High Energies Scattering in the AdS dual to “QCD”

Lattice 2007 --- August 3

Richard C. BrowerBoston University

Progress since Lattice 2006: “The Pomeron and Gauge/String Duality” by Brower, Polchinski,Strassler & Tan (BPST)

hep-th 0603115

(very) Few Related References Flat space:

‘tHooft, “Graviton Dominance in Ultra-High-Energy Scattering” PL B198 (1987).

Amati, Ciafaloni & Veneziano “Superstring Collisions at Plankian Energies”, PL B 197 (1987).

Bo Sundborg, “High-Energy Asymptotics: The one-loop string amplitude and resummation” NP B306 (1988)

AdS5: D’Hoker, Freedman, Mathur, Matusis & Rastelli, “Graviton exchange and complete 4-

point functions in the AdS/CFT correspondence” hep-th/9903196 v1

Cornalba, Costa, Penedones & Schiappa, “Eikonal Approximation in AdS/CFT: From Shock Waves to Four-Point Functions” hep-th/0611122 v1

Alday & Maldacena “Gluon scattering amplitudes at Strong coupling” hep-th/0705.0303 v1

Outline

Motivation

Dual 5-d Geometry of High Energy scattering

BFKL vs BPST Pomeron: ( log2(s) » ¸ = g2YM Nc )

Eikonal for AdS5 Gravity: ( ¸ >> log2 s )

Eikonalization + confinement ) Froissart Bound

Phenomenological Motivation Diffraction production will dominate LHC events.

Diffraction is a leading contender for the discovery of the Higgs!

What is its rate?

% of non-diffractive events fall like 1/Etot

LHC Diffractive Higgs: Forward Proton 420m Exp.

Of course Jets are often cleaner and Diffraction is still badly understood.

“Diffractive and Total Cross Section at Tevatron and LHC” (K. Goulianos hep-ex/0707.1055v1)

Theoretical Motivation

QCD obeys the (non-perturbative) Froissart theorem:

¾Tot(p+p ) X) = m-2p C(m¼/mp) log2(s/s0) + L

1. Is C(m¼/mp) >0 ? What is its value?2. What are the events that give C > 0?3. Does the AdS/CFT provide a generic mechanism C>0?4. Can one in principle compute C(m¼/mp) on then “lattice”?

Questions:

High Energy Elastic Scattering

p1

p2p3

p4

s = (p1 + p3)2

t = (p1 + p2)2

Optical Theorem:

Regge:

Nc ! 1 contributions

The Pomeron ´ the vacuum exchange contribution to scattering at high energies

at leading order in 1/Nc expansion.

where ¸ = g2Nc & gs = 1/Nc

Definition:

BFKL: Balitsky & Lipatov; Fadin,Kuraev,Lipatov‘75

Sum diagrams 1st order in g2 Nc and all orders (g2 Nc logs)n

gives cut starting at j0 = 1 + ¸ ln 2 /¼2.

Accidentally “planar” diagrams (e.g. Nc = 1) and conformal.

BKFL equation for 2 “reggized” gluon ladder is L = 2 SL(2,C) spin chain to one loop order .

BFKL is NOT a REGGE POLE! DIFFUSION “off shell k2 > 0” GLUON “virtuality”

k1

k2

k’1

k’2

ln s

t = - (k1 + k2)2

¸ = g2 Nc' 0

Moebius (aka SL(2,C)) invariance

1

3

2L

2-body Casimirs

AdS5/CFT Dictionary

The 5th dimension is conformal dilations

“Five” kinematical co-ordinate is size z / z’ of projectile/target

5 kinematical Parameters: 2-d Longitudinal p§ = p0 § p3 ' exp[ § log(s/¤qcd)] 2-d Transverse space: x’?- x? = b?

1-d Resolution: z = 1/Q (or z’ = 1/Q’)

°*(Q2)

b1

b2

b?

Boosting AdS5 to AdS3 isometries

with z = R2/r

BFKL:SL(2,C)

DIS : SLR(2,R)x SLL(2,R)

O(4,2) isometries

High energy Graviton exchange Kernel is AdS3 Green’s function

Strong Pomeron kernel: same structure in J-plane!

N = 4 SYM Leading Twist ¢(j) vs J=j

= 0 DGLAP(DIS moments)

= 0, BFKL

(0,2) T

j = j0 @ min

Eikonal Expansion

++ +

“sum” to get

Born term

1. sum of leading large s contribution for perturbative series.

2. propagation in a shock wave gravitational background of target. (‘tHooft’s method)

Again in AdS5 space can do it both ways.

We start with sources at the boundary and write down Witten (AdS5 Feynman) diagrams for the “S-matrix” with a “hardwall” IR regulator.

Two approaches to Eikonal Approximation

Witten Diagram Summation

AdS5 Eikonal Sum

We calculated explicitly the the box diagrams to seebeginning of series expansion in \chi. The kinematicsis basically the same as in Cheng-Wu’s classical paper from 1968.

Note: 3-d “impact” space

or Matrix eikonal

Shock Wave Eikonal Formulation

p-2

p+1

1.Solve linearized Einstein equ:

2.Propagate across shock:

IR cut-off or Confining Hard Wall Model(quick and dirty example of confining duals)

Large Sizes

String/Glueball

Add ConfinementIR wall!

Broken scale invariance in the 5th dimension

r ! 1r = rmin

r-

r-

r -4

Hadron/Glueball Massive Onium Currentr)

IR WALL

Kernel for hardwall at z =1

Khw/Kconf

z (z’ = 0.01)b?

Born Term for Hard Wall model

B.C.

Kconf(z,z,x?) - Khw(z,z,x?)Khw(z,z,x?)/Kconf(z,z,x?)

x?

z=w

x?

z=w

log(b)

Weak BFKL

AdS BFKL

AdS Gravity

log(s)

Theory Parameters: Nc & ¸ = g2 Nc

Concluding remarks

The KK modes represents a matrix version of eikonal formula like super string scattering of Amati, Ciafaloni and Veneziano ( “string bits are frozen” ).

Unitarization: Hardwall (confining) eikonal sum (probably) saturates the Froissart --- work in progress. (Brower, Strassler and Tan)

More central collisions require non-perturbative --- triple Regge, fan diagrams, black hole or plasma ball deconfinement region etc. See color glass condensate phase?

N = 4 SYM ´ AdS5 x S5 Open stings are Gluons dual to closed string Gravity.

D3-branes

Dynamics of N D3 branes at lowenergies is (Super) SU(N) YM.

Their mass curves the space (near horizon) into AdS5 and emits closed string (graviton)

ggravitons

A gluons

see “Total cross section at Tevatron and LHC” K. Goulianos hep ex/0707.1055v1