Appearance
8.3 — The Four Forces and How They Work
Chapter 8.2 listed the forces. This chapter explains where they come from, and the answer is one of the deepest ideas in physics: forces exist because of symmetries — specifically, because of a requirement that the laws of physics not depend on an arbitrary local choice of convention.
Demand that symmetry and the force carriers are not optional. They are forced into existence, with exactly the properties they turn out to have.
Gauge symmetry
Start with something familiar. Chapter 4.3 established that only differences in electric potential matter. Add 1000 volts to everything in a circuit and nothing changes — every current, every field, every measurement is identical.
That is a global symmetry: the physics does not care about the absolute zero of potential, as long as you shift it the same amount everywhere.
Now demand something stronger. What if you could shift the zero by a different amount at every point in space, independently, and still have the same physics?
That is local gauge symmetry, and it is a very demanding requirement. Naively it breaks everything: a wavefunction's phase appears in its derivatives, and if the phase varies from place to place, the derivatives pick up extra terms.
Work it through. Multiply the wavefunction by a position-dependent phase:
\psi \to e^{i\theta(x)}\psi
Chapter 7.3 established that a global phase is unobservable. But the Schrödinger equation contains \partial_\mu\psi, and:
\partial_\mu\left(e^{i\theta(x)}\psi\right) = e^{i\theta}\left(\partial_\mu\psi + i(\partial_\mu\theta)\psi\right)
The extra term i(\partial_\mu\theta)\psi ruins the invariance. The equation is no longer the same.
The fix. Introduce a new field A_\mu and replace the ordinary derivative with a covariant derivative:
D_\mu = \partial_\mu - iqA_\mu
and require that when the phase changes, the new field changes too:
A_\mu \to A_\mu + \frac{1}{q}\partial_\mu\theta
Now check. The unwanted term from the derivative is exactly cancelled by the change in A_\mu, and the equation is invariant.
What has just happened is remarkable. Demanding that a phase convention be locally arbitrary forced the existence of a new field. And that field's transformation rule, A_\mu \to A_\mu + \partial_\mu\theta/q, is exactly the gauge transformation of the electromagnetic potential known since Maxwell.
\boxed{\text{The photon exists because the electron's phase is locally arbitrary.}}
Electromagnetism is derived, not assumed. The symmetry group here is U(1) — the group of phases — and it has one generator, so there is one gauge field: the photon.
And the photon must be massless. A mass term in the Lagrangian looks like m^2A_\mu A^\mu, and under the gauge transformation this is not invariant. Gauge symmetry forbids a photon mass, which is why electromagnetism has infinite range and 1/r^2 holds exactly.
Yang–Mills: making the symmetry bigger
Chen Ning Yang and Robert Mills asked in 1954 what happens if the local symmetry is larger than a phase.
Take SU(2), which acts on a two-component object — mixing "up-type" and "down-type" into each other. Now the symmetry has three generators, so three gauge fields are forced.
Take SU(3), acting on three components. Eight generators, eight gauge fields.
The crucial difference from U(1): these groups are non-abelian, meaning the transformations do not commute. Rotating in one direction then another is not the same as doing it the other way round.
The consequence is enormous. The gauge fields themselves carry the charge of the symmetry. Gluons carry colour. W bosons carry weak charge. The photon, by contrast, carries no electric charge.
This makes the theory non-linear, because the force carriers interact with each other. Two photons pass straight through each other; two gluons scatter. Everything strange about the strong force follows from this.
Yang and Mills wrote the theory in 1954 and it had a fatal-looking problem: it predicted massless gauge bosons, and no massless strongly-interacting particles were seen. Pauli attacked the idea at a seminar so persistently that Yang sat down mid-talk. It took until 1971 for 't Hooft and Veltman to prove such theories are renormalisable, and then everything fell into place.
Quantum electrodynamics
QED is the U(1) theory: electrons and photons.
The single vertex, from which everything is built:
An electron line arrives, an electron line leaves, a photon line attaches. That one vertex, repeated and combined, gives every electromagnetic process there is.
The coupling strength is the fine structure constant:
\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \frac{1}{137.036}
Dimensionless, which is what makes it fundamental — it has the same value in any system of units, and any civilisation anywhere would measure the same number. Feynman called it "one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding".
Each vertex in a diagram contributes a factor of \sqrt{\alpha} to the amplitude, so each additional pair of vertices costs a factor of \alpha \approx 1/137 in probability. This is why perturbation theory works so well for QED — each successive correction is a hundred times smaller than the last, so a handful of terms gives twelve-figure accuracy.
Feynman diagrams as a tool
The rules for reading one:
- Time runs one way (usually left to right or bottom to top).
- Straight lines with arrows are fermions; an arrow against the time direction is an antiparticle.
- Wavy lines are photons, curly lines are gluons, dashed lines are W, Z or Higgs.
