High energy scattering of R−currents and AdS5/CFT4

Jan Kotanskiwith J. Bartels,A.-M. Mischler, V. Schomerus and M. Hentschinski

Hamburg University & DESY

March 4, 2009

AdS/CFT conjecture

[Maldacena, Witten, Gubser, Kebanov, Polykov ‘98]

Duality between:

Field theoryI CFTd

I N = 4 SYM theoryI Anomalous dimensionsI Correlation functionsI (of R-currents)I at weak coupling

String theoryI AdSd+1

I SupergravityI Energy of the IIB stringsI Correlation functionsI (of R-bosons)I at strong coupling

Issues of investigation:I Testing of the AdSd+1/CFTd correspondenceI Making use of it, i.e. prediction SYM theory at strong coupling

Regge limit of γ∗γ∗ scattering in QCD and N = 4 SYM

Scattering amplitude

〈λ1, λ2|Aa1a2a3a4 (s, t)|λ3, λ4〉 =

X

ji

ε(λ1)

j1(~p1)ε

(λ2)

j2(~p2)ε

(λ3)

j3(~p3)

∗ε(λ4)

j4(~p4)

∗Aa1a2a3a4j1 j2 j3 j4

(~pi )

as a correlation function of R−currents

i(2π)4δ

(4)(X

i

~pi )Aa1a2a3a4j1 j2 j3 j4

(~pi ) =

Z

0

@

4Y

i=1

d4xi e−i~pi ·~xi

1

A 〈Ja1j1

(~x1)Ja2j2

(~x2)Ja3j3

(~x3)Ja4j4

(~x4)〉

with polarizations λi = L,±.Regge limit

−t , |~pi |2 � s

t = −(~p1 +~p3)2, s = −(~p1 +~p2)

2

for the weak coupling:[Bartels,Roeck,Lotter ‘96],[Brodsky, Hautmann, Soper ‘97],[Bartels,Lublinsky ‘03],[Bartels,Mischler,Slavadore ‘08].for the strong coupling?

Ja2

j2(p2)

Ja1

j1(p1)

Ja4

j4(p4)

Ja3

j3(p3)

l1 − p1 l1 − k − p1 + q

l1 l1 − k

k q − k

l2 l2 + k

l2 − p2 l2 + k − q − p2

Kaluza-Klein reduction - 5d supergravity

Supergravity action reads

S =1

2κ2

Z

dd+1z√

g(−R + Λ) + Sm

Sm =1

2κ2

Z

dd+1z√

g»

14

FaµνFµνa +

ik24

√g

dabcεµνρσλFaµνFb

ρσAcλ − Aa

µJµa + . . .

–

Correlation functions

〈J(1)J(2) . . . J(n)〉CFT = ωnδn

δφ0(1) . . . δφ0(n)exp(−SAdS [φ[φ0]])

˛

˛

φ0=0 ,

with the metricds2 =

1x2

0

(dx20 + d~x2)

where d~x2 can be related to the Minkowski space by Wick rotation.The boundary of the Anti-de Sitter space is at x0 = 0.

Witten Diagrams

Graviton and boson exchange in the t−channel

Ja1

j1(~x1) J

a3

j3(~x3)

Ja2

j2(~x2) J

a4

j4(~x4)

z

w

Ja1

j1(~x1) J

a3

j3(~x3)

Ja2

j2(~x2) J

a4

j4(~x4)

z

w

R−Boson:

Gaa′

µν′ (z, w) = −δaa′

(∂µ∂ν′u)Gd=4,∆=3(u) + δaa′

∂µ∂ν′S(u)

Graviton:

Gµν;µ′ν′ (z, w) = (∂µ∂µ′u ∂ν∂ν′u+∂µ∂ν′u ∂ν∂µ′u) Gd=4,∆=4(u)+δµν δµ′ν′ H(u)+. . .

where G(u)∆,d = 2∆ Γ(∆)Γ(∆−d2 +

12 )

(4π)(d+1)/2Γ(2∆−d+1)ξ∆

2F1(∆2 , ∆+1

2 ;∆ − d2 + 1; ξ2) and

u =(z0−w0)2+(~z−~w)2

2w0z0= 1

ξ− 1 [Hoker,Freedman, Mathur, Matusis, Rastelli ‘99]

