References - Bogoliubov Laboratory of Theoretical Physics - JINR

Tyy = (1 − β2 )[β2 (1 + sin 2 θ) − 1] − β2 sin 4 θ
1+2β2 sin 2 θ − β4 − β4 sin 4 , (3)
θ
Txx = (1 − β2 )[β 2 (1 + sin 2 θ) − 1] + β 2 sin 4 θ
1+2β 2 sin 2 θ − β 4 − β 4 sin 4 θ
. (4)
Here the axis z is aligned along the positron momentum in the c.m. frame, the axis
x lies in the reaction plane and the axis y is directed along the normal to the reaction
plane ; β = v
c , v is the positron velocity in the c.m. frame ; 1 − β2 = me c 2
E+ ,whereE+ is
the positron ( or electron ) energy in the c.m. frame ; θ is the angle between the positron
momentum and the momentum of one of the photons in the c.m. frame.
Meantime, the differential cross section of the process γγ → e + e − in the c.m. frame
has the following form [1,2]:
dσ
dΩ = r2 1 − β
0
2
β
4
� 1+2β 2 sin 2 θ − β 4 − β 4 sin 4 θ
(1 − β 2 cos 2 θ) 2
�
, (5)
where r0 = e2
me c 2 .
The “trace” of the correlation tensor of the final (e + e − ) pair is determined by the
formula:
T = Txx + Tyy + Tzz =1−
4(1− β2 )
1+2β2 sin 2 θ − β4 − β4 sin 4 . (6)
θ
In doing so, the relative fraction of the triplet states is as follows [3]:
Wt =
T +3
4 =1−
1 − β2 1+2β2 sin 2 θ − β4 − β4 sin 4 , (7)
θ
and the relative fraction of the singlet state ( total spin S =0)equals
Ws =
1 − T
4 =1− Wt =
1 − β2 1+2β2 sin 2 θ − β4 − β4 sin 4 . (8)
θ
At β