A clean way to demonstrate how a Feynman diagram becomes equations is to use the tree-level scattering of two distinguishable charged fermions through exchange of a virtual photon:
[ e^-(p_1)+\mu^-(p_2)\longrightarrow e^-(p_3)+\mu^-(p_4). ]
The diagram is a shorthand for a perturbative contribution to the quantum amplitude—not literally a picture of particles’ paths. Each vertex and internal line maps to a factor prescribed by QED. arxiv
1. Start from QED
Use natural units, (\hbar=c=1), and metric convention (g^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)).
The QED Lagrangian for electrons and muons coupled to electromagnetism is
[ \mathcal L = \bar\psi_e(i\gamma^\mu\partial_\mu-m_e)\psi_e + \bar\psi_\mu(i\gamma^\mu\partial_\mu-m_\mu)\psi_\mu -\frac14 F_{\mu\nu}F^{\mu\nu} -\underbrace{e\,\bar\psi_e\gamma^\mu A_\mu\psi_e -e\,\bar\psi_\mu\gamma^\mu A_\mu\psi_\mu}{\mathcal L{\mathrm{int}}}, ]
where
[ F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu. ]
The interaction term,
[ \mathcal L_{\mathrm{int}}=-e\bar\psi\gamma^\mu A_\mu\psi, ]
says that a charged fermion can emit or absorb a photon. It produces the QED vertex factor
[ \boxed{-ie\gamma^\mu}. ]
The coupling satisfies
[ \alpha \equiv \frac{e^2}{4\pi}\approx \frac{1}{137}, ]
where (\alpha) is the fine-structure constant.
2. The diagram
With momenta labelled as above, the exchange channel is
electron: p1 ───►●────────► p3
│
│ q = p1 − p3
│
muon: p2 ───►●────────► p4
- Straight directed external lines: incoming/outgoing electron or muon states.
- Wavy internal line: a virtual photon.
- Each dot: an electromagnetic interaction vertex.
- Momentum conservation at the vertices implies
[ p_1+p_2=p_3+p_4, ]
and the momentum through the virtual photon can be chosen as
[ q=p_1-p_3=p_4-p_2. ]
Since the photon is internal, it need not obey the real-photon relation (q^2=0). Its off-shell four-momentum appears in the propagator denominator.
3. Translate diagram to factors
For a basic QED calculation in Feynman gauge, use:
| Diagram element | Mathematical factor |
|---|---|
| Incoming fermion with momentum (p), spin (s) | (u(p,s)) |
| Outgoing fermion | (\bar u(p,s)) |
| Electron/photon or muon/photon vertex | (-ie\gamma^\mu) |
| Internal photon carrying momentum (q) | (\displaystyle \frac{-ig_{\mu\nu}}{q^2+i\epsilon}) |
| Four-momentum conservation | (\displaystyle (2\pi)^4\delta^{(4)}!\left(\sum p_{\rm in}-\sum p_{\rm out}\right)) |
The (i\epsilon) prescription means (q^2\to q^2+i\epsilon); it specifies how the propagator’s pole is handled and encodes causal time ordering. The scalar propagator has the analogous form (i/(p^2-m^2+i\epsilon)). southampton.ac
4. Derive the amplitude
Apply the factors in the order of the fermion lines. For the electron line,
[ \bar u_e(p_3)\,(-ie\gamma^\mu)\,u_e(p_1). ]
For the muon line,
[ \bar u_\mu(p_4)\,(-ie\gamma^\nu)\,u_\mu(p_2). ]
For the exchanged photon,
[ \frac{-ig_{\mu\nu}}{q^2+i\epsilon}. ]
Multiplying them gives the (S)-matrix contribution:
[ i\mathcal M = \left[\bar u_e(p_3)(-ie\gamma^\mu)u_e(p_1)\right] \left[\frac{-ig_{\mu\nu}}{q^2+i\epsilon}\right] \left[\bar u_\mu(p_4)(-ie\gamma^\nu)u_\mu(p_2)\right]. ]
