The coupled Altarelli-Parisi (AP) equations for polarized singlet quark distribution and polarized gluon distribution, when considered in the small π‘₯ 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 π‘₯ (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 π‘₯β‰ˆ0.005 and as a result polarized gluon distribution Δ𝐺(π‘₯) is basically unconstrained at small π‘₯. There are theoretical arguments that polarized gluon distribution Δ𝐺(π‘₯) and the unpolarized gluon distribution 𝐺(π‘₯) are connected through the relation Δ𝐺(π‘₯)β‰ˆπ‘₯𝐺(π‘₯) at small π‘₯, but they cannot be verified due to lack of data. Precise measurement of the polarized structure function 𝑔1(π‘₯,𝑄2) and its logarithmic scale dependence can determine Δ𝐺(π‘₯) at small π‘₯ and thus it can reduce the extrapolation uncertainties of Δ𝐺(π‘₯) in the integral ∫10Δ𝐺(π‘₯,𝑄2)𝑑π‘₯ entering in the proton spin sum rule. EIC [4, 5] will allow for a determination of Δ𝐺(π‘₯) down to a very small value of 10βˆ’4 and it will eventually give the gluon contribution to the spin of the proton over all π‘₯ to about 10 percent accuracy. The same set of measurements will also provide a significantly better determination of the total quark contribution ΔΣ [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 π‘₯ and 𝑄2 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 π‘₯ in the NLO and we begin the process by using the boundary condition that the parton distribution vanishes at π‘₯=1 [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 π‘₯ and 𝑄2-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 π‘₯ 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 𝑔𝑝1(π‘₯,𝑄2) and as well as for neutron 𝑔𝑛1(π‘₯,𝑄2). 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 𝑑𝑓(𝑑)𝑑𝑑=π‘Ž1(𝑑)×𝑓(𝑑)+𝑏1(𝑑)×𝑔(𝑑),(2.1)𝑑𝑔(𝑑)𝑑𝑑=π‘Ž2(𝑑)×𝑓(𝑑)+𝑏2(𝑑)×𝑔(𝑑),(2.2) where π‘Ž1(𝑑), 𝑏1(𝑑), π‘Ž2(𝑑), and 𝑏2(𝑑) are known coefficients, 𝑓(𝑑) and 𝑔(𝑑) are unknown functions to be determined, and 𝑑 is the independent variable.

Equations (2.1) and (2.2) are analytically solvable if the coefficients π‘Žπ‘– and 𝑏𝑖, 𝑖=1,2, are constants, that is, independent of 𝑑 [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 π‘Žπ‘–(𝑑)=π‘Žπ‘–π‘‡(𝑑) and 𝑏𝑖(𝑑)=𝑏𝑖𝑇(𝑑), 𝑖=1,2 have got identical 𝑑 dependence 𝑇(𝑑).

Our system of (2.1) and (2.2) can be written as π‘‘π‘“π‘“ξ‚΅π‘Ž=𝑇(𝑑)1+𝑏1𝑔𝑓𝑑𝑑,π‘‘π‘”π‘”ξ‚΅π‘Ž=𝑇(𝑑)2𝑓𝑔+𝑏2𝑑𝑑.(2.3) Integrating (2.3), we have ξ€œξ‚΅π‘Žln𝑓=𝑇(𝑑)1+𝑏1π‘”π‘“ξ‚Άπ‘‘π‘‘βˆ’ln𝑐1ξ€œξ‚΅π‘Ž,(2.4)ln𝑔=𝑇(𝑑)2𝑓𝑔+𝑏2ξ‚Άπ‘‘π‘‘βˆ’ln𝑐2,(2.5) where 𝑐1 and 𝑐2 are constants of integration.

Substracting (2.5) from (2.4), we have 𝑓ln𝑔=ξ€œξ‚΅π‘Žπ‘‡(𝑑)1+𝑏1π‘”π‘“βˆ’π‘Ž2π‘“π‘”βˆ’π‘2𝑐𝑑𝑑+ln2𝑐1(2.6) or 𝑒ln𝑒0=ξ€œξ‚΅π‘Žπ‘‡(𝑑)1+𝑏1π‘’βˆ’π‘Ž2π‘’βˆ’π‘2𝑑𝑑,(2.7) where 𝑒0=𝑐2𝑐1,𝑓𝑒=𝑔.(2.8) Differentiating (2.7) with respect to 𝑑, we have π‘‘π‘’ξ€·π‘Žπ‘‘π‘‘=𝑇(𝑑)1𝑒+𝑏1βˆ’π‘Ž2𝑒2βˆ’π‘2𝑒(2.9) leading to 1π‘Ž2ξ€·πœ†π‘žβˆ’πœ†π‘ξ€Έξ‚Έπ‘‘π‘’π‘’βˆ’πœ†π‘βˆ’π‘‘π‘’π‘’βˆ’πœ†π‘žξ‚Ή=𝑇(𝑑)𝑑𝑑,(2.10) where πœ†π‘,π‘ž=𝑏2βˆ’π‘Ž1ξ€ΈΒ±ξ”ξ€·π‘Ž1βˆ’π‘2ξ€Έ2+4π‘Ž2𝑏1βˆ’2π‘Ž2.(2.11) Integrating (2.10), we have ξ€·lnπ‘’βˆ’πœ†π‘ξ€Έξ€·βˆ’lnπ‘’βˆ’πœ†π‘žξ€Έ=π‘Ž2ξ€·πœ†π‘žβˆ’πœ†π‘ξ€Έξ€œπ‘‡(𝑑)𝑑𝑑+ln𝑐,(2.12) where ln𝑐 is the integration constant.

From (2.12) we have 𝑓𝑒=𝑔=πœ†π‘βˆ’πœ†π‘žξ€Ίπ‘›βˆ«π‘‡ξ€»π‘exp(𝑑)π‘‘π‘‘ξ€Ίπ‘›βˆ«ξ€»,1βˆ’π‘exp𝑇(𝑑)𝑑𝑑(2.13) where 𝑛=π‘Ž2(πœ†π‘žβˆ’πœ†π‘).

