arXiv subjects

Michael S. Ruf

Publications and source records attributed to Michael S. Ruf.

At least 19 recordsLinked to original sources

arXiv2026

Spinning Black Hole Scattering at $\mathcal{O}(G^3 S^2)$: Casimir Terms, Radial Action and Hidden Symmetry

We resolve subtleties in calculating the post-Minkowskian dynamics of binary systems, as a spin expansion, from massive scattering amplitudes of fixed finite spin. In particular, the apparently ambiguous spin Casimir terms can be fully determined from the gradient of the spin-diagonal part of the amplitudes with respect to $S^2 = -s(s+1)\hbar^2$, using an interpolation between massive amplitudes with different spin representations. From two-loop amplitudes of spin-0 and spin-1 particles minimally coupled to gravity, we extract the spin Casimir terms in the conservative scattering angle between a spinless and a spinning black hole at $\mathcal{O}(G^3 S^2)$, finding agreement with known results in the literature. This completes an earlier study [Phys. Rev. Lett. 130 (2023), 021601] that calculated the non-Casimir terms from amplitudes. We also illustrate our methods using a model of spinning bodies in electrodynamics, finding agreement between scattering amplitude predictions and classical predictions in a root-Kerr electromagnetic background up to $\mathcal{O}(α^3 S^2)$. For both gravity and electrodynamics, the finite part of the amplitude coincides with the two-body radial action in the aligned spin limit, generalizing the amplitude-action relation beyond the spinless case. Surprisingly, the two-loop amplitude displays a hidden spin-shift symmetry in the probe limit, which was previously observed at one loop. We conjecture that the symmetry holds to all orders in the coupling constant and is a consequence of integrability of Kerr orbits in the probe limit at the first few orders in spin.

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arXiv2026

Three-Loop QCD Corrections to Scattering Amplitudes of a Higgs Boson and Three Partons

We present helicity amplitudes at three loops in the heavy-top quark effective theory of QCD for the scattering of a Higgs boson and three light partons with full-color dependence. To obtain these results, we use integration-by-parts identities implemented in the code Blade and evaluate the master integrals using canonical differential equations. We obtain compact analytic expressions in terms of generalized polylogarithms suitable for numerical evaluation. We observe that sub-leading color contributions at three loops are numerically as significant as leading-color terms. Our amplitudes are key ingredients for precision LHC phenomenology, for example in production cross sections involving a Higgs boson and a hadronic jet.

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arXiv2026

Classical Gravitational Scattering from the Ultraviolet and the Absence of Calabi-Yau Integrals in the Conservative Sector at $O(G^5)$

We explain why Calabi-Yau and complete elliptic integrals do not contribute to conservative observables at fifth post-Minkowskian order, despite appearing in intermediate steps. At even loop orders, conservative contributions are tied to terms proportional to the logarithm of the momentum transfer, which in dimensional regularization arise from singular regions. We show that in the classical limit, the integral classes responsible for Calabi-Yau and complete elliptic behavior are absent from the ultraviolet singular structures that generate the required logarithm. This perspective also suggests alternative strategies for analyzing the classical limit of multiloop integrals in the conservative sector at even loop orders.

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arXiv2026

Scattering Amplitudes and Conservative Binary Dynamics at $O(G^5)$ without Self-Force Truncation

We compute the complete potential-graviton contributions to the conservative radial action and scattering angle for two non-spinning bodies in general relativity, accurate through fifth order in Newton's constant and including second-order self-force (2SF) effects. The calculation is carried out in the scattering-amplitude framework, combining the double copy, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. To address a major computational bottleneck, we develop improved integration-by-parts algorithms that render calculations at this order tractable. The post-Minkowskian amplitude is presented as a series expansion, following the strategy used earlier in maximal supergravity. For the first self-force sector, which involves only polylogarithmic functions, we also provide a closed-form analytic expression. For the second self-force sector, as in earlier supergravity work, we find nontrivial cancellations among contributions related to integrals supported on Calabi-Yau geometry.

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arXiv2026

Second-order self-force potential-region binary dynamics at $O(G^5)$ in supergravity

We compute the potential-graviton contributions to the conservative scattering angle of two non-spinning bodies in maximal supergravity at fifth order in Newton's constant, including second-order self-force effects. Our goal is to tackle the challenging integrals arising at this order in Einstein gravity, but within the technically simpler framework of supergravity. The calculation employs the scattering-amplitude framework, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. The final result is expressed as a series expansion around the static limit, thereby avoiding the explicit evaluation of intricate special functions. This series solution for the master integrals applies, as well, to the corresponding computation in general relativity. Remarkably, we observe nontrivial cancellations among contributions associated with Calabi-Yau integrals, alongside a distinct contribution governed by a Heun differential equation.

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arXiv2025

Amplitudes, Supersymmetric Black Hole Scattering at $\mathcal{O}(G^5)$, and Loop Integration

We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in N = 8 supergravity at the fifth post-Minkowskian order to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present all master integrals and solve their associated differential equations necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in N = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in N = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.

