---
title: Fundamental Interactions and Force Carriers
module: Foundations
moduleNumber: 1
lessonNumber: 3
order: 103
summary: >
  Four interactions account for every force in nature: strong, electromagnetic,
  weak, and gravitational, in decreasing strength. Each is carried by a boson —
  the gluon, photon, W and Z, and the graviton — with a range fixed by the
  carrier's mass through the Yukawa relation, and a coupling constant that itself
  varies with distance.
topics: [Foundations]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 12 — Particle Physics; §12-2 Fundamental Interactions and the Force Carriers"
---

Every force observed in nature, from friction to the collapse of a stellar core,
is a manifestation of one of four fundamental interactions between elementary
particles. In decreasing order of strength they are the strong, electromagnetic,
weak, and gravitational interactions. Each interaction couples to a particular
kind of charge and is carried by one or more **force-carrier** particles, all of
which are bosons. This lesson sets out the four, the boson mediating each, and
the relation between a carrier's mass and the range of its force.

## Coupling constants

The "strength" of an interaction is quantified by a dimensionless **coupling
constant** that multiplies the space-dependent part of the potential energy. For
the Coulomb interaction of two charges, $U(r) = e^2/4\pi\varepsilon_0 r$, dividing
through by $\hbar c$ makes the multiplier dimensionless,

$$
\alpha = \frac{e^2}{4\pi\varepsilon_0\,\hbar c} \approx \frac{1}{137},
$$

the fine-structure constant already met in the Bohr model. It is the coupling
constant of the electromagnetic interaction. Making the constants dimensionless
lets every observer measure comparable values regardless of unit system.[^tl-strength]

> **Definition (Coupling constant).** The dimensionless number setting the
> probability amplitude for a particle to emit or absorb the boson that carries
> an interaction. In QED the electric charge $e = \sqrt{4\pi\varepsilon_0\hbar c\,\alpha}$
> is the amplitude for the photon-electron coupling, so the probability of an
> electromagnetic event scales as $e^2 \propto \alpha$.

Comparison across interactions is only approximate, since there is no unambiguous
common definition, but the relative magnitudes span more than thirty orders of
magnitude.

| Interaction | Carrier | Carrier mass $(\text{GeV}/c^2)$ | Source charge | Range (m) | Coupling | Time (s) |
| --- | --- | --- | --- | --- | --- | --- |
| Strong | gluon | $0$ | color | $10^{-15}$ | $\alpha_s \approx 1$ | $10^{-23}$ |
| Electromagnetic | photon | $0$ | electric | $\infty$ | $\alpha \approx \tfrac{1}{137}$ | $10^{-18}$ |
| Weak | $W^\pm$, $Z^0$ | $80,\ 91$ | weak | $10^{-18}$ | $\alpha_w \approx 10^{-5}$ | $10^{-16}$–$10^{-10}$ |
| Gravity | graviton | $0$ | mass | $\infty$ | $\alpha_g \approx 10^{-38}$ | — |

$$
% caption: Relative coupling strengths on a logarithmic scale, normalized to the
% strong interaction (~1): electromagnetic ~10^-2, weak ~10^-5, gravity ~10^-38.
% Thirty-three orders of magnitude separate the strong force from gravity.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % axis: log10(coupling), 0 at right (strong), down to -40 at left
  \draw[->, black] (0,-0.3) -- (0,4.6);
  \draw[black] (-0.15,0) -- (0,0);
  % baseline for bars at x from 0
  % strong: 10^0 -> length 4.0
  \draw[line width=6pt] (0,4.0) -- (4.0,4.0);
  \node[right, font=\scriptsize] at (4.1,4.0) {strong $\approx 1$};
  % EM: 10^-2 -> 3.8
  \draw[line width=6pt] (0,3.0) -- (3.8,3.0);
  \node[right, font=\scriptsize] at (3.9,3.0) {electromagnetic};
  % weak: 10^-5 -> 3.5
  \draw[line width=6pt] (0,2.0) -- (3.5,2.0);
  \node[right, font=\scriptsize] at (3.6,2.0) {weak};
  % gravity: 10^-38 -> 0.2
  \draw[line width=6pt] (0,1.0) -- (0.2,1.0);
  \node[right, font=\scriptsize] at (0.3,1.0) {gravity};
  \node[rotate=90, font=\scriptsize] at (-0.55,2.5) {increasing strength};
\end{tikzpicture}
$$

## The four interactions

Every fundamental interaction couples only to particles carrying its associated
charge. Some particles feel all four; others feel only some.

