A Classic Thesis Style - Johannes Gutenberg-Universität Mainz

126 theoretical background
taken into account and actually has the convenient effect of removing
the VT denominator of the transition rate per unit volume since:
(2π) 4 δ 4 (0)(2π) 4 δ 4 (pΛ + pK − pp − q) → VT(2π) 4 δ 4 (pΛ + pK − pp − q)
(A.22)
Next, we divide the transition rate per unit volume by the flux of
incident particles |Jinc| and by the number of target particles per unit
volume; which is 1/V according to our normalization. Finally, to get a
physical cross section, we must sum over a given group of final states.
The number of states for the final electron in a big box with periodic
boundary conditions is dN = V
(2π) 3 d 3 p ′ e and similarly for K + and Λ.
�
dσ =
V 3 d3p ′ e
(2π) 3
d3pK (2π) 3
d3pΛ (2π) 3
V
|Jinc| wfi
(A.23)
Along the electron direction, the incident flux times the remaining
volume factor is given by (me = 0):
JincV = Vrel
V ≈ 1 (A.24)
V
what allow us to write:
dσ =
or
where
and
� memP
EeEp
dσ = 4α2
q 4
|Mfi| 2 med 3 p ′ e
(2π) 3 E ′
d 3 p ′ e
2E ′ e
W µν = mpmΛ
(2π) 3
Lµν =
(2me) 2
2
1
d3pK (2π) 3EK LµνW
EeEp
µν
mΛd3pΛ (2π) 3 (2π)
EΛ
4 δ 4 (pΛ + pK − q − pP − q)
(A.25)
(A.26)
�
d3�pK d
2EK
3�pΛ δ
2EΛ
4 (pk + pΛ − q − pp) 1 �
µ +ν
J J
2
(A.27)
�
jµj + ν = 2(p ′ eµ peν + p′ 2
eνpeµ ) − gµνQ
(A.28)
Here we have already assumed that the electron beam and proton
target are unpolarized, and that we do not measure the polarization
of the outgoing particles. The summatory symbol, then, stands for
averaging over initial state polarizations and summing over final states.
In the final form of Lµν we have used standard trace techniques
for spin summation [94]. These two expression are manifestly Lorentz
second rank tensors. We follow the common practice to evaluate the