8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
1/45
The BFKL Pomeron in AdS/CFT
Lorenzo CornalbaUniversity of Milano Bicocca & Centro Fermi
The IST String Fest Lisbon, June 2009
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
2/45
Introduction I
High energy phenomena probe relevant features of interactions
In flat space string theory they are controlled by the leading Reggetrajectory of states
In perturbative YM theory (QCD and susy completions) dominated
by Pomeron exchangeHigh energy strings in holographic backgrounds will interpolatebetween these two regimes
We will present a general formalism valid both at weak and at strongcoupling
At large t Hooft coupling, analyze graviReggeon exchange in AdSand recover flat space interactions
At weak coupling, use holography to correctly analyze known pQCDresults
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
3/45
High Energy Scattering in Flat Space I
Scattering in flat space in the Regge limit s tSingle spin J exhange in the T-channel
Maximal spin dominates
J unbounded High s behavior controlled by leading Regge pole inthe complex J plane
Atree (s, t)
(t)
sj(t)
Strings in flat space
j(t) = 2 +2
t
Violation of unitarity at ultra-high energies
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
4/45
High Energy Scattering in Flat Space II
Partial wave decomposition in the S-channel
A J0
CJ
ts
e2iJ(s) (ImJ (s) 0)
In the Regge limit
A 2s
d3x eiqx+2i(s,x)
q2 = t|x| = 2J/s
Eikonal resummation of tree interaction
Atree 4is
d3x eiqx (s, x)
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
5/45
High Energy Scattering in Flat Space III
Eikonal limit when phase shift Atree (s, x) /s Gs/ |x| 1. Multigravireggeon exchange of linear gravitons
In AdS5 we expect similar physics with
Extra scale of the AdS radius Transverse space H3 instead ofE
3
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
6/45
High Energy Interactions in AdS I
AdS5 given by
X2 = 2 ( = 1)X M2 M4
Boundary points given by rays
Q2
= 0 Q Q
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
7/45
High Energy Interactions in AdS II
Boundary points behave as momenta for AdS interactions. External
wave functions
1
(2X Q)
d eiXQ 1
dual to CFT operator of dimension
Scalar amplitudesA (Qi)
depend on invariants
Qij = Qi + Qj2and are homogeneous
A ( , Qi, ) = iA ( ,Qi, )
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
8/45
High Energy Interactions in AdS III
Consider CFT correlator
O1 (Q1)O1 (Q3)O2 (Q2)O2 (Q4) = 1Q113Q224
A (cross ratios)
Cross ratios
Q13Q24
Q12Q34
Q14Q23
Q12Q34
Regge kinematics
Q13,Q24
0
Basic example in N= 4 SYMO1=Tr(Z2) O2 = Tr(W2)
A= 1 + N2
Aplanar +
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
9/45
T Channel View I
Conventions
X =
X+, X, x M2 M4
x =
x+, x, x M4
Scattering kinematics
Q1 = (0, 1, 0) Q3 = (q2, 1,q)Q2 = (1, q
2, q) Q4 =
(1, 0, 0)
with
q, q Future light cone M4
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
10/45
T Channel View II
Transverse conformal group
SO(3, 1) q, q vectorsSO(1, 1) q, q q, 1q
Cross ratios
2 = q2q2 cosh = q q
Regge kinematics
0 fixed
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
11/45
T Channel View III
Tchannel exchange TE,J (,) of a conformal block ofEnergy E+2
Spin J
Euclidean OPE for 0
TE,J 2+E
Scattering regime
TE,J 1J E ()E () propagator of energy 1 + E in H3
General spin J contribution to correlator
1J
d J () i ()
H3 = 1 + 2
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
12/45
T Channel View IV
Maximal spin dominates. If spin J unbounded resum contributions.Leading Regge pole at J = j() gives
Aplanar d 1j() () i ()
In N= 4 SYMj(, g)
(, g)
g2 = g2YMN
with j(, g) the spin of the twist two operator of dimension 2 + i
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
13/45
S Channel Eixonal Resummation I
Impact parameter representation
A |qq|4
