DIS and e+e- annihilation at strong coupling Yoshitaka Hatta (U. Tsukuba) Based on: arXiv:0710.2148 [hep-th]　with E. Iancu and A. Mueller arXiv:0804.4733 [hep-th] with T. Matsuo arXiv:0807.0098 [hep-ph] with T. Matsuo

Contents Introduction DIS in QCD DIS in N=4 SYM e+e- annihilation and jets in QCD Jets in N=4 SYM ? Thermal (exponential) hadron spectrum

Motivations Longstanding problems in the small- regime of QCD (Froissart’s theorem, soft vs. hard Pomerons, hadronization...) for which pQCD methods do not apply. How does the nature of scattering change as one goes from weak to strong coupling ?  N=4 SYM Possible applications to the phenomenology of `sQGP’ (finite-T ) at RHIC, or in the `unparticle’ sector at the LHC

Regge limit of QCD One of the most challenging problems of QCD is the high energy limit. Can we compute the total cross section, etc ? What are the properties of the final states ?

Experimental knowledge Regge trajectory intercept slope Quadratic relation between mass and spin of hadrons Interpretable as a spinning string with tension 0.5

Heuristic picture (life before QCD) Suppose scattering is dominated by the t-channel exchange of particles on some Regge trajectory. = spin-j =

Complex angular momentum Analytically continue the partial waves to the complex J-plane and pick up the leading Regge pole

Regge limit in string theory (flat space) The leading Regge trajectory (closed string) Amplitude dominated by the spin-2 graviton exchange

Deep inelastic scattering in QCD Two independent kinematic variables Photon virtuality Bjorken- Physical meaning : momentum fraction of the constituents (`partons’) High energy = small-

DIS structure functions Imaginary part of the forward Compton scattering amplitude Probe of the parton distribution in the target hadron.

The operator product expansion Twist-2, spin-j operators The moment sum rule Invert continued to arbitrary j

The BFKL Pomeron = + ‘Bootstrap’ but 1 Balitsky, Fadin, Kraev, Lipatov, `78 ‘Bootstrap’ = + but 1

`Phase diagram’ of QCD Saturation BFKL DGLAP

N=4 Super Yang-Mills SU(Nc) local gauge symmetry Conformal symmetry SO(4,2) The ‘t Hooft coupling doesn’t run. Global SU(4) R-symmetry  choose a U(1) subgroup and gauge it.

The correspondence (anomalous) dimension mass N=4 SYM at strong coupling is dual to weak coupling type IIB string theory on Spectrums of the two theories match Maldacena, `97 Our universe 5-th dim. SYM string (anomalous) dimension mass `t Hooft parameter curvature radius number of colors string coupling constant

DIS at strong coupling Polchinski, Strassler, `02 R-charge current excites metric fluctuations in the bulk, which then scatters off a dilaton (`glueball’) We are here Photon localized at large Dilaton localized at small Cut off the space at (mimic confinement)

High energy string S-matrix dilaton gauge boson vertex op. vertex op. Insert t-channel string states dual to twist-2 operators AdS version of the graviton Regge trajectory

Anomalous dimension in N=4 SYM `Pomeron vertex operators’ Twist—two operators Gubser, Klebanov, Polyakov, `02 Kotikov, Lipatov, Onishchenko, Velizhanin `05 Brower, Polchinski, Strassler, Tan, `06

Pomeron at strong coupling Photon wavefunction Dilaton wavefunction Pomeron = massive graviton in AdS with intercept Diffusion in the 5th dimension, important when

Beyond the single Pomeron exchange Graviton dominance — high energy behavior even worse than in QCD. Multiple Pomeron exchange (higher genus) goes like Keep Nc finite and study the onset of unitarity corrections. We shall be interested in the regime but to comply with unitarity Not suppressed when resummation

The real part strikes back The saddle point in the j—integration In DIS, typically

Resurrection of massless gravitons small large 2 4 6 2 4 6 graviton propagator in AdS

Eikonal approximation The real part much bigger than the imaginary part (unlike in QCD ! ) Typical in theories with gravity Large real part resummed by the eikonal approximation `t Hooft, `87 Amati, Ciafaloni, Veneziano `87 Can be extended to AdS Cornalba, Costa, Penedones, Schiappa `06 Brower, Strassler, Tan `07 Impact parameter Valid when the impact parameter is large. This is the case in DIS at large !

