Solution of non-singlet DGLAP evolution equation in leading order and next-to-leading order

(3)
(0 ) where P NS (x ) , P (1) NS ( x ) , ······ are non-singlet splitting functions in LO, NLO, ······
respectively. Appling all these and simplifying, the DGLAP evolution equations for non-singlet structure function in LO and NLO can be written as [6]
3
∂ F2NS (x, t ) 3 ∂ F2NS (x, t ) 3 NS −t + A f A (x )F2 (x, t ) + A f B (x ) = 0, ∂t ∂x 2 2
∂F2NS x, Q 2 = PNS x, Q 2 ⊗ F2NS x, Q 2 , ∂lnQ 2
(
)
(
)
(
)
(1)
where F2NS (x, Q 2 ) is the non-singlet structure function as the function of x and Q2, where x is the Bjorken variable and Q2 is the four momentum transfer in a DIS process. Here PNS(x, Q2) is the non-singlet kernel known pertarbatively up to the first few orders in coupling constant. With the notation
1 2 1+ω
2 1+ ω (1 + ω )
(
)
ω 1+ω
∫
dk 1 − k ln + 2(1 + ω ) ln (ω ) + 4(1 − ω ) , k k
with CA = CG = 3, C F ( and
= 102 −
)=
2 2 2
(
)
(1 + ω ) 67 π 11 − ln (ω ) + ln (ω ) + (1 − ω ) 9 3 3

40 + 2(1 + ω ) ln ω + 3 (1 − ω ) ,
PN f (ω ) =
and PA (ω ) =
5 2 1 + ω 2 − ln ω − − 2(1 − ω ) 3 3 1−ω
N f2 − 1 , Tf ( 2N f
) = 1 and
2
S
(t ) =
1 − 0t
1
lnt . 2 0t
0
= 11 −
2 Nf 3
-
1
38 N f are the one loop (LO) and two loop (NLO) corrections to the QCD 3
(t ) 2 {3 + 4ln (1 − x )}F NS (x, t ) + I NS (x, t ) = 0 ∂ F2NS − S 2 1 ∂t 2 3
and
∂ F2NS − ∂t
S
(4)
2
(t ) 2 {3 + 4ln (1 − x )}F NS (x, t ) + I NS (x, t ) −
2
2 [PF (ω ) − PA (ω )] + 1 C F C A [PG + PA (ω )] + C F TR N f PN f (ω ) , f (ω ) = C F 2
PF (ω ) = −
PG (ω ) =
40 1 2 1+ ω2 3 + 2ω ln ω − (1 + ω ) ln ω + (1 − ω ) , ln (ω ) ln (1 − ω ) − (1 − ω ) 3 2 1 − ω
Solution of non-singlet DGLAP evolution equation in leading and next-to-leading order at small-x by method of characteristics
R. Baishya1
Physics Department, J.N. College, Boko-781123, Assam, India
function and Nf being the flavour number. We can neglect β 1 for LO. Let us introduce the variable u = 1- ω and note that
x
=
x xu = x + . 1− u 1− u
Since x< ω <1, so 0<u<1-x, and hence the series is convergent for |u|<1. Now using Taylor’s expansion series, we can rewrite
(
)
(6)
I
NS 2
x (x, t ) = (x - 1)F (x, t )∫ f (ω )dω + ∫ f (ω )F2NS , t dω
NS 2 0 x
1
1
ω

(7)
and t = ln
Q2 , where ∧ is the QCD cut off parameter. Here ∧2
xu ∂F2NS (x, t ) x . F2NS , t = F2NS (x, t ) + 1− u ∂x
form
(8)
Using equation (8) in equation (6) and performing u-integrations, equation(4) becomes the
2 S
(Q ) , the strong
(2)
a (x ) ⊗ b (x ) ≡ ∫
we can write
x dy a (y )b , y y 0
1
PNS x, Q =
2
(
)
S
(Q ) P ( ) (x ) + (Q )
2 2 0 NS S
2


P (1) NS (x ) + Λ Λ ,
1 2
E-mail:- rjitboko@yahoo.co.in E-mail :- jks@tezu.ernet.in 1
parameter free solution and this is a more stringent test of our method in solving DGLAP evolution equations. The DGLAP evolution equation in the non singlet sector in the standard form [5-7] is given by
3
2 1

S 2
(t ) 2 I NS (x, t ) =

2
0
(5)
respectively, where
NS (x, t ) = I1
4 dω 3∫ 1−ω x
1
2 NS x NS 1 + ω F2 ω , t − 2F2 (x, t ) ,
xu x NS ,t F2NS , t = F2 x + 1−u ω
= F2NS (x, t ) + xu ∂ F2NS (x, t ) 1 xu + 1− u ∂x 2 1 − u
2
∂
2
F2NS (x, t ) ∂x 2
+Λ Λ
.
Since x is small in our region of discussion, the terms containing x2 and higher powers of x can be neglected. So we get
J. K. Sarma2
Physics Department, Tezpur University, Napam-784028, Assam, India The non-singlet structure functions have been obtained by solving Dokshitzer-GribovLipatov-Alterelli-Parisi (DGLAP) evolution equations in leading order (LO) and next-toleading order (NLO) at the small-x limit. Here a Taylor series expansion has been used and then the method of characteristics has been applied to solve the evolution equations. Results are compared with the Fermilab experiment E665 data and New Muon Collaboration (NMC) data. Keywords. DGLAP equation; small-x; method of characteristics; structure function; deep inelastic scattering (DIS) PACS numbers: 12.38-t; 12.39-x; 13.60.Hb In our recent report [1], we have explained the importance of DGLAP evolution equations and their solutions, and also the importance of the method of characteristics to solve those equations. There, we have solved singlet DGLAP evolution equation in LO and NLO at small-x [2] by using the method of characteristics and compared deuteron structure function thereby obtained with NMC data with satisfactory phenomenological success. In this brief report, we have solved non-singlet DGLAP evolution equation in LO and NLO at smallx by using the same method and compared non-singlet structure function thereby obtained with Fermilab experiment E665 data [3] and NMC data [4] with considerable phenomenological success. The non-singlet structure function is the combination of only valence quark densities and free of singlet quark and gluon densities. Therefore when solving the non-singlet DGLAP equation, we need not relate quark and gluon structure functions by a suitable function k(x) or by a constant as in the case of solution of the singlet DGLAP evolution equation given in our earlier report [1]. So, the solution of non-singlet equation is a