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 Total Downloads : 152
 Authors : R. Rajkhowa
 Paper ID : IJERTV2IS100878
 Volume & Issue : Volume 02, Issue 10 (October 2013)
 Published (First Online): 28102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Particular and Unique Solutions of Spin Dependent DGLAP Evolution Equations in Leading Order and Structure functions at smallx
R. Rajkhowa
Physics Department, T.H.B. College, Jamugurihat, Sonitpur, Assam.
Abstract
In this paper the spindependent singlet and nonsinglet structure functions have been obtained by solving DokshitzerGribovLipatovAltarelliParisi evolution equations in leading order in the small x limit. Here we have used Taylor series expansion and then the particular and unique solution to solve the evolution equations. We have also calculated t evolutions of deuteron, proton, neutron structure functions and xevolution of deuteron structure functions and the results are compared with the SLAC E143 Collaboration data.
PACS No.: 12.38. t, 12.39.x, 13.60.Hb, 13.88.+e
Keywords: Unique solution, Particular solution, Complete solution, AltarelliParisi equation, Structure function, Lowx physics.

Introduction
In our earlier works [14], we obtain particular and unique solutions of the spin independent DokshitzerGribovLipatov AltarelliParisi (DGLAP) evolution equations [58] for t and x evolutions of singlet and nonsinglet structure functions in leading order (LO) and nextto leading order (NLO) and hence tevolution of deuteron, proton, neutron, difference and ratio of proton and neutron and xevolution of deuteron in LO and NLO at lowx have been reported. The same technique can be applied to the spin dependent DGLAP evolution equations in leading order (LO). The present paper reports particular and unique solutions of spin dependent DGLAP evolution equations computed from complete solutions in leading order at lowx and calculation of t and x evolutions for singlet, nonsinglet, structure functions and hence tevolutions of deuteron, proton, neutron and x evolution of deuteron structure functions. Here, the integrodifferential spin dependent DGLAP evolution equations have been converted into first order partial differential equations by applying Taylor expansion in the smallx limit. Then they have been solved by standard analytical methods. The results of t and x evolutions are compared with the SLAC E143 Collaboration data.
In the present paper, section 1 is the introduction. In section 2 necessary theory has been discussed. Section 3 gives results and discussion, and section 4 is conclusion.

Theory
In polarized DIS experiments the asymmetry, A , of the cross sections for parallel and antiparallel orientations of the beam and target spins
A
measured, where and
are the cross sections for the opposite and same spin directions,
respectively. Similarly, the transverse asymmetry, determined from scattering of a longitudinally polarized beam on a transversely polarized target, is defined as
A
These asymmetries can be express in terms of longitudinal (A1) and transverse (A2) virtual photonnucleon asymmetries as
A D[ A A ]
and
A d[ A 1 y A ]
 1 2
where
2 1
2
2
,
,
2 y y2
D
Q 2(1 y) ,
1 y
, 2
2 2 and M
2(1 y)(1 R) y2
E y(2 y)
d D
1 y 2
4y x y
Q
The virtual photonnucleon asymmetries for the proton, neutron, and deuteron are defined as
p,n
p,n
2TL
d 0 2
TL TL
2
2
A1 1/ 2 3 / 2 , A
, A1
and
Ad 0 1
2
2
1 / 2
3 / 2
1/ 2
3 / 2
0 2
1/ 2
3/ 2
The longitudinal spindependent structure function g1(x) is defined as
g (x) 1 e2q (x)
1 2 i i
where
q (x) q q (x) q (x) q (x)
i i i i i
i
i
i
i
Here q and q (x) are the densities of quarks of flavor i with helicity parallel and
2
2
antiparallel to the nucleon spin. The spindependent structure functions g1(x, Q2) and g2(x, Q2) are related to the spinindependent structure function F2(x, Q2) as
F2 (x, Q )[ A1 (x, Q ) A (x, Q )]
F2 (x, Q )[ A1 (x, Q ) A (x, Q )]
2
2
F2 (x, Q
)[ A1 (x, Q
A (x, Q2 )
) 2 ]
2 2 2
2
and
g1
2x[1 R
(x, Q2 )]
g2
2x[1 R
(x, Q2 )]
(x, Q2 )
where
R L
is the ratio of the longitudinal and transverse virtual photon cross
T
T
sections.
