Classical Amplitudes and the Gravity Two-Body Problem

Modern on-shell methods can replace long collections of gauge-dependent gravitational diagrams by amplitudes assembled from physical factorization channels. The price of that compression is a precise dictionary: a quantum amplitude must be placed between localized states, its powers of $\hbar$ must be restored, its potential and radiation regions must be separated, and lower-order iterations must be removed before a classical two-body observable is identified. This chapter develops that dictionary from one-graviton exchange through loops, spin, and the double copy.

Helpful background. The Einstein Equation fixes the pure-gravity theory and coupling; Weak Metric Perturbations and Gauge Freedom supplies physical graviton exchange; Presymplectic Potentials and Currents motivates asymptotic charges as invariant comparison data; Relativistic Two-Body Dynamics places the amplitude method within the wider PM, PN, self-force, and effective-one-body problem.

Enter this chapter

The subject is the gravity-facing classical information in scattering amplitudes, not a replacement course in quantum scattering theory. The starting data are two massive asymptotic states with

$$p_A^2=-m_A^2, \qquad \sigma=-\frac{p_1\cdot p_2}{m_1m_2}\ge1,$$

and a spacelike impact parameter $b^\mu$ in the plane orthogonal to both incoming momenta. The desired outputs are an impulse, scattering angle, spin kick, radiated four-momentum, or waveform. A potential, loop integrand, color numerator, or infrared phase can help calculate those outputs, but none is automatically an observable.

Read the six pages in order, because each removes an ambiguity left by the preceding one:

1. Classical Limit of Quantum Scattering Amplitudes restores $\hbar$, localizes the incoming bodies, and identifies long-range nonanalytic terms without equating loop order with quantum order.
2. Tree Amplitudes and Leading Gravitational Deflection carries one-graviton exchange through normalization, Fourier transformation, and the Newtonian and massless-probe limits.
3. Loop Amplitudes and Classical Iterations separates a genuinely new PM kernel from repeated tree exchange, elastic infrared phases, and $\hbar$-suppressed corrections.
4. Unitarity Cuts for Classical Gravity constructs loop information from on-shell trees and then distinguishes cut kinematics from potential and radiation momentum regions.
5. Spin, Multipoles, and Massive On-Shell States matches quantum little-group states to a classical spin tensor, supplementary condition, centroid, and finite-size multipoles.
6. Double Copy and Classical Two-Body Observables assesses which gauge-theory constructions have survived observable-level gravitational checks and where state, contact, spin, and radiation dictionaries remain conditional.

The sequence moves from established low-order extraction to an active research boundary. Later methods do not weaken the earlier results: the tree deflection remains a sharp benchmark even when the loop or double-copy extension is unsettled.

One observable thread

Temporarily restore Planck’s constant while holding the hard momenta, masses, impact parameter, and classical spin fixed. Long-distance transfer and emitted massless momenta scale as

$$q^\mu=\hbar\bar q^\mu, \qquad k^\mu=\hbar\bar k^\mu, \qquad \hbar\to0.$$

Here $q$ is the soft variable integrated inside a perturbative observable, and $k$ is an individual radiated quantum’s momentum. Their scaling does not make the completed classical impulse or total radiated energy vanish. A fixed quantum spin label becomes spinless in this limit; a macroscopic spin requires $s\to\infty$ with $\hbar s$ fixed. These correlated limits determine which regions and state matrix elements contribute at classical order.

With $S=1+iT$ and relativistically normalized one-particle states, the impulse is

$$\Delta p_1^\mu = \langle\Psi\lvert S^\dagger P_1^\mu S-P_1^\mu \rvert\Psi\rangle.$$

Expanding it produces both a term linear in $T$ and a term quadratic in $T$. The latter combines with loop amplitudes and real-emission channels as required by unitarity. Kosower, Maybee, and O’Connell derive this wave-packet observable and its classical limit in Kosower, Maybee, and O’Connell 2019, §2, pp. 3–5, and §§3–5, pp. 5–45, PDF.

