10 - H1 - Desy

4 Theoretical framework
charged W ± boson. Figure 1.1 presents the simplest NC and CC Feymann diagrams with
a standard notation for the four momenta of the given particles provided in brackets. The
final state consists of hadronic system X and the scattered lepton being the electron in
case of NC interactions (ep → eX) and neutrino for CC interactions (ep → ν e X).
a) a) b) b)
a) b)
e(k) e(k ′ )
e(k) ν e (k ′ )
γ, Z 0 (q)
W(q)
p(P)
X
X
p(P)
X
X
Figure 1.1: Lowest order Feymann diagrams for the neutral current (a) and the charged
current (b) electron - proton scattering process.
For a fixed centre of mass energy √ s the kinematics of the reaction is fully given by
the four-momenta of the scattered particles. If k and k ′ denote the four-momenta of the
scattered lepton, P corresponds to the four-momentum of the incoming proton and both
electron and proton masses can be neglected, s can be expressed by
s = (k + P) 2 = 4E e E p , (1.3)
where E e and E p denote the energy of the electron and proton beams.
The four-momentum transfer q = k − k ′ defines the invariant mass of the virtual boson
√
q2 . Since q 2 has a negative value, the commonly used variable is a negative fourmomentum
transfer squared
Q 2 = −q 2 = −(k − k ′ ) 2 . (1.4)
The Q 2 is also referred to as a virtuality of the exchanged boson and in case of Q 2 ≈ 0
the exchanged boson is a quasi-real or on mass-shell particle.
The relative energy transfer of the electron in the proton rest frame is described by the
inelasticity y defined as
y = q · P
k · P . (1.5)
The Björken scaling variable x is defined as
x =
Q2
2P · q . (1.6)
If the struck parton is colinear to the proton, the Björken scaling variable x represents
the fraction of the four-momentum of the proton carried by the struck quark. Both x and
y take values between zero and one.