arXiv subjects

Gregor Kälin

Publications and source records attributed to Gregor Kälin.

At least 19 recordsLinked to original sources

arXiv2026

Nonlocal-in-time tail effects in gravitational scattering to fifth Post-Minkowskian and tenth self-force orders

Using the worldline effective field theory formalism, we derive the nonlocal-in-time conservative contributions arising from tail effects in gravitational scattering to fifth Post-Minkowskian (5PM) and tenth self-force (10SF) orders. The result features multiple polylogarithms of up to weight three. This challenging computation relies on state-of-the-art integration techniques, including a novel integration-by-parts algorithm: the Sparse Integral Reducer (SpideR). We find perfect agreement in the overlap with all existing literature through sixth post-Newtonian order. The results presented here provide a key ingredient for isolating the local-in-time component of the conservative two-body dynamics of binary inspirals at 5PM order.

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arXiv2025

Local-in-Time Conservative Binary Dynamics at Fifth Post-Minkowskian and First Self-Force Orders

We report the local-in-time conservative dynamics of nonspinning binary systems at fifth Post-Minkowskian (5PM) and first self-force (1SF) orders. This follows from an explicit calculation of the 5PM/1SF nonlocal-in-time tail-type contribution to the deflection angle via worldline effective field theory techniques. Proceeding as in [2403.04853], we subtract the nonlocal tail terms from the result in [2403.07781] and reconstruct a local-in-time Hamiltonian in isotropic gauge -- valid for generic orbits. For completeness, we reinstate the nonlocal terms relevant for elliptic-like motion up to 6PN/1SF in a small-eccentricity expansion. Via the connection between the (source) energy flux in [2210.05541] and tail effects, we also derive the SF-exact logarithmic-dependent part of the full 5PM bound Hamiltonian. Our results provide the most accurate description to date of the dynamics of bound compact objects within the framework of relativistic scattering computations.

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arXiv2024

Local-in-time Conservative Binary Dynamics at Fourth Post-Minkowskian Order

Leveraging scattering information to describe binary systems in generic orbits requires identifying local- and nonlocal-in-time tail effects. We report here the derivation of the universal (non-spinning) local-in-time conservative dynamics at fourth Post-Minkowskian order, i.e. ${\cal O}(G^4)$. This is achieved by computing the nonlocal-in-time contribution to the deflection angle, and removing it from the full conservative value in [2112.11296,2210.05541]. Unlike the total result, the integration problem involves two scales, velocity and mass ratio, and features multiple polylogarithms, complete elliptic and iterated elliptic integrals, notably in the mass ratio. We reconstruct the local radial action, center-of-mass momentum and Hamiltonian, as well as the exact logarithmic-dependent part(s), all valid for generic orbits. We incorporate the remaining nonlocal terms for elliptic-like motion to sixth Post-Newtonian order. The combined Hamiltonian is in perfect agreement in the overlap with the Post-Newtonian state of the art. The results presented here provide the most accurate description of gravitationally-bound binaries harnessing scattering data to date, readily applicable to waveform modelling.

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arXiv2023

Two-loop $\mathcal{N}=1$ SYM Amplitudes via SUSY Decomposition and Massive Spinor-Helicity

We obtain a color-kinematics-dual representation of the two-loop four-vector amplitude a general renormalizable massless $\mathcal{N}=1$ SYM theory, including internal matter as chiral supermultiplets. The integrand is constructed to be compatible with dimensional regularization and supersymmetry by employing two strategies (implicitly defining our regularization scheme): supersymmetric decomposition and matching to massive spinor-helicity amplitudes. All internal vector components inherit their $D$-dimensional properties by relating them to the previously constructed $D\leq6$, $\mathcal{N}=2$ SQCD amplitude using supersymmetric decomposition identities of individual diagrams. This leaves only diagrams with internal matter lines as unknown masters, which are in turn constrained on $D$-dimensional unitarity cuts by reinterpreting the extra-dimensional momentum components as masses for the chiral supermultiplets. We rely on the massive spinor-helicity formalism and massive on-shell $\mathcal{N}=1$ superspace, generalized here to complex masses. Finally, we extend the kinematic numerator algebra to include three-term identities that are dual to color identities linear in the matter Clebsch-Gordan coefficients, as well as two new optional identities satisfied by mass-deformed $\mathcal{N}=4$ and $\mathcal{N}=2$ SYM theories that preserve $\mathcal{N}=1$ supersymmetry. Altogether, these identities makes it possible to completely reduce the two-loop integrand to only two master numerators.

