Department of Physics

Applications of AdS/CFT in DIS

Anastasios Taliotis

Work done in collaboration with Javier Albacete and Yuri Kovchegov, arXiv:0806.1484 [hep-th]

Outline

•State the problem

•Traditional methods in QCD-regime of validity

•Solving the problem using AdS/CFT

•A number of solutions (quantum

corrections?)

•Predictions-comparisons

•Conclusions/ Summary

The problem: DISKinematics/notation

• Photons Virtuality (transverse momentum)Notation: Use interchangeably

• Bjorken-x ( ): A quark in the proton carries a fraction x of the total momentum P of the proton

• c.m. energy (s) –Rapidity (Y):

22 2

1 1~Q q q

x r

2

1ln ln ln

jB

sY s

x Q

jBx

x

| | 1/x x r Q

Traditional Methods

• The BFKL equation [Balitsky, Fadin, Kuraev, Lipatov]

• Glauber-Mueller model (multiple two-gluon

exchanges) [Glauber & Mueller ‘90]

• The BK equation [Balitsky, Kovchegov ‘00]

The BFKL equation• Resums ladder (single-pomeron) diagrams at high energy (Y~lns~ln1/x)• Turns out αsY= αsln1/x ~ 1 is the

resummation parameter that works

• vertex αs dYi~ln(1/xi) rapidityenhancement (xi=ki

+/p+). So all these terms must be resummed . • Cross section:

• Idea is similar to DGLAP: Evolution equation that resums large αslogQ using renormalization methods (integral/diff. equations). BFKL resums αsln1/x

2 31 2( ) ( ) ...n n

s s n s s nn n

Y C Y C

• Solution

• Obtain N(Y)=scat. Ampl. =

• Features of N (cross section):

* High energies (Y~lns): . Here defines the one pomeron

(gluon-ladder) intercept (corresponds to a single graviton exchange-we will revisit this).

* Violates Froissart bound (predicts at most log2(s) behavior at high energies ) derived from

optical theorem. Hence violates unitarity!

* Diffusion terms causes IR divergences (αs >>1) at high enough s and hence pQCD breaks!

( , ', )( , )s

f l l Ydk K k l f

Y

2( 1) ln / '( , ', ) ~

pc

Y l lYf l l Y e

1 2( ) ( ') ( , ', )l l f l l Y

1~ ps 1p

Elementary proof BFKL

Glauber-Mueller rescatterings-dipole (d) moving in a nucleus (A)

• N( ,b,Y=0)=

• Saturation scale (scale where N~1/2)

• Resummation param.

• Black disk:

2 2 2 1/3~1/ (1.2 )s sQ fm A

2 2 11 exp[ ( ) ln ]sQ b x

x

x

2 22 ( ( ), , 0) 2 1/sd bN Q b x Y R at x

( ( ), )sN Q b x

2 1/3s A

•Resums 2-gluon exchanges => no gluon ladders => no rapidity (Y) evolution

x

The BK equation Solves the dipole-nucleus (d-A) scattering problem

• Combines previous two models => has resummation parameters

• Assumptions:

-Y>>1 (works well at high energies)

-Choose frame where Nucleons are at rest (keep all evolution in the dipole)

-Large Nc (gluonic planar diagrams only)

2 1/3,s sY A

+ =

• Advantages:

-Works perfectly in the limits of pQCD (small αs).

-It saves unitarity at large ss (unlike BFKL/recall Froissart

bound at high energies).

-It reduces to the (successful) BFKL at small s.

• Disadvantage

Non linear (this is what saves unitarity)-hard to solve.

Dipole amplitude

ColourTransparency:By def.: no colour Interaction=>no scattering

Black disklimit,

22tot R

Solving BK equation yields dipole-target amplitudes like this:

Dense colour chargeat Saturation scale QsDef.: The scale where density of partons becomes high

2

1~

s

F

Q g

Dipole-Nucleus scattering from BK

•Solutions of BK equation for several

(fixed) rapidities Y. Saturation scale

Qs=1/rs is defined at dipole size rs such

that N(rs)=1/2 [Albacete et al ‘05].As Y increases, curves move to the left

and hence Qs increases (with Y=lns)

•Saturation scale is a function of

energy s (or Y since Y~lns). Similar

graphs will appear later in the

context of the AdS/CFT.

Summary of pQCD methods

We have seen three different ways how to deal with scattering problems

in pQCD. This means all methods assume αs( )<<1.

• BFKL – Works in lower energies (Y). Linear => easy to use/Fails at large Y.

