Abelian Higgs model

Abelian Higgs model
Name	Abelian Higgs model
Classification	Quantum field theory
Related	Higgs mechanism, Ginzburg–Landau theory, Superconductivity
Theorized	Peter Higgs, François Englert, Robert Brout, Philip Warren Anderson
Year	1964

Abelian Higgs model. The Abelian Higgs model is a relativistic quantum field theory that provides a fundamental framework for understanding spontaneous symmetry breaking and the generation of mass for gauge bosons. It serves as a simplified, U(1) gauge-invariant prototype for the more complex electroweak interaction mechanism in the Standard Model of particle physics. The model's mathematical structure is closely analogous to the phenomenological Ginzburg–Landau theory of superconductivity, bridging concepts in condensed matter physics and high-energy theory.

Introduction

The model was developed in the 1960s following groundbreaking work by Philip Warren Anderson in condensed matter and the independent proposals by Peter Higgs, François Englert, and Robert Brout. It represents a crucial step in the theoretical understanding of how gauge symmetries can be hidden, leading to massive force carriers, a concept essential for the Weinberg–Salam model. Its historical significance is underscored by the 2013 Nobel Prize in Physics awarded for the related discovery of the Higgs boson at the Large Hadron Collider.

Mathematical formulation

The dynamics are defined by a Lagrangian density coupling a complex scalar field to an Abelian gauge field, typically a U(1) connection. This Lagrangian is invariant under local U(1) gauge transformations, a symmetry fundamental to theories like quantum electrodynamics. The scalar field potential is chosen to have the characteristic "Mexican hat" shape, a feature central to the Landau theory of phase transitions. The minimal coupling between the fields is achieved via the covariant derivative, introducing an interaction term reminiscent of the coupling in the Proca action.

Spontaneous symmetry breaking

In the model's vacuum state, the scalar field acquires a non-zero vacuum expectation value, a process that spontaneously breaks the original U(1) gauge symmetry. This breaking is not explicit in the Lagrangian but arises from the ground state solution, analogous to the ferromagnetic phase in the Ising model. The choice of a specific vacuum value from a continuous manifold of degenerate states is an example of the Goldstone theorem, which predicts a massless Nambu–Goldstone boson. However, due to the gauge theory nature, this boson is absorbed in a distinctive manner.

Higgs mechanism

The would-be Nambu–Goldstone boson from symmetry breaking becomes the longitudinal polarization component of the gauge field, a process known as the Higgs mechanism. This grants mass to the gauge boson, transforming it from a massless particle like the photon into a massive one, similar to the W and Z bosons in the Standard Model. The remaining degree of freedom from the complex scalar field manifests as a massive spin-zero particle, the Higgs boson. This mechanism resolves the apparent conflict between gauge invariance and massive force carriers that troubled earlier theories like the Fermi theory of beta decay.

Physical implications

The model provides a quantum field theoretic description of the Meissner effect and the emergence of a finite penetration depth for magnetic fields in a superconductor, concepts formalized in the Ginzburg–Landau theory. In particle physics, it demonstrates how the Yukawa interaction can generate fermion masses when coupled to the Higgs field. The existence of topological soliton solutions, such as Abrikosov vortices or cosmic strings, has implications for both condensed matter systems and early universe cosmology following the Big Bang.

Extensions and related models

The non-Abelian generalization leads directly to the Glashow–Weinberg–Salam model of electroweak unification, which incorporates the SU(2) and U(1) gauge groups. The Higgs sector of the Minimal Supersymmetric Standard Model extends the framework with additional doublets. In the context of grand unified theories like SU(5) or cosmic inflation and is studied in the limit of strong coupling through techniques like lattice gauge theory simulations.

Category:Quantum field theory Category:Gauge theories Category:Higgs boson Category:Condensed matter physics