Equation (2.13) implies that we can write ξ‚Έπœ†π‘“(𝑑)=𝐾(𝑑)π‘βˆ’πœ†π‘žξ‚΅π‘›ξ€œ,ξ‚Έξ‚΅π‘›ξ€œ,𝑐exp𝑇(𝑑)𝑑𝑑𝑔(𝑑)=𝐾(𝑑)1βˆ’π‘exp𝑇(𝑑)𝑑𝑑(2.14) where 𝐾(𝑑) is a function of 𝑑 to be determined.

We now put (2.14) in (2.1) and obtain 𝑑𝐾(𝑑)𝐾=ξ€·ξ€·π‘›βˆ«π‘‡ξ€Έξ€Έπ‘Žexp(𝑑)𝑑𝑑+π‘πœ†π‘βˆ’πœ†π‘žξ€·π‘›βˆ«ξ€Έπ‘exp𝑇(𝑑)𝑑𝑑𝑇(𝑑)𝑑𝑑,(2.15) where ξ€·π‘Ž=π‘π‘›πœ†π‘žβˆ’π‘Ž1πœ†π‘žβˆ’π‘1ξ€Έ,𝑏=π‘Ž1πœ†π‘+𝑏1.(2.16) Integrating (2.15), we have 𝐾(𝑑)=𝐻0π‘Žexpξ‚Έξ‚΅1+𝑏1πœ†π‘ξ‚Άξ€œξ‚Ή,𝑇(𝑑)𝑑𝑑(2.17) where 𝐻0 is the constant of integration. From (2.14) we can now obtain the expression for 𝑓(𝑑) and 𝑔(𝑑).

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 πœ•Ξ”Ξ£(π‘₯,𝑑)=π›Όπœ•π‘‘π‘ (𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§Ξ”π‘ƒπ‘žπ‘žξ‚€π‘₯𝑧+𝛼ΔΣ(𝑧,𝑑)𝑠(𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§Ξ”π‘ƒπ‘žπ‘”ξ‚€π‘₯𝑧Δ𝐺(𝑧,𝑑),(3.1)πœ•Ξ”πΊ(π‘₯,𝑑)=π›Όπœ•π‘‘π‘ (𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§Ξ”π‘ƒπ‘”π‘žξ‚€π‘₯𝑧+𝛼ΔΣ(𝑧,𝑑)𝑠(𝑑)ξ€œ2πœ‹1π‘₯𝑑𝑧𝑧Δ𝑃𝑔𝑔π‘₯𝑧Δ𝐺(𝑧,𝑑),(3.2)πœ•Ξ”π‘žπ‘–(π‘₯,𝑑)=π›Όπœ•π‘‘π‘ (𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§Ξ”π‘ƒπ‘žπ‘žξ‚€π‘₯π‘§ξ‚Ξ”π‘žπ‘–+𝛼(𝑧,𝑑)𝑠(𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§Ξ”π‘ƒπ‘žπ‘”(π‘₯/𝑧)2𝑛𝑓Δ𝐺(𝑧,𝑑).(3.3)

The polarized splitting functions Δ𝑃𝑖𝑗(π‘₯) are defined as Δ𝑃𝑖𝑗(π‘₯)=Δ𝑃(0)𝑖𝑗𝛼(π‘₯)+𝑠(𝑑)2πœ‹Ξ”π‘ƒ(1)𝑖𝑗(π‘₯).(3.4)Δ𝑃(0)𝑖𝑗(π‘₯) and Δ𝑃(1)𝑖𝑗(π‘₯) are given in [14, 15].

Δ𝑃(0)𝑖𝑗(π‘₯) in the small π‘₯ limit are given as [16] Δ𝑃(0)π‘žπ‘ž4π‘₯=311+2ξ‚„,𝛿(1βˆ’π‘₯)Δ𝑃(0)π‘žπ‘”π‘₯=𝑛𝑓[],βˆ’1+2𝛿(1βˆ’π‘₯)Δ𝑃(0)π‘”π‘ž4π‘₯=3[],2βˆ’π›Ώ(1βˆ’π‘₯)Δ𝑃(0)𝑔𝑔π‘₯=34βˆ’136ξ‚„βˆ’π‘›π›Ώ(1βˆ’π‘₯)𝑓3𝛿(1βˆ’π‘₯)(3.5) and Δ𝑃(1)𝑖𝑗(π‘₯) in the small π‘₯ limit can be given as [14, 15] Δ𝑃(1)π‘žπ‘ž(π‘₯)=Ξ”π‘ƒπ‘žπ‘ž0𝑛𝑓+Ξ”π‘ƒπ‘žπ‘ž1𝑛𝑓lnπ‘₯+Ξ”π‘ƒπ‘žπ‘ž2𝑛𝑓ln2π‘₯,Δ𝑃(1)π‘žπ‘”(π‘₯)=Ξ”π‘ƒπ‘žπ‘”0𝑛𝑓+Ξ”π‘ƒπ‘žπ‘”1𝑛𝑓lnπ‘₯+Ξ”π‘ƒπ‘žπ‘”2𝑛𝑓ln2π‘₯,Δ𝑃(1)π‘”π‘ž(π‘₯)=Ξ”π‘ƒπ‘”π‘ž0𝑛𝑓+Ξ”π‘ƒπ‘”π‘ž1𝑛𝑓lnπ‘₯+Ξ”π‘ƒπ‘”π‘ž2𝑛𝑓ln2π‘₯,Δ𝑃(1)𝑔𝑔(π‘₯)=Δ𝑃𝑔𝑔0𝑛𝑓+Δ𝑃𝑔𝑔1𝑛𝑓lnπ‘₯+Δ𝑃𝑔𝑔2𝑛𝑓ln2π‘₯,(3.6) where Ξ”π‘ƒπ‘Žπ‘π‘–, π‘Ž=π‘ž,𝑔; 𝑏=π‘ž,𝑔 and 𝑖=0,1,2 are given in the appendix.

