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QCD evolution equations at small x

(A simple physical picture)

Wei ZhuEast China Normal University

KITPC 2012.07.

A simple physical picture

Two strange things that you have never heard

I. Strange behavior of QGP

II. A lost equation

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At small x, beyond impulse approximation

DGLAP amplitude (for gluon)

Impulse approximation

Review of QCD evolution equations at small x in a unified partonic framework

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Impulse app. Beyond Impulse app.

Small x

What will be 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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DGLAP

BFKL

GLR-MQ-ZRSGLR-MQ-ZRS

Nonperturbative correlation

Perturbative correlation

Balitsky, Fadin, Kuraev and Lipatov

Gribov, Levin and RyskinMueller and QiuZhu, Ruan Shen

Dokshitzer, Gribov, Lipatov, Altarelli and Parisi

DGLAPDGLAP

Real part

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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.

TOPT Cutting Rules

F.E. Close, J. Qiu and R.G. Roberts, Phys. Rev. D 40 (1989) 2820.

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);

TOPT-Cutting rule

1. List all possible TOPT diagrams with different cuts.

2. The contributions of the cut diagrams

have the identical integral kernel with only the following different factors R:

(a)The sign in the first factor is determined by the energy deficits;

(b)The second factor takes a value of 1/2 if the probe-vertex inserts in the initial line;

(c) function relates to the probe vertex.

BFKL

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BFKL

Comparing with the dipole picture

A strong assumption in the dipole approach is that the transverse size of the dipole is "frozen" during the interacting time.

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DGLAP----BFKL

DGLAP BFKL

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DGLAP----BFKL

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DGLAP

BFKL

GLR-MQ-ZRSGLR-MQ-ZRS

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GLR-MQ-ZRS

TOPT Cutting Rule

GLR-MQ-ZRS

Gribov, Levin and Ryskin , Mueller and Qiu

GLR-MQ vs ZRS

AGK cutting rule vs TOPT cutting rule

Abramovsky, Gribov and Kancheli, Cutting rule (1973)

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Different predictions

Only shadowing effect Shadosing

and

Antishadowing effects

Looking for the antishadowing effect

Test 1: EMC Effect

An alternative form of the GLR-MQ-ZRS equation

Test 2: Cronin Effect

Nuclear modification factor

Test 3:

Nuclear suppression factor

Independent of any energy loss models!

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arXiv:1012.4224W. Zhu, J.H. Ruan and F.Y. Hou

A rapid crossover from week energy

loss to strong energy loss at a universal critical energy of gluon jet Ec ∼ 10GeV

Predictions

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DGLAP BFKL

GLR-MQ-ZRS

??????

II. Looking for a lost equation

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Physical Pictures of

Present QCD Evolution Equations

• DGLAP

• BFKL

• GLR-MQ-ZRS=DGLAP+gluon fusion

• BK=BFKL+gluon fusion???

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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

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

GLR-MQ-ZRS BK

Beautiful Nature

Beautiful evolution equations

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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 BFKL

GLR-MQ-ZRS

???

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DGLAP BFKL

GLR-MQ-ZRS

???

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DGLAP-like----BFKL-like

New

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DGLAP BFKL

GLR-MQ-ZRS NEW MD-BFKL

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DGLAP

BFKL

GLR-MQ-ZRS

New

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MD-BFKL

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MD-BFKL Equation NEWUsing TOPT-cutting rule

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DGLAP BFKL

GLR-MQ-ZRS NEW MD-BFKL

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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 below 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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A unexpected solution

The unintegrated gluon distribution function F(x,k^2) 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,k^2) 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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Gluon disappearance at x<x_c

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Lyapunov exponents

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Why we haven’t found Chaosin

previous nonlinear QCD evolution equations?

General structure of QCD evolution equations

GLR-MQ-ZRS

Nonlinear and Non-singular

k_t is ordered and any oscillations in k_t space are suppressed.

BK

Nonlinear and Non-singular in its nonlinear part

Random and oscillations in k_t space are partly suppressed,

MD-BFKL

Nonlinear and Singular

Random and oscillations in k_t space are strong

1. Chaos solution is a general property in any nonlinear and regularized evolutionequations by virtual processes.

2. QCD evolution equations beyond DGLAP, BFKL, BK……at next order have nonlinear and singural structures.

3. We can meet Chaos in future evolution equations.

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