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arXiv2024

Global Bases for Nonplanar Loop Integrands, Generalized Unitarity, and the Double Copy to All Loop Orders

We introduce a constructive method for defining a global loop-integrand basis for scattering amplitudes, encompassing both planar and nonplanar contributions. Our approach utilizes a graph-based framework to establish a well-defined, non-redundant basis of integrands. This basis, constructed from a chosen set of non-redundant graphs together with a selection of irreducible scalar products, provides clear insights into various physical properties of scattering amplitudes and proves useful in multiple contexts, such as on-shell Ward identities and manifesting gauge-choice independence. A key advantage of our integrand basis is its ability to streamline the generalized unitarity method. Specifically, we can directly read off the coefficients of basis elements without resorting to ansatze or solving linear equations. This novel approach allows us to lift generalized unitarity cuts -- expressed as products of tree amplitudes -- to loop-level integrands, facilitating the use of the tree-level double copy to generate complete gravitational integrands at any loop order. This method circumvents the difficulties in identifying complete higher-loop-order gauge-theory integrands that adhere to the color-kinematics duality. Additionally, our cut-based organization is well-suited for expansion in hard or soft limits, aiding in the exploration of ultraviolet or classical limits of scattering amplitudes.

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arXiv2024

Absorptive Effects and Classical Black Hole Scattering

We describe an approach to incorporating the physical effects of the absorption of energy by the event horizon of black holes in the scattering amplitudes based post-Minkowskian, point-particle effective description. Absorptive dynamics are incorporated in a model-independent way by coupling the usual point-particle description to an invisible sector of gapless internal degrees-of-freedom. The leading order dynamics of this sector are encoded in the low-energy expansion of a spectral density function obtained by matching an absorption cross section in the ultraviolet description. This information is then recycled using the scattering amplitudes based Kosower-Maybee-O'Connell in-in formalism to calculate the leading absorptive contribution to the impulse and change in rest mass of a Schwarzschild black hole scattering with a second compact body sourcing a massless scalar, electromagnetic or gravitational field. The results obtained are in complete agreement with previous worldline Schwinger-Keldysh calculations and provide an alternative on-shell scattering amplitudes approach to incorporating horizon absorption effects in the gravitational two-body problem.

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arXiv2023

Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model

We compare numerical self-force results and analytical fourth-order post-Minkowskian (PM) calculations for hyperbolic-type scattering of a point-like particle carrying a scalar charge $Q$ off a Schwarzschild black hole, showing a remarkably good agreement. Specifically, we numerically compute the scattering angle including the full $O(Q^2)$ scalar-field self-force term (but ignoring the gravitational self-force), and compare with analytical expressions obtained in a PM framework using scattering-amplitude methods. This example provides a nontrivial, high-precision test of both calculation methods, and illustrates the complementarity of the two approaches in the context of the program to provide high-precision models of gravitational two-body dynamics. Our PM calculation is carried out through 4PM order, i.e., including all terms through $O(Q^2 G^3)$. At the fourth post-Minkowskian order the point-particle description involves two a-priori undetermined coefficients, due to contributions from tidal effects in the model under consideration. These coefficients are chosen to align the post-Minkowskian results with the self-force ones.

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arXiv2023

Conservative Binary Dynamics with a Spinning Black Hole at $\mathcal{O}(G^3)$ from Scattering Amplitudes

We compute the conservative two-body Hamiltonian of a compact binary system with a spinning black hole through $\mathcal{O}(G^3)$ to all orders in velocity, including linear and quadratic spin terms. To obtain our results we calculate the classical limit of the two-loop amplitude for the scattering of a massive scalar particle with a massive spin-1 particle minimally coupled to gravity. We employ modern scattering amplitude and loop integration techniques, in particular numerical unitarity, integration-by-parts identities, and the method of regions. The conservative potential in terms of rest-frame spin vectors is extracted by matching to a non-relativistic effective field theory. We also apply the Kosower-Maybee-O'Connell (KMOC) formalism to calculate the impulse in the covariant spin formalism directly from the amplitude. We work systematically in conventional dimensional regularization and explicitly evaluate all divergent integrals that appear in full- and effective-theory amplitudes, as well as in the phase-space integrals that arise in the KMOC formalism.

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arXiv2023

Conservative binary dynamics at order $O(α^5)$ in electrodynamics

We compute the potential-photon contributions to the classical relativistic scattering angle of two charged non-spinning bodies in electrodynamics through fifth order in the coupling. We use the scattering amplitudes framework, effective field theory, and multi-loop integration techniques based on integration by parts and differential equations. At fifth order, the result is expressed in terms of cyclotomic polylogarithms. Our calculation demonstrates the feasibility of the corresponding calculations in general relativity, including the evaluation of the encountered four-loop integrals.

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arXiv2021

Scattering Amplitudes, the Tail Effect, and Conservative Binary Dynamics at $O(G^4)$

We complete the calculation of conservative two-body scattering dynamics at fourth post-Minkowskian order, i.e. $O(G^4)$ and all orders in velocity, including radiative contributions corresponding to the tail effect in general relativity. As in previous calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves complete elliptic integrals, and polylogarithms with up to transcendental weight two. Using the amplitude-action relation, we obtain the radial action directly from the amplitude, and match the known overlapping terms in the post-Newtonian expansion.

