We develop the on-shell action formalism within Worldline Quantum Field Theory (WQFT) to describe scattering of spinning compact bodies in General Relativity in the post-Minkowskian (PM) expansion. The real on-shell action is constructed from vacuum diagrams with causal (retarded) propagators from which scattering observables such as momentum impulse and spin kick follow via Poisson brackets of the initial scattering data. Furthermore, we explore the implications of unitarity at the level of the worldline and show how generalised unitarity techniques can be adapted to WQFT to efficiently compute multi-loop contributions. Our work establishes a concrete link between WQFT and amplitude-based methods, elucidating how unitarity cuts ensure equivalence between the on-shell action derived from either approach. Extending the state-of-the-art, we complete the full on-shell action — including dissipative terms — at (formal) 3PM order and up to quartic spin interactions on both massive bodies.

Germany

Kays Haddad

Gustav Jakobsen

United Kingdom

Gustav Mogull

Jan Plefka

Unitarity and the On-Shell Action of Worldline Quantum Field Theory

<p>Using the worldline quantum field theory approach, we derive solutions to the equations of motion for spinning massive bodies up to quadratic order in spins. At leading post-Minkowskian (PM) order, these trajectories are obtained in the time domain, and at subleading order, they are obtained in the frequency domain. Our approach incorporates diagrammatic techniques and modern Feynman integration technologies and includes a family of loop integrals different to those seen in asymptotic PM calculations. Our results provide a new mechanism for computing the radiated angular momentum involved in gravitational scattering, which we reproduce at 2PM order up to linear spins. We have established a framework for computing higher-order effects to further extend the high-precision frontier in analytical gravitational wave physics and push predictions for the radiated angular momentum to higher perturbative orders.</p>

Kathrin Stoldt

Radiated angular momentum from spinning black hole scattering trajectories

We investigate the gravitational scattering of a spinning probe mass in a Kerr background using the worldline quantum field theory approach. This corresponds to the leading term (0SF) in the gravitational self-force expansion for the spinning two-body problem with large mass hierarchy. By reformulating the geodesic and Mathisson-Papapetrou-Dixon equations as a recursive Berends-Giele type equation known from multi-gluon scattering, we develop a novel integration-by-parts formalism on the worldline that enables systematic computation of scattering observables — specifically the impulse and spin kick — to arbitrary orders in Newton’s constant and spin. Here the transition to a position space formalism is key. We present explicit results up to and including the physical 7PM order, thereby incorporating all relevant higher-spin and higher-curvature terms on the worldline, advancing beyond previous calculations. This work represents an initial step to reconceptualise the gravitational self-force expansion through worldline quantum field theory.

Jitze Hoogeveen

Spinning the probe in Kerr with WQFT

Worldline quantum field theory (WQFT) has proven itself a powerful tool for classical two-body scattering calculations in general relativity. In this paper we develop a new worldline action involving bosonic oscillators, which enables the use of the WQFT formalism to describe massive compact bodies to all orders in their spins. Inspired by bosonic string theory in the tensionless limit, we augment traditional trajectory variables with bosonic oscillators capturing the spin dependence. We show its equivalence to the covariant phase space description of a spinning body in curved space and clarify the role of the spin-supplementary condition in a Hamiltonian treatment. Higher-spin Hamiltonians are classified to linear and quadratic order in curvature. Finally, perturbative computations at 1PM order for arbitrary powers and orientations of spin and at 2PM up to quartic spin order are performed, recovering results from the literature.

Spinning bodies in general relativity from bosonic worldline oscillators

The high-precision description of black hole scattering in classical general relativity using the post-Minkowskian (PM) expansion requires the evaluation of single-scale Feynman integrals at increasing loop orders. Up to 4PM, the scattering angle and the impulse are expressible in terms of polylogarithmic functions and Calabi-Yau (CY) twofold periods. As in QFT, periods of higher dimensional CY <math><mi>n</mi></math>-folds are expected at higher PM order. We find at 5PM in the dissipative leading order self-force sector (5PM-1SF) that the only nonpolylogarithmic functions are the K3 periods encountered before and the ones of a new hypergeometric CY threefold. In the 5PM-2SF sector further CY twofold and threefold periods appear. Griffiths transversality of the CY period motives allows one to transform the differential equations for the master integrals into <math><mi>ε</mi></math>-factorized form and to solve them in terms of a well-controlled function space, as we demonstrate in the 5PM-1SF sector.

