# Spin-Dependent Coupled Altarelli-Parisi Equations in the NLO and the Method of Successive Approximations

Authors: Ranjit Choudhury; D. K. Choudhury
Journal: ISRN High Energy Physics
 (2012)

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## Abstract

The coupled Altarelli-Parisi (AP) equations for polarized singlet quark distribution and polarized gluon distribution, when considered in the smallx 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.


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## Body

## 1. Introduction

The study of the evolution of the quark and gluon contributions at smallx (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 ∫01‍ΔG(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 Section2, 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 g1p(x,Q2) and as well as for neutron g1n(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(2.1)df(t)dt=a1(t)×f(t)+b1(t)×g(t),(2.2)dg(t)dt=a2(t)×f(t)+b2(t)×g(t),
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
(2.3)dff=T(t)(a1+b1gf)dt,dgg=T(t)(a2fg+b2)dt.
Integrating (2.3), we have
(2.4)ln⁡f=∫‍T(t)(a1+b1gf)dt-ln⁡c1,(2.5)ln⁡g=∫‍T(t)(a2fg+b2)dt-ln⁡c2,
where c1 and c2 are constants of integration.Substracting (2.5) from (2.4), we have
(2.6)ln⁡fg=∫‍T(t)(a1+b1gf-a2fg-b2)dt+ln⁡c2c1
or
(2.7)ln⁡uu0=∫‍T(t)(a1+b1u-a2u-b2)dt,
where
(2.8)u0=c2c1,u=fg.
Differentiating (2.7) with respect to t, we have
(2.9)dudt=T(t)(a1u+b1-a2u2-b2u)
leading to
(2.10)1a2(λq-λp)[duu-λp-duu-λq]=T(t)dt,
where
(2.11)λp,q=(b2-a1)±(a1-b2)2+4a2b1-2a2.
Integrating (2.10), we have
(2.12)ln⁡(u-λp)-ln⁡(u-λq)=a2(λq-λp)∫‍T(t)dt+ln⁡c,
where lnc is the integration constant.From (2.12) we have
(2.13)u=fg=λp-λqcexp⁡[n∫‍T(t)dt]1-cexp⁡[n∫‍T(t)dt],
where n=a2(λq-λp).Equation (2.13) implies that we can write
(2.14)f(t)=K(t)[λp-λqcexp⁡(n∫‍T(t)dt)],g(t)=K(t)[1-cexp⁡(n∫‍T(t)dt)],
where K(t) is a function of t to be determined.We now put (2.14) in (2.1) and obtain
(2.15)dK(t)K=(aexp⁡(n∫‍T(t)dt)+b)λp-λqcexp⁡(n∫‍T(t)dt)T(t)dt,
where
(2.16)a=c(nλq-a1λq-b1),b=a1λp+b1.
Integrating (2.15), we have
(2.17)K(t)=H0exp⁡[(a1+b1λp)∫‍T(t)dt],
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
(3.1)∂ΔΣ(x,t)∂t=αs(t)2π∫x1‍dzzΔPqq(xz)ΔΣ(z,t)+αs(t)2π∫x1‍dzzΔPqg(xz)ΔG(z,t),(3.2)∂ΔG(x,t)∂t=αs(t)2π∫x1‍dzzΔPgq(xz)ΔΣ(z,t)+αs(t)2π∫x1‍dzzΔPgg(xz)ΔG(z,t),(3.3)∂Δqi(x,t)∂t=αs(t)2π∫x1‍dzzΔPqq(xz)Δqi(z,t)+αs(t)2π∫x1‍dzzΔPqg(x/z)2nfΔG(z,t).The polarized splitting functionsΔPij(x) are defined as
(3.4)ΔPij(x)=ΔPij(0)(x)+αs(t)2πΔPij(1)(x).ΔPij(0)(x) and ΔPij(1)(x) are given in [14, 15].Δ
P
i
j
(
0
)
(
x
) in the small x limit are given as [16]
(3.5)ΔPqq(0)x=43[1+12δ(1-x)],ΔPqg(0)x=nf[-1+2δ(1-x)],ΔPgq(0)x=43[2-δ(1-x)],ΔPgg(0)x=3[4-136δ(1-x)]-nf3δ(1-x)
and ΔPij(1)(x) in the small x limit can be given as [14, 15]
(3.6)ΔPqq(1)(x)=ΔPqq0(nf)+ΔPqq1(nf)ln⁡x+ΔPqq2(nf)ln⁡2x,ΔPqg(1)(x)=ΔPqg0(nf)+ΔPqg1(nf)ln⁡x+ΔPqg2(nf)ln⁡2x,ΔPgq(1)(x)=ΔPgq0(nf)+ΔPgq1(nf)ln⁡x+ΔPgq2(nf)ln⁡2x,ΔPgg(1)(x)=ΔPgg0(nf)+ΔPgg1(nf)ln⁡x+ΔPgg2(nf)ln⁡2x,
where ΔPabi, a=q,g; b=q,g and i=0,1,2 are given in the appendix.α
s
(
t
), the running coupling constant of QCD in NLO, is defined as
(3.7)αs(t)=4πβ0t(1-β1ln⁡tβ02t),
where
(3.8)β0=11-2nf3,β1=102-38nf3
and nf is the number of active flavours.We define(3.9)T(t)=αs(t)2π.
To proceed further and to apply our formalism, we, as in [17], use the assumption
(3.10)T(t)2=T0T(t),
where T0 is a numerical parameter.


