On the 4th of July 2012, in a packed auditorium at CERN, two experiments announced that they had found a new particle at a mass of about 125\ \text{GeV}. Champagne corks flew, an 84-year-old Peter Higgs wiped away a tear, and the world's newspapers reeled off a nickname the physicists all hated: the God particle. What had actually been discovered was the last missing piece of the Standard Model — the ripple of a field that fills all of space and, in doing so, answers a question so basic it is easy to forget it needs an answer at all: why does anything have mass?

This page rests on a single sentence, worth reading twice: particles get their mass by interacting with a field that is switched on everywhere in empty space, and that field is switched on because of a phenomenon called spontaneous symmetry breaking. The underlying laws are perfectly symmetric; their lowest-energy state is not. Everything below — the wine-bottle potential, the massive W and Z, the massless photon, the boson found in 2012 — is a consequence of that one idea.

Spontaneous symmetry breaking: a symmetric law, a lopsided rest state

Balance a pencil perfectly on its point. The situation is completely symmetric — no direction around the vertical is special, and the laws governing the pencil don't prefer north over east. Yet the pencil cannot stay in that symmetric state: it is unstable, and it topples. The instant it falls it must pick some direction, and once it has fallen the symmetry is gone — there is now a definite direction the pencil points. Nobody chose the direction; the symmetry was broken spontaneously. The law kept its symmetry; the ground state did not.

This is the whole trick, transplanted from a pencil to a field. Consider a field \phi whose energy density — its potential — is

Read the two terms. The \lambda|\phi|^4 term is a positive bowl that always pushes the field back towards zero. But the sign in front of \mu^2|\phi|^2 is negative, so near \phi = 0 the potential slopes down as the field grows: the origin is a hill-top, not a valley. The field is unstable at zero, just like the upright pencil, and it rolls downhill to a nonzero value. Because V depends only on |\phi|, that lowest point is a whole circle of equally-good minima, and the field must pick one — spontaneously.

The Mexican hat

Plotted over the two real components of the complex field, V(\phi) looks like a sombrero, or the punt at the bottom of a wine bottle: a bump in the middle, a circular trough around it. The graph below is a vertical slice through that shape — take V(\phi) = -a\,\phi^2 + b\,\phi^4 along one axis. The central hill at \phi = 0 is the symmetric-but-unstable state; the two dips on either side are where the field actually comes to rest.

Drag the slider to change a (the depth of the negative \phi^2 term, i.e. \mu^2). At a = 0 the potential is a single symmetric bowl with its minimum at the origin — the symmetry is unbroken, the field sits at zero, and (as we will see) particles are massless. Turn a up and watch the middle push up into a bump while two minima slide outward: the symmetry breaks, and the field's rest value moves off zero.

In the real theory the trough sits at a field value written v \approx 246\ \text{GeV}, the vacuum expectation value. Empty space is not empty: it is filled, everywhere and always, with the Higgs field sitting at this nonzero value v. That single fact is what gives particles their mass.

Finding the minimum (worked example)

Where exactly does the field come to rest? Take the one-dimensional slice V(\phi) = -\mu^2\phi^2 + \lambda\phi^4 and do what you always do to find a minimum: set the derivative to zero.

Factor out \phi:

One solution is \phi = 0 — but that is the hill-top (a maximum, since the curvature there is negative). The physical minima come from the bracket:

So the field settles at |\phi| = \mu/\sqrt{2\lambda} \equiv v, a nonzero value fixed by the two constants in the potential. Plug in numbers: if \mu^2 = 8 and \lambda = 1 (in whatever units), then v = \sqrt{8/(2\cdot 1)} = \sqrt{4} = 2. The larger the negative curvature \mu^2, the further out the field sits; the steeper the quartic \lambda, the tighter it is pulled back in.

How the field hands out mass

Here is the payoff. A particle that couples to the Higgs field feels that ever-present background value v as it moves through space, and that constant interaction is exactly what a mass is. To an excellent conceptual approximation the mass a particle picks up is

Worked example. Suppose a particle has coupling g = 0.7 to a field with vacuum value v = 246\ \text{GeV}. Then its mass is m \sim g\,v = 0.7 \times 246 \approx 172\ \text{GeV} — right in the neighbourhood of the top quark. Halve the coupling to g = 0.35 and you halve the mass to about 86\ \text{GeV}. Mass is not an intrinsic label stamped on a particle; it is readout of how hard that particle grips the Higgs field.

This is the resolution of a real puzzle in the electroweak theory. The electroweak interaction starts with four massless force-carriers and a beautiful symmetry between them. When the Higgs field switches on to its value v, that symmetry breaks: three of the carriers eat part of the Higgs field and become the heavy W^+, W^- (about 80\ \text{GeV}) and Z (about 91\ \text{GeV}), while the fourth combination stays untouched and massless — the photon. One field, one broken symmetry, and suddenly the weak force is short-ranged while electromagnetism reaches across the universe.

The field versus the boson

Keep two things firmly apart. The Higgs field is the background — present everywhere, sitting at v, responsible for mass. The Higgs boson is a ripple in that field: poke the field hard enough (as the LHC does by smashing protons together) and it wobbles about its resting value, and that wobble is a particle. It is the difference between the calm surface of a pond (the field at rest) and a splash on it (a boson). The boson is the field's way of saying "I am really here" — which is why finding it was the decisive test.

And find it they did. The ATLAS and CMS detectors at CERN's Large Hadron Collider each independently saw an excess of events consistent with a new boson of mass m_H \approx 125\ \text{GeV}, announced on 4 July 2012. Higgs and François Englert — who had written the theory down in 1964, nearly half a century earlier — shared the 2013 Nobel Prize in Physics the following year.