Compton Amplitude Research Articles

From higher-spin gauge interactions to Compton amplitudes for root-Kerr

We develop massive higher-spin theory as a framework for describing dynamics of rotating compact objects, such as Kerr black holes. In this paper, we explore gauge interactions up to quartic order and corresponding Compton amplitudes of higher-spin massive objects coupled to electromagnetism and Yang-Mills theory. Their classical counterparts are known as root-Kerr gauge-theory solutions, whose amplitudes are closely related to those of Kerr black holes. We use three distinct approaches: (i) massive higher-spin gauge symmetry to introduce cubic interactions for all spins and the quartic interactions up to spin 3, which is implemented both off shell and via Ward identities; (ii) a chiral higher-spin approach to construct quartic Lagrangians with correct degrees of freedom to all spins; (iii) on-shell functional patterns before and after taking the classical limit to constrain the Compton amplitudes. As final results, we arrive at simple local formulae for the candidate root-Kerr Compton amplitudes both in the quantum regime and classical limit, to all orders in spin. This is a precursor to the gravitational Kerr case, which is presented in a follow-up paper.

Spinning binary dynamics in cubic effective field theories of gravity

We study the binary dynamics of two Kerr black holes with arbitrary spin vectors in the presence of parity-even and parity-odd cubic deformations of gravity. We first derive the tree-level Compton amplitudes for a Kerr black hole in cubic gravity, which we then use to compute the two-to-two amplitudes of the massive bodies to leading order in the deformation and the post-Minkowskian expansion. The required one-loop computations are performed using the leading singularity approach as well as the heavy-mass effective field theory (HEFT) approach. These amplitudes are then used to compute the leading-order momentum and spin kick in cubic gravity in the KMOC formalism. Our results are valid for generic masses and spin vectors, and include all the independent parity-even and parity-odd cubic deformations of Einstein-Hilbert gravity. We also present spin-expanded expressions for the momentum and spin kicks, and the all-order in spin deflection angle in the case of aligned spins.

Compton Amplitude for Rotating Black Hole from QFT.

We construct a candidate tree-level gravitational Compton amplitude for a rotating Kerr black hole, for any quantum spin s=0,1/2,1,…,∞, from which we extract the corresponding classical amplitude to all orders in the spin vector S^{μ}. We use multiple insights from massive higher-spin quantum field theory, such as massive gauge invariance and improved behavior in the massless limit. A chiral-field approach is particularly helpful in ensuring correct degrees of freedom, and for writing down compact off-shell interactions for general spin. The simplicity of the interactions is echoed in the structure of the spin-s Compton amplitude, for which we use homogeneous symmetric polynomials of the spin variables. Where possible, we compare to the general-relativity results in the literature, available up to eighth order in spin.

Spinning waveforms from the Kosower-Maybee-O’Connell formalism at leading order

We provide the analytic waveform in time domain for the scattering of two Kerr black holes at leading order in the post-Minkowskian (weak field, but generic velocity) expansion and up to fourth order in both spins. The result is obtained by the generalization of the Kosower-Maybee-O’Connell formalism to radiative observables, combined with the analytic continuation of the five-point scattering amplitude to complex kinematics. We use analyticity arguments to express the waveform directly in terms of the three-point coupling of the graviton to the spinning particles and the gravitational Compton amplitudes, completely bypassing the need to compute and integrate the five-point amplitude. In particular, this allows us to easily include higher-order spin contributions for any spinning compact body. Finally, in the spinless case, we find a new compact and gauge-invariant representation of the Kovacs-Thorne waveform. Published by the American Physical Society 2024

Reconstructing generalized parton distributions from the lattice off-forward Compton amplitude

We present a determination of the structure functions of the off-forward Compton amplitude H1 and E1 from the Feynman-Hellmann method in lattice QCD. At leading twist, these structure functions give access to the generalized parton distributions (GPDs) H and E, respectively. This calculation is performed for an unphysical pion mass of mπ=412 MeV and four values of the soft momentum transfer, t≈0,−0.3,−0.6,−1.1 GeV2, all at a hard momentum scale of Q¯2≈5 GeV2. Using these results, we test various methods to determine properties of the real-time scattering amplitudes and GPDs: (1) we fit their Mellin moments, and (2) we use a simple GPD ansatz to reconstruct the entire distribution. Our final results show promising agreement with phenomenology and other lattice results and highlight specific systematics in need of control. Published by the American Physical Society 2024

Color-kinematic numerators for fermion Compton amplitudes

We introduce a novel approach to compute Compton amplitudes involving a fermion pair inspired by Hopf algebra amplitude constructions. This approach features a recursive relation employing quasi-shuffle sets, directly verifiable by massive factorization properties. We derive results for minimal gauge invariant color-kinematic numerators with physical massive poles using this method. We have also deduced a graphical method for deriving numerators that simplifies the numerator generation and eliminates redundancies, thus providing several computational advantages.

