Abstract

The coupled Altarelli-Parisi (AP) equations for polarized singlet quark distribution and polarized gluon distribution, when considered in the small 𝑥 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 𝑥 (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 𝑥≈0.005 and as a result polarized gluon distribution Δ𝐺(𝑥) is basically unconstrained at small 𝑥. There are theoretical arguments that polarized gluon distribution Δ𝐺(𝑥) and the unpolarized gluon distribution 𝐺(𝑥) are connected through the relation Δ𝐺(𝑥)≈𝑥𝐺(𝑥) at small 𝑥, but they cannot be verified due to lack of data. Precise measurement of the polarized structure function 𝑔1(𝑥,𝑄2) and its logarithmic scale dependence can determine Δ𝐺(𝑥) at small 𝑥 and thus it can reduce the extrapolation uncertainties of Δ𝐺(𝑥) in the integral ∫10Δ𝐺(𝑥,𝑄2)𝑑𝑥 entering in the proton spin sum rule. EIC [4, 5] will allow for a determination of Δ𝐺(𝑥) down to a very small value of 10−4 and it will eventually give the gluon contribution to the spin of the proton over all 𝑥 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 𝑥 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 𝑥 in the NLO and we begin the process by using the boundary condition that the parton distribution vanishes at 𝑥=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 𝑥 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 𝑥 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(𝑥,𝑄2) and as well as for neutron 𝑔𝑛1(𝑥,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(𝑡)𝑑𝑡=𝑎1(𝑡)×𝑓(𝑡)+𝑏1(𝑡)×𝑔(𝑡),(2.1)𝑑𝑔(𝑡)𝑑𝑡=𝑎2(𝑡)×𝑓(𝑡)+𝑏2(𝑡)×𝑔(𝑡),(2.2) where 𝑎1(𝑡), 𝑏1(𝑡), 𝑎2(𝑡), and 𝑏2(𝑡) are known coefficients, 𝑓(𝑡) and 𝑔(𝑡) are unknown functions to be determined, and 𝑡 is the independent variable.

Equations (2.1) and (2.2) are analytically solvable if the coefficients 𝑎𝑖 and 𝑏𝑖, 𝑖=1,2, are constants, that is, independent of 𝑡 [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 𝑎𝑖(𝑡)=𝑎𝑖𝑇(𝑡) and 𝑏𝑖(𝑡)=𝑏𝑖𝑇(𝑡), 𝑖=1,2 have got identical 𝑡 dependence 𝑇(𝑡).

Our system of (2.1) and (2.2) can be written as 𝑑𝑓𝑓𝑎=𝑇(𝑡)1+𝑏1𝑔𝑓𝑑𝑡,𝑑𝑔𝑔𝑎=𝑇(𝑡)2𝑓𝑔+𝑏2𝑑𝑡.(2.3) Integrating (2.3), we have 𝑎ln𝑓=𝑇(𝑡)1+𝑏1𝑔𝑓𝑑𝑡−ln𝑐1𝑎,(2.4)ln𝑔=𝑇(𝑡)2𝑓𝑔+𝑏2𝑑𝑡−ln𝑐2,(2.5) where 𝑐1 and 𝑐2 are constants of integration.

Substracting (2.5) from (2.4), we have 𝑓ln𝑔=𝑎𝑇(𝑡)1+𝑏1𝑔𝑓−𝑎2𝑓𝑔−𝑏2𝑐𝑑𝑡+ln2𝑐1(2.6) or 𝑢ln𝑢0=𝑎𝑇(𝑡)1+𝑏1𝑢−𝑎2𝑢−𝑏2𝑑𝑡,(2.7) where 𝑢0=𝑐2𝑐1,𝑓𝑢=𝑔.(2.8) Differentiating (2.7) with respect to 𝑡, we have 𝑑𝑢𝑎𝑑𝑡=𝑇(𝑡)1𝑢+𝑏1−𝑎2𝑢2−𝑏2𝑢(2.9) leading to 1𝑎2𝜆𝑞−𝜆𝑝𝑑𝑢𝑢−𝜆𝑝−𝑑𝑢𝑢−𝜆𝑞=𝑇(𝑡)𝑑𝑡,(2.10) where 𝜆𝑝,𝑞=𝑏2−𝑎1±𝑎1−𝑏22+4𝑎2𝑏1−2𝑎2.(2.11) Integrating (2.10), we have ln𝑢−𝜆𝑝−ln𝑢−𝜆𝑞=𝑎2𝜆𝑞−𝜆𝑝𝑇(𝑡)𝑑𝑡+ln𝑐,(2.12) where ln𝑐 is the integration constant.

From (2.12) we have 𝑓𝑢=𝑔=𝜆𝑝−𝜆𝑞𝑛∫𝑇𝑐exp(𝑡)𝑑𝑡𝑛∫,1−𝑐exp𝑇(𝑡)𝑑𝑡(2.13) where 𝑛=𝑎2(𝜆𝑞−𝜆𝑝).

Equation (2.13) implies that we can write 𝜆𝑓(𝑡)=𝐾(𝑡)𝑝−𝜆𝑞𝑛,𝑛,𝑐exp𝑇(𝑡)𝑑𝑡𝑔(𝑡)=𝐾(𝑡)1−𝑐exp𝑇(𝑡)𝑑𝑡(2.14) where 𝐾(𝑡) is a function of 𝑡 to be determined.

