmn represents a Lorentz transformation with boost parameter v in the x1 quark direction and Tmn is given by
quark Tmn
1 x4 √ 0 λ = 12π 2 0 0
0 x2⊥
− x6
0 x21
2x1 x2 − 6 x −
2x1 x3 x6
−
2x1 x2 x6
x21 + x23 − x22 x6 −
2x2 x3 x6
0
2x1 x3 − 6 x . 2x2 x3 − 6 x 2 2 2 x1 + x2 − x3 x6
(163)
quark Tmn is the stress-energy tensor of a stationary heavy quark. Up to the overall multiplicative factor, it can be determined by the requirement that it is conserved and satisfies conformal symmetry. Thus, the leading short distance behavior of the near field of our quark is a boosted version of the stress-energy tensor of a stationary quark. At distances much shorter than the typical length scale of the fluid, the quark does not see the plasma it is moving through, and behaves as if it were in vacuum. Of more interest are the subleading corrections to the stress-energy tensor. These are given by
√ 2 λT v(x − vt) x2⊥ (−5 + 13v 2 − 8v 4 ) + (−5 + 11v 2 )(x − vt)2 hTtt i = √ 5/2 1 − v2 72 [x2⊥ (1 − v 2 ) + (x − vt)2 ] (164a) √ 2 2 2 2 2 λT v (x − vt) (1 − v )x⊥ + 2(x − vt) hTtx1 i = − √ (164b) 24[x2⊥ (1 − v 2 ) + (x − vt)2 ]5/2 1 − v2 √ 2 λT x⊥ (1 − v 2 )v 2 8x2⊥ (1 − v 2 ) + 11(x − vt)2 hTtx⊥ i = − √ (164c) 5/2 1 − v2 72 [x2⊥ (1 − v 2 ) + (x − vt)2 ] √ 2 λT v(x − vt) x2⊥ (8 − 13v 2 + 5v 4 ) + (11 − 5v 2 )(x − vt)2 hTx1 x1 i = √ (164d) 5/2 1 − v2 72 [x2⊥ (1 − v 2 ) + (x − vt)2 ] √ 2 λT v(1 − v 2 ) 8x2⊥ (1 − v 2 ) + 11(x − vt)2 hTx1 x⊥ i = √ (164e) 5/2 1 − v2 72 [x2⊥ (1 − v 2 ) + (x − vt)2 ]
38
S. S. Gubser et al.
√ 2 λT v(1 − v 2 )(x − vt) 5x2⊥ (1 − v 2 ) + 8x2⊥ hTx⊥ x⊥ i = − √ 5/2 1 − v2 72 [x2⊥ (1 − v 2 ) + (x − vt)2 ] √ 2 λT v(1 − v 2 )(x − vt)x2⊥ hTϕϕ i = − √ 2 1 − v 9 [r2 (1 − v 2 ) + (x − vt)2 ]5/2
(164f)
(164g)
where (x⊥ , ϑ) are polar coordinates for the x2 x3 plane. A strange feature of (164a) is that it exhibits a transition from a region of energy depletion behind the quark, to a region of energy depletion in front of it as the quarks velocity decreases. When v 2 > 5/8 there is a buildup of energy density ahead of it, forming a “bulldozer effect.” See Fig. 3. As it slows down extra lobe-like features appear until v 2 < 5/13 where the energy buildup is behind the quark, creating an “inverse-bulldozer” effect. See Fig. 4. Recall that the speed of sound in a conformal fluid is v 2 = 1/3, so that this transition occurs at velocities which are higher than the speed of sound. This indicates that the features we are seeing are not hydrodynamic in nature. A more detailed analysis of the deviation of the energy density from linearized hydrodynamics can be found in the literature.63 We will see in Sec. 8 that it is probably the near field of the stress-energy tensor which dominates high-angle emission of hadrons. It would certainly be interesting to understand the physical mechanism behind this near-field behavior. 7. Numerical Results for the Holographic Stress Tensor Expression (164) and the Fourier transform of (124) capture the near-field and farfield asymptotics of the stress-energy tensor. In 6.1.2, we have seen an indication that far from the moving source the energy-momentum tensor exhibits hydrodynamic behavior. In 6.2, we have seen that the near-field stress tensor exhibits nonhydrodynamic behavior with interesting features, like the multi-lobe structure in Figure 4. In the intermediate regime, there is a transition region between hydrodynamics and whatever short-distance physics governs the near field. To probe this region, one needs solutions to (44) for values of K where no analytic asymptotic treatment is available. We have obtained such solutions numerically. First, (52a), (52b), (52d), (52e), and (52g) were solved, and QA , QD , and QE were obtained.46 Then, the resulting momentum space stress-energy tensor was passed through an FFT to position space using a 1283 grid. Such a computation has been carried out for the energy density50, 51 and for the Poynting vector.52, 53 Consider the normalized energy density √ 1 − v2 K √ hT00 E= i, (165) (πT )4 λ K where hTmn i is defined in (89) as the stress-energy tensor of the system, minus the stress-energy of the thermal bath, minus the divergent delta-function contribution (92) at the position of the moving quark. It is convenient to decompose this rescaled
Energy Loss and the Gauge-String Duality
39
energy density into a Coulombic term, a near-field (large momentum) term, a farfield (small momentum) term, and a residual term: E = ECoulomb + EUV + EIR + Eres .
(166)
The Coulombic term represents the contribution coming from the near field of the quark: p K 2 − K12 v 2 v 2 K12 2 + (1 − v 2 ) , (167) ECoulomb = − 2 + v 24π 24π which is what we found in (162) converted to momentum space. It can be read off of the O(K) terms in (159). The far-field term, EIR , scales like O(K −5 ) at large momentum and asymptotes to (125a) at small momenta.h There are many possible expressions which satisfy the above criteria. Taking note of the resummation (168), we used EIR = −
1 3ivK1 (1 + v 2 ) − 3v 2 K12 1 3ivK1 (1 + v 2 ) − 3v 2 K12 + 2 2π K 2 − 3v 2 K1 − ivK 2 K1 2π K 2 − 3v 2 K12 − ivK 2 K1 + µ2IR
(168)
where µIR is a typical scale where (125a) stops being valid. We used µIR = 1. Similarly, EUV scales like O(K 1 ) at small momenta, and asymptotes to (159) at large momenta. To regulate the large momentum expressions in the IR we made the replacement −n/2 1 1 µ2UV = 1− 2 (169) K − K12 v 2 + µ2UV (K 2 − v 2 K12 )n/2 (K 2 − K12 v 2 + µ2UV )n/2
where the last term in (169) is expanded to order O(K −5 ). Similar to (168), µUV is a cutoff scale which we set to 1. After the replacement (169), the first two terms in a series expansion of the energy density take the form p (2 + v 2 ) K 2 − K12 v 2 + µ2UV 2K12 v 2 (1 − v 2 ) − (2 + v 2 )µ2UV p + ... . EUV = − + 24 48 K 2 − K12 v 2 µ2UV (170) Once EUV , EIR , and ECoulomb are known we can numerically compute Eres , which can be fed through a three-dimensional FFT with controllable errors because it is absolutely integrable. We then add back to the real space numerical expression for Eres the real space version of EIR , EUV , and ECoulomb to obtain the energy density in position space. The Fourier transform of the large momentum asymptotic expressions can be carried out using (133). As explained in Sec. 6.1.2, Fourier transforming the sound pole is difficult due to the cubic term in the denominator — the term associated with the shear viscosity. To convert the the sound pole structure of (168) to real space we first rewrote it as EIR = h Actually,
2 K⊥
A(K1 ) , + m(K1 )2
(171)
we require that it asymptote to (125a) up to a momentum-independent constant.
40
S. S. Gubser et al.
E for v= 0.58 0.1
0.08
Xp
0.06
0.04
0.02
0
-0.1
-0.05
0
X1
0.05
0.1
Fig. 3. (Color online) A contour plot of the energy density near the moving quark, with the bath and the Coulombic contributions subtracted.50 Red signifies energies above the background value of the plasma while blue signifies energies below the background value of the plasma. The black dot at X1 = Xp = 0 marks the location of the quark which is moving at a constant velocity of v = 0.58, just above the speed of sound. We work in dimensionless units where X1 = πT x1 and Xp = πT x⊥ . The dashed green line shows the presumed location of the Mach cone.
then Fourier transformed in the K⊥ direction using (133), and resorted to numerics to FFT the remaining K1 coordinate. This was carried out on a line with 1944 points with K1 ranging from −20 to 20. In Figures 3–5 we show the energy density (with the Coulombic field ECoulomb subtracted) at various spatial scales. The components of the energy flux can be treated in a similar manner: we define √ 1 − v2 K √ hT0i i (172) Si = − (πT )4 λ and decompose S~ into
S~ = S~Coulomb + S~UV + S~IR + S~res .
(173)
The Coulombic expression for the Poynting vector is given by the O(K) terms in (160) and (161). The small momentum expressions are given by S1 IR = − + S⊥ IR = −
1 i(1 + v 2 )K1 + vK 2 − 2v 3 K12 1 i(1 + v 2 )K1 + vK 2 − 2v 3 K12 + 2π K 2 − 3v 2 K12 − ivK 2 K1 2π K 2 − 3v 2 K12 − ivK 2 K1 + µ2IR 2v 1 + iK1 /4v 2v 1 + iK1 /4v − 2 2 π K − 4ivK1 π K − 4ivK1 + µ2IR
(174)
1 i(1 + v 2 )K⊥ + b2 K1 K⊥ ) 1 iK⊥ + + (regulators) 2π K 2 − 3v 2 K12 − ivK 2 K1 2π K 2 − 4ivK1
(175)
where we have set µIR = 1 and by “(regulators)” we mean terms containing the regulator µIR , analogous to those in (168) and (174). The large momentum expressions are given by applying (169) to (160) and (161). As was the case for the
Energy Loss and the Gauge-String Duality
41
E for v= 0.75 0.1
0.08
Xp
0.06
0.04
0.02
0
-0.05
-0.1
0
0.05
X1
0.1
Fig. 4. (Color 0 4online) A contour plot of the energy density near the moving quark, with the bath and the Coulombic contributions subtracted.50 Red signifies energies above the background value of the plasma while blue signifies energies below the background value of the plasma. The black 2 dot at X1 = Xp = 0 marks the location of the quark which is moving at a constant velocity of v = 0.75, well above the speed of sound. We work in dimensionless units where X1 = πT x1 and Xp = πT x⊥ . The dashed green line shows the presumed location of the Mach cone. 0
E for v = 0.75 14
12
Xp
10 8 6 4 2
0 -15
-10
-5
0
X1
5
10
15
Fig. 5. (Color online) A contour plot of the energy density far from the moving quark, with the bath and the Coulombic contributions subtracted.50 Red signifies energies above the background value of the plasma while blue signifies energies below the background value of the plasma. The black dot at X1 = Xp = 0 marks the location of the quark which is moving at a constant velocity of v = 0.75, well above the speed of sound. We work in dimensionless units where X1 = πT x1 and Xp = πT x⊥ . The dashed green line shows the presumed location of the Mach cone.
42
S. S. Gubser -0. et al.
Xp
S for v =0.75 14 12 10 8 6 4 2 0 -15
-10
-5
0
X1
5
10
15
Fig. 6. (Color online) Contour plot of the magnitude of the Poynting vector, with the Coulombic contribution subtracted.52 The magnitude of the Poynting vector goes from red (large) to white (zero) while the arrows show its direction. The dashed green line shows the presumed location of the Mach angle, and the blue line shows the location of the laminar wake — as dictated by its large distance asymptotics.
energy density, we used µUV = 1. The real space results for the Poynting vector for v 2 = 3/4 are shown in Fig. 6. 8. Hadronization, Jet-Broadening, and Jet-Splitting There is a significant gap between the results of Secs. 5–7 and experimental data. Before reviewing recent attempts to bridge this gap, let’s briefly summarize some of the relevant data. There are of course more authoritative summaries in the experimental literature.64–69 The data seem to reveal a phenomenon of “jet-splitting,” whereby an energetic parton traversing the medium deposits so much of its energy through high-angle emission that — with appropriate momentum cuts and subtractions — the extra particle production due to the parton is at a minimum in the direction of its motion, and reaches a maximum at an angle roughly 1.2 radians away. Jet-splitting is most simply illustrated through two-point histograms of the azimuthal angle ∆φ separating a pair of energetic hadrons close to mid-rapidity. To understand the phenomena better, it is useful to examine two landmark studies of these histograms: one from STAR64 and one from PHENIX.65 In the STAR analysis, fairly inclusive momentum cuts were considered: under one set of cuts, the less energetic of the two hadrons was required to have transverse momentum greater than 150 MeV/c. The resulting data show a peak for nearly collinear hadrons that is approximately the same shape for central gold-gold collisions as for proton-proton collisions: see Fig. 7. This “near-side jet” feature can reasonably be supposed to arise from vacuum fragmentation effects. The two-point
A
pa rs per tr gger: 1/N dNAB(d -jet)/d(∆φ)
Energy Loss and the Gauge-String Duality
0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0
0.5
(a) 0-5%
(b) 5-10%
(c) 10-20%
(d) 20-40%
(e) 40-60%
(f) 60-90%
1
1.5 2 ∆φ (rad)
2.5
3
0.5
1
1.5 2 ∆φ (rad)
2.5
43
3
Fig. 7. (Color online) Top: The STAR analysis64 shows substantial broadening of the awayside jet. Reprinted Fig. 1 with permission from J. Adams et al., Phys. Rev. Lett. 95, 152301 (2005), http://link.aps.org/abstract/PRL/v95/p152301. Copyright 2005 by the American Physical Society. Bottom: The PHENIX analysis65 shows jet-splitting for sufficiently central events. Reprinted Fig. 2 with permission from S. S. Adler et al., Phys. Rev. Lett. 97, 052301 (2006), http://link.aps.org/abstract/PRL/v97/p052301. Copyright 2006 by the American Physical Society.
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histogram also shows an “away-side jet” feature around ∆φ = π which is substantially broader for central gold-gold collisions than for proton-proton. Neither the “near-side jet” nor the “away-side jet” are reconstructed jets in the usual sense; instead, they are ideas that help explain the main features of histograms assembled from millions of events. It is usually assumed that the typical event contributing to the histograms involves a hard scattering event where one parton escaped the medium without much interaction, producing the highly energetic “trigger hadron,” and the other parton interacted substantially with the medium before generating an “associated hadron” in the vicinity of ∆φ = π. The upshot is that with inclusive momentum cuts, there is substantial broadening of the away-side jet, but not jet-splitting: associated hadron production is still maximized, or statistically indistinguishable from its maximum, at ∆φ = π. With tighter momentum cuts on the associated hadron, the data used in the particular STAR analysis under discussion show striking jet-broadening, but the scatter in the data is sufficient to prevent firm conclusions from being drawn — from this particular study — about whether there is jet-splitting. (Subsequent STAR analyses of both two- and three-point hadron correlators provide strong evidence in favor of jet-splitting.67–69 ) The PHENIX analysis65 is similar to the STAR analysis,64 but with more restrictive cuts: in particular, the less energetic hadron was required to have transverse momentum greater than 2.5 GeV/c. The resulting histograms, with a zero-yieldat-minimum (ZYAM) subtraction, show a distinct minimum in associated hadron production at ∆φ = π, with a broad maximum in the ballpark of ∆φ = π − 1.2. This jet-splitting persists down to roughly 50% centrality, meaning that it occurs for events where the impact parameter is less than about 10 fm. The ZYAM subtraction is an important part of the analysis, especially for the less central events. The reason a subtraction is needed is that for a non-central collision, there is an angular modulation of single-particle yields, approximately proportional to 1 + v2 cos 2φ, where the zero of φ coincides with the azimuthal direction of the impact parameter, and the elliptic flow coefficient v2 depends on the transverse momentum and species. Two-point hadron correlators receive a contribution from single-hadron yields. The ZYAM scheme is to subtract a multiple of the appropriate product of single-particle yields. The multiple is chosen so that the resulting histogram has one bin with zero net events, while all other bins have a positive net number of events. A natural hypothesis is that the high-angle emission leading to either jetsplitting or jet-broadening can be described in terms of a sonic boom in the medium.70, 71 Two related difficulties afflict this idea. First, it’s hard to get a sonic boom with a big enough amplitude to account for the data65 with reasonable rates of energy loss;45, 72 it should be noted however that not all investigators agree on this point,73 and that there are some phenomenological models based on sonic booms that fit the data.74, 75 Second, one usually finds a diffusion wake with comparable strength to the sonic boom.30, 52, 71, 76 At least in a static medium, it is hard to get jet-splitting in the presence of a significant diffusion wake.
Energy Loss and the Gauge-String Duality
45
To compute the relative strength of the diffusion wake and sonic boom for the heavy quark in the SYM theory, we go back to the conservation equation (99), 2 mn F n = izH lim Km hTK i.
(176)
~ K→0
In Sec. 6.1.2 we saw that at small K, the components of the stress-energy tensor may be decomposed into terms containing sound poles at K 2 ∼ 3K12 v 2 and terms mn associated with a wake which have a pole at K1 ∼ 0. If hTIR i is the leading, small K contribution to the stress-energy tensor, then we may decompose mn mn mn hTIR i = hTsound i + hTwake i,
(177)
where 3iK1 v iK1 iK2 iK3 √ iK1 iK1 v (πT )4 λ 1 + v2 0 0 mn i=−√ hTsound 2 2 2 2 iK2 0 iK1 v 0 1 − v 2π(K − 3K1 v ) iK3 0 0 iK1 v
(178)
and
0 √ 4 (πT ) λ i mn 1 i= √ hTwake 2 1 − v 2πK1 0 0
1 0 0 0
0 0 0 0
0 0 . 0
(179)
0
Since only the terms in (177) contribute to the total drag force in (176), this gives a natural division of the total drag force: n 2 mn Fsound = izH lim Km hTsound i ~ K→0
n 2 mn Fwake = izH lim Km hTwake i, ~ K→0
From (124), (125), (126), and (127), we find that 1 1 0 0 Fsound = − 2F0 Fwake = 1 + 2 F0 . v v
(180)
(181)
In our conventions, the zero component of F n gives us the total rate of change in the energy density plus any energy flux going out of the system. Thus, the ratio of energy going into sound waves to energy going into the wake is 1 + v 2 : −1 .
(182)
While sound modes carry energy away from the moving quark, the wake feeds energy in toward the quark. While this may seem counter-intuitive, in some sense it’s obvious: the diffusion wake consists of a flow of the medium forward toward the quark. The forward-moving momentum in the diffusion wake is the momentum deposited by the quark at earlier times. Qualitatively, (182) says that the diffusion wake and the sonic boom have comparable strength. When comparing (182) to the scenarios of energy loss in the literature,71, 72 one finds that (182) quantitatively matches a scenario where the relative strength of the wake is so large that it washes out features of jet-splitting associated with the sonic boom. It may be significant,
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associated hadron
φ
x2
x1
ϕ
x3
θ beamline
trigger hadron Fig. 8. (Color online) A sketch of the coordinate systems used to describe away-side hadron production. If the trigger hadron is at φ = 0, then a helpful relation at mid-rapidity (θ = π/2) is ϑ = π − φ, where ∆φ is the azimuthal separation between the trigger and associated hadrons.
however, that the medium is infinite and static, both in our work and in the linear hydrodynamic scenario71 that our results match onto at large length scales. In light of the difficulties in explaining the data with a “boom and wake” model, focused on the hydrodynamic regime, it is natural to investigate the effect on hadron production of the region of the medium close to the moving quark where hydrodynamics is inapplicable. This has been pursued in a series of works,63, 77–83 which we briefly summarize in the next few paragraphs. The first idea is to subtract away the leading order Coulombic contribution to hTmn i. Up to an overall multiplicative rescaling, these are the quantities we denoted ECoulomb and S~Coulomb in Sec. 7. The justification for this is that these fields describe the energy of the energetic parton itself, not the energy lost from it. The remaining energy density, which we will denote as sub , can be split up as sub = bath + ∆. (Note that in contrast to our definitions of E and its variants, sub explicitly includes the contribution from the ~sub , with the Coulombic contribution subtracted away, bath.) The Poynting vector S is non-zero only because of the presence of the quark. The basic plan is to use the ~sub ) into a Cooper-Frye algorithm84 to convert string theory predictions for (sub , S spectrum of hadrons. The Cooper-Frye algorithm is based on converting a fluid element at temperature T and with local four-velocity U m into hadrons according a Maxwell-
Energy Loss and the Gauge-String Duality
Boltzmann distribution in the local rest frame: Z µ dN =− dΣµ P µ eU Pµ /T f (pT , φ) = pT dpT dφ y=0 R3 Z ∞ Z 2π Z ∞ µ = x⊥ dx⊥ dϕ dx1 EeU Pµ /T , 0
0
47
(183)
−∞
where N is the number of hadrons, and we have set P m = pT
pT cos(π − φ) pT sin(π − φ) 0
and
(184)
U m = U 0 U 1 U⊥ cos ϕ U⊥ sin ϕ .
(185)
tanh y = cos θ .
(186)
Note that because of our choice of mostly plus signature, the energy of the hadron in the local rest frame of the fluid is −U m Pm . Also because of this choice of signature, we are obliged to include an explicit minus sign in the first integral expression of (183). To understand (183)–(185), it helps to refer to Fig. 8. The momentum of the associated hadron is P m , and (183) is written in the approximation that the associated hadron is massless — an excellent approximation since a typical hadron of interest is a pion with pT ∼ 3 GeV/c. The rapidity y is related to the angle from the beamline θ by
(Note that rapidity y has nothing to do with the depth coordinate y = z/zH used in previous sections.) The freeze-out surface is chosen to be a slice of constant x0 in (183). This is the best motivated choice for an infinite, asymptotically static medium. In an expanding medium, a more usual choice is a fixed-temperature surface with the temperature set close to the QCD scale. For isochronous freeze-out, the measure dΣm is simply dx1 dx2 dx3 1 0 0 0 , and in passing to the second line of (183) we have simply expressed the metric on R3 in radial coordinates. It is important to realize that the azimuthal angle ϕ around the direction of motion of the parton (assumed to be in the +x1 direction, as usual) is different from the azimuthal angle φ around the beamline. We have omitted in (183) a subtraction of the contribution of the bath to hadron production, which depends on pT but not on φ. We have also not attempted to normalize f (pT , φ): doing so would involve partitioning over the spectrum of hadrons. The information from string theory enters into (183) in two ways. First, the local four-velocity U m is the local rest frame of the medium, in which the Poynting vector vanishes. Second, the temperature T is the temperature in this local rest sub m n frame, deduced by plugging the energy density Tmn U U into the equation of mn state. Evidently, one needs all components of Tsub in order to precisely determine U m and T . This is a problem since only the m = 0 row of the stress tensor has been computed in full.52, 53 Another problem is that close to the quark, the medium is
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presumably far from equilibrium, so using Cooper-Frye seems somewhat perilous. We will return to a discussion of these two issues below. To understand how hydrodynamical and non-hydrodynamical effects contribute to the spectrum of produced hadrons, one must have some notion of where the boundary is between hydrodynamical and non-hydrodynamical regimes. This boundary is presumably not sharp. Three considerations have gone into identifying an appropriate boundary: (1) The non-hydrodynamical region can be chosen as the region where ∆/bath is less than some constant of order unity. For v = 0.9 ,
λ = 5.5 ,
N = 3,
TSYM = 200 MeV ,
(187)
a preferred choice is ∆/bath ≤ 0.3 .
(188)
Here and below, we will describe the region defined by (188) with the parameter choices (187) as the “Neck.” It extends roughly over −1 ≤ X1 ≤ 0.5 and 0 ≤ X⊥ ≤ 1.7. (For TSYM = 200 MeV, X = 1 corresponds to x = 1/πTSYM = 0.31 fm.) (2) The Neck region can be compared with the region where the Knudsen number exceeds some constant of order unity. An appropriate version of the Knudsen number in the current context is ~sub | |∇ · S . (189) Kn ≡ Γ ~sub | |S Here the sound attenuation length Γ is the same as the one discussed following (120): Γ = 4η/3sT = 1/3πT . For the choice of parameters (187), examination of the near-field expressions (164) shows that the region where Kn > ∼ 1/3 is somewhat bigger in the X1 direction than the Neck. However, corrections to ~sub may not be negligible for X1 and/or X⊥ the near-field approximation to S of order unity. (3) The Neck region can be compared with the region where the constitutive relamn tions of hydrodynamics break down. Given all components Tsub of the stress tensor, with the Coulomb field subtracted away, there is a straightforward procedure for testing the constitutive relations. First determine the local velocity field U m by passing to the local rest frame of the fluid. Let (T mn )L be the subtracted stress-energy tensor in the local rest frame. The energy density is read off immediately as (T 00 )L ; the pressure is deduced from the equation of state; and the shear viscosity contribution to the space-space parts of the stress tensor can be determined from U m and its gradient. Deviations of the spacespace components of (T mn )L from the combined contribution of pressure and shear viscosity are measures of the failure of hydrodynamics. A study78 of the near-field expressions (164) for a somewhat different choice of parameters from (187) (namely v = 0.99, λ = 3π, N = 3) concludes that
Energy Loss and the Gauge-String Duality
49
deviations from hydrodynamics are appreciable out as far as X ∼ 8. However, the near-field expressions definitely cannot be trusted at such large distances.i This analysis could therefore be considerably improved if all components of mn Tsub were computed directly from string theory. The main conclusion to draw from points 2 and 3 is that in the Neck region, the subtracted stress tensor is essentially unrelated to hydrodynamics. Instead, the physics may be presumed to be dominated by strong coherent color fields combined with responses of the medium to strong field gradients: hence the term “chromoviscous neck.” To return to hadronization: The Cooper-Frye integral over R3 can be split into the Neck region and the “Mach” region — which is everything else. In the Mach region, where the energy density comes mainly from the bath, a good approximation to the local rest frame can be found by setting ~ ~ = 3 Ssub . U 4 bath
(190)
In the neck region, this approximation is less reliable, but because space-space components of T mn are not available from a string theory calculation, it is hard to give a better motivated prescription for determining the local rest frame. With the choice (190), the result is that the Neck contribution to the Cooper-Frye integral leads to a distinctive double-peaked structure in f (pT , φ) for pT . This is remarkable when compared to the single-peaked structure emerging from a computation in a perturbative QCD framework based on Joule heating,81 which is similarly passed through the Cooper-Frye hadronization algorithm. See Fig. 9. The double-peaked structure from the Neck region of the trailing string stress tensor has nothing to do with the Mach cone. It doesn’t occur at the same angle: for example, at v = 0.58, the Mach angle ϑ = cos−1 cs /v is very nearly zero, but the Neck region still produces a double peak structure (not shown in Fig. 9) about a radian away from φ = π. When v is very close to 1, the double peaks get closer to φ = π. This is reminiscent of the structure observed at large K in Fourier space,46, 54 but the peaks observed in the predicted hadron spectra are more widely separated than the ones in Fourier space. While the hadronization studies63, 77–83 give valuable insight into the relation of the trailing string to high-angle hadron emission from an energetic parton, it is not claimed that the results are fully realistic, or that a direct comparison to untagged dihadron histograms, like the ones in Fig. 7, is justified. Let us review the potential difficulties. First, the trailing string describes an infinitely massive quark that propagates at a constant velocity through an infinite, static, thermal medium. Fluctuations leading to stochastic motions of finite mass quarks may significantly affect i A computation of the energy radiated from a moving quark53 shows that agreement with linearized hydrodynamics is already fairly good at X ∼ 5. Based on results of an earlier study,50 the onset of reasonable agreement with linearized hydro occurs near X = 3.
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Fig. 9. (Color online) Hadron production based on a Cooper-Frye hadronization of a perturbative QCD calculation and of trailing string results (AdS/CFT).81 Bath contributions have been subtracted away, and the curves have all been normalized to have the same maximum when the Cooper-Frye integral is carried out over the entire volume accessible to each computation. The Neck region for the perturbative QCD calculation is again defined as a region close to the quark where deviations from hydrodynamics are significant.
the results. Also, there may be an effective “form factor” for massive quarks that partially smears out the field close to the quark. In addition, it is not obvious that Cooper-Frye is justified, because the crucial effect comes from the non-equilibrium part of the medium (the Neck). Subtracting away the Coulomb field is certainly well-motivated physically, but it is possible to maintain some skepticism about whether it is the correct prescription in combination with Cooper-Frye. Finally, the approximation (190) to the local rest frame is imprecise in the Neck region. Despite these potentially serious issues, the punchline of the phenomenological studies63, 77–83 seems to us likely to be robust: the near-quark region has a substantially greater tendency toward high angle emission in the trailing string treatment
Energy Loss and the Gauge-String Duality
51
than in the perturbative QCD treatment based on Joule heating. Modulo concerns already expressed, the near-field contribution to high-angle emission is stronger than the contribution of the hydrodynamical regime, and it results in a significant double-hump structure, reminiscent of jet-splitting.
9. Conclusions Let us conclude by addressing the four main questions we raised in the introduction: (1) What is the rate of energy loss from an energetic probe? For√ a heavy quark moving at a velocity v, the drag force is Fdrag = v − π 2 λ T 2 √1−v . This is explained in more detail in Sec. 4. 2 (2) What is the hydrodynamical response far from the energetic probe? √ There is a sonic boom with Mach angle ϑ = cos−1 cs /v, where cs = 1/ 3 is the speed of sound dictated by conformal invariance. There is also a diffusion wake of comparable strength. These points are explained in Secs. 6.1.2 and 7. (3) What gauge-invariant information can be extracted using the gauge-string duality about the non-hydrodynamic region near the probe? The expectation values hT m0 i of the energy density and the Poynting vector of the gauge theory stress tensor have been computed with uniformly good accuracy across all length scales for several values of the velocity of the heavy quark, as we review in Sec. 7. Analytic approximations to the space-space components of hT mn i are also available at small length scales: see Sec. 6.2. (4) Do the rate and pattern of energy loss have some meaningful connection to heavy ion phenomenology? A suitable translation of parameters from SYM to QCD results in estimates of energy loss for c and b quarks which are not far from realistic, or which may be fully realistic. We summarize these estimates in Sec. 4. Studies of hadronization starting from the string theory predictions for the energy density and Poynting vector indicate that the trailing string leads to significant high-angle emission from the Neck region, close to the quark, suggestive of jet-splitting. We describe these studies in Sec. 8. Although it is premature to make detailed comparisons to data, it is clearly worthwhile to extend and refine both the string theory analysis and the phenomenological studies.
Acknowledgments We thank M. Gyulassy and J. Noronha for useful correspondence. This work was supported in part by the Department of Energy under Grant No. DE-FG0291ER40671 and by the NSF under award number PHY-0652782. Fabio De Rocha was also supported in part by the FCT grant SFRH/BD/30374/2006.
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Appendix A. Notation In this appendix we present short explanations of some of the nomenclature and mathematical notations used in the main text. xµ : The five spacetime coordinates of AdS5 -Schwarzschild, usually (t, x1 , x2 , x3 , z). m x , pm : The four-vectors for position and momentum in R3,1 . We use mostly plus signature, so (for example) η mn pm pn = −E 2 + p~2 = −m2 . ~x, p~: The three-vectors for position and momentum: spatial components of xm and pm . 2 x1 : This could mean either G1µ xµ = Lz2 x1 or η1ν xν = x1 . Our convention is to prefer the latter; likewise x2 = x2 and x3 = x3 . x⊥ : p The radial distance from the quark in the x2 , x3 plane: x⊥ = x22 + x23 . Occasionally we consider the two-vector ~x⊥ = (x2 , x3 ). z: This is the depth coordinate in AdS5 or AdS5 -Schwarzschild which is 0 at the boundary and has dimensions of length. The AdS5 metric 2 is ds2 = Lz2 (−dt2 + d~x2 + dz 2 ). zH : The depth of the horizon in AdS5 -Schwarzschild, related to the temperature by T = 1/πzH . y: Usually, a rescaled depth coordinate in AdS5 -Schwarzschild, defined by y = z/zH . But in Sec. 8 we use y to denote rapidity, i.e. tanh y = pz /E where pz is the momentum along the beampipe and E is the energy. r: We use r to indicate a radial separation in R3 . Some authors use r to denote the depth coordinate r = L2 /z in AdS5 . AdS5 : Five-dimensional anti-de Sitter space, the maximally symmetric negatively curved spacetime in 4 + 1 dimensions. Its metric is given by (3) with h = 1. SYM: An abbreviation for “N = 4 super-Yang-Mills theory in four dimensions,” which is the theory controlling the low-energy excitations of D3-branes. N : Usually, the number of colors: N = 3 in QCD. An exception is that in Sec. 8, we use N to indicate the number of hadrons predicted by the Cooper-Frye algorithm. 2 gYM : The gauge coupling of SYM, normalized so that gYM N = L4 /α02 , where N is the number of colors. gs : The gauge coupling of QCD. We also use αs = gs2 /4π. 2 λ: The ’t Hooft coupling, λ = gYM N. L: The radius of curvature of AdS5 . α0 : The Regge slope parameter of fundamental strings, see (14). κ: The five-dimensional gravitational coupling. Gµν : The spacetime metric of AdS5 or AdS5 -Schwarzschild. R: The Ricci scalar in AdS5 or AdS5 -Schwarzschild. We also use the
Energy Loss and the Gauge-String Duality
h:
T:
gαβ : σα : v: ξ: xµ∗ (σ): RAA :
τµν :
K τµν : hµν : hK µν : Axial gauge: Hmn :
A:
ψS :
53
Ricci tensor Rµν and the Riemann tensor Rαµβν . Our conventions are R = Gµν Rµν and Rµν = Gαβ Rαµβν , with signs arranged so that Rµν = − L42 Gµν in AdS5 of radius L. The “blackening function” for AdS5 -Schwarzschild, whose metric is 4 given in (3). It is given by h(z) = 1 − z 4 /zH . We sometimes think of h as a function of the depth z, and sometimes as a function of y = z/zH . h0 always means h0 (y) = −4y 3 . The temperature in the dual field theory, which is the same as the Hawking temperature of the dual black hole background. The temperature of the AdS5 -Schwarzschild background (3) is T = 1/πzH . The worldsheet metric of a string. Coordinates on the string worldsheet. The speed of a moving quark. Usually we take this motion to be in the +x1 direction. Gives the shape of the string that describes the quark in the fivedimensional geometry. See (13) and (20). The embedding function for a classical string in AdS5 -Schwarzschild. When no ambiguity is possible, we denote this embedding function more simply as xµ (σ). The nuclear modification factor, defined as the number of particles produced (usually at a particular value of pT and in a specified range of rapidity) in a collision of two nuclei with atomic number A, divided by the number produced in a proton-proton collision scaled up by the effective number of binary nucleon-nucleon collisions in the heavy-ion collision. The five-dimensional stress-energy tensor, not to be confused with the expectation value of the boundary stress-energy tensor hTmn i. For the trailing string, τµν is given in (42). The Fourier components of τµν . See (46) and (48). Small metric perturbations around AdS5 -Schwarzschild. The Fourier components of hµν defined by analogy with (46). A gauge choice for the metric perturbations where hµz = 0. The Fourier components of hK µν in axial gauge, up to a normalization factor. See (49). We think of the Hmn as functions of y = z/zH , not of z. The even tensor mode combination of metric perturbations in axial gauge: see (51a). Similar definitions for Bi , C, Di , and Ei follow, and gauge-invariant combinations B, D, and E can be found in (61)-(62). These are all functions of y = z/zH . The scalar master field. A master field is a gauge-invariant combination of metric fluctuations in AdS5 -Schwarzschild with a simple equation of motion. We encountered four other master fields: ψVeven , ψVodd , ψTeven , and ψTodd : see (67).
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√ αv : A recurring normalization factor given by αv = 1/ 2πα0 1 − v 2 . ~ The momentum conjugate to ~x/zH ; used as a dimensionless waveK: number to parameterize the three-dimensional Fourier space used to describe the medium’s response to the quark. ~ perpendicular to the motion K⊥ : The magnitude of the component of K of the quark. Rmn : The asymptotic values of Hmn at y = 0. The boundary condition Rmn = 0 says that the four-dimensional metric which the boundary gauge theory experiences is flat Minkowski space. Pmn : The coefficients of y 3 in a small y expansion of Hmn . See (76) and (77). They are related to the divergent contribution to the boundary stress-energy tensor given in (90). Qmn : The coefficients of y 4 in a small y expansion of Hmn when Rmn = 0. They are related to the expectation value of the boundary stressenergy tensor by (91). gmn : The metric of the boundary conformal field theory, usually set equal to the Minkowski metric ηmn with mostly plus signature. um : The four-velocity of a heavy quark moving through the thermal medium. hTmn i: The one-point function of the stress-energy tensor in SYM in the presence of the moving quark. We find it convenient to decompose it into three pieces given in (89). hydro (T )mn : A stress tensor which satisfies the hydrodynamic constitutive relations, (120). (T hydro,SYM)mn is the hydrodynamic contribution to the stress tensor of the SYM theory. K hTmn i: The Fourier modes of the contribution of the moving quark to the stress-energy tensor in the dual field theory minus the divergent piece corresponding to the infinite mass of the quark. See (89). It can be computed from Qmn through (95). fn : The source term for the energy-momentum tensor, iKm T mn = f n . Various superscripts specify which contribution of the energymomentum tensor is being sourced. For example, fnhydro sources (T hydro )mn . RX : The coefficient of the leading homogeneous solution for various linear combinations of Hmn and their derivatives, such as A, B, Bi , ψT , etc. Our boundary condition is RX = 0. PX : The analog of Pmn for various linear combinations of Hmn and their derivatives. QX : The analog of Qmn for various linear combinations of Hmn and their derivatives. See (102) and (108). Also, we use the notations QD = QD1 and QE = QE1 . pT : The component of momentum perpendicular to the beamline.
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ϑ: An angular coordinate in momentum space, sin ϑ = K1 /K, or in real space, sin ϑ = x1 /x. ϕ: The azimuthal angle around the direction of motion of an energetic quark. Our usual convention is that the energetic quark moves in the +x1 direction, and then tan ϕ = K3 /K2 or x3 /x2 . θ: The angle of a trajectory relative to the beam. θ = π/2 is midrapidity. φ: The azimuthal angle around the beam. The angular variable ∆φ in dihadron histograms is the separation in φ between two hadrons. I, K: Modified Bessel functions of the first and second kind. J is a Bessel function of the first kind, and L is a modified Struve function. Neck: The neck is the region near a moving quark where the response of the medium is non-hydrodynamical. In practice, for the choice of parameters (187), the Neck can be defined, as in (188), as the region where the energy density, excluding the Coulombic contribution, exceeds 1.3 times the asymptotic energy density of an infinite static bath. U m : The four-velocity of a fluid element, usually defined so that it vanishes in precisely the same Lorentz frame in which the Poynting vector vanishes. ~ The energy density and Poynting vector, rescaled to make them E, S: dimensionless, with contributions from the thermal bath excluded: see (165) and (172). E can be decomposed into a sum of contributions from the Coulomb field of the quark, subleading UV effects, IR effects, and a residual quantity Eres , as in (166). An analogous ~ decomposition can be performed on S. References 1. J. M. Maldacena, The large N limit of superconformal field theories and super-gravity, Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200. 2. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B428 (1998) 105–114, hep-th/9802109. 3. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, hep-th/9802150. 4. S. S. Gubser and A. Karch, From gauge-string duality to strong interactions: a Pedestrian’s Guide, 0901.0935. 5. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183–386, hep-th/9905111. 6. I. R. Klebanov, TASI lectures: Introduction to the AdS/CFT correspondence, hep-th/0009139. 7. E. D’Hoker and D. Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253. 8. S. S. Gubser, I. R. Klebanov, and A. W. Peet, Entropy and temperature of black 3-branes, Phys. Rev. D54 (1996) 3915–3919, hep-th/9602135.
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9. J. Dai, R. G. Leigh, and J. Polchinski, New connections between string theories, Mod. Phys. Lett. A4 (1989) 2073–2083. 10. J. Polchinski, Dirichlet-branes and Ramond–Ramond charges, Phys. Rev. Lett. 75 (1995) 4724–4727, hep-th/9510017. 11. E. Witten, Bound states of strings and p-branes, Nucl. Phys. B460 (1996) 335–350, hep-th/9510135. 12. J. H. Schwarz, Covariant field equations of chiral N = 2 D = 10 supergravity, Nucl. Phys. B226 (1983) 269. 13. G. Policastro, D. T. Son, and A. O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043, hep-th/0205052. 14. S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045, 0712.2456. 15. G. Policastro, D. T. Son, and A. O. Starinets, The shear viscosity of strongly coupled N = 4 supersymmetric Yang–Mills plasma, Phys. Rev. Lett. 87 (2001) 081601, hep-th/0104066. 16. S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C22 (2001) 379–394, hep-th/9803001. 17. C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe, Energy loss of a heavy quark moving through N = 4 supersymmetric Yang–Mills plasma, JHEP 07 (2006) 013, hep-th/0605158. 18. S. S. Gubser, Drag force in AdS/CFT, Phys. Rev. D74 (2006) 126005, hep-th/ 0605182. 19. J. Casalderrey-Solana and D. Teaney, Heavy quark diffusion in strongly coupled N = 4 Yang–Mills, hep-ph/0605199. 20. K. Peeters, J. Sonnenschein, and M. Zamaklar, Holographic melting and related properties of mesons in a quark gluon plasma, Phys. Rev. D74 (2006) 106008, hep-th/0606195. 21. H. Liu, K. Rajagopal, and U. A. Wiedemann, An AdS/CFT calculation of screening in a hot wind, hep-ph/0607062. 22. M. Chernicoff, J. A. Garcia, and A. Guijosa, The energy of a moving quark-antiquark pair in an N = 4 SYM plasma, hep-th/0607089. 23. J. Casalderrey-Solana and D. Teaney, Transverse momentum broadening of a fast quark in a N = 4 Yang–Mills plasma, JHEP 04 (2007) 039, hep-th/0701123. 24. S. S. Gubser, Momentum fluctuations of heavy quarks in the gauge-string duality, Nucl. Phys. B790 (2008) 175–199, hep-th/0612143. 25. C. P. Herzog, Energy loss of heavy quarks from asymptotically AdS geometries, JHEP 09 (2006) 032, hep-th/0605191. 26. P. Talavera, Drag force in a string model dual to large-N QCD, JHEP 01 (2007) 086, hep-th/0610179. 27. E. Caceres and A. Guijosa, Drag force in charged N = 4 SYM plasma, JHEP 11 (2006) 077, hep-th/0605235. 28. G. Michalogiorgakis, Heavy ion collisions, quasi-normal modes and non-linear sigma models, PhD thesis, Princeton University, 2007. 29. C. P. Herzog and A. Vuorinen, Spinning dragging strings, JHEP 10 (2007) 087, 0708.0609. 30. S. S. Gubser and A. Yarom, Universality of the diffusion wake in the gauge-string duality, Phys. Rev. D77 (2008) 066007, 0709.1089. 31. M. Chernicoff and A. Guijosa, Acceleration, energy loss and screening in stronglycoupled gauge theories, JHEP 06 (2008) 005, 0803.3070.
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54. J. J. Friess, S. S. Gubser, and G. Michalogiorgakis, Dissipation from a heavy quark moving through N = 4 super-Yang–Mills plasma, JHEP 09 (2006) 072, hep-th/0605292. 55. Y.-H. Gao, W.-S. Xu, and D.-F. Zeng, Wake of color fields in charged N = 4 SYM plasmas, hep-th/0606266. 56. J. Martinez-Morales, Eigentensors of the Lichnerowicz operator in Euclidean Schwarzschild metrics, Annalen der Physik 15 (2006), no. 9 653–662. 57. P. K. Kovtun and A. O. Starinets, Quasinormal modes and holography, Phys. Rev. D72 (2005) 086009, hep-th/0506184. 58. V. Balasubramanian and P. Kraus, A stress tensor for anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413–428, hep-th/9902121. 59. S. de Haro, S. N. Solodukhin, and K. Skenderis, Holographic reconstruction of spacetime and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595–622, hep-th/0002230. 60. S. S. Gubser and A. Yarom, Linearized hydrodynamics from probe-sources in the gauge-string duality, 0803.0081. 61. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 7th edn., 2007. Edited by A. Jeffrey and D. Zwillinger. 62. A. Yarom, The high momentum behavior of a quark wake, Phys. Rev. D75 (2007) 125010, hep-th/0702164. 63. J. Noronha, G. Torrieri, and M. Gyulassy, Near zone Navier–Stokes analysis of heavy quark jet quenching in an N = 4 SYM plasma, Phys. Rev. C78 (2008) 024903, 0712.1053. 64. STAR Collaboration, J. Adams et al., Distributions of charged hadrons associated with high transverse momentum particles in p p and Au + Au collisions at s(NN)**(1/2) = 200-GeV, Phys. Rev. Lett. 95 (2005) 152301, nucl-ex/0501016. 65. PHENIX Collaboration, S. S. Adler et al., Modifications to di-jet hadron pair correlations in Au + Au collisions at s(NN)**(1/2) = 200-GeV, Phys. Rev. Lett. 97 (2006) 052301, nucl-ex/0507004. 66. PHENIX Collaboration, A. Adare et al., Dihadron azimuthal correlations in Au+Au √ collisions at sN N = 200 GeV, Phys. Rev. C78 (2008) 014901, 0801.4545. 67. J. G. Ulery, Two- and Three-Particle Jet-Like Correlations, 0801.4904. 68. STAR Collaboration, B. I. Abelev et al., Indications of Conical Emission of Charged Hadrons at RHIC, 0805.0622. 69. J. G. Ulery, Conical emission in heavy ion collisions, J. Phys. G35 (2008) 104032, 0807.1613. 70. H. Stoecker, Collective flow signals the quark gluon plasma, Nucl. Phys. A750 (2005) 121–147, nucl-th/0406018. 71. J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney, Conical flow induced by quenched QCD jets, J. Phys. Conf. Ser. 27 (2005) 22–31, hep-ph/0411315. 72. A. K. Chaudhuri and U. Heinz, Effect of jet quenching on the hydrodynamical evolution of QGP, Phys. Rev. Lett. 97 (2006) 062301, nucl-th/0503028. 73. J. Casalderrey-Solana and E. V. Shuryak, Conical flow in a medium with variable speed of sound, hep-ph/0511263. 74. T. Renk and J. Ruppert, Mach cones in an evolving medium, Phys. Rev. C73 (2006) 011901, hep-ph/0509036. 75. T. Renk and J. Ruppert, The rapidity structure of Mach cones and other large angle correlations in heavy-ion collisions, Phys. Lett. B646 (2007) 19–23, hep-ph/0605330. 76. B. Betz et al., Universality of the Diffusion Wake from Stopped and Punch-Through Jets in Heavy-Ion Collisions, 0812.4401.
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QUARKONIUM AT FINITE TEMPERATURE
ALEXEI BAZAVOV Department of Physics, University of Arizona, Tucson, AZ 85721, USA ´ PETER PETRECZKY RIKEN-BNL Research Center and Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA ALEXANDER VELYTSKY∗ Enrico Fermi Institute, University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637, USA and HEP Division and Physics Division, Argonne National Laboratory, 9700 Cass Ave., Argonne, IL 60439, USA
We discuss properties of heavy quarkonium states at high temperatures based on lattice QCD and potential models. We review recent progress made in lattice calculations of spatial static quark anti-quark correlators as well as quarkonium correlators in Euclidean time. Recent developments in effective field theory approach and potential models are also discussed.
1. Introduction There was considerable interest in the properties and the fate of heavy quarkonium states at finite temperature since the famous conjecture by Matsui and Satz.1 It has been argued that color screening in medium will lead to quarkonium dissociation above deconfinement, which in turn can signal quark gluon plasma formation in heavy ion collisions. The basic assumption behind the conjecture by Matsui and Satz was the fact that medium effects can be understood in terms of a temperature dependent heavy quark potential. Color screening makes the potential exponentially suppressed at distances larger than the Debye radius and it therefore cannot bind the heavy quark and anti-quark once the temperature is sufficiently high. Based on this idea potential models at finite temperature with different temperature dependent potentials have been used over the last two decades to study quarkonium properties at finite temperature (see Ref. 2 for a recent review). It was not until recently that effective field theory approach, the so-called thermal pNRQCD, has been developed to justify the use of potential models at finite temperature.3 This approach, however, is based on the weak coupling techniques and will be discussed ∗ Current
address: Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA. 61
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in the next section. To understand the non-perturbative aspects of color screening lattice calculations of the spatial correlation functions of static quarks are needed. Recently a lot of progress has been made in this direction which will be the topic of Sec. 4. To prepare the reader for this in Sec. 3 we review the basics of lattice gauge theory. In principle it is possible to study the problem of quarkonium dissolution without any use of potential models. In-medium properties of different quarkonium states and/or their dissolution are encoded in spectral functions. Spectral functions are related to Euclidean meson correlation functions which can be calculated on the lattice. Reconstruction of the spectral functions from the lattice meson correlators turns out to be very difficult, and despite several attempts its outcome still remains inconclusive. One remarkable feature of the studies of the lattice meson correlators is their small temperature dependence despite the expected color screening. This seems to be puzzling. We will discuss the possible resolution of this puzzle in Sec. 6, while the current status of the lattice calculations of the Euclidean correlators and the corresponding meson spectral functions will be presented in Sec. 5. The summary and outlook will be given in Sec. 7. 2. pNRQCD at Finite Temperature There are different scales in the heavy quark bound state problem related to the heavy quark mass m, the inverse size ∼ mv and the binding energy mv 2 . Here v is the typical heavy quark velocity in the bound state and is considered to be a small parameter. Therefore it is possible to derive a sequence of effective field theories using this separation of scales (see Refs. 4 and 5 for recent reviews). Integrating out modes at the highest energy scale ∼ m leads to an effective field theory called non-relativistic QCD or NRQCD, where the pair creation of heavy quarks is suppressed by powers of the inverse mass and the heavy quarks are described by non-relativistic Pauli spinors.6 At the next step, when the large scale related to the inverse size is integrated out, the potential NRQCD or pNRQCD appears. In this effective theory the dynamical fields include the singlet S(r, R) and octet O(r, R) fields corresponding to the heavy quark anti-quark pair in singlet and octet states respectively, as well as light quarks and gluon fields at the lowest scale ∼ mv 2 . The Lagrangian of this effective field theory has the form ( nf Z ∇2r 1 a a µν X 3 † − Vs (r) S + q¯i iD / qi + d r Tr S i∂0 + L = − Fµν F 4 m i=1 +O
+
†
) n o ∇2r ~ S + S†~r · g E ~O − Vo (r) O + VA Tr O†~r · g E iD0 + m
o VB n † ~ O + O† O~r · g E ~ + ... . Tr O ~r · g E 2
(1)
Quarkonium at Finite Temperature
63
Here the dots correspond to terms which are higher order in the multipole expansion.5 The relative distance r between the heavy quark and anti-quark plays a role of a label, the light quark and gluon fields depend only on the center-of-mass coordinate R. The singlet Vs (r) and octet Vo (r) heavy quark potentials appear as matching coefficients in the Lagrangian of the effective field theory and therefore can be rigorously defined in QCD at any order of the perturbative expansion. At leading order Vs (r) = −
1 αs N 2 − 1 αs , Vo (r) = 2N r 2N r
(2)
and VA = VB = 1. The free field equation for the singlet field is ∇2 i∂0 + r − Vs (r) S(r, R) = 0, m
(3)
i.e. has the form of a Schr¨ odinger equation with the potential Vs (r). In this sense, potential models emerge from the pNRQCD. Note, however, that pNRQCD also accounts for interaction of the soft gluons which cannot be included in potential models, i.e. it can describe retardation effects. One can generalize this approach to finite temperature. However, the presence of additional scales makes the analysis more complicated.3 The effective Lagrangian will have the same form as above, but the matching coefficients may be temperature dependent. In the weak coupling regime there are three different thermal scales : T , gT and g 2 T . The calculations of the matching coefficients depend on the relation of these thermal scales to the heavy quark bound state scales.3 To simplify the analysis the static approximation has been used, in which case the scale mv is replaced by the inverse distance 1/r between the static quark and anti-quark. The binding energy in the static limit becomes Vo − Vs ≃ N αs /(2r). When the binding energy is larger than the temperature the derivation of pNRQCD proceeds in the same way as at zero temperature and there is no medium modifications of the heavy quark potential.3 But bound state properties will be affected by the medium through interactions with ultra-soft gluons, in particular, the binding energy will be reduced and a finite thermal width will appear due to medium induced singletoctet transitions arising from the dipole interactions in the pNRQCD Lagrangian3 (c.f. Eq. (1)). When the binding energy is smaller than one of the thermal scales the singlet and octet potential will be temperature dependent and will acquire an imaginary part.3 The imaginary part of the potential arises because of the singletoctet transitions induced by the dipole vertex as well as due to the Landau damping in the plasma, i.e. scattering of the gluons with space-like momentum off the thermal excitations in the plasma. In general, the thermal corrections to the potential go like (rT )n and (mD r)n ,3 where mD denotes the Debye mass. Only for distances r > 1/mD there is an exponential screening. In this region the singlet potential has
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a simple form αs −mD r 2 e + iCF αs T Vs (r) = −CF r rmD 2
Z
∞
dx
0
sin(mD r x) − CF αs (mD + iT ) , (x2 + 1)2
CF = (N − 1)/(2N )
(4)
The real part of the singlet potential coincides with the leading order result of the so-called singlet free energy.7 The imaginary part of the singlet potential in this limit has been first calculated in Ref. 8. For small distances the imaginary part vanishes, while at large distances it is twice the damping rate of a heavy quark.9 This fact was first noted in Ref. 10 for thermal QED. The effective field theory at finite temperature has been derived in the weak coupling regime assuming the separation of different thermal scales as well as ΛQCD . In practice the separation of these scales is not evident and one needs lattice techniques to test the approach. Therefore Sec. 4 will be dedicated to the study of static quarks at finite temperature on the lattice. To prepare the reader for this discussion some basics of the lattice gauge theory will be given in the next section. 3. Basics of Lattice Gauge Theory To study non-perturbative aspects of QCD we use lattice gauge theory.11 In this formalism a field theory is defined in a gauge-invariant way on a discrete space-time domain. This serves at least two purposes: a) to provide an ultra-violet cut-off for the theory, restricting highest momentum to π/a (a being the lattice spacing), and b) to evaluate the path integrals in the Euclidean formulation stochastically using importance sampling. On the lattice the fundamental degrees of freedom of a theory with local SU (N ) gauge symmetry are fermion fields ψx that reside on the sites of the lattice and carry flavor, color and Dirac indeces, which we suppress through the most of this paper, and gauge, bosonic degrees of freedom that in the form of SU (N ) matrices Ux,µ reside on links. Sites on a four-dimensional lattice are labeled with x ≡ (~x, t). The theory is defined by the partition function Z ¯ Z = DU DψDψ exp(−S) (5) where the action
S = Sg + Sf
(6)
contains gauge, Sg and fermionic, Sf parts. The latter part is bi-linear in fields and has the form ¯ ψ Sf = ψM
(7)
where M is the fermion matrix. In the simplest formulation the lattice gauge action can be written as X 1 (8) Sg = β 1 − TrUP , N P
Quarkonium at Finite Temperature
65
where UP is the so-called plaquette, a product of link variables along the elementary square and β = 2N/g 2 with g 2 being the bare gauge coupling. This is the Wilson gauge action.11 The explicit form of the fermion action that is often used in lattice QCD calculations will be discussed in Sec. 5.2. ˆ is given then by The expectation value of an operator O ˆ = hOi
1 Z
Z
¯ ˆ exp(−S). DU DψDψ O
(9)
Integration over the fermion fields (which are Grasmann variables) can be carried out explicitly: Z=
Z
DU det M [U ] exp(−Sg ) ≡
Z
DU exp(−Sef f ),
(10)
where Sef f = Sg − ln det M [U ] is the effective action. The fermion determinant det M [U ] describes the vacuum polarization effects due to the dynamical quarks and makes the effective action non-local in gauge variables. For this reason simulations with dynamical quarks are very resource demanding and the quenched approximation is often employed, where det M [U ] is set to 1. To evaluate the path integral (9) stochastically, an ensemble of NU gauge configurations, weighted with exp(−Sef f ), is generated using Monte Carlo or Molecular Dynamics techniques. The expectation value of the operator is then approximated by the ensemble average: ˆ ≃ hOi
NU 1 X Oi (U ), NU i=1
(11)
ˆ calculated on i-th configuration. (When where Oi (U ) is the value of the operator O, ˆ the operator O depends explicitly on the quark fields extra factors of M −1 appear as shown for a meson correlator below.) Consider a meson (quark anti-quark pair) operator of a general form ¯ x, t)ΓU(~x, ~y; t)ψ(~y , t), J(~x, ~y; t) = ψ(~
(12)
where Γ determines the spin structure and U is a gauge connection that corresponds to the excitations of the gluonic field. Dirac and color indeces in (12) are suppressed. The propagation of such meson from time t = 0 to t is described by the correlation function hJ(~x1 , ~y1 ; 0)J(~x2 , ~y2 ; t)i =
1 Z
Z
¯ DU DψDψ exp(−S)
¯ x1 , 0)ΓU(~x1 , ~y1 ; 0)ψ(~y1 , 0)ψ(~ ¯ y2 , t)Γ† U † (~x2 , ~y2 ; t)ψ(~x2 , t). (13) × ψ(~
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Again, integration over the quark fields can be carried out resulting in hJ(~x1 , ~y1 ; 0)J(~x2 , ~y2 ; t)i = hTr M −1 (~x2 , t; ~x1 , 0)ΓU(~x1 , ~y1 ; 0)M −1 (~y1 , 0; ~y2 , t)Γ† U † (~x2 , ~y2 ; t) i −1 − hTr M (~y1 , 0; ~x1 , 0)ΓU(~x1 , ~y1 ; 0) i × hTr M −1 (~x2 , t; ~y2 , t)Γ† U † (~x2 , ~y2 ; t) i.
(14)
The inverse of the fermion matrix, M −1 has meaning of a fermion propagator. Gauge transporters U(~x, ~y; t) can be taken as weighted sums of different paths connecting points ~x and ~ y . By choosing paths of certain shape or combinations of different paths it is possible to achieve a better overlap of the meson operator with a given state. One of the possibilities is to construct the gauge transporters by using APE smearing12 on spatial links: a link variable Ux,µ is replaced by a weighted average of itself and a sum of the 3-link paths connecting the same sites as Ux,µ : X † ′ Ux,µ → Ux,µ = (1 − 6c)Ux,µ + c Ux,µ Ux+ˆν ,µ Ux+ˆ (15) µ,ν . ν6=µ
This procedure can be applied iteratively. Then a gauge transporter U(~x, ~y ; t), taken as a product of smeared links, is equivalent to a weighted sum of differently shaped paths connecting the sites ~x and ~y. In the following we will consider meson correlators at finite temperature. The finite temperature is introduced by compactifying the the Eucliden time direction, i.e. T = 1/(Nτ a) with Nτ being the number of temporal time slices. Gauge fields and fermion fields obey periodic and anti-periodic boundary conditions in the temporal direction. In the next section we consider a spinless static quark anti-quark pair, Γ = I that can propagate only in time. In this case second term in (14) vanishes. In Sec. 5 we consider local meson operators with ~xi = ~yi , i = 1, 2 which means U(~x, ~y ; t) = I. 4. Correlation Functions of Static Quarks in Lattice Gauge Theory 4.1. Static meson correlators Consider static (infinitely heavy) quarks. The position of heavy quark anti-quark pair is fixed in space and propagation happens only along the time direction. In this limit the second term on the right hand side of (14) vanishes. We are interested in a spinless state and set Γ = I in this section. With respect to the color the meson can be in a singlet or adjoint state. These states are described by the following gauge connections U(~x, ~y ; t) = U (~x, ~y ; t), a
(16) a
U (~x, ~y ; t) = U (~x, ~x0 ; t)T U (~x0 , ~y; t),
(17)
where U (~x, ~y ) is a spatial gauge transporter, – the product of the gauge variables along the path connecting ~x and ~y , ~x0 is the coordinate of the center of mass of the meson and T a are the SU (N ) group generators.
Quarkonium at Finite Temperature
67
The meson operators are given then by Eq. (12) with ~x1 = ~x2 , ~y1 = ~y2 for static quarks ¯ x, t)U (~x, ~y; t)ψ(~y , t), J(~x, ~y ; t) = ψ(~ ¯ x, t)U (~x, ~x0 ; t)T a U (~x0 , ~y ; t)ψ(~y , t). J a (~x, ~y ; t) = ψ(~
(18) (19)
Substituting expressions (18) into Eq. (14) and noting that for a static quark the propagator M −1 (~x, 0; ~x, t) ∼ L(~x), where the temporal Wilson line L(~x) = QNτ −1 x,t),0 with U(~ x,t),0 being the temporal links, we get for the meson correlat=0 U(~ tors at t = 1/T : 1 ¯ x, ~y ; 1/T )i hJ(~x, ~y ; 0)J(~ N 1 = hTr L† (~x)U (~x, ~y ; 0)L(~y)U † (~x, ~y , 1/T ) i, N 2 NX −1 1 hJ a (~x, ~y; 0)J¯a (~x, ~y; 1/T )i Ga (r, T ) ≡ 2 N − 1 a=1 G1 (r, T ) ≡
=
1 hTrL† (x)TrL(y)i −1 1 − hTr L† (x)U (x, y; 0)L(y)U † (x, y, 1/T ) i, 2 N (N − 1) r = |~x − ~y |.
(20)
N2
(21)
The correlators depend on the choice of the spatial transporters U (~x, ~y ; t). Typically, a straight line connecting points ~x and ~y is used as a path in the gauge transporters, i.e. one deals with time-like rectangular cyclic Wilson loops. This object has been calculated at finite temperature in hard thermal loop (HTL) petrurbation theory in context of perturbative calculations of the singlet potential introduced in the previous section and quarkonium spectral functions.8,13,14 In the special gauge, where U (~x, ~y ; t) = 1 the above correlators give the standard definition of the singlet ¯ pair and adjoint free energies of a static QQ 1 hTr[L† (x)L(y)]i, N 1 hTrL† (x)TrL(y)i exp(−Fa (r, T )/T ) = 2 N −1 1 − hTr L† (x)L(y) i. N (N 2 − 1) exp(−F1 (r, T )/T ) =
(22)
(23)
The singlet and adjoint free energies can be calculated at high temperature in leading order HTL approximation7 resulting in (N 2 − 1)αs mD N 2 − 1 αs exp(−mD r) − , 2N r 2N 1 αs (N 2 − 1)αs mD exp(−mD r) − , Fa (r, T ) = 2N r 2N F1 (r, T ) = −
(24) (25)
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A. Bazavov, P. Petreczky & A. Velytsky
p with mD = gT (N/3 + Nf /6) being the leading order Debye mass and Nf is the number of quark flavors. At this order F1 and Fa are gauge independent or, in other words, do not depend on the choice of the parallel transporters U (~x, ~y; t). Note that at small distances (rmD ≪ 1) the singlet free energy F1 (r, T ) ≃ −
N 2 − 1 αs 2N r
(26)
is temperature independent and coincides with the zero temperature potential, while the adjoint free energy Fa (r, T ) ≃
N 1 αs − αs mD 2N r 2
(27)
depends on the temperature. The physical free energy of a static quark anti-quark pair, i.e. the one related to the work that has to be done to separate the static charges by certain distance is given by the thermal average of the singlet and adjoint free energies15 exp(−F (r, T )/T ) = =
1 N2 − 1 exp(−F1 (r, T )/T ) + exp(−Fa (r, T )/T ) 2 N N2 1 1 hTr [L(x)TrL(y)]i ≡ 2 G(r, T ). N2 N
(28)
This quantity is explicitly gauge independent. In leading order HTL approximation the free energy is F (r, T ) = −
(N 2 − 1) α2s exp(−2mD r). 8N 2 r2 T
(29)
The 1/r2 behavior is due to partial cancellation between the singlet and adjoint contribution15,16 and has been confirmed by lattice calculations in the intermediate distance regime above the deconfinement transition.17,18 Using the transfer matrix one can show that in the confined phase G1 (r, T ) =
∞ X
cn (r)e−En (r,T )/T ,
(30)
e−En (r,T )/T ,
(31)
n=1
G(r, T ) =
∞ X
n=1
where En are the energy levels of static quark and anti-quark pair.19 The coefficients cn (r) depend on the choice of U (x, y; t) entering the static meson operator in Eqs. (18-19). Since the color averaged correlator G(r, T ) corresponds to a gauge invariant measurable quantity it does not contain cn . The lowest energy level is the usual static quark anti-quark potential, while the higher energy levels correspond to hybrid potentials.20–23 Using multi-pole expansion in pNRQCD one can show that at short distances the hybrid potential corresponds to the adjoint potential
Quarkonium at Finite Temperature
69
up to non-perturbative constants.24 Indeed, lattice calculations of the hybrid potentials indicate a repulsive short distance part.20–23 Furthermore, the gap between the static potential and the first hybrid potential can be estimated fairly well at short distances in perturbation theory.25 If c1 = 1 the dominant contribution to Ga would be the first excited state E2 , i.e. the lowest hybrid potential which at short distances is related to the adjoint potential. In this sense Ga is related to static mesons with quark anti-quark in adjoint state. Numerical calculations show, however, that c1 is r-dependent and in general c1 (r) 6= 1. Thus Ga also receives contribution from E1 .19 The lattice data suggests that c1 approaches unity at short distances19 in accord with expectations based on perturbation theory, where c1 = 1 up to O(α3s ) corrections.24 Therefore at short distances, r ≪ 1/T the color singlet and color averaged free energy are related F (r, T ) = F1 (r, T ) + T ln(N 2 − 1). In the following we consider SU (2) and SU (3) gauge theories and refer to the adjoint state as triplet and octet, correspondingly. 4.2. Lattice results on static meson correlators Correlation function of static quarks have been extensively studied on the lattice since the pioneering work by Mclerran and Svetistky.15 Most of these studies, however, considered only the color averaged correlator, i.e. the correlation function of two Polyakov loops (for the most complete analysis see Ref. 26 and references therein). Since both color singlet and color octet degrees of freedom contribute to the Polyakov loop correlator it has large temperature dependence even at short distances and it is not a very useful quantity if we want to learn something about quarkonium properties at high temperatures. As this has been pointed out in Refs. 27 and 28 we need to know the quark anti-quark interactions in the singlet channel in order to learn about in-medium quarkonium properties. Therefore in recent years the singlet correlator has been computed on the lattice in SU (2) and SU (3) gauge theories18,29–32 as well as in full QCD with 3 and 2 flavors of dynamical quarks.33,34 Preliminary results also exist for 2+1 flavor QCD with physical value of the strange quark mass and light u, d-quark masses corresponding to pion mass of about 220M eV .35,36 All these calculations use Coulomb gauge definition of the singlet correlator, i.e. the definition (22) with Coulomb gauge fixing. The numerical results for SU (3) gauge theory are presented in Fig. 1 and compared with the zero temperature potential. Here we used the string Ansatz for the zero temperature potential π + σr, (32) V (r) = − 12r since this form gives a very good description of the lattice data in SU (3) gauge the√ ory.37 To convert lattice units to physical units the value σ = 420MeV has been used for the string tension. To remove the additive renormalization in the singlet free energy the lattice data have been normalized to V (r) given by Eq. (32) at the
70
A. Bazavov, P. Petreczky & A. Velytsky
F1(r,T) [GeV]
2 1.5 1 0.5
r [fm]
0.87Tc 0.91Tc 0.94Tc 0.98Tc 1.05Tc 1.50Tc 3.00Tc
1.5
2
0 -0.5 -1 0
0.5
1
2.5
3
Fig. 1. The color singlet free energy singlet free energy in quenched QCD (SU(3) gauge theory) calculated in Coulomb gauge.29,31,32 The solid line shows the zero temperature potential.
shortest distance available. As one can see from Fig. 1 the singlet free energy is temperature independent at short distances and coincides with the zero temperature potential. Below the deconfinement transition, T < Tc , it rises linearly, indicating confinement. The string tension at finite temperature is smaller than the zero temperature string tension σ. At large distances the singlet free energy is the same as the free energy determined from the Polyakov loop correlators and therefore the finite temperature string tension agrees with findings of Ref. 26. Medium effects set in at distance rmed ≃ 0.4 fm/(T /Tc ) and we see exponential screening at distance r > 1/T .32 In the intermediate distance regime 0.5 fm < r < 1.5 fm we see that the singlet free energy is enhanced relative to the zero temperature potential. This is not a real physical effect but an artifact of the calculations. Similar effect has also been seen in SU (2) gauge theory.18,19,30 At these distances the color singlet correlator is sensitive to the value of the coefficients cn (r) in Eq. (30). Below we will show that the enhancement of the singlet free energy is due to the fact that c1 (r) < 1 in this region. Numerical results for the singlet free energy in 2+1 flavor QCD are shown in Fig. 2. In this case a different normalization procedure has been used. The additive renormalization has been determined at zero temperature for each value of the lattice spacing used in the finite temperature calculations by matching the zero temperature potential to the string Ansatz (32) at distance r = r0 , with r0 being the Sommer scale.38 This additive renormalization then has been used for the singlet free energy. For detailed discussion of the calculation of the static
Quarkonium at Finite Temperature
1
71
F1(r,T) [GeV]
0.8 0.6 0.4 0.2
T/Tc = 0.82
0.89 0.97 1.02 1.09 1.58 1.95 2.54 3.29
0 -0.2 -0.4 -0.6 -0.8 0
0.5
1
1.5
2
2.5
r [fm] Fig. 2. The color singlet free energy in 2+1 flavor QCD.35,36 The solid line is the parametrization of the lattice data on the zero temperature potential from Ref. 39.
potential and its renormalization see Ref. 39. As in quenched QCD the singlet free energy is temperature independent at short distances and coincides with the zero temperature potential. At distances larger than the inverse temperature it is exponentially screened. The novel feature of the singlet free energy in full QCD is the string breaking, i.e. the fact that it approaches a constant at large separation. This happens because if the energy in the string exceeds the binding energy of a heavy-light meson the static quark and anti-quark are screened due to pair creation from the vacuum. This happens at distances of about 0.8 fm according to the figure. As temperature increases the distance where the singlet free energy flattens out becomes smaller and for sufficiently high temperatures turns out to be inversely proportional to the temperature. Thus string breaking smoothly turns into color screening as temperature increases. 4.3. Color singlet correlator in SU(2) gauge theory at low temperatures For a better understanding of the temperature dependence of the singlet correlator and its physical interpretation the values of the overlap factors cn (r) should be estimated. To extract the overlap factors we need to calculate the singlet correlators in the low temperature region, where only few energy levels contribute to the correlator. The calculations of the singlet correlator at low temperatures is difficult because of the rapidly deceasing signal to noise ratio. To overcome this difficulty
72
A. Bazavov, P. Petreczky & A. Velytsky
5.5
0.95Tc 0.76Tc 0.63Tc 0.54Tc 0.48Tc 0.42Tc T=0
5
F1(r,T)/σ1/2
4.5 4 3.5
F1(r,T)/σ1/2
4
3 2.5
3
2
2
r σ1/2
0
0.5
1
1.5
2
1.5 0
0.5
1
1.5
2
1/2
rσ
Fig. 3. The color singlet free energy in SU (2) gauge theory below the deconfinement temperature at β = 2.5 calculated on 323 × Nτ lattices. Also shown is the T = 0 potential. The inset shows the color singlet free energy from which the contribution from the matrix element T ln c1 has been subtracted.
it has been suggested to calculate Wilson loops with multilevel Luescher-Weisz algorithm40 instead of the Coulomb gauge fixing.41 In this case the color singlet correlator defined by Eq. (20) is an expectation value of the gauge-invariant Wilson loop with gauge transporter U (~x, ~y; t) being a product of the iteratively smeared spatial links. Numerical calculations have been performed in SU (2) gauge theory using standard Wilson gauge action.41 With the use of the multilevel algorithm it was possible to go down to temperatures as low as 0.32Tc not accessible in the previous studies. It is well known that smearing increases the overlap with the ground state by removing the short distance fluctuations in the spatial links.42 For this reason smearing also reduces the breaking of the rotational invariance to the level expected in the free theory. When no smearing is used the color singlet free energy, −T ln G1 (r, T ) shows a small but visible temperature dependence. The temperature dependence of the singlet free energy is significantly reduced when APE smearing is applied. The color singlet free energy for β = 2.5 and 10 APE smearings is shown in Fig. 3. As one can see from the figure the color singlet free energy shows very mild temperature dependence in the confined phase with noticeable temperature effects appearing only at T = 0.95Tc.
Quarkonium at Finite Temperature
73
1.2 c1(r) 1 0.8 0.6 0.4 0.2
Ns=24 Ns=32 Ns=24,APE10 Ns=32,APE10
0 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
r σ1/2 1.2 c1(r) 1 0.8 0.6 0.4 Ns=32 Ns=32, APE10 Ns=32, APE20
0.2 0.2
0.4
0.6
0.8 rσ
1
1.2
1/2
Fig. 4. The pre-exponential factor of the color singlet correlators as function of distance r for β = 2.5 (top) and β = 2.7 (bottom). Shown are results for unsmeared spatial links and 10 and 20 steps of APE smearing.
Consider the expansion (30). The dominant contribution comes from the ground state, so it is reasonable to fit the singlet correlator to the form G1 (r, T ) = c1 (r) exp(−E1 (r)/T ).
(33)
This allows to extract the matrix element c1 (r) using a simple exponential fit, which is shown in Fig. 4. When no APE smearing is used the value of c1 (r) strongly depends on the separation r. At small distances it shows a tendency of approaching unity as one would expect in perturbation theory. However, c1 (r) decreases with increasing distance r. At large distance its value is around 0.3 − 0.5. Similar results for c1 (r) have been obtained in the study of SU (2) gauge theory in 3 dimensions.19
74
A. Bazavov, P. Petreczky & A. Velytsky
When APE smearing is applied the r-dependence of the amplitude c1 (r) is largely reduced and its value is close to unity both for β = 2.5 and β = 2.7. For β = 2.7 we also see that increasing the number of smearing steps from 10 to 20 reduces the deviation of c1 (r) from unity. As discussed in Sec. 4.1 perturbation theory predicts that the deviation of c1 (r) from unity is of order α3s . Therefore it can be made arbitrarily small by going to sufficiently small distances. It is known, however, that lattice perturbation theory converges very poorly. The main reason for this has been identified with the short distance fluctuations of the link variables, which makes their mean value very different from unity.43 Smearing removes these short distance fluctuations and this is the reason why c1 (r) is much closer to unity when APE smearing is applied. Thus, almost the entire temperature dependence of the singlet free energy at dis√ tances 0.5 < r σ < 2 is due to the deviation of c1 from unity and can be largely reduced by applying APE smearing to the links in the spatial gauge connections. To further demonstrate this point in the inset of Fig. 3 we show the results for F1 (r, T ) + T ln c1 (r). Clearly no temperature dependence can be seen in this quan√ tity up to 0.95Tc, where we see temperature dependence at distances r σ ≥ 1.5 corresponding to the expected drop of the effective string tension. 4.4. Color singlet free energy in the deconfined phase The behavior of the color singlet free energy in the deconfined phase has been studied in Coulomb gauge18,29,32–34 and from cyclic Wilson loops.41 As discussed above at short distances it is temperature independent and coincides with the zero temperature potential. At large distances it approaches a constant F∞ (T ), which monotonically decreases with the temperature. The constant F∞ (T ) is the free energy of two isolated static quarks, or equivalently of a quark anti-quark pair at infinite separation. Its value is therefore independent of the definition of the singlet correlator G1 (r, T ) and is related to the renormalized Polyakov loop Lren (T ) = exp(−F∞ (T )/(2T )).29 At leading order F1 (r, T ) − F∞ (T ) is of Yukawa form (c.f. Eq. (24)). Therefore it is useful to define a quantity called the screening function S(r, T ) = r · (F1 (r, T ) − F∞ (T )).
(34)
This quantity shows exponential decay at distances r > 1/T in the deconfined phase both in pure gauge theories18,32 and full QCD.34 From its exponential decay the Debye screening mass has been determined and turns out to be about 40% larger than the leading order perturbative value. It is interesting to study the screening function using the free energy determined from cyclic Wilson loops and make comparison with Coulomb gauge results. This analysis has been recently done in Ref. 41 for SU (2) gauge theory. The behavior of the screening function at different temperatures is shown in Fig. 5. At short distances (rT < 0.5) the singlet free energy does not depend on the smearing level.
Quarkonium at Finite Temperature
75
1 S(r,T)
0.1
3.4Tc 2.3Tc 1.7Tc 1.4Tc 1.1Tc
0.01
0.001 0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
1/2
rσ 1
1.2Tc, β=2.3533 1.3Tc, β=2.5000 1.5Tc, β=2.4215 1.4Tc, β=2.7 1.7Tc, β=2.7
S(r,T)
3
0.1
m1/T
2.5 2 1.5 T/Tc
1 1
2
3
4
5
6
0.01 0
0.2
0.4
0.6
0.8
1
1.2
1.4
rT Fig. 5. The screening function S(r, T ) = r(F1 (r, T ) − F∞ (T )) in SU (2) gauge theory at different √ temperatures calculated for β = 2.7 as function of r σ (top) and as function of rT (bottom).
Furthermore, it is very close to the free energy calculated in Coulomb gauge. At large distances the screening function S(r, T ) shows an exponential decay determined by a temperature dependent screening mass m1 (T ), which is equal to the leading order Debye mass up to the non-perturbative g 2 corrections: m1 = mD + O(g 2 ).44,45 There is some dependence on the smearing level at larger distances which, however, disappears at high temperatures and with increasing the smearing level. In particular, for β 6 2.5 it turns out that there is no dependence on smearing level for 5 or more smearing steps. For β = 2.7 10–20 steps are needed, depending on the temperature. Fitting the large distance behavior of the screening function by an exponential form exp(−m1 (T )r) allows to determine the screening mass m1 (T ).
76
A. Bazavov, P. Petreczky & A. Velytsky
In the inset of Fig. 5 the color singlet screening masses extracted from the fits are shown in comparison with the results obtained in Coulomb gauge in Ref. 18. The solid line is the leading order Debye mass calculated using 2-loop gauge coupling g(µ = 2πT ) in M S-scheme. As we see from the figure the screening masses are smaller than those calculated in Coulomb gauge and agree well with the leading order perturbative prediction. 4.5. Color adjoint free energy The structure of Eqs. (20) and (21) shows that color adjoint correlator is given by a difference of the color averaged and singlet correlators. The color adjoint correlator in Coulomb gauge has been studied in pure gauge theory,7,18 3-flavor QCD33 and 2-flavor QCD.34 At low temperatures the color adjoint free energy turned out to be significantly smaller than the first hybrid potential contrary to the expectations. In fact at sufficiently large distance it was found to be identical to the singlet free energy. As has been pointed out in Ref. 19 this is due to the non-trivial r-dependence of the overlap factor c1 (r) and its deviation from unity. We have shown in Sec. 4.3 that deviations of the overlap factor c1 (r) from unity can be greatly reduced when one uses Wilson loops with the smeared spatial gauge connection. Therefore it is interesting to see how the adjoint free energy behaves in this approach. The numerical analysis has been done in SU (2) gauge theory41 therefore below we will refer to the adjoint free energy as the triplet free energy. If we assume that only two states contribute to the Eqs. (30) and (31), then from Eq. (21) it follows that 1 ∆E(r)/T , (35) F3 (r, T ) = E2 (r) − T ln 1 − c2 (r) + (1 − c1 (r))e 3 with ∆E(r) = E2 (r) − E1 (r). We have seen in Sec. 4.3 that the temperature dependence of the singlet free energy is quite small. In any case it is considerably smaller than the temperature dependence of the averaged free energy. Therefore the contribution of the excited states to G1 (r, T ) is quite small and it is reasonable to assume that c2 (r) ≪ 1. From the analysis of the multipole expansion we also expect that at small distances, c2 (r) ∼ (rΛQCD )4 .46 Thus, the temperature dependence of F3 (r, T ) and its deviation from the hybrid state E2 (r) is due to small deviation of c1 (r) from unity. At low temperatures, when ∆E ≫ T these small deviations are amplified by the exponential factor. This can be easily verified by subtracting the correction T ln(1 + 13 (1 − c1 )e∆E/T ) from the triplet free energy and assuming that E1 (r) is given by the ground state potential and E2 (r) is given by the first hybrid potential as calculated in Ref. 23. The numerical results are summarized in Fig. 6 which shows that after this correction is accounted for in the confined phase the triplet free energy at low temperatures agrees reasonably well with the first hybrid potential. As temperature increases more excited states contribute. In particular, at 0.76Tc the value of the triplet free energy can be accounted for by including the
Quarkonium at Finite Temperature
77
6.5 6 1.90Tc 1.27Tc 0.95Tc 0.76Tc 0.63Tc 0.48Tc 0.42Tc T=0
F3(r,T)/σ1/2
5.5 5 4.5 4 3.5 3 0
0.5
1
1.5
2
2.5
3
1/2
rσ
Fig. 6. The triplet free energy at different temperatures calculated at β = 2.5. The filled symbols correspond to calculations in Coulomb gauge. Also shown is the first hybrid potential calculated in Ref. 23.
next hybrid state.23 However, at 0.95Tc there are large temperature effects, which cannot be explained by including the contribution from only few excited states. In Fig. 6 we also show the triplet free energy above the deconfinement temperature compared to the calculations in Coulomb gauge.18 It turns out to be much smaller than in the confined phase and agrees well with Coulomb gauge results. This means that the small deviation of the overlap factor c1 (r) from unity is unimportant in this case. The triplet free energy monotonically decreases with increasing temperature as expected in HTL perturbation theory (c.f. Eq. (25)). In the limit of high temperatures and short distances, r ≪ 1/T we have E2 (r) = αs /(4r), ∆E(r) = αs /r, c2 (r) ≃ 0 and c1 (r) = 1 + O(α3s ). Therefore we can expand the logarithm in Eq. (35) to get F3 (r, T ) = +
1 αs + O(α3s T ) + O(αs mD ). 4 r
(36)
Thus the correction due to c1 (r) 6= 0 is much smaller than the expected leading order thermal effects in the triplet free energy. The strong temperature dependence of the adjoint free energy has been also observed in SU (3) gauge theory7 and in full QCD.33,34 Although the adjoint free energy is strongly temperature dependent even at short distances its r-dependence is almost the same as r-dependence of the singlet free energy. The derivative of the adjoint free energy with respect to r is identical within errors to that of the singlet free energy at short distances up to the Casimir factor, i.e. Fa′ (r, T )/F1′ (r, T ) = −1/(N 2 − 1).7
78
A. Bazavov, P. Petreczky & A. Velytsky
5. Quarkonium Spectral Functions 5.1. Meson correlators and spectral functions Now we focus the discussion on the relation between the Euclidean meson correlators and spectral functions at finite temperature. The zero temperature limit is straightforward. Most dynamic properties of the finite temperature system are incorporated in the spectral function. The spectral function σH (p0 , ~p) for a given mesonic channel H in a system at temperature T can be defined through the Fourier transform of the real time two point functions D> and D< or equivalently as the imaginary part of the Fourier transformed retarded correlation function,47 1 < (D> (p0 , ~p) − DH (p0 , p~)) 2π H 1 R = ImDH (p0 , ~p) π Z d4 p ip·x >(<) >(<) e DH (x0 , ~x) DH (p0 , p~) = (2π)4 σH (p0 , p~) =
(37)
> (x0 , ~x) = hJH (x0 , ~x), JH (0, ~0)i DH
< DH (x0 , ~x) = hJH (0, ~0), JH (x0 , ~x)i, x0 > 0
(38)
In essence σH is the Fourier transformation of the thermal average of the commutator [J(x), J(0)]. In the present paper we study local meson operators of the form (c.f. (12) with U = I) JH (t, x) = q¯(t, x)ΓH q(t, x)
(39)
with q a continuous real-time fermion position operator and ΓH = 1, γ5 , γµ , γ5 γµ , γµ γν
(40)
for scalar, pseudo-scalar, vector, axial-vector and tensor channels. The relation of these quantum number channels to different meson states is given in Table 1. Table 1. Meson states in different channels.
Γ
2S+1
γ5
1
γs
3
γs γs′
1
1 γ5 γs
LJ
JPC
uu
cc(n = 1)
cc(n = 2)
bb(n = 1)
bb(n = 2)
ηc
ηb
ηb′
ψ′
Υ(1S)
Υ(2S)
′
S0
−+
0
π
ηc
S1
1−−
ρ
J/ψ
P1
+−
1
b1
hc
hb
3
P0
0++
a0
χc0
χb0 (1P )
χb0 (2P )
3
P1
1++
a1
χc1
χb1 (1P )
χb1 (2P )
χc2
χb2 (1P )
χb2 (2P )
++
2
Quarkonium at Finite Temperature >(<)
The correlators DH (KMS) condition47
79
(x0 , ~x) satisfy the well-known Kubo-Martin-Schwinger
> < DH (x0 , ~x) = DH (x0 + i/T, ~x).
Inserting a complete set of states and using Eq. (41), one gets the expansion (2π)2 X −En /T (e ± e−Em /T ) σH (p0 , p~) = Z m,n × hn|JH (0)|mi|2 δ 4 (pµ − kµn + kµm )
(41)
(42)
where Z is the partition function, and k n(m) refers to the four-momenta of the state |n(m)i. A stable mesonic state contributes a δ function-like peak to the spectral function: σH (p0 , ~ p) = |h0|JH |Hi|2 ǫ(p0 )δ(p2 − m2H ),
(43)
where mH is the mass of the state and ǫ(p0 ) is the sign function. For a quasiparticle in the medium one gets a smeared peak, with the width being the thermal width. As one increases the temperature the width increases and at sufficiently high temperatures, the contribution from the meson state in the spectral function may be sufficiently broad so that it is not very meaningful to speak of it as a well defined state any more. The spectral function as defined in Eq. (42) can be directly accessible by high energy heavy ion experiments. For example, the spectral function for the vector current is directly related to the differential thermal cross section for the production of dilepton pairs:48 1 5α2em dW σV (p0 , ~p). (44) = 3 dp0 d p p~=0 27π 2 p20 (ep0 /T − 1)
Then presence or absence of a bound state in the spectral function will manifest itself in the peak structure of the differential dilepton rate. In finite temperature lattice calculations, one calculates Euclidean time propagators, usually projected to a given spatial momentum: Z GH (τ, p~) = d3 xei~p.~x hTτ JH (τ, ~x)JH (0, ~0)i (45) > This quantity is an analytical continuation of DH (x0 , ~p) > GH (τ, p~) = DH (−iτ, ~p).
(46)
Using this equation and the KMS condition one can easily show that GH (τ, p~) is related to the spectral function, Eq. (37), by an integral equation (see e.g. appendix B of Ref. 49): Z ∞ GH (τ, p~) = dωσ(ω, p~)H K(ω, τ ) 0
K(ω, τ ) =
cosh(ω(τ − 1/2T )) . sinh(ω/2T )
(47)
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This equation is the basic equation for extracting the spectral function from meson correlators. Equation (47) is valid in the continuum. Formally the same spectral representation can be written for the Euclidean correlator calculated on the lattice Glat ~). The corresponding spectral function, however, will be distorted by the H (τ, p effect of the finite lattice spacing. These distortions have been calculated in the free theory.50,51 When discussing the numerical results in following sections the subscript H denoting different channels for meson correlators and spectral functions will be omitted. 5.2. Lattice formulations for charmonium physics The quarkonium system at zero and finite temperature was studied in Refs. 52–55 using Wilson-type fermions # " X X c SW Wilson W ilson ¯ σµν Fµν ψ(x), (48) Sq = ψ(x) m0 + D 6 − 2 µ,ν x where the Dirac operator is defined as 1 DµWilson = ∇µ − γµ ∆µ 2
(49)
with 1 Uµ (x)ψ(x + µ) − Uµ† (x − µ)ψ(x − µ) 2 ∆µ ψ(x) = Uµ (x)ψ(x + µ) + Uµ† (x − µ)ψ(x − µ) − 2ψ(x) .
∇µ ψ(x) =
(50)
Furthermore, σµ,ν = {γµ , γν } and the field strength tensor is defined as i Fµν (x) = − [Qµν − Q†µν ] 2
(51)
4Qµν (x) = Uµ (x)Uν (x + µ ˆ)Uµ† (x + νˆ)Uν† (x) + Uν (x)Uµ† (x − µ ˆ + νˆ)Uν† (x − µ ˆ)Uµ (x − µ ˆ) + Uµ† (x − µ ˆ)Uν† (x − µ ˆ − νˆ)Uµ (x − µ ˆ − νˆ)Uν (x − νˆ) + Uν† (x − νˆ)Uµ (x − νˆ)Uν (x + µ ˆ − νˆ)Uµ† (x) .
(52)
The last term in the brackets in Eq. (48) helps to suppress O(a) lattice artifacts and is called the clover term. Formulations with cSW 6= 0 usually referred to as clover action. The standard Wilson action for fermions corresponds to cSW = 0. In the lattice literature usually the form with the hopping parameter κ = 1/(2am0 + 8) is √ used. In this form the fields need to be accordingly normalized ψ(x) → ψ(x)a−3/2 / 2κ. In Ref. 55 the anisotropic Fermilab formulation is used for the heavy quarks. The anisotropy allows to use fine temporal lattice spacings without significant increase in the computational cost. The study uses the quenched approximation and the
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standard Wilson action in the gauge sector for which the relation between the bare ξ0 and the renormalized anisotropy ξ = as /at is known in a wide range of the gauge coupling β = 6/g 2 .56 The anisotropic clover action57 is " X νs X Wilson ξ ¯ D 6 s Sq = ψ(x) m0 + νt D 6 Wilson + t ξ0 s x 1 − 2
t Csw
X s
Cs X σts Fts + sw σss′ Fss′ ξ0 s<s′
!#
ψ(x) .
(53)
Because the lattice spacings in space and time directions are different the spatial and temporal Dirac operators as well as the clover terms have different coefficients. By tuning the clover coefficients according to the Fermilab prescription57,58 it is possible to reduce O(at m0 ) discretization errors. We will refer to this formulation as anisotropic Fermilab formulation. In the following we will show results from Ref. 55, where anisotropic Fermilab formulation was used as well as results from the study that used isotropic clover action with non-perturbatively determined values of cSW .53 In the study with anisotropic Fermilab action two values for the renormalized anisotropy ξ = 2, 4 as well as β = 5.7, 5.9, 6.1, 6.5 were used. These parameters correspond to temporal lattice spacings a−1 t = 1.905−14.12 GeV. To set the scale for the lattice spacing the traditional phenomenological value r0 = 0.5 fm for the Sommer scale38 was used. The Sommer scale r0 has also been calculated for anisotropic Wilson action for β = 5.5 − 6.1.59 Alternatively one can estimate the lattice spacing from the difference between the mass of 1 P1 state and the spin averaged 1S mass: ∆M (1 P1 − 1S). To a very good approximation this mass difference is not affected by fine and hyperfine splitting and thus is not very sensitive to quenching errors. It was found that close to the continuum limit the lattice spacing determined from ∆M (1 P1 − 1S) is different from that determined from r0 by 10%57,60 if the phenomenological value r = 0.5 fm is used. Using the value of r0 = 0.469(7) determined in full QCD61 would give a value for ∆M (1 P1 − 1S) splitting which is closer to the experimental one, however, the ∆M (2S − 1S) splitting would be even further away from the experimental value.60 This problem is due to the quenched approximation. In the studies with isotropic clover action the following values of the lattice gauge couplings have been used β = 6.499, 6.640 and 7.192.53,62–64 These correspond to √ lattice spacings a−1 = 4.04, 4.86 and 9.72 GeV respectively if the value σ = 420MeV for the string tension is used. The continuum meson current in Eq. (39), JH is related to the lattice current as ¯ H ψ, JH = ZH a3s ψΓ
(54)
where ψ is the lattice quark field in Eq. (48). The renormalization constant ZH
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can be calculated in perturbation theory or non-perturbatively. For isotropic clover action this has been done with both methods (see Ref. 53 and references therein). 5.3. Bayesian analysis of meson correlators The obvious difficulty in the reconstruction of the spectral function from Eq. (47) is the fact that the Euclidean correlator is calculated only at O(10) data points on the lattice, while for a reasonable discretization of the integral in Eq. (47) we need O(100) degrees of freedom. The problem can be solved using Bayesian analysis of the correlator, where one looks for a spectral function which maximizes the conditional probability P [σ|DH] of having the spectral function σ given the data D and some prior knowledge H (for reviews see Refs. 65 and 66). Different Bayesian methods differ in the choice of the prior knowledge. One version of this analysis which is extensively used in the literature is the Maximum Entropy Method (MEM).67,68 It has been used to study different correlation functions in Quantum Field Theory at zero and finite temperature.52–54,62,65,68–79 In this method the basic prior knowledge is the positivity of the spectral function and the prior knowledge is given by the Shannon-Janes entropy Z σ(ω) . S = dω σ(ω) − m(ω) − σ(ω) ln m(ω) The real function m(ω) is called the default model and parametrizes all additional prior knowledge about the spectral functions, e.g. such as the asymptotic behavior at high energy.65,68 For this case the conditional probability becomes 1 2 (55) P [σ|DH] = exp − χ + αS , 2 with χ2 being the standard likelihood function and α a real parameter. Previously in the MEM analysis of the meson spectral functions the Bryan’s algorithm was used.67 A new algorithm was introduced in Ref. 55. It is worth to make connection between this method and the Bryan algorithm. In both cases the true problem is number-of-data dimensional — in more dimensions the problem would be underdetermined. To find the relevant subspace, the Bryan algorithm uses singular value decomposition, while the new algorithm finds the same relevant subspace by exact mathematical transformations. Although the method of identifying the subspace is different, the result is the same, and in both cases one proceeds with solving the original problem in this restricted subspace. The advantage of the new algorithm is that it is more stable numerically when one reconstructs quarkonium spectral functions at zero temperature. The comparison of the two algorithms was done for the pseudo-scalar spectral function for β = 6.1, ξ = 4.55 The problem with the Bryan algorithm is that it does not work well for charmonium correlators if the time extent is sufficiently large, which is the case at low temperatures; the iterative procedure does not always
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converge. For instance at β = 6.1, ξ = 4 and 163 × 96 lattice it was possible to get the spectral functions using the Bryan algorithm only when using τmax = 24 data points in the time direction. With the new algorithm there is no restriction on τmax which can be as large as Nτ /2. 5.4. Charmonium spectral functions at zero temperature 5.4.1. Pseudo-scalar and vector spectral functions at zero temperature In this subsection we discuss the results of Ref. 55 on charmonium spectral functions obtained using MEM. The zero temperature spectral functions for three different lattice spacings obtained by the new method from Ref. 55 are summarized in Fig. 7. Here the simple default model m(ω) = 1 (in units of the spatial lattice spacing) is used. To get a feeling for the statistical errors in the spectral functions its mean value in some interval I is calculated: R dωσ(ω) . (56) σ ¯ = IR dω I
Then the error on σ ¯ is calculated using standard jackknife method. These errors are shown in Fig. 7, where the length of the intervals are shown as horizontal error bars. As one can see from the figure, the ηc (1S) can be identified very well. The second peak is likely to correspond to excited states. Because of the heavy quark mass the splitting between different radial excitations is small and MEM cannot resolve different excitations individually but rather produces a second broad peak to which all radial excitation contribute. This can be seen from the fact that the amplitude 0.4 0.35
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0.05 0 5
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ω[GeV] Fig. 7. The pseudo-scalar spectral function at zero temperature for three finest lattice spacings. The horizontal line corresponds to the spectral function in the free massless limit at zero lattice spacing.