Covariant Compton Amplitudes in Gravity with Classical Spin

We develop a novel amplitude bootstrap technique manifestly free of unphysical poles for classically spinning particles interacting with gravitons utilizing only the double-copy and physical factorization limits. Combined with nonfactorization polynomial contact contributions from physical data for Kerr black holes, we can address high-spin-order covariant gravitational Compton amplitudes, identifying a pattern for the amplitude that we believe could extend to all orders in spin. Finally, we outline applications and outstanding questions. Published by the American Physical Society 2024

Black hole perturbation theory meets CFT2 : Kerr-Compton amplitudes from Nekrasov-Shatashvili functions

We present a novel study of Kerr Compton amplitudes in a partial wave basis in terms of the Nekrasov-Shatashvili (NS) function of the confluent Heun equation (CHE). Remarkably, NS-functions enjoy analytic properties and symmetries that are naturally inherited by the Compton amplitudes. Based on this, we characterize the analytic dependence of the Compton phase shift in the Kerr spin parameter and provide a direct comparison to the standard post-Minkowskian (PM) perturbative approach within general relativity (GR). We also analyze the universal large frequency behavior of the relevant characteristic exponent of the CHE—also known as the renormalized angular momentum—and find agreement with numerical computations. Moreover, we discuss the analytic continuation in the harmonics quantum number ℓ of the partial wave, and show that the limit to the physical integer values commutes with the PM expansion of the observables. Finally, we obtain the contributions to the tree-level, point-particle, gravitational Compton amplitude in a covariant basis through O(aBH8), without the need to take the superextremal limit for Kerr spin. Published by the American Physical Society 2024

Resummed spinning waveforms from five-point amplitudes

We compute the classical tree-level five-point amplitude for the two-to-two scattering of spinning celestial objects with the emission of a graviton. Using this five-point amplitude, we then turn to the computation of the leading-order time-domain gravitational waveform. The method we describe is suitable for arbitrary values of classical spin of Kerr black holes and does not require any expansion in powers of the spin. In this paper we illustrate it in the simpler case of the scattering of one Kerr and one Schwarzschild black hole. An important ingredient of our calculation is a novel form of the Compton amplitude with spinning particles including contact terms derived from matching to black-hole perturbation theory calculations. This ensures that our waveform is valid up to at least fourth order in the spin. Our method can be applied immediately to generate improved waveforms once higher-order contact terms in the Compton amplitude become available. Finally, we show the formula for the gravitational memory to all orders in the spin, which is in agreement with our results.

Scattering in black hole backgrounds and higher-spin amplitudes. Part II

We continue to investigate correspondences between, on the one hand, scattering amplitudes for massive higher-spin particles and gravitons in appropriate quantum-to-classical limits, and on the other hand, classical gravitational interactions of spinning black holes according to general relativity. We first construct an ansatz for a gravitational Compton amplitude, at tree level, constrained only by locality, crossing symmetry, unitarity and consistency with the linearized-Kerr 3-point amplitude, to all orders in the black hole’s spin. We then explore the extent to which a unique classical Compton amplitude can be identified by comparing with the results of the classical process of scattering long-wavelength gravitational waves off an exact Kerr black hole, determined by appropriate solutions of the Teukolsky equation. Up to fourth order in spin, we find complete agreement with a previously conjectured exponential form of the tree-level Compton amplitude. At higher orders, we extract tree-level contributions from the Teukolsky amplitude by an analytic continuation from a physical (a/GM < 1) to a particle-like (a/GM > 1) regime. Up to the sixth order in spin, we identify a unique conservative part of the amplitude which is insensitive both to the choice of boundary conditions at the black hole horizon and to branch choices in the analytic continuation. The remainder of the amplitude is determined modulo an overall sign from a branch choice, with the sign flipping under exchanging purely ingoing and purely outgoing boundary conditions at the horizon. Along the way, we make contact with novel applications of massive spinor-helicity variables pertaining to their relation to EFT operators and (spinning) partial amplitudes.