- **Strong interaction.** Acts between particles carrying **color charge**:
  quarks and gluons. It binds quarks into hadrons and, as a residual effect,
  holds nucleons in a nucleus. The carrier is the **gluon**, massless and spin-1.
  Unlike the photon, gluons carry color charge (one unit of a color and one of an
  anticolor), giving nine combinations that a group-theoretic technicality
  reduces to eight. Because they are colored, gluons couple to one another.
  Leptons carry no color and are blind to the strong force. The characteristic
  interaction time is $\sim 10^{-23}$ s, the time light takes to cross a nucleus.
- **Electromagnetic interaction.** Acts between all particles carrying electric
  charge or a magnetic moment. The carrier is the **photon**, massless, spin-1,
  and electrically neutral, so photons do not couple to one another. The range is
  infinite and the interaction time $\sim 10^{-18}$ s.
- **Weak interaction.** Acts between all quarks and leptons. The carriers are the
  charged $W^+$ and $W^-$ and the neutral $Z^0$, all spin-1. The $W^\pm$ change
  one quark flavor into another (they do not change lepton flavor); this drives
  beta decay. The range is $\sim 10^{-18}$ m, far shorter than the strong force,
  with interaction times from $10^{-16}$ to $10^{-10}$ s.
- **Gravitational interaction.** Acts between all particles with mass, the
  "gravitational charge." Its carrier, the spin-2 **graviton**, is expected to be
  massless and uncharged but has not been observed. At $\alpha_g \approx 10^{-38}$
  relative to the strong force, gravity is negligible between elementary
  particles.

The 1979 Nobel Prize recognized Glashow, Salam, and Weinberg for the
**electroweak theory**, which unifies the electromagnetic and weak interactions,
exactly a century after Maxwell unified electricity and magnetism. The
unification appears only at high particle energy, as the
[Standard Model](/particle-physics/electroweak-higgs/standard-model) lesson
develops.

$$
% caption: The primitive vertex for each interaction: a fermion emits the boson
% that carries the force. The weak vertex changes quark flavor; the strong vertex
% can change quark color.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % EM vertex
  \draw[->, thick] (0,0) -- (0.8,0.9);
  \draw[->, thick] (0.8,0.9) -- (1.6,1.8) node[above] {$e$};
  \node[below] at (0,0) {$e$};
  \draw[acc, thick, dashed] (0.8,0.9) -- (2.0,0.7) node[right, font=\scriptsize] {photon};
  \fill (0.8,0.9) circle (1.6pt);
  \node[font=\scriptsize] at (0.8,-0.6) {electromagnetic};
  % Weak vertex
  \begin{scope}[xshift=4.4cm]
    \draw[->, thick] (0,0) -- (0.8,0.9);
    \draw[->, thick] (0.8,0.9) -- (1.6,1.8) node[above] {$u$};
    \node[below] at (0,0) {$d$};
    \draw[acc, thick, dotted, line width=1pt] (0.8,0.9) -- (2.0,0.7) node[right, font=\scriptsize] {$W$};
    \fill (0.8,0.9) circle (1.6pt);
    \node[font=\scriptsize] at (0.8,-0.6) {weak (flavor change)};
  \end{scope}
  % Strong vertex
  \begin{scope}[xshift=8.8cm]
    \draw[->, thick] (0,0) -- (0.8,0.9);
    \draw[->, thick] (0.8,0.9) -- (1.6,1.8) node[above] {$q$};
    \node[below] at (0,0) {$q$};
    \draw[acc, thick, dash dot] (0.8,0.9) -- (2.0,0.7) node[right, font=\scriptsize] {gluon};
    \fill (0.8,0.9) circle (1.6pt);
    \node[font=\scriptsize] at (0.8,-0.6) {strong (color change)};
  \end{scope}
\end{tikzpicture}
$$