Mdxdx eiqxiqx e2i(x,x)
Cross ratios
S = |x| |x|
B impact parameter on H3
No interactionA = 1 = 0
Lattice sum
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
14/45
S Channel Eixonal Resummation II
For large E, J replace with M dxdx whereE =
Scosh (B/2) J =
Ssinh (B/2)
Phase shift determined eikonally by planar amplitude
1
N2d Sj()1 () i (B)
with
() = Vmin (,j()) () Vmin (,j())High energy unitarity
Im (S, B) 0Anomalous dimension 2/ of double trace O1 O2 ofdimension E and spin J for elastic AdS interactions
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
15/45
Momentum Space I
Correlator in momentum space
O1 (p1)O1 (p3)O2 (p2)O2 (p4)with
t , p2i fixed s AdS Poincar coordinates
x+ , x , x , rH3
External states
eisx eip1x f1 (r)
r2 1/p21
Amplitude
sdrd2x
r3 f1 (r) f3 (r) eip
x
dr d2x
r3 f2 (r) f4 (r) eip
x
e2i(S,B)
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
16/45
Momentum Space II
Phase shift depends on
S = srr
cosh B =1
2rrr2 + r2 + (x x)2
Bb
single
cross - ratio
Final answer
sd2b eipb e2i(s,b)
with
e2i(s,b) = s
dr
r3f1 (r) f3 (r)
dr
r3f2 (r) f4 (r) e2i(S,B)
f C l
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
17/45
Large t Hooft Coupling I
Gravitational interaction in AdS5 with
G 3
N2
Along O1 trajectory high energy states follow null geodesics in AdSparameterized by affine parameter x
+
and labelled by xand a pointx, r in H3 with propagator
1s
(x+j x+i )(xj xi ) H3(xj, rj|xi, ri)
21
34
i
j
L H f C li II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
18/45
Large t Hooft Coupling II
For graviton exchange phase given by tree level interaction betweengeodesics
(S, B) = i
dx+dx G3
S 2(B)
with 2 (B) propagator in H3 of scalar dimension 3
For g2 j(, g) 2
(, g) 2
4 + 2
Anomalous dimension of double trace O1 O2 of largedimension E and spin J
1
4N2
(E J)4
EJ Exact in 1/N2
I l di S i Eff I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
19/45
Including String Effects I
Regge trajectory of the graviton j(, g) with
g2 =4
2= g2YMN
Flat space limit
lim
jt, 2/
= 2 +
2
t
Energy momentum tensor j(2i, g) = 2
Decreasing intercept
j(, g) = 2 4 + 2
2g
W k t H ft C li I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
20/45
Weak t Hooft Coupling I
Amplitude Aplanar (,) computed to order g4
g2
221 (z, z) +
g4
1642 + 2zz z z
4zz21 (z, z) z, z = e
+ g4
164zz
z z2 (z, z)2 (1 z, 1 z)2 z
z 1 , zz 1
with
1 =zz
z z[2H2
logzzH1] Hp(z, z) = Lip(z)
Lip(z)
2 = 6H4 3logzzH3 + 12
log2zzH2
W k t H ft C li II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
21/45
Weak t Hooft Coupling II
In scattering regime obtain spin 1 Regge pole
Aplanar g4
822
sinh2 ()( 0)
corresponding to
j(, g) 1 (g 0) (, g)
i
g4
16
sinh
2
cosh3 2
S ll d BFKL P E h I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
22/45
Small g and BFKL Pomeron Exchange I
High energy hadron scattering (with s |t| QCD) at weak gdominated by hard perturbative Pomeron exchange
Quantum numbers of the vacuumTwogluon color singlet state with ladder interactions (reggeizedgluons)Spin
1 for g
0
j(, g) = 1 +g2
42
2 (1)
1 + i
2
1 i
2
+
BFKL picture for N= 4 SYM for g 0
Small g and BFKL Pomeron Exchange II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
23/45
Small g and BFKL Pomeron Exchange II
Amplitude (yi M gluon positions in transverse 2d space)
A H3
dy1dy3dy2dy4y413
y424
V (q, y1, y3) F (yi) V (q, y2, y4)
Pomeron propagator F (yi) sum of transverse partial waves of spin nand energy |n 1|+ 1. Only n = 0 term contributes for scalaramplitudes
d2
(1 + 2)2
Impact factor V (q, y1, y3) constrained by conformal symmetry to befunction of single cross ratio
u =x2y2
13
(
2x
y1) (
2x
y3)
Small g and BFKL Pomeron Exchange III
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
24/45
Small g and BFKL Pomeron Exchange III
Computable in perturbation theory
V u2 (1 (u))
Basis functions d V ()
H3
dy5