### 3.2. Solutions of AP Equations forΔΣ(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
(3.11)∂ΔΣ(x,t)∂t=T(t)(∫x1‍a1ΔΣ(x,t)+b1ΔG(x,t)T(t)+h1(x)∫x1‍dzzΔΣ(z,t)+h2(x)∫x1‍dzzln⁡zΔΣ(z,t)T(t)+h3(x)∫x1‍dzzln⁡2zΔΣ(z,t)+k1(x)∫x1‍dzzΔG(z,t)T(t)+k2(x)∫x1‍dzzln⁡zΔG(z,t)+k3(x)∫x1‍dzzln⁡2zΔG(z,t)),∂ΔG(x,t)∂t=T(t)(∫x1‍a2ΔΣ(x,t)+b2ΔG(x,t)T(t)+p1(x)∫x1‍dzzΔΣ(z,t)+p2(x)∫x1‍dzzln⁡zΔΣ(z,t)T(t)+p3(x)∫x1‍dzzln⁡2zΔΣ(z,t)+q1(x)∫x1‍dzzΔG(z,t)T(t)+q2(x)∫x1‍dzzln⁡⁡zΔG(z,t)+q3(x)∫x1‍dzzln⁡2zΔG(z,t)).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]:
(3.12)ΔΣ(x,t)|x=1=0,ΔG(x,t)|x=1=0.
With these substitutions (3.11) become
(3.13)∂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).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
(3.14)ΔΣ(x,t)=λ2Θ1eN1(nf)τ(t)-λ1Θ2eN2(nf)τ(t),ΔG(x,t)=Θ1eN1(nf)τ(t)-Θ2eN2(nf)τ(t),
where
(3.15)λ1=(b2-a1)+(a1-b2)2+4b1a2-2a2,λ2=(b2-a1)-(a1-b2)2+4b1a2-2a2,N1(nf)=a1+b1λ1+a2(λ2-λ1),N2(nf)=a1+b1λ1,τ(t)=∫‍T(t)dt
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 att=t0,
(3.16)ΔΣ(x,t)|t=t0=ΔΣ(x,t0),ΔG(x,t)|t=t0=ΔG(x,t0),
we can find out the constants of integration Θ1 and Θ2. With these the solutions after first approximation become
(3.17)ΔΣ1(x,t)=U10(x)exp⁡[N1(τ-τ0)]-U~10(x)exp⁡[N2(τ-τ0)],ΔG1(x,t)=V10(x)exp⁡[N1(τ-τ0)]-V~10(x)exp⁡[N2(τ-τ0)],
where
(3.18)U10(x)=λ2(ΔΣ(x,t0)-λ1ΔG(x,t0)(λ2-λ1)),U~10(x)=λ1(ΔΣ(x,t0)-λ2ΔG(x,t0)(λ2-λ1)),V10(x)=(ΔΣ(x,t0)-λ1ΔG(x,t0)(λ2-λ1)),V~10(x)=(ΔΣ(x,t0)-λ2ΔG(x,t0)(λ2-λ1)),τ0=(∫‍Tdt)|t=t0,
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
(3.19)∂ΔΣ(x,t)∂t=a1T(t)ΔΣ(x,t)+b1T(t)ΔG(x,t)+H10(x)T(t)eN1(τ-τ0)-H~10(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)-K~10(x)T(t)eN2(τ-τ0),
where H10(x), H~10, K10(x), and K~10(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
(3.20)ΔΣ2(x,t)=[U20(x)+U21(x)(τ-τ0)]exp⁡[N1(τ-τ0)]-[U~20(x)+U~21(x)(τ-τ0)]exp⁡[N2(τ-τ0)],(3.21)ΔG2(x,t)=[V20(x)+V21(x)(τ-τ0)]exp⁡[N1(τ-τ0)]-[V~20(x)+V~21(x)(τ-τ0)]exp⁡[N2(τ-τ0)].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 10GeV2 at NLO.Figure 2
x
Δ
G
(
x
,
Q
2
) at 10GeV2 at NLO.We again substituteΔΣ2(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
(3.22)∂ΔΣ(x,t)∂t=a1T(t)ΔΣ(x,t)+b1T(t)ΔG(x,t)+[H20(x)+H21(x)(τ-τ0)]T(t)eN1(τ-τ0)-[H~20(x)+H~21(x)(τ-τ0)]T(t)eN2(τ-τ0),∂ΔG(x,t)∂t=a2T(t)ΔΣ(x,t)+b2T(t)ΔG(x,t)+[K20(x)+K21(x)(τ-τ0)]T(t)eN1(τ-τ0)-[K~20(x)+K~21(x)(τ-τ0)]T(t)eN2(τ-τ0).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
(3.23)ΔΣ3(x,t)=[U30(x)+U31(x)(τ-τ0)+U32(x)(τ-τ0)22]exp⁡[N1(τ-τ0)]-[U~30(x)+U~31(x)(τ-τ0)+U~32(x)(τ-τ0)22]exp⁡[N2(τ-τ0)],ΔG3(x,t)=[V30(x)+V31(x)(τ-τ0)+V32(x)(τ-τ0)22]exp⁡[N1(τ-τ0)]-[V~30(x)+V~31(x)(τ-τ0)+V~32(x)(τ-τ0)22]exp⁡[N2(τ-τ0)].
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
(3.24)ΔΣn(x,t)=∑m=0n-1‍[Unm(x)(τ-τ0)mm!eN1(τ-τ0)-U~nm(x)(τ-τ0)mm!eN2(τ-τ0)],(3.25)ΔGn(x,t)=∑m=0n-1‍[Vnm(x)(τ-τ0)mm!eN1(τ-τ0)-V~nm(x)(τ-τ0)mm!eN2(τ-τ0)],
where Unm(x), U~nm(x), Vnm(x), and V~nm(x) are calculable functions of x. Equations (3.24) and (3.25) are our main results.