𝛼𝑠(𝑑), the running coupling constant of QCD in NLO, is defined as 𝛼𝑠(𝑑)=4πœ‹π›½0𝑑𝛽1βˆ’1ln𝑑𝛽20𝑑ξƒͺ,(3.7) where 𝛽0=11βˆ’2𝑛𝑓3,𝛽1=102βˆ’38𝑛𝑓3(3.8) and 𝑛𝑓 is the number of active flavours.

We define 𝛼𝑇(𝑑)=𝑠(𝑑).2πœ‹(3.9) To proceed further and to apply our formalism, we, as in [17], use the assumption 𝑇(𝑑)2=𝑇0𝑇(𝑑),(3.10) where 𝑇0 is a numerical parameter.

3.2. Solutions of AP Equations for ΔΣ(π‘₯,𝑑) and Δ𝐺(π‘₯,𝑑) in NLO

We first solve (3.1) and (3.2) for obtaining the approximate analytic expressions for ΔΣ(π‘₯,𝑑) and Δ𝐺(π‘₯,𝑑) in NLO. Using the assumption (3.10) and the small π‘₯ splitting functions in NLO, these equations can be written as πœ•Ξ”Ξ£(π‘₯,𝑑)ξ‚΅π‘Žπœ•π‘‘=𝑇(𝑑)1ΔΣ(π‘₯,𝑑)+𝑏1Δ𝐺(π‘₯,𝑑)+β„Ž1ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ΔΣ(𝑧,𝑑)+β„Ž2ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln𝑧ΔΣ(𝑧,𝑑)+β„Ž3ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln2𝑧ΔΣ(𝑧,𝑑)+π‘˜1ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧Δ𝐺(𝑧,𝑑)+π‘˜2ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln𝑧Δ𝐺(𝑧,𝑑)+π‘˜3ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln2ξ‚Ά,𝑧Δ𝐺(𝑧,𝑑)πœ•Ξ”πΊ(π‘₯,𝑑)ξ‚΅π‘Žπœ•π‘‘=𝑇(𝑑)2ΔΣ(π‘₯,𝑑)+𝑏2Δ𝐺(π‘₯,𝑑)+𝑝1(ξ€œπ‘₯)1π‘₯𝑑𝑧𝑧ΔΣ(𝑧,𝑑)+𝑝2(ξ€œπ‘₯)1π‘₯𝑑𝑧𝑧ln𝑧ΔΣ(𝑧,𝑑)+𝑝3ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln2𝑧ΔΣ(𝑧,𝑑)+π‘ž1ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧Δ𝐺(𝑧,𝑑)+π‘ž2ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln𝑧Δ𝐺(𝑧,𝑑)+π‘ž3ξ€œ(π‘₯)1π‘₯𝑑𝑧𝑧ln2ξ‚Ά.𝑧Δ𝐺(𝑧,𝑑)(3.11)π‘Ž1, 𝑏1, π‘Ž2, 𝑏2 are some known constants and β„Žπ‘–(π‘₯), π‘˜π‘–(π‘₯), 𝑝𝑖(π‘₯) and π‘žπ‘–(π‘₯), 𝑖=1,2,3 are known functions of π‘₯.

As described in [10], we obtain the first approximate solutions of (3.11) by replacing ΔΣ(π‘₯,𝑑) and Δ𝐺(π‘₯,𝑑) under the integrals appearing in the right-hand side of these equations by their boundary values at π‘₯=1 [11–13]: ||ΔΣ(π‘₯,𝑑)π‘₯=1||=0,Δ𝐺(π‘₯,𝑑)π‘₯=1=0.(3.12) With these substitutions (3.11) become πœ•Ξ”Ξ£(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)ΔΣ(π‘₯,𝑑)+𝑏1𝑇(𝑑)Δ𝐺(π‘₯,𝑑),πœ•Ξ”πΊ(π‘₯,𝑑)πœ•π‘‘=π‘Ž2𝑇(𝑑)ΔΣ(π‘₯,𝑑)+𝑏2𝑇(𝑑)Δ𝐺(π‘₯,𝑑).(3.13)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 ΔΣ(π‘₯,𝑑)=πœ†2Θ1𝑒𝑁1(𝑛𝑓)𝜏(𝑑)βˆ’πœ†1Θ2𝑒𝑁2(𝑛𝑓)𝜏(𝑑),Δ𝐺(π‘₯,𝑑)=Θ1𝑒𝑁1(𝑛𝑓)𝜏(𝑑)βˆ’Ξ˜2𝑒𝑁2(𝑛𝑓)𝜏(𝑑),(3.14) where πœ†1=𝑏2βˆ’π‘Ž1ξ€Έ+ξ”ξ€·π‘Ž1βˆ’π‘2ξ€Έ2+4𝑏1π‘Ž2βˆ’2π‘Ž2,πœ†2=𝑏2βˆ’π‘Ž1ξ€Έβˆ’ξ”ξ€·π‘Ž1βˆ’π‘2ξ€Έ2+4𝑏1π‘Ž2βˆ’2π‘Ž2,𝑁1𝑛𝑓=π‘Ž1+𝑏1πœ†1+π‘Ž2ξ€·πœ†2βˆ’πœ†1ξ€Έ,𝑁2𝑛𝑓=π‘Ž1+𝑏1πœ†1,ξ€œπœ(𝑑)=𝑇(𝑑)𝑑𝑑(3.15) and Θ1 and Θ2 are constants of integration. From now on we shall represent 𝑁𝑖(𝑛𝑓) by 𝑁𝑖 and 𝜏(𝑑) by 𝜏.

