arXiv subjects

Zvi Bern

Publications and source records attributed to Zvi Bern.

At least 19 recordsLinked to original sources

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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arXiv2025

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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arXiv2025

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

Observables and Unconstrained Spin Tensor Dynamics in General Relativity from Scattering Amplitudes

In a previous Letter, we showed that physical scattering observables for compact spinning objects in general relativity can depend on additional degrees of freedom in the spin tensor beyond those described by the spin vector alone. In this paper, we provide further details on the physics of these additional degrees of freedom, whose commutation relations and Poisson brackets are inherited from the underlying Lorentz symmetry, and on their consequence on observables. In particular, we give the waveform at leading order in Newton's constant and up to second order in the components of the spin tensor, and the conservative impulse, boost and spin kick, exhibiting spin magnitude change, through next-to-leading-order in Newton's constant and third order in the components of the spin tensor. We provide explicit examples -- a Newtonian two-particle bound system and a certain black-hole solution in an exotic matter-coupled gravitational theory -- that exhibit these degrees of freedom and are described by our four-dimensional and worldline field theories. We also discuss connections between these degrees of freedom and dynamical worldline multipole moments. We construct effective two-body Hamiltonians, we demonstrate explicitly that the extra degrees of freedom beyond the spin vector are necessary to describe the complete dynamics, and we explicitly remove certain unphysical singularities. Moreover, we show that the previously proposed eikonal (or radial action) formula correctly captures observables derived from the classical Hamiltonian. Finally, we comment on possible descriptions of the additional degrees of freedom from the perspective of Goldstone's theorem.

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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

Conservative Spin Magnitude Change in Orbital Evolution in General Relativity

We show that physical scattering observables for compact spinning objects in general relativity can depend on additional degrees of freedom in the spin tensor beyond those described by the spin vector alone. The impulse, spin kick, and leading-order waveforms exhibit such a nontrivial dependence. A signal of this additional structure is the change in the magnitude of the spin vector under conservative Hamiltonian evolution, similar to our previous studies in electrodynamics. These additional degrees of freedom describe dynamical mass multipoles of compact objects and decouple for black holes. We also show that the conservative impulse, spin kick and change of the additional degrees of freedom are encoded in the eikonal phase.

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arXiv2024

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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arXiv2023

Quantum Field Theory, Worldline Theory, and Spin Magnitude Change in Orbital Evolution

A previous paper~\cite{Bern:2022kto} identified a puzzle stemming from the amplitudes-based approach to spinning bodies in general relativity: additional Wilson coefficients appear compared to current worldline approaches to conservative dynamics of generic astrophysical objects, including neutron stars. In this paper we clarify the nature of analogous Wilson coefficients in the simpler theory of electrodynamics. We analyze the original field-theory construction, identifying definite-spin states some of which have negative norms, and relating the additional Wilson coefficients in the classical theory to transitions between different quantum spin states. We produce a new version of the theory which also has additional Wilson coefficients, but no negative-norm states. We match, through $\mathcal O(\alpha^2)$ and $\mathcal O(S^2)$, the Compton amplitudes of these field theories with those of a modified worldline theory with extra degrees of freedom introduced by releasing the spin supplementary condition. We build an effective two-body Hamiltonian that matches the impulse and spin kick of the modified field theory and of the worldline theory, displaying additional Wilson coefficients compared to standard worldline approaches. The results are then compactly expressed in terms of an eikonal formula. Our key conclusion is that, contrary to standard approaches, while the magnitude of the spin tensor is still conserved, the magnitude of the spin vector can change under conserved Hamiltonian dynamics and this change is governed by the additional Wilson coefficients. For specific values of Wilson coefficients the results are equivalent to those from a definite spin obeying the spin supplementary condition, but for generic values they are physically inequivalent. These results warrant detailed studies of the corresponding issues in general relativity.

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arXiv2023

Conservative binary dynamics at order $O(\alpha^5)$ 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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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

Supergravity amplitudes, the double copy and ultraviolet behavior

In this chapter, we present a scattering-amplitudes perspective on supergravity, and describe its application to the study of ultraviolet properties of supergravity theories at high loop orders. The basic on-shell tools that make such calculations feasible are reviewed, including generalized unitarity, color-kinematics duality, and the double-copy construction. We also outline the web of theories connected by the double copy. The results of various calculations of potential ultraviolet divergences are summarized. These include puzzling enhanced ultraviolet cancellations, for which no symmetry-based understanding currently exists, showing that there is much more to learn about the ultraviolet properties of supergravity theories. We finally comment on future calculations that should help resolve the puzzles.

