Physics Letters B 800 (2020) 135100
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Five-loop static contribution to the gravitational interaction potential
of two point masses
J. Blümlein
∗
, A. Maier, P. Marquard
Deutsches Elektronen–Synchrotron, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received
28 February 2019
Received
in revised form 4 November 2019
Accepted
11 November 2019
Available
online 14 November 2019
Editor: G.F.
Giudice
We compute the static contribution to the gravitational interaction potential of two point masses in
the velocity-independent five-loop (and 5th post-Newtonian) approximation to the harmonic coordinates
effective action in a direct calculation. The computation is performed using effective field methods based
on Feynman diagrams in momentum-space in d = 3 − 2ε space dimensions. We also reproduce the
previous results including the 4th post-Newtonian order.
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
The interpretation of the signals detected in gravitational wave
interferometers like LIGO and VIRGO [1]requires a very accurate
knowledge of the binary dynamics of large coalescing masses. This
also applies to the planned projects like INDIGO, LISA and the Ein-
stein
telescope [2]. The increasing improvements of the detector
sensitivity requires highly precise theoretical predictions.
Different
approaches are used as the effective one-body formal-
ism
[3–8], numerical relativity [9–11], the self-force formalism [12,
13],
the post-Newtonian (PN) [14–51] and post-Minkowskian ap-
proach
[6,7,48,52–62] and effective field theory methods [63,64];
for surveys see Refs. [65–70].
In
this letter we calculate a first contribution to the fifth post-
Newtonian
approximation: the five-loop static gravitational interac-
tion
potential between two non-spinning point masses. The two-
particle
force receives a series of different higher order corrections,
which can be parameterized by a small parameter , cf. Ref. [65],
F = F
N
+
∞
k=1
2k
F
kPN
+
4
F
SO
+
4
F
QO
+
5
F
RR
+
6
F
1PNSO
+
6
F
1PNQ O
+
6
F
OO
+
6
F
SS
+
6
F
TO
+... (1)
Here F
N
denotes Newton’s force [71], F
kP N
is the kth post-
Newtonian
force, F
RR
the 2.5 PN radiation reaction force, F
SO
and F
1PNSO
etc. are the spin-orbit coupling force and their post-
Newtonian
corrections, F
QO
and F
1PNQO
etc. are the quadrupole-
orbit
coupling force and its post-Newtonian corrections. F
OO
, F
SS
*
Corresponding author.
E-mail
address: Johannes.Bluemlein@desy.de (J. Blümlein).
and F
TO
denote the octupole-orbit coupling force, the spin-spin
coupling force, and tidal-orbit coupling force, respectively.
1
We will
concentrate here on the post-Newtonian corrections of the attrac-
tion
of two spinless masses in the following, using non-relativistic
gravitational fields obtained by a temporal Kaluza-Klein reduction
[99] followed by a Weyl rescaling [100]. The corresponding action
has been derived in Ref. [36]. In the representation time derivatives
of arbitrary order occur, which also introduce higher derivatives of
the accelerations a
1(2)
. Since the Lagrange density of General Rela-
tivity
is of second order, these terms shall be eliminated by adding
suitable double (multiple)-zero terms [23]. One aims at the terms
of the order
∼
G
k
N
r
k
m
l
1
m
k+1−l
2
, l ∈[1, k], (2)
where (k −1) labels the kth post-Newtonian approximation. Here
G
N
denotes Newton’s constant, r =|r| the distance of the two
masses and m
1(2)
are the two point masses.
We calculate the contribution to the fifth post-Newtonian ap-
proximation
in the static limit, i.e. leaving the velocity-dependent
contributions for a later work. The virial theorem [101] relates
m
1
v
2
1
+m
2
v
2
2
∼
G
N
r
m
1
m
2
on temporal average, where v
1(2)
denote
the velocities of the two point masses. Therefore velocity terms
have to be considered at this order in general. This also applies to
higher derivatives of the velocities, which can finally be mapped
to terms ∝ (G
k
N
/r
k
)v
l
i
by applying the equation of motion.
1
Radiation and spin effects are discussed in Refs. [22,72–81]and Refs. [82–98],
respectively. Radiative multipoles have been computed in various earlier work, re-
viewed
in e.g. Refs. [65–70].
https://doi.org/10.1016/j.physletb.2019.135100
0370-2693/
© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.