Now applying the input distributions at 𝑑=𝑑0, ||ΔΣ(π‘₯,𝑑)𝑑=𝑑0ξ€·=ΔΣπ‘₯,𝑑0ξ€Έ,||Δ𝐺(π‘₯,𝑑)𝑑=𝑑0ξ€·=Δ𝐺π‘₯,𝑑0ξ€Έ,(3.16) we can find out the constants of integration Θ1 and Θ2. With these the solutions after first approximation become ΔΣ1(π‘₯,𝑑)=π‘ˆ10𝑁(π‘₯)exp1ξ€·πœβˆ’πœ0βˆ’ξ‚π‘ˆξ€Έξ€»10𝑁(π‘₯)exp2ξ€·πœβˆ’πœ0,Δ𝐺1(π‘₯,𝑑)=𝑉10(𝑁π‘₯)exp1ξ€·πœβˆ’πœ0βˆ’ξ‚π‘‰ξ€Έξ€»10(𝑁π‘₯)exp2ξ€·πœβˆ’πœ0,ξ€Έξ€»(3.17) where π‘ˆ10(π‘₯)=πœ†2ΔΣπ‘₯,𝑑0ξ€Έβˆ’πœ†1Δ𝐺π‘₯,𝑑0ξ€Έξ€·πœ†2βˆ’πœ†1ξ€Έξƒͺ,ξ‚π‘ˆ10(π‘₯)=πœ†1ΔΣπ‘₯,𝑑0ξ€Έβˆ’πœ†2Δ𝐺π‘₯,𝑑0ξ€Έξ€·πœ†2βˆ’πœ†1ξ€Έξƒͺ,𝑉10(π‘₯)=ΔΣπ‘₯,𝑑0ξ€Έβˆ’πœ†1Δ𝐺π‘₯,𝑑0ξ€Έξ€·πœ†2βˆ’πœ†1ξ€Έξƒͺ,𝑉10(π‘₯)=ΔΣπ‘₯,𝑑0ξ€Έβˆ’πœ†2Δ𝐺π‘₯,𝑑0ξ€Έξ€·πœ†2βˆ’πœ†1ξ€Έξƒͺ,𝜏0=ξ‚΅ξ€œξ‚Ά||||𝑇𝑑𝑑𝑑=𝑑0,(3.18) and subscript 1 of ΔΣ1(π‘₯,𝑑) and Δ𝐺1(π‘₯,𝑑) in (3.17) refers to the first approximate solutions.

Now using the expressions (3.17) for ΔΣ1(π‘₯,𝑑) and Δ𝐺1(π‘₯,𝑑) in the places of ΔΣ(π‘₯,𝑑) and Δ𝐺(π‘₯,𝑑) appearing under the integrals in the right-hand side of (3.11), we have πœ•Ξ”Ξ£(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)ΔΣ(π‘₯,𝑑)+𝑏1𝑇(𝑑)Δ𝐺(π‘₯,𝑑)+𝐻10(π‘₯)𝑇(𝑑)𝑒𝑁1(πœβˆ’πœ0)βˆ’ξ‚π»10(π‘₯)𝑇(𝑑)𝑒𝑁2(πœβˆ’πœ0),πœ•Ξ”πΊ(π‘₯,𝑑)πœ•π‘‘=π‘Ž2𝑇(𝑑)ΔΣ(π‘₯,𝑑)+𝑏2𝑇(𝑑)Δ𝐺(π‘₯,𝑑)+𝐾10(π‘₯)𝑇(𝑑)𝑒𝑁1(πœβˆ’πœ0)βˆ’ξ‚πΎ10(π‘₯)𝑇(𝑑)𝑒𝑁2(πœβˆ’πœ0),(3.19) where 𝐻10(π‘₯), 𝐻10, 𝐾10(π‘₯), and 𝐾10(π‘₯) are known functions of π‘₯.

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 ΔΣ2ξ€Ίπ‘ˆ(π‘₯,𝑑)=20(π‘₯)+π‘ˆ21ξ€·(π‘₯)πœβˆ’πœ0𝑁exp1ξ€·πœβˆ’πœ0βˆ’ξ‚ƒξ‚π‘ˆξ€Έξ€»20ξ‚π‘ˆ(π‘₯)+21ξ€·(π‘₯)πœβˆ’πœ0𝑁exp2ξ€·πœβˆ’πœ0,ξ€Έξ€»(3.20)Δ𝐺2𝑉(π‘₯,𝑑)=20(π‘₯)+𝑉21ξ€·(π‘₯)πœβˆ’πœ0𝑁exp1ξ€·πœβˆ’πœ0βˆ’ξ‚ƒξ‚π‘‰ξ€Έξ€»20𝑉(π‘₯)+21ξ€·(π‘₯)πœβˆ’πœ0𝑁exp2ξ€·πœβˆ’πœ0.ξ€Έξ€»(3.21)π‘ˆ20(π‘₯), π‘ˆ21(π‘₯), 𝑉20(π‘₯), 𝑉21(π‘₯) and their tilde counterparts are known functions of π‘₯. 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 ΔΣ2(π‘₯,𝑑) and Δ𝐺2(π‘₯,𝑑) from (3.20), and (3.21) respectively, for the ΔΣ(π‘₯,𝑑) and Δ𝐺(π‘₯,𝑑) appearing in the integrals in the right-hand side of (3.11) and the resulting equations can be written as πœ•Ξ”Ξ£(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)ΔΣ(π‘₯,𝑑)+𝑏1+𝐻𝑇(𝑑)Δ𝐺(π‘₯,𝑑)20(π‘₯)+𝐻21ξ€·(π‘₯)πœβˆ’πœ0𝑇(𝑑)𝑒𝑁1(πœβˆ’πœ0)βˆ’ξ‚ƒξ‚π»20𝐻(π‘₯)+21ξ€·(π‘₯)πœβˆ’πœ0𝑇(𝑑)𝑒𝑁2(πœβˆ’πœ0),πœ•Ξ”πΊ(π‘₯,𝑑)πœ•π‘‘=π‘Ž2𝑇(𝑑)ΔΣ(π‘₯,𝑑)+𝑏2+𝐾𝑇(𝑑)Δ𝐺(π‘₯,𝑑)20(π‘₯)+𝐾21(ξ€·π‘₯)πœβˆ’πœ0𝑇(𝑑)𝑒𝑁1(πœβˆ’πœ0)βˆ’ξ‚ƒξ‚πΎ20(𝐾π‘₯)+21(ξ€·π‘₯)πœβˆ’πœ0𝑇(𝑑)𝑒𝑁2(πœβˆ’πœ0).(3.22)𝐻20(π‘₯), 𝐻21(π‘₯), 𝐾20(π‘₯) and 𝐾21(π‘₯) and their tilde versions as appearing in (3.22) are known functions of π‘₯.