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arXiv2022

Snowmass TF04 Report: Scattering Amplitudes and their Applications

The field of scattering amplitudes plays a central role in elementary-particle physics. This includes various problems of broader interest for collider physics, gravitational physics, and fundamental principles underlying quantum field theory. We describe various applications and theoretical advances pointing towards novel descriptions of quantum field theories. Comments on future prospects are included.

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arXiv2022

Effective Field Theory Islands from Perturbative and Nonperturbative Four-Graviton Amplitudes

Theoretical data obtained from physically sensible field and string theory models suggest that gravitational Effective Field Theories (EFTs) live on islands that are tiny compared to current general bounds determined from unitarity, causality, crossing symmetry, and a good high-energy behavior. In this work, we present explicit perturbative and nonperturbative $2 \to 2$ graviton scattering amplitudes and their associated low-energy expansion in spacetime dimensions $D\geq 4$ to support this notion. Our new results include a first nonperturbative example consisting of a $D=4$, $\mathcal{N}=1$ supersymmetric field theory that is coupled weakly to gravity. We show that this nonperturbative model lies on the same islands identified using four-dimensional perturbative models based on string theory and minimally-coupled matter circulating a loop. Furthermore, we generalize the previous four-dimensional perturbative models based on string theory and minimally-coupled massive spin-0 and spin-1 states circulating in the loop to $D$ dimensions. Remarkably, we again find that the low-energy EFT coefficients lie on small islands. These results offer a useful guide towards constraining possible extensions of Einstein gravity.

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arXiv2022

The SAGEX Review on Scattering Amplitudes, Chapter 2: An Invitation to Color-Kinematics Duality and the Double Copy

Advances in scattering amplitudes have exposed previously-hidden color-kinematics and double-copy structures in theories ranging from gauge and gravity theories to effective field theories such as chiral perturbation theory and the Born-Infeld model. These novel structures both simplify higher-order calculations and pose tantalizing questions related to a unified framework underlying relativistic quantum theories. This introductory mini-review article invites further exploration of these topics. After a brief introduction to color-kinematics duality and the double copy as they emerges at tree and loop-level in gauge and gravity theories, we present two distinct examples: 1) an introduction to the web of double-copy-constructible theories, and 2) a discussion on the application of the double copy to calculation relevant to gravitational-wave physics.

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arXiv2022

The SAGEX Review on Scattering Amplitudes

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments.

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arXiv2022

Binary Dynamics Through the Fifth Power of Spin at $\mathcal{O}(G^2)$

We use a previously developed scattering-amplitudes-based framework for determining two-body Hamiltonians for generic binary systems with arbitrary spin $S$. By construction this formalism bypasses difficulties with unphysical singularities or higher-time derivatives. This framework has been previously used to obtain the exact velocity dependence of the $\mathcal O(G^2)$ quadratic-in-spin two-body Hamiltonian. We first evaluate the $S^3$ scattering angle and two-body Hamiltonian at this order in $G$, including not only all operators corresponding to the usual worldline operators, but also an additional set due to an interesting subtlety. We then evaluate $S^4$ and $S^5$ contributions at $\mathcal O(G^2)$ which we confirm by comparing against aligned-spin results. We conjecture that a certain shift symmetry together with a constraint on the high-energy growth of the scattering amplitude specify the Wilson coefficients for the Kerr black hole to all orders in the spin and confirm that they reproduce the previously-obtained results through $S^4$.

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arXiv2021

Scalar QED as a toy model for higher-order effects in classical gravitational scattering

Quantum Electrodynamics (QED) serves as a useful toy model for classical observables in gravitational two-body systems with reduced complexity due to the linearity of QED. We investigate scattering observables in scalar QED at the sixth order in the charges (two-loop order) in a classical regime analogous to the post-Minkowskian expansion in General Relativity. We employ modern scattering amplitude tools and extract classical observables by both eikonal methods and the formalism of Kosower, Maybee, and O'Connell (KMOC). In addition, we provide a simplified approach to extracting the radial action beyond the conservative sector.

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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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