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Physics › Quantum Field Theory

Higgs Mechanism

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Electromagnetic Field Quantization (QED)Goldstone Theorem→→Electroweak UnificationHiggs Boson Discovery and Properties+1 more
higgs mass-generation gauge-boson-mass

Core Idea

The Higgs mechanism generates masses for gauge bosons through spontaneous breaking of a gauge symmetry. The would-be Goldstone bosons are "eaten" by the gauge bosons, becoming their longitudinal polarization components. The gauge bosons acquire mass while the theory remains renormalizable. The physical Higgs boson is the remaining massive scalar excitation.

Explainer

The Higgs mechanism is the process by which gauge bosons acquire mass through spontaneous symmetry breaking, without destroying gauge invariance or renormalizability. The simplest example is the abelian Higgs model: a U(1) gauge field A_mu coupled to a complex scalar phi with a Mexican hat potential. The Lagrangian is L = -1/4 F2 + |D_mu phi|^2 - V(phi), where D_mu = partial_mu - ieA_mu is the covariant derivative and V = -mu2|phi|^2 + lambda|phi|^4.

When phi acquires a vacuum expectation value <phi> = v/sqrt(2), the covariant derivative term |D_mu phi|^2 evaluated at the vacuum generates e2 v2 A_mu Amu / 2 -- a mass term for the gauge field with m_A = ev. The angular degree of freedom of phi (the would-be Goldstone boson) is absent from the physical spectrum in unitary gauge; it has been absorbed into the gauge field as its longitudinal polarization. The radial fluctuation remains as a massive scalar particle -- the Higgs boson with mass m_H = sqrt(2 lambda) v.

In the Standard Model, the electroweak gauge symmetry SU(2)_L x U(1)_Y is broken to U(1)_EM by a complex scalar doublet (four real components). Three Goldstone bosons are eaten by the W+, W-, and Z bosons, giving them masses. The fourth component remains as the physical Higgs boson, discovered at the LHC in 2012 with mass 125 GeV. The vacuum expectation value v = 246 GeV is fixed by the measured Fermi constant. The W and Z masses are then predictions: m_W = gv/2 approximately 80 GeV and m_Z approximately 91 GeV, in excellent agreement with experiment.

Fermion masses are also generated through the Higgs mechanism. Direct mass terms m psi-bar psi are forbidden by the chiral structure of the electroweak interaction (left- and right-handed fermions transform differently under SU(2)). Instead, Yukawa couplings y psi-bar_L phi psi_R connect the fermion fields to the Higgs doublet. When phi gets its vacuum expectation value, these become mass terms m_f = y_f v/sqrt(2). Each fermion's mass is proportional to its Yukawa coupling, which is a free parameter. The proof by 't Hooft and Veltman that theories with the Higgs mechanism are renormalizable was the theoretical foundation for the Standard Model.

Practice Questions 4 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → One-to-One Correspondence → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Making 10 as an Addition Strategy → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Iterated Integrals and Fubini's Theorem → Double Integrals in Cartesian Coordinates → Double Integrals in Polar Coordinates → Double Integrals in Polar Coordinates → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Triple Integrals in Cartesian Coordinates → Triple Integrals in Cylindrical and Spherical Coordinates → Change of Variables and the Jacobian Determinant → Applications of Triple Integrals: Volume and Mass → Vector Fields and Their Representations → Line Integrals of Vector Fields → Work and Circulation → Line Integrals of Scalar and Vector Functions → Fundamental Theorem for Line Integrals → Conservative Vector Fields → Conservative Vector Fields and Potential Functions → Curl and Divergence of Vector Fields → Curl and Divergence → Divergence Theorem → Electric Flux and Divergence Theorem → Gauss's Law: Integral Form and Meaning → Solving Problems with Gauss's Law → Conductors in Electrostatic Equilibrium → Capacitance and Capacitors → Dielectrics → Dielectric Constant and Relative Permittivity → Electric Field Inside Dielectric Materials → Dielectric Materials and Polarization → Dielectric Susceptibility and Permittivity → Energy Density in Electric Fields → Electric Current and Current Density → Electrical Resistance and Resistivity → Ohm's Law and Circuit Elements → Electromotive Force (EMF) and Batteries → Kirchhoff's Circuit Laws: Voltage and Current → DC Circuit Network Analysis Methods → Transient Response in RC Circuits → RC Circuits → LC and RLC Circuits → AC Circuits: Fundamentals → Impedance and Reactance → AC Power and Resonance → Electromagnetic Waves → Postulates of Special Relativity → Time Dilation → Length Contraction → Lorentz Transformation → Relativistic Velocity Addition → Relativistic Momentum and Energy → Mass-Energy Equivalence and E=mc² → Photons as Particles with Energy and Momentum → Planck-Einstein Relation: Energy and Frequency → Photoelectric Effect → The Photon: Light as Quanta → Compton Scattering → Wave-Particle Duality → de Broglie Wavelength → The Schrödinger Equation → State Vectors and Wavefunctions → Quantum Superposition → The Measurement Problem → Interpretations of Quantum Mechanics → Postulates of Quantum Mechanics → Observables and Quantum Operators → Commutators and Commutation Relations → Quantum Angular Momentum → Quantum Mechanical Treatment of Hydrogen → Solving the Schrödinger Equation for Hydrogen Atom → Quantum Numbers → Electron Configuration → Periodic Trends → Covalent Bonding → Electronegativity and Bond Polarity → Ionic Bonding → Lewis Structures → VSEPR Theory and Molecular Geometry → Molecular Geometry and Electron Pair Geometry → Molecular Polarity and Dipole Moments → Intermolecular Forces → States of Matter and Phase Changes: Melting, Boiling, and Sublimation → Gas Laws and the Ideal Gas Equation → Gas Stoichiometry and Volume-Volume Calculations → Thermochemistry and Enthalpy → Heat Capacity and Calorimetry → Entropy and Molecular Disorder → Spontaneity and ΔG → Entropy and Gibbs Free Energy → Chemical Equilibrium → Statistical Mechanics: Ensembles and the Boltzmann Distribution → Partition Function: Definition and Properties → Helmholtz Free Energy → Gibbs Free Energy → Phase Transitions: First Order and Second Order → Critical Phenomena and Critical Exponents → Landau Theory of Phase Transitions → Symmetry Breaking and Phase Transitions → Goldstone's Theorem and Gapless Modes → Goldstone Theorem → Higgs Mechanism

Longest path: 184 steps · 1245 total prerequisite topics

Prerequisites (2)

Goldstone Theoremhard Electromagnetic Field Quantization (QED)hard

Leads To (3)

Electroweak Unificationhard Higgs Boson Discovery and Propertieshard Standard Model Overviewhard