Proceeding in a similar manner we can obtain the solutions of (3.22) as ΔΣ3ξƒ¬π‘ˆ(π‘₯,𝑑)=30(π‘₯)+π‘ˆ31ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+π‘ˆ32ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22𝑁exp1ξ€·πœβˆ’πœ0βˆ’ξƒ¬ξ‚π‘ˆξ€Έξ€»30ξ‚π‘ˆ(π‘₯)+31ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+ξ‚π‘ˆ32ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22𝑁exp2ξ€·πœβˆ’πœ0,Δ𝐺3𝑉(π‘₯,𝑑)=30(π‘₯)+𝑉31ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+𝑉32ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22𝑁exp1ξ€·πœβˆ’πœ0βˆ’ξƒ¬ξ‚π‘‰ξ€Έξ€»30(𝑉π‘₯)+31(ξ€·π‘₯)πœβˆ’πœ0ξ€Έ+𝑉32(ξ€·π‘₯)πœβˆ’πœ0ξ€Έ22𝑁exp2ξ€·πœβˆ’πœ0.ξ€Έξ€»(3.23) The expressions for π‘ˆ3𝑖(π‘₯) and 𝑉3𝑖(π‘₯), 𝑖=0,1,2 and their tilde versions are calculable functions of π‘₯. Equations (3.23) are the third iterative solutions of (3.11). Proceeding in this way we can obtain the solutions after 𝑛 successive approximations, which can be written as ΔΣ𝑛(π‘₯,𝑑)=π‘›βˆ’1ξ“π‘š=0ξƒ¬π‘ˆπ‘›π‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁1(πœβˆ’πœ0)βˆ’ξ‚π‘ˆπ‘›π‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁2(πœβˆ’πœ0)ξƒ­,(3.24)Δ𝐺𝑛(π‘₯,𝑑)=π‘›βˆ’1ξ“π‘š=0ξƒ¬π‘‰π‘›π‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁1(πœβˆ’πœ0)βˆ’ξ‚π‘‰π‘›π‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁2(πœβˆ’πœ0)ξƒ­,(3.25) where π‘ˆπ‘›π‘š(π‘₯), ξ‚π‘ˆπ‘›π‘š(π‘₯), π‘‰π‘›π‘š(π‘₯), and ξ‚π‘‰π‘›π‘š(π‘₯) are calculable functions of π‘₯. Equations (3.24) and (3.25) are our main results.

3.3. Polarized Structure Functions 𝑔𝑝1(π‘₯,𝑄2), 𝑔𝑛1(π‘₯,𝑄2)

We have from (3.1) and (3.3) πœ•Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)=π›Όπœ•π‘‘π‘ (𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§Ξ”π‘ƒπ‘žπ‘žξ‚€π‘₯π‘§ξ‚Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑),(3.26) where Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)=Ξ”π‘žπ‘–1(π‘₯,𝑑)βˆ’2𝑛𝑓ΔΣ(π‘₯,𝑑).(3.27) We use the small π‘₯ approximation of splitting function Δ𝑃𝑖𝑗(π‘₯) and the assumption 𝑇(𝑑)2=𝑇0𝑇(𝑑) [17] (3.10). With these, (3.26) becomes πœ•Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)Ξ”Ξ£π‘žπ‘–ξ‚΅β„Ž(π‘₯,𝑑)+𝑇(𝑑)1(ξ€œπ‘₯)1π‘₯Ξ”Ξ£π‘žπ‘–(𝑧,𝑑)𝑧𝑑𝑧+β„Ž2(ξ€œπ‘₯)1π‘₯lnπ‘§Ξ”Ξ£π‘žπ‘–(𝑧,𝑑)𝑧𝑑𝑧+β„Ž3ξ€œ(π‘₯)1π‘₯ln2π‘§Ξ”Ξ£π‘žπ‘–(𝑧,𝑑)𝑧,𝑑𝑧(3.28) where π‘Ž1 and β„Žπ‘–(π‘₯), 𝑖=1,2,3 are known functions of π‘₯.

Now to obtain the first approximate solution of (3.28) we replace Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑) under the integrals appearing in the right-hand side of (3.28) by its boundary value at π‘₯=1 [11–13]: Ξ”Ξ£π‘žπ‘–(||π‘₯,𝑑)π‘₯=1=0.(3.29) With this substitution, (3.28) becomes πœ•Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑).(3.30) The solution of (3.30) can be given as Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)=πΆπ‘žπ‘–(π‘₯)𝑒𝑁3𝜏(𝑑),(3.31) where, πΆπ‘žπ‘–(π‘₯) is the π‘₯ dependent constant of integration, 𝑁3=π‘Ž1.

Now applying the input distribution at 𝑑=𝑑0, that is, Ξ”Ξ£π‘žπ‘–||(π‘₯,𝑑)𝑑=𝑑0=Ξ”Ξ£π‘žπ‘–ξ€·π‘₯,𝑑0ξ€Έ=Ξ”π‘žπ‘–ξ€·π‘₯,𝑑0ξ€Έβˆ’12𝑛𝑓ΔΣπ‘₯,𝑑0ξ€Έ,(3.32) the solution (3.31) can be written as ΔΣ1π‘žπ‘–(π‘₯,𝑑)=Ξ”Ξ£π‘žπ‘–ξ€·π‘₯,𝑑0𝑒𝑁3(𝜏(𝑑)βˆ’πœ(𝑑0)),(3.33) where the subscript 1 refers to the first approximate solution. Equation (3.33) is the first approximate solution of (3.28).