## Range and the Yukawa relation

The static potential produced by exchanging a boson of mass $m$ is the
time-independent Klein-Gordon solution,

$$
U(r) = \frac{A\, e^{-r/R}}{r}, \qquad R = \frac{\hbar}{mc},
$$

where $R$ is both the range of the force and the reduced Compton wavelength of
the mediating boson. A massless carrier ($m = 0$) gives $R = \infty$ and the
potential reduces to $U(r) = A/r$, the Coulomb form; this is why the
electromagnetic and gravitational forces have infinite range. A heavy carrier
gives a short range. Range and mediator mass are inversely related.

$$
% caption: Range of a force against the mass of its carrier, from the Yukawa
% relation R = hbar/(mc). Massless carriers give infinite range; the heavy weak
% bosons give the shortest range of the four.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, font=\scriptsize] {carrier mass};
  \draw[->, black] (0,0) -- (0,4.4) node[above, font=\scriptsize] {range};
  % curve R proportional to 1/m
  \draw[very thick] plot[domain=0.55:6.0, samples=60] (\x, {2.0/\x});
  % markers
  \fill (0.2,3.9) circle (2pt);
  \node[right, font=\scriptsize] at (0.3,3.95) {photon, gluon (massless)};
  \fill (5.3,0.38) circle (2pt);
  \node[above right, font=\scriptsize] at (5.0,0.5) {$W$, $Z$};
  \node[font=\scriptsize, align=center] at (3.6,2.6) {$R = \dfrac{\hbar}{mc}$};
\end{tikzpicture}
$$

> **Example (Range of the weak interaction).** The $Z^0$ mass is
> $91.16\ \text{GeV}/c^2$. Its Compton wavelength sets the range of the neutral
> weak interaction:
>
> $$
> R = \frac{\hbar c}{m_Z c^2}
> = \frac{197.3\ \text{eV·nm}}{91.16 \times 10^9\ \text{eV}}
> = 2.17 \times 10^{-18}\ \text{m} = 2.17 \times 10^{-3}\ \text{fm}.
> $$
>
> This is about $10^3$ times smaller than the range of the strong force.

## Running coupling constants

Coupling constants are not truly constant; they vary with the distance (or,
equivalently, the energy) at which the interaction is probed. The clearest case
is the electromagnetic charge. A point charge $q$ embedded in a dielectric
polarizes the surrounding molecules, whose negative ends screen $q$; a probe at
distance $r$ measures a reduced effective charge $q_{\text{eff}} = q/\kappa$, with
$\kappa$ the dielectric constant. Only inside the nearest molecular shell, closer
than the equilibrium separation $r_0$, does the probe measure the full $q$.

The QED vacuum does the same. A bare charge continually emits virtual photons
that briefly create electron-positron pairs; the virtual positrons are repelled
and the virtual electrons attracted, so the vacuum polarizes and partially
screens the charge. The role of $r_0$ is played by the electron Compton
wavelength $\lambda_C = h/m_e c = 2.43 \times 10^{-12}$ m. What is called "the
charge of the electron" is the fully screened value; probing closer reveals more
charge, so $\alpha$ increases at short distance.

$$
% caption: Vacuum polarization. Virtual pairs from the bare charge screen it, so
% the measured effective charge grows as the probe approaches, saturating at the
% bare value inside the electron Compton wavelength.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.2,0) node[right, font=\scriptsize] {distance $r$ from charge};
  \draw[->, black] (0,0) -- (0,4.0) node[above, font=\scriptsize] {effective charge};
  % curve: high near r0 (left), drops to plateau q/kappa (right)
  \draw[very thick] plot[domain=0.5:6.0, samples=60] (\x, {1.2 + 2.4/(\x)});
  % r0 marker
  \draw[black, dashed] (0.6,0) -- (0.6,3.8);
  \node[black, below, font=\scriptsize] at (0.6,0) {$r_0$};
  % plateau line
  \draw[black, dashed] (0,1.35) -- (6.0,1.35);
  \node[black, left, font=\scriptsize] at (0,1.35) {$\frac{q}{\kappa}$};
  \node[right, font=\scriptsize] at (0.75,3.5) {full charge $q$};
\end{tikzpicture}
$$

The strong and weak interactions run too, but differently. Because gluons carry
color and self-interact, the gluon loops _dilute_ the color charge at short
distance, so $\alpha_s$ _decreases_ as quarks approach. The running of the
couplings toward a possible common value underlies the grand-unification ideas of
the [beyond-Standard-Model](/particle-physics/beyond-standard-model/beyond-standard-model)
lesson.

