High energy scattering of R-currents and AdS5/CFT4 · AdS/CFT conjecture [Maldacena, Witten,...
Transcript of High energy scattering of R-currents and AdS5/CFT4 · AdS/CFT conjecture [Maldacena, Witten,...
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