Now using expression (3.33) for ΔΣ1π‘žπ‘–(π‘₯,𝑑) in the place of Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑) appearing under the integrals in the right-hand side of (3.28), we have πœ•Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)+π»π‘ž1(π‘₯)𝑇(𝑑)𝑒𝑁3(𝜏(𝑑)βˆ’πœ(𝑑0)),(3.34) where π»π‘ž1(π‘₯)=β„Ž1ξ€œ(π‘₯)1π‘₯Ξ”Ξ£π‘žπ‘–ξ€·π‘§,𝑑0𝑧𝑑𝑧+β„Ž2ξ€œ(π‘₯)1π‘₯lnπ‘§Ξ”Ξ£π‘žπ‘–ξ€·π‘§,𝑑0𝑧𝑑𝑧+β„Ž3ξ€œ(π‘₯)1π‘₯ln2π‘§Ξ”Ξ£π‘žπ‘–ξ€·π‘§,𝑑0𝑧𝑑𝑧.(3.35)

Following the method of variation [10] and using the input boundary condition (3.32), we have the second iterative solution of (3.28) as ΔΣ2π‘žπ‘–(π‘₯,𝑑)=Ξ”Ξ£π‘žπ‘–ξ€·π‘₯,𝑑0𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž1ξ€·(π‘₯)πœβˆ’πœ0𝑒𝑁3(πœβˆ’πœ0).(3.36) Equation (3.36) is an improvement over (3.33).

Again substituting ΔΣ2π‘žπ‘–(π‘₯,𝑑) from (3.36) for Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑) appearing under the integrals in the right-hand side of (3.28), we have πœ•Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)πœ•π‘‘=π‘Ž1𝑇(𝑑)Ξ”Ξ£π‘žπ‘–(π‘₯,𝑑)+π»π‘ž1(π‘₯)𝑇(𝑑)𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž2ξ€·(π‘₯)𝑇(𝑑)πœβˆ’πœ0𝑒𝑁3(πœβˆ’πœ0).(3.37) Proceeding in a similar manner we can obtain the solution of (3.37) as ΔΣ3π‘žπ‘–(π‘₯,𝑑)=Ξ”Ξ£π‘žπ‘–ξ€·π‘₯,𝑑0𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž1ξ€·(π‘₯)πœβˆ’πœ0𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž2ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22𝑒𝑁3(πœβˆ’πœ0),(3.38) where π»π‘ž2(π‘₯)=β„Ž1ξ€œ(π‘₯)1π‘₯π»π‘ž1(𝑧)𝑧𝑑𝑧+β„Ž2ξ€œ(π‘₯)1π‘₯𝐻lnπ‘§π‘ž1(𝑧)𝑧𝑑𝑧+β„Ž3ξ€œ(π‘₯)1π‘₯ln2π‘§π»π‘ž1(𝑧)𝑧𝑑𝑧.(3.39) Equation (3.38) is the solution of (3.28) after third approximation. Similarly, the fourth iterative solution will be ΔΣ4π‘žπ‘–(π‘₯,𝑑)=Ξ”Ξ£π‘žπ‘–ξ€·π‘₯,𝑑0𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž1ξ€·(π‘₯)πœβˆ’πœ0𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž2ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22𝑒𝑁3(πœβˆ’πœ0)+π»π‘ž3ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ36𝑒𝑁3(πœβˆ’πœ0),(3.40) whereπ»π‘ž3(π‘₯)=β„Ž1ξ€œ(π‘₯)1π‘₯π»π‘ž2(𝑧)𝑧𝑑𝑧+β„Ž2ξ€œ(π‘₯)1π‘₯𝐻lnπ‘§π‘ž2(𝑧)𝑧𝑑𝑧+β„Ž3ξ€œ(π‘₯)1π‘₯ln2π‘§π»π‘ž2(𝑧)𝑧𝑑𝑧.(3.41)

Proceeding in this way, we have the solution of (3.28) after 𝑛 approximation as Ξ”Ξ£π‘›π‘žπ‘–(π‘₯,𝑑)=π‘›βˆ’1ξ“π‘š=0π»π‘žπ‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁3(πœβˆ’πœ0),(3.42) where π»π‘žπ‘š(π‘₯)=β„Ž1ξ€œ(π‘₯)1π‘₯π»π‘ž(π‘šβˆ’1)(𝑧)𝑧𝑑𝑧+β„Ž2ξ€œ(π‘₯)1π‘₯𝐻lnπ‘§π‘ž(π‘šβˆ’1)(𝑧)𝑧𝑑𝑧+β„Ž3ξ€œ(π‘₯)1π‘₯ln2π‘§π»π‘ž(π‘šβˆ’1)(𝑧)𝑧𝑑𝑧.(3.43) Now using the expression for Ξ”Ξ£π‘š(π‘₯,𝑑) (3.24), the analytical expression for individual quark distributions Ξ”π‘žπ‘›π‘–(π‘₯,𝑑) after 𝑛 approximation can be given as Ξ”π‘žπ‘›π‘–(π‘₯,𝑑)=Ξ”Ξ£π‘›π‘žπ‘–1(π‘₯,𝑑)+2𝑛𝑓ΔΣ𝑛=(π‘₯,𝑑)π‘›βˆ’1ξ“π‘š=0ξƒ¬π‘ˆπ‘›π‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁1(πœβˆ’πœ0)βˆ’ξ‚π‘ˆπ‘›π‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁2(πœβˆ’πœ0)+π»π‘žπ‘šξ€·(π‘₯)πœβˆ’πœ0ξ€Έπ‘šπ‘’π‘š!𝑁3(πœβˆ’πœ0)ξƒ­.(3.44) We now use (3.44) and (3.25) (for Δ𝐺𝑛(π‘₯,𝑑)) to obtain the analytical expressions for 𝑔𝑝1(π‘₯,𝑑) and 𝑔𝑛1(π‘₯,𝑑) in NLO as 𝑔1𝑛1(π‘₯,𝑑)=2ξ“π‘žπ‘’2π‘žξ‚ΈΞ”π‘žπ‘›π‘–(π‘₯,𝑑)+Ξ”π‘žπ‘›π‘–+𝛼(π‘₯,𝑑)𝑠(𝑑)ξ€œ2πœ‹1π‘₯π‘‘π‘§π‘§ξ‚€Ξ”πΆπ‘žξ‚€π‘₯π‘§ξ‚ξ€·Ξ”π‘žπ‘›π‘–(𝑧,𝑑)+Ξ”π‘žπ‘›π‘–ξ€Έ(𝑧,𝑑)+Δ𝐢𝐺π‘₯𝑧Δ𝐺𝑛𝑖,(𝑧,𝑑)(3.45) where subscript 𝑛 indicates 𝑛 approximations and Ξ”πΆπ‘ž(π‘₯) and Δ𝐢𝐺(π‘₯) are called Wilson coefficients [15] given in the small π‘₯ limit as Ξ”πΆπ‘ž2(π‘₯)=3βˆ’43lnπ‘₯,Δ𝐢𝐺3(π‘₯)=2+12lnπ‘₯.(3.46) 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 𝑄2=1GeV2 [20]. We have taken 𝑛𝑓=3 and 𝑇0 of (3.10) to be 0.03 [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 π‘₯ region (3.11). However, the π‘₯-dependent parts of the solutions (3.23), namely, π‘ˆ3𝑖(π‘₯), ξ‚π‘ˆ3𝑖(π‘₯), 𝑉3𝑖(π‘₯), and 𝑉3𝑖(π‘₯) are two-fold integrations of certain hypergeometric and logarithmic functions (the 𝑛th approximate solutions involve (π‘›βˆ’1) fold such integrations).

