2.3 Symmetries and conservation laws in particle physics

Symmetries and conservation laws are the backbone of particle physics. They explain why certain interactions occur and others don't, shaping our understanding of the universe's fundamental workings. From energy conservation to charge preservation, these principles guide particle behavior and reactions.

The Standard Model, built on symmetry principles, unifies fundamental forces and particles. Through symmetry breaking and the Higgs mechanism, it explains particle masses and interactions. Understanding these concepts is crucial for grasping the deeper structure of the physical world.

Symmetries in Particle Physics

Fundamental Concepts of Symmetries

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• Symmetries in particle physics manifest as transformations leaving physical laws unchanged
• Noether's theorem connects continuous symmetries to conservation laws
  • Each continuous symmetry corresponds to a conserved quantity
• Principle of least action intertwines with symmetries and conservation laws
  • Path of physical system has stationary action
• Two main categories of symmetries exist in particle physics
  • Continuous symmetries (translations, rotations)
  • Discrete symmetries (parity, charge conjugation, time reversal)

Symmetries in the Standard Model

• Standard Model built on local gauge invariance principle
  • Laws of physics remain symmetric under certain transformations at each spacetime point
• Symmetry breaking mechanisms explain particle masses and force unification
  • Spontaneous symmetry breaking plays crucial role
• Gauge group SU(3) × SU(2) × U(1) describes fundamental interactions
  • SU(3) for strong force (Quantum Chromodynamics)
  • SU(2) × U(1) for electroweak force (unification of electromagnetic and weak forces)

Conservation Laws in Interactions

Energy and Momentum Conservation

• Energy conservation stems from time translation symmetry
• Momentum conservation arises from space translation symmetry
• These conservation laws govern all particle interactions
  • Determine allowed reaction products
  • Enable calculation of reaction rates and cross-sections

Charge and Quantum Number Conservation

• Electric charge conservation linked to global U(1) symmetry of electromagnetism
  • One of the most rigorously tested conservation laws
• Baryon number conservation explains proton stability
  • Total number of quarks minus antiquarks remains constant
  • Absence of proton decay (hydrogen atoms)
• Lepton number conservation governs lepton interactions
  • Total number of leptons minus antileptons stays constant
  • Allows muon decay ($\mu^- \rightarrow e^- + \bar{\nu}_e + \nu_\mu$)
  • Forbids processes like $\mu^- \rightarrow e^- + \gamma$

Angular Momentum and Flavor Conservation

• Angular momentum conservation related to space isotropy
  • Manifests in spin and orbital angular momentum conservation
• Flavor quantum numbers (strangeness, charm, beauty) approximately conserved
  • Preserved in strong and electromagnetic interactions
  • Can be violated in weak interactions (kaon decays)

Gauge Symmetries and the Standard Model

Gauge Symmetry Principles

• Gauge symmetries require gauge fields for local transformation invariance
• Local gauge invariance introduces gauge bosons as force carriers
  • Gluons for strong force
  • W and Z bosons for weak force
  • Photons for electromagnetic force

Quantum Chromodynamics and Electroweak Theory

• Quantum Chromodynamics based on SU(3) color gauge symmetry
  • Explains quark confinement
  • Predicts existence of color-neutral hadrons (protons, neutrons)
• Electroweak theory unifies electromagnetic and weak interactions
  • Based on SU(2) × U(1) gauge symmetry
  • Predicts existence of Higgs boson

Spontaneous Symmetry Breaking and Higgs Mechanism

• Spontaneous symmetry breaking in gauge theories explains gauge boson mass
  • Preserves local gauge invariance
• Higgs mechanism gives mass to W and Z bosons
  • Leaves photon massless
• Provides framework for understanding particle masses and interactions

Discrete Symmetries and Their Consequences

Parity and Charge Conjugation

• Parity (P) symmetry involves spatial inversion invariance
  • Violated in weak interactions
  • Leads to concept of left-handed and right-handed particles (neutrinos)
• Charge conjugation (C) symmetry relates particles to antiparticles
  • Also violated in weak interactions
  • Contributes to matter-antimatter asymmetry (baryon asymmetry of the universe)

Time Reversal and CPT Symmetry

• Time reversal (T) symmetry concerns time direction invariance
  • Violated in rare weak decay processes (K-meson decays)
• Combined CPT symmetry believed to be exact in nature
  • Implies identical masses and lifetimes for particles and antiparticles

Experimental Tests and Implications

• CP violation discovered in neutral kaon decays
  • Led to prediction and observation in B meson decays
  • Provides insights into matter-antimatter asymmetry
• V-A theory of weak interactions developed from discrete symmetry analysis
• Searches for electric dipole moments probe physics beyond Standard Model
  • Sensitive tests of fundamental symmetries (electron EDM experiments)