Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order Christoph Dlapa1Gregor K alin1Zhengwen Liu2 1Jakob Neef3 4and Rafael A. Porto1 1Deutsches Elektronen-Synchrotron DESY Notkestr. 85 22607 Hamburg Germany

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Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order

Christoph Dlapa,1Gregor K¨alin,1Zhengwen Liu,2, 1 Jakob Neef,3, 4 and Rafael A. Porto1

1Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

2Niels Bohr International Academy, Niels Bohr Institute,

University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark

3School of Mathematics and Statistics, University College Dublin, Belﬁeld, Dublin 4, Ireland, D04 V1W8

4Humboldt-Universit¨at zu Berlin, Zum Grossen Windkanal 2, D-12489 Berlin, Germany

We obtain the total impulse in the scattering of non-spinning binaries in general relativity at fourth

Post-Minkowskian order, i.e. O(G4), including linear, nonlinear, and hereditary radiation-reaction

eﬀects. We derive the total radiated spacetime momentum as well as the associated energy ﬂux.

The latter can be used to compute gravitational-wave observables for generic (un)bound orbits.

We employ the (‘in-in’) Schwinger-Keldysh worldline eﬀective ﬁeld theory framework in combination

with modern ‘multi-loop’ integration techniques from collider physics. The complete results are in

agreement with various partial calculations in the Post-Newtonian/Minkowskian expansion.

Introduction. Waveform models are an essential in-

gredient in data analysis and characterization of grav-

itational wave (GW) signals from compact binaries [1].

The level of accuracy plays a critical role, in particular for

future detectors such as LISA [2] and ET [3]. In order

to beneﬁt the most from the anticipated observational

reach [2–9], the modelling of GW sources must therefore

continue to develop—both through analytic methodolo-

gies [10–18] and numerical simulations [19–21]—in par-

allel with the expected increase in sensitivity with next-

generation GW interferometers.

Motivated by the Eﬀective-One-Body (EOB) for-

malism [22–27], the Boundary-to-Bound (B2B) dictio-

nary between unbound and bound observables [28–30],

and beneﬁting from powerful ‘multi-loop’ integrations

tools [31–61], signiﬁcant progress has been achieved in

recent years in our analytic understanding of (classical)

gravitational scattering in the Post-Minkowskian (PM)

expansion in powers of G(Newton’s constant); both

via eﬀective ﬁeld theory (EFT) [62–78] and amplitude-

based [79–98] methodologies. The PM regime incorpo-

rates an inﬁnite tower of Post-Newtonian (PN) correc-

tions at a given order in Gthat may increase the accuracy

of phenomenological waveform models [99, 100].

However, despite some notable exceptions [26, 73–

77, 79, 84, 97, 98, 101], the majority of the PM computa-

tions have so far impacted our knowledge of the conser-

vative sector, with potential interactions [66, 92] as well

as ‘tail eﬀects’ [67, 93] known to 4PM order. Yet, until

now, complete results had not been obtained at the same

level of accuracy. The purpose of this letter is there-

fore to report the total change of (mechanical) momen-

tum, a.k.a. the impulse, for the gravitational scattering of

non-spinning bodies—including all the hitherto unknown

linear, nonlinear and hereditary radiation-reaction dissi-

pative eﬀects—at O(G4), from which we derive the total

radiated spacetime momentum and GW energy ﬂux.

Building on pioneering developments in the PN regime

[102–109], the derivation proceeds via the EFT approach

in a PM scheme [62], extended in [76] to simultaneously

incorporate conservative and dissipative eﬀects via the

‘in-in’ Schwinger-Keldysh formalism [110–114]. As dis-

cussed in [76], the in-in impulse resembles the ‘in-out’

counterpart used in the conservative sector [66, 67], ex-

cept for its causal structure which entails the use of re-

tarded Green’s functions [76]. After adapting integration

tools to our problem, the calculation of the impulse is

mapped to a series of ‘three-loop’ mass-independent in-

tegrals. As in previous derivations [62, 63, 66, 67], the

latter are solved via the methodology of diﬀerential equa-

tions [32–38]. The relativistic two-body problem is then

reduced to obtaining the necessary boundary conditions

in the near-static limit. The boundary integrals are com-

puted using the method of regions [45], involving poten-

tial (oﬀ-shell) and radiation (on-shell) modes [102]. The

full solution is thus bootstrapped to all orders in the ve-

locities from the same type of calculations needed in the

EFT approach with PN sources [102, 109, 115–118]. As a

nontrivial check, by rewriting retarded Green’s functions

as Feynman propagators plus a reactive term [76], we

recover the value in [66, 67] for the Feynman-only (con-

servative) part. Agreement is also found in the overlap

with various PN derivations [26, 27, 119–124] and partial

PM results [98] obtained using the relations in [22].