To obtain the analytical forms of the fourth approximate solutions, we parametrize the results of the third iteration in the range 10βˆ’5βͺ•π‘₯βͺ•1 (the range of π‘₯ where LSS'05 numerical results for parton distributions are available) by the following effective functional forms: ΔΣ3(π‘₯,𝑑)=𝐿Σ(π‘₯)1+πœβˆ’πœ0ξ€Έ+ξ€·πœβˆ’πœ0ξ€Έ22𝑁exp1ξ€·πœβˆ’πœ0𝑁+exp2ξ€·πœβˆ’πœ0,Δ𝐺3(π‘₯,𝑑)=𝐿𝐺(π‘₯)1+πœβˆ’πœ0ξ€Έ+ξ€·πœβˆ’πœ0ξ€Έ22𝑁exp1ξ€·πœβˆ’πœ0𝑁+exp2ξ€·πœβˆ’πœ0,ξ€Έξ€»ξ€Έ(3.47) where 𝐿Σ(π‘₯)=𝛼1π‘₯𝛽1(1βˆ’π‘₯)𝛾1ξ€·1+𝛿1π‘₯+πœ‰1π‘₯πœ‚1ξ€Έ,𝐿𝐺(π‘₯)=𝛼2π‘₯𝛽2(1βˆ’π‘₯)𝛾2ξ€·1+𝛿2π‘₯+πœ‰2π‘₯πœ‚2ξ€Έ,𝛼1=0.396,𝛽1=0.693,𝛾1=3.046,𝛿1=2.86,πœ‰1=βˆ’12.049,πœ‚1𝛼=4.82,2=3.522,𝛽2=1.88,𝛾2=2.59,𝛿2=βˆ’2.81,πœ‰2=1.61,πœ‚2=0.043.(3.48) Using (3.47) we get the following approximate analytic forms after fourth approximation: ΔΣ4ξƒ¬π‘ˆ(π‘₯,𝑑)=40(π‘₯)+π‘ˆ41ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+π‘ˆ42ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22+π‘ˆ43ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ36𝑁exp1ξ€·πœβˆ’πœ0βˆ’ξƒ¬ξ‚π‘ˆξ€Έξ€»40ξ‚π‘ˆ(π‘₯)+41ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+ξ‚π‘ˆ42ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22+ξ‚π‘ˆ43(ξ€·π‘₯)πœβˆ’πœ0ξ€Έ36𝑁exp2ξ€·πœβˆ’πœ0,Δ𝐺4𝑉(π‘₯,𝑑)=40(π‘₯)+𝑉41ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+𝑉42ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22+𝑉43ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ36𝑁exp1ξ€·πœβˆ’πœ0βˆ’ξƒ¬ξ‚π‘‰ξ€Έξ€»40𝑉(π‘₯)+41ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ+𝑉42ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ22+𝑉43ξ€·(π‘₯)πœβˆ’πœ0ξ€Έ36𝑁exp2ξ€·πœβˆ’πœ0.ξ€Έξ€»(3.49)

The expressions for the π‘₯-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 𝑄2=10GeV2 (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 ΔΣ(𝑄2) and Δ𝐺(𝑄2) defined as 𝑄ΔΣ2ξ€Έ=ξ€œ10ΔΣπ‘₯,𝑄2𝑄𝑑π‘₯,Δ𝐺2ξ€Έ=ξ€œ10Δ𝐺π‘₯,𝑄2𝑑π‘₯(3.50) obtained from our solutions for ΔΣ(π‘₯,𝑄2) and Δ𝐺(π‘₯,𝑄2) at 𝑄2=10GeV2 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 10βˆ’5βͺ•π‘₯βͺ•1 (as in LSS'05), it may not be adequate in calculating the integrated quantities like ΔΣ(𝑄2) and Δ𝐺(𝑄2) which involve integrations of our solutions (3.49) in the range 0βͺ•π‘₯βͺ•1. (2)We obtained the solutions of AP equations with small π‘₯ approximation. The disagreement of the integrated quantities with the experimental values, as observed in Table 1, perhaps indicates that incorporation of high π‘₯ 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 π»π‘ž3(π‘₯) which can be obtained by evaluating the three integrals as given by (3.41). As such integrals involve several hypergeometric functions π»π‘ž2(π‘₯) along with logarithmic functions, to proceed for approximate fourth iterative analytical solution, we simplify it by performing the parametrizations in the range 10βˆ’5βͺ•π‘₯βͺ•1 (the range of π‘₯ where LSS'05 numerical results for parton distributions are available) to get the following effective expressions for π»π‘ž2(π‘₯)𝐻𝑒2(π‘₯)(for𝑒quark)=103.04π‘₯βˆ’0.15(1βˆ’π‘₯)8.33ξ€·1+0.477π‘₯βˆ’1.158π‘₯0.096ξ€Έ,𝐻𝑑2(π‘₯)(for𝑑quark)=βˆ’57.27π‘₯βˆ’0.19(1βˆ’π‘₯)9.47ξ€·1+0.524π‘₯βˆ’1.166π‘₯0.088ξ€Έ,𝐻𝑠2(π‘₯)(for𝑠quark)=βˆ’15.73π‘₯βˆ’0.064(1βˆ’π‘₯)7.63ξ€·1+0.393π‘₯βˆ’1.143π‘₯0.095ξ€Έ.(3.51)

We have obtained the fourth iterative solutions by using these effective expressions for π»π‘ž2(π‘₯), where π‘ž=𝑒,𝑑,𝑠 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 𝑄2=10GeV2 (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 π‘₯. In the high π‘₯ range our results however deviate from the numerical results probably due to the use of small π‘₯ splitting functions. Another observation is that while the numerical (LSS'05) result gives negative Δ𝑠(π‘₯), our solution for Δ𝑠(π‘₯) becomes positive beyond π‘₯β‰ˆ0.3.

