Early stages of the heavy ion collision

1. Early stages of the heavy ion collision V.P. – for expedion Riesengebirge 07 Based on paper by D. Kharzeev

2. Introduction • There is something puzzling in the data on hadron production in various processes, from e+e− annihilation and deep–inelastic scattering to heavy ion collisions: the relative abundances of different hadron species appear to follow the statistical distribution with a surprising accuracy • Moreover, at small transverse momenta the spectra of the produced hadrons also look approximately thermal. Based on hep-ph/0511354

3. Why not other mechanisms • While in heavy ion collisions it is possible to expect the emergence of statistical distributions as a result of intense re–interactions between the produced particles, this seems very implausible in e+e− annihilation at high energies √s, where the process of hadronization is stretched in space over a long distance ∼ √s/2µ2 and the density of produced hadrons is small (µ ∼ _QCD is an infrared cut-off describing the hadronization scale). • Statistical distributions may emerge as a result of the saddle–point approximation to the multi–particle phase space, when the dynamics is inessential and the production mechanism is ”phase space dominated”. However, in e+e− annihilation this mechanism can hardly be expected to work • the jet structure, the angular distributions of the produced hadrons, and inter–jet correlations point to the all–important role of QCD dynamics of gluon radiation, and thus the ”phase space dominance” cannot be invoked.

4. What’s the matter • emergence of statistical hadron abundances has something to do with the process of hadronization – in other words, with the way in which the QCD vacuum responds to the external color fields • To do this, we are forced to discuss the structure of the QCD vacuum. Little is known about it, but we do know that QCD vacuum is populated by the fluctuations of color fields, some of which are of semi–classical nature • It may therefore be useful to examine what is known at present about the interactions of charged quanta with background classical fields

5. Quantum fluctuations in classical background • first lead us to the discussion of quantum fluctuations in the background of the gravitational field of a black hole, where the quantum radiation appears to be thermal as shown by Hawking • It was demonstrated by Unruh that the Hawking phenomenon should be present in any non–inertial frame; indeed, this is required by Einstein’s Equivalence Principle • Then we will proceed to the process of electron–positron pair production in the background electric field, analyzed in QED by Schwinger. We will see that the Schwinger formula for the rate of e+e− pair production in a constant electric field allows for a simple statistical interpretation • for the case of a time–dependent field pulse, the spectra of the produced particles become thermal.

6. These two examples inspire the picture in which the thermal character of hadron production emerges from the interactions with the vacuum chromo–electric field, the strength of which in the color flux model is parameterized in terms of the string tension, or the slope of Regge trajectories b • we will see how the well–known formula for the Hagedorn temperature THagedorn = √6/4π 1/√b can be derived in this way

7. What happens if we create semi–classical color fields of strength exceeding the strength of the vacuum fields? • This can be achieved in the collisions of heavy ions at high energies, which are accompanied by a short pulse of the chromo–electric field E ∼ Q2s /g of duration τ ∼ 1/Qs here Qs is the saturation scale Qs in the Color Glass Condensate and g is the strong coupling.

8. such a strong color field induces the creation of partons with a distribution which is isotropic and thermal in a co–moving local frame, with an effective temperature Tin ≃ Qs/2π • For high enough energies and heavy enough nuclei, the value of Tin exceeds the Hagedorn temperature; the produced thermal matter is thus in the deconfined phase • The phase transition in this case can also be understood in a geometrical picture, in which the accelerationa ≃ Qsin the chromo–electric field determines the curvature of space in the non–inertial Rindler frame. This is mathematically analogous to the phase transition induced by the presence of a massive black hole.

9. Black hole evaporation • Hawking demonstrated that black holes evaporate by quantum pair production, and behave as if they have an effective temperature TH = κ/2π where κ = 4M is the acceleration of gravity at the surface of a black hole of mass M. • The thermal character of the black hole radiation stems from the presence of the event horizon, which hides the interior of the black hole from an outside observer. • The rate of pair production in the gravitational background of a black hole can be evaluated by considering the tunneling through the event horizon.