Graviton Amplitude - Regge limitThe graviton amplitude behaves as s2

IReggegrav = s2 δa1a3 δa2a4

2(2π)6δ

(2L)(~p1 + ~p2 + ~p3 + ~p4)

×

Z

dz0z20

X

m1=0,1

Wm1j1 j3

(~p1, ~p3)Km1 (z0|~p1|)Km1 (z0|~p3|)

×

Z

dw0w20

X

m2=0,1

Wm2j2 j4

(~p2, ~p4)Km2 (w0|~p2|)Km2 (w0|~p4|)

×δ(2)

(x1 − x3)δ(2)

(x2 − x4)G∆=3,d=2(u)

Ja1

j1(~x1) J

a3

j3(~x3)

Ja2

j2(~x2) J

a4

j4(~x4)

z

w

where δ(2L)(. . .) is the longitudinal two-dimensional Dirac delta,ξ =

2z0w0z20 +w2

0 +x234

and u = 1−ξ

ξand G∆=3,d=2(u) is the AdS3 scalar propagator

〈λ1|Wm1 (~p1, ~p3)|λ3〉 =

X

j1,j3

ε(λ1)

j1(~p1)ε

(λ3)

j3(~p3)

∗Wm1j1 j3

(~p1, ~p3)

≈ |~p1||~p3|(δm1,1(~ε(h1)

T · ~ε(h3)∗

T )δλ1,h1δλ3,h3

+ δm1,0δλ1,Lδλ3,L) ,

where hi are transverse polarizations and (~ε(h1)

T · ~ε(h3)∗

T ) ≈ −δh1h3 .The helicity is conserved.The similar exchange in the eikonal approach: [Cornalba,Costa,Penedores ‘07],[Brower,Strassler,Tan ‘08].

Dependence on the impact parameter (transverse distance)Integrals from the amplitude plotted as a function of x34 with |~pi | = 1.

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 1 2 3 4 5

PSfrag replacements

XTT (x34)XLT (x34)XLL(x34)

x34

Different lines correspond to different polarizations of R-bosons, namelyT for transverse and L for longitudinal.

I For x34 →∞ the amplitudes XTT , XTL = XLT , XTT vanish as 3225π

x−634 ,

6475π

x−634 , 128

225πx−6

34 , respectively.I For x34 → 0 the amplitude is logarithmically divergent.

Dependence on virtualities ratio rp = |~p1|/|~p2|

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

PSfrag replacements

XTT (rp )XLT (rp )XTL(rp )XLL(rp )

rp

Deep InelasticScattering

~p1

~p2 ~p4 = −~p2

~p3 = −~p1

〈λ1λ3|Iforwardgrav |λ2λ4〉 =

s2/|~p2|4

23(2π)12δ

a1a3 δa2a4 δ

(4)(~p1 + ~p2 + ~p3 + ~p4)

×

»

δλ1T δ

λ3T δ

λ2T δ

λ4T XTT (rp) + δ

λ1T δ

λ3T δ

λ2h2

δλ4h4

δh2h4XLT (rp)

+δλ2T δ

λ4T δ

λ1h1

δλ3h3

δh1h3XTL(rp) + δ

λ2h2

δλ4h4

δλ1h1

δλ3h3

δh1h3δh2h4

XLL(rp)

–

For rp →∞: 〈I forwardLL,TL 〉 → const

x2B

while 〈I forwardTT ,LT 〉 →

ln r−2p

x2B

with Bjorken xB.

Position of the vertices as a function of virtuality

The position of the integrand maximum in the (z0, w0)−space with|~p2| =fixed as a function of |~p1| and different R-boson polarizations.

