Conservative scattering of spinning black holes at fourth post-Minkowskian order

Using the ${\mathcal N}=1$ supersymmetric, spinning worldline quantum field theory formalism we compute the conservative spin-orbit part of the momentum impulse $\Delta p_i^\mu$, spin kick $\Delta S_i^\mu$ and scattering angle $\theta$ from the scattering of two spinning massive bodies (black holes or neutron stars) up to fourth post-Minkowskian (PM) order. These three-loop results extend the state-of-the-art for generically spinning binaries from 3PM to 4PM. They are obtained by employing recursion relations for the integrand construction and advanced multi-loop Feynman integral technology in the causal (in-in) worldline quantum field theory framework to directly produce classical observables. We focus on the conservative contribution (including tail effects) and outline the computations for the dissipative contributions as well. Our spin-orbit results agree with N$^3$LO post-Newtonian and test-body data in the respective limits. We also re-confirm the conservative 4PM non-spinning results.

Using the N = 1 supersymmetric, spinning worldline quantum field theory formalism we compute the conservative spin-orbit part of the momentum impulse ∆p µ i , spin kick ∆S µ i and scattering angle θ from the scattering of two spinning massive bodies (black holes or neutron stars) up to fourth post-Minkowskian (PM) order.These three-loop results extend the state-of-the-art for generically spinning binaries from 3PM to 4PM.They are obtained by employing recursion relations for the integrand construction and advanced multi-loop Feynman integral technology in the causal (in-in) worldline quantum field theory framework to directly produce classical observables.We focus on the conservative contribution (including tail effects) and outline the computations for the dissipative contributions as well.Our spin-orbit results agree with N 3 LO post-Newtonian and test-body data in the respective limits.We also re-confirm the conservative 4PM non-spinning results.
High-precision predictions for the gravitational waves emitted from the interaction of compact binaries are essential for data analysis of gravitational wave detectors [1][2][3][4][5][6].They are the prerequisites to address fundamental questions in astro-, gravitational, particle and nuclear physics through observations of gravitational waves.The third generation of detectors -LISA, Einstein Telescope and Cosmic Explorer [7], scheduled to go online in the 2030s -will reach an experimental accuracy that goes well beyond the present state-of-the-art in analytical and numerical gravitational wave physics [8,9].This situation has sparked a renewed effort to extend and innovate traditional approaches to the classical relativistic twobody problem.
In this Letter we provide the conservative, spin-orbit contributions to the impulse and spin kick at 4PM ac-curacy, together with the total scattering angle.These results provide the basis to refine effective one-body Hamiltonians and resummed scattering prescriptions for high-precision gravitational wave physics.Our worldline quantum field theory (WQFT) hinges on three innovations to the EFT approach for gravitational scattering: (i) quantizing both the worldline degrees of freedom and the gravitational field allows for a diagrammatic formulation of the classical perturbation theory yielding the observables as one-point functions of the worldline or gravitational fields [94], (ii) capturing the spin of the compact objects through a supersymmetric worldline theory [97], (iii) the Schwinger-Keldysh (in-in) initial value formulation of WQFT that induces the use of retarded propagators and a causality flow in the diagrammatic expansion [99].
