Quantum Corrections to Long-Range Potentials

At the leading one-loop order considered here, unknown local counterterms change analytic momentum polynomials. Massless loops also generate branch points—logarithms and square roots—that no local counterterm can cancel. Their Fourier transforms produce power-law tails with coefficients fixed by the light spectrum and lower-order couplings. A coordinate-space “potential” is a useful representation with a specified Born convention; the underlying on-shell amplitude is the invariant object.

Required background. Flat-Spacetime Quantum Field Theory supplies loop analyticity and Fourier transforms. Gravitational Actions and Boundary Terms supplies local counterterms.

Helpful background. Einstein Equations supplies the classical potential, Quantum Information supplies unitarity language, and Quantum Gravity supplies the EFT regime.

Analytic structure at small transfer

Use the mostly-plus metric. For a static exchange, $q^0=0$ and $q^2=\mathbf q^2\gt0$. Physical timelike cuts lie at $q^2\lt0$. With principal logarithm and square root, the low-transfer amplitude can contain

$$\mathcal M(q)= \frac{A}{q^2}+B\frac{1}{\sqrt{q^2-i0}} +C\log\!\left(\frac{q^2-i0}{\mu^2}\right) +P(q^2),$$

where $P$ is a polynomial. The massless pole produces the Newtonian force. The square root and logarithm arise from massless propagation. A local counterterm can change $P$, including its scale-dependent coefficients, but cannot remove either branch point.

The distinction is more reliable than labeling a diagram “quantum.” A one-loop graph can contain a classical square-root term as well as a quantum logarithm. Restoring Planck’s constant and performing the appropriate large-mass expansion separates their classical and quantum scaling.

Three-dimensional transform dictionary

With the inverse-transform convention carrying $(2\pi)^{-3}$ in momentum space, the noncontact identities for $r\gt0$ are

$$\operatorname{FT}^{-1}\!\left[\frac{1}{\mathbf q^2}\right] =\frac{1}{4\pi r},$$ $$\operatorname{FT}^{-1}\!\left[\frac{1}{\lvert\mathbf q\rvert}\right] =\frac{1}{2\pi^2r^2},$$

and

$$\operatorname{FT}^{-1}\!\left[\log\!\left(\frac{\mathbf q^2}{\mu^2}\right)\right] =-\frac{1}{2\pi r^3}.$$

Changing $\mu$ adds a constant in momentum space and therefore changes only a contact term at $r=0$. Likewise, a polynomial $P(\mathbf q^2)$ transforms to delta functions and derivatives at the source. For separated bodies, the inverse-square and inverse-cube tails are insensitive to those local shifts.

An independent dimensional check gives the same powers. In three spatial dimensions, a kernel with momentum dimension $-2$, $-1$, or $0$ transforms respectively as $r^{-1}$, $r^{-2}$, or $r^{-3}$, apart from contact distributions.

The logarithmic coefficient can be checked without a cutoff by analytic continuation. For $r\gt0$, define

$$F_\alpha(r)=\operatorname{FT}^{-1}\!\left[(\mathbf q^2)^{-\alpha}\right] =\frac{\Gamma(3/2-\alpha)}{4^\alpha\pi^{3/2}\Gamma(\alpha)} r^{2\alpha-3}.$$

Near $\alpha=0$, $F_\alpha(r)=\alpha/(2\pi r^3)+O(\alpha^2)$ away from the contact point. Because

$$-\left.\frac{\partial}{\partial\alpha}(\mathbf q^2)^{-\alpha}\right|_{\alpha=0} =\log\mathbf q^2,$$

differentiating gives $-1/(2\pi r^3)$. A reference scale inside the logarithm changes only the omitted contact distribution. This second route fixes both the radial power and sign of the separated-point identity.