10. Unruh effect • Unruh has found that a similar effect arises in a uniformly accelerated frame, where an observer detects an apparent thermal radiation with the temperature TU =a/2π (a is the acceleration). • The event horizon in this case emerges due to the existence of causally disconnected regions of space–time, conveniently described by using the Rindler coordinates.

11. The effects associated with a heat bath of usually are not easy to detect because of the smallness of the acceleration a in realistic experimental conditions. For example, for the acceleration of gravity on the surface of Earth g ≃ 9.8 m s−2 the corresponding temperature is only T ≃ 4 ×10−20 K. • If the energy spectrum of an accelerated observer is continuous, as is the case for a particle of mass m with a transverse (with respect to the direction of acceleration) momentum pT , we obtainthermal distribution in the ”transverse mass” mT =√( m2 + pT2) Wm(pT ) ∼ exp (−mT/TU )

12. An important example is provided by the dynamics of charged particles in external electric fields. Consider a particle of mass m and charge e in an external electric field of strength E. Under the influence of the Lorentz force, it moves with an acceleration a = eE/m, and the corresponding temperature is TU = a/2π. The Boltzman factor exp(−m/TU ) entering the particle creation rate in this case is WEm ∼ exp (−2 πm2/eE) differs from the classical Schwinger result for the rate of particle production in a constant electric field only by a factor of two in the exponent.

13. The rate of tunneling under the potential barrier in the quasi-classical approximation given by WEm = exp (−πm2/Ee) • This equation gives the probability to produce a particle and its antiparticle (each of mass m) out of the vacuum by a constant electric field E • The ratio of the probabilities to produce states of masses M and m is then WEM/WEm = exp ( −π (M2 − m2)/eE ) This relation allows a dual interpretation in terms of both Unruh and Schwinger effects

14. First, consider a detector with quantum levels m and M moving in a constant electric field. Each level is accelerated differently, however if the splitting is not large, M−m ≪ m we can introduce the average acceleration of the detector <a> = 2 eE/( M + m) • Substituting we arrive at WEM/WEm= exp ( 2 π (M − m)/ <a> ) • This expression is reminiscent of the Boltzmann weight in a heat bath with an effective temperatureT = <a>/2π. It implies that the detector is effectively immersed in a photon heat bath at temperature T ≈ eE/πm. This is the renown Unruh effect

15. It is important to remember that for a constant electric field the momentum distribution of the produced charged particles allows a statistical interpretation only in the transverse to the field direction • However for a short pulse of an electric field of duration τ ≤ m/(eE)the distribution becomes thermal in all three directions , physically, this happens because the field has not enough time to perform work on curving the momenta of the produced particles • The Hawking–Unruh interpretation therefore appears to capture an essentially non– perturbative dynamics

16. Consider a high–energy hadron of mass m and momentum P which interacts with an external field (e.g., another hadron) and transforms into a hadronic final state of invariant mass M ≫ m This transformation is accompanied by a change in the longitudinal momentum • and therefore by a deceleration; we assumed that the particle m is relativistic, with energy E ≃ p The probability for a transition to a state with an invariant mass M is given by transition amplitude density of hadronic final states

17. where b is the universal slope of the Regge trajectories, related to the string tension σ bythe relation σ = 1/(2πb)The unitarity dictates that the sum of the probabilities over all finite states Mshould be finite. This impose the following bound on the value ofacceleration a: Hagedorn temperature – the ”limitingtemperature of hadronic matter” derived traditionally from hadron thermodynamics.In our case it stems from the existence of a ”limiting acceleration” a0:

18. The meaning of the”limiting temperature”in hadron thermodynamics is well-known: above it, hadronic matter undergoes a phase transition into the deconfined phase, in which the quarks and gluons become the dynamical degrees of freedom To establish themeaning of the limiting acceleration, let us consider a dissociation of the incident hadron into a large number n ≫ 1 of partons. In this case the phase space density can be evaluated by the saddle point method (”statistical approximation”), with the result is determined by a typical parton momentum in the center-of-mass frame of the partonic configuration. When interpreted in partonic language,this implies a constant value of mean transverse momentum On the other hand, inthe parton saturation picture, the mean transverse momentum has to be associated withthe ”saturation scale” Qs determined by the density of partons in the transverse planewithin the wave functionof the incident hadron (or a nucleus).This leads to the phase space density The unitarity condition thus lead us to the acceleration a = Qs, which can exceed a0, and to the conclusion thatthe final partonic states are described by a thermal distribution with the temperature

