Small x physics and QCD evolution equations

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Small x physicsand QCD evolution equations

• Wei Zhu
• East China Normal University
• Wuhan 2008.12.4-6

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LHC - THE LARGE HADRON COLLIDER
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1
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Parton distribution function (PDF)
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The kinematic domains probed by the various
experiments, shown together with the partons that
they constrain
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• The gluon distributions of the nucleon cannot be
  extracted directly from the measured structure
  functions in deep inelastic scattering
  experiments.
• They mainly are predicted by using the QCD
  evolution equations.

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Many Interesting Subjects Relating to PDF

• Factorization

• Evolution Dynamics

• Shadowing, Anti-shadowing

• Saturation, Color Glass Condensation

• Higher Twist Effects

• Nuclear Effects

• Spin Problem, Polarized SFs
• Asymmetry of Quark Distributions
• Diffractive SFs
• Large Rapidity Gap
• Generalized (skewed) Parton Distributions

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All stories about small x physics are written by
using the QCD evolution equaitons
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Small X
Color Glass Condensation
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Fusion
Corrections of Gluon Fusion
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?
DGLAP or BFKL
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QCD Evolution Equations
BFKL (by Balitsky, Fadin, Kuraev and Lipatov)
GLR-MQ (by Gribov, Levin and Ryskin , Mueller and
Qiu)
Small x
DGLAP (by Dokshitzer, Gribov, Lipatov, Altarelli
and Parisi )
Modified DGLAP (by Zhu, Ruan and Shen)
JIMWLK (by Jalilian-Marian, Iancu, McLerran,
Weigert, Leonidov and Kovner)
Balitsky-Kovchegov equation
Various versions of the evolution equations based
on the color dipole picture
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The BK equation is equivalent to the leading part
of the decoupled JIMWLK equation

• A remarkable feature which emerges from the
  solution of the JIMWLK equation is that the
  scattering amplitude gradually approaches to a
  limit form. This behavior is called the
  saturation, where the gluon fusion balances with
  the gluon splitting.

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From saturation scale QS2, QCD evolution is
stopped
Saturation Scale
DGLAP
MD-DGLAP
JIMWLK
TRUE ?
Qs2 ?
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Some Predictions
The CGC approach for nucleus - nucleus collision
with the saturation of parton density.
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Glasma
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Physical Pictures ofPresent QCD Evolution
Equations

• DGLAP
• BFKL
• GLR-MQ-ZRSDGLAPgluon fusion
• BKBFKLgluon fusion???

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DGLAP
BFKL
Different face in different frame BK
GLR-MQ-ZRS
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BK in target rest frame and impact space
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BK in Bjorken frame and impact space
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DGLAP
BFKL
BK
GLR-MQ-ZRS
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• BK is inconsistent with DGLAP, BFKL and
  GLR-MQ-ZRS in physical picture.
• BK also is inconsistent with DGLAP, BFKL and
  GLR-MQ-ZRS in the equation form.

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DGLAP
BFKL
Different forms
GLR-MQ-ZRS
BK
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BK in impact space and scattering amplitude
BK in momentum space and UPDF
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DGLAP
BFKL
BK
GLR-MQ-ZRS
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J.H. Ruan, Z.Q. Shen and W. ZhuNuclear Physics
B760 (2007) 128.
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The comparisons of the numerical solutions for
the x-dependence of the unintegrated gluon
distribution using theBK-like equation (solid
curves) and BK equation (broken curves).
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For the k2-dependence of the unintegrated gluon
distribution.
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BK in Bjorken frame and momentum space
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In the GLR approach there is no systematic
resummation of small-x effects.

• Therefore, the LL(1/x)-resummation in the BK
  equation is incomplete and we need to consider
  the corrections of the gluon fusion to the
  BFKLequation at the LL(1/x) approximation
  completely in a new evolution equation

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Schematic kinematic regions of four evolution
equations
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DGLAP
BFKL
???
New
GLR-MQ-ZRS
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We try to derive a new modified BFKL equation,
which is consistent with DGLAP, BFKL and
GLR-MQ-ZRS.
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DGLAP amplitude (for gluon)
Impulse approximation

• At small x and fixed Q2, beyond impulse
  approximation

What will happen?
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• The correlations among the initial partons are
  neglected in the derivation of the DGLAP
  equation. This assumption is invalid in the
  higher density region of partons, where the
  parton wave functions begin tospatially overlap.
• The corrections of the correlations among
  initial gluons to the elementary amplitude at
  small x should be considered.
• We add a possible initial gluon to Fig.1a step
  by step.