PSfrag replacements

LL

LLLT

TL

TL

TT

yLLB = yTL

B TT , LT

|~p1| → 0

∞←|~p 1| →

0vTT

vTT

vTL

vTL

yB,LLyB,LT

vA = z0 |~p2|

vB = w0 |~p2|

Ja1

j1(~x1) J

a3

j3(~x3)

Ja2

j2(~x2) J

a4

j4(~x4)

z

w

The black circles denote points corresponding to |~p1| → 0 for TT and TL.For upper virtuality |~p1| → ∞ the upper vertex z0 → 0 [Polchinski, Strassler ‘02],[Hatta, Iancu, Mueller ‘08] and:

I for TT , LT : the lower vertex w0 → 0I for LL, TL: the lower vertex w0 → const

Boson exchange with Chern-Simons vertices

In the Regge limit the exchanged R−boson iswritten in terms of AdS3 scalar propagator.

The amplitude is proportional to s.

Ja1

j1(~x1) J

a3

j3(~x3)

Ja2

j2(~x2) J

a4

j4(~x4)

z

w

IReggeCS ≈ −s

da1a3a daa2a4

(2π)6δ

(2L)(~p1 + ~p2 + ~p3 + ~p4)δ

(2)(x1 − x3)δ

(2)(x2 − x4)|~p3||~p4|W

CSj1 j3 j2 j4

(~pi )

×

Z

dz0z20 K0(z0|~p1|)K1(z0|~p3|)

Z

dw0w20 K0(w0|~p2|)K1(w0|~p4|)G∆=2,d=2(u)

+

„

~p1 ↔ ~p3j1 ↔ j3

«

+

„

~p2 ↔ ~p4j2 ↔ j4

«

+

„

~p1 ↔ ~p3j1 ↔ j3

«

×

„

~p2 ↔ ~p4j2 ↔ j4

«

We have only transverse polarizations and helicity conservation

〈λ1λ2|WCS

(~pi )|λ3λ4〉 ≈ |~p1|2|~p2|

2 X

h1,h2,h3,h4

δλ1h1

δλ2h2

δλ3h3

δλ4h4

×((~ε(h1)

T · ~ε(h3)∗

T )(~ε(h2)

T · ~ε(h4)∗

T ) − (~ε(h3)∗

T · ~ε(h4)∗

T )(~ε(h1)

T · ~ε(h2)

T ))

= |~p1|2|~p2|

2 X

h1,h2,h3,h4

δλ1h1

δλ2h2

δλ3h3

δλ4h4

(2δh1h2− 1)δh1h3

δh2h4

Six-current correlators in the field theory

Scattering of a virtual photon on a weak bound nucleusor two virtual photons – Topology of pantss1 = (q + p1)

2, s2 = (q + p′2)

2, M2 = (q + p1 − p′1)

2, t1 = (p1 − p′1)

2,t2 = (p2 − p′

2)2, t = (q′ − q)2

the triple Regge limit

s1, s2 � M2 � −t1,−t2,−t

[Bartels, Wusthoff ‘95]

Two color channelsin QCD [Bartels,Hentschinski

‘09]

M2

s2

p1 t1 p′1 p2 t2 p′

2

s1

q′

t

q

Amplitude on the pants in QCD

Summing up all pant diagrams [Bartels,Hentschinski ‘09], i.e.

where we use a double line notation for the color part of the gluondiagram.

Amplitude on the pants in QCD

The amplitude for weak coupling

F(ω, ω1, ω2) = −D2(ω1) ⊗12 D2(ω2) ⊗341

Nc

“

[ω − ω1 − ω2]λV R + λ2Vpp

”

⊗D2(ω)

where color structure

Bootstrap eq.[Balitsky, Fadin, Kuraev, Lipatov ‘76-. . . ]

Triple Pomeron vertex[Bartels ‘93-. . . , Wusthoff ‘95]

I N = 4 SYM theory?I At strong coupling: 3-graviton exchange amplitude and other

Witten diagrams?

Summary

I Four-point functions at strong couplingI The leading one in Regge limit (∼ s2) is graviton exchange

I Dependence on the impact parameter and the ratio of virtualitiesI Position of vertices on the virtuality

I The boson exchange is of order sI Exchange can be written in terms of AdSd+1 scalar fields with:

∆ = d/2 + 2 for graviton ∆ = d/2 + 1 for boson diagramsI The amplitudes are real with conserved helicity

I Future: Six-point functions on pantsI Two color channels: three Pomeron vertex and reggeization diagramI At strong coupling one can also interprete them in terms of Witten

diagrams