(x, Q2 )
The polarized DGLAP evolution equation [9] in the standard form is given by
2
2
g1 (x, Q )
ln Q2
P(x, Q2 )
g1 (x, Q )
2
2
2
2
where g1(x, Q2) is the spindependent structure function as a function of x and Q2, where x is the Bjorken variable and Q2 is the fourmomentum transfer in a DIS process. Here P(x, Q2)
is the spindependent kernel known perturbatively up to the first few orders in
s (Q ) , the
strong coupling constant. Here represents the standard Mellin convolution, and the notation is given by
1 dy x
a(x) b(x) y a( y)b y
One can write
0
(Q2 )
(Q2 ) 2
P(x, Q2 ) s P(0) (x) s
P(1) (x)
2 2
where
P(0) (x) and
P(1) (x) are spindependent splitting functions in LO and NLO.
The singlet and nonsinglet structure functions [1012] are obtained from the polarized DGLAP evolution equations as
g S (x, t)
(t) 2
S 4 1 dw
2 S x S
* 1 s [
{3 4 ln(1 x)} g1 (x,t) {(1 w
)g1 ,t 2g1 (x,t)}]
t 2 3
3 x1 w
w
1 x
n (2w 1)G( , t)dw}] 0,
(1)
f x w
g NS (x,t)
s (t) 2
NS 4 1 dw
2 NS x NS
1
1
1
1
* 1
2
[ {3 4 ln(1 x)} g3
(x, t) {(1 w
3 1 w
)g1
, t 2g
w
(x, t)}] 0,
(2)
t
in LO.
x
Let us introduce the variable u = 1w and note that [13]
x x x
xu .
(3)
w 1 u
1 u
1
1
The series (3) is convergent for u<1. Since x<w<1, so 0<u<1x and hence the convergence criterion is satisfied. Now, using Taylor expansion method [1315] we can rewrite g S (x / w,t) as
1
1
g S (x / w,t) g S (x xu ),t
1 1 u
S u
g S (x,t)
1 2 u
2 2 g S (x,t)
g (x,t) x. 1 x 1 …..
1 1 u x
2 1 u
x2
1
1
which covers the whole range of u, 0<u<1x. Since x is small in our region of discussion, the terms containing x2 and higher powers of x can be neglected as our first approximation as
discussed in our earlier works [13].
g S (x / w,t) can then be approximated for smallx as
S S u
g S (x,t)
g (x / w,t) g (x,t) x. 1 .
(4)
1 1 1 u x
Similarly, G(x/w, t) can be approximated for smallx as
G(x / w,t) G(x,t) x.
u
1 u
G(x,t) . (5)
x
A
A
Using equations (4) and (5) in equation (1) and performing uintegrations we et
1
1
gS(x, t)
s (t)
(x)gS
(x, t) A
(x)G(x, t) A
gS(x, t)
(x) 1 A
(x)
G(x, t)
0.
(6)
t
Here
2 1 1 2
3 x
4 x
1
1
A (x) 2 {3 4 ln(1 x) (x 1)(x 3)}, 3
A (x) N
2 f
x(1 x),
A3(x) 2{x(1 x2) 2x ln( 1 )},
A4(x) N f x{(1 x)(2 x) ln x}.
3 x
We assume [13, 17]
1
1
G(x, t) = K(x) g S (x,t) . (7)
Here, K is a function of x. It is to be noted that if we consider Regge behaviour of singlet and gluon structure function, it is possible to solve coupled evolution equations for singlet and gluon structure functions and evaluate K(x) in LO and NLO. Otherwise this is a parameter to be estimated from experimental data. We take K(x) = k, axb, cedx, where k, a, b, c, d are
constants. Therefore equations (6) becomes
g S (x, t)
A f S
g S (x, t)
1 L1(x)g1 (x, t) L2 (x) 1 0.
(8)
t t
Here,
x
L ( x ) A ( x ) K( x )A ( x ) A ( x ) K( x ) , L2(x) = A3(x) + K(x) A4 (x) and Af = 4/(332nf).
