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

22 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

33 Motivating strongly coupled dynamics in HIC

44 Notation/Facts Proper time: Rapidity: Saturation scale : The scale where density of partons becomes high. QGP CGCCGC  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 valid for times t >> 1/Q s Bj Hydro g<<1; valid up to times  ~ 1/Q S.

55 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.

66 Introduction to AdS/CFT

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

88 Quantifying the Conjecture <exp z=0 ∫O φ 0 > CFT = Zs(φ |φ(z=0)= φo ) O is the CFT operator. Typically want <O 1 O 2 …O n > φ 0 =φ 0 (x 1,x 2,…,x d ) is the source of O in the CFT picture φ =φ (x 1,x 2,…,x d,z) is some field in string theory with B.C. φ (z=0)= φ 0 [Witten ‘98]

99 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 (x μ,z)= ∫dx ν' φ 0 (x μ’ )Δ(x μ – x μ’,z)

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

11 Holographic renormalization Know the SE Tensor of Gauge theory is given by So g μν acts as a source => in order to calculate T μν from AdS/CFT must find the metric. Metric has its eq. of motion i.e. Einsteins equations. Then by varying the Z s w.r.t. the metric at the boundary (once at z=0) can obtain. Example: de Haro, Skenderis, Solodukhin ‘00

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

13 I. AA: State/set up the problem

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

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

16 Field equations, AdS 5 & examples g μν T μν  Eq. of Motion (units L=1) for g ΜΝ (x M = x ±, x 1, x 2, z) is generally given ; empty space reduces  Empty & “Flat” AdS space: implies T μν =0 in QFT side  Empty but not flat AdS-shockwave: [Janik & Peschanski ’06] Then ~z 4 coef. implies = δ μ - δ ν - in QFT side

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.

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

19 Built up a perturbative approach Motivation: Knowing g MN in the forward light cone we automatically know T μν of QFT after the ion’s collision  just read it from ∂g MN (~z 4 coefficient). Know that T i ~μ i (i=1,2). Higher graviton exchanges; i.e. corrections to g MN should come with extra powers of μ 1 and μ 2 : μ 1 μ 2, μ 1 2 μ 2, μ 1 μ 2 2, … So reconstruct by expanding around the flat AdS: flat AdS, single shockwave(s), higher gravitons

20 Insight from Dim. Analysis, symmetries, kinematics & conservation Tracelessness + conservation T μν (x +, x - ) provide 3 equations. Also have x +  x - symmetry. Expect: For the case T i =μ i δ(x) shock- waves [μi]=3 and as energy density has [ε]=4 then we expect that the first correction to ε must be ε~ μ 1 μ 2 x + x - i.e.rapidity independent as diagram suggests.

21 Calculation/results Step 1.: Linearize field eq. expanding around η μν (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 z 6 t 1 (x - ) t 2 (x + ) with box the d'Alembertian in AdS 5. Step 3.: Solve them imposing (BC) causality. Find: h= z 4 h o (t 1 (x - ), t 2 (x + )) + z 6 h 1 (t 1 (x - ), t 2 (x + )). Step 4.: Use rest components of field eq. in order to determine rest components of g μν. Step 5.: Determine T μν by reading the z 4 coef. of g μν Conclude: Tμν has precisely the form we suspected for any t1, t2: Tμν is encoded in a single coefficient!

22 Particular sources (nucleus profile) Only need h o :. Encodes T μν. δ profiles: Get corrections: T + - ~T ┴┴ ~ h o ~μ 1 μ 2 τ 2 and T - - ~ μ 1 μ 2 (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]

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: T μν is encoded in a single coefficient h 0 (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.

24 Landau vs Bjorken Landau hydro: results from strong coupling dynamics at all times in the collision. While possible, contradicts baryon stopping data at RHIC. 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?

25 II. pA: State/set up the problem

26 pA collisions Eq. for transverse component: Diagrammatic Representation Scalar Propagator Multiple graviton ex. vertex ~ t 2 Initial Condition vertex cf. gluon production in pA collisions in CGC!

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

28 Calculation (δ profiles) Particular profiles: Diagram ressumation (all orders in μ 2 ) in the forward LC yields: Recalling the duality mapping: Finally recalling h o;ei encodes through yields to the results:

29 Results

30 Conclusions Not Bjorken hydro Indeed instead of T ┴ ┴ =p ~1/τ 4/3 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(μ 2 ;x - =α/2): (Landau Hydro??)

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

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…

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

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

Conclusions  Built perturbative expansion of dual geometry to determine Tµν ; applies for sufficiently early times: µτ 3 <<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~τ 2 log 2 τ-compare with ε~Qs 2 log 2 τ 35 [ Gubser ‘10 ] [Lappi, Fukushima]

36 Thank you