HU-EP-2232-RTG Linear Response Hamiltonian and Radiative Spinning Two-Body Dynamics Gustav Uhre Jakobsen1 2 3 and Gustav Mogull1 2 3

2025-04-27 0 0 879.17KB 24 页 10玖币

侵权投诉

HU-EP-22/32-RTG

Linear Response, Hamiltonian and Radiative Spinning Two-Body Dynamics

Gustav Uhre Jakobsen 1, 2, 3, ∗and Gustav Mogull 1, 2, 3, †

1Institut f¨ur Physik und IRIS Adlershof, Humboldt Universit¨at zu Berlin,

Zum Großen Windkanal 2, 12489 Berlin, Germany

2Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am M¨uhlenberg 1, 14476 Potsdam, Germany

3Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

Using the spinning, supersymmetric Worldline Quantum Field Theory formalism we compute the

momentum impulse and spin kick from a scattering of two spinning black holes or neutron stars up to

quadratic order in spin at third post-Minkowskian (PM) order, including radiation-reaction eﬀects

and with arbitrarily mis-aligned spin directions. Parts of these observables, both conservative and

radiative, are also inferred from lower-PM scattering data by extending Bini and Damour’s linear

response formula to include mis-aligned spins. By solving Hamilton’s equations of motion we also

use a conservative scattering angle to infer a complete 3PM two-body Hamiltonian including ﬁnite-

size corrections and mis-aligned spin-spin interactions. Finally, we describe mappings to the bound

two-body dynamics for aligned spin vectors: including a numerical plot of the binding energy for

circular orbits compared with numerical relativity, analytic conﬁrmation of the NNLO PN binding

energy and the energy loss over successive orbits.

The need for accurate waveform templates for com-

parison with gravitational wave signals coming from the

LIGO, Virgo and KAGRA detectors of binary merger

events [1–6] — and in the future LISA, the Einstein Tele-

scope and Cosmic Explorer [7] — has provoked enormous

interest in the gravitational two-body problem. One of

the most important physical properties inﬂuencing the

paths of massive objects following inspiral trajectories,

which as they accelerate produce gravitational waves, is

their spins. Accurately determining the spins of black

holes and neutron stars in binary orbits yields crucial

information about their origins: if the spins are approxi-

mately aligned with the orbital plane, then this suggests

formation of the binary system by slow accretion of mat-

ter; if they are mis-aligned (precessing), then this indi-

cates formation of the binary by a random capture event.

A fruitful path has been eﬀective ﬁeld theory (EFT)-

based methods, which tackle the inspiral stage of the

gravitational two-body problem using its natural sepa-

ration of length scales [8–12]: the size of the massive

bodies is far less than their separation, which in turn is

far less than their distance from us, the observer. Partial

results for the non-spinning two-body Hamiltonian are

available up to sixth post-Newtonian (PN) order [13–18];

in the spinning case a body-ﬁxed frame on the world-

line is often used [19–22], and results are available up to

N3LO in the spin-orbit sector [23–25] and in the spin-spin

sector [26–33].

However, an excellent alternative approach to the

bound two-body problem comes by way of studying two-

body scattering: here it is natural to deﬁne gauge-

invariant scattering observables in terms of the states

at past-/future-inﬁnity, where the gravitational ﬁeld

is weak. It is also natural here to adopt the post-

Minkowskian (PM) expansion in Newton’s constant G,

∗gustav.uhre.jakobsen@physik.hu-berlin.de

†gustav.mogull@aei.mpg.de

which resums terms from inﬁnitely high velocities in the

post-Newtonian (PN) series. One may use analytic con-

tinuation to directly produce PM observables for bound

orbits [34–37]; alternatively, conservative scattering ob-

servables may be used to infer a Hamiltonian for the

two-body system [38–44]. A more sophisticated version

of this strategy is to infer an eﬀective-one-body (EOB)

Hamiltonian [45–49], which may be extended to include

spin [50–54] and resums information from the test-body

limit.