1 1 2
4 x
The general solution [1516] of equation (8) is g (U, V) = 0, where g is an arbitrary function
and U (x, t,
g S ) = C1 and V (x, t,
g S ) = C2, where C1 and C2 are constants and they form a
1
1
1
1
solutions of equations
dx dt
dg S (x,t)
1 S .
(9)
f
f
A L (x) t
1
1
2

Af L (x)g1 (x,t)
Solving equation (9) we obtain
S
1 1
S S
L1 (x)
U x, t, g t exp
dx
and
V x,t, g g
(x,t) exp dx.
1
A L
f
f
2
(x)
1 1
L2 (x)
2. (a) Complete and Particular Solutions
Since U and V are two independent solutions of equation (9) and if and are arbitrary constants, then V = U + may be taken as a complete solution [1516] of equation (8). We take this form as this is the simplest form of a complete solution which contains both
the arbitrary constants and . In earlier works [17], considered a solution AU + BV = 0, where A and B are arbitrary constants. But that is not a complete solution having both the arbitrary constants as this equation can be transformed to the form V = CU, where C=A/B, i. e, the equation contains only one arbitrary constant. So, the complete solution
S L (x)
1 L (x)
g1 (x,t) exp 1 dx t exp 1 dx
(10)
f
f
L2 (x)
A L2 (x)
is a twoparameter family of surfaces, which does not have an envelope, since the arbitrary constants enter linearly [1516]. Differentiating equation (10) with respect to we get 0 = 1, which is absurd. Hence there is no singular solution. The one parameter family determined by taking = 2 has equation
S L (x)
1 1 2
g1 (x,t) exp 1 dx t exp
dx .
(11)
f
f
l2 (x)
A L (x)
2
Differentiating equation (11) with respect to , we get
1 1 1
Putting the
value of in equation (11), we get
t exp

Af
L2(x)
dx.
S 1 2 2
L (x)
(12)
g (x, t) t exp
1 dx,
1 4 Af L (x) L (x)
2 2
which is merely a particular solution of the general solution. Now, defining
1 2
L (x) 2 2
g S (x, t
)
t 2 exp 1 dx, at t = t0, where, t0 = ln (Q0 / ) at any lower value
1 0 4 0
Af L (x) L (x)
2 2
Q = Q0, we get from equation (12)
S S t 2
(13)
g (x,t) g (x,t ) ,
1 1 0 t 0
which gives the tevolution of singlet structure function
g S (x,t) . Again defining,
1
1
S 1 2 2
L (x)
we obtain from equation (12)
g (x , t) t exp
1 dx ,
1 0 4 Af L
(x)
L (x)
2
2 x x
0
x 2
L (x)
g S (x,t) g S (x
,t) exp
1
(14)
1 1 0
A
L (x)
L (x) dx
x f 2
2
0
1
1
which gives the xevolution of singlet structure function we get
g S (x,t) . Proceeding in the same way,
NS NS
t 2
(15)
g (x, t) g (x , t) .
1 1 0
t0
which give the tevolutions of nonsinglet structure functions in LO.
x 2 P(x)
g NS (x, t) g NS (x
, t) exp dx,
(16)
f
f
1 1 0
x
A Q(x) 0
Q(x)
which give the xevolutions of nonsinglet structure functions in LO.
For all these particular solutions, we take = 2. But if we take = and differentiate with respect to as before, we can not determine the value of . In general, if we take = y, we get in the solutions, the powers of (t/t0) and the numerators of the first term inside the integral sign be y/(y1) for t and xevolutions respectively in LO.
For phenomenological analysis, we compare our results with various experimental structure functions. Deuteron, proton and neutron structure functions can be written as
g d (x,t) 5 g S (x,t),
(17)
1 9 1
g p (x,t) 5 g S (x,t)

g NS (x,t),
(18)
1 18 1
18 1
g n (x,t) 5 g S (x,t) 3 g NS (x,t).
(19)
1 18 1
18 1
Now using equations (13) (14), (15) in equations (17), (18) and (19) we will get tevolutions of deuteron, proton, neutron and xevolution of deuteron structure functions at lowx as
d, p, n
d, p, n
t 2
(20)
g (x, t) g
(x, t
) ,
1 1 0 t 0
x 2
L (x)
gd (x, t) gd (x , t) exp
1
(21)
1 1 0
A
L (x)
L (x) dx
in LO for = 2.
x0 f 2
2
The determination of xevolutions of proton and neutron structure functions like that of deuteron structure function is not suitable by this methodology; because to extract the x evolution of proton and neutron structure functions, we are to put equations (14) and (16) in equations (18) and (19). But as the functions inside the integral sign of equations (14) and (16)
are different, we need to separate the input functions g S (x ,t) and g NS (x ,t) from the data
1 0 1 0
points to extract the xevolutions of the proton and neutron structure functions, which may contain large errors.

