1
Amplitude in Different Regions: Supplementary Material for Scattering Amplitudes, the Tail Effect,
and Conservative Binary Dynamics at
O
(
G
4
)
by Z. Bern, J. Parra-Martinez, R. Roiban, M. S. Ruf.
C.-H. Shen, M P. Solon, and M. Zeng.
In this appendix we present the scattering amplitudes in the (ppp) and (prr) regions separately. The contribution
from the (ppp) region already appeared in Ref. [31] and is given by
M
ppp
4
(
q
) =
G
4
M
7
ν
2
|
q
|
π
2
2
2
(
q
2
̃
μ
2
)
−
3
[
M
p
4
+
ν
(
M
t
4
+
M
π
2
4
+
M
rem
,
ppp
4
)]
(1)
+
∫
`
̃
I
4
r,
1
Z
1
Z
2
Z
3
+
∫
`
̃
I
2
r,
1
̃
I
r,
2
Z
1
Z
2
+
∫
`
̃
I
r,
1
̃
I
r,
3
Z
1
+
∫
`
̃
I
2
r,
2
Z
1
,
where
M
p
4
,
M
π
2
4
,
M
t
4
and the iteration integrals are as given in the main text in Eq. (3), and
M
π
2
4
+
M
rem
,
ppp
4
=
M
f
4
defined in Eq. (6) of Ref. [31]. The new result from the (prr) region is
M
prr
4
(
q
) =
G
4
M
7
ν
3
|
q
|
π
2
2
6
p
−
4
∞
(
q
2
̃
μ
2
)
−
3
(
−
M
t
4
+
M
rem
,
prr
4
)
.
(2)
The remainder functions in both regions are given by
M
rem
,x
4
=
r
x
8
+
r
x
9
log
(
σ
+1
2
)
+
r
x
10
arccosh(
σ
)
√
σ
2
−
1
+
r
x
11
log(
σ
) +
r
x
12
log
2
(
σ
+1
2
)
+
r
x
13
arccosh(
σ
)
√
σ
2
−
1
log
(
σ
+1
2
)
(3)
+
r
x
14
arccosh
2
(
σ
)
σ
2
−
1
+
r
x
15
Li
2
(
1
−
σ
2
)
+
r
x
16
Li
2
(
1
−
σ
1+
σ
)
+
r
x
17
1
√
σ
2
−
1
[
Li
2
(
−
√
σ
−
1
σ
+1
)
−
Li
2
(
√
σ
−
1
σ
+1
)]
+
r
x
18
1
√
σ
2
−
1
F
(
σ
)
,
where the relevant coefficients
r
x
i
in each region,
x
= ppp, prr, are given in Tables II and III respectively in terms of
the polynomials
g
i
in Table I of the paper. The transcendental function
F
(
σ
) in Eq. (3) above is defined as
F
(
σ
) = Li
2
(
1
−
σ
−
√
σ
2
−
1
)
−
Li
2
(
1
−
σ
+
√
σ
2
−
1
)
+ 3Li
2
(
√
σ
−
1
σ
+1
)
−
3Li
2
(
−
√
σ
−
1
σ
+1
)
+ 2 log
(
σ
+1
2
)
arccosh(
σ
)
,
(4)
and its coefficients cancel when both regions are combined.
r
ppp
8
=
1
144
(
σ
2
−
1)
2
σ
7
(
−
45 + 207
σ
2
−
1471
σ
4
+ 13349
σ
6
−
37566
σ
7
+ 104753
σ
8
−
12312
σ
9
−
102759
σ
10
−
105498
σ
11
+ 134745
σ
12
+ 83844
σ
13
−
101979
σ
14
+ 13644
σ
15
+ 10800
σ
16
)
r
ppp
9
=
1
4(
σ
2
−
1)
(
1759
−
4768
σ
+ 3407
σ
2
−
1316
σ
3
+ 957
σ
4
−
672
σ
5
+ 341
σ
6
+ 100
σ
7
)
r
ppp
10
=
1
24(
σ
2
−
1)
2
(
1237 + 7959
σ
−
25183
σ
2
+ 12915
σ
3
+ 18102
σ
4
−
12105
σ
5
−
9572
σ
6
+ 2973
σ
7
+ 5816
σ
8
−
2046
σ
9
)
r
ppp
11
=
2
σ
(
−
852
−
283
σ
2
−
140
σ
4
+ 75
σ
6
)
3(
σ
2
−
1)
r
ppp
12
= 4
g
1
−
7
g
2
−
3
4
g
4
r
ppp
13
= 0
r
ppp
14
=
−
σ
2
(
−
3 + 2
σ
2
)
2
8(
σ
2
−
1)
2
g
3
+ 2(
σ
2
−
1)
g
2
r
ppp
15
= 24
g
1
−
14
g
2
+ 2
g
3
−
3
2
g
4
r
ppp
16
=
−
g
4
r
ppp
17
=
−
σ
(
−
3 + 2
σ
2
)
σ
2
−
1
(8
g
1
−
4
g
2
)
r
ppp
18
=
σ
(
−
3 + 2
σ
2
)
2(
σ
2
−
1)
g
3
TABLE II. Functions specifying the amplitude in the ppp
region.
r
prr
8
=
1
144
σ
7
(
σ
2
−
1)
2
(
−
45 + 207
σ
2
−
1471
σ
4
+ 13349
σ
6
−
38704
σ
7
+ 24095
σ
8
−
52042
σ
9
+ 72647
σ
10
+ 55208
σ
11
−
78841
σ
12
−
17346
σ
13
+ 31259
σ
14
−
5004
σ
15
−
3600
σ
16
)
r
prr
9
=
1
12 (
σ
2
−
1)
(
−
4061 + 34464
σ
−
9133
σ
2
+ 9860
σ
3
+ 2025
σ
4
+ 5344
σ
5
−
1023
σ
6
−
900
σ
7
)
r
prr
10
=
1
24(
σ
2
−
1)
2
(
−
1237 + 5853
σ
−
16865
σ
2
+ 15653
σ
3
+ 3018
σ
4
−
9011
σ
5
+ 5540
σ
6
−
3165
σ
7
−
56
σ
8
+ 366
σ
9
)
r
prr
11
=
2
σ
(
−
852
−
283
σ
2
−
140
σ
4
+ 75
σ
6
)
3(
σ
2
−
1)
r
prr
12
=
−
4
g
1
+ 13
g
2
+
g
3
+
1
4
g
4
r
prr
13
=
−
8
σ
(
−
3 + 2
σ
2
)
(
σ
2
−
1)
g
1
r
prr
14
=
−
σ
2
(
−
3 + 2
σ
2
)
2
8(
σ
2
−
1)
2
g
3
−
2(
σ
2
−
1)
g
2
r
prr
15
=
−
40
g
1
+ 10
g
2
−
2
g
3
+
1
2
g
4
r
prr
16
=
−
g
4
r
prr
17
= 0
r
prr
18
=
−
σ
(
−
3 + 2
σ
2
)
2(
σ
2
−
1)
g
3
TABLE III. Functions specifying the amplitude in the
prr region.