Loop Amplitudes and Classical Iterations

A gravity loop can carry genuinely new classical information. It can also reproduce the effect of applying the tree interaction twice, generate a long-range infrared phase, and contain ordinary quantum corrections. These pieces coexist in the same amplitude. Extracting a higher-PM Hamiltonian therefore requires a matching calculation, while extracting an impulse or angle requires the corresponding unitarity or eikonal completion. Keeping the classical-looking loop term without removing lower-order iteration double counts the dynamics.

Required background. Events, Intervals, and Minkowski Geometry supplies invariant elastic kinematics; Weak Metric Perturbations and Gauge Freedom supplies the exchanged weak field. Helpful background. The Einstein Equation fixes the gravitational coupling; Presymplectic Potentials and Currents distinguishes canonical representations from invariant data; Relativistic Two-Body Dynamics supplies the PM expansion and its overlap with Hamiltonian methods.

Classical scaling inside a loop

Let the conservative transition operator and an effective interaction kernel be expanded as

$$T=\sum_{n\geq1}G^nT^{(n)}, \qquad V=\sum_{n\geq1}G^nV^{(n)}.$$

For separated spinless bodies in four spacetime dimensions, the long-range momentum dependence follows the pattern

$$T^{(1)}_{\mathrm{cl}}\sim\frac{1}{\mathbf q^2}, \qquad T^{(2)}_{\mathrm{cl,new}}\sim\frac{1}{\lvert\mathbf q\rvert}, \qquad T^{(3)}_{\mathrm{cl,new}}\sim\log\!\left(\frac{\mathbf q^2}{\mu^2}\right),$$

up to mass- and velocity-dependent coefficients. These structures Fourier transform to $1/r$, $1/r^2$, and $1/r^3$ behavior away from the origin. The displayed hierarchy identifies possible long-range terms; restored $\hbar$ counting is still needed because a quantum loop correction can also contain a logarithm.

The classical part of a loop comes from momenta much smaller than the hard masses and external three-momentum. In a nonrelativistic region analysis,

$$\ell^0\sim v\lvert\mathbf q\rvert, \qquad \lvert\boldsymbol\ell\rvert\sim\lvert\mathbf q\rvert$$

defines the potential region. The exchanged gravitons are off shell and nearly instantaneous, while an internal massive propagator approaches its mass shell. This is how a quantum loop integral produces a classical propagation history. Hard momenta instead probe the bodies’ Compton-scale structure and contribute local or $\hbar$-suppressed quantum terms.

At unrestricted velocity, the potential-region expansion is best understood as a device for generating the velocity series, followed by resummation and comparison with a relativistic evaluation. It must not be promoted to an exact separation of modes without that check. Bern and collaborators state the hierarchy, mode scalings, and classical graph criteria in Bern et al. 2019, §§2.1–2.3, pp. 5–12, PDF.

Born iteration and effective-theory matching

Choose a definite two-body Hilbert-space normalization and free Hamiltonian

$$H_0(\mathbf p)=E_1(\mathbf p)+E_2(\mathbf p), \qquad E_A(\mathbf p)=\sqrt{\mathbf p^2+m_A^2}.$$

The Lippmann–Schwinger equation is

$$T(E)=V+VG_0(E)T(E), \qquad G_0(E)=\frac{1}{E-H_0+i0}.$$

Expanding through $G^2$ gives the exact matching identity

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

In momentum space, the repeated lower-order interaction is

$$\begin{aligned} \langle\mathbf p'\lvert V^{(1)}G_0V^{(1)} \rvert\mathbf p\rangle &= \int\frac{d^d\boldsymbol\ell}{(2\pi)^d}\, \frac{ V^{(1)}(\mathbf p',\boldsymbol\ell) V^{(1)}(\boldsymbol\ell,\mathbf p) } { E-E_1(\boldsymbol\ell)-E_2(\boldsymbol\ell)+i0 }. \end{aligned}$$

Consequently,

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

This equation is meaningful only after fixing the transition-operator convention. With spatial states normalized by $(2\pi)^3\delta^3$, $S_{fi}=\delta_{fi}-2\pi i\delta(E_f-E_i)T_{fi}$, and the relativistic amplitude convention of the preceding page, elastic Born matching gives $T_{\rm Born}=-\mathcal M_{\rm tree}/(4E_1E_2)$. The minus sign turns the positive gravitational tree amplitude into an attractive potential. Momentum-dependent changes of basis alter the off-shell kernel and its iteration, while the complete on-shell scattering remains unchanged.

At one loop, box topologies contain precisely the enhanced, infrared-sensitive behavior expected from $V^{(1)}G_0V^{(1)}$. Triangle topologies supply the characteristic new $1/\lvert\mathbf q\rvert$ term. That mnemonic is useful at 2PM, but the matching identity—not a diagram label—is the general rule. Cheung, Rothstein, and Solon show at integrand level that the box coefficient factorizes into tree coefficients and that its low-energy singularity cancels the effective-theory iteration in Cheung, Rothstein, and Solon 2018, §§III–IV, pp. 3–4, Eqs. (20)–(24), PDF.

Super-classical terms and the eikonal logarithm

Before subtraction, a loop may scale more strongly in small $\lvert\mathbf q\rvert$ or in inverse powers of $\hbar$ than the genuinely new classical term. Such a “super-classical” contribution is not a new force law. It is the perturbative expansion of repeated lower-order evolution.

