Can Thermalization Happen at Small Coupling? Yuri Kovchegov The Ohio State University based on hep-ph/ and hep-ph/

Outline 1.First I will present a formal argument demonstrating that perturbation theory does not lead to thermalization and hydrodynamic description of heavy ion collisions. 2. Then I will give a simple physical argument showing that this is indeed natural: hydrodynamics may be achieved only in the large coupling limit of the theory.

Thermalization as Proper Time-Scaling of Energy Density Thermalization can be thought of as a transition between the initial conditions, with energy density scaling as with the proper time, to the hydrodynamics-driven expansion, where the energy density would scale as for ideal gas Bjorken hydro or as a different power of tau: however, hydro would always require for the power to be > 1:

Most General Boost Invariant Energy- Momentum Tensor The most general boost-invariant energy-momentum tensor for a high energy collision of two very large nuclei is which, due to gives cf. Bjorken hydro → Ifthen  No longitudinal pressure exists at early stages.

Most General Boost Invariant Energy- Momentum Tensor Deviations from the scaling of energy density, like are due to longitudinal pressure, which does work in the longitudinal direction modifying the energy density scaling with tau.  Non-zero longitudinal pressure ↔ deviations from If then, as, one gets.

Classical Fields Let us start with classical gluon fields produced in AA collisions. At the lowest order we have the following diagrams: The field is known explicitly. Substituting it into (averaging is over the nuclear wave functions) we obtain (McLerran-Venugopalan model)

Classical Fields At we can use the asymptotics of Bessel functions in to obtain (Bjorken estimate of energy density)  Lowest order classical field leads to energy density scaling as

Classical Fields: LO Calculation Q S   p  p 3  After initial oscillations one obtains zero longitudinal pressure with  p for the transverse pressure.

Classical Fields  All order classical gluon field leads to energy density scaling as from full numerical simulation by Krasnitz, Nara, Venugopalan ‘01 Classically there is no thermalization in AA.

Our Approach Can one find diagrams giving gluon fields which would lead to energy density scaling as Classical fields give energy density scaling as Can quantum corrections to classical fields modify the power of tau (in the leading late-times asymptotics)? Is there analogues of leading log resummations (e.g. something like resummation of the powers of  ln  ), “anomalous dimensions”? ?

Energy-Momentum Tensor of a General Gluon Field Let us start with the most general form of the “gluon field” in covariant gauge plug it into the expression for the energy momentum tensor keeping only the Abelian part of the energy-momentum tensor for now.

Energy Density of a General Gluon Field After some lengthy algebra one obtains for energy density defined as the following expression: We performed transverse coordinate averaging  function  f 1 is some unknown boost-invariant function, there are also f 2 and f 3.

Let us put k 2 =k’ 2 =kk’ =0 in the argument of f 1 (and other f’s). Integrating over longitudinal momentum components yields: As one can show i.e., it is non-zero (and finite) at any order of perturbation theory. Energy Density of a General Gluon Field

When the dust settles we get leading to We have established that  has a non-zero term scaling as 1/ . But how do we know that it does not get cancelled by the rest of the expression, which we neglected by putting k 2 =k’ 2 =kk’=0 in the argument of f 1 ?

Corrections to Energy Density For a wide class of amplitudes we can write with Then, for the 1 st term, using the following integral: we see that each positive power of k 2 leads to a power of 1/  such that the neglected terms above scale as  Corrections are subleading at large  and do not cancel the leading 1/  term. and Similarly one can show that the 2 nd term scales as

Corrections An analysis of corrections to the scaling contribution, can be summarized by the following approximate rules: (here  is the space-time rapidity)  The only tau-dependent corrections are generated by k 2,k’ 2 and (k+k’) 2. Since the k 2 =k’ 2 =(k+k’) 2 =0 limit is finite, corrections may come only as positive powers of, say, k 2. Using the first rule we see that they are suppressed by powers of 1/  at late times.

Non-Abelian Terms Now we can see that the non-Abelian terms in the energy-momentum tensor are subleading at late times: due to Bessel functions we always have such that the non-Abelian terms scale as and. They can be safely neglected. We have proven that at late times the hydrodynamic behavior of the system can not be achieved from diagrams, since

Generalizations: Rapidity-Dependent (“non-Boost Invariant”) Case We can generalize our conclusions to the rapidity-dependent distribution of the produced particles. First we note that in rapidity-dependent case Bjorken hydro no longer applies. However, in the rapidity-dependent hydro case we may argue that longitudinal pressure is higher than in Bj’s case leading to acceleration of particles in longitudinal direction and to energy density decreasing faster than in Bj case:

Rapidity-Dependent Case To prove that such proper time scaling can not be obtained from Feynman diagrams we note that rapidity-dependent corrections come in through powers of k + and k -. Since we need to worry only about powers of k +. Using we see that powers of k + do not affect the  dependence   logs are derivatives of powers, so the same applies to them) Therefore we get again and hydro appears to be unreachable in the rapidity-dependent case too.