• GM - is a toy model. Catches all basic features of a scattering amplitude N/Fails

to incorporate energy (Y) dependence (by construction).

• BK- Is successful to both lower s (reduces to BFKL) and higher s/Is nonlinear

and hard to solve analytically

• Other equations are:

-JIMWLK extremely complicated to solve.

-DGLAP resums large (αslogQ~1) using renormalization methods. Fails at large s.

2 21/Q x

Map of high energy QCD

• Question: at some x the saturation scale for a proton may be equal to the confinement scale. What happens there? One thing is certain: pQCD breaks down. Maybe AdS/CFT can help answer this question.

?

Applying AdS/CFT to DIS

DIS in the non-perturbative regime and AdS/CFT

• Deal with N =4 SUSY QCD.• Cross section• Looking for . Encodes all QCD effects.• Strategy: N=1-S S[Wilson loop] AdS/CFT.

* 2 2 2( , ) ( ) ( , , , )A Q Y QED d bN Q b Y z 2( , , )N Q b Y

Wilson loop hits Nucleus: light cone coordinates

Wilson Loop and the geometry of scattering

• S(xi┴,b,Y)=(1/Nc) RRee< W(b+r/2,b-r/2,Y) >;

• Average in all possible wavefunctions of nucleus.

• Suitable gauge-neglect gauge links at LC infinities:

W(x,y,Y)=tr[Uq(xin,xf,Y)Uq*(yin,yf,Y)*] where (note presence of 1/Nc in S)

• Integration runs from - to + close to x- LC with

θ<<1 (θ is defined by Y so that tan[π/2-θ] = exp[2Y], Y>>1)

( , , ) exp[ ]f

in

x

in f xU x x Y P ig dx A

2( , , , ) , & / 2Yin fx x x e x x x x x x b r fixed

q

q

x

3x

/ 2b r

/ 2b r

x 0x

3x

Set up the problem as a stringy problem

• Maldacena claims [Maldacena ’97, ‘98]

• Choose a reasonable background metric gμν that mimics the nucleus and argue that the configuration describes the collision.

• Mimic nucleus as smeared shockwave along (x+) LC axis.

(α is the extent of the nucleus in x- direction; μ has mass dim. cubed (more later))

~ NGiScW e

1det[ ( ) ], ( , ), 0,..., 4

2 'NGS d d g X X X X X

~ ( ) ( )T x a x

2 2/(2 ) ( ) ( )cT N x a xa

• Requirements:– The SE tensor of nucleus

must arise from the AdS duality: Tμν ∂gμν. – gμν should satisfy its own equation motion (Einstein's eq.)

Define z is the 5th dim. of AdS-both requirements satisfied choosing

Then Tμν is obtained from ∂gμν. Using the dictionary

22 2

2( , )

Lds g x z dx dx dz

z

24 / 0R L g

2

2 40

( , )( ) lim

2c

z

g x zNT x

z

24 2 2 2

22 ( ) ( )

Ldx dx x a x z dx dx dz

z a

[Janik & Peschanski ‘05]

x

z

• Agrees with our Tμν. Presence of Nc ensures a

non Nc suppressed perturbation of the (empty AdS

space) metric. Imagine Nc2 valence pointlike charges

moving along x+.

• Mimic the dipole ( ) as the end points of the

hanging string in the given background metric.

qq

4D Vs 5D configurationOur 4dworld

String stretching into the 5th dimension of AdS5 attached to a Wilson loop.

zSHOCKWAVE

x1=x

x3

Approximations/phenomenology:

- Static case (or else deal with highly non linear 2nd order DE). Corresponds to large extent nucleus along x+: A>>1 since can

show αα~p~p++AA1/31/3. p+ nucleus +momentum in the dipole’s rest frame.

ΛΛ some transverse scale characterizing the dipole and nucleus.

- Infinite transverse extend of homogeneous nucleus => no b dependence or angular dependence in transverse plane =>

- μ: Average over Tμν in ┴ plane by deduce

- c.m. Enegry (s).

1/3 1/32 / 2 / ~ /NR p A R p A p

1/3 2A p 21dx

S

2 2 2 2 2~ ( ) ~2

s p scale qq p identify s

( , , ) ( , ) ( , , ) ( , )q q q qW x x Y W r Y and hence N x x Y N r Y

• Wilson loop (hence scat. ampl.) is obtained by extremizing the SNG of string ( ) in the presence of shockwave ( ) .