For a conservative elastic calculation, one may instead match to a Hamiltonian or take the logarithm of an impact-parameter $S$ matrix,

$$S(b)=e^{2i\delta(b)}, \qquad \chi_{\rm 1PM}(E,J)=-\frac{2}{p_\infty}\frac{\partial\delta_1}{\partial b}, \qquad J=bp_\infty.$$

The displayed angle relation is the leading small-deflection formula. At higher orders, the Fourier-conjugate impact parameter and the incoming $J/p_\infty$ differ, as explained on Eikonal Methods and High-Energy Deflection. The logarithm separates new phase information from powers of lower-order exchange. In Hamiltonian language the corresponding operation is the Born subtraction

$$V^{(2)} = T_{\mathrm{full}}^{(2)} -V^{(1)}G_0V^{(1)}.$$

Cheung, Rothstein, and Solon display this subtraction and its one-loop realization in Cheung, Rothstein, and Solon 2018, §§III–V, pp. 3–5, especially Eqs. (20)–(24), PDF. An eikonal phase and a Hamiltonian use different intermediate variables, but after the normalization and subtraction conventions are aligned they must give the same angle.

Three dictionaries, one result

The pages can be organized around three distinct maps.

The classical-limit dictionary maps a quantum transition amplitude to a localized macroscopic process. It specifies the wave-packet window, the $q=\hbar\bar q$ scaling, the state normalization, and the soft hierarchy. A nonanalytic term such as $1/\mathbf q^2$ or $1/\lvert\mathbf q\rvert$ signals long range, but only restored $\hbar$ power counting determines whether it survives classically.

The dynamical dictionary maps amplitudes and cuts to new two-body evolution. At loop order, potential gravitons are off shell even though generalized unitarity places selected internal lines on shell to reconstruct the integrand. Region analysis and cut reconstruction answer different questions. The dynamical map also subtracts $V^{(1)}G_0V^{(1)}$, treats infrared phases consistently, and adds radiation-region contributions when the approximation requires them.

The body-and-theory dictionary identifies what scatters and what mediates the interaction. Quantum polarization tensors must be translated to a covariant classical spin with a declared supplementary condition. Kerr’s coefficients give one special infinite multipole tower; generic compact bodies have independent finite-size and tidal data. Likewise, a pure Yang–Mills double copy normally contains a dilaton and two-form as well as a graviton. Obtaining pure Einstein gravity requires a state projection or cancellation prescription.

These maps meet at an invariant. The leading spinless example is

$$\mathcal M_{\mathrm{tree}}(\mathbf q) = \frac{16\pi Gm_1^2m_2^2}{\mathbf q^2} (2\sigma^2-1),$$

which gives

$$\chi_{\mathrm{1PM}} = \frac{2Gm_1m_2}{J} \frac{2\sigma^2-1}{\sqrt{\sigma^2-1}}.$$

It reduces to the Newtonian weak-deflection angle and to $4GM/b$ for a massless probe. Damour obtains the same invariant from a direct classical trajectory calculation in Damour 2016, §V, pp. 6–7, Eqs. (55)–(61), PDF. That agreement is the chapter’s model for every higher-order claim.

Checks that bind the methods together

Each calculation should pass several checks that are independent of its preferred representation.

Dimensions and $\hbar$. After measures, wave packets, cuts, and iterations are assembled, an ordinary classical observable has a finite $\hbar^0$ limit. A leftover negative power commonly signals an unremoved lower-order iteration; a positive power identifies a quantum correction.

Gauge and state content. Reference vectors must cancel from a complete graviton cut. A double-copy construction must state whether its internal sum is pure graviton or graviton–dilaton–two-form. A correct result in dilaton gravity is not a failed GR result, but it cannot be quoted as pure GR without translation.

Mass shell and conservation. For each body,

$$2p_A\cdot\Delta p_A+(\Delta p_A)^2=0.$$

In conservative elastic scattering, $\Delta p_1+\Delta p_2=0$. With spin, the spin length and $p_A\cdot S_A=0$ must also be preserved, and orbital plus intrinsic angular momentum must balance. With radiation, the emitted four-momentum belongs in the conservation law.