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0.4 0.1 0.35 m(ω)/ω2
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ω [GeV] Fig. 8. The default model dependence of the pseudo-scalar spectral function at the finest lattice spacing (β = 6.5). In the inset the default models corresponding to different spectral functions are shown.
of the second peak (i.e. the area under the peak) is more than two times larger than the first one. Physical considerations tell us that it should be smaller than the first amplitude if it was a 2S state. When comparing amplitudes and peak positions from MEM analysis and from double exponential fits a very good agreement for the first peak and a fair agreement for the second peak are found. This gives confidence that at zero temperature charmonium properties can be reproduced well with MEM. For energies larger than 5 GeV one probably sees a continuum in the spectral functions which is distorted by finite lattice spacing. In particular the spectral function is zero above some energy which scales roughly as a−1 s . Note that for ω < 5 GeV the spectral function does not depend on the lattice spacing. One should control how the result depends on the default model. In Fig. 8 we show the spectral function for three different default models. One can see that the default model dependence is significant only for ω > 5 GeV. This is not surprising as there are very few time slices which are sensitive to the spectral functions at ω > 5 GeV, while the first peak is well determined by the large distance behavior of the correlator. The spectral function in the vector channel defined as 1X σii (ω), (57) σV (ω) = 3 i was also calculated and is shown in Fig. 9 for the three finest lattice spacings. The conclusions which can be derived from this figure are similar to the ones discussed above for the pseudo-scalar channel. The first peak corresponds to the J/ψ(1S) state, the second peak most likely is a combination of 2S and higher excited states,
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finally there is a continuum above 5 GeV which is, however, distorted by lattice artifacts. The similarity between the pseudo-scalar and vector channel is, of course, expected. The lower lying states in theses channels differ only by small hyperfine splitting.
5.4.2. Spectral functions for P states The spectral functions in the scalar, axial-vector and tensor channels which have the 1P charmonia as the ground state were also calculated in Ref. 55. The scalar spectral functions reconstructed using MEM are shown in Fig. 10. The first peak corresponds to χc0 state, but it is not resolved as well as the ground state in the pseudo-scalar channel. This is due to the fact that the scalar correlator is considerably more noisy than the pseudo-scalar or vector correlator. This can be understood as follows. For the heavy quark mass the contribution of the ground state in the scalar channel is suppressed as 1/m2 relative to the ground state contribution in the pseudo-scalar and vector channels, and therefore it is considerably smaller than the continuum contribution to the scalar correlator. For the two finest lattice spacings there is a second peak which may correspond to a combination of excited P states. Above ω > 5 GeV we see a continuum which is strongly distorted by lattice artifacts and probably also by MEM. The spectral functions in the axial-vector and tensor channels are shown in Fig. 11. They look similar to the scalar spectral functions. As in the scalar channel the first peak is less pronounced than in the case of S-wave charmonium spectral functions, and it corresponds to χc1 and hc state, respectively. The continuum part of the spectral function is again strongly distorted.
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Fig. 11. The axial-vector (left) and tensor (right) spectral functions at zero temperature for three lattice spacings.
5.5. Charmonium correlators at finite temperature The reconstruction of the spectral functions from lattice correlators is difficult already at zero temperatures. At finite temperature it is even more difficult to control the systematic errors in the spectral functions reconstructed from MEM. This is because with increasing temperature the maximal time extent τmax is decreasing as 1/T . Also the number of data points available for the analysis becomes smaller. Therefore other methods which can give some information about the change of the spectral functions as the temperature is increasing are often used. The temperature dependence of the spectral function will manifest itself in the temperature dependence of the lattice correlator G(τ, T ). Looking at Eq. (47) it is easy to see that the temperature dependence of G(τ, T ) comes from the temperature dependence of the
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G/Grecon
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1 0.95 β=6.5
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0.87Tc 1.07Tc 1.16Tc 1.39Tc 1.73Tc 2.31Tc 1.09Tc 1.20Tc 1.50Tc 1.99Tc 2.39Tc 2.99Tc 0.4
0.5
τ [fm] Fig. 12. The ratio G/Grecon for the pseudo-scalar channel for the two finer ξ = 4 lattices.
spectral function σ(ω, T ) and the temperature dependence of the kernel K(τ, ω, T ). To separate out the trivial temperature dependence due to K(τ, ω, T ) one calculates the reconstructed correlator53 Z ∞ Grecon (τ, T ) = dωσ(ω, T = 0)K(τ, ω, T ). (58) 0
If the spectral function does not change with increasing temperature we expect G(τ, T )/Grecon (τ, T ) = 1. In Ref. 55 an extensive study of the temperature dependence of this ratio for different channels at different lattice spacings was carried out. 5.5.1. The pseudo-scalar correlators First we present results from Ref. 55 for the temperature dependence of the pseudoscalar correlators. In Fig. 12 we show numerical results for G/Grecon on lattices with ξ = 4. We see almost no change in the pseudo-scalar correlator till temperatures as high as 1.2Tc. The temperature dependence of the pseudo-scalar correlator remains small for temperatures below 1.5Tc. Medium modifications of the correlator slowly turn on as we increase the temperature above this value. From the figures it is clear that the temperature dependence of the correlators is not affected significantly by the finite lattice spacing. The very small temperature dependence of the pseudo-scalar correlator suggests that the corresponding ground state ηc (1S) may survive till temperatures as high a 1.5Tc. The temperature dependence of the correlator found in this study is similar to that of Ref. 53, where isotropic lattices
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with very small lattice spacings, a−1 = 4.86, 9.72 GeV have been used. However, at temperatures higher than 1.5Tc the deviations of G/Grecon from unity become slightly larger than those found in Ref. 53. This is possibly due to the fact that cutoff effects are more important at higher temperatures. Thus despite similarities of the temperature dependence of the pseudo-scalar correlator to findings of Ref. 53 there are quantitative differences. One should note, however, statistical errors and systematic uncertainties are larger in the analysis presented in Ref. 53 than in Ref. 55. In the study on isotropic lattices the ratio G/Grecon starts to depend more strongly on the temperature only around 3Tc.53 5.5.2. The P-wave correlators Next we present the temperature dependence of the scalar, axial-vector and tensor correlators corresponding to P -states. The τ dependence of the scalar correlator was studied on ξ = 2, 4 lattices.55 The numerical results on fine lattices with ξ = 4 are shown in Fig. 13. We see some differences in G/Grecon calculated at β = 6.1 and β = 6.5. Thus the cutoff dependence of G/Grecon is larger in the scalar channel than in the pseudo-scalar one. For β = 6.1 and ξ = 4 the calculations were done on 243 × 24 lattice to check finite volume effects. The corresponding results are shown in Fig. 13 indicating that the finite volume effects are small. On the finest lattice the enhancement of the scalar correlator is very similar to that found in calculations done on isotropic lattices,53 but small quantitative differences can be identified. In Figs. 14 and 15 we show the temperature dependence of the axial-vector and tensor correlators respectively for ξ = 4. Qualitatively their behavior is very similar to the scalar correlator but the enhancement over the zero temperature result is larger. The results for the axial-vector correlators again are very similar to those published in Ref. 53. The difference in G/Grecon calculated at β = 6.1 and β = 6.5 are smaller than in the scalar channel. The large increase in the scalar, axial-vector and tensor correlators may be interpreted as indicator of strong modification of the corresponding spectral function and, possibly the dissolution of 1P charmonia states. However, as we will see in the next sections the situation is more complicated. It has been noticed in Ref. 80 that the increase in G/Grecon may be due to the zero mode contribution. 5.5.3. The vector correlator The numerical results for the vector correlator of Ref. 55 are shown in Fig. 16 for ξ = 4. As one can see from the figures the temperature dependence of G/Grec is different from the pseudo-scalar case and this ratio is larger than unity for all lattice spacings. Similar results have been obtained on isotropic lattices.63 The enhancement of the vector correlator is due to the presence of the transport contribution in the spectral function49,81 and will be discussed in Sec. 5.10. Since the vector current is conserved the vector correlator also caries information about
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Fig. 16. The ratio G/Grecon for the vector channel for the two finer ξ = 4 lattices.
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the transport of heavy quarks in the plasma. This shows up as a peak in the spectral function at zero energy, leading to the observed enhancement in G/Grecon . 5.6. Charmonium spectral functions at finite temperature In Sec. 5.4 we have seen that using MEM one can reconstruct well the main features of the spectral function, in particular the ground state properties. At finite temperature the situation becomes worse because the temporal extent is decreasing. The maximal time separation is τmax = 1/(2T ). As a consequence it is no longer possible to isolate the ground state well. Also the number of available data points becomes smaller. While the later limitation can be overcome by using smaller and smaller lattice spacings in time direction the former limitation is always present. Therefore we should investigate systematic effects due to limited extent of the temporal direction. It appears that the pseudo-scalar channel is the most suitable case for this investigation as at zero temperature it is well under control and there is no contribution from heavy quark transport. To estimate the effect of limited temporal extent in Ref. 55 the spectral function at zero temperature is calculated considering only the first Ndata time-slices in the analysis for β = 6.5, ξ = 4. The result of this calculation is shown in Fig. 17 where Ndata = 80, 40, 20 and 16. The last two values correspond to the finite temperature lattices in the deconfined phase. In this case we see the first peak quite clearly. As one can see from the figure already for Ndata = 40 and τmax = 0.56 fm the second peak corresponding to radial excitation is no longer visible and the first peak becomes significantly broader. The position of the first peak, however, is unchanged. As the number of data points is further decreased (Ndata = 20, 16) we see further broadening of the first peak and a small shift of the peak position to higher energies. These systematic effects should be
Ndata= 80, τmax=1.12fm
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taken into account when analyzing the spectral functions at finite temperature. Therefore when studying the spectral functions at finite temperature we always compare with the zero temperature spectral functions reconstructed with the same number of data points and τmax as available at that temperature. In Fig. 18 the spectral functions at different temperatures are shown together with the zero temperature spectral functions.55 As a default model for T = 1.2Tc and T = 1.5Tc m(ω) = 0.01 is used. For T = 2.0Tc m(ω) = 1 is used since the use of m(ω) = 0.01 resulted in numerical problems in the MEM analysis. The figure shows that the pseudo-scalar spectral function is not modified till 1.5Tc within the errors of the calculations. This is consistent with the conclusions of Ref. 54, 53. One should note, however, that it is difficult to make any conclusive statement based on the shape of the spectral functions as this was done in the above mentioned works. The dependence of the reconstructed spectral functions on the default model m(ω) is much stronger at finite temperature. The spectral functions were reconstructed using different types of default models. For all temperatures T ≤ 1.5Tc the difference between the finite temperature spectral function and the zero temperature one is very small compared to the statistical errors for all default models considered here. In particular a use is made of the default models constructed from the high energy part of the lattice spectral functions calculated at zero temperature as this was done in Ref. 53. The idea is that at sufficiently high energy the spectral function is dominated by the continuum and is temperature independent. Therefore it is suitable to provide the prior knowledge, i.e. the default model. With this default model the spectral functions were calculated at T = (1.07 − 1.5)Tc .55 Very little temperature dependence of the spectral functions was found, see Fig. 19. Note, however, that for this choice of the default model no clear peak can be identified in the spectral functions if T ≥ 1.2Tc. The spectral function in the vector channel is also calculated.55 The results are shown in Fig. 20 for the default model m(ω) = 0.01. As this was already discussed in the previous section the basic difference between the pseudo-scalar and vector spectral functions at finite temperature is the presence of the transport peak at ω ≃ 0. The difference of the temperature dependence of the vector and pseudoscalar correlators is consistent with this assumption. The vector spectral function reconstructed with MEM shows no evidence of the transport peak at ω ≃ 0. On the other hand the spectral function at 1.2Tc differs from the zero temperature spectral function, in particular the first peak is shifted to smaller ω values. We believe that this is a problem of the MEM analysis which cannot resolve the peak at ω ≃ 0 but instead mimics its effect by shifting the J/ψ peak to smaller ω. Also at 2.4Tc the spectral function extends to smaller ω values than in the pseudo-scalar correlator which again indicates some structure at ω ≃ 0. The analysis of the vector spectral functions using other choices for the default model always indicates that the spectral functions at finite temperature differs from the zero temperature spectral functions and extend to significantly smaller ω values.
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20
0
5
10 15 ω[GeV]
20
Fig. 19. The pseudo-scalar spectral function at different temperatures together with the zero temperature spectral functions reconstructed using default model coming from the high energy part of the zero temperature spectral function.
5.7. Charmonium correlators and spectral functions at finite momenta So far we reviewed charmonia at zero spatial momentum, i.e. charmonia at rest in the heatbath’s rest frame. It is certainly of interest to study the temperature dependence of correlators and spectral functions at non-zero spatial momentum. Such a study has been done using isotropic lattices with lattice spacing a−1 = 4.86 GeV and 9.72 GeV.63 It has been found that the pseudo-scalar correlators are enhanced compared to the zero temperature correlators for non-vanishing spatial momenta, see Fig. 21. Furthermore, the enhancement of vector correlator at finite spatial momentum is larger than at zero spatial momentum. In Ref. 55 the finite momentum pseudo-scalar correlators are calculated on anisotropic lattices at β = 6.1, ξ = 4 for different temperatures. The differences in G/Grecon calculated in this work and in Refs. 53 and 63 are present already at zero momentum and are presumably due to finite lattice spacing errors. Apart from this the momentum dependence of the pseudo-scalar correlators is similar to the findings of Refs. 53 and 63. It would be interesting to see if the difference in the temperature dependence of the correlators at zero and finite spatial momenta is due to a contribution to the spectral functions below the light cone at finite temperature.50,51
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0.14 0.12 T=0, Ndata=20
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ω[GeV] Fig. 20. The vector spectral function for β = 6.5 and Nt = 40, 20 corresponding to temperatures 1.2Tc and 2.4Tc . In the analysis the default model m(ω) = 0.01 has been used.
5.8. Bottomonium spectral functions at zero temperature The use of Fermilab formulation described in the previous sections allows for a study of bottomonium for the same range of lattice spacings. Usually bottomonium is studied using lattice NRQCD (see e.g. Ref. 82). First study of bottomonium within the relativistic framework was presented in Ref. 83. More recently it was studied in Ref. 55, where as before the lattice spacing is fixed by the Sommer scale r0 , assuming r0 = 0.5 fm. Note that the lattice spacings determined from r0 are about 20% smaller than in Ref. 83 where it was determined from bottomonium 1 P1 − 1S mass splitting. As the result the new estimates of the Υ mass are smaller than those in Ref. 83. However, one finds good agreement if the values of the lattice spacing quoted in Ref. 83 are used to calculate the physical masses.
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Fig. 21. Comparison of correlators at 1.1 Tc and 1.5 Tc for pseudoscalar and (transverse) vector charmonia with those reconstructed from spectral function at 0.75 Tc .
0.16
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ω[GeV] Fig. 22. The pseudo-scalar bottomonium spectral function at zero temperature for different lattice spacings.
Using MEM the spectral functions in different channels for three lattice spacings were analyzed. In Fig. 22 we show the spectral functions in the pseudo-scalar channel. Since the physical quark mass is different at different lattice spacings the horizontal scale was shifted by the difference of the calculated Υ-mass and the corresponding experimental value. We can see that the first peak in the spectral function corresponds to the ηb (1S) state and its position is independent of the lattice spacing. The remaining details of the spectral functions are cut-off dependent and we cannot distinguish the excited states from the continuum. The position and the amplitude of the first peak in the spectral functions is in good agreement with the results of simple exponential fit. As in the charmonium case the maximal
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0.1 at-1= 5.88GeV, ξ=2
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at-1=10.89GeV, ξ=4
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0.02
0 8
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12
14 16 ω[GeV]
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20
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Fig. 23. The scalar bottomonium spectral function at zero temperature for different lattice spacings.
energy ωmax for which the spectral function is non-zero scales approximately as a−1 s . Similar results have been obtained in the vector channel. The spectral function in the scalar channel is shown in Fig. 23. As it was the case for charmonium the correlators in this channel are more noisy than in the pseudo-scalar channel and as a result it is more difficult to reconstruct the spectral function. Nevertheless we are able to reconstruct the χb0 state which is the first peak in the spectral function. The peak position and the amplitude are in reasonable agreement with the result of the simple exponential fit. 5.9. Bottomonium at finite temperature Besides calculating the bottomonium spectral function at zero temperature Ref. 55 presents a study of the temperature dependence of bottomonium correlators to see medium modification of bottomonia properties. In Fig. 24 we show G/Grecon for vector and pseudo-scalar channel at different lattice spacings. This ratio appears to be temperature independent and very close to unity up to quite high temperatures. This is consistent with the expectation that 1S bottomonia are smaller than 1S charmonia and thus are less affected by the medium. They could survive till significantly higher temperatures. Compared to charmonium case the difference between the pseudo-scalar and vector channels is smaller. This is also expected as the transport contribution which is responsible for this difference is proportional to ∼ exp(−mc,b /T ), and thus is much smaller for bottom quarks (see the discussion in the next section). Similar temperature dependence of the pseudo-scalar bottomonium correlator has been found in calculations with isotropic clover action.84 The temperature dependence of the scalar correlator is shown in Fig. 25. Contrary to the pseudo-scalar and vector correlators it shows strong temperature
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1 0.1
0.2
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τ [fm] Fig. 24. The ratio G/Grecon in the pseudo-scalar (top) and vector channels at different lattice spacings.
dependence and G/Grecon is significantly larger than unity already at 1.1Tc . Again, similar enhancement in G/Grecon has been observed in isotropic lattice calculations.84 The spectral functions were reconstructed at finite temperature for β = 6.3. For the pseudo-scalar channel the results are shown in Fig. 26. As in the charmonium case we compare the finite temperature spectral function with the zero temperature spectral function obtained with the same number of data points and time interval. As expected the spectral function shows no temperature dependence within errors. On the other hand it was not possible to reliably reconstruct the scalar spectral function at finite temperature due to numerical problems. Presumably much more statistics is needed to get some information about the scalar spectral function.
G/Grecon
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β=5.9
1.25Tc 1.66Tc 2.49Tc 3.32Tc
β=6.1
1.16Tc 1.39Tc 1.73Tc
2
99
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1 3 2
G/Grecon
1 1.15Tc 1.54Tc 1.85Tc 2.31Tc
β=6.3
3 2 1 0.1
0.2
0.3
τ [fm] Fig. 25. The ratio G/Grecon in the scalar channel for different lattice spacings.
0.1 T=0, Ndata=16 0.08 σ(ω)/ω
2
T=1.15Tc
0.06 0.04 0.02 0 8
10
12
14
16
18
20
22
ω[GeV] Fig. 26. The pseudo-scalar bottomonium spectral function at finite temperature.
5.10. Zero modes contribution At finite temperature quarkonium spectral functions contain information about states containing a quark anti-quark pair as well as scattering of the external probe off a heavy quark from the medium. The later gives a contribution to the spectral function below the light cone (ω < k). In the limit of zero momentum it becomes χi (T )ωδ(ω) in the free theory. The generalized susceptibilities χi (T ) were calculated in Ref. 51 in the free theory. Interaction with the medium leads to the broadening of the delta function, which becomes a Lorentzian with a small width.81 Because
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the quark anti-quark pair contributes to the spectral function at energies ω > 2m it is reasonable to separate quarkonium spectral function into two terms64 i i σ i (ω, T ) = σhigh (ω, T ) + σlow (ω, T ).
(59)
i Here σhigh (ω, T ) is the high energy part of the spectral functions which is nonzero only for ω > 2m and describes the propagation of bound or unbound quark i anti-quark pairs. On the other hand σlow (ω, T ) receives the dominant contribution at ω ≃ 0. Because the width of the peak at ω ≃ 0 is small the later gives an almost constant contribution to the Euclidean correlator, which is called the zero mode contribution. We can write an analogous decomposition for the Euclidean correlator
Gi (τ, T ) = Gihigh (τ, T ) + Gilow (τ, T ).
(60)
To a very good approximation Gilow (τ, T ) = χi (T )T , i.e. constant. In the previous sections we used the ratio G(τ, T )/Grecon (τ, T ) to study the temperature dependence of the correlators. This dependence comes separately from the high energy part and low energy part, which gives the zero mode contribution. The zero mode contribution is absent in the derivative of the correlator with respect to τ . Therefore one can study the temperature dependence of the correlators induced by change of bound state properties and/or its dissolution by considering the ratio of the derivatives of the correlators G′ (τ, T )/G′recon (τ, T ). The temperature dependence of scalar (13) and axial-vector correlators (14) has been presented in previous sections in terms of G/Grecon . It is temperature independent in the confined phase and shows large enhancement in the deconfined phase. This large enhancement is present both in charmonium and bottomonium correlators. To eliminate the zero mode contribution the derivative of the correlators and the corresponding ratio G′ (τ, T )/G′recon (τ, T ) have been calculated in Ref. 64 using isotropic lattices. The numerical results for this ratio at β = 7.192 are shown in Fig. 27. The results from anisotropic lattices55 are also shown. There is good agreement between the results obtained from isotropic and anisotropic lattices. As one can see from Fig. 27 G′ (τ, T )/G′recon (τ, T ) shows very little temperature dependence and is close to unity. This means that almost the entire temperature dependence of the scalar and axial-vector correlators is due to zero mode contribution and Ghigh (τ, T ) is temperature independent. The temperature dependence of the S-states seen in previous sections is also greatly reduced and the agreement between isotropic and anisotropic calculations is better for the ratio of derivatives. For the pseudo-scalar channel this behavior can be explained by the presence of a small negative zero mode.64 Since the high energy part of the quarkonium correlators turns out to be temperature independent to very good approximation one can assume that Gihigh (τ, T ) ≃ Girecon (τ, T ). Then the low energy part of the correlators, i.e. the zero mode contribution can be evaluated as Gilow (τ, T ) = Gi (τ, T ) − Girecon (τ, T ).64 Current lattice
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Fig. 27. The ratio of the derivatives G′ (τ, T )/G′recon (τ, T ) in the scalar channel (left) and axialvector channel (right) calculated at β = 7.192. The results from anisotropic lattice calculations at β = 6.555 are also shown (open symbols).
data are not precise enough to see a clear τ dependence in Gilow (τ, T ) and it is compatible with being a constant. Therefore, one could take the value of low energy part of the correlator at the midpoint as an estimate for the zero mode contribution, Gilow (τ = 1/(2T ), T ) = T χi (T ). The zero mode contribution is related to the propagation of single (unbound) heavy quark in the medium. Therefore, it is natural to describe its temperature dependence in terms of a quasi-particle model with effective temperature dependent heavy quark masses. Since the temporal component of the vector correlator has no high energy part, i.e. Gvc0 (τ, T ) = −T χvc0 (T ) = −T χ(T ) it is most suitable for fixing the effective quark mass. Matching the lattice data to the free theory expression for χ(T ) one can determine the effective quasiparticle masses meff .64 The results of this analysis are shown in Fig. 28 for different values of the constituent heavy quark mass, including the bottom quarks, in terms of thermal mass correction δmeff (T ) = meff (T )−m. As one can see from the figure the thermal mass correction decreases monotonically with increasing the constituent quark mass and increasing temperature for bottom quarks it is not incompatible with the leading-order perturbative prediction δmeff = −
4 g 2 (T ) mD , 3 4π
(61)
with mD = g(T )T being the perturbative Debye mass (Nf = 0 because we work in the quenched approximation). Having determined the effective heavy quark mass we can study the zero mode contribution in other channels. If the quasi-particle model is correct the zero mode contribution should be a function of meff /T only. Therefore in Fig. 29 the temperature dependence of zero mode contribution for the vector and axial-vector channel is shown as function of T /meff . Indeed, all the lattice data seem to fall on one curve within errors, which agrees with the quasi-particle model prediction shown
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Fig. 28. The thermal mass correction as function of the temperature. The dashed line and the band correspond to the perturbative prediction of the thermal mass correction.
Fig. 29. The thermal velocity of heavy quarks (left) and the zero mode contribution to the axial3 ax 2 vector correlator Gax low /T = χ /T (right). The lines show the prediction of the quasi-particle model with meff (T ). The open symbols show the thermal velocity squared estimated on anisotropic lattices.55
as the black lines. For the vector channel we show the data in terms of the ravc0 tio Gvc , where Gvc low (T )/G low (T ) is the sum over all spatial components. This is because this quantity does not depend on the renormalization64 and has simple 2 physical interpretation in terms of averaged thermal velocity squared vth . The later follows from the fact that due to the large quark mass the Boltzmann approximation can be used and we have Z Z 2 Gvc 3 p −Ep /T 3 −Ep /T 2 low (T ) ≃ d p e d pe = vth . (62) Gvc0 (T ) Ep2 The zero mode contribution in the vector channel has also been studied on anisotropic lattices55 and in Fig. 29 we also show the corresponding results for thermal velocity squared.
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6. Potential Models at Finite Temperature Quarkonium properties at finite temperature have been studied in potential models since the famous paper of Matsui and Satz1 (for recent review see Ref. 2). The basic idea behind this is that color screening will modify the potential, which becomes short range and cannot support bound states of heavy quarks at sufficiently high temperatures. As discussed in Sec. 2 this approach can be justified if there is a separation of the scales related to binding energy and other scales in the problem, like the inverse size of the bound states and the temperature. Close to the QCD transition this separation of scales is not obvious, however, the lattice calculation of static quark correlators, discussed in Sec. 4 show that screening effects are very strong already in the transition region. In fact, correlation functions of static quark anti-quark pair show significant temperature modifications already at distances similar to quarkonium size. Therefore, we may expect the most of quarkonium bound states dissolve in the deconfined phase at temperatures close to the transition temperature. This seemingly contradicts to the small temperature dependence of quarkonium correlators and spectral functions discussed in the previous section. Due to the heavy quark mass quarkonium spectral functions can be calculated in the potential approach by relating the spectral functions in the threshold region to the non-relativistic Green function14,85,86 6 (63) σ(ω) = K ImGnr (~r, r~′ , E)|~r=r~′ =0 , π σ(ω) = K
6 1 ~ ·∇ ~ ′ Gnr (~r, r~′ , E)| ~′ , Im∇ ~ r =r =0 π m2c
(64)
for S-wave, and P -wave quarkonia, respectively. Here E = ω − 2m. The pre-factor K accounts for relativistic and radiative corrections. The non-relativistic Green function is calculated from the Schr¨odinger equation with a delta-function on the right hand side. Away from the threshold the spectral function is matched to the perturbative result. It has been shown that this approach can provide a fair description of quarkonium correlators at zero temperature.87 Other approaches to calculate quarkonium spectral functions in potential models were proposed in Refs. 49, 88–91. Close to the transition temperature the perturbative calculations of the potential are not applicable. Therefore in Ref. 85 the singlet free energy calculated in quenched lattice QCD has been used to construct the potential. The drawback of this approach is that there is no one to one correspondence between the singlet free energy and the potential in the non-perturbative domain. Therefore in Ref. 85 several possible forms of the potential compatible with the lattice data have been considered. Furthermore, lattice calculations do not have much information about the imaginary part of the potential. Therefore the imaginary part of the potential has been approximated by a small constant term. The results of the calculations of the charmonium and bottomonium spectral functions for S-wave are shown in Fig. 30. In the case of charmonium all bound states are melted at temperatures higher than 1.2Tc. We see, however a significant threshold enhancement, i.e. near
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ω [GeV]
Fig. 30. The charmonium (top) and bottomonium (bottom) spectral functions at different temperatures. For charmonium we also show the spectral functions from lattice QCD obtained from the MEM at 1.5Tc . The error-bars on the lattice spectral function correspond to the statistical error of the spectral function integrated in the ω-interval corresponding to the horizontal error-bars. The insets show the corresponding ratio G/Grecon together with the results from anisotropic lattice calculations.55 For charmonium, lattice calculations of G/Grecon are shown for T = 1.2Tc (squares), 1.5Tc (circles), and 2.0Tc (triangles). For bottomonium lattice data are shown for T = 1.5Tc (circles) and 1.8Tc (triangles).
the threshold the spectral function is much larger than in the free case. In the bottomonium case only the ground state survives in the deconfined phase. At temperatures above 2Tc we see the melting of the ground state as well. Quarkonium spectral functions calculated using the perturbative potential, where the imaginary part is fully taken into account, show similar qualitative features.14,86 Here too a significant threshold enhancement is seen. The imaginary part of the potential plays an important role in weakening the bound state peak
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or transforming it to mere threshold enhancement. Let us note that the next-toleading order perturbative correction to the quarkonium spectral functions also give rise to significant enhancement in the threshold region.92 From the spectral functions we can calculate the quarkonium correlation functions in Euclidean time G(τ, T ) and compare them to the available lattice data. This comparison is shown in Fig. 30 for the ratio G(τ, T )/Grecon (τ, T ). As one can see from the figure the melting of bound states does not lead to large change in the Euclidean correlation functions. The ratio G(τ, T )/Grecon (τ, T ) calculated in the potential model is flat and temperature independent in agreement with lattice calculations. This means that threshold enhancement can compensate for melting of bound states in terms of the Euclidean correlators. Spectral functions of P -wave quarkonium have been also calculated14,85 and show significant threshold enhancement as well. As the consequence the ratio of the derivatives G′ (τ, T )/G′recon (τ, T ) is temperature independent and close to unity85 in agreement with lattice calculations shown in the previous section. The analysis discussed above has been done in quenched QCD. This is because only in quenched QCD we have sufficiently precise lattice calculations of quarkonium correlators. Potential model calculations of the spectral functions have been extended to 2+1 flavor QCD using the lattice data discussed in Sec. 4. These calculations show that all quarkonium states except the ground state bottomonium dissolve in the quark gluon plasma.93 The upper limits on the dissociation temperatures for different quarkonium states obtained in this analysis are given in Table 2. Table 2. Upper bound on the dissociation temperatures for different quarkonium states in 2+1 flavor QCD.93
State
χc
ψ′
J/ψ
Υ′
χb
Υ
Tdis
≤ Tc
≤ Tc
1.2Tc
1.2Tc
1.3Tc
2Tc
7. Conclusion In this paper we discussed quarkonium properties in quark gluon plasma. We discussed how color screening can be studied non-perturbatively on the lattice using spatial correlation functions of static quark and anti-quark. We discussed the importance of singlet and adjoint (triplet for SU (2) or octet for SU (3)) degrees of freedom for understanding the temperature dependence of static meson (quarkantiquark) correlators. It has been shown how color singlet and adjoint degrees of freedom can be defined in the effective field theory framework, the so-called thermal pNRQCD. We have seen that the singlet correlators of static quark and anti-quark show strong screening effects at distances comparable to quarkonium size. Thus it is natural to expect that most quarkonium states melt in quark gluon plasma. Quite surprisingly lattice calculations of quarkonium correlators in Euclidean time show
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very little temperature dependence. We reviewed the current status of these calculations in quenched QCD. It turns out that the spatial lattice spacing should be smaller than (4 GeV)−1 to have full control over the discretization errors. Therefore the extension of these calculations to full QCD pioneered in Ref. 94 will be quite difficult. The only source of significant temperature dependence of quarkonium correlators is the zero mode contribution which is not related to bound states but to transport properties of heavy quarks in the medium.81 We have shown that this contribution can be well described by a quasi-particle model with temperature dependent heavy quark mass. We have presented the calculations of quarkonium spectral functions in potential model and have shown that all quarkonium states, except the ground state bottomonium melt in the deconfined phase. We have also shown how the seemingly existing contradiction of strong color screening leading to quarkonium melting and very weak temperature dependence of quarkonium correlators can be resolved within potential models. It turns out that strong threshold enhancement of quarkonium spectral functions can compensate the absence of bound state and result in Euclidean correlarors, which are almost temperature independent. The fact that no charmonium bound states can exist in quark gluon plasma has important consequences for describing charmonium production in heavy ion collisions. In absence of bound states heavy quarks in the plasma can be treated quasi-classically using Langevin dynamics. In this scenario the residual correlation between the heavy quark and anti-quark, visible in the threshold enhancement and the finite life time of the plasma play an important role.95 It turns out that using these ideas and the interactions of heavy quark determined in lattice QCD it is possible to uderstand the J/ψ suppression pattern at RHIC.95 Acknowledgements This work was supported by U.S. Department of Energy under Contract No. DEAC02-98CH10886. The work of A.B. was supported by grants DOE DE-FC0206ER-41439 and NSF 0555397. A.V. work was supported by the Joint Theory Institute funded together by Argonne National Laboratory and the University of Chicago, and in part by the U.S. Department of Energy, Division of High Energy Physics and Office of Nuclear Physics, under Contract DE-AC02-06CH11357. References 1. T. Matsui and H. Satz, J/psi suppression by quark-gluon plasma formation, Phys. Lett. B178, 416 (1986). 2. A. Mocsy, Potential Models for Quarkonia (2008), eprint 0811.0337 [hep-ph]. 3. N. Brambilla, J. Ghiglieri, A. Vairo, and P. Petreczky, Static quark-antiquark pairs at finite temperature, Phys. Rev. D78, 014017 (2008). 4. N. Brambilla et al., Heavy quarkonium physics (2004), eprint hep-ph/0412158. 5. N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77, 1423 (2005).
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32. O. Kaczmarek, F. Karsch, F. Zantow, and P. Petreczky, Static quark anti-quark free energy and the running coupling at finite temperature, Phys. Rev. D70, 074505 (2004). 33. P. Petreczky and K. Petrov, Free energy of a static quark anti-quark pair and the renormalized Polyakov loop in three flavor QCD, Phys. Rev. D70, 054503 (2004). 34. O. Kaczmarek and F. Zantow, Static quark anti-quark interactions in zero and finite temperature QCD. I: Heavy quark free energies, running coupling and quarkonium binding, Phys. Rev. D71, 114510 (2005). 35. O. Kaczmarek, Screening at finite temperature and density, PoS CPOD07, 043 (2007). 36. RBC Collaboration, work in progress. 37. S. Necco and R. Sommer, The N(f) = 0 heavy quark potential from short to intermediate distances, Nucl. Phys. B622, 328–346 (2002). 38. R. Sommer, A New way to set the energy scale in lattice gauge theories and its applications to the static force and alpha-s in SU(2) Yang–Mills theory, Nucl. Phys. B411, 839–854 (1994). 39. M. Cheng et al., The QCD equation of state with almost physical quark masses, Phys. Rev. D77, 014511 (2008). 40. M. Luscher and P. Weisz, Locality and exponential error reduction in numerical lattice gauge theory, JHEP 09, 010 (2001). 41. A. Bazavov, P. Petreczky, and A. Velytsky, Static quark anti-quark pair in SU(2) gauge theory, Phys. Rev. D78, 114026 (2008). 42. S. P. Booth et al., Towards the continuum limit of SU(2) lattice gauge theory, Phys. Lett. B275, 424–428 (1992). 43. G. P. Lepage and P. B. Mackenzie, On the viability of lattice perturbation theory, Phys. Rev. D48, 2250–2264 (1993). 44. A. K. Rebhan, The Non-Abelian Debye mass at next-to-leading order, Phys. Rev. D48, 3967–3970 (1993). 45. A. K. Rebhan, Non-Abelian Debye screening in one loop resummed perturbation theory, Nucl. Phys. B430, 319–344 (1994). 46. A. Vairo, private communication (2008). 47. M. Le Bellac, Thermal Field Theory, Cambridge monographs on mathematical physics (Cambridge Univ. Press, Cambridge, 1996). 48. E. Braaten, R. D. Pisarski, and T.-C. Yuan, Production of soft dileptons in the quarkgluon plasma, Phys. Rev. Lett. 64, 2242 (1990). 49. A. Mocsy and P. Petreczky, Quarkonia correlators above deconfinement, Phys. Rev. D73, 074007 (2006). 50. F. Karsch, E. Laermann, P. Petreczky, and S. Stickan, Infinite temperature limit of meson spectral functions calculated on the lattice, Phys. Rev. D68, 014504 (2003). 51. G. Aarts and J. M. Martinez Resco, Continuum and lattice meson spectral functions at nonzero momentum and high temperature, Nucl. Phys. B726, 93–108 (2005). 52. T. Umeda, K. Nomura, and H. Matsufuru, Charmonium at finite temperature in quenched lattice QCD, Eur. Phys. J. C39S1, 9–26 (2005). 53. S. Datta, F. Karsch, P. Petreczky, and I. Wetzorke, Behavior of charmonium systems after deconfinement, Phys. Rev. D69, 094507 (2004). 54. M. Asakawa and T. Hatsuda, Jpsi and etac in the deconfined plasma from lattice QCD, Phys. Rev. Lett. 92, 012001 (2004). 55. A. Jakovac, P. Petreczky, K. Petrov, and A. Velytsky, Quarkonium correlators and spectral functions at zero and finite temperature, Phys. Rev. D75, 014506 (2007).
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HEAVY QUARKS IN THE QUARK-GLUON PLASMA
RALF RAPP Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843-3366, USA [email protected] HENDRIK VAN HEES Institut f¨ ur Theoretische Physik, Justus-Liebig-Universit¨ at Giessen, D-35392 Giessen, Germany [email protected]
Heavy-flavor particles are believed to provide valuable probes of the medium produced in ultrarelativistic collisions of heavy nuclei. In this article we review recent progress in our understanding of the interactions of charm and bottom quarks in the Quark-Gluon Plasma (QGP). For individual heavy quarks, we focus on elastic interactions for which the large quark mass enables a Brownian motion treatment. This opens a unique access to thermalization mechanisms for heavy quarks at low momentum, and thus to their transport coefficients in the quark-gluon fluid. Different approaches to evaluate heavy-quark diffusion are discussed and compared, including perturbative QCD, effective potential models utilizing input from lattice QCD and string-theoretic estimates in conformal field theories. Applications to heavy-quark observables in heavy-ion collisions are realized via relativistic Langevin simulations, where we illustrate the important role of a realistic medium evolution to quantitatively extract the heavy-quark diffusion constant. In the heavy quarkonium sector, we briefly review the current status in potential-model based interpretations of correlation functions computed in lattice QCD, followed by an evaluation of quarkonium dissociation reactions in the QGP. The discussion of the phenomenology in heavy-ion reactions focuses on thermal model frameworks paralleling the open heavy-flavor sector. We also emphasize connections to the heavy-quark diffusion problem in both potential models and quarkonium regeneration processes.
1. Introduction The investigation of strongly interacting matter constitutes a major challenge in modern nuclear and particle physics. Of particular interest are phase changes between hadronic and quark-gluon matter, similar to the one which is believed to have occurred in the early Universe at a few microseconds after its birth. While the theory of the strong force is by now well established in terms of Quantum Chromodynamics (QCD),1–3 two of its major manifestations in the world around us — the confinement of quarks and gluons and the generation of hadronic masses — are subject of vigorous contemporary research. Both phenomena occur at energy-momentum scales of Q2 . 1 GeV2 where the QCD coupling constant is rather large, αs & 0.3, and therefore perturbation theory is not reliable and/or applicable. In a hot and dense medium at sufficiently large temperature (T ) and/or 111
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quark chemical potential (µq ), one expects the finite-size hadrons to be dissolved into a deconfined Quark-Gluon Plasma (QGP) where the condensates underlying hadronic-mass generation have melted. Numerical simulations of lattice-discretized QCD (lQCD) at finite temperature predict the phase change from hadronic to quark-gluon matter to occur at a “pseudo-critical” temperature of Tc ' 200 MeV.4 This appears to be a rather small scale for a “perturbative QGP” (pQGP) of weakly interacting quarks and gluons to be realized, even though the computed energy density matches that of an ideal (non-interacting and massless) QGP within 20% or so for T & 1.2 Tc. In the laboratory, one hopes to create a QGP by colliding heavy atomic nuclei at ultrarelativistic energies, with a center-of-mass energy per colliding nucleon pair √ well above the nucleon rest mass, s/A MN . If the energy deposition in the reaction zone is large enough, and if the interactions of the produced particles are strong enough, the notion of an interacting medium may be justified, despite its transient nature. This notion has been convincingly verified in nuclear collision experiments over the last ∼ 25 years at the Super-Proton-Synchrotron (SPS) at the European Organization for Nuclear Research (CERN)5 and at the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory (BNL).6 Transverse-momentum (pT ) spectra of different hadron species in the low-pT regime (pT . 2–3 GeV) reveal that the produced medium explodes collectively reaching expansion velocities in excess of half the speed of light. In the high-pT regime (pT & 5 GeV), which in the heavy-ion environment became available at RHIC for the first time, hadron spectra are suppressed by up to a factor of ∼ 5 relative to p–p collisions, indicative for a strong absorption of high-energy partons traversing the medium. 7 The inclusive production of charm-quark bound states (J/ψ mesons) is suppressed by a factor of 3-5 at both SPS and RHIC, indicative for their dissolution in the medium (possibly related to deconfinement).8–10 A large excess of electromagnetic radiation (photons and dileptons) is observed, indicative for medium temperatures around 200 MeV and a “melting” of the ρ-meson resonance (possibly related to hadronic mass de-generation).11,12 A more differential analysis of hadron spectra in non-central Au–Au collisions at RHIC reveals a large elliptic asymmetry of the collective flow (“elliptic flow”): the spatial asymmetry of the initial nuclear overlap zone is converted into an opposite asymmetry in the final hadron pT spectra. Within a hydrodynamic modeling of the exploding fireball this observation requires a rapid thermalization and a very small viscosity of the interacting medium. 13–16 Only then can spatial pressure gradients build up fast enough to facilitate an effective conversion into azimuthal asymmetries in the energy-momentum tensor of the system. The agreement of hydrodynamic predictions with elliptic-flow data at RHIC led to the notion of an “almost perfect liquid”, with a ratio of viscosity to entropy density close to a conjectured lower bound of any quantum mechanical system.17 The microscopic mechanisms underlying these rather remarkable transport properties are yet to be determined. In this context, heavy quarks (charm and
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bottom, Q = c and b) and their bound states (charmonia and bottomonia) are recognized as particularly suitable probes of the medium produced in ultrarelativistic heavy-ion collisions (URHICs).18,a In the present article we will attempt to review the current status of the theory and phenomenology of this promise. Let us first focus on the sector of individual heavy quarks (open heavy flavor). The fact that their masses are well above the typical temperature of the system, mQ T , has at least three important implications: ¯ production process is essentially restricted to primordial (1) The (hard) QQ N –N collisions,20 i.e., re-interactions in the subsequently evolving medium are not expected to change the number of heavy quarks (reminiscent of the “factorization theorem” of perturbative QCD21 ); this is borne out experimentally by a scaling of c¯ c production, Nc¯c , with the number of binary N –N collisions, Ncoll , at different collision centralities.22 (2) The thermal relaxation time of heavy quarks ought to be larger than for light quarks, parameterically by a factor ∼ mQ /T ≈ 5–20. With a lightquark and gluon thermalization time of τq,g ' 0.3–1 fm/c (as indirectly inferred from hydrodynamic modeling at RHIC) and an estimated QGP lifetime of τQGP ' 5 fm/c in central Au–Au collisions, one expects τc (τb ) to be on the same order as (significantly larger than) τQGP . Thus, charm (and especially bottom) quarks are not expected to reach thermal equilibrium, but their re-interactions should impart noticeable modifications on the initial momentum spectrum (less pronounced for bottom). The final heavy-quark (HQ) spectra may therefore encode a “memory” of the interaction history throughout the evolving fireball, by operating in between the limits of thermalization and free streaming. (3) The theoretical task of describing HQ interactions is amenable to a diffusion treatment, i.e., Brownian motion of a heavy test particle in a bath of a lightparticle fluid. Nonrelativistically, the typical thermal momentum of a heavy quark is p2th ' 3mQ T T 2 , and therefore much larger than the typical momentum transfer from the medium, Q2 ∼ T 2 . This allows to expand the Boltzmann equation in momentum transfer to arrive at a Fokker–Planck description of HQ diffusion in the QGP, which directly yields the pertinent transport coefficients as well. The above three points provide a well-defined framework to construct in-medium HQ interactions in QCD matter and test them against observables in URHICs (quantitative comparisons additionally require to account for effects of hadronization of the quarks, as well as reinteractions in the hadronic medium). The Fokker– a The
(weak-decay) lifetime of the top quark of ∼ 0.1 fm/c is too short to render it a viable probe in URHICs; thus, heavy quarks will exclusively refer to charm and bottom in this article. Strange quarks are in between the heavy- and light-quark limit, forming their own complex of valuable observables.19
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Planck approach is readily implemented for the case of elastic p + Q → p + Q scattering off partons in the medium (p = q, q¯, g).23–28 In the light-hadron sector, however, the large suppression of high-pT spectra is believed to be largely caused by radiative energy loss of high-energy partons traversing the QGP, i.e., mediuminduced gluon radiation of type q + g → q + g + g.7,29,30 Even in the low-pT regime, perturbative 2, 3 ↔ 3 scattering processes have been suggested to facilitate the rapid thermalization required by phenomenology (albeit in connection with rather large coupling constants of αs ' 0.5).31 The situation could be quite different in the HQ sector. In the low-momentum limit, gluon-Bremsstrahlung of a heavy quark is suppressed32 and the dominant momentum-transfer reaction is elastic scattering.26 As is well known from classical electrodynamics, the radiative energy loss of a muon is suppressed relative to an electron by a mass ratio (me /mµ )4 . In perturbative QCD (pQCD) it is currently an open question at what momentum scale radiative energy loss of a heavy quark takes over from the collisional one (which, most likely, will depend on additional parameters such as temperature, path length, etc.). In fact, this may not even be a well-defined question since nonperturbative processes at moderate momentum transfers may supersede perturbative ones before the elastic part of the latter dominates over the radiative one. The relation between perturbative and nonperturbative interactions is one of the key issues to be addressed in this review. Experimental signatures for the modifications of HQ spectra in URHICs are currently encoded in single-electron (e± ) spectra associated with the semileptonic decays of charm and bottom hadrons, D, B, Λc , . . . → eνX. These measurements require a careful subtraction of all possible “photonic” sources of electrons, such as photon conversions in the detector material, Dalitz decays of π and η, vector-meson decays, and others. The modifications of the “non-photonic” electron spectra (associated with heavy-flavor decays) in Au–Au collisions are then quantified by the stane dard nuclear modification factor, RAA , and elliptic flow coefficient, v2e . The available √ RHIC data in semicentral and central Au–Au collisions at sN N = 200 GeV exe hibit a substantial elliptic flow of up to v2 ' 10% and a large high-pT suppression e down to RAA ' 0.25, respectively.22,33–35 Both values are quite comparable to those for light hadrons (the pion v2 reaches somewhat higher, to about 15%). Radiative energy-loss models36 based on perturbative QCD cannot explain the e± data. These data were, in fact, instrumental37 in reconsidering elastic scattering as a significant source of parton energy loss in the QGP.25–27,38 The combination of pQCD elastic and radiative scattering does not suffice either to reproduce the observed suppression once a realistic bottom component is accounted for in the electron spectra. 38 Elastic scattering based on nonperturbative interactions, as proposed in Refs. 25 and 28, simultaneously accounts for the e± elliptic flow and suppression reasonably well.22 This has reinforced the hope that HQ observables provide the promised precision tool to characterize transport properties of the “strongly coupled QGP” m (sQGP). E.g., if a clear mass hierarchy in thermal relaxation times, τQ ∝ TQ τq (as well as τb /τc = mb /mc ) emerges from a quantitative analysis of URHIC data, it
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would be suggestive for a universal behavior of light- and heavy-quark transport in the QGP. However, there is still a substantial way to go before such a program can be realized, as discussed below.b It should not be surprising if in-medium properties of open heavy flavor, especially at low momentum, are closely related to medium modifications of heavy quarkonia. The latter have a long history as “probes” of the QGP in heavy-ion collisions, especially as potential indicators of the deconfinement transition, cf. Refs. 8– 10 for a broad up-to-date coverage of this topic. In particular, progress in finitetemperature lattice QCD41–43 has triggered vigorous reconsideration of the question whether quarkonia, especially their ground states, can survive in the QGP significantly above the critical temperature. These developments include the application of potential models at finite temperature, coupled with the hope that heavy-quark free energies as computed in thermal lQCD can serve as a model-independent input for the low-energy heavy-quark interaction. If charmonium binding indeed remains sufficiently strong in the QGP to support bound states up to rather high temperatures, it is conceivable that the underlying interaction is of a more general relevance and therefore also operative in heavy-light45 and maybe even light-light44 systems. Especially in the former case, from the point of view of elastic (on-shell) scattering of a heavy quark in the medium, the conditions for momentum transfer are comparable to the heavy-heavy interaction governing quarkonium properties. Since low-momentum HQ interactions determine their transport properties, one immediately recognizes an intimate relation between HQ transport and in-medium quarkonia. These connections are also being exploited in the analysis of thermal lQCD computations of quarkonium correlation functions.46 In addition to the binding properties, the inelastic reaction rates of quarkonia with surrounding partons or hadrons are a key ingredient for a quantitative description of their spectral function in QCD matter (also here “quasi-elastic” scattering of thermal partons with a heavy quark inside the quarkonium bound states may play an important role, especially if the binding energy becomes small47 ). A good control over all of these aspects is mandatory to utilize quarkonium properties as diagnostic tool in heavy-ion collisions and eventually deduce more general properties of the medium produced in these reactions. As in the open heavy-flavor sector, this has to be built on a solid knowledge of the space-time history of nuclear collisions, as well as of the initial conditions on quarkonium spectra. The latter aspect could be more involved than for single heavy quarks, since (a) measurements in p–A collisions show that cold-nuclear-matter (CNM) effects from the incoming nuclei (e.g., the so-called nuclear absorption) affect the primordial charmonium number significantly (e.g., with up to 60% suppression for J/ψ at SPS energies when extrapolated to central Pb– Pb collisions); (b) the bound-state formation time introduces another rather long time scale (soft energy scale) which is easily of the order of (or longer than) the bA
recent review article39 addresses similar topics but from a more elementary perspective; see also Ref. 40.
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thermalization time of the medium (at least for charmonia and excited bottomonia at RHIC energies and higher). Charmonium suppression beyond the level of CNM effects has been established in semi-/central Pb–Pb and Au–Au collisions at SPS48 and RHIC,49 respectively. An intriguing finding is that the observed suppression pattern and magnitude is very comparable at SPS and RHIC, despite the different collision energies which lead to substantial variations in, e.g., light-hadron observables (most notably a factor ∼ 2 larger charged particle rapidity density and stronger collective phenomena at RHIC). This “degeneracy” was predicted47 as a consequence of charmonium regeneration mechanisms:50–52 a stronger suppression in the hotter and denser medium at RHIC is compensated by the coalescence of c and c¯ quarks in the QGP and/or at hadronization (the c¯ c production cross section at RHIC is about a factor of ∼ 100 larger than at SPS energies). While an “extra” source of charmonia increases the complexity of pertinent observables in heavy-ion reactions, it also provides another, rather direct, connection between the open and hidden heavy-flavor sectors. The secondary charmonium yield from c–¯ c coalescence necessarily carries imprints of the charm-quark distributions, both in its magnitude (softer c-quark spectra are expected to result in larger coalescence probabilities) and in its momentum spectra (including elliptic flow). A comprehensive theoretical and phenomenological analysis of open and hidden heavy flavor is thus becoming an increasingly pressing and challenging issue. As a final remark on quarkonia in this introduction, we point out that bottomonium production in heavy-ion reactions is less likely to receive regeneration contributions (at least at RHIC and possibly neither at LHC). In addition, the increase in bottomonium binding energies (compared to charmonia) render them rather sensitive probes of color screening which strongly influences its dissociation rates.53 Bottomonia thus remain a promising observable to realize the originally envisaged “spectral analysis of strongly interacting matter”.54 Our review is organized as follows: In Sec. 2 we outline the theoretical framework of evaluating HQ diffusion in equilibrium QCD matter. We first recall basic steps in setting up the HQ diffusion equation (Sec. 2.1) which determines the time evolution of the HQ distribution function in terms of pertinent transport coefficients based on elastic scattering amplitudes. This is followed by a discussion of several microscopic approaches to calculate the HQ friction and diffusion coefficients in the QGP: perturbative QCD (Sec. 2.2) at leading (2.2.1, 2.2.2, 2.2.3) and next-to-leading order (2.2.4) as well as for three-body scattering (2.2.5); nonperturbative calculations (Sec. 2.3) implementing resonance-like correlations in the QGP using HQ effective theory (Sec. 2.3.1), in-medium T -matrices with HQ potentials estimated from thermal lattice QCD (Sec. 2.3.2), or collisional-dissociation mechanisms of heavy mesons (Sec. 2.3.3); and string-theoretic evaluations based on the conjectured correspondence to conformal field theories (Sec. 2.4). The variety of the proposed approaches calls for an attempt to reconcile the underlying assumptions
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and basic interactions (Sec. 2.5). This is followed by a discussion of inelastic (radiative) energy-loss calculations and their relation to elastic ones (Sec. 2.6). We briefly consider interactions of open heavy-flavor hadrons in hadronic matter (Sec. 2.7). In Sec. 3 we discuss applications of HQ diffusion to URHICs using relativistic Langevin simulations of the Fokker–Planck equation within an expanding finite-size thermal medium (Sec. 3.1). A realistic description of the latter (utilizing hydrodynamics, transport models or suitable parameterizations thereof) is an essential prerequisite to enable a quantitative extraction of transport properties of the QCD medium (Sec. 3.2). Further ingredients are reliable initial conditions (possibly modified by nuclear effects) and the conversion of quarks to hadrons (Sec. 3.3). Implementations of different HQ diffusion coefficients in various space-time models are quantitatively analyzed in terms of the resulting HQ spectra at RHIC, in particular their nuclear modification factor and elliptic flow (Sec. 3.4). Including effects of hadronization (as well as semileptonic electron decays), a quantitative comparison of these calculations to single-electron spectra at RHIC is conducted (Sec. 3.5). We emphasize the importance of a consistent (simultaneous) description of pt spectra and elliptic flow. Only then can these observables be converted into a meaningful (albeit preliminary) estimate of charm- and bottom-quark diffusion coefficients in the QGP. We finish the discussion on open heavy flavor with an attempt to utilize these coefficients for a schematic estimate of the ratio of shear viscosity to entropy density in the QGP (Sec. 3.6). In Sec. 4 we elaborate on theoretical and phenomenological analyses of quarkonia in medium and their production in heavy-ion collisions. We first address spectral properties of quarkonia in equilibrium matter (Sec. 4.1); Euclidean correlation functions computed in lattice QCD with good precision have been analyzed in terms of potential models based on screened HQ potentials (Sec. 4.1.1). The interplay of color screening and parton-induced dissociation reactions has important consequences for the evaluation of quarkonium dissociation widths (Sec. 4.1.2). In light of the charmonium equilibrium properties the current status of the phenomenology in heavy-ion collisions is discussed (Sec. 4.2). First, quarkonium transport equations are introduced along with their main ingredients, i.e., dissociation widths and equilibrium numbers using relative chemical equilibrium at fixed HQ number (Sec. 4.2.1); this is followed by model comparisons to J/ψ data at SPS and RHIC, scrutinizing suppression vs. regeneration mechanisms and their transversemomentum dependencies (Sec. 4.2.2), and a brief illustration of predictions for Υ production at RHIC. In Sec. 5 we recollect the main points of this article and conclude. 2. Heavy-Quark Interactions in QCD Matter At an energy scale of the (pseudo-) critical QCD transition temperature, the large charm- and bottom-quark masses imply that the HQ diffusion problem is a nonrelativistic one (unless initial conditions bring in an additional large scale). In the weak-coupling regime this further implies that the dominant interactions of the
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heavy quark are elastic scattering (gluon radiation is suppressed by an extra power in αs and cannot be compensated by a large momentum transfer as could be the case for a fast quark; see, e.g., the discussion in Ref. 26). It turns out, however, that the perturbative expansion of the charm-quark diffusion coefficient, evaluated using thermal field theory, is not well convergent even for a strong coupling constant as low as αs = 0.1.55 Thus, non-perturbative methods, e.g., resummations of large contributions or interactions beyond perturbation theory, are necessary to improve the estimates of HQ diffusion. This is not surprising since transport coefficients usually involve the zero-momentum limit of correlation functions rendering them susceptible to threshold effects which may increase with the mass of the particles. A simple example of such kind are Coulomb-like bound states (e.g., heavy quarkonia), where the binding energy increases with increasing HQ mass, B ∝ α2s mQ , to be compared to thermal effects, e.g., at a scale ∼ gT for Debye screening (to leading order in g) or at ∼ T for inelastic dissociation reactions with thermal partons. An interesting question in this context is whether potential models are a viable means to evaluate HQ interactions in the QGP. If a suitable formulation of a potential at finite temperature can be established, a promising opportunity arises by extracting these from first principle lattice computations of the HQ free energy. In the heavyquarkonium sector such a program has been initiated a few years ago56–60 with fair success, although several open questions remain.58,61–63 If applicable, potential models have the great benefit of allowing for nonperturbative solutions utilizing Schr¨ odinger or Lippmann–Schwinger equations; the calculated scattering amplitudes can then be straightforwardly related to transport coefficients. A key issue in this discussion is the transition to the (ultra-)relativistic regime, which becomes inevitable in applications to experiment toward high momentum. While relativistic kinematics can be readily accounted for, the opening of inelastic (radiative) channels poses major problems. However, here the contact to perturbative calculations may be possible and provide a valuable interface to match the different regimes, at least parametrically (e.g., in the limit of a small coupling constant and/or high temperature). This reiterates the importance of identifying the common grounds of seemingly different calculations for HQ properties in medium. We start the discussion in this Section by setting up the Brownian Motion framework for heavy quarks in the QGP (Sec. 2.1). The main part of this Section is devoted to the evaluation of the Fokker–Planck transport coefficients. We focus on elastic interactions, classified into (various levels of) perturbative (Sec. 2.2) and nonperturbative approaches (Secs. 2.3 and 2.4). As we will see, there is considerable conceptual overlap in the calculations available in the literature, the main difference being that they are carried out in different approximation schemes (Sec. 2.5). Our presentation also encompasses inelastic reactions with an additional gluon in the final and/or initial state, i.e., radiative energy-loss calculations within perturbative QCD (Sec. 2.6). This raises the issue of their relative magnitude compared to elastic interactions which has recently received considerable re-consideration even for light
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quarks and gluons. Finally, we address interactions of hadrons carrying charm or bottom in hadronic matter (Sec. 2.7). Bottom-up extrapolations in temperature (or density) in the hadronic world are useful complements to top-down ones in the QGP, to reveal qualitative trends of, e.g., the HQ diffusion coefficient toward T c . 2.1. Heavy-quark diffusion in the quark-gluon plasma As emphasized in the Introduction, an attractive feature in analyzing HQ motion in a QGP is the ensuing simplification to a Brownian motion framework.23 The latter is characterized by a Fokker–Planck equation where HQ interactions are conveniently encoded in transport coefficients. These, in turn, are readily related to underlying (elastic) scattering matrix elements on light partons in the QGP which allow for direct comparisons of microscopic models of the HQ interaction (as elaborated in subsequent sections). Starting point for the derivation of the Fokker–Planck equation23 is the Boltzmann equation for the HQ phase-space distribution, fQ , p ∂ ∂ ∂ + +F fQ (t, x, p) = C[fQ ], (1) ∂t ωp ∂x ∂p where ωp =
q
m2Q + p2 denotes the energy of a heavy quark with three-momentum
p, F is the mean-field force, and C[fQ ] summarizes the collision integral which will be analyzed in more detail below. In the following, mean-field effects will be neglected, and by integration over the fireball volume, Eq. (1) simplifies to an equation for the momentum distribution, ∂ fQ (t, p) = C[fQ ], ∂t
(2)
where fQ (t, p) =
Z
d3 xfQ (t, x, p).
(3)
The collision integral on the right-hand side of Eq. (2) encodes the transition rate of heavy quarks due to collisions into and out of a small momentum cell d3 p around the HQ momentum p, Z C[f ] = d3 k[w(p + k, k)fQ (p + k) − w(p, k)fQ (p)] . (4) Here w(p, k) is the transition rate for collisions of a heavy quark with heat-bath particles with momentum transfer k, changing the HQ momentum from p to p − k. Accordingly the first (gain) term in the integral describes the transition rate for HQ scattering from a state with momentum p + k, into a state with momentum p, while the second (loss) term the scattering out of the momentum state p.
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The transition rate, w, can be expressed through the cross section of the collision processes in the heat bath. For elastic scattering of a heavy quark with momentum p on a light quark in the heat bath with momentum q, one finds Z dσ d3 q fq,g (q)vrel (p, q → p − k, q + k), (5) w(p, k) = γq,g (2π)3 dΩ where fq,g are the Fermi or Bose distributions for thermal light quarks or gluons, and γq = 6 or γg = 16 the respective spin-color degeneracy factors. The relative velocity is defined as p (p · q)2 − (mQ mq )2 , (6) vrel = ωQ ωq where p = (ωp , p) and q = (ωq , q) are the four momenta of the incoming heavy and light quark, respectively. Upon expressing the invariant differential cross section, P dσ/dΩ, in Eq. (5) in terms of the spin-color summed matrix element, |M|2 , the collision term, Eq. (4), takes the form Z Z Z 1 d3 q 0 d3 q d3 p 0 1 X C[fQ ] = |M|2 3 3 3 2ωp (2π) 2ωq (2π) 2ωp0 (2π) 2ωq0 γQ × (2π)4 δ (4) (p + q − p0 − q 0 )[fQ (p0 )fq,g (q 0 ) − fQ (p)fq,g (q)]
(7)
with k = p − p0 = q 0 − q. The key approximation is now that the relevant momentum transfers to the heavy quark obey |k| |p|. This enables to expand the HQ momentum distribution function, fQ , and the first argument of the transition rate, w, in the collision integral, Eq. (4), with respect to k up to second order,c w(p + k, k)fQ (p + k, k) ' w(p, k)fQ (p) + k +
∂ [w(p, k)fQ (p)] ∂p
1 ∂2 ki kj [w(p, k)fQ (p)] 2 ∂pi ∂pj
(8)
(i, j = 1, 2, 3 denote the spatial components of the 3-vectors, with standard summation convention for repeated indices). The collision integral then simplifies to Z ∂ 1 ∂2 3 C[fQ ] ' d k ki + ki kj [w(p, k)fQ (p)] , (9) ∂pi 2 pi pj i.e., the Boltzmann equation (2) is approximated by the Fokker–Planck equation, ∂ ∂ ∂ fQ (t, p) = Ai (p)fQ (t, p) + [Bij (p)fQ (t, p)] . (10) ∂t ∂pi ∂pj c According
to the Pawula theorem64 any truncation of the collision integral at finite order is only consistent with fundamental properties of Markov processes if the truncation is made at the 2nd -order term.
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The drag and diffusion coefficients are given according to Eq. (9) by Z Ai (p) = d3 kw(p, k)ki , Bij (p) =
1 2
Z
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(11)
d3 kw(p, k)ki kj .
For an isotropic background medium, especially in the case of (local) equilibrium (implying that the coefficients are defined in the local rest frame of the heat bath), rotational symmetry enables to simplify the coefficients to Ai (p) = A(p)pi , k
Bij (p) = B0 (p)Pij (p) + B1 (p)Pij⊥ (p) ,
(12)
where the projection operators on the longitudinal and transverse momentum components read k
Pij (p) =
pi pj , p2
Pij⊥ (p) = δij −
pi pj . p2
(13)
Implementing these simplifications into the collision integral, Eq. (7), the scalar drag and diffusion coefficients in Eq. (12) are given by integrals of the form Z Z Z d3 q 0 1 d3 q d3 p 0 1 X 0 hX(p )i = |M|2 2ωp (2π)3 2ωq (2π)3 2ωp0 (2π)3 2ωq0 γQ g,q × (2π)4 δ (4) (p + q − p0 − q 0 )fq,g (q)X(p0 ) . In this notation, the coefficients can be written as pp0 A(p) = 1 − 2 , p 1 (p0 p)2 02 B0 (p) = p − , 4 p2 1 (p0 p)2 0 2 B1 (p) = − 2p p + p . 2 p2
(14)
(15)
Note that Eq. (14) includes the sum over gluons and light quarks (u, d, s). The physical meaning of the coefficients becomes clear in the non-relativistic approximation of constant coefficients, γ ≡ A(p) = const and D ≡ B0 (p) = B1 (p) = const, in which case the Fokker–Planck equation further simplifies to ∂ ∂ fQ (t, p) = γ [pi fQ (t, p)] + D∆p fQ (t, p) . ∂t ∂pi
(16)
E.g., for an initial condition fQ (t = 0, p) = δ (3) (p − p0 ) ,
(17)
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the solution takes the form of a Gaussian distribution, o−3/2 n γ γ [p − p0 exp(−γt)]2 [1 − exp(−2γt)] exp − . fQ (t, p) = 2πD 2D 1 − exp(−2γt)
(18)
From the equation for the mean momentum,
hpi = p0 exp(−γt) ,
(19)
one sees that γ determines the relaxation rate of the average momentum to its equilibrium value, i.e., it is a drag or friction coefficient. The standard deviation of the momentum evolves according to
3D 2 p2 − hpi = [1 − exp(−2γt)] , γ
(20)
i.e., D is the momentum-diffusion constant, describing the momentum fluctuations. In the limit t → ∞, Eq. (18) approaches the (non-relativistic) Boltzmann distribution, 3/2 2πD γp2 fQ (t, p) = exp − . (21) γ 2D Since in thermal equilibrium the heavy quarks have to obey an equilibrium distribution with the temperature, T , of the heat bath, the drag and diffusion coefficients should satisfy the Einstein dissipation-fluctuation relation, D = mQ γT .
(22)
The relativistic Fokker–Planck equation will be discussed in Sec. 3.1 in connection with its formulation in terms of stochastic Langevin equations. We note that the spatial diffusion coefficient, Ds , which describes the broadening of the spatial distribution with time,
2 2 x (t) − hx(t)i ' 6Ds t , (23) is related to the drag and momentum-diffusion coefficient through Ds =
T T2 = . mQ γ D
(24)
2.2. Perturbative QCD approaches In a first step to evaluate HQ diffusion in a QGP perturbation theory has been applied, thereby approximating the medium as a weakly interacting system of quark and gluon quasiparticles. Such a treatment is expected to be reliable if the temperature is large enough for the typical momentum transfers, Q2 ∼ T 2 , to be in the perturbative regime, Q2 ≥ 2 GeV2 or so. This is most likely not satisfied for matter conditions realized at SPS and RHIC. For more realistic applications to experiment several amendments of the perturbative approach have been suggested which are discussed subsequently (focusing again on elastic HQ scattering on light partons).
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g
g
g
g
Q
Q
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g
q
q
Q
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Q
Fig. 1. Feynman diagrams for leading-order perturbative HQ scattering off light partons.
2.2.1. Schematic leading order The initial estimates of equilibration times and energy loss of heavy quarks in the QGP23 have started from the leading-order (LO) perturbative diagrams involving the minimum of two strong-interaction vertices, as displayed in Fig. 1. Pertinent matrix elements65 figuring into Eq. (14) in the vacuum have been computed in Ref. 65. The dominant contribution arises from gluon t-channel exchange, i.e., the 3rd and 4th diagram in Fig. 1. For forward scattering, the gluon propagator develops the well-known infrared singularity which has been regularized by introducing a Debye-screening mass, G(t) =
1 1 → , t t − µD
µD = gT ,
(25)
√ where g = 4παs denotes the strong coupling constant. Even for a value as large as αs = 0.4, and at a temperature of T = 300 MeV (typical for the early stages in heavy-ion collisions at RHIC), the thermal relaxation time, τeq = 1/γ, for charm (bottom) quarks turns out around ∼ 15(40) fm/c (and therefore much larger than a typical QGP lifetime of ∼ 5 fm/c at RHIC), see, e.g., right panel of Fig. 7 (in Ref. 24 the corrections due to quantum-equilibrium distributions (Bose/Fermi) have been investigated and found to be small). Note that with the above gluon propagator, the pertinent total HQ-parton cross section is parametrically given by σQp ∝ α2s /µ2D , i.e., it essentially increases only linearly in αs (p = q, q¯, g). 2.2.2. Leading order with hard thermal loop resummation In Ref. 26, the schematic introduction of the Debye mass into the t-channel gluonexchange propagator has been extended by a LO hard-thermal loop (HTL) calculation of the charm-quark drag and diffusion coefficients in the QGP. In this approach, the screening of the gluon propagator in the t-channel diagrams (Fig. 1) is realized by inserting the HTL gluon propagator for the region of small momentum exchange. In Coulomb gauge, with q = |q|, this propagator is given by Gµν (ω, q) = −
δµ0 δν0 δij − qi qj /q 2 + 2 , + Π00 q − ω 2 + ΠT
q2
(26)
R. Rapp & H. van Hees 0.8 Nf = 0 0.7 0.6 Nf = 2
0.5
ηD(p) / ηD(p= 0)
D (αs2 T)
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2.2
mD/T = 0.5 mD/T = 1 .0 mD/T = 1 .5
2 1.8 1.6 1.4 1.2
0.4 Nf = 3 0.3
1
d p∝p dt
0.8
Nf = 6
0.2
0.6
d p∝v dt
0.4
0.1 0 0
0.2
0.5
1
1 .5
2
2.5 mD/T
0 0
0.5
1
1.5
2
2.5
3 p/M
Fig. 2. (Color online) HQ transport coefficients in HTL improved perturbation theory. 26 Left panel: spatial diffusion coefficient at p = 0 as a function of an independently varied Debye mass, mD ≡ µD , figuring into the t-channel gluon exchange propagator, for different quark-flavor content of the (Q)GP. Right panel: momentum dependence of the drag coefficient, ηD (p) ≡ A(p), for three values of µD in t-channel gluon exchange; the lower curve, with dp/dt ∝ v, resembles a calculation in the non-relativistic limit (M ≡ mc = 1.4 GeV in the x-axis label denotes the charm-quark mass).
where the i, j ∈ {1, 2, 3} denote the spatial components of µ, ν ∈ {0, 1, 2, 3}. The HTL self-energies read 2 ω ω(q 2 − ω 2 ) q+ω 2 ΠT (ω, q) = µD + ln − iπ , 2q 2 4q 3 q−ω (27) q+ω ω 2 ln − iπ . Π00 (ω, q) = µD 1 − 2q q−ω For small energy transfers, ω, and a slowly moving heavy quark, v 1, only the time component of the propagator contributes to the squared matrix elements which in this limit reduces to the Debye-screened Coulomb-like propagator, Eq. (25). Figure 2 shows the spatial diffusion coefficient and the momentum dependence of the drag coefficient resulting from this calculation. Compared to the screening description with a constant Debye mass, the drag coefficient shows a slight increase for an intermediate range of momenta (cf., e.g., the pQCD curves in Fig. 12). 2.2.3. Leading order with running coupling As indicated in the Introduction, the current data situation at RHIC does not allow for an understanding of the electron data in terms of LO pQCD with reasonably small coupling constant (say, αs ≤ 0.4). This was a motivation for more recent studies,66,67 augmenting the LO pQCD framework in search for stronger effects. Two basic amendments have been introduced. First, the idea of Ref. 26 of introducing a reduced screening mass in the gluon propagator was made more quantitative.
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Starting from an ansatz for the screened gluon propagator, Gr (t) ∝
1 . t − rµ2D
(28)
the objective is to obtain an estimate for the constant r (it was denoted κ in Refs. 66 and 67; we changed the notation to avoid conflicts in what follows below). This has been done in analogy to a corresponding QED calculation,68,69 by requiring that the energy loss of a high-energy quark obtained in a LO-pQCD calculation with the screened propagator, Eq. (28), matches a calculation where for low momentum transfers, |t| < |t∗ |, the HTL propagator, Eq. (26), and for |t| > |t∗ | the perturbative gluon propagator, Eq. (25), is used; |t∗ | is a momentum-transfer scale between g 2 T 2 and T 2 . The QED calculation68,69 yields an energy loss which is independent of the matching scale |t∗ |, while this is not the case in QCD. This problem is treated by introducing an infrared-regulator mass into the hard part of the energy-loss integrals involving the t-channel exchange-matrix elements, chosen such that the dependence on |t∗ | is weak for |t∗ | < T 2 (the validity range of the HTL approximation). This translates into effective values for the r coefficient in Eq. (28) of r ' 0.15–0.2. Second, a running strong coupling constant is introduced well into the nonperturbative regime but with an infrared-finite limit. The justification for such a procedure70 is that it can account for (low-energy) physical observables (e.g., in e+ e− annihilation71 ) in an effective way. The parameterization adopted in Refs. 66 and 67 is based on an extrapolation of Ref. 70 into the spacelike regime, ( for Q2 ≤ 0 4π L−1 − 2 αeff (Q ) = (29) β0 1/2 − π −1 arctan(L+ /π) for Q2 > 0, where β0 = 11−2Nf /3, Nf = 3, and L± = ln(±Q2 /Λ2 ). The pertinent substitution in the t-channel gluon-exchange matrix elements amounts to αeff (t) α → , t t−µ ˜2
(30)
where the regulator mass is chosen as µ ˜ 2 ∈ [1/2, 2]˜ µ2D , while the Debye-screening mass is determined self-consistently from the equation Nc Nf 2 µ ˜D = + 4πα(−˜ µ2D )T 2 . (31) 3 6 To find the optimal value for the regulator mass a similar strategy of matching the energy loss with a Born approximation has been employed, using the substitution, Eq. (30), in the t-channel diagrams, with a HTL calculation along the same lines as summarized above for the calculation with non-running αs . The results for the drag coefficients for charm quarks under the various model assumptions described above are depicted in Fig. 3. Changing the screening mass from the standard Debye mass, µD , to that reproducing the HTL energy loss, with r = 0.15 in Eq. (28),
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αS µ2 A 0.3 m2D B αS (2πT ) m2D C αS (2πT ) 0.15 × m2D D running (Eq. (29)) m ˜ 2D E running (Eq. (29)) 0.2 × m ˜ 2D F running (Eq. (29)) 0.11 × 6π αeff (t) T 2
line form dotted thin dashed thin full thin dashed bold full bold dashed dotted bold
figure color black black black red red purple
Fig. 3. (Color online) The drag coefficient as a function of HQ three-momentum in the amended pQCD scheme with reduced infrared regulator and running coupling constant (left panel). 66,67 The corresponding legend (right panel) details the different parameter choices in the calculation.
increases the drag coefficient by a factor of 2. In view of the large reduction in r this appears to be a rather moderate effect. This is simply due to the fact that the change mostly enhances forward scattering which is little effective in thermalizing (isotropizing) a given momentum distribution. Implementing the running-coupling scheme with a small screening mass yields a substantial enhancement by a factor of ∼ 5. 2.2.4. Next-to-leading order The rather large values of the coupling constant employed in the calculations discussed in the previous sections imminently raise questions on the convergence of the perturbative series. This problem has been addressed in a rigorous next-to-leadingorder (NLO) calculation for the HQ momentum-diffusion coefficient, κ = 2D, in Refs. 55 and 72. This work starts from the definition of κ as the mean squared momentum transfer per unit time, which in gauge theories is given by the timeintegrated correlator of color-electric-field operators connected by fundamental Wilson lines: Z
g2 a b κ= dtTrH W † (t, 0)Eia (t)TH W (t, 0)Eib (0)TH ; (32) 3dH W (t; 0) denotes a fundamental Wilson line running from t0 = 0 to t along the a static trajectory of the heavy quark, TH are the generators of the gauge group in the representation of the heavy quark and dH its dimension. In leading order this reduces to a Wightman-two-point function of A0 fields at zero frequency, i.e., in the usual real-time propagator notation, κ'
CH g 2 3
Z
d3 p 2 >00 p G (ω = 0, p) , (2π)3
(33)
with CH = 4/3 the Casimir operator of the HQ representation. The integral is IR regulated by HTL corrections, i.e., a Debye mass, µ2D = g 2 T 2 (Nc + Nf /2)/3. In
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αs 0.01
Q
0.05
0.1
0.2
0.3
0.4
0.6
P
R
Q
P
0.5
R (B)
Q P
P Q
0.4
κ/g 4 T 3
(A)
Next-to-leading order Leading order Truncated leading order
0.3 0.2 0.1 0
(C)
(D)
0.5
1
1.5
2
2.5
gs
Fig. 4. NLO calculations for HQ diffusion in the QGP.55 Left panel: NLO diagrams for the momentum-diffusion coefficient, κ; the double line represents the heavy quark, all propagators are soft and HTL resummed, and all vertices include HTL vertices. Right panel: comparison of LO to NLO result for κ as a function of the strong coupling, αs .
the left panel of Fig. 4 the NLO corrections to the LO result, Eq. (33), are depicted in terms of Feynman diagrams. The double line represents the heavy quark, and all propagators and vertices include HTL corrections, leading to a gauge invariant expression as it should be the case for an observable quantity like κ. The diagrams are evaluated in Coulomb gauge within the closed-time path (real-time) Keldysh formalism of thermal quantum-field theory (TQFT). The real part of diagram (A) provides a correction to the Debye mass. Diagrams (C) and (D) take into account real and virtual corrections by additional soft scattering or plasmon emission/absorption of the light or heavy scatterer, respectively. Diagram (B) represents interference between scattering events occurring on the light scatterer’s and on the heavy quark’s side. Contrary to naive power counting, the NLO calculation provides O(g) corrections due to scattering with soft gluons with momentum, q ' µD , and due to overlapping scattering events, dominated by t-channel Coulombic scatterings involving soft momentum transfers, ' µD ∝ gT . The right panel of Fig. 4 shows that the NLO correction to κ is positive, i.e., the momentum-diffusion coefficient becomes larger compared to the LO calculation. The convergence is poor even for rather small coupling constants. A rigorous resummation scheme to cure this behavior is not known to date, especially to establish convergence in the typical range of coupling constants under conditions in relativistic heavy-ion collisions, αs ' 0.3–0.4. In Ref. 72 the investigation of NLO corrections is extended to the weak-coupling limit of N = 4 supersymmetric Yang-Mills (SYM) theory. Also in this case the perturbative series turns out to be poorly convergent, even for low couplings. 2.2.5. Three-body elastic scattering Another step in the (would-be) perturbative hierarchy are three-body collisions, which are expected to become increasingly important at high parton density. An
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(a)
(b)
(c)
Fig. 5. Different topological classes of diagrams for three-body elastic scattering of a heavy quark (thick lines) off light quarks and antiquarks (thin lines).73
attempt to assess the effects of three-body elastic scattering for HQ diffusion has been conducted in Ref. 73, with pertinent Feynman diagrams as depicted in Fig. 5. Special care has to be taken in regularizing contributions from diagrams with intermediate particles going on-shell; these can lead to divergent real parts in the scattering amplitude and represent successive two-body scatterings (rather than genuine three-body scattering). Therefore, in Ref. 73 the intermediate quark lines in diagram (a) and (b) are supplemented with an in-medium collisional width, and only the real part of their propagator is kept in the evaluation of the diagrams. For three-body elastic processes involving one or two gluons, it has been assumed that the dominant contributions arise from diagrams with similar topology as diagram (b) in Fig. 5 for Qqq scattering; all other contributions are neglected. To compare with two-body gluo-radiative inelastic scattering the LO diagrams have been used to evaluate matrix elements for Qq → Qqg, Q¯ q → Q¯ qg and Qg → Qgg processes. Within this scheme, at temperatures T = 200–300 MeV, three-body elastic scattering processes are estimated to contribute to the c- and b-quark friction coefficients with a magnitude comparable to two-body elastic scattering. Again, this raises the question of how to control the perturbative series for HQ diffusion. As a by-product, the friction coefficient for radiative scattering, Qp → Qpg, was estimated to exceed the one from elastic two-body scattering for HQ momenta p & 12 GeV (for both charm and bottom). 2.3. Non-perturbative interactions The evidence for the formation of a strongly coupled QGP (sQGP) at RHIC has motivated vigorous theoretical studies of the possible origin of the interaction strength (see, e.g., Ref. 74 for a recent review). In particular, several lattice QCD computations of hadronic correlation functions at finite temperature have found indications that hadronic resonances (or bound states) survive up to temperatures of twice the critical one or more (for both a gluon plasma (GP) and a QGP),41,42,75,76 cf. also Sec. 4.1.1 of this article. Pertinent spectral functions (extracted from Euclidean correlators using probabilistic methods, i.e., the maximum entropy method) exhibit ¯ and q q¯ channels. The consequences of hadronic resoresonance peaks in both QQ nances in the QGP for HQ transport have been elaborated in Refs. 25, 28, 77 and 45. The starting point in Refs. 25, 28 and 77 is the postulate that heavy-light quark
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(Q¯ q ) resonances, i.e., “D” and “B” mesons, persist in the QGP. In Refs. 25 and 28 this has been realized within an effective resonance model for Q–¯ q scattering (Sec. 2.3.1) while in Ref. 77 HQ fragmentation into mesons and their subsequent momentum broadening was considered (Sec. 2.3.3). The phenomenological success of these models (cf., e.g., the right panel of Fig. 26) called for a more microscopic evaluation of the heavy-light quark correlations. This was realized in Ref. 45 where in-medium heavy-light quark T -matrices were computed with interaction potentials estimated from HQ free energies in lattice QCD (cf. Sec. 2.3.2). This approach is the direct analog to the potential models used in the heavy quarkonium context (cf. Sec. 4.1.1). We finish this section with a brief discussion of a recent suggestion to extract information on HQ diffusion more directly from thermal lattice QCD (Sec. 2.3.4), which would constitute a valuable benchmark for both perturbative and non-perturbative calculations. 2.3.1. Effective Q¯ q-resonance model The heavy-light quark resonance model25 has been set up by combining HQ effective theory (HQET) with chiral symmetry in the light-quark sector, q = (u, d), based on the Lagrangian, / / 5 1+v 0 0 ∗1+v c − q¯γ Φ c + h.c. LDcq =LD + Lc,q − iGS q¯Φ0 2 2 1 + v/ 1 + v/ − GV q¯γ µ Φ∗µ c − q¯γ 5 γ µ Φ1µ c + h.c. , (34) 2 2 written in the charm sector (an equivalent one in the bottom sector follows via the replacements c → b and D → B for the HQ and resonance fields, respectively; v: HQ four-velocity). The pertinent free Lagrangians read 0 Lc,q = c¯(i/ ∂ − mc )c + q¯ i/ ∂q , ∗ LD0 = (∂µ Φ† )(∂ µ Φ) + (∂µ Φ0 ∗† )(∂ µ Φ∗0 ) − m2S (Φ† Φ + Φ∗† 0 Φ0 )
(35)
1 2 ∗† ∗µ Φ∗µν + Φ†1µν Φµν − (Φ∗† + Φ†1µ Φµ1 ) . 1 ) + mV (Φµ Φ 2 µν Φ and Φ∗0 denote the pseudoscalar and scalar meson fields (corresponding to D and D0∗ mesons) which are assumed to be degenerate chiral partners (mass mS ) as a consequence of chiral restoration in the QGP. The same reasoning applies to the vector and axialvector states (mass mV ), Φ∗µ and Φ1µ (corresponding to D ∗ and D1∗ ). HQ spin symmetry furthermore asserts the degeneracy of spin-0 and -1 states with identical angular momentum, implying mS = mV and the equality of the coupling constants, GS = GV . In the strange-quark sector only the pseudoscalar (Ds ) and vector (Ds∗ ) resonance states are considered (i.e., chiral symmetry is not imposed).
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q
D k
l−k l
D
D
k
s
u
c
c
c
c
q¯
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c
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q
Fig. 6. Left panel: one-loop diagram representing the D-meson self-energy in the QGP within the effective resonance model.25 Right and middle panels: elastic Q¯ q and Qq scattering diagrams for s- and u-channel resonance exchange, respectively.
The boson-resonance propagators are dressed with heavy-light quark selfenergies at the one-loop level (cf. left panel of Fig. 6). To leading order in HQET, in accordance with spin symmetry, the self-energies for the vector/axialvector resonances are given by ΠD∗ ,µν = (vµ vν − gµν )ΠD (s),
(36)
where s = p2 denotes the meson’s four momentum, and ΠD is the self-energy of the pseudoscalar/scalar resonances. Its imaginary part reads Im ΠD (s) = −
3G2 (s − m2c )2 Θ(s − m2c ) , 8π s
(37)
while the real part is calculated from a twice-subtracted dispersion relation with the wave-function and mass counter terms adjusted such that the following renormalization conditions hold, (ren)
∂s ΠD
(s)|s=0 = 0 ,
(ren)
Re ΠD
(s)|s=m2D = 0 .
As an alternative regularization scheme, dipole form factors, 2 2Λ2 , F (|q|) = 2Λ2 + q 2
(38)
(39)
have been supplemented to simulate finite-size vertices of the resonance model, (ff)
Im ΠD (s) = Im ΠD (s)F 2 (|q|) , (40) √ with |q| = (s − m2c )/(2 s). In this scheme, the real part is calculated from an unsubtracted dispersion relation, while the bare resonance mass is adjusted to obey the second renormalization condition in Eq. (38). With charm- and bottom-quark masses of mc = 1.5 GeV and mb = 4.5 GeV, the physical resonance masses are adjusted to mD = 2 GeV and mB = 5 GeV, respectively. This is in approximate accordance with earlier T -matrix models of heavy-light quark interactions.78,79 Likewise, the coupling constant, G, is adjusted
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12
60
τ[fm]
σ [mb]
8
charm, pQCD charm, reso (Γ=0 4 GeV) + pQCD bottom, pQCD bottom, reso (Γ=0 4 GeV) + pQCD
80
reso s-channel reso u-channel pQCD qc scatt. pQCD gc scatt.
10
6
40
4 20
2 0 1.5
2
3 2.5 1/2 s [GeV]
3.5
4
02
0 25
03 T [GeV]
0 35
04
Fig. 7. (Color online) Left panel: total HQ scattering cross sections off light partons in LO pQCD (blue lines) and within the effective resonance model (red lines). Right panel: thermalization times, τ = 1/A(p = 0), for charm and bottom quarks in LO pQCD with αs = 0.4 and Debye-screening mass µD = gT , compared to the results from the resonance+pQCD model, as a function of QGP temperature.
such that the resonance widths vary as ΓD,B = 0.4 . . . 0.75 GeV. The resulting heavy-light quark scattering matrix elements (cf. middle and right panels of Fig. 6) have been injected into Eq. (15) to calculate HQ drag and diffusion coefficients. In the left and right panel of Fig. 7 we compare the total HQ elastic scattering cross sections and resulting thermal relaxation times, τeq = 1/A(p = 0), of the resonance model with LO pQCD (cf. the diagrams in Fig. 1). Although the total cross sections are not very different in magnitude, the thermalization times decrease by around a factor of ∼ 3–4 when adding resonant scattering, for all temperatures T = 1–2 T c . The main reason for this behavior is that s-channel Q¯ q scattering is isotropic in the rest frame of the resonance, while the pQCD cross section is largely forward-peaked (t-channel gluon exchange), and thus produces a much less efficient transport cross section (which encodes an extra angular weight). The charm-quark equilibration c times in the resonance+pQCD model, τeq = 2–10 fm/c, are comparable to the expected QGP lifetime at RHIC of around τQGP ' 5 fm/c. Thus, at least for charm quarks, substantial modifications of their pt spectra towards local equilibrium in the flowing medium can be expected. The consistency of the Fokker–Planck approach can be checked with the dissipation-fluctuation relation, Eq. (77), at p = 0, cf. left panel of Fig. 8. For the forward-peaked pQCD-matrix elements, the relation is fulfilled within 3%, while with the isotropic resonance scattering deviations reach up to 11% in the renormalization scheme and up to 26% in the formfactor-cutoff scheme at the highest temperatures considered (T = 400 MeV). Note however, that for a typical thermal evolution at RHIC, average fireball temperatures above T = 250 MeV are only present within the first fm/c39 ; below this temperature, the deviations are less than 5% for all cases. The right panel of Fig. 8 illustrates that (for identical resonance
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05
form factor Λ=1 GeV renormalized 02
04 A[1/fm]
D/(γmQ)[GeV]
0 25
charm: total (renormalized) charm: pQCD bottom: total charm: total (form factor)
03
T=250 MeV
0 15 01
T=200 MeV
02 0 05
01 02
03 T[GeV]
0 25
0 35
0
04
1
05
15
p[GeV]
Fig. 8. Left panel: check of the dissipation-fluctuation relation at p = 0 for c and b quarks with and without resonance interactions and in the renormalization and form-factor cutoff schemes. Right panel: momentum dependence of the drag coefficient in the renormalization and the formfactorcutoff scheme, where the coupling constants have been chosen in both schemes as to obtain a resonance width Γ = 0.4 GeV.
02
0 06
resonances Γ=0 3 GeV resonances Γ=0 4 GeV resonances Γ=0 5 GeV pQCD: αs=0 3 pQCD: αs=0 4 pQCD: αs=0 5
T=200 MeV
0 05 2
B0[GeV /fm]
0 15
A[1/fm]
0 07
resonances Γ=0 3 GeV resonances Γ=0 4 GeV resonances Γ=0 5 GeV pQCD: αs=0 3 pQCD: αs=0 4 pQCD: αs=0 5
T=200 MeV
01
0 04 0 03 0 02
0 05 0 01
0
1
05 p[GeV]
15
0
1
05
15
p[GeV]
Fig. 9. Drag (left panel) and transverse-diffusion coefficient (right panel) for pQCD and resonance model with varying interactions strengths as a function of the HQ momentum at a temperature of T = 200 MeV.
widths) the formfactor regularization scheme leads to somewhat larger (smaller) friction coefficients at low (high) momentum than the renormalization scheme. In Fig. 9 the momentum dependence of the drag and transverse diffusion coefficients is depicted using either resonance-scattering or pQCD-matrix elements. Resonance scattering becomes relatively less efficient for higher HQ momenta since the center-of-mass energy in collisions with thermal light antiquarks increasingly exceeds the resonance pole. The variations of the coefficients with the strong coupling constant in the pQCD scattering-matrix elements or the resonance-coupling constant in the effective resonance-scattering model are rather moderate. This is due to compensating effects of an increase of the matrix elements with α2s or G4 , on the one hand, and the accordingly increased Debye-screening mass for pQCD scattering or the broadening of the resonances widths, on the other hand.
Heavy Quarks in the Quark-Gluon Plasma
q, q¯ T
=
V
+
V
=
Σglu
+
133
T
Q
Σ
T
Fig. 10. (Color online) Diagrammatic representation of the Brueckner many-body calculation for the coupled system of the T -matrix based on the lQCD static internal potential energy as the interaction kernel and the HQ self-energy.
