Anderson–Higgs mechanism

Anderson–Higgs mechanism
Cush · Public domain · source
Name	Anderson–Higgs mechanism
Field	Quantum field theory, Particle physics, Condensed matter physics
Discovered by	Philip W. Anderson, Peter Higgs, François Englert, Robert Brout, Gerald Guralnik, C. R. Hagen, Tom Kibble
Year	1962–1964
Related concepts	Spontaneous symmetry breaking, Gauge theory, Goldstone boson, Higgs boson, Standard Model

Anderson–Higgs mechanism. It is a fundamental process in quantum field theory whereby a gauge boson acquires mass through its interaction with a scalar field that has undergone spontaneous symmetry breaking. The mechanism is central to the Standard Model of particle physics, explaining the origin of mass for the W and Z bosons, and has profound analogies in condensed matter physics. Its prediction was confirmed by the discovery of the Higgs boson at the Large Hadron Collider.

Overview

The Anderson–Higgs mechanism provides a solution to the problem of giving mass to force carrier particles within a relativistic quantum field theory while preserving the underlying gauge invariance. In essence, when a scalar field permeating space acquires a non-zero vacuum expectation value, it breaks a symmetry of the system. The would-be massless Nambu–Goldstone boson from this breaking is "eaten" by a gauge boson, which becomes massive, a process sometimes called the "Higgs–Kibble mechanism". This framework is indispensable in the electroweak theory developed by Sheldon Glashow, Abdus Salam, and Steven Weinberg, and has parallels in phenomena like superconductivity described by the Ginzburg–Landau theory.

Theoretical basis

The theoretical foundation lies in the marriage of spontaneous symmetry breaking with Yang–Mills theory. In a symmetric state, gauge bosons are massless, as required by Lorentz invariance and gauge symmetry. Introducing a complex scalar field with a Mexican hat potential allows for a ground state that is not symmetric. The choice of a specific vacuum state breaks the symmetry, generating massless excitations along the flat direction of the potential, as per Goldstone's theorem. However, when the broken symmetry is a local gauge symmetry, these excitations become unphysical degrees of freedom that can be transformed away, or "gauged out", conferring mass on the gauge boson. This resolves the conflict between gauge invariance and massive force carriers.

Mathematical formulation

Mathematically, the simplest illustration is the Abelian Higgs model, a U(1) gauge theory. The Lagrangian density couples a complex scalar field φ to a gauge field A_μ via the minimal coupling prescription, with a potential V(φ) = μ²|φ|² + λ|φ|⁴. For μ² < 0, the potential minimum occurs at |φ| = v/√2, where v = √(−μ²/λ) is the vacuum expectation value. Expanding the field around this vacuum, φ(x) = (v + h(x))/√2 e^iθ(x), and choosing the unitary gauge (θ=0) absorbs the Goldstone mode θ into a redefinition of A_μ. The resulting Lagrangian shows a mass term (½ (ev)² A_μA^μ) for the gauge field, where e is the coupling constant, and a massive Higgs field h remains. The non-Abelian generalization in the Standard Model involves the SU(2) × U(1) gauge group and a complex Higgs doublet.

Physical consequences

The primary physical consequence is the generation of mass for the weak gauge bosons—the W^± and Z boson—which mediates the weak interaction, thereby explaining its short range. The photon of quantum electrodynamics remains massless as it corresponds to the unbroken U(1) symmetry of electromagnetism. The mechanism also gives mass to fundamental fermions like quarks and leptons via Yukawa couplings to the Higgs field. In condensed matter physics, an analogous effect occurs in superconductors described by BCS theory, where the photon inside the material behaves as if it has mass, leading to the Meissner effect and the expulsion of magnetic fields.

Experimental evidence

Indirect evidence accumulated for decades through precise measurements of weak interaction parameters at facilities like LEP at CERN and the SLC, which confirmed the massive nature of the W and Z bosons and their consistency with the Standard Model. The conclusive direct evidence was the discovery of the Higgs boson itself, announced on July 4, 2012, by the ATLAS and CMS collaborations at the Large Hadron Collider. The observed particle's properties—its spin, parity, and interactions—match the predictions for the Standard Model Higgs boson, cementing the validity of the mechanism. This discovery led to the Nobel Prize in Physics in 2013 being awarded to Peter Higgs and François Englert.

Historical development

The concept has roots in condensed matter physics. In 1962, Philip W. Anderson, in work on superconductivity and plasma physics, suggested that in a relativistic context, a plasmon could become massive, akin to a gauge boson acquiring mass. The fully relativistic quantum field theory version was independently developed in 1964 by three groups: François Englert and Robert Brout at the Université libre de Bruxelles; Peter Higgs at the University of Edinburgh; and Gerald Guralnik, C. R. Hagen, and Tom Kibble at Imperial College London. Higgs was the first to explicitly note the surviving massive scalar particle. The mechanism's incorporation into the electroweak theory by Steven Weinberg and Abdus Salam in 1967, building on Sheldon Glashow's work, established it as the cornerstone of the Standard Model. Category:Quantum field theory Category:Particle physics Category:Condensed matter physics Category:Mechanisms (physics)