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arXiv2023

Machine Learning Post-Minkowskian Integrals

We study a neural network framework for the numerical evaluation of Feynman loop integrals that are fundamental building blocks for perturbative computations of physical observables in gauge and gravity theories. We show that such a machine learning approach improves the convergence of the Monte Carlo algorithm for high-precision evaluation of multi-dimensional integrals compared to traditional algorithms. In particular, we use a neural network to improve the importance sampling. For a set of representative integrals appearing in the computation of the conservative dynamics for a compact binary system in General Relativity, we perform a quantitative comparison between the Monte Carlo integrators VEGAS and i-flow, an integrator based on neural network sampling.

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arXiv2023

Bootstrapping the relativistic two-body problem

We describe the formalism to compute gravitational-wave observables for compact binaries via the effective field theory framework in combination with modern tools from collider physics. We put particular emphasis on solving the "multi-loop" integration problem via the methodology of differential equations and expansion by regions. This allows us to "bootstrap" the two-body relativistic dynamics in the Post-Minkowskian (PM) expansion from boundary data evaluated in the near-static (soft) limit. We illustrate the procedure with the derivation of the total spacetime impulse in the scattering of non-spinning bodies to 4PM (three-loop) order, i.e. ${\cal O}(G^4)$, including conservative and dissipative effects.

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arXiv2022

Radiation Reaction and Gravitational Waves at Fourth Post-Minkowskian Order

We obtain the total impulse in the scattering of non-spinning binaries in general relativity at fourth Post-Minkowskian order, i.e. ${\cal O}(G^4)$, including linear, nonlinear, and hereditary radiation-reaction effects. We derive the total radiated spacetime momentum as well as the associated energy flux. The latter can be used to compute gravitational-wave observables for generic (un)bound orbits. We employ the ("in-in") Schwinger-Keldysh worldline effective field theory framework in combination with modern "multi-loop" integration techniques from collider physics. The complete results are in agreement with various partial calculations in the Post-Newtonian/Minkowskian expansion.

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arXiv2022

Radiation-Reaction in the Effective Field Theory Approach to Post-Minkowskian Dynamics

We extend the Post-Minkowskian (PM) effective field theory (EFT) approach to incorporate conservative and dissipative radiation-reaction effects in a unified framework. This is achieved by implementing the Schwinger-Keldysh "in-in" formalism and separating conservative and non-conservative terms according to the formulation in [1210.2745], which we show promotes Feynman's $i0$-prescription and cutting rules to a prominent role at the classical level. The resulting integrals, involving both Feynman and retarded propagators, can be bootstrapped to all orders in the velocity via differential equations with boundary conditions including potential and radiation modes. As a paradigmatic example we provide an ab initio derivation of the classical solution to the scattering problem in general relativity to ${\cal O}(G^3)$. For the sake of completeness, we also reproduce the leading order radiation-reaction effects in classical electrodynamics.

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arXiv2022

From Boundary Data to Bound States III: Radiative Effects

We extend the "boundary-to-bound" (B2B) correspondence to incorporate radiative as well as conservative radiation-reaction effects. We start by deriving a map between the total change in observables due to gravitational wave emission during hyperbolic-like motion and in one period of an elliptic-like orbit, which is valid in the adiabatic expansion for non-spinning as well as aligned-spin configurations. We also discuss the inverse problem of extracting the associated fluxes from scattering data. Afterwards we demonstrate, to all orders in the Post-Minkowskian expansion, the link between the radiated energy and the ultraviolet pole in the radial action in dimensional regularization due to tail effects. This implies, as expected, that the B2B correspondence for the conservative sector remains unchanged for local-in-time radiation-reaction tail effects with generic orbits. As a side product, this allows us to read off the energy flux from the associated pole in the tail Hamiltonian. We show that the B2B map also holds for non-local-in-time terms, but only in the large-eccentricity limit. Remarkably, we find that all of the trademark logarithmic contributions to the radial action map unscathed between generic unbound and bound motion. However, unlike logarithms, other terms due to non-local effects do not transition smoothly to quasi-circular orbits. We conclude with a discussion on these non-local pieces. Several checks of the B2B dictionary are displayed using state-of-the-art knowledge in Post-Newtonian/Minkowskian theory.

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arXiv2022

Dynamics of Binary Systems to Fourth Post-Minkowskian Order from the Effective Field Theory Approach

We present the contribution from potential interactions to the dynamics of non-spinning binaries to fourth Post-Minkowskian (4PM) order. This is achieved by computing the scattering angle to ${\cal O}(G^4)$ using the effective field theory approach and deriving the bound radial action through analytic continuation. We reconstruct the Hamiltonian and center-of-mass momentum in an isotropic gauge. The (three-loop) integrals involved in our calculation are computed via differential equations, including a sector yielding elliptic integrals. Using the universal link between potential and tail terms, we also report: 1) The instantaneous energy flux at ${\cal O}(G^3)$; 2) The contribution to the 4PM unbound/bound radial action(s) depending on logarithms of the binding energy; 3) The (scheme-independent) logarithmic contribution to the 4PM non-local tail Hamiltonian for circular orbits. Our results in the potential region are in agreement with the recent derivation from scattering amplitudes. We also find perfect agreement in the overlap with the state-of-the-art in Post-Newtonian theory.