### 3.3. Polarized Structure Functionsg1p(x,Q2), g1n(x,Q2)

We have from (3.1) and (3.3)
(3.26)∂ΔΣqi(x,t)∂t=αs(t)2π∫x1‍dzzΔPqq(xz)ΔΣqi(x,t),
where
(3.27)ΔΣqi(x,t)=Δqi(x,t)-12nfΔΣ(x,t).
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
(3.28)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t)+T(t)(h1(x)∫x1‍ΔΣqi(z,t)zdz+h2(x)∫x1‍ln⁡zΔΣqi(z,t)zdz+T(t)T(t)+h3(x)∫x1‍ln⁡2zΔΣqi(z,t)zdz),
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]:
(3.29)ΔΣqi(x,t)|x=1=0.
With this substitution, (3.28) becomes
(3.30)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t).
The solution of (3.30) can be given as
(3.31)ΔΣqi(x,t)=Cqi(x)eN3τ(t),
where, Cqi(x) is the x dependent constant of integration, N3=a1.Now applying the input distribution att=t0, that is,
(3.32)ΔΣqi(x,t)|t=t0=ΔΣqi(x,t0)=Δqi(x,t0)-12nfΔΣ(x,t0),
the solution (3.31) can be written as
(3.33)ΔΣ1qi(x,t)=ΔΣqi(x,t0)eN3(τ(t)-τ(t0)),
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
(3.34)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t)+Hq1(x)T(t)eN3(τ(t)-τ(t0)),
where
(3.35)Hq1(x)=h1(x)∫x1‍ΔΣqi(z,t0)zdz+h2(x)∫x1‍ln⁡zΔΣqi(z,t0)zdz+h3(x)∫x1‍ln⁡2zΔΣqi(z,t0)zdz.Following the method of variation [10] and using the input boundary condition (3.32), we have the second iterative solution of (3.28) as
(3.36)ΔΣ2qi(x,t)=ΔΣqi(x,t0)eN3(τ-τ0)+Hq1(x)(τ-τ0)eN3(τ-τ0).
Equation (3.36) is an improvement over (3.33).Again substitutingΔΣ2qi(x,t) from (3.36) for ΔΣqi(x,t) appearing under the integrals in the right-hand side of (3.28), we have
(3.37)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t)+Hq1(x)T(t)eN3(τ-τ0)+Hq2(x)T(t)(τ-τ0)eN3(τ-τ0).
Proceeding in a similar manner we can obtain the solution of (3.37) as
(3.38)ΔΣ3qi(x,t)=ΔΣqi(x,t0)eN3(τ-τ0)+Hq1(x)(τ-τ0)eN3(τ-τ0)+Hq2(x)(τ-τ0)22eN3(τ-τ0),
where
(3.39)Hq2(x)=h1(x)∫x1‍Hq1(z)zdz+h2(x)∫x1‍ln⁡zHq1(z)zdz+h3(x)∫x1‍ln⁡2zHq1(z)zdz.
Equation (3.38) is the solution of (3.28) after third approximation. Similarly, the fourth iterative solution will be
(3.40)ΔΣ4qi(x,t)=ΔΣqi(x,t0)eN3(τ-τ0)+Hq1(x)(τ-τ0)eN3(τ-τ0)+Hq2(x)(τ-τ0)22eN3(τ-τ0)+Hq3(x)(τ-τ0)36eN3(τ-τ0),
where(3.41)Hq3(x)=h1(x)∫x1‍Hq2(z)zdz+h2(x)∫x1‍ln⁡zHq2(z)zdz+h3(x)∫x1‍ln⁡2zHq2(z)zdz.Proceeding in this way, we have the solution of (3.28) after n approximation as
(3.42)ΔΣnqi(x,t)=∑m=0n-1‍Hqm(x)(τ-τ0)mm!eN3(τ-τ0),
where
(3.43)Hqm(x)=h1(x)∫x1‍Hq(m-1)(z)zdz+h2(x)∫x1‍ln⁡zHq(m-1)(z)zdz+h3(x)∫x1‍ln⁡2zHq(m-1)(z)zdz.
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
(3.44)Δqni(x,t)=ΔΣnqi(x,t)+12nfΔΣn(x,t)=∑m=0n-1‍[Unm(x)(τ-τ0)mm!eN1(τ-τ0)-U~nm(x)(τ-τ0)mm!eN2(τ-τ0)∑m=0n-1‍+Hqm(x)(τ-τ0)mm!eN3(τ-τ0)].
We now use (3.44) and (3.25) (for ΔGn(x,t)) to obtain the analytical expressions for g1p(x,t) and g1n(x,t) in NLO as
(3.45)g1n(x,t)=12∑q‍eq2[∫x1‍Δqni(x,t)+Δq¯ni(x,t)12∑q‍eq2+αs(t)2π∫x1‍dzz(ΔCq(xz)(Δqni(z,t)+Δq¯ni(z,t))+ΔCG(xz)ΔGni(z,t))],
where subscript n indicates n approximations and ΔCq(x) and ΔCG(x) are called Wilson coefficients [15] given in the small x limit as
