AdS/CFT: Unification of Soft Pomeron and BFKL Confronting String Theory with High Energy Scattering Blois 2005: Chung-I Tan, Brown University Gauge/Gravity Duality Re-discovery of QCD String and Extra Dimensions Brower, Mathur, Tan, hep-th/ , Glueball Spectrum for QCD from AdS Supergravity duality R. Brower, J. Polchinski, M. Strassler, C-I Tan,, to appear.R. Brower, J. Polchinski, M. Strassler, C-I Tan, BFKL Kernel in String/Gauge Dual, to appear.

Outline Many Faces of QCD From ISR to HERA/Tevatron/LHC: Hard versus Soft Diffraction AdS/CFT: Re-birth of QCD-String Unification of BFKL with Soft Pomeron Odderon Glueballs in Deformed AdS Expectations: from Tevatron to LHC and Beyond

Many Faces of QCD

From s=10 GeV 2 to 10 6 GeV 2 Lower Energies: Soft Pomeron Shrinkage in forward peaks Higher Energies: BFKL Large angle scattering, Jets, etc. Large-N regularities

I. Review: QCD as String Asymptotic Freedom Confinement Force at Long Distance--Constant Tension/Linear Potential, Coupling increasing, Quarks and Gluons strongly bound “Stringy Behavior”

String Modes: Vibrational Modes and Rotational Modes

Dual Pion Amplitude (aka NS string) If   (m 2  ) = 0, then have Adler zeros in soft pion limit: A(p 1 ! 0) = 0, since s ! M 2 , t ! m 2  ( y Neveu-Schwarz “Quark model of dual pions”, 1971)

Death and Resurrection of QCD string (i) ZERO MASS STATE (gauge/graviton) (ii) SUPER SYMMETRY (iii) EXTRA DIMENSION 4+6 = 10 (iv) NO HARD PROCESSES! (totally wrong dynamics ) Stringy Rutherford Experiment At Wide Angle: s,-t,-u >> 1/  ’

Back to QCD: Need to give Mass to Graviton t=0t>0t<0 J Regge  (t) 2 ++ Graviton 1 -- Photon/Gluon Closed String Open String ?

Back to QCD: Need Hard Collisions Fixed Angle Dimensional Counting rules BFKL Etc.

II. String/Gauge Duality Early String Theories ==> Theory of Everything Maldacena ==> New Era for YM Theories

Maldacena’s String Counter Revolution Open String Closed String duality YM Wilson Loops Expectation Values in Bulk!

Scale Invariance and the 5 th dimension Strings (Gravity) in AdS 5 ´ (SUSY) Yang Mills String/Glueball Large Sizes

Approx. Scale Invariance and the 5 th dimension r ! 1 r = r min r-r- r-r- r  -4 Hadron Glueball Massive Onium Current  r) IR WALL

III. How to give Mass to Graviton? t=0t>0t<0 J Regge  (t) 2 ++ Graviton 1 -- Photon/Gluon Closed String Open String ?

4-Dim Massive Graviton 0= E 2 - (p p p p r 2 ) 5-Dim Massless Mode: If, due to Curvature in fifth-dim, p r 2  0, E 2 = (p p p 3 2 ) + M 2 E 2 = (p p p 3 2 ) + M 2 Four-Dimensional Mass:

IV. Wide Angle Scattering The 2-to-m glueball scattering amplitude T(p 1,p 2, , p m+2 ) for plane wave glueball: This is a check on the underlining universality of Maldacena's duality conjecture. scatter via the string(M-theory) amplitude: A(p i, r i, X i ) in the 10-d (or 11-d) bulk space (x,r,Y): AdS 5 £ X with IR cut-off on r > r min or 10-d IIB string theory AdS 7 £ S 4 $ Black Hole with horizon r = r min or 11-d M-theory. We now discuss two different approaches to the QCD string that both give the correct parton scaling formula.

10-d String theory Approach: Polchinski and Strassler Due to the Red Shift in the Warped Co-ordinate,  s = (R/r)  x,a plane wave glueball, exp[i x p], scatters with a local proper momentum, String is UV shifted in the YM’s IR. (This is the so called UV/IR connection.) THUS wide angle scattering IS exponentially suppressed in the region r 2 [r min,r scatt ] HOWEVER there is a small remaining amplitude at large r that that gives the correct conformal scaling of the naive parton model! E.g for a scalar glueball  » r -4 corresponding to n i = 4 for the YM operator, Tr[F 2 ], in exact agreement with the parton result.