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arXiv2022

Conservative Dynamics of Binary Systems at Fourth Post-Minkowskian Order in the Large-eccentricity Expansion

We compute the conservative dynamics of non-spinning binaries at fourth Post-Minkowskian order in the large-eccentricity limit, including both potential and radiation-reaction tail effects. This is achieved by obtaining the scattering angle in the worldline effective field theory approach and deriving the bound radial action via analytic continuation. The associated integrals are bootstrapped to all orders in velocities through differential equations, with boundary conditions in the potential and radiation regions. The large angular momentum expansion captures all the local-in-time effects as well as the trademark logarithmic corrections for generic bound orbits. Agreement is found in the overlap with the state-of-the-art in Post-Newtonian theory.

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arXiv2020

Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach

We derive the conservative dynamics of non-spinning binaries to third Post-Minkowskian order, using the Effective Field Theory (EFT) approach introduced in [2006.01184] together with the Boundary-to-Bound dictionary developed in [1910.03008, 1911.09130]. The main ingredient is the scattering angle, which we compute to ${\cal O}(G^3)$ via Feynman diagrams. Adapting to the EFT framework powerful tools from the amplitudes program, we show how the associated (master) integrals are bootstrapped to all orders in velocities via differential equations. Remarkably, the boundary conditions can be reduced to the same integrals that appear in the EFT with Post-Newtonian sources. For the sake of comparison, we reconstruct the Hamiltonian and the classical limit of the scattering amplitude. Our results are in perfect agreement with those in Bern et al. [1901.04424, 1908.01493].

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arXiv2020

Conservative Tidal Effects in Compact Binary Systems to Next-to-Leading Post-Minkowskian Order

Using the Effective Field Theory approach together with the Boundary-to-Bound map, we compute the next-to-leading order (NLO) Post-Minkowskian (PM) tidal effects in the conservative dynamics of compact binary systems. We derive the mass and current quadrupole and, for the first time, octupole corrections to the binding energy for circular orbits at ${\cal O}(G^3)$. Our results are consistent with the test-body limit as well as the existent Post-Newtonian literature. We also reconstruct a Hamiltonian incorporating tidal effects to NLO in the PM expansion and find complete agreement with the recent derivation of its quadrupolar part using the classical limit of scattering amplitudes.

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arXiv2020

From Boundary Data to Bound States

We introduce a -- somewhat holographic -- dictionary between gravitational observables for scattering processes (measured at the "boundary") and adiabatic invariants for bound orbits (in the "bulk"), to all orders in the Post-Minkowskian (PM) expansion. Our map relies on remarkable connections between the relative momentum of the two-body problem, the classical limit of the scattering amplitude and the deflection angle in hyperbolic motion. These relationships allow us to compute observables for generic orbits (such as the periastron advance $ΔΦ$) through analytic continuation, via a radial action depending only on boundary data. A simplified (more geometrical) map can be obtained for circular orbits, enabling us to extract the orbital frequency as a function of the (conserved) binding energy, $Ω(E)$, directly from scattering information. As an example, using the results in Bern et al. [1901.04424, 1908.01493], we readily derive $Ω(E)$ and $ΔΦ(J,E)$ to two-loop orders. We also provide closed-form expressions for the orbital frequency and periastron advance at tree-level and one-loop order, respectively, which capture a series of exact terms in the Post-Newtonian expansion. We then perform a partial PM resummation, using a "no-recoil" approximation for the amplitude. This limit is behind the map between the scattering angle for a test-particle and the two-body dynamics to 2PM. We show that it also captures a subset of higher order terms beyond the test-particle limit. While a (rather lengthy) Hamiltonian may be derived as an intermediate step, our map applies directly between gauge invariant quantities. Our findings provide a starting point for an alternative approach to the binary problem. We conclude with future directions and some speculations on the classical double copy.

