10 - H1 - Desy

1.2 Basics of electron - proton scattering 11
P qq P gq P gg P qg
Figure 1.7: Feymann diagrams for four splitting functions in the DGLAP evolution
equations.
1.2.4.2 BFKL evolution equations
The DGLAP equations neglect terms of the form log(1/x) which may become large as
x becomes small. Summation of such contributions leads to unintegrated gluon distributions
(dependent on the transverse momentum k T ), which obey the Balitsky-Fadin-
Kuraev-Lipatov (BFKL) [19, 20] approximation. In the framework of the unintegrated
gluon distribution, predictions for the measured cross sections are calculated using the
k T -factorisation theorem [21]. Cross sections are factorised into an off-shell (k T dependent)
partonic cross section and a k T -unintegrated parton distribution. The BFKL approximation
allows the summation of terms with leading powers of α s log 1 in the regime of very
x
low x and moderate Q 2 . In this approach, a strong ordering of the fractional longitudinal
momentum x i
x i ≪ x i−1 ≪ x i−2 ≪ ... ≪ x 1 (1.23)
and no ordering on the transverse momenta k T along the ladder are assumed. The resulting
BFKL evolution equation is given by
∂f g/p (x, kT 2)
∂ log(1/x) = 3α ∫ [
]
∞
s dk ′2
π k2 T
f g/p (x, k T ′2)
− f g/p(x, kT 2 )
T
0 k
T
′2 |k
T ′2 − k2 T | + f g/p(x, kT 2 √ ) . (1.24)
4k
′4
T
+ kT
4
The terms within brackets of equation 1.24 correspond to the real gluon emission and
virtual corrections, respectively. The BFKL gives the evolution of f g/p (x, kT 2 ) with respect
to small x. It can be solved for any small x and k T once it is known for a starting value.
For a fixed α s , the BFKL equation can be solved and the result is
f(x, k 2 T ) = F(x, k2 T ) ( x
x 0
) −λ
, (1.25)
where λ = 3αs 4log2. Hence, this approximation predicts that the gluon density increases
π
proportional to (1/x) −λ as x decreases.
1.2.4.3 CCFM evolution equations
Both the DGLAP and the BFKL methods only sum over one particular leading behaviour
of the evolution to obtain results. A complete (infinite order) calculation should take into
account both, the terms in log(Q 2 ) and log(1/x) and sum over them. To accomplish
this, Ciafaloni [22] and Catani, Fiorani and Marchesini [23] introduced angular ordering