^_`a
_ba^cc!cdA*c9%
c9,
?%
! NLO 5
"#7 &<7 )9 :; * 4/04 )5
56
7 * 1 ,3+
,$ ) #-.* 1,2%&'( )"#$ "
1
% % #$! ! " 2
! " ) #%&$ '( 3
! 4
%* %* ! 5
>?
E .B$ C
D @&/ &: ' A$ NLO >? #-
:;< ='- =- 9 #
$ -87 DGLAP #,-. /
0% #66. +)
@A % 97
NLO9:% 9 :$
.% IJ" .
#% G-% >? @&/ F! #-
B!
=- 9!
+: + 9 '
%
#
$ .-7 M<-% #N
O 9:% #"'
:$
.% +7L - #% ; -% B< 9 9K- .
#% @B ",7 BC -D
:;< =- >J) #- :;< O
$ =
9:$ .% ( R
: .B$ 9:$
.% +
< - EQ$ D -
' C* P$ :;< -
9
. #-O "'
$ #$
R% EU7 -K-% #
%'( %( B$ T
.- -O S % #-
Extracting the polarized parton distribution functions at NLO approximation
,using the laplace transforms
F.Taghavi-Shahri 1,2; S. Atashbar Tehrani 1,3; A.Mirjalili 4; M. M. Yazdanpanah 5
1
School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM),Tehran,Iran
2
Department of Physics, Ferdowsi University of Mashhad,, Mashhad, Iran
3
Department of Physics, Azad University, Yazd Branch, ran
4
Physics Department, Yazd University, Yazd, Iran
5
Faculty of Physics, Shahid Bahonar University of Kerman, Kerman,Iran
Abstract
We analytically solve the DGLAP evolution equations at the NLO approximation, using the Laplace transformation. Then
we achieve to the polarized parton densities. For this propose we need to the Laplace transformation of the parton splitting
functions at the related approximation. Extending the calculations to the NLO approximation requires the required
recurrence relations in terms of the expanding parameters. Up to the desired accuracy, the iteration steps of the recurrence
relations can be stopped. y accessing to the polarized parton densities, we can calculate the parton contributions to the
nucleon spin. Following that we are able to compute the polarized proton structure functions in terms of the polarized
parton densities. The results are in good agreement with the available experimental data and phenomenological models.
PACS No.
( 13)
9* (DIS) J*
YZ #"* %'(
%77
- EQ$ EO
$ #X-* P$ V-*
V-* EQ$ 9
='- - # O
$ BO
' 967K ' P% 9A,-% 9:$
.% BO
'%
P$ EU7 -]-6" V-* E 9' B*)
' % O
$ E E E W* #-X ( '
$
.B$ - 9K- -% '
^__`
_ba^cc!cdA*c9%
c9,
?%
4S
w
D s (Q ) w ln Q
2
2
" %'( '-% 9 . B7 EQ$ EO
$ -]-6"
GG( x, Q 2 ) GFS (GPgqLO D s (Q 2 ) NLO
GPgq )( x, Q 2 ) GG (GPggLO 4S
D s (Q 2 ) NLO
GPgg )( x, Q 2 )
4S
#,-. /
0% +) 9 * #% I& ; 9:$
.% #-
='- =- ef$ 'Q DGLAP
-
E ' C* P$ ( R 9:% E6% +:
(3)
1
/
0% @&/ +: Q Ln h% h R
)
x
&: BN
O E ' P #% S 9 Q F! ' k-!
I& 9,
?% E #6:< * 9% .
- EQ$
/
0% #66. +) 9 @&/ &: ' A$ :;< ='- =- P
B$ NLO >? DGLAP
.P* 9$
.% -
:9* P* #% A$ @&/
Q
³
L[ GF ( w)GH (Q w)dw; s]
0
L[GF (Q ); s]L[GH (Q ); s]
(4)
#) /
/
0% ((1)9,
0%) 9A,-%
:( #% B$ @&/ +:
Q
w
GFNS (Q ,W )
GFNS ( w,W )e (Q w) (GPqqLO (Q w) 0
wW
D s (Q 2 ) NLO
GPqq (Q w))dw
4S
³
(5)
:B$ ' -N 9 @&/ F! /
9,
0% l-K
W2
B$
GI NS (s )
GI NS ( s )
=- .
