⚛️Particle Physics Unit 6 – Higgs Mechanism and Symmetry Breaking

Fundamental Concepts

• Quantum field theory (QFT) provides a framework for describing fundamental particles and their interactions
• In QFT, particles are excitations of underlying fields that permeate all of spacetime
• Gauge symmetries play a crucial role in the Standard Model of particle physics (electroweak and strong interactions)
• Gauge bosons mediate the fundamental forces (photons for electromagnetic, gluons for strong, W and Z bosons for weak)
• Higgs mechanism explains the origin of mass for elementary particles through interactions with the Higgs field
• Spontaneous symmetry breaking occurs when a system's lowest energy state does not exhibit the full symmetry of its equations

Symmetry in Physics

• Symmetries are transformations that leave the physical laws unchanged (translation, rotation, reflection)
• Noether's theorem links continuous symmetries to conservation laws (energy, momentum, charge)
  • Time translation symmetry corresponds to energy conservation
  • Spatial translation symmetry corresponds to momentum conservation
• Gauge symmetries describe the invariance of a system under local transformations of fields
• Gauge theories have been incredibly successful in describing the fundamental interactions (electromagnetism, weak and strong forces)
• Symmetry breaking can occur when a system's ground state does not respect the full symmetry of its Lagrangian or Hamiltonian

Spontaneous Symmetry Breaking

• Spontaneous symmetry breaking (SSB) happens when a system's lowest energy state does not exhibit the full symmetry of its equations of motion
• In SSB, the system "chooses" one of the possible degenerate ground states, breaking the symmetry
• Examples of SSB include:
  • Ferromagnetism (spins align in a particular direction below Curie temperature)
  • Superconductivity (formation of Cooper pairs breaks U(1) gauge symmetry)
• Goldstone theorem states that SSB of a continuous symmetry leads to massless scalar particles called Goldstone bosons
• Higgs mechanism is a special case of SSB where the Goldstone bosons are "eaten" by gauge fields, giving them mass

The Higgs Field

• The Higgs field is a scalar field that permeates all of spacetime and interacts with certain particles
• It has a non-zero value everywhere, even in the vacuum state
• The Higgs field has a Mexican hat potential with a circle of minimum energy states
  • This shape allows for spontaneous symmetry breaking
• Excitations of the Higgs field are called Higgs bosons, which are massive scalar particles
• The Higgs field couples to the W and Z bosons, as well as fermions (quarks and leptons), giving them mass
• The strength of a particle's interaction with the Higgs field determines its mass

The Higgs Mechanism Explained

• The Higgs mechanism is a process by which certain particles acquire mass through interactions with the Higgs field
• It involves the spontaneous symmetry breaking of the electroweak gauge symmetry SU(2)_L × U(1)_Y
• Before symmetry breaking, the W and Z bosons are massless, and the electromagnetic and weak forces are unified
• The Higgs field has a non-zero vacuum expectation value (VEV), which breaks the electroweak symmetry
• The Goldstone bosons associated with the broken symmetry are "eaten" by the W and Z bosons, giving them mass
  • This process is called the "Higgs mechanism" or "Higgs-Kibble mechanism"
• The photon remains massless because the U(1)_EM symmetry is unbroken
• Fermions (quarks and leptons) also acquire mass through Yukawa interactions with the Higgs field

Particle Mass Generation

• The Higgs mechanism generates mass for the W and Z bosons, as well as fermions (quarks and leptons)
• The mass of a particle is proportional to its coupling strength to the Higgs field
  • Stronger coupling results in a larger mass
• The Higgs VEV, denoted as v, is approximately 246 GeV
• W boson mass: $m_W = \frac{1}{2}gv$, where g is the weak coupling constant
• Z boson mass: $m_Z = \frac{1}{2}\sqrt{g^2 + g'^2}v$, where g' is the hypercharge coupling constant
• Fermion masses: $m_f = \frac{y_f}{\sqrt{2}}v$, where $y_f$ is the Yukawa coupling for the fermion f
• The hierarchy of fermion masses (from neutrinos to the top quark) arises from their different Yukawa couplings

Experimental Discovery

• The Higgs boson was the last undiscovered particle of the Standard Model
• Its existence was predicted in the 1960s by Peter Higgs, Francois Englert, and others
• The search for the Higgs boson was a major goal of the Large Hadron Collider (LHC) at CERN
• In July 2012, the ATLAS and CMS collaborations announced the discovery of a new particle consistent with the Higgs boson
  • The discovered particle had a mass of approximately 125 GeV
• Further measurements confirmed that the new particle's properties matched those predicted for the Higgs boson
• The discovery of the Higgs boson completed the Standard Model and confirmed the mechanism for particle mass generation

Implications and Future Research

• The discovery of the Higgs boson was a triumph for the Standard Model and the Higgs mechanism
• It confirmed the existence of the Higgs field and its role in generating particle masses
• The Higgs boson's properties (mass, spin, couplings) are being studied in detail to test the Standard Model predictions
• Precision measurements of the Higgs boson's couplings could reveal deviations from the Standard Model, hinting at new physics
• The Higgs field's potential and its stability at high energies have implications for the fate of the universe
• Extensions of the Standard Model (supersymmetry, composite Higgs) predict additional Higgs bosons or modify the Higgs sector
• Future colliders (HL-LHC, ILC, FCC) will continue to study the Higgs boson and search for new phenomena related to the Higgs field