We also record the values of the strange quark contribution βˆ«Ξ”π‘†=10(Δ𝑠(π‘₯)+Δ𝑠(π‘₯))𝑑π‘₯ (Table 2) towards the spin of the proton at 𝑄2=10GeV2 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 𝑔𝑝1(π‘₯,𝑑) and 𝑔𝑛1(π‘₯,𝑑) 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 π‘₯ 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 π‘₯ 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 ΔΣ(π‘₯,𝑑) and polarized gluon density Δ𝐺(π‘₯,𝑑) in the small π‘₯ 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 π‘₯ region. We have also given the analytical expressions for the individual polarized quark densities Ξ”π‘ž(π‘₯,𝑑) and using them we have obtained the expressions for the polarized structure functions 𝑔𝑝1(π‘₯,𝑑) and 𝑔𝑛1(π‘₯,𝑑). Our results are found to be compatible with those obtained numerically (LSS'05). It is possible that such agreement will improve if the assumption 𝑇(𝑑)2=𝑇0𝑇(𝑑) (3.10) can be removed and it will be our attempt in the future communication.


ConsiderΞ”π‘ƒπ‘žπ‘ž0𝑛𝑓=ξ‚΅πœ‹23ξ‚ΆπΆβˆ’92𝐹+ξ‚΅223βˆ’πœ‹1823ξ‚ΆπΆπΉπΆπ΄βˆ’49𝐢𝐹𝑇𝑓,Ξ”π‘ƒπ‘žπ‘ž1𝑛𝑓=βˆ’5𝐢2𝐹+236πΆπΉπΆπ΄βˆ’83𝐢𝐹𝑇𝑓,Ξ”π‘ƒπ‘žπ‘ž2𝑛𝑓=βˆ’3𝐢2𝐹2+πΆπΉπΆπ΄βˆ’2𝐢𝐹𝑇𝑓,Ξ”π‘ƒπ‘žπ‘”0𝑛𝑓=ξ‚΅2πœ‹23ξ‚ΆπΆβˆ’22𝐹𝑇𝑓+ξ‚΅24βˆ’2πœ‹23𝐢𝐴𝑇𝑓,Ξ”π‘ƒπ‘žπ‘”1𝑛𝑓=2πΆπ΄π‘‡π‘“βˆ’9𝐢𝐹𝑇𝑓,Ξ”π‘ƒπ‘žπ‘”2𝑛𝑓=βˆ’πΆπΉπ‘‡π‘“βˆ’2𝐢𝐴𝑇𝑓,Ξ”π‘ƒπ‘”π‘ž0𝑛𝑓=βˆ’172𝐢2𝐹+419πΆπΉπΆπ΄βˆ’169𝐢𝐹𝑇𝑓,Ξ”π‘ƒπ‘”π‘ž1𝑛𝑓=βˆ’2𝐢2𝐹+4𝐢𝐹𝐢𝐴,Ξ”π‘ƒπ‘”π‘ž2𝑛𝑓=𝐢2𝐹+2𝐢𝐹𝐢𝐴,Δ𝑃𝑔𝑔0𝑛𝑓=97𝐢182π΄βˆ’769πΆπ΄π‘‡π‘“βˆ’10𝐢𝐹𝑇𝑓,Δ𝑃𝑔𝑔1𝑛𝑓=293𝐢2π΄βˆ’43πΆπ΄π‘‡π‘“βˆ’10𝐢𝐹𝑇𝑓,Δ𝑃𝑔𝑔2𝑛𝑓=4𝐢2π΄βˆ’2𝐢𝐹𝑇𝑓,(A.1) where 𝐢𝐹=4/3, 𝐢𝐴=3 and 𝑇𝑓=𝑛𝑓𝑇𝑅=𝑛𝑓/2.

Therefore, 𝐻3(π‘₯)=β„Ž1ξ€œ(π‘₯)1π‘₯𝐿Σ(𝑧)𝑧𝑑𝑧+β„Ž2ξ€œ(π‘₯)1π‘₯𝐿Σ(𝑧)𝑧ln𝑧𝑑𝑧+β„Ž3ξ€œ(π‘₯)1π‘₯𝐿Σ(𝑧)𝑧ln2𝑧𝑑𝑧+π‘˜1ξ€œ(π‘₯)1π‘₯𝐿𝐺(𝑧)𝑧𝑑𝑧+π‘˜2ξ€œ(π‘₯)1π‘₯𝐿𝐺(𝑧)𝑧ln𝑧𝑑𝑧+π‘˜3ξ€œ(π‘₯)1π‘₯𝐿𝐺(𝑧)𝑧ln2𝐾𝑧𝑑𝑧,3(π‘₯)=𝑝1ξ€œ(π‘₯)1π‘₯𝐿Σ(𝑧)𝑧𝑑𝑧+𝑝2ξ€œ(π‘₯)1π‘₯𝐿Σ(𝑧)𝑧ln𝑧𝑑𝑧+𝑝3ξ€œ(π‘₯)1π‘₯𝐿Σ(𝑧)𝑧ln2𝑧𝑑𝑧+π‘ž1ξ€œ(π‘₯)1π‘₯𝐿𝐺(𝑧)𝑧𝑑𝑧+π‘ž2ξ€œ(π‘₯)1π‘₯𝐿𝐺(𝑧)𝑧ln𝑧𝑑𝑧+π‘ž3ξ€œ(π‘₯)1π‘₯𝐿𝐺(𝑧)𝑧ln2𝑧𝑑𝑧.(A.2)