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arXiv2021

Scattering Amplitudes and Conservative Binary Dynamics at ${\cal O}(G^4)$

Using scattering amplitudes, we obtain the potential contributions to conservative binary dynamics in general relativity at fourth post-Minkowskian order, ${\cal O}(G^4)$. As in previous lower-order calculations, we harness powerful tools from the modern scattering amplitudes program including generalized unitarity, the double copy, and advanced multiloop integration methods, in combination with effective field theory. The classical amplitude involves polylogarithms with up to transcendental weight two and elliptic integrals. We derive the radial action directly from the amplitude, and determine the corresponding Hamiltonian in isotropic gauge. Our results are in agreement with known overlapping terms up to sixth post-Newtonian order, and with the probe limit. We also determine the post-Minkowskian energy loss from radiation emission at ${\cal O}(G^3)$ via its relation to the tail effect.

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arXiv2021

Gravitational Bremsstrahlung from Reverse Unitarity

We compute the total radiated momentum carried by gravitational waves during the scattering of two spinless black holes at the lowest order in Newton's constant, $\mathcal O(G^3)$, and all orders in velocity. By analytic continuation into the bound state regime, we obtain the ${\cal O}(G^3)$ energy loss in elliptic orbits. This provides an essential step towards the complete understanding of the third-post-Minkowskian binary dynamics. We employ the formalism of Kosower, Maybee, and O'Connell (KMOC) which relates classical observables to quantum scattering amplitudes and derive the relevant integrands using generalized unitarity. The subsequent phase-space integrations are performed via the reverse unitarity method familiar from collider physics, using differential equations to obtain the exact velocity dependence from near-static boundary conditions.

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arXiv2021

Radiative Classical Gravitational Observables at $\mathcal O(G^3)$ from Scattering Amplitudes

We compute classical gravitational observables for the scattering of two spinless black holes in general relativity and $\mathcal N {=} 8$ supergravity in the formalism of Kosower, Maybee, and O'Connell (KMOC). We focus on the gravitational impulse with radiation reaction and the radiated momentum in black hole scattering at $\mathcal O(G^3)$ to all orders in the velocity. These classical observables require the construction and evaluation of certain loop-level quantities which are greatly simplified by harnessing recent advances from scattering amplitudes and collider physics. In particular, we make use of generalized unitarity to construct the relevant loop integrands, employ reverse unitarity, the method of regions, integration-by-parts (IBP), and (canonical) differential equations to simplify and evaluate all loop and phase-space integrals to obtain the classical gravitational observables of interest to two-loop order. The KMOC formalism naturally incorporates radiation effects which enables us to explore these classical quantities beyond the conservative two-body dynamics. From the impulse and the radiated momentum, we extract the scattering angle and the radiated energy. Finally, we discuss universality of the impulse in the high-energy limit and the relation to the eikonal phase.

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arXiv2021

Extremal black hole scattering at O(G^3): graviton dominance, eikonal exponentiation, and differential equations

We use $\mathcal N=8$ supergravity as a toy model for understanding the dynamics of black hole binary systems via the scattering amplitudes approach. We compute the conservative part of the classical scattering angle of two extremal (half-BPS) black holes with minimal charge misalignment at $\mathcal O(G^3)$ using the eikonal approximation and effective field theory, finding agreement between both methods. We construct the massive loop integrands by Kaluza-Klein reduction of the known $D$-dimensional massless integrands. To carry out integration we formulate a novel method for calculating the post-Minkowskian expansion with exact velocity dependence, by solving velocity differential equations for the Feynman integrals subject to modified boundary conditions that isolate conservative contributions from the potential region. Motivated by a recent result for universality in massless scattering, we compare the scattering angle to the result found by Bern et. al. in Einstein gravity and find that they coincide in the high-energy limit, suggesting graviton dominance at this order.

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arXiv2020

Universality in the classical limit of massless gravitational scattering

We demonstrate the universality of the gravitational classical deflection angle of massless particles through O(G^3) by studying the high-energy limit of full two-loop four-graviton scattering amplitudes in pure Einstein gravity as well as N >= 4 supergravity. As a by-product, our first-principles calculation provides a direct confirmation of the massless deflection angle in Einstein gravity proposed long ago by Amati, Ciafaloni and Veneziano, and is inconsistent with a recently proposed alternative.

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arXiv2018

Quantum effective action for degenerate vector field theories

We calculate the divergent part of the one-loop effective action in curved spacetime for a particular class of second-order vector field operators with a degenerate principal part. The principal symbol of these operators has the structure of a longitudinal projector. In this case, standard heat-kernel techniques are not directly applicable. We present a method which reduces the problem to a nondegenerate scalar operator for which standard heat-kernel techniques are available. Interestingly, this method leads to the identification of an effective metric structure in the longitudinal sector. The one-loop divergences are compactly expressed in terms of invariants constructed from this metric.

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