[Paper Review] From Scattering Amplitudes to Classical Physics: Universality, Double Copy and Soft Theorems

Yilber Fabian Bautista, Alfredo Guevara|arXiv (Cornell University)|Mar 29, 2019

Quantum Mechanics and Applications2 references44 citations

TL;DR

The paper develops a covariant multipole expansion for massive-particle scattering with photon/graviton emission, reveals universality and soft-exponentiation, and establishes a classical double copy for radiation, with applications to memory effects and gravitational radiation from QED Bremsstrahlung.

ABSTRACT

We introduce a covariant Multipole Expansion for the scattering of a massive particle emitting photons or gravitons in $D$ dimensions. We find that these amplitudes exhibit very powerful features such as universality, soft exponentiation, orbit and spin multipoles, etc. Using ${ m{SO}}(D)$ representation theory we show that the photon and graviton amplitudes are related via a new double copy procedure for massive spinning states. All these features are then promoted to properties of the observables arising in the classical version of such theories. Focusing on radiation, we provide two main applications: 1) An exponential Soft Theorem relating conservative effects and gravitational radiation to all orders in $ω$; whose leading order directly leads to the $D=4$ Memory Effect. 2) A classical double copy to evaluate gravitational radiation from QED Bremsstrahlung, matching previous classical computations and extending them to spin-quadrupole order.

Motivation & Objective

• Motivate a covariant framework to describe scattering of massive bodies emitting massless fields across D dimensions.
• Uncover universality and soft-theorem structures in photon and graviton amplitudes, including orbit and spin multipoles.
• Demonstrate a double copy relation between photon and graviton amplitudes in the massive, spinning case.
• Translate quantum amplitudes into classical observables such as radiation and memory effects.
• Provide classical double-copy prescriptions to compute gravitational radiation from QED Bremsstrahlung results.

Proposed method

• Introduce a covariant multipole expansion for A_n^h,s describing transitions of a massive spin-s state emitting n-2 massless particles.
• Use SO(D) representation theory to relate photon and graviton amplitudes via a new double copy operation for massive spinning states (A_n^{ph,s} odot A_n^{ph, ilde s} = K_n A_n^{gr,s+ ilde s}).
• Represent 3-pt seeds A_3^s,ph in a spin-covariant form with J^{μν} operators and exponential maps e^{ω J}, linking to Lorentz transformations.
• Expand A_4 and A_5 in terms of orbit multipoles and spin multipoles, showing how soft expansion encodes conservative vs radiative contributions.
• Derive explicit forms for spin-dependent multipoles up to quadrupole order, including Weyl-tensor structures and Q^{μν} projections, and connect to known Lagrangian results in D≥4.
• Promote results to classical observables by taking the ħ→0 limit and using cut-structure of M_4 and M_5 to obtain radiation and memory formulas.
• Demonstrate a classical double-copy for radiative observables by translating photon numerators into gravity via the n_gr = n_ph ⊙ n_ph − tr(n_ph ⊙ n_ph) rule.

Experimental results

Research questions

• RQ1 How can a covariant multipole expansion capture universality and soft-theorem structure in the scattering of massive particles emitting photons or gravitons?
• RQ2 What is the proper double-copy operation that relates massive spinning photon and graviton amplitudes in arbitrary dimensions?
• RQ3 How do orbit and spin multipoles organize the soft expansion and correspond to classical radiation multipoles (dipole, quadrupole, etc.)?
• RQ4 Can the framework yield explicit expressions for classical observables such as radiation fields and memory effects directly from quantum amplitudes?
• RQ5 Do the results extend to higher spins and dimensions consistently, and how do they relate to known Lagrangian and Kerr-type descriptions?

Key findings

• A covariant multipole expansion reveals universality and soft-exponentiation in massive A_n^h,s amplitudes across D dimensions.
• Photon and graviton amplitudes are connected by a new classical double-copy procedure for massive spinning states, implemented via an ω- and J^{μν}-dependent exponential map.
• Orbit multipoles and spin multipoles organize the soft expansion, with leading terms recovering dipole radiation and gravitational quadrupole radiation, and higher orders describing additional multipoles.
• Explicit spin-dependent seeds lead to expressions for A_3^{gr,s} that reproduce Weyl-tensor couplings and Kerr-like stress-energy structures in appropriate limits.
• Classical observables such as the gravitational radiation from M_4 and M_5, memory effects, and the 1PM scattering angle can be obtained from the soft-exponential formalism, with results matching known computations in four dimensions and extending to D>4.
• A purely classical double-copy formula is established at the level of radiative observables, showing that gravity radiative data can be constructed from Maxwell (photon) data without introducing dilaton/axion states at the considered order.