Department of Physics Applications of AdS/CFT in DIS Anastasios Taliotis Work done in collaboration with Javier Albacete and Yuri Kovchegov, arXiv: [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: DIS Kinematics/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):

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 α s Y= α s ln1/x ~ 1 is the resummation parameter that works vertex α s dY i ~ln(1/x i ) rapidity enhancement (x i =k i + /p + ). So all these terms must be resummed. Cross section: Idea is similar to DGLAP: Evolution equation that resums large α s logQ using renormalization methods (integral/diff. equations). BFKL resums α s ln1/x

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 log 2 (s) behavior at high energies ) derived from optical theorem. Hence violates unitarity! * Diffusion terms causes IR divergences (α s >> 1) at high enough s and hence pQCD breaks! 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: Resums 2-gluon exchanges => no gluon ladders => no rapidity (Y) evolution

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 N c (gluonic planar diagrams only) +=

Advantages: -Works perfectly in the limits of pQCD (small α s ). s -It saves unitarity at large s (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 Colour Transparency: By def.: no colour Interaction=>no scattering Black disk limit, Solving BK equation yields dipole-target amplitudes like this: Dense colour charge at Saturation scale Qs Def.: The scale where density of partons becomes high

Dipole-Nucleus scattering from BK Solutions of BK equation for several (fixed) rapidities Y. Saturation scale Q s =1/r s is defined at dipole size r s such that N(r s )=1/2 [Albacete et al ‘05]. As Y increases, curves move to the left and hence Q s 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 α s ( )<<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 (α s logQ~1) using renormalization methods. Fails at large s.

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.

Wilson loop hits Nucleus: light cone coordinates

Wilson Loop and the geometry of scattering R eS(x i ┴,b,Y)=(1/N c ) R e ; Average in all possible wavefunctions of nucleus. Suitable gauge-neglect gauge links at LC infinities: W(x,y,Y)=tr[Uq(x in, x f,Y)Uq*(yi n,y f,Y)*] where (note presence of 1/N c in S) Integration runs from -  to +  close to x - LC with θ >1)

Set up the problem as a stringy problem Maldacena claims [Maldacena ’97, ‘98] Choose a reasonable background metric g μν 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))

Requirements: –The SE tensor of nucleus must arise from the AdS duality: T μν  ∂g μν. –g μν should satisfy its own equation motion (Einstein's eq.)  Define z is the 5 th dim. of AdS-both requirements satisfied choosing  Then T μν is obtained from ∂g μν. Using the dictionary [ Janik & Peschanski ‘05]

Agrees with our T μν. Presence of N c ensures a non N c suppressed perturbation of the (empty AdS space) metric. Imagine N c 2 valence pointlike charges moving along x +. Mimic the dipole ( ) as the end points of the hanging string in the given background metric.

4D Vs 5D configuration Our 4d world String stretching into the 5 th dimension of AdS 5 attached to a Wilson loop. z SHOCK WAVE x 1 =x x3x3

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 α~p + A 1/3 show α~p + A 1/3. p + nucleus +momentum in the dipole’s rest frame. Λ Λ 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 T μν in ┴ plane by deduce -c.m. Enegry (s).

Wilson loop (hence scat. ampl.) is obtained by extremizing the S NG of string ( ) in the presence of shockwave ( ). Subtract self interactions. Consider N of DIS from string theory

Calculation-Results String Trajectory z(x), By symmetry z max at x=0. Then Evaluate action/Subtract infinities (at z=0)

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.

A tale of solutions The three Branches z max Re and Im parts of z max 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! Tale of solutions/Quantum corrections

Predictions Guided 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 & III

I. Strictly Im branch : the dipole amplitude Branch of z max gives a physical N(r,Y): Note that it stops moving to the left at very high energy! This branch gives the following saturation scale, defined by requiring that N(r=1/Q S, 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 )  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? α p = 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 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). saturation of saturation Both predict saturation of saturation. α p =1.5Conjecture pomeron intersept α p =1.5 Summary

Thank you

Back-up slide String Trajectory z(x), By symmetry z max at x=0. Then Evaluate action/Subtract infinities dt^2 component of g μν changes sign at z=z h =1/√s