The B2B dictionary [28–30] allows us to connect scat-

tering data to observables for bound states via analytic

continuation. However, similarly to the lack of perias-

tron advance at 3PM [28, 29, 63], the symmetries of the

problem yield a vanishing coeﬃcient for the radiated en-

ergy integrated over a period of elliptic-like motion at

4PM, trivially recovered by the B2B map. Neverthe-

less, since nonlinear radiation-reaction eﬀects do not con-

tribute to the integrated radiated energy at O(G4), we

can then derive the GW ﬂux in an adiabatic approxi-

mation [30]. This allows for the computation of radiative

observables for generic (un)bound orbits through balance

equations, as in the EOB approach [22], thus including

an inﬁnite series of velocity corrections.

arXiv:2210.05541v1 [hep-th] 11 Oct 2022

Figure 1. In-in Feynman topologies to 4PM order. The arrows

indicate the ﬂow of (retarded) time. The crosses represent the

location of the derivative in the impulse in (3). The last two

diagrams are the only ‘self-energies’ needed at 4PM [76].

The EFT in-integrand. Following the Schwinger-

Keldysh formalism [110–114] adapted to the EFT ap-

proach in [76, 104], the eﬀective action is obtained via

aclosed-time-path integral involving a doubling of the

metric perturbation (h±

µν ) as well as the worldline (xα

a,±)

degrees of freedom, schematically,

eiSeff [xa,±]=ZDh+Dh−ei{SEH[h±]+Spp[h±,xa,±]},(1)

with SEH and Spp the closed-path version of the Einstein-

Hilbert and point-particle worldline actions, respectively.

We ignore here spin degrees of freedom and ﬁnite-size ef-

fects (see [62, 65]). We also restrict ourselves to the clas-

sical regime and therefore the path integral in (1) is com-

puted in the saddle-point approximation—keeping only

connected ‘tree-level’ Feynman diagrams of the gravita-

tional ﬁeld(s)—with the compact objects treated as ex-

ternal nonpropagating sources.

In this scenario, the matrix of (causal) propagators is

given by the Keldysh representation:

KAB(x−y) = i0−∆adv(x−y)

−∆ret(x−y) 0 ,(2)

with A, B ∈ {+,−} and ∆ret/adv the standard re-

tarded/advanced Green’s functions. The impulse, e.g.

for particle 1, then follows from [76]

∆pµ

1=−ηµν Z∞

−∞

dτ1

δSeﬀ[xa,±]

δxν

1,−(τ1)PL

Gn∆(n)pµ

1,(3)

to all PM orders, where the subscript ‘PL’ stands for the

Physical Limit:{xa,−→0, xa,+→xa}[103]. As for

the conservative sector [66, 67], we must also include it-

erations from lower order solutions to the equations of

motion. The latter are obtained from the eﬀective ac-

tion in the same physical limit. The diagrams needed to

O(G4) are depicted in Fig. 1. Mirror images (not shown)

must also be computed. See [76] for details.

The impulse is further decomposed into scalar integrals

in the perpendicular and longitudinal directions, i.e. for

particle 1 (and likewise for particle 2)

∆(n)pµ

1=c(n)

bµ

bn+1

bnX

c(n)

1ˇuaˇuµ

a,(4)

with bµ≡bµ

1−bµ

2the impact parameter, b≡p−bµbµ,

and ˆ

bµ≡bµ/b. We use the notation [97]

ˇuµ

1≡γuµ

2−uµ

γ2−1,ˇuµ

2≡γuµ

1−uµ

γ2−1, γ ≡u1·u2,(5)

with ua’s the incoming velocities, b·ua= 0, u2

a= 1, and

ˇua·ub=δab. Ignoring absorption, the preservation of

the on-shell condition, p2

a=m2

a, implies 2pa·∆pa=

−(∆pa)2, which serves as a nontrivial consistency check.

Integration. Similarly to the derivations in [66, 67],

but incorporating the key distinction between Feynman

and retarded propagators, the components of the impulse

can be reduced to diﬀerent families of integrals,

i=1

dd`i

πd/2

δ(`i·uai)

(±`i·u/

ai−i0)ni

k=1

Dνk

,(6)

restricted by Dirac-δfunctions. Following [66, 67], the

`i=1,2,3’s are the loop momenta, ni, νkare integers, and

ai∈ {1,2}, with u/

1=u2,u/

2=u1. In contrast to the

conservative part, the Dk’s are now various sets of re-

tarded/advanced propagators, e.g. {(`0±i0)2−`2, . . .},

consistent with causality. The same constraints as before

apply on the external data [63], such that the relevant

integrals can only depend on γ. As in our previous cal-

culations [66, 67], we conveniently introduce the param-

eter x, deﬁned through the relation γ≡(x2+1)/2x[95],

and compute these integrals by using dimensional regu-

larization (in d≡4−2dimensions) and the method of

diﬀerential equations [32–38].