(3.46)ΔCq(x)=23-43ln⁡x,ΔCG(x)=32+12ln⁡x.
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 Section4, 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), U~3i(x), V3i(x), and V~3i(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 range10-5⪕x⪕1 (the range of x where LSS'05 numerical results for parton distributions are available) by the following effective functional forms:
(3.47)ΔΣ3(x,t)=LΣ(x)[1+(τ-τ0)+(τ-τ0)22](exp⁡[N1(τ-τ0)]+exp⁡[N2(τ-τ0)]),ΔG3(x,t)=LG(x)[1+(τ-τ0)+(τ-τ0)22](exp⁡[N1(τ-τ0)]+exp⁡[N2(τ-τ0)]),
where
(3.48)LΣ(x)=α1xβ1(1-x)γ1(1+δ1x+ξ1xη1),LG(x)=α2xβ2(1-x)γ2(1+δ2x+ξ2xη2),α1=0.396,β1=0.693,γ1=3.046,δ1=2.86,ξ1=-12.049,η1=4.82,α2=3.522,β2=1.88,γ2=2.59,δ2=-2.81,ξ2=1.61,η2=0.043.
Using (3.47) we get the following approximate analytic forms after fourth approximation:
(3.49)ΔΣ4(x,t)=[U40(x)+U41(x)(τ-τ0)+U42(x)(τ-τ0)22+U43(x)(τ-τ0)36]exp⁡[N1(τ-τ0)]-[U~40(x)+U~41(x)(τ-τ0)+U~42(x)(τ-τ0)22-+U~43(x)(τ-τ0)36]exp⁡[N2(τ-τ0)],ΔG4(x,t)=[V40(x)+V41(x)(τ-τ0)+V42(x)(τ-τ0)22+V43(x)(τ-τ0)36]exp⁡[N1(τ-τ0)]-[V~40(x)+V~41(x)(τ-τ0)+V~42(x)(τ-τ0)22-+V~43(x)(τ-τ0)36]exp⁡[N2(τ-τ0)].The expressions for thex-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) atQ2=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 ofΔΣ(Q2) and ΔG(Q2) defined as
(3.50)ΔΣ(Q2)=∫01‍ΔΣ(x,Q2)dx,ΔG(Q2)=∫01‍ΔG(x,Q2)dx
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 range10-5⪕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 smallx 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 ofΔ∑(Q2) and ΔG(Q2) obtained from this work in NLO after four iterations at Q2=10GeV2.
1st iter.
2nd iter.
3rd iter.
4th iter.
Exp. values
Δ
∑
(
Q
2
)
0.348
0.427
0.646
0.631
0.35
±
0.06
(COMPASS atQ2=3GeV2)
0.33
±
0.04
(HERMES atQ2=5GeV2)
Δ
G
(
Q
2
)
0.134
0.342
0.458
0.535
0.29
±
0.32
(From all fits of present data)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)(3.51)Hu2(x)(foruquark)=103.04x-0.15(1-x)8.33(1+0.477x-1.158x0.096),Hd2(x)(fordquark)=-57.27x-0.19(1-x)9.47(1+0.524x-1.166x0.088),Hs2(x)(forsquark)=-15.73x-0.064(1-x)7.63(1+0.393x-1.143x0.095).We have obtained the fourth iterative solutions by using these effective expressions forHq2(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 Q2=10GeV2.Figure 4
x
Δ
d
(
x
) in NLO at Q2=10GeV2.Figure 5
x
Δ
s
(
x
) in NLO at Q2=10GeV2.We also record the values of the strange quark contributionΔS=∫01‍(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 ofΔS obtained from this work in NLO after four iterations at Q2=10GeV2.
1st iter.
2nd iter.
3rd iter.
4th iter.
LSS′06 (MS¯)
DSSV (MS¯)
Δ
S
-
0.045
-
0.049
-
0.0145
-
0.033
-
0.126
±
0.010
-
0.114We now use our formalism in the next to leading order (NLO) to calculate the structure functionsg1p(x,t) and g1n(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 forxg1p(x,Q2) in NLO at Q2=10GeV2.Figure 7
Plot forxg1n(x,Q2) in NLO at Q2=10GeV2.


