# Nonlinear Correction to the Longitudinal Structure Function at Small

###### Abstract

We computed the longitudinal proton structure function , using the nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-parisi (NLDGLAP) evolution equation approach at small . For the gluon distribution, the nonlinear effects are related to the longitudinal structure function. As, the very small behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. We show, the strong rise that is corresponding to the linear QCD evolution equations, can be tamed by screening effects. Consequently, the obtained longitudinal structure function shows a tamed growth at small . We computed the predictions for all detail of the nonlinear longitudinal structure function in the kinematic range where it has been measured by collaboration and compared with computation Moch, Vermaseren and Vogt at the second order with input data from MRST QCD fit.

###### pacs:

13.85Hd, 12.38.Bx, 12.38.-t, 13.60.Hb,11.55Jy, 12.38.-t, 14.70.Dj^{†}

^{†}preprint: APS/123-QED

## .1 1 Introduction

The measurement of the longitudinal structure function
is of great theoretical importance, since it may
allow us to distinguish between different models describing the
QCD evolution at low-. In deep- inelastic scattering (DIS), the
structure function measurements remain incomplete until the
longitudinal structure function is actually measured [1].
At small values, the dominant contribution to
comes from the gluon operators. Hence a measurement of
can be used to extract the gluon structure
function and therefore the measurement of provides a
sensitive test of perturbative QCD [2-3]. As, at small , the
longitudinal structure function can be related to the gluon and
sea- quark distribution. The behavior of the longitudinal
structure function at small is given by the gluon behavior.
The gluon behavior is observed that governs the physics of high
energy processes in QCD. HERA shows [4-9] that the gluon
distribution function has a steep behavior in the small x region
(). This steep behavior is well described
in the framework of the DGLAP [10-12] evolution equations.

In DIS at moderate values of , the linear QCD evolution
equations lead to good description of this process. But at small
, the problem is more complicated since recombination processes
between gluons in a dense system have to be taken into account.
This strong rise can eventually violate unitarity and so it has to
be tamed by screening effects. These screening effects are
provided by multiple gluon interaction which lead to the nonlinear
terms in the DGLAP equations. These nonlinear terms reduce the
growth of the gluon distribution in this kinematic region where
is still small but the density of partons becomes so
large. Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ)[13-14]
performed a detailed study of this region. They argued that the
physical processes of interaction and recombination of partons
become important in the parton cascade at a large value of the
parton density, and that these shadowing corrections could be
expressed in a new evolution equation (the GLR-MQ
equation)[13-14]. The main characteristic of this equation is that
it predicts a saturation of the gluon distribution at very small
[15-16]. This equation was based on two processes in a
parton cascade:

i)The emission induced by the QCD vertex with
the probability which is proportional to where
is the density of gluon in
the transverse plane, is the target area, and is the size of the target which the gluons populate;

ii)The annihilation of a gluon by the same vertex
with the probability which is proportional to
, where is
probability
of the processes.

Therefore, to obtain a precise evidence of the shadowing correction in the HERA kinematic region, we consider the longitudinal structure function that directly dependence on the behavior of the gluon distribution. In this paper we estimate the shadowing correction to the longitudinal structure function behavior. We calculate this observable using the Altarelli- Marinelli equation [17-18]. The longitudinal structure function , projected from the hadronic tensor by combination of the metric and the spacelike momentum transferred by the virtual photon . Indeed, the longitudinal structure function is proportional to hadronic tensor as follows: where is the hadron momentum and is the hadronic tensor. In this relation we neglecting the hadron mass. The basic hypothesis is that the total cross section of a hadronic process can be written as the sum of the contributions of each parton type (quarks, antiquarks, and gluons) carrying a fraction of the hadronic total momentum. In the case of deep- inelastic- scattering it reads:

(1) |

where is the cross section corresponding to the parton and is the probability of finding this parton in the hadron target with the momentum fraction . Now, taking into account the kinematical constrains one gets the relation between the hadronic and the partonic structure functions:

(2) | |||||

where . Equation (3) expresses the hadronic structure functions as the convolution of the partonic structure function, which are calculable in perturbation theory, and the probability of finding a parton in the hadron which is a nonperturbative function. So, in correspondence with Eq.(3) one can write Eq.(1) as follows:

(3) |

where and are the singlet and nonsinglet quark distribution. and are the LO partonic longitudinal structure function corresponding to quarks and gluons, respectively [19-20]. At small the second term with the gluon density is the dominant one. Here the representation for the gluon distribution is used, where is the gluon density. After full agreement has been achieved, in the form of the gluon kernel , the standard collinear factorization formula for the longitudinal structure function at low reads:

(4) |

where kernel is defined by:

(5) |

and are the quark charges.

