The eikonal approximation for high-energy collisions, long familiar in the theory of potential scattering, is considered from the viewpoint of relativistic quantum field theory. We study, in particular, the Feynman amplitude $M(s, t)$ describing the scattering of two spin-0 particles, $a$ and $b$, interacting by the exchange of spin-0 mesons. We show that if ${M}_{n}(s, t)$, the contribution to $M(s, t)$ arising from all $n\mathrm{th}$-order Feynman diagrams in which exactly $n$ mesons are exchanged between $a$ and $b$, is written in an appropriately symmetrized way, and if the terms in any $a$ or $b$ particle propagator which are quadratic in the internal momenta are then dropped, the resulting expression, ${{M}_{n}}^{\mathrm{eik}}(s, t)$, may be evaluated in closed form, and the sum over $n$, which defines ${M}^{\mathrm{eik}}(s, t)$, may be carried out. The representation of ${M}^{\mathrm{eik}}(s, t)$ found in this way involves the exponential of a function $\ensuremath{\chi}$ of a relative space-time variable $x=({x}^{0}, \mathrm{x})$ and the external momenta; $\ensuremath{\chi}$ is a relativistic generalization of the eikonal ${\ensuremath{\chi}}_{\mathrm{pot}}$ familiar from the theory of high-energy potential scattering. ${M}^{\mathrm{eik}}(s, t)$ is both crossing-symmetric and time-reversal-invariant. In the static limit (${m}_{b}\ensuremath{\rightarrow}\ensuremath{\infty}$), $\ensuremath{\chi}$ tends to ${\ensuremath{\chi}}_{\mathrm{pot}}$ for the appropriate Yukawa potential and ${f}^{\mathrm{eik}}=\ensuremath{-}\frac{{M}^{\mathrm{eik}}}{8\ensuremath{\pi}\ensuremath{\surd}s}$ has a limiting form ${{f}_{\mathrm{pot}}}^{\mathrm{eik}}$, which we also derive directly from the theory of potential scattering; for small scattering angles, ${{f}_{\mathrm{pot}}}^{\mathrm{eik}}$ coincides with the standard result. The amplitude for particle-antiparticle scattering is studied in the same model. It is shown that the eikonal ${\ensuremath{\chi}}_{2}(x)$ associated with the contribution of all annihilation-type diagrams has a logarithmic singularity at $x=0$ whose coefficient is proportional to $\ensuremath{\alpha}(t)+1$, where $\ensuremath{\alpha}(t)$ is the Regge-trajectory function obtained from the asymptotic behavior of the ladder-type diagrams alone. Another connection with Regge behavior is made by showing that the summation of a certain infinite class of radiative corrections to the lowest-order $\ensuremath{\gamma}\ensuremath{-}e$ Compton amplitude gives rise, in our eikonal approximation, to an eikonal $\ensuremath{\chi}(x)$ which has a similar logarithmic singularity with strength $1+\ensuremath{\beta}(t)$; here $\ensuremath{\beta}(t)$ is the trajectory function, introduced less directly in earlier work, which reproduces the major part of the spectrum of positronium on setting $\ensuremath{\beta}(t)=l=n\ensuremath{-}1$. A generalization of a simple algebraic identity used in the derivation of the above results, in the form of an integral representation, permits their extension to the case where one or more particles are off the mass shell. This is illustrated by a computation of an eikonal-type approximation to the Green's function for a relativistic particle moving in an external scalar field and by the summation of an infinite class of contributions to the vertex function in the model referred to above. The possibility of applying an off-shell eikonal approximation to the analysis of production processes is emphasized.

Eikonal Approximation in Quantum Field Theory

We describe a systematic framework for finding the conservative potential of compact binary systems with spin based on scattering amplitudes of particles of arbitrary spin and effective field theory. An arbitrary-spin formalism is generally required in the classical limit. By matching the tree and one-loop amplitudes of four spinning particles with those of a suitably-chosen effective field theory, we obtain the spin1-spin2 terms of a two-body effective Hamiltonian through O(G^2) and valid to all orders in velocity. Solving Hamilton's equations yields the impulse and spin changes of the individual bodies. We write them in a surprisingly compact form as appropriate derivatives of the eikonal phase obtained from the amplitude. It seems likely this structure persists to higher orders. We also point out various double-copy relations for general spin.

Spinning Black Hole Binary Dynamics, Scattering Amplitudes and Effective Field Theory

We introduce on-shell variables for Heavy Particle Effective Theories (HPETs) with the goal of extending Heavy Black Hole Effective Theory to higher spins and of facilitating its application to higher post-Minkowskian orders. These variables inherit the separation of spinless and spin-inclusive effects from the HPET fields, resulting in an explicit spin-multipole expansion of the three-point amplitude for any spin. By matching amplitudes expressed using the on-shell HPET variables to those derived from the one-particle effective action, we find that the spin-multipole expansion of a heavy spin-$s$ particle corresponds exactly to the multipole expansion (up to order $2s$) of a Kerr black hole, that is, without needing to take the infinite spin limit. Finally, we show that tree-level radiative processes with same-helicity bosons emitted from a heavy spin-$s$ particle exhibit a spin-multipole universality.