## 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
(3.1)∂ΔΣ(x,t)∂t=αs(t)2π∫x1‍dzzΔPqq(xz)ΔΣ(z,t)+αs(t)2π∫x1‍dzzΔPqg(xz)ΔG(z,t),(3.2)∂ΔG(x,t)∂t=αs(t)2π∫x1‍dzzΔPgq(xz)ΔΣ(z,t)+αs(t)2π∫x1‍dzzΔPgg(xz)ΔG(z,t),(3.3)∂Δqi(x,t)∂t=αs(t)2π∫x1‍dzzΔPqq(xz)Δqi(z,t)+αs(t)2π∫x1‍dzzΔPqg(x/z)2nfΔG(z,t).The polarized splitting functionsΔPij(x) are defined as
(3.4)ΔPij(x)=ΔPij(0)(x)+αs(t)2πΔPij(1)(x).ΔPij(0)(x) and ΔPij(1)(x) are given in [14, 15].Δ
P
i
j
(
0
)
(
x
) in the small x limit are given as [16]
(3.5)ΔPqq(0)x=43[1+12δ(1-x)],ΔPqg(0)x=nf[-1+2δ(1-x)],ΔPgq(0)x=43[2-δ(1-x)],ΔPgg(0)x=3[4-136δ(1-x)]-nf3δ(1-x)
and ΔPij(1)(x) in the small x limit can be given as [14, 15]
(3.6)ΔPqq(1)(x)=ΔPqq0(nf)+ΔPqq1(nf)ln⁡x+ΔPqq2(nf)ln⁡2x,ΔPqg(1)(x)=ΔPqg0(nf)+ΔPqg1(nf)ln⁡x+ΔPqg2(nf)ln⁡2x,ΔPgq(1)(x)=ΔPgq0(nf)+ΔPgq1(nf)ln⁡x+ΔPgq2(nf)ln⁡2x,ΔPgg(1)(x)=ΔPgg0(nf)+ΔPgg1(nf)ln⁡x+ΔPgg2(nf)ln⁡2x,
where ΔPabi, a=q,g; b=q,g and i=0,1,2 are given in the appendix.α
s
(
t
), the running coupling constant of QCD in NLO, is defined as
(3.7)αs(t)=4πβ0t(1-β1ln⁡tβ02t),
where
(3.8)β0=11-2nf3,β1=102-38nf3
and nf is the number of active flavours.We define(3.9)T(t)=αs(t)2π.
To proceed further and to apply our formalism, we, as in [17], use the assumption
(3.10)T(t)2=T0T(t),
where T0 is a numerical parameter.