2.3.2. In-medium T -matrix with lQCD-based potentials The idea of utilizing HQ free energies computed in lattice QCD to extract a driving kernel for heavy-light quark interactions in the QGP has been carried out in Ref. 45, with the specific goal of evaluating HQ diffusion. Since the latter is, in principle, determined by low-energy HQ interactions, the potential-model framework appears to be suitable for this task. Moreover, with a potential extracted from lQCD, the calculation could be essentially parameter-free. Currently, however, such an approach bears significant uncertainty, both from principle and practical points of view, e.g., whether a well-defined potential description can be constructed in medium80–82 and, if so, how to extract this information from, say, the HQ free energy. In the vacuum, both questions have been answered positively,83,84 thus validating the 30 year-old phenomenological approaches using Cornell potentials for heavy quarkonia, which provide a very successful spectroscopy.85 The potential approach has been extended to heavy-light mesons in Refs. 86 and 87. A Brueckner-like in-medium T -matrix approach for heavy-light quark scattering in the QGP has been applied in Ref. 45, diagrammatically represented in Fig. 10. The underlying (static) two-body potential has been identified with the internal energy U1 (r, T ) = F1 (r, T ) + T S1 (r, T ) = F1 (r, T ) − T
∂F1 (r, T ) , ∂T
(41)
extracted from two lQCD computations of the color-singlet HQ free energy above Tc , for quenched88 and two-flavor89 QCD (pertinent parameterizations are given in Refs. 90 and 44, hereafter referred to as [Wo] and [SZ], respectively). This choice (rather than, e.g., the free energy) provides an upper limit for the interaction strength.44,58,90,91 To use Eq. (41) as a potential in a T -matrix calculation, the internal energy has to be subtracted such that it vanishes for r → ∞, V1 (r, T ) ≡ U1 (r, T ) − U1 (r → ∞, T ) ,
(42)
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which is dictated by the convergence of the T -matrix integral in momentum space. It is suggestive to interpret the asymptotic value U1∞ ≡ U1 (r → ∞, T ) as an inmedium HQ-mass, 1 mQ (T ) = m0Q + U1∞ , 2
(43)
where m0Q denotes the bare HQ mass (e.g., m0c ' 1.25 GeV92 for the bare c-quark mass). However, close to Tc , the values for U1∞ (T ) extracted from lQCD calculations develop a rather pronounced peak structure,93,94 which renders a mass interpretation problematic. Progress in understanding these properties is closely connected with the proper identification of the potential. First lQCD estimates of the inmedium HQ mass (extracted by relating zero-mode contributions to quarkonium correlators to the HQ susceptibility) indicate a moderate increase when approaching Tc from above.46 In Ref. 45 constant (average) in-medium charm- and bottom-quark masses of mc = 1.5 GeV and mb = 4.5 GeV, respectively, have been employed. The T -matrix approach is readily generalized to color-configurations other than the singlet channel of the qQ pair. The complete set of color states for Q¯ q (singlet and octet) and Qq (anti-triplet and sextet) pairs has been taken into account assuming Casimir scaling as in pQCD, 1 V8 = − V1 , 8
V¯3 =
1 V1 , 2
1 V 6 = − V1 , 4
(44)
which is, in fact, supported by finite-T lQCD.95,96 To augment the static (colorelectric) potentials with a minimal relativistic (magnetic) correction for moving quarks,97 the so-called Breit correction as known from electrodynamics98 has been implemented via the substitution Va → Va (1 − α ˆ1 · α ˆ2 ),
(45)
where α ˆ 1,2 are quasiparticle velocity operators. The above constructed heavy-light potentials can now be resummed in a twobody scattering equation. In accordance with the static nature of the potentials, it is appropriate to use a three-dimensional reduction of the full four-dimensional BetheSalpeter equation. This leads to the well-known ladder series which is resummed by the Lippmann–Schwinger (LS) integral equation for the T -matrix In the q–Q center-of-mass (CM) frame it takes the form Z d3 k 0 0 Ta (E; q , q) = Va (q , q) − Va (q 0 , k) GqQ (E; k) (2π)3 × Ta (E; k, q) [1 − fq (ωkq ) − fQ (ωkQ )] .
(46)
The driving kernel (potential) can now be identified with the Fourier transform of the coordinate-space potential extracted from lQCD, Z Va (q 0 , q) = d3 r Va (r) exp[i(q − q 0 ) · r] (47)
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in a given color channel, a ∈ {1, ¯ 3, 6, 8}. The concrete form of the intermediate q–Q (or q¯–Q) propagator, GqQ (E, k), depends on the reduction scheme of the underlying Bethe-Salpeter equation. It has been verified45,91 that, e.g., the Thompson99 and Blancenbecler-Sugar100 scheme lead to very similar results in the present context (as was found for nucleon-nucleon scattering). In the former, the two-particle propagator is given by GqQ (E, k) =
1 E−
(ωkq
+
iΣqI )
− (ωkQ + iΣQ I )
,
(48)
where E and k denote the CM energy and relative momentum of the qQ pair, respectively. The quasi-particle widths are chosen as Γq,Q = −2Σq,Q = 200 MeV, I I and the light quark masses as constant at mq = 0.25 GeV, with on-shell energies q (49) ωkq,Q = m2q,Q + k 2 .
The latter figure into the Pauli blocking factor with Fermi-Dirac distributions, fq,Q (ω q,Q ) =
1 exp(ω q,Q /T ) + 1
(50)
(at the considered temperatures their impact is negligible). The solution of the T -matrix Eq. (46) is facilitated by a an expansion into partial waves, l, X Va (q 0 , q) = 4π (2l + 1) Va,l (q 0 , q) Pl [cos ∠(q, q 0 )] , l
0
Ta (E; q , q) = 4π
X
(2l + 1) Ta,l (E; q 0 , q) Pl [cos ∠(q, q 0 )] ,
(51)
l
which yields a one-dimensional LS equation, Z 2 Ta,l (E; q 0 , q) = Va,l (q 0 , q) + dkk 2 Va,l (q 0 , k) GQq (E; k) π × Ta,l (E; k, q) [1 − fF (ωkQ ) − fF (ωkq )] ,
(52)
for the partial-wave components, Ta,l , of the T -matrix. Equation (52) can be solved numerically by discretization and subsequent matrix-inversion with the algorithm of Haftel and Tabakin.101 The resulting S-wave (l = 0) T -matrices indeed show resonance structures in a QGP in the channels where the potential is attractive, i.e., in the meson (color-singlet) and diquark (color-antitriplet) channels. The pertinent peaks in the imaginary part of the T -matrix develop close to the Q–q threshold, and melt with increasing temperature at around 1.7 Tc and 1.4 Tc , respectively (cf. left panel of Fig. 11). In the repulsive channels, as well as for P -waves, the T -matrices carry much reduced (non-resonant) strength. The increasing strength in the meson and diquark channels (the latter relevant for baryon binding) when approaching Tc from above is suggestive for “pre-hadronic” correlations toward the hadronization transition.
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6 5
0
4 ΣQ (MeV)
-2
-Im T (GeV )
50
sing, T=1 2 Tc sing, T=1 5 Tc trip, T=1 2 Tc trip, T=1 5 Tc
s wave
3 2
-50 -100 T=1 1 Tc, real T=1 1 Tc, imag T=1 4 Tc, real T=1 4 Tc, imag T=1 8 Tc, real T=1 8 Tc, imag
-150
1 -200
0 0
1
2 Ecm (GeV)
3
4
0
1
2 p (GeV)
3
4
Fig. 11. (Color online) Results of a Brueckner-type approach for c-quarks in a QGP 45 based on a potential corresponding to the internal energy extracted from quenched lattice QCD. 88,90 Left panel: imaginary part of the in-medium T matrices for S-wave c–q scattering in color-singlet and -triplet channels at two different temperatures; right panel: real and imaginary parts of c-quark selfenergies based on the T -matrices in the left panel.
The next step is to use the T -matrices to compute the light-quark contribution to HQ self-energies, i.e., the last diagram in the second line of the Brueckner scheme illustrated in Fig. 10. In a given color-channel, a, the T -matrix induced self-energy is given by Z 2 k dkdx dSI da ΣQ [fF (ωk ) + fB (ω + ωk )] Ta (E; p, k) , (53) (ω, p) = a 6 4π 2 where dSI = 4(2l + 1)Nf denotes the spin-isospin and angular momentum degeneracy of all Qq (or Q¯ q) configurations (assuming spin and light-flavor symmetry) and da the color degeneracy of channel a; the factor 1/6 averages over the incoming HQ color-spin degrees of freedom. The resulting charm-quark selfenergies (summed over all light quarks and antiquarks) are displayed in the right panel of Fig. 11. One finds rather small corrections to the HQ mass (presented by the real part of Σ), but the imaginary parts are substantial, Γc = −2 Im Σc ' 100–300 MeV for temperatures T = 1.1–1.8 Tc (with the largest values attained close to Tc ). These values were the motivation for the choice of input widths in the propagator, Eq. (48), of the T -matrix equation, thus providing a rough self-consistency. The in-medium mass corrections, on the other hand, are associated with the gluonic contribution to the HQ self-energy (corresponding to the first term in the lower line Fig. 10), which have not been calculated explicitly in Ref. 45, but roughly represent the (average) (1) asymptotic values of the HQ potential, U∞ (as discussed above). The final step is to implement the T -matrix elements into a calculation of HQ drag and diffusion coefficients via Eq. (15); one finds X
|M|2 =
64π (s − m2q + m2Q )2 (s − m2Q + m2q )2 s2 X × Nf da (|Ta,l=0 (s)|2 + 3|Ta,l=1 (s) cos(θcm )|2 ) . a
(54)
Heavy Quarks in the Quark-Gluon Plasma 0 15
0 15
T-matrix [Wo]: 1 1 Tc T-matrix [Wo]: 1 5 Tc T-matrix [Wo]: 1 8 Tc pQCD: 1 1 Tc pQCD: 1 4 Tc pQCD: 1 8 Tc
T-matrix [SZ]: 1 1 Tc T-matrix [SZ]: 1 5 Tc T-matrix [SZ]: 1 8 Tc pQCD: 1 1 Tc pQCD: 1 4 Tc pQCD: 1 8 Tc
01 Α (1/fm)
Α (1/fm)
01
0 05
137
0 05
0
0 0
1
2
3
4
5
0
1
p (GeV)
2
3
4
5
p (GeV)
Fig. 12. (Color online) The drag coefficients at different temperatures, using the parameterization of the HQ potential from [Wo] (left panel) and [SZ] (right panel) compared to LO pQCD with αs = 0.4 and µD = gT .
The resulting friction coefficients are summarized in Fig. 12 as a function of momentum for three temperatures and for two potential extractions from lQCD. 44,90 Generally, the Qq T -matrix based coefficients are largest at low HQ momentum, as to be expected from the resonance formation close to threshold. The values exceed the LO-pQCD coefficients at small temperatures and for both potentials by a factor of ∼ 3–5. At higher temperatures the enhancement reduces considerably, to a factor of less than 2 in the [SZ] potential and to essentially equal strength for the [Wo] potential.d In fact, the coefficients computed with the [SZ] potential have a slightly (30%) increasing trend with T , while the [Wo] potential leads to a decreasing trend. This difference needs to be scrutinized by future systematic comparisons of lQCD input potentials. Compared to the resonance model (cf., e.g., left panel of Fig. 9), the T -matrix calculations yield quantitatively similar results at temperatures not too far above Tc but become smaller at higher T due to resonance melting (which is presumably a more realistic feature of a non-perturbative interaction strength). 2.3.3. Collisional dissociation of heavy mesons in the QGP In Ref. 77, a so-called reaction-operator (GLV) approach has been applied to resum multiple elastic scatterings of a fast Q¯ q pair. The quenching of heavy quarks in a QGP is calculated by solving coupled rate equations for the fragmentation of c and b quarks into D and B mesons and their dissociation in the QGP. The main mechanisms for HQ energy loss are collisional broadening of the meson’s transverse d Recall
that the LO-pQCD calculations employ a rather large coupling (α s = 0.4), and are dominated by scattering off gluons in the heat bath; thus a minimal merging of the gluon sector with the T -matrix calculations would consist of adding the gluonic part of LO-pQCD; this procedure is adopted below whenever combined results are shown or utilized. In principle, a non-perturbative treatment should also be applied to HQ-gluon scattering.
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momentum and the distortion of its intrinsic light-cone wave function. The latter is modeled in a Cornell-type potential ansatz87 (note that this bears some similarity to the T -matrix approach discussed in Sec. 2.3.2). This in-medium HQ/heavy-meson fragmentation/dissociation mechanism leads to comparable high-pt suppression for B and D mesons, which is quite contrary to perturbative calculations for both collisional and radiative energy loss (where the suppression of b quarks is significantly less pronounced than for c quarks). This feature largely results from the much smaller formation times of B-mesons compared to D-mesons, leading to a faster fragmentation-dissociation cycle for b quarks/B mesons. 2.3.4. Estimates of HQ diffusion in lattice QCD It has recently been suggested that, unlike in the case of other transport coefficients (e.g., the shear viscosity), the HQ diffusion coefficient might be amenable to a determination within lQCD, based on an analytic continuation of the color-electricfield correlator along a Polyakov loop102 (see also Ref. 103 for earlier related work). The starting point of these considerations is the spectral function of the HQ current correlator, h Z Z i 1 µ ν 3 J (t, x), J (0, 0) , (55) ρµν (ω) = dt exp(iωt) d x Q V 2 Q
µ where JQ denotes the HQ current operator in the Heisenberg picture. The spatial diffusion coefficient, Ds , can be extracted from this spectral function by the pole position at ω = −iDs k2 , where k is the HQ momentum. The condition for a pole leads to the Kubo relation
Ds =
3 X 1 ρii V (ω) lim , 00 3χ ω→0 i=1 ω
where χ00 is the conserved-charge susceptibility Z
0 1 00 0 χ = d3 x JQ (t, x)JQ (0, 0) . T
(56)
(57)
For a heavy quark, the spectral function, Eq. (55), is expected to develop a sharp peak around ω = 0 which can be described by a Lorentzian function close to this point. The width of this function is given by the drag coefficient, which obeys the fluctuation-dissipation relations, discussed in Sec. 2.1. Using HQ effective theory techniques it is shown that in the static limit the momentum-diffusion coefficient, 0 κ = 2D, is given by a correlator involving color-electric fields and JQ operators whose Euclidean analogue can be mapped to an expectation value involving Wilson lines and color-electric fields, similar to Eq. (32). This purely gluonic correlation function can in principle be evaluated in lQCD. Another lattice-based approach to assess HQ diffusion has been suggested in Ref. 104 in terms of (discretized) classical gauge theory. The limitation of this
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139
approach is set by the thermal (hard) scale ∼ πT where quantum theory suppresses excitations. Since the HQ thermalization rate is expected to be governed by the electric screening scale ∼ gT , this limitation may not be severe for small and moderate coupling (gT T ), and thus allow for valuable insights. First, it has been verified that, in the weak coupling limit, the discretized (nonperturbative) classical computation indeed agrees well with pQCD. Upon increasing the coupling strength, the classical lattice results for the HQ momentum diffusion coefficient increasingly exceed the LO perturbative result, by about an order of magnitude for a moderate coupling strength corresponding to αs ' 0.2. Next, the NLO term (with slightly increased strength to account for HTL effects), as calculated in Ref. 55, has been added to the LO calculation which extends the agreement of pQCD with the classical lattice results to larger (but still weak) coupling. For αs ' 0.2 the increase over LO amounts to a factor ∼ 2, which means that the classical lattice result remains substantially larger (by a factor of ∼ 5) than the NLO value. Besides reconfirming the poor convergence of pQCD, this also suggests that the perturbative series is not alternating but that higher order terms keep increasing the value of the HQ momentum diffusion coefficient. Semi-quantitatively, such an enhancement is in the ball park of the factor ∼ 3–4 found in the effective resonance model (Sec. 2.3.1) or T -matrix approach (Sec. 2.3.2). 2.4. String theoretical evaluations of heavy-quark diffusion The conjectured correspondence between certain classes of string theories, formulated in five-dimensional Anti-de-Sitter space (AdS5 ), and gauge theories with conformal invariance (conformal field theory, CFT) has opened interesting possibilities to address nonperturbative aspects of QCD. The so-called AdS/CFT correspondence implies a “duality” of a weakly coupled gravity to a strongly coupled supersymmetric (and conformal) gauge theory, specifically N = 4 SU(Nc ) super-YangMills (SYM) theory. This connection has been exploited to formulate the problem of HQ diffusion at finite temperature and extract an “exact” nonperturbative result for HQ transport coefficients in the SYM plasma.105–107 The translation to QCD matter is beset with several caveats,108 e.g., the particle content of the SYM medium is quite different compared to the QGP. While this may be corrected for by a suitable rescaling of the temperature by matching, e.g., the energy densities, e a more problematic difference is the absence of a scale (other than temperature) in conformal SYM. Thus, the latter does not possess a breaking of scale invariance, a running coupling constant, confinement nor spontaneous chiral symmetry breaking, and consequently no notion of a critical temperature, either. Thus SYM is quite different from QCD in the zero- and low-temperature regimes. However, at sufficiently high T , where the QCD medium deconfines its fundamental charges, e This
procedure works quite well when comparing quantities in quenched and unquenched lattice QCD computations, e.g. for the critical temperature.
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the resemblance to SYM might be much closer. E.g., the pressure in SYM in the strong coupling limit amounts to about 75% of the Stefan-Boltzmann limit, close to what is found in thermal lattice QCD for a wide range above Tc . In addition, the finding of an extremely small shear viscosity in strongly coupled SYM, η/s = 1/4π (conjectured to be a universal lower bound),17 and the apparently low-viscosity QCD medium deduced from the success of hydrodynamic models at RHIC, is another good reason to further pursue exact nonperturbative calculations in SYM for quantities that are relevant for RHIC phenomenology. If nonperturbative effects in the strongly coupled QGP at moderate temperatures, T = 1–2 Tc , are ultimately connected to the presence of the phase change(s) (and thus inherently to the critical temperature as a relevant scale), the CFT-QCD connection would not be a rigorous one. But even in this case, the nonperturbative computation of transport coefficients of a strongly coupled system at a given reference temperature “not too close” to Tc should provide useful insights. The first step in computing HQ diffusion for CFT is the introduction of a heavy quark into the conformal field theory. This can be achieved by either introducing a heavy charge via breaking the gauge group from Nc + 1 to Nc (which, strictly speaking, generates (2Nc + 1) “Higgsed” “W ” bosons),109 or by adding a finitemass N = 2 hypermultiplet with charges in the fundamental representation as a “probe” of the CFT medium. In either case, the pertinent object on the 4dimensional boundary of the 5-dimensional AdS space represents a fundamental charge. In Refs. 105 and 106, the HQ drag has been evaluated by computing its momentum degradation, dp/dt = −γp, through the force on the trailing string, resulting in a friction (or drag) coefficient, γAdS/CFT
√ 2 π λTSYM , = 2mQ
(58)
2 where λ = gSYM Nc denotes the ’t Hooft coupling constant. Alternatively, in Ref. 107 the problem was formulated focusing on the diffusion term. For time scales longer than the thermal relaxation time of the medium, but short compared to the HQ relaxation time, the fluctuation term in the Langevin equation (63) dominates over the drag term. The evaluation of the noise (or force) correlator is then carried out via the fluctuations of the string, resulting in a noise coefficient which is directly related to the diffusion coefficient (cf. Eqs. (65) and (74) below). Furthermore, the latter can be related to the friction coefficient using the Einstein relation, Eq. (22); it turns out that the result is identical to Eq. (58), which also verifies that a Langevin process consistent with the fluctuation-dissipation theorem applies in the SYM theory (see, however, Refs. 110 and 111, where the applicability of the Langevin framework in AdS/CFT for high-momentum quarks is discussed). The square-root dependence of γAdS/CFT on the coupling constant λ clearly characterizes its nonperturbative nature; in this sense it is parametrically large for comparatively small coupling constants. The temperature dependence is rather “conventional”, as to be expected
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since there are no additional scales in the problem (the HQ mass in the denominator implies the standard suppression of the HQ relaxation rate by ∼ T /mQ ). The next question is how to convert the result into a (semi-) quantitative estimate for the QCD plasma. Naively, one may just insert the values of the strong coupling constant, gs , and QGP temperature, T , for gSYM and TSYM , respectively. A more suitable identification probably consists of matching physical quantities which leads to somewhat different parameter values. E.g., in Ref. 108, comparable temperatures were identified by matching the energy densities (ε) of the QGP and SYM-plasma. Since the latter has a factor ∼ 3 larger particle content (degeneracy factor), one has a smaller temperature at the same ε, TSYM ' T /31/4 . For the coupling constant, one can exploit the fact that in AdS/CFT the potential between a heavy charge and anticharge is essentially of Coulomb-type, both at zero 109 and finite temperature.112,113 In the latter case, the potential goes to zero at some finite range, characteristic for Debye-screening behavior. This range can be used to identify the length scale in comparison to typical screening radii of heavy-quark free energies as computed in thermal lattice QCD (although some ambiguity remains).108 Matching the magnitude of the potentials at the screening radius then allows for a matching of the coupling constants. This leads to significantly smaller values for λ (by a factor of 3-6) than the naive identification with αs = 0.5. In connection with the redefined temperature, the improved AdS/CFT-based estimate for the HQ friction coefficient in QCD amounts to γ ' 0.3–0.9 c/fm at T = 250 MeV, which is significantly smaller than the “naive” estimate of ∼ 2 c/fm. 2.5. Comparison of elastic diffusion approaches In view of the recent proliferation of seemingly different approaches to evaluate HQ transport coefficients in the QGP it becomes mandatory to ask to what extent they are related and encode similar microscopic mechanisms.114 It turns out that all of the approaches discussed above incorporate a color Coulomb-type interaction. This is rather obvious for the T -matrix approach, where the input potentials from lattice QCD clearly exhibit the Coulomb part at sufficiently small distance (including effects of color screening). The one-gluon exchange in pQCD (which is the dominant contribution to HQ rescattering, recall the two right diagrams in Fig. 1), also recovers the Coulomb potential in the static limit (color screening enters via the Debye mass in the spacelike gluon-exchange propagator). The collisional dissociation mechanism involves the Cornell potential for the D- and B-meson wave functions and thus incorporates a Coulomb interaction as well; the emphasis in this approach is on formation-time effects essentially caused by the different (free) binding energies of D and B mesons. In addition, the confining part of the Cornell potential may play a role (as in the T -matrix approach). Finally, in conformal field theory (AdS/CFT), the absence of any scale promotes the Coulomb potential to the unique form of a potential, V (r) ∝ 1/r (this is the only way of generating a quantity with units of energy). On the other hand, scale-breaking effects are present in the
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QCD-based approaches in terms of a running coupling constant (pQCD), while the Cornell and lQCD-based potentials additionally feature linear terms ∝ σr where the string tension introduces a further (nonperturbative) scale. In fact, in Ref. 94 it has been argued, based on an analysis of lQCD results for the heavy-quark free (and internal) energy, that “remnants” of the confining force play a prominent role for temperatures not too far above Tc (e.g., for heavy quarkonium binding). If one assumes the prevalence of the Coulomb interaction, the obvious first question is with what strength (coupling constant) it figures into the different approaches, which should be fairly straightforward to determine. A more involved issue is to scrutinize the underlying approximation schemes and their applicability. E.g., perturbative approaches with large (running) coupling constants have poor (if any) control over higher-order corrections. As usual in such situations, diagrams with large contributions should be identified and resummed (which is, of course, a non-trivial task, e.g., maintaining gauge invariance); it would be illuminating to extract a static gluon-exchange (Coulomb) potential for a given set of parameters. The T -matrix approach performs a resummation of the ladder series of a static (color-electric) potential; magnetic interactions are implemented in a simplified manner using the Breit current-current interaction from electrodynamics. It has been verified that for large center-of-mass energies, the qQ T -matrix recovers the result for perturbative scattering. However, a number of effects are neglected and need to be scrutinized, including the interactions with gluons beyond pQCD, retardation, extra gluon or particle/antiparticle emission (e.g., in a coupled channel treatment) and the validity (and/or accuracy) of a potential approach at finite temperature (this issue will reappear in the context of heavy quarkonia in Sec. 4). In the collisional dissociation approach, it would be interesting to explore medium effects in the employed potential (i.e., on the mesonic wave function). Ideally, by improving on specific assumptions in a given approach, an agreement would emerge establishing a common result. Explicit connections with the AdS/CFT results are more difficult to identify. Maybe it is possible to push the T -matrix approach into a regime of “large” coupling, or study the existence and properties of (D and B) bound states in the string theory setting. In Fig. 13 we summarize the drag coefficients as function of momentum (for 3 temperatures, left panel) and temperature (for p = 0, right panel) resulting from the approaches discussed above, i.e., (i) leading-order pQCD calculations with fixed αs = 0.4 and Debye-screening mass, µD = gT , in the t-channel gluon-exchange contributions to the matrix elements for elastic gQ and qQ scattering, (ii) in-medium T -matrix calculations using lQCD-based qQ potentials, augmented by the leading-order pQCD matrix elements for elastic gQ scattering,45 (iii) pQCD calculations with running αs and reduced screening mass,66,67 and (iv) the AdS/CFT correspondence matched to QCD108 with γQCD = CT 2 /mQ for C = 1.5–2.6.115
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At all temperatures, the T -matrix approach, (ii), produces significantly more HQ interaction strength than LO pQCD, (i), while for T > 0.2 GeV the thermalization rate for the T -matrix is a factor of ∼ 2–4 less than for AdS/CFT, (iv), or for LO-pQCD with running coupling and reduced infrared regulator, (iii). Close to Tc ' 180 MeV, however, the 3 approaches (ii), (iii) and (iv) are not much different and share overlap around γ' 0.2 c/fm. The spread in the numerical results reiterates the necessity for systematic checks as indicated above. Finally, one can convert the drag coefficients into estimates of other HQ transport coefficients of the QGP. Within the Fokker–Planck approach the spatial diffusion coefficient, Ds , is directly related to the drag coefficient, γ, as given by Eq. (24). Figure 14 shows the dimensionless quantity 2πT Ds for charm (left panel) and bottom quarks (right panel) as a function of temperature for LO pQCD, LO pQCD with running coupling and reduced infrared regulator, effective resonance model and T -matrix approach. The former three are fairly constant as a function of temperature while the T -matrix approach exhibits a significant increase with temperature, indicating maximal interaction strength close to Tc . This originates from the increasing potential strength (decrease in color-screening) with decreasing temperature, enhancing resonance correlations at lower temperature. It is tempting to interpret this feature as a precursor phenomenon of hadronization. However, its robustness needs to be checked with a broader range of lattice potentials. We recall that the internal-energy based potentials probably provide an upper estimate for the strength of the interaction. It is interesting to note that for all approaches the results for b quarks coincide with the ones for c quarks within ∼ 20–30%. The largest deviation is seen in the T -matrix approach, where the (spatial) diffusion coefficient is smaller for b quarks than for c quarks (B-meson resonances survive until higher temperatures than D resonances). This is qualitatively similar to what has been
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found for the collisional dissociation mechanism, where the relative enhancement of the b-quark energy loss (compared to charm) is due to smaller B-meson formation times. Since the latter are related to larger B-meson binding energies, the dynamical origin of the smaller Ds for b quarks appears to be of similar origin as in the T -matrix approach. The relative magnitudes of the various approaches reflect what we discussed before for the drag coefficient. 2.6. Collisional versus radiative energy loss For slowly moving heavy quarks in the QGP, the parametrically dominant interaction is elastic scattering. However, at high pT , radiative scattering is believed to eventually become the prevailing energy-loss mechanism. It is currently not known at which pT this transition occurs. Therefore, it is important to assess the relative importance between elastic and inelastic scattering processes in the medium, even at the level of perturbative scattering only. Toward this purpose, we first recollect basic results on the gluon-Bremsstrahlung mechanism for light-parton, and then HQ, energy loss in the QGP, followed by a direct comparison to collisional energy loss for heavy quarks. A seminal perturbative treatment of gluo-radiative energy loss (E-loss) of highenergy partons in the QGP has been given in Refs. 116 and 117 (BDMPS). The medium is modeled as static scattering centers which implies that the E-loss is purely radiative. The key finding is that the E-loss due to multiple in-medium scattering of a high-energy parton grows as L2 , where L is the path length of the parton traversing the medium. The static scattering centers, at positions xi , are described by screened Coulomb potentials, Vi (q) =
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The range of the potentials is assumed to be small compared to the mean free path, λ, of the scattered parton, i.e., 1/µD λ. In this case successive scatterings can be considered as independent, thus enabling an eikonal approximation for the elastic scattering on static centers, i.e., a classical propagation of the particle with energy E µD , undergoing independent kicks, thereby radiating Bremsstrahlung gluons. In analogy to the QED case an important ingredient is the coherent resummation of the multiple-scattering Bremsstrahlung amplitudes (“Landau-Pomeranchuk-Migdal effect”) which can be formulated as a diffusion equation for the effective scattering amplitudes (or pertinent currents). The total radiative E-loss of a high-energy parton traversing a medium of path length L is then given by ∆E =
αs 2 qˆL , 2
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where qˆ is the diffusion coefficient for transverse-momentum broadening in scat tering off the static scattering centers, qT2 = qˆL. Perturbative calculations of the transport coefficient result in a value of about qˆ ' 1 GeV/fm2 at typical energy densities of ' 10 GeV/fm3 (translating into T ' 250 MeV) relevant for the QGP at RHIC.118 It turns out, however, that the description of high-momentum pion suppression at RHIC in the BDMPS formalism requires an approximate ten-fold increase of the perturbative value for qˆ.29 Recent calculations of perturbative Eloss including both elastic and radiative contributions within a thermal-field-theory framework indicate that collisional E-loss may be significant even for high-pT light partons.30 This would imply a reduction of the value required for qˆ from RHIC phenomenology. An early calculation119 of radiative charm-quark E-loss, −dE/dx, in the QGP has found that it dominates over the elastic one down to rather small momenta, p ≤ 2 GeV.24 In Ref. 32 it has been pointed out that the application of radiative E-loss to heavy quarks leads to the appearance of the so-called “dead cone”, i.e., the suppression of forward gluon radiation for Θ < mQ /E, where Θ denotes the direction of motion of the gluon with energy E, relative to the direction of the HQ momentum.32 It has been predicted that the reduced HQ E-loss leads to a heavyto-light hadron ratio above one in the high-pT regime accessible at RHIC. Within the BDMPS model, extended to heavy quarks, it has been argued,120 however, that medium-induced gluon radiation tends to fill the dead cone. As will be discussed in Sec. 3.5, a similar value for qˆ as in the light-hadron sector is necessary to come near the observed suppression of high-pT electrons from HQ decays in terms of radiative E-loss alone.36 The BDMPS formalism for light partons has been generalized to resum an expansion of gluo-radiative parton E-loss in the GP with opacity, n ¯ = L/λ, employˆ n is ing a so-called reaction operator approach121 (GLV). A reaction operator R th constructed that relates the n power in a opacity-inclusive radiation probability distribution to classes of diagrams of order n−1. This results in a recursion relation for the radiation probability distribution, corresponding to a certain resummation
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to all orders in opacity, which can be implemented in Monte-Carlo simulations for jet quenching. The GLV reaction-operator method for light-parton radiative E-loss in the QGP has been extended to heavy quarks in Ref. 122 (DGLV), implementing the kinematical suppression of gluon radiation by the HQ mass in the “dead cone”. A direct study of the relative magnitude of collisional (elastic) and radiative pQCD HQ E-loss in the GP has been undertaken in Ref. 38. For the elastic E-loss of a parton with color Casimir constant, CR , the leading logarithm expression in an ideal QGP with Nf effective quark flavors at temperature T , dE el Nf = CR πα2s T 2 1 + f (v) ln(Bc ), (61) dx 6 has been used. In an ultrarelativistic gas of massless partons the jet-velocity function is given by 1 1 2 1+v f (v) = 2 v + (v − 1) ln , (62) v 2 1−v while estimates for Bc are taken from Refs. 123, 124, 68 and 69. The different values for Bc obtained in these models are considered as reflecting theoretical uncertainties. The radiative E-loss within the DGLV reaction-operator approach is calculated in Ref. 122 based on Refs. 121 and 125. The left panel of Fig. 15 compares pQCD radiative and collisional E-loss for various quark flavors (masses) at high p T > 5 GeV
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in a gluon plasma (GP) with T ' 240 MeV. For light and charm (bottom) quarks the elastic E-loss is comparable to the radiative one up to pT ' 10(20) GeV, and still significant above. The right panel of Fig. 15 reiterates that, within pQCD, collisional E-loss is an essential component in calculating the suppression of lightparton and especially HQ spectra at RHIC. Note that the relative importance of collisional E-loss is expected to increase if non-perturbative effects become relevant (which predominantly figure toward lower pT ), or if the GP is replaced by a QGP. 2.7. D mesons in the hadronic phase To complete the discussion of open charm in QCD matter we briefly address medium modifications of charm hadrons in hadronic matter. Pertinent studies may be divided into calculations for cold nuclear matter as well as for hot meson matter. Early studies of D-mesons in cold nuclear matter focused on possible mass shifts due to scalar and vector mean fields acting on the light-quark content of the meson.126 At normal nuclear matter density %N ≡ % = 0.16 fm−3 , attractive mass shifts of up to − 100 MeV have been reported for D + and D0 mesons (where both mean fields contribute with the same sign) while the mass change of the D − and ¯ 0 turned out to be small due to a cancellation of the mean fields. Similar findings D have been reported in QCD sum rule calculations127 where the (isospin-averaged) D-meson mass is reduced by about − 50 MeV, mostly as a consequence of the reduction in the light-quark condensate. Rather different results are obtained in microscopic calculations of D-meson selfenergies (or spectral functions) based on coupled channel T -matrices for DN scattering in nuclear matter.128,129 These calculations incorporate hadronic many-body effects, most notably DN excitations into charm-baryon resonances not too far from the DN threshold, e.g., Λc (2593) and Σc (2625), as well as charm exchange into πΛc and πΣc channels. In Ref. 128 separable meson-baryon interactions have been employed with parameters constrained to dynamically generate the Λc (2593) state. Since the in-medium D-meson spectral function figures back into the T -matrix, one is facing a selfconsistency problem (much like for the heavy-light quark T -matrix discussed in Sec. 2.3.2). Selfconsistent calculations including nucleon Pauli blocking and dressing of intermediate pion and nucleon propagators result in D-meson spectral functions with a significant broadening of up to ΓD ' 100 MeV but a rather small shift of the peak position of about − 10 MeV (for %N ≡ % = 0.16 fm−3 ). In Ref. 129, a somewhat stronger coupling of DN to the Λc (2593) results in a stronger collective DN −1 Λc (2593) mode (about 250 MeV below the free D-meson mass) and a pertinent level repulsion which pushes up the “elementary” D-peak by ∼ 30 MeV. Also in this calculation the broadening is significant, by about ∼ 50 MeV. The D − was found to be rather little affected, neither in mass nor in width. Investigations in the selfconsistent coupled-channel framework have been extended to a nucleon gas at finite temperature130 with a more complete treatment of DN scattering, cf. left and middle panels of Fig. 16. The thermal motion of nucleons implies that a larger kinematic regime in the center-
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of-mass (cm) energies in the scattering amplitude is probed (compared to T = 0). For the real parts this leads to a further averaging of the positive and negative parts of the amplitude, while the imaginary parts are negative definite (some loss of interaction strength may occur in channels with resonances close to threshold). More quantitatively, at normal nuclear matter density, the resulting mass shifts are ¯ decreasing to about 10–20 MeV at T = 0 (attractive for D and repulsive for D), half (or less) at T = 150 MeV. On the other hand, the D-meson width is around ¯ width is small at T = 0 100 MeV at both zero and finite temperature, while the D but increases to about 30 MeV at T = 150 MeV. Medium modifications of D-mesons in a hot pion gas have been studied in Ref. 131. The main idea in this work is to implement the recently discovered scalar D0∗ (2310) and axialvector D10 (2430) states as chiral partners of the pseudoscalar D and vector D∗ (2010) mesons, respectively. Their large widths of 200–400 MeV are primarily attributed to S-wave pion decays into D and D ∗ . In a thermal pion gas, D0∗ (2310) and D10 (2430) therefore act as strong resonances in Dπ and D ∗ π scattering, which have been treated in Breit-Wigner approximation. In addition, Dwave resonances, D1 (2420) and D2∗ (2460), have been accounted for. The resulting collisional widths of D and D ∗ reach up to about ∼ 40–60 MeV for temperatures around T = 175 MeV, while the mass shifts are attractive up to − 20 MeV. It can be expected that (e.g., in a selfconsistent calculation) the inclusion of medium
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effects on the resonances (e.g., chiral partners are expected to approach degeneracy towards chiral restoration) will lead to a reduction in the mass shift (not so much for the widths). Above the critical temperature, it is then natural to switch to a quark-based description, i.e., c + q scattering, much like in the effective resonance model discussed in Sec. 2.3.1. When combining the effects of pion and nucleon scattering on D-mesons in hot hadronic matter, their total width at temperatures around Tc adds up to Γtot D (T = 180 MeV) ' 150 MeV. This is only by about a factor of ∼ 2 smaller than what was found for c-quarks at T = 1.1 Tc in the T -matrix approach for c–q scattering, cf. right panel of Fig. 11. Since it can be expected that other excited hadrons contribute to D-meson rescattering (albeit with less strength), D-meson transport properties may not be much different from those of c-quarks in the QGP, at least at temperatures close to Tc . It is therefore of considerable interest to employ in-medium D-meson T -matrices to evaluate heavy-flavor transport coefficients in hadronic matter. 3. Heavy-Quark Observables in Relativistic Heavy-Ion Collisions One of the main motivations for the vigorous theoretical studies of HQ diffusion in the QGP is the possibility of utilizing HQ observables in ultrarelativistic heavy-ion collisions as a quantitative probe of the matter produced in these reactions. If the latter reaches approximate local thermal equilibrium, such applications can be performed by solving the Fokker–Planck equation for a heavy quark diffusing within a collectively expanding background medium with space-time dependent temperature and flow field (applicable for “sufficiently slow” charm and bottom quarks). This is typically realized with a Monte-Carlo simulation using a test-particle ansatz for an equivalent stochastic Langevin equation. In such a formulation, a direct relation between the input in terms of a (temperature-dependent) HQ diffusion coefficient and the modifications of HQ spectra in the evolution can be established. In addition, the Langevin formulation admits an efficient implementation of the dissipationfluctuation relation for relativistic kinematics. Alternatively, the modifications of HQ spectra in URHICs have been evaluated by implementing test particles into numerical transport simulations of the background medium.132–134 This, in principle, accounts for non-equilibrium effects in the medium evolution (which could be particularly relevant for high-pt particles in the bulk), but the connection to the diffusion concept becomes less direct (HQ cross sections need to be evaluated in an equilibrium medium to extract “equivalent” diffusion coefficients). However, when analyzing theoretical predictions of HQ spectra we also compare to results of transport models for the bulk evolution. We start our presentation in this Section by briefly outlining how the Fokker– Planck equation is implemented into numerical simulations based on a relativistic Langevin process (Sec. 3.1). This is followed by a discussion of different models
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for the background medium in relativistic heavy-ion collisions, where we focus on thermal models including hydrodynamics and expanding fireballs (Sec. 3.2). The main task here is to provide realistic benchmarks for the conversion of transport coefficients into modifications of HQ spectra. This furthermore requires the definition of controlled initial conditions for the HQ spectra (Sec. 3.3), usually taken from p–p collisions, possibly augmented by nuclear effects (in particular a “Cronin” p t broadening). Available Langevin simulations combining different inputs are quantitatively compared at the level of the final-state HQ spectra resulting from the QGP and “mixed” phases in central and semicentral Au–Au collisions at RHIC (Sec. 3.4). Even though HQ spectra are not observable, they provide the cleanest theoretical level of comparison, before further processing through hadronization, electron decay and charm/bottom composition occurs. The latter three steps are necessary to enable comparisons to currently available electron data (Sec. 3.5), and thus arrive at an empirical estimate of the HQ diffusion coefficient characterizing the QCD medium produced at RHIC. In a more speculative step, the extracted HQ transport coefficient may be used to schematically estimate the ratio of shear viscosity to entropy density (Sec. 3.6), which has recently received considerable attention in connection with viscous hydrodynamic simulations at RHIC. 3.1. Relativistic Langevin simulations The Fokker–Planck equation, introduced in Sec. 2.1, is equivalent to an ordinary stochastic differential equation. Neglecting mean-field effects of the medium, the force acting on the heavy particle is divided into a “deterministic” part, describing its average interactions with the light particles in the medium (friction or drag), and a “stochastic” part, taking into account fluctuations around the average on the level of the standard deviation. Thus the relativistic equations of motion for a heavy quark become a coupled set of stochastic differential equations, which for an isotropic medium can be written in the form pj dxj = dt , E (63) √ dpj = −Γpj dt + dtCjk ρk , where E = (m2Q + p2 )1/2 , and Γ and Cjk are functions of (t, x, p) with j, k = 1, 2, 3; they are related to the transport coefficients A and B (discussed in the previous section) below. Γ and Cjk describe the deterministic friction (drag) force and the stochastic force in terms of independent Gaussian-normal distributed random variables ρ = (ρ1 , ρ2 , ρ3 ), 3 ρ2 1 exp − , (64) P (ρ) = 2π 2 In the limit dt → 0, the covariance of the fluctuating force is thus given by D E (fl) (fl) Fj (t)Fk (t0 ) = Cjl Ckl δ(t − t0 ) .
(65)
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However, with these specifications the stochastic process is not yet uniquely defined, but depends on the specific choice of the momentum argument of the covariance matrix, Cjk , in Eq. (63),135 i.e., the definition of the stochastic integral. Usual schemes are given by the pre-point Ito, the mid-point Stratonovic-Fisk, and the post-point Ito (or H¨ anggi-Klimontovich136) interpretation of the stochastic integral. We can summarize all these realizations of the stochastic process by specifying the actual momentum argument in the covariance matrix by Cjk → Cjk (t, x, p + ξdp) ,
(66)
where ξ = 0, 1/2, 1 corresponds to the pre-point Ito, the mid-point Stratonovic, and the post-point Ito realizations, respectively. The equation for the corresponding phase-space distribution function can be found by calculating the average change of an arbitrary phase-space function, g(x, p), with time. According to Eq. (63), with the specification Eq. (66) of the stochastic process, we find ∂g ∂Cjk ∂g pj + −Γpj + ξ Clk hg(x + dx, p + dp) − g(x, p)i = ∂xj E ∂pj ∂pl 2 1 ∂ g + Cjl Ckl dt + O(dt3/2 ) . (67) 2 ∂pj ∂pk Here, the arguments of both, Γ and Cjk , have to be taken at (t, x, p) since the corrections are of the neglected order, O(dt3/2 ). In the derivation of this equation the statistical properties of the random variables ρi , implied by Eq. (64), hρj i = 0 ,
hρj ρk i = δjk ,
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have been used. It follows that the average of an arbitrary phase-space function is by definition given by the phase-space distribution function for the heavy particle (in our context a heavy quark), fQ (t, x, p), e.g., Z Z ∂ d hg(x, p)i = d3 x d3 p g(x, p) fQ (t, x, p) . (69) dt ∂t After integrations by parts, and since Eq. (67) holds for any function g, one finally arrives at the Fokker–Planck equation, ∂fQ pj ∂fQ ∂ ∂Cjk 1 ∂2 + = Γpj − ξClk fQ + (Cjl Ckl fQ ) . (70) ∂t E ∂xj ∂pj ∂pl 2 ∂pj ∂pk The drag term, i.e., the first term on the right-hand side of this equation, depends on the definition of the stochastic integral in terms of the parameter ξ. Comparison with Eq. (10) shows that, independent of the choice of ξ, the covariance matrix is related to the diffusion matrix by p p k ⊥ + 2B1 Pjk , Cjk = 2B0 Pjk (71)
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while the friction force is given by Γpj = Apj − ξClk
∂Cjk . ∂pl
(72)
Numerical investigations have shown that the drag and diffusion coefficients inferred from microscopic models according to Eqs. (15) in general do not warrant a good agreement of the long-time limit of the solution to the Fokker–Planck evolution with the relativistic equilibrium J¨ uttner-Boltzmann distribution (where the temperature is given by the background medium). The problem is with the longitudinal diffusion coefficient, B1 , which induces an overestimate of the corresponding fluctuating forces. Thus, one typically adjusts the drag coefficient by choosing B 1 in Eq. (70) to satisfy the asymptotic equilibration condition.26,137,138 To find the dissipation-fluctuation relation, imposed by the equilibration condition, we first study the heavy quark’s motion in a heat bath in thermal equilibrium in its rest frame. Then the momentum distribution of the heavy quarks should become a J¨ uttner-Boltzmann distribution, eq fQ (p) ∝ exp (−E/T )
(73)
with the temperature, T , imposed by the heat bath. For a Langevin process with B0 = B1 = D, i.e., p (74) Cjk = 2D(E)δjk ,
where the diffusion coefficient has been written as a function of the heavy quark’s energy, E, the equilibration condition for a given parameter ξ in Eq. (66) is obtained by using Eqs. (73) and Eq. (74) in Eq. (70): A(E)ET − D(E) + T (1 − ξ)D 0 (E) = 0 .
(75)
Since the drag and diffusion coefficients are usually given numerically, the most convenient update rule for the Langevin process is achieved by setting ξ = 1, i.e., using the post-point Ito (H¨ anggi-Klimontovich) rule for the stochastic integral in Eq. (63) and imposing the simple relativistic dissipation-fluctuation relation, D = AET .
(76)
This guarantees the proper approach of the heavy quark’s phase-space distribution to the appropriate equilibrium distribution with the temperature imposed by the heat bath. For the more general form of the covariance matrix, Eq. (71), the post-point Ito value, ξ = 1, has been chosen in Ref. 28, and the longitudinal diffusion coefficient is set to B1 = AET ,
(77)
Heavy Quarks in the Quark-Gluon Plasma
153
while the drag coefficient A as well as the transverse diffusion coefficient, B0 , are used as given by Eq. (15) for the various microscopic models for HQ scattering in the QGP. Comparing to Eq. (75), one finds that this is equivalent to the strategy followed in Ref. 26 of using the prepoint-Ito rule, ξ = 0, but to adjust the drag coefficient according to the dissipation-fluctuation relation Eq. (75). 3.2. Background medium in heavy-ion collisions For HQ transport coefficients computed in an equilibrium QGP, the natural and consistent framework to describe the evolving medium in heavy-ion collisions are hydrodynamic simulations, formulated in the same (thermodynamic) variables. This choice is further rendered attractive by the success of ideal hydrodynamics in describing bulk observables at RHIC, in particular pT spectra and elliptic flow of the most abundant species of hadrons.13–16 The agreement with meson and baryon spectra typically extends to pT ' 2–3 GeV, respectively. At the parton level, this converts into a momentum of pt ' 1 GeV, which approximately coincides with the “leveling-off” of the experimentally observed v2 (pt ) (at higher momenta hydrodynamics overestimates the elliptic flow). On the one hand, this appears as a rather small momentum in view of the ambition of describing HQ spectra out to, say, pt ' 5 GeV. However, one should realize that (a) more than 90% of the bulk matter is comprised of light partons with momenta below pt ' 1 GeV, and (b) the velocity of a pt = 5 GeV charm quark (with mc = 1.5 GeV) is very similar to a pt = 1 GeV light quark (with mq = 0.3 GeV). This suggests that most of the interactions of a pt = 5 GeV charm quark actually occur with soft light partons (which are well described by a hydrodynamic bulk). This has been verified by explicit calculations139 and is, after all, a prerequisite for the applicability of the Fokker–Planck approach (i.e., small momentum transfer per collision). On the other hand, one may be concerned that the overestimate of the experimentally observed elliptic flow at intermediate and high pt within hydrodynamics may exaggerate the HQ elliptic flow in Langevin simulations. This is, however, not necessarily the case, since the transfer of v2 from the bulk to the heavy quark critically depends on the light-parton phase space density (the drag coefficient is proportional to it); since the hydrodynamic spectra fall significantly below the experimental ones at higher pT , the phase space density of the hydrodynamic component is relatively small. It is therefore not clear whether the (small) fraction of thermalized particles at high pt implies an overestimate of the total v2 ; this may be judged more quantitatively by comparing to transport calculations. In parallel to hydrodynamic descriptions of the bulk medium, expanding fireball models have been employed. These are simplified (and thus convenient) parameterizations of a full hydrodynamic calculation in terms of an expanding volume and spatial flow-velocity field, but otherwise based on similar principles and variables. E.g., entropy conservation is used to convert a given volume into a temperature via an underlying equation of state (EoS). The reliability of a fireball model
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Table 1. Survey of parameters figuring into hydrodynamic and fireball evolutions employed in the Langevin simulations of HQ spectra for semicentral Au–Au collisions at RHIC, corresponding to MT,26,140 HGR,28,45,142 AHH115,141 and GA.13,67
MT
HGR
AHH
GA
τ0 [fm/c ]
1.0
0.33
0.6
0.6
T0 [MeV]
265 (max)
340 (avg)
250 (avg)
330 (max) 260 (avg)
Tc [MeV]
165
180
170
165
initial spatial
wounded nucleon
isotropic
lin. comb. of Ncoll and Npart
lin. comb. of Ncoll and Npart
b [fm]
6.5
7
5.5
7
bulk-v2
5
5.5%
3%
4.75%
τFB [fm/c ]
∼9
∼5
∼9
∼7
QGP-EoS
massless (Nf = 3)
massless (Nf = 2.5)
massless (Nf = 3)
massless (Nf = 3)
HQ Int.
pQCD HTL
pQCD+reso pQCD+T-mat
AdS/CFT
pQCD run. αs
mc,b [GeV]
1.4
1.5, 4.5
1.5, 4.8
1.5, 5.1
crucially hinges on a realistic choice of the parameters, in connection with a proper description of the final state in terms of particle production and collective flow. In principle, a fireball model offers some additional flexibility in varying the evolution, which may provide useful checks of the sensitivity to specific aspects of the expansion. Several key parameters of thermal medium evolution models employed in HQ Langevin simulations are compiled in Table 1. The starting point of both hydro and fireball models are the initial conditions of the thermal medium, characterized by a formation time when the medium is first assumed to be (locally) equilibrated. At RHIC, typical formation times are estimated to be in the range of τ0 ' 0.3–1 fm/c. With a total entropy fixed to reproduce the measured rapidity density of hadrons at a given centrality, e.g., at impact parameter b ' 7 fm/c, these formation times translate into average initial temperatures of T0 ' 250–350 MeV. If the entropy density scales as s ∝ T 3 , one can roughly compare the initial conditions in different approaches using S = s0 V0 and V0 ∝ τ0 . E.g., an initial T0max = 265 MeV based on τ0 = 1 fm/c increases by a factor of 31/3 upon decreasing τ0 = 0.33 fm/c, resulting in T0max ' 382 MeV; if the number of light flavors in the EoS is reduced, T0 increases as well; e.g., T0avg = 260 MeV based on τ0 = 0.6 fm/c and Nf = 3 (as in Ref. 67) increases to T0avg = 260 MeV (0.6/0.33)1/3 (47.5/42.25)1/3 ' 330 MeV for τ0 = 0.33 fm/c and Nf = 2.5 (reasonably consistent with Ref. 28, cf. Table 1). The QGP-dominated evolution lasts for about 2–4 fm/c, followed by a mixed phase
Heavy Quarks in the Quark-Gluon Plasma
155
of similar duration at a critical temperature Tc ' 165–180 MeV. The effects of a continuous (cross-over) transition, as well as of the hadronic phase, have received little attention thus far, but are not expected to leave a large imprint on HQ observables. After all, the cross-over transition found in lQCD exhibits a marked change in energy density over a rather narrow temperature interval. A more important aspect is the consistency between the EoS used to extract the temperature of the bulk evolution and the corresponding degrees of freedom figuring into the calculation of the HQ transport coefficients. In hydrodynamical backgrounds used thus far26,67,115 the evolution is described with a 2+1-dimensional boost-invariant simulation with an ideal massless-gas EoS. The initial state is typically determined by distributing the entropy in the transverse plane according to the wounded nucleon density. Unfortunately, the impact parameters in current Langevin simulations vary somewhat, which is particularly critical for the magnitude of the subsequently developed elliptic flow. The value of the critical temperature has some influence on the QGP lifetime (lower temperatures leading to larger QGP duration), as does the hadron-gas EoS (more hadronic states imply a larger entropy density at T c and thus a reduced duration of the mixed phase). The termination point of the evolution (beginning, middle or end of mixed phase) is rather significant, especially for the HQ v2 which needs about 5 fm/c to build up most of its magnitude. In Refs. 28 and 45 the medium is parameterized, guided by the detailed hydrodynamic calculations of Ref. 142, as a homogeneous thermal elliptic “fire cylinder” of volume V (t). The QGP temperature is determined via the QGP entropy density, s, under the assumption of isentropic expansion (total S = const), s=
4π 2 3 S = T (16 + 10.5Nf ) . V (t) 90
(78)
The thus inferred temperature is used in Eq. (15) to compute the friction coefficients, A, and transverse diffusion coefficient, B0 , with the longitudinal diffusion coefficient fixed by the dissipation-fluctuation relation, Eq. (77). In the mixed phase at Tc = 180 MeV the QGP drag and diffusion coefficients are scaled by a factor ∝ %2/3 to account for the reduction in parton densities (rather than using hadronic calculations). Special care has to be taken in the parameterization of the elliptic flow in noncentral Au–Au collisions: the contours of constant flow velocity are taken as confocal ellipses in the transverse plane with the pertinent transverse flow set consistently in perpendicular direction. The time evolution of the surface velocity of the semi-axes of the elliptic fire cylinder parameterizes the corresponding results of the hydrodynamic calculations in Ref. 142, in particular the time-dependence of the elliptic-flow parameter, v2 , for the light quarks. The parameters are adjusted such (s) that the average surface velocity reaches v⊥ = 0.5c and the anisotropy parameter v2 = 5.5% at the end of the mixed phase. Finally, the velocity field is specified by scaling the boundary velocity linearly with distance from the center of the fireball, again in accordance with the hydrodynamic calculation.142
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-2
-3
10
1/(2πpT) dN/dpT [a u ]
STAR D * STAR prelim D (×2 5) c-quark (mod PYTHIA) c-quark (CompHEP)
-4
10
2
1/(2 πpT) d N/dpT dy [a u ]
10
0
10
-5
10
-6
10
d+Au √sNN=200 GeV
10
10 10
STAR (prel, pp) STAR (prel, d+Au/7 5) STAR (pp) D→e B→e sum
±
e
-5
-6
10 10 10 10
-7 -8 -9
-10
σbb/σcc= 4.9 x10
-3
-11
-7
10 0
-3
-4
1
2
3
4 5 pT[GeV]
6
7
8
10 0
1
2
3
4 pT [GeV]
5
6
7
8
Fig. 17. (Color online) Fits of D- and D ∗ -meson pT spectra in 200AGeV d-Au collisions at RHIC with a modified PYTHIA simulation (left panel) and the corresponding non-photonic singleelectron pt spectra in p–p and d-Au collisions.144 The missing yield of high-pT electrons is fitted with the analogous B-meson decay spectra, thus fixing the bottom-charm ratio at σ b¯b /σc¯c ' 4.9 · 10−3 .
3.3. Initial conditions and hadronization The Langevin simulations of HQ diffusion in the QGP require initial conditions for charm- and bottom-quark phase-space distributions. For the spatial part of the initial distribution in the transverse plane all calculations adopt binary-collision scaling following from a Glauber model, reflecting a hard process for the primordial production mechanism. Furthermore, all calculations thus far focus on a limited rapidity window around midrapidity, where the longitudinal distribution is assumed to be uniform in space-time rapidity. As for the initial HQ pt spectra, Ref. 26 employs a fit to a leading-order parton-model calculation from the CompHEP package, 143 dN 1 ∝ δ(η − y) 2 , 2 dydηd pt (pt + Λ2 )α
(79)
with α = 3.5 and Λ = 1.849 GeV. In Refs. 28 and 45, PYTHIA results for cquark spectra have been tuned to reproduce available D-meson spectra in d-Au collisions at RHIC (assuming δ-function fragmentation, cf. left panel of Fig. 17). The pertinent semileptonic single-electron decay spectra approximately account for p–p and d-Au spectra up to pT = 4 GeV; the missing part at higher pT is then supplemented by B-meson contributions. This procedure results in a crossing of the D- and B-meson decay electrons at pT ' 5 GeV and a cross-section ratio of σb¯b /σc¯c ' 4.9 · 10−3 (see right panel of Fig. 17), which is within the range of pQCD predictions.145 With initial conditions and bulk medium evolution in place, one can evolve HQ phase-space distributions through the QGP (and mixed phase) of a heavy-ion collision. The final HQ spectra, however, require further processing before comparisons to observables can be made. First, one has to address the hadronization of the HQ spectra into charm and bottom hadrons (D, D ∗ , Λc etc.). Two basic mechanisms
Heavy Quarks in the Quark-Gluon Plasma
157
have been widely considered in hadronic collisions, i.e., fragmentation of an individual quark and recombination with an extra quark from the environment. In general, the former is mostly applicable for high-energy partons while the latter requires a sufficient overlap of the mesonic wave function with the phase-space density of surrounding quarks and is therefore more relevant toward lower momentum. The fragmentation of a quark is implemented by applying the factorization theorem of QCD.21 At large transverse momenta, the production process of a parton occurs on a short time scale, τprod ' 1/pt , while hadronization occurs at the considerably larger time scale τhad ' 1/ΛQCD . Thus the production cross section for a hadron can be factorized into an elementary parton-production cross section (hard process) and a phenomenological universal transition probability distribution, Dh/i (z), for a parton i of momentum pi to convert into a hadron with momentum fraction z = ph /pi ≤ 1. For light quarks and gluons the fragmentation functions, Dh/i , are rather broad distributions around z ' 0.5, but for heavy quarks they become rather sharply peaked toward z = 1 and are sometimes even approximated by a δ-function, D(z) = δ(1 − z). The mechansim of recombination of a produced quark with other quarks or antiquarks in its environment (e.g., the valence quarks of the colliding hadrons) has first been introduced in the late 1970’s to explain flavor asymmetries in π and K meson production in hadronic collisions at forward rapidities.146 In particular, the recombination idea has been rather successful in describing flavor asymmetries in the charm sector,147,148 even close to midrapidity. In the context of heavy-ion collisions, quark coalescence models, applied at the hadronization transition, provide a simple and intuitive explanation for the observed constituent-quark number scaling (CQNS) of the elliptic flow of light hadrons149,150 and the (unex0 ¯ pectedly) large baryon-to-meson ratios (e.g., p/π ' 1 or (Λ + Λ)/(4K S ) ' 1.3 in central 200 AGeV Au–Au collisions at RHIC) at intermediate transverse momenta (2 GeV . pT . 5 GeV).151–154 CQNS refers to a scaling property of the hadronic elliptic flow, v2,h (pT ), in terms of a universal function v2,q (pT /n) = v2,h (pT )/n, where n denotes the number of constituent quarks in a given hadron, h. CQNS naturally emerges from the recombination of approximately comoving quarks and antiquarks in a collectively flowing medium. Thus, within this picture, v2,q (pt ) is interpreted as a universal elliptic flow of the quarks with transverse momentum pt at the moment of hadronization (typically assumed to be the quark-hadron transition at Tc ). It can be expected that the phenomenologically very successful coalescence concept also applies in the HQ sector of heavy-ion collisions.155,156 Note that, unlike quark fragmentation, quark recombination adds momentum and elliptic flow to the produced hadron (through the quark picked up from the environment). At this point it might be instructive to reiterate a conceptual connection between the quark coalescence model and the idea of resonance correlations in the QGP. The latter were found to be an efficient mechanism for arriving at a small HQ diffusion constant, both within the effective resonance model (Sec. 2.3.1) and within the T -matrix approach (Sec. 2.3.2). Especially in the T -matrix approach, the
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resonance correlations were found to strengthen toward the expected hadronization transition, and thus provide a natural emergence of heavy-light quark coalescence at Tc . These ideas have recently been implemented in a resonance-based description of the coalescence process in kinetic theory based on a Boltzmann equation. 157 This approach improves instantaneous coalescence formulations in that it respects energy conservation and establishes a well-defined equilibrium limit in the coalescence process (i.e., the thermal distribution for the formed meson). Subsequently, resonance-recombination has been combined with “realistic” quark phase-space distributions, as generated in relativistic Langevin simulations.158 In particular, it was found that CQNS could be recovered under the inclusion of space-momentum correlations in the quark phase-space distributions. In Refs. 28 and 45 HQ spectra at RHIC have been hadronized in a combined coalescence plus fragmentation scheme. For the hadronization of, e.g., charm quarks into D mesons one obtains the D-meson spectra as coal tot dND dNcfrag dND = + . dy d2 pT dy d2 pT dy d2 pT
(80)
For the first term on the right-hand side, the quark-coalescence model of Ref. 155 has been employed, where the pT spectrum of a D meson follows from a convolution of light anti-quark and charm-quark phase-space distributions, fq¯,c , as coal dND = gD dyd2 pT
Z
p · dσ (2π)3
Z
d3 qfD (q, x)fq¯(pq¯, rq¯)fc (pc , rc ) .
(81)
Here, p = pq¯ + pc is the D-meson momentum and gD a combinatorial factor accounting for color-neutrality and spin averaging. The D-meson Wigner function, fD (q, x), is assumed as a double Gaussian in relative momentum pc − pq¯ and size, r c − rq¯, and dσ is the hyper-surface element 4-vector of the hadronization volume. The charm-quark distribution corresponds to the Langevin output at the end of the mixed phase of the fireball model, while the light-quark distributions are taken from previous applications of the coalescence model to light-hadron observables at RHIC.152 This represents a parameter-free conversion of HQ distributions into heavy-meson spectra (note that the final state of the expanding fireball model28 has been matched to the parameterization of collective velocity and elliptic flow for the light-quark distributions in the coalescence model152 ). The coalescence mechanism does not exhaust all heavy quarks in the hadronization process, especially toward higher pt (where the light-quark phase-space density becomes small). Therefore, the remaining heavy quarks are hadronized using fragmentation, which for simplicity is treated in δ-function approximation (as has been done in connection with the initial conditions). The formation of baryons containing heavy quarks (e.g., Λc ) has been neglected since it has been found to give only small contributions, i.e., Λc /D 1. Quantitative refinements should, however, include these processes, see, e.g., Refs. 159 and 160.
Heavy Quarks in the Quark-Gluon Plasma
159
Finally, the comparison to electron spectra requires to compute semileptonic decays of heavy-flavor hadrons. Thus far, these have been approximated in threebody kinematics, e.g., D → eνK. An important finding in this context is that the resulting electron v2 (pT ) traces the one of the parent meson rather accurately,155,161 implying that electron spectra essentially carry the full information on the heavymeson v2 . In the pT spectra, the decay electrons appear at roughly half of the momentum of the parent meson. It has also been pointed out159,160 that Λc baryons have a significantly smaller branching fraction into electrons (about 4–5%) than D mesons (7% and 17% for neutral and charged D’s, respectively). Thus, in case of a large Λc /D enhancement, a net electron “loss” could mimic a stronger suppression than actually present at the HQ level. In fact, even variations in the neutral to charged chemistry from p–p to A–A collisions148 could be quantitatively relevant. 3.4. Model comparisons of heavy-quark spectra at RHIC We are now in position to conduct quantitative comparisons of diffusion calculations using transport simulations for HQ spectra in 200 AGeV Au–Au collisions at RHIC. We focus on Langevin simulations but also allude to Boltzmann transport models. The modifications of the initial spectra are routinely quantified in terms of the nuclear modification factor, RAA , and elliptic-flow parameter, v2 , defined by RAA (pt ; b) =
v2 (pt ; b) =
AA dNQ (b)/dpt , pp Ncoll (b) dNQ /dpt
R
dN AA (b)
dφ dptQdydφ cos(2φ) , R dN AA (b) dφ dptQdydφ
(82)
AA respectively; dNQ (b)/dpt denotes the HQ pt spectrum in an A–A collision at impp pact parameter, b, which is scaled by the spectrum dNQ /dpt from p–p collisions times the number of binary nucleon-nucleon collisions, Ncoll (b) (to account for the same number of heavy quarks). Thus, any deviation of RAA from one indicates nuclear effects (from the QGP but possibly also in the nuclear initial conditions or from the pre-equilibrium stages). The elliptic-flow parameter, v2 (pt ), is the second Fourier coefficient in the expansion of the (final) momentum distributions in the azimuthal angle, φ, relative to the reaction plane (x–z plane) of the nuclear collision. At midrapidity, where the “directed” flow (v1 ) is expected to vanish, the v2 coefficient is the leading source of azimuthal asymmetries. A non-zero v2 is only expected to occur in noncentral A–A collisions due to an “almond”-shaped nuclear overlap zone (with a long (short) axis in y (x) direction). Typical sources for a nonzero elliptic flow are a path-length difference for absorption of particles traversing the reaction zone or an asymmetry in the collective (hydrodynamic) flow due to stronger pressure gradients across the short axis. Both effects convert the initial spatial anisotropy, v2 , in a positive momentum anisotropy in the particle pt -spectra.
R. Rapp & H. van Hees 1.4 1.2 1
D (2πT) = 1.5 D (2πT) = 3 D (2πT) = 6 D (2πT) = 12 D (2πT) = 24
(a)
v2 (pT)
RAA
160
0.22 0.2 0.18
LO QCD
(b)
0.16 0.14
0.8
0.12 0.1
0.6
0.08
0.4
0.06 0.04
0.2
0.02
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 pt (GeV)
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 pt (GeV)
Fig. 18. (Color online) Nuclear modification factor (left panel) and elliptic flow (right panel) of charm quarks as a function of transverse momentum in semicentral (b = 6.5 fm) Au–Au collisions using a hydrodynamic evolution of the bulk medium at RHIC.26 The calculations are performed for HTL-improved LO-pQCD scattering with variable strong coupling and fixed Debye-screening mass in t-channel gluon-exchange scatteringi (µD = 1.5 T ). The relation of the spatial diffusion coefficient, Ds (denoted D in the figure legend), to the strong coupling constant, αs , is given by the approximate relation 2πT Ds ≈ 6(0.5/αs )2 .
While the former mechanism is usually associated with high-pt particles (typical leading to a v2 < 5%), the latter is driven by collective expansion due to thermal pressure mostly applicable to low-pt particles (with significantly larger v2 values, in excess of 5%). Since in the Langevin simulations heavy quarks are assumed to be exclusively produced in primordial N –N collisions (i.e., their number is conserved subsequently), the HQ RAA can be simply calculated as the ratio of the HQ pt distribution function at the moment of hadronization to the initial distribution (taken from p–p collisions), fQ (thad , pt ) , fQ (t0 , pt ) R dφfQ (thad , pt , φ) cos(2φ) , v2 (pt ) = fQ (thad , pt ) RAA =
(83)
while the v2 is computed using its definition given above. The next five figures (18-22) encompass calculations of RAA and v2 in semicentral Au–Au collisions at RHIC for the following approaches: (i) Figure 18 [MT]26 displays Langevin simulations for c quarks (with the prepoint Ito realization of the stochastic integral) using a hydrodynamic evolution for b = 6.5 fm; the HQ drag and diffusion coefficients are based on LO hardthermal loop scattering matrix elements with variable αs but fixed Debye screening mass.
Heavy Quarks in the Quark-Gluon Plasma
15 v2 [%]
RAA
20
c, reso (Γ=0 4-0 75 GeV) c, pQCD, αs=0 4 b, reso (Γ=0 4-0 75 GeV)
1.5
161
1
10
0.5
c, reso (Γ=0.4-0.75 GeV) c, pQCD, αs=0.4 b, reso (Γ=0.4-0.75 GeV) Au-Au √s=200 GeV (b=7 fm)
5 Au-Au √s=200 GeV (b=7 fm)
0 0
1
2 3 pt [GeV]
4
5
0 0
1
2 3 pt [GeV]
4
5
Fig. 19. (Color online) The HQ RAA (left panel) and v2 (right panel) for semicentral (b = 7 fm) Au–Au collisions at RHIC within the effective resonance + pQCD model compared to results from LO √ pQCD elastic scattering only with αs = 0.4 and corresponding Debye-screening mass µD = 4παs T .
(ii) Figure 19 [HGR]28 displays Langevin simulations for c and b quarks (with the post-point Ito (H¨ anggi-Klimontovich) realization) using a thermal fireball expansion for b = 7 fm; the HQ drag and diffusion coefficients are based on the effective resonance+pQCD model25 for variable resonance width (coupling strength) and αs = 0.4 in the pQCD part. (iii) Figure 20 [AHH]115 displays Langevin simulations for c quarks (with the prepoint Ito realization) using a hydrodynamic expansion for b = 5.5 fm; the HQ drag and diffusion coefficients are based on the strong-coupling limit with AdS/CFT correspondence with a variable coupling strength estimated from matching to QCD.108 (iv) Figure 21 [HMGR]45 displays Langevin simulations as under (ii) but with HQ transport coefficients based on the T -matrix+pQCD approach for two lQCDbased input potentials. (v) Figure 22 [Mol]134 displays Boltzmann transport simulations using a covariant transport model for b = 8 fm; the HQ interactions are modeled by schematic LO pQCD cross sections, including upscaling by “K factors”. Before going into details, let us try to extract generic features of the calculations. In all cases there is a definite correlation between a reduction in RAA (pt > 3 GeV) and an increase in v2 (pt ), i.e., both features are coupled to an increase in interaction strength (decrease in the spatial HQ diffusion coefficient). Furthermore, the v2 (pt ) shows a typical, essentially linear, increase reminiscent of a quasi-thermal regime followed by a saturation characteristic for the transition to a kinetic regime. In all Langevin calculations the saturation for charm quarks occurs at about p t = 2–3 GeV. For the largest interaction strength considered (Ds ' 1/(2πT )), the left panels of Figs. 18 and 20 even suggest a turnover of v2 (at this point one should recall that all calculations displayed in this section utilize elastic scattering only which is expected to receive appreciable corrections at high pt due to radiative
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1.4
b=5.5fm, f =0.5 0
γ =0.3 (charm) γ =1.0 γ =3.0 γ =30.0
1.2
0.8
b=5.5fm, f =0.5 0
γ =0.3 (charm) γ =1.0 γ =3.0 γ =30.0
vc2
RcAA
1
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 0
0.6 0.4 0.2 0 0
1
2
3
4 5 6 7 pout [GeV]
8
9 10
1
2
3
t
4 5 6 7 pout [GeV]
8
9 10
t
Fig. 20. (Color online) RAA (left) and v2 (right) of charm quarks resulting from hydrodynamic √ simulations of b = 5.5 fm Au–Au ( sN N = 200 GeV) collisions using AdS/CFT-motivated charmquark diffusion constants with variable strength parameter, γ,115 which corresponds to the constant C in Eq. (84).
Table 2. Overview of model approaches (1st column) and input parameters (2nd column: spatial charm-quark diffusion coefficient, 3rd column: nuclear impact parameter) for Langevin simulations of charm-quark spectra in Au–Au collisions at RHIC; selected values for the resulting elliptic flow (v2max ' v2 (pt = 5 GeV)) and nuclear modification factor are quoted in columns 4 and 5. The last two rows represent charm-quark transport calculations in a transport model for the bulk.
Model [Ref.] hydro + LO-pQCD26 hydro + LO-pQCD
26
fireball + LO-pQCD28
Ds (2πT )
b [fm]
v2max
RAA (pt = 5 GeV)
24
6.5
1.5%
0.7
6
6.5
5%
0.25
∼ 30
7
2%
0.65
28
∼6
7
6%
0.3
115
21
7.1
1.5–2%
∼ 0.7
hydro + “AdS/CFT” (84)115
2π
7.1
4%
∼ 0.3
∼ 30
8
∼ 2%
∼ 0.65
∼7
8
10%
∼ 0.4
fireball + reso+LO-pQCD hydro + “AdS/CFT” (84)
transport + LO pQCD
134
transport + LO pQCD134
processes). On the other hand, for pt = 2–3 GeV the nuclear modification factor is still significantly falling, leveling off only at larger pt ' 5–6 GeV. As expected, bottom quarks exhibit much reduced effects for comparable diffusion constants due to their factor ∼ 3 larger mass (see Fig. 19 and lower panels in Fig. 21). Next, we attempt more quantitative comparisons. Some representative numbers for the resulting RAA and v2 values are compiled in Table 2.114 First we compare the LO-pQCD calculations for HQ diffusion in the hydrodynamic and fireball backgrounds corresponding to Figs. 18 and 19, respectively; for a comparable
Heavy Quarks in the Quark-Gluon Plasma 16
15
T-matrix [Wo] T-matrix [SZ] Reso, Γ=0 4-0 75 GeV pQCD, αs=0 4
14 12
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spatial diffusion coefficient, Ds ' 24–30/(2πT ), both calculations show a maximal v2 of about 2% and a RAA (pt = 5 GeV) ' 0.7 (recall the smaller b = 6.5 fm in [MT] vs. 7 fm in [HGR] which may lead to somewhat smaller v2 , and the smaller T0 = 265 MeV [MT] vs. 340 MeV in [HGR] which entails somewhat less suppression). For the [AHH] hydro calculation with an AdS/CFT-motivated ansatz for the HQ friction constant, γ=C
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RAA (pt = 5 GeV) ' 0.25–0.3. The 4% value for Ref. 115 [AHH] will increase somewhat if the hydro evolution is run to the end of the mixed phase rather than terminated in the middle of the mixed phase (this is supported by the discussion in Sec. 3.5.2). We also note that in the fireball model of Ref. 28 [HGR] the inclusive (pt -integrated) v2 at the end of the mixed phase was adjusted to the experimentally observed light-particle v2 ' 5.5–6% at an impact parameter of b = 7 fm, i.e., it presumably includes an extra 20% of bulk-v2 compared to the hydrodynamic calculations.f Such an amount is typically built up in the subsequent hadronic phase of hydrodynamic evolutions and thus not present in pertinent HQ simulations within a QGP (+ mixed) phase. We now make some comments specific to individual calculations. The Langevin calculations using HQ T -matrix interactions (supplemented with pQCD scattering off gluons) shown in Fig. 21 are rather close to the effective resonance model, even though they do not involve tunable parameters. However, they are still beset with substantial uncertainty, as indicated by the use of two different input potentials (in addition, the use of the free energy, F1 , instead of the internal energy, U1 , as potential significantly reduces the effects). One also notices that the v2 at low pt is very similar to the resonance model while the suppression at high pt is somewhat less pronounced. This is so since the T -matrix transport coefficients (a) fall off stronger with 3-momentum (the resonant correlations are close to the Q–q threshold), and (b) decrease with increasing temperature (resonance melting). The latter combines with the facts that the suppression is primarily built up in the very early stages (where the T -matrix is less strong) while the bulk v2 takes a few fm/c to build up (at which point the T -matrix has become stronger). Furthermore, the T -matrix calculations lead to stronger medium effects on b quarks than the effective resonance model; this reflects the stronger binding due to the mass effect in the T -matrix calculation. A principal limitation of the Langevin approach is the treatment of fluctuations which are by definition implemented in Gaussian approximation. The latter arises due to enforcing the dissipation-fluctuation relation (mandatory to ensure the HQ distributions to approach equilibrium) which tends to underestimate the momentum fluctuations especially at high momentum, compared to a full transport calculation. This leads to an overestimate of the quenching effect at high pt even for the same average energy loss. One may assess these limitations more quantitatively by comparing to Boltzmann simulations including partonic phases,133,134 an example of which is displayed in Fig. 22 for charm quarks in b = 8 fm Au–Au collisions at RHIC. The baseline LO-pQCD calculations indicated by the crosses in Fig. 22, labeled by “1.33 mb”, may be compared to the fireball-Langevin simulations represented by the blue lines in Fig. 19. In both calculations the underlying elastic parton-HQ cross sections correspond to a strong coupling constant of αs ' 0.4. f This
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The quenching and elliptic flow come out quite similar in both calculations at least up to pt ' 5 GeV, especially when accounting for the slightly different centrality. E.g., in the Boltzmann treatment, the RAA for pt = 5 GeV charm quarks is about 0.6–0.7 with a v2 of a few percent. For a four-fold increase of the cross section (which would roughly correspond to a reduction of Ds (2πT ) from ∼ 30 to ∼ 7), one finds RAA (pt = 5 GeV) ' 0.4 and a maximum v2 of close to 10%. While the latter value is somewhat larger than the Langevin predictions, the agreement is not too bad. Finally, Fig. 23 shows results from an exploratory calculation in the Langevin approach where HQ drag and diffusion coefficients from elastic scattering in the effective resonance model (cf. Sec. 2.3.1) are combined with induced gluon radiation in the DGLV E-loss formalism (cf. Sec. 2.6).163 One of the uncertainties in
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this calculation is the extrapolation of the radiative E-loss into the low-momentum regime, where it still contributes rather substantially; e.g., the elliptic flow of charm quarks is increased over elastic pQCD+resonance model by ca. 40%, and even more (ca. 100%) relative to pQCD elastic scattering only. Another limitation is the above mentioned caveat in Langevin theory of underestimating the (E-loss) fluctuations implying an overestimate of the quenching at high pt . This can also be seen when comparing to the pQCD radiative E-loss calculations,7,38,164 where the gluon radiation is treated microscopically within an opacity expansion. A consistent merging of radiative and elastic processes in HQ transport thus remains a challenging task. 3.5. Heavy-meson and electron observables To compare to observables, the HQ spectra discussed in the preceding section need to be converted into spectra of color-neutral final-state particles. At the minimal level, this requires hadronization into charm and bottom mesons and baryons. Thus, a measurement of identified HQ hadrons constitutes the most direct way to make contact with theoretical predictions. Currently, the richest source of information on HQ spectra in Au–Au collisions at RHIC are single-electron (e± ) spectra, which, after the subtraction of sources coupling to a photon (“photonic sources”), are associated with semileptonic decays of HQ hadrons. As discussed in Sec. 3.3, the decay electrons largely preserve the modifications of the parent hadron spectra, albeit shifted in pt (by roughly a factor of ∼ 2). The more severe complication is the composition of the e± spectra, most notably the partition into charm and bottom parents.165 Since the heavier bottom quarks are, in general, less affected by the medium, their contribution significantly influences the resulting e± spectra. Unless otherwise stated, the calculations discussed below include a “realistic” input for the charm/bottom partition, i.e., either in terms of pQCD predictions for p–p spectra or via empirical estimates from D-meson and electron spectra in p–p and dAu. Within the current theoretical and experimental uncertainties, both procedures agree, with an expected crossing of charm ad bottom electrons at pt ' 3–6 GeV in p–p collisions at RHIC energy. Almost all of the approaches for computing HQ diffusion and/or energy loss introduced in Sec. 2 have been applied to e± data at RHIC. We organize the following discussion into (mainly perturbative) E-loss calculations (usually applied within a static geometry of the nuclear reaction zone) as well as perturbative and nonpertubative diffusion calculations using Langevin simulations for an expanding medium. 3.5.1. Energy-loss calculations Radiative energy loss (E-loss) of high-energy partons in the QGP is believed to be the prevalent mechanism in the suppression of light hadrons with high pT ≥ 6 GeV. It turns out that the application of this picture to the HQ sector (Sec. 2.6) cannot account for the observed suppression in the non-photonic e± spectra.
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In the DGLV formalism, the high-pT e± suppression due to radiative E-loss of c and b quarks falls short of the data by about a factor of 3, cf. left panel of Fig. 24. 38 This led the authors to consider elastic E-loss (see also Refs. 27 and 167) which was found to be comparable to the radiative one out to the highest electron pT measured thus far (∼ 10 GeV). Their combined effect still underestimates the measured suppression by about a factor of ∼ 2 for pt > 4 GeV. Similar findings were reported within the BDMPS approach: for a transport coefficient of qˆ = 14 GeV2 /fm,g the e± spectra cannot be reproduced either, unless an unrealistic assumption of neglecting the bottom contribution is made, cf. middle panel of Fig. 24.36 Both E-loss calculations36,38 are performed for a static (time-averaged) medium of gluons, with fragmentation as the sole mechanism for hadronization. This is expected to be a good approximation at high pT . Processes leading to an energy gain in the spectra, e.g., due to drag effects or coalescence with a light quark, are not included. Such processes lead to an increase in the pT of the final-state hadron and thus to an increase in the electron RAA , which would augment the discrepancy with data. The neglect of the diffusive term becomes particularly apparent in the elliptic flow. In the E-loss treatment the only source of an azimuthal asymmetry in the pT spectra in non-central Au–Au collisions is the spatial geometry of the overlap zone: particles traveling along the short axis are less likely to be absorbed g This
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than those moving along the long axis of the approximately elliptic reaction zone. The positive v2 generated by this mechanism amounts to up to a few percent and significantly falls short of the observed electron v2 , see right panel in Fig. 24. As an alternative mechanism, the collisional dissociation of D and B mesons from HQ fragmentation in the QGP (Sec. 2.3.3) has been implemented into an E-loss calculation.77,h A rather striking prediction of this calculation is that the shorter formation time of B mesons leads to stronger suppression than for D mesons above hadron momenta of pt ' 15 GeV at RHIC, cf. left panels in Fig. 25. This turns out to be an important ingredient in the successful reproduction of the e± suppression data as shown in the right panel of Fig. 25. 3.5.2. Langevin simulations The importance of elastic scattering for HQ diffusion and E-loss has been emphasized, prior to quantitative measurements of e± spectra, in Refs. 25 and 26, albeit within rather different realizations of the underlying HQ interaction (recall Secs. 2.3.1 and 2.2.2, respectively).
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The HQ spectra of the effective resonance + LO-pQCD model25 (using Langevin simulations within an expanding fireball, recall Fig. 19) have been converted into e± spectra utilizing a combined coalescence/fragmentation scheme at Tc followed by heavy-meson three-body decays.28,168 The predicted e± spectra and elliptic flow show approximate agreement with 2005 PHENIX33,166 and STAR169 data, see upper left and right panel of Fig. 26, respectively. Compared to the results for LO pQCD interactions only (blue lines), the resonance interactions (red bands) turn out to be instrumental in generating the required suppression and elliptic flow (see upper panels of Fig. 26). LO-pQCD scattering alone, even with a strong coupling of αs = 0.4, does not produce sufficient coupling to the bulk medium to suppress the primordial quark spectra, nor to drag the heavy quarks along with the collective flow of the expanding fireball. The effect of heavy-light quark coalescence is illustrated by a calculation where only fragmentation is used as a hadronization mechanism (lower panels in Fig. 26). In this case, the shape of the e± RAA and the magnitude
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of the v2 are not well reproduced. Coalescence processes add both momentum and v2 to the meson (and thus to the e± ) spectra, i.e., the suppression becomes smaller. It is furthermore instructive to compare the LO-pQCD results with fragmentation only (lower left panel in Fig. 26) to the pQCD E-loss calculations, especially to the elastic DGLV results where αs = 0.3 has been used (left panel of Fig. 24). The suppression level in the pertinent electron RAA is quite comparable for a rather large range in pT . The increasing trend in the Langevin calculations for pT & 5 GeV is presumably due to the dominant b-quark contribution (which is barely suppressed even in the resonance model up to pT & 5 GeV, see left of Fig. 19). Let us also estimate the impact of radiative contributions on the resonance model. Within DGLV the electron suppression due to radiative E-loss alone amounts to about 0.6–0.8 for pT ' 4–10 GeV. Upon multiplying the RAA for the resonance+pQCD model in the upper left panel of Fig. 26 with this factor, the result would be compatible with current RHIC data. The PHENIX collaboration has conducted a comprehensive comparison of their 2006 e± data22 to theoretical calculations predicting both RAA and v2 ,28,36,171 cf. left panel of Fig. 27. The interpretation reiterates some of the main points made above: (i) the missing drag in (radiative) E-loss calculations entails a substantial underprediction of the v2 ; (ii) Langevin calculations using elastic scattering require rather small HQ diffusion coefficients, Ds (2πT ) ' 4-6, to be compatible with the observed
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level of suppression and elliptic flow, and, (iii) quark coalescence improves the simultaneous description of these two observables. The heavy-light quark T -matrix approach, based on input potentials estimated from thermal lattice-QCD, has been applied within the same Langevin-fireball + coalescence/fragmentation scheme as the effective resonance model.45 The pertinent e± spectra (cf. right panel of Fig. 27) exhibit a comparable level of agreement with current RHIC data22,34,35 as in the resonance model, with a similar uncertainty due to different extractions of the HQ internal energy. Although the T -matrix calculations involve, in principle, no tunable parameters, the inherent theoretical uncertainties are appreciable (e.g., in the definition of the in-medium potential in terms of internal or free energy). Let us, however, recall a rather general feature of the T -matrix approach which was visible already at the level of the HQ spectra in Fig. 21: the weak temperature dependence of, e.g., the friction coefficient implies that the HQ coupling to the medium remains rather strong in the later QGP and mixed phase stages of the evolution. Since the bulk v2 is largest in these later stages, while the suppression largely occurs in the first 1-2 fm/c,39 the T -matrix interactions generate relatively more v2 than suppression compared to, e.g., the resonance model (or, alternatively: for the same v2 , the suppression in the T -matrix approach is smaller). This traces back to the increasing color-Debye screening with increasing temperature, which leads to a gradual melting of the resonance correlations and a marked increase of the spatial diffusion constant, Ds /(2πT ), with temperature (recall Fig. 14). Such a temperature dependence appears to improve the consistency in the simultaneous description of the e± RAA and v2 , but more precise data are needed to scrutinize this feature (including d-A collisions to quantify the Cronin effect, which could increase the RAA without noticably affecting v2 ). Finally, we reproduce in Fig. 28 selected results of the Boltzmann-transport approach of HQ diffusion67 with the background medium described by the hydrodynamical model of Ref. 13; hadronization is treated in a combined coalescence+fragmentation approach similar to the one in Refs. 28 and 45. The HQ interactions in the QGP are implemented via the elastic pQCD scattering amplitudes described in Sec. 2.2.3. The left panels in Fig. 28 refer to a model with fixed coupling constant, αs (2πT ), at given temperature and standard screening mass (r = 1). A large K factor of K = 12 is needed to simultaneously reproduce the electron RAA and v2 data. The final elliptic flow is found to be rather sensitive to the late QGP stages of the evolution, favoring hadronization at the end of the mixed phase (i.e., at a small transition energy-density, trans ); this is consistent with the findings of Ref. 39 and the above discussion of the T -matrix interaction. It thus corroborates that anisotropic matter-flow can only be transferred to the heavy quarks if the latter is sufficiently large, while E-loss (reflected in high-pT suppression) is mostly effective when the fireball density is high. The simulations find little impact of the initial-state Cronin effect on the final heavy-quark v2 , but the suppression is somewhat reduced primarily for e± momenta of pT ' 1–3 GeV. The right panel
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Fig. 28. (Color online) Boltzmann-transport model results for electron R AA (upper panels) and v2 (lower panels)67 in a hydrodynamic evolution using different versions of LO-pQCD HQ interactions, compared to PHENIX data22,35 in 200 AGeV Au–Au collisions at RHIC. The curves in the left panels are computed with fixed αs (2πT ) at given temperature, conventional infrared regulator (˜ µ2D = rµ2D with r≡κ = 1) and large K-factors; the curves in the right panels are computed with a running αs , reduced IR regulator (r = 0.2) and reduced K factors of 1.5–2 (or 2–3), as represented by the bands. trmax or trmin indicate freezeout at the beginning or end of the mixed phase, respectively.
of Fig. 28 illustrates that similar results can be achieved with smaller K-factors, K = 1.5–2, if the pQCD cross sections are augmented by a running coupling, αs (t) (t: 4-momentum transfer in the elastic scattering process), and a small infrared regulator, µ ˜ 2D = rµ2D with r = 0.2, in t-channel gluon-exchange scattering. 3.6. Viscosity? In this section we utilize the quantitative estimates for the HQ diffusion coefficient as extracted from current RHIC data to obtain a rough estimate of the ratio of shear viscosity to entropy density, η/s, in the QGP. This quantity has received considerable attention recently since (a) it allows to quantify deviations from the predictions of ideal fluid dynamics for observables like the elliptic flow, and (b) conformal field theories in the strong coupling limit are conjectured to set a universal lower bound for any liquid, given by η/s = 1/(4π),17 referred to as KSS bound. In
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the following we will bracket the estimates derived from HQ observables by using relations of Ds and η/s in the weak- and strong-coupling limit (the latter assumed to be given by the AdS/CFT correspondence). Following the discussion in Sec. 2.4, the strong coupling limit in the AdS/CFT framework results in a (spatial) HQ diffusion constant of Ds ' 1/(2πT ). Combining this with the lower bound of the viscosity-to-entropy-density quoted above, one obtains 1 η = T Ds . s 2
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Finally, equating the transport mean-free path λtr to the mean-free time τtr and taking into account the delay due to the mass effect of the heavy quark on the thermalization time, τQ ≈ τtr T /mQ , one finds η 1 ≈ T Ds . s 5
(88)
In comparison to the “strong-coupling” estimate within AdS/CFT, Eq. (86), the shear viscosity appears to be underestimated when the kinetic theory for a dilute gas is applied to liquids. These estimates are now applied to several of the HQ diffusion calculations discussed above, see the left panel of Fig. 29. Since η/s ∝ Ds (2πT ), the main features of Fig. 14 are transmitted to η/s, in particular the weak temperature dependence of the LO-pQCD calculations and the effective resonance model. Of course, the absolute values of these calculations differ considerably. A different behavior is only found for the T -matrix+pQCD model, which suggests a transition from a strongly coupled regime close to Tc to relatively weak coupling above ∼ 2 Tc . In fact, the uncertainty band has been constructed as follows: for the lower limit, the weak-coupling estimate Eq. (88) is used; for the upper limit, the strong-coupling limit estimate, Eq. (86), at T = 0.2 GeV is linearly interpolated with the LOpQCD weak-coupling limit at T = 0.4 GeV (the strong-coupling estimate for T matrix+pQCD overshoots the LO-pQCD result at this temperature). As for the spatial diffusion constant, the increase of η/s with temperature is related to colorDebye screening of the lQCD-based potentials which entails a gradual melting of the
174 2 15
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Fig. 29. (Color online) The ratio of shear viscosity to entropy density, η/s. Left panel: schematic estimates using charm-quark diffusion constants based on (a) schematic LO pQCD elastic scattering (αs = 0.4) in the weakly interacting limit, Eq. (88) (dashed line), (b) pQCD elastic scattering with running coupling constant and small IR regulator (band enclosed by dash-dotted lines using the weak- and strong-coupling limits), (c) the effective resonance + pQCD model in the strongcoupling limit, Eq. (86) (band enclosed by long-dashed lines for Γ = 0.4–0.75 GeV), and (d) the lattice-QCD potential based T -matrix approach augmented by pQCD scattering off gluons (band enclosed by solid lines constructed from the weak- and strong-coupling limits). Right panel: lattice QCD computations in a gluon plasma174 compared to results inferred from perturbation theory.175,176
dynamically generated resonances in the heavy-light quark T -matrix. It is tempting to interpret the decrease of η/s when approaching Tc from above as a precursorphenomenon of hadronization and thus connected to the phase transition itself. It remains to be seen whether a similar mechanism is operative in the light-quark and/or gluon sector (three-body interactions are unlikely to produce this due to the decrease in particle density when approaching Tc from above). Such a behavior is rather general in that it has been observed around phase-transition points for a large variety of substances, see, e.g., the discussion in Refs. 177 and 178. Finally we show in the right panel of Fig. 29 a quenched lQCD computation of η/s.174 The error bars are appreciable but the results tend to favor η/s values which are below LO-pQCD calculations. The specific pQCD result included in this plot employs a next-to-leading logarithm calculation for the shear viscosity 176 and a self-consistent hard-thermal-loop calculation for the entropy density.175 It is rather close to the schematic LO calculation (using αs = 0.4) in the left panel of Fig. 29.
4. Heavy Quarkonia in Medium In recent years it has become increasingly evident that observables in the heavyquarkonium and open heavy-flavor sectors are intimately connected. In the original picture of charmonium suppression as a probe of color-screening in hot and dense QCD matter179 there are no obvious such connections. Several recent developments have changed this situation. Thermal lattice QCD calculations find
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that charmonium correlation functions are remarkably stable up to temperatures of ∼ 2 Tc or higher, suggestive for the survival of the ground state (ηc , J/ψ) well into the QGP. This interpretation is supported by probabilistic extractions of the pertinent quarkonium spectral functions. It implies that quarkonia can not only dissociate but also regenerate in the QGP.i It immediately follows that the yield and spectra of regenerated quarkonia are, in principle, sensitive to the abundance and momentum spectra of open-charm states in the system. E.g., for a fixed total charm number in the system, a softening of the heavy-quark spectra is expected to increase c–¯ c overlap in phase space and thus enhance the probability for charmonium formation. At the same time, elliptic flow of charm quarks will imprint itself on regenerated charmonia. Furthermore, HQ interactions with light quarks (and possibly gluons) may be closely related to the interaction (or potential) between two heavy quarks. E.g., the T -matrix approach discussed in the previous section is directly based on potentials which are extracted from the HQ free energy computed in lattice QCD. As we argued there, this approach to evaluate HQ diffusion has several attractive features, both theoretically (it may provide maximal interaction strength in the vicinity of Tc ) and phenomenologically (it describes current HQ observables at RHIC fairly well). In the remainder of this section we address several aspects of quarkonia in medium and in heavy-ion collisions with a focus on connections to the open heavyflavor sector. More extensive reviews on quarkonia in medium have recently been given in Refs. 8–10, which we do not attempt to reproduce here. In Sec. 4.1 we give a brief review of theoretical issues in the understanding of in-medium quarkonium spectral properties, in terms of thermal lattice QCD results for correlation and spectral functions and their interpretation using effective potential models (Sec. 4.1.1). The latter are employing input potentials extracted from heavy-quark free energies computed in lattice QCD, thus enabling, in principle, an internal consistency check, provided a suitable potential can be defined. While color screening is a key medium effect in the potentials (governing the binding energy of the bound states), a quantitative assessment of spectral functions requires the inclusion of finite-width effects induced by dissociation reactions and possibly elastic scattering (Sec. 4.1.2). In Sec. 4.2 we elaborate on recent developments in describing heavy-quarkonium production in heavy-ion collisions. The main focus is on transport models which track the dissociation and regeneration of charmonia (and bottomonia) through the QGP, mixed and hadronic phases (Sec. 4.2.1), complemented by a brief discussion of initial conditions as affected by cold-nuclear-matter (CNM) effects (shadowing, Cronin effect and nuclear absorption). This is followed by an assessment of the current status of charmonium phenomenology at SPS and RHIC.
i Note
that higher dissociation rates also imply higher formation rates.
