Elementary Particles §

Step 1: three gauge factors, three interactions. The gauge group has three factors, each corresponding to one interaction:

1. SU(3) — the strong interaction (QCD). Acts on the “color” degree of freedom of quarks. The gauge bosons are the eight gluons.
2. SU(2) — the weak isospin. Acts on left-handed fermion doublets. The gauge bosons are W¹, W², W³ (which mix to give W⁺, W⁻, and a component of Z⁰).
3. U(1) — hypercharge. Acts on all fermions with a charge assignment. The gauge boson B mixes with W³ to give the photon (γ) and the Z⁰ boson.

Step 2: charge from quantum numbers. Each fermion carries a weak isospin $T_3$ and a hypercharge $Y$. The electric charge is the combination $Q = T_3 + Y/2$ (the Gell-Mann–Nishijima relation). This single formula is what fixes the entire charge spectrum of the Standard Model.

Worked example. The left-handed quark doublet $(u_L, d_L)$ has hypercharge $Y = 1/3$. The up quark has $T_3 = +1/2$, so $Q_u = 1/2 + (1/3)/2 = 2/3$. The down quark has $T_3 = -1/2$, so $Q_d = -1/2 + (1/3)/2 = -1/3$. The left-handed lepton doublet $(\nu_L, \ell_L)$ has $Y = -1$: the neutrino gets $Q_\nu = 1/2 + (-1)/2 = 0$ and the charged lepton $Q_\ell = -1/2 + (-1)/2 = -1$.

Step 3: electroweak symmetry breaking. Before symmetry breaking the gauge group is the full SU(2)×U(1). After the Higgs field acquires a vacuum expectation value (see the Higgs boson section), the unbroken subgroup is U(1)ₑₘ, whose gauge boson is the photon. The W⁺, W⁻, and Z⁰ acquire mass; the photon remains massless. The general mechanism (spontaneous symmetry breaking, the “eaten” degrees of freedom) is developed in the gauge-symmetry section of the QFT page; here we only use its consequence for the particle content.

Authoritative sources

• Gell-Mann & Nishijima, “The Conserved Vector Current Hypothesis” (1956)
• Glashow, “Partial-symmetries of weak interactions” (1961)

Example problem.

Step 1: the three generations. Each generation contains one quark doublet and one lepton doublet (plus right-handed singlets). The particle content is:

(Charges in units of the proton charge e.)

Step 2: quarks vs. leptons.

• Quarks carry color charge (triplet under SU(3)) and are confined: they never appear in isolation, only in color-neutral hadrons (see nuclear physics).
• Leptons do not carry color charge and are never confined.
• Neutrinos are neutral and have tiny masses (see the neutrinos subsection).

Step 3: the mass hierarchy. Within each type, mass increases sharply across generations: the top quark ($m_t \approx 173$ GeV) is about 10⁵ times heavier than the up quark, and the tau is about 3500 times heavier than the electron. The reason for this hierarchy is not explained by the Standard Model.

Authoritative sources

• Particle Data Group, Review of Particle Physics

Example problem.

Step 1: why they are special. The minimal Standard Model contains only left-handed neutrinos and no right-handed partners, so no renormalizable mass term can be written for them. Neutrino oscillation experiments (Super-Kamiokande, SNO, 2002) show that a neutrino produced as one flavor (νₑ, ν_μ, ν_τ) is later detected as another, which is only possible if the flavor states are superpositions of states with different non-zero masses.

Step 2: what this means. The mass eigenstates (ν₁, ν₂, ν₃) differ from the flavor eigenstates by a mixing matrix (the PMNS matrix), exactly analogous to the CKM mixing of quarks. The measured mass-squared differences are $\Delta m^2 \sim 10^{-3}$ and $\sim 10^{-5}$ eV², so neutrino masses are at most a few tenths of an eV — many orders of magnitude below every other fermion.

Step 3: open questions. Whether neutrinos are Dirac or Majorana particles (i.e. whether the antineutrino is the same particle as the neutrino), and the absolute mass scale, are still open. These questions sit at the boundary with cosmology, where the sum of neutrino masses affects structure formation.