## 3.2. Solutions of AP Equations forΔΣ(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
(3.11)∂ΔΣ(x,t)∂t=T(t)(∫x1‍a1ΔΣ(x,t)+b1ΔG(x,t)T(t)+h1(x)∫x1‍dzzΔΣ(z,t)+h2(x)∫x1‍dzzln⁡zΔΣ(z,t)T(t)+h3(x)∫x1‍dzzln⁡2zΔΣ(z,t)+k1(x)∫x1‍dzzΔG(z,t)T(t)+k2(x)∫x1‍dzzln⁡zΔG(z,t)+k3(x)∫x1‍dzzln⁡2zΔG(z,t)),∂ΔG(x,t)∂t=T(t)(∫x1‍a2ΔΣ(x,t)+b2ΔG(x,t)T(t)+p1(x)∫x1‍dzzΔΣ(z,t)+p2(x)∫x1‍dzzln⁡zΔΣ(z,t)T(t)+p3(x)∫x1‍dzzln⁡2zΔΣ(z,t)+q1(x)∫x1‍dzzΔG(z,t)T(t)+q2(x)∫x1‍dzzln⁡⁡zΔG(z,t)+q3(x)∫x1‍dzzln⁡2zΔG(z,t)).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]:
(3.12)ΔΣ(x,t)|x=1=0,ΔG(x,t)|x=1=0.
With these substitutions (3.11) become
(3.13)∂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).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
(3.14)ΔΣ(x,t)=λ2Θ1eN1(nf)τ(t)-λ1Θ2eN2(nf)τ(t),ΔG(x,t)=Θ1eN1(nf)τ(t)-Θ2eN2(nf)τ(t),
where
(3.15)λ1=(b2-a1)+(a1-b2)2+4b1a2-2a2,λ2=(b2-a1)-(a1-b2)2+4b1a2-2a2,N1(nf)=a1+b1λ1+a2(λ2-λ1),N2(nf)=a1+b1λ1,τ(t)=∫‍T(t)dt
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 att=t0,
(3.16)ΔΣ(x,t)|t=t0=ΔΣ(x,t0),ΔG(x,t)|t=t0=ΔG(x,t0),
we can find out the constants of integration Θ1 and Θ2. With these the solutions after first approximation become
(3.17)ΔΣ1(x,t)=U10(x)exp⁡[N1(τ-τ0)]-U~10(x)exp⁡[N2(τ-τ0)],ΔG1(x,t)=V10(x)exp⁡[N1(τ-τ0)]-V~10(x)exp⁡[N2(τ-τ0)],
where
(3.18)U10(x)=λ2(ΔΣ(x,t0)-λ1ΔG(x,t0)(λ2-λ1)),U~10(x)=λ1(ΔΣ(x,t0)-λ2ΔG(x,t0)(λ2-λ1)),V10(x)=(ΔΣ(x,t0)-λ1ΔG(x,t0)(λ2-λ1)),V~10(x)=(ΔΣ(x,t0)-λ2ΔG(x,t0)(λ2-λ1)),τ0=(∫‍Tdt)|t=t0,
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
(3.19)∂ΔΣ(x,t)∂t=a1T(t)ΔΣ(x,t)+b1T(t)ΔG(x,t)+H10(x)T(t)eN1(τ-τ0)-H~10(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)-K~10(x)T(t)eN2(τ-τ0),
where H10(x), H~10, K10(x), and K~10(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
(3.20)ΔΣ2(x,t)=[U20(x)+U21(x)(τ-τ0)]exp⁡[N1(τ-τ0)]-[U~20(x)+U~21(x)(τ-τ0)]exp⁡[N2(τ-τ0)],(3.21)ΔG2(x,t)=[V20(x)+V21(x)(τ-τ0)]exp⁡[N1(τ-τ0)]-[V~20(x)+V~21(x)(τ-τ0)]exp⁡[N2(τ-τ0)].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 10GeV2 at NLO.Figure 2
x
Δ
G
(
x
,
Q
2
) at 10GeV2 at NLO.We again substituteΔΣ2(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
(3.22)∂ΔΣ(x,t)∂t=a1T(t)ΔΣ(x,t)+b1T(t)ΔG(x,t)+[H20(x)+H21(x)(τ-τ0)]T(t)eN1(τ-τ0)-[H~20(x)+H~21(x)(τ-τ0)]T(t)eN2(τ-τ0),∂ΔG(x,t)∂t=a2T(t)ΔΣ(x,t)+b2T(t)ΔG(x,t)+[K20(x)+K21(x)(τ-τ0)]T(t)eN1(τ-τ0)-[K~20(x)+K~21(x)(τ-τ0)]T(t)eN2(τ-τ0).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
(3.23)ΔΣ3(x,t)=[U30(x)+U31(x)(τ-τ0)+U32(x)(τ-τ0)22]exp⁡[N1(τ-τ0)]-[U~30(x)+U~31(x)(τ-τ0)+U~32(x)(τ-τ0)22]exp⁡[N2(τ-τ0)],ΔG3(x,t)=[V30(x)+V31(x)(τ-τ0)+V32(x)(τ-τ0)22]exp⁡[N1(τ-τ0)]-[V~30(x)+V~31(x)(τ-τ0)+V~32(x)(τ-τ0)22]exp⁡[N2(τ-τ0)].
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
(3.24)ΔΣn(x,t)=∑m=0n-1‍[Unm(x)(τ-τ0)mm!eN1(τ-τ0)-U~nm(x)(τ-τ0)mm!eN2(τ-τ0)],(3.25)ΔGn(x,t)=∑m=0n-1‍[Vnm(x)(τ-τ0)mm!eN1(τ-τ0)-V~nm(x)(τ-τ0)mm!eN2(τ-τ0)],
where Unm(x), U~nm(x), Vnm(x), and V~nm(x) are calculable functions of x. Equations (3.24) and (3.25) are our main results.