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On-shell heavy particle effective theories

Using $$ \mathcal{N} $$
 = 8 supergravity as a theoretical laboratory, we extract the 3PM gravitational eikonal for two colliding massive scalars from the classical limit of the corresponding elastic two-loop amplitude. We employ the eikonal phase to obtain the physical deflection angle and to show how its non-relativistic (NR) and ultra-relativistic (UR) regimes are smoothly connected. Such a smooth interpolation rests on keeping contributions to the loop integrals originating from the full soft region, rather than restricting it to its potential sub-region. This task is efficiently carried out by using the method of differential equations with complete near-static boundary conditions. In contrast to the potential-region result, the physical deflection angle includes radiation-reaction contributions that are essential for recovering the finite and universal UR limit implied by general analyticity and crossing arguments. We finally discuss the real emission of massless states, which accounts for the imaginary part of the 3PM eikonal and for the dissipation of energy-momentum. Adopting a direct approach based on unitarity and on the classical limit of the inelastic tree-level amplitude, we are able to treat $$ \mathcal{N} $$
 = 8 and General Relativity on the same footing, and to complete the conservative 3PM eikonal in Einstein’s gravity by the addition of the radiation-reaction contribution. We also show how this approach can be used to compute waveforms, as well as the differential and integrated spectra, for the different radiated massless fields.

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Springer : The eikonal approach to gravitational scattering and radiation at $ \mathcal{O} $(G$^{3}$)

Using the recently established formalism of a worldline quantum field theory description of the classical scattering of two spinless black holes, we compute the far-field time-domain waveform of the gravitational waves produced in the encounter at leading order in the post-Minkowskian (weak field but generic velocity) expansion. We reproduce the previous results of Kovacs and Thorne in a highly economic way. Then, using the waveform, we extract the leading-order total radiated angular momentum and energy (including differential results). Our work may enable crucial improvements of gravitational-wave predictions in the regime of large relative velocities.

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Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory.

We study the gravitational radiation emitted during the scattering of two spinless bodies in the post-Minkowskian Effective Field Theory approach. We derive the conserved stress-energy tensor linearly coupled to gravity and the classical probability amplitude of graviton emission at leading and next-to-leading order in the Newton's constant $G$. The amplitude can be expressed in compact form as one-dimensional integrals over a Feynman parameter involving Bessel functions. We use it to recover the leading-order radiated angular momentum expression of Damour [arXiv:2010.01641]. Upon expanding it in the relative velocity between the two bodies $v$, we compute the total four-momentum radiated into gravitational waves at leading-order in $G$ and up to order $v^8$, finding agreement with Herrmann et al. [arXiv:2101.07255]. Our results also allow to investigate the zero frequency limit of the emitted energy spectrum.

Gravitational Bremsstrahlung in the Post-Minkowskian Effective Field Theory

In this work we derive for the first time the complete gravitational cubic-in-spin effective action at the next-to-leading order in the post-Newtonian (PN) expansion for the interaction of generic compact binaries via the effective field theory for gravitating spinning objects, which we extend in this work. This sector, which enters at the fourth and a half PN (4.5PN) order for rapidly-rotating compact objects, completes finite-size effects up to this PN order, and is the first sector completed beyond the current state of the art for generic compact binary dynamics at the 4PN order. At this order in spins with gravitational nonlinearities we have to take into account additional terms, which arise from a new type of worldline couplings, due to the fact that at this order the Tulczyjew gauge for the rotational degrees of freedom, which involves the linear momentum, can no longer be approximated only in terms of the four-velocity. One of the main motivations for us to tackle this sector is also to see what happens when we go to a sector, which corresponds to the gravitational Compton scattering with quantum spins larger than one, and maybe possibly also get an insight on the inability to uniquely fix its amplitude from factorization when spins larger than two are involved. A general observation that we can clearly make already is that even-parity sectors in the order of the spin are easier to handle than odd ones. In the quantum context this corresponds to the greater ease of dealing with bosons compared to fermions.

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Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order

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We derive the static Schwarzschild-Tangherlini metric by extracting the classical contributions from the multiloop vertex functions of a graviton emitted from a massive scalar field. At each loop order the classical contribution is proportional to a unique master integral given by the massless sunset integral. By computing the scattering amplitudes up to three-loop order in general dimension, we explicitly derive the expansion of the metric up to the fourth post-Minkowskian order $O({G}_{N}^{4})$ in four, five and six dimensions. There are ultraviolet divergences that are cancelled with the introduction of higher-derivative nonminimal couplings. The standard Schwarzschild-Tangherlini is recovered by absorbing their effects by an appropriate coordinate transformation induced from the de Donder gauge condition.

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Schwarzschild-Tangherlini metric from scattering amplitudes in various dimensions

We compute the mass and current quadrupole tidal corrections to the four-momentum and energy flux radiated during the scattering of two spinless bodies, at leading order in G and at all orders in the velocities, using the effective field theory worldline approach. In particular, we derive the conserved stress-energy tensor linearly coupled to gravity generated by the two bodies, including tidal fields, and the waveform in direct space. The integral is solved using scattering amplitude techniques. We show that our expressions are consistent with existing results up to the next-to-next-to-leading order in the post-Newtonian expansion.

http://link.aps.org/pdf/10.1103/PhysRevLett.129.121101

Stavros Mougiakakos

Papers

Gravitational Bremsstrahlung in the Post-Minkowskian Effective Field Theory

Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order

Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order

Schwarzschild-Tangherlini metric from scattering amplitudes in various dimensions

Gravitational Bremsstrahlung with Tidal Effects in the Post-Minkowskian Expansion.