#### Abstract

The coupled Altarelli-Parisi (AP) equations for polarized singlet quark distribution and polarized gluon distribution, when considered in the small limit of the next to leading order (NLO) splitting functions, reduce to a system of two first order linear nonhomogeneous integrodifferential equations. We have applied the method of successive approximations to obtain the solutions of these equations. We have applied the same method to obtain the approximate analytic expressions for spin-dependent quark distribution functions with individual flavour and polarized structure functions for nucleon.

#### 1. Introduction

The study of the evolution of the quark and gluon contributions at small (Bjorken variable) towards the spin of the proton through Altarelli-Parisi equations (AP) [1–3] is an important area of DIS. There are no data below and as a result polarized gluon distribution is basically unconstrained at small . There are theoretical arguments that polarized gluon distribution and the unpolarized gluon distribution are connected through the relation at small , but they cannot be verified due to lack of data. Precise measurement of the polarized structure function and its logarithmic scale dependence can determine at small and thus it can reduce the extrapolation uncertainties of in the integral entering in the proton spin sum rule. EIC [4, 5] will allow for a determination of down to a very small value of and it will eventually give the gluon contribution to the spin of the proton over all to about 10 percent accuracy. The same set of measurements will also provide a significantly better determination of the total quark contribution [5].

The Jacobi polynomial method [5–9] is one of the important methods for obtaining the solutions of spin-dependent Altarelli-Parisi equations. The main advantage of this method is that it allows us to factorize the and dependence of the structure function in a manner that allows an efficient parameterization and evolution of the structure function. The method of successive approximations is used to solve the integral equation [10]. In such a method one begins with a crude approximation to a solution by using an initial condition and improves it step by step by applying a repeatable operation (Picard's method of successive approximations). In this work, we have applied this method for obtaining the solutions of the spin-dependent integrodifferential coupled Altarelli-Parisi equations at small in the NLO and we begin the process by using the boundary condition that the parton distribution vanishes at [11–13]. We have shown that the application of this method in solving these equations results in solutions which appear as summation of series, each term of which is the product of and -dependent functions.

We have structured our work as follows: in Section 2, we have given a method of solution of a system of two first order linear homogeneous differential equations with variable coefficients under certain conditions. In Section 3, we have shown that in the small limit of the splitting functions and under some reasonable approximation of the coupling constant, the AP equations for polarized parton distributions become two first order simultaneous linear nonhomogeneous integrodifferential equations. By using the method described in Section 2 and the method of variation [10], we have shown that the solutions can be improved through successive approximations. The same procedure is applied to obtain the approximate analytical expressions for polarized quark distributions with individual flavour and using them we have obtained the expressions for polarized structure functions for proton and as well as for neutron . We have compared our solutions with some numerically obtained solutions.

#### 2. Method of Solving a System of FLHE

A system of two first order linear homogenous differential equations (FLHE) with variable coefficients can be given as where , , , and are known coefficients, and are unknown functions to be determined, and is the independent variable.

Equations (2.1) and (2.2) are analytically solvable if the coefficients and , , are constants, that is, independent of [10]. But, as noted in [10], there is no general method of solving such a system of equations when the coefficients are not constant.

Here we shall present a method of solving (2.1) and (2.2) when and , have got identical dependence .

Our system of (2.1) and (2.2) can be written as Integrating (2.3), we have where and are constants of integration.

Substracting (2.5) from (2.4), we have or where Differentiating (2.7) with respect to , we have leading to where Integrating (2.10), we have where is the integration constant.

From (2.12) we have where .

Equation (2.13) implies that we can write where is a function of to be determined.

We now put (2.14) in (2.1) and obtain where Integrating (2.15), we have where is the constant of integration. From (2.14) we can now obtain the expression for and .

#### 3. Spin-Dependent AP Equations and Polarized Structure Functions in NLO

##### 3.1. Altarelli-Parisi Equations

The coupled Altarelli-Parisi equations [1–3] for polarized singlet quark density, polarized gluon density, and polarized individual quark density are given as

The polarized splitting functions are defined as and are given in [14, 15].

in the small limit are given as [16] and in the small limit can be given as [14, 15] where , ; and are given in the appendix.