- Every vertex contributes a coupling factor.
- Internal lines are virtual and are not observable.
- Energy and momentum are conserved at every vertex.
Worked reading: electron–electron scattering. Two electrons come in, exchange a photon, and leave. The amplitude is proportional to e^2 = 4\pi\alpha, and squaring gives a probability proportional to \alpha^2. Compute the full expression and out comes the Coulomb force of Chapter 4.1, including its 1/r^2 behaviour.
Higher orders. Add a loop — a photon emitted and reabsorbed — and you get a correction of relative size \alpha/\pi \approx 0.0023. That is exactly the leading term in the electron's anomalous magnetic moment from Chapter 7.7, computed by Schwinger in 1948 and carved on his tombstone.
Getting to twelve figures required going to five loops, involving 12,672 diagrams, computed over decades. The agreement with experiment is the best in science.
Important caveat, repeated from Chapter 7.10: a Feynman diagram is a term in a mathematical series, not a picture of what happens. No electron follows those lines.
Quantum chromodynamics
QCD is the SU(3) theory: quarks and gluons, with colour as the charge.
Three colours (red, green, blue) and three anticolours. Eight gluons, each carrying a colour–anticolour combination.
The vertices:
- Quark emits or absorbs a gluon, changing its colour.
- Gluon emits a gluon — the three-gluon vertex, which has no QED analogue.
- Four-gluon vertex — likewise.
Those last two are why the strong force is completely unlike electromagnetism.
Asymptotic freedom
The coupling constant is not constant. It depends on the energy at which you measure it, and this running is a real, measured effect.
In QED, the vacuum screens charge. Virtual electron–positron pairs polarise around a charge, reducing what you see from far away. Get closer and you penetrate the screening, so the effective charge grows.
\alpha(\text{low energy}) = \frac{1}{137}, \qquad \alpha(91\ \text{GeV}) = \frac{1}{128}
Measured, and it matches prediction.
In QCD the opposite happens. Virtual quark pairs screen, as in QED, but virtual gluon pairs antiscreen — because gluons carry colour, they spread it outwards rather than concentrating it. With three colours and six quark flavours, the gluon effect wins.
\boxed{\alpha_s\ \text{decreases as energy increases}}
This is asymptotic freedom, found by Gross, Wilczek and Politzer in 1973, and it won the 2004 Nobel Prize.
Measured values:
| Energy | \alpha_s |
|---|---|
| 1 GeV | \sim0.5 |
| 91 GeV (Z mass) | 0.118 |
| 1 TeV | \sim0.09 |
Two consequences, and they run in opposite directions.
At high energy the quarks inside a proton behave almost as free particles, which is exactly what the SLAC deep inelastic scattering of Chapter 8.1 found and could not otherwise explain. Perturbation theory works, and LHC predictions are computable.
At low energy the coupling becomes large and perturbation theory fails completely. Ordinary nuclear physics — the binding of protons and neutrons — cannot be computed by summing diagrams. It requires lattice QCD, in which spacetime is replaced by a grid and the equations are solved numerically on supercomputers. Lattice calculations now reproduce the proton's mass from first principles to about 1 %, which is one of the great computational achievements in physics.
Confinement
Try to pull a quark out of a proton.
Between two electric charges, the field lines spread out and the force falls as 1/r^2. Between two colour charges, the gluons attract each other, so the field lines are pulled into a narrow tube — a flux tube — of roughly constant cross-section.
A tube of constant cross-section means constant energy per unit length, so:
V(r) \approx \sigma r, \qquad \sigma \approx 1\ \text{GeV/fm}
The potential grows without limit. The force between quarks is roughly constant at about 160,000 newtons — the weight of sixteen tonnes, between two subatomic particles, and it does not decrease with distance.
So separating them takes infinite energy? No. Long before that, the energy in the tube exceeds 2m_qc^2, and the tube snaps by creating a new quark–antiquark pair at the break.
Result: you get two mesons, not two free quarks.
Pull harder and you make more mesons. This is exactly like trying to isolate a north magnetic pole by cutting a magnet (Chapter 4.5): you get two magnets. Quarks are confined, permanently, and no free quark has ever been observed. Searches for fractional charge in matter have set limits of about one quark per 10^{20} nucleons.
And this is why particle collisions produce jets. Knock a quark out of a proton at high energy and it flies off, the flux tube stretches, pairs are created, and the result is a narrow spray of hadrons all moving in roughly the quark's original direction. A jet is the visible fingerprint of a quark, and reading jets is how the top quark and the Higgs were found.
The weak force
The weak force is the strangest of the four, and it broke a principle everybody thought was absolute.
Beta decay, at the quark level:
d \to u + W^- \to u + e^- + \bar{\nu}_e
which turns a neutron (udd) into a proton (uud).