We now put (2.14) in (2.1) and obtain 𝑑𝐾(𝑡)𝐾=𝑛∫𝑇𝑎exp(𝑡)𝑑𝑡+𝑏𝜆𝑝−𝜆𝑞𝑛∫𝑐exp𝑇(𝑡)𝑑𝑡𝑇(𝑡)𝑑𝑡,(2.15) where 𝑎=𝑐𝑛𝜆𝑞−𝑎1𝜆𝑞−𝑏1,𝑏=𝑎1𝜆𝑝+𝑏1.(2.16) Integrating (2.15), we have 𝐾(𝑡)=𝐻0𝑎exp1+𝑏1𝜆𝑝,𝑇(𝑡)𝑑𝑡(2.17) where 𝐻0 is the constant of integration. From (2.14) we can now obtain the expression for 𝑓(𝑡) and 𝑔(𝑡).

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 𝜕ΔΣ(𝑥,𝑡)=𝛼𝜕𝑡𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑞𝑞𝑥𝑧+𝛼ΔΣ(𝑧,𝑡)𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑞𝑔𝑥𝑧Δ𝐺(𝑧,𝑡),(3.1)𝜕Δ𝐺(𝑥,𝑡)=𝛼𝜕𝑡𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑔𝑞𝑥𝑧+𝛼ΔΣ(𝑧,𝑡)𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑔𝑔𝑥𝑧Δ𝐺(𝑧,𝑡),(3.2)𝜕Δ𝑞𝑖(𝑥,𝑡)=𝛼𝜕𝑡𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑞𝑞𝑥𝑧Δ𝑞𝑖+𝛼(𝑧,𝑡)𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑞𝑔(𝑥/𝑧)2𝑛𝑓Δ𝐺(𝑧,𝑡).(3.3)

The polarized splitting functions Δ𝑃𝑖𝑗(𝑥) are defined as Δ𝑃𝑖𝑗(𝑥)=Δ𝑃(0)𝑖𝑗𝛼(𝑥)+𝑠(𝑡)2𝜋Δ𝑃(1)𝑖𝑗(𝑥).(3.4)Δ𝑃(0)𝑖𝑗(𝑥) and Δ𝑃(1)𝑖𝑗(𝑥) are given in [14, 15].

Δ𝑃(0)𝑖𝑗(𝑥) in the small 𝑥 limit are given as [16] Δ𝑃(0)𝑞𝑞4𝑥=311+2,𝛿(1−𝑥)Δ𝑃(0)𝑞𝑔𝑥=𝑛𝑓[],−1+2𝛿(1−𝑥)Δ𝑃(0)𝑔𝑞4𝑥=3[],2−𝛿(1−𝑥)Δ𝑃(0)𝑔𝑔𝑥=34−136−𝑛𝛿(1−𝑥)𝑓3𝛿(1−𝑥)(3.5) and Δ𝑃(1)𝑖𝑗(𝑥) in the small 𝑥 limit can be given as [14, 15] Δ𝑃(1)𝑞𝑞(𝑥)=Δ𝑃𝑞𝑞0𝑛𝑓+Δ𝑃𝑞𝑞1𝑛𝑓ln𝑥+Δ𝑃𝑞𝑞2𝑛𝑓ln2𝑥,Δ𝑃(1)𝑞𝑔(𝑥)=Δ𝑃𝑞𝑔0𝑛𝑓+Δ𝑃𝑞𝑔1𝑛𝑓ln𝑥+Δ𝑃𝑞𝑔2𝑛𝑓ln2𝑥,Δ𝑃(1)𝑔𝑞(𝑥)=Δ𝑃𝑔𝑞0𝑛𝑓+Δ𝑃𝑔𝑞1𝑛𝑓ln𝑥+Δ𝑃𝑔𝑞2𝑛𝑓ln2𝑥,Δ𝑃(1)𝑔𝑔(𝑥)=Δ𝑃𝑔𝑔0𝑛𝑓+Δ𝑃𝑔𝑔1𝑛𝑓ln𝑥+Δ𝑃𝑔𝑔2𝑛𝑓ln2𝑥,(3.6) where Δ𝑃𝑎𝑏𝑖, 𝑎=𝑞,𝑔; 𝑏=𝑞,𝑔 and 𝑖=0,1,2 are given in the appendix.

𝛼𝑠(𝑡), the running coupling constant of QCD in NLO, is defined as 𝛼𝑠(𝑡)=4𝜋𝛽0𝑡𝛽1−1ln𝑡𝛽20𝑡,(3.7) where 𝛽0=11−2𝑛𝑓3,𝛽1=102−38𝑛𝑓3(3.8) and 𝑛𝑓 is the number of active flavours.

