118
5 General-Relativistic Matrix Kinetic Theory
where
C
e
γ,rr :=
1
2
d 3 k 2
(2π) 3 2k 2
d 3 p 1
(2π) 3 2E p1
d 3 p 2
(2π) 3 2E p2
(2π)
4 δ
(4) (k + p 1 − k 2 − p 2 )
×
f s2s
2
f r2r
2
f
◦
s
1 s1 f
◦
r r A(γ r2 , e s2 → γ r , e s1 ) A
∗ (γ r
2
, e s
2
→ γ r , e s
1
)
− f s1s
1
f rr f
◦
s
2 s2 f
◦
r
2 r2 A(γ r , e s1 → γ r2 , e s2 ) A
∗ (γ r , e s
1
→ γ r
2
, e s
2
)
, (5.10.10)
C
γ
e,ss :=
1
2
d 3 p 2
(2π) 3 2E p2
d 3 k 1
(2π) 3 2k 1
d 3 k 2
(2π) 3 2k 2
(2π)
4 δ
(4) ( p + k 1 − p 2 − k 2 )
×
f r2r
2
f s2s
2
f
◦
r
1 r1 f
◦
s s A(γ r2 , e s2 → γ r1 , e s ) A
∗ (γ r
2
, e s
2
→ γ r
1
, e s )
− f r1r
1
f ss f
◦
r
2 r2 f
◦
s
2 s2 A(γ r1 , e s → γ r2 , e s2 ) A
∗ (γ r
1
, e s → γ r
2
, e s
2
)
, (5.10.11)
C
e
e,ss :=
1
4
d 3 p 2
(2π) 3 2E p2
d 3 p 3
(2π) 3 2E p3
d 3 p 4
(2π) 3 2E p4
(2π)
4 δ
(4) ( p + p 2 − p 3 − p 4 )
×
f s3s
3
f s4s
4
f
◦
s
2 s2 f
◦
s s A(e s3 , e s4 → e s , e s2 ) A
∗ (e s
3
, e s
4
→ e s , e s
2
)
− f s2s
2
f ss f
◦
s
3 s3 f
◦
s
4 s4 A(e s , e s2 → e s3 , e s4 ) A
∗ (e s , e s
2
→ e s
3
, e s
4
)
, (5.10.12)
C
p
e,ss :=
1
2
d 3 p 2
(2π) 3 2E p2
d 3 q 1
(2π) 3 2E q1
d 3 q 2
(2π) 3 2E q2
(2π)
4 δ
(4) ( p + q 1 − p 2 − q 2 )
×
f t2t
2
f s2s
2
f
◦
t
1 t1 f
◦
s s A(e s2 , p t2 → e s , p t1 ) A
∗ (e s
2
, p t
2
→ e s , p t
1
)
− f t1t
1
f ss f
◦
t
2 t2 f
◦
s
2 s2 A(e s , p t1 → e s2 , p t2 ) A
∗ (e s , p t
1
→ e s
2
, p t
2
)
, (5.10.13)
while C
p
γ , C
γ
p , C
p
p and C
e
p are the same as C
e
γ , C
γ
e , C
e
e and C
p
e , respectively, but with
the electrons and protons interchanged. We also remind that
f
◦
rr := δ rr + f rr ,
f
◦
ss := δ ss − f ss ,
f
◦
tt := δ tt − f tt . (5.10.14)
Invoking the polarization vectors
a
r ≡
a
r (
k) for photons and the wave-functions u s ≡
u s (
p) and u t ≡ u t ( q) for electrons and protons, respectively, that were introduced
in Sect. 5.8, the involved scattering amplitudes to lowest order read
iA(γ r2 , e s2 → γ r1 , e s1 ) :=
+
=
i
2
e
2
r 1
a
r 2
b ¯
u s 1
γ
a
( p 1 +
k 1 + m e ) γ
b
k 1 · p 1
−
γ
b
( p 2 −
k 1 + m e ) γ
a
k 1 · p 2
u s 2
≡
i
2
e
2
r 1
a
r 2
b ¯
u s 1
γ
a
k 1 − 2 p
a
1
γ
b
k 1 · p 1
+
γ
b
k 1 γ
a
+ 2 p
a
2
k 1 · p 2
u s 2
≡ ie
2
r 1
a
r 2
b ¯
u s 1 A
ab
(k 1 , k 2 , p 1 , p 2 ) u s 2 ,
(5.10.15)