GI NS
W2
LO
1
4S
(s) NLO
W2
GI NS
(s)
W
h% h :[1-6]B$ ' < ' 9N&O -; * I
1
W (Q0 , Q ) {
4S
/
0%
2
2
Q2
³D
s (Q c
2
)d ln Q c 2 Q
Ln
Q0 2
1
x
F! ' J #, +JA #/
0% 9* DGLAP
# @&/ +: ' d .P #% (Q ,W ) F! 9 ( x, Q 2 )
9 W
+JA /
0% U
' S 9 Q '
='- =- >J) :K /
0% $ 9 % #,
.B$ +) +
< I 9 9* - #% +: -
9,
9 B"'
@&/ @-0% +: ' BJ!
* 9A,-% NLO >? .P* A$
eWGINS ( s )GF 0 NS ( s)
GFNS ( s,W )
E FA
( x, Q 2 ) F!
(6)
='- L&i 9* :;< -
='- /
9; 9*
=
-O *-* #!j V-*
W
³ D s (W c) dW c . :- #% M0 9-
/
0% ' A$ :;< -]-6" ='-
DGLAP
:( #% B$ 9-
0
=- E 9:$
.% . G
: #!
=- @&/ +:
:L - #/-i i
f 9* 9!" -N NLO >?
.B$ -O ( ( ' G-% s F! 9, ='- =- EJ B,
) E :;< ='- =- - #% @&/ @-0% +: ' A$
.( B$ #!j *-* 4S
w
D s (Q ) w ln Q 2
2
GFNS ( x, Q 2 ) GFNS (GPqqLO D s (Q 2 ) NLO
GPqq )( x, Q 2 )
4S
4S
w
2
D s (Q ) w ln Q
2
2 G FS ( x , Q )
(1)
LO
G FS (G Pqq 2
G5D A > H D s (Q )
4S
NLO
2
LO
)( x , Q ) G G (G Pqg NLO
2
)( x , Q )
G Pqq
2
D s (Q )
4S
G Pqg
2
^___
_ba^cc!cdA*c9%
c9,
?%
I ' A$ - #% (11) (10) /
0% '
#-
:;< ='- =- 9A,-% 9:$
.% /
p .B!
#f$
GG ( s,U ) GFS ( s,U ) @&/ +: ' A$ 9 '
:;< -]-6" =- 9:$
.%
E P U 9 W ' S 9 Q ' # @&/
I ' A$ *S% .
#% @&/ F! (iteration) I ' A$ 9 '
-K E!
F!
I -O 96 BJ #66. s +) 0% 9 .- -O #/-i J NLO >? :$
.% .P
NLO
GM g
#-r6" Wf B!
=- 9!
+: + Y0%
NLO
GT g
C'/ 9* #% #
l-K 9 #
B$
.J +7L 9 B:J
DGLAP /
0% a(W )
D s (W )
4S
UGFS ( s, U ) Gf S0 ( s)
+) ' %( B$ - O
$ =
I' ' ( Gg 0 ( s ) :;< ='- =- .B$ A$ #D #66.
NLO
GI S
GI S ( s )GFS ( s, U ) LO
( s ) L[a(W )Gf S ( s, W );U ] GT S ( s )GG ( s, U ) LO
NLO
GT S
( s ) L[a(W )Gg ( s,W );U ]
(7)
B
t %
9- s F! 9, @
?% >J) 9* - O
$ =
#6* I' ' '
-% E -K-% #D B$ 9- ='- =-
:B$ %( B$ O
$ =
Gq( s, Q02 )
N qK q (1 c q
³
f
0
e svGq ( x
e v , Q02 )dv
s aq
s a q bq 1
) B( s a q , bq 1)
.
#% 6 =
B q {u v , d v , q, g} /
9; 9*
M0 :- -O 9- (S,U) F!