Soft vs Hard in M-QCD (RCB & CIT hep-th/Tan )   Red Shift: Proper Length:  s = (r/R)  x Local Momentum: p local  = (R/r) p  Therefore at wide angles! A string (  ’ R 2 s/r 2,  ’ R 2 t/r 2 ) » exp[ - R 2 s log(s) /r 2 ] A Regge region is an average like:

Hard vs Soft Regge in M-QCD (RCB & C-I Tan hep-th/Tan ) Hard IR region: BFKL-like Pomeron with almost flat cut in the j-plane Soft: IR region: r ' r min, gives Regge pole with slope  ' qcd »  ' R 3 /r 3 min The ``shrinkage'' is caused the soft stringy ``form factor'' in impact parameter:

V: BFKL vs Soft Pomeron t=0t>0t<0 J BFKL continuum Regge  (t) 2 ++ Glueball Strong Coupling: Brower, Polchinski, Strassler, Tan Weak Coupling:

BFKL Summation

Diffusion in Impact Space

BFKL vs Soft Pomeron Perturbative QCD Short-Distance  BFKL (0) ~ 1.4 Increasing Virtuality No Shrinkage of elastic peak Fixed-cut in t Diffusion in Virtuality Non-Perturbative Long-distance: Confinement  P (0) ~ 1.08 Fixed trans. Momenta Shrinkage of Elasstic Peak: ~1/ log s  ’(0) ~ 0.3 Gev -2 Diffusion in impact space

Hard versus Soft Diffraction in AdS (Lightcone Derivation) With X + = 

The Schwarz-Christoffel trans maps the upper half plane (a) into the light-cone strip  + i  (b):

Reduction to 1-d Path Integral where

b 12 b 34 b X1X1 X2X2 Rapidity y = log(s/s 0 ) and t = - q 2 ? Regge Behavior is diffusion for time log(s) in impact parameter space (and AdS radial space) exp[ -  ’ q 2 ? log(s) ]  exp[ - b 2 /(  ’ log(s))] Boosts shrink size of “hadronic string”

Diffusion in AdS 5 u=-log Z----translational invariance

VI: Strong Coupling Pomeron V(u) = -t e -u 0 < u < 1 Attractive for t >0, Regge Pole + BFKL cut t < 0 only scattering state for BFKL, with t < 0 t >0 V(u) u Hard Wall at r = r min

BFKL-Soft Pomeron Unification t=0t>0t<0 J BFKL continuum Regge  (t) 2 ++ Glueball Strong Coupling: Weak Coupling:

Leading J-Plane Singularities H(t)  = 1/2 (2-J) 

Leading J-Plane Singularities for C=-1 H odd (t)  = 1/2 (1-J) 

VII. Glueballs at g 2 N c = 1 Strong coupling Dual to Gravity

Deformed AdS-Metric: Confinement Deformed AdS Space, Massive Graviton Soft Pomeron and Tensor Glueball QCD String Tension Breaking Supersymmetry Witten’s proposal---AdS/BH metric

AdS/BH Metric

Deformed AdS and confinement

Gravity vs Y.M. on Branes

IIA Classification of QCD_4 G  G ,11 G 11,11 m 0 (Eq.)A ,11 A  m 0 (Eq.) G ij 2 ++ C i 1 ++ (-)  (T 4 ) B ij 1 +- C (-) (N 4 ) G i  1 -+ (-) C  (V 4 ) B i  1 -- (-) C ij  (M 4 ) G  (S 4 ) G   (L 4 ) States from 11-d G MN States from 11-d A MNL Subscripts to J PC refer to P  = -1 states

Tensor Glueball/Graviton Wave functions n=1n=3 n=8Potential n=0 Randall-Sundram graviton r min r

Lattice Data vs AdS Confining Gauge Theory at  ’ = 0 R. Brower, S. Mathur, and C-I Tan, hep-th/ , “Glueball Spectrum of QCD from AdS Supergravity Duality”.

Pomeron in Gauge/Gravity Duality Pomeron as Massive Graviton! Intercept “lowered” from 2! Interpolate with BFKL due to “hard” at AdS Bdry.

Summary Unified Description of BFKL-Soft Pomeron in AdS/CFT ---Heterotic Pomeron (E. M. Levin & C-I Tan) Improved Confining Background Massless Quarks, Chiral Symmetry, …….. ………….. Beyond strong coupling or Strong Gravity Beyond Perturbative String Interactions …….