The Worldline Quantum Field Theory (WQFT) is a

new formalism for producing gravitational scattering ob-

servables [55–63]. It builds on the highly successful PM-

based worldline EFT approach [64], which has been used

to produce scattering observables at 3PM [65–67] and

4PM orders [68–71]; the worldline EFT has also produced

gravitational Bremsstrahlung and radiative observables

including tidal eﬀects and spin [72–75]. The WQFT goes

a step further by quantizing worldline degrees of free-

dom, which bypasses the need for intermediate oﬀ-shell

objects such as the eﬀective action. A supersymmetric

extension to the worldline accounts for quadratic-in-spin

eﬀects [57,58], conveniently avoiding the typical use of

a body-ﬁxed frame. In Ref. [61] we used the WQFT to

produce conservative scattering observables — the mo-

mentum impulse ∆pµ

iand spin kick ∆Sµ

i— at 3PM or-

der.

In this paper, we upgrade these observables to in-

clude radiation-reaction (dissipative) eﬀects, using the

Schwinger-Keldysh in-in formalism [76–80] that has re-

cently been incorporated into both the WQFT and PM-

based worldline EFT frameworks [63,67]. Our results

conﬁrm the radiated four-momentum Pµ

rad recently pre-

dicted with the worldline EFT approach [75]. Given these

new observables, we postulate and conﬁrm an extension

to Bini and Damour’s linear response relation [81–83]

which allows us to predict terms in the conservative and

radiative parts of the full scattering observables, depend-

ing on their behavior under the time-reversal operation

arXiv:2210.06451v3 [hep-th] 19 Jul 2023

vµ

i→ −vµ

i. The extension holds for arbitrary spin orien-

tations, and goes beyond linear response.

The WQFT is inspired by QFT amplitudes-based

methods for tackling the classical two-body problem [84–

87]. These build on well-honed strategies for deriving

scattering amplitudes [88–92] and performing the asso-

ciated loop integrals [93,94]. In the non-spinning case

a slew of two-body results have been produced at 3PM

order (two loops) [95–102] and at 4PM order (three

loops) [71,103,104]. There has also recently been work

on N-body scattering and potentials [105]. Radiation-

reaction eﬀects have been incorporated [106–112], and an

in-in style formalism for directly producing observables

has been introduced [113–116]. To handle spin, higher-

spin ﬁelds are used [117–126] and results have been pro-

duced at 2PM order at quadratic [52,127,128], quar-

tic [129] and higher orders in spin [130–132]. Similar

results have also been achieved with the closely related

heavy-particle EFT [133–138].

Most notably, a 3PM quadratic-in-spin Hamiltonian

has now been derived using amplitudes-based meth-

ods [139], involving spin on one of the two massive bodies

only and without ﬁnite-size corrections. In this paper,

using the conservative scattering observables derived in

Ref. [61] we both conﬁrm this result and extend it to

include spin-spin eﬀects and ﬁnite-size corrections rele-

vant for neutron stars. Quite remarkably, we ﬁnd that

knowledge of a single scattering angle suﬃces to com-

pletely determine the Hamiltonian, also when arbitrarily

mis-aligned spin vectors are involved.

Our paper is structured as follows. In Section Iwe re-

view the dynamics of spinning massive bodies, including

their description up to quadratic order in spin in terms of

an N= 2 supersymmetric worldline action. We demon-

strate how, with a suitable SUSY shift, we can switch

between the canonical and covariant spin-supplementary

conditions (SSCs). In Section II we review the Schwinger-

Keldysh in-in formalism in the context of WQFT, and

in Section III put it to use deriving the complete 3PM

quadratic-in-spin momentum impulse ∆pµ

1and spin kick

∆Sµ

1including radiation-reaction eﬀects. We present the

results schematically, demonstrate how one may intro-

duce scattering angles for mis-aligned spins, and perform

various consistency checks.