## 3.3. Polarized Structure Functionsg1p(x,Q2), g1n(x,Q2)

We have from (3.1) and (3.3)
(3.26)∂ΔΣqi(x,t)∂t=αs(t)2π∫x1‍dzzΔPqq(xz)ΔΣqi(x,t),
where
(3.27)ΔΣqi(x,t)=Δqi(x,t)-12nfΔΣ(x,t).
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
(3.28)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t)+T(t)(h1(x)∫x1‍ΔΣqi(z,t)zdz+h2(x)∫x1‍ln⁡zΔΣqi(z,t)zdz+T(t)T(t)+h3(x)∫x1‍ln⁡2zΔΣqi(z,t)zdz),
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]:
(3.29)ΔΣqi(x,t)|x=1=0.
With this substitution, (3.28) becomes
(3.30)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t).
The solution of (3.30) can be given as
(3.31)ΔΣqi(x,t)=Cqi(x)eN3τ(t),
where, Cqi(x) is the x dependent constant of integration, N3=a1.Now applying the input distribution att=t0, that is,
(3.32)ΔΣqi(x,t)|t=t0=ΔΣqi(x,t0)=Δqi(x,t0)-12nfΔΣ(x,t0),
the solution (3.31) can be written as
(3.33)ΔΣ1qi(x,t)=ΔΣqi(x,t0)eN3(τ(t)-τ(t0)),
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
(3.34)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t)+Hq1(x)T(t)eN3(τ(t)-τ(t0)),
where
(3.35)Hq1(x)=h1(x)∫x1‍ΔΣqi(z,t0)zdz+h2(x)∫x1‍ln⁡zΔΣqi(z,t0)zdz+h3(x)∫x1‍ln⁡2zΔΣqi(z,t0)zdz.Following the method of variation [10] and using the input boundary condition (3.32), we have the second iterative solution of (3.28) as
(3.36)ΔΣ2qi(x,t)=ΔΣqi(x,t0)eN3(τ-τ0)+Hq1(x)(τ-τ0)eN3(τ-τ0).
Equation (3.36) is an improvement over (3.33).Again substitutingΔΣ2qi(x,t) from (3.36) for ΔΣqi(x,t) appearing under the integrals in the right-hand side of (3.28), we have
(3.37)∂ΔΣqi(x,t)∂t=a1T(t)ΔΣqi(x,t)+Hq1(x)T(t)eN3(τ-τ0)+Hq2(x)T(t)(τ-τ0)eN3(τ-τ0).
Proceeding in a similar manner we can obtain the solution of (3.37) as
(3.38)ΔΣ3qi(x,t)=ΔΣqi(x,t0)eN3(τ-τ0)+Hq1(x)(τ-τ0)eN3(τ-τ0)+Hq2(x)(τ-τ0)22eN3(τ-τ0),
where
(3.39)Hq2(x)=h1(x)∫x1‍Hq1(z)zdz+h2(x)∫x1‍ln⁡zHq1(z)zdz+h3(x)∫x1‍ln⁡2zHq1(z)zdz.
Equation (3.38) is the solution of (3.28) after third approximation. Similarly, the fourth iterative solution will be
(3.40)ΔΣ4qi(x,t)=ΔΣqi(x,t0)eN3(τ-τ0)+Hq1(x)(τ-τ0)eN3(τ-τ0)+Hq2(x)(τ-τ0)22eN3(τ-τ0)+Hq3(x)(τ-τ0)36eN3(τ-τ0),
where(3.41)Hq3(x)=h1(x)∫x1‍Hq2(z)zdz+h2(x)∫x1‍ln⁡zHq2(z)zdz+h3(x)∫x1‍ln⁡2zHq2(z)zdz.Proceeding in this way, we have the solution of (3.28) after n approximation as
(3.42)ΔΣnqi(x,t)=∑m=0n-1‍Hqm(x)(τ-τ0)mm!eN3(τ-τ0),
where
(3.43)Hqm(x)=h1(x)∫x1‍Hq(m-1)(z)zdz+h2(x)∫x1‍ln⁡zHq(m-1)(z)zdz+h3(x)∫x1‍ln⁡2zHq(m-1)(z)zdz.
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
(3.44)Δqni(x,t)=ΔΣnqi(x,t)+12nfΔΣn(x,t)=∑m=0n-1‍[Unm(x)(τ-τ0)mm!eN1(τ-τ0)-U~nm(x)(τ-τ0)mm!eN2(τ-τ0)∑m=0n-1‍+Hqm(x)(τ-τ0)mm!eN3(τ-τ0)].
We now use (3.44) and (3.25) (for ΔGn(x,t)) to obtain the analytical expressions for g1p(x,t) and g1n(x,t) in NLO as
(3.45)g1n(x,t)=12∑q‍eq2[∫x1‍Δqni(x,t)+Δq¯ni(x,t)12∑q‍eq2+αs(t)2π∫x1‍dzz(ΔCq(xz)(Δqni(z,t)+Δq¯ni(z,t))+ΔCG(xz)ΔGni(z,t))],
where subscript n indicates n approximations and ΔCq(x) and ΔCG(x) are called Wilson coefficients [15] given in the small x limit as
(3.46)ΔCq(x)=23-43ln⁡x,ΔCG(x)=32+12ln⁡x.
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 Section4, 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), U~3i(x), V3i(x), and V~3i(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 range10-5⪕x⪕1 (the range of x where LSS'05 numerical results for parton distributions are available) by the following effective functional forms:
