Department of Physics HIC from AdS/CFT Anastasios Taliotis

1. Department of Physics HIC from AdS/CFT Anastasios Taliotis Work done in collaboration with Javier Albacete and Yuri Kovchegov, arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph], arXiv:1004.3500 [hep-ph] (published in JHEP and Phys. Rev. C) 1

2. Outline Motivating strongly coupled dynamics Introduction to AdS/CFT I. AA: State/set up the problem Attacking the problem using AdS/CFT Predictions/comparisons/conclusions/Summary II. pA: State/set up the problem Predictions/Conclusions III. Transverse Dynamics-a quick look 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. Introduction to AdS/CFT 6

7. Scales & Parameters Type IIB superstring N =4 SYM SU(Nc) Q. gravity & fields Q. strings Clas. fields & part. Clas. Strings => (Ignore QM / small ) => Large Nc => (Ignore extended objects/small ) => Large λ 7

8. Quantifying the Conjecture [Witten ‘98] <exp z=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 8

9. How to use the correspondence • Take functional derivatives on both sides. LHS gives correlation functions. RHS is the machine that computes them (at any value of coupling!!). • Must write fields φ (that act as source in the CFT) as a convolution with a boundary to bulk propagator: φ(xμ,z)= ∫dxν'φ0 (xμ’)Δ(xμ – xμ’,z) 9

10. φ(xμ,z) being a field of string theory must obey some equation of motion; say □φ=0. Then Δ(xμ – xμ’,z) is specified solving □Δ=δ(xμ – xμ’) δ(z) Note: • Usually approximate string theory by SUGRA and hence Zs by a single point (saddle point); we approximated the large coupling gauge problem with a point of string theory!! Once we know Zs, we are done; can compute anything in CFT. 10

11. Holographic renormalization de Haro, Skenderis, Solodukhin ‘00 Example: • 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. • Then by varying the Zs w.r.t. the metric at the boundary (once at z=0) can obtain < Tμν >. 11

12. Holographic renormalization • 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. I. AA: State/set up the problem 13

14. Rmrks: • Deal with N=4 SYM theory • Coupling is tuned large and remains large at all times • Forget previous results of pQCD 14

15. Strategy Initial Tµν phenomenology Initial Geometry Evolve Einst. Eq. AdS/CFT Dictionary Dynamical Geometry Dynamical Tµν (our result) 15

16. Field equations, AdS5 & examples gμνTμν • Eq. of Motion (units L=1) for gΜΝ(xM = x±, x1, x2, z) is generally given ; empty space reduces • Empty & “Flat” AdS space: impliesTμν=0in QFT side • Empty but not flat AdS-shockwave: [Janik &Peschanski ’06] Then ~z4 coef. implies<Tμν(xμ)>= δμ - δν - < Tμν(x-)>in QFT side 16

17. Single nucleus Single shockwave Choose T-- (x-) a localized function along x- but not along ┴ plane. So take μ is associated with the energy carried by nucleus ([μ]=3). May represent the shockwave metric as a single vertex: a graviton exchange between the source (the nucleus living at z=0; the boundary of AdS) and point X in the bulk which gravitational field is measured. 17

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

19. Built up a perturbative approach • Motivation: Knowing gMN in the forward light cone we automatically know Tμν of QFT after the ion’s collision  just read it from ∂gMN (~z4 coefficient). • Know that Ti ~μi (i=1,2). Higher graviton exchanges; i.e. corrections to gMN should come with extra powers of μ1andμ2: μ1μ2, μ12μ2, μ1μ22, … • So reconstruct by expanding around the flat AdS: flat AdS, single shockwave(s), higher gravitons 19

20. Insight from Dim. Analysis, symmetries, kinematics & conservation Tracelessness + conservation Tμν(x+, x-) provide 3 equations. Also have x+ x- symmetry. Expect: For the case Ti =μiδ(x) shock-waves [μi]=3 and as energy density has [ε]=4 then we expect that the first correction to ε must be ε~ μ1μ2x+ x- i.e.rapidity independent as diagram suggests. 20

21. Calculation/results • Step 1.:Linearize field eq. expanding around ημν (partial DE with w.r.t. x+,x-, z with non constant coef.). • Step 2.:Decouple the DE. In particular g(2)┴┴=h(x+,x-, z ) obeys: □h=8/3 z6 t1(x-)t2 (x+) with box the d'Alembertianin AdS5. • Step 3.:Solve them imposing (BC) causality. Find: h= z4 ho(t1(x-) ,t2(x+)) + z6 h1(t1(x-) ,t2(x+)). • Step 4.:Use rest components of field eq. in order to determine rest components of gμν. • Step 5.:Determine Tμν by reading the z4 coef. of gμν • Conclude:Tμν has precisely the form we suspected for any t1, t2: Tμν is encoded in a single coefficient! 21

