AdSCFT: Unification of Soft Pomeron and BFKL

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AdS/CFT Unification of Soft Pomeron and BFKL
Blois 2005 Chung-I Tan, Brown University
Confronting String Theory with High Energy
Scattering

• 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,
  BFKL Kernel in String/Gauge Dual, to appear.

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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

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Many Faces of QCD
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From s10 GeV2 to 106 GeV2

• Lower Energies
• Soft Pomeron
• Shrinkage in forward peaks
• Higher Energies
• BFKL
• Large angle scattering, Jets, etc.

Large-N regularities
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I. Review QCD as String

• Asymptotic Freedom

Confinement
Force at Long Distance--Constant
Tension/Linear Potential, Coupling increasing,
Quarks and Gluons strongly bound ltgt Stringy
Behavior
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String Modes Vibrational Modes and Rotational
Modes
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Dual Pion Amplitude (aka NS string)
If ??(m2?) 0, then have Adler zeros in soft
pion limit A(p1 ! 0) 0, since s
! M2p , t !m2p
(y Neveu-Schwarz Quark model of dual pions,
1971)
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Death and Resurrection of QCD string
(i) ZERO MASS STATE (gauge/graviton) (ii) SUPER
SYMMETRY (iii) EXTRA DIMENSION 46 10 (iv)
NO HARD PROCESSES! (totally wrong dynamics)
Stringy Rutherford Experiment
At Wide Angle s,-t,-u gtgt 1/a
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Back to QCD Need to give Mass to Graviton
Regge ?(t)
J
Closed String
2 Graviton
?
Open String
1-- Photon/Gluon
t0
tgt0
tlt0
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Back to QCD Need Hard Collisions

• Fixed Angle Dimensional Counting rules
• BFKL
• Etc.

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II. String/Gauge Duality

• Early String Theories gt Theory of Everything
• Maldacena gt New Era for YM Theories

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Maldacenas String Counter Revolution
Open String ltgt Closed String duality
YM Wilson Loops lt---gt Expectation Values in Bulk!
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Scale Invariance and the 5th dimension
Large Sizes
String/Glueball
Strings (Gravity) in AdS5 (SUSY) Yang Mills
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Approx. Scale Invariance and the 5th dimension
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III. How to give Mass to Graviton?
Regge ?(t)
J
Closed String
2 Graviton
?
Open String
1-- Photon/Gluon
t0
tgt0
tlt0
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4-Dim Massive Graviton
5-Dim Massless Mode
0 E2 - (p12 p22 p3 2 pr2)
If, due to Curvature in fifth-dim, pr2 ? 0,
Four-Dimensional Mass
E2 (p12 p22 p32) M2
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IV. Wide Angle Scattering
The 2-to-m glueball scattering amplitude
T(p1,p2,L, pm2) for plane wave glueball
scatter via the string(M-theory) amplitude A(pi,
ri, Xi) in the 10-d (or 11-d) bulk space (x,r,Y)
We now discuss two different approaches to the
QCD string that both give the correct parton
scaling formula.

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

This is a check on the underlining universality
of Maldacena's duality conjecture.
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10-d String theory Approach Polchinski and
Strassler
Due to the Red Shift in the Warped Co-ordinate ,
D s (R/r) D x ,a plane wave glueball, expi x
p, scatters with a local proper momentum,
String is UV shifted in the YMs 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 f r-4 corresponding
to ni 4 for the YM operator, TrF2, in exact
agreement with the parton result.
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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
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Hard vs Soft Regge in M-QCD (RCB C-I Tan
hep-th/Tan 0207144)
Soft IR region r ' rmin, gives Regge pole
with slope a'qcd a' R3/r3min
The shrinkage'' is caused the soft stringy
form factor'' in impact parameter
Hard IR region BFKL-like Pomeron with almost
flat cut in the j-plane
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V BFKL vs Soft Pomeron
Weak Coupling
Strong Coupling Brower, Polchinski,
Strassler, Tan
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BFKL Summation
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Diffusion in Impact Space
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BFKL vs Soft Pomeron

• Perturbative QCD
• Short-Distance
• aBFKL (0) 1.4
• Increasing Virtuality
• No Shrinkage of elastic peak
• Fixed-cut in t
• Diffusion in Virtuality

• Non-Perturbative
• Long-distance Confinement
• aP(0) 1.08
• Fixed trans. Momenta
• Shrinkage of Elasstic Peak lttgt 1/ log s
• a(0) 0.3 Gev-2
• Diffusion in impact space

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Hard versus Soft Diffraction in AdS (Lightcone
Derivation)
With X ?
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The Schwarz-Christoffel trans maps the upper
half plane (a) into the light-cone strip ? i?
(b)
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Reduction to 1-d Path Integral
where
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Regge Behavior is diffusion for time log(s) in
impact parameter space (and AdS radial space)
Rapidity y log(s/s0) and t - q2?
Boosts shrink size of hadronic string
exp - ? q2? log(s) ? exp - b2/(? log(s))
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Diffusion in AdS5
u-log Z----translational invariance
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VI Strong Coupling Pomeron

• V(u) -t e-u 0 lt u lt 1
• Attractive for t gt0, Regge Pole
• BFKL cut
• t lt 0 only scattering state for BFKL, with

V(u)
t lt 0
u
t gt0
Hard Wall at r rmin
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BFKL-Soft Pomeron Unification
Weak Coupling
Strong Coupling
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Leading J-Plane Singularities

• H(t) Y ?1/2 (2-J) Y

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Leading J-Plane Singularities for C-1

• Hodd(t) Y ?1/2 (1-J) Y

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VII. Glueballs at g2Nc 1
Strong coupling Dual to Gravity
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Deformed AdS-Metric Confinement
Wittens proposal---AdS/BH metric
Breaking Supersymmetry

• Deformed AdS Space, Massive Graviton
• Soft Pomeron and Tensor Glueball
• QCD String Tension

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AdS/BH Metric
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Deformed AdS and confinement
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Gravity vs Y.M. on Branes
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IIA Classification of QCD_4
States from 11-d GMN
States from 11-d AMNL
Subscripts to JPC refer to Pt -1 states
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Tensor Glueball/Graviton Wave functions
n1
n3
n0
r
rmin
n8
Potential
Randall-Sundram graviton
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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.
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Pomeron in Gauge/Gravity Duality

• Pomeron as Massive Graviton!
• Intercept lowered from 2!
• Interpolate with BFKL due to hard at AdS Bdry.

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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
• .