Yang–Mills existence and the mass gap

1Forces are fields, and ripples are particles

In modern physics every force is carried by a field: something with a value at every point in space and time. The ripples in a field are particles. The electromagnetic field is the familiar example. Maxwell wrote down its equations, and its ripples are photons, the particles of light.

Photons have no mass. That is why light can cross the universe, and why the electric force between two charges fades as $1/r^2$ but never reaches zero. In the widget, the grey $1/r$ curve is this kind of massless pull.

Swap in a carrier with mass $m$ and the picture changes. The pull now looks like the Yukawa form, with an extra exponential factor that kills it off. The distance over which it dies, the range, is about $\hbar/(mc)$. Here $\hbar$ is the reduced Planck constant and $c$ is the speed of light. A useful number is $\hbar c \approx 197.3$ MeV·fm, where 1 fm (a femtometre) is $10^{-15}$ m.

2Gauge symmetry, and the twist Yang and Mills added

Maxwell's theory has a hidden freedom called gauge symmetry. You can relabel the field at every point in space, each point independently, and nothing physical changes. It sounds like bookkeeping, but this freedom fixes the form of the equations almost completely.

In 1954 Yang and Mills asked what happens with a richer symmetry, one whose relabellings do not commute. "Do A, then B" gives a different result from "do B, then A". Mathematicians call such symmetries non-abelian. The symmetry is described by a gauge group, such as $SU(2)$ or $SU(3)$, groups of small matrices.

The result is striking. In Maxwell's theory, photons carry no electric charge, so they pass straight through each other. In Yang–Mills theory the carriers carry the very charge they transmit, so they attract one another. That is the commutator term $g[A_\mu, A_\nu]$ in the field strength.

3Gluons and the strong force

For the strong nuclear force the gauge group is $SU(3)$, and the carriers are called gluons. They stick to quarks, the particles inside protons and neutrons, and they also stick to each other. The full theory with quarks is called QCD, quantum chromodynamics. The Clay problem is about the simpler theory with gluons only, called pure Yang–Mills, for any compact simple gauge group.

The classical Yang–Mills equations describe massless waves that travel at light speed, exactly like light. Yet the real strong force behaves nothing like light. Its lightest particles are heavy, and the force dies off within about the width of a proton.

Pure-gluon computer simulations even predict particles made of nothing but gluons, called glueballs. In 1999 Colin Morningstar and Mike Peardon computed the lightest one at about $1730 \pm 50 \pm 80$ MeV, with the next at about 2400 MeV. Put 1730 MeV into the widget and the range comes out near 0.11 fm.

4What the mass gap says

Picture a ladder of every energy the quantum theory allows. The vacuum, empty space, sits on the bottom rung at zero. The mass gap says the next rung is not right next to zero. It sits a definite distance $\Delta$ above it.

So creating anything at all costs at least $\Delta$ of energy, which is the mass of the lightest particle. A heavy lightest particle means a short-range force. The mass gap is the precise mathematical version of "the strong force does not reach beyond the nucleus".

Jaffe and Witten, who wrote the official problem, show what the gap implies. Measurements made far apart become unrelated exponentially fast, with a decay rate set by $\Delta$. In the formula below, $O$ is any measurement and $C$ is any number smaller than $\Delta$.

5Where the gap comes from: dimensional transmutation

Here is the puzzle. The classical equations contain no mass at all. So where could a mass come from? Physicists believe it is built by quantum effects through a process called dimensional transmutation: the coupling $g$, which has no units, gets traded for an energy scale $\Lambda$.

In the formula below, $\mu$ is the energy at which you measure $g$, and $b_0$ is a fixed positive number set by the gauge group. Look at the shape: $e^{-1/(2b_0 g^2)}$. As $g$ shrinks towards zero, this goes to zero faster than any power of $g$.

That matters because most physics calculations expand in powers of a small coupling: a term in $g^2$, then $g^4$, and so on. Every term in that series sees a gap of exactly zero. The gap is non-perturbative: no amount of expanding can ever reveal it.

6Why you weigh what you do

This is not only abstract. A proton weighs about 938 MeV. The quarks inside it have only a few MeV of mass between them. Almost all of a proton's mass is energy of the gluon field and the motion of the quarks, the same physics that produces the mass gap.

So most of the mass of ordinary matter, including you, comes from the effect this problem is about. A related effect is confinement: quarks and gluons are never seen alone. Pull two quarks apart and the energy stored between them grows until a new quark–antiquark pair pops out. Jaffe and Witten list the mass gap and confinement as two separate properties, closely linked but not the same.

7What "exists" means to a mathematician

Physicists use Yang–Mills theory every day, so in what sense does it not exist? A physicist can compute with a recipe. A mathematician needs an object: a precisely defined space of states, an energy operator $H$, and fields that satisfy a list of rules, the Wightman or Osterwalder–Schrader axioms, which pin down what a quantum field theory is.

The construction must also behave correctly at very short distances, matching asymptotic freedom: the discovery in 1973, by David Gross and Frank Wilczek and independently by David Politzer, that the coupling gets weaker at shorter distances. It won the 2004 Nobel Prize in Physics.

No interacting quantum field theory in four spacetime dimensions has ever been built to this standard. Not one. Jaffe and Witten call Yang–Mills the natural first candidate, precisely because asymptotic freedom tames the short distances.