Diffractive dissociation Dominant source of inelasticity from cuts between pomerons. Tidal force stretches a string, causing excitation. Giddings, `06 flat space Amati, et al, `87 AdS Only mildly suppressed, comparable to the elastic phase

`Phase diagram’ of QCD Saturation BFKL DGLAP

Phase diagram at strong coupling

Jets in QCD Observation of jets in `75 marks one of the most striking confirmations of QCD and the parton picture Average angular distribution reflecting fermionic degrees of freedom (quarks)

Fragmentation function Count how many hadrons are there inside a quark. Feynman-x First moment average multiplicity Second moment energy conservation

Evolution equation The fragmentation functions satisfy a DGLAP-type equation Take a Mellin transform Timelike anomalous dimension (assume )

Timelike anomalous dimension in QCD Lowest order perturbation ~ Soft singularity Resummation Angle-ordering Basseto, Ciafaloni, Marchesini, ‘80 Mueller, `81

Inclusive spectrum ‘Hump-backed’ shape consistent with pQCD with coherence. Roughly an inverse Gaussian in peaked at Mueller, Dokshitzer, Fadin, Khoze largel-x small-x

e+e- annihilation in strongly coupled N=4 SYM Hofman, Maldacena Y.H., Iancu, Mueller, Y.H., Matsuo arXiv:0803.1467 [hep-th] arXiv:0803.2481 [hep-th] arXiv:0804.4733 [hep-th] arXiv:0807.0098 [hep-ph] 5D Maxwell equation Want to compute at strong coupling But there is no corresponding operator in gauge theory…

DIS vs. e+e- : crossing symmetry Parton distribution function Fragmentation function Bjorken variable Feynman variable

Reciprocity respecting evolution DGRAP equation The two anomalous dimensions derive from a single function Dokshitzer, Marchesini, Salam, ‘06 is consistent with the Gribov-Lipatov reciprocity Confirmed up to three loops (!) in QCD Mitov, Moch, Vogt, `06 Application to AdS/CFT Basso, Korchemsky, ‘07 Assume this is valid at strong coupling and see where it leads to.

Average multiplicity at strong coupling crossing Y.H., Matsuo ‘08 c.f. in perturbation theory, c.f. heuristic argument Y.H., Iancu, Mueller ‘08

Jets at strong coupling? in the supergravity limit The inclusive distribution is peaked at the kinematic lower limit Rapidly decaying for 1 At strong coupling, branching is so fast and complete. There are no partons at large-x !

Energy correlation function Hofman, Maldacena `08 Energy distribution is spherical for any Correlations disappear as weak coupling strong coupling All the particles have the minimal four momentum~ and are spherically emitted. There are no jets at strong coupling !

Thermal hadron spectrum Identified particle yields are well described by a thermal model The model works in e+e- annihilation, hadron collisions, and heavy-ion collisions Becattini, Chliapnikov, Braun-Munzinger et al. The origin of the exponential law is not understood, though there are many speculations (QGP? Hawking radiation?) around…

A statistical model Bjorken and Brodsky, ‘70 Total cross section given by the one-loop result ! Cross section to produce exactly n ‘pions’ Assume Then Strikingly similar to the AdS/CFT predictions !

Computing the matrix element string amplitude 5D photon AdS metric 5D scalars ~ Since is large, do the saddle point approximation in the z-integral

The saddle point has an imaginary part. Probably absent once the decay width is Included in the propagators. Einhorn, ‘77 This leads to for ‘mesons’ for `partons’

Summary DIS and e+e- accessible at strong coupling The amplitude is dominantly real, final states mostly diffractive (unlike in QCD…) There are no jets, multiplicity rises linearly with energy (unlike in QCD…) The multiplicity ratio exhibits ‘‘thermal’’ behavior (like in QCD?)