(b) Unique Solutions
Due to conservation of the electromagnetic current, g1
must vanish as Q2 goes to zero
[1819]. Also R0 in this limit. Here R indicates ratio of longitudinal and transverse cross sections of virtual photon in DIS process. This implies that scaling should not be a valid concept in the region of very lowQ2. The exchanged photon is then almost real and the closesimilarity of real photonic and hadronic interactions justifies the use of the Vector Meson Dominance (VMD) concept [2021] for the description of F2. In the language of perturbation theory, this concept is equivalent to a statement that a physical photon spends part of its time as a bare, pointlike photon and part as a virtual hadron [19]. The power and beauty of explaining scaling violations with field theoretic methods (i.e., radiative corrections in QCD) remins, however, unchallenged in as much as they provide us with a framework for the whole xregion with essentially only one free parameter [22]. For Q2 values much larger than 2, the effective coupling is small and a perturbative description in terms of quarks and gluons interacting weakly makes sense. For Q2 of order 2, the effective coupling is infinite and we cannot make such a picture, since quarks and gluons will arrange themselves into strongly bound clusters, namely, hadrons [18] and so the perturbation series breaks down at smallQ2 [18]. Thus, it can be thought of as marking the boundary between a world of quasifree quarks and gluons, and the world of pions, protons, and so on. The value of is not predicted by the theory; it is a free parameter to be determined from experiment. It should expect that it is of the order of a typical hadronic mass [18]. Since the value of is so small
we can take at Q = ,
g S (x,t) 0
due to conservation of the electromagnetic current [19].
1
1
This dynamical prediction agrees with most adhoc parameterizations and with the data [22]. Using this boundary condition in equation (10) we get = 0 and
1 L (x)
g S (x,t) t exp 1 dx.
(22)
1 Af L2 (x)
L2 (x)
1
L (x) 2 2
Now, defining
g S (x,t
) t
exp 1 dx, at t = t0, where t0 = ln (Q0 / ) at
1 0 0
Af L2 (x)
L2 (x)
any lower value Q = Q0, we get from equations (22)
g S (x, t) g S (x, t
t
t
) ,
(23)
1 1 0
t0
1
1
which gives the tevolutions of singlet structure function same way we get
g S (x,t)
in LO. Proceeding in the
x 1
L (x)
g S (x, t) g S (x
, t) exp
1
(24)
1 1 0
A
L (x)
L (x) dx
x0 f 2 2
which gives the xevolutions of singlet structure function F2S(x, t) in LO. Similarly, we get for nonsinglet structure functions
g NS (x,t) g NS (x,t
t
t
) .
(25)
1 1 0
t 0
x 1 P(x)
g NS (x, t) g NS (x
,t) exp
dx,
(26)
f
f
1 1 0
x
A Q(x) 0
Q(x)
which give the t and xevolutions of nonsinglet structure functions in LO.
Therefore corresponding results for tevolution of deuteron, proton, neutron structure functions and xevolution of deuteron structure function are
d, p, n
d, p, n
t
(27)
g (x, t) g (x, t ) ,
1 1 0 t 0
x 1
L (x)
gd (x, t) gd (x , t) exp
1
(28)
1 1 0 A
L (x)
L (x) dx
in LO.
x0 f 2
2
Already we have mentioned that the determination of xevolutions of proton and neutron structure functions like that of deuteron structure function is not suitable by this methodology. It is to be noted that unique solutions of evolution equations of different structure functions are same with particular solutions for y maximum (y = ) in = y relation. The procedure we follow is to begin with input distributions inferred from experiment and to integrate the evolution equations (24) and (26) numerically.

Results and Discussion
In the present paper, we have compared the results of t evolutions of spindependent deuteron, proton and neutron structure functions in LO with different experimental data sets measured by the SLACE143 [23] collaboration. The SLACE143 collaborations data sets give the measurement of the spindependent structure function of deuteron, proton and neutron in deep inelastic scattering of spindependent electrons at incident energies of 9.7,
16.2 and 29.1 GeV on a spindependent Ammonia target. Data cover the kinematical x range
0.024 to 0.75 and Q2range from 0.5 to 10 GeV2.