The same statement appears cleanly in impact-parameter space. If the elastic $S$ matrix in a fixed channel is written schematically as

$$S(b)=\exp\!\left[2i\delta(b)\right], \qquad \delta=G\delta_1+G^2\delta_2+\cdots,$$

then the $G^2$ term in $S-1$ contains both $2i\delta_2$ and the fixed iteration $-2\delta_1^2$. Taking the logarithm separates the new eikonal phase from the square of tree exchange. A potential match, an eikonal logarithm, and an observable impulse therefore remove the same repeated physics in different representations.

In the observable approach, write $S=1+i\mathcal T$; this $\mathcal T$ includes the energy-conserving distribution and is distinct from the Lippmann–Schwinger $T$ above. The cancellation involves both the one-loop amplitude and the term quadratic in the tree transition matrix:

$$\Delta p_1^\mu = \langle\Psi\lvert i(P_1^\mu\mathcal T-\mathcal T^\dagger P_1^\mu) +\mathcal T^\dagger P_1^\mu\mathcal T \rvert\Psi\rangle.$$

Kosower, Maybee, and O’Connell exhibit the cancellation among box, crossed-box, and cut-box contributions in a complete next-to-leading impulse in Kosower, Maybee, and O’Connell 2019, §5.2.3–§5.2.6, pp. 38–45, PDF. The lesson is structural even though their explicit example is electromagnetic: unitarity completion is part of the classical limit.

Infrared phases and radiation regions

Long-range forces do not admit an infrared-finite plane-wave $S$ matrix term by term. Dimensional regularization may expose poles and logarithms, while a small graviton mass or finite-time prescription reorganizes them differently. The iteration term carries the same low-energy singularity as the full theory because both describe identical long-distance propagation. Their difference, or the derivative of the eikonal phase that yields the angle, is the relevant conservative quantity.

It is incorrect to discard every imaginary term. Some imaginary pieces are pure elastic phases and exponentiate; others are fixed by unitarity cuts; still others represent real radiation or absorption. The diagnosis depends on the observable and boundary conditions. When no radiation is kinematically included in the sector under study, the conservative kernel is obtained after removing the elastic iteration. When radiation is allowed, particle impulse plus radiated four-momentum must satisfy total balance.

Potential modes are sufficient for the conservative spinless 2PM example, but this does not remain true at arbitrary order. Radiation modes satisfy $\ell^0\sim\lvert\boldsymbol\ell\rvert$ and generate dissipative effects as well as time-symmetric tail contributions. Near-zone and radiation-region expansions can overlap, so a subtraction prescription and scale cancellation are required. A higher-PM result that retains only the potential region must state why radiation, tails, and overlap terms are absent at that order.

A practical subtraction pattern

Suppose a regulated one-loop result has the schematic long-range form

$$T_{\mathrm{full}}^{(2)}(\mathbf q) = I_{\mathrm{Born}}(\mathbf q;\delta_{\rm IR},\mu) +\frac{C_{\mathrm{cl}}(\sigma,m_1,m_2)} {\lvert\mathbf q\rvert} +\hbar\,C_{\mathrm{q}}(\sigma) \log\!\left(\frac{\mathbf q^2}{\mu^2}\right) +\cdots.$$

Here dimensional regularization uses $d=3-2\delta_{\rm IR}$ spatial dimensions and scale $\mu$. The term $I_{\mathrm{Born}}$ includes the enhanced infrared phase and is completely determined by $V^{(1)}G_0V^{(1)}$ in the chosen basis. Matching removes it. The second term is the new 2PM classical kernel. The explicitly $\hbar$-suppressed logarithm is a quantum long-range correction, and the ellipsis includes contact terms or higher powers of transfer. This decomposition is schematic: coefficients and even the allocation of finite pieces to $I_{\mathrm{Born}}$ depend on normalization and basis, while the final scattering angle does not.

Three checks should be applied together. The enhanced term must cancel exactly against the lower-order iteration. The remaining Hamiltonian must reproduce the same invariant angle as the eikonal or impulse calculation. Any regulator scale must cancel from that angle at the retained order, unless a physical tail scale with a separately identified radiation contribution remains.

Common pitfalls

Keeping the whole loop as a new potential. This counts the tree interaction twice and generally leaves an infrared phase in a supposedly conservative Hamiltonian.

Dropping boxes by topology alone. At 2PM their enhanced part is iterative, but topology is not a substitute for matching, especially with spin, higher operators, or higher loops.

Calling every imaginary part unphysical. Elastic phases, cut terms, emitted radiation, and absorption have different meanings. Decide using unitarity and the observable’s boundary conditions.

Exercise

Let $T^{(1)}=V^{(1)}$ and suppose a calculation produces

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

Find $V^{(2)}$, and show directly that using $GV^{(1)}+G^2W$ in the Lippmann–Schwinger equation reproduces $T_{\mathrm{full}}$ through $G^2$.

Solution

Order-by-order matching gives

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

Now set $V=GV^{(1)}+G^2W$. Iterating once,

$$\begin{aligned} T &=V+VG_0V+O(G^3)\\ &=GV^{(1)} +G^2\left[W+V^{(1)}G_0V^{(1)}\right] +O(G^3), \end{aligned}$$

which equals the assumed full result. If one had instead set $V^{(2)}=T_{\mathrm{full}}^{(2)}$, the term $V^{(1)}G_0V^{(1)}$ would appear once inside $V^{(2)}$ and a second time from solving the scattering equation.

The next page explains how on-shell cuts construct the loop information needed before this subtraction: Unitarity Cuts for Classical Gravity.

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.
• Kosower, David A., Ben Maybee, and Donal O’Connell, “Amplitudes, Observables, and Classical Scattering”, Journal of High Energy Physics 2019, 137 (2019); Open PDF.