## A worked weak decay

The free neutron decays via the weak interaction with a half-life of about
10.4 min:

$$
n \to p + e^- + \bar{\nu}_e.
$$

At the quark level a down quark emits a $W^-$ and becomes an up quark, turning
$udd$ into $uud$; the $W^-$ then decays to an electron and an electron
antineutrino. The flavor change is the signature of the charged weak
interaction.

$$
% caption: Feynman diagram of neutron beta decay. A down quark emits a W boson
% and becomes an up quark, converting the neutron to a proton; the W decays to an
% electron and an antineutrino.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % neutron quarks at bottom, proton at top
  \node[left, font=\scriptsize] at (-0.2,-0.2) {neutron $udd$};
  \node[left, font=\scriptsize] at (-0.2,3.2) {proton $uud$};
  % two spectator quarks straight up
  \draw[->, thick] (0.4,0) -- (0.4,3.0);
  \draw[->, thick] (1.1,0) -- (1.1,3.0);
  % the d quark converting to u at a vertex
  \draw[->, thick] (1.8,0) -- (2.4,1.5);
  \draw[->, thick] (2.4,1.5) -- (1.8,3.0) node[above, font=\scriptsize] {$u$};
  \node[below, font=\scriptsize] at (1.8,0) {$d$};
  \fill (2.4,1.5) circle (1.6pt);
  % W boson out to the right
  \draw[acc, thick, dotted, line width=1pt] (2.4,1.5) -- (3.9,1.5) node[midway, above, font=\scriptsize] {$W$};
  \fill (3.9,1.5) circle (1.6pt);
  % W decays to electron and antineutrino
  \draw[->, thick] (3.9,1.5) -- (5.2,2.5) node[right, font=\scriptsize] {$e$};
  \draw[->, thick] (5.2,0.5) -- (3.9,1.5);
  \node[right, font=\scriptsize] at (5.2,0.5) {antineutrino};
\end{tikzpicture}
$$

> **Example (Strong-interaction cross section).** The cross section for a
> reaction is roughly the area of a circle whose radius is the interaction range.
> For the strong interaction, $R_S \approx 10^{-15}$ m gives
>
> $$
> \sigma_S \approx \pi R_S^2 = \pi(10^{-15}\ \text{m})^2 \approx 3.1 \times 10^{-30}\ \text{m}^2 = 31\ \text{mb},
> $$
>
> tens of millibarns, consistent with measured strong-interaction cross sections.
> The weak interaction, with a range $10^3$ times smaller, gives cross sections
> smaller by $\sim 10^6$, which is why weak processes are rare.

## Interaction times and identifying the force

The characteristic interaction time is the clearest experimental fingerprint of
which force drives a decay. A particle held within the range of a force for less
than its interaction time is unlikely to interact through it; a particle that
decays through a force does so on that timescale.

| Interaction | Interaction time (s) | Example lifetime |
| --- | --- | --- |
| Strong | $10^{-23}$ | resonance particles |
| Electromagnetic | $10^{-18}$ | $\pi^0 \to \gamma\gamma$, $\sim 10^{-16}$ s |
| Weak | $10^{-10}$ | $\Lambda^0 \to p + \pi^-$, $\sim 10^{-10}$ s |

A particle with an anomalously long lifetime for its mass is decaying through a
weaker force than expected, the signature of a conservation law forbidding the
faster channel. Which interactions can drive a given decay is decided by the
[conservation laws](/particle-physics/symmetries/conservation-laws-symmetries)
of the next lesson.

[^tl-strength]: Tipler & Llewellyn, §12-2 — coupling constants and interaction strengths, including the fine-structure constant as the electromagnetic coupling and the Yukawa range-mass relation.