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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• …….