22. Particular sources (nucleus profile) • Only need ho: . Encodes Tμν. • δ profiles: Get corrections: T+ -~T┴┴ ~ ho ~μ1 μ2τ2 and T- - ~ μ1 μ2(x+)2 • Step profiles: Here δ’s are smeared; • At the nucleus will run out of momentum and stop! [Grumiller, Romatschke ’08] [Albacete, Kovchegov, Taliotis’08] 22

23. Conclusions/comparisons/summary • Constructed graviton expansion for the collision of two shock waves in AdS. Goal is obtain SE tensor of the produced strongly-coupled matter in the gauge theory. Can go to any finite order. Lower order hold for early times. • LO agress with [Grumiller, Romatschke ‘08]. NLO and NNLO corrections have been also performed. • They confirm: Tμν is encoded in a single coefficient h0(x+,x-). Also come with alternate sign. • Likely nucleus stops. A more detailed calculation (all order ressumation in A) in pA[Albacete, Taliotis, Yu.K. ‘09] confirms it. • Possibly have Landau hydro. However its Bjorken hydro that describes (quite well) RHIC data. 23

24. Landau vs Bjorken Bjorken hydro: describes RHIC data well. The picture of nuclei going through each other almost without stopping agrees with our perturbative/CGC understanding of collisions. Can we show that it happens in AA collisions using field theory or AdS/CFT? Landau hydro: results from strong coupling dynamics at all times in the collision. While possible, contradicts baryon stopping data at RHIC. 24

25. II. pA: State/set up the problem 25

26. pA collisions Diagrammatic Representation Initial Condition vertex cf. gluon production in pA collisions in CGC! Scalar Propagator Eq. for transverse component: Multiple graviton ex. vertex ~ t2 26

27. Eikonal Approximation &Diagrams Resummation • Nucleus is Lorentz-contracted and so are small; hence ∂+is large compared to ∂- and ∂z. • This allows to sub the vertices and propagators with effectives and simplify problem. For more see [Kovchegov, Albacete, Taliotis’09]. • Apprxn applies for 27

28. Calculation (δ profiles) • Particular profiles: • Diagram ressumation (all orders in μ2) in the forward LC yields: • Recalling the duality mapping: • Finally recalling ho;ei encodes <Tμν> through yields to the results: 28

29. Results 29

30. Conclusions • Not Bjorken hydro Indeed instead of T┴ ┴=p ~1/τ4/3 it is found that • Not (any other) Ideal Hydrodynamics either Indeed, from and considering μ=ν=+ deduce that T++ >0; however T++ is found strictly negative! • Proton stopping in pA also For AA, it was found earlier that with estimation stopping time estimated by . Same result recovered here by considering the total T++and expanding to O(μ2;x-=α/2): (Landau Hydro??) 30

31. Proton Stopping(Landau Hydro??) T++ X+ 31

32. Future Work • Use CGC as initial condition in order to evolve the metric to later times! Ambiguities Many initial metrics give same initial condition. Choose the simplest? • Include transverse dynamics? Very hard but… 32

33. Recent WorkarXiv: 1004.3500v1[hep-th] - [Taliotis] Snapshot of the collision at given proper time τ • Causality separates evolution in a very intuitive way! • General form of SE tensor: For given proper time τ it has the form

34. Eccentricity-Momentum Anisotropy Momentum Anisotropy εx= εx (x≡τ/b) (left) and εx = εx(1/x) (right) for intermediate x≡τ/b . Agrees qualitatively with [Heinz,Kolb, Lappi,Venugopalan,Jas,Mrowczynski] 34

35. Conclusions • Built perturbative expansion of dual geometry to determine Tµν ; applies for sufficiently early times: µτ3<<1. • Tµν evolves according to causality in an intuitive way! Also Tµν is invariant under . • Our exact formula (when applicable) allows as to compute Spatial Eccentricity and Momentum Anisotropy . • When τ>>r1 ,r2 have ε~τ2 log 2 τ-compare with ε~Qs2log 2τ [Gubser ‘10] [Lappi, Fukushima] 35

36. Thankyou 36