fundamental interactions and elementary-particle families

In modern physics the whole variety of interactions reduces to four fundamental ones: gravitational, electromagnetic, strong, and weak. Each is carried by the exchange of mediator particles — gauge bosons: the photon $\gamma$ for the electromagnetic, gluons $g$ for the strong, the $W^\pm$ and $Z$ bosons for the weak, while gravity is carried by tensor perturbations of the metric (the hypothetical graviton $G$). The relative strengths are conveniently remembered as $\text{strong} : \text{electromagnetic} : \text{weak} : \text{gravitational} \approx 1 : 10^{-2} : 10^{-6} : 10^{-39}$ at typical energies of order a GeV. The range of the strong interaction is $\sim 10^{-15}$ m and of the weak interaction $\sim 10^{-18}$ m, whereas the electromagnetic and gravitational interactions have unlimited range and fall off as $1/r^2$. By spin, particles divide into fermions (half-integer spin, obeying the Pauli principle), which make up matter — quarks and leptons — and bosons (integer spin), which carry interactions or give mass (the Higgs boson $H$). The quarks $u, d, s, c, b, t$ have fractional charges $+2/3$ and $-1/3$, are never observed in isolation (confinement), and form hadrons: baryons made of three quarks (for example $p = uud$) and mesons made of a quark–antiquark pair (for example $\pi^+ = u\bar d$).

The proton consists of the quarks $uud$, the neutron of $udd$. Compute the charge of each particle in units of $e$ and state which family they belong to and which interaction holds them together.

The muon decays as $\mu^- \to e^- + \bar\nu_e + \nu_\mu$ with a lifetime of $2.2\times10^{-6}$ s, while the neutral pion decays as $\pi^0 \to \gamma\gamma$ with a lifetime of $8\times10^{-17}$ s. Determine which interactions are responsible, name the mediators, and explain the difference in lifetimes.

Estimate the range of the weak interaction from the $W$-boson mass $m_W c^2 = 80$ GeV using $R \approx \hbar c / (m c^2)$ with $\hbar c = 197$ MeV·fm. Compare it with the range of the strong interaction estimated from the pion mass $m_\pi c^2 = 140$ MeV, and explain why a neutrino with an energy of order 1 MeV hardly interacts with matter.

A safe self-study activity based on databases (no measurements with radiation sources are performed).

1. Open the online resource PDG Live (pdg.lbl.gov) or an interactive particle table and an open particle-physics simulation (for example, a detector event display).
2. Fill in a table for the particles $p$, $n$, $\pi^+$, $K^-$, $e^-$, $\mu^-$, $\nu_e$, $\gamma$, $W^-$, $H$: type (fermion/boson), family (quark, lepton, gauge boson, scalar), charge in units of $e$, baryon and lepton numbers, spin.
3. Check the conservation of charge, baryon number, and lepton number in the reactions $n \to p + e^- + \bar\nu_e$ and $\pi^+ \to \mu^+ + \nu_\mu$.
4. For each interaction, state its mediator, relative strength, and range, and then explain in one sentence which interaction dominates in each of the reactions you checked.

To show that the four fundamental interactions differ in mediator, relative strength, and range, and that elementary particles divide by spin into matter fermions (quarks, leptons) and mediator bosons (plus the Higgs scalar).

Two coordinated diagrams on a single sheet.

(a) Interaction table. Rows — gravitational, electromagnetic, strong, weak; columns — mediator (mass, spin), relative strength, range, example process. Inside the cells, the symbols $G$, $\gamma$, $g$, $W^\pm/Z$ and the numbers $1 : 10^{-2} : 10^{-6} : 10^{-39}$.

(b) Particle "tree". On the left the fermion branch with spin $1/2$: quarks $u, d, s, c, b, t$ with charges $+2/3$ and $-1/3$; leptons $e, \mu, \tau, \nu_e, \nu_\mu, \nu_\tau$. On the right the boson branch: gauge $g$, $\gamma$, $W^\pm$, $Z$ (spin 1) and the scalar $H$ (spin 0). Arrows from the quarks show composite hadrons: $p = uud$, $n = udd$, $\pi^+ = u\bar d$.

A basketball of mass $m = 0{,}62\ \mathrm{kg}$ falls from a height $h = 1{,}5\ \mathrm{m}$. Earth pulls it with a force $F_g = mg \approx 6{,}1\ \mathrm{N}$. Just before impact the speed is $v = \sqrt{2gh} \approx 5{,}4\ \mathrm{m/s}$ and the momentum is $p = mv \approx 3{,}4\ \mathrm{kg\cdot m/s}$. If the ball stops in $\Delta t = 0{,}02\ \mathrm{s}$, the floor exerts an average force $F = \frac{\Delta p}{\Delta t} \approx 1{,}7\times10^{2}\ \mathrm{N}$ — 28 times its weight.

The scenario uses two fundamental interactions. The gravitational interaction acts on the ball's mass and sets $F_g = mg$. The contact force comes from the electromagnetic interaction: electron shells of atoms in the ball and floor repel, preventing the bodies from passing through each other. The atoms themselves are made of quarks (inside protons and neutrons) and leptons (electrons).

A smartphone's glass screen contains transparent electrodes. Each electrode has a capacitance of about $C = 10\ \mathrm{pF}$ at a voltage $V = 1{,}8\ \mathrm{V}$. The charge on such an electrode is $Q = CV \approx 1{,}8\times10^{-11}\ \mathrm{C}$, which corresponds to about $N = Q/e \approx 1{,}1\times10^{8}$ electrons. When a finger — full of water and mobile ions — comes within fractions of a millimetre, the capacitance changes by about $1\ \mathrm{pF}$ (roughly $10\%$), and the controller registers a touch.

The sensor's operation is purely the electromagnetic interaction. Its carrier is the photon — the gauge boson of the electromagnetic field. Charges in the electrodes and in the finger interact at a distance, changing the capacitance. The electrons that create the charge and current in the sensor belong to the lepton family.

The Sun radiates a power $L = 3{,}8\times10^{26}\ \mathrm{W}$. By $E = mc^2$, this means a mass $m = \frac{L}{c^2} \approx 4{,}2\times10^{9}\ \mathrm{kg}$ is converted into energy every second. Since fusing four protons into a helium nucleus converts about $0{,}66\%$ of the mass into energy, the Sun consumes roughly $6\times10^{11}\ \mathrm{kg}$ of hydrogen per second. The reaction begins with a proton turning into a neutron: $p + p \rightarrow d + e^+ + \nu_e$, where $d$ is a deuteron, $e^+$ a positron, and $\nu_e$ an electron neutrino.

Two short-range interactions are at work here. The weak interaction turns a proton into a neutron by emitting a $W$ boson; without it the $pp$ chain would not start. The strong interaction, carried by gluons, binds quarks inside protons and neutrons and glues them into a deuteron and helium, overcoming the electric repulsion of protons. The neutrino and positron are leptons, while protons and neutrons are baryons built from quarks.