Next, in Section IV we upgrade the linear response

relation to mis-aligned spin directions, generating both

conservative and radiative terms from the full 3PM scat-

tering observables ∆pµ

1and ∆Sµ

1. In Section Vwe

use the conservative scattering observables, and in par-

ticular the scattering angle, to build a complete 3PM

quadratic-in-spin Hamiltonian. Finally, in Section VI we

discuss unbound-to-bound mappings for the speciﬁc case

of aligned spins: we generate the binding energy for cir-

cular orbits, both numerically and analytically and up

to 4PN order, and produce plots of the binding energy

as a function of the orbital frequency close to merger

— comparing our results with numerical relativity. We

also determine the energy radiated per orbit using an ap-

propriate analytic continuation [37]. In Section VII we

conclude.

I. SPINNING MASSIVE BODIES

A pair of black holes or neutron stars interacting

through a gravitational ﬁeld in D-dimensional Einstein

gravity are described by

S=SEH[gµν ] + Sgf [gµν ] +

i=1

S(i)[gµν , xµ

i, ψa

i],(1)

where SEH is the Einstein-Hilbert action (κ=√32πG),

SEH =−2

κ2ZdDx√−g R , (2)

Sgf is a gauge-ﬁxing term and S(i)are the two worldline

actions. Up to quadratic order in spin [57,58]

S(i)

=−Zdτih1

2gµν ˙xµ

i˙xν

i+i¯

ψi,aDψa

Dτi+1

2Rabcd ¯

ψa

iψb

i¯

ψc

iψd

+CE,iEi,ab ¯

ψa

iψb

iPi,cd ¯

ψc

iψd

ii,(3)

where the projector is Pi,ab := ηab −eaµebν ˙xµ

i˙xν

i/˙x2

i,ηab

is the (mostly-minus) Minkowski metric and Ei,ab :=

Raµbν ˙xµ

i˙xν

i/˙x2

i. The ﬁnite-size multipole moment coef-

ﬁcients CE,i are deﬁned such that CE,i = 0 for black

holes, and

Dψa

Dτi

:= ˙xµ

i∇µψa

i=˙

ψa

i+ ˙xµ

iωµabψi,b .(4)

The two bodies with masses mihave positions xµ

i(τi); the

complex anticommuting ﬁelds ψa

i(τi), deﬁned in a local

frame ea

µ(x) with gµν =ea

µeb

νηab, encode spin degrees of

freedom.

The worldline action (3) enjoys a global N= 2 super-

symmetry:

δxµ

i=i¯ϵiψµ

i+iϵi¯

ψµ

i, δψa

i=−ϵiea

µ˙xµ

i−δxµ

iωµabψb

i,(5)

with constant SUSY parameters ϵiand ¯ϵi=ϵ†

i. As shown

in Ref. [58], these shifts are generated by the conserved

supercharges ˙xi·ψiand ˙xi·¯

ψi. There is also a U(1)

symmetry:

δψa

i=iαiψa

i, δ ¯

ψa

i=−iαi¯

ψa

i, δxµ

i= 0 ,(6)

generated by the conserved charge ψi·¯

ψi. Lastly,

reparametrization invariance of the worldlines in τiim-

plies

˙x2

i= 1+Rabcd ¯

ψa

iψb

i¯

ψc

iψd

i+2CE,iEi,ab ¯

ψa

iψb

iPi,cd ¯

ψc

iψd

i(7)

is also preserved. As ˙x2

i̸= 1 generically along the world-

lines this implies that τiare not the proper times; how-

ever, as we are generally only interested in the asymptotic

behavior this subtlety will not be important.