One of the striking discoveries at HERA is the steep rise of the gluon distribution function with decreasing value [6]. Indeed, considering the HERA data, as is shown, , where is the Pomeron intercept mines one. As the value of the gluon density becomes so large that the annihilation of gluons becomes important. So, this singular behavior is tamed by the shadowing effects. The strategy in this paper is based on the Regge- like behavior of the gluon distribution function that tamed with the shadowing correction. We assume this behavior as:

(6) |

We note that at , shadowing gluon distribution (sh.) and unshadowing gluon distribution (unsh.) behavior are equal. At the small behavior of the shadowing gluon distribution assumed to be[21-24]:

(7) |

where is the value of the gluon which would saturate the unitarity limit in the leading shadowing approximation. Based on this behavior, the shadowing exponent of the gluon distribution can be determined as,

(8) | |||||

here
,
and being the number of
active quark flavors (), also
t=ln,
=ln (that is the
QCD cut- off parameter, i.e.,
). The value of depends on how the gluon ladders couple to the
proton, or on how the gluons are distributed within the proton.
will be of the order of the proton radius
if the gluons are spread
throughout the entire nucleon, or much smaller
if gluons are concentrated in
hot- spot [25] within the proton. This equation (Eq.9) gives the
shadowing exponent of the shadowing gluon distribution function at
the scale . In order to solve this equation [26]
we take with respect to
that is the input unshadowing gluon
distribution that
take from QCD parametrisation.

Applying the dominant shadowing gluon distribution (i.e.Eq.7), in order to calculate of the shadowing longitudinal structure function at small to equation (5). After integration, we find that:

(9) |

where

(10) |

The shadowing gluon distribution function should be defined in this equation (GLR-MQ equation) as:

(11) | |||||

where we used the modified gluon evolution equation arise from fusion of two gluon ladders. In this equation the first term is the standard DGLAP result that is linear into the parton distribution functions. Since we are interesting to evolution of the longitudinal structure function with respect to nonlinear corrections, we can easily perform this behavior using the following equation:

(12) |

where the derivative of the shadowing gluon distribution with respect to is given by Eq.12. Since we have assumed that the gluon have the Regge behavior at low as controlled by shadowing corrections, we can easily solve this equation with respect to this behavior. We find the derivative of shadowing structure function with respect to as:

(13) |

where

(14) | |||

and

(15) | |||

Therefore, the following equation is a formula to extracted the shadowing longitudinal structure function, using the shadowing gluon distribution exponent and the shadowing gluon distribution determined in [26] at small as a function of value with respect to the initial conditions at ,as

(16) |

where

(17) |

and

(18) |

in this equation is a constant and dependence to the initial
conditions at and . The results are shown in
Fig.1 for at hot- spot point with
.

In Fig.1, the values of the nonlinear longitudinal structure
functions are compared with the experimental data[6,27].
Also, we compared our results with
predictions of up to NLO in perturbative QCD [28-32] that
the input densities is given by MRST parameterizations [33]. The simple conclusions,
which could be obtained from this figure, is the following. First
of all, our results at hot- spot for constant give
values comparable of the shadowing longitudinal structure functions that are comparing with
the experimental data. They grow both with the rapidity .
Secondly, our data show that shadowing longitudinal structure
function increase as x decreases, that its corresponding with PQCD
fits at low x, but this behavior tamed with respect to nonlinear
terms at GLR-MQ equation. Consequently, our results based on the
Regge- like behavior of the shadowing gluon distribution function
slower evolution of the shadowing longitudinal structure function
in the nonlinear case at small values of . These effects show
that their strong rise are taming in accordance with unitarity of
the
description by the interactions between gluons.

In conclusion, in this paper we have obtained the effects of
adding the nonlinear GLR-MQ corrections to the DGLAP evolution
equation and especially the shadowing effects to the longitudinal
structure function at low . We saw that the gluon recombination
effects are expected to play an increasingly important role. These
effects that arise from fusion of the two gluon ladders, slow down
the evolution of the gluons from the standard DGLAP behavior. We
show that the obtained results for the shadowing longitudinal
structure function at small- have a power- like behavior. As
this growth tamed by the shadowing effects. This implies that the
dependence of the shadowing
longitudinal structure function at low is consistent with a
power law, , for fixed .
This behavior is associated with the exchange of an object known
as the hard Pomeron and also exponent of the
shadowing longitudinal structure function defined
as a polynomial function with respect to .

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## .2 Figure captions

Fig 1:The values of the shadowing longitudinal structure function
at with
(square) that accompanied with model error by solving the
GLR-MQ evolution equation
that compared with H1 Collab. data
(up and down triangle). The error on the H1
data is the total uncertainty of the determination of
representing the statistical, the systematic and the model errors added in quadrature.
Circle data are the MVV prediction [31-33]. The solid line is the
NLO QCD fit to the H1 data for and
.