, the running coupling constant of QCD in NLO, is defined as where and is the number of active flavours.

We define To proceed further and to apply our formalism, we, as in [17], use the assumption where is a numerical parameter.

##### 3.2. Solutions of AP Equations for and in NLO

We first solve (3.1) and (3.2) for obtaining the approximate analytic expressions for and in NLO. Using the assumption (3.10) and the small splitting functions in NLO, these equations can be written as , , , are some known constants and , , and , are known functions of .

As described in [10], we obtain the first approximate solutions of (3.11) by replacing and under the integrals appearing in the right-hand side of these equations by their boundary values at [11–13]: With these substitutions (3.11) become Equations (3.13) are two first order simultaneous linear homogeneous differential equations with variable coefficients. We solve these equations by the method described in Section 2 and find the solutions as where and and are constants of integration. From now on we shall represent by and by .

Now applying the input distributions at , we can find out the constants of integration and . With these the solutions after first approximation become where and subscript of and in (3.17) refers to the first approximate solutions.

Now using the expressions (3.17) for and in the places of and appearing under the integrals in the right-hand side of (3.11), we have where , , , and are known functions of .

The solutions of the homogeneous parts of (3.19), that is, the solutions of the first order linear coupled homogeneous equation (3.13) can be obtained by the method described earlier and the solutions are given as (3.14). Now, to obtain the solutions of the nonhomogeneous coupled equation (3.19) we apply the method of variation [10]. Thus the solutions of (3.19) can be given as , , , and their tilde counterparts are known functions of . Equations (3.20) and (3.21) are the second iterative solutions of (3.11) and in comparison to the first iterative solutions (3.17), they are closer to the numerical results as seen from Figures 1 and 2.

We again substitute and from (3.20), and (3.21) respectively, for the and appearing in the integrals in the right-hand side of (3.11) and the resulting equations can be written as , , and and their tilde versions as appearing in (3.22) are known functions of .

Proceeding in a similar manner we can obtain the solutions of (3.22) as The expressions for and , and their tilde versions are calculable functions of . Equations (3.23) are the third iterative solutions of (3.11). Proceeding in this way we can obtain the solutions after successive approximations, which can be written as where , , , and are calculable functions of . Equations (3.24) and (3.25) are our main results.

##### 3.3. Polarized Structure Functions ,

We have from (3.1) and (3.3) where We use the small approximation of splitting function and the assumption [17] (3.10). With these, (3.26) becomes where and , are known functions of .

Now to obtain the first approximate solution of (3.28) we replace under the integrals appearing in the right-hand side of (3.28) by its boundary value at [11–13]: With this substitution, (3.28) becomes The solution of (3.30) can be given as where, is the dependent constant of integration, .

Now applying the input distribution at , that is, the solution (3.31) can be written as where the subscript refers to the first approximate solution. Equation (3.33) is the first approximate solution of (3.28).

Now using expression (3.33) for in the place of appearing under the integrals in the right-hand side of (3.28), we have where

Following the method of variation [10] and using the input boundary condition (3.32), we have the second iterative solution of (3.28) as Equation (3.36) is an improvement over (3.33).

Again substituting from (3.36) for appearing under the integrals in the right-hand side of (3.28), we have Proceeding in a similar manner we can obtain the solution of (3.37) as where Equation (3.38) is the solution of (3.28) after third approximation. Similarly, the fourth iterative solution will be where

Proceeding in this way, we have the solution of (3.28) after approximation as where Now using the expression for (3.24), the analytical expression for individual quark distributions after approximation can be given as We now use (3.44) and (3.25) (for ) to obtain the analytical expressions for and in NLO as where subscript indicates approximations and and are called Wilson coefficients [15] given in the small limit as Equations (3.44) and (3.45) are our main results.