Supersymmetric in-in WQFT formalism.-The effective worldline theory of spinning bodies (Kerr BHs or NSs) with masses m i and space-time coordinates x µ i (τ ) on a general D-dimensional space-time with metric g µν is described up to quadratic order in spin by an N = 2 supersymmetric worldline theory [97].As we are focusing on the spin-orbit (linear-in-spin) dynamics here, the N = 1 incarnation of this theory will suffice: The real anti-commuting vectors ψ a i (τ ) are defined in a flat tangent space using the vierbein e µ a and Dψ a i Dτ = ψa i + ẋµ ω µ ab ψ i,b with the spin-connection ω µ ab (our metric is mostly minus).We work in D = 4 − 2ǫ dimensions with S EH the bulk Einstein-Hilbert action including a gauge-fixing term; the process of dimensional regularisation, wherein we ultimately send ǫ → 0, is aided by only this part of the full action needing to be lifted to D dimensions.The ψ a i (τ ) carry the spin degrees of freedom with the spin tensors We expand the fields around their respective backgrounds: the metric g µν = η µν + κh µν , with κ = √ 32πG, and the worldlines where is the covariant impact parameter.We also introduce the Lorentz factor γ = v 1 • v 2 and the relative velocity v = γ 2 − 1/γ.Causal observables including radiative effects arise from the Schwinger-Keldysh (in-in) formalism applied to WQFT [99] where one doubles the fields: normalized such that 1 = 1 and with { } (n) denoting the (n)'th copy of the doubled fields.The key property we exploit is that the WQFT tree-level one-point functions solve the classical equations of motion.Moreover, the computation of one-point functions of in-in WQFT reduces to the use of retarded propagators combined with the standard in-out WQFT Feynman rules [99].This formalism yields an efficient QFT-based scheme to solve the classical equations perturbatively.
Conservative observable can in turn be defined by neglecting all interactions between h (1) µν .This may be achieved by using the in-in formalism only for the worldlines while keeping the in-out formalism for the gravitons and projecting on the real part of observables [123,124].This separation of conservative effects at 4PM has proven its efficiency for the non-spinning results [89,101].
WQFT Feynman rules.-The graviton propagator in de Donder gauge with Feynman prescription reads with P µν;ρσ := η µ(ρ η σ)ν − 1 D−2 η µν η ρσ while the worldline propagators associated with z µ i and ψ ′µ i read, respectively ( The arrow on the propagators indicates the momentum or energy flow on the retarded propagators.Importantly, the Feynman graviton propagators reflect our focus on conservative observables.Full dissipative results may be obtained by using retarded propagators instead.The Feynman vertices of the spinning WQFT to lower multiplicities have been exposed in [97].The generic worldline vertex couples n gravitons to m worldline fields and reads where k µ = n i=1 k µ i is the total outflowing fourmomentum and the dotted outgoing line symbolizes the background parameters {b µ , v µ , Ψ µ } of Eq. ( 2).We see that only energy is conserved on the worldline.The bulk graviton vertices are generic.At 4PM order we need the worldline vertices V n|m above for {n = 1, . . ., 4; m = 0, . . ., 5 − n}, and the 3-,4-,5-graviton vertices.
where we have Fourier transformed to momentum space.Both observables are given as the sum of all diagrams at a given PM order with one outgoing Z µ i line with vanishing energy.The spin kick is subsequently derived from the kick of the Grassmann variable as in Ref. [62].
Integrand generation.-The 4PM impulse and spinkick integrands are generated recursively via Berends-Giele type relations.The one-point functions for the worldline "super-fields" Z i = {z i , ψ ′ i } and for the graviton are represented as Their recursive definitions follow from the Schwinger-Dyson equations and are depicted in Fig. 1.Spelling this out systematically to order G 4 allows for an algorithmic construction of the integrand: in our case, we efficiently inserted Feynman rules into the generated trees using FORM [125].There are 201 graphs contributing to the 4PM impulse in the non-spinning case, 529 with spin and 253 contributing to the 4PM spin kick.
Reduction to scalar integrals.-A generic 4PM diagram after performing the worldline energy integrals via the δ-functions in Eq. ( 6) takes the form where the D i are either linear or massless propagators depending on the loop momenta ℓ i , velocities v i and momentum transfer q.The numerators num[ℓ i ] are polynomial in loop momenta.Tensor reduction of num[ℓ i ] to scalar integrals is performed by expanding the loop momenta on a basis dual to v µ i and q µ , as demonstrated in the 3PM case [98].The only dimensionful quantity in the 3-loop ℓ i integral is the momentum transfer q µ .Hence, |q| = −q 2 may be scaled out, and the remaining 3-loop integrals depend only on the Lorentz factor γ.
