In pair production , a photon creates an electron positron pair. In the process of photons scattering in air (e.g. in lightning discharges), the most important interaction is the scattering of photons at the nuclei of atoms or molecules . The full quantum mechanical process of pair production can be described by the quadruply differential cross section given here:
d
4
σ
=
Z
2
α
fine
3
c
2
(
2
π
)
2
ℏ
|
p
+
|
|
p
−
|
d
E
+
ω
3
d
Ω
+
d
Ω
−
d
Φ
|
q
|
4
×
×
[
−
p
−
2
sin
2
Θ
−
(
E
−
−
c
|
p
−
|
cos
Θ
−
)
2
(
4
E
+
2
−
c
2
q
2
)
−
p
+
2
sin
2
Θ
+
(
E
+
−
c
|
p
+
|
cos
Θ
+
)
2
(
4
E
−
2
−
c
2
q
2
)
+
2
ℏ
2
ω
2
p
+
2
sin
2
Θ
+
+
p
−
2
sin
2
Θ
−
(
E
+
−
c
|
p
+
|
cos
Θ
+
)
(
E
−
−
c
|
p
−
|
cos
Θ
−
)
+
2
|
p
+
|
|
p
−
|
sin
Θ
+
sin
Θ
−
cos
Φ
(
E
+
−
c
|
p
+
|
cos
Θ
+
)
(
E
−
−
c
|
p
−
|
cos
Θ
−
)
(
2
E
+
2
+
2
E
−
2
−
c
2
q
2
)
]
.
{\displaystyle {\begin{aligned}d^{4}\sigma &={\frac {Z^{2}\alpha _{\textrm {fine}}^{3}c^{2}}{(2\pi )^{2}\hbar }}|\mathbf {p} _{+}||\mathbf {p} _{-}|{\frac {dE_{+}}{\omega ^{3}}}{\frac {d\Omega _{+}d\Omega _{-}d\Phi }{|\mathbf {q} |^{4}}}\times \\&\times \left.\\\end{aligned}}}
with
d
Ω
+
=
sin
Θ
+
d
Θ
+
,
d
Ω
−
=
sin
Θ
−
d
Θ
−
.
{\displaystyle {\begin{aligned}d\Omega _{+}&=\sin \Theta _{+}\ d\Theta _{+},\\d\Omega _{-}&=\sin \Theta _{-}\ d\Theta _{-}.\end{aligned}}}
This expression can be derived by using a quantum mechanical symmetry between pair production and Bremsstrahlung .
Z
{\displaystyle Z}
atomic number ,
α
f
i
n
e
≈
1
/
137
{\displaystyle \alpha _{fine}\approx 1/137}
fine structure constant ,
ℏ
{\displaystyle \hbar }
reduced Planck constant and
c
{\displaystyle c}
speed of light . The kinetic energies
E
k
i
n
,
+
/
−
{\displaystyle E_{kin,+/-}}
E
+
,
−
{\displaystyle E_{+,-}}
momenta
p
+
,
−
{\displaystyle \mathbf {p} _{+,-}}
E
+
,
−
=
E
k
i
n
,
+
/
−
+
m
e
c
2
=
m
e
2
c
4
+
p
+
,
−
2
c
2
.
{\displaystyle E_{+,-}=E_{kin,+/-}+m_{e}c^{2}={\sqrt {m_{e}^{2}c^{4}+\mathbf {p} _{+,-}^{2}c^{2}}}.}
Conservation of energy yields
ℏ
ω
=
E
+
+
E
−
.
