We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons. We show how an exponentiation of Cachazo-Strominger soft factors, acting on massive higher-spin amplitudes, can be used to find spin contributions to the aligned-spin scattering angle, conjecturally extending previously known results to higher orders in spin at one-loop order. The extraction of the classical limit is accomplished via the on-shell leading-singularity method and using massive spinorhelicity variables. The three-point amplitude for arbitrary-spin massive particles minimally coupled to gravity is expressed in an exponential form, and in the infinite-spin limit it matches the effective stress-energy tensor of the linearized Kerr solution. A four-point gravitational Compton amplitude is obtained from an extrapolated soft theorem, equivalent to gluing two exponential three-point amplitudes, and becomes itself an exponential operator. The construction uses these amplitudes to: 1) recover the known tree-level scattering angle at all orders in spin, 2) recover the known one-loop linear-in-spin interaction, 3) match a previous conjectural expression for the one-loop scattering angle at quadratic order in spin, 4) propose new one-loop results through quartic order in spin. These connections link the computation of higher-multipole interactions to the study of deeper orders in the soft expansion. arXiv:1812.06895v3 [hep-th] 9 Sep 2019 Contents 1 Introduction 1 2 Multipole expansion of three-and four-point amplitudes 6 3 Scattering angle as Leading Singularity 17 4 Discussion 27 A Three-point amplitude with spin-1 matter 29 B Spin tensor for spin-1 matter 30 C Angular-momentum operator 32Recently, a classical version of the soft theorem up to sub-subleading order has been used by Laddha and Sen [22] to derive the spectrum of the radiated power in black-hole scattering with external soft graviton insertions. This relies on the remarkable fact that conservative and non-conservative long-range effects of interacting black holes can be computed from the scattering of massive point-like sources [23][24][25][26]. Indeed, rotating black holes can be treated via a spin-multipole expansion, the order 2s of which can be reproduced by scattering spins minimally coupled particles exchanging gravitons [27], as illustrated in figure 1a. The matching between these amplitudes with spin and a non-relativistic potential for black-hole scattering has been performed explicitly in the post-Newtonian (PN) framework [27][28][29].Here we present a complementary picture to the one of [22] by employing the soft theorem in the conservative sector (i.e. no external gravitons), focusing on rotating black holes and at the same time extending the soft factor in (1.1) to higher orders in the soft expansion. This is achieved in the following way: It was shown by one of the authors in [29] that the classical ( -independent) piece of the spin-s amplitude can be extracted from a covariant Holomorphic Classical...
We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on CP 1 , to higher-dimensional projective spaces CP k−1 . The standard, k = 2 Mandelstam invariants, s ab , are generalized to completely symmetric tensors s a 1 a 2 ...a k subject to a 'massless' condition s a 1 a 2 ···a k−2 b b = 0 and to 'momentum conservation'. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all 'biadjoint amplitudes' for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.
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