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

3.2. Solutions of AP Equations for ΔΣ(𝑥,𝑡) and Δ𝐺(𝑥,𝑡) in NLO

We first solve (3.1) and (3.2) for obtaining the approximate analytic expressions for ΔΣ(𝑥,𝑡) and Δ𝐺(𝑥,𝑡) in NLO. Using the assumption (3.10) and the small 𝑥 splitting functions in NLO, these equations can be written as 𝜕ΔΣ(𝑥,𝑡)𝑎𝜕𝑡=𝑇(𝑡)1ΔΣ(𝑥,𝑡)+𝑏1Δ𝐺(𝑥,𝑡)+ℎ1(𝑥)1𝑥𝑑𝑧𝑧ΔΣ(𝑧,𝑡)+ℎ2(𝑥)1𝑥𝑑𝑧𝑧ln𝑧ΔΣ(𝑧,𝑡)+ℎ3(𝑥)1𝑥𝑑𝑧𝑧ln2𝑧ΔΣ(𝑧,𝑡)+𝑘1(𝑥)1𝑥𝑑𝑧𝑧Δ𝐺(𝑧,𝑡)+𝑘2(𝑥)1𝑥𝑑𝑧𝑧ln𝑧Δ𝐺(𝑧,𝑡)+𝑘3(𝑥)1𝑥𝑑𝑧𝑧ln2,𝑧Δ𝐺(𝑧,𝑡)𝜕Δ𝐺(𝑥,𝑡)𝑎𝜕𝑡=𝑇(𝑡)2ΔΣ(𝑥,𝑡)+𝑏2Δ𝐺(𝑥,𝑡)+𝑝1(𝑥)1𝑥𝑑𝑧𝑧ΔΣ(𝑧,𝑡)+𝑝2(𝑥)1𝑥𝑑𝑧𝑧ln𝑧ΔΣ(𝑧,𝑡)+𝑝3(𝑥)1𝑥𝑑𝑧𝑧ln2𝑧ΔΣ(𝑧,𝑡)+𝑞1(𝑥)1𝑥𝑑𝑧𝑧Δ𝐺(𝑧,𝑡)+𝑞2(𝑥)1𝑥𝑑𝑧𝑧ln𝑧Δ𝐺(𝑧,𝑡)+𝑞3(𝑥)1𝑥𝑑𝑧𝑧ln2.𝑧Δ𝐺(𝑧,𝑡)(3.11)𝑎1, 𝑏1, 𝑎2, 𝑏2 are some known constants and ℎ𝑖(𝑥), 𝑘𝑖(𝑥), 𝑝𝑖(𝑥) and 𝑞𝑖(𝑥), 𝑖=1,2,3 are known functions of 𝑥.

As described in [10], we obtain the first approximate solutions of (3.11) by replacing ΔΣ(𝑥,𝑡) and Δ𝐺(𝑥,𝑡) under the integrals appearing in the right-hand side of these equations by their boundary values at 𝑥=1 [11–13]: ||ΔΣ(𝑥,𝑡)𝑥=1||=0,Δ𝐺(𝑥,𝑡)𝑥=1=0.(3.12) With these substitutions (3.11) become 𝜕ΔΣ(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ(𝑥,𝑡)+𝑏1𝑇(𝑡)Δ𝐺(𝑥,𝑡),𝜕Δ𝐺(𝑥,𝑡)𝜕𝑡=𝑎2𝑇(𝑡)ΔΣ(𝑥,𝑡)+𝑏2𝑇(𝑡)Δ𝐺(𝑥,𝑡).(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(𝑥,𝑡)=𝜆2Θ1𝑒𝑁1(𝑛𝑓)𝜏(𝑡)−𝜆1Θ2𝑒𝑁2(𝑛𝑓)𝜏(𝑡),Δ𝐺(𝑥,𝑡)=Θ1𝑒𝑁1(𝑛𝑓)𝜏(𝑡)−Θ2𝑒𝑁2(𝑛𝑓)𝜏(𝑡),(3.14) where 𝜆1=𝑏2−𝑎1+𝑎1−𝑏22+4𝑏1𝑎2−2𝑎2,𝜆2=𝑏2−𝑎1−𝑎1−𝑏22+4𝑏1𝑎2−2𝑎2,𝑁1𝑛𝑓=𝑎1+𝑏1𝜆1+𝑎2𝜆2−𝜆1,𝑁2𝑛𝑓=𝑎1+𝑏1𝜆1,𝜏(𝑡)=𝑇(𝑡)𝑑𝑡(3.15) and Θ1 and Θ2 are constants of integration. From now on we shall represent 𝑁𝑖(𝑛𝑓) by 𝑁𝑖 and 𝜏(𝑡) by 𝜏.

Now applying the input distributions at 𝑡=𝑡0, ||ΔΣ(𝑥,𝑡)𝑡=𝑡0=ΔΣ𝑥,𝑡0,||Δ𝐺(𝑥,𝑡)𝑡=𝑡0=Δ𝐺𝑥,𝑡0,(3.16) we can find out the constants of integration Θ1 and Θ2. With these the solutions after first approximation become ΔΣ1(𝑥,𝑡)=𝑈10𝑁(𝑥)exp1𝜏−𝜏0−𝑈10𝑁(𝑥)exp2𝜏−𝜏0,Δ𝐺1(𝑥,𝑡)=𝑉10(𝑁𝑥)exp1𝜏−𝜏0−𝑉10(𝑁𝑥)exp2𝜏−𝜏0,(3.17) where 𝑈10(𝑥)=𝜆2ΔΣ𝑥,𝑡0−𝜆1Δ𝐺𝑥,𝑡0𝜆2−𝜆1,𝑈10(𝑥)=𝜆1ΔΣ𝑥,𝑡0−𝜆2Δ𝐺𝑥,𝑡0𝜆2−𝜆1,𝑉10(𝑥)=ΔΣ𝑥,𝑡0−𝜆1Δ𝐺𝑥,𝑡0𝜆2−𝜆1,𝑉10(𝑥)=ΔΣ𝑥,𝑡0−𝜆2Δ𝐺𝑥,𝑡0𝜆2−𝜆1,𝜏0=||||𝑇𝑑𝑡𝑡=𝑡0,(3.18) and subscript 1 of ΔΣ1(𝑥,𝑡) and Δ𝐺1(𝑥,𝑡) in (3.17) refers to the first approximate solutions.

