Tree-Level Gravitational Amplitudes

Tree amplitudes expose the physical content hidden by covariant gauge fixing. Conserved stress tensors remove longitudinal propagator terms, massless poles factorize into lower-point amplitudes, and the nonrelativistic pole reproduces Newton’s potential. This worked example scatters two distinct massive scalars by one-graviton exchange, then checks gauge invariance, factorization, and the soft limit.

Required background. Flat-Spacetime Quantum Field Theory supplies LSZ amplitudes and Born matching. Weak-Field Gravity supplies the Newtonian limit. Gravitational Actions and Boundary Terms supplies the source coupling.

Helpful background. Einstein Equations supplies the classical comparison, while Quantum Information supplies unitarity and factorization language.

Scalar stress-tensor vertex

Use $g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$, $\kappa^2=32\pi G$, and the convention

$$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_m}{\delta g^{\mu\nu}}.$$

The leading interaction is $+\kappa h_{\mu\nu}T^{\mu\nu}/2$. Between on-shell scalar states of mass $m$ and momenta $p$ and $p'$, define

$$T^{\mu\nu}(p,p')= p^\mu p'^{\nu}+p^\nu p'^{\mu} -\eta^{\mu\nu}(p\cdot p'+m^2).$$

With $q=p'-p$ and $p^2=p'^2=-m^2$, direct substitution gives

$$q_\mu T^{\mu\nu}=0.$$

The matter–graviton vertex is therefore $i\kappa T^{\mu\nu}/2$. Its normalization follows from the same stress-tensor definition as the classical Einstein equation; changing the sign convention for $T_{\mu\nu}$ requires changing both places.

One-graviton exchange

Let particles 1 and 2 exchange momentum $q$. In harmonic gauge, the four-dimensional propagator numerator gives

$$i\mathcal M_t= \left(\frac{i\kappa}{2}T_1^{\mu\nu}\right) \frac{-iP_{\mu\nu,\rho\sigma}}{q^2-i0} \left(\frac{i\kappa}{2}T_2^{\rho\sigma}\right),$$

or

$$\mathcal M_t= \frac{\kappa^2}{4(q^2-i0)} \left(T_1^{\mu\nu}T_{2\,\mu\nu}-\frac12T_1T_2\right).$$

Any longitudinal gauge-dependent numerator term vanishes because $q_\mu T_a^{\mu\nu}=0$. This is the explicit on-shell Ward check. The same covariant source and propagator construction follows from the gravitational Feynman rules developed by DeWitt DeWitt 1967, §§5–7.

Near $q^2=0$, the amplitude has a simple pole. Its residue is the product of two on-shell scalar–scalar–graviton three-point amplitudes summed over the two physical graviton helicities. This pole factorization is required by unitarity. A local four-scalar counterterm can add a polynomial in $q^2$, but it cannot change the residue of the massless exchange pole.

Newtonian limit

For slowly moving particles, $T_a^{00}\simeq2m_a^2$, $T_a\simeq-2m_a^2$, and spatial components are suppressed. The numerator becomes

$$T_1^{\mu\nu}T_{2\,\mu\nu}-\frac12T_1T_2 \simeq2m_1^2m_2^2.$$

For spacelike transfer $q^0\simeq0$, $q^2\simeq\mathbf q^2$, so

$$\mathcal M_{\mathrm{NR}}(\mathbf q) \simeq\frac{16\pi Gm_1^2m_2^2}{\mathbf q^2}.$$

Relativistically normalized external states obey the Born relation

$$\widetilde V(\mathbf q)=-\frac{\mathcal M_{\mathrm{NR}}}{4m_1m_2} =-\frac{4\pi Gm_1m_2}{\mathbf q^2}.$$

Fourier transformation yields $V(r)=-Gm_1m_2/r$. This independently checks the sign of the source coupling, the factor $\kappa^2=32\pi G$, the trace subtraction, and the external-state normalization. Omitting the $-T_1T_2/2$ term doubles the force.