The integration problem then resembles the steps al-

ready performed for the computation in [66, 67], except

for a few notable diﬀerences. First of all, as before we

use integration-by-parts (IBP) relations [39–44] and re-

duce (6) to a basis of master integrals. Because of the

fewer number of symmetries of the in-in integrand, the

algebraic manipulations become a bit more involved than

with Feynman-only propagators. But more importantly,

the boundary conditions in the near-static limit γ'1

must be computed in terms of retarded/advanced Green’s

functions. For this purpose, we resort to the method of

regions and the expansion into potential and radiation

modes. The potential-only part was obtained in [66],

and recovered here from the full solution. For radia-

tion modes, the same type of integrals appearing in PN

derivations [109], combined with leftover integrals over

potential-only modes at one and two loops, are suﬃcient

to bootstrap the entire answer. See [61] for more details.

Total impulse. Inputing the values of the boundary master integrals and translating from xto γspace, we ﬁnd

c(4)tot

π=−3h1m1m2(m3

1+m3

64(γ2−1)5/2+m2

1m2

2(m1+m2)21h2E2γ−1

γ+1 

32(γ−1)pγ2−1+

3h3K2γ−1

γ+1 

16 (γ2−1)3/2−

3h4Eγ−1

γ+1 Kγ−1

γ+1 

16 (γ2−1)3/2+π2h5

8pγ2−1+h6log γ−1

2

16 (γ2−1)3/2

3h7Li2qγ−1

γ+1 

(γ−1)(γ+ 1)2−

3h7Li2γ−1

γ+1 

4(γ−1)(γ+ 1)2+m3

1m2

2h8

48 (γ2−1)3+pγ2−1h9

768(γ−1)3γ9(γ+ 1)4+h10 log γ+1

2

8 (γ2−1)2−h11 log γ+1

2

32 (γ2−1)5/2+h12 log(γ)

16 (γ2−1)5/2

−h13 arccosh(γ)

8(γ−1)(γ+ 1)4+h14 arccosh(γ)

16 (γ2−1)7/2−

3h15 log γ+1

2log γ−1

γ+1 

8pγ2−1+

3h16 arccosh(γ) log γ−1

γ+1 

16 (γ2−1)2−

3h17Li2γ−1

γ+1 

64pγ2−1−3

32pγ2−1h18Li21−γ

γ+ 1

+m2

1m3

23h15 log 2

γ−1log γ+1

2

8pγ2−1+3h16 log γ−1

2arccosh(γ)

16 (γ2−1)2+h19

48 (γ2−1)3+h20

192γ7(γ2−1)5/2+h21 log γ+1

2

8 (γ2−1)2+h22 log γ+1

2

16 (γ2−1)3/2+h23 log(γ)

2 (γ2−1)3/2

−h24 arccosh(γ)

16 (γ2−1)3+h25 arccosh(γ)

16 (γ2−1)7/2−3h26 arccosh2(γ)

32 (γ2−1)7/2+3h27 log2γ+1

2

2pγ2−1+3h28 log γ+1

2arccosh(γ)

16 (γ2−1)2+

h29Li21−γ

γ+1 

4pγ2−1+

3h30Li2γ−1

γ+1 

8pγ2−1,

c(4)tot

1ˇu1=9π2h31m1m2

2(m1+m2)2

32 (γ2−1) +2h32m1m2

2m2

1+m2

2

(γ2−1)3+m2

1m3

24h33

3 (γ2−1)3−8h34

3 (γ2−1)5/2+8h35 arccosh(γ)

(γ2−1)3−16h36 arccosh(γ)

(γ2−1)3/2,

c(4)tot

1ˇu2=−m4

1m2 9π2h31

32 (γ2−1) +2h32

(γ2−1)3!+m3

1m2

2−4h37

3 (γ2−1)3+h38

705600γ8(γ2−1)5/2+π2h39

192 (γ2−1)2+h40 arccosh(γ)

6720γ9(γ2−1)3+32h41 arccosh(γ)

3 (γ2−1)3/2

−8h42 arccosh2(γ)

(γ2−1)2+32h43 arccosh2(γ)

(γ2−1)7/2+h44 log(2) arccosh(γ)