Now using the expressions (3.17) for ΔΣ1(𝑥,𝑡) and Δ𝐺1(𝑥,𝑡) in the places of ΔΣ(𝑥,𝑡) and Δ𝐺(𝑥,𝑡) appearing under the integrals in the right-hand side of (3.11), we have 𝜕ΔΣ(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ(𝑥,𝑡)+𝑏1𝑇(𝑡)Δ𝐺(𝑥,𝑡)+𝐻10(𝑥)𝑇(𝑡)𝑒𝑁1(𝜏−𝜏0)−𝐻10(𝑥)𝑇(𝑡)𝑒𝑁2(𝜏−𝜏0),𝜕Δ𝐺(𝑥,𝑡)𝜕𝑡=𝑎2𝑇(𝑡)ΔΣ(𝑥,𝑡)+𝑏2𝑇(𝑡)Δ𝐺(𝑥,𝑡)+𝐾10(𝑥)𝑇(𝑡)𝑒𝑁1(𝜏−𝜏0)−𝐾10(𝑥)𝑇(𝑡)𝑒𝑁2(𝜏−𝜏0),(3.19) where 𝐻10(𝑥), 𝐻10, 𝐾10(𝑥), and 𝐾10(𝑥) are known functions of 𝑥.

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 ΔΣ2𝑈(𝑥,𝑡)=20(𝑥)+𝑈21(𝑥)𝜏−𝜏0𝑁exp1𝜏−𝜏0−𝑈20𝑈(𝑥)+21(𝑥)𝜏−𝜏0𝑁exp2𝜏−𝜏0,(3.20)Δ𝐺2𝑉(𝑥,𝑡)=20(𝑥)+𝑉21(𝑥)𝜏−𝜏0𝑁exp1𝜏−𝜏0−𝑉20𝑉(𝑥)+21(𝑥)𝜏−𝜏0𝑁exp2𝜏−𝜏0.(3.21)𝑈20(𝑥), 𝑈21(𝑥), 𝑉20(𝑥), 𝑉21(𝑥) and their tilde counterparts are known functions of 𝑥. 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.


Figure 1 
∑ 𝑥 Δ ( 𝑥 , 𝑄 2 ) at 1 0 G e V 2 at NLO.

Figure 2 
𝑥 Δ 𝐺 ( 𝑥 , 𝑄 2 ) at 1 0 G e V 2 at NLO.

We again substitute ΔΣ2(𝑥,𝑡) and Δ𝐺2(𝑥,𝑡) from (3.20), and (3.21) respectively, for the ΔΣ(𝑥,𝑡) and Δ𝐺(𝑥,𝑡) appearing in the integrals in the right-hand side of (3.11) and the resulting equations can be written as 𝜕ΔΣ(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ(𝑥,𝑡)+𝑏1+𝐻𝑇(𝑡)Δ𝐺(𝑥,𝑡)20(𝑥)+𝐻21(𝑥)𝜏−𝜏0𝑇(𝑡)𝑒𝑁1(𝜏−𝜏0)−𝐻20𝐻(𝑥)+21(𝑥)𝜏−𝜏0𝑇(𝑡)𝑒𝑁2(𝜏−𝜏0),𝜕Δ𝐺(𝑥,𝑡)𝜕𝑡=𝑎2𝑇(𝑡)ΔΣ(𝑥,𝑡)+𝑏2+𝐾𝑇(𝑡)Δ𝐺(𝑥,𝑡)20(𝑥)+𝐾21(𝑥)𝜏−𝜏0𝑇(𝑡)𝑒𝑁1(𝜏−𝜏0)−𝐾20(𝐾𝑥)+21(𝑥)𝜏−𝜏0𝑇(𝑡)𝑒𝑁2(𝜏−𝜏0).(3.22)𝐻20(𝑥), 𝐻21(𝑥), 𝐾20(𝑥) and 𝐾21(𝑥) and their tilde versions as appearing in (3.22) are known functions of 𝑥.