Limits and symmetry. The spinless result must be symmetric under interchanging the two bodies. The slow-motion, test-body, zero-spin, and—where meaningful—massless-probe limits provide discriminating comparisons. Spurious Compton poles and regulator scales must cancel from the final observable.

Independent route. A cut calculation should reproduce a discontinuity; a potential should reproduce an angle; a spin amplitude should reproduce the SSC and Kerr multipoles in the black-hole specialization. Bern and collaborators demonstrate how unitarity, double-copy tree data, region integration, effective-theory matching, and scattering-angle checks fit together at spinless 3PM in Bern et al. 2019, §§2–6, pp. 5–42, and §§9–11, pp. 82–100, PDF.

Scope and onward routes

The chapter assumes weak, large-impact-parameter scattering in asymptotically flat spacetime. Individual pages state when they additionally assume point particles, no spin, elastic kinematics, a conservative time-symmetric sector, a particular PM order, or a specified gravity theory. Strong-field capture, merger, and a general quantum-gravity $S$ matrix are outside this scope.

For the invariant PM description of impulses, radial action, radiation, and bound-orbit continuation, continue to Post-Minkowskian Dynamics and Scattering. For permanent multipoles, tidal operators, dissipation, matching, and scale dependence of extended bodies, continue to Worldline Effective Field Theory. Those chapters use amplitude information but give the body model and dynamical regime their own systematic treatment.

Review the chapter

Consider a one-loop small-transfer amplitude with four schematic terms:

$$T^{(2)}(\mathbf q) = I_{\mathrm{iter}}(\mathbf q) +\frac{C_{\mathrm{cl}}}{\lvert\mathbf q\rvert} +\hbar C_{\mathrm q}\log(\mathbf q^2/\mu^2) +C_{\mathrm{loc}}\mathbf q^2.$$

Classify the terms for a separated conservative 2PM observable. What additional information is needed if $C_{\mathrm{cl}}$ was obtained by a double copy of spinning gauge-theory states?

Solution

$I_{\mathrm{iter}}$ is fixed by repeated tree exchange. It must be retained while matching and then canceled against $V^{(1)}G_0V^{(1)}$, or removed by taking the eikonal logarithm. The $1/\lvert\mathbf q\rvert$ term is the candidate new long-range 2PM kernel. It is accepted only after its contribution to an invariant angle or impulse passes the mass-shell, symmetry, and limiting checks.

The logarithm displayed with an explicit factor of $\hbar$ is a quantum long-range correction in this bookkeeping and does not enter the ordinary classical 2PM result. The polynomial term is local: its Fourier transform has support at coincidence and cannot change the force between separated point particles, though it can renormalize a contact operator.

For the spinning double copy, one must also specify the gravitational state projection, including any dilaton or two-form; the generalized-gauge and local Compton completion; the large-spin $\hbar s$ limit; the SSC and polarization transport; and the multipole coefficients defining Kerr or another body. Finally, lower-order spin iterations must be subtracted and the resulting spin kick or angle compared with a direct calculation in the same gravitational theory.

References

• Bern, Zvi, Clifford Cheung, Radu Roiban, Chia-Hsien Shen, Mikhail P. Solon, and Mao Zeng, “Black Hole Binary Dynamics from the Double Copy and Effective Theory”, Journal of High Energy Physics 2019, 206 (2019); Open PDF.
• Cheung, Clifford, Ira Z. Rothstein, and Mikhail P. Solon, “From Scattering Amplitudes to Classical Potentials in the Post-Minkowskian Expansion”, Physical Review Letters 121, 251101 (2018); Open PDF.
• Damour, Thibault, “Gravitational Scattering, Post-Minkowskian Approximation and Effective-One-Body Theory”, Physical Review D 94, 104015 (2016); Open PDF.
• Kosower, David A., Ben Maybee, and Donal O’Connell, “Amplitudes, Observables, and Classical Scattering”, Journal of High Energy Physics 2019, 137 (2019); Open PDF.