A. Background Symmetries

Fields are perturbatively expanded around their back-

ground values at past inﬁnity:

gµν (x) = ηµν +κhµν (x),(8a)

xµ

i(τi) = bµ

i+τivµ

i+zµ

i(τi),(8b)

ψa

i(τi)=Ψa

i+ψ′a

i(τi),(8c)

where pµ

i=mivµ

iis the initial momentum; the initial

value of the spin tensor is given by

Sab

i=−2imi¯

Ψ[a

iΨb]

i.(9)

The antisymmetrization [ab] includes a factor 1/2 — note

that this normalization of the spin tensor diﬀers from

that used in Refs. [57,58,61]. The vierbein is similarly

expanded as

µ=ηaν ηµν +κ

2hµν −κ2

8hµρhρν+O(κ3),(10)

which allows us to drop the distinction between space-

time µ, ν, . . . and local frame a, b, . . . indices. The global

N= 2 SUSY in the far past is

δbµ

i=i¯ϵiΨµ

i+iϵi¯

Ψµ

i, δvµ

i= 0 , δΨµ

i=−ϵivµ

⇒δSµν

i= 2p[µ

iδbν]

i.(11)

To ﬁx these symmetries we ﬁnd it convenient to enforce

the covariant spin-supplementary condition (SSC):

pi·Ψi= 0 =⇒pi,µSµν

i= 0 ,(12)

Using the reparametrization symmetry we also enforce

i= 1 and b·vi= 0, where bµ=bµ

2−bµ

1is the impact

parameter pointing from the ﬁrst to the second massive

body. Finally, γ=v1·v2; we will also make use of unit-

normalized “hatted” variables, e.g. ˆ

bµ=bµ/|b|.

The total initial angular momentum of the system is

Jµν =Lµν +Sµν

1+Sµν

Lµν = 2b[µ

1pν]

1+ 2b[µ

2pν]

2,(13)

where Lµν is the orbital component. In this context,

we see that the background symmetries (11) correspond

simply to invariance of the system’s total angular mo-

mentum under shifts in the origins of the two bodies bµ

We can also shift the center of our coordinate system

xµ→xµ+aµ, in which case Lµν →Lµν + 2a[µPν]as

discussed in e.g. Ref. [140]. The orbital and spin an-

gular momentum vectors, deﬁned speciﬁcally in D= 4

dimensions, are invariant under these shifts:

Lµ:= 1

2ϵµνρσLνρ ˆ

Pσ=−1

Eϵµνρσbνpρ

1pσ

2,(14a)

Sµ

i:= miaµ

i=1

2ϵµνρσSνρ

ivσ

i,(14b)

where

Pµ=pµ

1+pµ

2,(15a)

pµ=m1m2

E2(γm1+m2)vµ

1−(γm2+m1)vµ

2,(15b)

are respectively the total and center-of-mass (CoM) mo-

mentum, pµ= (0,p∞). Here E=|P|=MΓ =

Mp1+2ν(γ−1) is the energy in the CoM frame, M=

m1+m2,ν=µ/M =m1m2/M2are the total mass

and symmetric mass ratio; p∞=|p∞|=µpγ2−1/Γ is

the center-of-mass momentum. With the covariant SSC

choice (12) the total angular momentum Jµis given by

Jµ:=1

2ϵµνρσJνρ ˆ

Pσ

=Lµ+X

ivi·ˆ

P Sµ

i−Si·ˆ

P vµ

i.(16)

Notice that Jµ̸=Lµ+Sµ

1+Sµ

2, which is due to Sµ

ibeing

deﬁned in their respective inertial frames vµ

irather than

the center-of-mass frame ˆ

Pµ.

