Transcript High Energies Scattering in the AdS dual to “QCD”

High Energies Scattering in the
AdS dual to “QCD”
Richard C. Brower
Boston University
Lattice 2007 --- August 3
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 4point 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:

Eikonalization + confinement ) Froissart Bound
( ¸ >> log2 s )
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
Questions:
1.
2.
3.
4.
Is C(m¼/mp) >0 ? What is its value?
What are the events that give C > 0?
Does the AdS/CFT provide a generic mechanism C>0?
Can one in principle compute C(m¼/mp) on then “lattice”?
High Energy Elastic Scattering
Optical Theorem:
s = (p1 + p3)2
p3
p2
p1
p4
t = (p1 + p2)2
Regge:
Nc ! 1 contributions
Definition:
The Pomeron ´ the vacuum exchange
contribution to scattering at high energies
at leading order in 1/Nc expansion.
where ¸ = g2Nc
& gs = 1/Nc
BFKL: Balitsky & Lipatov; Fadin,Kuraev,Lipatov‘75
t = - (k1 + k2)2
¸ = g2 Nc' 0
k’2
k2
k’1
k1
ln s
 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”
Moebius (aka SL(2,C)) invariance
1
L
2
3
2-body Casimirs
AdS5/CFT Dictionary
The 5th dimension is conformal
dilations
“Five” kinematical co-ordinate is size z / z’ of
projectile/target
b2
2
°*(Q )
b?
b1
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’)
Boosting AdS5 to AdS3 isometries
with z = R2/r
O(4,2) isometries
DIS :
SLR(2,R)x SLL(2,R)
BFKL:
SL(2,C)
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)
(0,2) T  
 = 0, BFKL
j = j0 @ min 
Eikonal Expansion
Born term
+
“sum” to get
+
+
Two approaches to Eikonal Approximation
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.
Witten Diagram Summation
AdS5 Eikonal Sum
Note: 3-d “impact” space
or Matrix eikonal
We calculated explicitly the the box diagrams to see
beginning of series expansion in \chi. The kinematics
is basically the same as in Cheng-Wu’s classical paper from
1968.
Shock Wave Eikonal Formulation
1.Solve linearized Einstein equ:
2.Propagate across shock:
p+1
p-2
IR cut-off or Confining Hard Wall Model
(quick and dirty example of confining duals)
Large Sizes
Add Confinement
IR wall!
String/Glueball
Broken scale invariance in the 5th dimension
(r)
Hadron/Glueball
Massive Onium
Current
r -4
r-
IR WALL
r-
r = rmin
r !1
Kernel for hardwall at z =1
Khw/Kconf
b?
z (z’ = 0.01)
Born Term for Hard Wall model
z=w
x?
z=w
x?
Khw(z,z,x?)/Kconf(z,z,x?)
B.C.
Kconf(z,z,x?) - Khw(z,z,x?)
Theory Parameters: Nc & ¸ = g2 Nc
log(s)
Weak BFKL
AdS BFKL
AdS Gravity
log(b)
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.
Dynamics of N D3 branes at low
energies is (Super) SU(N) YM.
Their mass curves the space (near horizon)
into AdS5 and emits closed string (graviton)
ggravitons
A gluons
D3-branes
see “Total cross section at Tevatron
and LHC” K. Goulianos
hep ex/0707.1055v1