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BFKL
DGLAP
Modified BFKL
GLR-MQ-ZRS
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QCD, Parton Model, Tree Level


a
b
c


e
d
2


f
Beyond impulse approximation
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Some Results
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MD-BFKL Equation NEW
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DGLAP
BFKL
NEW MD-BFKL
GLR-MQ-ZRS
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DGLAP
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Infrared divergences

• The evolution kernel has singularities ,which
  relate to the emission or absorption of
  quantawith zero momentum.
• Since a correct theory is IR safe, the IR
  divergences are cancelled by combining real-and
  virtual-soft gluon emissions.

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BFKL
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GLR-MQ-ZRS
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MD-BFKL
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TOPT

• W. Zhu, Nucl. Phys. B551, 245 (1999).
• W. Zhu and J.H. Ruan, Nucl. Phys. B559,
  378(1999).
• W. Zhu, Z.Q. Shen and J.H. Ruan, Nucl. Phys.
  B692, 417 (2004)

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Time ordred perturbation theory

• The sum of cut graphs is necessary not only for
  infrared safety, but also for collecting the
  leading contributions and restoring unitarity.
• The TOPT-cutting rules are proposed to present
  the simple connections among the relating
  cut-diagrams including real- and virtual-diagrams.

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DGLAP-----BFKL

• At leading approximation, the elementary
  amplitudes of BFKL and DGLAP equations share the
  same evolution kernel-gluon splitting.
• But in the BFKL-initial state, the initial
  state in the BFKL equation can connect with the
  gluon splitting vertex by two different ways.
• The factor 1/k2 in the DGLAP evolution
  kernel becomes
• k2/k2/(k'-k)2 in the BFKL equation
  because of the
• contributions of the interference diagrams.

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DGLAP----BFKL
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GLR----MD-BFKL
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DGLAP
BFKL
NEW MD-BFKL
GLR-MQ-ZRS
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• Once the DGLAP, BFKL and GLR-MQ-ZRS
  equations are determined,
• the form of the MD-BFKL equation is fixed.

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Solutions of the MD-BFKL equation

• A stronger shadowing suppresses the gluon density
  and even leads to the gluon disappearance bellow
  the saturation region.
• This unexpected effect is caused by a chaotic
  solution of the new equation

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Input distribution
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Gluon disappearance at xltx_c
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A unexpected solution

• The unintegrated gluon distribution function
  F(x,k2) in the MD-BFKL equation begins its
  smooth evolution under suppression of gluon
  recombination like the solution of the BK
  equation.
• When x comes to a critical x_c, F(x,k2)
  will oscillate aperiodically and the shadowing
  effect suddenly increases .
• This stronger shadowing breaks the balance
  between the gluon fusion and splitting.

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• We find that this is the characteristic
  feature of Chaos random and sensitive to the
  initial conditions.

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Lyapunov exponents
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The increase of the new particle events the
increasing energy s will be stopped due to the
gluondisappearance when xlt x_c.
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Will chaos effect we demonstrated in the
MD-BFKL equationdisappear after further
corrections are considered?

• The fact that chaos appear in the MD-BFKL
  equation firstly is related to the singularities
  in its nonlinear terms.
• The possible QCD corrections to the MD-BFKL will
  probably make singularities of this nonlinear
  equation more complicated.

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• We could expect that more interesting chaos
  phenomena will appear in the new MD-BFKL
  equation.

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In summary

• The corrections of the gluon fusion to the BFKL
  equation in a unified partonic framework are
  studied.
• This modified BFKLequation predicts a stronger
  shadowing, which suppresses the gluon density and
  even leads to the gluon disappearance at a very
  small x region. We conclude that this unexpected
  effect is caused by a chaotic solution of the
  nonlinear evolution equation.

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Can a chaos solution in QCD evolution equation
restrain high energy collider physics?