8 (γ2−1)2+

3h45 Li2γ−1

γ+1 −4Li2qγ−1

γ+1 

16 (γ2−1)2

3h46 log γ+1

2arccosh(γ)−2Li2pγ2−1−γ

8 (γ2−1)2−

h47 Li2−γ−pγ2−12−2 log(γ) arccosh(γ)

16 (γ2−1)2+m2

1m3

2−2h48

45 (γ2−1)3

+h49

1440γ7(γ2−1)5/2+π2h50

48 (γ2−1)2+h51 arccosh(γ)

480γ8(γ2−1)3−16h52 arccosh(γ)

5 (γ2−1)3/2−16h53 arccosh2(γ)

(γ2−1)2−32h54 arccosh2(γ)

(γ2−1)7/2−h55 log(2) arccosh(γ)

4 (γ2−1)2

h56 Li2γ−1

γ+1 −4Li2qγ−1

γ+1 

32 (γ2−1)2+

h57 log 2

γ+1 arccosh(γ) + 2Li2pγ2−1−γ

4 (γ2−1)2+

h58 Li2−γ−pγ2−12−2 log(γ) arccosh(γ)

8 (γ2−1)2.

(7)

See Table I for the list of hi(γ) polynomials.

The impulse for the second particle follows by exchanging

1↔2 in the masses, incoming velocities, and impact

parameter. As expected from the calculations in [66, 67],

the complete results feature dilogarithms (Li2(z)), and

complete elliptic integrals of the ﬁrst (K(z)) and second

(E(z)) kind.

Conservative. As shown explicitly in [76], a (time-

symmetric) conservative contribution can be identiﬁed

by rewriting retarded Green’s functions in terms of Feyn-

man propagators plus a reactive term, and keeping the

real part of the Feynman-only piece, i.e. ∆pµ

cons ≡R∆pµ

with imaginary terms canceling out against counter-parts

from the reactive terms. Performing these steps in the

full in-in integrand, and associated boundary conditions

entering in the total impulse, we readily recover the

conservative results in [66, 67] including potential and

radiation-reaction tail eﬀects.

Dissipative. As discussed in [76], the terms stemming

oﬀ of the mismatch between Feynman and retarded prop-

agators incorporate dissipative eﬀects. Needless to say,

these terms can also be read oﬀ directly from the total

result by subtracting the conservative part. Following

the analysis in [66, 67], we disentangle the various pieces

according to factors of v−2

∞, with v∞≡pγ2−1, which

signal the presence of an on-shell mode.

Starting with a single radiation mode we encounter

instantaneous dissipative eﬀects at linear order in the

radiation-reaction. The latter are odd under time re-

versal and contribute to the ˆ

band ˇuadirections. We ﬁnd

agreement in the overlap with known partial results in

linear-response theory in the PN literature [26, 27, 119–

123], as well as with the (odd) contribution to the c(4)

coeﬃcient recently derived in [98]. All of the remaining

radiative terms involve two radiation modes. After re-

moving the Feynman-only (radiative) conservative pieces

[66, 67], the leftovers contain hereditary as well as non-

linear radiation-reaction dissipative eﬀects. The former

enters both in the longitudinal and perpendicular direc-

tions, whereas the latter contributes only to the total ra-

diated perpendicular momentum at this order.1We also

ﬁnd perfect consistency with known nonlinear and hered-

itary results in the PN expansion [120–124].

See the supplemental material and ancillary ﬁle for ex-

plicit expressions.

1At this point, however, we cannot distinguish whether nonlinear

radiation-reaction terms are due to either eﬀects at second order

in the linear radiation-reaction force or truly nonlinear gravita-

tional corrections.

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RadiationReactionandGravitationalWavesatFourthPost-MinkowskianOrderChristophDlapa,1GregorKalin,1ZhengwenLiu,2,1JakobNeef,3,4andRafaelA.Porto11DeutschesElektronen-SynchrotronDESY,Notkestr.85,22607Hamburg,Germany2NielsBohrInternationalAcademy,NielsBohrInstitute,UniversityofCopenhagen,Blegdamsvej17,DK...

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Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order Christoph Dlapa1Gregor K alin1Zhengwen Liu2 1Jakob Neef3 4and Rafael A. Porto1 1Deutsches Elektronen-Synchrotron DESY Notkestr. 85 22607 Hamburg Germany.pdf

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Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order Christoph Dlapa1Gregor K alin1Zhengwen Liu2 1Jakob Neef3 4and Rafael A. Porto1 1Deutsches Elektronen-Synchrotron DESY Notkestr. 85 22607 Hamburg Germany

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