1 Antishadowing effect in the unitarized BFKL equation Jianhong Ruan, Zhenqi Shen, Jifeng Yang and Wei Zhu East China Normal University

2 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.

3 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

4 *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

5 AGK or TOPT ? AGK cutting rule----GLR-MQ equation TOPT---Modified DGLAP equation W. Zhu, Nucl. Phys. B551, 245 (1999). Shadowing and Antishadowing effects

6 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.

7 k_T-factorization schema 2. The evolution equation incorporating shadowing and antishadowing effects

8

9

10

11

12

13

14 The one step evolution containing the gluon recombination

15 The cut diagrams of the gluon recombination kernels for modified DGLAP equation

16 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/ ).

17

18 (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.

19 The recombination of two unitegratedgluon distribution functions

20 is more complicated than An approximative model We use the kernel to replace the kernel

21 The contributions of two correlatedunintegrated distribution functions F^(2) to the measured(integrated ) distribution G via the recombination processes are

22

23 Combining withthe BFKL equation, we obtain a unitarized BFKL equation

24 The gluon distribution becomes flatter near the saturation limit

25 3. Numerical analysis

26

27

28

29

30

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

32

33

34 Eq. (23) reduces to the BK equation (in the impactparameter-independent case) at the suturation limit

35

36

37

38

39

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

41

42 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.

43 Our equation BK equation