5 General-Relativistic Matrix Kinetic Theory
where
C
e
γ,rr :=
1
2
d 3 k 2
(2π) 3 2k 2
d 3 p 1
(2π) 3 2E p1
d 3 p 2
(2π) 3 2E p2
(2π)
4 δ
(4) (k + p 1 − k 2 − p 2 )
×
f s2s
2
f r2r
2
f
◦
s
1 s1 f
◦
r r A(γ r2 , e s2 → γ r , e s1 ) A
∗ (γ r
2
, e s
2
→ γ r , e s
1
)
− f s1s
1
f rr f
◦
s
2 s2 f
◦
r
2 r2 A(γ r , e s1 → γ r2 , e s2 ) A
∗ (γ r , e s
1
→ γ r
2
, e s
2
)
, (5.10.10)
C
γ
e,ss :=
1
2
d 3 p 2
(2π) 3 2E p2
d 3 k 1
(2π) 3 2k 1
d 3 k 2
(2π) 3 2k 2
(2π)
4 δ
(4) ( p + k 1 − p 2 − k 2 )
×
f r2r
2
f s2s
2
f
◦
r
1 r1 f
◦
s s A(γ r2 , e s2 → γ r1 , e s ) A
∗ (γ r
2
, e s
2
→ γ r
1
, e s )
− f r1r
1
f ss f
◦
r
2 r2 f
◦
s
2 s2 A(γ r1 , e s → γ r2 , e s2 ) A
∗ (γ r
1
, e s → γ r
2
, e s
2
)
, (5.10.11)
C
e
e,ss :=
1
4
d 3 p 2
(2π) 3 2E p2
d 3 p 3
(2π) 3 2E p3
d 3 p 4
(2π) 3 2E p4
(2π)
4 δ
(4) ( p + p 2 − p 3 − p 4 )
×
f s3s
3
f s4s
4
f
◦
s
2 s2 f
◦
s s A(e s3 , e s4 → e s , e s2 ) A
∗ (e s
3
, e s
4
→ e s , e s
2
)
− f s2s
2
f ss f
◦
s
3 s3 f
◦
s
4 s4 A(e s , e s2 → e s3 , e s4 ) A
∗ (e s , e s
2
→ e s
3
, e s
4
)
, (5.10.12)
C
p
e,ss :=
1
2
d 3 p 2
(2π) 3 2E p2
d 3 q 1
(2π) 3 2E q1
d 3 q 2
(2π) 3 2E q2
(2π)
4 δ
(4) ( p + q 1 − p 2 − q 2 )
×
f t2t
2
f s2s
2
f
◦
t
1 t1 f
◦
s s A(e s2 , p t2 → e s , p t1 ) A
∗ (e s
2
, p t
2
→ e s , p t
1
)
− f t1t
1
f ss f
◦
t
2 t2 f
◦
s
2 s2 A(e s , p t1 → e s2 , p t2 ) A
∗ (e s , p t
1
→ e s
2
, p t
2
)
, (5.10.13)
while C
p
γ , C
γ
p , C
p
p and C
e
p are the same as C
e
γ , C
γ
e , C
e
e and C
p
e , respectively, but with
the electrons and protons interchanged. We also remind that
f
◦
rr := δ rr + f rr ,
f
◦
ss := δ ss − f ss ,
f
◦
tt := δ tt − f tt . (5.10.14)
Invoking the polarization vectors
a
r ≡
a
r (
k) for photons and the wave-functions u s ≡
u s (
p) and u t ≡ u t ( q) for electrons and protons, respectively, that were introduced
in Sect. 5.8, the involved scattering amplitudes to lowest order read
iA(γ r2 , e s2 → γ r1 , e s1 ) :=
+
=
i
2
e
2
r 1
a
r 2
b ¯
u s 1
γ
a
( p 1 +
k 1 + m e ) γ
b
k 1 · p 1
−
γ
b
( p 2 −
k 1 + m e ) γ
a
k 1 · p 2
u s 2
≡
i
2
e
2
r 1
a
r 2
b ¯
u s 1
γ
a
k 1 − 2 p
a
1
γ
b
k 1 · p 1
+
γ
b
k 1 γ
a
+ 2 p
a
2
k 1 · p 2
u s 2
≡ ie
2
r 1
a
r 2
b ¯
u s 1 A
ab
(k 1 , k 2 , p 1 , p 2 ) u s 2 ,
(5.10.15)