The specific choice of three δ(ℓ k • v i k ) functions in Eq. ( 9) follows the mass dependence of a given diagram, which scales as m 1 m 2 m i1 m i2 m i3 .Diagrams are thereby grouped into two categories: test-body contributions with mass dependence m 4  1 m 2 or m 1 m 4 2 and comparablemass contributions m 3 1 m 2 2 , m 2 1 m 3 2 -see Fig. 2. For the conservative impulse we can easily reconstruct the m 1 m 4 2 and m 2 1 m 3 2 components using ∆p µ 1,cons = −∆p µ 2,cons , the impulse on the second body being given simply by relabeling the two worldlines.When computing ∆ψ µ 1,cons no similar relation exists; however, the integrals in opposing mass sectors are also related by a trivial relabeling.
Integral families and reduction to masters.-There are three integral families that need to be reduced to master integrals.The first 4PM family is (i = 1, 2) with the propagators (j = 1, 2, 3 and k = 1, 2): and 1 = 2, 2 = 1.The I [1] and I [2] families contribute to the test-body and comparable-mass regimes respectively.The other 4PM family is given by

FIG. 2:
Examples of comparable-mass graphs with mass dependence m 2 1 m 3 2 contributing to the 4PM calculation.One should attach an outgoing worldline to any worldline node and apply the resulting causality flow.The corresponding scalar integrals feature as top sectors in the differential equations: All graphs can be described by the J family (11), except for the last graph belonging to the I family (10).The first two graphs give rise to the elliptic functions in the final result.The second-to-last graph is non-zero only in the (PR+RP) region and therefore does not contribute to the conservative results in this paper.
Each family splits into two branches: even (b-type) or odd (v-type) in the number of worldline propagators.
In the non-spinning impulse, these integrals multiply terms proportional to b µ , v µ i respectively (16).Using integration-by-parts (IBP) relations [126][127][128][129] we reduce the families to 23 master integrals for the I-b and I-v types each as well as 64 of J-b type and 66 of J-v type.The complete spinning impulse computation (including dissipation) results in approximately 10 5 integrals for reduction to scalar masters.
Differential equations.-To solve for the master integrals we employ the method of canonical differential equations (DEs) [54].Each master integral family is grouped into a vector I ordered according to the number of active propagators.The DE in x = γ − γ 2 − 1 reads d I/dx = M (ǫ, x) I with a lower-block triangular matrix M (ǫ, x).Finding a transformation matrix T that brings us to a canonical basis with an ǫ factorized DE d Ĩ/dx = ǫA(x) Ĩ is a highly involved procedure in which we employ the packages [130][131][132][133].The resulting symbol alphabet is {x, 1 + x, 1 − x, 1 + x 2 }, and we encounter elliptic integrals in the J-b family [103,130].
Fixing boundary conditions .-Boundary conditions on the master integrals are determined in the static limit (γ → 1, v → 0) using the method of regions [82,101,134,135] to expand the integrand in v. Regions in the static limit are characterized by different scalings of the bulk graviton loop momenta with potential (P) and radiative (R) modes defined by relative scalings of their spacial and timelike components: Only gravitons which may go on-shell can be radiative and there are at most two of these defining the three regions: (PP), (RR) and (PR+RP).The regions (PP) and (PR+RP) are purely conservative and dissipative respectively, while the (RR) region carries both kinds of effects.In the (PP) region the integrals reduce to test-body integrals described by the I [1] family (10); in the (RR) region they reduce to double-mushroom integrals like the first graph of Fig. 3. Either way, the boundary integrals are independent of γ and thus functions only of D = 4 − 2ǫ.
In our conservative observables we include only the (PP) and (RR) regions as the (PR+RP) region generates terms odd in v.
