High Energy Scattering and String /Gauge Duality Evidence for QCD as an Effective String Theory Maldacena’s Counter Revolution: Gauge/Gravity Duality Re-discovery of QCD String and Extra Dimensions Glueball Spectrum; Hard vs Soft Pomeron in AdS Background Stretched Strings Confronting String Theory with High Energy Scattering ISMD2004, Sonoma State, California Chung-I Tan, Brown University

QCD in Strong Coupling and HE Scattering Maldacena, hep-th/ , The Large N Limit of Superconformal Field Theory and Supergravity Brower, Mathur, Tan, hep-th/ , Glueball Spectrum for QCD from AdS Supergravity duality Polchinski and Strassler, hep-th/010974, Hard Scattering and Gauge/String Duality; hep-th/ , Deep Inelastic Scattering and Gauge/String Duality; Polchinski and Susskind, hep-th/ , String Theory and the Size of hadrons Brodsky and de Teramond, hep-th/ , Light-Front Hadron Dynamics and AdS/CFT Correspondence Brower and Tan, hep-th/ , Hard Scattering in the M-theory dual for the QCD String Brower, Lowe and Tan, hep-th/ , Hagedorn transition for strings on pp-wave ad tori with chemical potentials Brower and Tan, in preparation, Stretched String in an AdS Black hole Background and Light cone quantization Some References

I. Evidence for 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

Tests for String Excitations Open String Mesons Closed String Glueballs String Junction Baryons Open String with Fixed Sources Can be tested using Lattice Gauge Theory: Discrete sites ---> Finite Variables Positive Euclidean Action ---> Numerical Simulation Phenomenological tests ISMD proceedings last 30 years

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

“Failure” of Superstrings in Flat-Spacetime (1973~1998) Zero mass states: (gauge/graviton) Extra dimensions: D=4+6=10 Supersymmetries: No Hard Scattering:

AdS/CFT Duality (1998--) Strong-Weak Dual Example: Ising-model--(high-low temp.) 4d “Gauge” dual to “Geometry” in higher dim Absence of 4 dim-Graviton Wilson loop expectation ~ sum over surfaces in the bulk with loop in boundary 4d YM Theories at weak coupling is dual to higher dim String Theories with deformed AdS Background.

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

Wilson Loop in AdS/CFT For rectangular loop: Extension into minimum surface in r-dir Surface into AdS space Zero Tension for QCD String---> Unacceptable

III. Re-discovering QCD String Witten’s proposal---AdS/BH metric Deformed AdS Space, Massive Graviton Soft Pomeron and Tensor Glueball QCD String Tension Breaking Supersymmetry

Witten Proposal

AdS/BH metric

VI. QCD in Strong Coupling and HE Scattering Stretched String--Transverse Vibrational Modes in deformed AdS. IR Physics--Glueballs as AdS Gravitons at strong coupling. UV Physics--Parton counting rules for wide-angle scattering. Stringy Deconfinement. Fat vs Thin Strings. Hagedorn Transition (aka deconfinement ?) Pomeron as Massive Graviton. BFKL Pomeron, Regge, DIS, Froissart Bound, ………..

QCD after Brane Revolution Effective degrees of freedom---massless fields of type IIA String theory Confinement by AdS/BH background New chapter for non-perturbative QCD Weak Coupling --> Perturbative QCD :unchanged Confinement --> Dual description in AdS : weak field approx.

Physics at Different QCD Scales in AdS Space

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

Glueball Spectrum R. Brower, S. Mathur, and C-I Tan, hep-th/ , “Glueball Spectrum of QCD from AdS Supergravity Duality”.

QCD Rutherford Experiment At WIDE ANGLES QCD exhibits power law behavior: where n=  i n i is the number of ``partons'' in external lines. Actually QCD is only conformal up to small asymptotic freedom logs. The OPE gives in terms of the lowest twist  i.

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 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.

Summary on Hard Scattering (3) Compared with lowest order perturbative results: (1) AdS 5 Hard Scattering (Polchinski-Strassler): WHY is it same QCD perturbative result with g 2 N ! (g 2 N)^2? (2) AdS 7 Hard Scattering (Brower-Tan): WHY does this only depend on the string tension?

string tension AdS5, e.g. Polchinski/Strassler AdS7 M-theory, e.g., Brower/Tan

Soft vs Hard Regge Scattering Similar arguments can be applied to the Regge limit: s >> -t Dominant scattering at large r, gives a BFKL-like Pomeron with almost flat ``trajectory'' (actually a cut in the j-plane) The IR region, r ' r min, gives soft Regge pole with slope  ' qcd »  ' R 3 /r 3 min The ``shrinkage'' of the Regge peak is caused the soft stringy ``form factor'' in impact parameter:

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

Stretched String in deformed AdS background Lattice evidence for string excitations Stretched String Excitations in AdS

M. Luscher and P. Weisz, J. H. E. Physics, 07 (2002) 328 K. J. Junge, J. Kuti, C. Morningstar, hep-lat/

Wilson Loop as Surface Sum Searching for evidence of String!

Minimum Surface for Wilson Loop

QCD String a la Polyakov

Energy of Transverse Oscillations Linear Potential:

Excited states (Semi-classical limit) At large L:  0 (z) ! 1,  (z) ! 1 except near end points:  0 (z) = V 2 (z)/V 2 (0)

Radial (longitudinal) Mode Near r = r min (or y = 0), V(y) ' r 2 min + const y 2 THE SPECTRUM IS GAUGE INVARIANT Choose a gauge with fluctuations only in longitudinal (X 3 ) or radial (Y) or normal to classical surface, etc X 3 = z +  or Y = y cl (z) + , etc except at end points  E = (d+1) M GB + O(1/L)

V. Future Developments Improved Confining Background Challenges: BFKL, Froissart bound, Massless Quarks, Chiral Symmetry, …….. ………….. Beyond strong coupling or Strong Gravity. See you at next ISMD!