In fig.1, we present our results of tevolutions of spindependent deuteron structure function for the representative values of x given in the figures for y = 2 (solid lines) and y maximum (dashed lines) in = y relation in LO. Data points at lowestQ2 values in the figures are taken as input to test the evolution equation. Agreement is found to be excellent.
In fig.2, we present our results of tevolutions of spindependent proton structure function for the representative values of x given in the figures for y = 2 (solid lines) and y maximum (dashed lines) in = y relation in LO. Data points at lowestQ2 values in the figures are taken as input to test the evolution equation. Agreement is found to be excellent.
Fig.1: Results of tevolutions of deuteron structure functions (solid lines for y = 2 and dashed lines for y maximum in = y relation) for the representative values of x in LO for SLACE 143 data. For convenience, value of each data point is increased by adding i, where i = 1, 2, 3. are the numberings of curves counting from the bottom of the lowermost curve as the 1st order. Data points at lowestQ2 values in the figures are taken as input.
In fig.3, we present our results of tevolutions of spindependent neutron structure function deuteron, proton for the representative values of x given in the figures for y = 2 (solid lines) and y maximum (dashed lines) in = y relation in LO. Data points at lowestQ2 values in the figures are taken as input to test the evolution equation. Agreement is found to be excellent
Unique solutions of tevolution for structure functions are same with particular solutions for y maximum (y = ) in = y relation in LO.
For a quantitative analysis of xdistributions of structure functions, we calculate the integrals that occurred in equations (21) and (28) for Nf = 4. In fig.4 we present our results of
xdistribution of deuteron structure functions F2d for K(x) = constant (solid lines), K(x) = axb
(dotted lines) and for K(x) = ce dx (dashed lines), where a, b, c and d are constants and for
Fig.2: Results of tevolutions of proton structure functions (solid lines for y = 2 and dashed lines for y maximum in = y relation) for the representative values of x in LO for SLACE 143 data. For convenience, value of each data point is increased by adding i, where i = 1, 2, 3 are the numberings of curves counting from the bottom of the lowermost curve as the 1st order. Data points at lowestQ2 values in the figures are taken as input.
representative values of Q2 given in each figure, and compare them with SLACE143 collaborations deuteron lowx low Q2 data[26]. In each the data point for x value just below
0.102 has been taken as input F2d (x 0, t). If we take K(x) = 30, then agreement of the result with experimental data is found to be excellent. On the other hand if we take K(x)= axb , then
agreement of the results with experimental data is found to be good at a = 43, b = 1. Again if we take K(x) = ce dx , then agreement of the results with experimental data is found to be good at c = 7, d = 1. For xevolutions of deuteron structure function, results of unique solutions and results of particular solutions have not any significance difference in LO.
Fig.3: Results of tevolutions of neutron structure functions (solid lines for y = 2 and dashed lines for y maximum in = y relation) for the representative values of x in LO for SLACE 143 data. For convenience, value of each data point is increased by adding i, where i = 1, 2, 3. are the numberings of curves counting from the bottom of the lowermost curve as the 1st order. Data points at lowestQ2 values in the figures are taken as input.
In fig.5, we present the sensitivity of our results for different constan values of K(x). We observe that at K(x) = 30, agreement of the results with experimental data is found to be excellent. If value of K(x) is increased, the curve goes downward direction and if value of K(x) is decreased, the curve goes upward direction. But we observe that difference of the curves is very small and all these curves are overlapped. But the nature of the curve is similar.
In fig.6, we present the sensitivity of our results for different values of a at fixed value of b. Here we take b = 1. We observe that at a = 43, agreement of the results with experimental data is found to be excellent. If value of a is increased the curve goes upward
Fig.4: Results of xevolutions of deuteron structure functions for K(x) = k (constant) (solid lines), K(x) = axb(dotted lines) and K(x) = cedx (dashed lines) in LO for representative values of Q2 given in each figure, and compare them with SLACE143 data. In each graph, the data point for xvalue just below 0.102 has been taken as input to test the evolution equation. Agreement of the results are excellent at k = 30, a = 43, b = 1, c = 7, d = 1. For convenience, value of each data point is increased by adding i, where i = 1, 2,3 are the numberings of curves counting from the bottom of the lowermost curve as the 1st order. Data points at lowestQ2 values in the figures are taken as input.
direction and if value of a is decreased the curve goes downward direction. But the nature of the curve is similar.
