Tree Amplitudes and Leading Gravitational Deflection

At first post-Minkowskian order, one exchanged graviton already contains the complete relativistic weak-deflection law for two nonspinning point masses. The calculation is a useful calibration because every part of the amplitude dictionary can be checked independently: its slow-motion limit gives the Newtonian potential, its transverse Fourier transform gives the impulse, and its massless-probe limit gives Einstein’s leading light deflection. The amplitude is the input; the scattering angle is the classical observable.

Required background. Events, Intervals, and Minkowski Geometry supplies invariant two-body kinematics; Weak Metric Perturbations and Gauge Freedom supplies the linearized graviton and its coupling. Helpful background. The Einstein Equation fixes the $8\pi G$ normalization; Presymplectic Potentials and Currents motivates comparison through invariant boundary data; Relativistic Two-Body Dynamics places the result in the PM and eikonal expansions.

One-graviton exchange

Take two distinguishable, minimally coupled massive scalars as proxies for structureless bodies. With

$$S_{\mathrm{grav}} = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R, \qquad p_A^2=-m_A^2,$$

define

$$\sigma=-\frac{p_1\cdot p_2}{m_1m_2}, \qquad q^\mu=p_1^{\prime\mu}-p_1^\mu.$$

In the center-of-momentum frame at leading order in the small deflection, $q^0=0$ and $q^2=\mathbf q^2\gt 0$. Using relativistically normalized external states, the long-range part of the tree amplitude is

$$\mathcal M_{\mathrm{tree}}(\mathbf q) = \frac{16\pi Gm_1^2m_2^2}{\mathbf q^2} \left(2\sigma^2-1\right) +O(\mathbf q^0).$$

The $O(\mathbf q^0)$ terms are analytic at zero transfer. They can change contact operators or a chosen potential representation, but they cannot alter a separated-body deflection. This formula is the site-signature translation of the scalar exchange result in Bjerrum-Bohr et al. 2018, “Scalar Interaction Potentials,” pp. 3–5, Eqs. (16), (25), and (28), PDF: their mostly-minus convention has $p_A^2=+m_A^2$ and a spacelike transfer with negative four-square.

Gauge-dependent pieces of the graviton propagator disappear because the on-shell scalar stress tensors are conserved. That cancellation is an amplitude-level check, not a statement that a coordinate-space potential is unique. The surviving numerator depends only on $\sigma$, and is symmetric under $1\leftrightarrow2$.

Newtonian normalization

Before deriving a relativistic angle, verify the external-state normalization. For a conservative instantaneous interaction, the leading nonrelativistic matching is

$$\widetilde V^{(1)}(\mathbf q) = -\frac{\mathcal M_{\mathrm{tree}}(\mathbf q)} {4E_1E_2}.$$

At small speed, $E_A\to m_A$ and $\sigma\to1$, so

$$\widetilde V^{(1)}(\mathbf q) \longrightarrow -\frac{4\pi Gm_1m_2}{\mathbf q^2}.$$

With

$$\int\frac{d^3\mathbf q}{(2\pi)^3} \frac{e^{i\mathbf q\cdot\mathbf r}}{\mathbf q^2} =\frac{1}{4\pi r},$$

one recovers $V(r)=-Gm_1m_2/r$. A missing factor of $4E_1E_2$, or importing a formula with a different one-particle normalization, fails this check immediately. The systematic amplitude-to-potential matching and its field-basis dependence are discussed in Cheung, Rothstein, and Solon 2018, §§II–V, pp. 2–5, PDF.

Eikonal phase and transverse impulse

Let $E=E_1+E_2$ and let $p_\infty=\lvert\mathbf p\rvert$ be the magnitude of either particle’s momentum in the center-of-momentum frame. The invariant identity

$$Ep_\infty = \sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2} =m_1m_2\sqrt{\sigma^2-1}$$

will remove frame-dependent quantities at the end. At leading order, define the impact-parameter eikonal phase by

$$\delta_1(b) = \frac{1}{8Ep_\infty} \int\frac{d^2\mathbf q}{(2\pi)^2}\, e^{i\mathbf q\cdot\mathbf b}\, \mathcal M_{\mathrm{tree}}(\mathbf q).$$

This is the half-phase convention $S(b)=e^{2i\delta(b)}$; the full exponent $2\delta_1$ therefore carries the often-quoted prefactor $1/(4Ep_\infty)$.

The transfer is integrated over the two-plane orthogonal to the incoming momenta. Infrared regularization gives

$$\int\frac{d^2\mathbf q}{(2\pi)^2} \frac{e^{i\mathbf q\cdot\mathbf b}}{\mathbf q^2} =-\frac{1}{2\pi}\log\!\left(\frac{b}{b_0}\right),$$

where $b_0$ is an arbitrary infrared length. Hence

$$\delta_1(b) = -\frac{Gm_1m_2(2\sigma^2-1)} {\sqrt{\sigma^2-1}} \log\!\left(\frac{b}{b_0}\right).$$

The phase depends on $b_0$, but the force and angle do not. Stationary phase relates the transfer magnitude to $-\partial_b(2\delta_1)$. To leading order in the deflection,

$$\chi_{\mathrm{1PM}} =-\frac{2}{p_\infty}\frac{\partial\delta_1}{\partial b} = \frac{2Gm_1m_2}{J} \frac{2\sigma^2-1}{\sqrt{\sigma^2-1}}, \qquad J=bp_\infty.$$

This is the positive magnitude of the angle. If $b^\mu$ points from body 2’s incoming line to body 1’s, attraction gives the transverse impulse

$$\Delta p_{1\perp}^{\mu} = -\frac{2Gm_1m_2(2\sigma^2-1)} {\sqrt{\sigma^2-1}} \frac{b^\mu}{b^2}, \qquad \Delta p_{2\perp}^{\mu}=-\Delta p_{1\perp}^{\mu}.$$

Damour derives the same result directly from linearized general relativity and expresses it as an invariant function of $s$ and $J$ in Damour 2016, §V, pp. 6–7, Eqs. (55)–(61), PDF. Agreement between that trajectory calculation and the amplitude transform is the decisive check; agreement of two gauge-dependent potentials would be weaker.

