Download Thesis in Pdf Format - Theoretical Nuclear Physics and ...

6 Chapter 2. Reaction Observables and Kinematics
transverse polarization of the absorbed photon, respectively. This separation is well
known as the Rosenbluth decomposition [24]. Exclusive reactions from composite
systems in which one or more particles are detected in coincidence with the scattered
electron, allow to extract additional information about the target system. At
present, the availability of high-duty electron facilities make these coincidence experiments
involving the analysis of polarization degrees of freedom much easier than
in previous times.
In discussing A + e → B + e ′ + p processes, one faces the possibility of polarizing
the incident and scattered wave, as well as the hadrons in the initial and final
channel. In this work we will follow the conventions for the A(⃗e, ⃗e ′ ⃗p)B kinematics
and observables introduced by Donnelly and Raskin in Refs. [25, 26].
The four-momenta of the incident and scattered electron are labeled as K µ (ɛ, ⃗ k)
and K ′µ (ɛ ′ , k ⃗′ ). The electron momenta ⃗ k and k ⃗′ define the scattering plane. The
four-momentum transfer is given by q µ = K µ − K ′µ = K µ A−1 + Kµ f − Kµ A , where
K µ A and Kµ A−1
are the four-momenta of the target and residual nucleus, respectively,
while K µ f
is the four-momentum of the ejected nucleon. Also, qµ = (ω, ⃗q), where
the three-momentum transfer ⃗q = ⃗ k − k ⃗′ = ⃗ k A−1 + ⃗ k f − ⃗ k A , and the energy transfer
ω = ɛ − ɛ ′ = E A−1 + E f − E A , are defined in the standard manner. The xyz
coordinate system is chosen such that the z-axis lies along the momentum transfer
⃗q, the y-axis lies along ⃗ k × k ⃗′ and the x-axis lies in the scattering plane; the reaction
plane is then defined by ⃗ k f and ⃗q, as in Fig. 2.1.
We now discuss processes in which a polarized electron with helicity h impinges
on a nucleus and induces the knockout of a single nucleon, leaving the residual
nucleus in a specific discrete state. The Feynman diagram corresponding to this
process is shown in Fig. 2.2. The Bjorken-Drell convention [27] for the γ matrices
and Dirac spinors is followed. Accordingly, the normalization condition for the Dirac
plane waves, characterized by a four-momentum K µ and spin-state S µ , is
ū(K µ , S µ )u(K µ , S µ ) = 1 . (2.1)
The electron charge is denoted by −e, and the virtual photon is represented by the
propagator D F (Q) µν = −g µν /Q 2 , with Q 2 ≡ −q µ q µ ≥ 0.
The differential scattering cross section in the laboratory frame can then be
written as [25, 26, 27] :
dσ = 1 β
m e
ɛ
∑
if
|M fi | 2 m e d 3 k ⃗′
M A−1 d 3 ⃗ kA−1 M f d 3 ⃗ kf
ɛ ′ (2π) 3 E A−1 (2π) 3 E f (2π) 3
×(2π) 4 δ (4) (K µ + K µ A − K′µ − K µ A−1 − Kµ f
), (2.2)
where β = | ⃗ k|/ɛ = |⃗v e | and where ∑ corresponds to the appropriate average over
initial states and sum over final states as will be discussed below. The corresponding