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4.1. Spectral properties of quarkonia in the QGP 4.1.1. Lattice QCD and potential models The phenomenological Cornell potential85 for the interaction between two heavyquark (color-) charges in the color-singlet channel, VQQ ¯ (r; T = 0) = −
4 αs + σr , 3 r
(89)
has been very successful in reproducing the vacuum spectroscopy of charmonium and bottomonium bound states. It consists of a (color-) Coulomb plus a (linear) confining part with a very limited number of parameters, i.e., a strong coupling constant, αs , and string tension, σ (in addition, an effective HQ mass, mQ , needs to be specified). Subsequent developments have put this framework on a more rigorous footing by showing that (a) the potential description can be recovered as a low-energy effective theory of QCD with heavy quarks,83,84 and (b) lattice QCD computations of the color-singlet heavy-quark free energy, F1 , have found excellent agreement with the functional form (and parameters) of the Cornell potential.180 ¯ bound-state properties in the QGP have suppleEarly calculations181 of Q–Q mented the Cornell potential by a phenomenological ansatz for color screening of both the Coulomb and confining parts, VQQ ¯ (r; T ) =
4α σ s −µD (T )r 1 − e−µD (T )r − e . µD (T ) 3r
(90)
The key quantity carrying the temperature dependence is the Debye screening mass, µD (∝ gT in thermal pQCD). Already at that time the possibility was established that ground-state charmonia (and even more so bottomonia) can survive until temperatures (well) above Tc . More recently, quantitative lQCD computations of the finite-temperature color-singlet free energy of a HQ pair, F1 (r; T ), have become available, see, e.g., Fig. 30. The results nicely illustrate the color-screening effect and its gradual penetration to smaller distances. When inserting the in-medium free energy as an improved estimate of the finite-temperature HQ potential into a Schr¨ odinger equation, the “melting” temperature of the J/ψ (ψ 0 , χc ) was found to be just above (below) Tc .183 Further progress in thermal lQCD came with the computation of heavy quarkonium correlation functions, Gα (τ, r) = hhjα (τ, r)jα† (0, 0)ii
(91)
(also referred to as temporal correlators), as a function of Euclidean time, τ . j α represent the creation/annihilation operators of a hadronic current of given quantum numbers, α. In the pseudoscalar and vector charmonium channels (corresponding to c–¯ c S-waves with ηc and J/ψ states, respectively), the Euclidean correlators were found to exhibit a surprisingly weak temperature dependence up to ∼ 2 Tc , even
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Fig. 30. (Color online) Free energy of a static color-singlet HQ pair as computed in lattice QCD for Nf = 393 (left) and Nf = 2182 flavors (right). The critical temperature is Tc = 193(170) MeV for the Nf = 3(2) calculation, and the string tension typically amounts to σ 1/2 ' 420 MeV = 1/(0.45 fm).
at large τ , suggestive for rather stable bound states. The temporal correlators are related to the physical spectral function, σα (E, p; T ), via Gα (τ, p; T ) =
Z∞
dE σα (E, p; T ) K(E, τ ; T )
(92)
0
with a thermal integral kernel K(E, τ ; T ) =
cosh[E(τ − 1/2T )] . sinh[E/2T ]
(93)
Equation (92) implies that the extraction of the spectral function from the Euclidean correlator requires a nontrivial integral inversion. Especially at finite T , where periodic boundary conditions limit the information on Gα (τ, p; T ) to a finite interval, 0 ≤ τ ≤ 1/T , and for a finite number of τ points, the unambiguous inversion to obtain σα (E, p; T ) becomes an ill-defined problem. However, using probabilistic methods (in particular the so-called maximum entropy method (MEM)), a statistical reconstruction of σα (E, p; T ) is possible and has been applied.41,42 The approximate constancy of the temporal correlators leads to spectral functions with rather stable ground-state peaks corroborating the notion of surviving ground states well above Tc . To resolve the apparent discrepancy with the low dissociation temperature found in the potential model discussed above, it has been suggested to employ as potential the internal rather than the free energy, which are related via F (r; T ) = U (r; T ) − T S(r; T ) .
(94)
Especially in the color-singlet channel, the (positive) entropy contribution rises sig¯ separation, r, thus producing “deeper” potentials (cf. Fig. 31) nificantly with Q–Q
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U1 [MeV]
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Fig. 31. HQ free energy in the color-singlet channel computed in thermal N f = 2 lattice QCD (left)180 and corresponding numerically extracted internal energy (right). 94
entailing stronger binding. Consequently, pertinent evaluations of quarkonium spectra lead to larger dissociation temperatures, which seemingly agree better with the lQCD spectral functions. These assertions have been made more quantitative 56–63 by employing potential models to calculate in-medium spectral functions, perform the straightforward integral in Eq. (92) and compare to the rather precise temporal correlators from lQCD. It is important to realize that the Euclidean correlators involve the pertinent spectral function at all energies. In Ref. 56 the inmedium bound-state spectrum obtained from a Schr¨ odinger equation (using either a screened Cornell potential or lQCD internal energies) has been combined with a ¯ continuum above threshold. No agreement with perturbative ansatz for the Q–Q lQCD correlators could be established. In Refs. 57 and 58 the importance of rescat¯ continuum was emphasized and implemented tering effects for the interacting Q–Q ¯ spectral functions. In Ref. 57 continuum correlainto the calculations of the Q–Q tions were implemented via Gamov resonance states in Breit–Wigner approximation, while in Ref. 58 a thermodynamic T -matrix approach was employed, Tα (E) = Vα +
Z
d3 k ¯ Vα GQQ¯ (E; k) Tα (E) [1 − fQ (ωkQ ) − fQ (ωkQ )] , 3 (2π)
(95)
exactly as introduced in the context of HQ diffusion in Sec. 2.3.2, recall Eq. (46). The T -matrix approach enables a consistent treatment of bound and continuum states on equal footing, as well as the implementation of medium effects (selfenergies) into the intermediate two-particle propagator, GQQ¯ , recall Eq. (48). Pertinent results for S- and P -wave charmonium spectral functions, using the internal energy extracted from the Nf = 3 free energy93 (left panel of Fig. 30), are shown in Fig. 32 for a constant (T -independent) charm-quark mass of mc = 1.7 GeV. One clearly recognizes the reduction in binding energy as a result of color screening by the upward moving bound-state peak position with increasing temperature (the c¯ c threshold is fixed at 2mc = 3.4 GeV). Also note that nonperturbative rescattering effects close to and above threshold induce a substantial enhancement in
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¯ spectral function over the non-interacting continuum (indicated by the the Q–Q red long-dashed lines), an effect which is of prime importance in the calculation of HQ diffusion in q–Q scattering as well. When applied to the calculation of Euclidean correlators58 in Eq. (92), the upward shift of low-energy strength due to the moving bound states in the S-wave spectral function shown in the left panel of Fig. 32 entails a suppression of G(τ ) with temperature which disagrees with the weak temperature dependence found in lQCD. Another important ingredient to understand the behavior of the correlators is the temperature dependence of the HQ mass. Schematically, the, say, J/ψ boundstate mass may be written as mJ/ψ = 2m∗c − εB .
(96)
This illustrates that a small (large) binding energy, εB , can be compensated by a small (large) effective quark mass in a way that the ground-state mass stays approximately constant. Indeed, when interpreting the asymptotic value of the inmedium potential as an effective HQ mass correction, m∗c = m0c + ∆mc ,
∆mc ≡ X(r = ∞; T )/2 ,
(97)
with X = U or F (or an appropriate combination thereof), the use of U implies strong binding with large effective quark masses while the use of F leads to weak binding with small m∗c (recall from Fig. 31 that the “U -potential” is deeper than the “F -potential” but features a larger asymptotic value for r → ∞). Consequently, it has been found that reasonable agreement with lQCD correlators can be achieved with different spectral functions, covering a rather large range of dissociation temperatures, e.g., slightly above Tc using screened Cornell potentials similar to the free
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Fig. 33. (Color online) S-wave charmonium spectral functions (η c or J/ψ) computed employing: (i) a screened Cornell potential within a nonrelativistic Green’s function approach (left), 61,62 and (ii) a lQCD-based Nf = 3 internal energy93 within a T -matrix approach.58 Both calculations account for in-medium charm-quark masses, and approximately reproduce the weak temperature dependence of the temporal correlators computed in lQCD up to temperatures of at least 2T c (see, e.g., inset in the left panel). In calculation (i) the bound-state has disappeared at temperatures below 1.5Tc , while in calculation (ii) it is visible up to ∼ 2.5 Tc .
energy (F1 ),61,62 ∼ 1.5 Tc using a linear combination of free and internal energy,57,63 or up to ∼ 2.5 Tc when using internal energies.58j , cf. Fig. 33. To resolve this redundancy, it will be necessary to develop independent means of determining the in-medium quark mass and the appropriate quantity to be identified with the HQ potential. First estimates of the HQ mass from thermal lQCD have been obtained by approximating the HQ number susceptibility within a quasiparticle model with effective quark mass.46 The results suggest a rather moderate temperature variation of the latter, which deviates significantly from the perturbative predictions up to T '3 Tc . In Refs. 80–82 hard-thermal-loop and HQ effective theory techniques have been applied to derive the leading terms in a perturbative and HQ mass expansion of a finite temperature potential. An interesting finding of these investigations is that the potential develops an imaginary part in the medium which arises from the Landau damping of the exchanged gluons, representing a decay channel of the HQ bound state. A more general discussion of the in-medium decay width of heavy quarkonia, which plays a central role for phenomenology in heavy-ion collisions, is the subject of the following section. The impact of finite-width effects on charmonium correlators has been studied within the T -matrix approach in Ref. 58, by implementing an imaginary part into the charm-quark propagators. A broadening of charmonium spectral functions j We
do not address here the issue of so-called zero-mode contributions to quarkonium correlators, which arise on the lattice due to the periodic boundary conditions in temporal direction (and physically correspond to HQ scattering).184 These contributions are essential to obtain quantitative agreement with the lQCD correlators in all mesonic quantum-number channels except the pseudoscalar one (ηc ); they are rather straightforward to implement in quasi-particle approximation.
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leads to an enhancement of the temporal correlators (due to additional strength at lower energies), which, however, is only a few percent for a charmonium width on the order of ΓΨ ' 100 MeV. On the one hand, such a value for the width is phenomenologically significant, as it implies that about 60% of the charmonia decay within a time of 2 fm/c. On the other hand, for larger widths, their impact on the correlators should be accounted for in quantitative comparisons to lQCD “data”. 4.1.2. Dissociation widths The spectral width of a quarkonium state propagating through matter can, in principle, receive contributions from elastic and inelastic reactions with the medium particles. Elastic scattering affects the momentum distribution of the quarkonium while inelastic interactions change its abundance (via dissociation or formation). More formally, the quarkonium acquires a complex selfenergy which can be expressed via the in-medium scattering amplitude, MΨi , folded over the (thermal) distribution, fi , of the medium particles, ΣΨ (p) =
XZ i
d3 k fi (ωi (k); T ) MΨi (p, k) . (2π)3 2ωi (k)
(98)
The real part of ΣΨ characterizes in-medium changes of the quarkonium mass while the imaginary part determines its width, ΓΨ (E) = −2 Im ΣΨ (E).k Most of the attention thus far has been directed to the inelastic reactions (rather than elastic scattering). Using the optical theorem to relate the imaginary part of the forward scattering amplitude to the cross section, one arrives at the well-known expression ΓΨ =
XZ i
d3 k diss fi (ωk , T ) vrel σΨi (s) , (2π)3
(99)
where vrel denotes the relative velocity of the incoming particles and s = (p + k)2 the squared center-of-mass energy of the Ψ–i collision. The first evaluation of the inelastic quarkonium reaction cross section with gluons was conducted in Refs. 185 and 186. Employing Coulomb wave functions for the quarkonium bound state, ¯ is the analog of photo-dissociation of the leading-order process, Ψ + g → Q + Q, hydrogen, see left panel of Fig. 34. For an S-wave Ψ bound state with binding energy εB , the cross section is given as a function of incoming gluon energy, k0 , by 2π σgΨ (k0 ) = 3
32 3
2
mQ εB
1/2
1 (k0 /εB − 1)3/2 . m2Q (k0 /εB )5
(100)
¯ level via in-medium effects on the addition, mass and width changes are induced at the Q–Q ¯ potential and direct Ψ → Q + Q ¯ decays, respectively. These effects can be accounted for, Q–Q ¯ T -matrix, Eq. (95). e.g., in the underlying Q–Q
k In
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Fig. 34. Diagrams for quarkonium-dissociation reactions via parton impact; left panel: gluodissociation185,186 ; right panel: quasifree dissociation.47
This expression has a rather pronounced maximum structure with the peak cross section reached for a gluon energy, k0max = 10 7 εB . The applicability of the gluodissociation formula should be reasonable for the free bottomonium ground state (εΥ ' 1 GeV), but borderline for J/ψ (εJ/ψ ' 0.6 GeV). Taken at face value for QGP temperatures of T = 300–400 MeV, where the typical thermal energy of (massless) gluons is around k0 = 1 GeV, the convolution of the gluo-dissociation cross section with a thermal gluon distribution function in Eq. (99) results in an inelastic J/ψ width (lifetime) of ΓJ/ψ ' 150–400 MeV (τJ/ψ ' 0.5–1.3 fm/c), see the dashed line in the right panel of Fig. 35. The situation changes if the quarkonium binding energy decreases due to colorscreening as discussed in the previous section (or for excited charmonia which are weakly bound even in vacuum). In this case, the peak of the gluo-dissociation cross section moves to smaller energies and becomes rather narrow; the loss of phase space can be basically understood by the fact that for a loosely bound Ψ state, the absorption of an on-shell gluon on an (almost) on-shell quark is kinematically impossible (suppressed). Consequently, with decreasing binding energy, the cross section has less overlap with the thermal gluon spectrum,47,187 leading to a decreasing width with temperature (cf. dotted line in the left panel of Fig. 35). This unphysical behavior signals the presence of other inelastic processes taking over. In Ref. 47 the so-called “quasifree” dissociation mechanism has been suggested, J/ψ+p → c+¯ c+p, where a thermal parton (p = g, q, q¯) scatters “quasi-elastically” off an individual heavy quark in the bound state (see right panel of Fig. 34). The p–Q scattering amplitude has been evaluated in leading-order (LO) perturbation theory,65 including thermal parton and Debye masses and slightly modified kinematics due to the small but finite binding energy. While naively of next-to-leading order (NLO) in αs compared to gluo-dissociation, the additional outgoing parton opens a large phase space rendering the quasifree process significantly more efficient for weakly bound states. Therefore, it readily applies to excited states as well, enabling a treatment of all charmonia on an equal footing (which is essential even for J/ψ observables, since the latter receive significant feed-down contributions from χc and ψ 0 states, see Sec. 4.2.1 below). For a coupling constant of αs ' 0.25, the quasifree dissociation rate reaches ΓJ/ψ = 100–200 MeV for temperatures, T = 300–400 MeV (cf. solid line in the left panel of Fig. 35). These values could be substantially
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183
QG
ΓΨ (MeV)
100
10
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quasifree (εB ) vac gluo-diss (εB ) med
gluo-diss (εB ) 1
0 15
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Fig. 35. (Color online) Parton-induced dissociation widths for J/ψ at rest in a QGP as a function of temperature. Left panel: gluo-dissociation width based on the cross section, Eq. (100), with free (dashed line) and in-medium decreasing binding energy (dotted line), as well as “quasifree dissociation” width (J/ψ + p → c + c¯ + p) with in-medium decreasing εB (solid line).47,187 Right panel: full NLO calculation for (a) quark- and (b) gluon-induced dissociation for different thermal parton masses.188
enhanced if non-perturbative p–Q scattering mechanisms are operative, much like the ones discussed in Sec. 2.3. A complete NLO calculation for parton-induced charmonium destruction has recently been carried out in Ref. 188, including the effects of in-medium reduced binding energies. The right panels of Fig. 35 show the results for quark- and gluon-induced breakup of the J/ψ for αs = 0.5 and different (fixed) thermal parton masses. For temperatures around 250 MeV, the sum of both contributions (ΓJ/ψ ' 350 MeV) is about a factor ∼ 4 larger than the quasifree results (ΓJ/ψ ' 80 MeV) in the left panel, calculated for αs = 0.25.47 Thus, there is good agreement between these two calculations, since the rate is basically ∝ α 2s (for T = 250 MeV the T -dependent Debye-mass in Ref. 47, µD = gT , amounts to µD ' 440 MeV). The three-momentum dependence of the dissociation rate of a moving quarkonium, ΓΨ (p), has been calculated for full NLO and the quasifree rates in Refs. 189 and 190, respectively. In both calculations a weak increase of the rate with increasing three-momentum is found. Since the quasifree cross section is essentially constant, this increase is caused by the increasing flux of thermal partons encountered by the moving bound state. A similar result has been obtained in a calculation employing the AdS/CFT correspondence191 (recall Sec. 2.4 for more details on this framework and its caveats). On the other hand, gluo-dissociation leads to a rather pronounced decrease of the dissociation rate with increasing three-momentum, since the pertinent cross section is peaked at relatively low energies and falls off rapidly at large center-of-mass energies, s = (p + k)2 . Of course, if gluo-dissociation becomes ineffective, its three-momentum dependence becomes immaterial. A decreasing pdependence should also be expected in quasifree dissociation if non-perturbative
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(resonance-like) q–Q interactions are operative (recall Sec. 2.3), since the pertinent √ cross sections are concentrated at small s as well. To utilize quarkonium observables as a probe of QGP formation in URHICs, it is mandatory to have good control over the modifications of quarkonia in hadronic matter, in particular their inelastic reaction rates. From current lattice QCD calculations it is very difficult to extract information on excited states (say, ψ 0 ). In addition, the results in the P -wave (scalar and axialvector) channels (corresponding to χc,0 and χc,1 , respectively) are sensitive to the so-called zero-mode contributions which are not directly related to the bound-state properties (as briefly mentioned in a previous footnote). Potential models find that ψ 0 , χc,0 and χc,1 “melt” close to or even below Tc , suggesting substantial modifications in the hadronic phase. Even for the J/ψ, hadronic dissociation may lead to significant suppression (in addition to suppressed feed-down from ψ 0 and χc states). One main obstacle in a reliable assess¯ ment of these reactions is that low-energy reactions of the type h+J/ψ → D+ D+X constitute a nonperturbative problem with little experimental information available to constrain effective models. An initial estimate of quarkonium dissociation by light hadrons has been obtained by using the gluo-dissociation cross section, Eq. (100), convoluted over the gluon distribution inside hadrons.192 Since the latter is rather soft (k0 ' 0.1 GeV), the gluon energy is in general not sufficient to break up the J/ψ, leading to a (lowinel energy) cross section of order σhJ/ψ ' 0.1 mb. Quark-exchange reactions,193 e.g., ¯ (including excited D mesons in the in meson-induced breakup, h + J/ψ → D + D final-state), are presumably more relevant. Effective quark models predict dissociation cross sections of order 1-2 mb, see, e.g., Refs. 194 and 195. An alternative approach is to construct effective hadronic models, pioneered in Ref. 196. Guiding principles are basic symmetries including gauge invariance for vector mesons (J/ψ, ρ) as well as flavor symmetries, most notably SU(4) (albeit explicitly broken by the charm-quark mass) and chiral symmetry which is operative in interactions with (pseudo-) Goldstone bosons (π and K).197–202 The main uncertainty in these models remains a reliable determination of the cutoff scales figuring into the hadronic vertex form factors. With cutoff values of around 1 GeV, the agreement with quark models is quite reasonable; dissociation reactions induced by ρ mesons appear to be the most important channel. Their thermal density in hadronic matter (i.e., for temperatures of ∼ 180 MeV) is not very large, so that the total J/ψ width does not exceed a few MeV, and therefore is substantially smaller than in the QGP. E.g., for a total hadron density of %h = 3%0 and a thermally averaged cross section diss of hσhJ/ψ i = 1 mb (corresponding to significantly larger peak cross sections), a diss rough estimate for the dissociation rate gives Γdiss hJ/ψ = hσhJ/ψ vrel i %h ' 5 MeV. An interesting possibility to constrain effective hadronic vertices in a rather modelindependent way is to use QCD sum rules.200,203 Pertinent estimates yield, e.g., diss a thermally averaged πJ/ψ dissociation cross section of hσπJ/ψ vrel i = 0.3 mb at T = 150 MeV, in the same range as the above estimate. We finally remark that
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in-medium effects, e.g., modified spectral distributions of D-mesons, can increase the final-state phase space and lead to an appreciable increase of the dissociation rate.204 This is especially pertinent to excited charmonia like the ψ 0 , whose mass ¯ threshold, so that a slight reduction (or broadening) in the is close to the free D D ¯ 205 D-meson mass can open the direct decay channel, ψ 0 → DD. 4.2. Quarkonium production in heavy-ion collisions Similar to the open heavy-flavor sector, a key objective (and challenge) in the quarkonium sector is to connect their equilibrium properties to observables in heavy-ion collisions, and eventually deduce more general insights about basic properties of QCD matter, e.g., color de-/confinement and Debye screening. Since quarkonium states in heavy-ion collisions are even more rare than individual heavy quarks, it is suitable to adopt a transport treatment for their distribution functions in a realistic “background medium” whose evolution is not affected by the heavy quarks or quarkonia. The connection between observables extracted from the distribution function (after its “transport” through the medium) and the equilibrium properties discussed in the previous section is given by the coefficients and equilibrium limit of the transport equation, as elaborated in the following section, 4.2.1. The current status in comparing various model implementations to charmonium data at SPS and RHIC will be assessed in Sec. 4.2.2. 4.2.1. Quarkonium transport in heavy-ion collisions The (classical) Boltzmann equation describing the time evolution of the phase-space distribution function, fΨ (r, τ ; p), of an (on-shell) quarkonium state, Ψ (with energy p0 = ωp = (p2 + m2Ψ )1/2 ), may be written as pµ ∂µ fΨ (r, τ ; p) = −ωp ΓΨ (r, τ ; p) fΨ (r, τ ; p) + ωp βΨ (r, τ ; p)
(101)
(a mean-field term has been neglected assuming that the real part of the Ψ selfenergy is small). When focusing on inelastic reactions, ΓΨ (r, τ ; p) represents the dissociation rate discussed in Sec. 4.1.2 above, which governs the loss term, i.e., the first term on the right-hand-side (rhs) of Eq. (101). The (r, τ ) dependence of ΓΨ typically converts into a temperature dependence via the fireball evolution of √ a heavy-ion reaction for given projectile/target (A/B), collision energy ( s) and impact parameter (b). The second term on the rhs of Eq. (101) is the gain term accounting for the formation of quarkonia. For a 2 → 2 process (as, e.g., realized ¯ → g + Ψ), it takes the form206 via the inverse of gluo-dissociation, Q + Q βΨ (p; r, τ ) =
1 2p0
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To ensure detailed balance, the cross section figuring into the formation probabilform ity, WQgΨ vrel 4 ωpQ ωpQ , has to be the same (up to a kinematic and ¯ ¯ (s) = σQQ→gΨ Q statistical factor) as the one used in the dissociation rate, Eq. (99). For reactions ¯ → p + Ψ), the microscopic beyond 2 ↔ 2 (such as the quasifree process, p + Q + Q evaluation of the gain term becomes more involved. Note the explicit dependence ¯ on the HQ phase-space distribution functions, f Q,Q , in Eq. (102), whose modifications in heavy-ion reactions are the central theme in Sec. 3 of this review. The temperature-dependent step function in Eq. (102) signifies the limit set by the dissociation temperature, Tdiss , above which a well-defined Ψ state no longer exists and thus formation reactions are not meaningful. It is both instructive and useful for practical applications to simplify the gain term by integrating out its spatial and three-momentum dependence. This is possible under the assumption of a homogeneous medium and thermally equilibrated HQ distribution functions; one obtains207 dNΨ eq = −ΓΨ (NΨ − NΨ ), dτ
(103)
which now clearly exhibits detailed balance in terms of the approach to the equieq librium limit, NΨ , of the state Ψ. The latter quantity is given by Z d3 p eq eq 2 NΨ = VFB nΨ (mΨ ; T, γQ ) = dΨ γQ fΨ (ωp ; T ) , (104) (2π)3 carrying the explicit dependence on the (in-medium) quarkonium mass, mΨ (which, ¯ binding energy and in-medium HQ in turn, depends on a combination of Q–Q mass, cf. Eq. (96)); dΨ denotes the spin degeneracy of the Ψ state and VFB the (time-dependent) fireball volume. The appearance of a HQ fugacity, γQ = γQ¯ , owes ¯ its origin to the (theoretically and experimentally supported) postulate that Q Q production is restricted to hard N –N collisions upon initial nuclear impact. The number NQQ¯ = NQ = NQ¯ of heavy anti-/quarks is then conserved in the subsequent fireball evolution (separately for charm and bottom), which is achieved by introducing γQ at given fireball volume and temperature into the thermal densities of open and hidden HQ states, i.e., NQQ¯ =
X eq 1 I1 (Nop ) 2 Nop + VFB γQ nΨ (T ) , 2 I0 (Nop )
(105)
Ψ
for either charm (Q = c) or bottom (Q = b). The thermal open-charm (bottom) number, Nop , depends on whether one is evaluating it in terms of individual quark ∗ states, Nop = VFB γQ 2neq for a (weakly interacting) QGP), Q (mQ , T ) (appropriate P eq or in terms of hadronic states, Nop = VFB γQ α nα (T, µB ). This is, in principle, a nontrivial issue, since both hadronic and partonic evaluations of Nop can be subject to corrections, see, e.g., Refs. 207, 53 and 208. In the hadronic phase one expects the spectral functions of D mesons, Λc baryons, etc. to undergo significant medium effects, e.g., reduced masses and/or increased widths. In the partonic phase,
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especially close to Tc , it is not inconceivable that hadronic bound states are still present and thus an approximation with weakly interacting quasiquarks may not be an accurate one. Even within a quasiquark description, significant uncertainty is associated with the value of the HQ mass adopted in the calculation of Nop and thus in the quantitative determination of γQ . The general trend is that for a given temperature, volume and NQQ¯ , a smaller value for m∗Q results in a larger eq value for neq Q and thus in a smaller value for γQ , which, in turn, reduces NΨ quadratically. The underlying physics is that of relative chemical equilibrium: for a given number of heavy anti-/quarks, the latter preferentially occupy the states of the lowest energy. In the simplest case, where a quasiquark description applies and the Ψ mass is given by the expression (96), the Ψ number is essentially determined eq by its binding energy (larger εB implying larger NΨ ). The gain term as written in Eq. (102) is, strictly speaking, only applicable in the quasiquark approximation. If additional resonances are present in the medium (e.g., D-meson resonances or cq diquark states), additional reaction channels would have to be included in a coupled rate-equation framework to account for the competition of these resonances to harbor c quarks. In the simplified treatment given by Eq. (103), this competition eq is included via the c-quark fugacity figuring into NΨ . A slightly different view on regeneration and suppression processes in the QGP is advocated in Ref. 209, based on the strongly coupled nature of the QGP (sQGP) as produced at SPS and RHIC (i.e., at not too high temperatures). It is argued that a small charm-quark diffusion constant (cf. Secs. 2 and 3) inhibits the separation of the produced c and c¯ pair in the sQGP. In connection with the survival of J/ψ bound states well above Tc (as, e.g., in the right panel of Fig. 33), this enhances the probability for a produced c¯ c pair to bind into a charmonium state (relative to p–p collisions). In particular, this approach accounts for the possibility that the pairwise produced c and c¯ quarks do not explore the entire fireball volume as usually assumed in equilibrium models. Such an effect has also been implemented in a more simplified manner in the thermal-rate equation approach of Refs. 207 and 190 in terms of a time-dependent correlation volume. Let us briefly discuss the initial conditions for the quarkonium distribution functions. Starting point are measured quarkonium spectra in p–p collisions. In a heavy-ion collision, these are subject to modifications before the medium can be approximated with a thermal evolution. “Pre-equilibrium” effects may be distin¯ guished according to whether they occur before or after the hard QQ-production process takes place. The former include nuclear modifications of the parton distribution functions generically denoted as “shadowing”, as well as pt broadening (Cronin effect) attributed to a scattering of the projectile/target partons on their ¯ way through the target/projectile nucleus prior to the fusion reaction into Q– Q. In a random-walk picture, the accumulated transverse momentum is approximated by ∆p2t = agN hli, where hli is an average nuclear path length of both gluons before the hard scattering, and agN parameterizes the transverse-momentum kick per path length in gluon-nucleon scattering. Both “pre-fusion” effects are in principle
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¯ production process. In p–p collisions, universal, i.e., not directly linked to the QQ ¯ a fraction of 1–2% of c¯ c pairs (∼ 0.1% of bb pairs) develop a correlation that leads to the formation of a charmonium (bottomonium) state.210 In nuclear collisions ¯ pair with passing-by nucleons can destroy inelastic collisions of the produced QQ this correlation. This so-called nuclear absorption may be parameterized by an efΨN l fective absorption-cross section, σabs . As is well-known, the finite (and different) ¯ pair interacting with a nucleon formation times of quarkonia imply that the QQ does, in general, not represent a fully formed quarkonium, but rather a preresonance state. A microscopic description of nuclear absorption is therefore a rather challenging task.211,212 At a minimal level, finite formation times imply that the values for effective nuclear absorption cross sections should be expected to depend √ on collision energy ( s), rapidity (y) and bound-state quantum numbers (since different binding energies imply different formation times). A careful measurement and systematic interpretation of quarkonium suppression in p–A collisions, where the formation of a thermal medium is not expected (at least at SPS and RHIC), is therefore an inevitable prerequisite for quantitative interpretations of heavy-ion data, see, e.g., Refs. 213–215 for recent work. Pre-equilibrium effects not only modify the momentum dependence of the quarkonium distribution functions but also their spatial dependence.
4.2.2. Quarkonium phenomenology in heavy-ion collisions As discussed in the Introduction, there is ample evidence for both chemical and thermal equilibration in the low-pt regime of (bulk) particle production in ultrarelativistic heavy-ion collisions. A well defined set of thermodynamic variables characterizing the temperature evolution and flow fields of the fireball greatly facilitates the comparison of independent calculations for quarkonium production and maintains direct contact to their in-medium properties in equilibrated QCD matter. In this section we therefore focus on rate-equation approaches implemented into thermal background media. We recall that the experimental quarkonium yields usually include feed-down contributions due to decays of higher resonances. E.g., for J/ψ production in p–p collisions about 30% (10%) of the observed number arises from decays of χc (ψ 0 ) states.216,217 The standard assumption in heavy-ion collisions is that primordial production fractions of excited states scale as in p–p collisions, but subsequent suppression (and/or regeneration) will change these ratios (due to different inelastic cross sections at all stages). This needs to be taken into account for realistic comparisons to heavy-ion data (unless otherwise stated, it is included in the theoretical l Nuclear
absorption typically occurs at a rather large Ψ–N center-of-mass energy (comparable to √ the s of primordial N –N collisions) and is therefore in a very different energy regime than the low-energy hadronic absorption cross section relevant for the later hadron-gas stage of the fireball evolution.
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Fig. 36. (Color online) J/ψ production at SPS in the thermal rate-equation approach 190,207 (left panels) and the statistical hadronization model221 (right panels). Upper panels display the centrality dependence of the inclusive J/ψ yield (normalized to Drell-Yan production or the number of binary N –N collisions), lower panels the average J/ψ transverse momentum (squared in the left panel).
models discussed below). After thermal freezeout, the decay branchings are assumed to be as in vacuum (since the J/ψ lifetime after freezeout (ca. 2000 fm/c) is about a factor of ∼ 200 larger than the fireball lifetime, in-medium dilepton decays contribute a small fraction to the spectrum observed in the detectors). √ Let us start by analyzing J/ψ production in Pb–Pb( s = 17.3 AGeV) collisions at SPS in the context of NA50 data,48,218–220 cf. Fig. 36. The left panels display the outcome of thermal-rate equation calculations,190,207 where quasifree dissociation rates in the QGP (cf. left panel of Fig. 35) and hadronic SU(4) cross sections for meson-induced dissociation in the HG are evolved over an expanding fireball model (adjusted to empirical information on hadron production and flow velocities). The prevalent effect is identified as suppression in the QGP, controlled by an effective strong coupling constant, αs ' 0.25, in the quasifree rate. This value is adjusted to reproduce the suppression level in central collisions (where the average initial temperature amounts to about T0 ' 210 MeV). Regeneration is a rather small effect, based on a p–p open-charm cross section of σc¯c = 5.5 µb distributed over
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two fireballs (the covered rapidity window amounts to ∆y = 3.6, resulting in a rapidity density of dσc¯c /dy ' 1.53 µb. A recent compilation222 of charm production at fixed-target energies finds a total cross section of σc¯c ' 3.6–5.2 µb; with an experimental rapidity width of around ∆y = 2,223 the resulting rapidity density is approximately dσc¯c /dy = (2.2 ± 0.5) µb). In addition, a correction for incomplete charm-quark thermalization has been implemented207 via a kinetic relaxation time (τceq ) reducing the equilibrium J/ψ number. The lower left panel of Fig. 36 suggests that the centrality dependence of the average J/ψ’s transverse-momentum squared, hp2t i, is largely governed by the Cronin effect as extracted from experimental p–A data.219 The quasifree charmonium dissociation rates, which increase with threemomentum,190 lead to a slight suppression of hp2t i for central collisions. The right panels in Fig. 36 are calculated within the statistical hadronization model,221 assuming that all primordial charmonia are suppressed and production entirely occurs at the critical temperature for hadronization based on relative chemical equilibrium of open- and hidden-charm hadrons (with Nc¯c fixed as in Eq. (105)). This also implies that the charm-quark momentum distributions are kinetically equilibrated. With a rapidity density for the p–p charm cross section of dσc¯c /dy = 5.7 µb the NA50 data can be reproduced reasonably well. This input c¯ c number is larger by a factor of ∼ 4 compared to the input in the left panels, which accounts for most of the difference to the regeneration yield in the rateequation calculation for central collisions (remaining discrepancies are largely due to the c-quark relaxation correction which becomes more pronounced toward more peripheral collisions).m The interpretation of the J/ψ’s average pt is also rather different, in that it entirely stems from a thermal source (with moderate collective flow) in the vicinity of Tc (the resulting hp2t i is quite consistent with the regeneration component in the lower left panel of Fig. 36). NA50 has also measured ψ 0 production.224,225,227 Using p–A collisions, the exψ0 tracted nuclear absorption cross section has been updated227 to σnuc = (7.7 ± J/ψ 0.9) mb, which is significantly larger than for J/ψ, σnuc = (4.2 ± 0.5) mb. In Pb–Pb √ collisions ( s = 17.3 AGeV), the ratio ψ 0 /(J/ψ) is suppressed substantially already in rather peripheral collisions, cf. Fig. 37.224,225 Within the thermal rate-equation approach,207 this behavior cannot be explained by inelastic reactions in the QGP alone, since very little (if any) QGP is formed in peripheral Pb–Pb collisions at SPS. However, hadronic dissociation of the ψ 0 can account for the suppression pattern, but only if in-medium effects are included (cf. left panel of Fig. 37),207,226 ¯ threshold which accelerates ψ 0 suppression due specifically a reduction of the D D ¯ (a similar effect can result from a to the opening of the direct decay mode, ψ 0 → DD broadening of the D-meson spectral functions as discussed in Sec. 2.7). The updated (larger) ψ 0 nuclear absorption cross section227 also plays a significant role in the quantitative description of the low-centrality data. The statistical hadronization m The
charm ensemble at SPS is in the canonical limit, Nop 1, for which I1 (Nop )/I0 (Nop ) ' 0.5Nop in Eq. (105), and thus Nψ ∝ Nc¯c .
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model predicts a flat ψ 0 /(J/ψ) ratio, given by the thermal densities at the hadronization temperature. The shape and magnitude of the calculated ratio is rather consistent with the NA50 data for central and semicentral collisions where hadronization from a QGP can be expected to be applicable; deviations occur for more peripheral centralities. Thus, the ψ 0 /(J/ψ) ratio does not provide a clear discrimination of regeneration- and suppression-dominated scenarios at SPS. One of the controversies in the interpretation of the NA50 data has been whether they feature any “sharp” drop in their centrality dependence, e.g., around ET ' 35 GeV (or Npart ' 120) in the upper left (right) panel of Fig. 36. Such a drop has been associated with a threshold behavior for QGP formation resulting in an abrupt “melting” of the χc states due to color screening229 (recall that χc feeddown presumably makes up ∼ 30% of the inclusive J/ψ yield). The investigation of this question was one of the main objectives of the successor experiment of NA50, NA60, where J/ψ production in a medium size nuclear system (In-In) has been measured.228 Figure 38 compares the NA60 J/ψ data as a function of centrality to three theoretical predictions,187,229,230 all of which reproduce the NA50 data reasonably well. The predictions of the thermal rate-equation approach187,207 roughly reproduce the onset and magnitude of the suppression (except for the most central data points); the threshold-melting scenario229 misplaces the onset of the suppression (which in the data is below Npart = 100, contrary to the Pb–Pb system) and the comover calculation230 overpredicts the suppression. The leveling-off (or even
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√ Fig. 38. (Color online) NA60 data for the centrality dependence of J/ψ production in In-In ( s = 228 17.3 AGeV) collisions, compared to theoretical predictions based on (from top to bottom): (a) the threshold-melting scenario229 (dashed line), (b) the thermal rate-equation approach187 (dash-dotted line) and (c) the comover suppression approach230 (dotted line). Nuclear absorption effects have been divided out of the data and calculations based on measured suppression in p–A collisions.
increasing trend) of the data for Npart ≥ 150 is somewhat unexpected and deserves further study. The thermal rate-equation framework has been used to predict J/ψ production at RHIC.207,n With updates190 for the experimental input (a smaller nuclear absorption cross section232 and a larger J/ψ number in p–p collisions49 which figures into the denominator of the nuclear modification factor and leads to a relative reduction of the regeneration yield), an approximate agreement with current PHENIX data on the centrality dependence of inclusive J/ψ production and pt spectra emerges, see left panels of Fig. 39 (the underlying charm cross section in p–p, σc¯c = 570 µb, translates into dσc¯c /dy ' 100 µb, consistent with PHENIX measurements233,234 ). The main features of this interpretation are an about equal share of (suppressed) primordial and regenerated J/ψ’s in central Au–Au collisions (where the average initial temperature is about T0 = 370 MeV), as well as a significant reduction of the average p2t due to secondary production, as compared to primordial production with an estimated Cronin effect (the latter is not yet accurately determined from p–A data). Consequently, the regeneration component is concentrated in the low-pt regime of the spectra. The dependence on the kinetic relaxation time for c quarks is rather moderate while the inclusion of the momentum dependence in the quasifree dissociation rate190 has little effect.
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Fig. 39. (Color online) J/ψ production in 200 AGeV Au–Au collisions at RHIC in the transport approach of Ref. 190 (left panels, using an expanding thermal fireball with reaction rates based on the quasifree dissociation process) and Ref. 231 (right panels, using a hydrodynamic evolution with reaction rates based on the gluo-dissociation process). The upper (lower) panels display the inclusive J/ψ RAA (average p2t ) as a function of participant-nucleon number. The data are from the PHENIX collaboration.49
The right panels of Fig. 39 show the results of the rate-equation approach of Refs. 206 and 231, where gluo-dissociation rates in the QGP are convoluted over a 2+1-dimensional hydrodynamic evolution (employing a charm cross section of dσc¯c /dy = 120 µb, in line with PHENIX data233 ); reactions in the hadronic phase are neglected. It is very encouraging that the results are in good agreement with the ones in the left panels190 which are obtained with similar physics input but in a different realization (e.g., fireball vs. hydro but with comparable initial temperatures (T0 ' 350 MeV in central collisions) and charm cross section, etc.). The centrality dependence of the inclusive J/ψ yield shows a slight step-structure in the upper right panel,231 induced by different dissociation temperatures for J/ψ (Tdiss = 320 MeV) and χc , ψ 0 (Tdiss = Tc = 165 MeV) above which the suppression is assumed to be practically instantaneous (similar findings have been reported in Ref. 235). The hp2t i of the regenerated component is somewhat smaller in Ref. 231 compared to Ref. 190, since in the former it is computed via continuous regeneration based on Eq. (102), while in the latter the pt spectra of the regenerated component are approximated with a thermal blast wave at Tc .
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Recent experimental data on J/ψ production in Cu–Cu collisions indicate that the nuclear modification factor tends to increase at high pt > 5 GeV.236,237 Such a trend is not present in QGP suppression calculations based on the quasifree dissociation rate which increases with the three-momentum of the J/ψ.190 However, the inclusion of charmonium-formation time effects238–240 (which reduce the dissociation rate due to time dilation in the development of the hadronic wave packet) J/ψ and the contributions from bottom feed-down241 can lead to an increase of RAA 242 at high pt . An initially promising signature to discriminate suppression and c–¯ c coalescence mechanisms is the elliptic flow of charmonia (which also provides a close connection to, and thus consistency check with, the collective flow of open charm). If only suppression mechanisms are operative, the azimuthal asymmetry of the charmonium momentum distributions entirely develops from the path length differences caused by the long vs. the short axis of the almond shaped nuclear overlap zone. As for open charm (recall middle panel of Fig. 24), this effect is rather small, generating a maximal v2 (pt ) of up to 2–3%.243 On the other hand, for c–¯ c coalescence, the charmonium bound state inherits up to twice the c-quark v2 155,157 at the time of formation, especially if the c–¯ c binding energy is small (in that case little v2 is carried away by an outgoing light parton). This effect is further maximized if the coalescence occurs late in the evolution, e.g., at the hadronization transition (where most of the elliptic flow is believed to have built up). However, according to the above discussion, the charmonium regeneration yield is mostly concentrated at rather low pt < 3 GeV, where the magnitude of the c-quark v2 (pt /2) is not very large (this is a consequence of the mass ordering of v2 , whose rise in pt is shifted to larger values for heavier particles), recall, e.g., the right panel of Fig. 19. Consequently, within the transport models displayed in Fig. 39, the net v2 (pt ) of regenerated and primordial J/ψ’s combined does not exceed 2–3%206,242,244 and would therefore be difficult to discriminate from primordial production only. An interesting question, which thus far has received little attention, concerns elastic interactions of charmonia in the medium and whether they could contribute to their v2 in heavy-ion collisions. Elastic interactions should become more relevant as the binding energy of the charmonium increases, rendering them more compact objects which are less likely to break up. Interestingly, the NA60 collaboration has reported a rather large inclusive (pt > 0.5 GeV) elliptic flow of v2 = (6.8 ± 4)% √ for J/ψ’s in semicentral In-In ( s = 17.3 AGeV) collisions at the SPS.245 This observation will be difficult to explain based on dissociation reactions alone. Next, we address the rapidity dependence of J/ψ production at RHIC, where PHENIX measurements in the dielectron channel at central rapidity, |y| < 0.35, and in the dimuon channel, at |y| = 1.2–2.2, indicate a maximum in RAA (y) around y = 0 for central and semicentral collisions. This trend can be nicely reproduced by the statistical hadronization model as a consequence of the underlying c-quark distributions (cf. left and middle panel in Fig. 40) which are expected to be narrower
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than for bulk particle production, implying higher c-quark densities and thus larger charmonium yields at central rapidity. This dependence is more difficult to explain in the thermal rate-equation approaches246 where only about 50% (or less) of the J/ψ’s originate from c–¯ c regeneration. In addition, the charm ensemble at RHIC is not yet fully in the grand canonical limito for which I1 /I0 → 1 in Eq. (105), and thus the sensitivity on the c-quark density (fugacity) is less pronounced. Furthermore, the thermal suppression of the primordial component exhibits an opposite trend, being slightly less suppressed at forward y due to reduced light-particle production. It is quite conceivable that cold-nuclear-matter effects imprint significant rapidity dependencies on the primordial component (e.g., a stronger shadowing and/or nuclear absorption at forward |y|).213,214 In Ref. 213, e.g., stronger nuclear absorption of J/ψ’s at forward rapidity at RHIC has been found as a consequence of different production mechanisms which probe different kinematics in the nuclear parton distribution functions. Recent calculations of charmonium production within microscopic transport models for the bulk-medium evolution can be found, e.g., in Refs. 247 and 248. The results are generally quite reminiscent of the rate-equation calculations discussed above. In particular, the description of RHIC data requires the inclusion of regeneration interactions. This is also true for the so-called comover approach, which has been extended to include charmonium formation reactions in Ref. 249. The interplay of suppression and regeneration leads to interesting consequences for the excitation function of charmonium production. The approximate degeneracy of J/ψ suppression by about a factor of ∼ 3 in central A–A collisions at both SPS and RHIC has been anticipated in the two-component model of Ref. 47, with √ a rather flat behavior for s = 17–200 AGeV. This degeneracy is expected to √ √ be lifted at higher (LHC, s = 5500 AGeV) and lower (FAIR, s = 8 AGeV) o In
part, this is due to a finite correlation volume introduced for c¯ c quarks in the approach of Refs. 207 and 190.
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collision energies. At LHC, the statistical hadronization model predicts the inclusive J/ψ RAA in central Pb–Pb collisions to recover the level in p–p collisions, i.e., RAA (Npart = 350) → 1, based on a p–p open-charm cross section of dσc¯c /dy = 640 µb. At FAIR energies, on the other hand, statistical production is small,208 while transport250 and rate-equation approaches predict about a factor of two suppression, mostly dominated by nuclear absorption. The effective nuclear absorption cross section will thus be an essential quantity to be determined in p–A reactions at FAIR. Finally, let us turn to bottomonium production, which adds several new aspects compared to charmonium production: (i) the binding energies of bottomonium states are larger by about a factor of ∼ 2 which opens a wider window to study their dependence on color screening (due to larger dissociation temperatures) and makes them more robust in the hadronic phase; (ii) at given collision energy, the number c ones (e.g., by about a of b¯b pairs is substantially smaller than the number of c¯ factor of ∼ 200 at RHIC28,145 ) (iii) bottom quarks are less susceptible to changes in their momentum distributions due to their factor ∼ 3 larger mass (as discussed in Secs. 2 and 3 of this review). The latter 2 points suggest that regeneration processes play less of a role.p Early analyses of Υ production in heavy-ion collisions have focused on the pt -dependence of suppression scenarios where instantaneous dissociation above a critical energy density has been combined with formation-time effects, both at LHC54,251,252 and RHIC.252 The opposite limit of secondary production alone has been evaluated in the statistical hadronization model.221 The thermal rate-equation approach, Eq. (103), has been applied to Υ production in Ref. 53, in analogy to the charmonium sector as displayed in the left panels of Figs. 36 and 39. The time evolution of the Υ(1S) yield in central Au–Au collisions at RHIC has been calculated for the following two scenarios (as shown in Fig. 41): in the first one (left panel), reduced in-medium binding energies (according to solutions of a Schr¨ odinger equation with a color-screened Cornell potential181 ) are combined with quasifree dissociation (and formation) reactions; in the second one, the gluo-dissociation process is applied to the Υ(1S) with vacuum binding energy (assuming mb = 5.28 GeV in connection with εB = 1.1 GeV). One finds that color-screening enables a ∼ 40% suppression of the Υ(1S) within the first 1–2 fm/c (where the average medium temperature is above 200 MeV), with insignificant eq contributions from regeneration since NΥ is too small. On the other hand, with its vacuum binding energy, the Υ(1S) is basically unaffected at RHIC. This suggests a very promising sensitivity of Υ(1S) production to color-screening at RHIC (of course, the observed Υ(1S) yield contains feed-down contributions, amounting to ca. 50% in p–p collisions; this underlines the importance of measuring the
p However,
care has to be taken in deducing that this renders Υ regeneration irrelevant at RHIC, since (a) the bottom ensemble is in the canonical limit, and (b) the regeneration yield needs to be compared to the primordial yield: in p–p collisions the ratio Υ/(b¯b) ' 0.1% is about a factor of 10 smaller than in the charm sector where J/ψ/(c¯ c) ' 1%.
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5. Conclusions Heavy-quark physics is an increasingly useful and adopted tool in the theoretical analysis of hot and dense QCD matter and its study in ultrarelativistic collisions of heavy nuclei. Key reasons for this development lie in a combination of exciting new data becoming available not only for quarkonia but also for open heavy-flavor observables, together with attractive features of charm and bottom quarks from a theoretical point of view. These features are, of course, rooted in the large scale introduced by the heavy-quark (HQ) mass, which enables the use of expansion techniques, most notably HQ effective theories and Brownian motion for HQ diffusion in a QGP fluid. Furthermore, the production of heavy quarks in nuclear reactions is presumably restricted to primary N –N collisions which renders HQ spectra a calibrated probe of the medium over the entire range of transverse momentum. The latter provides a unique opportunity for a comprehensive investigation of the QCD medium at all scales, ranging from diffusion physics in the low-pt limit to a “standard” hard probe at pt mQ . Heavy quarks thus connect transport coefficients in the QGP and observables in ultrarelativistic heavy-ion collisions in the arguably most direct way. Finally, relations between the open and hidden heavy-flavor sectors promise valuable mutual constraints, both theoretically and phenomenologically. In this review we have largely focused on aspects of soft physics for HQ propagation and binding in the QGP.
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Interactions of slowly moving heavy quarks in a QGP are dominated by elastic scattering off thermal partons. A perturbative expansion of these interactions, specifically for the HQ diffusion coefficient, shows poor convergence for coupling constants believed to be relevant for a QGP as formed in heavy-ion collisions. Several options of amending the perturbative treatment have been suggested, e.g., a reduced screening mass or running coupling constant at low momentum transfer. While increasing the interaction strength, they inevitably face the problem of little control over higher order “corrections”. Nonperturbative approaches have been put forward which can, in principle, overcome this problem by a (partial) resummation of large contributions. E.g., a potential-based T -matrix approach characterized by a scattering equation becomes particularly promising if the input interaction can be specified in a model-independent way, i.e., from thermal lattice QCD. Currently, open questions remain as to the validity of the potential approach at finite temperature, and a suitable definition of the potential from the HQ free energy. Here, a close connection between the open and hidden heavy-flavor sectors emerges via the same low-energy interaction operative for HQ diffusion and quarkonium bound states. Qualitatively, one finds that, if ground-state quarkonia survive until temperatures well above Tc , potential scattering of heavy quarks in the QGP builds up resonance-like correlations which are instrumental in obtaining a small HQ diffusion coefficient close to Tc . As an alternative nonperturbative approach, HQ diffusion has been estimated in the strong-coupling limit of conformal field theory (CFT) using a conjectured correspondence to string theory in Anti-de-Sitter (AdS) space. With CFT parameters adapted to resemble QCD, the resulting HQ diffusion constant is comparable to the T -matrix approach close to Tc , but is approximately constant with increasing temperature while the T -matrix interaction weakens and ultimately approaches pQCD estimates. Quantitative applications of the Brownian-motion framework to HQ observables at RHIC critically hinge on a reliable description of the background medium evolution. The latter specifies the ambient conditions in the (approximately) thermal bath including its temperature and collectivity, whose magnitudes directly impact the nuclear modification and elliptic flow of HQ spectra. A survey of available calculations indicates that the translation of a given HQ diffusion coefficient into suppression and elliptic flow of HQ spectra is currently at the ∼ 50% accuracy level. This needs to be further scrutinized and improved. In line with the theoretical expectations for HQ diffusion in a QGP at T = 1–2 Tc , the current data call for significantly stronger interactions than provided by LO pQCD. A simultaneous and consistent evaluation of spectra (RAA (pt )) and elliptic flow (v2 (pt )) is pivotal to this conclusion. A non-negligible role is played by the hadronization process: heavy-light quark coalescence processes seem to improve the experimentally observed correlation of a rather large v2 and a moderately suppressed RAA in the electron spectra at moderate peT ≤3 GeV. More robust conclusions will require a better knowledge of the Cronin effect figuring into the initial conditions for the HQ spectra. A theoretically appealing aspect of quark coalescence is its close relation to resonance correlations
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in the QGP, which could be at the origin of the nonperturbative interaction strength in HQ diffusion. From the experimental side, important discrimination power will come with an explicit measurement of D mesons, to explicitly separate the bottom contibution (present in the electron spectra). Theoretical investigations should take advantage of the opportunity to predict angular correlation measurements which will become feasible soon. We believe that charm data at SPS energy would constitute a valuable complement to RHIC data, which could help in deciding how much of the bulk flow (and suppression) can be imparted on charm quarks in a medium at smaller temperatures. In the heavy-quarkonium sector we have started with a brief synopsis of current applications of potential models in medium. At this stage, the comparison of calculated spectral functions to Euclidean correlation functions computed in lattice QCD suggests that scenarios with either strong binding and rather large in-medium HQ mass, or weak binding and smaller HQ mass, are viable. On the contrary, inelastic reaction rates are rather sensitive to the binding energy and thus to the strength of color-Debye screening, especially for bottomonia. This translates into a promising discrimination power of bottomonium suppression measurements at RHIC and LHC. The phenomenology of charmonia is presumbaly more involved; e.g., kinetic rate-equation calculations for J/ψ production in central Au–Au collisions at RHIC indicate that the number of surviving primordial J/ψ’s is comparable to the number of secondary produced ones via c–¯ c coalescence in the QGP (and/or at hadronization). Such an interpretation is consistent with J/ψ pT spectra where the coalescence yield is concentrated at low pT , thus reducing the average p2T compared to primordial production. Kinetic approaches furthermore suggest that regeneration is subleading at the SPS, and that the observed suppression occurs at energy densities above the critical one. Deeper insights will follow when advanced theoretical models meet future precision data, including rapidity dependencies, elliptic flow, excited charmonia and much needed constraints from d-Au collisions at RHIC (to pin down cold-nuclear-matter effects). An extended excitation function via a RHIC energy scan, LHC and FAIR will further disentangle suprression and regeneration effects. We emphasize again that the presence of regeneration mechanisms would imply valuable connections to the open heavy-flavor sector, as coalescing heavy quarks necessarily imprint their kinematics on HQ bound states. In summary, we believe that in-medium HQ physics will continue as a challenging but rewarding forefront research field for many years to come, with ample opportunities for surprises, insights and progress. Acknowledgments We are indebted to D. Cabrera, V. Greco, C.M. Ko, M. Mannarelli, I. Vitev and X. Zhao for productive and enjoyable collaboration on various aspects of the topics discussed in this review. We furthermore thank J. Aichelin, P. Gossiaux, W. Horowitz and P. Petreczky for illuminating discussions. This work has been
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VISCOUS HYDRODYNAMICS AND THE QUARK GLUON PLASMA
DEREK A. TEANEY Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA [email protected]
One of the most striking results from the Relativistic Heavy Ion Collider is the strong elliptic flow. This review summarizes what is observed and how these results are combined with reasonable theoretical assumptions to estimate the shear viscosity of QCD near the phase transition. A data comparison with viscous hydrodynamics and kinetic theory calculations indicates that the shear viscosity to entropy ratio is surprisingly small, η/s < 0.4. The preferred range is η/s ' (1 ↔ 3) × 1/4π.
1. Introduction One of the most striking observations from the Relativistic Heavy Ion Collider (RHIC) is the very large elliptic flow.1, 2 The primary goal of this report is to explain as succinctly as possible precisely what is observed and how the shear viscosity can be estimated from these observations. The resulting estimates3–12 indicate that the shear viscosity to entropy ratio η/s is close to the limits suggested by the uncertainty principle,13 and the result of N = 4 Super Yang–Mills (SYM) theory at strong coupling14, 15 η 1 = . s 4π These estimates imply that the heavy ion experiments are probing quantum kinetic processes in this theoretically interesting, but poorly understood regime. Clearly a complete understanding of nucleus-nucleus collisions at high energies is extraordinarily difficult. We will attempt to explain the theoretical basis for these recent claims and the uncertainties in the estimated values of η/s. Additionally, since the result has raised considerable interest outside of the heavy ion community, this review will try to make the analysis accessible to a fairly broad theoretical audience. 1.1. Experimental overview In high energy nucleus-nucleus collisions at RHIC approximately ∼ 7000 par√ ticles are produced in a single gold-gold event with collision energy, s = 200 GeV/nucleon. Each nucleus has 197 nucleons and the two nuclei are initially length contracted by a factor of a hundred. The transverse size of the nucleus is 207
208
D. A. Teaney
RAu Beam
θ
RAu γ
Z Fig. 1. Overview of a heavy ion event. In the left figure the two nuclei collide along the beam axis usually labeled as Z. At RHIC the nuclei are length contracted by a factor of γ ' 100. The right figure shows the collision vertex of a typical event as viewed in a schematic particle detector and shows a few of the thousands of charged particle tracks recorded per event. The angle θ is usually reported in pseudo-rapidity variables as discussed in the text.
RAu ∼ 5 fm and the duration of the event is roughly of order ∼ RAu /c. Figure 1 shows the pre-collision geometry. Also shown is a schematic of the collision vertex and a schematic particle detector. Usually the two nuclei collide off-center at impact parameter b and oriented at an angle ΨRP with respect to the lab axes as shown in Fig. 2. During the collision the spectator nucleons (see Fig. 2) continue down the beam pipe, leaving behind an excited almond shaped region. The impact parameter b is a transverse vector b = (bx , by ) pointing from the center of one nucleus to the center of the other. As discussed in Sec. 2 both the magnitude and direction of b can be determined on an event by event basis. We will generally work with reaction plane coordinates X and Y rather than lab coordinates. The elliptic flow is defined as the anisotropy of particle production with respect to the reaction plane (see Fig. 2 and Fig. 3) 2 pX − p2Y , (1) v2 ≡ p2X + p2Y or the second Fourier coefficient of the azimuthal distribution, hcos(2(φ − ΨRP ))i. Elliptic flow can also be measured as a function of transverse momentum pT = p p2X + p2Y by expanding the differential yield of particles in a Fourier series 1 dN 1 dN = (1 + 2v2 (pT ) cos 2(φ − ΨRP ) + . . .) . pT dydpT dφ 2πpT dydpT
(2)
Here ellipses denote still higher harmonics, v4 , v6 and so on. In addition the flow can be measured as a function of impact parameter, particle type, and rapidity. For
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209
Y Spectators
X ΨRP
b Spectators
Fig. 2. A schematic of the transverse plane in a heavy ion event. Both the magnitude and direction of the impact parameter b can be determined on an event by event basis. X and Y label the reaction plane axes and the dotted lines indicate the lab axis. ΨRP is known as the reaction plane angle.
a mid-peripheral collision (b ' 7 fm) the average elliptic flow hv2 i is approximately 7%. This is surprising large. For instance, the ratio of particles in the X direction to the Y is 1 + 2v2 : 1 − 2v2 ' 1.3 : 1. At higher transverse momentum the elliptic flow grows and at pT ∼ 1.5 GeV elliptic flow can be as large as 15%. 1.2. An interpretation of elliptic flow The generally accepted explanation for the observed flow is illustrated in Fig. 3. Since the pressure gradient in the X direction is larger than in the Y direction, the nuclear medium expands preferentially along the short axis of the ellipse. Elliptic flow is such a useful observable because it is a rather direct probe of the response of the QCD medium to the high energy density created during the event. If the mean free path is large compared to the size of the interaction region, then the produced particles will not respond to the initial geometry. On the other hand, if the transverse size of the nucleus is large compared to the interaction length scales involved, hydrodynamics is the appropriate theoretical framework to calculate the response of the medium to the geometry. In a pioneering paper by Ollitrualt, the elliptic flow observable was proposed and analyzed based partly on the conviction that ideal hydrodynamic models would vastly over-predict the flow.16, 17 However, calculations based on ideal hydrodynamics do a fair to reasonable job job in reproducing the observed elliptic flow.18–22 This has been reviewed elsewhere.23, 24 Nevertheless, the hydrodynamic interpretation requires that the relevant mean free paths and relaxation times be small compared to the nuclear sizes and expansion rates. This review will assess the consistency of the hydrodynamic interpretation by categorizing viscous corrections. The principle tool is viscous hydrodynamics which needs to be extended into the relativistic domain in order to
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Y
1 0 1 1
X
Fig. 3. The conventional explanation for the observed elliptic flow. The spectators continue down the beam pipe leaving behind an excited oval shape which expands preferentially along the short axis of the ellipse. The finally momentum asymmetry in the particle distribution v2 reflects the response of the excited medium to this geometry. The dot with transverse coordinate x = (x, y) is illustrated to explain a technical point in Sec. 2.
address the problems associated with nuclear collisions. This problem has received considerable attention recently and progress has been achieved both at a conceptual25–29 and practical level.5, 9–11, 30, 31 Generally macroscopic approaches, such as viscous hydrodynamics, and microscopic approaches, such as kinetic theory, are converging on the implications of the measured elliptic flow.6, 12, 29, 32–34 There has never been an even remotely successful model of the flow with η/s > 0.4. Since η/s is a measure of the relaxation time relative to ~/kB T (see Sec. 3), this estimate of η/s places the kinetic processes measured at RHIC in an interesting and fully quantum regime. 2. Elliptic Flow
Measurements and Definitions
The goal of this section is to review the progress that has been achieved in measuring the elliptic flow. This progress has produced an increasingly self-consistent hydrodynamic interpretation of the observed elliptic flow results. This section will also collect the various definitions which are needed to categorize the response of the excited medium to the initial geometry. 2.1. Measurements and definitions As discussed in the introduction (see Fig. 2) both the magnitude and direction of the impact parameter can be determined on an event by event basis. The magnitude of the impact parameter can be determined by selecting events with definite multiplicity for example. For instance, on average the top 10% of events with the highest multiplicity correspond to the 10% of events with the smallest impact parameter. Since the cross section is almost purely geometrical in this energy range
Viscous Hydrodynamics and the Quark Gluon Plasma
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this top 10% of events may be found by a purely geometrical argument. This line of reasoning gives that the top 10% of events are produced by collisions with an impact parameter in the range 0 < b < b∗ ,
where
10% =
πb∗ 2 , σtot
(3)
and σtot ' π(2RA )2 is the total inelastic cross section. After categorizing the top 10% of events we can categorize the top 10-20% of events and so on. The general relation is 2 b ' % Centrality . (4) 2RA Here we have neglected fluctuations and many other effects. For instance there is a very small probability that an event with impact parameter b = 4 fm will produce the same multiplicity as an event with b = 0 fm. A full discussion of these and many other issues is given in Ref. 35. The end result is that the magnitude of the impact parameter b can be determined to within half a femtometer or so.36 Now that the impact parameter is quantified, a useful definition is the number of participating nucleons (also called “wounded” nucleons). The number of nucleons per unit volume in the rest frame of the nucleus is ρA (x − xo , z), were x − xo is the transverse displacement from a nucleus centered at xo , and z is the longitudinal direction. These distributions are known experimentally and are reasonably modeled by a Woods-Saxon form.35 The number of nucleons per unit transverse area is Z ∞ TA (x − xo ) = dz ρA (x − xo , z) . (5) −∞
Then, after reexamining Fig. 3, we find that the probability that a nucleon at x = (x, y) will suffer an inelastic interaction passing through the right nucleus centered b/2 = (+b/2, 0) is 1 − exp (−σNN TA (x − b/2)) , where σNN ' 40 mb is the inelastic nucleon-nucleon cross section. The number of nucleons which suffer an inelastic collision per unit area is then dNp = TA (x⊥ + b/2) [1 − exp (−σNN TA (x⊥ − b/2))] dxdy + TA (x⊥ − b/2) [1 − exp (−σNN TA (x⊥ + b/2))] .
(6)
Finally, the total number of participants (i.e. the the number of nucleons which collide) is Z dN Np = dx dy . (7) dxdy
For a central collision of two gold nuclei the number of participants Np ' 340 nearly equals the total number nucleons in the two nuclei, N = 394, leaving about fifty
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b/(2RA)
1 0.9
1.0 0.8
0.6
0.5
0.3 0.2
0.0
0.8 0.7
Rrms/RA
0.6 0.5 2
2 ε =
0.4 0.3 0.2 0.1 0
0
0.2
0.4
0.6
0.8
1 max
Np/N p
Fig. 4. The standard Glauber eccentricity s,part as a function of the number of participants. Npmax ' 340 is the maximum number of participants in a central AuAu event and RA ' 6.3 fm is the gold radius. The top axis shows the translation between impact parameter and participants. The root mean square radius Rrms and the standard Glauber eccentricity are given in Eq. (13) and Eq. (15).
spectators. By comparing the top axis in Fig. 4 to the bottom axis, the relationship between participants, impact parameter b, and centrality can be determined. The reaction plane angle, ΨRP , is also determined experimentally. Here we will describe the Event Plane method which is conceptually the simplest. Assume first that the reaction plane angle is known. Then the particle distribution can be expanded in harmonics about the reaction plane dN ∝ 1 + 2v2 cos(2(φ − ΨRP )) + . . . (8) dφ If the number of particles is very large one could simply make a histogram of the angular distribution of particles in an event with respect to the lab axis. Then the reaction plane angle could be determined by finding where the histogram is maximum. This is the basis of the event plane method.37 For all the particles in the event we form the vector ! X X ~ = (Qx , Qy ) = (9) Q cos 2φi , sin 2φi . i
i
R Using the continuum approximation, Qx ' dφ dN/dφ cos(2φ), we can estimate ~ the reaction plane angle ΨRP , from the Q-vector ~ Q ≡ (cos(2Ψ2 ), sin(2Ψ2 )) ' (cos(2ΨRP ) , sin(2ΨRP )) . ~ |Q|
(10)
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Then we can estimate the elliptic flow as v2obs ' hcos(2(φi − Ψ2 ))i. The estimated angle Ψ2 differs from ΨRP due to statistical fluctuations. Consequently v2obs will be systematically smaller than v2 since Ψ2 is not ΨRP . This leads to a correction to the estimate given above which is known as the reaction plane resolution. The final result, after considering the dispersion of Ψ2 relative to the true reaction plane angle ΨRP is v2 =
v2obs R
where
R = hcos 2(Ψ2 − ΨRP )i .
(11)
In practice the resolution parameter R is estimated by dividing a given event into sub-events and looking at the dispersion in Ψ2 between different sub-events. There is a lot more to the determination of the event plane in practice. Fortunately the various methods have been reviewed recently.37 An important criterion for the validity of these methods is that the magnitude of elliptic flow be large compared to statistical fluctuations v22
1 . N
(12)
For v2 ' 7% and N ' 500 we have N v22 ' 2.5. Since this number is not particularly large the simple method described above is not completely adequate in practice. The resolution parameter is R ' 0.7 in the STAR experiment. At the LHC, estimates suggest that the resolution parameter R could be as large as38 R ' 0.95. Current methods use two particle, four particle, and higher cummulants to remove the effects of correlations and fluctuations. These advances are discussed more completely in Sec. 2.3 and have played an important role in the current estimates of the shear viscosity. The current measurements provide a unique theoretical opportunity to study systematically how hydrodynamics begins to develop in mesoscopic systems. We would like to measure the response of nuclei to the geometry. To this end, we categorize the overlap region with an asymmetry parameter s,part
2 y − x2 . (13) s,part = 2 hy + x2 i Traditionally the average h. . .i is taken with respect to the number of participants in the transverse plane, for example Z
2 1 dNp y − x2 = dxdy (y 2 − x2 ) . (14) Np dx dy
We will explain the “s,part” label shortly; for the moment we return to Fig. 4, which plots the asymmetry parameter versus centrality and also shows the the root mean square radius p Rrms = hx2 + y 2 i , (15)
which is important for categorizing the size of viscous corrections.
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Au+Au 200 GeV 60% - 70% 50% - 60% 40% - 50% 30% - 40% 20% - 30% 10% - 20% 5% - 10%
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Fig. 5. Elliptic flow v2 (pT ) as measured by the STAR collaboration39, 40 for different centralities. The measured elliptic flow has been divided by the eccentricity – hydro ≡ s,part in this work. The curves are ideal hydrodynamic calculations based on Refs. 23 and 21 rather than the viscous hydrodynamics discussed in much of this review.
2.2. Interpretation We have collected the essential definitions of , centrality, and v2 , and are now in a position to return to the physics. The scaled elliptic flow v2 / measures the response of the medium to the initial geometry. Figure 5 shows v2 (pT )/ as a function of centrality, 0-5% being the most central and 60-70% being the most peripheral. Examining this figure we see a gradual transition from a weak to a strong dynamic response with growing system size. The interpretation adopted in this review is that this change is a consequence of a system transitioning from a kinetic to a hydrodynamic regime. There are several theoretical curves based upon calculations of ideal hydrodynamics20, 21 which for pT < 1 GeV approximately reproduce the observed elliptic flow in the most central collisions. Since ideal hydrodynamics is scale invariant (for
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Transverse Momentum pT (GeV/c) Fig. 6. A comparison of the elliptic flow of pions and protons to the elliptic flow of the multi-strange φ and Ω− hadrons.43
a scale invariant equation of state) the expectation is that the response v2 / of this theory should be independent of system size or centrality. This reasoning is borne out by the more elaborate hydrodynamic calculations shown in the figure. On the other hand, the data show a gradual transition as a function of increasing centrality, rising towards the ideal hydrodynamic calculations in a systematic way. These trends are captured by models with a finite mean free path.41 The data show other trends as a function of centrality. In more central collisions the linearly rising trend, which resembles the ideal hydrodynamic calculations, extends to larger and larger transverse momentum. We will see in Sec. 5 that viscous corrections to ideal hydrodynamics grow as p 2 ` mfp T , T L
(16)
where L is a characteristic length scale. Thus these viscous corrections restrict the applicable momentum range in hydrodynamics.4 In more central collisions, where `mfp /L is smaller, the transverse momentum range described by hydrodynamics extends to increasingly large pT . These qualitative trends are reproduced by the more involved viscous calculations discussed in Sec. 6. To conclude this section, we turn to Fig. 6 which compares the elliptic protons and pions to the flow of the multi-strange hadrons Ω− and φ. (These hadrons have valence quark content sss and s¯ s respectively.) The important point is that the Ω− is nearly twice as heavy as the proton and more importantly, does not have a strong resonant interaction analogous to the ∆. For these reasons the hadronic relaxation time of the Ω− is expected to be much longer than the duration of the heavy ion event.42 Nevertheless the Ω shows nearly the same elliptic flow as the protons. This
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CYM, m = 0.2 GeV CYM, m = 0.5 GeV Glauber, Npart Glauber, Ncoll
0.6 0.5
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Fig. 7. Figure from Ref. 48 showing various estimates for the initial eccentricity in heavy ion collisions. The physics of the KLN eccentricity is described in the text. In the KLN model the eccentricity is expected to increase by about 20% when going from RHIC to the LHC.41
provides fairly convincing evidence that the majority of the elliptic flow develops during a deconfined phase which hadronizes to produce a flowing Ω− baryon. 2.3. The eccentricity and fluctuations Clearly much of the interpretation of elliptic flow relies on a solid understanding of the eccentricity. There are several issues here. First there is the theoretical uncertainty in this average quantity. For example, so far we have defined the “standard Glauber participant eccentricity” in Eq. (13). An equally good definition is provided by collision scaling. For instance, one measure used in heavy ion collisions is the number of binary nucleon-nucleon collisions per transverse area d2 Ncoll = σNN TA (x + b/2)TA (x − b/2) , dxdy
(17)
Then the eccentricity is defined with this Ncoll weight in analogy with Eq. (13). Figure 7 shows the “standard Glauber Ncoll eccentricity”. Another more sophisticated model is provided by the KLN model which is based on the ideas of gluon saturation and the Color Glass Condensate (CGC)44, 45 as implemented in Refs. 46 and 47. This model is a safe upper bound on what can be expected for the eccentricity from saturation physics and is also shown in Fig. 7. We can not describe the details of this model and its implementation here. However, the physical reason why this model has a sharper eccentricity is the readily understood: the center of one nucleus (nucleus A) passes through the edge of the other nucleus (nucleus B). Since the density of gluons per unit area in the initial wave function is larger in the center of a nucleus relative to the edge, the typical momentum scale of nucleus A
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(∼ Qs,A ) is larger nucleus B (∼ Qs,B ). It is then difficult for the long wavelength (low momentum) gluons in B to liberate the short wavelength gluons in A. The result is that the production of gluons falls off more quickly near the x edge relative to the y edge making the eccentricity larger. Clearly this physics is largely correct although the magnitude of the effect is uncertain. Another CGC estimate of is based on classical simulations of Yang–Mills fields. The simulations include similar saturation physics but model the production and non-perturbative sectors differently. The eccentricity from these simulations is also shown in Fig. 7 and is similar to the Ncoll eccentricity.48 Thus the predictions of the KLN model seem to be a safe upper bound for the eccentricity in heavy ion collisions. Note that an important phenomenological consequence of the the KLN model is that the eccentricity grows with beam energy and is expected to increase about 20% from the RHIC to the LHC.41 Another important aspect in heavy ion collisions when interpreting the elliptic flow data is fluctuations in the initial eccentricity. These fluctuations are not accounted for in Fig. 7. The history is complicated and is reviewed in Refs. 49 and 37. There are fluctuations in the initial eccentricity of the participants especially in peripheral AuAu and CuCu collisions. Thus rather than using the continuum approximation given in Eq. (13) it is better to implement a Monte-Carlo Glauber calculation and estimate the eccentricity using the “participant plane eccentricity”. Figure 8 illustrates the issue: In a given event the ellipse is tilted and the eccentricity depends on the distribution of participants. This event by event eccentricity is denoted P P in the literature. Clearly the experimental goal is to extract the response coefficient C relating the elliptic flow to the eccentricity on an event by event basis v2 = CP P .
y’
(18)
y x’
x
Fig. 8. A figure from Ref. 50 illustrating the participant plane eccentricity P P in a single event.
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If the flow methods measured hv2 i, then we could simply divide the measured flow to determine the response coefficient, C = hv2 i / hP P i. The PHOBOS collaboration deciphered the confusing CuCu data by recognizing the need for P P and following this procedure.51 However, it was generally realized (see in particular. Ref. 52) that the elliptic flow methods do not measure preciselyphv2 i. Some methods (such as two particle correlations v2 {2}) are sensitive to hv22 i, while other methods (such as the event plane method v2 {EP }) measure something closer to hv2 i. What precisely the event plane method measures depends on the reaction plane resolution in a known way.49 So just dividing the measured flow by the average participant eccentricity is not entirely correct. The appropriate quantity to divide by depends on the method.50, 52, 53 In a Gaussian approximation for the eccentricity fluctuations this can be worked out analytically. p For instance, the two hv22 i), should be divided particle correlation elliptic flow v {2} (which measures 2 p 2 by hP P i . An important corollary of this analysis is that v2 {4} (v2 measured from four particle correlations) can be divided by s of Eq. (13) to yield a good estimate of the coefficient C. This is the policy adopted in Fig. 5. Unfortunately, in the most peripheral AuAu bins and in CuCu the Gaussian approximation is poor due to strong correlations amongst the participants.54 These correlations arise because participants come in pairs and every participant is associated with another participant in the other nucleus. Presumably the last centrality bin in Fig. 5 could be moved up or down somewhat due to non-Gaussian corrections of this sort. With a complete understanding of what each method measures, Ref. 49 was able to make a simple model for the fluctuations and non-flow and show that hv2 i measured by the different methods are compatible to an extremely good precision. This work should be extended to the CuCu system where non-Gaussian fluctuations are stronger and ultimately corroborate the PHOBOS analysis.51, 54 This is a worthwhile goal because it will clarify the transition into the hydrodynamic regime.8 2.4. Summary In this section we have gone into considerable experimental detail – perhaps more than necessary to explain the basic ideas. The reason for this lengthy summary is because the trends seen in Fig. 5 were not always so transparent. The relatively coherent hydrodynamic and kinetic interpretation of the observed elliptic flow (which was previewed in Sec. 2.2 and which is discussed more completely below) is the result of careful experimental analysis. 3. The Shear Viscosity in QCD In this section we will discuss thermal QCD in equilibrium with the primary goal of collecting various theoretical estimates for the shear viscosity in QCD. The prominent feature of QCD at finite temperature is the presence of an approximate phase transition from hadrons to quarks and gluons. The Equation of State (EoS) from lattice QCD calculations is shown in Fig. 9, and the energy
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p4 asqtad p4 asqtad
4 2
T [MeV]
0 100 150 200 250 300 350 400 450 500 550 Fig. 9. Figure from Ref. 55 illustrating the energy density and pressure by T 4 of QCD computed with Nτ = 8 lattice data. (In this figure is energy density e(T ) and the pressure p is denoted with P throughout this review.) SB /T 4 ≡ eSB /T 4 is the energy density of a free three flavor massless QGP (see text).
density e(T ) shows a rapid change for the temperature range, T ' 170 − 220 MeV. As estimated in Sec. 4, the transition region is directly probed during high energy heavy ion collisions. Well below the phase transition, the gas of hadrons is very dilute and the thermodynamics is dominated by the measured particle spectrum. For instance the number of pions in this low temperatures regime is nπ = dπ
Z
d3 p 1 , 3 E /T p (2π) e −1
(19)
p where Ep = p2 + m2π and dπ = 3 counts the three fold isospin degeneracy, + − 0 π , π , π , in the spectrum. If all known particles are included up to a mass mres < 2.5 GeV, the resulting Hadron Resonance Gas (HRG) equation of state does a reasonable job of reproducing the thermodynamics up to about T ' 180 MeV. However, the validity of this quasi-particle description is unclear above a temperature of Ref. 56, T ' 140 MeV. As the temperature increases, the hadron wave functions overlap until the medium reorganizes into quark and gluon degrees of freedom. Well above the transition the QCD medium evolves to a phase of massless quarks and gluons. The energy density is approximately described by the StefanBoltzmann equation of state eglue = dglue
Z
d3 p Ep , 3 E p (2π) e /T − 1
equark = dquark
Z
d3 p Ep , 3 E p (2π) e /T + 1
(20)
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where dglue = 2×8 counts spin and color, and dquark = 2×2×3×3 counts spin, antiquarks, flavor, and color. Performing these integrals we find, eSB = eglue + equark ' 15.6 T 4 as illustrated by the line in the top-right corner of the figure. We have described the particle content well above and well below the transition. Near the approximate phase transition the validity of such a simple quasi-particle description is not clear. The transition is a rapid cross-over where hadron degrees of freedom evolve into quark and gluon degrees of freedom rather than a true phase transition. All correlators change smoothly, but rapidly, in a temperature range of T ' 170 − 210 MeV. From a phenomenological perspective the smoothness of the transition suggests that the change from quarks to hadrons should be thought of as a soft process rather than an abrupt change. Lattice QCD simulations have determined the equation of state rather well. However, in addition to the equation of state, we need to estimate the transport coefficients to assess whether the heavy ion reactions produce enough material, over a large enough space-time volume to be described in thermodynamic terms. The shear and bulk viscosities govern the transport of energy and momentum and are clearly the most important. Later in Sec. 4 and Sec. 5 we will describe the role of shear viscosity in the reaction dynamics. In this section we summarize the shear viscosities found in various theoretical computations which will place these dynamical conclusions in context. A good way to implement this theoretical summary is to form shear viscosity to entropy ratio,15 η/s. To motivate this ratio we remark that it seems difficult to transport energy faster than a quantum time scale set by the inverse temperaturea , τquant ∼
~ . kB T
A sound wave propagating with speed cs will diffuse (or spread out) due to the shear viscosity. Linearized hydrodynamics shows that this process is controlled by the momentum diffusion coefficient, Dη ≡ η/(e+P), where e+P is the enthalpy (see for example Ref. 57). Noting that the diffusion coefficient has units of (distance)2 /time, a kinetic theory estimate for the diffusion process yields Dη ≡
η 2 ∼ vth τR , e+P
(21)
2 where τR is the particle relaxation time and vth ∼ c2s is the particle velocity. Dividing 2 by vth and using the thermodynamic estimates 2 sT ∼ evth ∼ P ∼ n kB T ,
(22)
η ~ τR ∼ τR T ∼ . s kB τquant
(23)
we see that
a In
this paragraph we will restore ~ and the Boltzmann constant, kB .
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Therefore, η/s is the ratio between the medium relaxation time and the quantum time scale τquant in units of ~/kB , i.e. a measure of the transport time in “natural units”. In the dilute regime the ratio between the medium relaxation time and the quantum time scale is long and kinetic theory can be used to calculate the shear viscosity to entropy ratio. First we consider a simple classical massless gas with particle density n and a constant hard sphere cross section σo . The equation of state of this gas is e = 3P = 3nT and the shear viscosity is computed using kinetic theory58 η ' 1.2
T , σo
(24)
The entropy is s = (e + P)/T and the resulting shear to entropy ratio is T η ' 0.3 . s nσo
(25)
In what follows, this calculation will provide a qualitative understanding of more sophisticated kinetic calculations. In the dilute hadronic regime, η/s was calculated in Ref. 59 using measured elastic cross sections for a gas of pions and kaons. In the ππ phase shifts there is a prominent ρ resonance, while in the πK channel there is a prominent K ∗ resonance. Thus the equation of state of this gas is well modeled by an ideal gas of π, K, ρ and K ∗ .60, 61 The viscosity of this mixture was computed in Ref. 59 and the current author digitized this viscosity, computed the entropy, and determined the η/s ratio. This is shown in Fig. 10. Slightly larger values were obtained in Ref. 56 which also estimated the range of validity for hadronic kinetic theory, T < ∼ 140 MeV. Finally a more involved Kubo analysis of the UrQMD hadronic transport model62 (which includes many resonances) is also displayed in Fig. 10. At asymptotically high temperatures the coupling constant αs is weak and the shear viscosity can be computed using perturbation theory. Initially, only 2 → 2 elastic scattering was considered, and the shear viscosity was computed in a leading log plasma with self consistent screening.64 Later it was recognized65, 66 that collinear Bremsstrahlung processes are important for the calculation of shear viscosity and this realization ultimately resulted in a complete leading order calculation.63 We can estimate η/s in the perturbative plasma using Eq. (24) with s ∝ T 3 and σ ∝ α2s /T 2 , 1 η ∼ 2. s αs
(26)
The final result from a complete calculation has the form η 1 = 2 F (mD /T ) , s αs
(27)
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T(MeV) Fig. 10. (Color Online) A compilation of values of η/s. The results from Prakash et al are from Ref. 59 and describe a meson gas of pions and kaons (and indirectly K ∗ and ρ) computed with measured cross sections. The black points are based on a Kubo analysis of the UrQMD code which includes many higher resonances.62 The red lines are different implementations of the AMY (Arnold, Moore, Yaffe) calculation of shear viscosity.63 In each curve the Debye scale is fixed mD = 2T . In the dashed red curves the (one loop three flavor) running coupling is taken at the scale µ. In the solid red curves αs is kept fixed. The two loop running coupling is shown with µ = 2πT for comparison and the two loop µ = πT (not shown) is similar to the one loop µ = 2πT result. In the AMY curves, changing the Debye mass by ±0.5T changes η/s by ∼ ±30%. Finally the thin dashed line indicates a simple model discussed in the text with `mfp = 1/T .
where F (mD /T ) is a function of the Debye mass which was computed for mD /T small and then extrapolated to more realistic values.63 There are many scales in the problem and it is difficult to know what precisely to take for the Debye mass and the coupling constant. At lowest order in the coupling, the Debye mass is67 Nc Nf + g2T 2 , (28) m2D = 3 6 but this is too large to be considered reliable. For definiteness we have evaluated the leading coupling constant in Eq. (27) at a scale of πT and set the Debye mass to mD = 2T . The resulting value of η/s is shown in Fig. 10. Various other alternatives are explored in the figure and underscore the ambiguity in these numbers. Clearly all of the calculations presented have a great deal of uncertainty around the phase transition region. On the hadronic side there are a large number of
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inelastic reactions which become important. On the quark gluon plasma side, the strong dependence on the Debye scale and the coupling constant is disconcerting. It is very useful to have a strongly coupled theory where the shear viscosity to entropy ratio can be computed exactly. In strongly coupled N = 4 SYM theory with a large number of colors, η/s can be computed using gauge gravity duality and yields the result14, 15 1 η = . s 4π
(29)
From the perspective of heavy ion physics this result was important because it showed that there exist field theories where η/s can be this low. Although N = 4 has no particle interpretation, we note that extrapolating Eq. (24) by setting `mfp = 1/nσo = 1/πT yields a value for η/s which is approximately equal to the SYM result. In Fig. 10 we have displayed this numerology with `mfp = 1/T for clarity. There are many aspects of transport coefficients which have not been reviewed here. For instance, there is an ongoing effort to determine the transport coefficients of QCD from the lattice.68, 69 While a precise determination of the transport coefficients is very difficult,57, 70, 71 the lattice may be able to determine enough about the spectral densities to distinguish the orthogonal pictures represented by N = 4 SYM theory and kinetic theory.68 This is clearly an important goal and we refer to Ref. 72 for theoretical background. Also throughout this review we have emphasized the shear viscosity and neglected the bulk viscosity. This is because on the hadronic side of the phase transition the bulk viscosity is a thousand times smaller than the shear viscosity in the regime where it can be reliably calculated.59 Similarly on the high temperature QGP side of the phase transition the bulk viscosity is also a thousand times smaller than shear.73 However, near a second order phase transition the bulk viscosity can become very large.74–76 Nevertheless the rapid cross-over seen in Fig. 9 is not particularly close to a second order phase transition and universality arguments can be questioned (see Ref. 55 for a discussion in the context of the chiral susceptibility.) Given the ambiguity at this moment it seems prudent to leave the bulk viscosity to future review. 4. Hydrodynamic Description of Heavy Ion Collisions In the previous sections we analyzed the phase diagram of QCD and estimated the transport coefficients in different phases. In this section we will study the hydrodynamic modeling of heavy ion collisions. In Sec. 4.2 we will consider ideal hydrodynamics and assume that the mean free paths are small enough to support this interpretation. Subsequently we will study viscous hydrodynamics in Sec. 4.3. Sec. 4.4 will analyze the ratio of the viscous terms to the ideal terms and use the estimates of the transport coefficients given above to assess the validity of the hydrodynamic interpretation. Sec. 4.6 will discuss the recent advances in interpreting the hydrodynamic equations beyond the Navier
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Stokes limit. This work will lay the foundation for the more detailed hydrodynamic models presented in Sec. 6. 4.1. Ideal hydrodynamics The stress tensor of an ideal fluid and its equation of motion are simply T µν = euµ uν + P∆µν ,
∂µ T µν = 0 ,
(30)
where e is the energy density, P(e) is the pressure, and uµ = (γ, γv) is the four velocity. Here we will use the metric (−, +, +, +) and define the projection tensor, ∆µν = g µν + uµ uν , with uµ uµ = −1 and ∆µν uµ = 0. This decomposition of the stress tensor is simply a reflection of the fact that in the local rest frame of a thermalized medium the stress tensor must have the form, diag(e, P, P, P). In developing viscous hydrodynamics we will define two derivatives which are the time derivative D, and the spatial derivatives ∇µ in the local rest frame D ≡ uµ ∂µ ,
∇µ ≡ ∆µν ∂µ .
(31)
Using ∂µ = −uµ D + ∇µ and uµ Duµ = 0, the ideal equations of motion can be written De = −(e + P)∇µ uµ ,
(32)
∇ P . e+P
(33)
Duµ = −
µ
The first equation says that the change in energy density is due to the PdV work or equivalently that entropy is conserved. To see this we associate ∇µ uµ with the fractional change in volume per unit time in the co-moving frame, dV /V = dt × ∇µ uµ , and use the thermodynamic identity, d(eV ) = T d(sV ) − PdV . The second equation says that the acceleration is due to the gradients of pressure. The enthalpy plays the role of the mass density in a relativistic theory. 4.2. Ideal Bjorken evolutions and three dimensional estimates In this section we will follow an analysis due to Bjorken77 and apply ideal hydrodynamics to heavy ion collisions. Bjorken’s analysis was subsequently extended in important ways.13, 78, 79 In a high energy heavy ion collisions the two nuclei pass through each other and the partons are scarcely stopped. This statement underlies much of the interpretation of high energy events and an enormous amount of data is consistent with this assumption. For a time which is short compared to the transverse size of the nucleus, the transverse expansion can be ignored. Given that the nuclear constituents pass through each other, the longitudinal momentum is much much larger than the transverse momentum. Because of this scale separation there is a strong identification between the space-time coordinates
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and the typical z momentum. For example a particle with typical momentum pz and energy E will be found in a definite region of space time vz =
pz z ' . E t
(34)
This kinematics is best analyzed with proper time and space-time rapidity variablesb , τ and ηs p 1 t+z 2 2 . ηs ≡ log τ ≡ t −z , 2 t−z
At a proper time τ particles with rapidity y are predominantly located at space time rapidity ηs y≡
1 pz + E 1 t+z log ' log ≡ ηs . 2 E − pz 2 t−z
(35)
Figure 11 illustrates these coordinates and shows schematically the identification between ηs and y. At an initial proper time τo , there is a collection of particles predominantly moving with four velocity uµ in each space-time rapidity slice 0 u + uz 1 log ' ηs . (36) 2 u0 − uz The beam rapidity at RHIC is ybeam ' 5.3 and therefore roughly speaking the particles are produced in the space-time rapidity range −5.3 < ηs < 5.3. It is important to realize that (up to about a unit or so) each space-time rapidity slice is associated with a definite angle in the detector. For ultra-relativistic particles E ' p we have 1 p + pz 1 1 + cos θ ηs ' y ' log = log ≡ ηpseudo , (37) 2 p − pz 2 1 − cos θ where a particular θ is shown in Fig. 1. The measured pseudo-rapidity distribution of charged particles is shown in Fig. 12. We can estimate the energy in a unit of pseudo-rapidity by taking hEi ' 0.5 GeV as the energy per particle. Then the energy in a pseudo-rapidity unit is dE dNch ' hEi × 1.5 ' 3.0 GeV × (Np /2) , dηpseudo dηpseudo where (Np /2) ' 170 is the number of participant pairs in a central event. The factor of 1.5 has been inserted to account for the fact that there are approximately equal numbers of π + , π − and π 0 (the most abundant particle) but only π + and π − b Here η denotes the space time rapidity, η s pseudo denotes the pseudo-rapidity (see below), η denotes the shear viscosity. In raised space time indices in τ, ηs coordinates we will omit the “s” when confusion can not arise, e.g. π ηη = π ηs ηs .
time
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Spectator Nucleons
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0 -3
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z Fig. 11. A figure motivating for the Bjorken model. The space between the dashed lines of constant ηs are referred to as a space-time rapidity slice in the text. Lines of constant proper time τ are given by the solid hyperbolas. The collection of particles in the ηs = 0 rapidity slice is indicated by the small arrows for the central (ηs = 0) rapidity slice only. The solid arrows indicates the average four velocity uµ in each slice. The spectators are those nucleons which do not participate in the collision and lie along the light cone.
5 200 GeV 0-6% 200 GeV 35-40% 19.6 GeV 0-6% 19.6 GeV 35-40%
Au+Au dNch/dη/〈Npart/2〉
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Fig. 12. The measured charged particle pseudo-rapidity distribution dNch /dηpseudo for different beam energies divided by the number of participant pairs, Np /2. Np /2 ' 170 for a central (0-6%) √ AuAu collision. This review focuses on s = 200 GeV/nucleon.
are counted in dNch /dηpseudo . This estimate agrees reasonably with the measured dET /dηpseudo ' 3.2 GeV × Np /2 from Ref. 80. Bjorken used these kinematic ideas to estimate the initial energy density in the ηs = 0 rapidity slice at an initial time, τo ' 1 fm. The estimate is based on the fairly
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well supported assumption that the energy which finally flows into the detector dET /dηpseudo largely reflects the initial energy in a given space-time rapidity slice 1 ∆E 1 ∆E 1 dET ' , (38) ' A ∆z Aτo ∆ηs Aτo dηpseudo GeV ' 5.5 3 . (39) fm In the last line we have estimated the area of a gold nucleus as A ' 100 fm2 , taken τo ' 1 fm, and used the measured dET /dηpseudo . This estimate is generally considered a lower limit since during the expansion there is PdV work as the particles in one rapidity slice push against the particles in another rapidity slice13, 78, 79 (See Fig. 11). Using the equation of state in Fig. 9 we estimate an initial temperature, T (τo ) ' 250 MeV. As mentioned above this estimate is somewhat low for hydrodynamic calculations and a more typical temperature is T ' 310 MeV, which has roughly twice the Bjorken density.24 As seen in Fig. 12, the distribution of the energy density e(τo , ηs ) in spacetime rapidity is not uniform. In the Color Glass Condensate (CGC) picture for instance, the final distribution of multiplicity is related to the x distribution of partons inside the nucleus.44 Bjorken made the additional simplifying assumption that the energy density is uniform in space-time rapidity, i.e. e(τo , ηs ) ' e(τo ). With this simplification, the identification between the fluid and space time rapidities remains fixed as the fluid flows into the forward light cone. We have discussed the motivation for the Bjorken model. Formally the model consists of the following ansatz for the hydrodynamic variables Bj '
e(t, x) = e(τ ) ,
uµ (t, x) = (u0 , ux , uy , uz ) = (cosh(ηs ), 0, 0, sinh(ηs )) .