Proceeding in a similar manner we can obtain the solutions of (3.22) as ΔΣ3𝑈(𝑥,𝑡)=30(𝑥)+𝑈31(𝑥)𝜏−𝜏0+𝑈32(𝑥)𝜏−𝜏022𝑁exp1𝜏−𝜏0−𝑈30𝑈(𝑥)+31(𝑥)𝜏−𝜏0+𝑈32(𝑥)𝜏−𝜏022𝑁exp2𝜏−𝜏0,Δ𝐺3𝑉(𝑥,𝑡)=30(𝑥)+𝑉31(𝑥)𝜏−𝜏0+𝑉32(𝑥)𝜏−𝜏022𝑁exp1𝜏−𝜏0−𝑉30(𝑉𝑥)+31(𝑥)𝜏−𝜏0+𝑉32(𝑥)𝜏−𝜏022𝑁exp2𝜏−𝜏0.(3.23) The expressions for 𝑈3𝑖(𝑥) and 𝑉3𝑖(𝑥), 𝑖=0,1,2 and their tilde versions are calculable functions of 𝑥. 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(𝑥,𝑡)=𝑛−1𝑚=0𝑈𝑛𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁1(𝜏−𝜏0)−𝑈𝑛𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁2(𝜏−𝜏0),(3.24)Δ𝐺𝑛(𝑥,𝑡)=𝑛−1𝑚=0𝑉𝑛𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁1(𝜏−𝜏0)−𝑉𝑛𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁2(𝜏−𝜏0),(3.25) where 𝑈𝑛𝑚(𝑥), 𝑈𝑛𝑚(𝑥), 𝑉𝑛𝑚(𝑥), and 𝑉𝑛𝑚(𝑥) are calculable functions of 𝑥. Equations (3.24) and (3.25) are our main results.

3.3. Polarized Structure Functions 𝑔𝑝1(𝑥,𝑄2), 𝑔𝑛1(𝑥,𝑄2)

We have from (3.1) and (3.3) 𝜕ΔΣ𝑞𝑖(𝑥,𝑡)=𝛼𝜕𝑡𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝑃𝑞𝑞𝑥𝑧ΔΣ𝑞𝑖(𝑥,𝑡),(3.26) where ΔΣ𝑞𝑖(𝑥,𝑡)=Δ𝑞𝑖1(𝑥,𝑡)−2𝑛𝑓ΔΣ(𝑥,𝑡).(3.27) We use the small 𝑥 approximation of splitting function Δ𝑃𝑖𝑗(𝑥) and the assumption 𝑇(𝑡)2=𝑇0𝑇(𝑡) [17] (3.10). With these, (3.26) becomes 𝜕ΔΣ𝑞𝑖(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ𝑞𝑖ℎ(𝑥,𝑡)+𝑇(𝑡)1(𝑥)1𝑥ΔΣ𝑞𝑖(𝑧,𝑡)𝑧𝑑𝑧+ℎ2(𝑥)1𝑥ln𝑧ΔΣ𝑞𝑖(𝑧,𝑡)𝑧𝑑𝑧+ℎ3(𝑥)1𝑥ln2𝑧ΔΣ𝑞𝑖(𝑧,𝑡)𝑧,𝑑𝑧(3.28) where 𝑎1 and ℎ𝑖(𝑥), 𝑖=1,2,3 are known functions of 𝑥.

Now to obtain the first approximate solution of (3.28) we replace ΔΣ𝑞𝑖(𝑥,𝑡) under the integrals appearing in the right-hand side of (3.28) by its boundary value at 𝑥=1 [11–13]: ΔΣ𝑞𝑖(||𝑥,𝑡)𝑥=1=0.(3.29) With this substitution, (3.28) becomes 𝜕ΔΣ𝑞𝑖(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ𝑞𝑖(𝑥,𝑡).(3.30) The solution of (3.30) can be given as ΔΣ𝑞𝑖(𝑥,𝑡)=𝐶𝑞𝑖(𝑥)𝑒𝑁3𝜏(𝑡),(3.31) where, 𝐶𝑞𝑖(𝑥) is the 𝑥 dependent constant of integration, 𝑁3=𝑎1.

Now applying the input distribution at 𝑡=𝑡0, that is, ΔΣ𝑞𝑖||(𝑥,𝑡)𝑡=𝑡0=ΔΣ𝑞𝑖𝑥,𝑡0=Δ𝑞𝑖𝑥,𝑡0−12𝑛𝑓ΔΣ𝑥,𝑡0,(3.32) the solution (3.31) can be written as ΔΣ1𝑞𝑖(𝑥,𝑡)=ΔΣ𝑞𝑖𝑥,𝑡0𝑒𝑁3(𝜏(𝑡)−𝜏(𝑡0)),(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 ΔΣ1𝑞𝑖(𝑥,𝑡) in the place of ΔΣ𝑞𝑖(𝑥,𝑡) appearing under the integrals in the right-hand side of (3.28), we have 𝜕ΔΣ𝑞𝑖(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ𝑞𝑖(𝑥,𝑡)+𝐻𝑞1(𝑥)𝑇(𝑡)𝑒𝑁3(𝜏(𝑡)−𝜏(𝑡0)),(3.34) where 𝐻𝑞1(𝑥)=ℎ1(𝑥)1𝑥ΔΣ𝑞𝑖𝑧,𝑡0𝑧𝑑𝑧+ℎ2(𝑥)1𝑥ln𝑧ΔΣ𝑞𝑖𝑧,𝑡0𝑧𝑑𝑧+ℎ3(𝑥)1𝑥ln2𝑧ΔΣ𝑞𝑖𝑧,𝑡0𝑧𝑑𝑧.(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 ΔΣ2𝑞𝑖(𝑥,𝑡)=ΔΣ𝑞𝑖𝑥,𝑡0𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞1(𝑥)𝜏−𝜏0𝑒𝑁3(𝜏−𝜏0).(3.36) Equation (3.36) is an improvement over (3.33).

