Department of Physics HIC with Dynamics ┴ from Evolving Geometries in AdS

1. Department of Physics HIC with Dynamics┴ from Evolving Geometries in AdS arXiv: 1004.3500 [hep-th], AnastasiosTaliotis Partial Extension of arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph] (published in JHEP and Phys. Rev. C) [ Albacete, Kovchegov, Taliotis] 1

2. Outline Motivating strongly coupled dynamics in HIC AdS/CFT: What we need for this work State/set up the problem Attacking the problem using AdS/CFT Predictions/comparisons/conclusions/Summary Future work 2

3. Motivating strongly coupled dynamics in HIC 3

4. Notation/Facts Proper time: Rapidity: Saturation scale : The scale where density of partons becomes high. valid for times t >> 1/Qs Bj Hydro QGP • CGC describes matter distribution due to classical gluon fields and is rapidity- independent ( g<<1, early times). • Hydro is a necessary condition for thermalization. Bjorken Hydro describes successfully particle spectra and spectral flow. Is g??>>1 at late times?? Maybe; consistent with the small MFP implied by a hydro description. • No unified framework exists that describes both strongly & weakly coupled dynamics g<<1; valid up to times t ~ 1/QS. CGC 4

5. Goal: Stress-Energy (SE) Tensor • SE of the produced medium gives useful information. • In particular, its form (as a function of space and time variables) allows to decide whether we could have thermalization i.e. it provides useful criteria for the (possible) formation of QGP. • SE tensor will be the main object of this talk: we will see how it can be calculated by non perturbative methods in HIC. 5

6. Most General Rapidity-Independent SE Tensor The most general rapidity-independent SE tensor for a collision of two transversely large nuclei is (at x3 =0) which, due to gives • We will see three different regimes of p3 6

7. Early times : τQs <<1 • CGC III. Much later times:τQs >>>1 Bjorken Hydrodynamics • Later times : τ>~1/Qs • CGC 0 p(τ) Isotropization thermalization [Free streaming] [Lappi ’06 Fukushima ’07: pQCD] [Talıotıs ’10: AdS/CFT] • Classical gluon fields • Pert. theory applies • Describes RHIC data well • (particle multiplicity dN/dn) • Classical gluon fields • Pert. Theory applies • Energy is conserved • Hydrodynamic description • Does pert. Theory apply?? • Describes data successfully • (spectra dN/d2pTdn for K, ρ, n & elliptic flow) • [Heınz et al] [Krasnitz, Nara,Venogopalan, Lappi, Kharzeev, Levin, Nardi] 7

8. If then, as , one gets . Bjorken Hydro & strongly coupled dynamics Deviations from the energy conservation are due to longitudinal pressure, P3 which does work P3dV in the longitudinal direction modifying the energy density scaling with tau. • It is suggested that neither classical nor quantum gluonic or fermionic fields can cause the transition from free streaming to Bjorken hydro within perturbation theory.[Kovchegov’05] • On the other hand Bjorken hydro describe simulations satisfactory. • Conclude that alternative methods are needed! 8

9. AdS/CFT: What we need for this work 9

10. Quantifying the Conjecture <expz=0∫O φ0>CFT =Zs(φ|φ(z=0)= φo) O is the CFT operator. Typically want <O1 O2…On> φ0 =φ0 (x1,x2,… ,xd) is the source of O in the CFT picture φ=φ(x1,x2,… ,xd,z) is some field in string theory with B.C. φ(z=0)=φ0 10

11. Holographic renormalization [Witten ‘98] Example: • Quantifying the Conjecture<expz=0∫O φ0>CFT =Zs(φ|φ(z=0)= φo) • Know the SE Tensor of Gauge theory is given by • So gμν acts as a source => in order to calculate Tμν from AdS/CFT must find the metric. Metric has its eq. of motion i.e. Einsteins equations. 11

12. Holographic renormalization de Haro, Skenderis, Solodukhin ‘00 • Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as with z the 5th dimension variable and the 4d metric. • Expand near the boundary (z=0) of the AdS space: • Using AdS/CFT can show: , and 12

13. State/set up the problem 13

14. Strategy Initial Tµν phenomenology Initial Geometry Evolve Einstein's Eq. AdS/CFT Dictionary Dynamical Geometry Dynamical Tµν (our result)

15. Field equations, AdS5 shockwave; ∂gMNTμν • Eq. of Motion (units L=1) for gΜΝ(xM = x±, x1, x2, z) is generally given • AdS-shockwave with bulk matter: [Janik &Peschanski’06] Then ~z4coef. implies<Tμν(xμ)> ~ -δμ + δν + µlog(r1) δ(x+) in QFT side Corresponding bulk tensor JMN : 15

16. Single nucleus Single shockwave The picture in 4d is that matter moves ultrarelativistically along x- according to figure. Einstein's equations are satisfied trivially except (++) component; it satisfies a linear equation: □(z4t1)=J++ This suggests may represent the shockwave metric as a single vertex: a graviton exchange between the source J++(the nucleus living at z=0; the boundary of AdS) and point XM in the bulk which gravitational field is measured. 16

17. 4D Picture of Collision

18. Superposition of two shockwavesNon linearities of gravity ? Higher graviton ex. Due to non linearities 18 Flat AdS One graviton ex.

19. Back-to-Back reactions for JMN • In order to have a consistent expansion in µ2 we must determine • We use geodesic analysis • Bulk source J++(J--) moves in the gravitational filed of the shock t1(t2) • Important: is conserved iff b≠0 Self corrections to JMN

20. Calculation/results • Step 1: Choose a gauge:Fefferman-Graham coordinates • Step 2:Linearize field eq. expanding around 1/z2ηMN (partial DE with w.r.t. x+,x-, z with non constant coef.). • Step 3:Decouple the DE. In particular all components g(2)µν obey: □g(2)µν =A(2)µν(t1(x-) ,t2 (x+) ,J) with box the d'Alembertianin AdS5. • Step 4:Solve them imposing (BC) causality-Determine the GR • Step 5:Determine Tμν by reading the z4coef. of gμν • Side Remark:Gzzencodes tracelessness of Tµν Gzνencode conservation of Tµν 20

21. The Formula for Tµν

22. Eccentricity-Momentum Anisotropy Momentum Anisotropy εx= εx(x) (left) and εx= εx(1/x) (right) for intermediate . Agrees qualitatively with [Heinz,Kolb, Lappi,Venugopalan,Jas,Mrowczynski]

23. Conclusions • Built perturbative expansion of dual geometry to determine Tµν; applies for sufficiently early times: µτ3<<1. • Tµνevolves according to causality in an intuitive way! There is a kinematical window where is invariant under . • Our exact formula (when applicable) allows as to compute Spatial Eccentricity and Momentum Anisotropy . [Gubser ‘10]

24. When τ>>r1,r2 have ε~τ2log2 τ-compare with ε~Q2slog2 τ • Despite J being localized, it still contributes to gµν and so to Tµν not only on the light-cone but also inside. • Impact parameter is required otherwise violate conservation of JMN and divergences of gµν. Not a surprise for classical field theories. • Our technique has been applied to ordinary (4d) gravity and found similar behavior for gµν. • A phenomenological model using the (boosted) Woods-Saxon profile: Taliotis’10 MS thesis.dept. of Mathematics, OSU [Gubser,Yarom,Pufu ‘08] Note symmetry under when b=0; [Gubser’10] • For τ> r1,r2

25. Thankyou

26. Supporting slides

27. O(µ2) Corrections to Jµν Remark: These corrections live on the forward light-cone as should!

28. Field Equations