(40)
The model is invariant under boosts in the z direction. Thus given a physical quantity at mid-rapidity (ηs = 0), one can determine this quantity at all other rapidities by a longitudinal boost. We will use curvilinear coordinates where81 xµ = (τ, x⊥ , ηs ) ,
gµν = diag(−1, 1, 1, τ 2 ) ,
(uτ , ux , uy , uη ) = (1, 0, 0, 0) . (41) In this coordinate system boost invariance implies that everything is independent of ηs . To interpret a tensorial component in these coordinates, we multiply √ by gηη = τ for every raised ηs index, and subsequently associate the product with the corresponding cartesian component at mid-rapidity. For example, τ 2 T ηη = T zz |ηs =0 = P. Similarly, τ uη = uz |ηs =0 = 0 for boost invariant flow. Substituting the boost invariant ansatz (Eq. (40)) into the conservation laws yields the following equation for the energy densityc de e+P =− . (42) dτ τ c A quick way to derive this is to work in a neighborhood of z = η = 0 where uz ' z/t. Substituting s this approximate form into ∂µ T µν = 0 in cartesian coordinates, quickly yields Eq. (42) with the replacement t → τ .
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Multiplying this equation by the volume of a space time rapidity slice V = τ ∆ηs A (see Fig. 11) we find d(e τ ∆ηs A) = −P d(τ ∆ηs A) ,
(43)
and we can interpret this result13 as saying that the energy energy per unit spacetime rapidity (e τ ∆ηs A) decreases due to the PdV work. It is for this reason that the Bjorken estimate eBj (which assumes that the r.h.s. of Eq. (43) equals zero) should be considered a lower bound. In general, assuming Bjorken scaling (Eq. (40)) and the conservation laws, but not assuming local thermal equilibrium, one finds e + T zz de =− , dτ τ
(44)
where T zz ≡ τ 2 T ηη is the effective longitudinal pressure. Viscous corrections will modify T zz from its equilibrium value of P. Returning to the equilibrium case, Eq. (42) can be solved for a massless ideal gas equation of state (e = 3P ∝ T 4 ) and the time dependence of the temperature is τ 1/3 o T (τ ) = To , (45) τ where To is the initial temperature. The temperature decreases rather slowly as a function of proper time during the initial one dimensional expansion. This will turn out to be important when discussing equilibration. For a massless ideal gas, the entropy is s = (e + P)/T ∝ T 3 and decreases as τo s(τ ) = so . (46) τ Now we discuss what happens when the initial energy density distribution is not uniform in rapidity. Due to pressure gradients in the longitudinal direction, there is some longitudinal acceleration. This changes the strict identification between the space time rapidity and the fluid rapidity given in Eq. (36). It also changes the temperature dependence given above. One way to quantify this effect is to look at the results of 3D ideal hydrodynamic calculations and study the differences between R 2 the initial energy distribution R 2 in space-time rapidity d x⊥ e(τo , x⊥ , ηs ) and the final energy distribution, d x⊥ e(τf , x⊥ , ηs ). Generally, the final distribution in space-time rapidity is similar to the initial distribution in space-time rapidity.18, 82 Therefore, the effect of longitudinal acceleration is unimportant until late times. The nuclei have a finite transverse size, RAu ∼ 6 fm. After a time of order
RAu , c the expansion becomes three dimensional. To estimate how the temperature evolves during the course of the resulting 3D expansion, consider a sphere of radius R which expands in all three directions. The radius and volume increase as τ∼
R∝τ,
V ∝ τ 3.
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Since for an ideal expansion the total entropy in the sphere is constant, the entropy density decreases as 1/τ 3 and the temperature decreases as s∝
1 , τ3
T ∝
1 . τ
(47)
Here we have estimated how the entropy decreases during a one and three dimensional expansion of an ideal massless gas. Now if during the course of the collision there are non-equilibrium processes which generate entropy that ultimately equilibrates, the temperature of this final equilibrated gas will be larger than if the expansion was isentropic. Effectively the temperature will decrease more slowly. To estimate this effect in a one dimensional expansion, we imagine a free streaming gas where the longitudinal pressure is zero. Then from Eq. (42) we have e de ∼ . dτ τ
(48)
In the sense discussed above, this equation may be integrated to estimate that the temperature and entropy decrease as T ∝
1 τ 1/4
,
s∝
1 τ 3/4
.
(49)
Similarly in a three dimensional expansion we can estimate how entropy production will change the powers given in Eq. (47). Again consider a sphere of radius R which expands in all three directions, such that R ∝ τ and V ∝ τ 3 . For a free expansion without pressure the total energy in the sphere is constant, and the energy density decreases as 1/τ 3 . Similarly, we estimate that the temperature and entropy density decrease as T ∝
1 , τ 3/4
s∝
1 . τ 9/4
(50)
In summary we have estimated how the temperature and entropy density depend on the proper time τ during the course of an ideal and non-ideal 1D and 3D expansion. This information is recorded in Table 1. These estimates are also nicely realized in actual hydrodynamic simulations. Figure 13 shows the dependence of Table 1. Dependence of temperature and entropy as a function of time in a 1D and 3D expansion. The indicated range, for instance 1/3 ÷ 1/4, is an estimate of how extreme non-equilibrium effects could modify the ideal power from 1/3 to 1/4.
Quantity
1D Expansion
3D Expansion
T
1/3÷1/4 1 τ
1÷3/4 1 τ
s ∝ T3
1÷3/4 1 τ
3÷9/4 1 τ
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D. A. Teaney
100
Cu+Cu, b=0 fm EOS I
r=0 fm
τ
r=3fm
−1
X0 5
s(fm )
10 -3
**
1
0.1
viscous (1+1)-d hydro ideal (1+1)-d hydro viscous (0+1)-d hydro ideal (0+1)-d hydro
1
τ(fm/c)
**
10
Fig. 13. Figure from Ref. 9 showing the entropy density (s) in CuCu simulations as a function of proper time τ using ideal and viscous hydrodynamics. The top set of lines shows the entropy in the center of the nucleus-nucleus collision, (r = 0 fm), and the bottom set of lines shows the analogous curves closer to the edge (r = 3 fm). During an initial one dimensional expansion the entropy density decreases as s ∝ 1/τ . Subsequently the entropy decreases as s ∝ 1/τ 3 when the expansion becomes three dimensional at a time, τ ∼ 5 fm. The lines labeled by (0 + 1) ideal and (0 + 1) viscous are representative of the ideal and viscous Bjorken results Eq. (42) and Eq. (55) respectively.
entropy density as a function of proper time τ . The figure indicates that the entropy decreases as 1/τ during an initial one dimensional expansion and subsequently decreases as 1/τ 3 when the expansion becomes three dimensional at a time of order ∼ 5 fm. These basic rules will be useful when estimating the relative size of viscous terms in what follows. 4.3. Viscous Bjorken evolution and three dimensional estimates This section will analyze viscosity in the context of the Bjorken model with the primary goal of assessing the validity of hydrodynamics in heavy ion collisions. In viscous hydrodynamics the stress tensor is expanded in all possible gradients. Using lower order equations of motion any time derivatives of conserved quantities can be rewritten as spatial derivatives. First the stress tensor is decomposed into ideal and viscous pieces µν T µν = Tideal + π µν + Π∆µν ,
(51)
µν where Tid is the ideal stress tensor (Eq. (30)) and Π is the bulk stress. π µν is the symmetric traceless shear tensor and satisfies the orthogonality constraint, π µν uν = 0. The equations of motion are the conservation laws ∂µ T µν = 0 together with a constituent relation. The constituent relation expands π µν and Π
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in terms gradients of the conserved charges T 00 and T 0i or their thermodynamic conjugates, temperature T and four velocity uµ . To first order in this expansion, the equations of motion are ∂µ T µν = 0 ,
π µν = −ησ µν ,
Π = −ζ∇µ uµ ,
(52)
where η and ζ are the shear and bulk viscosities respectively, and we have defined the symmetric traceless combination 2 σ µν = ∇µ uν + ∇ν uµ − ∆µν ∇λ uλ . 3 For later use we also define the bracket h. . .i operation
(53)
1 µα νβ 1 ∆ ∆ (Aαβ + Aβα ) − ∆µν ∆αβ Aαβ , (54) 2 3 which takes a tensor and renders it symmetric, traceless and orthogonal to uµ . Note that σ µν = 2 h∂ µ uν i. We now extend the Bjorken model to the viscous case following Ref. 13. The bulk viscosity is neglected in the following analysis and we refer to Sec. 3 for a more complete discussion. Substituting the Bjorken ansatz (Eq. (40)) into the conservation laws and the associated constituent relation (Eq. (52)) yields the time evolution of the energy density hAµν i ≡
e + P − 43 η/τ de =− . (55) dτ τ The system is expanding in the z direction and consequently the pressure in the z direction is reduced from its ideal value. Formally this arises due to the gradient ∂z uz = 1/τ and the constituent relation Eq. (52) 4η . (56) 3τ Thus during a viscous Bjorken expansion the system will do less longitudinal work than in the ideal case. T zz = P −
4.4. The applicability of hydrodynamics and η/s Comparing the viscous equation of motion Eq. (55) to the ideal equation of motion Eq. (42), we see that the hydrodynamic expansion is controlled by η 1 1. e+P τ
(57)
This is a very general result and is a function of time and temperature. Using the thermodynamic relation e + P = sT , we divide this condition into a constraint on a medium parameter η/s and a constraint on an experimental parameter 1/τ T η s |{z}
medium parameter
×
1 τ T |{z}
experimental parameter
1.
(58)
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If the experimental conditions are favorable enough, it is appropriate to apply hydrodynamics regardless of the value of η/s. This is the case for sound waves in air where although η/s is significantly larger than the quantum bound, hydrodynamics remains a good effective theory. However, for the application to heavy ion collisions, the experimental conditions are so unfavorable that only if η/s is close to the quantum bound will hydrodynamics be an appropriate description. For instance, we estimated the experimental condition in Sec. 4.1 1 1 fm 300 MeV . (59) = 0.66 τo To τo To Here we have evaluated this experimental parameter at a specific initial time τo and will return to the time evolution of these estimates in the next section. In Sec. 3 we estimated the medium parameter η/s and can now place these results in context 1 fm 300 MeV η/s 1. (60) 0.2 0.3 τo To From this condition we see that hydrodynamics will begin to be a good approximation for η/s < ∼ 0.3 or so. This estimate is borne out by the more detailed calculations presented in Sec. 6. Reexamining Fig. 10, we see that the value of η/s ' 0.3 is at the low end of the perturbative QGP estimates given in the figure and it is difficult to reconcile the observation of strong collective flow with a quasi-particle picture of quarks and gluons. Thus the estimates of η/s coming from the RHIC experiments, which are based on the hydrodynamic interpretation of the observed flow, should be accepted only with considerable care. 4.5. Time evolution In the previous section we have estimated the applicability of hydrodynamics at a time τo ≈ 1 fm. In this section we will estimate how the size of the viscous terms depends on time. For this purpose we will keep in mind a kinetic theory estimate for the shear viscosity η∼
T , σ
(61)
and estimate how the gradient expansion parameter in Eq. (55) depends on time. We will contrast a conformal gas with σ ∝ 1/T 2 (e.g. perturbation theory or N = 4 SYM) to a gas with fixed cross section, σ = σo . There are clearly important scales in the quark gluon plasma as the medium approaches the transition point. For instance spectral densities of current-current correlators near the transition point show a very discernible correlation where the ρ meson will form in the hadron phase.69, 83, 84 Thus the intent of studying this extreme limit with a constant cross section is to show some of the possible effects of these scales. Further the constant cross section kinetic theory has been used to analyze the centrality dependence of elliptic flow.8
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First consider a theory where the temperature T is the only scale and also consider a 1D Bjorken expansion. The shear viscosity is proportional to T 3 and the enthalpy scales as T 4 , so the hydrodynamic expansion parameter scales as 1 1 η 1 ∼ ∼ 2/3 . (e + P) τ τT τ
(62)
In the last step we have used the fact that for a scale invariant gas undergoing an ideal Bjorken expansion the temperature decreases as 1/τ 1/3 . In general if we have some non-equilibrium processes which produce entropy during the course of the expansion, the temperature will decrease more slowly than estimated in the ideal gas case – see Table 1. The result is that we do not expect the temperature to decrease more slowly than 1/τ 1/4 , and we can estimate that the hydrodynamic expansion parameter evolves as 1 1 1 η ∝ ∝ 2/3÷3/4 . (e + P) τ τT τ
(63)
Thus during a 1D expansion of a conformal gas the system will move closer to equilibrium. Compare this scale invariant theory to a gas with a very definite cross section σo . For a constant σo the hydrodynamic expansion parameter evolves as 0÷1/4 η 1 1 1 ∝ ∝ . (64) (e + P) τ sστ τ Thus, with a constant cross section, the gas will move neither away nor toward equilibrium as a function of time. Non-equilibrium physics will make the matter evolve slowly toward equilibrium. Now we will compute the analogous effects for a three dimensional expansion. In the conformal case η ∝ T 3 and T ∝ τ1 , so that the final result is 0÷1/4 1 η 1 1 ∝ ∝ . (65) (e + P) τ τT τ Thus a conformal gas expanding isentropically in three dimensions also moves neither away nor towards equilibrium, though entropy production will cause it to slowly equilibrate. Similarly for gas with a constant cross section the hydrodynamic parameter evolves as η 1 1 ∝ ∝ τ 2÷5/4 . (66) (e + P) τ sστ In estimating this last line we have used Table 1. Thus we see that a gas with fixed cross-sections which expands in three dimensions very rapidly breaks up. The preceding results are summarized in Table 2. Essentially the heavy ion collision proceeds along the following line of reasoning. First, there is a one dimensional expansion where the temperature is the dominant scale in the problem. The parameter which controls the applicability of hydrodynamics η/[(e + P)τ ] decreases as a
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D. A. Teaney
Table 2. Dependence of the hydrodynamic expansion parameter η/[(e + P)τ ] as a function of time for two different functional forms for η (η ∝ T and η ∝ T 3 ) and two expansion types (1D and 3D). A range of powers is given; the first power corresponds to ideal hydrodynamics and the second power corresponds to an estimate of non-equilibrium evolution.
Model η ∝ T3 η∝
T σo
1D Expansion η ∝ (e + P)τ
3D Expansion
2/3÷3/4 1 τ
η ∝ (e + P)τ
0÷1/4 1 τ
η ∝ (e + P)τ
0÷1/4 1 τ
η ∝ τ 2÷5/4 (e + P)τ
function of time; hydrodynamics gets better and better, evolving according to the upper left corner of Table 2. As the system expands and cools toward the transition region additional scales enter the problem. Typically at this point τ ∼ 4 fm/c the expansion also becomes three dimensional. The system then enters the lower right corner of Table 2 and very quickly the nucleus-nucleus collision starts to break up. We note that it is necessary to introduce some scale into the problem in order to see this freezeout process. For a conformal liquid with η ∝ T 3 the system never freezes out even for a 3D expansion. This can be seen by looking at the upper-right corner of the table and noting that the hydrodynamic expansion parameter behaves as η ∝ (e + P)τ
0÷1/4 1 , τ
(67)
and therefore approaches a constant (or slowly equilibrates) at late times. From this discussion we see that the temperature dependence of the shear viscosity is ultimately responsible for setting the duration of the hydrodynamic expansion. 4.6. Second order hydrodynamics In the previous sections we developed the first order theory of relativistic viscous hydrodynamics. In the first-order theory there are reported instabilities which are associated with the gradient expansion.85 Specifically, in the first-order theory the stress tensor is instantly specified by the constituent relation and this leads to acausal propagation86 and ultimately the instability. Nevertheless, it was generally understood that one could write down any relaxation model which conserved energy and momentum and which included some notion of entropy, and the results of such a model would be indistinguishable from the Navier Stokes equations.87–89 Many hydrodynamic models were written down90–94 starting with a phenomenological model by Israel and Stewart90, 95 and M¨ uller.96 For example in the authors own work the strategy was to write down a fluid model (based on Ref. 93) which relaxed on some time scale to the Navier Stokes equations, solve these model equations on the computer, and finally to verify that the results are independent of the details of
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the model.10 Thus the goal was to solve the Navier Stokes equations and to estimate the effects of higher order terms. Recently an important work by R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov (hereafter BRSSS) clarified and classified the nature of these higher order terms.25 An important impetus for this work came from the AdS/CFT correspondence.25, 28, 97 Many of the fluid models discussed above were motivated by kinetic theory. However, in the strongly coupled N = 4 plasma, kinetic theory is not applicable, and the precise meaning of these models was vague. BRSSS determined precisely in what sense these second order viscous equations are theories and in what sense they are models. Simultaneously the Tatta group completed the calculation of the second order transport coefficients in N = 4 SYM theory and clarified the hydrodynamic nature of black branes in the process.28 The spirit of the BRSSS analysis is the following: (1) Write the stress tensor as an expansion in all possible second order gradients of conserved charges and external fields which are allowed by the symmetries. The transport coefficients are the coefficients of this gradient expansion. (2) In this expansion temporal derivatives can be rewritten as spatial derivatives using lower order equations of motion. (3) The conservation laws ∂µ T µν = 0 and the associated constituent relation dictates the dynamics of the conserved charges in the presence of the external field. By adjusting the transport coefficients, this dynamics will be able to reproduce all the retarded correlators of the microscopic theory. In general, for a theory with conserved baryon number there are many terms. By focusing on a theory without baryon number and also assuming that the fluid is conformally invariant, the number of possible second order terms is relatively small. The classification of gradients in terms of their conformal transformation properties was very useful, both theoretically and phenomenologically. At a theoretical level there are a manageable number of terms to write down. At a phenomenological level the gradient expansion converges more rapidly when only those second order terms which are allowed by conformal invariance are included (see Sec. 6). Subsequently when additional conformal breaking terms are added, the conformal classification provides a useful estimate for the size of these terms, i.e. quantities that scale as Tµµ = e − 3p should be estimated differently than those that scale as energy density itself. In retrospect, this classification is an “obvious” generalization of the first order Navier-Stokes equations. Proceeding more technically, in analogy to the constituent relation of the NavierStokes theory Eq. (52), BRSSS determine that the possible forms of the gradient expansion in a conformal liquid are π
µν
1 µν = −ησ + ητπ hDσ i + σ ∂·u d−1 D E D E D E + λ1 σ µλ σ νλ + λ2 σ µλ Ωνλ + λ3 Ωµλ Ωνλ , µν
µν
(68)
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D. A. Teaney
where the vorticity tensor is defined as 1 Ωµν = ∆µα ∆νβ (∂α uβ − ∂β uα ) , (69) 2 and d = 4 is the number of space-time dimensions. Conformal invariance forces a particular combination of second derivatives to have a single coefficient 1 µν µν σ ∂·u . (70) τπ hDσ i + d−1
The time derivative Dσ µν may be expanded out using lower order equations of motion if desired. The constituent relation (Eq. (68)) and the conservation laws form the second order equations of motion of a conformal fluid. They are precisely analogous to the first order theory. As in the first order case, these equations are also acausal. To circumvent this issue, BRSSS (following the spirit of earlier work by Israel and Stewart90, 95 and M¨ uller96 ) promote the constituent relation to a dynamical equation for the viscous components of the stress tensor π µν . Using the lower order relation π µν = −ησ µν , the (conformal) dependence of η on temperature η ∝ T d−1 , and the ideal equation of motion Eq. (32), the following equation arises for π µν d π µν = −ησ µν − τπ hDπ µν i + π µν ∇ · u d−1 E D λ1 D µ νλ E λ2 D µ νλ E + 2 π λπ π λΩ (71) − + λ3 Ωµλ Ωνλ . η η From a numerical perspective the resulting equation of motion is now first order in time derivatives, hyperbolic and causal. The modes in this (and similar) models have been studied in Refs. 89, 86 and 25. Nevertheless it should be emphasized that the domain of validity of the resulting equations is still the same as Eq. (68), i.e. the hydrodynamic regime. Thus for instance the second order equations should be used in a regime where |π µν + ησ µν | |ησ µν | . Outside of this regime there is no guarantee that entropy production predicted by this model will be positive during the course of the evolution.25 It should also be emphasized that this is not a unique way to construct a hydrodynamic model which reduces to Eq. (68) in the long wavelength limit — see Ref. 72 for an example discussed in these terms. What is guaranteed is that any conformal model or dynamics (such as conformal kinetic theory26, 27 or the dynamics predicted by AdS/CFT25, 28 ) will be expressible in the long wavelength limit in terms of the gradient expansion given above. There is an important distinction between the first and second order theories.72, 87, 88 In the first order theory, the ideal motion is damped, and there are corrections to the ideal motion of order the inverse Reynolds number η `mfp vth ∆t ∆t ∼ , (72) Re−1 ≡ 2 (e + P)L L L
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where L is the characteristic spatial dimension of the system, ∆t is the time of observation, and vth is a typical quasi-particle velocity. Thus for sufficiently long times the viscous corrections become large and must be resummed by solving the Navier-Stokes equations to capture the damping of the fluid motion. Once this is done however, the remaining higher order terms (which are captured by the second order theory) are uniformly small and modify the Navier-Stokes solution by an amount of order `2mfp /L2 . Often this makes these higher order terms difficult to measure in normal laboratory liquids.72 For completeness we record the model equations which have been discussed in the heavy ion literature.5, 9, 29, 30 (1) The first of these is the simplified Israel-Stuart equation, π µν = −ησ µν − τπ hDπ µν i .
(73)
Since the derivatives do not appear as the combination hDπ µν i +
d π µν ∇ · u , d−1
(74)
but rather involve hDπ µν i separately, this model does not respect conformal invariance. (2) The second model is the full Israel-Stewart equation which has the following form98 1 τπ ρ ηT ν) π µν = −ησ µν − τπ hDπ µν i + π µν ∂ρ u + 2τπ π α(µ Ω α , (75) 2 τπ ηT d ν) → −ησ µν − τπ hDπ µν i + π µν ∇ · u + 2τπ π α(µ Ω α . (76) d−1 In the last line we have used the conformal relation, ηT /τπ ∝ T d+1 and equation of motion, D(ln T ) = −1/(d − 1)∇ · u . The model is equivalent to taking λ1 = λ3 = 0 and λ2 = −2ητπ . There has been some effort to compute the coefficients of the gradient expansion both at strong and weak coupling. The gradient expansion in Eq. (68) implies that the relative size of the coefficients is (`mfp /L)2 , and this is of order [η/(e + P)]2 in a relativistic theory. The strong coupling results25, 28 are listed in Table 3. At weak coupling, the results of kinetic calculations are also listed in the table26 (see also Ref. 27). In kinetic theory the physics of these higher order terms stems from the streaming terms and the collision integrals.25, 26 To first order in the gradients, the distribution is modified from its equilibrium form n → n + δf ,
(77)
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D. A. Teaney
Table 3. Compilation of values of (rescaled) second order transport quantities (ητπ , λ1 , λ2 , λ3 ). All numbers in this table should be multiplied by η2 /(e + P). The complete strong coupling results are from an amalgamation of Ref. 25 and Ref. 28. The weak coupling results are from Ref. 26 and the relaxation time approximation was studied in Ref. 25 and clarified in Ref. 26. Hydrodynamic simulations of the heavy ion event are not sensitive to these values. In a theory where λ1 = ητπ the second order corrections to a viscous 0 + 1 dimensional Bjorken evolution vanish.
Quality
N = 4 SYM
ητπ
4 − 2 ln(2) ' 2.61
5.9 to 5.0
(due to g)
λ1
2
5.2 to 4.1
(due to g)
λ2
−4 ln(2) ' −2.77
−11.8 to −10
λ3
0
QCD Kinetic Theory
(≡ −2ητπ )
0
Relaxation Time 6 6 −12
(≡ ητπ ) (≡ −2ητπ ) 0
where δf ∝ pi pj σij — see Sec. 5. Substituting this correction back into the Boltzmann equation P µ ∂µ f = −C[f ] ,
(78)
leads to several terms which are responsible for the second order corrections to hydrodynamics. We enumerate these contributions: (1) The τπ and λ2 terms are the result of streaming of the first viscous correction P µ ∂µ δf and do not involve the collision integral. The common origin of these terms ultimately explains the relation between them, λ2 = −2ητπ . (2) The contribution to λ1 (the visco-elastic ππ term) reflects the streaming P µ ∂µ δf and the non-linearities of the collision integral, C[δf ]. (3) Finally the vorticity-vorticity term does not appear on the LHS of the Boltzmann equation and therefore this term vanishes in kinetic theory.25–27 In the strong coupling limit the absence of a vorticity-vorticity coupling is not understood. In the relaxation time approximation discussed in Sec. 5 (with τR ∝ Ep ) the coefficient τπ is readily calculated with linearized kinetic theory for a massless gas25, 26 ητπ = 6
η2 , e+P
(79)
The kinetic theory relations λ2 = −2ητπ and λ3 = 0 are respected for the same reasons as the full theory. Also in the relaxation time approximation one finds, λ1 = ητπ . In the full kinetic theory the difference λ1 − ητπ reflects the deviation from the quadratic ansatz discussed Sec. 5, and to a much lesser extent the nonlinearities of the collision integral. Nevertheless the relation λ1 = ηηπ almost holds indicating the dominance of the streaming term. Overall the relaxation time approximation provides a good first estimate of these coefficients in kinetic theory.
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This is important because the second order corrections to a 0 + 1 Bjorken expansion vanish if λ1 = ητπ — see below. From a practical perspective a majority of simulations have used the full IsraelStewart equations5, 11, 30 and treated τπ as a free parameter, varying [η/(e + P)τπ ] down from the relaxation time value by a factor of two. While it is gratifying that higher order transport coefficients can be computed and classified, the final phenomenological results (see Sec. 6) are insensitive to the precise value of all 10, 30, 31 second order terms for η/s < Thus, the full hydrodynamic simulations ∼ 0.3. corroborate the estimate given in Sec. 4.4 for the range of validity of hydrodynamics. 4.7. Summary We have discussed various orders in the gradient expansion of hydrodynamics. Here we would like to summarize these results for a 0 + 1 dimensional Bjorken evolution. The equation of motion for the Bjorken expansion is e + T zz de =− , dτ τ
(80)
where T zz ≡ τ 2 T ηη is the stress tensor at mid space-time rapidity, ηs = 0. The stress tensor through second order is13, 25, 31 T zz = P −
4η 8 + (λ1 − ητπ ) 2 . 3τ 9τ
(81)
Each additional term reflects one higher order in the hydrodynamic expansion parameter [η/(e + p)τ ] discussed in Sec. 4.4. We have made use of the intermediate results 4 2 2 σ µν = diag σ τ τ , σ xx , σ yy , τ 2 σ ηη = 0, , ,− , (82) 3τ 3τ 3τ
and
µλ ν 4 4 8 σ σλ = diag 0, − 2 , − 2 , 2 . 9τ 9τ 9τ
(83)
Notice in Eq. (81) that there is a cancellation between the relaxation terms ∼ ητπ Dσ and the visco-elastic response,31 λ1 σσ. In kinetic theory the difference ητπ − λ1 is determined primarily from the deviation of δf from the quadratic ansatz (see Sec. 5). Thus ητπ is expected to be approximately equal to λ1 due to the overall kinematics of the streaming term,26 P µ ∂µ δf . Examining Table 3 we see that for a relaxation time approximation ητπ = λ1 and the second order corrections to a 0 + 1 Bjorken expansion of a conformally invariant fluid vanish! This cancellation is partially present for the full kinetic theory and for the strongly interacting theory. Figure 14 shows the how the different orders in Eq. (81) influence the evolution of the energy density for a conformally invariant equation of state (P = e/3 = sT /4) and various values of η/s. For definiteness we have used the N = 4 ratios for the
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D. A. Teaney
τ e / τ o eo
1 0.9
η/s =0.3
0.8
η/s =0.2
0.7
η/s =0.1
0.6
η/s =0 Navier Stokes 2nd Order
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 τ/τo Fig. 14. The energy density (×τ ) relative to the initial energy density (×τo ) for a 0+1 dimensional Bjorken expansion. The temperature is To ' 300 MeV and τo ' 1 fm, so that 1/τo To ' 0.66. The second order correction is smaller than expected due to a cancellation between the relaxation term ∼ ητπ Dσ and the viscoelastic term,31 ∼ λ1 σσ. In a relaxation time approximation the second order correction vanishes (see text).
second order transport coefficients but this makes little difference since only the combination λ − ητπ matters — see Table 3. Finally we have used the estimates of Sec. 4.4 for the initial temperature To and time τo 1 = 0.66 τo To
1 fm τo
300 MeV To
.
Generally the effect of second order terms is small (due to the cancellation) and the value of the first order terms drive the correction to the ideal evolution.
5. Kinetic Theory Description In Sec. 4 we discussed various aspects of viscous hydrodynamics as applied to heavy ion collisions. Since ultimately the experiments measure particles, there is a need to convert the hydrodynamic information into particle spectra. This section will provide an introduction to the matching between the kinetic and hydrodynamic descriptions. This will be important when comparing the hydrodynamic models to data in Sec. 6. In addition, since Sec. 3 discussed various calculations of the shear viscosity in QCD, this section we will sketch briefly how these kinetic calculations are performed. Good summaries of this set of steps are provided by Refs. 99, 73 and 100.
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In kinetic theory the spectrum of particles in a volume Σ is given by the CooperFrye formula101 Z 1 d3 N dΣµ P µ f (−P · u) . (84) E 3 = d p (2π)3 Σ Note that when Σ is a three volume at fixed time, dΣµ = (dV, 0, 0, 0), and this formula reduces to the traditional result. Four vectors are denoted with capitol letters P µ = (Ep , p), and the equilibrium distribution function is denoted with n(−P · u) =
1 . exp(−P · u/T ) ± 1
(85)
We will also use a suffix notation, np = n(−P · U ) and fp = f (−P · u). The distribution function obeys the Boltzmann equation Z ∂t fp + vp · ∂x fp = − Γ12→34 (f1 f2 − f3 f4 ) , (86) 234
where vp = ∂Ep /∂p = p/Ep , and we have assumed 2 → 2 scattering with classical statistics for simplicity, fp = exp(−Ep /T ). The momenta are labeled as f2 = fp2 and f1 = fp1 with p1 ≡ p. The integral over the phase space is abbreviated Z Z 3 d p2 d3 p3 d3 p4 = , (87) (2π)3 (2π)3 (2π)3 234 and the transition rate Γ12→34 for 2 → 2 scattering is related to the usual Lorentz 2 invariant matrix element |M| by 2
Γ12→34 =
|M| (2π)4 δ 4 (P1 + P2 − P3 − P4 ) . (2E1 )(2E2 )(2E3 )(2E4 )
(88)
The generalization of what follows to a multi-component gas with quantum statistics is left to Ref. 99. During a viscous evolution the spectrum will be modified from its ideal form f = np + δfp ,
(89)
and this has important phenomenological consequences.4 The modification of the distribution function depends on the details of the microscopic interactions. In a linear approximation the deviation is proportional to the strains and can be calculated in kinetic theory. When the most important strain is shear, the deviation δf is proportional σij . Traditionally we parameterize the viscous correction to the distribution in the rest frame of the medium byd χ(|p|) ˆip ˆ j σij . δfp = −np χ(|p|) p d When
quantum statistics are taken into account this should be written ˆ j σij , δfp = −np (1 ± np ) χ(|p|)ˆ pi p
where the overall minus is introduced because in the Navier-Stokes theory π µν = −ησµν .
(90)
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Then the stress tensor in the local rest frame is Z d3 p pi pj ij ij ij T = pδ − ησ = [np + δfp ] . (2π)3 Ep
(91)
Substituting Eq. (90) for δfp and using rotational symmetry we have Z 2 d3 p p2 η= χ(|p|)np . 15 (2π)3 Ep
(92)
Thus we see that the form of the viscous correction to the distribution function determines the shear viscosity. To calculate the transport coefficients the Boltzmann equation is analyzed in the rest frame of a particular location xo . In a neighborhood of this point the temperature and flow fields are uµ (x, t) ' (1, ui (x, t)) ,
T (x, t) ' To + δT (x, t) ,
(93)
where ui (xo , t) = δT (xo , t) = 0. The equilibrium distribution function in this neighborhood is Ep pi ui (x, t) o o n(−P · u) ' np + np δT (x, t) + , (94) To2 To where we have used the short hand notation, nop = exp(−Ep /To ). We can now substitute the distribution function into the Boltzmann equation and find an equation the δf . The left hand side of the Boltzmann equation involves gradients, and therefore only the equilibrium distribution needs to be considered. Substituting Eq. (94) into the l.h.s. of Eq. (86), using the ideal equations of motion ∂t ui = −
∂ iP , (e + P)
(95)
∂t e = −(e + P)∂i ui ,
(96)
and several thermodynamic relationships de , dT n 1 1 d(µ/T ) = dP + d = 0, e+P T (e + P) T cv =
(97) (98)
we find thate np ∂t fe + vp · ∂x fe = Ep e We
"
Ep2 (e + P) |p|2 − 3T T T cv
!
pi pj ∂i u + σij 2T i
#
.
(99)
have tacitly assumed that the dispersion curve E(p) does not depend on the temperature. This is fine as long as we are not considering the bulk viscosity.73
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The result is proportional to two strains ∂i ui and σij which are ultimately responsible for the bulk and shear viscosities respectively. For a massless conformal gas we have |p|2 = Ep2 , T cv = 4e and e + p = (4/3)e. The result is that the term proportional to ∂i ui vanishes and consequently the bulk viscosity is zero in this limit. Subsequently, we will consider only the modifications due to the shear viscosity and refer to Sec. 3 for a more complete discussion of bulk viscosity. In Eq. (99), the l.h.s. of the Boltzmann equation is evaluated at the point xo . We also evaluate the r.h.s. of the Boltzmann equation at the point xo to linear order in δf Z i j δf2 δf3 δf4 o o δf (p) o p p Γ12→34 np n2 + o − o − o . (100) np σij = − 2T Ep nop n2 n3 n4 234 In writing this equation we have made use of the detailed balance relation n1 n2 Γ12→34 = n3 n4 Γ12→34 .
(101)
Eq. (100) should be regarded as a matrix equation for the distribution function δf (p). Although δf (p) or χ(p) can be determined numerically by straight forward discretization and matrix inversion, a variational method is preferred in practice.64, 99 After determining δf (p) or χ(|p|) the shear viscosity can be determined from Eq. (92). Inverting the integral equation in Eq. (100) requires a detailed knowledge of the microscopic interactions. Lacking such detailed knowledge, one can resort to a relaxation time approximation, writing the Boltzmann equation as ∂t f + vp · ∂x f = −
1 (−P · U ) δf , τR (−P · U ) Ep
(102)
where τR (−P ·U ) is a momentum dependent relaxation time. In the local rest frame this reduces to 1 ∂t f + vp · ∂x f = − δf . (103) τR (Ep ) By fiat the correction to the distribution function is simple δf = −np
τR (Ep ) i j p p σij . 2T Ep
(104)
First we consider the case where the relaxation time grows with energy τR (Ep ) = Const ×
Ep T
δf = −Const × np
pi pj σij 2T 2
(105)
The form of this correction is known as the quadratic ansatz and was used by all hydrodynamic simulations so far.5, 10, 30, 31 Substituting this form into Eq. (92) one determines that for a classical gas of arbitrary mass the constant is4 η Const = , with τR (Ep ) ∝ Ep . (106) e+P
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For a Bose or Fermi gas we have the replacement np → np (1 ± np ) and Eq. (106) is approximate holding at the few percent level. For a mixture of different classical particles with one common relaxation time Eq. (106) also holds. In practical simulations this correction is written covariantly and the phenomenological field π µν = −ησµν is used, leading to the final result δf =
1 P µ P ν πµν . 2(e + P)T 2
(107)
The quadratic ansatz may seem arbitrary, but it is often a good model of collisional energy loss and weak scattering. For instance an analysis of the leading-log Boltzmann equation (along the lines of Eq. (100)) shows that the quadratic ansatz describes the full results to 10-15% accuracy.99 However, this agreement is in part an artifact of the leading-log, or soft scattering, approximation. For example, in the leading-log plasma the energy loss of a “high” pT quark from the bath is given by102 dp = dt
Nf CF g 4 T 2 Nc + log (T /mD ) . 2 24π
(108)
From this formula we see that the energy loss is constant at high momentum and therefore the relaxation time scales as τR ∝ p in agreement with Eq. (105). In reality Eq. (108) is decidedly wrong at large momentum where radiative energy loss becomes increasingly significant and can shorten the relaxation time. Indeed when collinear radiation is included in the Boltzmann equation the quadratic ansatz becomes increasingly poor.103 In Sec. 6 we will consider a relaxation time which is independent of energy as an extreme alternative τR ∝ Const ,
(109)
and explore the phenomenological consequences of this ansatz. As discussed in Sec. 6, the differential elliptic flow v2 (pT ) is sensitive to the form of these corrections, while the integrated v2 is constrained by the underlying hydrodynamic variables, and is largely independent of these details. This last remark should be regarded with caution as it has not been fully quantified. 6. Viscous Hydrodynamic Models of Heavy Ion Collisions At this point we are in a position to discuss several viscous hydrodynamic models which have been used to confront the elliptic flow data. To initiate discussion, we show simulation results for v2 (pT ) from Luzum and Romatschke in Fig. 15. Comparing the simulation to the “non-flow corrected” data for pT < ∼ 1.5 GeV, we can estimate an allowed range for the shear viscosity, η/s ≈ 0.08 ↔ 0.16. Below we will place this estimate in context by culling figures from related works.
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Glauber
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η/s=0.08
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Fig. 15. Figure from Ref. 31 which shows how elliptic flow depends on shear viscosity. The theory curves are most dependable for pT < ∼ 1.5 GeV and should be compared to the “non-flow corrected” data. The Glauber and CGC initial conditions have different eccentricities as described in the text.
A generic implementation of viscous hydrodynamics consists of several parts: (1) At an initial time τo the energy density and flow velocities are specified. For Glauber initial conditions, one takes for example e(τo , x⊥ ) ∝
dNcoll , dx dy
(110)
where the overall constant is adjusted to reproduce the multiplicity in the event. The simulations assume Bjorken boost invariance with the ansatz e(τ, x⊥ , η) ≡ e(τ, x⊥ ) ,
(111)
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uµ (τ, x⊥ , η) = (uτ , ux , uy , uη ) = (uτ (τ, x⊥ ), ux (τ, x⊥ ), uy (τ, x⊥ ), 0) . (112) In cartesian coordinates uz = uτ sinh(ηs ) and ut = uτ cosh(ηs ). The calculations typically assume zero transverse flow velocity at the initial time τo ux (τo , x⊥ ) = uy (τo , x⊥ ) = 0 ,
uτ (τo , x⊥ ) = 1 .
The strains are taken from the Navier stokes theory for example 4η 2η 2η µν ττ xx yy 2 ηη , , − , = 0, π (τo , x⊥ ) = diag π , π , π , τ π 3τ 3τ 3τ
(113)
(114)
and reflect the traceless character of shear stress. (2) The equations of motion are then solved. Viscosity modifies the hydrodynamic variables, T and uµ , and also modifies off diagonal components of the stress tensor through the viscous corrections π µν . (3) A “freezeout” condition is specified either by specifying a freezeout temperature or a kinetic condition. During the time evolution a freezeout surface is constructed. For instance the freezeout surface in Fig. 15 is the space-time three volume Σ where Tfo ' 150 MeV. (4) Finally, in order to compare to the data, particle spectra are computed by matching the hydrodynamic theory onto kinetic theory. Specifically, on the freezeout surface final particle spectra are computed using Eq. (125). Roughly speaking this “freezeout” procedure is equivalent to running the hydro up to a particular proper time τf or temperature Tf and declaring that the thermal spectrum of particles at that moment is the measured particle spectrum. There are many issues associated with each of these items. The next subsections will discuss them one by one. 6.1. Initial conditions First we note that the hydrodynamic fields are initialized at a time τ0 ' 1 fm/c, which is arbitrary to a certain extent. Fortunately, both in kinetic theory and hydrodynamics the final results are not particularly sensitive this value.31, 104 Also, all of the current simulations have assumed Bjorken boost invariance. While this assumption should be relaxed, past experience with ideal hydrodynamics shows that the mid-rapidity elliptic flow is not substantially modified.18 Above we have discussed one possible initialization of the hydro which makes the energy density proportional to the number of binary collisions, e.g. the Glauber curves of Fig. 15. Another reasonable option is to make the entropy proportional to the number of participants10 s(τ0 , x⊥ ) ∝
dNp . dx dy
(115)
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As a limit one can take the CGC model discussed in Sec. 2. Finally it is generally assumed that the initial transverse flow is zero ux (τ0 , x⊥ ) = uy (τ0 , x⊥ ) = 0 .
(116)
This assumption should probably lifted in future calculations and a more reasonable (but still small) estimate is given in Ref. 105. Examining Figs. 15 and 7 we see see that there is a significant and predictable linear dependence on the eccentricity. When extracting the shear viscosity from the data, this uncertainty in the eccentricity leads to a factor of two uncertainty in the final results for the shear viscosity. As emphasized in Sec. 2, the CGC model should be thought of as an upper limit to the anisotropy that can be produced in the initial state. Therefore, the uncertainty in η/s is probably not as large as dispersion in the curves would indicate. In ideal hydrodynamics, the spread in v2 (pT ) resulting from the different initializations specified by Eq. (115) and Eq. (110) was studied,106 and is small compared to the difference between the Glauber and CGC curves in Fig. 15. Once the initial conditions for the temperature and the flow velocities are specified, the off diagonal components of the stress tensor π µν are determined by the spatial gradients in T and uµ . To second order this is given by Eq. (68) and there is no ambiguity in this result. Time derivatives may be replaced with spatial derivatives using lower order equations of motion to second order accuracy. In the phenomenological theory π µν is promoted to a dynamical variable. Clearly the appropriate initial condition for this variable is something which deviates from −ησ µν by second order terms. However, the extreme choice π µν = 0 was studied to estimate how initial non-equilibrium effects could alter the final results. This is just an estimate since the relaxation of these fields far from equilibrium is not well captured by hydrodynamics. On the other hand, comparisons with full kinetic theory simulations show that the Israel-Stewart model does surprisingly well at reproducing the relaxation dynamics of the full simulation.29 From a practical perspective, even with this extreme choice π µν = 0, the stress tensor relaxes to the expected form π µν = −ησ µν relatively quickly. The result is that v2 (pT ) is insensitive to the different initializations of π µν . This can be seen clearly from Fig. 16. 6.2. Corrections to the hydrodynamic flow Once the initial conditions are specified, the equations of motion can be solved. First we address the size of the viscous corrections to the temperature and flow velocities. The magnitude of the viscous corrections depends on the size of the system and the shear viscosity. Figure 17 shows a typical result for the AuAu system. As seen from the figure the effect of viscosity on the temperature and flow velocities is relatively small, of order ∼ 15% for η/s ' 0.2. An explanation for this result is the following:107 In the first moments of the collision the system is expanding in the longitudinal direction and the pressure in
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D. A. Teaney
Cu+Cu, b=7 fm, SM-EOSQ, π
v2
−
viscous hydro viscous hydro (flow anisotropy only) ideal hydro
0.4
mn
initialized by π =2ησ mn initialized by π =0
0.3
mn
0.2
0.1
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1
2
3
pT(GeV)
16
Viscous Euler
14
Velocity
2
e τ (GeV/ fm )
Fig. 16. Figure from Ref. 9 studying the independence of the final results on the initialization of π µν . Note that due to different metric and symmetrzation conventions the Navier Stokes limit is 2ησmn rather than −ησµν adopted here.
Viscous
1
Euler 0.8
12
τ=1
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8 6
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0
1
2
3
4
5
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Fig. 17. A central AuAu simulation with an ideal gas equation of state p = e/3 and η/s = 0.2 which compares the viscous and Euler evolution.107 The left figure shows the energy density (×τ ) for different times. The right figure shows the velocity for different times.
the longitudinal directions is reduced τ 2 T ηη = P −
4η . 3τ
(117)
At first sight, this means that the system cools more slowly and indeed this is initially true. However, since the shear tensor is traceless there is an increase in the
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transverse pressure which is uniform in all directions T xx = T yy = P +
2η , 3τ
(118)
and which ultimately increases the radial flow. Since the radial flow is larger in the viscous case, the system ultimately cools faster. Having discussed the dependence of T and uµ , we turn to a quantity which largely dictates the final elliptic flow – the momentum anisotropy.81 The momentum anisotropy is defined asf R 2 d x⊥ (T xx − T yy ) hT xx − T yy i 0 R = ep = . (119) hT xx + T yy i d2 x⊥ (T xx + T yy ) Figure 18 illustrates how this momentum anisotropy increases as a function of time in the CuCu and AuAu systems. Although the flow fields T (x⊥ , τ ) and uµ (x⊥ , τ ) are quite similar between the ideal and viscous cases, the ideal anisotropy is significantly reduced by viscous effects. The reason for this reduction is because the viscous stress tensor anisotropy, T xx − T yy , involves the difference Πxx − Πyy , in addition to the temperature and flow velocities. This additional term is ultimately responsible for the deviation of e0p between the ideal and viscous hydrodynamic calculations. At later times there is some modification of e0p , due to the flow itself, but this is dependent on freezeout. The deviation ∆Π = Πxx − Πyy will have important phenomenological consequences in determining the viscous correction to the elliptic flow spectrum. 6.3. Convergence of the gradient expansion Figure 18 also compares the simplified Israel-Stewart equation Eq. (74) to the full Israel-Stewart equation Eq. (75) as a function of the relaxation time parameter τπ . The result supports much of the discussion given in Sec. 4.6. In the roughest approximation neither the simplified Israel-Stewart equation nor the full IsraelStewart equation depend on the relaxation time parameter τπ . When an ideal gas equation of state is used the dependence on τπ is stronger especially for the simplified Israel-Stewart equation.30 Note also that the dependence on τπ is stronger in the smaller CuCu system than in AuAu. However, in the conformally invariant f Note
there is a misprint in the original definition of e0p in Ref. 81. Eq. (3.2) of that work used a double bracket notation indicating an energy density weight which should only be a single bracket as above. This double bracket definition (rather than Eq. (119)) was used subsequently in Ref. 10 which is why the associated curve differs from other recent works.30, 31 Further the 0 definition should perhaps include R R a factor of u so that the integrals would be boost invariant, i.e. dΣµ uµ (T xx − T yy ) = τ ∆ηs d2 x u0 (T xx − T yy ). However, typically the integral is dominated µνρ near the center where u0 ' 1. An alternative definition Shydro is suggested in Sec. 6.5.
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(e)
3
3
e0=30 GeV/fm , τ0=0.6fm/c
e0=30 GeV/fm , τ0=0.6fm/c
0.1
εp,
εp,
01
(f)
Au+Au, b=7fm, EOS L
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0 05
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4 τ−τ0 (fm/c)
6
ideal hydro viscous hydro full I-S eqn. viscous hydro simplified I-S eqn
8
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5
τ−τ0 (fm/c)
10
Fig. 18. Figure from Ref. 30 comparing the development of the flow anisotropy e0p (Eq. (119)) in viscous hydrodynamics relative to the ideal hydrodynamics. The lower band of curves are all representative of viscous hydro and differ only in how the second order corrections are implemented. The anisotropy differs from ideal hydro because the anisotropy involves the viscous difference, Πxx − Πyy .
full Israel-Stewart equation the dependence on τπ is negligible, indicating that the result is quite close to the first order Navier Stokes theory. The more rapid convergence of the gradient expansion in conformally invariant fluids is a result of the fact that the derivatives in the conformal case come together as 4 µν µν τπ hDπ i + π ∇ · u . 3 We have selected one figure out of many.10, 30, 31 The result of the analysis is that the flow fields and v2 (pT ) are largely independent of the details of the second order terms at least for η/s < ∼ 0.3. For this range of parameters, hydrodynamics at RHIC is an internally consistent theory. 6.4. Kinetic theory and hydrodynamic simulations Clearly viscous hydrodynamics is an approximation which is not valid at early times and near the edge of the nucleus. This failure afflicts the current viscous calculations at a practical level right at the moment of initialization. For instance, the longitudinal pressure T zz ≡ τ 2 T ηη = P −
4 η , 3 τ0
(120)
eventually becomes negative near the edge of the nucleus indicating the need to transition to a kinetic description.108 (Note that P ∝ T 4 while η ∝ T 3 , so at sufficiently low temperatures the viscous term is always dominant regardless of the magnitude of η/s.) The current calculations simply limit the size of this correction through the phenomenological Israel-Stewart model. For example, one approach would be to take τ 2 T ηη = P + τ 2 Πηη ,
(121)
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with τ 2 Πηη
4 η − 3 τo = −0.9 P
while 4/3 η/τ0 < 0.9 P
.
(122)
otherwise
This ad-hoc fix is clearly not nice and points to the larger problem of freezeout which is difficult to address with hydrodynamics itself. Freezeout is the colloquial term for the transition from a hydrodynamic to a kinetic regime and is impossible to separate cleanly from the viscosity itself in a realistic nucleus-nucleus collision. Clearly as the shear viscosity is made smaller and smaller, a larger and larger space time volume is described by hydrodynamics. To estimate the size of the relevant space time region we remark that hydrodynamics is valid when the relaxation time τR is much smaller than the inverse expansion rate, τR ∂µ uµ 1. Therefore, in the simulations one can estimate the region of validity by monitoring the expansion rate relative to the relaxation time.109, 110 Specifically, freezeout is signaled when 1 η ∂µ uµ ∼ . P 2
(123)
This combination of parameters can
2be motivated
2 from the kinetic theory estimates.111 The pressure is P ∼ e v th , with vth the typical quasi-particle 2 velocity. The viscosity is of order η ∼ e vth τR with τR the relaxation time. Thus hydrodynamics breaks down when η 1 ∂µ uµ ∼ τR ∂µ uµ ∼ . P 2
(124)
Figure 19 estimates the space-time region described by viscous hydrodynamics. Examining this figure we see that the time duration of the hydrodynamic regime is a relatively strong function of η/s at least for a conformal gas. In reality the behavior of the shear viscosity near the transition region will control when the hydrodynamics will end. Clearly the surface to volume ratio in Fig. 19 is not very small. Hydrodynamics is a terrible approximation near the edge. There is a need for a model which smoothly transitions from the hydrodynamic regime in the center to a kinetic or free streaming regime at the edge. Near the phase transition kinetic theory may not be a good model for QCD, but it has the virtue that it gracefully implements this hydro to kinetic transition. Although the interactions and the quasi-particle picture of kinetic theory may be decidedly incorrect, this is unimportant in the hydrodynamic regime. In the hydrodynamic regime the only properties that determine the evolution of the system are the equation of state, P(e), and the shear viscosity and bulk viscosities, η(e) and ζ(e). In the sense that kinetic theory provides a reasonable guess as to how the surface to volume ratio influences the forward evolution, these models can be used to estimate the shear viscosity and the estimate may be more reliable than
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15
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ηs=0.05
T HMeVL 375 295 215 135 55 ηs=0.133
Τ HfmcLL ηs=0.4 5
0
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r HfmL Fig. 19. This figure estimates for a conformal gas with equation of state p = e/3 and constant η/s the space time region described by viscous hydrodynamics.10 The contours are where (η/P) ∂µ uµ = 0.6 for different values of η/s. For the smallest value η/s = 0.05 the system freezes out at a time of order ∼ 40 fm. This unrealistically long time reflects the conformal nature of the gas as discussed in Sec. 4.5. For comparison we have shown the (η/P) ∂µ uµ = 0.225 contour for η/s = 0.05.
the hydrodynamic models. A priori one should demand that the kinetic models have the same equation of state and the same shear viscosity as expected from QCD, η ∝ T 3 . For instance in a kinetic model of massless particles with a constant cross section σo (such as studied in Refs. 8, 34 and 29) the shear viscosity scales as η ∝ T /σo . This difference with QCD should be kept in mind when extracting conclusions about the heavy ion reaction. Further, many transport models conserve particle number, which is an additional conservation law not inherent to QCD; this also changes the dynamics. Keeping these reservations in mind we examine Fig. 20 from Ref. 11. This figure shows a promising comparison between kinetic theory with a constant cross section (σo = Const) and a viscous hydrodynamic calculation with η ∝ T /σo . The case with σ ∝ τ 2/3 will not be discussed in this review, but is an attempt to mimic a fluid which has η ∝ T 3 . What is exciting about this figure is the fact that the hydrodynamic conclusions are largely supported by the results of a similar kinetic theory. This gives considerable confidence that surface to volume effects are small enough that the hydrodynamic conclusions presented in Fig. 15 are largely unchanged by particles escaping from the central region. More formally, the opacity is large enough to support hydrodynamics. There have been other kinetic calculations which are working towards extracting η/s from the heavy ion data.12, 32, 33 In particular Refs. 7, 6, 112 and 113 used a kinetic theory implementation of 2 → 2 and 2 → 3 interactions motivated by weak
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Fig. 20. Elliptic flow from Ref. 11 for a massless classical gas with a constant cross section in kinetic theory (Transp. σ = const) and viscous hydrodynamics (Hydro σ = const). The σ ∝ τ 2/3 case is not discussed in this review but is an effort to simulate a gas with η ∝ T 3 .
coupling QCD.33 The simulation also calculates the Debye scale self consistently, i.e. in equilibrium one sets m2D ∝ g 2 T 2 and T changes with time. Out of equilibrium this mass scale mD is determined from the momentum distribution of particles. Consequently this model respects the symmetry properties of high temperature QCD, i.e. the model has η ∝ T 3 and does not conserve particle number. For the model parameter αs = 0.3 ↔ 0.5 (which is only schematically related to the running coupling) the shear to entropy ratio is η/s = 0.16 ↔ 0.08.112, 113 The model (known as BAMPS) is conformal and never freezes out as discussed in Sec. 4.5. The current implementation of BAMPS simply stops the kinetic evolution when the energy density reaches a critical value, ec ' 0.6 ↔ 1.0 GeV/fm3 . This is an abrupt way to introduce a needed scale into the problem and schematically approximates the rapid variation of the shear viscosity in this energy density range. Figure 21 shows the time development of elliptic flow in this model which can reproduce the observed flow only for η/s = 0.16 ↔ 0.08. The time development of v2 seen in Fig. 21 shows that it is very difficult to separate precisely the shear viscosity in the initial stage from the freezeout process controlled by ec . Clearly the transition from a hydrodynamic regime to a kinetic regime is important to clarify in the future. In the meantime most hydrodynamic groups have invoked an ad-hoc freezeout prescription. In Refs. 5, 31, 30 and 9 the hydrodynamic codes were run until a typical freezeout temperature, Tfo ' 150 MeV. Technically the freezeout surface is constructed by marching forward in time and triangulating the space-time surface with constant temperature. In Ref. 11 a surface of constant
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part
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>
Fig. 21. (Top) The development of elliptic flow v2 as a function of time in the BAMPS model.7 (Bottom) The final elliptic flow as a function of centrality. The shear viscosity to entropy ratio η/s corresponding to the model parameter αs = 0.3 ↔ 0.6 was estimated in Ref. 113, 112 and is η/s = 0.16 ↔ 0.08. The evolution is stopped when the energy density reaches a critical value of ec .
particle density was chosen n ' 0.365/fm3 , and in Ref. 10 the chosen surface was motivated by the kinetic kinetic condition in Eq. (123). Ideally this could be improved by dynamically coupling the hydrodynamic evolution to a kinetic description or by simulating the entire event with a kinetic model which closely realizes the equation of state and transport coefficients used in the hydrodynamic simulations.
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6.5. Particle spectra Finally we turn to the particle spectra in viscous hydrodynamics. Ideally the system would evolve through the approximate phase transition down to sufficiently low temperatures where the dynamics could be described either with viscous hydrodynamics or with the kinetic theory of a Hadron Resonance Gas (HRG). In reality this does not seem particularly likely since the system is already expanding three dimensionally and the scales are approximately fixed (see Sec. 4.5). The estimates of the shear viscosity to entropy ratio in a hadronic gas are reliable for T < ∼ 130 MeV and do not support this optimistic picture (see Sec. 3). In seems quite unlikely that there is equilibrium evolution in the HRG below a temperature of T ' 150 MeV. Clearly the dynamics is extremely complex during the quark-hadron transition which takes place for an energy density of e ' 0.5 ↔ 1.2 GeV/fm3 . In this range, the temperature changes by only ∆T ' 20 MeV. However, the hydrodynamic simulations evolve this complicated region for a significant period of time, τ ' 4 fm ↔ 7 fm. This transition region can be seen from the inflection in the AuAu plots in Fig. 18. The pragmatic approach to this complexity is to compute the quasi-particle spectrum of hadrons at a temperature of T ' 150 MeV. Since the HRG describes the QCD thermodynamics well, this pragmatism is fairly well motivated. The approach conserves energy and momentum and when viscous corrections are included also matches the strains across the transition. In ideal hydrodynamics simulations the subsequent evolution of the hadrons has been followed with hadronic cascade models.18, 19, 22 The result of these hybrid models is that the hadronic rescattering is essentially unimportant for the v2 (pT ) observables presented here. Technically, the procedure is the following: along the freezeout surface the spectrum of particles is computed with the Cooper-Frye formula Z da dN a dΣµ P µ f a (−P · u/T ) , (125) E 3 = d p (2π)3 Σ where a labels the particle species, the distribution function is,
f a (−P · u) = na (−P · u/T ) + δf a (−P · u/T ) ,
(126)
and da labels the spin-isospin degeneracy factor for each particle included (see Sec. 5). In practice, the Boltzmann approximation is often sufficient. In Ref. 31 all particles were included up to mass of mres < 2.0 GeV and then subsequently decayed. In other works a simple single species gas was used to study various aspects of viscous hydrodynamics divorced from this complex reality.10, 11 All of the viscous models used the quadratic ansatz discussed in Sec. 5, writing the change to the distribution function of the a-th particle type as f a → na + δf a ,
(127)
with δf a given by δf a =
1 na (1 ± na )P µ P ν πµν . 2(e + P)T 2
(128)
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0.5
0.4
v2
0.3
Ideal
Au-Au (b=7 fm) EOS Ideal Gas ε0=27.7 GeV/fm3 T0=0.342 GeV Viscous (η/s=0.08, τη=3η/sT)
fo only
0.2
fo + δf
0.1
0 0
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1
1.5 2 pT [GeV]
2.5
3
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Fig. 22. v2 (pT ) based on Refs. 10 and 114 showing the v2 (pT ) with (fo + δf ) and and without (fo only) the viscous modification of the distribution function. The result depends to a certain extent on freezeout and the freezeout temperature here is Tfo = 130 MeV.
Before continuing we review the elements that go into a complete hydrodynamic calculation. First initial conditions are specified (see Sec. 6.1) ; then the equations are solved with the viscous term (see Sec. 6.2) ; after this a freezeout surface is specified (see Sec. 6.4 for the limitations of this); finally we compute spectra using Eq. (125) and Eq. (128). This particle spectra can ultimately be compared to the observed elliptic flow. With this oversight we take a more nuanced look at Fig. 15. To separate the viscous modifications of the flow variables (T and uµ ) and the viscous modifications of the distribution function, we turn to Fig. 22. Examining this figure we see that a significant part of the corrections due to the shear viscosity are from the distribution function rather than the flow. Although the magnitude of the flow modifications depends on the details of freezeout, this dependence on δf is the typical and somewhat distressing result. We emphasize however that it is inconsistent to drop the modifications due to δf . The result of dropping the δf means that the energy and momentum of the local fluid cell T µν uν is matched by the particle content, but the off diagonal strains π µν are not reproduced. The assumption underlying the comparison of viscous hydrodynamics to data is that the form of these off diagonal strains is largely unmodified from the Navier Stokes limit during the freezeout process. As the particles are “freezing out” this is a reasonable assumption. However, since these strains are only partially constrained by conservation laws, this assumption needs to be tested against kinetic codes as already emphasized above. This freezeout problem is clearly an obstacle to a reliable extraction of η/s from the data.
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CGC
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v2
η/s=0.08
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0.04 η/s=0.24
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200 NPart
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Fig. 23. Dependence of elliptic flow versus some centrality from Ref. 31. The lines show the results of viscous hydrodynamics and the colored squares show the anisotropy of the stress tensor 0.5 e0p (see Eq. (119)) for different values of η/s. The sensitivity to the quadratic ansatz is estimated in the text and corresponds to half the difference between the red (η/s = 0.08) and blue (η/s = 0.16) curves. The heuristic rule v2 ' 0.5 e0p is motivated in the text.
Although the dependence on δf in v2 (pT ) is undesirable, the viscous modifications of the integrated elliptic flow v2 largely reflects the modifications to the stress tensor itself. The observation is that the stress anisotropy e0p (see Eq. (119)) determines the average flow according to a simple rule of thumb31, 81 1 0 e . (129) 2 p Figure 23 shows e0p and v2 as a function of centrality. The figure supports this rule and suggests that sufficiently integrated predictions from hydrodynamics do not depend on the detailed form of the viscous distribution. To corroborate this conclusion we turn to an analysis originally presented by Ollitrault16 and subsequently generalized to the viscous case.10, 115 First we parameterize the single particle spectrum dN/dpT with an exponential and v2 (pT ) as linearly rising, i.e. v2 '
1 dN ' Ce−pT /T , pT dpT
and
v2 ∝ pT .
(130)
With this form one finds quite generally that the p2T weighted elliptic flow is twice the average v2
2 px − p2y . 2v2 ' A2 ≡ 2 (131) px + p2y
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The p2T weighted elliptic flow has a much closer relationship to the underlying hydrodynamic variables. Indeed we will show how this simple rule of thumb arises and that it is largely independent of the details of δf . To this end, we evaluate the sphericity tensor which will have a simple relationship to A2 Z d3 p (132) S µνρ = SIµνρ + SVµνρ = P µ P ν P ρ [n(−P · u) + δf (−P · u)] (2π)3 Ep Then using the Cooper-Frye formula (with dΣµ = (τ dηs d2 x⊥ , 0, 0, 0)) the asymmetry at any given moment in proper time is R 2 d x⊥ (S 0xx − S 0yy ) A2 = R 2 . (133) d x⊥ (S 0xx + S 0yy ) The sphericity tensor consists of an ideal piece and a viscous piece f → n + δf ,
S µνρ → SIµνρ + SVµνρ .
(134)
First we consider the ideal piece and work in a classical massless gas approximation for ultimate simplicity. The tensor is a third rank symmetric tensor and can be decomposed as SIµνρ = A(T )uµ uν uρ + B(T ) (uµ g νρ + perms) . Here A(T ) and B(T ) are thermodynamic functions and are given by Z A d3 p p2 =B= np = (e + P)T . 6 (2π)3 3
(135)
(136)
For Bose and Fermi gases this relation between A(T ), B(T ), and the enthalpy is approximate. Thus the ideal piece of the sphericity tensor is largely constrained by thermodynamic functions. The viscous piece is largely constrained by the shear viscosity. As discussed in Sec. 5 we parameterize the δf with χ(p) δf = −np
χ(p) P µ P ν σµν . (P · U )2
(137)
While the precise form of the viscous correction χ(p) depends on the details of the microscopic interactions, it is constrained by the shear viscosity Z 2 d3 p η= p χ(p)np . (138) 15 (2π)3 Substituting the viscous parameterization into the definition of the sphericity we find S 0xx = −C(T ) u0 σ xx + 2ux σ x0 , (139) ' −C(T )u0 σ xx ,
(140)
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where C(T ) is C(T ) =
2 15
Z
d3 p 2 p χ(p)np . (2π)3
(141)
For simplicity we have assumed that the flow is somewhat non-relativistic so that ux π x0 is O(v 2 ) compared to u0 π xx . To get a feeling for how sensitive the results are to the quadratic ansatz we will work with a definite functional form χ(p) = Const × p2−α .
(142)
In a relaxation time approximation discussed in Sec. 5, α = 0 corresponds to a relaxation time which increases with linearly p while α = 1 corresponds to a relaxation time independent of p. Substituting this ansatz we find C(T ) ' (6 − α)T η , where
0 < α < 1.
(143)
Having assembled the ingredients we can write down an approximate formula for A2 R 2 d x⊥ T u0 (e + P)(ux ux − uy uy ) + (1 − α6 ) (π xx − π yy ) , A2 ' R 2 d x⊥ T u0 (e + P)(ux ux + uy uy ) + (e + P)/3 + (1 − α6 ) (π xx + π yy ) (144) with 0 < α < 1. This is the desired formula which expresses the observed elliptic flow in terms of the hydrodynamic variables. To reiterate, the coefficient α changes the functional form the viscous distribution function and 0 < α < 1 is a reasonable range — α = 0 is the usual quadratic ansatz. It is useful to compare this formula to the definition of e0p R 2 d x⊥ [(e + P)(ux ux − uy uy ) + (π xx − π yy )] 0 ep = R 2 . (145) d x⊥ [(e + P)(ux ux + uy uy ) + 2P + (π xx + π yy )]
Thus while e0p is not exactly equal to the A2 of Eq. (144), it is close enough to explain the heuristic rule, 2v2 ' e0p . The overall symmetries and dimensions of the sphericity tensor suggests a definition for an analogous quantity in hydrodynamics 1 µνρ Shydro = T (e + P)uµ uν uρ + (e + P)(uµ g νρ + perms) + (uµ π νρ + perms) . 6 (146) In fact in a classical massless gas approximation with the quadratic ansatz, the analysis sketched above shows that µνρ Shydro =
1 µνρ S . 6
(147)
The important point for this review is that from Eq. (144) we see that the integrated elliptic flow is relatively insensitive to the quadratic form for the viscous distribution function. More specifically the uncertainty is ∼ 15% of Aideal − Aviscous , i.e. about 2 2
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half the difference between the blue (η/s = 0.16) and red curves (η/s = 0.08) in Fig. 23. However, the estimate shows that changing the quadratic ansatz will move the curves systematically higher increasing the preferred value of η/s to a certain extent. The quadratic ansatz (which has been used by all hydrodynamic simulations) can be a poor approximation when collinear emission processes are included in the Boltzmann description.103 Clearly addressing more completely the uncertainties associated with the particle content and microscopic interactions in Fig. 23 is a task for the future. Nevertheless, it does seem that sufficiently integrated quantities will reflect rather directly the bulk properties of the hydrodynamic motion in a way that can be quantified. 7. Summary and Outlook The elliptic flow data presented in Figs. 5 and 6 provide strong evidence for hydrodynamic evolution in a deconfined phase of QCD. It is difficult to think of other QCD based mechanisms which could explain the measured elliptic flow. This is because the nuclear geometry has a size RAu ∼ 4 fm, which is large compared to the momentum scales in the problem p ∼ 0.6 GeV. Given this large size, the response of the nuclei to this initial geometry must develop over a relatively long time, τ ' 4 fm/c. Hydrodynamics is the appropriate framework to describe the collective motion over these long time scales. We have discussed three advances which have corroborated the hydrodynamic interpretation of the flow. The first advance is experimental and a brief overview is provided in Sec. 2. These measurements now show quite clearly that the hydrodynamic response is from a deconfined phase (see Fig. 6). Furthermore, the measurements also show that the flow decreases in smaller systems in a systematic way (see Fig. 5). The second advance is in viscous hydrodynamics. There has been important conceptual progress in understanding hydrodynamics beyond the Navier-Stokes order (see Sec. 4.6). The important result of this analysis (see Fig. 14) is that for a Bjorken expansion there are generic kinematic cancellations between the second order terms which reduce their relative importance. In fact in a 0 + 1D Bjorken expansion of a conformal gas in the relaxation time approximation, the second order corrections vanish, while in other more general kinetic theories the second order corrections almost vanish. This understanding of second order hydrodynamics has spurred additional progress in viscous simulations of the heavy ion event (see Sec. 6). It is satisfying that viscous hydrodynamics simulations are in better agreement with the data than ideal hydrodynamics and naturally explain several trends in the elliptic flow. For example, viscosity explains the fall off of v2 at high momentum (see Fig. 22 and Fig. 5) and the fall off of v2 in more peripheral collisions (see Fig. 5 and Fig. 23). Many of these trends were previously reproduced by transport models3 and it is exciting to see that they are now reproduced with a macroscopic approach.
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The final advance has been in kinetic theory simulations. Kinetic theory simulations smoothly interpolate between the hydrodynamic regime and free streaming. Since hydrodynamics is universal (i.e. it only depends on P(e) and η(e)), the kinetic theory results can be used to estimate how still higher gradients would influence the hydrodynamic conclusions. Ideally the results of these simulations would be largely independent of the details of the interactions. The kinetic simulations have demonstrated the ability to reproduce the viscous hydrodynamics (see Fig. 20)), validating the hydrodynamic assumption that the dynamics near the edge does not significantly influence the time development of elliptic flow. In addition, various advances have allowed these kinetic codes to simulate a fluid which does not conserve particle number and which has a shear viscosity, η ∝ T 3 (see Fig. 21). Thus, the estimates of η/s from these simulations are complementary to the macroscopic approach, and must be accepted even if theoretical prejudice rejects the microscopic details. Examining the figures in this review, we see that there is consensus between hydrodynamic and kinetic models on several points:3–12 • Shear viscosity is needed to reproduce trends in the data. This consensus has yet to translate to an agreed upon lower bound on η/s but probably will in the not too distant future. • It is impossible to reproduce the elliptic flow if the shear is too large: η/s > ∼ 0.4 .
(148)
This bound is quite safe and has been found by all groups which have tried to reproduce the observed flow. • The preferred value of η/s is (see Fig. 15 and Fig. 21) η/s ' (1 ↔ 3) ×
1 . 4π
To reduce these constraints further several items need to studied and quantified. • At the end of the collision kinetic assumptions need to be made. The uncertainties involved with the particle content and the quadratic ansatz to the distribution need to be quantified to a much greater extent than has been done so far. The theoretical motivations and limitations of the quadratic ansatz have been discussed in Sec. 5. In Sec. 6.5 we have taken the nascent steps to quantify these uncertainties with an independent analysis of Fig. 23 due to Luzum and Romatschke. • The nucleus-nucleus collision terminates as the system enters the transition region and starts expanding three dimensionally. Some estimates for this transition process have been given in Sec. 4.5. Clearly the transition region sets a very definite scale e ' 0.6 ↔ 1.2 GeV/fm3 which can not be ignored. It is difficult to separate the rapid η(e) dependence in this region from the shear
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viscosity well into the QGP phase (See Fig. 21). This scale influences the dependence of v2 on centrality and hampers an extraction of η/s from this size dependence. It will be important to categorize in a model independent way (e.g. η(e)) how this scale influences the final flow and the final estimates of η/s. Finally, while this review has focused squarely on elliptic flow, the short transport time scales estimated here have implications for a large number of other observables — energy loss,1, 2 the ridge and Mach cones,116–118 heavy quarks,119, 120 and many more. Ultimately these observables will provide a more complete picture of strongly coupled dynamics near the QCD phase transition. Acknowledgments I gratefully acknowledge help from Raimond Snellings, Kevin Dusling, Paul Romatschke, Peter Petreczky, Denes Molnar, Guy Moore, Pasi Huovinen, Tetsufumi Hirano, Peter Steinberg, Ulrich Heinz, Thomas Schaefer, and Carsten Greiner. Any misstatements and errors reflect the shortcomings of the author. I also am grateful to the organizers and participants of the Nearly Perfect Fluids Workshop which clarified the appropriate content of this review. D.T. is supported by the U.S. Department of Energy under an OJI grant DE-FG02-08ER41540 and as a Sloan Fellow. References 1. J. Adams et al. [STAR Collaboration], Nucl. Phys. A 757, 102 (2005) [arXiv:nuclex/0501009]. 2. K. Adcox et al. [PHENIX Collaboration], Nucl. Phys. A 757, 184 (2005) [arXiv:nuclex/0410003]. 3. D. Molnar and M. Gyulassy, Nucl. Phys. A 697, 495 (2002) [Erratum-ibid. A 703, 893 (2002)] [arXiv:nucl-th/0104073]. 4. D. Teaney, Phys. Rev. C 68, 034913 (2003) [arXiv:nucl-th/0301099]. 5. P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, 172301 (2007) [arXiv:0706.1522 [nucl-th]]. 6. Z. Xu, C. Greiner and H. Stocker, Phys. Rev. Lett. 101, 082302 (2008) [arXiv:0711.0961 [nucl-th]]. 7. Z. Xu and C. Greiner, Phys. Rev. C 79, 014904 (2009) [arXiv:0811.2940 [hep-ph]]. 8. H. J. Drescher, A. Dumitru, C. Gombeaud and J. Y. Ollitrault, Phys. Rev. C 76, 024905 (2007) [arXiv:0704.3553 [nucl-th]]. 9. H. Song and U. W. Heinz, Phys. Rev. C 77, 064901 (2008) [arXiv:0712.3715 [nuclth]]. 10. K. Dusling and D. Teaney, Phys. Rev. C 77, 034905 (2008) [arXiv:0710.5932 [nuclth]]. 11. D. Molnar and P. Huovinen, J. Phys. G 35, 104125 (2008) [arXiv:0806.1367 [nuclth]]. 12. G. Ferini, M. Colonna, M. Di Toro and V. Greco, Phys. Lett. B 670, 325 (2009) [arXiv:0805.4814 [nucl-th]]. 13. P. Danielewicz and M. Gyulassy, Phys. Rev. D 31, 53 (1985).
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113. Z. Xu and C. Greiner, Phys. Rev. Lett. 100, 172301 (2008) [arXiv:0710.5719 [nuclth]]. 114. K. Dusling, private communication. 115. D. Teaney, and K. Dusling, in progress. 116. S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 97, 052301 (2006) [arXiv:nucl-ex/0507004]. 117. B. Alver et al. [PHOBOS Collaboration], arXiv:0812.1172 [nucl-ex]. 118. B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett. 102, 052302 (2009) [arXiv:0805.0622 [nucl-ex]]. 119. B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett. 98, 192301 (2007) [arXiv:nucl-ex/0607012]. 120. A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 98, 172301 (2007) [arXiv:nucl-ex/0611018].
HADRON CORRELATIONS IN JETS AND RIDGES THROUGH PARTON RECOMBINATION
RUDOLPH C. HWA Institute of Theoretical Science and Department of Physics University of Oregon, Eugene, OR 97403-5203, USA [email protected]
Hadron correlations in jets, ridges and opposite dijets at all pT above 2 GeV/c are discussed. Since abundant data are available from RHIC at intermediate p T , a reliable hadronization scheme at that pT range is necessary in order to relate the semihard partonic processes to the observables. The recombination model is therefore first reviewed for that purpose. Final-state interaction is shown to be important for the Cronin effect, large B/M ratio and forward production. The effect of semihard partons on the medium is then discussed with particular emphasis on the formation of ridge with or without trigger. Azimuthal anisotropy can result from ridges without early thermalization. Dynamical path length distribution is derived for any centrality. Dihadron correlations in jets on the same or opposite side are shown to reveal detail properties of trigger and antitrigger biases with the inference that tangential jets dominate the dijets accessible to observation.
1. Introduction Among the many properties of the dense medium that have been studied at RHIC, the nature of jet-medium interaction has become the subject of particular current interest.1,2 Jet quenching, proposed as a means to reveal the effect of the hot medium produced in heavy-ion collisions on the hard parton traversing that medium,3,4 has been confirmed by experiments5,6 and has thereby been referred to as a piece of strong evidence for the medium being a deconfined plasma of quarks and gluons.7–10 By the time of Quark Matter 2006 the frontier topic has moved beyond the suppression of single-particle distribution at high pT and into the correlation of hadrons on both the near side and the away side of jets.11,12 The data on dihadron and trihadron correlations are currently analyzed for low and intermediate pT , so the characteristic of hydrodynamical flow is involved in its interplay with semihard partons propagating through the medium. The physics issues are therefore broadened from the medium effects on jets to include also the effect of jets on medium. Theoretical studies of those problems can no longer be restricted to perturbative QCD that is reliable only at high pT or to hydrodynamics that is relevant only at low pT . In the absence of any theory based on first principles that is suitable for intermediate pT , phenomenological modeling is thus inevitable. A sample of some of the papers published before 2008 are given in Refs. 13–31. 267
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Hadron correlation at intermediate pT involves essentially every complication that can be listed in heavy-ion collisions. First, there is the characteristic of the medium created. Then there is the hard or semihard scattering that generates partons propagating through it. The interaction of those partons with the medium not only results in the degradation of the parton momentum, but also gives rise to ridges in association with trigger jets and to broad structure on the away side. Those features observed are in the correlations among hadrons, so hadronization is an unavoidable subprocess that stands between the partonic subprocess and the detected hadrons. Any realistic model must deal with all aspects of the various subprocesses involved. Without an accurate description of hadronization, observed data on hadron correlation cannot be reliably related to the partonic origin of ridges and jet structure. It is generally accepted that fragmentation is the hadronization subprocess at high pT , as in lepton-initiated processes. At intermediate and lower pT recombination or coalescence subprocess (ReCo) in heavy-ion collisions has been found to be more relevant.32–35 Despite differences in detail, the three formulations of ReCo are physically very similar. In Refs. 33 and 34 the descriptions are 3-dimensional and treat recombination and fragmentation as independent additive components of hadronization. In Ref. 35 the formulation is 1D on the basis that acollinear partons have low probability of coalescence, and is simple enough to incorporate fragmentation as a component of recombination (of shower partons) so that there is a smooth transition from low to high pT . Since the discussions on jet-medium interaction in the main part of this review are based largely on the formalism developed in Ref. 35 that emphasizes the role of shower partons at intermediate pT , the background of the subject of recombination along that line is first summarized along with an outline of how non-trivial recombination functions are determined. Some questions raised by critics, concerning such topics as entropy and how partons are turned into constituent quarks, are addressed. More importantly, how shower partons are determined is discussed. Large baryon-to-meson ratio observed in heavy-ion collisions is a signature of ReCo, since the physical reason for it to be higher than in fragmentation is the same in all three formulations.32–34 The discussion here that follows the formulation of the recombination model (RM) by Hwa-Yang should not be taken to imply less significance of the other two, but only the limits of the scope of this review. Considerable space is given to the topics of the Cronin effect (to correct a prevailing misconception) and to forward production at low and intermediate pT in Sec. 3. The large B/M ratio observed at forward production cements the validity of recombination so that one can move on to the main topic of jet-medium interaction. The two aspects of the jet-medium interaction, namely, the effect of jets on the medium and that of the medium on jets, are discussed in Secs. 4 and 6, respectively. In between those two sections we insert a section on azimuthal anisotropy because semihard jets can affect what is conventionally referred to as elliptic flow at low p T and also because ridge formation can depend on the trigger azimuth at intermediate
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pT . Much theoretical attention has been given in the past year to the phenomena of ridges on the near side and double hump on the away side of triggers at intermediate pT .36 – 50 Our aim here is not to review the various approaches of those studies, but to give an overview of what has been accomplished on these topics in the RM. The focus is necessary in order to cover a range of problems that depend on a reliable description of the hadronization subprocess. This review is complementary to the one given recently by Majumder,51 which emphasizes the region of pT much higher than what is considered here, so that factorized fragmentation can be applied. Due to space limitations this review cannot go into the mathematical details of either the basic formalism or the specific problems. Adequate referencing is provided to guide the interested reader to the original papers where details can be found. The discussions will mainly be qualitative, thus rendering an opportunity to describe the motivations, assumptions and physical ideas that underlie the model calculations. For example, the shower parton is an important ingredient in this approach that interpolates between what are soft (thermal-thermal) and hard (shower-shower), but we have neither space nor inclination to revisit the precise scheme in which the shower-parton distributions are derived from the fragmentation functions. The concept of thermal-shower recombination and its application to intermediate-pT physics are more important than the numerical details. Similarly, we emphasize the role that the ridges play (without triggers) in the inclusive distributions of single particles because of the pervasiveness of semihard scattering, the discussion of which can only be phenomenological. Attempts are made to distinguish our approach from conceptions and interpretations that are generally regarded as conventional wisdom. Some examples of what is conventional are: (1) Cronin effect is due to initial-state transverse broadening; (2) large B/M ratio is anomalous; (3) azimuthal anisotropy is due to asymmetric high pressure gradient at early time; (4) recombination implies quark number scaling (QNS) of v2 ; (5) dijets probe the medium interior. In each case evidences are given to support an alternative interpretation. In (4), it is the other way around: QNS confirms recombination but the breaking of QNS does not imply the failure of recombination. Other topics are more current, so no standard views have been developed yet. Indeed, there exist a wide variety of approaches to jet-medium interaction, and what is described here is only one among many possibilities. 2. Hadronization by Recombination 2.1. A historical perspective In the 70s when inclusive cross sections were beginning to be measured in hadronic processes the only theoretical scheme to treat hadronization was fragmentation for lepton-initiated processes for which the interaction of quark were known to be the basic subprocess responsible for multiparticle production. The same fragmentation process was applied also to the production of high-pT particles in hadronic collisions.52 Local parton-hadron duality was also invoked as a way to avoid focusing on
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the issue of hadronization.53 In dual parton model where color strings are stretched between quarks and diquarks, the fragmentation functions (FFs) are attached to the ends of the strings to materialize the partons to hadrons, even if one of the ends is a diquark.54 However, at low pT in pp collisions quarks are not isolated objects in the parton model since there are gluons and wee-partons at small x,55 so the justification for the confinement of color flux to a narrow string is less cogent than at high pT . A more physically realistic description of hadronization seemed wanting. The first serious alternative to fragmentation against the prevalent scheme for hadronization was the suggestion that pion production at low pT in pp collisions can be treated by recombination.56 The simple equation that describes it is Z dx1 dx2 dN π = Fqq¯(x1 , x2 )Rπ (x1 , x2 , x) , (1) x dx x1 x2 where Fqq¯(x1 , x2 ) is the q q¯ distribution, taken to be the product Fq (x1 )Fq¯(x2 ) of the q and q¯ distributions already known at the time among the parton distributions of a proton. The recombination function (RF) Rπ (x1 , x2 , x) contains the momentum conserving δ(x1 + x2 − x) with a multiplicative factor that is constrained by the counting rule developed for quarks in hadrons. That simple treatment of hadronization turned out to produce results that agreed with the existing data very well. The next important step in solidifying the treatment of recombination is the detailed study of the RF. If RF is circumscribed by the characteristics of the wave function of the hadron formed, then it should be related to the time-reversed process of describing the structure of that hadron. In dealing with that relationship it also becomes clear that the distinction between partons and constituent quarks must be recognized and then bridged — a problem that has puzzled some users of the RM even in recent years. Since hadron structure is the basis for RF, it became essential to have a description of the constituents of a hadron in a way that interpolates between the hadronic scale and the partonic scale. It is in the context of filling that need that the concept of valons was proposed.57 The origin of the notion of constituent quarks (CQs) is rooted in solving the bound-state problem of hadrons. However, in describing the structure of a nucleon in deep inelastic scattering the role of CQs seems to be totally absent in the structure functions F, such as νW2 (x, Q2 ). The two descriptions are not merely due to the difference in reference frames, CQs being in the rest frame, the partons in a high-momentum frame. Also important is that the bound state is a problem at the hadronic scale, i.e. low Q2 , while deep inelastic scattering is at high Q2 . The two aspects of the problem can be connected by the introduction of valons as the dressed valence quarks, i.e., each being a valence quark with its cloud of gluons and sea quarks which can be resolved only by high-Q2 probes. At low Q2 the internal structure of a valon cannot be resolved, so a valon becomes what a CQ would be in the momentum-fraction variable in an infinite-momentum frame. Thus the valon distribution in a hadron is the wave-function squared of the CQs, whose structure
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functions are described by pQCD at high Q2 . Note that the usual description of Q2 -evolution by DGLAP has no prescription within the theory for the boundary condition at low Q2 . That distribution at low Q2 is precisely what the valon distribution specifies. In summary, the structure function F h (x, Q2 ) of a hadron is a convolution of the valon distribution Gv/h (y) and the structure function F v (z, Q2 ) of a valon58 XZ 1 dyGv/h (y)F v (x/y, Q2 ) , (2) F h (x, Q2 ) = ν
x
where y is the momentum fraction (not rapidity) of a valon in the hadron h. The first description of the properties of Gv/h (y) is given in Refs. 57 and 58, derived from the early data F h (x, Q2 ). More recent determination of Gv/h (y) is described in Ref. 59 where more modern parton distribution functions have been used.60 Gv/h (y) is the single-valon inclusive distribution in hadron h, and is the appropriate integral of the exclusive distribution, Gv/π (y1 , y2 ) for pion and Gv/p (y1 , y2 , y3 ) for proton. More specifically, Gv/π (y1 , y2 ) is the absolute square of the pion wave function hv1 (y1 )v2 (y2 )|πi in the infinite-momentum frame. Once we have that, it is trivial to get the RF for pion (i.e., by complex conjugation), since it is the time-reversed process. Thus for pion and proton, we have x x x1 x2 2 1 , , (3) Rπ (x1 , x2 , x) = 2 Gv/π x x x x x x x1 x2 x3 2 3 1 Rp (x1 , x2 , x3 , x) = G , , , (4) v/p x3 x x x
where the factors on the RHS are due to the fact that the RFs are invariant distributions defined in the phase space element Πi dxi /xi , whereas Gv/h are non-invariant defined in Πi dyi , as seen in (2). The exclusive distribution Gv/h contains the moP mentum conserving δ( i yi −1). For pion there is nothing else, but for other hadrons the prefactors are given in Refs. 59 and 61. Having determined the RF, the natural question next is how partons turn into valons before recombination in a scattering process. Let us suppose that we can calculate the multi-parton distribution F (x1 , x2 ) for a q and q¯ moving in the same direction, whether at low or high pT . If their momentum vectors are not parallel, with relative transverse momentum larger than the inverse hadronic size, then the probability of recombination is negligible. Relative longitudinal momentum need not be small, since the RF allows for the variation in the momentum fractions, just as the partons in a hadron can have various momentum fractions. Now, as the q and q¯ move out of the interaction region, they may undergo color mutation by soft gluon radiation as well as dress themselves with gluon emission and reabsorption with the possibility of creating virtual q q¯ pairs, none of which can be made precise without a high Q2 probe. The net effect is that given enough time before hadronization the quarks convert themselves to valons with essentially the same original momenta, assuming that the energy loss in vacuum due to soft gluon radiation is negligible
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(even though color mutation is not negligible). For that reason we may simply write F (x1 , x2 )R(x1 , x2 , x) as multiplicative factors, as done in (1), while treating F (x1 , x2 ) as the distribution of partons and R(x1 , x2 , x) as the RF of valons. The detail of this is explained in Ref. 58. The question of entropy conservation has been raised at times, especially by those with experience in nuclear physics. In elementary processes, such as q+ q¯ → π, unlike a nuclear process p + n → d, the color degree of freedom is important. Since a pion is colorless, the q and q¯ that recombine must have opposite color. If they do not, they cannot travel in vacuum without dragging a color flux tube behind them. The most energy-efficient way for them to evolve is to emit soft gluons thereby mutating their color charges until the q q¯ pair becomes colorless and recombine. Such soft processes leave behind color degrees of freedom from the q q¯ system whose entropy is consequently not conserved. It is therefore pointless to pursue the question of entropy conservation in recombination, since the problem is uncalculable and puts no constraint on the kinematics of the formation of hadrons. Besides, the entropy principle should not be applied locally. A global consideration must recognize that the bulk volume is increasing during the hadronization process, and thus this compensates any decrease of local entropy density. After the extensive discussion given above on the RF, we have come to the point of being able to assert that the main issue about recombination is the determination of the multi-parton distribution, such as Fqq¯(x1 , x2 ) in (1), of the quarks that recombine. Related to that is the question about the role of gluons which have to hadronize also. By moving the focus to the distributions of partons that hadronize, the investigation can then concentrate on the more relevant issues in heavy-ion collisions concerning the effect of the nuclear medium. 2.2. Shower partons At low pT in the forward direction the partons that recombine are closely related to the low-Q2 partons in the projectile. It is a subject to be discussed in a following section. At intermediate and high pT the partons are divided into two types: thermal (T) and shower (S). The former contains the medium effect; the latter is due to semihard and hard scattered partons. The consideration of shower partons is a unique feature of our approach to recombination, which is empowered by the possibility to include fragmentation process as SS or SSS recombination. The jet-medium interaction is taken into account at the hadronization stage by TS recombination, although at an earlier stage the energy loss of the partons before emerging from the medium is another effect of the interaction that is, of course, also important. A quantitative theoretical study of that energy loss in realistic heavy-ion collisions at fixed centrality cannot be carried out and compared with data without a reliable description of hadronization. At intermediate pT there is no evidence that fragmentation is applicable because the baryon/meson ratio would be too small, as we shall describe in the next section.
