P
RAMANA c Indian Academy of Sciences Vol. 53, No. 4—journal of October 1999
physics pp. 701–706
The longitudinal structure function
FL(x;Q 2
)
and the gluon distribution
G(x;Q2)at low
xABHIJEET DAS and A SAIKIA
Department of Physics, Gauhati University, Guwahati 781 014, India
Department of Physics, Arya Vidyapeeth College, Guwahati 781 016, India MS received 9 April 1999
Abstract. We obtain a relation between the longitudinal structure functionFL (x;Q
2
),F2 (x;Q
2
)
andG(x;Q2)at small x, using the formalism recently reported by one of the authors [2]. We also obtain a relation betweenFL(x;Q2),F2(x;Q2)and its slope(dF2(x;Q2))=(dlnQ2). This provides us with the determination of the longitudinal structure functionFL
(x;Q 2
)fromF2 (x;Q
2
)
data and hence extract the gluon distributionG(x;Q2).
Keywords. Longitudinal structure function; gluon distribution; QCD; lowx. PACS Nos 12.38 Bx; 12.38 Cy
1. Introduction
The longitudinal structure functionFL (x;Q
2
)comes as a consequence of the violation of Callan–Gross relation and is defined as
F
L (x;Q
2
)=F
2 (x;Q
2
) 2xF
1 (x;Q
2
):
In the naive parton model,FL
=0for spin1=2giving the Callan–Gross relation. How- ever, in the QCD inspired quark-parton model,FL
6=0and is very sensitive to the QCD effects.
The QCD prediction for the longitudinal structure functionFLis [1]
F
L (x;Q
2
)=
s (Q
2
)
"
4
3 Z
1
x dy
y
x
y
2
F
2 (y;Q
2
)
+2 X
i e
2
i Z
1
x dy
y
x
y
2
1 x
y
G(y;Q 2
)
#
; (1)
e
ibeing the electric charge of theith quark in the unit of proton charge.
At smallx(x<10 3), the dominant contribution comes from the gluon and this makes the measurement ofFL
(x;Q 2
)an almost direct measure of the gluon distribution. How- ever, FL is very difficult to measure experimentally since its contribution to the cross- section is suppressed by they2factor. The relation betweenFLandF2becomes important for such purpose because the structure functionF2 has been studied extensively experi- mentally as well as theoretically [7,8,5] so such studies onF2 can be used not only to extractFLbut alsoG.
In the present work, we evaluate the longitudinal structure functionFLusing the frame- work of [2] and thereby extract the gluon densityG(x;Q2)from it. We also obtain a relation amongFL,F2 and(dF2
=dlnQ 2
). This will facilitate the determination of FL
fromF2and(dF2
=dlnQ 2
)data.
The paper is organised as follows. Inx2, we outline the formalism and obtain various results. Inx3, we compare our analytical result with those available in the literature [3].
Finally we conclude inx4.
2. Formalism
Puttingz=1 (x=y)in (1), we get for four flavours,
F
L (x;Q
2
)= 4
s (Q
2
)
3 Z
1 x
0
dz(1 z)F
2
x
1 z
;Q 2
+ 5
3 Z
1 x
0
dzz(1 z)G
x
1 z
;Q 2
= 4
s (Q
2
)
3
I
1 +
5
3 I
2
: (2)
We now evaluate the two integralsI1 andI2 in (2) using the framework of [2]. The functionF2
(x=(1 z))is expanded about a pointz =where0<<1 xsuch that
jz j<R,Rbeing the radius of convergence. Thenth term in the integrand ofI1will be
T
n
=(z ) n 1
F (n 1)
2
(n 1)!
(1 z); (3)
where
F (n 1)
2
=
@F
2 (
x
1 z )
@z n 1
z=
:
Here we suppressed theQ2 dependence ofF2. The above series is integrated term by term and we get at lowx
A
n
= Z
1 x
0 T
n dz =
n 1
X
r=0 a
r
(1 x) n r
n r
(1 x) n r+1
n r+1
n 1
X
r=0 a
r
1
n r
1
n r+1
; (4)
where
a
r
=
( ) r
r!(n r 1)!
F (n 1)
2 :
The complete integrated series is obtained by evaluating the termsA1,A2,A3,:::. The integralI1is then given by
I
1
=A
1 +A
2 +A
3 +
= 1
2 F
2
x
1
+ 1
6 F
(1)
2
x
1
+ 1
24 F
(2)
2
x
1
+
= 1
2
F
2
x
1
+a
+R (a)
; (5)
whereais a parameter andR (a)is given by
R (a)=
1
3 a
F (1)
2
x
1
+
1
12 a
2
2!
F (2)
2
x
1
+
1
60 a
3
3!