In fig.7, we present the sensitivity of our results for different values of b at fixed value of a. Here we take a = 43, we observe that at b = 1, agreement of the results with experimental data is found to be excellent. If value of b is increased or decreased then the curve goes downward direction. The nature of the curve is dissimilar.
In fig.8, we present the sensitivity of our results for different values of c at fixed value of d. Here we take d = 1. We observe that at c = 7, agreement of the results with experimental data is found to be excellent. If value of c is increased the curve goes downward direction and if value of c is decreased the curve goes upward direction. But the nature of the curve is not similar.
Fig.5: Sensitivity of our results for different constant values of K(x).
Fig.6: Sensitivity of our results for different values of a at fixed values of b = 1.
Fig.7: Sensitivity of our results for different values of b at fixed values of a = 43.
Fig.8: Sensitivity of our results for different values of c at fixed values of d = 1.
In fig.9, we present sensitivity of our results for different values of d at fixed value of c. Here we take c = 7, we observe that at d = 1, agreement of the results with experimental
data is found to be excellent. If value of d is increased then the curve goes upward direction and if value of d is decreased the curve goes downward direction. Here also the nature of the curve is similar.
From our above discussion, it has been observed that we cannot establish a unique relation between singlet and gluon structure functions i.e. a unique expression for K(x) in equation (7) by this method, K(x) in the forms of a constant, an exponential function of x or a power in x can equally produced required xdistribution of deuteron structure functions. But unlike many parameter input xdistribution functions generally used in the literature, our method required only one or two such parameter. The explicit form of K(x) can actually be obtained only by solving coupled GLDAP evolution equations for singlet and gluon structure functions, and works are going on in this regard.
Traditionally the GLDAP equations provide a means of calculating the manner in which the parton distributions change at fixed x as Q2 varies. This change comes about because of
Fig.9: Sensitivity of our results for different values of d at fixed values of c = 7.
the various types of parton branching emission processes and the xdistributions are modified as the initial momentum is shared among the various daughter partons. However the exact rate of modifications of xdistributions at fixed Q2 can not be obtained from the GLDAP equations since it depends not only on the initial x but also on the rate of change of parton distributions
with respect to x, dnF(x)/ dxn (n = 1 to ), up to infinite order. Physically this implies that that at highx, the parton has a large momentum fraction at its disposal and as a result it radiates partons including gluons in innumerable ways, some of them involving complicated QCD mechanisms. However for lowx, many of the radiation processes will cease to occur due to momentum constraints and the xevolutions get simplified. It is then possible to visualize a situation in which the modification of the xdistribution simply depends on its initial value and its first derivative. In this simplified situation, the GLDAP equations give information on the shapes of the xdistribution as demonstrated in this paper. The clearer testing of our results of xevolution is actually the equation (25) which is free from the additional assumption equation (16). But nonsinglet data is not sufficiently available in lowx to test our result.

Conclusion
We solve spin dependent DGLAP evolution equation in LO using Taylor expansion method and derive t and xevolutions of various spin dependent structure functions and compare them with global data with satisfactory phenomenological success. It has been observed that though we have derived a unique tevolution for deuteron, proton, neutron structure functions in LO, yet we can not establish a completely unique xevolution for deuteron structure function in LO due to the relation K(x) between singlet and gluon structure functions. K(x) may be in the forms of a constant, an exponential function or a power function and they can equally produce required xdistribution of deuteron structure functions. But unlike many parameter arbitrary input xdistribution functions generally used in the literature, our method requires only one or two such parameters. On the other hand, we observed that the Taylor expansion method is mathematically simpler in comparison with other methods available in the literature. Explicit form of K(x) can actually be obtained only by solving coupled DGLAP evolution equations for singlet and gluon structure functions. Though we study LO evolution equation for spin structure function, we hope that it can be extendable to NLO also. So we see that this simple method may have a wide application in solving DGLAP evolution equations.
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