B. Canonical Spin Variables

We also ﬁnd it useful to introduce canonical variables

[50,51,141] which are designed to ensure that

Jµ=Lµ

can +Sµ

1,can +Sµ

2,can ,(17)

and P·Lcan =P·Si,can = 0. The canonical spin vectors

Sµ

i,can are given by a boost of the covariant spin vectors

Sµ

ito the center-of-mass frame:

Sµ

i,can := Λµν(vi→ˆ

P)Sν

=Sµ

i−ˆ

P·Si

γi+ 1(ˆ

Pµ+vµ

i),

(18)

where γi=ˆ

P·viis the time component of vµ

iin the

center-of-mass frame. To ensure preservation of the total

angular momentum Jµ(16), we have

Lµ

can =Lµ+

i=1 (γi−1)Sµ

i+ˆ

P·Si

γi+1 (ˆ

Pµ−γivµ

i).(19)

The canonical impact parameter bµ

can — in terms of which

Lµ

can =−E−1ϵµνρσbν

canpρ

1pσ

2— is related to bµby a spe-

ciﬁc SUSY shift (11):

ϵi=ˆ

P·Ψi

γi+ 1 .(20)

We then have, with Ei=γimi

bµ

i,can =bµ

i+1

Ei+mi

Sµν

iˆ

Pν,(21a)

Ψµ

i,can = Ψµ

i−ˆ

P·Ψi

γi+ 1 vµ

i,(21b)

and can conﬁrm that the the canonical spin tensor

Sµν

i,can =−2imi¯

Ψ[µ

i,canΨν]

i,can satisﬁes the canonical Pryce-

Newton-Wigner SSC [142–144]:

(ˆ

P+vi)·Ψi,can = 0 =⇒(ˆ

Pµ+vi,µ)Sµν

i,can = 0 .(22)

The canonical spin vector is then also given by

Sµ

i,can =1

2ϵµνρσSνρ

i,can ˆ

Pσ,(23)

and has a vanishing time component in the center-of-

mass frame: P·Si,can = 0. This will be useful when we

construct a Hamiltonian in Section V.

II. WQFT IN-IN FORMALISM

Complete observables including both conservative and

radiative contributions are produced from WQFT us-

ing the Schwinger-Keldysh in-in formalism [76–80]. For-

mally this involves doubling the degrees of freedom in

our theory: hµν → {h1µν , h2µν },zµ

i→ {zµ

1i, zµ

2i}and

ψ′µ

i→ {ψ′µ

1i, ψ′µ

2i}. Observables are deﬁned in terms of a

path integral including two copies of the action:

⟨O⟩in−in := ZD[hAµν , zµ

Ai, ψ′µ

Ai]ei(S[{}1]−S∗[{}2])O,(24)

where A= 1,2 and we use the shorthand {}A:=

{gAµν , xµ

Ai, ψµ

Ai}. The boundary conditions on hAµν ,zµ

and ψ′µ

Ai are that all ﬁelds equate at future inﬁnity,

h1µν (t= +∞,x) = h2µν (t= +∞,x),(25a)

zµ

1i(τi= +∞) = zµ

2i(τi= +∞),(25b)

ψ′µ

1i(τi= +∞) = ψ′µ

2i(τi= +∞),(25c)

and vanish at past inﬁnity:

h1µν (t=−∞,x) = h2µν (t=−∞,x) = 0 ,(26a)

zµ

1i(τi=−∞) = zµ

2i(τi=−∞)=0,(26b)

ψ′µ

1i(τi=−∞) = ψ′µ

2i(τi=−∞)=0.(26c)

This entangling of the boundary conditions gives rise to

oﬀ-diagonal terms in the propagator matrices involving

the doubled ﬁelds. For full details, see Ref. [63].

Fortunately, when performing calculations there is no

need to double degrees of freedom in this way. The key

insight of Ref. [63] was that tree-level single-operator ex-

pectation values (24) are produced using precisely the

same Feynman rules as in the in-out formalism, but with

retarded propagators pointing towards the outgoing line.

