Unitarity Cuts for Classical Gravity

Generalized unitarity constructs a gravity loop from products of on-shell tree amplitudes. For the classical two-body problem, this replaces a large collection of gauge-dependent Feynman diagrams by a smaller set of gauge-invariant cut conditions. The simplification is powerful but conditional: a cut determines analytic information in the quantum amplitude, not yet a classical observable. Momentum-region power counting, iteration subtraction, and an invariant final check are still required.

Required background. Events, Intervals, and Minkowski Geometry supplies mass shells and invariant phase space; Weak Metric Perturbations and Gauge Freedom supplies physical graviton polarizations and stress-tensor conservation. Helpful background. The Einstein Equation fixes the gravitational state content and coupling; Presymplectic Potentials and Currents clarifies gauge-invariant boundary data; Relativistic Two-Body Dynamics supplies the PM observable that the cut calculation must reproduce.

The two-graviton cut

For elastic scattering of two massive bodies, a one-loop $t$-channel cut places two exchanged massless lines on shell. Schematically,

$$\begin{aligned} \operatorname{Cut}_t\mathcal M^{\mathrm{1\,loop}} &= \sum_{\lambda_1,\lambda_2} \mathcal M_{\mathrm L}^{\mathrm{tree}} \left(p_1,p_1';\ell_1^{\lambda_1},\ell_2^{\lambda_2}\right)\\ &\qquad\times \mathcal M_{\mathrm R}^{\mathrm{tree}} \left(p_2,p_2';-\ell_1^{-\lambda_1},-\ell_2^{-\lambda_2}\right), \end{aligned}$$

with

$$\ell_1^2=\ell_2^2=0, \qquad \ell_1+\ell_2=q.$$

The sum is over physical intermediate states. The distributional identity behind a discontinuity is

$$\frac1{x+i0}-\frac1{x-i0}=-2\pi i\delta(x).$$

For the site’s signature, a scalar Feynman propagator can be written $-i/(\ell^2+m^2-i0)$; its overall factor and the orientation of the discontinuity must be carried through the cut together. A physical positive-energy massless state has measure $d^4\ell\,\theta(\ell^0)\delta(\ell^2)/(2\pi)^3$. In the physical scattering region $q$ is spacelike, so two real future-directed null vectors cannot sum to $q$. A $t$-channel cut is continued from its physical crossed channel, or solved algebraically with complex momenta. It does not describe two gravitons emitted in this encounter.

The tree factors here are gravitational Compton amplitudes: a graviton scatters from a massive body. They include all pole diagrams and contact terms required by gauge invariance.

Each tree amplitude is invariant under replacing a graviton polarization by a pure-gauge tensor once the complete set of tree diagrams and contact terms has been included. Reference-vector dependence must therefore cancel in their product. This makes a cut an excellent test of a proposed integrand. The general relation between discontinuities, multidimensional residues, and leading singularities is developed in Cachazo and Guevara 2017, §2, pp. 4–15, PDF.

The cut fixes coefficients of integral functions whose propagators can all be exposed by that cut. A spanning collection of cuts is needed because one cut does not generally determine every contact term or numerator component. Two candidate integrands that agree on a selected cut may disagree on another cut or by terms invisible in strictly four-dimensional kinematics.

Classical information selected by the cut

The long-range conservative contribution is nonanalytic in the small spacelike transfer $q$. At one loop, the relevant two-graviton cut crosses the $t$ channel and separates the two massive matter lines. Its product of gravitational Compton trees determines triangle and box integral coefficients.

For the spinless 2PM problem, the important behaviors are:

• A triangle with one massive and two massless propagators produces the new classical $1/\lvert\mathbf q\rvert$ term.
• A box can scale more strongly, but its enhanced part is the iteration of one-graviton exchange. It belongs to the eikonal exponent or to $V^{(1)}G_0V^{(1)}$, not to a new $V^{(2)}$.
• Bubble, rational, and polynomial terms are analytic or too soft in the large-angular-momentum counting to affect the separated-body 2PM potential.

Cachazo and Guevara show that the relevant triangle leading singularities reproduce the classical one-loop contribution, while the potential still requires tree-iteration subtraction, in Cachazo and Guevara 2017, §3, pp. 16–22, especially Eqs. (3.17)–(3.23), PDF. The statement is deliberately restricted to the stated one-loop, long-range problem. At higher loops, with finite-size operators, or with radiation, terms invisible to a simple triangle mnemonic can matter.

Bern and collaborators formulate a scalable set of classical cut criteria in Bern et al. 2019, §2.3, pp. 9–12, PDF. A useful cut separates the two matter species onto opposite sides; every independent potential loop has a massive line close enough to its mass shell to supply the energy residue; self-energy-like exchanges attached only to one matter line do not produce the desired two-body potential; and numerator terms that are too high order in the soft loop momenta are quantum or local. These are power-counting results, not universal graph deletion rules.

Cut kinematics versus momentum regions

Three notions of “on shell” must be kept distinct. External bodies are physical asymptotic states. Cut internal lines are placed on shell to reconstruct the analytic amplitude, often at complex momenta. Potential gravitons in the later region expansion are off shell, with energy parametrically smaller than spatial momentum. There is no contradiction: the cut determines integral coefficients, and the integration or region analysis determines how those integrals contribute to the classical limit.