A scattering-amplitude potential convention

For two minimally coupled spinless masses, with gravitons as the only massless loop species in this result, define a scattering potential by taking the nonrelativistic three-dimensional Fourier transform of the complete on-shell amplitude. In this convention the long-range terms are

$$V(r)=-\frac{Gm_1m_2}{r} \left[ 1+3\frac{G(m_1+m_2)}{r} +\frac{41}{10\pi}\frac{G}{r^2} +\cdots \right].$$

The second term inside the brackets is a classical post-Minkowskian correction; it comes from the $1/\lvert\mathbf q\rvert$ nonanalyticity. The $41G/(10\pi r^2)$ term is the leading quantum correction in units with $\hbar=1$; restoring units attaches one power of $\hbar$. The complete full-amplitude calculation, its potential definition, and the result appear in Bjerrum-Bohr, Donoghue, and Holstein 2003, §2.1, §4, Eq. (44).

The coefficient $3$ is not a standalone observable. Subtracting the second Born iteration to construct a bound-state Hamiltonian reshuffles the classical static term together with momentum-dependent operators, and a coordinate or canonical transformation can reshuffle it again. The quantum term is unchanged by that Born subtraction at this order, but a fully specified potential still requires a coordinate and Hamiltonian convention. The complete scattering amplitude, bound-state energy, or scattering angle computed consistently is invariant.

This source distinction matters historically. Donoghue’s original 1994 one-particle-reducible calculation printed a different coefficient; the later full-amplitude calculation identified the earlier vertex-algebra problem and obtained $41/(10\pi)$ Bjerrum-Bohr, Donoghue, and Holstein 2003, §§1, 3.5, 4. The robust low-energy claim is the nonanalytic on-shell amplitude: unknown local ultraviolet counterterms cannot change its separated-point coefficient once the light spectrum and observable convention are fixed.

Checks and limits

Several checks discriminate the result. The Newtonian term scales as $G$ and $1/r$. The classical correction scales as $G^2(m_1+m_2)/r^2$ and survives the classical limit after the masses and kinematics are scaled consistently. The quantum term scales as $G^2/r^3$ in the full potential and vanishes when $\hbar\to0$ is restored.

Adding another massless field changes the branch-cut discontinuity and therefore its long-range coefficient; “universal” refers to a declared light spectrum and EFT order. Giving the particles responsible for a cut a mass moves that cut to a threshold: its direct separated-point contribution is exponentially suppressed at large $r$, while its low-transfer expansion renormalizes local operators. At higher orders, those matched operators can appear inside light-field loops and affect subleading power-law tails. Heavy physics is therefore not excluded from every nonanalytic coefficient at all orders.

The expansion requires $Gm_a/r\ll1$, $G/r^2\ll1$, separation larger than body sizes and Compton wavelengths appropriate to the point-particle approximation, and a controlled subtraction of Born iterations. At $r$ near the Schwarzschild radius or Planck length, higher orders are not suppressed.

A final scale check compares successive brackets directly. The classical correction relative to Newton is $G(m_1+m_2)/r$, while the quantum correction is $G/r^2$. For macroscopic bodies at large separation the former can dominate enormously even though both are small. Calling every $G^2$ term “quantum” would therefore misclassify the post-Newtonian contribution. Loop topology, restored powers of $\hbar$, and the mass dependence must be checked together.

Exercises

Exercise. Classify the spatial support of $a+b\mathbf q^2+c/\lvert\mathbf q\rvert+d\log(\mathbf q^2/\mu^2)$. Which coefficients can unknown local matching change, and what are the separated-point radial powers of the remaining terms?

Solution

The $a$ and $b\mathbf q^2$ terms are analytic. They transform to a delta function and its Laplacian, so local counterterms or matching can change them. The $c/\lvert\mathbf q\rvert$ term is nonanalytic and gives $c/(2\pi^2r^2)$. The logarithm is also nonanalytic and gives $-d/(2\pi r^3)$ for $r\gt0$, plus a scheme-dependent contact term. Unknown short-distance physics cannot alter $c$ or $d$ without adding or changing degrees of freedom that are light at the transfer scale.

References

• Bjerrum-Bohr, N. E. J., John F. Donoghue, and Barry R. Holstein. “Quantum Gravitational Corrections to the Nonrelativistic Scattering Potential of Two Masses.” Physical Review D 67 (2003): 084033; erratum 71 (2005): 069903. DOI. Open manuscript.