Again substituting ΔΣ2𝑞𝑖(𝑥,𝑡) from (3.36) for ΔΣ𝑞𝑖(𝑥,𝑡) appearing under the integrals in the right-hand side of (3.28), we have 𝜕ΔΣ𝑞𝑖(𝑥,𝑡)𝜕𝑡=𝑎1𝑇(𝑡)ΔΣ𝑞𝑖(𝑥,𝑡)+𝐻𝑞1(𝑥)𝑇(𝑡)𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞2(𝑥)𝑇(𝑡)𝜏−𝜏0𝑒𝑁3(𝜏−𝜏0).(3.37) Proceeding in a similar manner we can obtain the solution of (3.37) as ΔΣ3𝑞𝑖(𝑥,𝑡)=ΔΣ𝑞𝑖𝑥,𝑡0𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞1(𝑥)𝜏−𝜏0𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞2(𝑥)𝜏−𝜏022𝑒𝑁3(𝜏−𝜏0),(3.38) where 𝐻𝑞2(𝑥)=ℎ1(𝑥)1𝑥𝐻𝑞1(𝑧)𝑧𝑑𝑧+ℎ2(𝑥)1𝑥𝐻ln𝑧𝑞1(𝑧)𝑧𝑑𝑧+ℎ3(𝑥)1𝑥ln2𝑧𝐻𝑞1(𝑧)𝑧𝑑𝑧.(3.39) Equation (3.38) is the solution of (3.28) after third approximation. Similarly, the fourth iterative solution will be ΔΣ4𝑞𝑖(𝑥,𝑡)=ΔΣ𝑞𝑖𝑥,𝑡0𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞1(𝑥)𝜏−𝜏0𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞2(𝑥)𝜏−𝜏022𝑒𝑁3(𝜏−𝜏0)+𝐻𝑞3(𝑥)𝜏−𝜏036𝑒𝑁3(𝜏−𝜏0),(3.40) where𝐻𝑞3(𝑥)=ℎ1(𝑥)1𝑥𝐻𝑞2(𝑧)𝑧𝑑𝑧+ℎ2(𝑥)1𝑥𝐻ln𝑧𝑞2(𝑧)𝑧𝑑𝑧+ℎ3(𝑥)1𝑥ln2𝑧𝐻𝑞2(𝑧)𝑧𝑑𝑧.(3.41)

Proceeding in this way, we have the solution of (3.28) after 𝑛 approximation as ΔΣ𝑛𝑞𝑖(𝑥,𝑡)=𝑛−1𝑚=0𝐻𝑞𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁3(𝜏−𝜏0),(3.42) where 𝐻𝑞𝑚(𝑥)=ℎ1(𝑥)1𝑥𝐻𝑞(𝑚−1)(𝑧)𝑧𝑑𝑧+ℎ2(𝑥)1𝑥𝐻ln𝑧𝑞(𝑚−1)(𝑧)𝑧𝑑𝑧+ℎ3(𝑥)1𝑥ln2𝑧𝐻𝑞(𝑚−1)(𝑧)𝑧𝑑𝑧.(3.43) Now using the expression for ΔΣ𝑚(𝑥,𝑡) (3.24), the analytical expression for individual quark distributions Δ𝑞𝑛𝑖(𝑥,𝑡) after 𝑛 approximation can be given as Δ𝑞𝑛𝑖(𝑥,𝑡)=ΔΣ𝑛𝑞𝑖1(𝑥,𝑡)+2𝑛𝑓ΔΣ𝑛=(𝑥,𝑡)𝑛−1𝑚=0𝑈𝑛𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁1(𝜏−𝜏0)−𝑈𝑛𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁2(𝜏−𝜏0)+𝐻𝑞𝑚(𝑥)𝜏−𝜏0𝑚𝑒𝑚!𝑁3(𝜏−𝜏0).(3.44) We now use (3.44) and (3.25) (for Δ𝐺𝑛(𝑥,𝑡)) to obtain the analytical expressions for 𝑔𝑝1(𝑥,𝑡) and 𝑔𝑛1(𝑥,𝑡) in NLO as 𝑔1𝑛1(𝑥,𝑡)=2𝑞𝑒2𝑞Δ𝑞𝑛𝑖(𝑥,𝑡)+Δ𝑞𝑛𝑖+𝛼(𝑥,𝑡)𝑠(𝑡)2𝜋1𝑥𝑑𝑧𝑧Δ𝐶𝑞𝑥𝑧Δ𝑞𝑛𝑖(𝑧,𝑡)+Δ𝑞𝑛𝑖(𝑧,𝑡)+Δ𝐶𝐺𝑥𝑧Δ𝐺𝑛𝑖,(𝑧,𝑡)(3.45) where subscript 𝑛 indicates 𝑛 approximations and Δ𝐶𝑞(𝑥) and Δ𝐶𝐺(𝑥) are called Wilson coefficients [15] given in the small 𝑥 limit as Δ𝐶𝑞2(𝑥)=3−43ln𝑥,Δ𝐶𝐺3(𝑥)=2+12ln𝑥.(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 𝑥 region (3.11). However, the 𝑥-dependent parts of the solutions (3.23), namely, 𝑈3𝑖(𝑥), 𝑈3𝑖(𝑥), 𝑉3𝑖(𝑥), and 𝑉3𝑖(𝑥) are two-fold integrations of certain hypergeometric and logarithmic functions (the 𝑛th approximate solutions involve (𝑛−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 10−5⪕𝑥⪕1 (the range of 𝑥 where LSS'05 numerical results for parton distributions are available) by the following effective functional forms: ΔΣ3(𝑥,𝑡)=𝐿Σ(𝑥)1+𝜏−𝜏0+𝜏−𝜏022𝑁exp1𝜏−𝜏0𝑁+exp2𝜏−𝜏0,Δ𝐺3(𝑥,𝑡)=𝐿𝐺(𝑥)1+𝜏−𝜏0+𝜏−𝜏022𝑁exp1𝜏−𝜏0𝑁+exp2𝜏−𝜏0,(3.47) where 𝐿Σ(𝑥)=𝛼1𝑥𝛽1(1−𝑥)𝛾11+𝛿1𝑥+𝜉1𝑥𝜂1,𝐿𝐺(𝑥)=𝛼2𝑥𝛽2(1−𝑥)𝛾21+𝛿2𝑥+𝜉2𝑥𝜂2,𝛼1=0.396,𝛽1=0.693,𝛾1=3.046,𝛿1=2.86,𝜉1=−12.049,𝜂1𝛼=4.82,2=3.522,𝛽2=1.88,𝛾2=2.59,𝛿2=−2.81,𝜉2=1.61,𝜂2=0.043.(3.48) Using (3.47) we get the following approximate analytic forms after fourth approximation: ΔΣ4𝑈(𝑥,𝑡)=40(𝑥)+𝑈41(𝑥)𝜏−𝜏0+𝑈42(𝑥)𝜏−𝜏022+𝑈43(𝑥)𝜏−𝜏036𝑁exp1𝜏−𝜏0−𝑈40𝑈(𝑥)+41(𝑥)𝜏−𝜏0+𝑈42(𝑥)𝜏−𝜏022+𝑈43(𝑥)𝜏−𝜏036𝑁exp2𝜏−𝜏0,Δ𝐺4𝑉(𝑥,𝑡)=40(𝑥)+𝑉41(𝑥)𝜏−𝜏0+𝑉42(𝑥)𝜏−𝜏022+𝑉43(𝑥)𝜏−𝜏036𝑁exp1𝜏−𝜏0−𝑉40𝑉(𝑥)+41(𝑥)𝜏−𝜏0+𝑉42(𝑥)𝜏−𝜏022+𝑉43(𝑥)𝜏−𝜏036𝑁exp2𝜏−𝜏0.(3.49)