Albrecht Klemm

Christoph Nega

Benjamin Sauer

Calabi-Yau periods for black hole scattering in classical general relativity

We compute the fifth post-Minkowskian (5PM) order contributions to the scattering angle and impulse of classical black hole scattering in the conservative sector at first self-force order using the worldline quantum field theory formalism. This challenging four-loop computation required the use of advanced integration-by-parts and differential equation technology implemented on high-performance computing systems. Use of partial fraction identities allowed us to render the complete integrand in a fully planar form. The resulting function space is simpler than expected: In the scattering angle, we see only multiple polylogarithms up to weight three and a total absence of the elliptic integrals that appeared at 4PM order. All checks on our result, both internal—cancellation of dimensional regularization poles and preservation of the on-shell condition—and external—matching the slow-velocity limit with the post-Newtonian (PN) literature up to 5PN order and matching the tail terms to the 4PM loss of energy—are passed.

Mathias Driesse

Gustav Uhre Jakobsen

CERN

Johann Usovitsch

Conservative Black Hole Scattering at Fifth Post-Minkowskian and First Self-Force Order

We determine the adiabatic tidal contributions to the radiation reacted momentum impulse <math><mrow><mi>Δ</mi><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>μ</mi></mrow></msubsup></mrow></math> and scattering angle <math><mi>θ</mi></math> between two scattered massive bodies (neutron stars) at next-to-next-to-leading post-Minkowskian order. The state-of-the-art three-loop (fourth post-Mikowskian order) worldline quantum field theory toolkit using dimensional regularization is employed to establish the classical observables. We encounter divergent terms in the gravitoelectric and gravitomagnetic quadrupolar sectors necessitating the addition of postadiabatic counterterms in this classical theory. This leads us to include also the leading postadiabatic tidal contributions to the observables. The resulting renormalization group flow of the associated postadiabatic Love numbers is established and shown to agree with a recent gravitoelectric third post-Newtonian analysis in the nonrelativistic limit.

Tidal effects and renormalization at fourth post-Minkowskian order

We compute the radiation reacted momentum impulse <math><mrow><mi>Δ</mi><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>μ</mi></mrow></msubsup></mrow></math>, spin kick <math><mi>Δ</mi><msubsup><mi>S</mi><mi>i</mi><mi>μ</mi></msubsup></math>, and scattering angle <math><mi>θ</mi></math> between two scattered spinning massive bodies (black holes or neutron stars) using the <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric worldline quantum field theory formalism up to fourth post-Minkowskian (4PM) order. Our calculation confirms the state-of-the-art nonspinning results, and extends them to include spin-orbit effects. Advanced multiloop Feynman integral technology including differential equations and the method of regions are applied and extended to deal with the retarded propagators arising in a causal description of the scattering dynamics. From these results we determine a complete set of radiative fluxes at subleading PM order: the 4PM radiated four-momentum and, via linear response, the 3PM radiated angular momentum, both again including spin-orbit effects.

Dissipative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order

Using the <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric, spinning worldline quantum field theory formalism, we compute the conservative spin-orbit part of the momentum impulse <math><mrow><mi>Δ</mi><msubsup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow><mrow><mi>μ</mi></mrow></msubsup></mrow></math>, spin kick <math><mi>Δ</mi><msubsup><mi>S</mi><mi>i</mi><mi>μ</mi></msubsup></math>, and scattering angle <math><mi>θ</mi></math> 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 multiloop 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 next-to-next-to-next-to-leading-order post-Newtonian and test-body data in the respective limits. We also reconfirm the conservative 4PM nonspinning results.

Yingxuan Xu

Conservative Scattering of Spinning Black Holes at Fourth Post-Minkowskian Order

We exhibit an initial-value formulation of the worldline quantum field theory (WQFT) approach to the classical two-body problem in general relativity. We show that the Schwinger-Keldysh (in-in) formalism leads to purely retarded propagators in the evaluation of observables in the WQFT. Integration technology for retarded master integrals is introduced at third post-Minkowskian (3PM) order. As an application we compute the complete radiation-reacted impulse and radiated four momentum for the scattering of two non-spinning neutron stars including tidal effects at 3PM order, as well as the leading (2PM) far-field gravitational waveform.