• Subtract self interactions. Consider

1det[ ( ) ]

2 'NGS d d g A X Xa

0 1 2 3 4( , ) ( , , 0, 0, ( )),X t x X t X x X X X z x

[ ( ) ( 0)]( , ln ) 1 Re[ ]NG NGi S SN r Y s e

2 / 2A s

( / 2) 0z x r

2 2~1/qq Q r

qq

N of DIS from string theory

Calculation-Results• String Trajectory z(x),

• By symmetry zmax at x=0. Then

• Evaluate action/Subtract infinities (at z=0)

3 3 4 4max max/ 2 / 3 (1/ 2,3 / 4,7 / 4, / )ox r c rz z F z z

2 4max max1oc r z s z

/ 2 0r x

2 2

3max max

2( , ) 1 Re[exp{ [ 2 ]}]

( , ) ( , )2o

o

c rN r s i s

z r s z r sc

Six saddle points for SUGRA

• Three (plus their negatives = six) generally complex saddle points of SUGRA => six different amplitudes N.

• Maybe

???• Before answering, study the nature of these

branches.

6

1

1 ( )ii

N saddle

A tale of solutionsThe three Branches zmax

• Re and Im parts of zmax given from its cubic equation as a function of r (transverse dipole size) at fixed energy (s).

I.

II.

III.

• Argument: Consistency with first principles indicates the right branch(es)

• Mathematical consistency.

(i) Integrals of real variables may have complex saddle points (ex. Airy Integrals).

(ii) In a sequence of saddle points a subset may dominate (Stoke’s phenomenon) .

(iii) So far we approximated the whole string theory action by six points!! Maybe Quantum corrections, i.e. functional dets ( ) filter out the meaningful solution!

†X DXe dX

Tale of solutions/Quantum corrections

PredictionsGuided by physics

• Our guide are the first

principles. Choose

physical solutions.

• N 0 as r 0.• N 1 as r (black disk limit)• N > 0 and N(r,Y) monotonic function of r, Y.

=r

Investigation of the branches

I, II & IIII, II & III

I. Strictly Im branch : the dipole amplitude

• Branch of zmax gives a physical N(r,Y):

Note that it stopsmoving to the leftat very high energy!

This branch gives the following saturation scale, defined by requiring that N(r=1/QS, s)=0.5.

Saturation of saturation [pQCD: Kharzeev, Levin, Nardi ’07]

Basic features of I. branch

At lower energies ( ) we find . Identifying this behavior with single pomeron (corresponds to single graviton exchange) exchange obtain the pomeron intercept (BFKL predicts ) 1~ ps

2 1r s 1/ 2~N s

sQ1/3~sQ A

Saturation ( ) of saturation. At high s saturation becomes unexpended. In this limit can show .

All expected asymptotics (large and small r, monotonicity). Can shown is true for all the parameter space.

Does not map to Maldacena’s solution in the lim where space becomes empty AdS (nucleus absence). Should it map?

ααpp = 1.5 = 1.5

II & III branches• II branch gives negative scattering

amplitude (discard).

• III branch looks like figure. (Relaxing monotonicity condition)

Predicts pomeron intercept ap=2-agrees with [Brower, Polchinski &

Strassler]. Is this the Brower et al solution? They do talk about elastic dominance which means N=2… maybe yes.

• Also predicts saturation of saturation.

• While no fundamental principle seems to prohibit oscillations, to me they seem very unphysical.

• Does map to Maldacena solution.

Conclusions Predictions

2/3A Saturation of saturation

• Seen equations/models in pQCD

• Worked out DIS problem in context of AdS/CFT.Calculated the QCD contribution of process.

• Model: photon = dipole hanging string in background of a shockwave the nucleus.

• Found two meaningful scat. ampl. N(r,s). Both predict saturation of saturationsaturation of saturation.

• Conjecture pomeron intersept ααpp=1.5=1.5

Summary

Thank you

Back-up slide• String Trajectory z(x),

• By symmetry zmax at x=0. Then

• Evaluate action/Subtract infinities

3 3 4 4max max/ 2 / 3 (1/ 2,3 / 4,7 / 4, / )ox r c rz z F z z

2 4max max1oc r z s z

max

2 41/2 2 4

max 0 02 24 4max

2 1( , ) { 1 }hz zren

NG

dz s z dzS z s z

z zz z

2 2

3max max

2 2 1[ ],

2o

hho

c rz

z z zc s

/ 2 0r x

dt^2 component of gμν changes sign at z=zh=1/√s