The expressions for the 𝑥-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 𝑄ΔΣ2=10ΔΣ𝑥,𝑄2𝑄𝑑𝑥,Δ𝐺2=10Δ𝐺𝑥,𝑄2𝑑𝑥(3.50) obtained from our solutions for ΔΣ(𝑥,𝑄2) and Δ𝐺(𝑥,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 10−5⪕𝑥⪕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 0⪕𝑥⪕1. (2)We obtained the solutions of AP equations with small 𝑥 approximation. The disagreement of the integrated quantities with the experimental values, as observed in Table 1, perhaps indicates that incorporation of high 𝑥 effect is important.

Table 1 
Values of Δ∑(𝑄2) and Δ𝐺(𝑄2) obtained from this work in NLO after four iterations at 𝑄2=10GeV2.

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 𝐻𝑞3(𝑥) which can be obtained by evaluating the three integrals as given by (3.41). As such integrals involve several hypergeometric functions 𝐻𝑞2(𝑥) along with logarithmic functions, to proceed for approximate fourth iterative analytical solution, we simplify it by performing the parametrizations in the range 10−5⪕𝑥⪕1 (the range of 𝑥 where LSS'05 numerical results for parton distributions are available) to get the following effective expressions for 𝐻𝑞2(𝑥)𝐻𝑢2(𝑥)(for𝑢quark)=103.04𝑥−0.15(1−𝑥)8.331+0.477𝑥−1.158𝑥0.096,𝐻𝑑2(𝑥)(for𝑑quark)=−57.27𝑥−0.19(1−𝑥)9.471+0.524𝑥−1.166𝑥0.088,𝐻𝑠2(𝑥)(for𝑠quark)=−15.73𝑥−0.064(1−𝑥)7.631+0.393𝑥−1.143𝑥0.095.(3.51)

We have obtained the fourth iterative solutions by using these effective expressions for 𝐻𝑞2(𝑥), where 𝑞=𝑢,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 𝑥. In the high 𝑥 range our results however deviate from the numerical results probably due to the use of small 𝑥 splitting functions. Another observation is that while the numerical (LSS'05) result gives negative Δ𝑠(𝑥), our solution for Δ𝑠(𝑥) becomes positive beyond 𝑥≈0.3.


Figure 3 
𝑥 Δ 𝑢 ( 𝑥 ) in NLO at 𝑄 2 = 1 0 G e V 2 .

Figure 4 
𝑥 Δ 𝑑 ( 𝑥 ) in NLO at 𝑄 2 = 1 0 G e V 2 .

Figure 5 
𝑥 Δ 𝑠 ( 𝑥 ) in NLO at 𝑄 2 = 1 0 G e V 2 .