This is the only force that changes flavour. Nothing else can turn one kind of quark into another, and without it there would be no radioactivity, no fusion in the Sun, and no way to make the heavy elements.
Parity violation
Parity is mirror reflection. Until 1956 everybody assumed the laws of physics look the same in a mirror.
Lee and Yang pointed out in 1956 that this had been tested for the strong and electromagnetic forces and never for the weak one.
Chien-Shiung Wu tested it within months. She cooled cobalt-60 to 0.01 K and aligned the nuclear spins with a magnetic field, then measured the direction of the emitted electrons.
If parity held, electrons would come out equally in both directions along the spin axis.
They did not. More electrons came out opposite to the nuclear spin than along it. Parity is violated, and maximally.
The physical statement: the weak force couples only to left-handed particles and right-handed antiparticles. Handedness here means the projection of spin along the direction of motion. A right-handed electron does not feel the weak force at all.
This is genuinely bizarre. There is no known reason for nature to prefer one handedness, and it is the only place in physics where a mirror-image world would behave differently. Pauli's reaction on hearing the result: "I cannot believe God is a weak left-hander."
Lee and Yang received the Nobel Prize in 1957, within a year — one of the fastest awards ever. Wu, who did the experiment, did not, which is widely regarded as one of the clearest injustices in the prize's history.
CP violation
If mirror reflection alone is violated, perhaps mirror plus swapping matter for antimatter (CP) is preserved.
Cronin and Fitch found in 1964 that it is not. Neutral kaons decay in a way that violates CP by about 0.2 %. They received the Nobel Prize in 1980.
This matters more than its size suggests. Chapter 7.10 listed Sakharov's conditions for producing a matter-dominated universe, and CP violation is one of them. The Standard Model has it, and has about 10^{10} times too little.
CPT — charge, parity and time reversal together — is believed to be exact, and it is a theorem of any local relativistic quantum field theory. It is why a particle and its antiparticle must have exactly equal masses and lifetimes, tested to one part in 10^{18} for the neutral kaon system.
Electroweak unification
At energies above about 100 GeV, the electromagnetic and weak forces are one force.
The unification is not a metaphor. The theory has four gauge fields — three from SU(2) and one from U(1) — and the photon and the Z boson are mixtures of two of them:
\gamma = B\cos\theta_W + W^3\sin\theta_W
Z = -B\sin\theta_W + W^3\cos\theta_W
with \sin^2\theta_W = 0.2312, the weak mixing angle, measured to four decimal places.
Below 100 GeV the symmetry is broken by the Higgs field, the W and Z acquire mass, and the weak force becomes short-ranged and feeble. Chapter 8.4 explains the mechanism.
Glashow, Weinberg and Salam built this between 1961 and 1968 and shared the 1979 Nobel Prize.
Its predictions, all confirmed: the existence of the Z boson and hence neutral currents (found at CERN in 1973, in the Gargamelle bubble chamber), the W and Z masses (found 1983), and the relation m_W/m_Z = \cos\theta_W.
Grand unification, and why it is uncertain
The couplings run, and they run towards each other. Extrapolating:
- \alpha_s decreases with energy.
- \alpha_{\text{em}} increases.
- \alpha_{\text{weak}} increases slowly.
With the Standard Model alone, they nearly meet at about 10^{15} GeV — and miss.
With supersymmetry added, they meet much more precisely. This is the strongest indirect argument for supersymmetry, and it is why so many physicists expected the LHC to find it. It has not.
Grand unified theories predict proton decay, since they allow quarks to turn into leptons. The simplest model, SU(5), predicted a proton lifetime of about 10^{31} years.
Super-Kamiokande has looked and found nothing, setting a limit of >1.6\times10^{34} years for the main channel. Simple SU(5) is dead, killed by an experiment that found nothing at all — which is a good illustration that null results can be decisive.
Gravity remains outside all of this. It is not a gauge theory of the same kind, and Chapter 8.7 covers the attempts.
Where this shows up in your life
Nuclear reactors and radioactive dating run on the weak force.
The Sun runs on it: the first step, p+p \to d+e^++\nu_e, requires a proton to become a neutron, which is a weak process. That is why the Sun burns slowly enough to have supported four billion years of evolution.
Medical isotopes are produced and decay by weak interactions.
Quark confinement is why matter is made of protons and neutrons rather than free quarks, and therefore why chemistry exists.
And the strong force's asymptotic freedom is what makes LHC physics computable at all.
What the next chapter fixes
Gauge symmetry forbids mass terms for gauge bosons — and the W and Z are the heaviest particles in the table after the top quark. Something must break the symmetry without destroying the theory's consistency. Chapter 8.4 explains the mechanism, which required a new field filling all of space, took 48 years to confirm, and answers a question this book has been circling since Chapter 1.4: what actually is mass?