🔬Quantum Field Theory Unit 10 – Introduction to the Standard Model

What's the Standard Model?

• Theoretical framework describing the fundamental building blocks of matter and their interactions
• Combines quantum mechanics and special relativity to explain the behavior of subatomic particles
• Developed in the 1970s through collaborative efforts of physicists worldwide
• Includes three of the four fundamental forces (electromagnetic, weak, and strong interactions) but not gravity
• Organizes elementary particles into categories based on their properties and roles in interactions
• Highly successful in predicting and explaining experimental results with remarkable precision
• Provides a unified description of the electromagnetic and weak interactions through the electroweak theory
• Describes the strong interaction between quarks and gluons using quantum chromodynamics (QCD)

Fundamental Particles and Forces

• Matter particles are fermions with half-integer spin (quarks and leptons)
  • Quarks: up, down, charm, strange, top, bottom; participate in strong interactions
  • Leptons: electron, muon, tau, and their corresponding neutrinos; do not participate in strong interactions
• Force-carrying particles are bosons with integer spin (photon, W and Z bosons, gluons)
  • Photon: mediates electromagnetic interactions
  • W and Z bosons: mediate weak interactions
  • Gluons: mediate strong interactions between quarks
• Higgs boson is a scalar particle with zero spin; gives mass to other particles through the Higgs mechanism
• Fundamental forces have different strengths and ranges
  • Electromagnetic: infinite range, 1/137 relative strength
  • Weak: short range (~10^-18 m), 10^-5 relative strength
  • Strong: short range (~10^-15 m), 1 relative strength

Symmetries and Conservation Laws

• Symmetries play a crucial role in the Standard Model and lead to conservation laws
• Noether's theorem connects continuous symmetries to conserved quantities
• Gauge symmetries (local symmetries) are the foundation of the Standard Model
  • U(1) gauge symmetry for electromagnetism
  • SU(2) gauge symmetry for weak interactions
  • SU(3) gauge symmetry for strong interactions
• Global symmetries include:
  • Parity (P): spatial reflection symmetry; violated by weak interactions
  • Charge conjugation (C): particle-antiparticle symmetry; violated by weak interactions
  • Time reversal (T): reversal of motion; believed to be a fundamental symmetry of nature
  • CPT theorem states that the combination of C, P, and T is an exact symmetry of any Lorentz-invariant quantum field theory
• Approximate symmetries (chiral symmetry, isospin symmetry) provide insights into particle properties and interactions

Quantum Fields and Particle Interactions

• Particles are excitations of underlying quantum fields that permeate all of spacetime
• Interactions between particles are described by the exchange of virtual particles (force carriers)
• Feynman diagrams visually represent the mathematical expressions for particle interactions
  • External lines represent initial and final state particles
  • Internal lines represent virtual particle exchange
  • Vertices represent interaction points where particles are created or annihilated
• Coupling constants determine the strength of interactions at each vertex
• Perturbation theory is used to calculate the probabilities of particle interactions as a series expansion in powers of the coupling constant
• Higher-order Feynman diagrams involve loop corrections and contribute to the renormalization of physical quantities

Gauge Theories and Local Symmetries

• Gauge theories are based on the principle of local gauge invariance
• Local gauge symmetries require the introduction of gauge fields (force-carrying particles) to maintain invariance
• Gauge fields ensure the consistency of the theory and give rise to interactions
• Gauge transformations change the phase of the wavefunction locally while leaving the physical observables unchanged
• Gauge fixing is necessary to remove redundant degrees of freedom and define a consistent quantization procedure
• Faddeev-Popov ghost fields are introduced to preserve the unitarity of the theory in non-Abelian gauge theories (such as QCD)
• BRST symmetry is a generalization of gauge symmetry that includes ghost fields and ensures the consistency of the quantized theory

Electroweak Theory

• Unifies electromagnetic and weak interactions into a single framework
• Based on the SU(2)_L × U(1)_Y gauge symmetry group
  • SU(2)_L represents weak isospin symmetry
  • U(1)_Y represents weak hypercharge symmetry
• Introduces the concept of chirality, distinguishing between left-handed and right-handed fermions
• Electroweak symmetry breaking occurs through the Higgs mechanism
  • Higgs field acquires a non-zero vacuum expectation value
  • W and Z bosons gain mass, while the photon remains massless
• Weinberg-Salam model successfully predicted the masses of the W and Z bosons before their experimental discovery
• Electroweak precision tests have confirmed the validity of the theory to a high degree of accuracy

Quantum Chromodynamics (QCD)

• Theory of strong interactions between quarks and gluons
• Based on the SU(3) gauge symmetry group, with color charge as the conserved quantity
• Quarks come in three colors (red, green, blue), and gluons carry a combination of color and anti-color
• Gluons are self-interacting, leading to the phenomena of color confinement and asymptotic freedom
  • Color confinement: quarks are always bound within color-neutral hadrons (mesons and baryons)
  • Asymptotic freedom: strong interaction becomes weaker at high energies or short distances
• Running coupling constant of QCD decreases with increasing energy scale
• Perturbative QCD describes high-energy processes involving quarks and gluons
• Lattice QCD is a non-perturbative approach to solving QCD equations numerically on a discrete spacetime lattice

Limitations and Beyond the Standard Model

• Does not incorporate gravity, which is described by general relativity
• Neutrino masses and oscillations are not explained within the Standard Model framework
• Matter-antimatter asymmetry in the universe cannot be accounted for by the known CP violation in the Standard Model
• Dark matter and dark energy, which make up a significant portion of the universe's energy content, are not addressed
• Hierarchy problem: large discrepancy between the weak scale and the Planck scale, requiring fine-tuning of parameters
• Theories beyond the Standard Model aim to address these limitations:
  • Grand Unified Theories (GUTs): unify strong, weak, and electromagnetic interactions at high energies
  • Supersymmetry (SUSY): introduces a symmetry between fermions and bosons, providing a dark matter candidate and addressing the hierarchy problem
  • String theory: attempts to unify all forces, including gravity, by describing particles as vibrations of fundamental strings
• Ongoing experimental efforts (LHC, dark matter searches, neutrino experiments) aim to test and constrain theories beyond the Standard Model