Fundamental Interactions and 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, $U (r) = e^{2} /4 π ε_{0} r$ , dividing through by $ℏ c$ makes the multiplier dimensionless,

α = \frac{e ^{2}}{4 π ε _{0} ℏ 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.¹

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 $(GeV / c^{2})$	Source charge	Range (m)	Coupling	Time (s)
Strong	gluon	$0$	color	$1 0^{- 15}$	$α_{s} \approx 1$	$1 0^{- 23}$
Electromagnetic	photon	$0$	electric	$\infty$	$α \approx \frac{1}{137}$	$1 0^{- 18}$
Weak	$W^{\pm}$ , $Z^{0}$	$80, 91$	weak	$1 0^{- 18}$	$α_{w} \approx 1 0^{- 5}$	$1 0^{- 16}$ – $1 0^{- 10}$
Gravity	graviton	$0$	mass	$\infty$	$α_{g} \approx 1 0^{- 38}$	—

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.

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 $\sim 1 0^{- 23}$ 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 $\sim 1 0^{- 18}$ 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 $\sim 1 0^{- 18}$ m, far shorter than the strong force, with interaction times from $1 0^{- 16}$ to $1 0^{- 10}$ 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 $α_{g} \approx 1 0^{- 38}$ 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 lesson develops.

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.

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}, R = \frac{ℏ}{m c},

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.

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.

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 $q_{eff} = q / κ$ , with $κ$ 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 $λ_{C} = h / m_{e} c = 2.43 \times 1 0^{- 12}$ m. What is called the charge of the electron is the fully screened value; probing closer reveals more charge, so $α$ increases at short distance.

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.

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 $α_{s}$ decreases as quarks approach. The running of the couplings toward a possible common value underlies the grand-unification ideas of the beyond-Standard-Model 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^{-} + \overset{ν}{ˉ}_{e} .

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

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.

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	$1 0^{- 23}$	resonance particles
Electromagnetic	$1 0^{- 18}$	$π^{0} \to γ γ$ , $\sim 1 0^{- 16}$ s
Weak	$1 0^{- 10}$	$Λ^{0} \to p + π^{-}$ , $\sim 1 0^{- 10}$ 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 of the next lesson.

Tipler & Llewellyn, §12-2 — coupling constants and interaction strengths, including the fine-structure constant as the electromagnetic coupling and the Yukawa range-mass relation. ↩