High Energy Scattering and String /Gauge Duality Evidence for QCD as an Effective String Theory...

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 ScatteringConfronting 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/9711200, The Large N Limit of Superconformal Field Theory and Supergravity

• Brower, Mathur, Tan, hep-th/0003115, Glueball Spectrum for QCD from AdS Supergravity duality

• Polchinski and Strassler, hep-th/010974, Hard Scattering and Gauge/String Duality; hep-th/0219211, Deep Inelastic Scattering and Gauge/String Duality; Polchinski and Susskind, hep-th/0011204, String Theory and the Size of hadrons

• Brodsky and de Teramond, hep-th/0310227, Light-Front Hadron Dynamics and AdS/CFT Correspondence

• Brower and Tan, hep-th/0207144, Hard Scattering in the M-theory dual for the QCD String

• Brower, Lowe and Tan, hep-th/0211201, 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 ReferencesSome 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

dualityYM 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=1 n=3

n=8 Potential

n=0

Randall-Sundram gravitonrmin

r

Glueball Spectrum

R. Brower, S. Mathur, and C-I Tan, hep-th/0003115, “Glueball Spectrum of QCD from AdS Supergravity Duality”.

QCD Rutherford Experiment

At WIDE ANGLES QCD exhibits power law behavior:

where n= i ni 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 ScatteringThe 2-to-m glueball scattering amplitude T(p1,p2,L, pm+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(pi, ri, Xi) in the 10-d (or 11-d) bulk space (x,r,Y):

• AdS5 £ X with IR cut-off on r > rmin or 10-d IIB string theory• AdS7 £ S4$ Black Hole with horizon r = rmin or 11-d M-theory.

We now discuss two different approaches to the QCD string that both give thecorrect parton scaling formula.

10-d String theory ApproachDue 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 r2 [rmin,rscatt]

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 ni = 4 for the YM operator, Tr[F2], in exact agreement with the parton result.

Summary on Hard Scattering

(3) Compared with lowest order perturbative results:

(1) AdS5 Hard Scattering (Polchinski-Strassler):

WHY is it same QCD perturbative result with g2N ! (g2N)^2?

(2) AdS7 Hard Scattering (Brower-Tan):

WHY does this only depend on the string tension?

string tensionAdS5, e.g. Polchinski/Strassler

AdS7 M-theory, e.g., Brower/Tan

Soft vs Hard Regge ScatteringSimilar 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 ' rmin, gives soft Regge pole with slope 'qcd » ' R3/r3min

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/0207004

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) = V2(z)/V2(0)

Radial (longitudinal) Mode• Near r = rmin (or y = 0), V(y) ' r2

min + const y2

THE SPECTRUM IS GAUGE INVARIANT

•Choose a gauge with fluctuations only in longitudinal (X3) or radial (Y) or normal to classical surface, etc X3 = z + or Y = ycl(z) + , etc

except at end points

E = (d+1) MGB + 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!