In a slow-motion expansion the main regions scale as follows:

Region	Scaling	Classical role
Hard	$\ell^0\sim\lvert\boldsymbol\ell\rvert\sim m$	Short-distance quantum or local terms
Soft	$\ell^0\sim\lvert\boldsymbol\ell\rvert\sim\lvert\mathbf q\rvert$	Relativistic long-range region containing potential contributions
Potential	$\ell^0\sim v\lvert\mathbf q\rvert$, $\lvert\boldsymbol\ell\rvert\sim\lvert\mathbf q\rvert$	Near-zone conservative propagation and its iterations
Radiation	$\ell^0\sim\lvert\boldsymbol\ell\rvert\sim v\lvert\mathbf q\rvert$	Real radiation and time-symmetric tail effects at appropriate orders

The regions can overlap. Dimensional regularization makes many overlap integrals scaleless, but that fact must be shown in the problem at hand. At sufficiently high PM or PN order, radiation-region and near-zone pieces share a scale and only their sum is meaningful.

The classical $\hbar$ scaling supplies a parallel check. With $\ell^\mu=\hbar\bar\ell^\mu$ and $q^\mu=\hbar\bar q^\mu$, the cut measure, tree amplitudes, and propagators must combine to a finite observable. A coefficient that survives a cut but leaves an uncanceled negative power of $\hbar$ after the Born or eikonal iteration is included is not a new classical observable.

A reliable reconstruction workflow

First specify the theory whose states cross the cut. The double copy of pure Yang–Mills naturally contains a dilaton and two-form in addition to the graviton; a pure-Einstein calculation must project or cancel those states rather than call the full sum “gravity.”

Second build complete on-shell tree amplitudes. Factorization diagrams alone may be insufficient: gauge invariance and locality can require contact terms, especially for spin and higher multipoles.

Third declare the regulator and state sum. Four-dimensional helicity methods are efficient, but an $O(\epsilon)$ numerator multiplied by a $1/\epsilon$ infrared pole can leave a finite contribution. A calculation using $D=4-2\epsilon$ integration therefore needs either $D$-dimensional cuts or an argument showing why the missing components cannot affect the classical observable.

Fourth assemble an integrand that satisfies a spanning set of cuts. Generalized gauge transformations may move terms among diagrams without changing cuts; the integrand representation is not unique.

Fifth perform the classical region expansion and restore $\hbar$. Identify new long-range terms separately from hard quantum corrections and lower-order iterations.

Finally extract the impulse, eikonal angle, or matched Hamiltonian and compare it with an independent classical calculation. Cheung, Rothstein, and Solon carry out this sequence at one loop—unitarity coefficients, potential-region reduction, integrand subtraction, and a GR match—in Cheung, Rothstein, and Solon 2018, §§III–V, pp. 3–5, PDF.

Discriminating checks

Gauge reference vectors must cancel on every complete cut. Relabeling the two massive bodies must map the result by $m_1\leftrightarrow m_2$ and $q\to-q$. Spurious poles introduced by a parametrization must cancel in the assembled integrand or have zero residue. The soft-transfer scaling must reproduce the expected $1/\mathbf q^2$, $1/\lvert\mathbf q\rvert$, or logarithmic long-range structure at the appropriate PM order.

Cut completeness and classical relevance are separate checks. A term may be required for the quantum integrand but drop from the long-range classical observable; another term may be invisible on a restricted four-dimensional cut yet combine with an infrared pole. The safest final test is observable-level agreement: after iteration subtraction, the angle must match a Hamiltonian or direct general-relativistic trajectory calculation and must satisfy the Newtonian and test-body limits.

Common pitfalls

Treating a cut as emitted radiation. A generalized cut is an analytic reconstruction condition. Whether a radiation region contributes is decided by kinematics, boundary conditions, and the later region analysis.

Summing the wrong gravitational states. Squaring gauge-theory states can introduce a dilaton and two-form. Pure Einstein gravity requires an explicit state prescription.

Assuming four-dimensional cuts are automatically complete. Evanescent numerator terms can multiply dimensional poles. State why they are irrelevant or compute the needed $D$-dimensional information.

Exercise

Suppose the classical part of a one-loop integrand reduces after integration to

$$\mathcal M^{(2)} =C_{\Box}I_{\Box} +C_{\triangle,1}I_{\triangle,1} +C_{\triangle,2}I_{\triangle,2} +\text{analytic terms},$$

where $I_{\Box}$ has the enhanced scaling of two repeated tree exchanges and $I_{\triangle,A}\sim1/\lvert\mathbf q\rvert$. Which terms determine the new 2PM potential, and what further check is required before accepting them?

Solution

The box coefficient is fixed by the product of lower-order tree data, but its enhanced contribution represents the Born iteration. It must be retained while matching and then canceled against $V^{(1)}G_0V^{(1)}$. The two triangle terms supply the candidate new $1/\lvert\mathbf q\rvert$ kernel, including both mass assignments.

They are not accepted merely because they came from a cut. One must verify the graviton state sum and normalization, add both triangle assignments, perform the iteration subtraction in a declared basis, and show that the resulting invariant scattering angle agrees with an independent classical or Hamiltonian calculation. Analytic terms can be omitted only after confirming that they are contact terms for the stated point-particle long-range observable.

Spin changes both the tree building blocks and their contact terms. Continue to Spin, Multipoles, and Massive On-Shell States.

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.
• Cachazo, Freddy, and Alfredo Guevara, “Leading Singularities and Classical Gravitational Scattering”, Journal of High Energy Physics 2020, 181 (2020); 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.