Standard Model Interactions

Equivalent forms

annihilation

e^{+} + e^{-} \to \gamma + \gamma

beta decay

n \to p + e^{-} + \bar{\nu}_e \quad (d \to u + W^{-})

muon decay

\mu^{-} \to e^{-} + \bar{\nu}_e + \nu_{\mu}

pair production

\gamma \to e^{-} + e^{+} \;(\text{near a nucleus})

Feynman turned quantum field theory into pictures: every diagram is an integral, every squiggle a force carrier, and nothing that conserves the quantum numbers is forbidden.

Symbol	Name	SI unit	Dimension	Range
$e$	coupling constant (electric charge for QED)	C	I T	—
$\psi$	fermion field (electron, quark, …)	—	1	—
$A_{\mu}$	photon field (γ); W±, Z for the weak force; g for the strong force	—	1	—
$\gamma^{\mu}$	Dirac matrices coupling spin to spacetime	—	1	—

Discovery

Richard Feynman · Julian Schwinger · Sin-Itiro Tomonaga · 1948

At the 1948 Pocono conference, Feynman presented squiggly little pictures while Schwinger filled blackboards with operator algebra — both computing the same quantum electrodynamics. Freeman Dyson proved the approaches identical, and Feynman's diagrams won by sheer usability: any physicist could now draw a process and read off its probability. QED went on to predict the electron's magnetic moment to twelve decimal places — the most precisely verified theory in science. The weak and strong vertices completed the Standard Model by the 1970s.

Real-world applications

PET scanners image the body with $e^{+}e^{-} \to \gamma \gamma$ annihilation photons flying back-to-back.
Carbon dating and reactor physics run on the weak vertex (beta decay).
The Sun shines because $p \to n + e^{+} + \nu$ ₑ (weak interaction) lets protons fuse — the slowest step that gives the Sun ten billion years.
Pair production $\gamma \to e^{+}e^{-}$ limits the energy of gamma-ray astronomy and seeds particle showers in the atmosphere.
Every collider analysis at the LHC is a sum of Feynman diagrams compared against detected events.

Common misconceptions

“The photon in a diagram is a real photon.” — Internal lines are 'virtual': mathematical terms that can be off mass-shell; only external legs are observable particles.
“Antimatter moves backward in time literally.” — The arrow convention encodes charge conjugation in the math; antiparticles move forward in time like everything else.
“A single free photon can become $e^{+}e^{-}$ .” — Energy–momentum can't balance in vacuum; pair production needs a nucleus to absorb recoil.
“Lepton number conservation is exact.” — Neutrino oscillations already violate lepton flavor; whether total lepton number is exact is an open experimental question.

Experimental verification

QED's electron

g-2

agrees with experiment

to \sim 10^{-12}

; the W and Z bosons were found at CERN in 1983 with predicted masses; the Higgs completed the model in 2012; thousands of measured cross-sections at LEP, Tevatron and LHC match diagram calculations.

Derivation

The QED interaction term $-e\psi$ ̄ $\gamma ^\mu \psi A_\mu$ describes one elementary event: a charged fermion absorbs or emits a photon.
Perturbation theory expands any process in powers of the coupling; each term is a diagram built from vertices connected by propagators.
Conservation laws follow from symmetries (Noether): U(1) gauge symmetry → charge conservation; accidental symmetries → lepton and baryon number.
The weak vertex (quark–W) is the only one that changes particle type: $d \to u + W^{-}$ underlies all beta decay.
Amplitudes from all diagrams add (with interference); the rate follows from Fermi's golden rule.

Limiting cases

Low energy, QED only⟶ atomic physicsPhoton exchange between charges reduces to the Coulomb potential.

E ≫

M_W c^{2}

⟶ electroweak unificationAbove

\sim 100\,\mathrm{GeV}

the weak and electromagnetic interactions merge into one force.

What if…

What if baryon number weren't conserved?

Protons would decay. Grand unified theories predict exactly this (lifetime > $10^{34} yr)$ ; Super-Kamiokande has watched 50,000 tons of water for decades without one confirmed decay.

What if the weak force were as strong as electromagnetism?

Stellar fusion's first step would be fast instead of bottlenecked — stars would burn out in millennia, far too quickly for life to evolve.

1

Is n → p + e⁻ allowed? Check the books.

Given · Neutron

(Q=0

,

B=1

, Lₑ

=0)

→ proton

(Q=+1

,

B=1)

+ electron

(Q=-1

, Lₑ

=+1)

Find · Which conservation law forces the antineutrino into beta decay?

Steps

Charge: $0 \to +1 - 1 = 0 \checkmark$
Baryon number: $1 \to 1 \checkmark$
Electron-lepton number: $0 \to 0 + 1 = 1$ ✗ — violated!
$Add \nu$ ̄ₑ (Lₑ $= -1)$ : total Lₑ $= 1 - 1 = 0 \checkmark$ .

Result · Lepton number forces

n \to p + e^{-} + \nu

̄ₑ. Pauli postulated exactly this invisible particle in 1930 to save the bookkeeping — 26 years before anyone detected it.

2

Threshold energy for pair production

Given · Electron rest energy mₑ

c^{2} = 0.511\,\mathrm{MeV}

Find · Minimum photon energy

for \gamma \to e^{+} + e^{-}

near a heavy nucleus

Steps

Must create both rest masses: $E_\min = 2m$ ₑ $c^{2}$ .
$E_\min = 2 \times 0.511 = 1.022\,\mathrm{MeV}$ (the heavy nucleus absorbs momentum at negligible energy cost).

Result · 1.022 MeV — below this, gamma rays cannot materialize matter; above it, every gamma detector sees the 511 keV annihilation echo.

Dirac Equation→Fermi's Golden Rule→Pair Production Threshold→Fermi Beta Decay Spectrum→Yukawa Potential→