{\displaystyle \hbar \omega =E_{+}+E_{-}.}
The momentum
q
{\displaystyle \mathbf {q} }
virtual photon between incident photon and nucleus is:
−
q
2
=
−
|
p
+
|
2
−
|
p
−
|
2
−
(
ℏ
c
ω
)
2
+
2
|
p
+
|
ℏ
c
ω
cos
Θ
+
+
2
|
p
−
|
ℏ
c
ω
cos
Θ
−
−
2
|
p
+
|
|
p
−
|
(
cos
Θ
+
cos
Θ
−
+
sin
Θ
+
sin
Θ
−
cos
Φ
)
,
{\displaystyle {\begin{aligned}-\mathbf {q} ^{2}&=-|\mathbf {p} _{+}|^{2}-|\mathbf {p} _{-}|^{2}-\left({\frac {\hbar }{c}}\omega \right)^{2}+2|\mathbf {p} _{+}|{\frac {\hbar }{c}}\omega \cos \Theta _{+}+2|\mathbf {p} _{-}|{\frac {\hbar }{c}}\omega \cos \Theta _{-}\\&-2|\mathbf {p} _{+}||\mathbf {p} _{-}|(\cos \Theta _{+}\cos \Theta _{-}+\sin \Theta _{+}\sin \Theta _{-}\cos \Phi ),\end{aligned}}}
where the directions are given via:
Θ
+
=
∢
(
p
+
,
k
)
,
Θ
−
=
∢
(
p
−
,
k
)
,
Φ
=
Angle between the planes
(
p
+
,
k
)
and
(
p
−
,
k
)
,
{\displaystyle {\begin{aligned}\Theta _{+}&=\sphericalangle (\mathbf {p} _{+},\mathbf {k} ),\\\Theta _{-}&=\sphericalangle (\mathbf {p} _{-},\mathbf {k} ),\\\Phi &={\text{Angle between the planes }}(\mathbf {p} _{+},\mathbf {k} ){\text{ and }}(\mathbf {p} _{-},\mathbf {k} ),\end{aligned}}}
where
k
{\displaystyle \mathbf {k} }
In order to analyse the relation between the photon energy
E
+
{\displaystyle E_{+}}
Θ
+
{\displaystyle \Theta _{+}}
Ebert integrated the quadruply differential cross section over
Θ
−
{\displaystyle \Theta _{-}}
Φ
{\displaystyle \Phi }
d
2
σ
(
E
+
,
ω
,
Θ
+
)
d
E
+
d
Ω
+
=
∑
j
=
1
6
I
j
{\displaystyle {\begin{aligned}{\frac {d^{2}\sigma (E_{+},\omega ,\Theta _{+})}{dE_{+}d\Omega _{+}}}=\sum \limits _{j=1}^{6}I_{j}\end{aligned}}}
with
I
1
=
2
π
A
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
×
ln
(
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
−
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
(
Δ
1
(
p
)
+
Δ
2
(
p
)
)
+
Δ
1
(
p
)
Δ
2
(
p
)
−
(
Δ
2
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
−
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
(
Δ
1
(
p
)
−
Δ
2
(
p
)
)
+
Δ
1
(
p
)
Δ
2
(
p
)
)
×
[
−
1
−
c
Δ
2
(
p
)
p
−
(
E
+
−
c
p
+
cos
Θ
+
)
+
p
+
2
c
2
sin
2
Θ
+
(
E
+
−
c
p
+
cos
Θ
+
)
2
−
2
ℏ
2
ω
2
p
−
Δ
2
(
p
)
c
(
E
+
−
c
p
+
cos
Θ
+
)
(
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
)
]
,
I
2
=
2
π
A
c
p
−
(
E
+
−
c
p
+
cos
Θ
+
)
ln
(
E
−
+
p
−
c
E
−
−
p
−
c
)
,
I
3
=
2
π
A
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
×
ln
(
(
(
E
−
+
p
−
c
)
(
4
p
+
2
p
−
2
sin
2
Θ
+
(
E
−
−
p
−
c
)
+
(
Δ
1
(
p
)
+
Δ
2
(
p
)
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
−
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
)
)
(
(
E
−
−
p
−
c
)
(
4
p
+
2
p
−
2
sin
2
Θ
+
(
−
E
−
−
p
−
c
)
+
(
Δ
1
(
p
)
−
Δ
2
(
p
)
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
−
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
)
)
−
1
)
×
[
c
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
p
−
(
E
+
−
c
p
+
cos
Θ
+
)
+
[
(
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
)
(
E
−
3
+
E
−
p
−
c
)
+
p
−
c
(
2
(
(
Δ
1
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
)
E
−
p
−
c
+
Δ
1
(
p
)
Δ
2
(
p
)
(
3
E
−
2
+
p
−
2
c
2
)
)
]
[
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
]
−
1
+
[
−
8
p
+
2
p
−
2
m
2
c
4
sin
2
Θ
+
(
E
+
2
+
E
−
2
)
−
2
ℏ
2
ω
2
p
+
2
sin
2
Θ
+
p
−
c
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
+
2
ℏ
2
ω
2
p
−
m
2
c
3
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
]
[
(
E
+
−
c
p
+
cos
Θ
+
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
]
−
1
+
4
E
+
2
p
−
2
(
2
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
−
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
(
Δ
1
(
p
)
E
−
+
Δ
2
(
p
)
p
−
c
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
2
]
,
I
4
=
4
π
A
p
−
c
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
+
16
π
E
+
2
p
−
2
A
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
2
,
I
5
=
4
π
A
(
−
(
Δ
2
(
p
)
)
2
+
(
Δ
1
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
