The Higgs field and spontaneous symmetry breaking

The Standard Model describes every fundamental interaction between particles by gauge theories with the combined symmetry group SU(3) × SU(2) × U(1). An element of SU(n) is an $n \times n$ complex matrix that is unitary — its inverse equals its Hermitian conjugate — and special, with determinant 1. SU(3) belongs to the strong interactions, SU(2) to the weak. U(1) consists of $1 \times 1$ unitary matrices: complex numbers of modulus 1,

[The lecturer remarks that U(1) "could be an SU(1)" and so needs no S. Strictly, demanding determinant 1 of a $1 \times 1$ matrix leaves only the number 1, so SU(1) is trivial; U(1) is the group of all phases.]

Where U(1) hides in electromagnetism

The gauge symmetry you learn in classical electromagnetism does not obviously look like multiplication by a phase. Classically, the potentials $A_\mu$ (time component $-\phi$, space components $\vec A$) can be shifted by the gradient of any function $\Lambda(x)$ without changing $\vec E$ and $\vec B$ (with $c = 1$; the lecturer allows that a factor of $c$ may be missing). The connection appears in quantum electrodynamics. The electron is described by the Dirac field $\psi(x)$ (a spinor field), and under a change of gauge it acquires a local phase:

with $e$ the fundamental charge. (The overall sign in the exponent depends on conventions.) The classical gauge function is, up to the charge, the phase of the electron field — and that is what it means to call electrodynamics a U(1) gauge theory. The strong and weak interactions work the same way, except that the phase factor becomes a $3 \times 3$ or $2 \times 2$ complex matrix. Matrices need not commute, so for these interactions the order of gauge transformations matters: they are non-Abelian.

A student asks for the physical significance of being non-Abelian. The lecturer couldn't give a quick answer, but later suggested a partial one: the gauge bosons of the weak force have mass, which he connected with the non-Abelian structure. [That link is loose: the gluons are non-Abelian too and massless. The W and Z masses come from the symmetry breaking described below.]

There are strong experimental and theoretical reasons to believe that the interactions have not always looked as different as they do today. When the universe was much hotter and denser, the different symmetries began to look the same. The key example is the electroweak part, SU(2) × U(1): SU(2) is associated with the carriers of the weak force — the Z, W⁺ and W⁻ — and U(1) with the photon. The equations for these bosons look very similar, except that the weak bosons are massive. In a "perfect" world — the lecturer pictures everything at a balmy $10^{16}$ GeV, with all the vector bosons exchanging freely — the electric and weak interactions would be governed by one unbroken symmetry. That they are separate today is due to spontaneous symmetry breaking, now established by the discovery of the Higgs boson at the LHC.

[Simplifications here. The electromagnetic U(1) is not simply a U(1) inside SU(2): the photon (and likewise the Z) is a mixture of one SU(2) direction and the separate "hypercharge" U(1). Above the electroweak scale the unbroken symmetry is still the two-factor SU(2) × U(1); a single group appears only with grand unification. And electroweak symmetry breaks at energies around 100 GeV; $10^{16}$ GeV is the much higher scale of grand unification.]

A direction in an abstract space

To picture the breaking, the lecturer compares SU(2) to the rotations of a sphere. [Strictly, SU(2) is the double cover of the rotation group SO(3): two SU(2) matrices correspond to each rotation.] With perfect SU(2) symmetry every direction looks the same. Breaking it picks out one direction as special, and only the symmetry of rotating about that direction — a U(1) — survives. The "directions" are not in physical space but in the abstract space of gauge fields.

A student asks how fixing a value can break SU(2) but leave U(1). The lecturer's answer: if SU(2) is like symmetry on a sphere, breaking it is like picking out a circle on the sphere — you can still move freely around that circle, which is a U(1) symmetry. He adds a count: the Higgs doublet has four real parameters (two complex numbers), and the breaking adds one constraint. [More precisely, the condition $|h| =$ const is one constraint on four real numbers, so the minima form a 3-sphere. And with SU(2) acting alone, any non-zero $h$ would break SU(2) completely; the U(1) of electromagnetism survives only because the hypercharge U(1) also acts on $h$, as noted above.]

