High Energies Scattering in the AdS dual to “QCD” Lattice August 3 Richard C. Brower Boston University Progress since Lattice 2006: “The Pomeron and Gauge/String Duality” by Brower, Polchinski,Strassler & Tan (BPST) hep-th

(very) Few Related References  Flat space: ‘tHooft, “Graviton Dominance in Ultra-High-Energy Scattering” PL B198 (1987). Amati, Ciafaloni & Veneziano “Superstring Collisions at Plankian Energies”, PL B 197 (1987). Bo Sundborg, “High-Energy Asymptotics: The one-loop string amplitude and resummation” NP B306 (1988)  AdS 5 : D’Hoker, Freedman, Mathur, Matusis & Rastelli, “Graviton exchange and complete 4- point functions in the AdS/CFT correspondence” hep-th/ v1 Cornalba, Costa, Penedones & Schiappa, “Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions” hep-th/ v1 Alday & Maldacena “Gluon scattering amplitudes at Strong coupling” hep-th/ v1

Outline  Motivation  Dual 5-d Geometry of High Energy scattering  BFKL vs BPST Pomeron: ( log 2 (s) » ¸ = g 2 YM N c )  Eikonal for AdS 5 Gravity: ( ¸ >> log 2 s )  Eikonalization + confinement ) Froissart Bound

Phenomenological Motivation Diffraction production will dominate LHC events. Diffraction is a leading contender for the discovery of the Higgs! What is its rate? % of non-diffractive events fall like 1/E tot LHC Diffractive Higgs: Forward Proton 420m Exp. Of course Jets are often cleaner and Diffraction is still badly understood.

“Diffractive and Total Cross Section at Tevatron and LHC” (K. Goulianos hep-ex/ v1)

Theoretical Motivation QCD obeys the (non-perturbative) Froissart theorem: ¾ Tot (p+p ) X) = m -2 p C(m ¼ /m p ) log 2 (s/s 0 ) +  1.Is C(m ¼ /m p ) >0 ? What is its value? 2.What are the events that give C > 0? 3.Does the AdS/CFT provide a generic mechanism C>0? 4.Can one in principle compute C(m ¼ /m p ) on then “lattice”? Questions:

High Energy Elastic Scattering p1p1 p2p2 p3p3 p4p4 s = (p 1 + p 3 ) 2 t = (p 1 + p 2 ) 2 Optical Theorem: Regge:

N c ! 1 contributions The Pomeron ´ the vacuum exchange contribution to scattering at high energies at leading order in 1/N c expansion. where ¸ = g 2 N c & g s = 1/N c Definition:

BFKL: Balitsky & Lipatov; Fadin,Kuraev,Lipatov‘75  Sum diagrams 1 st order in g 2 N c and all orders (g 2 N c logs) n gives cut starting at j 0 = 1 + ¸ ln 2 / ¼ 2.  Accidentally “planar” diagrams (e.g. N c = 1 ) and conformal.  BKFL equation for 2 “reggized” gluon ladder is L = 2 SL(2,C) spin chain to one loop order.  BFKL is NOT a REGGE POLE! DIFFUSION “off shell k 2 > 0” GLUON “virtuality” k1k1 k2k2 k’ 1 k’ 2 ln s t = - (k 1 + k 2 ) 2 ¸ = g 2 N c ' 0

Moebius (aka SL(2,C)) invariance L 2-body Casimirs

AdS 5 /CFT Dictionary The 5 th dimension is conformal dilations

“Five” kinematical co-ordinate is size z / z’ of projectile/target 5 kinematical Parameters: 2-d Longitudinal p § = p 0 § p 3 ' exp[ § log(s/¤ qcd )] 2-d Transverse space: x’ ? - x ? = b ? 1-d Resolution: z = 1/Q (or z’ = 1/Q’) °*(Q 2 ) b1b1 b2b2 b?b?

Boosting AdS 5 to AdS 3 isometries with z = R 2 /r BFKL: SL(2,C) DIS : SL R (2,R)x SL L (2,R) O(4,2) isometries

High energy Graviton exchange Kernel is AdS 3 Green’s function Strong Pomeron kernel: same structure in J-plane!

N = 4 SYM Leading Twist ¢(j) vs J=j = 0 DGLAP (DIS moments)  = 0, BFKL (0,2) T   j = j min 

Eikonal Expansion +++ “sum” to get Born term

1.sum of leading large s contribution for perturbative series. 2.propagation in a shock wave gravitational background of target. (‘tHooft’s method) Again in AdS 5 space can do it both ways. We start with sources at the boundary and write down Witten (AdS 5 Feynman) diagrams for the “S-matrix” with a “hardwall” IR regulator. Two approaches to Eikonal Approximation

Witten Diagram Summation

AdS 5 Eikonal Sum We calculated explicitly the the box diagrams to see beginning of series expansion in \chi. The kinematics is basically the same as in Cheng-Wu’s classical paper from Note: 3-d “impact” space or Matrix eikonal

Shock Wave Eikonal Formulation p-2p-2 p+1p+1 1.Solve linearized Einstein equ: 2.Propagate across shock:

IR cut-off or Confining Hard Wall Model (quick and dirty example of confining duals) Large Sizes String/Glueball Add Confinement IR wall!

Broken scale invariance in the 5 th dimension r ! 1 r = r min r-r- r-r- r  -4 Hadron/GlueballMassive OniumCurrent  r) IR WALL

Kernel for hardwall at z =1 K hw /K conf z (z’ = 0.01) b?b?

Born Term for Hard Wall model B.C. K conf (z,z,x ? ) - K hw (z,z,x ? ) K hw (z,z,x ? )/K conf (z,z,x ? ) x?x? z=w x?x?

log(b) Weak BFKL AdS BFKL AdS Gravity log(s) Theory Parameters: N c & ¸ = g 2 N c

Concluding remarks  The KK modes represents a matrix version of eikonal formula like super string scattering of Amati, Ciafaloni and Veneziano ( “string bits are frozen” ).  Unitarization: Hardwall (confining) eikonal sum (probably) saturates the Froissart --- work in progress. (Brower, Strassler and Tan)  More central collisions require non-perturbative --- triple Regge, fan diagrams, black hole or plasma ball deconfinement region etc. See color glass condensate phase?

N = 4 SYM ´ AdS 5 x S 5 Open stings are Gluons dual to closed string Gravity. D3-branes Dynamics of N D3 branes at low energies is (Super) SU(N) YM. Their mass curves the space (near horizon) into AdS 5 and emits closed string (graviton) g  gravitons A  gluons

see “Total cross section at Tevatron and LHC” K. Goulianos hep ex/ v1