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

2. 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

3. 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

4. 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 ?

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

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

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

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

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

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

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

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

13. `Phase diagram’ of QCD
Saturation
BFKL
DGLAP

14. 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.

15. 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

16. 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)

17. 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

18. 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

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

20. 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

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

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

23. 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 !

24. 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

25. `Phase diagram’ of QCD
Saturation
BFKL
DGLAP

26. Phase diagram at strong coupling

27. 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)

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

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

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

31. 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

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

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

34. 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.

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

36. 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 !

37. 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 !

38. 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…

39. 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 !

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

41. 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’

42. 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?)