What breaks the symmetry is the Higgs field, which fills all of spacetime. At very high energy it is perfectly symmetric — every direction in gauge-field space looks the same. As the universe expands and cools, the Higgs field moves to a value that picks out a direction. Concretely, it is a complex doublet:

The Higgs field has an energy density, its potential energy $V$. Energy density is physical — if you could scoop up a sample of the early universe and measure it, the answer couldn't depend on your choice of gauge — so $V$ must be gauge invariant. The only invariant combination of the fields is their total magnitude:

Compare a spherically symmetric problem in electrostatics, where the potential can depend only on how far you are from the centre.

All the magic is in the shape of $V$. For spontaneous symmetry breaking, its minimum must lie at a non-zero value of $|h|$ — say 1 in some units. Early on, when everything is hot, the field sits where the energy is high; as the universe cools, it moves to the minimum. But $|h| = 1$ does not fix the field. Put it all into $h_1$, real; or all into $h_2$, imaginary; or split it between them: there are infinitely many configurations with the same magnitude and the same energy. The field settles into one of them at random. That is spontaneous symmetry breaking.

The usual analogy is water freezing. Liquid water is rotationally symmetric; when it freezes, crystal planes form in some particular orientation and the rotational symmetry is lost. Here the interactions start out symmetric in gauge-field space, and by settling on particular values of $h_1$ and $h_2$ the field defines a "crystal plane" — a preferred direction.

A student asks how one could ever measure which direction in gauge space was picked. The lecturer calls this a deep question that can't be fully answered: research into electroweak symmetry breaking is a hot topic precisely because there are details we don't yet understand. Another asks whether gravity has a gauge symmetry. It does, but it fits in very differently: classically, the gauge symmetries of general relativity are coordinate transformations; at the quantum level there is no simple way to say it, apart from the promising but speculative framework of string theory.

What follows from the breaking is a subject for a quantum field theory course; the lecturer sketches three consequences.

1. Electrons and neutrinos become different. With the Higgs field zero, an electron and a neutrino obey exactly the same equation — there is no difference between them. Once $h_1$ and $h_2$ settle, one combination of the underlying fields acquires a mass and an electric charge, and becomes the electron. (Asked about the other generations, the lecturer thinks the same holds — and later says he is very confident that the mass mechanism applies to the muon and tau; the existence of further generations presumably reflects another, even more poorly understood, broken symmetry.) In the minimal Standard Model, neutrinos come out with exactly zero mass. We now know they have mass — though only bounds on its size are known — so something is missing; masses can be put in by hand, but that is not fully satisfying, and many experiments are probing the neutrino sector.

2. Quarks and leptons acquire rest mass. This is what made the newspapers. What does mass mean in a field theory? Work out the spectrum of oscillations of the field; each frequency $\omega$ corresponds to an energy $\hbar\omega$, and if the frequencies have a lower bound above zero, the theory has particles with non-zero rest mass:

When the Higgs field is zero, the spectrum for quark and lepton fields goes all the way down to zero frequency. When it is non-zero, the interaction between, say, the electron field and the Higgs field shifts the spectrum up, so the frequency can never fall below a minimum: every quark and charged lepton picks up mass.

3. In grand unified theories, the universe fills with magnetic monopoles. This last, surprising consequence is the subject of the next page.

• In quantum electrodynamics, a gauge transformation multiplies the electron field by a local phase $e^{ie\Lambda(x)}$: electromagnetism is a U(1) gauge theory. The weak and strong forces replace the phase with $2 \times 2$ and $3 \times 3$ matrices (non-Abelian).
• The weak and electromagnetic symmetries look different today because of spontaneous symmetry breaking.
• The Higgs field is a complex doublet transforming under SU(2); its energy density depends only on $|h|^2$, with the minimum at $|h| \neq 0$. As the universe cools the field settles at random into one of infinitely many equivalent minima.
• Consequences: electrons and neutrinos become distinct, quarks and charged leptons acquire mass ($m = \hbar\omega_\text{min}/c^2$), and — in grand unified theories — magnetic monopoles form.