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J.Sheibani@stu.yazd.ac.ir

Corresponding author: A.Mirjalili@yazd.ac.ir

Atashbart@gmail.com

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In this article, using Laplace transformation, an analytical solution is obtained for the DGLAP evolution equation at the next-to-leading order of perturbative QCD. The technique is also employed to extract, in the Laplace

QCD factorization theorems

To access the PDFs and then nPDFs, it is required to get the solution of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations

In this paper, the required analysis has been performed using sequential Laplace transforms, which lead us to an analytical solution of the DGLAP evolution equations at NLO approximation. For this purpose, singlet, nonsinglet, and individual gluon distributions inside the nucleus are analytically calculated. We present our results for the valence quark distributions

The remainder of this paper consists of the following sections: In Sec.

The singlet

It is now possible to discuss briefly the method which is based on the Laplace transformation technique to extract analytical solutions for the parton distribution functions, using the DGLAP evolution equations. The evolution equations presented in Eqs.

In continuation, the leading-order splitting functions of the PDFs, presented in Refs.

and

The next-to-leading order splitting functions

Now if we intend to get the solution for the nonsinglet part,

To amend the notations which are used in the article, it should be noted that at the leading order approximation,

We should use the variable

To calculate the parton distribution in nuclear media, we would need to have the parton distributions for a free proton. To achieve this, it is required to use a set of PDFs at the input scale

Parton distribution for free proton at

On the other hand, using a number of parameters, the nPDFs are specified at a fixed

Using a

The following functional forms would be assumed for the weight function in Eq.

The following nPDFs can be obtained, considering the weight function in Eq.

The

For the case of isoscalar nuclei in which the number of protons and neutrons in a nucleus are equal to each other, valence quarks as well as antiquarks would have similar distributions. But since in heavy nuclei the number of the neutrons is larger than the number of protons

As in Ref.

In order to be able to do the required calculations for iron (Fe), calcium (Ca), carbon (C), helium (He), and deuterium (D) nuclei, we need the relevant weight functions which are presented in the following relations in which the effects of shadowing, antishadowing, Fermi motion, and the EMC regions are included

In Fig.

Weight function for Fe, Ca, C, He, and D nuclei, subleveled by (a) to (d) for different types of PDFs.

Parton distribution for

Based on the Laplace transform technique, we perform here an analytical calculation of the nuclear structure function

As before,

Based on the analytical solution for the DGLAP evolution equations, using the Laplace transformation technique, we shall first present in this section our results that have been obtained for the parton distribution functions after which the nuclear structure function ratio

Parton distribution function for

Parton distribution function for

In Fig.

EMC effect for

EMC effect for

EMC effect for

It is seen that our analytical solutions, based on the inverse Laplace transform technique at the NLO approximation for the nuclear structure function over a wide range of

EMC effect for

Finally, in Fig.

EMC effect for

EMC effect for

The results for NLO decoupled analytical evolution equations for singlet

The authors are indebted to F. Olness for giving the required grid data. A.M. acknowledges Yazd University for providing facilities to do this project.