Collecting constants and contracting the Lorentz indices:
[ i\mathcal M = i\,\frac{e^2}{q^2+i\epsilon} \left[\bar u_e(p_3)\gamma^\mu u_e(p_1)\right] \left[\bar u_\mu(p_4)\gamma_\mu u_\mu(p_2)\right]. ]
Therefore, with the common convention in which the (S)-matrix contains (i\mathcal M),
[ \boxed{ \mathcal M = \frac{e^2}{q^2+i\epsilon} \left[\bar u_e(p_3)\gamma^\mu u_e(p_1)\right] \left[\bar u_\mu(p_4)\gamma_\mu u_\mu(p_2)\right] } ]
together with the overall conservation factor
[ (2\pi)^4\delta^{(4)}(p_1+p_2-p_3-p_4). ]
This has a useful interpretation:
[ J_e^\mu=\bar u_e(p_3)\gamma^\mu u_e(p_1), \qquad J_{\mu}^\nu=\bar u_\mu(p_4)\gamma^\nu u_\mu(p_2), ]
so that
[ \mathcal M = \frac{e^2}{q^2+i\epsilon}\,J_e^\mu J_{\mu,\mu}. ]
In words: the electron current produces a virtual photon, the photon propagates, and the muon current absorbs it.
5. From amplitude to an observable
A single diagram yields an amplitude, not directly a probability. For an unpolarized scattering experiment, one averages over initial spins and sums over final spins:
[ \overline{|\mathcal M|^2} = \frac14\sum_{\text{spins}}|\mathcal M|^2. ]
The spin sums are reduced using
[ \sum_s u(p,s)\bar u(p,s)=\slashed p+m, \qquad \slashed p\equiv \gamma^\mu p_\mu. ]
Thus,
[ \overline{|\mathcal M|^2} = \frac{e^4}{4(q^2)^2} \operatorname{Tr} \left[ (\slashed p_3+m_e)\gamma^\mu (\slashed p_1+m_e)\gamma^\nu \right] \operatorname{Tr} \left[ (\slashed p_4+m_\mu)\gamma_\mu (\slashed p_2+m_\mu)\gamma_\nu \right]. ]
The differential cross section for a generic (2\to2) process is then
[ d\sigma = \frac{1}{4\sqrt{(p_1\cdot p_2)^2-m_e^2m_\mu^2}} \, \overline{|\mathcal M|^2} \, d\Phi_2, ]
where the two-body Lorentz-invariant phase-space element is
[ d\Phi_2 = (2\pi)^4\delta^{(4)}(p_1+p_2-p_3-p_4) \prod_{f=3,4} \frac{d^3\mathbf p_f}{(2\pi)^3\,2E_f}. ]
This workflow—draw all permitted diagrams at a chosen perturbative order, write (\mathcal M), sum amplitudes, square the total, then integrate phase space—is the standard diagram-to-prediction pipeline. arxiv
Why this is a good demonstration
Electron–muon scattering is simpler than electron–electron scattering because the final particles are distinguishable. For (e^-e^-\to e^-e^-), there is an additional exchange diagram, and the total amplitude becomes
[ \mathcal M_{\text{total}} = \mathcal M_t-\mathcal M_u, ]
with the relative minus sign coming from exchanging identical fermions. That interference is physically important, but it obscures the core “vertex × propagator × vertex” idea for a first example.
At low momentum transfer, the photon factor (1/q^2) becomes the momentum-space origin of the familiar long-range Coulomb interaction. In the nonrelativistic static limit, (q^0\simeq0), so (q^2\simeq-\mathbf q^2), and Fourier transforming a (1/\mathbf q^2) dependence yields a (1/r) potential. Thus the diagrammatic QED calculation connects directly back to
[ V(r)=\frac{e^2}{4\pi r} = \frac{\alpha}{r}, ]
for like charges, with the appropriate sign determined by the charges involved.