Generalizations: Including Quarks We can repeat the same procedure for quark fields: starting from and, repeating the steps similar to the above, we obtain for the leading contribution to energy density at late times which leads to → No hydrodynamics for quarks either!

Energy Density Scaling It appears that the corrections to the leading energy scaling are suppressed by powers of . Therefore, any set of Feynman diagrams gives which means that longitudinal pressure is zero at small coupling and ideal (non-viscous) hydrodynamic description of the produced system can not result from perturbation theory!

Semi-Physical Interpretation Is this “free streaming”? A general gluon production diagram. The gluon is produced and multiply rescatters at all proper times. The dominant contribution appears to come from all interactions happening early. → Not free streaming in general, but free streaming dominates at late times.

Physical Argument Assume thermalization does take place and the produced system is described by Bjorken hydro. Let us put the QCD coupling to zero, g=0, starting from some proper time past thermalization time, i.e., for all    th. Put g=0 here. g>0

Puzzle  From full QCD standpoint, if we put g=0 the system should start free-streaming, leading to  However, as the equation of state reduces to that of an ideal gas,, Bjorken hydrodynamics, described by, leads to !? In the g=0 limit the system still does work in the longitudinal direction!?

Resolution The problem is with Bjorken hydrodynamics: the ideal gas equation of state,, assumes a gas of particles non-interacting with each other, but interacting with a thermal bath (e.g. a box in which the gas is contained, or an external field). Indeed, in a heavy ion collision there is no such external thermal bath, and the ideal gas equation of state is not valid for produced system.  Bjorken hydrodynamics is not the right physics in the g→0 limit and hence can not be obtained perturbatively.

I’m not saying anything new! Non-equilibrium viscosity corrections modify the energy-momentum tensor: Danielewicz Gyulassy ‘85 In QCD shear viscosity is divergent in the g→0 limit Arnold, Moore, Yaffe ‘00 invalidating ideal Bjorken hydrodynamics!

I’m not saying anything new! Therefore, in the g→0 limit, Bjorken hydrodynamics gets an o(1) correction, which tends to reduce the longitudinal pressure, putting it in line with free streaming: (Of course, divergent shear viscosity implies that other non-equilibrium corrections, which come with higher order derivatives of fluid velocity, are likely to also become important, giving finite pressure in the end.)

Is Bjorken Hydrodynamics Impossible?  Above we showed that scaling receives no perturbative corrections. Thus the answer to the above question may be “no, it is not possible”.  Alternatively, one may imagine an ansatz like It gives free streaming in the g→0 limit without any perturbative corrections, and reduces to Bjorken hydrodynamics if g→∞. In this case hydrodynamics is a property of the system in the limit of large coupling! Then the answer is “yes, it is possible”.

Conclusions Bad news: perturbative thermalization appears to be impossible. Weakly interacting quark-gluon plasma can not be produced in heavy ion collisions. Good news: non-perturbative thermalization is possible, leading to creation of strongly coupled plasma, in agreement with RHIC data. However, non-perturbative thermalization is very hard to understand theoretically (AdS/CFT?). More bad news: I know an easier problem – quark confinement. ☺

Backup Slides

Space-time picture of the Collision 1.First particles are produced: Initial Conditions 2.Particles interact with each other and thermalize forming a hot and dense medium - Quark-Gluon plasma. 3.Plasma cools, undergoes a confining phase transition and becomes a gas of hadrons. 4.The system falls apart: freeze out.

Thermalization: Bottom-Up Scenario  Includes 2 → 3 and 3 → 2 rescattering processes with the LPM effect due to interactions with CGC medium (cf. Wong).  Leads to thermalization over the proper time scale of  Problem: Instabilities!!! Evolution of the system may develop instabilities. (Mrowczynski, Arnold, Lenaghan, Moore, Romatschke, Randrup, Rebhan, Strickland, Yaffe) However, it is not clear whether instabilities would speed up the thermalization process. They may still lead to isothropization, generating longitudinal pressure needed for hydrodynamics to work. Baier, Mueller, Schiff, Son ‘00

Energy Loss in Instanton Vacuum An interesting feature of the (well-known) energy loss formula is that, due to an extra factor of  in the integrand, it is particularly sensitive to medium densities at late times, when the system is relatively dilute. At such late times instanton fields in the vacuum may contribute to jet quenching as much as QGP would: (assuming 1d expansion, see the paper for more realistic estimates)