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Antishadowing effect in the unitarized BFKL equation

Jianhong Ruan, Zhenqi Shen, Jifeng Yang and Wei Zhu

East China Normal University

Nuclear Physics B 760 (2007) 128–144

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1.Introduction

DGLAP

BFKL

GLR-MQ

(by Gribov, Levin and Ryskin , Mueller and Qiu)

Modified DGLAP (by Zhu, Ruan and Shen)

BK equation

(by Balitsky-Kovchegov)

JIMWLK

(by Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov

and Kovner)Modified BFKL

(by Ruan, Shen,Yang and Zhu)

Evolution to large Q^2

Evolution to small x

(by Balitsky, Fadin, Kuraev and Lipatov)

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

),(),,( 22 QxxgQxxq

),(),,( 22TT kxNkxf

……

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Modified DGLAP = DGLAP

+ shadowing ( - )

Modified BFKL = BFKL

+ shadowing ( - )

Nuclear Physics B 760 (2007) 128

Nuclear Physics B 551 (1999) 245;

Nuclear Physics B 559 (1999) 378

+ antishadowing ( + )

+ antishadowing ( + )

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Where are the negative shadowing corrections from ?

---the gluon recombination

The suppression to the gluon splitting comes from its inverse process

The negative screening effect in the recombination process originally occurs in the interferant cut-diagrams of the recombination amplitudes.

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The antishadowing effect always coexists with

the shadowing effect in the QCD

recombination processes

Where are the positive antishadowing corrections from ?

------A general conclusion of the momentum conservation

According to k_T-factorization schema

2. The evolution equation incorporating

shadowing and antishadowing effects

the perturbative evolution kernel

the probe -parton cross section∗

nonperturbative unintegrated gluon distribution function

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Thus, the connection between and

The relation of the unintegrated gluon distribution with the integrated gluon distribution is

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In the evolution along the transverse momentum

In the evolution along the longitudinal momentum

At LL(k^2) A

→0

At LL(x) A

→0

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the DGLAP equation at small x for the gluon distribution

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The modified DGLAP equation

the TOPT calculation give (at DLLA)

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the modified DGLAP equation combining DGLAP dynamics at small x

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 unitegrated gluon distribution functions

is more complicated than

BFKLMDK

DGLAPMDK

An approximate model

DGLAPMDK BFKLMDK =

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the contributions of two correlated unintegrated distribution functions to the measured (integrated) distribution G via the recombination processes

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Combining with the BFKL equation, we obtain a unitarized BFKL equation

The gluon distribution becomes flatter near the saturation limit

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3. Numerical analysis

input distribution at 20 10x

where

The way to get

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(a) The unintegrated gluon distribution function F(x,k2) as the function of x for different values of k2. The solid-,point- and dashed-curves are the solutions of Eq. (20) with ηFmin = 0.001, Eq. (23) and BFKL equation, respectively

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(b) Similar to (a) but as the function of k2 for different values of x. The possible solutions of Eq. (20) should lie between the solid and point curves.

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4. Discussions

• (1) Compare the BK equation, which is originally written in the transverse coordinator space for the scattering amplitude.

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Which reduces to the BK equation (in the impact parameter-independent case) at the saturation limit

With the input distribution

where

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(a) The normalized scattering amplitude N(r,x) as the function of x for different values of r. The solid-, point- and dashed-curves are the solutions of Eq. (32) with ηNmin = 0.01, the BK equation (33) and the linear part of Eq. (32), respectively. The possible solutions of Eq. (32) should lie between the solid and point curves.

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(b) Similar to (a)but as the function of r for different values of x. The solid-, point- and dashed-curves are the solutions of Eq. (32) with ηNmin = 0.01, the BK equation (33) and the linear part of Eq. (32), respectively. The possible solutions of Eq. (32) should lie between the solid and point curves.

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A possible solution of Eq. (32) with ηN = 0.1

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A possible solution of Eq. (32) with ηN = 0.1

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(2) Nuclear shadowing and antishadowing with the MD- BFKL equation

(Jianhong Ruan, Zhenqi Shen, and Wei Zhu)

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Input distribution

We take

we use the well known F2N(x,Q2)-data of a free proton to determine the parameters

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5. Conclusions (1) 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.

(2) the nuclear shadowing and antishadowing effects are explained by unitarized BFKL equation. The Q2- and x-variations of the nuclear parton distributions are detailed based on the level of the unintegrated gluon distribution. In particular, the asymptotical behavior of the unintegrated gluon distribution in various nuclear targets are studied. We find that the geometric scaling in the expected saturation range is violated. Our results in the nuclear targets are insensitive to the input distributions if the parameters are fixed by the data of a free proton.

Thank You!