19. The same result can be obtained by considering the acceleration a = gE/mof a partonwith off-shellnessm ≡ √p2 ≃ Qsin an external color field gE ≃ Q2s . It is interestingto note that to exceed the limiting acceleration, and thus the limiting Hagedorn temperature for the produced hadronic matter, one has to build up strong color fields, exceeding gE ∼ 1/b This is achieved by parton saturation in the Color GlassCondensate, when at sufficiently high energies and/or large mass numbersof the colliding nuclei Parton saturation in the initial wave functions thus seems to be anecessary pre-requisite for the emergence of a thermal deconfined partonic matter in the final state

20. The thermal distribution is built over the time period of • apparent thermalization originates from the presence of the eventhorizon in an accelerating frame: • the incident hadron decelerates in an external color field,which causes the emergence of the causal horizon. • Quantum tunneling through this eventhorizon then produces a thermal final state of partons, in complete analogy with thethermal character of quantum radiation from black holes.

21. Space-time picture of the HI collision hep-ph/0511354

22. In conventional space–time picture of a relativistic heavy ion collision the colliding heavy ions approach the interaction region along the light cone from x = t = −∞ and x = −t = ∞. The partons inside the nuclei in the spirit of the collinear factorization approach are assumed to have a vanishing transverse momentum kT , have a zero virtuality k2 = −k2T = 0, and thus are also localized on the light cone at ±x = t The collision at x = t = 0 produces the final state particles with transverse momenta pT which according to the uncertainty principle approach their massshell at a proper time

23. Consider now the case when partons in the wave functions of the colliding nuclei havenon-vanishing transverse momenta, as in the Color Glass Condensate picture where theirtransverse momenta are on the order of the saturation scale Qs. In this case the partonsare space–like k2 = −k2T and are located OFF the light cone As the colliding nucleiapproach each other, these partons begin to interact; note that since they are space–like,their interactions are acausal, and are responsible for the breakdown of factorization forthe parton modes with kT ≤ Qs The interactions of partons with the color fields of another nucleus decelerate them, with a typical acceleration|a| ≃ QsThe space–time trajectories of the partons are thus given by hyperbolae in Minkowski space, or by the lines with a fixed value of in Rindler space

24. For partons moving in the left (L) and right (R) sectors of space–time with an acceleration the light cone surfaces represent the event horizon of the future (F) The information from the futureis hidden from them, and the sector F is classically disconnected from L and R However,as discussed above, the future sector F can be reached from the left L and right R sectorsby quantum tunneling

25. this does not imply any ”bouncing” of the colliding nuclei - the on–shell particles, as well as the high–momentum valence partons from the colliding nuclei are transparent for each other - The soft components of the parton wave functions however do interact strongly. Nevertheless, the whole picture at first glancelooks completely orthogonal to the conventional parton model. Indeed, in parton modelthe Weizsacker–Williams gluon fields surrounding the valence quarks are transverse, with (p is the momentum of the quark). The corresponding ”equivalent gluons”are almost on mass shell, and no longitudinal chromo–electric field is present: the gluonfield tensor is flat in the longitudinal direction: F+− = 0 (”+” and ”−” refer to thelight–cone components). However, a more careful analysis reveals that this picture is not complete: the configuration of the gluon field produced in the sector ”F” of is characterized by F+− 6= 0, with a substantial longitudinal chromo–electric field Ez In the Color Glass Condensate picture, the strength of the field is and the duration of the pulse is

26. Summary Note that in the conventional string model picture, the produced chromo–electric field ispurely longitudinal, with the strength proportional to the string tension. This exhibits apossible continuity between the string and parton approaches to multi–particle production,and suggests the existence of the minimal allowed value of the saturation momentumQs min. Based on the arguments given above, one is led to the conclusion that • statistical features of multi–particle production may emerge as a consequence of the Hawking–Unruh effect.Theacceleration, and the emergence of the corresponding event horizon for partons, is caused by the pulse of chromo–electric field which accompanies inelastic interactions at high energies. • For hadron interactions at moderate energies, the effective temperature appears equal to the Hagedorn temperature • Once the strength of the chromo–electric field E ∼ Q2s /g exceeds a critical value the partons are produced with an effective temperature T > THagedorn, • i.e. in a deconfined state.