We obtain () i
4V () tanh
2
V ()
with
V () = V () =g2
2
1
cosh
2
Saturation at Weak Coupling I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
25/45
Saturation at Weak Coupling I
BFKL trajectory
j(, g) = 1 + o(g2)
(S, B) and () imaginary
Vanishing momentum transfer t = 0
s Q2 Q2p21 = p
23
p22 = p
24
Focus on cross section
drr3
f1 (r) f3 (r) r 1/Qdr
r3f2 (r) f4 (r) r 1/Q
d2b Re 1 e
2i(S,B) (s, Q, Q)
Saturation at Weak Coupling II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
26/45
Saturation at Weak Coupling II
Integral over impact parameter (with S = s/QQ)
(s, Q, Q) 1QQ
|lnQ/Q| dB sinh BRe
1 e2i(S,B)For large B one has || 1. Phase shift 1 along saturation line
d elnS(j()1)B(1+i)
so that
B
ln S
B
Bs (S) lnS
= 0.06 g2 + 0.14 (exp. value)
Saturation at Weak Coupling III
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
27/45
Saturation at Weak Coupling III
Deep Saturation
|lnQ/Q| Bs (s/QQ)
Approximate black disk
1QQ
Bs(S)|lnQ/Q| dB sinh B 1 B
ln S
B
Dominant Contribution
When Bs lnS then
c
s
+
s
c
Q
Q
+
Q
with c, c, from r, r integrals
Applications to DIS I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
28/45
Applications to DIS I
O1 E&M current (photon)
O2 proton
Kinematics
s Q2/xQ related to confinement scale &mass of proton
(simulate confinement with wavefunction in r)
Cross section for small x is
Q2F2
x, Q2
Geometric Scaling I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
29/45
Geometric Scaling I
Near saturation |lnQ/Q| Bs (s/QQ)cross section reads
d2b Im (S, B) 1
QQN2 d () sQQ
j()1
QQi
B
ln S
B
At saddle point
1
Q2 (1+is)
12
=
Q
Qs
2with Qs = Q
2
1
x
21
Geometric Scaling II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
30/45
Geometric Scaling II
In deep saturation
1
QQ QxQ
1
Q2 12
Specific dependence on the scaling variable
Experimental evidence
0.001
0.01
0.1
1
10
0.0001 0.001 0.01 0.1 1 10 100
fixed experimentally to be 0.138 0.021
Fit to DIS Data I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
31/45
Fit to DIS Data I
Expression for F2
Q2, x
cQQ
QxQ
+ Q
xQ
cQ
Q
+
Q
Q/x
Q
105
104
103
102
10
1
10-1
1 10 102
GeV
GeV
(i)(ii)
(iii)
1
Weak coupling
Q> Qmin 0.7 1 GeV2 Inside saturation
lnQ
xQ > lnQ
Q (Q 0.2 1 GeV)3 Asymptotic linear regime
Q
x
Q
10 ( 3)
Fit to DIS Data II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
32/45
Minimize mean square deviation against experimental and simulateddata
0.126
c 0.13c
0.14
1 (GeV)
Match experimental data inrather large kinematical
range with 6% accuracy
0.5 < Q2 < 10
x< 102
Real Data I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
33/45
0.5 1.0 2.0 4.0
Q [GeV]
0.5 1.0 2.0 4.0
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
0.2
0.4
0.6
0.8
F2
/Q
Y = 3.6Y = 3.8
Y = 4.0Y = 4.2
Y = 4.6
Y = 4.4
Y = 4.8Y = 5.0
Simulated Data I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
34/45
0.5 1.0 2.0 4.0
Q [GeV]
0.5 1.0 2.0 4.0
0.2
0.4
0.6
0.2
0.4
0.6
F2
/Q
Y = 3.6Y = 3.8
Y = 4.0Y = 4.24.4=Y
Y = 4.8Y = 5.0 6.4=Y
0.2
0.4
0.6
Comments on Dipole Formalism I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
35/45
p
Dipole phase shift
(s, r, r,b) D (s, r, r,b)
Bb b
single
cross - ratio
two
cross - ratios
Representation ofDd () sj()1 Ti (r, r,b)
where
Ti (r, r,b) |r| |r| /b21+i for |r| , |r| |b|
Comments on Dipole Formalism II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
36/45
For r, r
|b
|B lnb2/rr
d Sj()1(rr/b)1+i
D does not satisfy unitarity constraints (no asymptotic dipolestates)
Even if one assumes saturation at ImD 1, a simple exponentialsaddling for D is not possible for general r, r,b
For |r| , |r| |b| one obtains only the first term in . A pure blackdisk is then a poor approximation of experimental data
Impact Factors for Spin 1 Operators I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
37/45
AdS graviton trajectory corresponds to n = 0 BFKL trajectory.
