Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT

8/3/2019 Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT

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The BFKL Pomeron in AdS/CFT

Lorenzo CornalbaUniversity of Milano Bicocca & Centro Fermi

The IST String Fest Lisbon, June 2009


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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


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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


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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)


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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


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High Energy Interactions in AdS I

AdS5 given by

X2 = 2 ( = 1)X M2 M4

Boundary points given by rays

Q2

= 0 Q Q


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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, )


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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 +


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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


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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


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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


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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


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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


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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


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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)


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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

QQ

s

QQ

+

s

QQ

c

Q

Q

+

Q

with c, c, from r, r integrals

Applications to DIS I


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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


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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

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Lorenzo Cornalba- The BFKL Pomeron in AdS/CFT