# Higgs mechanism

The Higgs mechanism, originally discovered by the British physicist Peter Higgs (building on a previous suggestion by Philip Anderson in condensed matter physics), is the mechanism that gives mass to all elementary particles in particle physics. It makes the W boson different from the photon, for example. It can be understood as an elementary case of tachyon condensation where the role of the tachyon is played by a scalar field called the Higgs field. The massive quantum excitation of the Higgs field is also called the Higgs boson.

Actually, this mechanism was anticipated by Ernst Stückelberg in 1957 before Higgs. See the Stückelberg action for more details.

The breakthrough of Higgs (which was independently discovered by Brout and Englert at the Université Libre de Bruxelles) was to give mass to a vector boson, a.k.a. gauge boson, by coupling it to a scalar field. This was done in the context of a spontaneous symmetry breaking model, of the type constructed by Yoichiro Nambu and others in an attempt to explain the strong interactions. (These sorts of models were also inspired by work in condensed matter theory, notably by Lev Landau and Vitaly Ginzburg).

For an example of spontaneous symmetry-breaking, imagine a complex scalar field whose value at each point in space is

[itex]H(x,y,z).[itex]

Consider giving the field a potential energy of the form

[itex]V(x,y,z) = \Big(|H(x,y,z)|^2 - v^2\Big)^2[itex]

integrated over space. It is non-negative, and there is a continuous manifold of minima at

[itex]|H|^2 = v^2.[itex]

What this means in less technical terms is that the potential energy density, as a function of [itex]H[itex], looks like the bottom of a wine bottle: a hump in the middle and a circular valley around it. (One visualizes the complex field value as a 2-dimensional plane, the Argand diagram, and the potential as the height above the plane.)

The point [itex]H = 0[itex] is symmetric with respect to the U(1) symmetry that changes the complex phase of [itex]H[itex] as

[itex]H \rightarrow e^{i \theta}H[itex]

(and more generally, with respect to SU(2) x U(1) electroweak symmetry, for example), it is disfavored energetically. The Higgs field will roll "down the hill" and settle to a stable value

[itex]H=v e^{i \phi}[itex]

for some randomly chosen value of [itex]\phi[itex]. This induces an asymmetry of the vacuum, in the sense that the ground state is not invariant under the U(1) symmetry, which transforms one value of [itex]\phi[itex] to a different one.

The problem in using a spontaneous symmetry-breaking model in particle physics is that, according to a theorem of Jeffrey Goldstone, it predicts a massless scalar particle, which is the quantum excitation along the direction of [itex]\phi[itex], a so-called Nambu-Goldstone boson. There is no potential energy cost to move around the bottom of the circular valley, so the energy of such a particle is pure kinetic energy, which in quantum field theory implies that its mass is zero. But no massless scalar particles were detected.

A similar problem in Yang-Mills theory, a.k.a. nonabelian gauge theory, was the existence of massless gauge bosons, which (apart from the photon) were also not detected. It was Higgs' insight that when you combined a gauge theory with a spontaneous symmetry-breaking model, the two problems solved themselves rather elegantly. Higgs had found a loophole in the Goldstone theorem: when you couple the scalar to the gauge theory, the massless [itex]\phi[itex] mode of the Higgs combines with the vector boson to form a massive vector boson.

Higgs' original article presenting the model was rejected by Physical Review Letters when first submitted, apparently because it didn't predict any new detectable effects. So he added a sentence at the end, mentioning that it implies the existence of one or more new, massive scalar bosons, which don't form complete representations of the symmetry. These are the Higgs bosons.

Before the symmetry-breaking, all elementary particles (except the Higgs boson itself) are massless and the symmetry is unbroken, much like the rotational symmetry of a pencil that stands on its tip. However, the scalar field spontaneously slides from the point of maximum energy in a randomly chosen direction into a minimum - much like the pencil that eventually falls. This implies that the original symmetry is broken and elementary particles - such as the leptons, quarks, W boson, and Z boson acquire nonzero masses. The origin of the masses can be interpreted as a result of the interactions of the other particles with the "Higgs ocean".

The Higgs mechanism was incorporated into modern particle physics by Steven Weinberg and is an essential part of the Standard Model.

A slightly more technical presentation of the Higgs mechanism, which presumes at least an elementary knowledge of quantum field theory, is reviewed in the article on the Yukawa interaction.

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