What about n 1 trajectories ?Consider as external states spin 1 operators
OA1 OA2
Impact factorVmn (q, y1, y3)
with
symmetric in m, n and y1, y3vanishing weight in q, y1 , y3
Full amplitude
1
|q|21 |q|22
H3
dy1dy3dy2dy4
y413
y424
Vmn (q, y1, y3) F (yi) Vmn (q, y2, y4)
Impact Factors for Spin 1 Operators II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
38/45
Transverse SO(3, 1) conformal symmetry implies
Vmn =5
i=1
fi (u) Fmni
with
Fmn1 =
mn
Fmn2 =qmqn
q2
Fmn3 = 1
2qm
yn1
q
y1
+yn3
q
y3
12
qn
ym1
q
y1
+ym3
q
y3
Fmn4 = q2
4
ym1
yn1
(q y1)2 q
2
4
ym3
yn3
(q y3)2
Fmn5 =ym1
yn3
+ ym3
yn1
y13
Impact Factors for Spin 1 Operators III
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
39/45
Conserved current with 1 = 3 and
qm
1q6
Vmn = 0
Projection on n = 0 and n = 2 trajectory
Vmn = Vmn0 + Vmn2
Construct n = 2 part using basis functions
Vmn2
d T ()
H3dy5
Spin 2 propagator from q to y5 is unique and one can verify that
qmVmn2 = 0 qmV
mn2 = 0 V
mn2 mn = 0
Impact Factors for Spin 1 Operators IV
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
40/45
Remaining four structures are n = 0 terms
Vmn0 =4
i=1
Dmni Si (u)
with
Dmn1 = mn qmqn
q2
Dmn2 =qmqn
q2
Dmn3 = qm
qn + qn
qm
Dmn4 = q
2 2
qmqn+
qm
qn+ qn
qm
1
3
mn q
mqn
q2
q2q
Impact Factors for Spin 1 Operators V
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
41/45
Easy to determine S1, S2, S3
S1 = f1 +1
6f4 +
1 2u6
f5
S2 = f1 + f2 2f3 12
f4 12u
f5
S3 = du
4u2(2uf3 + uf4 + f5)
Complex to disentangle S4 and the n = 2 contribution
Vmn
=
Dmn4 S4 + V
mn2
with
qmVmn = 0 V
mn mn = 0
Impact Factors for Spin 1 Operators VI
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
42/45
Use mVmn2
= 0 to determine
( 3) S4 =
du
4u2
3uf4 + 5f5 + u
2(3u 2)f4 + u(u 2)f5
with
= 4u
2
(1 u)d2
du2 4u2 d
du
To determine Vmn2
note that Vmn can be viewed as an infinitesimalvariation of the metric on H3. The n = 0 term is then a combineddiffeomorphism and Weyl transformation. Therefore, theinfinitesimal Cotton tensor
Cabc (q, y1, y3)
due to a metric fluctuation Vmn will only come from the n = 2 term
Impact Factors for Spin 1 Operators VII
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
43/45
Due to conformal invariance and the symmetries of the Cottontensor, we may construct a single Cotton function
C (u) =
q22(2y1 q) y13 y
a1 y
b3 y
c1 Cabc
given explicitly by
C = (1 u)u2
(3u 1)f4 + u(3u 2)f4 +u2
2(u 1)f4
f5 + (1
2u)f5
u
2(u
1)f5 From C one can immediately deduce T ()
Examples in N= 4 SYM I
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
44/45
Basic spin 1 operators with 1 = 3 in the free limit
Tr (m)
Tr
iDm
j
+ c Tr
ijm
TriDm j + c
Tr ijm
Only the Rsymmetry current in the 15 of SO(6) is chiral and hasprotected dimension. The other spin 1 operators (in the 15 and 1 ofSO(6)) acquire dimension in the g2 limitVery simple computation (compared to standard momentum space
techniques) allows to compute
Impact factor for scalar quark current Tr
Dm
3u2 Fmn4 + 2u3 Fmn5
Examples in N= 4 SYM II
8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT
45/45
Impact factor for quark current Tr(
m
)2u3 Fmn1 + 4u
3 Fmn5
The n = 0 contributions are different. The n = 2 contributions areidentical with Cotton function
C = 36u3 (1 u) (1 2u)Reinserting the SO(6) factors one has that the Rsymmetry currenthas no overlap with the n = 2 trajectory
Basic Conjecture (tested already in more cases) : The SUGRAchirally protected states in N= 4 SYM interact uniquely with then = 0 Pomeron / graviton trajectory