We also record the values of the strange quark contribution ∫Δ𝑆=10(Δ𝑠(𝑥)+Δ𝑠(𝑥))𝑑𝑥 (Table 2) towards the spin of the proton at 𝑄2=10GeV2 in different iterations to be compared with the experimental values [21, 22].

Table 2 
Values of Δ𝑆 obtained from this work in NLO after four iterations at 𝑄2=10GeV2.

We now use our formalism in the next to leading order (NLO) to calculate the structure functions 𝑔𝑝1(𝑥,𝑡) and 𝑔𝑛1(𝑥,𝑡) 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).


Figure 6 
Plot for 𝑥 𝑔 𝑝 1 ( 𝑥 , 𝑄 2 ) in NLO at 𝑄 2 = 1 0 G e V 2 .

Figure 7 
Plot for 𝑥 𝑔 𝑛 1 ( 𝑥 , 𝑄 2 ) in NLO at 𝑄 2 = 1 0 G e V 2 .

4. Conclusion

QCD analysis of the quark and gluon contributions towards the spin of the nucleon in the small 𝑥 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 𝑥 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(𝑥,𝑡) and polarized gluon density Δ𝐺(𝑥,𝑡) in the small 𝑥 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 𝑥 region. We have also given the analytical expressions for the individual polarized quark densities Δ𝑞(𝑥,𝑡) and using them we have obtained the expressions for the polarized structure functions 𝑔𝑝1(𝑥,𝑡) and 𝑔𝑛1(𝑥,𝑡). 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(𝑡)2=𝑇0𝑇(𝑡) (3.10) can be removed and it will be our attempt in the future communication.

Appendix

ConsiderΔ𝑃𝑞𝑞0𝑛𝑓=𝜋23𝐶−92𝐹+223−𝜋1823𝐶𝐹𝐶𝐴−49𝐶𝐹𝑇𝑓,Δ𝑃𝑞𝑞1𝑛𝑓=−5𝐶2𝐹+236𝐶𝐹𝐶𝐴−83𝐶𝐹𝑇𝑓,Δ𝑃𝑞𝑞2𝑛𝑓=−3𝐶2𝐹2+𝐶𝐹𝐶𝐴−2𝐶𝐹𝑇𝑓,Δ𝑃𝑞𝑔0𝑛𝑓=2𝜋23𝐶−22𝐹𝑇𝑓+24−2𝜋23𝐶𝐴𝑇𝑓,Δ𝑃𝑞𝑔1𝑛𝑓=2𝐶𝐴𝑇𝑓−9𝐶𝐹𝑇𝑓,Δ𝑃𝑞𝑔2𝑛𝑓=−𝐶𝐹𝑇𝑓−2𝐶𝐴𝑇𝑓,Δ𝑃𝑔𝑞0𝑛𝑓=−172𝐶2𝐹+419𝐶𝐹𝐶𝐴−169𝐶𝐹𝑇𝑓,Δ𝑃𝑔𝑞1𝑛𝑓=−2𝐶2𝐹+4𝐶𝐹𝐶𝐴,Δ𝑃𝑔𝑞2𝑛𝑓=𝐶2𝐹+2𝐶𝐹𝐶𝐴,Δ𝑃𝑔𝑔0𝑛𝑓=97𝐶182𝐴−769𝐶𝐴𝑇𝑓−10𝐶𝐹𝑇𝑓,Δ𝑃𝑔𝑔1𝑛𝑓=293𝐶2𝐴−43𝐶𝐴𝑇𝑓−10𝐶𝐹𝑇𝑓,Δ𝑃𝑔𝑔2𝑛𝑓=4𝐶2𝐴−2𝐶𝐹𝑇𝑓,(A.1) where 𝐶𝐹=4/3, 𝐶𝐴=3 and 𝑇𝑓=𝑛𝑓𝑇𝑅=𝑛𝑓/2.

Therefore, 𝐻3(𝑥)=ℎ1(𝑥)1𝑥𝐿Σ(𝑧)𝑧𝑑𝑧+ℎ2(𝑥)1𝑥𝐿Σ(𝑧)𝑧ln𝑧𝑑𝑧+ℎ3(𝑥)1𝑥𝐿Σ(𝑧)𝑧ln2𝑧𝑑𝑧+𝑘1(𝑥)1𝑥𝐿𝐺(𝑧)𝑧𝑑𝑧+𝑘2(𝑥)1𝑥𝐿𝐺(𝑧)𝑧ln𝑧𝑑𝑧+𝑘3(𝑥)1𝑥𝐿𝐺(𝑧)𝑧ln2𝐾𝑧𝑑𝑧,3(𝑥)=𝑝1(𝑥)1𝑥𝐿Σ(𝑧)𝑧𝑑𝑧+𝑝2(𝑥)1𝑥𝐿Σ(𝑧)𝑧ln𝑧𝑑𝑧+𝑝3(𝑥)1𝑥𝐿Σ(𝑧)𝑧ln2𝑧𝑑𝑧+𝑞1(𝑥)1𝑥𝐿𝐺(𝑧)𝑧𝑑𝑧+𝑞2(𝑥)1𝑥𝐿𝐺(𝑧)𝑧ln𝑧𝑑𝑧+𝑞3(𝑥)1𝑥𝐿𝐺(𝑧)𝑧ln2𝑧𝑑𝑧.(A.2)