Why the impact-parameter result is physical

The logarithm in $\delta_1$ is sometimes mistaken for an observable divergence. It is instead the expected phase of a force that extends to arbitrarily large distance. Changing $b_0$ adds a constant to the phase; it cannot change its derivative, the stationary momentum transfer, or the scattering angle. This is the impact-parameter counterpart of the fact that a Coulomb-like plane-wave amplitude needs infrared care even when its classical force is perfectly well defined.

The flux factor is equally important. In the convention used here, $1/(8Ep_\infty)$ converts the relativistically normalized quantum amplitude into the half-phase $\delta$, while $1/(4Ep_\infty)$ converts it into the full exponent phase $2\delta$. Neither is adjustable after the state normalization and the definition of $S(b)$ have been declared. Combining the chosen factor with $Ep_\infty=m_1m_2\sqrt{\sigma^2-1}$ is what turns the answer into a function of invariant asymptotic data. Omitting the flux factor gives the Newtonian limit the wrong mass dependence and spoils the light-deflection limit.

At this order the impulse lies in the two-dimensional plane transverse to both incoming momenta. A longitudinal correction is nevertheless required at the next order because the outgoing body must remain on shell:

$$(p_1+\Delta p_1)^2=-m_1^2 \quad\Longrightarrow\quad 2p_1\cdot\Delta p_1+(\Delta p_1)^2=0.$$

The leading transverse impulse has $p_1\cdot\Delta p_1=0$ and is $O(G)$; its square is $O(G^2)$. Thus a compensating longitudinal piece first appears at $O(G^2)$. This is a useful warning for loop calculations: retaining only a transverse-looking new term at the next PM order can violate the mass-shell constraint even if its transverse coefficient is correct.

Relativistic and Newtonian limits

In the slow relative-motion limit,

$$\sigma=1+\frac{v_\infty^2}{2}+O(v_\infty^4), \qquad p_\infty=\mu v_\infty+O(v_\infty^3), \qquad \mu=\frac{m_1m_2}{m_1+m_2}.$$

The angle becomes

$$\chi_{\mathrm{1PM}} = \frac{2G(m_1+m_2)}{bv_\infty^2} +O(v_\infty^0),$$

which is the weak-deflection limit of a Newtonian hyperbola. This check tests both the relativistic numerator and the flux factor.

The formula also has a smooth massless-probe limit when written before dividing by the masses:

$$\chi_{\mathrm{1PM}} = \frac{2G}{J} \frac{2(p_1\cdot p_2)^2-m_1^2m_2^2} {\sqrt{(p_1\cdot p_2)^2-m_1^2m_2^2}}.$$

For a massless probe of energy $\omega\ll M$ scattering from a body of mass $M$ initially at rest, $-p_1\cdot p_2=M\omega$ and $J=b\omega[1+O(\omega/M)]$. Negligible recoil then gives

$$\chi_{\mathrm{light}}=\frac{4GM}{b}.$$

The small recoil qualification matters: for finite $\omega/M$, the center-of-mass energy is $\sqrt{s}=\sqrt{M^2+2M\omega}$ and $p_\infty=M\omega/\sqrt{s}$, so the center-of-mass 1PM angle is $4G\sqrt{s}/b$, not exactly $4GM/b$. The massless limit must be taken in the invariant form; sending the probe mass to zero while holding the massive-particle variable $\sigma$ fixed is not the fixed-energy limit.

Scope of the result

The derivation assumes pointlike nonspinning bodies, elastic conservative scattering, asymptotically flat kinematics, and a small displayed 1PM angle. The last condition requires $GM/(bv_\infty^2)\ll1$ at low speed and $G\sqrt{s}/b\ll1$ at high energy; $GM/b\ll1$ alone is insufficient across both limits. The result is exact in velocity but only first order in $G$. It omits iterations, new loop information, radiation, absorption, tides, and spin. Continue to Loop Amplitudes and Classical Iterations for the separation of repeated tree exchange from new classical physics.

Exercise

Starting from the invariant 1PM angle, recover both the Newtonian weak-deflection result and the massless-probe result. State which quantities must be held fixed in each limit.

Solution

For the Newtonian limit, hold $b$, $m_1$, $m_2$, and the relative speed $v_\infty$ fixed. Use $\sqrt{\sigma^2-1}=v_\infty+O(v_\infty^3)$, $2\sigma^2-1=1+O(v_\infty^2)$, and $J=b\mu v_\infty+O(v_\infty^3)$. Then

$$\chi = \frac{2Gm_1m_2}{b\mu v_\infty^2} +O(v_\infty^0) = \frac{2G(m_1+m_2)}{bv_\infty^2} +O(v_\infty^0).$$

For the massless-probe limit, hold $\omega$, $b$, and $M$ fixed with $\omega/M\ll1$, and send the probe rest mass to zero. The invariant ratio tends to $2M\omega$, while $J\simeq b\omega$, giving $\chi\simeq4GM/b$. The probe hierarchy is what permits recoil to be neglected. In both limits the arbitrary eikonal scale $b_0$ disappears after differentiating.

References

• Bjerrum-Bohr, N. E. J., Poul H. Damgaard, Guido Festuccia, Ludovic Planté, and Pierre Vanhove, “General Relativity from Scattering Amplitudes”, Physical Review Letters 121, 171601 (2018); 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.