)
(
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
)
×
[
ℏ
2
ω
2
p
−
2
E
+
c
p
+
cos
Θ
+
[
E
−
[
2
(
Δ
2
(
p
)
)
2
(
(
Δ
2
(
p
)
)
2
−
(
Δ
1
(
p
)
)
2
)
+
8
p
+
2
p
−
2
sin
2
Θ
+
(
(
Δ
2
(
p
)
)
2
+
(
Δ
1
(
p
)
)
2
)
]
+
p
−
c
[
2
Δ
1
(
p
)
Δ
2
(
p
)
(
(
Δ
2
(
p
)
)
2
−
(
Δ
1
(
p
)
)
2
)
+
16
Δ
1
(
p
)
Δ
2
(
p
)
p
+
2
p
−
2
sin
2
Θ
+
]
]
[
(
Δ
2
(
p
)
)
2
+
4
p
+
2
p
−
2
sin
2
Θ
+
]
−
1
+
2
ℏ
2
ω
2
p
+
2
sin
2
Θ
+
(
2
Δ
1
(
p
)
Δ
2
(
p
)
p
−
c
+
2
(
Δ
2
(
p
)
)
2
E
−
+
8
p
+
2
p
−
2
sin
2
Θ
+
E
−
)
E
+
−
c
p
+
cos
Θ
+
−
[
2
E
+
2
p
−
2
{
2
(
(
Δ
2
(
p
)
)
2
−
(
Δ
1
(
p
)
)
2
)
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
8
p
+
2
p
−
2
sin
2
Θ
+
[
(
(
Δ
1
(
p
)
)
2
+
(
Δ
2
(
p
)
)
2
)
(
E
−
2
+
p
−
2
c
2
)
+
4
Δ
1
(
p
)
Δ
2
(
p
)
E
−
p
−
c
]
}
]
[
(
Δ
2
(
p
)
E
−
+
Δ
1
(
p
)
p
−
c
)
2
+
4
m
2
c
4
p
+
2
p
−
2
sin
2
Θ
+
]
−
1
−
8
p
+
2
p
−
2
sin
2
Θ
+
(
E
+
2
+
E
−
2
)
(
Δ
2
(
p
)
p
−
c
+
Δ
1
(
p
)
E
−
)
E
+
−
c
p
+
cos
Θ
+
]
,
I
6
=
−
16
π
E
−
2
p
+
2
sin
2
Θ
+
A
(
E
+
−
c
p
+
cos
Θ
+
)
2
(
−
(
Δ
2
(
p
)
)
2
+
(
Δ
1
(
p
)
)
2
−
4
p
+
2
p
−
2
sin
2
Θ
+
)
{\displaystyle {\begin{aligned}I_{1}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln \left({\frac {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}{-(\Delta _{2}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}}\right)\\&\times \left,\\I_{2}&={\frac {2\pi Ac}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}\ln \left({\frac {E_{-}+p_{-}c}{E_{-}-p_{-}c}}\right),\\I_{3}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln {\Bigg (}{\Big (}(E_{-}+p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{-}-p_{-}c)+(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)\\&-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}{\Big (}(E_{-}-p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(-E_{-}-p_{-}c)\\&+(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}^{-1}{\Bigg )}\\&\times \left}{\Big }^{-1}\\&+{\Big }{\Big }^{-1}\\&+\left.{\frac {4E_{+}^{2}p_{-}^{2}(2(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}-4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})(\Delta _{1}^{(p)}E_{-}+\Delta _{2}^{(p)}p_{-}c)}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}}\right],\\I_{4}&={\frac {4\pi Ap_{-}c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)}{(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}+{\frac {16\pi E_{+}^{2}p_{-}^{2}A(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}},\\I_{5}&={\frac {4\pi A}{(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\\&\times \left\right.\\&+p_{-}c{\Big ]}{\Big }^{-1}\\&+{\frac {2\hbar ^{2}\omega ^{2}p_{+}^{2}\sin ^{2}\Theta _{+}(2\Delta _{1}^{(p)}\Delta _{2}^{(p)}p_{-}c+2(\Delta _{2}^{(p)})^{2}E_{-}+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\\&-{\Big \}{\Big ]}{\Big }^{-1}\\&-\left.{\frac {8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{+}^{2}+E_{-}^{2})(\Delta _{2}^{(p)}p_{-}c+\Delta _{1}^{(p)}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\right],\\I_{6}&=-{\frac {16\pi E_{-}^{2}p_{+}^{2}\sin ^{2}\Theta _{+}A}{(E_{+}-cp_{+}\cos \Theta _{+})^{2}(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\end{aligned}}}
and
A
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2
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{\displaystyle {\begin{aligned}A&={\frac {Z^{2}\alpha _{fine}^{3}c^{2}}{(2\pi )^{2}\hbar }}{\frac {|\mathbf {p} _{+}||\mathbf {p} _{-}|}{\omega ^{3}}},\\\Delta _{1}^{(p)}&:=-|\mathbf {p} _{+}|^{2}-|\mathbf {p} _{-}|^{2}-\left({\frac {\hbar }{c}}\omega \right)+2{\frac {\hbar }{c}}\omega |\mathbf {p} _{+}|\cos \Theta _{+},\\\Delta _{2}^{(p)}&:=2{\frac {\hbar }{c}}\omega |\mathbf {p} _{i}|-2|\mathbf {p} _{+}||\mathbf {p} _{-}|\cos \Theta _{+}+2.\end{aligned}}}
This cross section can be applied in Monte Carlo simulations. An analysis of this expression shows that positrons are mainly emitted in the direction of the incident photon.
