10 - H1 - Desy

10 Theoretical framework
1.2.4.1 DGLAP evolution equations
The Dockshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [14–18]
define the way in which quark and gluon momentum distribution evolve with the scale of
the interaction. Within DGLAP approach a strong ordering of the transverse momenta
k T,i
Q 2 ≫ k 2 T,i ≫ k2 T i −1 ≫ k T,i−2 . . . ≫ k T1 ≫ Q 2 0 , (1.14)
and a soft ordering of the fractional longitudinal momenta x i
x i < x i−1 < x i−2 . . . < x 1 (1.15)
are assumed. Here, Q 2 0 is the virtuality of the parton at the start of the emission cascade
and Q 2 is the virtuality of the exchanged photon. The DGLAP evolution equations are
given by
∂f q/p (x, Q 2 )
=
∂ logQ 2
∫ 1
= α s(Q 2 )
2π x
∂f g/p (x, Q 2 )
=
∂ logQ 2
= α s(Q 2 )
2π
∫ 1
x
dy
y
dy
y
[ ( )
( )]
x x
f q/p (y, Q 2 )P qq + f g/p (y, Q 2 )P qg ,
y
y
[ ( )
( )]
x x
f q/p (y, Q 2 )P gq + f g/p (y, Q 2 )P gg ,
y
y
(1.16)
(1.17)
where q and g denote quark and gluon density function respectively
( )
and P ij are splitting
functions of a parton i to parton j with the momentum fraction shown in figure 1.7.
Both DGLAP equations assume massles partons and are hence only valid for gluons and
light quarks (u, d and s). The splitting functions are given in leading order by
P qq (z) = 4 ( ) 1 + z
2
+ 2δ(1 − z), (1.18)
3 (1 − z) +
P qg (z) = 1 (
z 2 + (1 − z) 2) , (1.19)
2
P gq (z) = 4 ( ) 1 + (1 − z)
2
, (1.20)
3 z
(
z
P gg (z) = 6 + 1 − z ) ( 11
+ z(1 − z) +
(1 − z) + z
2 − n )
f
δ(1 − z), (1.21)
3
where the notation (F(x)) + defines a distribution such that for any sufficiently regular
test function f(x),
∫ 1
0
dxf(x) (F(x)) +
=
∫ 1
0
x
y
dx (f(x) − f(1))F(x). (1.22)