%
4A Gf S0 ( s) )9, @
?% #-
='- =- E!
a0 a1e b1W
UGG( s,U ) Gg 0 (s)
NLO
GI g
LO
GT g
GI g ( s )GG ( s, U ) LO
( s ) L[a(W )Gg ( s,W );U ] ( s )GF ( s, U ) NLO
GT g ( s )
L[a(W )Gf ( s,W );U ]
(8)
A$ @&/ &: 9:$
.% ' 9; ' A$ -O $ 9,
0% E (8) (7) p :9-;
#!j V-* :;< ='- =- T
1 + + .B$ ( NLO >? -]-6" #66. +) ' %( B$ - :;< O
$ =
2
9J
?% 3 + .B$ ( #D ( 9J
?%
.B$ C
D #$
R% =
E
1
L[a(W )Gf ( s,W );U ]
¦ a Gf (s,U b )
j
j
j 0
1
L[a(W )Gg ( s,W );U ]
¦ a Gg (s,U b ); b
j
j
0
0
j 0
(9)
>? DGLAP /
0% ' 9,
0% B
:P U s F! NLO
(U GI S )GFS ( s,U ) GT S ( s )GG( s,U )
Gf S0 (s) a1 (GI SNLO ( s )Gf ( s, U b1 ) GT SNLO ( s )Gg ( s, U b1 ))
(10)
GT g ( s )GF ( s,U ) (U
GI g ( s ))GG(s,U )
Gg 0 (s) a1 (GI gNLO ( s )Gf (s,U b1 ) GT gNLO ( s )Gg (s,U b1 ))
(11)
^__^
_ba^cc!cdA*c9%
c9,
?%
P$ P
< #-
:;< ='- =- 9:$
.% ' d R
u% .P
- EQ$ -
' C* : P Q 2 4GeV 2 'u v (Q 2
4GeV 2 )
'd v (Q 2
4GeV 2 )
1
³ Gu ( x, Q )dx
³ Gd ( x, Q )dx
³ Gd ( x, Q )dx
³ Gd ( x, Q )dx
v
0
' q (Q 2
4GeV 2 )
'g (Q 2
4GeV 2 )
1
0
1
0
2
0.928
2
0.342
v
0
1
2
0.0661
2
3.393
v
v
" p$- %( B$ T
#-O v
; T
E
.[8-13] #
%'( -r
"
D %I
=- ef$ #66. +) E, 9,
?% E #66. +) ' %( B$ #-
:;< ='- =- :1 +
NLO >?
DGLAP /
0% ' A$ #-
:;< ='-
T
.B$ " 9] NLO >? @&/ &: v!- :;< O
$ =- #-
:;< ='- =- -K-% #$
,% #D #-O J
.
,67
[1]
M. M. Block, L. Durand, P. Ha, and D.W. McKay, Eur.Phys. J. C
69, 425 (2010).
[2] M. M. Block, L. Durand, P. Ha, and D.W. McKay, Phys.Rev. D 84,
094010 (2011).
[3] M. M. Block, L. Durand, P. Ha, and D.W. McKay, Phys.Rev. D 83,
v0cv054009 (2011).
[4] M. M. Block, Eur. Phys. J. C 65, 1 (2010).
[5] M. M. Block, Eur. Phys. J. C 68, 683 (2010).
[6] F. Taghavi-Shahri, A. Mirjalili and M. M. Yazdanpanah,
#66. +) ' %( B$ O
$ =
9J
?% :2 +
.NLO >? - [7] #D
Eur. Phys. J. C 71, 1590 (2011).
[7] K. Abe et al. [E143 collaboration], Phys. Rev. D 58
(1998) 112003 [arXiv:hep-ph/9802357].
[8] Y. Goto et al. [Asymmetry Analysis Collaboration],
Phys. Rev. D 62, 034017 (2000) [hep-ph/0001046].
[9] M. Gluck, E. Reya, M. Stratmann and W. Vogelsang,
Phys. Rev. D 63, 094005 (2001) .
[10] M. Gluck, E. Reya, M. Stratmann and W. Vogelsang,
Phys. Rev. D 63, 094005 (2001) .
[11] E. Leader, A. V. Sidorov and D. B. Stamenov, Phys. Rev.
D 75, 074027 (2007).
[12] D. de Florian, G. A. Navarro and R. Sassot, Phys. Rev.
D 71, 094018 (2005).
[13] A. Airapetian et al. [HERMES Collaboration], Phys.
Rev. D 75, 012007 (2007).
#66. +) ' - %( B$ O
$ =
9J
?% :3 +
. [8-13]#$
R % [7] #D

Download Report