Reaching 4PM order introduces the physical phenomenon of tail effects [89,101].In the 4PM contribu-FIG.3: Two examples of graphs contributing in the (RR) region but not the (PP) region.
tions to an observable X (impulse, spin kick or scattering angle) poles in ǫ = 4−D 2 appear in the (PP) and conservative (RR) contributions: higher-order terms being finite as ǫ → 0. Non-analytic dependence on v −4ǫ in the (RR) region is a direct consequence of the velocity scaling of the two radiative gravitons.The cancelation of these poles when assembling X cons introduces logarithmic velocity dependence: the dots indicating terms that are rational in γ 2 − 1 in the static limit.
Results.-We begin with ∆p (4)µ i,cons , the G 4 component of the impulse ∆p µ i,cons .It may be decomposed as ∆p where the basis vectors and spin structures ρ There are five and eight elements in ρ and the much simpler set The first line of Eq. ( 17) includes transcendental weight-1 functions, the second and third lines weight-2 functions and the final line quadratic combinations of elliptic functions of the first and second kind.The barred coefficients c(σ) l and dσ l may be obtained from the unbarred ones by relabeling using ∆p The G 4 component of the spin kick ∆S (4)µ i,cons admits a similar decomposition, involving the same functions F (b,v) α but a different set of basis vectors and spin structures: and takes the schematic form Here e (σ) l (γ) and f α,l (γ) and their barred counterparts are rational functions (again, up to integer powers of γ 2 − 1).For the full expression we refer the reader to the ancillary file.
As checks on these two observables we have confirmed: (i) the cancellation of all 1/ǫ poles occurring between the (PP) and (RR) regions; (ii) conservation of p 2 i , S 2 i and the N = 1 global supercharge Q i = p i •ψ i .While the first two only check the simpler terms carrying F We also define the total scattering angle θ for generic spin configurations as with p ∞ = m 1 m 2 γ 2 − 1/E, total energy E = |p µ 1 + p µ 2 | and total mass M = m 1 + m 2 , n and m counting PM and spin orders respectively.The 4PM spin-orbit contribution is θ (4,1)  cons where the test-body contributions (second line) agree with the geodesic motion in a Kerr background [136].
Here we use the mass parameters ν = m 1 m 2 /M 2 and δ = (m 2 −m 1 )/M and we have defined s ± = −(a 1 ±a 2 )• L. The 32 polynomial functions h α (γ) are given in the supplementary material (23).We have checked this result against the corresponding N 3 LO PN [39,41] literature, and found agreement by taking the PN expansion.The tail term P (4) θ (γ) of the scattering angle is simply related to the 3PM radiated energy E (3) rad as follows: where J = p ∞ |b| is the initial angular momentum.This equation follows the pattern derived in Ref. [137] and constitutes another non-trivial check of our results.All of our results are included in an ancillary file attached to the arXiv submission of this Letter.
Outlook.-Having produced a complete set of 4PM linear-in-spin conservative scattering observablesand successfully compared them with N 3 LO spin-orbit PN [39,41]-our next step will be upgrading them to include dissipative effects, as has already been done in the non-spinning case [102,103].This will require two changes to our setup: retarded graviton propagators in place of time-symmetric Feynman (see Ref. [99]) and incorporation of the (PR+RP) regions when fixing boundary conditions on master integrals.Notwithstanding the added complexity, quadratic-in-spin order is also an achievable target -corresponding N 3 LO quadraticin-spin PN results are already available [38,40].In the near future we also seek to use these results to describe bound orbits, the main obstacle being the aforementioned tail effect [89,101].Recent Numerical Relativity simulations of spinning black holes on hyperbolic-like orbits [138] also offer us future numerical comparisons of the scattering angle θ.

SUPPLEMENTARY MATERIAL
Scattering angle.-The 32 rational functions h ± α (γ) (up to integer powers of γ 2 − 1) appearing in the spin-orbit contribution to the scattering angle (21) take the explicit form

FIG. 1 :
FIG.1: Berends-Giele type recursion relation to construct Z µ i (ω) and hµν (k) perturbatively.The causality flow is always from the Zi and h blobs to the outgoing line.They are equivalent to the PM-expanded geodesic and Einstein equations.