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The fragmentation function (FF), D(x), is a phenomenological quantity whose Q2 evolution is calculable in pQCD; however, at some low Q2 before evolution the distribution in x is parametrized by fitting the data. With that reality in mind it is reasonable to consider an alternative way of treating the FF, one that builds in more dynamical content by regarding fragmentation as a recombination process. That is, if we replace the LHS of Eq. (1) by the invariant function xDiπ (x), then the corresponding two-parton distribution in the integrand on the RHS is the product distributions of two shower partons in a jet initiated by a parton of type i. To be specific, consider, for example, the fragmentation of gluon to pion Z x2 dx1 dx2 q q¯ π Rqπq¯(x1 , x2 , x) , (5) S (x1 )Sg xDg (x) = x1 x2 g 1 − x1 where the q and q¯ distributions in a shower initiated by the gluon are the same, but their momentum-fraction dependencies are such that if one (x1 ) is a leading quark, the other (x2 ) has to be from the remainder (1 − x1 ) of the parton pool. With xDgπ (x) being a phenomenological input, it is possible to solve (5) numerically to obtain Sgq (z). It has been shown in Ref. 62 that there are enough FFs known from analyzing leptonic processes to render feasible the determination of various shower parton distributions (SPDs), which are denoted collectively by Sij with i = q, q¯, g and j = q, s, q¯, s¯, where q can be either u or d. If in i the initiating hard parton is an s quark, it is treated as q. That is not the case if s is in the produced shower. The parameterization of Sij has the form Sij (z) = Az a (1 − z)b (1 + cz d ) ,
(6)
where the dependence of A, a, etc., on i and j are given in a Table in Ref. 62. Those parameters were determined from fitting the FFs at Q = 10 GeV, and have been used for all hadronization processes without further consideration of their dependence on pT . It should be recognized that those shower partons are not to be identified with the ones due to gluon radiation at very high virtuality calculable in pQCD, which is not applicable for the description of hadronization at low virtuality. To sum up, in the conventional approach the FF is treated as a black box with a parton going in and a hadron going out, whereas in the RM we open up the black box and treat the outgoing hadron as the product of recombination of shower partons, whose distributions are to be determined from the FFs. Once the SPDs are known, one can then consider the possibility that a shower parton may recombine with a thermal parton in the vicinity of a jet and thus give a more complete description of hadronization at intermediate-pT region, especially in the case of nuclear collisions. The SPDs parametrized by (6), being derived from the meson FFs, open up the question of what happens if, instead of j ¯j 0 → M , three quarks in the shower recombine, e.g., uud → p, or uds → Λ. A self-consistent scheme of hadronization would have to demand that the formation of baryons is a possibility in a fragmentation process and that the SPDs already determined should give an unambiguous prediction of what baryon FFs are. The calculation has been carried out in Ref. 63
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where the results for g → p and g → Λ in gluon jets are in good agreement with data64 without the use of any adjustable parameters. To be able to relate meson and baryon FFs is an attribute of our formalism for hadronization that has not been achieved in other theoretical approaches, and provides further evidence that the SPDs are reliable for use at the hadronization scale. 2.3. Parton distributions before recombination In the study of shower partons in a jet we have assumed the validity of the approximation that the fragmentation process is essentially one dimensional. One may question whether the recombination process in a nuclear collision for a hadron produced at high pT may necessitate a 3D consideration, since two different length scales seem to be involved, one being that of the hadron produced, the other being the size of nuclear medium at hadronization time. Indeed, recombination schemes formulated in 3D have been proposed, and various groups have independently found satisfactory results that are similar to one another.32–34 The essence of recombination is, however, not in the 2D transverse plane normal to the direction of hadron momentum because if the coalescing parton momenta are not roughly parallel, then the relative momentum would have a large component in that transverse plane. If that component is larger than the inverse of the hadron size, then the two (or three) partons cannot recombine. Thus partons from regions of the nuclear medium that are far apart cannot form a hadron, rendering the concern over different length scales in the problem inessential. Only collinear partons emanating from the same region of the dense medium can recombine. For that reason the 1D formulation of recombination is adequate, as simple as expressed in (1). If one asks why the relative momentum can be large in the hadron direction, but not transverse to it, the answer lies in the foundation of the parton model where the momentum fraction can vary from 0 to 1, while the transverse parton momentum kT is limited to ∼ (hadron radius)−1 . The RFs in (3) and (4) are related to the 1D wave function in that framework. Having justified the 1D formulation of recombination, let us now focus on the distributions of the recombining partons at low pT , and later at high pT . Since pQCD cannot be applied to multiparticle production at low pT , our consideration of the problem is based on Feynman’s parton model, which was originally proposed for hadron production at low pT .55 In a pp collision there are valence and sea quarks and gluons whose x-distributions at low Q2 are known.60 Without hard scattering their momenta carry them forward, and they must hadronize in the fragmentation region of the initial proton. RM has provided a quantitative treatment of the single-hadron inclusive distribution in xF not only for pp, but also for all realistic hadronic collisions.56,58,61 What is to be remarked here is how gluons hadronize. In an incident proton as in other hadrons, the gluons carry about half the momentum of the host. Since gluons cannot hadronize by themselves, but can virtually turn to q q¯ pairs in the sea, we require that all gluons be converted to the sea quarks (thus
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saturating the sea) before recombination. This idea was originally suggested by Duke and Taylor,65 and was implemented quantitatively in the valon model in Refs. 58 and 61. The simplest way to achieve that is to increase the normalization of the q q¯ sea quarks without changing their x distributions so that the total momentum of the valence (unchanged) and sea quarks (enhanced) exhausts the initial momentum of the hadron without any left over for gluons. With the Fqq¯(x1 , x2 ) thus obtained, the use of (1) results in an inclusive π distribution that agrees with data in both normalization and x spectrum.58,61 Using the appropriate valon distributions of pion and kaon, the success extends beyond p → π ± to π + → π − , K + → π ± , and π + → K ± in hadronic collisions at low pT . In nuclear collisions there is the additional complication arising from momentum degradation when partons traverse nuclear medium. It is a subject that will be brought up in Sec. 3.4. When pT is not small, then there has to be a semihard or hard scattering at the partonic level so that a parton with kT > 3 GeV/c has to be created. In that case shower partons are developed in addition to the thermal partons, so the partons before recombination can be separated into the following types: TT+TS+SS for mesons and TTT+TTS+TSS+SSS for baryons.35 The thermal partons have kT mainly < 2 GeV/c. If one has a reliable scheme to calculate the thermal partons, then their kT distributions can, of course, be used in the recombination equation. It does not mean that hydrodynamics is a necessary input in the RM. In pA collisions, for instance, hydrodynamics is not reliable, yet the Cronin effect can be understood in the RM for both proton and pion production without associating the effect with initial-state scattering — a departure from the conventional thinking that will be discussed in the next section. In most applications reviewed here, the distributions of thermal partons are determined from fitting the data at low pT , and are then used in the RM to describe the behavior of hadrons at pT & 3 GeV/c. When we consider correlation at a later section, careful attention will be given to the enhancement of thermal partons due to the energy loss of a semihard or hard parton passing through the nuclear medium. It is only in the framework of a reliable hadronization scheme can one learn from the detected hadrons the nature of jet-medium interaction, as aspired in jet tomography. 3. Large Baryon/Meson Ratios 3.1. Intermediate pT in heavy-ion collisions A well-known signature of the RM is that the baryon/meson (B/M) ratio is large — larger than what is customarily expected in fragmentation. The p/π ratio of the FFs, i.e., Dp/q (x)/Dπ/q (x), is at most 0.2 at x ' 0.3, and is much lower at other values of x.64 However, for inclusive distributions in heavy-ion collisions at RHIC the ratio Rp/π is as large as ∼ 1 at pT ' 3 GeV/c,66,67 as shown in Fig. 1. Thus hadronization at intermediate pT cannot be due to parton fragmentation. Three groups (TAM, Duke and Oregon) have studied the problem in the Recombination/Coalescence (ReCo) model32–35 and found large Rp/π in agreement with
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Fig. 1. Comparison of baryon/meson ratio from STAR data to theoretical curves from ReCo. 33–35 Left panel is from Ref. 66; right panel is from Ref. 67.
the data. The underlying reason that is common in all versions of ReCo is that for p and π at the same pT the three quarks that form the p has average momentum pT /3, while the q and q¯ that form the π has pT /2. Since parton distributions are suppressed severely at increasing kT , there are more quarks at pT /3 than at pT /2, so the formation of proton is not at a disadvantage compared to that of a pion despite the difference in the RFs. For either hadron the recombination process is at an advantage over fragmentation because of the addivity of momenta. Fragmentation suffers from two penalties: first, the initiating parton must have a momentum higher than pT , and second, the FFs are suppressed at any momentum fraction, more for proton than for pion. Thus the yield from parton fragmentation is lower compared to that from parton recombination at intermediate pT , even apart from the issue of B/M ratio. When faced with the question why baryon production is so efficient, the proponents of pion fragmentation regard it as an anomaly. Despite efforts to explain the enhancement in terms of baryon junction,68,69 the program has not been successful in establishing it as a viable mechanism for the formation of baryons.70 From the point of view of ReCo there is nothing anomalous. A simple way to understand the pT dependence of Rp/π (pT ) is to consider the 1D formulation of ReCo given in Ref. 35, where the invariant distributions of meson and baryon production are expressed as ! Z Y 2 M dqi 0 dN p = Fqq¯(q1 , q2 )RM (q1 , q2 , p) , (7) dp q i i=1 p
0 dN
B
dp
=
Z
3 Y dqi q i=1 i
!
F3q (q1 , q2 , q3 )RB (q1 , q2 , q3 , p) ,
(8)
in which all quarks are collinear with the hadron momentum p. We assume that the rapidity y is ≈ 0, so the transverse momenta are the only essential variables, for which the subscripts T of all momenta are therefore omitted, for brevity. Mass
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effect at low pT renders the approximation poor and the 1D description inadequate. However, in order to gain a transparent picture analytically, let us ignore those complications and assume provisionally that all hadrons are massless. Then the experimental observation of exponential behavior of the pT distribution of pions at low pT , i.e., dN π /pdp ∝ exp(−p/T ), implies that the thermal partons behave as T (q) = q
dN th = Cqe−q/T , dq
(9)
where C has dimension (GeV)−1 , and Rπ given in (3) is dimensionless. When thermal partons dominate Fqq¯ and F3q , the multiparton distributions can be written as products: T (q1 )T (q2 ) and T (q1 )T (q2 )T (q3 ), respectively. It is then clear from the dimensionlessness of the quantities in (7) and (8) that with the proton distribution having C 3 dependence, as opposed to the pion distribution being ∝ C 2 , the p/π ratio has the property Rp/π (p) =
dN p /pdp ∝ Cp , dN π /pdp
(10)
so long as thermal recombination dominates. This linear rise with p is the behavior seen in Fig. 1, although the mass effect of proton makes it less trivial in pT . Nevertheless, this simple feature is embodied in the more detailed computation until shower partons become important for pT > 3 GeV/c.35 From the above analysis which should apply to any baryon and meson, it follows that the ratios Λ/K and Ω/φ should also increase with pT in a way similar to p/π. Such behaviors have indeed been observed by STAR,71,72 as have been obtained in theoretical calculation.73 Taken altogether, it means that without TS and TTS recombination the B/M ratios would continue to rise with pT . But the data all show that the ratios peak at around pT ≈ 3 GeV/c. In the RM the bend-over is due to the increase of the TS component of the meson earlier than the TTS component of the baryon, since two thermal partons in the latter have more weight than the single thermal parton in the former. The shower parton distribution S(q) in heavy-ion collisions is a convolution of the hard parton distribution fi (k) and the S distribution derived from FF, discussed in Sec. 2.2, i.e., XZ S(q) = ζ dkkfi (k)Si (q/k). (11) i
fi (k) is the transverse-momentum distribution of hard parton i at midrapidity and contains the shadowing effect of the parton distribution in nuclear collisions. A simple parametrization of it is given in Ref. 74 as follows fi (k) = K
A , (1 + k/B)n
(12)
where K = 2.5 and A, B, n are tabulated for each parton type i for nuclear collisions at RHIC and LHC. The parameter ζ = 0.07 is the average suppression factor that
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can be related to the nuclear modification factor RAA , and was denoted by ξ in Ref. 35 and other references thereafter. Since fi (k) has a power-law dependence on k, so does S(q2 ) on q2 in contrast to the exponential behavior of the thermal partons, T (q1 ). This upward bending of S(q2 ) relative to T (q1 ) is the beginning of the dominance of TS and TTS components over TT and TTT components, resulting in a peak in the B/M ratio at around pT ∼ 3 GeV/c. Detailed descriptions of these calculations are given in Refs. 35 and 73. We add here that the effort made to consider the shower partons before recombination is motivated by our concern that a hard parton with high virtuality cannot hadronize by coalescing with a soft parton with low virtuality. The introduction of shower partons is our way to bring the effects of hard scattering to the hadronization scale. At the same time the formalism does not exclude fragmentation by a hard parton, since SS and SSS recombination at high pT are equivalent to fragmentation but in a language that has dynamical content at the hadronization scale. One could ask how the RM can be applied reliably in the intermediate-pT region before the shower partons were introduced. The approach adopted in Ref. 32 does not involve the determination of the hard parton distribution by perturbative calculation, but uses the pion data as input to extract the parton distribution at the hadronization scale at all pT in the framework of the RM. It is on the basis of the extracted parton distribution (which must in hindsight contain the shower partons) that the proton inclusive distribution is calculated. Thus the procedure is self-consistent. The result is that the p/π ratio is large at pT ∼ 3 GeV/c in agreement with data; furthermore, it was a prediction that the ratio would decrease as pT increases beyond 3 GeV/c, as confirmed later by data shown in Fig. 1. 3.2. Cronin effect The conventional explanation of the Cronin effect,75 i.e., the enhancement of hadron spectra at intermediate pT in pA collisions with increasing nuclear size, is that it is due to multiple scattering of projectile partons as they propagate through the target nucleus, thus acquiring transverse momenta, and that a moderately large-k T parton hadronizes by fragmentation.76 The emphasis has been on the transverse broadening of the parton in the initial-state interaction (ISI) and not on the finalstate interaction (FSI). In fact, the Cronin effect has become synonymous to ISI effect in certain circles. However, that line of interpretation ignores another part of the original discovery75 where the A dependence of hadrons produced in pA collisions, when parameterized as dN (pA → hX) ∝ Aαh (pT ) , dpT
(13)
has the property that αp (pT ) > απ (pT ) for all pT measured. That experimental result alone is sufficient to invalidate the application of fragmentation to the hadronization process, since if the A dependence in (13) arises mainly from the ISI,
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π
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Fig. 2. Central-to-peripheral ratios for the production of pion (left panel) and proton (right panel) in dAu collisions. Data are from Ref. 77 and lines are from Refs. 78 and 79.
where the multiply-scattered parton picks up it kT , then the transverse broadening of that parton should have no knowledge of whether the parton would hadronize into a proton or a pion, so αh should be independent of the hadron type h. A modern version of the Cronin effect is given in terms of the central-toperipheral nuclear modification factor for dAu collisions at midrapidity C 1/Ncoll dN h /pT dpT (C) h RCP , (14) (pT ) = P 1/Ncoll dN h /pT dpT (P )
where C and P denote central and peripheral, respectively, and Ncoll is the average number of inelastic N N collisions. If hadronization is by fragmentation, which is a factorizable subprocess, the FFs for any given h should cancel in the ratio of p h (14), so RCP should be independent of h. However, the data show that RCP (pT ) > π RCP (pT ) for all pT > 1 GeV/c when C = 0–20% and P = 60–90% centralities.77 See Fig. 2. Clearly ISI is not able to explain this phenomenon, which strongly suggests the medium-dependence of hadronization. The data further indicate that the pT h dependence of RCP (pT ) peaks at pT ∼ 3 GeV/c for both p and π, reminiscent of the p/π ratio at fixed centrality in AuAu collisions although the C/P ratio for dAu collisions is distinctly different. Hadron production at intermediate pT and η ∼ 0 in dAu collisions can be treated in the RM in a similar way as for AuAu collisions. Although no hot and dense medium is produced in a dAu collision, so thermal partons are not generated in the same sense as in AuAu collisions, nevertheless soft partons are present to give rise to the low-pT hadrons. For notational uniformity we continue to refer to them as thermal partons. We apply the same formalism developed in Ref. 35 to the dAu problem and consider the TT+TS+SS contributions to π production (TTT+TTS+TSS+SSS for p). The thermal T distribution is determined by fitting the pT spectra at pT < 1 GeV/c for each centrality; the shower-parton S distribution is calculated as before but without nuclear suppression. Unlike the dense
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thermal system created in AuAu collisions, the T distribution in this case is weaker; its parameters C and T (inverse slope) that correspond to the ones in Eq. (9) are smaller. Furthermore, C decreases with increasing peripherality, while T remains unchanged at 0.21 GeV/c.78 Thus thermal-shower recombination becomes important at pT & 1 GeV/c, which is earlier than in AuAu collisions. As a consequence, π RCP (pT ) becomes > 1 at pT > 1 GeV/c. That is the Cronin effect, but not due to ISI. The same situation occurs for proton production, only stronger.79 The calculated results for the inclusive distributions of both π and p agree well with data p π h at all centralities, hence also RCP (pT ) and RCP (pT ). Figure 2 shows RCP (pT ) for 77 C = 0–20% and P = 60–90% in dAu collisions for h = π and p; the lines are the p π results obtained in the RM.78,79 The reason for RCP > RCP can again be traced to 3-quark recombination for p and only 2 quarks for π. When pT is large, fragmenh tation dominates (i.e. SS and SSS), and both RCP approach 1, since FFs cancel and the yields are normalized by Ncoll . No exotic mechanism need be invoked to explain the p production process. FSI alone is sufficient to provide the underlying physics for the Cronin effect. 3.3. Forward production in dAu collisions Hadron production at forward rapidities in dAu collisions was regarded as a fertile ground for exposing the physics of ISI, especially saturation physics,80,81 since the nuclear effect in the deuteron fragmentation region was thought to cause minimal FSI. It was further thought that the difference in nuclear media for the Au side (η < 0) and the d side (η > 0) would lead to backward-forward asymmetry in particle yield in such a way as to reveal a transition in basic physics from multiple scattering in ISI for η . 0 to gluon saturation for η > 0. The observation by BRAHMS82 that RCP decreases with increasing η was regarded as an indication supporting that view.83 That line of thinking, however, assumes that FSI is invariant under changes in η so that any dependence on η observed is a direct signal from ISI. Such an assumption is inconsistent with the result of a study of forward production in dAu collisions in the RM, where both RCP (pT , η) and RB/F (pT ) are shown to be well reproduced by considering FSI only.84 Any inference on ISI from the data must first perform a subtraction of the effect of FSI, and just as in the case of the Cronin effect there is essentially nothing left after the subtraction. In Sec. 3.2 the Cronin effect at midrapidity (η ≈ 0) is considered. The extension to η > 0 along the same line involves no new physics. However, it is necessary to determine the η dependencies of the soft and hard parton spectra at various centralities. For the soft partons, use is made of the data on dN/dη to modify the normalization of T (q, η) already determined at η = 0. For the hard partons, modified parametrizations of their distributions fi (kT , η) are obtained from leading order minijet calculations using the CTEQ5 pdf 85 and the EKS98 shadowing.86 A notable feature of the result is that fi (kT , η) falls rapidly with kT as η increases, especially near the kinematical boundary kT = 8.13 GeV/c and η = 3.2. Thus TS
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(b)
Fig. 3. (a) RCP (pT , η) for forward production. Data are from Ref. 82 and lines from Ref. 84. (b) Back/Forward ratio RB/F (pT ) for 0.5 < |η| < 1.0 that shows agreement of theoretical result84 with preliminary data87 for pT < 2 GeV/c but not for higher pT , but later data88 (inset) show agreement for pT > 2 GeV/c also.
and SS components are negligible compared to TT at large η for any pT and any centrality, even though the TT component is exponentially suppressed. In central collisions there is the additional suppression due to momentum degraduation of the forward partons going through the nuclear medium of the target Au. Putting the various features together leads to the ratio RCP (pT , η) shown in Fig. 3(a), where the data are from Ref. 82 and the curves from the calculation in Ref. 84. It is evident that the decrease of RCP (pT , η) at pT > 2 GeV/c as η increased from η = 0 to η = 3.2 is well reproduced in the RM. Only one new parameter is introduced to describe the centrality and η dependence of the inverse slope T of the soft partons, but no new physics has been added. The suppression of RCP (pT , η) at η > 1 is due mainly to the reduction of the density of soft partons in the forward direction, where hard partons are suppressed. Extending the consideration to the backward region and using the same T (η) extrapolated to η < 0, the backward/forward ratio of the yield can be calculated. 84 For η = ±0.75 corresponding to the data of STAR at 0.5 < |η| < 1.0 and 0–20% centrality,87 the calculated result on RB/F for π + + π − + p + p¯ is shown by the solid line in Fig. 3(b). While it agrees with the data very well for pT < 2 GeV/c, it is noticably lower than the data for all charged particles for pT > 2 GeV/c. However, more recent data on RB/F (pT ) for π + +π − +p+ p¯,88 shown in the inset of Fig. 3(b), exhibit excellent agreement with the same theoretical curve that should be regarded as a prediction. The fact that RB/F is > 1 for all pT measured may be regarded as a proof against initial transverse broadening of partons, since forward partons of d have more nuclear matter of Au to go through than the backward partons of Au. Thus if ISI is responsible for the acquisition of pT of the final-state hadrons, then RB/F should be < 1. The data clearly indicate otherwise.
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3.4. Forward production in AuAu collisions Theoretical study of hadron production in the forward direction in heavy-ion collision is a difficult problem for several reasons. The parton momentum distribution at low Q2 and large momentum fraction x in nuclear collisions is hard to determine, especially when momentum degradation that accounts for what is called “baryon stopping” cannot be ignored. Furthermore, degradation of high-momentum partons in the nuclear medium implies the regeneration of soft partons at lower x; that is hard to treat also. The use of data as input to constrain unknown parameters is unavoidable; however, existent data have their own limitations. Measurement at fixed η cannot be used to provide information on xF dependence unless pT is known.89 Measurement of both η and pT has been limited to charged hadrons90 that cannot easily be separated into baryons and mesons. For these various reasons forward production in AA collisions has not been an active area of theoretical investigation. However, there are gross features at large η that suggest important physics at play and deserve explanation. PHOBOS data show that particles are detected at η 0 > 0 where η 0 is the shifted pseudorapidity defined by η 0 = η − ybeam .89 It is significant because it suggests that 0 if hpT i is not too small, it corresponds to xF > 1, where xF = (pT /mp )eη . Instead of violation of momentum conservation, the interpretations in the RM is that a proton can be produced in the xF > 1 region, if three quarks from three different nucleons in the projectile nucleus, each with xi < 1, recombine to form a nucleon with xF = P 91 That kinematical region is referred to as transfragmentation region i xi > 1. (TFR), which is not accessible, if hadronization is by fragmentation. The theoretical calculation in the RM involves an unknown parameter, κ, which quantifies the degree of momentum degradation of low-kT partons, in the forward direction. For κ in a reasonable range, not only can nucleons be produced continuously across the xF boundary, but also can p/π ratio attain an amazingly large value.91 BRAHMS has determined the pT distribution of all charged particles at η = 3.2.90 For hpT i = 1 GeV/c, the corresponding values of xF for pion and proton are, respectively, 0.4 and 0.54. Taking the preliminary value of the p¯/p ratio at 0.05 into account, it is possible to estimate the value of κ and then calculate the pT distribution of p + p¯ + π + + π − .92 The p/π ratio was predicted to be ∼ 1 at pT ∼ 1 GeV/c. However, at QM2008 the more recent data on Rp/p was reported to have a ¯ 93 lower value at 0.02 and on Rp/π a higher value at ∼ 4 at pT ∼ 1 GeV/c.94 Those new data prompted a reexamination of the problem in the RM; with appropriate changes in the treatment of degradation, regeneration and transverse momentum, the very large p/π ratio can be understood.95 Since p, p¯ and π production at large η depends sensitively on q and q¯ distributions, which in turn depend strongly on the dynamical process of momentum degradation and soft-parton regeneration (the parameterization of which requires phenomenological inputs), the procedure in Ref. 95 is to use Rp/π and Rp/p as input ¯ in order to determine κ and then calculate the x distributions of the hadrons. At
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(a)
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(b)
Fig. 4. (a) Proton/pion ratio in forward production at η = 3.2 showing agreement between data 94 and solid line from the RM;95 the dashed line is the contribution from the longitudinal components at fixed η. (b) Comparison of the pT distribution of charged particles at η = 3.2 from BRAHMS90 with calculated result from the RM.95
fixed η the x and pT distributions are related. It turns out that the result on the x distribution leads to a large contribution to the pT distribution of Rp/π (pT ) shown by the dashed line in Fig. 4(a). The additional enhancement shown by the solid line arises from the mass dependence of the inverse slopes Th due to flow. While the ratio Rp/π is insensitive to the absolute normalizations of the yields, the inclusive distribution of all charged particles is not. In Fig. 4(b) is shown the good agreement between the calculated result and the data in both normalization and shape with no extra parameters beyond κ already fixed. It should be noted that the p/π ratio, shown in Fig. 4(a), is extremely large at η = 3.2 and modest pT < 2 GeV/c. The underlying physics is clearly the suppression of q¯ at medium x and the enhancement of p due to 3q recombination, where the (valence) quarks are from three different nucleons in the projectile. No other hadronization mechanisms are known to be able to reproduce the data on the large Rp/π at large η.
3.5. Recombination of adjacent jets at LHC So far we have considered only the physics at RHIC energies and the recombination of thermal and shower partons, either between them or among themselves. At RHIC high-pT jets are rare, so the shower partons are from one jet at most in an event. At LHC, however, high-pT jets are copiously produced for pT < 20 GeV/c. When the jet density is high, the recombination of shower partons in neighboring jets becomes more probable and can make a significant contribution to the spectra of hadrons in the 10 < pT < 20 GeV/c range, high by RHIC standard, but intermediate at LHC. If that turns out to be true, then a remarkable signature is predicted and is easily measurable: the p/π ratio will be huge, perhaps as high as 20.96
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If a hard parton of momentum kT is produced, shower partons in its jet with P momenta qi are limited by the constraint i qi < kT , so that the recombination of those shower partons can produce a hadron with momentum pT not exceeding kT . However, if there are two adjacent jets with hard-parton momenta k1T and k2T , then to form a hadron at pT from shower partons in those two jets, neither k1T nor k2T need to be larger than pT , so the rate of such a process would be higher. Furthermore, to form a proton at pT the shower parton qi can be lower than those for pion formation at the same pT , so kiT can be even lower. Thus Rp/π in 2-jet recombination can be much higher than the ratio in 1-jet fragmentation. The probability for 2-jet recombination, however, also depends on the overlap of jet cones, since the coalescing shower partons must be nearly collinear. That overlap decreases with increasing kiT , so there is a suppression factor in the SS or SSS recombination integral that depends on the widths of the jet cones. Using some reasonable estimates on all the factors involved, it is found that Rp/π can be between 5 and 20 in the range 10 < pT < 20 GeV/c, decreasing with increasing pT .96 Although exact numbers are unreliable, the approximate value of Rp/π is about 2 orders of magnitude higher than what is expected in the usual scenario of fragmentation from single hard partons. The origin of the large Rp/π at LHC discussed above is basically the same as that for forward production in AuAu collisions at RHIC. In both cases it is the multisource supply of the recombining partons that enhances the proton production. At large pT at LHC there are more than one jet going in the same direction; at large pL at RHIC there are more than one nucleon going in the forward direction. In the latter case we already have data supporting our view that Rp/π should be large as shown in Fig. 3(a). It would be surprising that our prediction of large Rp/π at LHC turns out to be untrue. 4. Ridgeology
Phenomenology of Ridges
In the previous section the topics of discussion have been exclusively on the singleparticle distributions in various regions of phase space. Everywhere it is found that the B/M ratio is large when pT is in the intermediate range. We now consider twoparticle correlations, on which there is a wealth of data as a result of the general consensus in both the experimental and theoretical communities that more can be learned about the dense medium when one studies the system’s effect on (and response to) penetrating probes. The strong interaction between energetic partons and the medium they traverse, resulting in jet quenching, is the underlying physics that can be revealed in the jet tomography program.4,97 To calibrate the medium effect theoretically, it is necessary to have a reliable framework in which to do calculation from first principles, and that is perturbative QCD. Although many studies in pQCD have been carried out to learn about the modification of jets in dense medium in various approximation schemes,51,98 they are mainly concerned with the effect of the medium on jets at high pT , and the results can only be
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compared with data on single-particle distributions, such as RAA (pT ). The response of the medium to the passage of hard partons is not what can be calculated in pQCD, since it involves soft physics. That is, however, the physical origin of most of the characteristics in the correlation data. An understanding of that response is one of the objectives of studying correlations. Without the reliable theory to describe correlation, especially at low to intermediate pT where abundant data exist, it becomes necessary to use phenomenological models to relate various features of correlation. When all the features can consistently be explained in the framework of a model, then one may feel that a few parameters are a small price to pay for the elucidation of the jet-medium interaction. On two-particle correlation the most active area in recent years has been the use of triggers at intermediate or high pT to select a restricted class of events and the observation of associated particles at various values of η and φ relative to the trigger.2,99 Among the new features found, the discovery of ridges on the near side has stimulated extensive interest and activities.100 We review in this section only those aspects in which recombination plays an important role, which in turn makes inferences on the origin of the ridges. We start with a summary of the experimental facts. 4.1. Experimental features of ridges The distribution of particles associated with a trigger at intermediate pT exhibits a peak at small ∆η and ∆φ sitting on top of a ridge that has a wide range in ∆η, where ∆η and ∆φ are, respectively, the differences of η and φ of the associated particle from those of the trigger.99,101 A 2D correlation function in (∆η, ∆φ) first shown by Putschke101 at QM06 is reproduced here in Fig. 5(a). STAR has been able to separate the ridge (R) from the peak (J), where J refers to Jet, although both are features associated with jets. The structure shown in Fig. 5(a) is for 3 < ptrig < 4 GeV/c and passoc > 2 GeV/c in central AuAu collisions. The ridge T T yield integrated over ∆η and ∆φ decreases with decreasing Npart , until it vanishes at the lowest Npart corresponding to pp collisions, so R depends strongly on the nuclear medium. That is not the case with J. On the other hand, R is also strongly correlated to jet production, since the ridge yield is insensitive to ptrig T . Thus the ridge is a manifestation of jet-medium interaction. Putschke further showed101 that the ridge yield is exponential in its dependence on passoc and that the slope in the semi-log plot is essentially independent of ptrig T T . That is shown by the solid lines in Fig. 5(b). The inverse slope parameterized by T is slightly higher than T0 of the inclusive distribution, also shown in that figure. Since the pT range in that figure is between 2 and 4 GeV/c, we know from singleparticle distribution that the shape of the inclusive spectrum is at the transition from pure exponential on the low side to power-law behavior on the high side. The last data point at passoc = 4 GeV/c being above the straight line is an indication T of that. Thus the value T0 of the pure exponential part for the bulk is lower than
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(b) (a)
Fig. 5. (a) Jet structure from Ref. 101 for charged particles associated with a trigger. (b) Dependence of ridge yield101 on passoc for various ptrig T T .
what that straight line suggests. The exponential behavior of R should be taken to mean that the particles in the ridge are emitted from a thermal source. Usually thermal partons are regarded as begin uncorrelated. In the case of R they are all correlated to the semihard parton that initiates the jet. We thus interpret the observed characteristics as indicating that the ridge is from a thermal source at T , enhanced by the energy lost by the semihard parton transversing the medium at T0 . The B/M ratio of particles in the ridge is found to be even higher than the same ratio of the inclusive distributions in AuAu collisions at 200 GeV. More specifically, (p + p¯)/(π + + π − ) in R for ptrig > 4 GeV/c and 2 < passoc < ptrig is about 1 at T T T 102 pT = 4 GeV/c. In contrast, that ratio in J is more than 5 times lower. There is indication that the Λ/K ratio in the ridge is just as large.71 As discussed in Sec. 3, it is hard to find any way to explain the large B/M ratio outside the framework of recombination. Since the exponential behavior in pT implies the hadronization of thermal partons, the application of recombination very naturally gives rise to large B/M ratio, as we have seen in Sec. 3.1. Putting together all the experimental features discussed above, we can construct a coherent picture of the dynamical origin of the R and J components of the jet structure, although no part of it can be rigorously proved for lack of a calculationally effective theory of soft physics. There are several stages of the dynamical process. (i) A hard or semihard scattering takes place in the medium resulting in a parton directed outward in the transverse plane at midrapidity. Because of energy loss to the medium, those originating in the interior are not able to transverse the medium as effectively as those created near the surface. That leads to trigger bias. (ii) Whatever the nature of the jet-medium interaction is, the energy lost from the semihard parton goes to the enhancement of the thermal energy of the partons in the near vicinity of the passing trajectory. Those enhanced thermal partons
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are swept by the local collective movement outward whether or not the flow can be described by equilibrated hydrodynamics initially. (iii) Since the initial scattering takes place at |η| < 0.7, which is the pseudorapidity range of the trigger acceptance, the shower (S) partons associated with the jet are restricted to the same range of η. However, the enhanced thermal partons that interact strongly with the medium can be carried by the high-η initial partons that they encounter on the way out and be boosted to higher η. Thus the distribution of the enhanced thermal partons is elongated in ∆η, but not in ∆φ because the expansion of the bulk system is in longitudinal and radial directions, not in the azimuthal direction. Consequently, the hadronization of the enhanced thermal partons has the shape of a ridge. (iv) In terms of recombination the ridge is formed by TT and TTT recombination, while the peak J is formed largely by TS and TTS (or TSS) recombination, and possibly also by fragmentation (SS and SSS), depending on pT and centrality. Since the J component involves S, it is restricted to a narrow cone in ∆η and ∆φ. An initial attempt to incorporate all these properties in the RM was made in Ref. 15 before the ridge data were reported in QM06.101 By the time of QM08 ridgeology has become an intensely studied subject, as evidence by the talks in Ref. 100. 4.2. Recombination of enchanced thermal partons Although the properties of ridges described in the above subsection are derived from events with triggers, it should be recognized that ridges are present with or without triggers. That is because the ridges are induced by semihard scattering which can take place whether or not a hadron in a chosen pT range is used to select a subset of events. Experimentally, it is known that the peak and ridge structure is seen in auto-correlation where no triggers are used.103 The implication is that the ridge hadrons are pervasive and are always present in the single-particle spectra. Hard scattering of partons can occur at all virtuality Q2 , with increasing probability at lower and lower Q2 . When the parton kT is < 3 GeV/c, the rate of such semihard scattering can be high, while the time scale involved is low enough (∼ 0.1 fm/c) to be sensitive to the initial spatial configuration of the collision system. Thus for noncentral collisions there can be nontrivial φ dependence, which we shall discuss in Sec. 5. Hadron formation does not take place until much later, so it is important to bear in mind the two time scales involved in ridgeology. Ridges are the hadronization products of enhanced thermal partons at late time, which are stimulated by semihard parton created at early time. In the absence of a theoretical framework to calculate the degree of enhancement due to energy loss, we extract the characteristics of the thermal distributions from the data. Although hydrodynamics may be a valid description of the collective flow after local thermal equilibrium is established, it does not take semihard scattering into consideration and assumes
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fast thermaliztion without firmly grounded justification. If the semihard scattering occurs in the interior of the dense medium, the energy of the scattered parton is dissipated in the medium and contributes to the thermalization of the bulk (B). That process may take some time to complete. If the semihard scattering occurs near the surface of the medium, its effect can be detected as J + R in these events selected by a trigger with the trigger direction not far from the local flow direction, a point to be discussed in more detail later in Sec. 4.4. Inclusive distribution averages over all events without triggers, including all manifestation of hard and semihard scatterings; hence, it is the sum of B + R + J. Since J is associated with the shower partons (S), we identify J with the recombination of TS+SS for the mesons and TTS+TSS+SSS for the baryons, leaving TT+TTT for B +R. Thus the exponential behavior of the thermal partons is revealed in the exponential behavior of B + R in pT , for which we emphasize the inclusion of the ridge contribution to the inclusive distribution. In noncentral collisions the ridges are not produced uniformly throughout all azimuth,104 so dN/dpT that averages over all φ has varying proportions of B and R contributions depending on centrality. To be certain that we can get a measure of the R contribution independent of φ, we focus on only the most central collisions in this and the next subsections. Continuing to use the notation kT for the transverse momentum of the semihard parton at the point of creation in the medium, qT for that at the point of exit from the medium, and pT for the hadron outside, we have for thermal partons the distribution given in (9) just before recombination. Our first point to stress here is that the inverse slope T in (9) includes the effects of both B and R. Putting that expression into (7) where one takes Fqq¯(q1T , q2T ) = T (q1T )T (q2T ) and being more explicit with the RF for pion in (3), i.e., q1T q2 q1 q2 + T −1 , Rπ (q1T , q2T , pT ) = T 2 T δ pT pT pT
(15)
(16)
one obtains35 dNπB+R C 2 −pT /T = e , pT dpT 6
(17)
although in 2004 no one was aware of the existence of ridges. From the data105 for identified hadrons, one can fit the π + distribution for 0–5% centrality in the range 1 < pT < 3 GeV/c and get T = 0.3 GeV/c. This value is slightly lower than the one given in Ref. 101 which takes the slope of the inclusive distribution in the range 2 < pT < 3.5 GeV/c. Reference 105 provides data for K and p also, which have the same value of T as above for 1 < pT < 3 GeV/c, thus confirming that the exponential behaviors of the hadronic spectra can be traced to the common thermal distribution in (9) through recombination. At lower pT the spectra for K
Hadron Correlations in Jets and Ridges Through Parton Recombination
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and p deviate from exponential behavior because of mass effect, which can largely be taken into account by using ET instead of pT , where ET (pT ) = mT − m0 ,
mT = p2T + m20
1/2
,
(18)
m0 being the hadron rest mass. Thus we write for all hadrons 0 dNhB+R = Ah (pT )e−ET (pT )/Th , pT dpT
(19)
where Aπ (pT ) = C 2 /6 is a constant for pion, but for proton Ap (pT ) = C 3 A0 p2T /p0 where A0 is a numerical factor that arises from the wave functions (valon distribution) of the proton.35 Note that the inverse slope is now denoted by Th0 , since the data105 show dependence on hadron type when the distributions are plotted as functions of ET . Furthermore, Tp0 is found to depend on centrality, which is a feature that can be understood in the RM as being due to the non-factorizability of the thermal parton distributions of uud at very peripheral collisions where the density of thermal partons is low.106 For central collisions, Tp0 = 0.35 GeV. We summarize the empirical results for π and p as follows: Tπ0 = 0.3 GeV,
(20)
Tp0 = 0.35(1 − 0.5c) GeV,
(21)
where c denotes % centrality, e. g., c = 0.1 for 10%. We shall hereafter use Th0 to denote the inverse slope in ET for B + R, and Th for that in ET for B only, i.e., dNhB = Bh (pT ) = Ah (pT )e−ET (pT )/Th . pT dpT
(22)
It is hard to find data that describes the bulk contribution only, since the effect of semihard scattering cannot easily be filtered out. Indeed, as pT → 0, there is no operational way without using trigger to distinguish B(pT ) from all inclusive. For that reason the prefactor Ah (pT ) in (22) is the same as that in (19). In events with trigger above a threshold momentum, semihard partons with lower momenta than that threshold can contribute to R; it becomes a part of the background, which is experimentally treated as B. Thus the only meaningful way to isolate R quantitatively is by use of correlation, while accepting the difficulty of separating B and R outside the momentum ranges where the correlated particles are measured. Another way of stating that attitude is to accept the experimental paradigm of regarding the mixed events as a measure of the background (hence, by definition, the bulk), and treating R as only that associated with a trigger. Our cautionary point to make is that such a background can contain untriggered ridges. In practice, one can take the difference between (19) and (22) and identify it as the ridge yield
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i h 00 0 dNhR = Rh (pT ) = Ah (pT )e−ET (pT )/Th 1 − e−ET (pT )/Th , pT dpT
(23)
where 1 1 ∆Th 1 = , − 0 = 00 Th Th Th Th Th0
∆Th = Th0 − Th .
(24)
If ∆Th Th , then the quantity in the square bracket makes a small correction to the exp [−ET (pT )/Th0 ] behavior, and one can determine Th0 from the data. The only data available that address the ridge distribution are in Ref. 101 where the associated particles are in the range 2 < passoc < 4 GeV/c, exhibiting an approxiT mately exponential behavior. It is shown in Ref. 106 by using the data for trigger momentum in the range 4 < ptrig < 5 GeV/c that with ∆Th = 45 MeV in (24) T the ridge distributions can be well fitted. The expression for Rh (pT ) in (23) has no explicit dependence on ptrig T , as is roughly the case with the data. It does have strong dependence on passoc , which is pT in (23). Experimental exploration of the T lower passoc region would provide further validation that (23) needs. The physics T basis for that distribution is the recombination of thermal partons given in (9). 4.3. Trigger from the ridge We have discussed above the observation of ridge in triggered events, but to have a trigger from the ridge seems to put the horse behind the cart. There must be a phenomenological motivation for that role reversal. Let us start with the single-pion inclusive distribution that shows an exponential decrease in pT followed by a power-law behavior. The boundary between the two regions is at ∼ 2 GeV/c. We have associated the exponential region to TT recombination and the power-law region to TS+SS. We have also discussed the contribution to T from the enhanced thermal partons arising from the medium response to semihard partons. In order to be able to investigate the TT component better without the interference from the shower contribution so that one can examine the B + R components cleanly, it would be desirable to be able to push the TS+SS components out of the way. That is not possible with the light u and d quarks, but not impossible with the s quark, since the heavier quark is suppressed in hard scattering. If one observes the hadrons formed from only the s quarks, either φ or Ω, one finds exponential behavior at all pT measured, which in the case Ω extends to as high as 5.5 GeV/c.107 – 113 The absence of any indication of up-bending of the distributions clearly suggests that the source of the s quarks is thermal in nature and that no shower partons participate in the formation of φ and Ω. That problem is studied in Ref. 114 along with K and Λ production. Indeed, the data can be well reproduced by TT for φ, TTT for Ω, TT+TS for K, and TTT+TTS+TSS for Λ.
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Since s quarks in S make insignificant contribution to Ω production for pT < 6 GeV/c, and since thermal partons are uncorrelated, it is reasonable to expect that the Ω observed has no correlated particles. It was therefore predicted 115 that if Ω in 3 < pT < 6 GeV/c is treated as a trigger particle, there should be no associated particles appearing as a peak in ∆φ on the near side. Within a year that prediction was falsified by a report at QM06 showing that there is a near-side peak after background subtraction in ∆φ for Ω in the range 2.5 < ptrig < 4.5 GeV/c in T 116 central AuAu collisions with charged particles in the range 1.5 < passoc < ptrig T T . The data created a dilemma: is Ω created by a jet or not? If it is, why is the pT distribution strictly exponential with no hint of jet characteristics? If it is not, why is there a ∆φ peak in azimuthal correlation? The dilemma became known as the Ω puzzle.117 The resolution of that puzzle is in the recognition that both the trigger Ω and the associated particles are in the ridge, first conjectured in Ref. 117 and later quantified in Ref. 118. Jets are involved, since without jets there can be no ridge. But not all jet structures exhibit a prominent peak above a ridge. It depends on the trigger particle and the ranges of ptrig and passoc . Consider the jet yield compared to T T trig the ridge yield at 3 < pT < 6 GeV/c in AuAu collisions at 0–10% centrality.71,119 The J/R ratio at passoc ∼ 1.2 GeV/c decreases as the trigger particle goes from h T ¯ For Λ/Λ ¯ trigger and unidentified charged h associated, the to KS0 and then to Λ/Λ. −1 ¯ trigger must J/R ratio is ≤ 10 for |∆η|J < 0.7 and |∆η|R < 1.7. Since the Λ/Λ contain an s quark which is absent in the shower, the participating s quark must be a thermal parton. For ptrig near 3 GeV/c, thermal s quark around 1 GeV/c or less T can be quite abundant, and the initiating semihard parton need not be very hard. With passoc as low as ∼ 1 GeV/c, the light hadrons in R dominate over those in J, so T J/R is small. As the strangeness in the trigger increases, more thermal s quarks are involved with less dependence on shower. J/R is likely to be even smaller, although present data with Ω trigger lack statistics to show the ∆η distribution. If the jet structure shows mainly a ridge with negligible peak (J) in ∆η, we have referred to it as a phantom jet,117 i.e., a Jet-less jet. The corresponding ∆φ distribution should then be dominantly R. Since the initiating semihard partons are either gluon or light quarks, the usual jet structure may still be seen if the trigger particle is ordinary. But for Ω trigger the structure is very different. Now, we can address the Ω puzzle. The enhanced thermal partons generated by the semihard parton contain s as well as u and d quarks. Three s quarks anywhere in the ridge can recombine to form a trigger Ω. Other quarks, in particular the light quarks, in the ridge can form associated particles. The pool of enhanced thermal partons are all correlated to the semihard parton in every event selected by the Ω trigger, so the associated particles are all restricted to |∆φ| < 1. Since the s quarks that form the Ω are thermal, the Ω spectrum in pT is exponential. Hence the Ω puzzle is solved. The trigger can be from the ridge. The above description outlines a detailed calculation of the ∆φ distribution of charged hadrons produced in association with Ω.118 The background is calculated
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(a)
(b)
Fig. 6. (a) (Left panel: Data119 on associated particle distribution in ∆φ for three hyperon triggers at 2.5 < ptrig < 4.5 GeV/c and 1.5 < passoc < ptrig T T T . (b) Right panel: Near-side ridge yield associated with Ω trigger. The lines are from calculations in the RM118 for particles associated with Ω trigger, with the solid and dashed lines for slightly different normalization constants of the Gaussian peak.
in the RM using previous parametrizations, and the height agrees with the data. That is important, since the ridge signal is less than 4% of the background height. Two adjustable parameters are used to fit the ∆φ distribution of the ridge, but then the yield/trigger is calculated as a function of ptrig without further unknowns T in the model, and the result is in agreement with the data,71,119 as shown in Fig. 6. The solid and dashed curves in the two panels of that figure are the results of the calculation using two values of the strength of enhanced thermal partons that differ by only 1%, yet the height of the ridge varies by about 20%. That is because the ridge is the difference between large numbers of B + R and B. Such accuracy is beyond the scope of any dynamical theory to achieve. The phenomenological approach adopted in Ref. 118 has been the only one that offers a quantitative understanding of the Ω problem. Recently, PHENIX has shown data120 that can be interpreted as support for the notion of trigger from the ridge. At ptrig < 4 GeV/c (for unidentified trigger) T the per-trigger yield of the associated particles is found to be less than expected from fragmentation, and the “dilution” effect is attributed to the increase of the number of triggers due to soft processes. In our language fragmentation is the SS component, and the TS component due to medium effect gives an increase already over fragmentation for pT < 6 GeV/c. The additional dilution effect at pT < 4 GeV/c is due to TT recombination, which is enhanced by the ridge contribution. 4.4. Dependence of ridge formation on trigger azimuth So far our consideration in ridgeology has been concerned mainly with the dependence on pT . Now, let us turn to azimuthal correlation, although our discussion of the main topic of azimuthal anisotropy of single-particle distributions is deferred
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to the following section. Our present focus is on the correlation between the directions of the trigger and ridge particles in the transverse plane, a topic that can be discussed prior to φ anistropy because it is mainly a problem in ridge formation. The subject was stimulated by the report104 that the ridge yield depends on the azimuthal angle φs between the trigger angle φT and the reaction plane ΨRP , even for nearly central collisions, but especially for noncentral collisions. Since geometry is an important factor that influences the φs dependence, it is necessary to treat carefully the initial configuration of the problem: (a) the point of origin (x0 , y0 ) of the semihard parton in the almond-shaped overlap region, (b) the angle φs of the parton’s trajectory, (c) the density of the medium, D(x, y), along that trajectory, and (d) the point of exit from the medium. In the approximation that the medium does not expand very much during the time that the semihard parton near the surface traverses the medium, it is not difficult to calculate the path length, but it is much more difficult to calculate the energy loss that depends on D(x, y). Even if there is a reliable way to account for the effect of the medium on the semihard parton, there is no known way to translate that to the effect of the parton on the medium. The enhancement of the thermal partons that lead to the ridge particles takes time to develop, during which the medium expands. Local flow direction depends on where the enhanced thermal partons are in the overlap, which evolves into elliptical geometry. Each of the various parts of the process can be represented by a factor that can be expressed in terms of variables that have reasonable physical relevance. Without entering into the details that are described in Ref. 121 we can outline the essence here. Let P (x0 , y0 , t) denote the probability of detecting a parton emerging from the medium, where t is the path length measured from the initial point (x0 , y0 ) to the surface along a straight-line trajectory at angle φs . Let C(ψ(x, y), φs ) be a function that describes the correlation between the enhanced thermal partons along the flow direction ψ(x, y) and the semihard parton direction φs . Finally, let Γ(x, y, φ) describe the fluctuation of the angle φ of a ridge particle from the average flow direction. For fixed (x0 , y0 ) and φs , the ridge-particle distribution in φ is then R(φ, φs , x0 , y0 ) = N P (x0 , y0 , t)t
Z
1
dξD(xξ , yξ )C(xξ , yξ , φs )Γ(xξ , yξ , φ) , (25) 0
where N is a normalization constant related to the rate of production of the ridge particle and ξ is the fraction of the path length t along the trajectory starting at 0 at (x0 , y0 ) and ending at 1. For the observed distribution it is necessary to integrate over all (x0 , y0 ). Not every semihard parton included in that integration gets out of the medium to generate a particle that triggers the event. The ridge distribution per trigger is therefore normalized by the probability of the ridge-generating parton emerging from the medium R dx0 dy0 R(φ, φs , x0 , y0 ) , (26) R(φ, φs ) = R dx0 dy0 P (x0 , y0 , t)
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R. C. Hwa
where the integration is over the entire region of initial overlap. There is no explicit dependence on ptrig and passoc in (25), but the parameters specifying the different T T factors in the equation do. We leave the momenta fixed in the ranges 3 < ptrig <4 T 104 GeV/c and 1.5 < passoc < 2.0 GeV/c, as specified in the experiment, and focus T only on the dependencies on φ and φs . The most important factor in (25) is C(x, y, φs ), which is parametrized as (φs − ψ(x, y))2 , (27) C(x, y, φs ) = exp − 2λ where λ specifies the Gaussian width of the correlation between the directions of semihard parton φs and local flow ψ(x, y). If ψ(x, y) is close to φs for most of the points (x, y) along the trajectory, then the thermal partons enhanced by successive soft emissions are carried by the flow along the same direction. The effects reinforce one another and lead to the formation of a ridge in a narrow cone. On the other hand, if ψ(x, y) is very different from φs , then the enhanced soft partons are dispersed over a range of surface area, so their hadronization leads to no pronounced effect. That is the essence of the correlated emission model (CEM).121 The probability P (x0 , y0 , t) depends, in addition to the nuclear thickness at (x0 , y0 ), the survival probability S(t), which is exponentially behaved in t. Most of the semihard partons that can emerge from the medium are created in a layer near the surface. In Ref. 121 the thickness of that layer is set to be approximately R A /4. A more detailed study of that will be described in Sec. 6 below. The fluctuation distribution Γ(x, y, φ) turns out not to influence the final result in any sensitive way. A pictorial description of the origin of φs dependence of ridge yield in CEM is shown in the left panel of Fig. 7, where (a) φs ∼ 0 and (b) φs ∼ 70◦ in noncentral
Fig. 7. Left panel: schematic sketches of trigger directions (thin arrows) and flow directions (thick arrows) in noncentral collisions. Right panel: Data104 on the ridge yield/trigger vs φs for (a) 0–5% and (b) 20–60% centrality. The lines are from CEM121 with common normalization adjusted to fit the left-most data point in (a).
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collisions. The trigger directions φs are represented by the thin arrows, while the flow direction ψ(x, y) are depicted by the thick arrows that are normal to the surface. When the two types of arrows match, the reinforcement leads to a strong ridge. In (a) the matching condition is met where the density is higher than in (b) where the matching arrows occur near the top of the overlap. Thus the former case for φs ∼ 0 has higher ridge yield than in the latter case for larger φs . That difference becomes more drastic as the collision centrality becomes more peripheral. The results on the ridge yield integrated over φ are compared to the data in the right panel of Fig. 7. In (a) of that panel the normalization is adjusted to fit the data point at the lowest φs and the shape of the φs dependence is adjusted by varying the parameter λ in (27). The value determined is λ = 0.11; it corresponds to a correlation cone of half-width ∼ 20◦ . The curve in part (b) of that panel for 20–60% centrality is obtained without any further adjustment. Although the agreement with data is not perfect, the curve does reproduce the trend of steep descend, followed by a region of flatness or even a small up-bending. The behavior at large φs is due to the additional contribution of semihard partons from the left-half ellipse at high y0 . Without being integrated over φ, the ridge distribution R(φ, φs ) given by (25) and (26) describes the dependence of the ridge yield on ∆φ = φ − φs . An interesting discovery upon careful study of R(φ, φs ) is that its dependence on ∆φ is not symmetric across the ∆φ = 0 point, depending on the sign of φs . In Fig. 8(a) is shown its behavior for six positive values of φs . Not shown are the mirror images of those curves across ∆φ = 0 for corresponding negative values of φs . Since the measurement104 averages over both positive and negative values of φs , there is no observable asymmetry in the data. The reason for the asymmetry in Fig. 8(a) is that for φs > 0 most of the points along the trajectory have ψ(x, y) < φs , since ψ(x, y) is generally normal to the surface. Hence, the ridge particles are mostly at ∆φ < 0. The reverse is true for φs < 0. The left shift of the peaks in ∆φ in Fig. 8(a) has been confirmed by STAR recently.122
(a)
(b)
Fig. 8. (a) Left shift of the ridge yield in ∆φ for φs > 0 in CEM.121 (b) Inside-outside asymmetry function defined in Eq. (29).
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R. C. Hwa
Another way to test the asymmetry is to measure a quantity A(φs ) called insideoutside asymmetry function.121 To that end, define for 0 < φs < π/2 Z φs Z φs +1 Y+ (φs ) = dφR(φ, φs ), Y− (φs ) = dφR(φ, φs ) (28) φs −1
φs
and for −π/2 < φs < 0 reverse the definition. Then for any φs define A(φs ) =
Y+ (φs ) − Y− (φs ) . Y+ (φs ) + Y− (φs )
(29)
This asymmetry function should always be positive whether there is a left shift for φs < 0 or a right shift for φs > 0. By reflection symmetry it vanishes at φs = 0 and ±π/2. In CEM the properties of A(φs ) at two centralities are shown in Fig. 8(b). STAR reported very recently at QM09 that such an asymmetry has indeed been found to exist in the data on the ridges for various φs .122,123 5. Azimuthal Anisotropy The azimuthal dependence of single-particle distribution has been studied ever since the beginning of heavy-ion physics.124 – 127 Hydrodynamical model at low pT 128 – 132 and ReCo at intermediate pT 33,34,133,134 have been able to describe the data on elliptic flow very well. The only points worthy of comments here are those outside the realm of what has been covered in the references given above. There are then only three points: (a) Is early thermalization necessary? (b) What is the role of the shower partons? (c) At what pT does the quark number scaling begin to break down? 5.1. Effects of ridge formation at low pT The usual hydrodynamical treatment of elliptic flow assumes rapid thermalization with initial time of expansion set at τ0 = 0.6 fm/c. Such an early time of equilibration has never been shown to be the consequence of any dynamical process that is firmly grounded and commonly accepted. The question then is whether the azimuthal anisotropy can be driven initially not by pressure gradient at τ < 1 fm/c, but by some other mechanism that is sensitive to the early spatial configuration. That mechanism is suggested in Ref. 135 to be semihard scattering, the rate of which can be high for parton kT around 2–3 GeV/c, while the time scale involved is low (∼ 0.1 fm/c). Semihard partons created near the surface of the nuclear overlap can lead to a continuous range of ridges that is shaped by the initial geometry. The effects of such ridges are not considered in the usual studies in hydrodynamics, but should not be ignored. It is shown in Refs. 135 and 106 that the azimuthal anisotropy of the ridges produces the observed v2 at low pT without the use of hydrodynamics. Elliptic flow at some point of the expansion of the system may well be describable by hydrodynamics, but early thermalization is not required.
Hadron Correlations in Jets and Ridges Through Parton Recombination
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For dN/dφ of single-particle distribution, no triggers are used and ridges are generated by many semihard partons produced in each event. Semihard scatterings take place throughout the overlap, but only those occurring near the surface and directed normal to the surface can lead to the development of ridges, as discussed in Sec. 4.4. If a drastic simplification is made to require ridge particles to be also directed only in the the directions normal to the almond-shaped initial boundary in the transverse plane, then the φ distribution of all ridges, R(pT , φ), is restricted to the region φ ∈ R, which is a set of angles defined by |φ| < Φ(b)
and
|π − φ| < Φ(b) ,
(30)
where Φ(b) = cos−1 (b/2RA ).135 Fluctuation from the restricted range is, of course, possible, not only because the semihard partons can have any scattered angle, but also because the ridge particles can fluctuate from the directions of the parton trajectories. It is shown in the Appendix of Ref. 106 that the region R is nevertheless a good approximation even when all those effects plus elliptic geometry are taken into account, provided that the inaccuracy in the regions around |φ| ∼ π/2 for noncentral collisions is unimportant — which is indeed the case, since the density at the upper and lower tips of the ellipse is low so ridge production there is suppressed. Thus in the box approximation the ridge distribution is R(pT , φ) = R(pT )Θ(φ) ,
(31)
Θ(φ) = θ(Φ − |φ|) + θ(Φ − |π − φ|) .
(32)
where
In the assumption that the above anisotropy from the ridge is the only φ dependence in dN/dφ, replacing the usual assumption of rapid thermalization and the consequent pressure gradient, the bulk medium is then isotropic and the single-particle distribution at low pT can be written in the form dN = B(pT ) + R(pT )Θ(φ) . pT dpT dφ
(33)
The normalized second harmonic in φ can then be calculated analytically, yielding v2 (pT , b) = hcos 2φib =
sin 2Φ(b) . πB(pT )/R(pT ) + 2Φ(b)
(34)
At low pT the first term in the denominator is much larger than the second, so (34) is reduced to the even simpler formula v2 (pT , b) '
R(pT ) sin 2Φ(b) , πB(pT )
(35)
This is such a compact formula that its validity should be checked regardless of its derivation for the sake of having available a simple description of what is usually called elliptic flow after extensive computation.
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R. C. Hwa
The pT distributions of B(pT ) and R(pT ) are given, respectively, in (22) and (23). The latter can be written in a form that factors out B(pT ) so that R/B has the simple form 00 R(pT ) = eET (pT )/Th − 1 , B(pT )
(36)
which becomes ET (pT )/Th00 at low pT , since Th00 is large. It then follows from (35) that for small ET v2h (ET , b) =
ET sin 2Φ(b) . πTh00
(37)
Thus the initial slope in ET depends only on Th00 (which sets the scale) and the geometric factor sin 2Φ(b). That factorizability is in agreement with the data for pion, but not so well for proton for mid-central to peripheral collisions. That is because Tp00 develops a b dependence when the centrality c is above 0.3, as can be seen in (21). While Tπ00 is essentially constant Tp00 (c) can be approximated by a polynomial in c106 Tπ00 = 1.7 GeV,
Tp00 = 2.37 (1 − 1.05c + 0.26c2 ) GeV .
(38)
For ET ∼ 1 GeV the full expression in (36) should be used in (35), instead of the small ET approximation in (37). The results for π and p are shown in Fig. 9, and reproduce the data124 very well. The reason why Tp0 (c) decreases with increasing peripherality, resulting in similar trend in Tp00 (c), is that 3-quark recombination is more difficult when the thermal partons become less abundant at lower medium density as the collisions get more peripheral. At fixed pT ∼ 0.5 GeV/c the b dependences of v2h for h = π and p are studied in Ref. 106 in which it is shown that the characteristics of the data are well captured by the simple formula in (34).
(a)
(b)
Fig. 9. (a) Pion v2 and (b) proton v2 , calculated in the RM106 by taking ridge effect into account. The data are from Ref. 124.
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5.2. Effects of shower partons at intermediate pT As pT is increased to above 2 GeV/c, it is necessary to consider the role played by the shower partons, which introduce φ dependence due to jet quenching of hard parton that depends on path length in the medium. The dominance of TS + SS recombination over TT leads to a change in the pT dependence of v2 (pT ). The shower parton distribution S(q), given in (11), due to a hard parton produced at momentum k, is for central collisions and is averaged over all φ. Now for noncentral collisions with φ anisotropy, that formula needs to √ be generalized. Assuming that the energy loss of a hard parton is proportional to k 0 where k 0 is the initial parton momentum,34,136 one can write ∆k = k 0 − k in the form √ ˆ φ) k 0 , (39) ∆k = (b)`(b, ˆ φ) is the normalized path length and (b) is the energy-loss coefficient where `(b, that depends, apart from geometrical factors, a parameter 0 that is to be determined. After solving (39) for k 0 , and replacing ζfi (k) in (11) by fi (k 0 ) at the shifted momentum, one can keep the first two non-vanishing terms in the harmonic expansion of fi (k 0 (k, b, φ)) and get fi (k 0 (k, b, φ)) = fi (k) [g0 (k, b) + 2g2 (k, b) cos 2φ] ,
(40)
where g0 and g2 can be determined explicitly in terms of 0 , k and b.106 The shower contribution to the single-pion distribution, when averaged over all φ, may be written in the symbolic form dNπT S+SS (b) 1 = g0 (k, b)fi (k) ⊗ (T S + SS) . pT dpT 2π
(41)
The contribution of the shower component to v2π (pT , b) is v2π,sh (pT , b) =
g2 (k, b)fi (k) ⊗ (T S + SS) . g0 (k, b)fi (k) ⊗ (T S + SS)
(42)
The thermal component v2π,th (pT , b) is as given in (34) and (36). The overall v2π is obtained from the above with the help of an interpolating function W (pT , b) v2π (pT , b) = v2π,th (pT , b)W (pT , b) + v2π,sh (pT , b) [1 − W (pT , b)] ,
(43)
where W (pT , b) =
TT , T T + T S + SS
(44)
with T T being the thermal, and T S +SS the shower, components of the φ-averaged dNπ /pT dpT . By fitting the single-pion distribution at 0–10% centrality over the range 1 < pT < 6 GeV/c using all three T T + T S + SS terms, 0 is found to be 0.55 GeV1/2 . It is then possible to calculate v2π (pT , b) without any further adjustment; the result is shown in Fig. 10(a). The data points for ET > 1 GeV are from
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(a)
(b)
Fig. 10. (a) Left panel: pion v2 and (b) Right panel: proton v2 , both at higher pT where the effects of shower partons are taken into account. The data are Refs. 126 and 137.
Ref. 137. The saturation of v2π in that range is thus interpreted in the RM as being due to the shower partons, where W (pT , b) is suppressed and g2 is much smaller than g0 at high k. For proton v2 the general procedure in the calculation is similar to that for pion, except for an additional thermal parton to incorporate. There is also the complication of b dependence in the inverse slope Tp0 in (21). Taking them all into account the result for v2p is shown in Fig. 10(b). Comparison with data126,137 is acceptable, although more accurate data are needed to check the calculated results at high ET and b. What is learned from this study is that the main source of φ anisotropy is the path-length dependence of jet quenching, which is parametrized by one unknown 0 that is determined by fitting the inclusive distribution at one value of pT . The characteristics of v2h (pT , b) in Fig. 10 are obtained without any more adjustable parameters. It should be noted that we have refrained from using the term “elliptic flow”, except in reference to past work based on hydrodynamics. In Sec. 5.1 the emphasis is on the effect of ridges due to semihard scattering that are not taken into account by early-time hydrodynamics, although the exponential behavior of the thermal partons may well be the result of late-time hydrodynamical flow. Then in this section hard scattering is incorporated in the treatment of jet-medium interaction through TS recombination before fragmentation dominates. Such interaction is outside hydrodynamics, so the overall characteristics of v2h (pT , b) are not the properties of flow. In Ref. 34 the effects of jet quenching on the hard partons are also considered, but since only the fragmentation of those partons is included, the transitional contribution from TS interaction is not explicitly taken into account. That turns out to be important in the intermediate pT region.
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5.3. Breaking of quark number scaling Quark number scaling (QNS) has long been regarded as a signature of recombination33,34,133 and has been verified in a number of experiments.124,126,127 It has been regarded as a statement of the universality of v2h (pT /nq )/nq , where nq is the number of constituent quarks in the hadron h. It is based on the assumption of factorizability of the distribution of the quarks that recombine, i.e., the multiquark distributions in (7) and (8) (but with φi dependencies included) can be written as Fnq (q1 , φ1 ; q2 , φ2 ; · · · ) =
nq Y
i=1
Fi (qi , φi ) =
nq Y
Fi (qi )[1 + 2v2i (qi ) cos 2φi ] .
(45)
i=1
Coupled with the assumption that the RF has the simple form δ(qi − pT /nq ), it then follows trivially from (7) and (8) that v2h (pT ) = nq v2q (pT /nq ) ,
(46)
if the v2i (qi ) of all quarks in (45) are the same, denoted by v2q (q). From our discussions throughout this paper it is clear that none of the above assumptions are valid under close examination. TT+TS+SS for pion and TTT+TTS+TSS+SSS for proton are obviously not factorizable. Even at low p T where only the recombination of thermal partons is important, the inverse slopes Th0 , given in (20) and (21), are not the same for π and p. Consequently, the R/B ratios for π and p are different, as seen in (36) and (38), resulting in different v2h (ET , b). Furthermore, the wave functions of π and p are very different, since the pion is a tightly bounds state of the constituent quarks, while the proton is loosely bound. That means the momentum fractions of the quarks (valons) are not 1/2 for pion and 1/3 for proton. It is then a very rough approximation to write the momentum conservation δ-function, δ(Σqi − pT ), as δ(nq qT − pT ) with a common qT . At intermediate pT where shower partons become important, we have seen that they acquire the φ dependence of the hard parton, given in (40), so v2S (qi ) for the shower is different from v2T (qj ) for the thermal parton. Even if TS and TTS contributions dominate, one can at best, by ignoring all other complications, have v2M (pT ) = v2T (q1 ) + v2S (q2 ),
v2B (pT ) = v2T (q1 ) + v2T (q2 ) + v2S (q3 ) .
(47)
They would not lead to QNS, as expressed in (46). Most data in support of QNS are for minimum bias and at low ET . It is shown in Ref. 106 that the calculated result at ET /nq < 0.5 GeV does exhibit QNS in agreement with the data, but the scaling is broken above that. The breaking of QNS is due primarily to the nonequivalence of the φ dependencies of the thermal and shower partons. The important point to stress here is that the breaking of QNS at intermediate ET does not imply the failure of recombination (in fact, it is expected), but the validity of QNS at lower ET does confirm recombination as the proper hadronization process.
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6. Hadron Correlation in Dijet Production In Sec. 4 the subject of discussion is ridgeology, which is the study of the effects of jets on the medium. In this section we consider the reverse, i.e., the effects of the medium on the jets for pT not so extremely high as to exclude TS recombination. In triggered events the ridge is the broad pedestal on top of which sits the peak. The structure of that peak, when compared to the jet structure in pp collisions, reveals the medium effect on hard partons. Since the peak is restricted to a narrow cone around the trigger direction, it is natural in the RM to associate it with shower partons, while the ridge being broad is associated with the thermal partons. Hadrons closely correlated to a trigger particle as jets exhibit peaks on both the same and away sides.138 – 142 How the structures of the jets on the two opposite sides differ from each other is a strong indication of the difference in energy losses in the two jets, since their path lengths in the medium are generally different. In realistic collisions even when centrality is chosen to be within a narrow range, the path lengths can vary significantly depending on the location and angle of a scattered hard parton. Thus a careful study of the properties of the near- and away-side jets must start with finding a good description of the variation of energy loss within each class of centrality. 6.1. Distribution of dynamical path length By dynamical path length we mean not only the geometrical length of a trajectory, but also the medium effect along that trajectory. To quantify that in an analytic expression, we need to revisit the single-particle distribution discussed earlier for central collision, but now formulated in a way appropriate for any centrality. For pion production we have for y ∼ 0 Z p π p C 2 −p/T 1 X dq dNπ c = e + 2 Fi (q) TS(q, p) + Di , pdp 6 p i q q q
(48)
where all momenta are in the transverse plane with the subscript T omitted. The first term on the right side is from (17) for TT recombination; the centrality dependence of C is given in Ref. 143. The first term in the square bracket is for TS recombination which will be detailed below, and the second term is the fragmentation function that is equivalent to SS recombination. Fi (q) is the distribution of parton i at the surface of the medium with q denoting the momentum of the hard parton there. It differs from the distribution fi (k) given in (12), that describes the hard parton with momentum k at the point of creation. In Secs. 2 and 3 where only central collisions are considered, an average suppression factor ζ is used, as shown in (11). We now replace ζfi (k) by a path-dependent term related to Fi (q) by Fi (q) =
Z
L 0
dt L
Z
dkkfi (k)G(q, k, t) ,
(49)
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where G(q, k, t) is a degradation factor that describes the decrease of parton momentum from k to q as the hard parton traverses a distance t through the medium.144 L is the average maximum length of that trajectory. In the limit L → 0, Fi (q) should become the parton distribution function Fi (k) for pp collisions. For energy loss we seek a form that is consistent, on one hand, with ∆E ∝ L as suggested in Ref. 97 for 1D expansion, and on the other hand, with hdE/dLi ∝ E in Ref. 145 for 6 < E < 12 GeV. A reasonable approximation of the differential energy loss is then ∆E = β∆L , E
(50)
which translates to our variables as k − q = kβt .
(51)
at small t with β being an adjustable parameter. For larger t we exponentiate the above to q = ke−βt ,
(52)
and let G(q, k, t) take the simple form G(q, k, t) = qδ(q − ke−βt ) .
(53)
The δ function can be broadened to account for fluctuations, but we shall take (53) as an adequate approximation of the complicated processes involved in the parton-medium interaction, the justification of which rests ultimately on how well the calculated result can agree with the data for 2 < pT < 12 GeV/c. Using (53) in (49) yields 1 Fi (q) = βL
Z
qeβL
dkkfi (k) ,
(54)
q
which shows explicitly how fi (k) is transformed to Fi (q) by the nuclear effect parametrized by βL, while fi (k) itself contains the hidden modifications due to such effects as Ncoll dependence and shadowing.74 Since fi (k) is dNihard /kdkdy|y=0 , (54) becomes in the limit of L → 0 the invariant distribution for hard parton i production in pp collision. In heavy-ion collisions Eq. (11), written for centrality ∼ 0%, can now be improved to the form Z dq S(q1 ) = Fi (q)Sij (q1 /q) (55) q for any centrality with L being the geometrical path length. We can now write the TS recombination term in (48) as Z Z 0 dq1 j q1 c p) = Si dq2 C¯j e−q2 /T Rj¯j (q1 , q2 , p) , (56) TS(q, q1 q where Rj¯j is the RF given in (16).
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Equation (48) is now totally specified, relating the observable pion spectrum to the nuclear parameter βL through Fi (q) given in (54). We emphasize that because all three components of TT+TS+SS recombination are included in (48) it can describe the pion distribution at all pT using only one parameter βL for each centrality. Figure 11(a) shows the fits of the π 0 distributions for nine bins of centrality,146 with a different value of βL used for each centrality.144 Those values of βL are shown by the nine points in Fig. 11(b) (ignoring the line for the moment). Evidently, the agreement with data is excellent over such a wide range of pT and centrality c. Apart from concluding that the RM works well, the jet-medium interaction has been effectively summarized by one phenomenological function βL(c), which could not have been extracted from the data without a reliable way to relate energy loss to the suppression of pion production at all pT through appropriate hadronization. Having obtained βL(c) we now go a step further to inquire what kind of variation of the dynamical path length can the nuclear overlap generate for any given centrality. That is, for a fixed c, βL(c) is the average of a variable ξ over a probability distribution P (ξ, c) that describes the likelihood that a particular trajectory occurs at centrality c with an effective energy loss such that Z βL(c) = dξξP (ξ, c). (57) Thus ξ plays the role of βL except that it can vary from 0 to a maximum for every fixed initial elliptic geometry depending on the initial point and orientation of the
(b) (a)
(c)
Fig. 11. (a) Pion spectra146 for various centralities lowered by a factor of 0.2 for each step of increase of 10%. The solid lines are fits in the RM144 with one parameter for each centrality c. ¯ that is the average Those parameters are shown by dots in (b), which are fitted by the curve ξ(c) of the dynamical path length ξ over P (ξ, c) defined in Eq. (58). (c) ξ distributions for various c.
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¯ hard parton. So ξ is the dynamical path length of a trajectory whose average ξ(c) is βL(c). In Ref. 144 P (ξ, c) is chosen to have the form P (ξ, c) = N ξ(ξ0 − ξ)αc ,
(58)
where N normalizes the total probability to 1. The two parameters ξ0 and α are adjusted to fit the nine points in Fig. 11(b), with the result shown by the curve that renders an excellent fit for ξ0 = 5.42,
α = 15.2 .
(59)
Thus (58) is a very efficient way to describe the energy loss effect for all centralities. The shapes of P (ξ, c) are shown in Fig. 11(c) for 6 values of c, exhibiting the expected peak that decreases with increasing c or shrinking ellipse. In view of the difficulty of deriving βL(c) from first principles, let alone P (ξ, c), it is very convenient to have the path-dependent quenching effect be represented by the simple description in (58) and (59). The single-pion distribution at midrapidity for centrality c can now be expressed as Z dNπ (c, ξ) dNπ (c) = dξP (ξ, c) , (60) pdp pdp where (48) is to be used for dNπ (c, ξ)/pdp, provided that Fi (q) in it is not as given in (54), but with βL replaced by ξ, and, of course, fi (q) scaled by Ncoll (c). It should be clear that (48) has basically two parts: Fi (q) that describes the hard parton part and the rest that describes the hadronization part. We have in this subsection incorporated the centrality-dependent energy-loss effect on Fi (q) by the use of just two dimensionless parameters given in (59). Having successfully formulated the treatment of the single-hadron distribution, we are now ready to proceed to the study of dihadron correlation in near- and away-side jets. 6.2. Near-side and away-side yields per trigger For dihadron correlation many momentum vectors of partons and hadrons are involved. To depict clearly their relationships with one another, we show in Fig. 12 a pictorial representation of all of them. The near side is on the right and the away side on the left. The vectors k, q, and pt are, respectively, the momenta of the initiating hard parton, of the same parton as it leaves the medium, and of the trigger hadron. The associated hadron on the same side is labeled by pa . On the away side the corresponding momenta are k 0 , q 0 , and pb , there being no trigger on that side. If there are no transverse momenta in the beam partons, k and k 0 should be equal and opposite in every event; however, their averages over triggered events may not be the same, so it is better to label them distinctly from the start. The momenta of interest in the following are such that the hadronic pT are in the range 2 < pT < 10 GeV/c but with special emphasis on pT < 6 GeV/c.
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Fig. 12. (Color online) A sketch of momentum vectors of partons (in red) and hadrons (in blue) with near side being on the right and away side on the left.
For two pions in the same jet, neither of which are in the ridge, we can leave out the TT contribution, and write as a generalization of (48) X Z dq pt π pt dNππ (ξ) 1 c c − p t , pa ) TS(q = Fi (q, ξ) TS(q, pt ) + Di pt pa dpt dpa (pt pa )2 i q q q ) p p p p p p a t a a t a π π c − p a , pt ) D , +TS(q + 2 D2 , (61) q i q qj q q
where D2 (z1 , z2 ) is the dihadron fragmentation function.144 For notational brevity (61) is for a fixed ξ, the averaging over P (ξ, c) being a process that can be applied when c is fixed. For two pions on opposite sides the recoil parton must be considered explicitly, so (49) should be generalized to ξ
Fi0 (q, q 0 , ξ) =
Z
=
1 ξ
0
Z
βdt ξ qeξ q
Z
dkkfi (k)G(q, k, t)G(q 0 , k 0 ,
dkkfi (k)qq 0 δ(qq 0 − kk 0 e−ξ ) .
ξ − t) β (62)
with k 0 = k. Thus the dipion distribution for pb on the side away from pt is X Z dq dq 0 1 dNππ (ξ) 0 0 c p t ) + pt D π pt = F TS(q, (q, q , ξ) pt pb dpt dpb (pt pb )2 i q q0 i q i q c 0 , pb ) + pb Dπ0 pb . (63) × TS(q q0 i q0
The δ function in (62) restricts the integration of q 0 in (63) to the range from qe−ξ to qeξ that correspond to the hard scattering point being on the near-side boundary to the far-side boundary. The yield per trigger at fixed centrality can now be obtained for the near side as R dξP (ξ, c)dNππ (ξ)/pt pa dpt dpa near R Yππ (pt , pa , c) = , (64) dξP (ξ, c)dNπ (ξ)/pt dpt
Hadron Correlations in Jets and Ridges Through Parton Recombination
(a)
307
(b)
Fig. 13. (a) Near-side jet yield per trigger vs trigger momentum pt and associated particle momentum pa . (b) Away-side jet yield per trigger vs pt and pb . away and similarly for Yππ (pt , pb , c) with dNππ (ξ)/pt pb dpt dpb being used in the numerator. Since every factor is already specified, it remains only for the computation to be carried out. The results are shown in Fig. 13(a) for near side and (b) for away side. The three sheets for c = 0.05, 0.35, and 0.86 are separated by a factor of 10 between sheets for clarity’s sake. Superficially, the two figures may look similar in a vertical scale that spans over 4 orders of magnitude, but there are significant differences that only by closer examination can one learn from them the nature of the medium effects on the jets.
6.3. Medium effects on dijets Let us first cut the two figures in Fig. 13 by three fixed-pt planes at pt = 4, 6, and 8 GeV/c, and show the results in Fig. 14(a) and (b) for c = 0.05 and 0.35. Note (i) the near dependence of Yππ (pa ) on pa is more sensitive to pt than that of the dependence away of Yππ (pb ) on pb ; (ii) the increase of the yield with pt is more pronounced for the near near side than for the away side; and (iii) Yππ has negligible dependence on c, but away Yππ increases by roughly a factor of 2 when c changes from 0.05 to 0.35. Let us discuss these three features separately. On item (i) one can determine the near-side average inverse slope Ta of the approximate exponential behavior in the range 2 < pa < 4 GeV/c for c = 0.05, for which there are data. The result is shown by the line in Fig. 15(a), and agrees well with the data.71,101,147 Evidently, the spectrum of the associated particles in a jet becomes harder as the trigger momentum increases, as one would expect. It should be remarked that the line in Fig. 15(a) is for pions, while the data are for all charged particles. However, since pions dominate in jet peaks (unlike the ridges), the comparison is not unreasonable. The inverse slope Tb for the away-side jet at c = 0.05 for the same ranges of pt and pb is shown in Fig. 15(b), exhibiting a lower value compared to Ta and a mild decrease with pt . There exist no suitable data
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Fig. 14. (a) Left panel: near-side jet yield per trigger vs pa for fixed pt and c. Data points are from Ref. 6.11. (b) Right panel: the same for away side vs pb .
(a)
(b)
Fig. 15. Inverse slopes of associated particles on the (a) near side and (b) away side. Data in (a) are from Refs. 101 and 147.
for comparison. The strong difference between Ta and Tb has good reasons, as will be discussed below. Two data points from Ref. 148 are included in Fig. 14(a), the details of which are discussed in Ref. 144. They lend support to the theoretical curve in both magnitude and pa dependences. Note (ii) about the pt dependence of the yield is related to note (i) about the near away pt dependencies of the shapes in pa and pb . First of all, Yππ is larger than Yππ in magnitude, meaning that there is more suppression on the away side than on the near side. To quantify that interpretation let ξ be fixed at 2.9 corresponding to the maximum probability for c = 0.05 so that the suppression effect is not partially hidden by averaging over ξ, which can vary from 0 to ξ0 . Let the suppression factor for the near side be defined by Γnear (pT ) = he−βt ipT = hq/kipT ,
(65)
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309
where the average is performed over dNπ /pT dpT given in (48), and the last expression follows from (53). Γnear (pT ) gives a measure of the fraction of momentum retained after energy loss reduces k to q. For the away side the suppression factor is defined by Γaway (pt , pb ) = he−ξ+βt ipt ,pb ,
(66)
where −ξ + βt is equivalent to −β(L − t) if ξ is denoted by βL for a fixed medium length L, in which t is the portion from the hard-scattering point to the near-side surface, and L − t is the distance to the away-side surface. The average in (66) is done using the dihadron distribution given in (63).
(a)
(b)
Fig. 16. Suppression factors for the (a) near side and (b) away side.
The calculated results for Γnear (pT ) and Γaway (pt , pb ) are shown in Figs. 16(a) and 16(b). They clearly indicate that Γnear (pT ) is much larger than Γaway (pt , pb ) with the implication that there is less suppression on the near side than on the away side. The physics of the phenomenon is clear: at large pT the point of creation of hard parton with large k is predominantly close to the near-side surface in order to minimize energy loss with the consequence that the distance to the away-side surface is longer, thus more suppression for pb on that side. Since Γnear (pT ) saturates at around 0.85, only 15% of the parton energy is lost to the medium on the near side. The corresponding hβti is less than 0.2, so hβti/ξ = hti/L ≈ 0.065, meaning that the hard partons are created within a layer of thickness ∼ 13% of L from the surface. That is a quantitative description of trigger bias. On the other hand, the behavior of Γaway (pt , pb ) reveals the opposite: at fixed pb it decreases with increasing pt , implying more suppression as the hard-scattering point is pulled closer to the near-side surface. That is antitrigger bias. At fixed pt , Γaway (pt , pb ) increases with pb , since higher pb demands higher q 0 , which can be satisfied only if hL − ti is reduced or hk 0 i increased, actually both. Lowering hL − ti is, of course, a way to
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R. C. Hwa
reduce energy loss by pulling the scattering point closer to the away-side surface. But the increase of hk 0 i with pb is another aspect of antitrigger bias, whose details are described in Ref. 144. Event-by-event momentum conservation requires k 0 = k; however, when averaged over all events, hk 0 i depends on pb , while hki does not. In general, hk 0 i is far greater than hki because any trigger favors shorter hti, and any finite pb pushes up hk 0 i/hki though not hk 0 /ki. Because of the difficulty of producing an associated particle on the away side relative to one on the near side, Tb is lower and decreases with increasing pt in Fig. 15(b), while Ta is higher and increases with pt in Fig. 15(a). Now on note (iii) about centrality dependence Fig. 14(b) shows that the depenaway dencies of Yππ (pt , pb , c) on pt and pb are essentially the same whether c = 0.05 or 0.35, except that the magnitude of the yield is increased due to the reduction of path length at higher c. For the near side there is essentially no dependence of near Yππ (pt , pa , c) on c. That is shown more explicitly in Fig. 17, where the lower three lines are for pt = 4 GeV/c and the upper line is for pt = 6 GeV/c. The solid lines are for integrated yields with 2 < pa < 4 GeV/c. The near independence on c is a manifestation of the trigger bias, since the hard-parton production point, being restricted to a layer roughly 13% of L just inside the near-side surface, is insensitive to how large the main body of the medium is. Actually, the decrease of the TS component with c balances the increase of the SS component with c so that the net yield being their sum is approximately constant in c. The data points in Fig. 17 support the calculated result in both the magnitude and the c dependence near of Yππ (pt , c) when pa is integrated from 2 to 4 GeV/c.101 In summary the discussion above gives quantitative demonstration that the trigger bias is the preference for the hard-scattering point to be close to the nearside surface and that the antitrigger bias is the consequence: hk 0 i is much larger
Fig. 17. Near-side yield per trigger vs centrality c. The solid lines are for integrated yields with 2 < pa < 4 GeV/c. The data points are from Ref. 101.
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(a)
311
(b)
Fig. 18. (a) The ratio hq 0 i/hk 0 i at the symmetry point p = pt = pb for four values of centrality. (b) Distribution of associated pion (pb ) in the away-side jet for six values of pion trigger momentum (pt ) in GeV/c for ξ fixed at 2.9.
than hki, and even larger than pt or pb . Those are the properties of the events selected by a trigger at pt with an associated particle on either the near or away side.
6.4. Symmetric dijets and tangential jets We now follow the summary comment at the end of Sec. 6.3 with the question on what if we select events with symmetric dijets where pt = pb . Instead of studying the properties of a third particle in association with the two trigger particles (for which we need trihadron correlation function), we can nevertheless learn a great deal from examining closely various calculable quantities in the dihadron correlation problem. Let p be the momentum of the symmetric dijets, p = pt = pb . One can calculate hk 0 i(p, c) and hq 0 i(p, c) for various centralities, even though they are not directly measurable. The result is that they both increase almost linearly with p and that there is essentially no dependence on c. Their ratio hq 0 i/hk 0 i is therefore approximately constant in p with a value of about 0.8 as shown in Fig. 18(a). That behavior is similar to the property of Γnear (p), shown in Fig. 16(a), which, according to (65), is also hq/ki. It is important to bear in mind that hq 0 i and hk 0 i are averages over the two-particle distribution dNππ (c)/pt pb dpt dpb with pt = pb , given in (63) for fixed ξ, followed by averaging over P (ξ, c), whereas hqi, hki and hq/ki are averages over the single-particle distribution dNπ /pdp, as noted after (65). The near-side averages know nothing about the away-side analysis, so Γnear (p) describes only the suppression associated with trigger bias. The fact that the suppression on the away side is about the same as on the near side, when the average hq 0 /k 0 i is over symmetric dijets and hq/ki is averaged over near-side jet, and that it is true for any centrality, has only one important implication: the dijets are created very near the surface on both sides. It means that the symmetric dijets are dominated
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by tangential jets, which behave similarly at all centralities. That is a striking conclusion. Hard scattering can, of course, occur anywhere in the overlap region. But those partons from the interior lose most of their momenta on the way out. Those that are created on the near side would not give rise to symmetric dijets. Thus only those created along the rim and directed tangentially to the surface can lead to events with pt = pb . There is some experimental evidence to support this finding. In 2jet + 1 correlation studied149 it is found that the third particle does not show any ridge structure 2/3 and that the centrality dependence goes as Npart . The latter means that the dijets are created near the surface, while the former means that the dijets are tangential because we have already seen in Sec. 4.4 that ridge production depends on the local flow direction to match the trigger direction due to the correlation function in (27). The flow direction near the surface is normal to the surface and is therefore normal to tangential jets. In the experimental analysis149 the two jets have pT cuts at pT1 > 5 GeV/c and pT2 > 4 GeV/c, so they are not exactly symmetrical. If ridge formation from those dijets is to be discovered in the future, it would pose a serious challenge to the treatment of ridgeology in Sec. 4. But so far the data are in accord with the results both here and on ridges. 6.5. Unsymmetric dijets and tomography The conclusion in the last section on the dominance in symmetric dijets by tangential jets leads one to consider the only option left for probing the dense medium by parton jets apart from using direct γ, and that is the study of away-side jets in unsymmetric dijets. If hard or semihard partons created near the surface are most responsible for the near-side jet, then the away-side jet should experience fully the effect of the medium. That is indeed the case when one calculates hq 0 i and hk 0 i for various values of pt and pb at fixed ξ. It is found in Ref. 144 that for ξ = 2.9, hq 0 i/hk 0 i is in the range of 0.15 to 0.3 for 4 < pt < 10 GeV/c and 2 < pb < 6 GeV/c, and gives a quantitative measure of the degree of energy loss. Unfortunately, that cannot be checked experimentally, since only centrality can be selected in realistic collisions, not the dynamical path length ξ. At fixed centrality the value of ξ can fluctuate over a wide range, as can be ¯ is 2.9, but the characteristic of the seen in Fig. 11(c). For c = 0.05 the average ξ(c) away-side jet is dominated by the lower ξ portion of the range. The consequence away is that Yππ (pt , pb , c) does not depend sensitively on pt , as Fig. 14(b) has already indicated for any fixed c. That is in sharp contrast from the case where ξ is fixed away at 2.9, and Yππ (pt , pb , ξ) decreases by an order of magnitude as pt is decreased from 9 to 4 GeV/c at any fixed pb , as shown in Fig. 18(b). That diminishing yield is because hki is lower at lower pt and the energy loss by the recoil parton traversing the thick medium results in reduced probability of producing a pion at fixed pb . That is not the case in Fig. 14(b). At fixed c, the decrease of pt does not lead to away significantly lower Yππ (pt , pb , c) because the hard-scattering point is already in a
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Fig. 19. (a) Left panel: average dynamical path length of recoil parton directed toward the away side. (b) Right panel: ratio of the average parton momenta of recoil parton at the surface to that at the creation point.
region of lower ξ to minimize energy loss. Lowering pt increases the yield at fixed pb due to softer parton, but the number of triggers is also higher, so the yield per trigger remain nearly unchanged. In other words, allowing ξ to be small even at small c removes energy loss as a decisive factor in the problem. That is what makes dijet tomography ineffective as a probe to learn about a medium that has no fixed thickness. Further insight can be gained by studying hβt0 i at fixed c, where t0 denotes the distance from the hard-scattering point to the away-side surface. Figure 19(a) shows hβt0 i vs pb for various pt and c.144 A general impression from that figure is that hβt0 i is low, less than approximately 0.4. That is much lower than βL = 2.9, which is the average dynamical path length determined from fitting the singlepion distribution from c = 0.05 [see Fig. 11(b)]. The height of that distribution at large pT [see Fig. 11(a)] is what renders RAA ≈ 0.2, a number that pQCD calculations aim to obtain. The fit in Ref. 144 is achieved by setting βL = 2.9 in order to obtain the correct normalization for dNπ /pt dpt at large pT , for which the contributions from all partons, near or far, hard or semihard, are counted. The result that hβt0 i βL shown in Fig. 19(a) is therefore indicative of the fact that conditional probability with pt and pb fixed is highly restricted compared to the inclusive probability. With ht0 i being much less than L at fixed c, one is led to conclude that the unsymmetric dijets are also produced near the surface and are essentially tangential, as with symmetric dijets. Thus the medium interior is not probed. This conclusion is distinctively different from the case where ξ is fixed. Figure 16(b) has shown that the suppression on the away side can be large (hence Γaway small) for unsymmetric dijets at ξ = 2.9. In that case hβt0 i would not be small, as we now have for any fixed c.
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Experimental data149 on slightly asymmetric dijets at high pT show no essential difference in the structures of the two jets and are in support of the findings in Ref. 144. Another way to come to the same conclusion is to study the energy loss of the away-side jet. In Fig. 19(b) is shown hq 0 i/hk 0 i for c = 0.05 and 0.35.144 It is the fraction of average momentum retained by the hard or semihard parton after traversing the medium on the away side. Being around 0.8 implies that only about 20% of the parton momentum is lost to the medium, not much more than the fractional energy loss on the near side. At fixed pt Fig. 19(b) shows an increase of hq 0 i/hk 0 i with increasing pb because the hard scattering point is pulled more to the away side, but it shows a decrease of the ratio at increasing pt since the point is then pulled to the near side. This push-and-pull effect of pt and pb is clearly what one expects in the oppositely-directed jets when the path lengths on the two sides are comparable. At the symmetry point pt = pb = 4 GeV/c, hq 0 i/hk 0 i for c = 0.05 is only slightly lower than its value for c = 0.35, implying strongly that the fractional energy loss on both sides remains about the same regardless of centrality. That can only mean that the hard partons are created near the surface and directed tangentially. Making pt 6= pb in unsymmetric dijets does not change hq 0 i/hk 0 i drastically. Thus so long as the medium thickness cannot be controlled, there seems to be no useful tomography that can be done with parton-initiated dijets. If the away-side jets are dominated by those created near the away-side surface (such as the tangential jets), then the events triggered by direct γ on the near side are also likely to be dominated by those where the hard scattering takes place near the away-side surface (not just tangential) so long as an associated particle with significant pb is required on the away side. The π-triggered and γ-triggered distributions, IAA , of the associated particles should be roughly the same. There is some experimental evidence for that similarity.150 It is important to note that the above comment is for tomography only, i.e., medium effect on jets. The jet effect on medium, such as Mach cone, is a difference matter that depends on different physics and may well reveal properties of the medium interior. 7. Conclusion In this review many problems in heavy-ion collisions have been examined in many parts of the phase space. The good agreement between theoretical calculation and experimental data in almost all cases cannot but affirm that the successes of the theoretical approach adopted cannot be all fortuitous and that the interpretations given to the physical origins of the measured phenomena are not without some degree of realism. In some cases there are no apparent alternative schemes to explain the data. Of course, a phenomenological model is not a theory based on first principles, but its usefulness should not be overlooked when its scope is out of reach by any theory commonly accepted.
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In the first part of this review, mainly in Secs. 2 and 3, we have summarized the essential basis of the recombination model, answering some questions raised by critics, and pinning down some of the parameters, while pushing the frontier to kinematical regions no other models have attempted, and establishing parton recombination as a universal hadronization mechanism. Then in the second and main part recombination is used in an essential way to relate what are observed about jets and ridges to two complementary aspects about the dense medium: the effects of hard or semihard partons on the medium and the converse. Those aspects of the medium effects cannot be made empirically relevant unless there is a reliable description of hadronization for all pT where correlation data exist. Some of the new insights gained that are of particular current interest are: (a) (b) (c) (d) (e)
correlation between trigger and ridge (Sec. 4.4), independence of φ anisotropy on fast thermalization (Sec. 5.1), recombination does not guarantee quark number scaling (Sec. 5.3), different properties of the near- and away-side jets (Sec. 6.3), dominance of tangential jets in symmetric and unsymmetric dijets (Secs. 6.4 and 6.5).
