The Standard Model

The Standard Model is the accepted theory of elementary particles. It combines three parts: the quark model of hadron structure, the electroweak theory unifying the electromagnetic and weak interactions, and quantum chromodynamics (QCD) for the strong interaction. It is second only to QED in the precision of its predictions. This lesson assembles the three parts: how SU(3) symmetry organizes the hadrons and led to quarks, how color and QCD confine them, and how spontaneous symmetry breaking gives the weak bosons their mass.

The eightfold way and quarks

The accelerators of the 1950s produced a flood of hadrons. The most successful classification, the eightfold way, was proposed by Gell-Mann and Ne'eman in 1961. Hadrons of the same spin and parity are grouped into supermultiplets plotted in the hypercharge-isospin plane. The scheme rests on the Lie group SU(3), whose lowest multiplets contain $1,3,8,10,\dots$ members; the eight-member octets give the method its name.

The eightfold way predicted a gap. In the spin-$\frac{3}{2}$ baryon decuplet, nine of ten members were known in 1961; the tenth, the ${\Omega }^{-}$, was predicted with definite mass and spin and discovered in 1964, an exact parallel to Mendeleev's periodic-table gaps.

The regularities and the missing SU(3) triplet led Gell-Mann and Zweig in 1964 to propose quarks as the constituents of all hadrons.

Because the quantum numbers add, a hadron's properties are arithmetic. The $uds$ combination has charge $\frac{2}{3}-\frac{1}{3}-\frac{1}{3}=0$ and baryon number $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1$. Every hadron in the tables is built from the six quarks and their antiquarks.

Direct evidence for quarks comes from deep inelastic scattering: electrons or muons of $15$–$200$ GeV strike a nucleon and scatter at large angles off pointlike spin-$\frac{1}{2}$ constituents much smaller than the nucleon, exactly as Rutherford's alpha scattering revealed the nucleus. The same experiments raised the proton spin crisis: polarized scattering shows the three valence quarks account for only 20 to 30 percent of the proton's spin, with the rest carried by quark orbital angular momentum, gluon spin, and a polarized sea of virtual quark-antiquark pairs. The naive picture in which three quark spins add to $\frac{1}{2}$ is incomplete, and the origin of the proton's spin remains unsettled.

The charm quark arrived in 1974 with the $J/\psi$ meson ($c\bar{c}$), whose lifetime of $10^{-20}$ s was extraordinarily long for its mass. The tau lepton (1975), the upsilon ($b\bar{b}$, giving the bottom quark), and the top quark (1995, at $176\text{GeV}/c^2$ the heaviest known particle) completed three matched generations of quarks and leptons.

Color and QCD

The early quark model had two problems: no free quarks were found, and some baryons violated the exclusion principle. The ${\Delta }^{++}$ has spin $\frac{3}{2}$ and content $uuu$, three identical fermions in the same state. Greenberg's solution gave each quark flavor a new threefold color charge (red, blue, green), so the three $u$ quarks differ in color and the exclusion principle is respected.

QCD is the theory of the strong interaction between quarks and gluons, built on the model of QED. The mediator is the gluon, massless and spin-1 like the photon, but with a decisive difference: gluons carry color (one color and one anticolor), so unlike the neutral photon they interact with one another. Gluon self-interaction produces two effects on the running coupling ${\alpha }_s$:

• Asymptotic freedom. Gluon loops dilute the color charge at short distance, so ${\alpha }_s$ decreases as quarks approach. Inside a nucleon the quarks move nearly freely, as deep inelastic scattering confirms.
• Confinement. At large separation the potential grows without bound.

One model potential captures both:

with $k\approx 1\text{GeV/fm}$. The Coulomb-like first term dominates at small $r$; the linear term dominates at large $r$, so the force $F=-\nabla V\to k$ (constant) rather than falling to zero.

Confinement explains why free quarks are never seen. Pulling a quark from a hadron adds energy to the color field; when enough accumulates, a quark-antiquark pair is created rather than a lone quark liberated. The new quark and antiquark bind with the fragments, producing hadrons and never a free quark. This is the origin of the virtual pions in the Yukawa picture of the nuclear force.

The electroweak theory

The electroweak theory treats the electromagnetic and weak interactions as two faces of one interaction. Above $\sim 100$ GeV it is mediated by four massless bosons of equal footing: a triplet $W^{+}$, $W^0$, $W^{-}$ and a singlet $B^0$. At ordinary energy the symmetry is spontaneously broken: the observed photon is one combination of $W^0$ and $B^0$, and the $Z^0$ is the orthogonal combination.

The symmetry-breaking agent is the Higgs field, requiring a new spin-0 boson, the Higgs boson. Particles acquire mass by interacting with the Higgs field: the photon, which does not couple to it, stays massless, while the $W^{\pm }$ and $Z^0$ reach $\sim 80$–$91\text{GeV}/c^2$. The Higgs boson's rest energy is of order $1$ TeV, produced in high-energy proton collisions and a primary target of the Large Hadron Collider.

The model assembled

The Standard Model's fundamental particles are the leptons and quarks, each in three generations, plus the force carriers.

• Matter particles. Six leptons and six quarks, all spin-$\frac{1}{2}$ fermions obeying the exclusion principle.
• Force carriers. The photon, the $W^{\pm }$ and $Z^0$, and eight gluons — all integer-spin bosons. The graviton lies outside the model.
• The Higgs. A spin-0 boson whose field gives the other particles mass.

Every force is one of four basic interactions, and a particle feels an interaction only if it carries the corresponding charge: electric charge (quarks and charged leptons), weak charge (all quarks and leptons), color charge (quarks and gluons), and mass (all particles). Hadrons are the color-neutral bound states; leptons exist only as isolated particles. The strong interaction acts in two ways: the fundamental color interaction between quarks, mediated by gluons, and the residual strong interaction between color-neutral nucleons, mediated by meson exchange, analogous to the residual electromagnetic force that binds neutral atoms into molecules.

Decays are tracked at the quark level by following flavor changes, each mediated by the weak $W$ boson when strangeness changes.

The Standard Model leaves questions open: why four interactions, why their strengths differ so widely, and what lies at higher energy. These drive the beyond-Standard-Model program.¹