A Classic Thesis Style - Johannes Gutenberg-Universität Mainz

8 introduction
there was almost no information for researchers who wanted to make
use of these new type of detectors.
Appendix A presents a detailed derivation of the general structure
of kaon electroproduction cross-section. This departure from standard
schematic presentations typically encountered in experimental thesis
is the result of my personal interest in theoretical physics and it
attempts to provide a useful reference for a better understanding of
the presented results and a unified source for future students. It is the
hope of this author that the effort to compile and digest the material
used for this derivation had resulted in a pedagogical presentation at
a level easily understandable for experimentalists.
1.3 kinematics of e + p → e ′ +k + + y and cross-section structure
In the standard terminology used in inelastic scattering, hyperon
electroproduction off the proton, where the outgoing electron and kaon
are detected in coincidence and the produced Λ or Σ 0 is reconstructed,
is called an exclusive reaction [7].
Full kinematical characterization of the reaction is given by the
following five four momenta:
• pe = (Ee, �pe) for the incident electron.
• p ′ e = (E ′ e, �p ′ e) for the scattered electron.
• pp = (mp,�0) for the proton target.
• pK = (Ek, �pk) for the produced kaon.
• pY = (EY, �pY) for the unobserved hyperon.
where the components of the four vectors have been given in the lab
frame where the proton is at rest.
Energy-momentum conservation for this reaction expressed by pe +
pp = p ′ e + pY + pK imposes 4 conditions upon the 20 kinematical
variables. Since all particles are on their mass shell (p 2 = m 2 ), five
extra degrees of freedom are eliminated. Electron and proton initial
states are under the experimentalist control leaving finally only 20 −
4 − 5 − 6 = 5 variables free. The scattered electron transfers energy and
momentum to the proton by interchanging a virtual photon with four
momentum qµ = peµ − p′ eµ = (ω, �q). The Lorentz invariant formed
by squaring this four vector is usually called in the literature the four
momentum transfer or simply the momentum transfer. It is easy to
prove that for relativistic electrons in the lab frame we have:
Q 2 = −(qµ) 2 ≈ 4EeE ′ 2 θe
e sin
2
(1.1)