1 QCD evolution equations at small x (A simple physical picture) Wei Zhu East China Normal University KITPC A simple physical picture

Two strange things that you have never heard I. Strange behavior of QGP II. A lost equation

3 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

4 Impulse app. Beyond Impulse app. Small x What will be happen?

5 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.

6 DGLAP BFKL GLR-MQ-ZRS Nonperturbative correlation Perturbative correlation Balitsky, Fadin, Kuraev and Lipatov Gribov, Levin and Ryskin Mueller and Qiu Zhu, Ruan Shen Dokshitzer, Gribov, Lipatov, Altarelli and Parisi

DGLAP

Real part

10 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) 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 2

18 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.

23

DGLAP----BFKL

DGLAP BFKL

27 DGLAP----BFKL

29 DGLAP BFKL GLR-MQ-ZRS

31 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)

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: W. 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

69 DGLAP BFKL GLR-MQ-ZRS ??? II. Looking for a lost equation

70 Physical Pictures of Present QCD Evolution Equations DGLAP BFKL GLR-MQ-ZRS=DGLAP+gluon fusion BK=BFKL+gluon fusion???

71 BK in target rest frame and impact space

72 BK in Bjorken frame and impact space

73 DGLAPBFKL GLR-MQ-ZRS BK

74 BK in impact space and scattering amplitude BK in momentum space and UPDF

75 DGLAPBFKL GLR-MQ-ZRS BK

Beautiful Nature Beautiful evolution equations

77 We try to derive a new modified BFKL equation, which is consistent with DGLAP, BFKL and GLR-MQ-ZRS.

78 DGLAPBFKL GLR-MQ-ZRS ???

79 DGLAPBFKL GLR-MQ-ZRS ???

80 DGLAP-like----BFKL-like

New

83 DGLAPBFKL GLR-MQ-ZRS NEW MD-BFKL

84 DGLAP BFKL GLR-MQ-ZRS New

85 MD-BFKL

86

87 MD-BFKL Equation NEW Using TOPT-cutting rule

88 DGLAPBFKL GLR-MQ-ZRS NEW MD-BFKL

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

90 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

91 Input distribution

92 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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94 Gluon disappearance at x<x_c

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Lyapunov exponents

97

Why we haven’t found Chaos in 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 evolution equations 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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