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Antishadowing effect in the unitarized BFKL equation

Jianhong Ruan, Zhenqi Shen, Jifeng Yang and Wei Zhu

East China Normal University

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Abstract

• A unitarized BFKL equation incorporating shadowing and antishadowing corrections of the gluon recombinationis proposed. This equation near the saturation limit reduces to the Balitsky-Kovchegov evolution equation. We find that the influence of the antishadowing effect to the pre-asymptotic form of the gluon distribution is un-negligible.

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1. Introduction

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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*Where are from the negative corrections ?

**The suppression to the gluon splitting comes from

its inverse process---the gluon recombination.

***The negative screening effect in the recombination

process originally occurs in the interferant

cut-diagrams of the recombination amplitudes

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AGK or TOPT ?

AGK cutting rule----GLR-MQ equation

TOPT---Modified DGLAP equation

W. Zhu, Nucl. Phys. B551, 245 (1999).

Shadowing and Antishadowing effects

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The antishadowing effect always coexists with the shadowing effect in the QCD recombination

processes ----A general conclusion of the momentum

conservation

Similar antishadowing effect should

exist in any unitarized BFKL equations.

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k_T-factorization schema

2. The evolution equation incorporating

shadowing and antishadowing effects

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The one step evolution containing

the gluon recombination

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The cut diagrams of the gluon

recombination kernels for

modified DGLAP equation

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At DLLA

W. Zhu and J.H. Ruan, Nucl. Phys. B559, 378 (1999);

W. Zhu and Z.Q. Shen, HEP. \& NP. 29, 109 (2005)

(arXiv:hep-ph/0406213).

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(i) The momentum conservation of partons is restored in a complete modified DGLAP equation;

• (ii) Because of the shadowing and antishadowing effects in the modified DGLAP equation have different kinematic regions, the net effect depends not only on the local value of the gluon distribution at the observed point, but also on the shape of the gluon distribution when the Bjorken variable goes from x to x/2.

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The recombination of two unitegratedgluon distribution functions

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is more complicated than

An approximative model

We use the kernel

to replace the

kernel

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The contributions of two correlatedunintegrated distribution functions F^(2) to the

measured(integrated ) distribution G via the recombination processes are

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Combining withthe BFKL equation, we obtain a unitarized BFKL equation

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The gluon distribution becomes flatter near the saturation limit

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3. Numerical analysis

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4. Discussions

• (1) Compare our nonlinear evolution equation with theBK equation, which is originally written in the transverse coordinator space for the scattering amplitude.

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Eq. (23) reduces to the BK equation (in the impactparameter-independent

case) at the suturation limit

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(2) Comparing with the Gotsman- Levin- Maor-Naftali model, Nucl.Phys. A750 (2005) 39

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5. Conclusions

• We presented the correction of the gluon recombination to the BFKL equation and it leads to a new unitarized nonlinear evolution equation, which incorporates both shadowing and antishadowing effects. The new equation reduces to the BK equation near the saturation limit. The numerical solution of the equation shows that the influence of the antishadowing effect to the pre-asymptotic form of the the gluon distribution is un-negligible.

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Our equation BK equation