Recursion in the classical limit and the neutron-star Compton amplitude

We study the compatibility of recursive techniques with the classical limit of scattering amplitudes through the construction of the classical Compton amplitude for general spinning compact objects. This is done using BCFW recursion on three-point amplitudes expressed in terms of the classical spin vector and tensor, and expanded to next-to-leading-order in ћ by using the heavy on-shell spinors. Matching to the result of classical computations, we find that lower-point quantum contributions are, in general, required for the recursive construction of classical, spinning, higher-point amplitudes with massive propagators. We are thus led to conclude that BCFW recursion and the classical limit do not commute. In possession of the classical Compton amplitude, we remove non-localities to all orders in spin for opposite graviton helicities, and to fifth order in the same-helicity case. Finally, all possible on-shell contact terms potentially relevant to black-hole scattering at the second post-Minkowskian order are enumerated and written explicitly.

Moments and power corrections of longitudinal and transverse proton structure functions from lattice QCD

We present a simultaneous extraction of the moments of ${F}_{2}$ and ${F}_{L}$ structure functions of the proton for a range of photon virtuality, ${Q}^{2}$. This is achieved by computing the forward Compton amplitude on the lattice utilizing the second-order Feynman-Hellmann theorem. Our calculations are performed on configurations with two different lattice spacings and volumes, all at the $SU(3)$ symmetric point. We find the moments of ${F}_{2}$ and ${F}_{L}$ in good agreement with experiment. Power corrections turn out to be significant. This is the first time the ${Q}^{2}$ dependence of the lowest moment of ${F}_{2}$ has been quantified.

Scattering of gravitational waves off spinning compact objects with an effective worldline theory

We study the process, within classical general relativity, in which an incident gravitational plane wave, of weak amplitude and long wavelength, scatters off a massive spinning compact object, such as a black hole or neutron star. The amplitude of the asymptotic scattered wave, considered here at linear order in Newton's constant $G$ while at higher orders in the object's multipole expansion, is a valuable characterization of the response of the object to external gravitational fields. This amplitude coincides with a classical ($\ensuremath{\hbar}\ensuremath{\rightarrow}0$) limit of a quantum four-point (object and graviton in, object and graviton out) gravitational Compton amplitude, at the tree (linear-in-$G$) level. Such tree-level Compton amplitudes are key building blocks in generalized-unitary-based approaches to the post-Minkowskian dynamics of binaries of spinning compact objects. In this paper, we compute the classical amplitude using an effective worldline theory to describe the compact object, determined by an action functional for translational and rotational degrees of freedom, including couplings of spin-induced higher multipole moments to space-time curvature. We work here up to the levels of quadratic-in-spin quadrupole and cubic-in-spin octupole couplings, respectively involving Wilson coefficients ${C}_{2}$ and ${C}_{3}$. For the special case ${C}_{2}={C}_{3}=1$ corresponding to a black hole, we find agreement through cubic-in-spin order between our classical amplitude and previous conjectures arising from considerations of quantum scattering amplitudes. We also present new results for general ${C}_{2}$ and ${C}_{3}$, anticipating instructive comparisons with results from effective quantum theories.

The massless three-loop Wilson coefficients for the deep-inelastic structure functions F2, FL, xF3 and g1

We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for the structure functions F2(x, Q2), FL(x, Q2), xF3(x, Q2) in the overline{textrm{MS}} scheme and the polarized Wilson coefficients of the structure function g1(x, Q2) in the Larin scheme up to three-loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We work in the Larin scheme in the case of contributing axial-vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarized three-loop Wilson coefficients are calculated for the first time. As a by-product we also obtain the quarkonic three-loop anomalous dimensions from the O(1/ε) terms of the unrenormalized forward Compton amplitude. Expansions for small and large values of the Bjorken variable x are provided.

Classical Gravitational Spinning-Spinless Scattering at O(G^{2}S^{∞}).

Making use of the recently derived, all-spin, opposite-helicity Compton amplitude, we calculate the classical gravitational scattering amplitude for one spinning and one spinless object at O(G^{2}) and all orders in spin. By construction, this amplitude exhibits the spin structure that has been conjectured to describe Kerr black holes. This spin structure alone is not enough to fix all deformations of the Compton amplitude by contact terms, but when combined with considerations of the ultrarelativistic limit we can uniquely assign values to the parameters remaining in the even-in-spin sector. Once these parameters are determined, much of the spin dependence of the amplitude resums into hypergeometric functions. Finally, we derive the eikonal phase for aligned-angular-momentum scattering.

