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Can a chaos solution in QCD evolution equation restrain high

energy collider physics?

朱伟 , 沈祯祺 , 阮建红East China Normal University

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LHC - THE LARGE HADRON COLLIDER

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),( 21 QxG

),( 22 QxG

New

Physics

YT es

kx 2,1

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The kinematic domains probed by the various experiments, shown together with the partons that they constrain

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Gluon distribution and unintegrated gluon distribution

• 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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0

2,QxG

x

? DGLAP or BFKL

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QCD 演化方程研究现状

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QCD Evolution Equations

DGLAP(by Dokshitzer, Gribov, Lipatov, Altarelli and Parisi )

Small x

BFKL (by Balitsky, Fadin, Kuraev and Lipatov)GLR-MQ (by Gribov, Levin and Ryskin , Mueller and Qiu)

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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0x

DGLAP

BFKL

Corrections of Gluon Fusion

toDGLAP GLR-MQ-ZRS

BFKL BK

Fusion

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DGLAP Equation

• Dokshitzer-Gribov-Lipatov 1975

• Altarelli-Parisi 1977

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BFKL Equation

• Balitsky-Fadin-Kuraev-Lipatov 1975-1978

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GLR-MQ-ZRS Equation

• Gribov-Levin-Ryskin 1983

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Where is the GLR equations?

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Mueller-Qiu 1986

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Zhu-Ruan-Shen 1999-2005

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BK Equation

• Balitsky-Kovchegov 1996-2000

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JIMWLK Equation

• Jalilia-Marian, Iancu,McLerra,Weigert,Leonidov and Kovne 1997-2001

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Multi pole Wilson lineBare gluon

Nuclear classical QCD field

(W-W field)

x

y

z

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Impulse app. Beyond Impulse app.

Saturation

GLR-MQ-ZRS equationDGLAP equations JIMWLK equations

Color Glass Condensation

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From saturation scale QS2, QCD evolution is

stoppedSaturation Scale

Qs2 ?

TRUE ?

MD-DGLAP

JIMWLK

DGLAP

2ln k

kd

kxdxg2

2 ),(

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一个人人可以想到的理论框架

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At small x and fixed Q2, beyond impulse approximation

DGLAP amplitude (for gluon)

Impulse approximation

NQ xx

What will happen?

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DGLAP

BFKL

GLR-MQ-ZRS

Modified BFKL

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QCD, Parton Model, Tree Level

+ +

++

+ +……2

a b c

d e

fBeyond impulse approximation

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αs, twist-2, a2 + virtual diagrams

BFKLαs, twist-2, b2 + virtual diagrams

DGLAP

αs2, twist-4, c2 +2ae + virtual

diagrams

GLR-MQ-ZRS

αs2, twist-4, d2 +2bf + virtual

diagrams MD-BFKL

Some Results

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难点

红外安全

如何计祘所有切割图，包括虚图？

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我们掌握的 " 绝技 "

• 基於时序微扰理论的切割法则

• 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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MD-BFKL Equation NEW

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Consistency of four evolution equations

• The DGLAP and BFKL equations have the same evolution dynamics, which evolve on the Q^2-axis and the k_T-plane,respectively.

• The MD-DGLAP and MD-BFKL equations also have the same evolution dynamics , which evolve on the Q^2-axis and the k_T-plane, respectively.

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Schematic kinematic regions of four evolution equations

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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 bellower the saturation region.

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Lyapunov exponents

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The rapid increase of the gluon density with increasing energy will be stopped due to the gluon

disappearance when x<x_c

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Discussions

• Many higher order corrections areneglected, such as possible mixture of the operators with different twists, the NLL (next leading logarithmic) and NLO (next leading order) corrections, singularities from non-perturbativeparts in the factorization procedure.

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Will the chaos effect in the MD-BFKL equation disappear after

further corrections are considered?

• We could expect that more interesting chaos phenomena will appear in new MD-BFKL equation.These phenomena will most be interested to high energy physics and nonlinear science.

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In summary

• 胶子分布函数是研究高能强子碰撞物理的必需知识。

• QCD 演化方程是预言胶子分布函数的可靠工具。

• DGLAP, BFKL, GLR-MQ-ZRS, BK 方程 是常用的四个方程。. 我们在一个更普遍的部分子图像中导出了

第五个 QCD 演化方程 MD-BFKL 方程 .

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在 MD-BFKL 方程中首次出现了非线性方程中常见的 Choas.

• 它预言在某一临界 x 值胶子突然消失 , 从而中断高能强子碰撞中新粒子事例随能量的增加。