(3.47)ΔΣ3(x,t)=LΣ(x)[1+(τ-τ0)+(τ-τ0)22](exp⁡[N1(τ-τ0)]+exp⁡[N2(τ-τ0)]),ΔG3(x,t)=LG(x)[1+(τ-τ0)+(τ-τ0)22](exp⁡[N1(τ-τ0)]+exp⁡[N2(τ-τ0)]),
where
(3.48)LΣ(x)=α1xβ1(1-x)γ1(1+δ1x+ξ1xη1),LG(x)=α2xβ2(1-x)γ2(1+δ2x+ξ2xη2),α1=0.396,β1=0.693,γ1=3.046,δ1=2.86,ξ1=-12.049,η1=4.82,α2=3.522,β2=1.88,γ2=2.59,δ2=-2.81,ξ2=1.61,η2=0.043.
Using (3.47) we get the following approximate analytic forms after fourth approximation:
(3.49)ΔΣ4(x,t)=[U40(x)+U41(x)(τ-τ0)+U42(x)(τ-τ0)22+U43(x)(τ-τ0)36]exp⁡[N1(τ-τ0)]-[U~40(x)+U~41(x)(τ-τ0)+U~42(x)(τ-τ0)22-+U~43(x)(τ-τ0)36]exp⁡[N2(τ-τ0)],ΔG4(x,t)=[V40(x)+V41(x)(τ-τ0)+V42(x)(τ-τ0)22+V43(x)(τ-τ0)36]exp⁡[N1(τ-τ0)]-[V~40(x)+V~41(x)(τ-τ0)+V~42(x)(τ-τ0)22-+V~43(x)(τ-τ0)36]exp⁡[N2(τ-τ0)].The expressions for thex-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) atQ2=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 ofΔΣ(Q2) and ΔG(Q2) defined as
(3.50)ΔΣ(Q2)=∫01‍ΔΣ(x,Q2)dx,ΔG(Q2)=∫01‍ΔG(x,Q2)dx
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 range10-5⪕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 smallx 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 ofΔ∑(Q2) and ΔG(Q2) obtained from this work in NLO after four iterations at Q2=10GeV2.
1st iter.
2nd iter.
3rd iter.
4th iter.
Exp. values
Δ
∑
(
Q
2
)
0.348
0.427
0.646
0.631
0.35
±
0.06
(COMPASS atQ2=3GeV2)
0.33
±
0.04
(HERMES atQ2=5GeV2)
Δ
G
(
Q
2
)
0.134
0.342
0.458
0.535
0.29
±
0.32
(From all fits of present data)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)(3.51)Hu2(x)(foruquark)=103.04x-0.15(1-x)8.33(1+0.477x-1.158x0.096),Hd2(x)(fordquark)=-57.27x-0.19(1-x)9.47(1+0.524x-1.166x0.088),Hs2(x)(forsquark)=-15.73x-0.064(1-x)7.63(1+0.393x-1.143x0.095).We have obtained the fourth iterative solutions by using these effective expressions forHq2(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 Q2=10GeV2.Figure 4
x
Δ
d
(
x
) in NLO at Q2=10GeV2.Figure 5
x
Δ
s
(
x
) in NLO at Q2=10GeV2.We also record the values of the strange quark contributionΔS=∫01‍(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 ofΔS obtained from this work in NLO after four iterations at Q2=10GeV2.
1st iter.
2nd iter.
3rd iter.
4th iter.
LSS′06 (MS¯)
DSSV (MS¯)
Δ
S
-
0.045
-
0.049
-
0.0145
-
0.033
-
0.126
±
0.010
-
0.114We now use our formalism in the next to leading order (NLO) to calculate the structure functionsg1p(x,t) and g1n(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 forxg1p(x,Q2) in NLO at Q2=10GeV2.Figure 7
Plot forxg1n(x,Q2) in NLO at Q2=10GeV2.


## 4. Conclusion

QCD analysis of the quark and gluon contributions towards the spin of the nucleon in the smallx 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 Section2 we have given a method for solving a system of two first order linear homogeneous differential equations with variable coefficients.In Section3 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 g1p(x,t) and g1n(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.



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*Source: 943871-2012-08-05.xml*