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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 885 (2014) 571–582

www.elsevier.com/locate/nuclphysb

Analysis of the small-x behavior of gluon distribution

and a search for gluon recombination

Mayuri Devee

∗

, J.K. Sarma

HEP Laboratory, Department of Physics, Tezpur University, Tezpur, Assam, 784 028, India

Received 28

April 2014; received in revised form 30 May 2014; accepted 9 June 2014

Available

online 12 June 2014

Editor: Hong-Jian

Abstract

The

behavior of gluon distributions in the small-x region are semi-numerically predicted by using the

nonlinear Gribov–Levin–Ryskin and Mueller–Qiu (GLR–MQ) evolution equation with the nonlinear shad-

owing

term incorporated. It is interesting to observe that gluon recombination at the twist-4 level lowers the

rapid growth of gluon densities with decreasing x compared to the linear DGLAP evolution. The nonlinear

gluon density functions are calculated and compared with different parameterizations as well as models.

Results show that our predictions for the gluon density function are comparable with different parame-

terizations.

We make a deliberate attempt to explore the effect of nonlinear or shadowing corrections and

we observe that nonlinear effects play a vital role in the kinematic region of small-x (x ≤ 10

−3

) and Q

(2 ≤ Q

≤ 20 GeV

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

1. Introduction

At small-x gluons become the most copious partons in hadrons and can dictate the behavior

of hadronic cross sections through their initial distribution. The x and Q

dependence of gluons

can be predicted employing the tools of perturbative QCD through the standard QCD evolution

equations at the twist-2 level such as Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) [1]

and Balitsky–Fadin–Kuraev–Lipatov (BFKL) [2] equations. Both these equations prognosticate

Corresponding author.

E-mail

address: deveemayuri@gmail.com (M. Devee).

http://dx.doi.org/10.1016/j.nuclphysb.2014.06.005

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP

572 M. Devee, J.K. Sarma / Nuclear Physics B 885 (2014) 571–582

an abrupt rise of the gluon densities towards small-x which is perceived in the deep inelastic

scattering (DIS) experiments at HERA [3] as well. The number of quark and gluon distributions

increases towards large-Q

evolution but they remain dilute with their sizes proportional to the

transverse resolution scale 1/Q. But it is expected that the growth of gluon densities saturates at

a given time due to Froissart bound [4]. The Froissart bound indicates that the total cross section

does not grow faster than the log

arithm squared of the energy i.e., σ

total

(ln s)

, where m

is the scale of the range of the strong force. At very high energies, one can get into the region of

smaller and smaller values of x and, under these situations, the gluon recombination being more

effectual balances gluons splitting at some point. Literally, at small-x the growth of the gluon

distribution is incredibly enunciated and as we m

ove towards small-x at ﬁxed Q

the number of

gluons of ﬁxed size 1/Q increases and they will necessarily start to overlap spatially. Thus the

sharp growth of gluon distribution is eventually subdued because of the correlative interactions

between gluons. This process is commonly alluded to as saturation of gluon density and it occurs

when chances of tw

o gluons recombining into one is as prodigious as the chances for a gluon

splitting into two. The pioneering ﬁnding of the geometrical scaling in HERA data [5] as well

as the existence of geometrical scaling in the production of comprehensive jets in the LHC data

[6] suggest that the phenomenon of saturation occurs in nature. Man

y phenomenological and

theoretical efforts have been made in the latest decades exploring this subject [7–13].

The DGLAP evolution equations can delineate the available experimental data in a decent

manner considering a large domain of x and Q

with appropriate parameterizations. Accord-

ingly, the approximate analytical solutions of the DGLAP evolution equations have been studied

in recent years with signiﬁcant phenomenological success [14–17] and the present authors have

also pursued such an approach with reasonable phenomenological success [18]. However, at

small-x, gluon recombination processes leads to nonlinear corrections to the linear DGLAP

volution equations due to multiple gluon interactions. These nonlinear terms tame the abrupt

growth of the gluon distribution in the kinematic region where α

continues to be small but the

density of gluons becomes very high. The ﬁrst perturbative QCD (pQCD) calculations reporting

the fusion of two gluon ladders into one were carried out by Gribov, Levin and Ryskin (GLR)

[19] and by Mueller and Qiu (MQ) [20,21]. They insinuated that these nonlinear corrections

due to gluon recombination could be depicted in a ne

w evolution equation with an additional

nonlinear term quadratic in the gluon density. This equation, widely known as the GLR–MQ

equation, can be regarded as the updated version of the usual DGLAP equations with the correc-

tions for gluon recombination. The GLR–MQ evolution equation deals with all fa

n diagrams, that

is, all workable 2 → 1 ladder recombinations, in the double leading logarithmic approximation

(DLLA). This equation aggregates the contributions of gluon recombination diagrams using the

Abramovsky–Gribov–Kancheli (AGK) cutting rule [22]. Later, Mueller investigated the contri-

butions of multi-parton correlations by using the Glauber model and at the twist-4 approximation

the Glauber–Mueller equation simpliﬁes to the GLR–MQ equation.

The study of the GLR–MQ equation is e

xceedingly important for understanding the non-

linear effects of gluon–gluon fusion due to the high gluon density at small enough x as

well as for the determination of the saturation momentum. Moreover, other nonlinear evolu-

tion equations relevant at high gluon densities such as the Modiﬁed-DGLAP (Md-DGLAP)

[23], Balitsk

y–Kovchegov (BK) [24] and Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–

Kovner (JIMWLK) [25,26] equations have been derived and studied in the recent years. Note that

the BK and JIMWLK equations are both based on BFKL evolution, whereas, the Md-DGLAP

equation is based on DGLAP evolution where the applications of the AGK cutting rule in the

GLR–MQ equation wa

s argued in a more general consideration.

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