Authoritative sources

• Super-Kamiokande, “Observation of Neutrino Oscillations” (1998)
• PDG, Neutrino Masses and Mixing

Step 1: the rule. Charge conjugation flips the sign of every additive charge while leaving mass and spin unchanged. The antiparticle of the electron is the positron $e^+$ (charge +1, same mass 0.511 MeV); the antiparticle of the up quark is the anti-up quark $\bar{u}$ (charge −2/3, anti-color).

Step 2: which particles are self-conjugate. A particle can be its own antiparticle only if all its additive charges vanish. The photon, the Z⁰, and (in the minimal model) the neutrinos are neutral; the photon and Z⁰ are self-conjugate. The Higgs boson is also its own antiparticle.

Step 3: annihilation. When a particle meets its antiparticle they can annihilate into bosons, conserving all charges. An electron and positron annihilate into two photons ($e^+ e^- \to \gamma \gamma$); a quark and antiquark can annihilate into gluons or a photon. This is how matter and antimatter are created and destroyed in equal amounts in the early universe (see cosmology).

Step 4: CPT. The CPT theorem guarantees that a particle and its antiparticle have exactly equal mass and lifetime. Any measured difference would signal a breakdown of the Standard Model’s foundations.

Authoritative sources

• Dirac, “A Quantum Theory of the Electron” (1928)
• CPT theorem

Example problem.

Step 1: the four carriers.

• Gluons (8): massless, carry color charge themselves (adjoint representation of SU(3)), mediate the strong force. Self-interaction of gluons leads to confinement and asymptotic freedom.
• W⁺, W⁻ (mass ≈ 80.4 GeV/c²): mediate charged-current weak interactions (e.g., beta decay).
• Z⁰ (mass ≈ 91.2 GeV/c²): mediates neutral-current weak interactions.
• Photon (γ): massless, mediates electromagnetism.

Step 2: charged vs. neutral currents. The W bosons change the electric charge of the particles they connect (a charged current): a down quark becomes an up quark. The Z⁰ and the photon connect particles of the same charge (neutral currents). This distinction is what makes the weak interaction able to change particle identity, as in beta decay.

Step 3: beta decay as a worked process. In beta-minus decay, a down quark emits a W⁻ and becomes an up quark; the W⁻ then decays into an electron and an electron antineutrino.

The field-theoretic description of each force (Maxwell equations, the strong and weak fields) is in the Interactions category; here the same forces appear as the exchange of these bosons.

Authoritative sources

• Feynman, “The Theory of Positrons” (1949)
• Particle Data Group, Review of Particle Physics

Example problem.

Step 1: the mechanism (formalism). The Higgs field is an SU(2) doublet with hypercharge $Y = 1$. Its potential has the “Mexican hat” shape, and its nonzero vacuum expectation value $v \approx 246$ GeV breaks the electroweak symmetry. The full derivation — the Mexican-hat potential, the three “eaten” degrees of freedom that become the longitudinal modes of W⁺, W⁻, Z⁰, and the Yukawa mass formula $m_f = y_f v/\sqrt{2}$ — is developed in the gauge-symmetry section of the QFT page. Here we keep only the phenomenology.

Step 2: the mass spectrum. The Higgs couples to every fermion with a strength (Yukawa coupling) $y_f$ proportional to its mass: $m_f = y_f v/\sqrt{2}$. The top quark has the largest coupling ($y_t \approx 1$), the electron the smallest ($y_e \approx 3 \times 10^{-6}$). The W and Z masses are $m_W = gv/2 \approx 80.4$ GeV and $m_Z \approx 91.2$ GeV.

Step 3: discovery. The Higgs boson was discovered at the LHC in 2012 by the ATLAS and CMS collaborations via its decay channels $H \to \gamma\gamma$ and $H \to ZZ^* \to 4\ell$. Its measured mass of 125.25 GeV is consistent with the electroweak precision predictions.

Authoritative sources

• ATLAS, “Observation of a new particle in the search for the Standard Model Higgs boson” (2012)
• CMS, “Observation of a new boson at a mass of 125 GeV” (2012)

Example problem.

A particle is observed with the following quantum numbers: electric charge $Q = -1/3$, color triplet, weak isospin doublet (left-handed), and it participates in strong interactions. Identify the particle and state which gauge bosons it couples to.