Physic Labs

Particle physics

Feynman diagrams and the S-matrix

Feynman diagrams encode perturbative terms in scattering amplitudes; the S-matrix connects incoming and outgoing asymptotic states, while vertex factors, propagators, and loop integrals determine amplitudes.

Interactions turn free states into scattering states. The S-matrix collects transition amplitudes; expanding in couplings yields diagrams, each representing a definite integral rather than a classical particle trajectory.

S=Texp⁡ ⁣[−i∫dt HI(t)],⟨f∣S∣i⟩=δfi+i(2π)4δ4(pf−pi)MfiS=T\exp\!\left[-i\int dt\,H_I(t)\right],\qquad \langle f|S|i\rangle=\delta_{fi}+i(2\pi)^4\delta^4(p_f-p_i)\mathcal{M}_{fi}

Definition: Definition and physical meaning

External legs denote incoming/outgoing particles; internal lines are propagators; vertices follow interaction rules. The amplitude sums diagrams at the required order, integrates loop momenta, and includes symmetry factors.

Adjust the parameters and drag the figure to explore the model; readouts are quantitative illustrations under the stated assumptions.

Structure and interpretation

At lowest-order QED, e⁺e⁻→μ⁺μ⁻ has one s-channel diagram: electron–positron and muon–antimuon currents connected by a photon propagator. Near m_Z include Z exchange and interference; one drawing is not the full prediction.

Example: Quantitative example

Count couplings in the tree-level QED diagram for e⁺e⁻→μ⁺μ⁻: it has two fermion–photon vertices, each contributing e. Thus M∝e² and σ∝|M|²∝e⁴∝α².

Solution

Squaring the amplitude gives four powers of e; since α=e²/(4π), the cross section scales as α² apart from kinematics. Loops add higher-order corrections.

A scattering amplitude is obtained by adding contributions with the same initial and final states; one diagram is not selected as the “real path.” At tree level only finitely many diagrams are usually needed, while each loop adds an internal-momentum integral. Squaring the amplitude and applying the appropriate spin averages yields a cross section that can be compared with event counts.

Labels on lines and vertices are part of the calculation rules, not decoration. Each vertex conserves four-momentum, and each propagator depends on the internal four-momentum and particle mass; external legs are put on shell. For a closed loop, unknown momenta must be integrated, while the optical theorem relates the amplitude's imaginary part to the total probability for physical channels. These checks connect diagrammatic calculations to unitarity.

Quick check

What is the exchanged line in tree-level QED e⁺e⁻→μ⁺μ⁻?

What is a Feynman diagram?

References

  1. Michael E. Peskin and Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory
  2. Steven Weinberg (1995). The Quantum Theory of Fields, Volume I