DIS in AdS

1
DIS in AdS

• Yuri Kovchegov
• The Ohio State University
• Work done in collaboration with Javier Albacete
  and Anastasios Taliotis, arXiv0806.1484 hep-th

2
Outline

• Motivation
• AdS/CFT techniques
• Calculation
• Results
• Conclusions

3
Motivation
4
DIS at weak coupling
x-

• At weak coupling, DIS at small-x is related to a
  Wilson loop going through the target (aka the
  dipole model)

5
Non-linear evolution equation

• At small coupling, dipole-target amplitude is
  calculated using the non-linear (BK) evolution
  equation

I. Balitsky, 96, HE effective lagrangian Yu. K.,
99, large NC QCD

• We also know
• the kernel at NLO order (Balitsky, Chirilli 07)
• running coupling corrections to the LO kernel
  (Balitsky 06, Yu.K. and Weigert 06)

6
Dipole amplitude
Solving BK equation yields dipole-target
amplitudes like this
Black disk limit,
Color transparency
1/QS
7
Dipole amplitude

• Heres a plot of dipole amplitude N(r,Y) as a
  function of r for different rapidities Y given by
  the solution of running-coupling BK equation.
• As saturation scale increases with Y, the curve
  moves to the left.

8
Map of high energy QCD
?

• Question at some (not too large) x the
  saturation scale for a proton may be equal to the
  confinement scale. What happens there? Maybe
  AdS/CFT can help answer this question.

9
Pomeron intercept

• The linear part of BK equation is the famous BFKL
  equation. It is known that NLO BFKL correction to
  the pomeron intercept is large and negative.
• Heres the plot of the intercept as a function of
  the coupling (from Brower, Polchinski, Strassler
  and Tan, hep-th/0603115) for N4 SYM

LO BFKL
AdS/CFT
Can this be confirmed using the dipole amplitude?
NLO BFKL
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AdS/CFT techniques and formulation of the problem
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AdS/CFT Approach
z0
Our 4d world
5d (super) gravity lives here in the AdS space
5th dimension
AdS5 space a 5-dim space with a cosmological
constant L -6/L2. (L is the radius of the AdS
space.)
z
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AdS/CFT Correspondence (Gauge-Gravity Duality)
Large-Nc, large lg2 Nc N4 SYM theory in our 4
space-time dimensions
Weakly coupled supergravity in 5d anti-de Sitter
space!

• Can solve Einstein equations of supergravity in
  5d to learn about energy-momentum tensor in our
  4d world in the limit of strong coupling!
• Can calculate Wilson loops by extremizing string
  configurations.
• Can calculate e.v.s of operators, correlators,
  etc.

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Calculating Wilson loops using AdS/CFT
correspondence
z

• To calculate a Wilson loop, need to
• find the extremal (classical) string
  configuration
• find the corresponding Nambu-Goto action SNG
• then the expectation value of the Wilson loop is
  given by

Maldacena, 98
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Target in AdS/CFT

• The target nucleus is a shock wave in 4d with
  the energy-momentum tensor

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Holographic renormalization
de Haro, Skenderis, Solodukhin 00

• Energy-momentum tensor is dual to the metric in
  AdS. Using Fefferman-Graham coordinates one can
  write the metric as
• with z the 5th dimension variable and
  the 4d metric.
• Expand near the boundary of the
  AdS space
• For Minkowski world
  and with

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Shock wave in AdS
The metric of a shock wave in AdS corresponding
to the ultrarelativistic nucleus in 4d is
Janik, Peschanksi 05
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The problem

• We want to calculate the dipole-nucleus forward
  scattering amplitude N(r,Y) in AdS. (From here on
  r x-.)
• We will work in the rest frame of the dipole
• The amplitude N(r,Y) is given by the expectation
• value of the Wilson loop, calculated by
• extremizing the string attached to the loop
• in the background of the shock wave metric

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What we want to calculate
Our 4d world
z
SHOCKWAVE
String stretching into the 5th dimension of AdS5
attached to a Wilson loop.
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Calculation
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Static approximation

• Delta-functions are unwieldy. We will smear the
  shock wave
• Here
  and
• Nothing interesting happens to the string outside
  the shock wave. Hence we need to calculate the
  action deficit inside the shock wave only. For
  large enough nuclei (large A, e.g. neutron stars)
  this means static approximation the string does
  not move once sufficiently deep inside the shock
  wave.