The 2PM Hamiltonian for binary Kerr to quartic in spin

From the S-matrix of spinning particles, we extract the 2 PM conservative potential for binary spinning black holes up to quartic order in spin operators. An important ingredient is the exponentiated gravitational Compton amplitude in the classical spin-limit for all graviton helicity sectors. The validity of the resulting Hamiltonian is verified by matching to known lower spin order results, as well as direct computation of the 2PM impulse and spin kicks from the eikonal phase and that from the test black hole scattering based on Mathisson-Papapetrou-Dixon equations.

Searching for Kerr in the 2PM amplitude

The classical scattering of spinning objects is well described by the spinor-helicity formalism for heavy particles. Using these variables, we derive spurious-pole-free, all-spin opposite-helicity Compton amplitudes (factorizing on physical poles to the minimal, all-spin three-point amplitudes) in the classical limit for QED, QCD, and gravity. The cured amplitudes are subject to deformations by contact terms, the vast majority of whose contributions we can fix by imposing a relation between spin structures — motivated by lower spin multipoles of black hole scattering — at the second post-Minkowskian (2PM) order. For QED and gravity, this leaves a modest number of unfixed coefficients parametrizing contact-term deformations, while the QCD amplitude is uniquely determined. Our gravitational Compton amplitude allows us to push the state-of-the-art of spinning-2PM scattering to any order in the spin vectors of both objects; we present results here and in the supplementary material file 2PMSpin8Aux.nb up to eighth order in the spin vectors. Interestingly, despite leftover coefficients in the Compton amplitude, imposing the aforementioned relation between spin structures uniquely fixes some higher-spin parts of the 2PM amplitude.

The Compton Amplitude, lattice QCD and the Feynman-Hellmann approach

A major objective of lattice QCD is the computation of hadronic matrix elements. The standard method is to use three-point and four-point correlation functions. An alternative approach, requiring only the computation of two-point correlation functions is to use the Feynman-Hellmann theorem. In this talk we develop this method up to second order in perturbation theory, in a context appropriate for lattice QCD. This encompasses the Compton Amplitude (which forms the basis for deep inelastic scattering) and hadron scattering. Some numerical results are presented showing results indicating what this approach might achieve.

Compton black-hole scattering for s ≤ 5/2

Quantum scattering amplitudes for massive matter have received new attention in connection to classical calculations relevant to gravitational-wave physics. Amplitude methods and insights are now employed for precision computations of observables needed for describing the gravitational dynamics of bound massive objects such as black holes. An important direction is the inclusion of spin effects needed to accurately describe rotating (Kerr) black holes. Higher-spin amplitudes introduced by Arkani-Hamed, Huang and Huang at three points have by now a firm connection to the effective description of Kerr black-hole physics. The corresponding Compton higher-spin amplitudes remain however an elusive open problem. Here we draw from results of the higher-spin literature and show that physical insights can be used to uniquely fix the Compton amplitudes up to spin 5/2, by imposing a constraint on a three-point higher-spin current that is a necessary condition for the existence of an underlying unitary theory. We give the unique effective Lagrangians up to spin 5/2, and show that they reproduce the previously-known amplitudes. For the multi-graviton amplitudes analogous to the Compton amplitude, no further corrections to our Lagrangians are expected, and hence such amplitudes are uniquely predicted. As an essential tool, we introduce a modified version of the massive spinor-helicity formalism which allows us to conveniently obtain higher-spin states, propagators and compact expressions for the amplitudes.

Generalized parton distributions from the off-forward Compton amplitude in lattice QCD

We determine the properties of generalised parton distributions (GPDs) from a lattice QCD calculation of the off-forward Compton amplitude (OFCA). By extending the Feynman-Hellmann relation to second-order matrix elements at off-forward kinematics, this amplitude can be calculated from lattice propagators computed in the presence of a background field. Using an operator product expansion, we show that the deeply-virtual part of the OFCA can be parameterised in terms of the low-order Mellin moments of the GPDs. We apply this formalism to a numerical investigation for zero-skewness kinematics at two values of the soft momentum transfer, $t = -1.1, -2.2 \;\text{GeV}^2$, and a pion mass of $m_{\pi}\approx 470\;\text{MeV}$. The form factors of the lowest two moments of the nucleon GPDs are determined, including the first lattice QCD determination of the $n=4$ moments. Hence we demonstrate the viability of this method to calculate the OFCA from first principles, and thereby provide novel constraint on the $x$- and $t$-dependence of GPDs.