Tree-Level Worked Examples

This page walks the full computational pipeline — Dyson series → Wick's theorem → Feynman rules → LSZ → cross section — for concrete processes at tree level (no loops), where every amplitude is finite and closed-form. It complements the standalone Compton scattering calculation with simpler scalar and QED examples, and introduces Mandelstam variables and crossing.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Mandelstam variables

For a $2\to 2$ process $p_1+p_2\to p_3+p_4$, the Lorentz-invariant kinematics are packaged in the three Mandelstam variables:

$$s=(p_1+p_2{)}^2,t=(p_1-p_3{)}^2,u=(p_1-p_4{)}^2,$$

satisfying $s+t+u={\sum }_i{m}_i^2$. Here $s$ is the center-of-mass energy squared (the invariant mass of the initial state), while $t$ and $u$ are momentum transfers. Amplitudes are naturally written as functions $\mathcal{M}(s,t,u)$, and the three channels correspond to the three ways an internal particle can be exchanged.

Example 1: ${\varphi }^4$ scattering $\varphi \varphi \to \varphi \varphi$

The simplest possible amplitude. From Wick's theorem, the single four-point vertex gives, at first order in $\lambda$,

$$i\mathcal{M}=-i\lambda ⟹\overline{|\mathcal{M}{|}^2}={\lambda }^2.$$

There is no internal line and no angular dependence — the scattering is isotropic. The differential cross section from the master formula (identical particles, mass $m$, center-of-mass) is

$$\frac{d\sigma }{d\Omega }=\frac{{\lambda }^2}{64{\pi }^{2}s}\cdot \frac{|{\mathbf{p}}_f|}{|{\mathbf{p}}_i|}=\frac{{\lambda }^2}{64{\pi }^{2}s},$$

and the total cross section, with a symmetry factor $\frac{1}{2}$ for identical final particles, is $\sigma ={\lambda }^2/(32\pi s)$. This is the "hydrogen atom" of QFT calculations: every ingredient appears once, at its simplest.

Example 2: Yukawa theory $\psi \psi \to \psi \psi$

Couple a Dirac fermion to a real scalar mediator via ${\mathcal{L}}_{\text{int}}=-g\bar{\psi }\psi \varphi$. At tree level, fermion–fermion scattering proceeds by scalar exchange in the $t$- and $u$-channels (the two ways to connect identical external fermions), giving

$$i\mathcal{M}=(-ig{)}^2\left[\frac{\bar{u}(p_3)u(p_1)\bar{u}(p_4)u(p_2)}{t-m_{\varphi }^2}-\frac{\bar{u}(p_4)u(p_1)\bar{u}(p_3)u(p_2)}{u-m_{\varphi }^2}\right],$$

with the relative minus sign enforced by Fermi statistics (antisymmetry under exchange of the identical final fermions — a direct consequence of the anticommutator quantization). In the non-relativistic limit this amplitude reproduces the Yukawa potential $V(r)=-\frac{g^2}{4\pi }\frac{e^{-m_{\varphi }r}}{r}$ — the original point of Yukawa's theory: a massive mediator gives a short-range force of range $1/m_{\varphi }$, while a massless mediator ($m_{\varphi }\to 0$) gives the long-range Coulomb form.

Example 3: QED $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$

The cleanest QED cross section, and the backbone of $e^{+}{e}^{-}$-collider physics. At tree level a single $s$-channel photon is exchanged. Using the QED Feynman rules:

$$i\mathcal{M}=[\bar{v}(p_2)(-ie{\gamma }^{\mu })u(p_1)]\frac{-i{\eta }_{\mu \nu }}{s}[\bar{u}(p_3)(-ie{\gamma }^{\nu })v(p_4)].$$

Spin-averaging turns $\overline{|\mathcal{M}{|}^2}$ into a product of two gamma-matrix traces. In the high-energy limit ($s\gg m_e^2,m_{\mu }^2$) they evaluate to

$$\overline{|\mathcal{M}{|}^2}=2e^4\frac{t^2+u^2}{s^2}=e^4(1+{\cos }^2\theta ),$$

giving the celebrated angular distribution and total cross section

$$\boxed{\frac{d\sigma }{d\Omega }=\frac{{\alpha }^2}{4s}(1+{\cos }^2\theta ),\sigma =\frac{4\pi {\alpha }^2}{3s},}$$

with $\alpha =e^2/4\pi$. The $1/s$ falloff and the total $\sigma =4\pi {\alpha }^2/3s$ are textbook benchmarks, and the ratio of hadronic to muonic cross sections $R=\sigma (e^{+}{e}^{-}\to \text{hadrons})/\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})$ is a classic measurement of the number of quark colors — see collider observables.

Crossing symmetry

The three examples above are related by crossing: the same analytic function $\mathcal{M}(s,t,u)$ describes different physical processes depending on which invariants are positive. Moving a particle from the initial to the final state (with $p\to -p$) exchanges channels:

Crossing is a consequence of the analyticity of amplitudes (the same property underlying the optical theorem) and means a single calculation yields several cross sections. The Compton amplitude and pair annihilation $e^{+}{e}^{-}\to \gamma \gamma$ are similarly crossing-related.

Summary

Each follows the identical pipeline: draw diagrams, apply rules for $i\mathcal{M}$, square and spin-average, integrate over phase space.

Where this leads

• A full QED example with all traces: Compton scattering.
• Higher orders and their divergences: loop integrals.
• Comparison to data: collider measurements.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 5.1–5.4.
• Halzen & Martin, Quarks and Leptons, Ch. 6.
• Srednicki, Quantum Field Theory, Ch. 11, 60.