27. CGC backup slide hep-ph/0602063 The CGC approach is based on two ideas. The firstone is the structure of the parton cascade at high energy which is shown in Fig. The main contributionto the high energy scattering is given by a parton fluctuation in which all partons are strongly ordered intime The time structure of the parton cascade for a fast particle (nucleus) in the target rest frame

28. CGC The typical lifetime of this fluctuation at high energy of the projectile ε is large and is proportional to The interaction with the target of the size R destroys the coherence of the parton wave function ofthe projectile.The typical time, which is needed for this, is of the order of and it is much smaller than the lifetime of all faster partons in our fluctuation Therefore, this interaction cannot change the momentum distribution of the fast parton in the projectile wave function The influence ofthe target mostly manifests itself in the loss of coherence for majority of the partons; changes in momentaoccur only for a few very slow (‘wee’) partons The ‘wee’ parton part of the wave function together withinteraction with target could be factorized out while the energy dependence and distributions of the fast partons should not depend on the properties of the target. The CGC approach adds a very essential new idea to the parton cascade picture. Since all partonswith rapidity larger than y live longer than the parton with rapidity y, for a dense system suchas a nucleus they can be considered as the source of the classical field that emits a gluon with rapidity ydominant source of parton production is the longitudinal background field

29. hep-ph/0510232 apparent thermalization of quarks andgluons, leading to success of perfect fluid hydrodynamics in describing heavyion collisions at RHIC, can only be attributed to the non-perturbative QCDeffects notcaptured by Feynman diagrams. hep-ph/0608173 I review recent analytical and numerical advances in the study of non-equilibrium quark-gluonplasma physics. I concentrate on studies of the dynamics of plasmas which are locallyanisotropic in momentum space. In contrast to locally isotropic plasmas such anisotropic plasmashave a spectrum of soft unstable modes which are characterized by exponential growth oftransverse (chromo)-magnetic fields at short times. Parametrically the instabilities provide thefastest method for generation of soft background fields and dominate the short-time dynamicsof the system.

30. hep-ph/0506179 Hydrodynamic description requires a local thermodynamic equilibrium of the system understudy but an approximate hydrodynamic behaviour is already manifested when a momentumdistribution of liquid components is not of equilibrium form but merely isotropic. While theprocess of equilibration is relatively slow, the parton system becomes isotropic rather fast due tothe plasma instabilities. Azimuthal fluctuations observed in relativistic heavy-ion collisions areargued to distinguish between a fully equilibrated and only isotropic parton system producedin the collision early stage. The mechanism of filamentation instability

31. nucl-th/0401056 The real fields emerging in heavy-ion collisions act at the time interval comparable with the Compton scale. Since the dynamics of particle creation in time-dependent homogeneous fields differs from that of stationary fields [17], it is essential to properly modify the system description at early stage of nuclear collisions It is shown that dynamics of vacuum particle creation at time scale compatible with the inverse mass of particle depends essentially on the specific characteristics of field configuration (a form and a duration of the field pulse). In particular, the Schwinger-like regime of particle creation might not be realized at the typical time scale of QGP formation, _0 ∼ 1 fm/c. As a consequence, production of heavy strange quarks becomes more abundant. The role of string fusion and reduction of quark masses due to the chiral symmetry restoration which can alter the strangeness production is studied. Within the proposed dynamical scenario the mass reduction mechanism appears to be more effective for the enhancement of strangeness yield than the formation of color rope. Our study shows that the time evolution picture of the gluon field should be incorporated consistently into the color flux tube model for quantitative description of parton production in ultrarelativistic heavy ion collisions