F (3)
2
x
1
+: (6)
LetR (a)=0ata=a0. This gives,
I
1
= 1
2 F
2
x
1
+a
0
:
Now,F2 will be sensitive to the variation ofx only when(x=(1 ) ) a0 :So, whenx!0,!a0. Thus,
I
1
= 1
2 F
2
x
1
; (7)
wheresatisfies the condition
1
3
F (1)
2
x
1
+
1
12
2
2!
F (2)
2
x
1
+
1
60
2
3!
F (3)
2
x
1
+=0: (8)
We may have an error of nearly 2% in reaching condition (8) from (6) fora0:5(and
( 0.5, 0)) [2].
The integralI2can be evaluated following the above steps and is found to be
I
2
=G
x
1
; (9)
wheresatisfies the condition
1
2
G (1)
x
1
+
3
20
2
2!
G (2)
x
1
+
1
30
3
3!
G (3)
x
1
+=0: (10) UsingI1andI2(from (7) and (9)), the expression forFLcan be written as
F
L (x;Q
2
)= 2
s (Q
2
)
3
F
2
x
1
;Q 2
+ 5
9 G
x
1
;Q 2
(11) using (2).
Using (1), the gluon distribution can be expressed as
G(x;Q 2
)= 3
5 6
3
4
s
F
L
(1 )x;Q 2
1
2 F
2
1
1 x;Q
2
: (12) This relation can be used to extract the gluon density from the knowledge ofF2andFL.
Again, from [2], we have
dF
2 (x;Q
2
)
dlnQ 2
= 20
9
s (Q
2
)
4
G
x
1 0
; (13)
where0satisfies the condition
1
2
0
G (1)
x
1 0
+
7
40
02
2!
G (2)
x
1 0
+
11
240
03
3!
G (3)
x
1 0
+=0: (14) From (13) we get
G(x;Q 2
)=
9
5
s (Q
2
)
dF
2
(1 0
)x;Q 2
dlnQ 2
: (15)
Inserting (15) in (11) we get
F
L (x;Q
2
)=
2
s (Q
2
)
3
"
F
2
x
1
;Q 2
+
s (Q
2
)
dF
2
1 0
1 x;Q
2
dlnQ 2
#
: (16)
This relation gives usFLdirectly fromF2and(dF2
=dlnQ 2
)data.
3. Results and discussion
To evaluateandwe assumeF2andGto have the general formNx (1 x)Æ(1+x) which at lowxis of the formx . We thereby evaluateandnumerically from (8) and (10) by considering the first three terms of the series. The values ofand for different values ofare shown in table 1.
Table 1.
Error in Error in
evaluating evaluating
0.5 0.329535648 3:0210 7 0.515975005 0
0.3 0.331923586 4:2 10 7 0.532081394 2:510 7
0.08 0.338208103 2:8 10 6 0.5514730411 0
With these values ofand, relation (12) for the gluon distribution becomes
G(x;Q 2
)= 3
5 6
3
4
s
F
L
(0:484024995x;Q 2
)
1
2 F
2
(0:721925026x;Q 2
)
for =0:5;
G(x;Q 2
)= 3
5 6
3
4
s
F
L
(0:467918606x;Q 2
)
1
2 F
2
(0:700396835x;Q 2
)
for =0:3;
G(x;Q 2
)= 3
5 6
3
4
s
F
L
(0:448526958x;Q 2
)
1
2 F
2
(0:677746223x;Q 2
)
for =0:08: (17) A similar relation has been obtained by Cooper–Sarkar et al [3] which can be written as
G(x;Q 2
)= 3
5 5:8
3
4
s
F
L
(0:417x;Q 2
) 1
1:97 F
2
(0:75x;Q 2
)
: (18) In the derivation of (18), the integrand is Taylor expanded aroundx=0. It has already been found [2] that such an expansion is unstable for variation of. This fact becomes apparently clear when we look at the origin of the co-efficients ofFL andF2and their arguments in [6]. The presented method overcomes such defect.
The phenomenological application of the presented method is in progress.
4. Conclusion
We have obtained a simple analytical relation for the longitudinal structure functionFLin terms ofF2and(dF2
=dlnQ 2
). We also obtained a relation from which the gluon density
G(x;Q 2
)can be extracted from a knowledge ofFLandF2. The agreement between our relation (17) and (18) from [3] indicates the validity of the framework of [2] to evaluate the convolution integralsI1andI2. We note that the kernels associated with these integrals are not singular within the range of integration which supports our claim in [4] that the method of Taylor expansion works in evaluating convolution integrals when the associated kernels are not singular within the range of integration.
Acknowledgement
One of the authors, AD, gratefully acknowledges financial support from the Department of Science and Technology, Government of India.
References
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