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Three extrema

• We extremized the Nambu-Goto cation and obtained
  3 different extrema, each of them corresponding
  to complex-valued string coordinates!
• The string configurations are
• characterized by their maximum
• extent labeled zmax .
• zmax is given by the following equation
• with the center of
  mass energy
• and c0 a constant.

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Three extrema

• The three extremal string configurations are
  labeled by the index n0,1,2 corresponding to
  three roots of the cubic equation for zmax2

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Results
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A tale of two solutions

• Two of the branches give physically meaningful
  results.
• By physically meaningful we imply N(r,Y) such
  that
• N(r,Y) ? 0 as r ? 0 (color transparency)
• N(r,Y) ? 1 as r or Y ?8 (unitarity)
• N(r,Y) is a monotonic non-negative function of r
  and Y
• Below I will show plots resulting from several
  rather simple analytic expressions for N(r,Y)
  that we obtained.

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Solution A the dipole amplitude

• n1 branch of zmax gives a physical N(r,Y)

Note that it stops moving to the left at very
high energy!
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Solution A the saturation scale

• n1 branch of zmax gives the following saturation
  scale, defined by requiring that N(r1/QS, s)0.5

QS is constant for Bjorken x lt 0.2! Saturation of
saturation?
(cf. Kharzeev, Levin, Nardi 07)
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Solution B

• Taking complex conjugate of the n2 branch would
  also give a physical amplitude N(r,Y) iff we
  modify the prescription for calculating the
  amplitude
• No justification other than it works. Something
  similar is used in quasi-classical quantum
  mechanics

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Solution B the dipole amplitude

• the complex conjugate of the n2 branch gives a
  physical N(r,Y)

It also stops moving to the left at very high
energy!
Note that N0 for small r!
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Solution B the saturation scale

• the complex conjugate of the n2 branch gives the
  saturation scale

QS is also constant for Bjorken x lt 0.2!
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Conclusions

• We have obtained the forward dipole-nucleus
  scattering amplitude N(r,Y) in the large-Nc limit
  of N4 SYM and large t Hooft coupling.
• The amplitude N(r,Y) is unitary and saturates to
  the black disk limit at large r and/or large Y.
  The saturation scale is independent of energy at
  high energy (small Bjorken x).
• This freezing of the saturation scale is often
  successfully implemented in phenomenological
  applications of CGC.
• What happens to the map of high energy QCD?

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Map of high energy QCD at large coupling
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Backup Slides
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Other works the amplitude

• From the works of Brower, Polchinski, Strassler
  and Tan 06 and Janik and Peschanski 00 one
  would expect N(r,s) s at moderate energies.
• Our Solution A gives N(r,s) s1/2 instead!
• Can we reconcile the two approaches?
• We can take the string configuration from
  Solution B and use the standard prescription for
  calculating Wilson loopsThis gives N(r,s)
  s2 at small/moderate s!
• This may be due to a 2-graviton exchange at LO.
  This is different from JP and BPST, but only by a
  square.
• But look at the plot of N as a function of r

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N(r,s) for n2 branch with standard prescription
for Wilson loops calculation
While no fundamental principle seems to prohibit
oscillations, to me they seem very unphysical.
35
Other works the saturation scale

• Hatta, Iancu, and Mueller 07 get QS21/x.
• Dominguez, Marquet, Mueller, Wu, and Xiao 08 get
  QSconst, just like us.
• We can reproduce the results of Hatta, Iancu and
  Mueller 07 by
• using n2 branch with standard prescription for N
  to write
• equating this expression to 1 to obtain QS
• However, when N1 this expression no longer
  applies. Defining QS by requiring that N(r1/QS,
  s)0.5 we get QSconst again!