References
Ebert, U. , Angular distribution of Bremsstrahlung photons and of positrons for calculations of terrestrial gamma-ray flashes and positron beams, Atmos. Res. (2014), vol. 135-136, pp. 432-465
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is the 
the 
the 
the 
of the positron and electron relate to their total energies 
and 
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of the 
is the momentum of the incident photon.
and the emission angle 
between photon and positron, Köhn and 
and 
. The double differential cross section is:
![{\displaystyle {\begin{aligned}I_{1}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln \left({\frac {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}{-(\Delta _{2}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}-{\sqrt {(\Delta _{2}^{(p)})^{2}+4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})+\Delta _{1}^{(p)}\Delta _{2}^{(p)}}}\right)\\&\times \left,\\I_{2}&={\frac {2\pi Ac}{p_{-}(E_{+}-cp_{+}\cos \Theta _{+})}}\ln \left({\frac {E_{-}+p_{-}c}{E_{-}-p_{-}c}}\right),\\I_{3}&={\frac {2\pi A}{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}}\\&\times \ln {\Bigg (}{\Big (}(E_{-}+p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{-}-p_{-}c)+(\Delta _{1}^{(p)}+\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)\\&-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}{\Big (}(E_{-}-p_{-}c)(4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(-E_{-}-p_{-}c)\\&+(\Delta _{1}^{(p)}-\Delta _{2}^{(p)})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)-{\sqrt {(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}})){\Big )}^{-1}{\Bigg )}\\&\times \left}{\Big }^{-1}\\&+{\Big }{\Big }^{-1}\\&+\left.{\frac {4E_{+}^{2}p_{-}^{2}(2(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}-4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})(\Delta _{1}^{(p)}E_{-}+\Delta _{2}^{(p)}p_{-}c)}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}}\right],\\I_{4}&={\frac {4\pi Ap_{-}c(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)}{(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}}}+{\frac {16\pi E_{+}^{2}p_{-}^{2}A(\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}}{((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})^{2}}},\\I_{5}&={\frac {4\pi A}{(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})((\Delta _{2}^{(p)}E_{-}+\Delta _{1}^{(p)}p_{-}c)^{2}+4m^{2}c^{4}p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\\&\times \left\right.\\&+p_{-}c{\Big ]}{\Big }^{-1}\\&+{\frac {2\hbar ^{2}\omega ^{2}p_{+}^{2}\sin ^{2}\Theta _{+}(2\Delta _{1}^{(p)}\Delta _{2}^{(p)}p_{-}c+2(\Delta _{2}^{(p)})^{2}E_{-}+8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\\&-{\Big \}{\Big ]}{\Big }^{-1}\\&-\left.{\frac {8p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+}(E_{+}^{2}+E_{-}^{2})(\Delta _{2}^{(p)}p_{-}c+\Delta _{1}^{(p)}E_{-})}{E_{+}-cp_{+}\cos \Theta _{+}}}\right],\\I_{6}&=-{\frac {16\pi E_{-}^{2}p_{+}^{2}\sin ^{2}\Theta _{+}A}{(E_{+}-cp_{+}\cos \Theta _{+})^{2}(-(\Delta _{2}^{(p)})^{2}+(\Delta _{1}^{(p)})^{2}-4p_{+}^{2}p_{-}^{2}\sin ^{2}\Theta _{+})}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89082d84daae3b60c3793714c522e974deaa7739)