##### 3.4. Results and Discussion

Among the several analyses that have included all or most of the present world data on polarized structure functions [18–23] we have used here LSS'05 NLO (MS) input distributions (set-1) at [20]. We have taken and of (3.10) to be [17].

Initially, as described in Section 4, we have worked out up to third approximation and obtained the solutions after third approximation (3.23) of the approximate Altarelli-Parisi equations at small region (3.11). However, the -dependent parts of the solutions (3.23), namely, , , , and are two-fold integrations of certain hypergeometric and logarithmic functions (the th approximate solutions involve fold such integrations).

To obtain the analytical forms of the fourth approximate solutions, we parametrize the results of the third iteration in the range (the range of where LSS'05 numerical results for parton distributions are available) by the following effective functional forms: where Using (3.47) we get the following approximate analytic forms after fourth approximation:

The expressions for the -dependent functions appearing in (3.49) are calculable.

We now compare our solutions after fourth approximation with the LSS'05 numerical results (NLO (MS), set-1) at (Figures 1 and 2).

We observe that with more and more iterations our analytic solutions come closer to the LSS'05 numerical results.

However, the values of and defined as obtained from our solutions for and at in NLO are found to be higher than the corresponding experimental values [23–28].

There may be two sources from where some errors have crept in.(1)As the parametrizations were done only in the range (as in LSS'05), it may not be adequate in calculating the integrated quantities like and which involve integrations of our solutions (3.49) in the range . (2)We obtained the solutions of AP equations with small approximation. The disagreement of the integrated quantities with the experimental values, as observed in Table 1, perhaps indicates that incorporation of high effect is important.

In the case of polarized quark distributions with individual flavour also, we have initially worked up to third iterative solutions (3.38). The fourth iterative solution (3.40) contains terms like which can be obtained by evaluating the three integrals as given by (3.41). As such integrals involve several hypergeometric functions along with logarithmic functions, to proceed for approximate fourth iterative analytical solution, we simplify it by performing the parametrizations in the range (the range of where LSS'05 numerical results for parton distributions are available) to get the following effective expressions for

We have obtained the fourth iterative solutions by using these effective expressions for , where as shown by (3.51).We now compare our work for polarized flavour specific quark distributions with the LSS'05 NLO (MS), set-1 numerical results at (Figures 3, 4, and 5). We have observed that increasing iterations bring our solutions for individual quark densities closer to the numerical results at small . In the high range our results however deviate from the numerical results probably due to the use of small splitting functions. Another observation is that while the numerical (LSS'05) result gives negative , our solution for becomes positive beyond .

We also record the values of the strange quark contribution (Table 2) towards the spin of the proton at in different iterations to be compared with the experimental values [21, 22].

We now use our formalism in the next to leading order (NLO) to calculate the structure functions and as given by (3.45) and compare them with the LSS'05 numerical results (Figures 6 and 7). It is observed that our approximate analytical results are compatible with that obtained numerically (LSS'05).

#### 4. Conclusion

QCD analysis of the quark and gluon contributions towards the spin of the nucleon in the small region is very important for a clear understanding of the spin structure of the nucleon and this is mainly done through the Altarelli-Parisi [1–3] evolution equations. In this work we have given a formalism based on the method of successive approximations, for obtaining analytical solutions in the next to leading order (NLO), valid in small region.

In Section 2 we have given a method for solving a system of two first order linear homogeneous differential equations with variable coefficients.

In Section 3 we have obtained approximate analytical solutions of Altarelli-Parisi equations for the polarized singlet quark density and polarized gluon density in the small limit at NLO by using a method described in Section 2 along with the method of iteration. It is observed that, with increasing number of iterations, the solutions approach the numerical results, specifically in the small region. We have also given the analytical expressions for the individual polarized quark densities and using them we have obtained the expressions for the polarized structure functions and . Our results are found to be compatible with those obtained numerically (LSS'05). It is possible that such agreement will improve if the assumption (3.10) can be removed and it will be our attempt in the future communication.

#### Appendix

Consider where , and .

Therefore,