Soft emission and factorization

Add an outgoing graviton of momentum $k$ and polarization $\varepsilon_{\mu\nu}$, then take $k\to0$ while the hard momenta remain fixed. Attaching the graviton to each external line produces the universal leading behavior

$$\mathcal M_{n+1}(k)= \frac{\kappa}{2}\sum_a\eta_a \frac{p_a^\mu p_a^\nu\varepsilon_{\mu\nu}}{p_a\cdot k} \mathcal M_n+O(k^0),$$

with $\eta_a=+1$ for outgoing and $-1$ for incoming legs. Weinberg proves this leading factor for arbitrary hard processes Weinberg 1965, pp. B516–B524. Replacing $\varepsilon_{\mu\nu}$ by $k_\mu\zeta_\nu+k_\nu\zeta_\mu$ reduces the soft factor to $\kappa\zeta\cdot\sum_a\eta_ap_a$, which vanishes by momentum conservation.

The soft theorem constrains an amplitude but does not determine its finite $O(k^0)$ terms. Likewise, pole factorization fixes residues but permits contact interactions. Gauge invariance, locality, and factorization together sharply restrict tree amplitudes; they do not remove every EFT coefficient at higher derivative order.

Regime and adverse checks

The calculation assumes asymptotically flat states, weak coupling $Gs\ll1$, distinct scalar species so that only the displayed transfer channel is needed, and momentum transfer outside strong-gravity impact parameters. Identical particles require exchange diagrams; spin adds stress-tensor structures; radiation demands an inclusive infrared-safe observable.

Useful adverse checks are crossing the external legs, replacing a polarization by a pure-gauge tensor, taking the nonrelativistic and soft limits in both orders, and verifying the residue with explicit physical helicities. The forward limit $q^2\to0$ is infrared singular because gravity is long ranged; it is not an ultraviolet failure.

The example also separates classical and quantum statements. A tree graph contains no loop factor and its long-range pole reproduces a classical field, even though it was derived with quantum external states. Quantum corrections begin with loops, but powers of $G$ alone do not identify them: nonlinear classical iterations can also carry higher powers of $G$. One must track the loop order, powers of Planck’s constant before setting it to one, and the nonanalytic momentum structure used in the classical limit.

Contact terms provide another discriminating check. Adding a local operator changes the amplitude by a polynomial in the Mandelstam invariants. It can modify short-distance scattering and higher powers of momentum, but it cannot manufacture the $1/\mathbf q^2$ pole responsible for the $1/r$ force. The pole’s location and residue are therefore more universal than the complete tree amplitude at arbitrarily high energy.

Exercises

Exercise. Starting from the scalar stress tensor above, prove $q_\mu T^{\mu\nu}=0$ for $q=p'-p$. Then use the nonrelativistic matrix elements to recover the coefficient of $\widetilde V(\mathbf q)$.

Solution

On shell,

$$q\cdot p=p\cdot p'+m^2, \qquad q\cdot p'=-(p\cdot p'+m^2).$$

Substitution gives

$$q_\mu T^{\mu\nu} =(p\cdot p'+m^2)(p'^\nu-p^\nu-q^\nu)=0.$$

For each slow particle, $T^{00}\simeq2m^2$ and $T\simeq-2m^2$, so the spin-two numerator is $2m_1^2m_2^2$. Multiplying by $\kappa^2/(4\mathbf q^2)$ gives $16\pi Gm_1^2m_2^2/\mathbf q^2$. Dividing by $-4m_1m_2$ in the Born relation yields $\widetilde V=-4\pi Gm_1m_2/\mathbf q^2$, whose position-space transform is the attractive Newtonian potential.

References

• DeWitt, Bryce S. “Quantum Theory of Gravity. II. The Manifestly Covariant Theory.” Physical Review 162 (1967): 1195–1239. DOI.
• Weinberg, Steven. “Infrared Photons and Gravitons.” Physical Review 140 (1965): B516–B524. DOI.