AdS/CFT: Unification of Soft Pomeron and BFKL

Confronting String Theory with High Energy ScatteringConfronting 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/0003115, Glueball Spectrum for QCD from AdS Supergravity duality•R. Brower, J. Polchinski, M. Strassler, C-I Tan, R. Brower, J. Polchinski, M. Strassler, C-I Tan, BFKL Kernel in String/Gauge Dual, to appear., 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 GeV2 to 106 GeV2

• 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 (m2) = 0, then have Adler zeros in soft pion limit:

A(p1 ! 0) = 0, since s ! M2 , t !m2

(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=0 t>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

dualityYM Wilson Loops <---> Expectation Values in Bulk!

Scale Invariance and the 5th dimension

Strings (Gravity) in AdS5 ´ (SUSY) Yang Mills

String/Glueball

Large Sizes

Approx. Scale Invariance and the 5th dimension

r ! 1r = rmin

r-

r-

r -4

Hadron Glueball

Massive Onium Currentr)

IR WALL

III. How to give Mass to Graviton?

t=0 t>0t<0

J

Regge (t)

2++

Graviton

1--

Photon/Gluon

Closed String

Open String

?

4-Dim Massive Graviton

0= E0= E22 - (p - (p1122 + p + p22

22 + p + p33 22 + p + prr22))

5-Dim Massless Mode:5-Dim Massless Mode:

If, due to Curvature in fifth-dim, pIf, due to Curvature in fifth-dim, prr22 0, 0,

EE22 = (p = (p1122 + p + p22

22 + p + p3322) + M) + M22

Four-Dimensional Four-Dimensional Mass:Mass:

IV. 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 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 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.

Soft vs Hard in M-QCD (RCB & CIT hep-th/Tan 0207144)

Red Shift: Proper Length: s = (r/R) x Local Momentum: plocal

= (R/r) p

Therefore at wide angles!

Astring( ’ R2 s/r2, ’ R2 t/r2) » exp[ - R2 s log(s) /r2]

A Regge region is an average like:

Hard vs Soft Regge in M-QCD (RCB & C-I Tan hep-th/Tan 0207144)

Hard IR region: BFKL-like Pomeron with almost flat cut in the j-plane

Soft: IR region: r ' rmin, gives Regge pole with slope 'qcd » ' R3/r3min

The ``shrinkage'' is caused the soft stringy ``form factor'' in impact parameter:

V: BFKL vs Soft Pomeron

t=0 t>0t<0

J

BFKLcontinuum

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: <|t|> ~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

b12

b34

bX1

X2

Rapidity y = log(s/s0) and t = - q2?

Regge Behavior is diffusion for time log(s) in impact parameter space(and AdS radial space)

exp[ - ’ q2? log(s) ] exp[ - b2/(’ log(s))]

Boosts shrink size of “hadronic string”

Diffusion in AdS5

u=-log Z----translational invarianceu=-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 = rmin

BFKL-Soft Pomeron Unification

t=0 t>0t<0

J

BFKLcontinuum

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

Hodd(t) =1/2 (1-J)

VII. Glueballs at g2Nc = 1

Strong coupling Dual to Gravity

Deformed AdS-Metric: ConfinementDeformed 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 G11,11 m0 (Eq.) A,11 A m0(Eq.)

Gij

2++

Ci

1++(-)

0++ 4.7007 (T4)

Bij

1+-

C123

0+-(-)

7.3059(N4)

Gi

1-+(-)

C

0-+ 5.6555 (V4)

Bi

1--(-)

Cij

1-- 9.1129(M4)

G

0++ 2.7034(S4)

G

0++ 10.7239(L4)

States from 11-d GMN States from 11-d AMNL

Subscripts to JPC refer to P = -1 states

Tensor Glueball/Graviton Wave functions

n=1 n=3

n=8 Potential

n=0

Randall-Sundram gravitonrmin

r

Lattice Data vs AdS Confining Gauge Theory at ’ = 0

R. Brower, S. Mathur, and C-I Tan, hep-th/0003115, “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• …….