If there is to be one unifying conclusion to be made as a result of these findings, it is that most observables on jets and ridges are due to hard or semihard partons created near the surface. Partons created in the interior of the medium that lead to dijets are not able to compete effectively with those that have shorter distance to traverse. So long as the observables allow the parton creation points to include regions that offer the partons a choice of paths of least resistance, they will take it and dominate. That is the reality faced by experiments that can only fix centrality, not medium thickness. With that recognition the efficacy of jet tomography is called into question. Unlike X-ray scanning of organic or inorganic substances, there is no control of the sources of the hard partons, so the necessary averaging process underweighs the contribution from the region of the medium that one wants to learn most about. The above comment refers to parton-initiated dijets. Of course, for single hadron at large pT the nuclear modification factor RAA has long been used as a measure of energy loss in dense medium in experiments and in theory. As soon as a condition is imposed on the detector of another particle on the away side, the region of the system probed in changed. If that hadron’s pT is low, it may be in the doublehumped shoulder region, which does provide some information on what the effect of the away-side jet is on the medium. However, to learn about the effect of the main body of the medium on the jet from dihadron correlation is more difficult. Ridgeology addresses a different set of problems, quite distinct from dijets. Ridges are stimulated by jets, but are not a part of the jets that are characterized by the participation of shower partons. Considerable attention has recently been drawn to the study of ridges. At this point there is no consensus in their theoretical interpretation. The connection between the ridges found in triggered
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events101,151 and those found in autocorrelation without triggers103,152 is in our opinion tenuous, until the pT dependence of the ridge is clarified. Minijets153 peaking at yt ∼ 2.8 correspond to hpT i ∼ 1.2 GeV/c, which is much lower than the pT range of the triggers used in ridge analysis. The ridge found in autocorrelation has no pT cut, while those found in triggered events have passoc > 2 GeV/c.101 T Nevertheless, there may exist a connection between autocorrelation and the ridges without trigger discussed in Sec. 4.2. Another phenomenon of some current interest is the observation of extended ridge at large ∆η.151 Before one concludes that such a long-range correlation can only arise from the mechanism of strings (or color flux tubes) being stretched between forward- and backward-going quarks, it seems prudent to allow firstly the possibility of other types of early-time dynamics, and secondly a broader view of the problem of particle production at large |η|. In particular, one should consider the role played by the hard parton that leads to both the trigger and the ridge. Moreover, one should consider the issue of large p/π ratio at η = 3.2 (see Sec. 3.4) as a part of the solution of the bigger problem. If the observed large p/π ratio is a feature of the final-state interaction (FSI), one should not regard the ridge phenomenon as a manifestation of the initial-state interaction (ISI) only without taking into consideration also the effects of FSI. As we have noted at a number of places throughout this review, hadronization is an important link between the observables and partonic dynamics. The characteristic of ridge formation in azimuthal correlation discussed in Sec. 4.4 is a good example of the interplay between ISI and FSI. It would be surprising if the same does not hold true for the correlation in rapidity. The considerations given in this review to jets and ridges may not be directly relevant at LHC where jets are copiously produced — unless pT is extremely high. At RHIC the background to a rare jet is thermal, but at LHC the background to a high-pT jet, say at pT ∼ 100 GeV/c, includes many other lower-pT jets. The admixture of thermal and shower partons in the background introduces new complications to the notion of enhanced thermal partons, and renders what is simply conceived for RHIC inadequate for LHC. That new energy frontier will indeed open up a wide new horizon. Note Added in Proof A recent study of the azimuthal and centrality dependencies of jet production has shed more light on the subject discussed in Sec. 6.1 and therefore on the nuclear modification factor RAA .154 The dynamical path length ξ considered in (57) and (58) is averaged over φ and its properties are determined phenomenologically as shown in Fig. 11. In Ref. 154 a theoretical study of the dynamical path length for every given pair of φ and c is carried out, taking into account the medium density along the hard parton’s trajectory as well as the probability of producing a hard parton at the creation point. Although the same symbol ξ is used, it is a very different quantity describing the degradation of the parton momentum k by
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q = ke−ξ . What is noteworthy is that there is a scaling behavior of RAA (pT , φ, c) ¯ c), i.e., the dependence on φ and c exhibits when expressed in terms of the mean ξ(φ, ¯ a universal behavior in ξ. The PHENIX data155 on π 0 RAA are shown in Fig. 20 for two pT ranges and five bins of centrality; each set of symbols has six evenly-spaced φ values from 0 to π/2, appearing from left to right in the order of increasing φ. The solid lines are the results of the theoretical calculation in the recombination π model. An approximate but simple description of the scaling behavior is RAA ≈ 154 ¯ exp(−2.6ξ). A similar behavior is found also in the away-side yield per trigger for back-to-back dijets. On the subject of recombination of adjacent jets at LHC discussed in Sec. 3.5, further study has shown that pion production at pT ∼ 10 GeV/c due to 2-jet recombination can lead to spectacular deviation from the prediction of 1-jet fragmentation. Figure 21 shows RAA at LHC for two different values of the 2-jet overlap
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104. A. Feng, (for STAR Collaboration), talk given at Quark Matter 2008, Jaipur, India (2008), J. Phys. G: Nucl. Part. Phys. 35 (2008) 104082. 105. C. Adler et al., (STAR Collaboration), Phys. Rev. Lett. 90 (2003) 082302. 106. C. B. Chiu, R. C. Hwa and C. B. Yang, Phys. Rev. C 78 (2008) 044903. 107. J. Adams et al., (STAR Collaboration), Phys. Lett. B 612 (2005) 181. 108. S. S. Adler et al., (PHENIX Collaboration), Phys. Rev. C 72 (2005) 014903. 109. J. Adams et al., (STAR Collaboration), Phys. Rev. Lett. 92 (2004) 182301. 110. S. L. Blyth (for STAR Collaboration), J. Phys. G 32 (2006) S461; 34 (2007) S933. 111. B. I. Abelev et al., (STAR Collaboration), Phys. Rev. Lett. 99 (2007) 112301. 112. J. Adams et al. (STAR Collaboration), Nucl. Phys. A 757 (2005) 102. 113. J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 98 (2007) 062301 . 114. R. C. Hwa and C. B. Yang, Phys. Rev. C 75 (2007) 054904. 115. R. C. Hwa, BNL/RIKEN Workshop on Strangeness in Collisions, February 2006 (unpublished); R. C. Hwa and C. B. Yang, Nucl-th/0602024 (unpublished first version of Ref. 114). 116. J. Bielcikova (for STAR Collaboration), J. Phys. G: Nucl. Part. Phys. 34 (2007) S929. 117. R. C. Hwa, J. Phys. G: Nucl. Part. Phys. 34 (2007) S789. 118. C. B. Chiu and R. C. Hwa, Phys. Rev. C 76 (2007) 024904. 119. J. Bielcikova (for STAR Collaboration), Eur. Phys. J. C 49 (2007) 321. 120. A. Adare et al., (PHENIX Collaboration), Phys. Rev. C 78 (2008) 014901. 121. C. B. Chiu and R. C. Hwa, Phys. Rev. C 79 (2009) 034901. 122. P. Netrakanti, parallel talk given at Quark Matter 2009, Knoxville, TN. 123. J. Putschke, plenary talk given at Quark Matter 2009, Knoxville, TN. 124. J. Adams et al., (STAR Collaboration), Nucl. Phys. A 757 (2005) 102; Phys. Rev. C 72 (2005) 014904. 125. K. Adcox et al., (PHENIX Collaboration), Nucl. Phys. A 757 (2005) 184; Phys. Rev. C 69 (2004) 024904. 126. B. I. Abelev et al., (STAR Collaboration), Phys. Rev. C 75 (2007) 054906. 127. A. Adare et al., (PHENIX Collaboration), Phys. Rev. Lett. 98 (2007) 162301. 128. D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett. 86 (2001) 4783. 129. P. Houvinen, P. F. Kolb, U. Heinz, P. V. Ruuskanen and S. A. Voloshin, Phys. Lett. B 503 (2001) 58. 130. P. F. Kolb, U. Heinz, P. Houvinen, K. J. Eskola and K. Tuominen, Nucl. Rev. A 696 (2001) 197. 131. P. F. Kolb and U. Heinz, in Quark-Gluon Plasma 3, eds. R. C. Hwa and X.-N. Wang (World Scientific, Singapore, 2004), p. 634. 132. U. Heinz, arXiv:0901.4355. 133. D. Moln´ ar and S. A. Voloshin, Phys. Rev. Lett. 91 (2003) 092301. 134. V. Greco and C. M. Ko, Phys. Rev. C 70 (2004) 024901. 135. R. C. Hwa, Phys. Lett. B 666 (2008) 228. 136. R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, J. High Energy Phys. 0109 (2001) 033. 137. PHENIX data plot, id p0636 (www.phenix.bnl.gov). 138. J. Adams et al., (STAR Collaboration), Phys. Rev. Lett. 95 (2005) 152301. 139. S. S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 97 (2006) 052301. 140. J. Adams et al., (STAR Collaboration), Phys. Rev. Lett. 97 (2006) 162301. 141. A. Adare et al. (PHENIX Collaboration), Phys. Rev. Lett. 98 (2007) 232302. 142. A. Adare et al., (PHENIX Collaboration), Phys. Rev. C 77 (2008) 011901 (R). 143. R. C. Hwa and Z. Tan, Phys. Rev. C 72 (2005) 024908; 72 (2005) 057902.
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150. 151. 152. 153. 154. 155.
ELLIPTIC FLOW: A STUDY OF SPACE-MOMENTUM CORRELATIONS IN RELATIVISTIC NUCLEAR COLLISIONS
PAUL SORENSEN Physics Department, Brookhaven National Laboratory, Building 510, Upton, NY 11973, USA [email protected]
Here I review measurements of v2 , the second component in a Fourier decomposition of the azimuthal dependence of particle production relative to the reaction plane in heavy-ion collisions. v2 is an observable central to the interpretation of the subsequent expansion of heavy-ion collisions. Its large value indicates significant space-momentum correlations, consistent with the rapid expansion of a strongly interacting Quark Gluon √ Plasma. Data is reviewed for collision energies from sN N = 2 to 200 GeV. Scaling observations and comparisons to hydrodynamic models are discussed.
1. Introduction Collisions of heavy nuclei have been exploited for decades to search for and study the transition of hadronic matter to quark gluon plasma.1,2 In these collisions, the extended overlap area, where the nuclei intersect and initial interactions occur, does not possess sphrerical symmetry in the transverse plane. Rather, for noncentral collisions, the overlap area is roughly elliptic in shape. If individual nucleonnucleon collisions within the interaction region are independent of each other (e.g. point-like) and no subsequent interactions occur, this spatial anisotropy will not be reflected in the momentum distribution of particles emitted from the interaction region. On the other hand, if the initial interactions are not independent, or if there are subsequent interactions after the initial collisions, then the spatial anisotropy can be converted into an anisotropy in momentum-space. The extent to which this conversion takes place allows one to study how the system created in the collision of heavy nuclei deviates from a point-like, non-interacting system. The existence and nature of space-momentum correlations is therefore an interesting subject in the study of heavy ion collisions and the nature of the matter created in those collisions.3,4 Figure 1 shows an illustration of the possible stages of a heavyion collisions starting with some initial energy density deposited at mid-rapidity, followed by a QGP expansion, a hadronization phase boundary, a kinetic freeze-out boundary and finally the observation of particle trajectories in a detector. One can consider a number of ways to study space-momentum correlations: e.g. two-particle correlations5 and HBT.6,7 In this review we discuss elliptic flow v2 ; an observable that has been central in the interpretation of heavy-ion data 323
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Fig. 1. A schematic diagram of the expansion after an ultra-relativistic heavy-ion collision.
and QGP formation.8 Given the predominantly elliptic shape of the initial overlap region, it is natural to ask whether this shape also shows up in the distribution of particles in momentum-space. Figure 2 shows a schematic illustration of the conversion of coordinate-space anisotropy to anisotropy in momentum-space. The left panel shows the position of nucleons in two colliding nuclei at the moment of impact. The overlap region is outlined and shaded. A Fourier decomposition can be used to describe the azimuthal dependence of the final triple momentum-space distributions:9 d3 N 1 dN = × [1 + 2v1 cos(φ − Ψ) + 2v2 cos(2(φ − Ψ)) + . . . ] , pT dpT dyd(φ − Ψ) 2π pT dpT dy (1) where φ is the azimuth angle of the particle, y the longitudinal rapidity variable, pT the transverse momentum, and Ψ is the reaction plane angle defined by the vector connecting the centers of the two colliding nuclei. Positive v2 implies that more particles are emitted along the short axis of the overlap region. To study the extent to which space-momentum correlations develop in heavy-ion collisions, one can measure the second component v2 and compare it to the initial spatial eccentricity.3,10 The right panel of Fig. 2 shows the final azimuthal distribution
Review of Elliptic Flow v2
y
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py pT (GeV/c) 1.5
x
1.0 0.75 0.5 0.25
px
√ Fig. 2. Schematic illustrations of a sN N = 200 GeV Au+Au collision with a 6 fm impact parameter. The left panel shows the nucleons of the two colliding nuclei with an ellipse outlining the approximate interaction region. The right panel shows a momentum-space representation of v 2 . The average radius of each successive ring represents the pT of the particles while the anisotropy of the ring represents the magnitude of v2 . The highest pT particles (outer-ring) exhibit the strongest v2 while the lowest pT particles (inner-ring) exhibit a vanishingly small v2 .
of particles in momentum-space. The curves represent the anisotropy at different pT values measured in 200 GeV Au+Au collisions11 : i.e. f (pT , φ) = pT ∗ (1 + 2v2 (pT ) cos(atan2(py , px ))). The goal of v2 measurements is to study how the initial spatial anisotropy in the left panel is converted to the momentum-space anisotropy in the right panel. In this review, a summary of v2 data for different colliding systems, different center-of-mass energies, and different centralities is given. This review will focus on results from the first four years of operation of the Relativistic Heavy Ion Collider (RHIC). We start with a brief discussion of the beam energy dependence of v2 and some ideas about what physics might be relevant. Even before considering physics scenarios to explain how a space-momentum correlation develops, one can see that to interpret v2 it is important to understand the initial geometry and how it varies with the collision centrality and system-size. Since, the concept of the reaction-plane is so central to the definition of v2 and eccentricity is so central to it’s interpretation, I discuss the two in a sub-section below. Then a review of RHIC data is provided. This will include the dependence of v2 on center-of-mass energy, centrality, colliding system, pseudo-rapidity, pT , particle mass, constituent quark number and various scaling laws. In the following section, I will discuss comparisons to models and the emergence of the hydrodynamic paradigm at RHIC. Particular emphasis is given to uncertainties in the model comparisons. In that section I will also discuss current attempts to extract viscosity and future directions of investigation. Voloshin, Poskanzer, and Snellings recently wrote a review article12 on collective phenomena in non-central nuclear collisions that deals with a similar subject matter. That article provides valuable detail on technical aspects of measuring v2 . In this review I will attempt to avoid duplicating that work by discussing interpretations of v2 more extensively and refer the reader to that review where appropriate.
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0.1
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in-plane 0 out-ofplane
FOPI EOS E895 E877 CERES NA49 STAR Phenix Phobos
-0.05
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Ebeam/A (GeV) Fig. 3. The beam energy dependence of elliptic flow measurements. The RHIC and E877 data are for charged hadrons, independent of species. The NA49 data is for charged pions. The E895 data are for protons and the FOPI data are for atomic number Z=1. At each energy, the sample of particles is close to the total charge. Positive v2 values indicate that particles tend to be more aligned with the reaction-plane (in-plane). RHIC and SPS data suggest a smooth trend of in-plane √ √ v2 growing with log( snn ) above Ebeam /A ≈ 20 GeV or sN N ≈ 6 GeV.
1.1. Two decades in time and five decades in beam energy Positive values of v2 imply that particles tend to be produced more abundantly in the x-direction than in the y-direction. This is referred to as in-plane flow. Figure 3 shows v2 measured in an interval of beam energies covering five orders of magnitude.13–22 For Ebeam /A ranging from approximately 0.12 − 5 GeV √ (1.92 < snn < 3.3 GeV), v2 is negative. For this energy range, spectator protons and neutrons are still passing the interaction region while particles are being produced. Their presence inhibits particle emission in the in-plane direction leading to the phenomenon termed squeeze-out. At still lower energies, v2 is positive as the rotation of the matter leads to fragments being emmitted in-plane. At this energy beam rapidity and mid-rapidity are essentially indistinguishable with ybeam < 0.41 units. In-plane flow: As the beam energy is increased, the nuclei become more Lorentz contracted and the time it takes the spectators to pass each other decreases.
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It was predicted by Ollitrault3 that at high enough beam energy, the squeeze-out phenomena would cease and v2 would take on positive values. Positive v2 values √ were measured at the AGS for energies above Ebeam /A = 5 GeV ( snn = 3.3 GeV). √ For energies above Ebeam /A ≈ 20 GeV ( snn = 6.3 GeV), v2 exhibits a steady log√ linear increase: v2 ≈ 0.01 + 0.0042 log(Ebeam /A) or v2 ≈ 0.008 + 0.0084 log( snn ) where the data represented are from intermediate impact parameter A+A collsions. It appears therefore that RHIC v2 data may be part of a smooth trend that began at SPS energies. This trend was noted previously at least once.23 Understanding the physics that underlies that trend is one of the challenges of heavy-ion physics. One class of models that has provided an illustrative reference for heavy-ion collisions are hydrodynamic models which are used to model the expansion the matter remaining in the fireball after the initial collisions.24–33 This model can be used to determine how matter with a vanishingly small mean free path would convert the initial eccentricity into v2 . These models typically treat all elliptic flow as arising from the final state expansion rather than from some initial state effects. 34–37 In the hydrodynamic models, large pressure gradients in the in-plane direction lead to a preferential flow of matter in the in-plane direction. In this review, we will use hydrodynamic models as a convenient reference. Other models providing a valuable reference for measurements include hadronic and partonic cascades and transport models.38–46 Additionally, the blast-wave model provides a successful parametrization of low pT heavy-ion data, including v2 , HBT, and spectra in terms of several freeze-out parameters.30,47
1.2. Initial geometry: the reaction plane and eccentricity In the collision of two symmetric nuclei, a unique vector (the y-axis) can be defined by applying the right-hand-rule to the momentum vector of one nucleus and the vector pointing to the center of the other nucleus. The y-axis is a pseudovector. The reaction-plane is then the plane perpendicular to the y-axis containing the points at the center of the two nuclei. The reaction-plane and the right-handed coordinate system are illustrated in Fig. 4. The figure contains perspective illustrations of two nuclei approaching with an impact parameter of 6 fm. The impact parameter is the distance between the centers of the two nuclei at the moment of their closest approach. The two nuclei in this illustration are Lorentz contracted by a Lorentz gamma factor of 10 which roughly corresponds to the appropriate gamma for top SPS energies. The reaction-plane is not directly observed in experiments, however, and this introduces a systematic uncertainty into the measurement of v2 . One often relies instead on indirect observations to estimate v2 .48–52 For example, when forming two particle azimuthal correlations such as d(φdN , a non-zero v2 value will lead 1 −φ2 ) to a modulation in ∆φ = φ1 − φ2 of the form 1 + 2hv22 i cos(2∆φ). Figure 5 shows the correlation function for hadrons produced at mid-rapidity at RHIC.53,54 The
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y y
y
y
x z
x
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z
z x
x
y
y
y z x
z
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√ Fig. 4. Schematic illustrations of a sN N = 200 GeV Au+Au collision with a 6 fm impact parameter. The right-handed coordinate systems defined by the momentum of the nucleus (here the z-axis) crossed with the vector pointing to the center of the approaching nucleus (the x-axis) is shown for each nucleus. The reaction-plane is the plane normal to the y-axis, containing the centers of the two colliding nuclei. The nucleons are distributed inside the nucleus according to a Woods-Saxon distribution. The nuclei are Lorentz contracted in the z-direction.
panels show different centralities. The area normalized correlation function is R AB Y (∆φ) Y AB (∆φ) dN AB R Mixed C(∆φ) ≡ Same × ∝ (2) AB AB d(∆φ) YMixed (∆φ) YSame (∆φ)
AB AB where YSame (∆φ) and YMixed (∆φ) are, respectively, the uncorrected yields of pairs in the same and in mixed events within each data sample. C(∆φ) shows a clear cos(2∆φ) dependence. We note here that what is measured in these correlation functions is hv22 i = hv2 i2 + σv22 in anticipation of a discussion of v2 fluctuations. v2 will not be the only contribution to the azimuthal dependence of the twoparticle azimuthal correlations. Other processes that are not related to the reactionplane can give rise to structures in the shape of the two-particle ∆φ distribution as well. These non-reaction plane contributions are commonly called “non-flow”. The subject of non-flow is an important one and will be discussed throughout this review. The contribution of non-flow can be seen more clearly by looking at very peripheral collisions or by selecting high momentum particles to increase the chance that a particular pair of hadrons are correlated to a hard scattered parton (jet). Figure 6 shows the correlation function for higher momentum particles.55 The solid line shows what a pure v2 correlation would look like.56 The difference between those curves and the data are often taken as a measurement of jet correlations. 55,57,58
Review of Elliptic Flow v2
Fig. 5. Charged hadron correlation functions in Au+Au collisions at centrality intervals and two pT ranges.
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√ snn = 200 GeV for two
Even if the reaction-plane were known with precision, there is no first principles calculation of the initial matter distribution in the overlap region, so the eccentricity is uncertain. Various models can be used to calculate the initial spatial eccentricity which can then be compared to v2 . Defining the y-axis according to the right-handrule, the eccentricity εs is traditionally calculated as: εs =
hy 2 − x2 i , hx2 + y 2 i
(3)
where the average represents a weighted mean. Other eccentricity definitions have also been considered.59 The weights can be some physical quantity in a model such as energy or entropy density, or simply the position of nucleons participating in the collision. One popular method for calculating the eccentricity is to use a Monte Carlo Glauber model. Details can be found in a recent review.60 In that model, a finite number of nucleons are distributed in a nucleus according to a Woods-Saxon distribution. Then two nuclei are overlaid with a fixed impact parameter and the x and y positions of the participating nucleons is determined based on whether the nucleons overlap in the transverse plane; each nucleon is considered to be a √ disk with an area determined by the s dependent nucleon-nucleon cross-section. The x and y coordinates of the participating nucleons are then used to calculate the eccentricity. Those nucleons that do not participate in this initial interaction are called spectators. One can anticipate that due to the finite number of nucleons
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Fig. 6. Charged hadron correlations for a variety of centrality intervals. The correlation function is formed between two samples of hadrons based on their pT : the “trigger” particle sample is selected with 2.5 GeV/c < pT < 4.0 GeV/c while the “associated” particle sample is selected with 1 GeV/c < pT < 2.5 GeV/c.
in this model, the initial geometry will fluctuate. Other models used to determine the initial matter distribution including HIJING,61 NEXUS,62 and Color Glass Condensate models63–66 also reach the same conclusions; the initial overlap region is expected to be lumpy rather than smooth. Figure 7 shows the gluon density in the transverse plane which is probed by a 0.2 fm quark-antiquark dipole at two √ different x values in the IPsat CGC model63 (x = 2pT / sN N is 10−5 in the left panel and 10−3 in the right panel). The lumpiness is immediately apparent. Until recently however,67,68 v2 data was compared almost exclusively to calculations assuming an infinitely smooth initial matter distribution (for example in initializing a hydrodynamic expansion). Improving on that approximation may be d2 N important for understanding the shape expected for the d∆ηd∆φ distribution.5,69 This distribution is also investigated in heavy ion collisions in order to search for jets. In any scenario where space-momentum correlations develop, the correlations dN distribution and fluctuations in the initial geometry can be manifested in the d∆φ and understanding these correlations is important for interpreting heavy-ion collisions. Fluctuations in the initial geometry have also led to the idea of measuring particle distributions relative to the participant-plane rather than the reactionplane.59,70 The participant-plane is defined by the major axis of the eccentricity
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Fig. 7. Gluon density in the transverse plane when the nucleus is probed at different x values by a 0.2 fm quark-antiquark dipole in the IPsat CGC model.
εstd
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Fig. 8. The distribution of eccentricity on an event-by-event basis when calculated relative to the reaction-plane (left) and the participant plane (right).
which, due to fluctuations, can deviate from the reaction-plane. The eccentricity relative to the participant-plane is a positive definite quantity and is always larger than the eccentricity relative to the reaction-plane; the participant-plane is defined by rotating to the axis that maximizes the eccentricity. Figure 8 shows the eventby-event distribution of the standard eccentricity (left panel) and the participant eccentricity (right panel) as a function of impact parameter determined from a Monte-Carlo Glauber calculation. The fluctuations in this model are large as illustrated by the widths of the distributions. The relationship between the different definitions of eccentricity and their fluctuations are explained clearly in two recent papers.59,71 Different models for the initial matter distribution yield different estimates of hεi p and hε2 i. The deviations in the hεi for different models can be of the order of 30% and strongly centrality dependent.72 That uncertainty in hεi leads to an inherent uncertainty when comparing models to v2 /ε. This level of uncertainty becomes
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important when attempting to estimate transport properties of the matter based on comparisons of the observed v2 to the initial ε. 2. Review of Recent Data √ The first paper published on RHIC data was on elliptic flow in sN N = 130 GeV Au+Au collisions.18 Figure 9 shows that data on the centrality dependence of v2 . The values of v2 reach a maximum of approximately 6% for peripheral collisions where the initial eccentricity of the system is largest. That value is 50% larger than the values reached at SPS energies21 and the v2 values are a factor of two larger than the those predicted by the RQMD transport-cascade model.42 For central collisions, the measurements approach the zero mean-free-path limit estimated from the eccentricity shown in the figure as open boxes. The boxes represent the eccentricity scaled by 0.19 (bottom edge of the boxes) and 0.25 (top edge of the boxes). Those values are chosen to represent the typical conversion of eccentricity √ to v2 in hydrodynamic models. At lower sN N energies, the RQMD model provided a better description of the data, while hydrodynamic models significantly over-predicted the data. The conclusion based on this early comparison, therefore,
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Fig. 9. First measurements of v2 versus centrality at RHIC. Positive values are observed which are largest for events with the largest eccentricity and decrease for more central, symmetric collisions. The trend with centrality clearly indicates a space-momentum correlation driven by the eccentricity of the initial overlap zone. Centrality is expressed in terms of the observed multiplicity of a give event relative to the highest multiplicity observed nch /nmax . Data are compared to eccentricity scaled by 0.19 (bottom edge of the boxes) and 0.25 (top edge of the boxes). The values are chosen to represent the typical conversion of eccentricity to v2 in a hydrodynamic model.
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(1/S)dNch /dy Fig. 10. v2 scaled by a Monte Carlo Glauber model calculation of the initial overlapp eccentricity. The ratio is plotted versus transverse particle density (1/S)dNch /dy, where S = π hx2 ihy 2 i is a weighted average area calculated with the same model as the eccentricity. Data are taken from √ different sN N values and different centralities. Plotted in this format, the data suggest v 2 /ε for different energies and overlap geometries is determined by the transverse particle density, and approaches a zero mean-free-path hydrodynamic limit for most central top energy collisions at RHIC. This conclusion is not universally accepted and is still being investigated.
was that heavy-ion collisions approximately satisfy the assumptions made in the hydrodynamic models: 1) zero mean-free-path between interactions, and 2) early local thermal equilibrium.33 These conclusions remain at the center of scientific debate in the heavy-ion community. In Fig. 10, v2 is scaled by model calculations of the initial eccentricity and 21 plotted versus transverse particle density S1 dN This facilitates comparisons of v2 dy . √ across different sN N energies, collision centralities and system-sizes.10,41 For the case of ballistic expansion of the system — that is an expansion for which the produced particles escape the initial overlap zone without interactions — v2 should only reflect the space-momentum correlations that arise from the initial conditions. Those can exist in the case that the initial interactions are not pointlike34 but rather involve cross-talk between different N + N interactions within the overlap zone. The opposite extreme from the ballistic expansion limit is the zero mean-free-path limit represented by ideal hydrodynamic models. Lacking a length scale, the zero mean-free-path models should not depend on system-size and instead should be a function of density. The measurements of v2 are expected to rise from values near the ballistic expansion limit and asymptotically approach the zero mean-free-path limit as the
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density of the system is increased. Data in Fig. 10 exhibit such a behavior with the most central collisions at full RHIC energy apparently becoming consistent with the hydrodynamic model. This conclusion however depends on the model calculations for the initial eccentricity and on the assumption that the observed v2 dominantly arises from an expansion phase where anisotropic pressure gradients are the origin of the space-momentum correlations. Different models for the eccentricity yield ε results that deviate both in their centrality dependence and in their overall magnitude. Reasonable models for the eccentricity can easily give magnitudes 30% larger than those used in Fig. 10 with a stronger centrality dependence. The ratio v2 /ε can therefore be smaller than what is shown and have a different shape.72 √ Given this level of uncertainty, the conclusion that heavy-ion collisions at sN N = 200 GeV approximately satisfy the assumptions made in the hydrodynamic models i.e. early local thermal equilibrium and interactions near the zero mean-free path limit, would be more convincing if an asymptotic approach to a limiting value were observed. Rather, for the eccentricity calculation used in Fig. 10, the data suggest a nearly linear rise with no indication of asymptotic behavior. In the hydrodynamic picture, one might also expect that v2 /ε versus S1 dN dy will be sensitive to the equation-of-state of the matter formed during the expansion phase. Since v2 is expected to reflect space-momentum correlation developed due to pressure gradients and S1 dN dy is a measure of the transverse particle density, 1 dN v2 /ε versus S dy could be considered as a proxy for the pressure versus energy density or the equation-of-state. It’s difficult to identify in the data the features that are expected in the equation-of-state. Figure 11 shows the equation-of-state calculated in a recent lattice QCD calculation.73 The onset of the QGP phase is RHIC
LHC
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Fig. 11. The QCD equation-of-state (pressure over energy density versus the fourth power of the energy density) as determined in Lattice calculations.
Review of Elliptic Flow v2
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seen to lead to an increase in the pressure as the energy density is increased above a critical value. The energy density in heavy-ion collisions is often estimated from the Bjorken formula74 : Bj =
1 dET Aτ dy
(4)
T which depends only on the experimentally accessible quantity dE dy , the overlap area of the nuclei and τ , the unknown formation time which is often assumed to be 1 fm. The Bjorken estimate for the energy density is closely related to the transverse particle density (1/S)(dN/dy).
2.1. Differential elliptic flow In addition to studying how v2 integrated over all particles depends on the centrality √ or sN N of the collision, one can study how v2 depends on the kinematics of the produced particles (differential elliptic flow). Figure 12 shows the centrality and
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Fig. 12. v2 (η) for 200 GeV Au+Au collisions for three centrality intervals. The inset shows the ratio of v2 (η) for central collisions over peripheral collisions. Open symbols are data reflected to negative η.
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pseudo-rapidity dependence of v2 for 200 GeV Au+Au collisions.75 v2 is largest at mid-rapidity where the transverse particle density is largest and then falls off at larger |η| values. This behavior is therefore consistent with the trends seen in integrated v2 where v2 /ε appears to increase with increasing transverse particle density. The fall off of v2 (η) with increasing |η| is common to the three centrality intervals studied. The inset of the figure shows the ratio of v2 in peripheral over central collisions. Within errors the ratio is flat indicating a similar shape for all centralities with v2 (η) only changing by a scale factor. Scaling of v2 (η) for different energies and system sizes will be discussed in a later section. Figure 13 shows v2 for a variety of particle species as a function of their transverse momentum pT .76–81 In the region below pT ∼ 2 GeV/c, v2 follows mass ordering with heavier particles having smaller v2 at a given pT . Above this range, the mass ordering is broken and the heavier baryons take on larger v2 values. A hydrodynamic model for v2 (pT ) is also shown which describes the v2 in the lower pT region well. This mass ordering is a feature expected for particle emission from a boosted source. In the case that particles move with a collective velocity, more massive particles will receive a larger pT kick. As the particles are shifted to higher pT , the lower momentum regions become depopulated with a larger reduction
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in the direction with the largest boost (in-plane). This reduction reduces v2 at a given pT , with the reduction largest for more massive particles. Note that this does not imply that the more massive particles have a smaller integrated v2 value, and in fact the opposite is true. Figure 14 shows v2 for identified particles integrated over all pT .82,83 The integration shows that v2 increases with particle mass. This is because the more massive particles have a larger hpT i and v2 is generally increasing with pT in the pT region where the bulk of the particles are produced. The hydrodynamic model also exhibits this trend.
2.1.1. Identified particle v2 (pT ): RHIC versus SPS √ Figure 15 shows pion and proton v2 from sN N = 62.4 Au+Au81 and 17.3 GeV Pb+Pb collisions.21 The centrality intervals have been chosen similarly for the 17.3 GeV and 62.4 GeV data. The STAR data at 62.4 GeV are measured within the pseudo-rapidity interval |η| < 1.0 and the 17.3 GeV data are from the rapidity interval 0 < y < 0.7. These intervals represent similar y/ybeam intervals. It has been shown that v2 data for pions and kaons at 62.4 GeV are similar to 200 GeV data; the 62.4 GeV data only tending to be about 5% smaller than the 200 GeV data. Appreciable differences are seen between the 17.3 GeV and 62.4 GeV data. At pT > 0.5 GeV/c, for both pions and protons, the v2 values measured at 62.4 GeV
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are approximately 10%–25% larger than those measured at 17.3 GeV. Although the magnitude of v2 is different at the lower energy, the systematics of the particle-type dependencies are similar. In particular, pion v2 and proton v2 cross over each other √ at pT near 1.7 GeV/c for sN N = 17.3, 62.4 and 200 GeV data. Due to the limited kinematic range covered by the 17.3 GeV data, it’s not possible to determine if the v2 of baryons at pT > 2 GeV becomes larger than that for the lighter mesons. The increase in the magnitude of v2 from 17.3 GeV to 62.4 GeV and the similarity of 62.4 GeV v2 to 200 GeV v2 has been taken as a possible indication for the onset of a limiting behavior.84 In a collisional picture, a saturation of v2 could √ indicate that for sN N at and above 62.4 GeV the number of collisions the system constituents experience in a given time scale can be considered large and that hydrodynamic equations can therefore be applied. Hydrodynamic model calculations of v2 depend on the model initialization and the poorly understood freeze-out assumptions. As such, rather than comparing the predicted and measured values at one energy, the most convincing way to demonstrate that a hydrodynamic limit √ has been reached may be to observe the onset of limiting behavior with sN N . For this reason, v2 measurements at a variety of center-of-mass energies are of interest. Figure 15 shows that when the 17.3 and 62.4 GeV v2 (pT ) data are compared within similar |y|/ybeam intervals, the differences between v2 (pT ) within the data sets may be as small as 10%–15%. As such, a large fraction of the deviation between the SPS data and hydrodynamic models arises due to the wide rapidity range covered by those measurements (v2 approaches zero as beam rapidity is approached75),
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increased hpT i values at RHIC and the larger v2 values predicted for the lower colliding energy by hydrodynamic models.
2.2. High pT At higher pT , v2 no longer rises with pT and the mass ordering is broken. Above pT ∼ 2 GeV/c the more massive baryons exhibit a larger v2 than the mesons. While the pion and kaon v2 reach a similar maximum of v2 ≈ 0.14 at pT ≈ 2.5 GeV/c, the baryon v2 continues to rise until it reaches a maximum of v2 ≈ 0.20 at pT ≈ 4.0 GeV/c. For still larger pT , the v2 values exhibit a gradual decline until v2 for all particles is consistent with v2 ≈ 0.10 at pT ≈ 7 GeV/c. Fine detail cannot yet be discerned at pT > 7 due to statistical and systematic uncertainties. At these higher pT values one expects that the dominant process giving rise to v2 is jet-quenching85 where hadron suppression is larger along the long axis of the overlap region than along the short axis.86–88 For very large energy loss, the value of v2 should be dominated by the geometry of the collision region. Figure 16 shows a comparison of v2 data89 for 3 < pT < 6 GeV/c compared to several geometric models.90 This comparison seems to indicate that v2 in this intermediate pT range is still too large to be related exclusively to quenching. In the higher pT regions, significant azimuthal structure will arise from jets, so non-flow correlations are thought to be significant in this region.58 These effects have been studied in several ways. The four-particle cumulant v2 has been studied as a function of pT and the ratio of the four- and two-particle cumulants v2 {4}/v2 {2} is found to decrease with increasing pT .91,92 This decrease is identified with a gradual
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Fig. 17. The left panel shows the ratio v2 {4}/v2 {2} for the 0%–70% centrality interval in 200 GeV Au+Au collisions. The ratio falling below unity indicates the importance of non-flow and v 2 fluctuations. The larger reduction at high pT seems indicative of an increase in non-flow due to jets. The right panel shows the centrality dependence of the v2 {4} and v2 measured with respect to the event plane in 130 GeV Au+Au collisions. The inset shows the ratio of the two.
increase in the contribution of jets to v2 {2} (see Fig. 17 left panel). The four-particle cumulant suppresses contributions due to intra-jet correlations but the statistical errors of the measurement are larger. One can also suppress jet structure in the v 2 measurement by implementing a ∆η cut in the pairs of particles being used in the analysis.11 In this case, a high pT particle is correlated with other particles in the event that are separated by a minimum ∆η. This method relies on the assumption that jet correlations do not extend beyond a given ∆η range. Interactions of jets with the medium in nuclear collisions however can change the structure of jets and extend the correlations in ∆η beyond the widths observed in p + p collisions.93 This method therefore is not guaranteed to eliminate non-flow from jets. The problem of measuring v2 without non-flow and of measuring modifications of jet structure by the medium are entirely coupled. If one is known, the other is trivial. Other methods for suppressing non-flow include measuring correlations between particles at mid-rapidity and and an event-plane determined from particles observed at forward rapidity.94 In the extreme and the most effective case, the event-plane was reconstructed from spectator neutrons in a Zero-Degree Calorimeter to measure v2 of produced particles near η = 0. An extension of analyses based on the change in correlations across various rapidity intervals is the analysis of the two dimensional correlation landscape for two-particle correlations e.g. d2 N/∆φ∆η.95 After unfolding the two particle correlations one can attempt to identify various structures with known physics such as jets, resonance decay, or HBT based on their width in η and φ. The remaining cos(2∆φ) structure can then be used to estimate hv2 i2 + σv22 . This method will be discussed below.
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p T (GeV/c) Fig. 18. Second harmonic azimuthal correlations in p + p, d+Au, and Au+Au collisions. The quantity hΣi cos(n(φpT − Φi ))i = M hv2 v2 (pT )i + M δ2 facilitates comparisons between different systems particularly where a reaction-plane may not be well defined.
As we study progressively smaller systems the connection between the nucleusnucleus reaction-plane and the azimuthal structure breaks down. In the limit that one proton from each nucleus participates in the interaction, the reaction-plane defined by the colliding protons will not necessarily be related to the reaction-plane defined by the vector connecting the centers of the colliding nuclei. In order to facilitate a comparison between the pT dependence of azimuthal correlations in large systems and small systems, the scalar product huQ∗ i is used where u = ei2φ and Q∗ = e−i2Ψ .89 The mean of uQ∗ therefore yields a quantity that depends on hv2 v2 (pT )i and non-flow as follows: huQ∗ i = hΣi cos(n(φpT − Φi ))i = M hv2 v2 (pT )i + M δ2 ,
(5)
where M is the multiplicity used in the sum. Figure 18 shows this quantity for p+p, d+Au, and three Au+Au centrality intervals. The p+p and d+Au data are repeated in each of the panels. The most peripheral Au+Au collisions are shown in the left panel. The centrality bin shown is not usually presented since trigger inefficiencies for low multiplicity events makes it difficult to define the actual centrality range sampled. In this case, the data has been published in order to compare uQ∗ between the most peripheral sample of events and p + p collisions. The data in Au+Au has a similar shape and magnitude as the data in p + p. This suggests that peripheral collisions are dominated by the same azimuthal structure as p + p collisions; an
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observation consistent with two-particle ∆η, ∆φ correlations.96 The data from midcentral Au+Au collisions shown in the middle panel however, exhibit a magnitude and shape clearly different than p+p collisions. While uQ∗ for p+p, d+Au and very peripheral Au+Au collisions rises monotonically with pT , for mid central Au+Au collisions, the data rises to a maximum at pT = 3 GeV/c and then falls. For central collisions shown in the right panel, a similar feature is seen with data rising to a maximum at pT = 3 GeV/c, then falling until pT = 6 GeV/c, where it begins rising again. This second rise is presumably a manifestation of non-flow at high pT in central collisions. These data suggest that azimuthal structure in Au+Au collisions above pT = 6 GeV/c is dominated by jets. This is also consistent with the conclusions reached by examining the particle type dependence of v2 and RCP .78 In Fig. 18 the p + p data is replotted in each panel to facilitate a comparison between the shape and magnitude in p + p to that in Au+Au. In the absence of jet-quenching however, non-flow at high pT is expected to scale with the number binary nucleon-nucleon collisions Nbinary . The plotting format in Fig. 18 on the other hand, assumes that δ2 ∝ 1/M rather than Nbin /M 2 as would be expected for hard scattering. The multiplicity has been shown to scale as (1 − xhard )Npart + xhard Nbinary with xhard ≈ 0.11.97,98 This is referred to as the two-component model. In order to compare azimuthal structure in Au+Au collisions to Nbinary scaling of p + p collisions we can form a ratio in analogy with RAA for single hadrons: ∗
uQ RAA =
huQ∗ iAA /MAA Nbinary huQ∗ ipp /Mpp
(6)
where MAA and Mpp are the multiplicities in A + A and p + p collisions with MAA taken according to the two component model. In the case that jet production in Au+Au collisions scales with the number of binary collisions, as hard processes uQ∗ uQ∗ are expected to, RAA should be unity. The right panel of Fig. 19 shows RAA for charged hadrons in 0%−−5% central Au+Au collisions. For comparison, RAA from uQ∗ single particle charged hadron spectra is also shown in the figure. RAA first rises abruptly with pT to a maximum of 2 at pT ≈ 0.5 GeV/c and then falls to a value of uQ∗ 0.25 at pT ≈ 5 GeV/c. At pT > 5 GeV/c RAA is similar to RAA . This shows that jet-quenching suppresses the charged hadron spectra, and the azimuthal structure by a similar amount; confirming that the single hadron suppression is indeed related uQ∗ to jet-quenching. RAA is complimentary to studies of IAA , the ratio of dihadron correlations in Au+Au and p + p collisions.57 We note the presence of what appears to be a local minimum and local maximum pT ≈ 1.5 and 2.0 GeV/c respectively. It is not clear if this is a real feature or simply an artifact largely caused by the shape of the p + p data. In the case that it is a real feature, it is possibly related to the changing particle composition in Au+Au collisions where baryons with larger v2 values become more prominent. At pT = 3 GeV/c, baryons and mesons in p + p collisions are created in the proportion 1:3 while at the same pT in central Au+Au collisions the proportion is approximately uQ∗ 1:1. RAA will be an interesting quantity to investigate for identified particles. One
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can anticipate a quark number dependence at intermediate pT as seen in RCP and v2 . 2.3. Multiply strange hadrons and heavy flavor The build-up of space-momentum correlations throughout the collision evolution is cumulative. Information about space-momentum correlations developed during a Quark-Gluon-Plasma phase can be masked by interactions during a later hadronic phase. For studying a QGP phase, it is useful to use a probe that is less sensitive to the hadronic phase. Multi-strange hadrons have hadronic cross-sections smaller than the equivalent non-strange hadrons, and the v2 values measured for hadrons such as φ-mesons (ss) and Ω-baryons (sss) are therefore thought to be more sensitive to a quark-gluon-plasma phase than to a hadronic phase.32 Figure 20 shows v2 (pT ) for the φ-meson.99,100 The v2 rises with pT and reaches a maximum of approximately 15% at pT near 2 GeV/c. At intermediate pT , the φmeson v2 appears to follow a trend similar to the other meson KS0 . This observation suggests that either the φ-meson cross section is larger than anticipated or rescattering during the hadronic phase does not contribute significantly to v2 . The latter possibility requires that v2 is established prior to a hadronic phase, suggestive of development of v2 during a QGP phase.
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v2 0.25 0.2
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pT (GeV/c) Fig. 20. v2 (pT ) for the φ meson. The φ is composed of an ss pair and is expected to have a smaller hadronic cross-section than non-strange, or singly-strange hadrons. The φ v 2 is compared to KS0 and Λ v2 .
Measurements of multiply strange hadrons are interesting because they should be less coupled to the matter in a hadronic phase and therefore a better reflection of the QGP phase. Heavy quarks on the other hand (e.g. charm and bottom quarks) may be less coupled to even the QGP matter.101,102 It’s not a priori obvious that heavy quarks will couple significantly to the medium and be influenced by its apparent expansion. The extent to which they do couple to the medium should be reflected in how large v2 for heavy flavor hadrons becomes and how much the nuclear modification (RAA ) deviates from unity. Precision measurements of Heavy Flavor mesons or baryons are not yet available from the RHIC experiments. As a proxy for identifying D-mesons, the STAR and PHENIX experiments have measured non-photonic electrons.103,104 Non-photonic electrons are generated from the weak-decays of heavy flavor hadrons and after various backgrounds have been accounted for can, with some caveats,105 be used to infer the RAA and v2 of Dmesons. The top panel (a) of Fig. 21 shows RAA for non-photonic electrons.106,107 Prior to the measurement of non-photonic electron RAA , it was expected that heavy-flavor hadrons would be significantly less suppressed than light flavor hadrons. These expectations based on a decrease in the coupling of charm quarks to the medium because of the dead-cone effect,101 are contradicted by the data; At pT ≈ 5 GeV/c, non-photonic electrons are as suppressed as pions. This suppression suggests a
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stronger than expected coupling of charm quarks to the medium. This coupling apparently also leads to significant v2 as seen in Fig. 21 (b). Also shown in the figure is a calculation of v2 and RAA based on a Langevin model.108 In that model, the strength of the energy loss and momentum diffusion of charm quarks is characterized in terms of a diffusion coefficient (D). RAA and v2 for charm quarks is then computed for several values of D. Two of these values are shown in Fig. 21. Although neither curve provides an entirely satisfactory simultaneous description of v2 and RAA , the comparison suggests that the diffusion coefficient is large. This comparison only achieves rough agreement, but the calculation illustrates the sensitivity of heavy flavor hadrons to transport coefficients of the QGP and v2 is an important quantity to measure for these hadrons. This is also in agreement with H. van Hees, et al.109 where coalescence at the hadronization phase boundary is also considered and found to help improve the agreement with data.
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2.4. Fluctuations and correlations Comparisons between data and models are complicated by uncertainties in the initial eccentricity and by uncertainties in the data. Estimating transport quantities from the data may require a precision comparison between eccentricity and v 2 so it is important to reduce the uncertainties in both. As discussed in the previous sections, a CGC model of the initial conditions yields eccentricity values typically 30% larger than a Gluaber model while the fluctuations (σε ) are still of the same width. The ratio of σε /ε in a CGC model is therefore smaller than in a Glauber model.65,66 One can expect the statistical fluctuations in eccentricity to show up as dynamical fluctuations in v2 measurements. Measuring the dynamic v2 fluctuations in conjunction with hv2 i can therefore provide an additional constraint on the initial conditions.110–113 Several methods have been employed for measuring v2 and the various methods have different dependencies on non-flow correlations and v2 fluctuations.59,68,71,114–118 The differences between these measurements give information on non-flow correlations and v2 fluctuations. If one uses a two particle correlation to estimate v2 , then one finds v2 {2}2 = hcos(2(φi − φj ))i = hv2 i2 + σv22 + δ2 where the average is over all unique pairs of particles. v2 is the single particle anisotropy with respect to the reaction plane v2 = hcos(2(φ − Ψ))i and δ2 is the non-flow parameter which summarizes the contributions to hcos(2(φi − φj ))i from correlations not related to the reaction plane. If one uses a 4-particle cumulant v2 {4} calculation, then for most cases the non-flow term will be suppressed by large combinatorial factors and v2 fluctuations will contribute with the opposite sign. For Gaussian fluctuations, v2 {4}2 ≈ hv2 i2 − σv22 .71 Without knowing δ or σv2 , one cannot determine the exact value of hv2 i. Rather, hv2 i2 could lie anywhere between v2 {4}2 and (v2 {2}2 + v2 {4}2)/2. It is advantageous to confront various models with the data that is experimentally accessible. The difference between the two- and four-particle cumulants in the case of Gaussian v2 fluctuations is: v2 {2}2 − v2 {4}2 ≈ δ2 + 2σv22 .
(7)
The term δ + 2σv22 is also approximately equivalent to the non-statistical width of the distribution of the length of the flow vector distribution (dN/d|q2 |) and is 2 called σtot . The flow vector for the nth harmonic is defined as qn,x = Σi cos(nφi ) 2 and qn,y = Σi sin(nφi ). Figure 22 shows σtot extracted from the difference be11 tween the two- and four-particle cumulants. In the case of Gaussian fluctuations, 2 higher cumulants such as v2 {6} are equal to v2 {4}. In this case, the quantity σtot and v2 {2}2 summarizes the information available experimentally from the second harmonic flow vector distribution. No more information can be accessed without applying more differential techniques or by making assumptions about the shape or centrality dependence of flow, non-flow, or flow fluctuations. An example of a more differential analysis is also shown in Fig. 22, where two particle correlations have
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been fit in ∆φ-∆η space.119 Terms identified with various non-flow sources have been included with the fit and the remaining cos(2∆φ) modulation is then identified as v2 {2D}2 . In the case that the sources of non-flow are correctly parametrized, δ = v2 {1D}2 −v2 {2D}2 , where v2 {1D}2 is hcos(2∆φ)i integrated over all azimuthal structure. Then v2 {2D}2 = hv2 i2 + σv22 . This procedure is discussed below. Even without attempting to disentangle flow fluctuations from non-flow correlations, the assumption that non-flow is a positive quantity (consistent with 2 v2 {1D}2 − v2 {2D}2 ) can be used with σtot to provide an upper limit on v2 fluctuations σv22 <
2 σtot . 2
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To facilitate a comparison between this limit and models of the initial eccentricity σ σε the upper limit on hvv22i is compared to hεi in the model. To form this ratio appropriately, the same assumptions should be made for σv2 and hv2 i i.e. zero non-flow. 2 σv22 σtot /2 v2 {2}2 − v2 {4}2 < = hv2 i2 (v2 {2}2 + v2 {4}2 )/2 v2 {2}2 + v2 {4}2
(9)
This upper limit is shown in Fig. 23 and compared to several models of σε /ε. The models include two Glauber Monte Carlo models;121 one using the coordinates of participating nucleons to calculate the eccentricity, the other using the coordinates of constituent quarks confined inside the nucleons. The constituent quark Monte
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Carlo Glauber Model (cqMCG)112 treats the nucleus as 3 × A constituent quarks grouped in clusters of three confined to the size of a hadron. This increases the number of participants by roughly a factor of three, reducing the fluctuations in eccentricity. The correlations between the constituent quarks required by confinement partially counteract this effect since those correlations act to broaden the eccentricity distribution. The net effect, however, is a narrowing of the distribution. Also shown is a Color Glass Condensate (CGC) based model which yields eccentricity values 30% larger than the Glauber models leading to a reduction of σε /ε. The Monte Carlo Glauber model based on the eccentricity of nucleons already exhausts 2 most of the width σtot = δ2 + 2σv22 . This shows that the statistical width of the eccentricity fluctuations in the Glauber model already accounts for almost all of the non-statistical width of the flow vector distribution thus leaving little room for other sources of fluctuations and correlations. This is particularly challenging since non-flow has been neglected in setting the upper limit and the only v2 fluctuations considered are those arising from eccentricity fluctuations. We have therefore neglected fluctuations that would arise during the expansion phase.119,120,122 One can write the total width including these terms: 2 σε 2 σtot = δ2 + 2 v2 + 2σv22 ,dyn , (10) ε where the middle term in the right-hand-side is the v2 fluctuations from eccentricity fluctuations and the final term is v2 fluctuations from the expansion phase. The
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middle term arises from the approximation that to first order σv2 /v2 = σε /ε. This approximation is prevalent in the literature. The last term can be related to the Knudsen number of the matter during the expansion.122 Measurements demonstrating the existence of non-flow or dynamic v2 fluctuations (σv2 ,dyn ) therefore would challenge the model. The CGC or cqMCG models provide a more likely description since σε /ε is smaller in those models than the upper limit on σv2 /hv2 i. The 2 upper limit on σv2 provided by σtot provides a valuable test, therefore, for models of the initial eccentricity and can help to reduce the uncertainty on ε; an essential component in extracting meaning from the value of v2 . 2.4.1. Two-dimensional correlations and v2 {2D} One way to study non-flow contributions to two-particle correlations is to measure the correlations as a function of ∆η and ∆φ.95 This allows different sources of two particle correlations to be studied where each source is identified by its characteristic dependence on ∆φ and ∆η. Additional information can be obtained by including information about the charge-sign dependence of the correlations. An example of such an analysis is shown in Fig. 24. The figure displays four panels. The top panels show the correlation density ρ − ρref ∆ρ = √ , √ ρref ρref
(11)
where ρ is the pair density and ρref is the product of the single particle densities. This normalization is chosen to search for deviations of the correlations in large systems from those in small systems. If Au+Au collision were simply a superposition √ of p + p collisions for example, ∆ρ/ ρref would be the same in p + p and Au+Au collisions. This figure was produced based on simulated data which was tuned to match real data in very peripheral Au+Au or p+p collisions (top left) and 20%–30% central Au+Au collisions (top right).119,123 The correlations in these two systems have been found to be very different. The p + p collisions exhibit structures characteristic of fragments from string breaking (a narrow ridge at ∆η = 0 independent of ∆φ) and fragments from semi-hard scattered partons or mini-jets. These mini-jets yield a two-dimensional Gaussian correlation at 0, 0, and a broad ridge at ∆φ = π. The away-side jet can sweep over a wide ∆η range since the partons can have a momentum within the proton or Au nucleus. For semi-central and central Au+Au collisions, the correlation landscape is drastically different with the most prominent feature being v2 giving rise to a clear cos(2∆φ) shape. If the shape of the various non-flow terms structures is well understood, the correlation landscape can be fit and hv22 i can be extracted, independent of the sources of non-flow. This procedure depends on having an accurate description of the shape of the non-flow sources. Some of these are easily identified based on their charge dependence or their characteristic shapes. Other sources may be less easily identifiable though, particularly if they become modified by the medium in Au+Au collisions.
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√ Fig. 24. The top panels show simulated data for the two-particle correlation measure ∆ρ/ ρref . The simulations are tuned to yield data similar to p + p collisions (left) and Au+Au collisions (right). The bottom left panel shows v2 estimated from the 2- and 4-particle cumulants. The √ shaded regions show the v2 derived by integrating over all structure in ∆ρ/ ρref (upper shaded √ region) or by performing a 2-Dimensional fit to the surface of ∆ρ/ ρref (lower shaded region). The bottom right panel shows the 2- and 4-particle cumulant data transformed to compare directly to √ ∆ρ[2]/ ρref : {2} = nv2 {2}2 and {4} = nv2 {4}2 .
The bottom panels of Fig. 24 show a proof-of-principle extraction of hv22 i based p 2 on the simulated data. The left panel shows hv2 i while the right panel shows the √ per-particle measure ∆ρ[2]/ ρref . Also shown are the two-particle and four-particle cumulant data. The upper hatched region in the bottom left panel shows hv22 i =
2π √ ∆ρ[2]/ ρref n
(12)
extracted without separating out non-flow; called v22 {1D}. n is the multiplicity of measured tracks. The lower hatched region shows the same without including
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the non-flow structures in the calculation; called v22 {2D}. Data are plotted versus 2Nbinary /Npart so that most central collisions are on the right. This procedure makes use of the two-particle correlations landscape to separate different contributions to the azimuthal structure. The fit procedure does require some assumptions be made in order to separate non-flow from v2 .123 These include that v2 and σv2 are independent of ∆η and that higher harmonics of vn and σvn do not contribute. Relaxing those assumptions may make it difficult to distinguish between non-flow and more complicated correlations related to the reaction plane. More information can be made use of however, by examining the charge, sign dependence and particletype dependence of the various correlation structures. This is therefore a promising method for disentangling v2 and non-flow. 2.5. Scaling observations Elliptic flow measurements represent an extensive data set. v2 has been measured for 0.1 < pT < 12 GeV/c, for −5 < η < 5, for mesons from the pion to the φ and J/ψ, for baryons from the proton to the Ω, and for transverse particle densities 3 < 1 dN S dy < 30. And yet, given the complexity of heavy-ion collisions and such a large data-set, the measurements exhibit many surprisingly simple features. These we can summarize in terms of simple scaling observations where a large amount of data is found to behave in a regular and simple way when plotted versus the appropriate variable. The observation of a particular scaling then motivates the question: why does the data only depend on that variable? These scaling observations, therefore, not only allow us to summarize large amounts of data in a simple form, but they also suggest simple physical explanations for the data with perhaps deeper implications. In this section I review several observed scaling laws. 2.5.1. Longitudinal scaling Figure 12 shows the centrality dependence of v2 (η).75 Although more detailed measurements at small η show that v2 is approximately independent of η for |η| < 1,11 the data extending to larger η exhibit a nearly triangular shape: having a maximum √ at η = 0 with a nearly linear decrease with |η|. A similar shape is seen for sN N = 124 200, 130, 62.4, and 19.6 GeV Au+Au collisions. √ Figure 25 shows v2 (|η| − ybeam ) for sN N from 19.6 to 200 GeV; one order of √ magnitude in sN N . The data are for the 40% most central collisions. Ideally the xaxis would display y − ybeam but data on identified particle v2 spanning such a large range of rapidity are not available. One finds that within errors, all data lie on a single curve. This suggests a smooth variation of the development of space-momentum correlations from forward rapidity to mid-rapidity. This scaling observation also implies that the value of v2 obtained at mid-rapidity is a smooth function of ybeam √ or equivalently of log( sN N ); consistent with the smooth trend seen in Fig. 3 for √ v2 above sN N of approximately 5-10 GeV. An energy scan at RHIC extending
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Fig. 25. v2 (η − ybeam ) for a variety of energies. The centrality interval used is 0% − −40%. Plotted in this format, data at all energies fall on one curve. This scaling is also seen in dN/dy(η − y beam ) and is referred to as limiting fragmentation. v2 is largest at middle rapidity (η = 0 or η − ybeam = −ybeam ) and vanishes at η = ybeam (η − ybeam = 0).
√ down to sN N = 5 GeV will make it possible to investigate this trend with better precision and with a single detector, eliminating many systematic uncertainties.125 This simple trend may be confirmed with more precision or perhaps deviations will point to a softest point in the equation-of-state.126,127 2.5.2. Kinetic energy and constituent quark number scaling At low pT , v2 is ordered by mass with heavier particles having a smaller v2 value at a given pT value.76,77 This ordering is indicative of particle emission from a boosted source with the boost larger in the in-plane direction than the out-of-plane direction. Indeed, blast-wave fits implementing this scenario agree very well with the data in this region.47 It is also found that in this same pT region, when v2 is plotted versus mT − m0 all data fall on a common line.128 mT − m0 is the particles transverse kinetic energy and sometimes labeled KET .129 Figure 26 shows v2 versus mT − m0 for particles ranging in mass from the pion with mass of 0.1396 GeV/c2 to the Ξ with mass of 1.321 GeV/c2 . The measurement is made for the 0–80% centrality interval in 200 GeV Au+Au collisions. Similar scaling has also been demonstrated for 62.4 GeV collisions.81 The data exhibit obvious trends. At low mT − m0 , v2 values for all particles rise linearly with no apparent differences between the particles with different masses. Near mT − m0 = 0.8 GeV/c2 , v2 (mT − m0 ) for mesons and baryons diverges. The meson v2 begins to saturate, obtaining a maximum value of 14–15% near mT − m0 = 2.5 GeV/c2 . The baryon v2 continues to rise, obtaining a maximum value of approximately 19–20%
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Fig. 26. v2 for a variety of particles plotted versus mT − m0 where m2T = p2T + m20 , and m0 is the rest mass of the particle. mT − m0 is also the transverse kinetic energy of the particle KET .
at mT − m0 = 3 to 3.5 GeV/c2 . The relative masses of the baryons and mesons do not seem to be relevant, rather the number of constituent quarks in the hadron determines the v2 values in this range. The mass dependence can be better checked using the φ-meson which has a mass slightly larger than that of the proton. The statistical significance of the φ v2 is limited but measurements seem to indicate that the φ lies closer to the mesons than to the baryons i.e. closer to the particles with a common number of constituent quarks than to particles with a common mass.99 The observation of the quark-number dependence of v2 at intermediate pT led to speculation that hadron formation through the coalescence of dressed quarks at the hadronization phase boundary could lead to an amplification of v2 with baryons getting amplified by a factor of 3 while mesons were amplified by a factor of 2.130–140 This picture was subsequently strengthened by the observation that a similar quark-number dependence arises in RCP 78,141 : the ratio of the single particle spectra in central collisions to that in peripheral collisions. At intermediate pT the RCP values for various particle species are also grouped by the number of constituent quarks, with baryons having a larger RCP . The larger RCP for baryons signifies that baryon production increases with collision centrality faster than meson production; an observation consistent with the speculation that hadrons from Au+Au collisions are formed by coalescence such that baryon production becomes easier as the density of the system increases. The more general and less model dependent statement is that the baryon versus meson dependence arises from high density and therefore most likely from multi-quark or gluon effects or sometimes
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called “higher twist” effects. The combination of large baryon v2 and large baryon RCP also immediately eliminates a class of explanations attempting to describe one or the other observation: e.g. originally it was speculated that the larger R CP for baryons might be related to a smaller jet-quenching for jets that fragment to baryons than for jets that fragment to mesons. This explanation would lead to a smaller baryon v2 and is therefore ruled out by the larger v2 for baryons. The same can be said for color transparency models142 which would account for the larger baryon RCP in this pT region but would predict a smaller baryon v2 . Color transparency may still be relevant to the particle type dependencies at pT > 5 where RCP for protons is slightly larger than RCP for pions143 and the v2 measurements are not yet precise enough to conclude whether the baryon v2 is also smaller than the meson v2 . This is a topic that needs to be studied further. In a coalescence picture, the final momentum of the observed hadron would depend on the momentum of the coalescing constituent quarks. The exact dependence is not known but a relatively good scaling of v2 for KS0 and Λ was found when v2 /n was plotted as a function of pT /n. Such a scaling implies that the momentum of the hadron is simply the sum of the momenta of the coalescing quarks. Figure 27 shows v2 /n versus pT /n for KS0 -mesons and Λ-baryons. The scaling appears to be good throughout the whole pT range but part of this perception is due to the decrease of v2 for both particles at small pT . When a ratio is taken between the v2 /n(pT /n) values, a clear deviation from scaling is seen in the lower pT region. A combination of the mT − m0 scaling in Fig. 26 and the v2 /n scaling in Fig. 27 will lead to a good scaling over the whole measured momentum range; since v2 (mT − m0 ) for all particles fall on a single line at low mT − m0 , dividing the x- and y-axis by n will
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Transverse Momentum pT /n (GeV/c) Fig. 27. v2 for KS0 and Λ scaled by the number of their constituent quarks (n) and plotted versus pT /n. The data appear to fall on a universal curve which has been taken as an indication of hadron formation via coalescence of quarks from a flowing medium.
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Fig. 28. A more detailed study of quark number scaling in Au+Au collisions. In the left panels (a, c, and e) v2 /n is shown versus (mT − m0 )/n for three centrality classes. Hydrodynamic models are also shown for comparison. Data are fit to a single curve. In the right panels (b, d and f) the ratio of the data and hydro model to the fit function are shown.
not destroy that scaling seen in Fig. 26. Plotting v2 /n versus (mT − m0 )/n should therefore provide a good scaling across a large kinematic range. Figure 28 shows v2 /n versus (mT − m0 )/n for 200 GeV Au+Au collisions in three different centrality intervals.80 Data for KS0 -mesons, Λ-baryons and Ξ-baryons are shown. The left panels show the data with a hydrodynamic calculation and a fitting function. The phenomenologically motivated function v2 /n =
a + bx + cx2 a −(x−d) − 2 1 + exp e
(13)
with x = (mT − m0 )/n, describes the data well for the three centralities. The function captures the rise then saturation and steady decline seen in the data. We
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Table 1. The fit parameters describing the curves in Fig. 28.
Centrality
a
40%–80% 10%–40% 0%–10%
20.0e-02 16.4e-02 8.96e-02
Fit parameters for Eq. (13) b c d 0.0 −4.53e-03 −4.08e-03
0.0 0.0 0.0
−1.19e-02 2.61e-02 6.52e-02
e 2.37e-01 2.40e-01 2.70e-01
ascribe no physical meaning to the function or the five fit parameters but simply use it as a convenient reference. The right panels show the ratio of the data and the hydro model to the fit function. For reference, the fit parameters are shown in Table 1. The data is in good agreement with the fit function for all centralities while this hydro model calculation does not agree well with the data in any centrality. There is a systematic deviation from the ideal n scaling at (mT − m0 )/n > 0.8 GeV/c2 with KS0 mesons having slightly larger v2 /n values than Λ baryons. This deviation from ideal scaling was predicted based on the inclusion of higher fock states in the hadrons or the inclusion of a finite width in the hadron wave function.144,145 Deviations can also arise in a hadronic phase when the hadronic cross sections are relevant. In the case that hadronic cross-sections are an important factor, higher statistics data for Ω baryons and φ mesons should deviate from their respective groups. We also note that hadronic cascade models also obtain approximate v2 /n scaling due to the use of the additive quark model for hadronic cross-sections.146 On the other hand, these models under-predict the integrated v2 by a factor of two. We also note that non-flow contributions can affect the scaling observed in this range and the particle-type dependence of non-flow sources is still being investigated. In Fig. 29 we investigate the breaking of ideal scaling in more detail with data integrated over a larger centrality interval. While this reduces the statistical uncertainty, it also introduces uncertainties due to the large centrality bin width. In particular, when particle yields have different centrality dependencies, the average eccentricity of events producing a particle can deviate from particle to particle. For example, the enhancement of baryons in central collisions will mean that the average baryon comes from a more central event than the average meson. Given the decrease of v2 with centrality, this can lead to a decrease of baryon v2 simply due to the wide centrality bin. Although there are caveats and systematic errors still to be quantified, we note that the baryons in Fig. 29 appear to lie systematically and significantly below the mesons. Self-similar curves are fit to mesons and baryons. The curves appear to describe the data. We note that the two self-similar curves shown in Fig. 29 can be nearly unified if we replace n with n+1. This demonstrates that the naive constituent quark scaling is violated to the extent that baryon v 2 is actually closer to 4/3 the meson v2 rather than 3/2. The connection of the baryon versus meson dependence and the number of constituent quark scaling appears to
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(mT -m0)/n [GeV/c ] Fig. 29. v2 /n versus (mT − m0 )/n for minimum bias Au+Au collisions. The quark number scaling appears to be violated when integrating over a wide centrality bin. Mesons and Baryons are fit with the same functional form but with different parameters. The scaling is violated to the extent that n + 1 would give a better agreement i.e. baryon v2 is closer to four-thirds the meson v2 rather than three-halves.
not be as directly connected to the number of constituent quarks as originally conceived. Whether this is indicative of higher fock states, the wave-function of the hadrons, an as yet un-accounted for experimental systematic error, or something else is yet to be determined. The systematic uncertainties based on the particle-type dependence of non-flow are still being investigated. 2.5.3. System-size scaling The system-size dependence of v2 can be studied by looking at the centrality dependence of v2 or by colliding smaller nuclei. Ideal hydro predictions, having a zero mean-free-path assumption, should be independent of the system-size. In this case, given the same eccentricity, the v2 should be independent of system size. One can try to account for the change in eccentricity by dividing v2 by eccentricity from a model but this introduces a large amount of uncertainty. Another approach is to study the shape of v2 (pT ) to see if that varies.147 The left panel of Fig. 30 shows v2 measured in Au+Au and Cu+Cu collisions for several centrality intervals.129 In the right panel, v2 (pT ) is scaled by 3.1 times the mean v2 for that data set. 3.1hv2 i was taken as a proxy for the eccentricity of the collision system, and this proxy is not inconsistent with models of eccentricity which are quite uncertain. What is best demonstrated by this scaling, is that although the magnitude of v2 changes significantly for the different centralities and systems, the shape of v2 (pT ) is very similar.
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Transverse Momentum pT (GeV/c) Fig. 30. Left panel: v2 versus pT in Au+Au and Cu+Cu collisions for four centrality intervals. Right panel: the same v2 scaled by 3.1 times the pT integrated v2 . The scaling demonstrates that the shape of v2 (pT ) is approximately independent of centrality and system size. A function is fit to the Au+Au data and shown as a solid orange line.
The invariance of v2 (pT ) with system-size can be taken as an indication that the viscosity of the expanding medium created in heavy-ion collisions can not be large when v2 is established; Large viscous effects should introduce a system-size dependence to v2 (pT ) with viscosity causing v2 to saturate at lower pT values in the smaller system.147 Hydrodynamic calculations including viscosity confirm this idea.148–153 To look more carefully for a system-size dependence in the shape of v2 we plot the ratio of the scaled data to a curve fit to the Au+Au data. The results are shown in Fig. 31. The Cu+Cu data systematically deviate from the Au+Au data. The pT dependence of the ratio indicates that the Cu+Cu data begins to saturate before the Au+Au data. This leads to a ratio that first rises then falls. This would happen the other way around if the Au+Au data saturated first. The uncertainties in the figure are large but the shapes are still significantly different. The system-size dependence of v2 (pT ) may be a valuable tool for estimating the viscosity of the matter created in heavy-ion collisions. Although the data on v2 includes many particle types, a wide kinematic range in pT and η, a variety of system-sizes and a wide range in center-of-mass energy, we’ve been able to identify several regular features of the data. These include a √ nearly linear rise of v2 at mid-rapidity with log( sN N ): √ v2 = 0.008 + 0.0084 log( sN N )
(14)
for 0%–20% central Au+Au or Pb+Pb collisions and a pT , mass and particle-type
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Transverse Momentum pT (GeV/c) Fig. 31. The v2 (pT )/3.1hv2 i data from the right panel of Fig. 30 scaled by a function fit to the Au+Au data. This figure illustrates that their seems to be a significant difference between the shape of v2 (pT ) in Au+Au and Cu+Cu collisions with the Cu+Cu collisions exhibiting a more abrupt turn-over – i.e. Cu+Cu data first rises faster with pT then falls faster at pT > 1 GeV/c.
dependence that can be parametrized by v2 /n =
a + bx + cx2 a −(x−d) − , 2 1 + exp e
(15)
where x = (mT −m0 )/n, while v2 (η) to good approximation decreases linearly from it’s maximum at mid rapidity to beam rapidity. This linear rise may be a trivial √ consequence of the log( snn ) dependence of mid-rapidity v2 or vice-versa. 3. Confronting the Hydrodynamic Paradigm with RHIC Data We have discussed the hydrodynamic model extensively in this review as a convenient reference for how well the matter produced in heavy-ion collision converts spatial deformation into momentum space anisotropy. Hydrodynamic models of heavy-ion collisions have many uncertainties. These include, uncertain initial conditions, uncertain thermalization times, and uncertain freeze-out conditions. A successful description of data using a hydrodynamic model offers the promise of not
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protons 0.5
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Fig. 32. The four panels show pT spectra and v2 (top and bottom) for pions and protons (left and right). Data from 200 GeV Au+Au collisions is compared to a variety of hydrodynamic models. Most models do not agree with pT spectra and v2 simultaneously.
only establishing the attainment of local equilibrium but also the promise of providing information on the Equation-of-State of the matter and its transport properties. The uncertainty in the models, however, are large and it has not yet been possible to extract this desired information with satisfactory certainty. In addition, the possibility that significant v2 arises from initial-state effects34,37 could call into question the applicability of hydrodynamics and the need for prolific final-state rescattering. Measurements of two particle correlations, which have often been interpreted as arising from mini-jets,96,119 need to be reconciled with the idea of a locally thermalized matter with extensive final-state rescattering. If the hydrodynamic models and data are irreconcilable, the paradigm will, of course, have to be abandoned. To check for consistency with hydrodynamic models,29,30,154–156 the PHENIX collaboration created a comprehensive comparison between heavy-ion data on p T spectra and v2 /ε.8 The inclusion of a comparison to HBT data was hampered by the lack of predictions from some of the models. The comparison to pT spectra and v2 is shown in Fig. 32. The left panels show pT spectra with pions in the top panel and protons in the bottom. The right panels show v2 for the same particles. The combination of data on v2 and spectra provide a stringent test for the models as
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some models can reproduce one quantity but only by adjusting parameters in such a way that the agreement with other observables is spoiled. The models shown in the figure differ in several ways. Models that include a phase transition and a QGP phase are shown with solid lines while models without a pure QGP phase are shown as dotted lines. Including this phase transition acts to reduce the value of v2 since the equation of state is soft during the transition. This means that the speed of sound drops (in these models to zero), so that conversion of coordinate space eccentricity to momentum space anisotropy is halted during the phase transition. In the case that the models, do approximately match the pion spectra and v2 , the most directly observable consequence of the lack of a phase transition is on the proton spectra and proton v2 . The proton spectra end up being too soft, and the splitting between proton and pion v2 is reduced with the proton v2 becoming larger. This is somewhat counter-intuitive but is a consequence of fixing the parameters to match central data. The models also differ in their treatments of the final hadronic stage. The calculations from Teaney et al. include a hybrid model that uses a hadronic cascade (RQMD) for the final hadronic evolution. Hirano and Kolb do not use such an afterburner but allow the particle abundances to stop changing at a temperature above the temperature at which they stop interacting; chemical freeze-out happens before kinetic freeze-out. Huovinen on the other hand, maintains chemical and kinetic equilibrium throughout the expansion. These different treatments have very important consequences for the particle-type dependence of the pT spectra and v2 . Huovinen’s treatment can reproduce the v2 for pions and protons, but only at the expense of under-predicting the number of protons; a direct consequence of maintaining chemical equilibrium until the final freeze-out at a relatively low temperature. The only model which compares well to all the data is Teaney’s model including a QGP phase, a phase transition, and a hadronic phase modeled with RQMD. Such a hybrid model adds significantly to the number of tunable parameters as compared for example to Huovinen’s model. On the other hand, the Teaney model shows that some particle types are less affected by the hadronic phase and therefore less sensitive to some of the uncertainty in freeze-out prescription. Figure 33 shows the Teaney calculation with Hydro only versus Hydro+RQMD. The particle species least affected by the inclusion of a hadronic afterburner, are the φ-meson and the Ω-baryon. This arises presumably from the small hadronic cross-section for these hadrons. This suggests high-statistics measurements for these particles are a viable way to avoid uncertainties in the effects of hadronic re-scattering. Besides the uncertainty in the freeze-out prescription, there is uncertainty on the eccentricity of the expanding fire-ball at the start of the conjectured hydrodynamic evolution. Figure 34 shows a hybrid hydro+cascade model compared to v 2 data.72 Two model curves are shown: one with a Color-Glass-Condensate (CGC) initial eccentricity,66 the other with a Monte-Carlo-Glauber (MCG) eccentricity. As discussed previously the CGC eccentricity is larger than the MCG eccentricity; this
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leads to an over-prediction for v2 . On the other hand, this hybrid model does not include viscous effects in the QGP phase so the difference between the hybrid+CGC prediction could be related to viscosity. In fact, since viscosity acts to reduce v 2 , the hybrid+MCG curve shows that there is no room for viscosity in this model. This violates the lower bound on viscosity derived based on quantum mechanical arguments158 and also later from string theory.159 Clearly, to estimate the viscosity allowed, or required by the data, the uncertainty on the initial conditions must be 2 reduced. As discussed previously, the measured quantity σtot = δ2 + 2σv22 provides a sensitive test of the models of the initial conditions and needs to be carefully compared to the hydrodynamic model predictions with various initial conditions. 3.1. Transport model fits An approach to circumvent the uncertainties in the hydrodynamic models has been outlined in Refs. 160 and 161 where v2 /ε is fit as a function of S1 dN dy . The fit function is used to infer how close the data come to a saturated value in the collisions with the highest density achieved. The fit function is constrained by how v2 /ε should
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approach the high density and the low density limits. One can construct different equations but in a transport code, the following was found to represent the approach to the zero mean-free-path limit well:162 v2 v sat 1 = 2 ε ε 1 + K/K0
(16)
where K is the Knudsen number and K0 is a constant of order one. Figure 35 shows the data and fit in the left panel and the inferred Knudsen number in the right panel. Based on this procedure it is found that RHIC v2 data are still some 20% below the saturation value anticipated within the fit function. This conclusion however, not only depends on the assumptions built into the transport model approach but also the centrality dependence of the eccentricity. The Color Glass Condensate model for example predicts a stronger centrality dependence for the eccentricity than the Monte Carlo Glauber model. As a consequence, this fit implies that if the initial conditions at RHIC are described by the CGC model, then the v2 data is closer to its saturation limit than if the MCG gives the correct description. This is counter intuitive and opposite to the conclusions reached based on real hydrodynamic calculations, which indicate that the larger CGC initial eccentricity allows more room for viscous effects in the QGP phase.148,150 The transport based fit circumvents the actual solving of hydrodynamics but the conclusions are dependent on the centrality dependence of the initial eccentricity which is strongly model dependent. The fit also includes the speed of sound as a free parameter. This
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(1/S)(dN/dy) (mb ) Fig. 35. Left panel: v2 in Au+Au and Cu+Cu collisions scaled by eccentricity calculated in a CGC framework and plotted versus (1/S)(dN/dy). The fit function and hydrodynamic limit are explained in the text. Right panel: The effective Knudsen extracted from the data and fit in the left panel.
effectively leads to an equation of state which has no phase transition but which is allowed to vary in the fit. A complimentary and perhaps better method for accessing the Knudsen number and the viscosity is to study the shape change of v2 (pT ) for different system-sizes which avoids the uncertainty in the eccentricity.147 This is a work currently in progress. 3.2. Viscous hydrodynamics The apparent success of ideal hydrodynamic models to describe the gross features of RHIC data has led to the inference of small viscosity and the claim of the discovery of the perfect liquid at RHIC. The perfect liquid announcement was listed as the top physics story of 2005 by the American Institute of Physics and was widely covered in the popular press. Much recent work has gone towards including viscous effects in hydrodynamic calculations so that the viscosity can be more accurately estimated.148–153 Figure 36 shows one such calculation. The top panel shows results when the hydrodynamic evolution starts from Glauber initial conditions while the bottom panel shows the case of CGC initial conditions. The results that come closest to the STAR data81 are given by η/s = 0.08 for the Glauber initial conditions and η/s = 0.16 for the CGC initial conditions. The STAR non-flow corrected data
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Fig. 36. Hydrodynamic calculations including viscous effects. The top panel shows v 2 (pT ) for the case that the initial conditions are described with a Monte Carlo Glauber model. The bottom panel is based on Color Glass Condensate initial conditions. The curves show results for different values of η/s, the ratio of shear viscosity to entropy.
are from Fig. 4 of Ref. 80. The 10%–40% central data from that reference was scaled to account for the difference between the 10%–40% centrality interval and the 0%–80% (minbias) centrality interval. The larger η/s inferred based on the CGC model arises from the larger initial eccentricity which leaves more room for viscous effects that tend to reduce the v2 . This contradicts the conclusions drawn from the transport model inspired fit, which allows the equation-of-state to change for the two different initial conditions. The pT dependence of the data is also better captured in the larger viscosity CGC scenario. The larger viscosity inferred from the
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CGC initial conditions gives a more pronounced turn over of v2 (pT ) which better describes the pT dependence of v2 . The comparison shown in Fig. 36 shows that hydrodynamic models including viscosity have a good chance of reproducing RHIC 1 2 where 4π is data as long as the shear viscosity to entropy ratio η/s is less than 4π the conjectured lower limit. 3.3. Fluctuating initial conditions The comparison of v2 and other RHIC data to hydrodynamic models seems to indicate that when viscous corrections are included, a successful description of the data may be possible. There is uncertainty in this comparison, however, related to uncertainties in the initial conditions and in the freeze-out prescription. The uncertainty in the initial conditions can be addressed experimentally with measurements of v2 fluctuations which in turn require an understanding of non-flow correlations; The experimentally accessible information appears to reduce to v 2 {2}2 and v2 {2}2 −v2 {4}4 = δ2 +2σv22 . An alternative approach may be for hydrodynamic models to predict v2 {2} and v2 {4} by including correlations and fluctuations in the models. Progress has been made in this direction. Early work relating to the effect of fluctuations in the initial conditions on hydrodynamic calculations was carried out using the NeXSPheRIO hydrodynamic model.67,163 The initial eccentricity fluctuations were indeed found to lead to v2 fluctuations as shown in Fig. 37. Later it was suggested that correlations in the initial conditions could lead to vn fluctuations of even and odd orders of n that would manifest themselves as non-sinusoidal, apparently non-trivial, two-particle correlations as seen in the RHIC data. 69,164 Subsequent work following through on this idea shows that hydrodynamic models with fluctuating initial conditions do lead to two-particle correlations with structure beyond a simple cos(2∆φ) shape.165 The correlation structure arising from the fluctuations in the initial conditions is shown in the right panel of Fig. 37. The model exhibits many of the features seen in the data including a jet-like peak, a near-side ridge, and an away-side ridge shifted away from ∆φ = π. All this structure arises without the explicit inclusion of jets in the model. The apparently exotic correlations do not appear in the model when a smooth initial condition is used. This calculation illustrates the importance of accounting for fluctuations in the initial conditions when interpreting the correlation landscape. It also demonstrates that complex interactions between jets and the medium, including mach-cones, are not needed to explain the correlations data nor is the concept of mini-jets necessarily required. In light of the NeXSPheRIO calculations, the highly structured correlation landscape at RHIC should not necessarily be taken as an invalidation of the hydrodynamic models. The correlations may simply reflect the need to abandon certain approximations, including the approximation of infinitely smooth initial conditions. Besides comparing to two-particle correlation data, these models can be used to calculate v2 {2} and v2 {4} to directly compare to data. It will be interesting to see how
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the correlation landscape in this model depends on the parameters of the model, in particular, the thermalization time and the freeze-out time. The connection of vn fluctuations (related to two-particle correlations) to the lifetime of the system was first pointed out by Mishra et al.166 In that reference the authors also introduce p the anaology between hvn2 i fluctuations and the power spectrum of the Cosmic Microwave Background Radiation.
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3.4. Addressing uncertainties Uncertainties in the freeze-out prescription and the effects of the hadronic phase can be experimentally addressed through precise measurements of φ-mesons and Ω-baryons. Models indicate that due to their small hadronic cross-sections, these hadrons are minimally influenced by the hadronic phase and reflect well the QGP phase. In addition, heavy flavor hadrons may help determine or provide a crosscheck for the transport properties of the QGP. Another approach to extracting the viscosity is by studying the shape of v2 (pT ) versus system size. This approach does not rely on a model for the initial eccentricity. Uncertainties in the eccentricity and the initial conditions can be reduced through measurements of v2 fluctuations and two-particle correlations. These studies are ongoing. One can also measure v n fluctuations for arbitrary n value. These are of course related to the two-particle correlation landscape which has already been extensively studied at RHIC. It will be of great interest to see how the correlation landscape predicted in hydrodynamic models with fluctuating initial conditions changes depending on the model parameters. The correlations data may help constrain quanties like the lifetime of the system. The studies listed above, along with a beam-energy scan at RHIC and the first data from LHC, will allow for more progress in understanding the matter created in heavy-ion collisions and its subsequent evolution.
4. Summary In this review, v2 measurements were presented as a method for studying spacemomentum correlations in heavy-ion collisions. The measurements of v2 indicate the eccentricity in the initial overlap region is transferred efficiently to momentumspace. At top RHIC energy, the conversion is near that expectated from zero meanfree-path hydrodynamic predictions. The comparisons of data to hydrodynamics, however, depends on model calculations of the initial eccentricity. Several models for the initial eccentricity have been discussed. The mass, and pT dependence of v2 at pT < 1 GeV/c is found to be consistent with emission from a boosted source. Above that, the particle type dependence of v2 exhibits a dependence on the number of constituent quarks in the hadron, with baryons obtaining v2 values larger than mesons. The relationship between two-particle correlations, v2 , and v2 fluctuations has also been discussed. Calculations showing that some of the structures in two-particle correlations can be ascribed to fluctuations in the initial conditions, have been reviewed. Measurements of correlations and v2 fluctuations can therefore be used to constrain models for the initial conditions. These constraints, along with improved measurements of the shape of v2 (pT ) as a function of system-size, improved measurements of φ and Ω v2 , measurements of v2 for heavy-flavor hadrons, measurements at LHC energies, and a beam-energy scan at RHIC will further improve our understanding of the properties of the matter created in heavy-ion collisions.
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PREDICTIONS FOR THE HEAVY-ION PROGRAMME AT THE LARGE HADRON COLLIDER
´ NESTOR ARMESTO Departamento de F´ısica de Part´ıculas and Instituto Galego de F´ısica de Altas Enerx´ıas, Universidade de Santiago de Compostela, 15706 Santiago de Compostela, Spain [email protected]
I review the main predictions for the heavy-ion programme at the Large Hadron Collider (LHC) at CERN, as available in early April 2009. I begin by remembering the standard claims made in view of the experimental data measured at the Super Proton Synchrotron (SPS) at CERN and at the Relativistic Heavy Ion Collider (RHIC) at the BNL. These claims will be used for later discussion of the new opportunities at the LHC. Next I review the generic, qualitative expectations for the LHC. Then I turn to quantitative predictions: First I analyze observables which characterize directly the medium produced in the collisions — bulk observables or soft probes: multiplicities, collective flow, hadrochemistry at low transverse momentum, correlations and fluctuations. Second, I move to calibrated probes of the medium i.e. typically those whose expectation in the absence of any medium can be described in Quantum Chromodynamics (QCD) using perturbative techniques (pQCD), usually called hard probes. I discuss particle production at large transverse momentum and jets, heavy-quark and quarkonium production, and photons and dileptons. Finally, after a brief review of pA collisions, I end with a summary and a discussion about the potentiality of the measurements at the LHC — particularly those made during the first run — to further substantiate or, on the contrary, disproof the picture of the medium that has arisen from the confrontation between the SPS and RHIC data, and theoretical models.
1. Introduction The experimental programme for the study of ultra-relativistic heavy-ion collisions started in 1986 at the Super Proton Synchrotron (SPS) at CERN. It accelerated protons and ions (up to Pb), at plab ≤ 158 GeV per nucleon in the case of Pb.a The next step was the Relativistic Heavy Ion Collider (RHIC) at the BNL, which √ began in 2000, accelerating protons and ions up to AuAu collisions at sN N = 200 GeV. Both experimental programmes have allowed for the extraction of important conclusions about the properties of the strongly interacting matter produced in such collisions.1–5 The next step in the near future, apart from RHIC upgrades6 and the energy and collision species scan at the SPS,7 is the heavy-ion programme at the Large Hadron Collider (LHC) at CERN.8 It will accelerate ions as heavy as Pb (A = 208, a Natural
units ~ = c = 1, and kB = 1 will be used throughout this manuscript. 375
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Z = 82), with energies √
sN N = 2
Z × 7 TeV ≃ 5.5 TeV for PbPb, A
(1)
with a total center-of-mass energy of 1.15 PeV. The nominal peak luminosity will be L0 = 1027 cm−2 s−1 , with hLi/L0 = 0.5 and a estimated running time 106 s/year.b Collisions of other ions and asymmetric collisions like pPb9 are possible, the latter with a shift in the center-of-mass rapidity with respect to the rapidity in the laboratory given by δy =
1 Z1 A2 ln 2 Z2 A1
(2)
A2 1 for A Z1 A Z2 B collisions. While the first proton beams circulated along the LHC ring in September 2008 and the first pp collisions are expected for autumn 2009, the first PbPb collisions are only expected for the second half of 2010. Three out of the four large experiments at the LHC: ALICE, ATLAS and CMS, will measure PbPb collisions.10–13 While ALICE is a dedicated experiment to nucleus-nucleus collisions, both ATLAS and CMS will offer detector capabilities complementary to each other and to ALICE. They will provide a wide range of measurements covering all the main relevant observables in heavy-ion collisions. Measurements in pp and nucleus-nucleus collisions at roughly the same unexplored top energy will be, for the first time, performed using the same accelerator and detectors. The increase in center-of-mass energy of almost a factor 30 with respect to RHIC, together with the complementary detector capabilities, will offer new measurements with respect to those presently available. As evident examples:
• The yield of particles with large mass or transverse momentum — hard probes9,14–16 — will be much more abundant, and some of them will be measured for the first time (with high statistics) in heavy-ion collisions, like Υ or Z 0 + jet production (see Fig. 1, taken from Refs. 17 and 18). • Calorimeter capabilities of ATLAS and CMS12,13 will allow for measuring jets both at central and non-central rapidities.c ALICE will also be equipped with an electromagnetic calorimeter.18 • The kinematical coverage of the parton densities inside proton and nuclei will greatly exceed that available at the SPS and RHIC (see Fig. 2, taken from Refs. 9 and 21). The aim of this review is to present a comprehensive compilation of the existing predictions for the heavy-ion programme at the LHC, not to discuss the current b These
numbers are to be compared with those for pp collisions: L0 = 1034 cm−2 s−1 , with hLi/L0 closer to 1 and an estimated running time 8 · 106 s/year. c Jets in heavy-ion collisions have been measured for the first time by the STAR Collaboration at RHIC.19,20
σ (mb)
Predictions for the Heavy-Ion Programme at the Large Hadron Collider
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cc
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ϒ
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pmax = s/2 exp(- η) T
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LHC, pA HQ
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tion
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tion,
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ZDC, LHCf TOTEM RPs ALFA RPs FP420
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4
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ALICE
CASTOR, T2
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LHC from HQ
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yQ
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ATLAS,CMS
3
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dp /dη (GeV/c)
hard = A2 σ hard ) in PbPb minimum Fig. 1. Left: Cross sections for various hard processes (σPbPb pp √ bias collisions in the range sNN = 0.01 ÷ 14 TeV. Figure taken from Ref. 17. Right: Expected annual yields in the ALICE EMCal acceptance for various hard processes for minimum bias PbPb collisions at 5.5 TeV. Figure taken from Ref. 18.
-1
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Fig. 2. Left: Approximate pT − η coverage of current and proposed spectrometers and calorimeters at the LHC. Figure taken from 21 . Right: Resolution power (Q2 ) × momentum fraction (x) coverage of the SPS, RHIC and LHC experiments for parton densities (grey bands and solid lines), compared with the regions covered by previous lepton-nucleus and proton-nucleus experiments (colored markers). Figure taken from Ref. 9.
interpretation of available experimental data. Nevertheless, I will briefly indicate the main, ‘standard’ claims extracted from the experimental programmes at the SPS and RHIC. Let me stress that none of these claims are devoid of alternative explanations, and that their presentation will doubtlessly contain some personal
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bias. I will use them only to motivate the discussion of the new opportunities at the LHC and the discriminating power of the forthcoming measurements there. The standard claims at RHIC are the following (the reader may find extensive discussions and references to the relevant experimental data in Refs. 1–5, 22–25): • Multiplicities at RHIC are much lower than pre-RHIC expectations.26,27 The standard interpretation is that particle production in the collisions shows a large degree of coherence due to initial state effects. • The elliptic flow measured in the collisions can be well reproduced by calculations within ideal hydrodynamics with a very early thermalization (or isotropization) time and small room for shear viscosity. This is currently interpreted in terms of the creation of some form of matter which (nearly) equilibrates very early and behaves like a quasi-ideal fluid. • The yield of high transverse momentum particles of different species measured at RHIC is strongly depleted in comparison with the expectations of an incoherent superposition of nucleon-nucleon collisions (as suggested by the collinear factorization theorems and confirmed by experimental data on weakly interacting perturbative probes). This fact, named jet quenching, together with the absence of such depletion in dAu collisions, is understood as the creation of a partonic medium, very opaque to energetic partons traversing it. On the basis of these observations, it has been claimed that partonic matter, with an energy density larger than required by lattice QCD (see e.g. Refs. 28–30) for the phase transition from hadronic matter to the Quark-Gluon Plasma (QGP) to occur, has been formed. Such matter is extremely opaque to fast color charges traversing it, and its collective expansion closely resembles that of an ideal fluid. These two latter facts suggest that the produced matter is strongly coupled, which is in opposition to the naive picture of the QGP as an ideal parton gas and is not contradicted by lattice data which show some deviation from the Stefan-Boltzmann law and a finite value of the conformal anomaly up to temperatures larger than several times the deconfinement temperature. Many questions remain open both in the experiment (e.g. suppression of heavyflavor production or unbiased jet measurements, in nucleus-nucleus collisions) and on the theory side (can the observed phenomena be explained within pQCD or do they require strong coupling; what is the correct implementation and actual role of bulk and shear viscosity in hydrodynamical calculations; how can such an early isotropization be achieved; can the initial state — the nuclear wave function — be described by perturbative methods; how can we compute particle production in such a dense environment; etc.). In this review of predictions for the heavy-ion programme at the LHC.d I will classify them into different groups according to the following scheme: Those d Similar
efforts done for RHIC can be found in Refs. 26 and 27.
Predictions for the Heavy-Ion Programme at the Large Hadron Collider
379
observables which characterize the produced medium itself, which I will call bulk observables (or soft probes, as they refer to particles with momentum scales of the order of the typical momentum scale of the medium — the temperature if thermalization were achieved); and those whose expectation in the absence of any medium can be calculated by perturbative methods in QCD (pQCD) (thus characterized by a momentum scale much larger than both ΛQCD and the ‘temperature’ of the medium), commonly referred to as hard probes.9,14–16 Not being the subject of this review, I will provide few references to introduce the different subjects — I refer the reader to Refs. 31–33. I will start this review by some qualitative expectations for the LHC, based on simple arguments (in this respect see also Ref. 34). Through such discussion I aim to show how a single observable — charged multiplicity at mid-rapidity — strongly influences most other predictions. Then I will turn to detailed predictions on bulk observables. I will review those on multiplicities, collective flow, hadrochemistry at low transverse momentum, correlations and fluctuations. Next I will discuss hard and electromagnetic probes: particle production at large transverse momentum and jets, heavy quarks and quarkonia, and photons and dileptons.e Then I will review briefly pA collisions.9 I will conclude with a summary and a discussion about the potentiality of the measurements at the LHC — particularly those made during the first run — to further substantiate or, on the contrary, disproof the picture of the medium that has arisen from the SPS and RHIC. Most of the material that I will review is based on what was presented at the CERN Theory Institute on Heavy Ion Collisions at the LHC — Last Call for Predictions, held at CERN from May 14th to June 8th 2007, co-organized by Nicolas Borghini, Sangyong Jeon, Urs Achim Wiedemann and myself.35,36 I apologize in advance to those whose contributions I may unwillingly skip. I also apologize for not including any prediction for ultra-peripheral collisions (UPC) — see recent excellent reviews in Refs. 37 and 38. 2. Qualitative Expectations In principle, the reliability of the predictions for a given observable made within the framework of a given model is as good as the understanding of the existing experimental situation on that observable and related ones — provided the model contains the physical ingredients relevant for the extrapolation. It turns out that predictions for most observables, both for soft and hard probes, demand some parameter fixing which, in the most favorable case, can be related to a single measurable quantity. Such a quantity is usually the charged multiplicity at mid-rapidity or pseudorapidity which, in a more or less model-dependent way, can be related with energy densities, temperatures, etc. of the medium at some given time. e Concerning
photons, their production at low momentum cannot not be described within pQCD but they have customarily become part of the general item of hard and electromagnetic probes.
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In this Section, I will review some qualitative or semi-quantitative expectations for central PbPb collisions at the LHC. The aim here is not to provide realistic or definite numbers (actually I will be most conservative in the estimates, so very probably the quantities for the LHC are underestimated in comparison to those at RHIC), but more or less stringent bounds, and to show explicitly how different predictions become affected or determined by a single observable, namely charged multiplicity at mid-rapidity. Let me note that a collection of data-driven predictions can be found in Ref. 34. While this latter collection, in its aim to being as modelindependent as possible, is complementary to the one to be presented in the next Sections, it overlaps in spirit what will be presented here. In Table 1 I show the results within the Monte Carlo code39 for the number of participants, of collisions and the charged multiplicity at mid-rapidity and pseu√ dorapidity in central PbPb collisions at sN N = 5.5 TeV. While these quantities are obtained in the framework of a given simulator, they will serve for the purpose of illustration in this Section. They will also be employed to better allow a comparison among different predictions for multiplicities at mid-(pseudo)rapidity in Subsection 3.1.