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arXiv2020

From Boundary Data to Bound States II: Scattering Angle to Dynamical Invariants (with Twist)

We recently introduced in [1910.03008] a "boundary-to-bound" dictionary between gravitational scattering data and observables for bound states of non-spinning bodies. In this paper, we elaborate further on this (holographic) map. We start by deriving the following -- remarkably simple -- formula relating the periastron advance to the scattering angle: $ΔΦ(J,{\cal E}) =χ(J,{\cal E}) + χ(-J,{\cal E})$, via analytic continuation in angular momentum and binding energy. Using explicit expressions from [1910.03008], we confirm its validity to all orders in the Post-Minkowskian (PM) expansion. Furthermore, we reconstruct the radial action for the bound state directly from the knowledge of the scattering angle. The radial action enables us to write compact expressions for dynamical invariants in terms of the deflection angle to all PM orders, which can also be written as a function of the PM-expanded amplitude. As an example, we reproduce our result in [1910.03008] for the periastron advance, and compute the radial and azimuthal frequencies and redshift variable to two-loops. Agreement is found in the overlap between PM and Post-Newtonian (PN) schemes. Last but not least, we initiate the study of our dictionary including spin. We demonstrate that the same relation between deflection angle and periastron advance applies for aligned-spin contributions, with $J$ the (canonical) total angular momentum. Explicit checks are performed to display perfect agreement using state-of-the-art PN results in the literature. Using the map between test- and two-body dynamics, we also compute the periastron advance up to quadratic order in the spin, to one-loop and to all orders in velocity. We conclude with a discussion on the generalized "impetus formula" for spinning bodies and black holes as "elementary particles".

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arXiv2020

Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics

We develop an Effective Field Theory (EFT) formalism to solve for the conservative dynamics of binary systems in gravity via Post-Minkowskian (PM) scattering data. Our framework combines a systematic EFT approach to compute the deflection angle in the PM expansion, together with the 'Boundary-to-Bound' (B2B) dictionary introduced in [1910.03008, 1911.09130]. Due to the nature of scattering processes, a remarkable reduction of complexity occurs both in the number of Feynman diagrams and type of integrals, compared to a direct EFT computation of the potential in a PM scheme. We provide two illustrative examples. Firstly, we compute all the conservative gravitational observables for bound orbits to 2PM, which follow from only one topology beyond leading order. The results agree with those in [1910.03008, 1911.09130], obtained through the 'impetus formula' applied to the classical limit of the one loop amplitude in Cheung et al. [1808.02489]. For the sake of comparison we reconstruct the conservative Hamiltonian to 2PM order, which is equivalent to the one derived in [1808.02489] from a matching calculation. Secondly, we compute the scattering angle due to tidal effects from the electric- and magnetic-type Love numbers at leading PM order. Using the B2B dictionary we then obtain the tidal contribution to the periastron advance. We also construct a Hamiltonian including tidal effects at leading PM order. Although relying on (relativistic) Feynman diagrams, the EFT formalism developed here does not involve taking the classical limit of a quantum amplitude, neither integrals with internal massive fields, nor additional matching calculations, nor spurious ('super-classical') infrared singularities. By construction, the EFT approach can be automatized to all PM orders.

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arXiv2020

Infrared and transcendental structure of two-loop supersymmetric QCD amplitudes

Using a careful choice of infrared (IR) subtraction scheme, we demonstrate the cancellation of all terms with transcendental weights 0,1,2 from the finite part of the full-color two-loop four-gluon $\mathcal{N}=2$ supersymmetric QCD amplitude, with $N_f$ massless supersymmetric quarks. This generalizes the previously observed cancellation of weight-2 terms in the superconformal theory, where $N_f=2N_c$ for gauge group SU$(N_c)$. The subtraction scheme follows naturally both from general IR factorization principles and from an integrand-level analysis of divergences in this amplitude. The divergences are written in terms of scalar triangle integrals whose expressions are known to all orders in the dimensional regulator $ε=(4-D)/2$. We also present integrated expressions for the full-color two-loop four-point amplitudes with both matter and vectors on external legs in which lower-weight terms also cancel using an appropriate IR scheme. This provides us with values for the two-loop cusp, gluonic, and quark anomalous dimensions in $\mathcal{N}=2$ supersymmetric QCD, which are cross-checked between the three different amplitudes.

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arXiv2019

Two-loop $\mathcal{N}=2$ SQCD amplitudes with external matter from iterated cuts

We develop an iterative method for constructing four-dimensional generalized unitarity cuts in $\mathcal{N} = 2$ supersymmetric Yang-Mills (SYM) theory coupled to fundamental matter hypermultiplets ($\mathcal{N} = 2$ SQCD). For iterated two-particle cuts,specifically those involving only four-point amplitudes, this implies simple diagrammatic rules for assembling the cuts to any loop order, reminiscent of the rung rule in $\mathcal{N} = 4$ SYM. By identifying physical poles, the construction simplifies the task of extracting complete integrands. In combination with the duality between color and kinematics we construct all four-point massless MHV-sector scattering amplitudes up to two loops in $\mathcal{N} = 2$ SQCD, including those with matter on external legs. Our results reveal chiral infrared-finite integrands closely related to those found using loop-level BCFW recursion. The integrands are valid in $D\leq 6$ dimensions with external states in a four-dimensional subspace; the upper bound is dictated by our use of six-dimensional chiral $\mathcal{N} = (1,0)$ SYM as a means of dimensionally regulating loop integrals.

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