Gauge symmetry saved, mass endowed

In its original incarnation, the Higgs mechanism was designed to give mass to the gauge bosons that carry the weak force and to preserve a basic symmetry of the theory of weak interactions. Although many physicists may not be familiar with the gauge particles associated with the weak force, almost all are well acquainted with one gauge boson—the photon, which transmits the electromagnetic force. In quantum field theories, particles are expressed as fields; the photon field is a four-vector potential A^μ = (V/c, A_x, A_y, A_z) that incorporates the scalar (V) and vector (A) potentials, quantized versions of fields familiar from classical electrodynamics. Maxwell’s equations are invariant under the “gauge” transformation V → V − ∂Λ/∂t, A → A + ∇Λ for an arbitrary Λ(x,t). Similar gauge symmetries apply to the standard model of particle physics and generate the interactions between gauge bosons and other fundamental particles.

The electromagnetic force is long-range and the photon is massless. The weak force is short-range, and as a consequence its gauge carriers must be massive. It’s easy enough, in principle, to accommodate massive particles in a quantum field theory: Just include in the potential energy a term of the form ½m²A², where A² is the four-dimensional analogue of dotting the vector potential into itself. That quadratic mass term, however, breaks gauge invariance. Moreover, its inclusion in the quantum field theory leads to pathological infinities. Sometimes symmetries can tame such infinities, so theorists were motivated to try to preserve the symmetry under gauge transformations.

To do so, they introduced a complex field ϕ, which has a kinetic energy and a potential energy. A straightforward, well-known mathematical procedure instructed them how to do so in a way that preserves gauge invariance, and that algorithm leads to an interaction proportional to ϕ^∗ϕA². Ultimately, that interaction will be responsible for the gauge field’s mass; what’s missing is a final key ingredient: spontaneous symmetry breaking.

The theory of electrodynamics is rotationally invariant, but you’d never know that looking at a magnetic domain in a ferromagnet. There the atomic spins point in a particular, apparently preferred direction. Magnetic domains are exemplars of spontaneous symmetry breaking, the necessity of a system to choose a particular configuration from among many that are equally allowed by nature. The field ϕ also needs to make a choice. In the Higgs mechanism, its potential energy is given by V(ϕ) = λ(ϕ^∗ϕ – μ²)², which is illustrated in the figure as a function of the real and imaginary parts of ϕ. The potential has a ring of equally good minima: all points with a magnitude μ. Yet ϕ must choose to lie at a particular minimum; in the more precise language of quantum field theory, it must have a specific vacuum expectation value. The particular value chosen by the field doesn’t matter, so let’s set the value to be the real number μ and define a new field H by ϕ = H + μ. In terms of the new field H—the Higgs field—the interaction term ϕ^∗ϕA² = μ²A² + . . . . Voilà, the gauge-field potential energy now includes a term of the form ½m²A², indicative of a massive particle. The Higgs field, too, has a mass, proportional to μ√λ, inherited from the potential V(ϕ).

The symmetry structure of the full standard model is more complicated than in the simple model presented above, but the essentials of the Higgs mechanism are the same. Introduce a scalar field in a well-defined, gauge-invariant way. Once the field chooses a vacuum expectation value, its couplings to the fundamental fields of the theory give mass to the associated particles. The Higgs coupling constant λ, however, is not known a priori, and it cannot be related to fundamental parameters of other fields. Therefore, it is not possible to readily predict the Higgs mass.

The Higgs mechanism was not the first context in which particle physicists considered spontaneous symmetry breaking. By 1961 Jeffrey Goldstone had proved that a spontaneously broken quantum field theory necessarily includes a massless particle, now called a Goldstone boson. But the Higgs mechanism, because it involves gauge symmetries, violates the conditions for Goldstone’s result to apply. That allows the mechanism to perform its vital function of generating only massive particles.