The retarded graviton propagator is

µν ρσ

=iPµν;ρσ

k2+ sgn(k0)i0,(27)

where Pµν;ρσ := ηµ(ρησ)ν−1

D−2ηµν ηρσ and i0 denotes a

small positive imaginary part. For the worldline modes

zµ

iand ψ′µ

ithe retarded propagators are respectively

µν

=−iηµν

mi(ω+i0)2,(28a)

µν

=−iηµν

mi(ω+i0) .(28b)

The Feynman vertices are unchanged with respect to the

in-in formalism: for example, the single-graviton emis-

sion vertex from worldline iis

hµν (k)

=−imiκ

2eik·biδ

−

(k·vi)vµ

ivν

i+ikρSρ(µ

ivν)

2kρkσSρµ

iSνσ

i+CE,i

2vµ

ivν

ikρSiρσSiσλkλ,

(29)

where δ

−

(ω) := 2πδ(ω). At tree level the WQFT simply

provides a diagrammatic mechanism for solving the clas-

sical equations of motion in momentum space, and so

the use of retarded propagators ensures that boundary

conditions are ﬁxed in the far past.

III. RADIATIVE OBSERVABLES

Building on Ref. [61], we compute the momentum im-

pulse and change in ψµ

∆pµ

i:= [mi˙xµ

i]τi=+∞

τi=−∞ =−miω2⟨zµ

i(ω)⟩in−in|ω=0,(30a)

∆ψµ

i:= [ψµ

i]τi=+∞

τi=−∞ =−iω ⟨ψ′µ

i(ω)⟩in−inω=0 ,(30b)

but now also including radiation-reaction eﬀects. Using

the deﬁnitions of the spin tensor Sµν

i(9) and spin vector

Sµ

i(14b) we can then also derive the spin kick ∆Sµ

∆Sµν

i=−2imi¯

Ψ[µ

i∆ψν]

i+ ∆ ¯

ψ[µ

iΨν]

i+ ∆ ¯

ψ[µ

i∆ψν]

i,

∆Sµ

i=1

2miϵµνρσ(Sνρ

i∆pσ

i+∆Sνρ

ipσ

i+∆Sνρ

i∆pσ

i).(31)

We seek the 3PM components in a PM expansion:

∆X=X

Gn∆X(n),(32)

where ∆Xcould be any of these observables: ∆pµ

i, ∆Sµ

∆Sµν

ior ∆ψµ

The relevant Feynman diagrams for both calculations

are drawn in Fig. 1. These diagrams make no distinc-

tion between conservative and radiative eﬀects. As only

the m2

1m2

2component of ∆p(3)µ

1and the m1m2

2compo-

nent of ∆ψ(3)µ

1are speciﬁcally aﬀected by the inclusion

of radiation-reaction eﬀects we recompute these compo-

nents; for the rest, we simply bring forwards our previous

results from Ref. [61]. Integrands are assembled using

the WQFT Feynman rules [58] in D= 4−2ϵdimensions,

which involves integration on the momenta or energies

of all internal lines; vertices contain either energy- or

(a) (b) (c) (d) (e) (f) (g) (h)

(i) (j) (k) (l) (m) (n) (o) (p)

(q) (r) (s) (t) (u) (v) (w) (x)

(y) (z) (aa) (bb) (cc) (dd) (ee) (ﬀ)

FIG. 1: The 32 types of diagrams contributing to the m2

1m2

2components of ∆p(3)µ

1and the m1m2

2components of ∆ψ(3)µ

Diagrams (a)–(v) were already present in the conservative calculation [61], though their expressions are modiﬁed by the

inclusion of radiation; the mushrooms (w)–(ﬀ) are purely radiative, and did not appear in the strictly conservative

calculation [61]. Diagram (y) includes the same worldline propagator with opposite i0 prescriptions, and so belongs to the K

integral family (35). For brevity we use solid lines to represent both propagating deﬂection zµ

iand spin modes ψ′µ

momentum-conserving δ

−

-functions, whichever is appro-

priate. The energy integrals, corresponding to internal

propagation of zµ

ior ψ′

imodes, are trivial: conservation

of energy at the worldline vertices resolves them imme-

diately.