Table 1. Results in the Monte Carlo code in Ref. 39 for the mean impact parameter, number of participants and binary nucleon-nucleon collisions, and charged multiplicity at mid-(pseudo)rapidity, for different centrality classes defined by the number of √ participants, in central PbPb collisions at sNN = 5.5 TeV.
%
hbi (fm)
hNpart i
hNcoll i
dNch /dy|y=0
dNch /dη|η=0
0÷3
1.9
390
1584
3149
2633
0÷5
2.4
375
1490
2956
2472
0÷6
2.7
367
1447
2872
2402
0 ÷ 7.5
3.0
357
1390
2759
2306
0 ÷ 8.5
3.1
350
1354
2686
2245
0÷9
3.2
347
1336
2649
2214
0 ÷ 10
3.4
340
1303
2583
2159
For the purpose of fixing one reference centrality class, I will define it by a number of participants Npart = 350. In the following and unless otherwise stated, when referring to RHIC and the LHC I will be making reference to AuAu collisions at top RHIC energy, and PbPb collisions at the LHC, for a central centrality class defined by Npart = 350. Let me start with multiplicities, as they are a key observable which will determine many other predictions. As stated previously, expectations for other observables from collective flow to jet quenching, depend on the scaling of certain quantities e.g. initial energy density, which are related in some more or less direct way with the final multiplicity measured in the event. Thus, many predictions
Predictions for the Heavy-Ion Programme at the Large Hadron Collider
381
are provided for some specific values of parameters which may be linked with a multiplicity. Predictions for multiplicities can be discussed in the following way: A lower bound comes from the wounded nucleon model40 in which the multiplicity in nuclear collisions is expected to be proportional to the number of participant nucleons. This proportionality is also the limiting value expected by models which consider extremely strong shadowing effects. On the other hand, an upper limit can be set by the proportionality to the number of binary nucleon-nucleon collisions Ncoll , as expected both in models of particle production which suppose a dominance of hard, perturbative processes (using the collinear factorization theorem,41,42 inclusive particle production is proportional to the product of the fluxes of partons in projectile and target which in the totally incoherent limit is proportional to the number of nucleon-nucleon collisions) and in soft models of particle production in absence of shadowing corrections (see e.g. Ref. 43) through the cutting rules.44 On the basis of these considerations, the multiplicity can then be written in the following way (see also the discussions in Ref. 27): NN AA dNch 1−x dNch N (3) = + xN part coll , 0 < x < 1, dη η=0 dη η=0 2
with the superscript N N referring to nucleon-nucleon collisions — an average of pp, pn and nn.f Shadowing effects and energy-momentum constraints43 tend to decrease √ x. As an example, values extracted from RHIC data at sN N = 19.6 and 200 GeV45 are x ≃ 0.13. For nucleon-nucleon collisions, I will use the proton-(anti)proton data shown in Fig. 3. The three lines correspond to the parametrization of Sp¯ pS and Tevatron data by CDF46 NN dNch (CDF) = 2.5 − 0.25 ln sN N + 0.023 ln2 sN N , (4) dη η=0 to the parametrization in Ref. 47 NN dNch (ASW) = 0.47 (sN N )0.144 (Npart )0.089 = 0.50 (sN N )0.144 dη η=0
(5)
and to the PHOBOS parametrization in the contribution by Busza in Ref. 35, NN dNch (PHOBOS) = −0.5 + 0.39 ln sN N (6) dη η=0
(note that this parametrization was obtained from fits to nucleus-nucleus data and thus it was not intended to describe nucleon-nucleon data). I will also assume, as f At
large energies and at central rapidities, particle production should be determined by partons √ with small momentum fraction (which can be estimated using 2 → 1 kinematics as x ∼ mT / sNN , q with mT = p2T + m2 the transverse mass of the produced particle). At such small momentum fractions, isospin symmetry is expected to hold.
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suggested by RHIC data,45 that the energy and centrality dependences of charged particle yields at mid-rapidity decouple. Considering all this, I show in Table 2 some naive predictions for the LHC.g The predictions from the wounded nucleon model (x = 0) lie in the range 900 ÷ 1100, while those from a scaling with the number of collisions lie in the range 6800 ÷ 8400. The latter agree with the expectations in 1995 as shown in the ALICE Technical Proposal.49 The former roughly coincide with the expectations (1100,34 ) from limiting fragmentation (extended longitudinal scaling) and a self-similar trapezoidal shape of the η-distribution between RHIC and LHC energies. Let us note that, as discussed in Ref. 34, charged multiplicities larger than ∼ 1650 will be difficult to reconcile with limiting fragmentation.
dNNN ch /dη| 10
η=0
9
triangle: PYTHIA6.4 (pp)
8
square: PSM (pp)
7
solid: CDF (pp)
6
dashed: ASW (NN)
5
dotted: PHOBOS (NN)
4 3 2 1
102
3
10
4
Ecm (GeV) 10
Fig. 3. Charged multiplicity at mid-pseudorapidity in nucleon-nucleon collisions versus center-ofmass energy, from different parametrizations (CDF for p¯ p collisions 46 , ASW 47 and PHOBOS 35 for nucleon-nucleon collisions) and Monte Carlo simulators (PSM1.0 39 , and PYTHIA6.4 48 as shown in 12 , for pp collisions; these two points are included just for the purpose of illustration as they depend on the set of parameters used for the simulation).
Now one can try to estimate a lower bound for the energy density in this reference centrality class defined by Npart = 350. For this and for the forthcoming discussions in this Section, I will consider three possibilities for multiplicities: • Case I, the smallest one in Table 2, 900; • Case II, the maximum multiplicity allowed by limiting fragmentation,34 1650; • And Case III, a value of 2600 which is representative of the highest recent predictions for the LHC (see Subsection 3.1). g For the same centrality class defined by N part = 350, the corresponding charged multiplicity at η = 0 at top RHIC energy from the PHOBOS parametrization35 is 635.
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Table 2. Charged multiplicity at central pseudo-rapidity in PbPb collisions at LHC energy for Npart = 350 (Ncoll = 1354) from Eq. (3), for three different predictions of the corresponding multiplicity in pp collisions, see Fig. 3.
pp extrapolation
pp dNch /dη|η=0
ASW ASW
x
P bP b dNch /dη|η=0
5.97
0
1050
5.97
0.13
1950
ASW CDF
5.97 5.02
1 0
8100 900
CDF
5.02
0.13
1650
CDF
5.02
1
6800
PHOBOS
6.22
0
1100
PHOBOS
6.22
0.13
2050
PHOBOS
6.22
1
8400
I use the Bjorken estimate50 and the arguments about the average formation (proper) time for particle production, hτf orm i, in Ref. 2: measured 1 measured dN hǫi(hτf orm i) ≥ hmT i hτf orm iA dη η=0 measured hmmeasured i2 3 dNch T ≈ . (7) 2 πRA 2 dη η=0
2 In this equation hτf orm i ≈ hmmeasured i−1 , A = πRA is an upper bound for the T overlapping area for central collisions with RA = 1.12 A1/3 fm the nuclear radius, mT is the transverse mass and the super-index measured indicate that these are the final quantities measured in the detectors. For top RHIC energy, using hmmeasured i = 0.57 GeV as given by PHENIX2 (this quantity is weakly depenT measured dent on centrality), and taking dNch /dη|η=0 = 635 as given by the PHOBOS parametrization in Ref. 35, I get hǫi(hτf orm i = 0.35 fm) ≥ 12 GeV/fm3 . For the LHC, one has to estimate the increase in hmmeasured i with collision energy. For T √ that, I use the parametrization for hpT i ( s) by UA151 and adjust the hadron mass to get the value given by PHENIX i.e. √ √ √ s/GeV [GeV], hpT i s = 0.4 − 0.03 ln s/GeV + 0.0053 ln2 (8)
and m = 0.42 GeV. Then I get, for Case I, hǫi(hτf orm i = 0.29 fm) ≥ 22 GeV/fm3 . Therefore, the most conservative estimates for the LHC indicate a multiplicity increase of a factor 900/635 ≃ 1.4 and an increase of a factor ∼ 2 in energy density at formation time, with respect to top RHIC energy (or hǫi(hτf orm i = 0.29 fm) ≥ 42 and 66 GeV/fm3 for Cases II and III respectively). Now, and for the purpose of illustrating some qualitative behaviors, I turn to the eventual equilibration and dynamical evolution of the created system. For that
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I will use generic arguments based on the Bjorken ideal hydrodynamical scenario in one spatial dimension,50 see Refs. 33 and 52 for reviews of the hydrodynamical description of heavy-ion collisions. First, one needs the energy density at the time when hydrodynamical evolution is initialized, i.e. a thermalization or isotropization time. The estimates at RHIC lie in the range 0.17÷1 fm2–5,33,53 in ideal hydro (0.17 fm is the crossing time of two Au nuclei at RHIC), and similar values for studies RHIC including viscosity.54–56h I will take an intermediate time τtherm ≃ 0.6 fm as a reference value. To extrapolate to the LHC, it looks plausible that a system with larger density thermalizes faster. Using the ideas58,59 in the Color Glass Condensate (CGC, see the review in Ref. 33),i I will assume that the thermalization time scales like the inverse square root of the multiplicity at η = 0. Therefore one expects LHC hτtherm i ≃ 0.85, 0.62, 0.49 RHIC hτtherm i
(9)
for Cases I, II, III respectively. So the thermalization time at the LHC is τtherm . 0.5 fm. Assuming free streaming (ǫ ∝ 1/τ in the one-dimensional case) from formation time to thermalization, the corresponding lower bound for the energy density is 3
LHC hǫi(hτtherm i ≃ 0.5 fm) ≥ 12 GeV/fm ,
(10)
again factor ∼ 2 larger than the one obtained for RHIC. If one assumes that the thermalization time decreases with increasing particle density, then larger multiplicities at the LHC will favor smaller thermalization times and thus larger energy densities at thermalization. For example, in the model used for illustration, measured one-dimensional free streaming plus CGC, hǫi(hτtherm i ∝ (dNch /dη|η=0 )3/2 (modulo logarithmic corrections). Now I will consider the evolution of the system, in order to illustrate the typical scales for the different phases of the system. To do so, I assume an ultra-relativistic ideal gas of 3 light quarks and gluons in the deconfined phase, and of 8 pseudo-scalar mesons in the confined phase. Using Bjorken estimate (7) and the Stefan-Boltzmann law, the values of the relevant quantities at hτtherm i are given in Table 3. Then I consider the evolution of the system using the Bjorken ideal hydrodynamical scenario in one spatial dimension50 (see Ref. 60 for recent developments in 1+1 ideal hydrodynamics) for both the confined and deconfined phases, but with a free parameter α to mimic a larger dilution rate due to transverse expansion: h On
T T0
3
=
τ α 0
τ
,
τ 4α/3 ǫ 0 = , ..., ǫ0 τ
(11)
viscous hydrodynamics, see the recent review in Ref. 57. the CGC, the multiplicity is proportional to the saturation scale squared Q2s and its energy and centrality dependences roughly factorize, while the thermalization time is expected to be inversely proportional to Qs . i In
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Table 3. Values of the energy density, the temperature and the entropy density at hτtherm i for RHIC and the LHC. AA dNch /dη|η=0
hτtherm i (fm)
hǫi (GeV/fm3 )
Ti (GeV)
s0 (fm−3 ) 38
RHIC
635
0.6
6.8
0.241
LHC
900 (I)
0.51
12.6
0.281
60
LHC
1650 (II)
0.37
32.7
0.356
123
LHC
2600 (III)
0.30
64.4
0.422
204
with α = 1 corresponding to a pure longitudinal expansion. For a first-order phase transition, and assuming a deconfinement temperature of 170 MeV and a freeze-out temperature of 140 MeV, the evolution of the temperature is shown in Fig. 4. While the numbers shown in both the Figure and in Table 3 are most rough estimations, the plot illustrates some features common to more involved calculations: at the LHC the deconfined phase will last longer than at RHIC. The hadronic phase is not comparatively shorter than at RHIC (in this very schematic calculation using power-law evolutions of thermodynamical quantities), but its impact on some final observables (e.g. on photon or dilepton emission) could be expected to be smaller than at RHIC, due to the fact that the hadronic phase is restricted to the same range of temperatures but the partonic phase reaches higher T at the LHC than at RHIC. On the other hand, it clearly shows that the larger the multiplicities, the longer-lived the deconfined phase will be. Now I will focus on the dependence on multiplicity of the elliptic flow v2 integrated over transverse momenta.j According to general arguments, see e.g. Ref. 61, in the low-density limit the distortion of the azimuthal spectra with respect to the reaction plane XZ (and thus the elliptic flow v2 ) is proportional to the space anisotropy ǫx =
hy 2 − x2 i hy 2 + x2 i
(12)
and to the density of scattering centers (or particle density) in the transverse plane XY , 1 dNch v2 ∝ , (13) ǫx Sover dy y=0
with Sover the overlap area for a given centrality class and the average in (12) is done over the transverse energy density profile and, eventually, over the number of events. This relation is fulfilled by experimental data from lowest SPS to highest RHIC energies, see Ref. 62, and is illustrated in Fig. 5 left. It allows for a semi-quantitative relation between the multiplicity and the elliptic flow: I will assume that for AuAu or j The
discussion of the behavior of v2 (pT ) requires a parallel discussion of the hadronization process which is far more involved.
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Fig. 4. Temperature versus proper time in the Bjorken model for the four scenarios in Table 3 and for two values of α in Eq. (11): α = 1 and 1.2.
PbPb collisions at a given centrality class the spatial anisotropy, mainly determined by the geometry of the collision, and the overlap area are approximately the same and do not vary substantially with energy. Taking the slope of the experimental trend ∼ 0.005 and for a point lying at (22,0.16),k increases in multiplicity by factors 1.5, 2.5, 4l translates into increases in v2 /ǫx of ≃ 35, 100, 205% respectively. On the other hand, ideal hydrodynamics calculations64,65 indicate a saturation or limiting value of v2 /ǫ versus (1/Sover )dNch /dy|y=0 . The detailed value depends on the equation of state, on the details of initialization (see e.g. Ref. 66 for a study of the influence of different initial conditions on the spatial anisotropy) and hadronization prescription, and on the treatment of the confined phase. The inclusion of viscous effects further reduces such limiting value.54 Besides, as illustrated in Fig. 5 right, for a fixed initial spatial anisotropy, higher initial energy densities or temperatures imply larger density gradients which increase the final momentum
k These
values are roughly those of the experimental data (v2 = 0.051, ǫx = 0.319)63 for a 20÷30% centrality class, which corresponds53 to an impact parameter ∼ 7.5 fm and Npart ∼ 160 both for RHIC and the LHC. l These numbers are illustrative of the predictions for charged multiplicities at mid-pseudorapidity for Npart = 350 at the LHC, 900, 1650 and 2600 — Cases I, II and III respectively — compared with 635 at RHIC, and are applicable to other centralities provided the factorization between the centrality and energy dependences holds, see previous discussions.
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Fig. 5. Left: schematic plot showing the experimental trend (black) and the hydrodynamical limit (red line) of v2 /ǫx versus (1/Sover )dNch /dy|y=0 . Right: schematic behavior of the spatial ǫx and momentum ǫp anisotropies versus proper time τ for lower (dashed) and higher (solid lines) energy densities at a fixed initial space anisotropy.
anisotropy33,52 defined as ǫp =
hTxx − Tyy i , hTxx + Tyy i
(14)
and thus increase v2 /ǫx , as this momentum anisotropy is known65 to be related with the observed v2 ≃ ǫp /2. Numerical results53,67 within ideal hydrodynamics indicate increases in the transverse momentum integrated v2 at b ∼ 7.5 fm from RHIC to the LHC, ranging from ∼ 15%67 for charged multiplicities at mid-rapidity around 1200, to ∼ 40÷60% in Ref. 53 for twice this multiplicity for central PbPb collisions. Results in viscous hydro54 yield increases ∼ 10% for a charged multiplicity of 1800. Let us finally discuss very briefly the influence of multiplicities on the standard observable for jet quenching, namely single inclusive particle suppression usually studied through the nuclear modification factor, defined for a given particle k = h± (ch), π 0 , . . . as RAA (y, pT ) =
dNkAA dydpT dNkN N hNcoll i dydp T
,
(15)
with hNcoll i the average number of binary nucleon-nucleon collisions in the considered centrality class. A simple modelm to discuss this is the following: Let us assume m This
simplistic model, whose sole aim is allowing for a discussion of the competing effects of density increase and different biases, is by no means quantitative. For example, it does not consider in any detail geometrical biases like the surface bias, it does not take into account fragmentation and it assumes a pure power law behavior of the hadronic spectra which is true neither in data nor in pQCD. It is based on ideas developed in models of radiative energy loss in e.g. Refs. 68 and 69 but not restricted to these models — e.g. models with collisional energy loss also result in some probability of no energy loss.
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N. Armesto
for a fixed geometry (i.e. fixed length or eventual dynamical expansion) that partons can escape the medium without losing any energy with probability p0 , while they may lose some energy ∆E with probability 1 − p0 . Considering a spectrum ∝ 1/pnT (n = 8 roughly describes the spectrum in pp collisions at mid-rapidity at RHIC2 ) I get RAA (y, pT ) = p0 +
∆E 1 − p0 , ǫ= . (1 + ǫ)n pT
(16)
In Fig. 6 I show the ratio of nuclear modification factors at the LHC and at RHIC. For RHIC, I have chosen p0 = 0.1 and ǫ = 0.3 which produce a flat RAA (pT ) ≃ 0.21 which qualitatively corresponds with that observed for π 0 ’s in central AuAu collisions at RHIC for 10 GeV < pT < 20 GeV. To extrapolate to the LHC situation, I have chosen two values of n = 6, 5 and either the same values of p0 and ǫ, or these values modified by the expected ratio of multiplicities in Cases I and II (Case III is not illustrated for clarity of the plot), or a modification in which ∆E, and not ǫ, scales with multiplicity at mid-rapidity. Different options produce evidently different results (e.g. the flatter the spectrum, n = 5 compared to n = 6, the larger the ratio; the larger the multiplicity, the smaller the ratio), a fact which stresses the need of a control of the reference spectrum and of the geometry or
Fig. 6. Ratio of nuclear modification factors at the LHC and at RHIC from Eq. (16). Different line styles refer to different parameters p0 and ǫ or ∆E, see the legends in the plot, while lower and upper lines of each style correspond to spectral power-law exponents n = 6 and 5 respectively.
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389
dynamical behavior of the medium in order to extract quantitative conclusions about the medium properties from measurements of the nuclear modification factor. 3. Bulk Observables Now I turn to the predictions for observables which directly characterize the medium produced in the collisions. These bulk observables correspond to particles with momentum scales of the order of the typical scales of the medium — the temperature if thermalization is achieved — thus the name of soft probes that has been used to designate them. In the following, the use of names of authors will correspond usually to those predictions contained in the compilation,35,36 while those predictions not contained there will be referenced in the standard way. I refer the reader to the compilation35 for further information and model description of the former — a given contribution in Ref. 35 can be found by looking for the name of the authors in the Section devoted to the corresponding observable. In this Section I will review consecutively: multiplicities, collective flow, hadrochemistry at low transverse momentum, correlations and fluctuations. 3.1. Multiplicities Charged particle multiplicity at mid-(pseudo)rapidity is a first-day observable at the LHC. Many groups have produced such predictions, see a compilation in Fig. 7 where 25 predictions are shown.n Different groups provide predictions for different centrality classes. For a more accurate comparison, I re-scale them to a common observable (dNch /dη|η=0 ) and centrality class (hNpart i = 350) using the model.39 The re-scaling factors can be read off Table 1 and the corrected results found in Fig. 8. The re-scaling being made using a given model, its accuracy cannot be taken as very high, but it should reduce the uncertainties in the comparison to a 10% level. Let me start by describing briefly the different predictions presented in the plots. A rough classification, for mere organizational purposes, can be made into the following items: (1) Monte Carlo simulators of nuclear collisions. These models include many different physical ingredients to be combined in a consistent manner. They all take into account energy-momentum and quantum number conservation in a detailed way. While sometimes the physical ingredients are similar between different models, the details of the implementation lead to different results. • The PSM model39 contains a soft component with contributions from both the number of collisions and of participants which lead to the creation of nA
compilation containing a smaller number of predictions can be found in Ref. 36.
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N. Armesto
Charged multiplicity for η=0 in central PbPb at 5.5 TeV 25
20
15
10
5
0
Wolschin et al. Sarkisyan et al. Sa et al. Porteboeuf et al. Mitrovski et al. Lokhtin et al. Kharzeev et al. Jeon et al. Humanic Fujii et al. Eskola et al. El et al. Dias de Deus et al. Chen et al. Capella et al. Chaudhuri Bzdak Busza Bopp et al. Topor Pop et al. Armesto et al. Armesto et al. Arleo et al. Albacete Abreu et al.
0
dN/dη, 6% dN/dη, N =350 part dN/dη, 10% dN/dη, N =350 part dN/dη, 5% dN/dη, 5% dN/dη, N =350 part dN/dη, N =350 part dN/dη, 5% dN/dη, N =350 part dN/dη, 5% dN/dη, 3% dN/dη, N =350 part dN/dη, b<3 fm dN/dη, N =350 part dN/dη, N =350 part dN/dη, b=3 fm dN/dη, N =350 part dN/dη, 10% dN/dη, 5% dN/dη, N =350 part dN/dη, N =350 part dN/dη, 10% dN/dη, 6% dN/dy, N =346.6 part
1000 2000 3000 4000 5000 6000
Fig. 7. Predictions for multiplicities in central Pb-Pb collisions at the LHC. On the left the name of the authors can be found. On the right, the observable and centrality definition is shown. The error bar in the points reflects the uncertainty in the prediction. See the text for explanations.
color strings, satisfying roughly Eq. (3) with x calculable within the model and tending to 1 as energy constraints becomes less and less important with increasing energy. It also contains a hard component using standard pQCD, in which nuclear parton densities (npdf’s) are used. Finally, string fusion is introduced as a collective mechanism in the soft component. ¯ 35,70 model contains a soft component proportional to the • The HIJING/BB number of participants, and a hard component proportional to the number of collisions which also considers npdf’s. It includes mechanisms for baryon number transport from the fragmentation to the central rapidity regions (string junctions), and introduces collectivity through an enhanced string tension — color ropes. The different predictions reflect the uncertainties in the increased string tension. • The DPMJET model35,71 is similar to the PSM, but it includes string
Predictions for the Heavy-Ion Programme at the Large Hadron Collider
dNch/dη|
η=0
25
20
15
10
5
0
in PbPb at sNN=5.5 TeV for Npart=350
Wolschin et al. Sarkisyan et al. Sa et al. Porteboeuf et al. Mitrovski et al. Lokhtin et al. Kharzeev et al. Jeon et al. Humanic. Fujii et al. Eskola et al. El et al. Dias de Deus et al. Chen et al. Capella et al. Chaudhuri Bzdak Busza Bopp et al. Topor Pop et al. Armesto et al. Armesto et al. Arleo et al. Albacete Abreu et al.
0
391
1000
corr., RDM CQM + Landau hydro corr., PACIAE EPOS corr., UrQMD corr., HYDJET++ saturation data driven, limiting frag. corr., NN superposition fcBK evolution corr., EKS98+geom. sat. corr., BAMPS percolation corr., AMPT+gluon shad. DPM+Gribov shad. log. extrap. corr., wounded diq. mod. data driven, limiting frag. corr., DPMJET III corr., HIJING/BB v2.0 PSM geom. scaling corr., log. extrap. corr., rcBK evolution corr., logistic evol. eq.
2000
3000
4000
5000
6000
Fig. 8. Predictions for multiplicities in central Pb-Pb collisions at the LHC. On the left the name of the authors can be found. On the right, I indicate whether a correction has been applied or not, and provide a brief indication of some key ingredients in the model. The error bar in the points reflects the uncertainty in the prediction. See the text for explanations.
junction transport, percolation of strings as a collective mechanism and the strong shadowing proposed for the soft sector in Ref. 43. • The AMPT model35,72 considers a parton cascade initialized by HIJING73 with subsequent hadronization via strings and a hadron transport. The different predictions correspond to the different npdf’s used. • The HYDJET++ model74 contains a soft, thermalized component which is treated hydrodynamically, and a hard component treated through PYTHIA (and PYQUEN, see Subsection 4.1). The error bar corresponds to a variation of the minimum transverse momentum for the hard component from 7 GeV (larger multiplicity) to 10 GeV (smaller multiplicity). • The UrQMD model75 contains a soft component, and a hard component through PYTHIA,48 with a detailed space-time evolution of the pre-hadronic and hadronic degrees of freedom.
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N. Armesto
• The EPOS model35,76 contains similar ideas to those of PSM and DPMJET but aims to account for energy-momentum conservation at the level of the cross sections (usually the cross sections are computed ignoring energy-momentum constraints which are applied a posteriori on the mechanism of particle production), and contains a detailed model for the soft-hard transition, for the treatment of the hadronic remnants and a separation between a dense core, eventually treated via hydrodynamical evolution, and a dilute corona which hadronizes via strings. • The PACIAE model77 contains a parton cascade initialized by PYTHIA with hadronization via string formation and decay and a hadron transport. Collective effects are introduced through the increase of the string tension which, in this case, produces both a harder spectrum in transverse momentum and higher masses — as in previous approaches which consider increased string tensions — and an enhancement of particle production with large longitudinal momentum. (2) Models based on saturation ideas. • Abreu et al.35,78 is based on a non-linear, logistic evolution equation which resembles the Balitsky-Kovchegov (BK) equation in high-density QCD (see the review in Ref. 33), for fixed-size dipoles. It admits an analytic solution and shows limiting fragmentation for some restricted parameter space. • Albacete35,79 is a prediction based on the running-coupling BK equation, with multiplicities computed through the use of kT -factorization and local parton-hadron duality (LPHD). The error bar reflects the uncertainties in the extrapolation coming from the freedom to fix the parameters at RHIC. • Armesto et al.35,47 is a prediction based on the extension of the geometric scaling observed in lepton-proton collisions to proton-nucleus and nucleus collisions, and on LPHD. It provides a pocket formula for multiplicities, Eq. (5), in which the energy and centrality dependences explicitly factorize. The error bar reflects the uncertainties in the nuclear size dependence of the saturation scale extracted from lepton-nucleus data. • Eskola et al.35 use a pQCD approach supplemented with a geometric saturation ansatz. The obtained multiplicities and energy densities are used as input for an ideal hydrodynamical calculation. • Fujii et al.35 use the fixed-coupling BK equation plus limiting fragmentation together with kT -factorization and LPHD. The error bar corresponds to the different initial conditions for evolution. • Kharzeev et al.35,80 use the saturation ideas together with kT -factorization and LPHD. The error bar corresponds to the different options in which the saturation scale grows with energy or saturates.
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(3) Data-driven predictions. • Arleo et al.35 is a logarithmic extrapolation of multiplicities at RHIC which is used as input for ideal hydrodynamical calculations at low transverse momentum coupled to pQCD at large transverse momentum. • Busza35 is a data-driven extrapolation based on the logarithmic increase of particle densities from SPS to RHIC and on the factorization of energy and geometry dependences. • Chaudhuri55 is a data-driven extrapolation based on the logarithmic increase of multiplicities at RHIC which is used as input for a viscous hydrodynamical calculation. • Jeon et al.35 is a data-driven extrapolation based on limiting fragmentation (considering not only the slope of the pseudorapidity distribution near beam rapidity but also the curvature) and the logarithmic increase of particle densities from SPS to RHIC. A small, not visible error bar is due to the different choice of parameters in the fits to existing data. (4) Others. • Bzdak81 uses a variant of the wounded nucleon model40 in which the relevant degrees of freedom are not nucleons but quarks and diquarks. In order to obtain predictions for multiplicity, the results in the model for the number of wounded quarks and diquarks have been supplemented by the multiplicities for nucleon-nucleon collisions in Table 2, with the error bar reflecting the uncertainty in the latter. • Capella et al.35 is a soft model in which the multiplicity gets contributions from both the number of collisions and of participants, supplemented with a very strong shadowingo related with diffraction in lepton-proton collisions.43 • Dias de Deus et al.35 use a model in which the multiplicity, proportional to the number of collisions, is decreased by a geometric factor, given by two-dimensional continuum percolation and related with the fraction of transverse area occupied by the overlapping sources of particles (strings). • El et al.35,82 is a parton cascade initialized by CGC conditions. The parton cascade includes both 2 ↔ 2 and 2 ↔ 3 processes and uses LPHD to relate the output of the cascade with the final multiplicities. The error bar reflects the uncertainty in the extrapolation of the saturation scale in the CGC initial conditions from RHIC to LHC energies. • Humanic83 is a superposition model based on a geometrical ansatz to determine the number of pp collisions, which are modeled through PYTHIA. A space-time picture of hadronization is also included which allows a link to a hadron cascade. o This
very strong shadowing corresponds to ideas very close to those of saturation but formulated in a soft domain in which pQCD techniques are not applicable and phenomenological models are required.
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N. Armesto
• Sarkisyan et al.84 is a model based on the constituent quark model which leads to a participant-like picture similar to that in the model of Bzdak, with the energy deposition in the collision considered within Landau hydrodynamics (see e.g. Ref. 85 for a recent review). • Wolschin et al.35,86 is a relativistic diffusion equation in rapidity and time of the Fokker-Planck type. The error bar reflects the uncertainties in the extrapolation of the diffusion parameters from RHIC to the LHC. From the plots one can conclude that most predictions lie in the range 1000 ÷ 2000. It should be noted than a value lower than 1000 could be, depending on the corresponding value for pp collisions,p in conflict with participant scaling. On the other hand, a value larger than 2000 will be a challenge for saturation physics. Monte Carlo simulators, due to their complexity, do not include yet many recent theoretical developments, e.g. none implements saturation effects. This might be the reason why they tend to give the largest values. Finally, multiplicities show a decreasing tendency with time from 1995,49 through pre-RHIC predictions,27 until now. This is due to the inclusion of collective effects which imply a large degree of coherence in particle production like saturation, strong color fields, percolation, or strong gluon shadowing. This strong coherence can be understood as a decrease in the number of sources which contribute independently to multiparticle production. Proposals to find evidence of these ideas in correlations will be discussed in Subsection 3.4. They would also leave an imprint in multiplicity distributions, see Bopp et al. in Ref. 35. Also the pseudorapidity distributions are informative: for example, the model by Abreu et al.35,78 shows a extremely wide plateau in rapidity (along ∼ 8 units). Now I review the predictions for baryon transport. Since this is an important observable from the point of view of the hadrochemistry, it could be included in Subsection 3.3. But it is also a global characteristic of the collision which goes beyond the single number, dNch /dη|η=0 , mainly discussed so far. The general prediction for the net proton number (p −¯ p) at mid-rapidity is smaller than 4 for central PbPb collisions at the LHC, to be compared with the value 5 ÷ 8 in central AuAu at top RHIC energy.87 This is so in models of different kinds, ranging from approaches with the baryon junction mechanism or other baryon ¯ DPMJET or the EPOS model, hydrodynamical transport effects, as HIJING/BB, models like Eskola et al., the diffusion equation of Wolschin et al., the statistical model of Rafelski et al., see Subsection 3.3, the saturation model in Ref. 88 (see Fig. 9 for the energy evolution of the mean rapidity shift hδyi = |hynet baryon i − ybeam | in this approach), or the model90 based on string formation with momentum fractions taken from parton distribution functions and string fragmentation through the Schwinger mechanism.
p pp
collisions at the same energy as PbPb may not occur before several year of successful pp data-taking. Therefore, for the first runs an interpolation between p¯ p collisions at Tevatron and pp collisions at LHC energies could be required.
December 18, 2009 10:47 WSPC/INSTRUCTION FILE
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Fig. 9. Mean rapidity shift of net baryons as a function of beam rapidity ybeam in the model in Ref. 88. Solid and dashed lines correspond to different options for the behavior of the saturation scale. The solid straight line shows the prediction for the position of the fragmentation peaks. The star at ybeam ≃ 8.5 is the prediction for central PbPb collisions at the LHC. Experimental data can be found in Ref. 89. Figure taken from Ref. 88.
Note that I have focused on predictions for charged multiplicities at mid-rapidity well covered by all heavy-ion detectors at the LHC and, thus, a true first-day observable. Predictions for the total charged multiplicity in all phase space also exist, see e.g. Busza in Ref. 35 (or even Ref. 91 for predictions in strongly coupled super-symmetric Yang-Mills theories computed through the use of the AdS/CFT correspondence). 3.2. Collective flow Here I turn to collective flow — another first-day observable. Specifically, I will discuss elliptic flow at mid-rapidity, both integrated over and as a function of transverse momentum. In this subsection I will concentrate on elliptic flow for hadrons, while that for photons will be discussed in the corresponding Subsection 4.3. Concerning the pT -integrated v2 , the expectation both from data-driven estimations and from more involved, model-dependent calculations, is for it to increase when going from RHIC to the LHC. Such increase, as discussed in Section 2 and in Ref. 34, looks stronger in data-driven extrapolations than in hydrodynamical models. Numerical results53,67 within ideal hydrodynamics indicate increases in the transverse momentum integrated v2 at b ∼ 7.5 fm from RHIC to the LHC, ranging from ∼ 15% for charged multiplicities at mid-rapidity around 1200,67 to ∼ 40÷60% in Ref. 53 for twice this multiplicity in central PbPb. Results in viscous hydro54 yield increases ∼ 10% for a charged multiplicity of 1800. Let me note that by viscous hydro I mean calculations considering shear viscosity but neglecting bulk viscosity. Studies on the impact of the latter are at the very beginning.92–94
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hydro fixed coupling hydro running coupling σ=5.5 mb fixed coupling σ=3.3 mb running coupling Phobos
0.14 0.12
v2
0.1 0.08 0.06 0.04 0.02 0 50
100
150
200 250 Npart
300
350
400
Fig. 10. v2 versus Npart at RHIC (lower line) and at the LHC (upper lines), for different values of the parameters in Eq. (17). The normalization is not determined in the model. Experimental data are from PHOBOS, Ref. 96. Figure taken from Ref. 35.
Concerning the difference between ideal hydrodynamics and non-ideal scenarios, a consequence of a larger density of the medium is that the ideal hydrodynamical behavior will be better fulfilled at the LHC than at RHIC,95 as a higher density implies a smaller mean free path and a faster thermalization. Thus the behavior of the medium at the LHC is expected to be closer to that of an ideal fluid than at RHIC, if one assumes that the medium at RHIC shows only partial thermalization i.e. that the mean free path is not yet much smaller than the system size. This is illustrated in Fig. 10, where v2 ∝
σ dNtot 1 ǫx √ , , K −1 = 1 + K/0.7 Sover dy 3
(17)
with K the Knudsen number, Sover the overlap area, σ the typical cross section √ between constituents of the medium and 1/ 3 comes from the speed of sound of an ideal ultra-relativistic gas. The ideal hydrodynamical limit is reached for K → 0, and Sover , ǫx and dNch /dy are provided through initial conditions, see Drescher et al. in Ref. 35 for details. Now I turn to v2 (pT ). First I will discuss the expectations within the framework of hydrodynamical models. From the matching of pQCD with hydrodynamical spectra, see e.g. Eskola et al. in Ref. 35 and 53, hydrodynamical calculations are expected to be valid up to larger transverse momentum, pT < 3 ÷ 4 GeV, at the LHC than at RHIC. In ideal hydrodynamical calculations, a very similar v2 (pT ) at RHIC and at the LHC is expected for pions at pT < 2 GeV, while the v2 (pT ) for protons is expected smaller (as illustrated in Fig. 11), see Bluhm et al., Kestin et al., Eskola et al.35 and Refs. 53, 67 and 97 for computations corresponding to initial charged multiplicities which range, in central PbPb, from ∼ 1200 in Kestin et al.
Predictions for the Heavy-Ion Programme at the Large Hadron Collider
0.4
π+ (S0=117 π+ (S0=271 p (S0=117 p (S0=271
0.35 0.3
397
fm-3) fm-3) fm-3 ) fm-3)
v2
0.25 0.2 0.15 0.1 π
0.05
+
p
0 0
0.5
1
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2 2.5 pT (GeV)
3
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4
Fig. 11. v2 versus pT from ideal hydrodynamical calculations for different entropy densities corresponding to RHIC and LHC situations, for positive pions and protons, for b = 7 fm. Figure from Ref. 67.
to ∼ 2300 in Eskola et al. Note that even a decrease of v2 (pT ) does not necessarily imply a decrease in the pT -integrated v2 — actually all models show the opposite behavior — as it has to be convoluted which a pT -spectrum which is harder at the LHC than at RHIC. If fact the small increase of v2 (pT ) shown in Ref. 53, less than 10%, translates into a much larger increase of pT -integrated v2 than other predictions, as discussed above. On the other hand, the available calculation within viscous hydrodynamics55 shows a decrease. The difference between different predictions comes not only from the different initial conditionsq (as illustrated in Ref. 53), but also from details of the calculations (made in either two53 or three (all others) spatial dimensions), the equation of state in both the confined and deconfined phases and its matching (see e.g. Bluhm et al. for a study of the influence of the equation of state), the treatment of the hadronic phase, the hadronization procedure (e.g. a statistical method in Ref. 97), etc. Now I focus on other approaches. The Monte Carlo simulators AMPT72 and EPOS76 give results35 at the LHC which are very close to those at RHIC for pions, while the former shows a decrease of v2 (pT ) for protons. The simulator in Ref. 83 gives sizably smaller v2 (pT ) at the LHC than at RHIC in spite of the fact that this model contains hadronic rescattering which, naively thinking, should increase v2 (pT ) due to the larger densities at the LHC. The parton cascade MPC by Molnar,35,99 which considers 2 ↔ 2 partonic collisions, provides interesting information on the relation between viscous hydrodyq For
example, the importance of the initial conditions for the application of hydrodynamical calculations has been recently discussed in Ref. 98.
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0.3
Au+Au, min. visc Pb+Pb, min. visc
v2
0.2 0.1 00
QLHC /QRHIC = 1.5 s s τ0 = 0.6 fm, b = 8 fm 1 2 pT [GeV]
3
Fig. 12. v2 versus pT from the MPC parton cascade of Molnar, for RHIC and LHC situations for b = 8 fm. Figure taken from Ref. 35.
namical calculations and transport results. The author chooses the parameters in the transport equation so as the shear viscosity is fixed to be η/s ≤ (4π)−1 (the equality corresponds to the so-called minimal viscosity bound100 ). The results show a decrease in v2 (pT ) when going from RHIC to the LHC, see Fig. 12, with all dependence on multiplicity encoded in the relation of the saturation scale with multiplicity. Finally, the absorption model of Capella et al.35 considers the absorption of the produced particles moving along paths in the medium, with increasing absorption with increasing length of traversed matter. Such model predicts a strong increase when going from RHIC to the LHC, due to the increasing medium density (as indicated in the previous Subsection, this model predicts a charged multiplicity at mid-pseudorapidity ∼ 1800 for Npart = 350). Therefore, data on both pT -integrated v2 and on v2 (pT ), together with the measurements of the multiplicity, may help to verify whether the origin of the elliptic flow is a collective expansion (and thus thermalization or isotropization has been achieved and hydrodynamical models are applicable) or thermalization has been achieved only partially. In the latter case, a sizable increase is expected in v2 (pT ) for pT < 2 GeV, while in the former a decrease or a mild increase generically results. A sizable decrease would favor some viscosity effects, though the issue of the dependence on the initial conditions for hydrodynamical evolution should be settled for firm conclusions to be extracted. Note that, while finite temperature pQCD calculations (valid for T ≫ Tdec , see e.g. Ref. 101 and references therein) indicate that the shear viscosity to entropy ratio should increase with temperature, 1 η ∝ 2 , s αs (T ) ln α−1 s (T )
(18)
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T = 170 MeV 1
399
µB = 1 MeV
R
h/h K- / K + 0.8
+
Ω / Ω+
RHIC p/p
Ξ / ΞΛ/Λ p/p
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20 µB (MeV)
160
180
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Fig. 13. Antiparticle/particle ratios R as a function of µB for T = 170 MeV (left) and as a function of T for µB = 1 MeV (right). The horizontal line at 1 is meant to guide the eye. The √ p ¯/p ratio (averaged over the data of the 4 RHIC experiments at sNN = 200 GeV) is displayed (gray horizontal line) together with its statistical error (gray band). As illustrated, µB ≈ 27 MeV (dashed line) can be read off the Figure directly within the given accuracy (vertical gray band). Figure taken from Ref. 102.
with αs (T ) decreasing with increasing temperature, the behavior of this and other transport coefficients (like e.g. the bulk viscosity) for realistic temperatures close to the deconfinement temperature Tdec is not clear yet. 3.3. Hadrochemistry at low transverse momentum Hadrochemistry is a key observable to disentangle the mechanism of particle production. Statistical models constitute the most popular framework to discuss it. Within statistical models, predictions are done normally in the grand-canonical ensemble valid for large systems.r The relevant parameters, fireball temperature and baryochemical potential µB s are extrapolated from the results extracted at lower energies. The results obtained by different groups (Andronic et al. and Kraus et al. in Refs. 35 and 102) are shown in Fig. 13 and Table 4. It can be observed that even r In
the grand-canonical ensemble it is possible to predict the particle ratios without any reference to the total multiplicity i.e. to the total volume of the system. This case is one of the very few in which predictions can be done in absence of such information. s The strangeness suppression factor used at lower energies within the grand-canonical ensemble is 1 at RHIC energies and this value is assumed for the LHC.
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Table 4. Predictions of the thermal model for hadron ratios in central Pb+Pb collisions at LHC, for µB = 0.8 MeV and T = 161 MeV. The numbers in parentheses represent the error in the last digit(s) of the calculated ratios. Table taken from Andronic et al. in Ref. 35.
π − /π +
K − /K +
p¯/p
¯ Λ/Λ
¯ Ξ/Ξ
¯ Ω/Ω
1.001(0)
0.993(4)
−0.013 0.948+0.008
−0.011 0.997+0.004
−0.007 1.005+0.001
1.013(4)
p/π +
K + /π +
K − /π −
Λ/π −
Ξ− /π −
Ω− /π −
0.074(6)
0.180(0)
0.179(1)
0.040(4)
0.0058(6)
0.00101(15)
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Grand CanonicalPbPb Canonicalpp, RC=2 fm
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Λ/p
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Canonicalpp, RC=1 fm
K+/π+
Ratio
10-1
Fig. 14. Predictions for various particle ratios using different values for the cluster size RC . Figure taken from Ref. 102.
the p ¯ /p ratio takes values very close to 1 in the expected range of T ≃ 160÷175 MeV and µB ≃ 0 ÷ 6 MeV. Concerning these extrapolations, p ¯ /p is particularly sensitive to the value of µB , while the ratios of multi-strange baryons to non-strange particles are particularly sensitive to the temperature, see e.g. Ref. 102. For smaller systems, e.g. smaller nuclear sizes or peripheral collisions, the grandcanonical ensemble is not expected to provide a good description of particle production. While the traditional way of addressing the question for strangeness production is the use of a strangeness suppression factor — thus assuming a chemically non-equilibrated system, the proposal in Refs. 102 and 103 is to keep strangeness conservation in smaller volumes, called clusters. The effects on particle ratios of the consideration of clusters of different sizes can be seen in Fig. 14 and in Ref. 103. Another proposal within statistical models is the non-equilibrium scenario of Rafelski et al. in Ref. 104. This scenario shows a sensitivity to the total multiplicity
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Table 5. Predictions for particle yields at the LHC for different scenarios by Rafelski et al.: chemically equilibrated (second column), chemically non-equilibrated but with the same freeze-out temperature as the previous one (third column), and chemically non-equilibrated with a different temperature but for the same multiplicity as the previous nonequilibrated case. The ‘*’ refers to input values, while the subindex vis refers to values observable in the ALICE TPC (|η| < 0.9), S denotes the entropy, V the volume and b the baryon number. The slashes are used to give the particle yields with/without weak decays. Table taken from Rafelski et al. in Ref. 35.
T [MeV] dV /dy[ fm3 ] dS/dy dNch /dy|y=0 vis dNch /dy
140∗ 2036 7517 1150∗ 1351
140∗ 4187 15262 2351 2797∗
162∗ 6200∗ 18021 2430 2797
p b − ¯b (b + ¯b)/h− 0.1 · π ± K± φ Λ Ξ− Ω−
25/45 2.6 0.335 49/67 94 14 19/28 4 0.82
49/95 5.3 0.345 99/126 207 33 41/62 9.5 2.08
66/104 6.1 0.363 103/126 175 23 37/50 5.8 0.98
∆0 , ∆++ K0∗ (892) η η′ ρ ω f0
4.7 22 62 5.2 36 32 2.7
+ K+ /πvis Ξ− /Λvis Λ(1520)/Λvis Ξ(1530)0 /Ξ− φ/K+ K0∗ (892)/K −
0.165 0.145 0.043 0.33 0.15 0.236
9.3 48 136 11.8 73 64 5.5 0.176 0.153 0.042 0.33 0.16 0.234
13.7 52 127 11.5 113 104 9.7 0.148 0.116 0.060 0.36 0.13 0.301
in the central region and predicts, with respect to the chemically equilibrated one, an enhancement of multi-strange and single strange resonance yields, and a decrease of non-strange resonances (the prediction for net-baryon yields has been commented in the Subsection 3.1). Results can be seen in Table 5.35
−2 2
/p/ π0
d N/(2πpTdpTdy) (−0.5
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Fig. 15. Left: Ratio p ¯/π 0 at RHIC (black) and at the LHC (blue) from the percolation model of Cunqueiro et al., for central (upper lines) and peripheral (lower lines) collisions. Right: Transverse momentum spectra for various particle species in AMPT, Chen et al., at RHIC (lines) and at the LHC (symbols joined by lines), for b < 3 fm. Figures taken from Ref. 35.
The different scenarios for statistical production lead to marked differences in particle yields in heavy-flavor production, which will be commented on in Subsection 4.2. Now I turn to non-statistical models. These models mainly focus on the baryonto-meson ratios, whose large values measured at intermediate pT ∼ 3 GeV at RHIC,2–5 much larger than those measured in nucleon-nucleon collisions, constitute the (anti)-baryon anomaly which has triggered many new ideas. There are several available predictions: by ideal hydrodynamical models (Kestin et al.35,67 ), by recombination models as implemented in AMPT (Chen et al.35 ) and by models which consider a higher string tension like percolation models (Cunqueiro ¯ by Topor Pop et al.,35 see Figs. 15 and 16. In general, et al.35 ) and HIJING/BB hydrodynamical and recombination models predict larger baryon-to-meson ratios than models which consider an increased string tension in nucleus-nucleus collisions with respect to nucleon-nucleon. Let me comment that the percolation model of Cunqueiro et al.35 predicts a Cronin effect — a nuclear modification factor above one — for protons at midrapidity in central PbPb collisions. This is at variance with most extrapolations or theoretical expectations which predicts a disappearance of the Cronin effect with increasing collision energy, see e.g. Refs. 105 and 106. While every non-statistical prediction is linked to a multiplicity scenario, it is not so easy to see the effect of a variation of multiplicity on the results for hadrochemistry of different models. In principle, hydrodynamical and recombination models would benefit from a larger multiplicity due respectively to the larger applicability of hydrodynamics (larger duration of the hydrodynamical phase) and due the larger density in recombination models, with less restrictions due to finite density and volume. For models which consider a higher string tension (in nucleus-
Predictions for the Heavy-Ion Programme at the Large Hadron Collider
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S0 = 117 fm , nB,0 = 0.44 fm
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Fig. 16. Antiparticle/particle ratios (left) and transverse momentum spectra (right) for different particle species, in ideal hydrodynamics, for two scenarios corresponding to RHIC and the LHC. Figure taken from Ref. 67.
nucleus than in pp collisions), an increase in the string tension implies a reduction of multiplicity and an increase in baryon/strangeness production. Therefore, in these models an increase of multiplicity originating from a smaller string tension would imply a reduction of the effects characteristic of the enhanced string tension scenario. Obviously these most crude expectations can only be substantiated by further calculations for different multiplicity scenarios within the models. Finally, let me mention that the possibility of producing charmed exotic states in heavy-ion collisions at the LHC has also been addressed, see Lee et al. in Ref. 35. 3.4. Correlations Now I turn to correlations. First, I will indicate the predictions for the HanburyBrown-Twiss (HBT) interferometry (see the recent review107 ).
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PHENIX HKM, 0 30 GeV/fm3 (RHIC) 3 HKM, 0 70 GeV/fm (LHC) HKM, 0 110 GeV/fm3 (LHC)
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PHENIX HKM, 0 30 GeV/fm3 (RHIC) 3 HKM, 0 70 GeV/fm (LHC) HKM, 0 110 GeV/fm3 (LHC)
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1
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p [GeV] T
Fig. 17. Transverse momentum spectrum of pions and behavior of the pion HBT radii from Sinyukov et al., for different scenarios of initial energy densities. Figure taken from Ref. 35.
The generic expectation34 is that all HBT radii Rout , Rside and Rlong will increase when going from RHIC to the LHC. This is substantiated by several calculations using ideal hydrodynamics, like Frodermann et al. (with the transition out-of-plane to in-plane shape clearly reflecting in the radii), or in the hydrokinetic approach of Karpenko et al. and Sinyukov et al.35 (see Fig. 17). Both these calculations consider ideal hydrodynamics but different hadronization procedures, the latter intending to consider some out-of-equilibrium features. This is also the case in the calculations in the AMPT model by Chen et al. in Ref. 35 and the hydro+statistical model97,108 which combines a hydrodynamical behavior with hadronization through the statistical method. The corresponding results can be seen in Table 6. Within ideal hydrodynamics, the features of the HBT radii which are not in agreement with RHIC data — too large Rlong and Rout /Rside , and the behavior of Rout and Rside with the relative momentum of the pair — will also be present at the LHC. A piece of knowledge still missing in this context is the effect of viscosity on the HBT radii109 and the effects of pre-thermalization dynamics.110 On the other hand, partial thermalization, which implies a departure of the ideal hydrodynamical behavior,95 may also help to reduce the ratio Rout /Rside 111 in agreement with RHIC data. If this is the case, then the expectation that the situation at the LHC will be closer to ideal hydrodynamics will reflect in an increase
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405
Table 6. Predictions for the HBT radii at RHIC/LHC from two different models: Chen et al. in Ref. 35 for b = 0 and 0.3 < kT < 1.5 GeV, and the hydrodynamical plus statistical model in Ref. 97, for two pion multiplicity scenarios at the LHC, 558 and 1193.
RHIC/LHC
Chen et al.
Ref. 97
Rout
3.60/4.23
5.4/6.0÷6.5
Rside
3.52/4.70
4.3/5.3÷6.3
Rlong
3.23/4.86
6.1/7.6÷8.6
of this ratio when going from RHIC to the LHC. Besides, the inclusion of minijets modifies the behavior of the HBT radii and of the chaoticity parameter with respect to pure hydrodynamical predictions.74 Correlations can also be useful to clarify the mechanism of particle production. Correlations in rapidity were proposed long ago, see e.g. Ref. 112, as a measurement sensitive to the distribution of particle sources. More specifically, defining two rapidity intervals denoted by F and B with multiplicities nF and nB respectively, the correlation strength b (sometimes denoted as σF2 B ) is defined as hnF i(nB ) = a + bnB , b =
hnF nB i − hnF ihnB i DF2 B = . 2 DBB hn2B i − hnB i2
(19)
Predictions exist113 for such quantity at the LHC, see Fig. 18 and Dias de Deus et al.,35 in the framework of a two-step scenario which considers first the formation and interaction of particle emitters (coherent along large rapidity regions) which subsequently decay into the observed particles (see also Ref. 115). Many explanations try to address the existence of such long range correlations, see e.g. Ref. 116 and references therein. It has been linked to the so-called ridge phenomenon measured at RHIC (see e.g. Ref. 117): the existence of a two-particle correlation narrow in azimuth but extended along several units of pseudorapidity in AuAu collisions. While a quantitative description is missing, present qualitative explanations are based (e.g. Ref. 118) on the coupling of particle production correlated along a long rapidity range to the collective flow. An extended correlation as predicted by e.g. the two-step scenario mentioned above, together with the fact that the collective flow is expected to last longer at the LHC than at RHIC, should make this phenomenon more prominent at the LHC. 3.5. Fluctuations Many types of fluctuations have been proposed and analyzed as possible signatures of a phase transition in ultra-relativistic heavy-ion collisions: in multiplicity, charge, baryon number, transverse momentum, etc. The results at SPS and RHIC energies are not clear — the evidence of a non-statistical or non-trivial origin of fluctuations
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Fig. 18. Forward-backward correlation correlation strength for different values of the rapidity gap ∆η = ηF − ηB between the forward and backward windows at RHIC and at the LHC, from a two-step scenario. Preliminary data are from Ref. 114. Figure courtesy of the authors of Ref. 113.
at SPS and RHIC is still under debate — which has prevented predictions for the LHC. Available predictions are for the multiplicity fluctuations (Cunqueiro et al. in Ref. 35) quantified through the scaled variance of negative particles, h(n− )2 i − hn− i2 Σ2 (n− ) = hn− i hn− i
(20)
measured in a given rapidity interval δy. The predictions, shown in Fig. 19, indicate a non-monotonic behavior at some given number of participants (a change of slope at some Npart smaller with increasing energy) which is, in the framework of this model, indicative of the existence of a percolation phase transition. Note that in this model, as in others, multiplicity fluctuations are linked to those in transverse momentum. On the other hand, Torrieri in Ref. 35 proposes the use of fluctuations of particle ratios e.g. of kaons and pions, as measurements sensitive to the mechanism of particle dynamics: the fully equilibrated scenario of the grand-canonical ensemble should show a different behavior from the other ensembles or non-statistical scenarios. 4. Hard and Electromagnetic Probes In this Section I review the available predictions for those probes of the medium whose yields can be, in the absence of a medium, computed through perturbative techniques — hard probes. These are high transverse momentum particle production, heavy-quark and quarkonium production, and photon and dilepton produc-
Σ2(n )/
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407
SPS
4
RHIC
3.5
LHC
δy=1.5
3 2.5 2 1.5 1 0.5 0
50
100
150
200
250
300
350
400 Npar
Fig. 19. Scaled variance of negative particles versus the number of participants in PbPb at top SPS, AuAu at top RHIC, and PbPb at LHC energies, from bottom to top, in the percolation model of Cunqueiro et al. Figure taken from Ref. 35.
tion at large momentum or mass.t For photons and dileptons, i.e. electromagnetic probes, I will also consider their production at low momentum or mass, though their calculation lies, in principle, beyond the reach of perturbative techniques. Extensive studies on hard probes at the LHC can be found in Ref. 9 (pA collisions and benchmark studies), Ref. 14 (particle production at high transverse momentum and jets), Ref. 15 (heavy quarks and quarkonia) and Ref. 16 (photons and dileptons). 4.1. Particle production at large transverse momentum and jets The suppression of the yield of hadrons at large transverse momentum measured at RHIC2–5 — the jet quenching phenomenon — is one of the most important subjects of current research and debate in the field. It is most commonly quantified through the nuclear modification factor (15) and usually attributed to the energy loss of the leading parton which fragments onto the measured hadron, see e.g. the standard reviews in Ref. 33, 120 and the more recent one121 (more specific information about radiative energy loss which is the reference explanation can be found in Ref. 122, 123, and about studies of the energy loss in strongly coupled super-symmetric YangMills plasmas through the AdS/CFT correspondence in Ref. 124). One comment on the definition of the region that I call of large transverse momentum is in order. At RHIC, such region — usually taken at pT > 7 ÷ 10 GeV — is determined by that in which the characteristics of fragmentation become those in absence of any medium, i.e. in pp, and where fragmentation or hadronization t Concerning
the total charm and charmonium cross sections and their production at low momentum, doubts exist on whether they can be reliably computed in pQCD — either at fixed order or via resummation techniques — or not, see e.g. Ref. 119 for a discussion on the uncertainties for charm.
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is expected to be described by standard pQCD techniques so no collectivity in hadronization (e.g. recombination) seems to be required. More specifically, the baryon-to-meson anomaly disappears, the nuclear modification factor for different species becomes similar, etc. Note that this definition is not free from ambiguities as new effects included in the models (for example, in the transition from recombination to perturbative fragmentation) may shift it. At the LHC, due to the larger densities and larger expected collectivity, such region may start at larger pT than at RHIC, a question which only data will answer. I start by reviewing the predictions for the nuclear modification factor in central PbPb collisions at the LHC. In Fig. 20, I show 15 predictions for RAA at pT = 20, 50 GeV from different models. Differently from the case of multiplicities, where some easy re-scaling to a common centrality class was feasible, here such re-scaling is not possible as there is no simple relation between a change of density/multiplicity and the resulting energy loss and RAA . Therefore, I simply indicate in the figure the centrality definition or the multiplicity or energy density (with respect to that at RHIC) for which the predictions were computed. Different models use different parameters related with the medium density and the scattering strength of the parton with the medium. The most common one is the transport coefficient qˆ which can be related locally to the energy density through125 qˆ(x, y, z, τ ) = c · ǫ3/4 (x, y, z, τ ),
(21)
with c some constant which in pQCD is expected to be of order 1.u Other models use the gluon density, the energy density at thermalization time, the value of αs , etc. For descriptive purposes, the predictions can be classified into the following groups: (1) Models which consider only radiative energy loss (see Ref. 123 for a comparison among the theoretical basis of the different models). • Arleo et al.35,129 use fragmentation functions modified through their convolution with quenching weights — the probability of a given amount of energy loss — which are evaluated using a simplified radiation spectrum. The employed characteristic gluon frequency is ωc = 50 GeV. • Dainese et al.,35,130 the PQM model, use quenching weights calculated from the full radiation spectrum in the multiple soft scattering approximation and a static medium modeled by the initial overlap geometry. The different predictions correspond to different extrapolations of the transport coefficient from RHIC to LHC energies. u This
proportionality is expected in pQCD125 and also at strong coupling through the AdS/CFT correspondence, see Hong Liu in Refs. 35 and 126. Besides, the transport coefficient may acquire some energy dependence, see Casalderrey-Solana et al. in Ref. 35 and 127. Attempts have been essayed to compute it from first principles in QCD, see Antonov et al. in Refs. 35 and 128.
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Fig. 20. Predictions for the nuclear modification factor in central Pb-Pb collisions at the LHC, for pT = 20 (red filled symbols) and 50 (when available, blue open symbols) GeV. On the right, the name of the authors, the particle, centrality definition and some model explanation is shown. The error bar in the points reflects the uncertainty in the prediction. See the text for explanations.
• Renk et al.35,131 use, as the previous model, quenching weights calculated from the full radiation spectrum in the multiple soft scattering approximation, but with a hydrodynamical modeling of the medium and the relation (21). • Jeon et al.35,132 use a schematic model for the quenching weights which considers only an average energy loss. • Vitev35,133 uses the GLV model with quenching weights with gluon feedback and one-dimensional Bjorken expansion, and higher-twist shadowing of parton densities. The different predictions correspond to different extrapolations of the gluon density from RHIC to the LHC. • Wang et al.35,134 use a model for medium-modified fragmentation functions and compute the yields at next-to-leading order. A Bjorken
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expansion is considered. The error bars correspond to the different parametrization of nuclear shadowing employed in the pQCD calculations. (2) Models which consider radiative and elastic energy loss. • Qin et al.35,135 use the AMY model with radiative and collisional energy loss in a medium which is modeled through ideal hydrodynamics. The error bars correspond to different values of αs . • Wicks et al.35,136 use the GLV model for radiative energy loss whose quenching weights are convoluted with those from elastic energy loss. The error bars correspond to different extrapolations of the gluon density from RHIC to the LHC. • Lokhtin et al.,35,137 the PYQUEN model, is a implementation within PYTHIA of radiative energy loss which considers a mean radiative energy loss distributed among some gluons, which are then allowed to do vacuum final state radiation until the branching process stops, after which they scatter elastically. • Zakharov138 uses a model which consider quenching weights based on a single radiation spectrum in the multiple soft scattering approximation, plus elastic scattering, nuclear shadowing and Bjorken expansion of the medium. The error bars correspond to the different values at which the running coupling is frozen in the infra-red, and to considering a purely gluonic or a chemically equilibrated plasma. (3) Models with elastic energy loss plus parton conversions. Liu et al.35,139 consider production in pQCD with the possibility of elastic scattering in which conversion channels e.g. qg → gq or gg → q q¯, are included. This inclusion turns out to be of importance for the hadrochemistry at large transverse momentum, see below. For this model, the highest available pT for the predictions is 40, not 50 GeV. (4) Others. • Capella et al.35,140 use a comover absorption scenario in which energy gain and loss terms are implemented in one-dimensional rate equations, plus strong shadowing. The error bars correspond to the different kinematics (considering 2 → 1 or 2 → 2 processes) to evaluate the shadowing. • Cunqueiro et al.35,141 consider a scenario in which percolation induces an increase in the string tension and a strong modification of the distribution of particle sources. • Kopeliovich et al.35,142 consider a sudden hadronization scenario in which hadrons are created very soon and interact strongly with the produced medium. • Pantuev35,143 consider the medium as composed by a thin transparent corona and a totally opaque core, which can be alternatively interpreted in terms of a formation time for the QGP. Estimations of the variation of this time when going from RHIC to the LHC allow for the predictions.
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While no simple quantitative conclusion can be extracted from this variety of models, it can be claimed that those which implement radiative or collisional energy loss generically predict a nuclear modification factor between 0.15 ÷ 0.25 at pT = 20 GeV and increasing with increasing pT . Larger densities lead to larger suppressions, but the concrete value and the quantitative behavior with increasing pT are different for different models, a fact which is not only related with the theoretical model used for energy loss but also with the ‘embedding’ of such model in the medium. On the other hand, jets will be very abundantly produced at the LHC, see e.g. Refs. 14, 17 and 18 and Fig. 1 right. Provided the issues121 of jet reconstruction through some algorithm, background subtraction (see e.g. Ref. 144) and jet energy calibration are successfully addressed, they offer huge possibilities to verify the physical mechanism underlying the jet quenching phenomenon, both through the measurement of the RAA of jets (see Fig. 21 left) as well as more differential observables such as jet fragmentation functions (see Fig. 21 right), jet shapes, etc. For such studies, and for the study of particle correlations, new theoretical tools have to be developedv and implemented in Monte Carlo simulators (semiquantitative ideas were pioneered in Ref. 153, see e.g. Ref. 154 for a recent study). This is an ongoing effort with several groups involved and several Monte Carlo generators becoming gradually available: PYQUEN,137 Q-PYTHIA,155 JEWEL,156 YaJem,157 etc. v E.g.
the developments in the modified leading-logarithmic approximation (MLLA) approximation,145–149 the modifications of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution150,151 or the inclusion of elastic energy loss in the parton cascade.152
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Another aspect of great importance both to verify the origin of jet quenching and to understand the interplay between the energetic particles and the soft medium is hadrochemistry at large pT . First, within the MLLA approximation and modeling the medium-modification of the final state radiation pattern through a multiplicative constant in the collinear parts of the splitting functions,145 an enhancement of the ratio of baryons and strange mesons over pions due to medium effects is found within the fragmentation of a energetic parton,158 see Fig. 22. Second, the non-Abelian nature of radiative energy loss implies that gluons lose more energy than quarks due to their larger color charge — i.e. the value of the quadratic Casimir of the adjoint (3) and fundamental (4/3) representations in QCD, respectively. Therefore, hadrochemistry is affected, as different particles receive different contributions from the fragmentation of quarks and gluons and this relative contribution varies with particle momentum.w This can be seen in Fig. 23 left where Barnafoldi et al.35 show the results for particle ratios of the GLV energy loss with different opacities L/λ, with L the medium length and λ the mean free path of partons in the medium.
w In
this respect, it should be noted that some fragmentation functions e.g. those of protons are badly constrained from available experimental data in absence of any medium e.g. in e+ e− or pp. Therefore medium-modification studies are subject to an uncertainty that can only be resolved with a better knowledge of the vacuum fragmentation functions, see e.g. Ref. 159.
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Fig. 23. Left: Particle ratios versus transverse momentum in pQCD with GLV energy loss for different opacities, from Barnafoldi et al. Right: p/π + ratio at the LHC in pp, and in PbPb collisions with elastic energy loss and conversions, from Liu et al. Figures taken from Ref. 35.
Finally, conversions as discussed above in the model by Liu et al.35,139 also modify the hadrochemistry at large pT , see Fig. 23 right, and there is the possibility of recombination of partons from adjacent jets160 which may also increase the baryon-to-meson ratio at large transverse momentum. Now I focus on correlations at large transverse momentum and the disappearance of the backward azimuthal peak observed at RHIC. It is usually quantified through the hadron-triggered fragmentation functions for two hadrons h1 , h2 , a trigger particle h1 which defines the near side hemisphere (azimuthal angle 0) with ptrig and an associated particle h2 in the away side hemisphere (around azimuthal T angle π) with passo , reading T trig trig /ptrig DAA (zT = passo T T , pT ) = pT
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This quantity provides information on the conditional yield of particles in the backward hemisphere for a given trigger, and offers additional constraints (other than those coming from RAA (pT )) on the parameters characterizing the medium in models of energy loss.134 Predictions for the LHC exist, see Ref. 161 and Wang et al. in Refs. 35 and 134, Fig. 24. To conclude this Subsection, I will comment on one aspect for which no prediction is yet available, namely the wide structure observed in the backward azimuthal region when the pT of the associated particles is lowered to that of the particles in the bulk. The standard qualitative explanations, see Ref. 121 and references therein, go from deflection of jets in strong fields, to medium-induced radiation, Mach cones, Cherenkov radiation, etc. Recent experimental analysis162 seem to disfavor the
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deflection of jets. But the nature of such structure has not yet been unambiguously established, mainly because it is not yet clear how the jet energy is transferred to the expanding medium. Much effort is currently devoted to it, see e.g. Bauchle et al. and Betz et al. for Mach cones in a hydrodynamical medium, Dremin for Cherenkov radiation and Mannarelli et al. for the energy evolution of the angular structure of the energy deposition of jets, in Ref. 35. 4.2. Heavy quarks and quarkonia Heavy quark and quarkonium production and suppression are other standard hard probes, see the recent review.163 Beginning with heavy-quark production, it offers the possibility of testing the expected hierarchy of radiative energy loss:164,165 ∆E(gluons) > ∆E(light quarks) > ∆E(heavy quarks), with the first inequality coming from the different color factors (as discussed in the previous Subsection), and the second from the suppression of radiation due to the mass of the parent parton. Besides, collisional energy loss is expected to be more important for heavy quarks166 than for light partons, and the details of medium modeling affect more strongly energy loss for heavy quarks than for light partons (as a collection of static scattering centers, as a dynamical medium e.g. in Djordjevic et al. in Ref. 35 or in Ref. 167, etc.). The measurement by PHENIX and
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Fig. 25. Left: Nuclear modification factor for D and B mesons in the WHDG model for different gluon densities and αs = 0.4, from Wicks et al. Right: Ratio of nuclear modification factors for B over D mesons with radiative energy loss, for different transport coefficients and for the case where the mass effect on radiative energy loss is switched off, from Armesto et al. Figures taken from Ref. 35.
STAR168,169 of a nuclear modification factor much smaller than 1 for ‘non-photonic’ electrons (expected to come from the semi-leptonic decays of heavy flavors) has triggered a lot of activity. The LHC, with the new possibilities for heavy-flavor identification of beauty (and eventually of charm)10–12 and for the measurement of non-photonic electrons (with the possibility of separating charm and beauty via correlations170–172 ), together with the extended transverse momentum reach, offers an ideal testing ground for these ideas. In Figs. 25, 26 and 27, I show available predictions for PbPb collisions at the LHC. Specifically, in Fig 25 left, I show the results of the collisional plus radiative energy loss model of Wicks et al.136 which uses the DGLV model for radiative energy loss of heavy quarks, whose corresponding quenching weights are convoluted with those from elastic energy loss. The different lines correspond to different extrapolations of the gluon density from RHIC to the LHC. In Fig 25 right, I show the results from a purely radiative model by Armesto et al.173 which uses the quenching weights for heavy quarks computed in the multiple soft scattering approximation. The geometry in this model is considered as in the PQM model, see the previous Subsection. In this case, the results shown are not for the nuclear modification factor but for the ratio of nuclear modification factors of bottom and charm mesons, which clearly shows the mass effect on the energy loss in an accessible region of pT . In Fig. 26 left, the results of the model of Vitev174 are shown (see also Ref. 175). This model considers a very fast hadronization for heavy-flavored mesons, which then dissociate through collisions in the QGP. In Fig. 26 right van Hees et al.176 consider both radiative losses of the heavy quarks and their strong scattering with resonances in the plasma through a diffusion equation.
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In Fig. 27 left, the results of the collisional model of Ref. 177 are provided. This model considers elastic energy loss for fixed and running coupling constants, with the bands defined by different extrapolations of multiplicities. Finally, in Fig. 27 right, I show the results from Ref. 178 for the RCP (the nuclear modification factor defined not with respect to nucleon-nucleon collisions but with respect to a peripheral class of events) for muons coming both from semi-leptonic decays of heavy flavors and from decays of electro-weak bosons. For the former an energy loss model equivalent to that in Ref. 173 is used. The latter are expected to show no medium (hot matter) effect, thus this measurement contains its own self-calibration with respect to cold nuclear matter effects. All results presented here show a large suppression at pT ∼ 10 ÷ 20 GeV (they have been computed at mid-rapidity except those in Ref. 178 which has been done for the ALICE muon arm covering 2.5 < η < 4), and a gradual increase of the nuclear modification factor with pT . On the other hand, in a strongly coupled super-symmetric Yang-Mills plasma, the dominant energy loss mechanism for a heavy quark, computed through the use of the AdS/CFT correspondence, is a drag force (valid for small velocities of the heavy quark), see Refs. 121, 122 and 124. Calculations show177,179 that this drag force results in a nuclear modification factor much flatter with pT than in pQCD-based models with elastic or radiative energy losses. Besides, the effects of different shadowing mechanisms (Kopeliovich et al.35 ), of light-to-heavy conversions180 and of thermal production (see Chen et al. in Ref. 35) have been considered. Also, the possible characterization of the plasma through the de-correlation of D mesons coming from back-to-back charm-anticharm pairs181,182 is under investigation, see Fig. 28. Further, soft effects can modify the charm cross section with respect to usual expectations: an enhanced string tension as introduced ¯ 70 model leads to an enhancement of the charm cross section in in the HIJING/BB heavy-ion collisions183 which amounts to a 60 ÷ 70% at RHIC energies and to an order of magnitude at the LHC. Now I turn to the suppression of quarkonium — one of the canonical signatures of QGP formation since its proposal in 1986.184 Unfortunately, this signal is plagued with uncertainties from cold nuclear matter effects, both from the lack of knowledge of the nuclear parton densities (see Section 5) and the lack of understanding of nuclear absorption in cold, normal nuclear matter. The most recent phenomenological analysis185,186 indicate that the absorption cross section of J/ψ in nuclear matter is either constant or decreasing with increasing energy. On the contrary, theoretical models previous to RHIC data187,188 pointed to an increase of absorption with increasing energy. Although some progress189 has been made in understanding such unexpected feature, predictions for the LHC are still subject to large uncertainties. On the other hand, the behavior of quarkonia in a QGP is not clear either. Lattice data190 support a suppression pattern in which ψ ′ and χc melt just above the deconfinement temperature, while the J/ψ survives up to temperatures close to
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Fig. 28. DD correlation as a function of relative azimuth angle ∆φ with different drag parameter a for central Au+Au collisions at RHIC (left) and Pb+Pb collisions at LHC (right). The unit of the drag coefficient a is [10−6 (fm)−1 MeV−2 ]. No pT cut has been applied. The initial temperature T0 and thermalization time τ0 are, respectively, 340 MeV and 0.6 fm at RHIC and 610 MeV and 0.3 fm at LHC. Figure taken from Ref. 181.
2Tc and the Υ up to sizably larger T . Contrariwise, potential models, see Ref. 191 for a discussion, suggest that the J/ψ melts much closer to Tc than indicated by lattice results. The actual suppression pattern could be tested by the transverse momentum dependence of the suppression of different quarkonium states, as illustrated by Vogt in Ref. 35, see Fig. 29. There, clear differences can be seen between the suppression sequence of the different states e.g. χc suppression disappears at smaller pT than the J/ψ one if the dissociation temperature of J/ψ is close to Tc , while the opposite happens if the J/ψ dissociation temperature is much higher.x Another signature of the existence of a deconfined state of quarks would be an enhancement of the quarkonium yield due to a recombination process in which quarks and anti-quarks from the plasma form bound states. Apart from having being proposed as a justification for the baryon-to-meson anomaly and the scaling of v2 normalized to the quark number versus the quark kinetic energy observed at RHIC (see Refs. 2–5), such mechanism has been suggested to explain the apparent larger suppression at forward than at central rapidities measured at RHIC,193 an effect which goes in opposition to a density-driven suppression — the system is expected to be more dilute far from mid-rapidity. For the purpose of predictions, x While
the naive expectation is that the suppression disappears with increasing pT due to the smaller time that the bound state stays in the plasma, there are proposals192 that the suppression should reappear at larger pT due to the larger ‘effective’ T seen by the bound state when moving fast with respect to the medium.
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Fig. 29. The survival probabilities as a function of pT for the charmonium (left-hand side) and bottomonium (right-hand side) states for initial conditions at the LHC. The charmonium survival probabilities are J/ψ (solid), χc (dot-dashed) and ψ′ (dashed) respectively. The bottomonium survival probabilities are given for Υ (solid), χ1b (dot-dashed), Υ′ (dashed), χ2b (dot-dot-dashdashed) and Υ′′ (dotted) respectively. The top plots are for T0 = 700 MeV while the bottom are for T0 = 850 MeV. The left-hand sides of the plots for each state are for the lower dissociation temperatures, 1.1Tc for the J/ψ and 2.3Tc for the Υ while the right-hand sides show the results for the higher dissociation temperatures, 2.1Tc for the J/ψ and 4.1Tc for the Υ. Figure taken from Ref. 35.
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the largest uncertainty in the recombination mechanism comes from both the charm and the bulk multiplicities at LHC energies, see Fig. 30 left194 for the predictions for a fixed total multiplicity and different charm cross sections. The recombination mechanism could be tested by the different dependence of the quarkonium average transverse momentum on centrality with and without recombination, see Fig. 30 right.195 In this latter plot initial production correspond to the initial yield of J/ψ coming from the hard collisions (in which J/ψ is formed from c¯ c pairs from the same nucleon-nucleon collision), while in-medium formation corresponds to recombination. The parameter λ accounts for the difference of the transverse momentum of the J/ψ in pp and in pA, assuming for the latter a proportionality with the number of binary nucleon-nucleon collisions Ncoll , hp2T ipA = hp2T ipp + λ2 (Ncoll − 1).
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The only prediction which could be — to my knowledge — directly compared to data is that from Ref. 196. It illustrates the different effects that enter in the calculation: no absorption, strong nuclear shadowing (see Fig. 31 left), J/ψ suppression by comoving particles (for a charged multiplicity at mid-pseudorapidity around 1800, see Capella et al. in Subsection 3.1) and recombination with differpp pp 2 ent values of C(y) = (dNc¯ c /dy) /dNJ/ψ /dy. The different effects are illustrated in Fig. 31 right. The magnitude of the effect of recombination in this plot is much smaller than the one to be seen in Fig. 30 left, although the parameter fixing the
Fig. 31. Left: Nuclear modification factor of J/ψ and Υ versus the number of participant nucleons for PbPb collisions at the LHC with only cold nuclear matter effects (shadowing) and at different rapidities, from Bravina et al., with the bands reflecting different parametrizations for the gluon densities and different kinematics. Figure taken from Ref. 35. Right: Nuclear modification factor of J/ψ versus the number of participant nucleons for PbPb collisions at the LHC, with only the effects of shadowing and with comover suppression for different recombination strengths ranging from none to high (C = 0 ÷ 5), from Capella et al. Figure taken from Ref. 196.
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recombination varies in a similar range. This is due to the kinematic requirements imposed on the c¯ c for recombining, which are different in both models. Finally, let me indicate that heavy-flavor and quarkonium production can discriminate the different mechanisms of particle production. In fact, recombination can be formulated in the framework of statistical hadronization models. In the proposal by Rafelski et al.35,197 a sudden hadronization of the QGP could lead to strangeness over-saturation, which would imply an enhancement in the relative production of charmed-strange mesons and baryons over non-strange ones, with the corresponding diminution of the recombination probability for J/ψ formation. 4.3. Photons and dileptons
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description of photons in pQCD is far from being simple. There are contributions from the nuclear modification of parton densities, initial state energy loss, and final state energy loss which deviates the nuclear modification factor for direct photons (i.e. not coming from decays or conversions) from 1 — even in pA collisions, see Fig. 32 right, Vitev35 and Refs. 203 and 204. The elliptic flow coefficient v2 for photons is a most delicate measurement (a small signal affected by huge backgrounds) but offers great possibilities, as it is sensitive to the details of the mechanism of photon production. For example, the contributions from conversions (inverse Compton scattering205 ) is negative, see Fig. 33 top.206 It is also very sensitive, within hydrodynamical calculations, to the initial thermalization time.207–209 In Fig. 33 bottom, the different contributions to the
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photon v2 in the framework of ideal hydrodynamics are shown, from Chatterjee et al.35 The contribution from the QGP phase with respect to the hadronic phase is larger at the LHC than at RHIC, as expected. To conclude with photons, in Fig. 34 the different contributions (hard, thermaljet with and without energy loss, thermal, and fragmentation) to the transverse momentum spectrum of photons with pT > 8 GeV at the LHC, are shown210 (see also Ref. 211). The disentanglement of a thermal component on the background looks defying and demands a very detailed understanding of the background sources. Now I turn to dilepton production. Dileptons offer interesting information both in the low mass region M < 1 GeV and in the intermediate mass region 1 GeV < M < MJ/ψ , see e.g. Refs. 212, 213 and 214 and references therein. In the former they are expected to reflect the changes of resonances (masses, widths) in the medium. In the latter a window of sizable thermal emission has been speculated. In Fig. 35,35,215 the different contributions to the dilepton spectra at large (top) and small (bottom) transverse momentum is shown. The contribution from heavyquark decays seems to dominate all masses but M < 0.5 GeV, where hadronic contributions are very large. Therefore, the identification of thermal sources looks defying. A larger multiplicity should be linked with a larger temperature but also with a larger hadronic background. It seems that only a larger light multiplicity — if originating from a larger temperature — linked with a smaller heavy-quark cross section — leading to a smaller background — would improve the situation for detection of thermal dileptons in the intermediate mass region. All calculations shown until now for photons or dileptons in the low-pT region assume a very small thermalization time, < 1 fm, at which the system is isotropized/thermalized. The physical mechanisms which could make such fast
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isotropization feasible, are not understood yet. Therefore some authors have studied the possibility of a later isotropization time and a previous evolution in an anisotropic stage. For example, the authors in Refs. 216 and 217 consider a model in which the system evolves from an early formation time ∼ 0.1 fm in an anisotropic stage (a collisionally-broadened expansion) to an isotropization reached at 2 fm. They find a signal of such anisotropic behavior (depending on the kinematical cuts applied) in the enhancement of dileptons with large transverse momentum at y = 0, and a suppression of the pT -integrated yield (larger for forward rapidities), compared to the early isotropization scenario, see Fig. 36. Similar considerations for photons can be found in Ref. 218. To conclude, much information can be obtained from real and virtual photons at the LHC but an accurate understanding of backgrounds is required.y y The
ratio of real to virtual photons has been argued, Alam et al. in Refs. 35 and 219, to develop a plateau at transverse momentum greater than ∼ 2 GeV, quite insensitive to details of the model and reflecting the initial temperature of the system.
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5. pA Collisions While pA collisions will not take place until several successful data-taking heavyion runs have occurred, they offer a vast amount of information (see Ref. 9 and references therein) which finally may turn out to be essential for the interpretation of the PbPb data, as it was the case with dAu collisions at RHIC. They should establish the benchmark for the cold nuclear matter effects on top of which the eventual signals of a dense partonic stage are to be searched. I do not intend to give a full overview of all the possibilities of the pA programme, but rather focus on some selected aspects. First, pA collisions offer the possibility of constraining the nuclear parton densities in kinematical regions, see Fig. 2 right, which will not be explored in leptonnucleus collisions unless future colliders220,221 become eventually available. This is a key ingredient for hard probes, and the present situation of the parton densities in the x region of interest for the LHC (10−4 . x . 10−2 ) derived from DGLAP analysis (see e.g. the review222 or the recent work223 and references therein) is far from being satisfactory, see Fig. 37.223 As evident from this figure, the nuclear gluon densities at a low virtuality for x < 0.05 are very badly constrained.z The inclusion of pA data from the LHC in the fits can only improve this situation. On the other hand, isolated photons offer the possibility of a direct access to the gluon distribution (Fig. 38), see Arleo in Refs. 35 and 224, where it is shown that the nuclear modification factor for isolated photons closely follows the nuclear modifications factors for the gluon distribution and structure function F2 . z The
situation is similar for NLO analysis, see Ref. 223. On the other hand, DGLAP evolution reduces the uncertainties at larger scales.
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Another aspect which has raised large interest is the possibility to check the ideas of gluon saturation as proposed in the framework of the CGC, see the review in Ref. 33. Generically, the saturation scale which characterizes the momentum below which the gluon densities are expected to be maximal, is expected to increase with increasing rapidity or energy, reaching values in the range from 1 to several GeV in heavy-ion collisions at the LHC. Therefore saturation effects should become visible in a region usually considered within the range of applicability of pQCD. Specifically, the CGC predicts105,106 that the Cronin effect (the fact that
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Fig. 39. Left: RpA (y = 0) versus pT for pions in dAu collisions at RHIC (upper panel) and in pPb collisions at the LHC, by Kopeliovich et al. Solid and dashed lines correspond to the calculation with and without gluon shadowing respectively. Right: RpA for pions versus pT in dAu collisions at RHIC and in pPb collisions at the LHC, for different η, from Tuchin. Figures taken from Ref. 35.
the nuclear modification factor is larger than 1) observed at mid-rapidity in dAu collisions at RHIC disappears with increasing rapidity — as observed at RHIC, see Ref. 4 — and increasing energy. While there is no consensus on this suppression at forward rapidities at RHIC being a clear signal of saturation in the CGC, see e.g. Ref. 225 or Bravina et al. in Ref. 35 for an alternative approach to shadowing, pA collisions at the LHC offer the possibility of further tests. In Fig. 39 left, I show the predictions by Kopeliovich et al.35,226 in which the Cronin effect at mid-rapidity is still present in pPb collisions at the LHC. On the other hand, in Fig. 39 right predictions are shown within the CGC framework by Tuchin.35,227 While these predictions are for light flavors (see also De Boer et al.,35 or Refs. 228, 229 and 230 for predictions for Drell-Yan and photons), Tuchin also provides predictions for heavy-flavor production with similar features, namely a marked suppression of ratios both at mid- and forward rapidities in pPb collisions at the LHC. Clearly different scenarios should be discriminated by LHC data. Finally, in the framework of the CGC, the nuclear modification factor in pPb collisions at the LHC offers the possibility of establishing the relevance of different effects. For example, considering a running coupling instead of a fixed coupling in the CGC evolution equations at high energies or small momentum fractions x (the BK equation, see Ref. 33) leads231 to a nuclear modification factor at very large −(1−γ)/3 rapidities which goes from A ln A1/3 (γ ≃ 0.63, fixed coupling) to A−1/3 (running coupling, called total shadowing), with A the mass number of the nucleus. The same total shadowing is achieved when fluctuations (or pomeron loops, see Refs. 232 and 233) are included.234 Both effects are illustrated in Fig. 40.
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6. Summary and Discussion In this work I have reviewed the predictions for the heavy-ion programme at the LHC, as available in early April 2009. After an introduction I have discussed some qualitative expectations with the aim of illustrating how a single observable, namely charged multiplicity at mid-rapidity, influences predictions for the energy density and other thermodynamical quantities, the evolution of the system, predictions for elliptic flow (v2 ) or the nuclear modification factor (RAA ) in models of energy loss. Then I have turned to a compilation of results (additional information can be found in Refs. 9, 14–16, 27, 34–36). Referring to PbPb collisions at the LHC and, otherwise stated, to observables at mid-rapidity, a summary of what was presented is: (1) In Subsection 3.1 I have discussed the predictions for charged multiplicity at mid-pseudorapidity. Most predictions (for Npart ∼ 350, ∼ 10% more central collisions) now lie below 2000 — a value sizably smaller than pre-RHIC predictions,27 and they include a large degree of coherence in particle production through saturation, strong gluon shadowing, strong color fields, etc. On the other hand, the expectations for net protons at η = 0 are systematically below 4. (2) In Subsection 3.2 I have analyzed the results for elliptic flow in several models. pT -integrated v2 increases in all models when going from RHIC to the LHC, but this increase is usually smaller in hydrodynamical models than in naive expectations,34 and Section 2, and in some non-equilibrium, transport models. For v2 (pT ), hydrodynamical models indicate a value for pT . 2 GeV which is very close for pions, while a decrease is expected for protons. A strong decrease would be interpreted — once the initial conditions are settled — as an increase
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in viscous effects. On the other hand, non-equilibrium models generically result in an increase of v2 (pT ). In Subsection 3.3 predictions for hadrochemistry are reviewed. Different versions of the statistical models result in slightly different predictions, and nonequilibrium scenarios show distinctive features for resonance production. Hydrodynamical and recombination models predict large baryon-to-meson ratios at moderate pT . Approaches with strong color fields or percolation show Cronin effect for protons in central PbPb collisions at the LHC. In Subsection 3.4 I have reviewed the predictions for correlations. HBT radii are expected to increase from RHIC to the LHC. The predictive power of ideal hydrodynamics is reduced by the limitations that appear in its description of RHIC data. The role of viscosity in the hydrodynamical descriptions of HBT radii is still to be clarified. On the other hand, correlations in rapidity are expected to extend along large intervals and offer additional possibilities of constraining the multiparticle production mechanism. In Subsection 3.5 I have shown the existing predictions for multiplicity fluctuations — few predictions are available as the evidence of a non-statistical or non-trivial origin of fluctuations at SPS and RHIC is still under debate. Fluctuations also hold discriminative power between different mechanisms of particle production e.g. different statistical ensembles. In Subsection 4.1 I have enumerated the predictions for the nuclear modification factor for high-pT charged particles or pions in central collisions. They generically lie, for radiative or collisional energy loss models, in the range 0.15 ÷ 0.25 at pT = 20 GeV and increasing with increasing pT . Then I have commented on the possibilities of discriminating between the energy loss mechanism offered by jets, by hadrochemistry at large pT where several mechanisms like energy loss and parton conversions may be simultaneously at work, and by the study of correlations. In Subsection 4.2 results from different models with radiative or collisional energy loss for the nuclear modification factor of heavy flavors have been shown. They offer the possibility to further test the energy loss mechanism, as the energy diminution of a heavy quark traveling through the produced medium is different from that of a massless parton. On the other hand, predictions for quarkonium production are uncertain due to the lack of knowledge of both cold nuclear matter (nuclear parton densities and nuclear absorption) and hot nuclear matter (pattern of dissociation, recombination mechanism at work, etc.) effects. The identification of different quarkonium states and the large pT reach at the LHC, may help to settle the dissociation pattern and the role of recombination. But predictions for the nuclear modification factor of J/ψ are plagued with uncertainties due to e.g. nuclear shadowing or the c¯ c cross section for recombination. In Subsection 4.3 I have reviewed the available predictions for photon and dilepton production. While the large initial temperature or energy density implies a
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large yield of thermal real and virtual photons, the huge backgrounds make the disentanglement of a thermal component in the final spectrum challenging — a very precise knowledge of the pp baseline will be required. Effects beyond the usual equilibrium scenarios like anisotropies in the pre-equilibrium stage may modify the yields with respect to the early thermalization expectations. (9) In Section 5 I have analyzed the usefulness of the pA programme at the LHC for the purpose of reducing the uncertainties in the nuclear parton distributions which weaken the capabilities of hard probes to characterize the medium produced in the collisions. I have also discussed the possibilities of studies of high gluon density QCD through measurements of the nuclear modification factor in pPb collisions in a large rapidity interval. To put in context the predictions with respect to our current interpretation of existing data, let me draw a set of rough predictions for the LHC with respect to every ‘standard’ claim based on the experimental findings at RHIC (see Section 1): Finding at RHIC
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Obviously, neither the standard interpretations nor the predictions presented in this Table are free from problems and uncertainties, even more when the predictions tend to disagree with naive, data-driven expectations which would suggest multiplicities of order 1000, and sizably larger v2 (pT < 2 GeV) and smaller RAA than at RHIC. Finally, I find it tempting to speculate on possible scenarios based on the firstday measurement of charged particle production at mid-pseudorapidity in central PbPb collisions. Without any intention beyond showing how our understanding may become affected by the very first data and having in mind the present experimental situation and its ‘standard’ interpretation, three rough possibilities can be discussed: P bP b • A low multiplicity scenario, dNch /dη|η=0 < 1000, which would be close to the wounded nucleon model expectations and even smaller than most datadriven expectations. It would imply a extremely coherent particle production, difficult to describe even in saturation models. The conditions for collective flow would be relatively close to those at RHIC, and differentiating between naive extrapolations and hydrodynamical behaviors for v2 more involved, as their
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predictions would not be so different. On high transverse momentum particle production, the fact that the densities are close to RHIC ones, would imply that the difference e.g. in RAA from RHIC would be driven by the different transverse momentum spectra — the trigger bias, so the expectation would be an RAA larger than at RHIC for the same large transverse momentum (e.g. of the order or greater than 20 GeV). The low multiplicity implies a small background for jet and correlation studies. A small light multiplicity could also be a good scenario for recombination models for quarkonia (for a fixed heavy quark cross section). P bP b • An intermediate multiplicity scenario, 1000 < dNch /dη|η=0 < 2000 as predicted by most models with a large degree of coherence and by data-driven extrapolations. The differences between naive predictions and results of hydrodynamical models for v2 would be more noticeable. RAA should be more similar than at RHIC, for the same large transverse momentum, than in the previous scenario. P bP b • A large multiplicity scenario, 2000 < dNch /dη|η=0 . This scenario would defy naive extrapolations based on logarithmic increases and limiting fragmentation, and would be very problematic for saturation physics. Discriminating between naive predictions and results of hydrodynamical models for v2 should be relatively easy. In this case, a strong decrease of v2 at fixed small pT with respect to RHIC, would strongly suggest viscous effects. RAA at large pT , of the order or greater than 20 GeV, could be smaller than at RHIC for the same transverse momentum. Jet and correlation studies might be more defying due to the larger background. On the other hand, this scenario would imply larger temperatures and energy densities which may be welcome, even in spite of the larger background, for electromagnetic probes. To conclude, the heavy-ion programme at the LHC will offer most valuable information for improving our understanding of high-density QCD matter — and, in a wider context, on the behavior of the strong interaction at high energies — from the very first day of data taking. But it should be kept in mind that the usefulness of some observables will be restricted by our lack of knowledge of the pp and pA benchmarks, in particular to constrain the parton densities in nuclei. It seems plausible that a pA run will be needed — as it was the case at RHIC — in order to understand the effects of cold nuclear matter at LHC energies before strong conclusions about the heavy-ion programme can be drawn. A large amount of work has already been done to extrapolate existing models to the LHC situation. Still much work is needed in order to deal with some observables e.g. viscous hydrodynamical calculations or transport models for collective flow, or Monte Carlo tools for jet analysis, just to mention two obvious ongoing developments. The first LHC data will reduce much of the available freedom in model parameters. The more restricted model predictions done after those very first data will indicate, when confronted with subsequent data on other observables, whether
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the physics at the LHC is qualitatively similar to that at the SPS and RHIC or, on the contrary, new aspects appear which will require new ideas. Acknowledgements I thank J. Albacete, J. Alvarez-Muniz, F. Bopp, W. Busza, R. Concei¸cao, L. Cunqueiro, A. Dainese, J. Dias de Deus, A. El, D. d’Enterria, K. Eskola, V. Gon¸calves, U. Heinz, C.-M. Ko, I. Lokhtin, M. Martinez, J. G. Milhano, C. Pajares, V. Pantuev, T. Renk, E. Sarkisyan, V. Topor Pop, K. Tywoniuk, R. Venugopalan, I. Vitev, X. N. Wang, K. Werner and G. Wolschin for information on their predictions, P. Brogueira, J. Dias de Deus and J. G. Milhano for an updated version of a figure in their paper,113 and U. Heinz, A. Mischke, E. Saridakis, H. Song and D. Srivastava for useful comments. Special thanks are due to David d’Enterria, Guilherme Milhano, Carlos Pajares, Carlos Salgado and Konrad Tywoniuk for a critical reading of the manuscript. This work has been supported by Ministerio de Ciencia e Innovaci´ on of Spain under projects FPA2005-01963, FPA2008-01177 and a contract Ram´ on y Cajal, by Xunta de Galicia (Conseller´ıa de Educaci´on) and through grant PGIDIT07PXIB206126PR, and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). References 1. 2. 3. 4. 5. 6. 7.
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