QFT Bridge: Born Approximation and Tree Level

The first Born approximation connects a weak potential to a momentum-space scattering amplitude. A QFT exchange amplitude can sometimes be matched to that potential at low velocities. The connection explains why a massive mediator produces a finite-range interaction, while keeping the two amplitude normalizations distinct.

Required background. First Born Approximation derives the potential amplitude; the convention dictionary fixes its continuum normalization.

Born Approximation

For a local potential in three dimensions, use $\mathbf q=\mathbf k'-\mathbf k$ as wave-vector transfer; the momentum transfer is $\hbar\mathbf q$. With reduced mass $\mu_r$ and unit incident coordinate amplitude,

$$f_{\mathrm B}(\mathbf q) = - \frac{\mu_r}{2\pi\hbar^2} \int d^3r\, e^{-i\mathbf q\cdot\mathbf r} V(\mathbf r).$$

Thus the leading amplitude is proportional to the Fourier transform of the potential:

$$f_{\mathrm B}(\mathbf q) \propto \widetilde V(\mathbf q).$$

This is already a momentum-transfer description of scattering.

Tree-Level Analogy

In perturbative QFT, a tree-level exchange diagram gives an amplitude built from vertices and propagators without closed momentum loops. A single exchanged particle already requires two vertices, so first order in a matched potential need not mean first order in a fundamental QFT coupling. A scalar exchange often produces a momentum-space denominator of the schematic form

$$\frac{1}{\mathbf q^2+\mu^2}$$

in the nonrelativistic static limit. The Fourier transform of this structure gives a Yukawa potential:

$$V(r) \propto - \frac{e^{-\mu r}}{r}.$$

This is why the Born approximation is a useful bridge: potential scattering displays the same relation between spatial range and momentum-transfer dependence.

Yukawa Potential in the Born Approximation derives that transform, identifies $\mu=m_\phi c/\hbar$, and follows the resulting denominator through measurable nonrelativistic cross sections.

A Definite Matching Convention

In this section set $\hbar=c=1$ and use relativistic normalization $\langle\mathbf p'|\mathbf p\rangle=2E_{\mathbf p}(2\pi)^3\delta^3(\mathbf p'-\mathbf p)$. Define the invariant amplitude by

$$\langle f|S-I|i\rangle=i(2\pi)^4\delta^4(P_f-P_i)\mathcal M_{fi}.$$

For two distinguishable spinless particles of masses $m_1,m_2$, compare elastic low-velocity scattering in the center-of-mass frame. If $\widetilde V(\mathbf q)=\int d^3r\,e^{-i\mathbf q\cdot\mathbf r}V(r)$, the leading matching is

$$\widetilde V(\mathbf q)=-\frac{\mathcal M_{\mathrm{NR}}(\mathbf q)}{4m_1m_2}, \qquad f_{\mathrm B}=\frac{\mathcal M_{\mathrm{NR}}}{8\pi(m_1+m_2)}.$$

Here $\mu_r=m_1m_2/(m_1+m_2)$ relates the second equation to the Born formula. The cross-section check is immediate: $|f|^2$ agrees with $|\mathcal M|^2/(64\pi^2s)$ as $\sqrt s\to m_1+m_2$. Tong’s treatment of Yukawa exchange illustrates the equal-mass matching. Spin, exchange symmetry, recoil, and momentum-dependent interactions require additional structure. Do not restore isolated powers of $\hbar$ in this natural-unit matching without also converting the state and Fourier conventions.

Where the Analogy Stops

The Born approximation assumes a fixed potential and fixed particle number. QFT tree amplitudes include:

• relativistic normalization,
• spinor or polarization structure,
• antiparticles,
• crossing symmetry,
• particle production when allowed,
• gauge constraints,
• different multiparticle channels already at tree level.

Loops then introduce further corrections and renormalization.

A QFT tree diagram is not literally a classical particle exchanging a small object along a path. It is a term in an amplitude expansion.

Beyond Leading Order

Iterating the potential in the Lippmann–Schwinger equation produces higher Born terms. Born Series explains why potential iteration is not identical to QFT loop order. The correspondence becomes subtle because:

• potentials may already encode some relativistic effects;
• loops include virtual particle creation and renormalization;
• repeated potential exchange can require nonperturbative resummation;
• matching must avoid double counting.

Effective field theory handles this by matching low-energy observables order by order.

Exercises

1. Why does a short-range potential have a broad momentum-space transform?

Solution

A short-range function in position space requires many Fourier components to represent sharp spatial localization. Therefore its transform spreads over a wide range of momentum transfers. Conversely, a long-range potential has a transform concentrated at small momentum transfer.

2. What is the main physical analogy between the Born approximation and a tree-level exchange diagram?

Solution

Both can give the leading weak-scattering amplitude, but they count different interactions. The Born calculation uses one insertion of the effective potential. A one-mediator QFT exchange generally uses two elementary vertices and one propagator; matching their low-energy amplitude determines that potential.

References

• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• D. Tong, Quantum Field Theory, University of Cambridge lecture notes (2006), Chapter 3, “Interacting Fields,” Sections 3.5–3.6, author’s notes.
• S. Weinberg, The Quantum Theory of Fields, Vol. I: Foundations, Cambridge University Press, 1995.