Each graph has three unresolved four-momenta to inte-

grate over. The ﬁrst of these integrals is a Fourier trans-

form:

∆X(bµ, vµ

i, Sµν

=Zq

eiq·bδ

−

(q·v1)δ

−

(q·v2)∆X(qµ, vµ

i, Sµν

i),(33)

where qµis the total momentum exchanged from the sec-

ond to the ﬁrst worldline, and Rq:= RdDq/(2π)D. Here

we have implicitly deﬁned the momentum-space observ-

ables ∆X(qµ, vµ

i, Sµν

i), which are given as linear combi-

nations of two-loop Feynman integrals:

I(σ1;σ2;σ3)

n1,n2,...,n7[ℓµ1

1···ℓµn

1ℓν1

2···ℓνn

:= Zℓ1,ℓ2

−

(ℓ1·v2)δ

−

(ℓ2·v1)ℓµ1

1···ℓµn

1ℓν1

2···ℓνn

Dn1

1Dn2

2···Dn7

D1=ℓ1·v1+σ1i0, D2=ℓ2·v2+σ2i0,(34)

D3= (ℓ1+ℓ2−q)2+σ3sgn(ℓ0

1+ℓ0

2−q0)i0,

D4=ℓ2

1, D5=ℓ2

2, D6= (ℓ1−q)2, D7= (ℓ2−q)2.

These integrals with retarded propagators were discussed

at length in Ref. [63]: propagators D4–D7are prevented

from going on-shell by the requirement that ℓ1·v2=

ℓ2·v1= 0, so we can safely ignore their i0 prescriptions.

We also require

K(σ)

n1,n2,n3,n4,n5[ℓµ1···ℓµnkν1···kνn]

:= Zℓ,k

−

((k−ℓ)·v1)δ

−

(ℓ·v2)ℓµ1···ℓµnkν1···kνn

Dn1

1Dn2

2Dn3

3Dn4

4Dn5

D1=ℓ·v1+i0, D2=ℓ·v1−i0,(35)

D3=k2+σsgn(k0)i0, D4=ℓ2, D5= (ℓ−q)2,

which accounts for the possibility of a worldline propa-

gator appearing twice, but with diﬀerent i0 prescriptions

— diagram (y) in Fig. 1.

The subsequent integration steps were discussed in

Ref. [61], and are not substantially diﬀerent with the

inclusion of radiation-reaction eﬀects in the observables.

Tensorial two-loop integrals are reduced to scalar-type by

expanding on a suitable basis, and then reduced to mas-

ter integrals using integration-by-parts (IBP) identities.

Expressions for these master integrals were provided in

Ref. [63], and once the Fourier transform (33) has been

performed on the exchanged momentum qµwe are left

with the observables in Ddimensions. The scalar inte-

grals themselves have simple reality properties:

I(σ1;σ2;σ3)∗

n1,n2,...,n7= (−1)n1+n2I(σ1;σ2;σ3)

n1,n2,...,n7,

K(σ)∗

n1,n2,...,n5= (−1)n1+n2K(σ)

n1,n2,...,n5,(36)

i.e. they are either purely real or imaginary, depending

on whether they have an even or odd number of world-

line propagators respectively. While this implies that the

momentum-space observables ∆X(qµ, vµ

i, Sµν

i) are com-

plex functions, the Fourier transform (33) introduces ad-

ditional factors of i, giving rise to purely real observables

∆X(bµ, vµ

i, Sµν

i).

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HU-EP-22/32-RTGLinearResponse,HamiltonianandRadiativeSpinningTwo-BodyDynamicsGustavUhreJakobsen1,2,3,∗andGustavMogull1,2,3,†1Institutf¨urPhysikundIRISAdlershof,HumboldtUniversit¨atzuBerlin,ZumGroßenWindkanal2,12489Berlin,Germany2MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute),AmM¨...

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HU-EP-2232-RTG Linear Response Hamiltonian and Radiative Spinning Two-Body Dynamics Gustav Uhre Jakobsen1 2 3 and Gustav Mogull1 2 3

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