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# “Elementary Particles” Lecture 4

“Elementary Particles” Lecture 4. Niels Tuning Harry van der Graaf. Thanks. Ik ben schatplichtig aan: Dr. Ivo van Vulpen (UvA) Prof. dr. ir. Bob van Eijk (UT) Prof. dr. Marcel Merk (VU). Homework. Exercises Lecture 3:. Exercises Lecture 3: mass of Ω and decay of π 0.

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## “Elementary Particles” Lecture 4

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1. “Elementary Particles”Lecture 4 Niels Tuning Harry van der Graaf Niels Tuning (1)

2. Thanks • Ik ben schatplichtig aan: • Dr. Ivo van Vulpen (UvA) • Prof. dr. ir. Bob van Eijk (UT) • Prof. dr. Marcel Merk (VU) Niels Tuning (2)

3. Homework

4. Exercises Lecture 3: Niels Tuning (4)

5. Exercises Lecture 3: mass of Ω and decay of π0 π0→γ1γ2→e+e-e+e- 1232 MeV 1385 MeV 1533 MeV 1680 MeV Gell-Mann and Zweig predicted the Ω- Niels Tuning (5)

6. Exercises Lecture 3: Niels Tuning (6)

7. Exercises Lecture 3: (1) (3) (2) Niels Tuning (7)

8. Exercises Lecture 3: Niels Tuning (8)

9. Exercises Lecture 3: (a) (c) Niels Tuning (9)

10. Plan 11 Feb • Intro: Relativity and accelerators • Basis • Atom model, strong and weak force • Scattering theory • Hadrons • Isospin, strangeness • Quark model, GIM • Standard Model • QED • Parity, neutrinos, weak inteaction • QCD • e+e- and DIS • Higgs and CKM 1900-1940 18 Feb 1945-1965 4 Mar 1965-1975 18 Mar 1975-2000 22 Apr 2000-2013 13 May Niels Tuning (10)

11. Outlinefortoday: Interactions • Gauge invariance, and the Lagrangian • Electro-magnetic interaction • QED • Weak interaction • Parity violation • Strong interaction • QCD Niels Tuning (11)

12. Summary

13. Lecture 1: Accelerators & Relativity • Theory of relativity • Lorentz transformations (“boost”) • Calculate energy in collissions • 4-vector calculus • High energies needed to make (new) particles Niels Tuning (13)

14. Lecture 2: Quantum Mechanics & Scattering • Schrödinger equation • Time-dependence of wave function • Klein-Gordon equation • Relativistic equation of motion of scalar particles • Dirac equation • Relativistically correct, and linear • Equation of motion for spin-1/2 particles • Prediction of anti-matter Niels Tuning (14)

15. Lecture 2: Quantum Mechanics & Scattering • Scattering Theory • (Relative) probability for certain process to happen • Cross section • Fermi’s Golden Rule • Decay: “decay width” Γ • Scattering: “cross section” σ Scattering amplitude in Quantum Field Theory Classic a → b + c a + b → c + d Niels Tuning (15)

16. Lecture 3: Quarkmodel & Isospin • “Partice Zoo” not elegant • Hadrons consist of quarks • Observed symmetries • Same mass of hadrons: isospin • Slow decay of K, Λ: strangeness • Fermi-Dirac statistics Δ++,. Ω: color • Combining/decaying particles with (iso)spin • Clebsch-Gordan coefficients Niels Tuning (16)

17. Model elementary particles Tools Atom Strong force Mesons Leptons Quark model Baryons Quantum numbers Conservation Laws Standard Model Particles Forces Quantum- field theory & Local gauge invariance quarks/leptons ElectromagneticWeak Strong

18. Electro-Magnetism

19. Electro-magnetism • Towards a particle interacting with photon • Quantum Electro Dynamics, QED • Start with electric and magnetic fields J.C. Maxwell

20. Start: Classical electro-magnetism Maxwell equations • We wish to work relativistically • Can we formulate this in Lorentz covariant form? • Introduce a mathematical tool: Scalar potential also called φ:

21. Start: Classical electro-magnetism Maxwell equations • We wish to work relativistically • Can we formulate this in Lorentz covariant form? • Introduce a mathematical tool: • Note: • Choose: Scalar potential also called φ:

22. Start: Classical electro-magnetism Maxwell equations • We wish to work relativistically • Can we formulate this in Lorentz covariant form? • Introduce a mathematical tool: • Note: • Choose: Then automatically: Scalar potential also called φ:

23. Rewrite Maxwell Maxwell equations • Maxwell eqs. can then be written quite economically…:

24. Rewrite Maxwell Maxwell equations • Maxwell eqs. can then be written quite economically…: • Electromagnetic tensor • Unification of electromagnetism

25. Gauge Invariance

26. Gauge invariance: Classical Maxwell equations Physical Fields: Potentials: A = Vector potential ϕ= Scalar potential Invariant under: Gauge transformations Maxwell invariant  gauge symmetry

27. Gauge invariance: Classical Maxwell equations Physical Fields: Potentials: A = Vector potential ϕ= Scalar potential Invariant under: Gauge transformations Maxwell invariant  gauge symmetry

28. Not unique! Can choose extra constraints: Coulomb-gauge: Lorenz-gauge: Advantage: Lorentz-invariant

29. Not unique! Can choose extra constraints: Coulomb-gauge: Lorenz-gauge: (Ludvig) (Hendrik) Advantage: Lorentz-invariant Lorentz Lorenz Total confusion: Lorentz-Lorenz formula

30. Gauge invariance : QM Charged particle moving in electro-magnetic field: Classical A,φ Schrödinger eq. Pauli eq. (spin-1/2 in EM-field) Theory gauge invariant: (same physics with A, φ as with A’, φ’) ?

31. Rewrite and Wave equation (A,φ) Theory gauge invariant: (same physics with A, φ as with A’, φ’) ?

32. Rewrite and Wave equation (A,φ) Theory gauge invariant: (same physics with A, φ as with A’, φ’) ? Wave equation (A’,φ’) Yes! If

33. Local gauge symmetry Schrödinger equation (time-independent): Ψ’ stays solution of Schrödinger eq! Global phase:

34. Local gauge symmetry Schrödinger equation (time-independent): Ψ’ stays solution of Schrödinger eq! Global phase: Local phase: Ψ’ solution of Schrödinger eq?

35. Local gauge symmetry Schrödinger equation (time-independent): Ψ’ stays solution of Schrodinger eq! Global phase: Local phase: Ψ’ solution of Schrodinger eq? No: • (How) can you keep the Schrödinger equation invariant ?

36. Before going to Quantum Field Theory, lets remind ourselves of the Lagrange formalism

37. Lagrangian

38. Euler-Lagrange • Euler-Lagrange equations: • Least-action, or Hamilton’s principle: Niels Tuning (38)

39. Lagrangian and Equation of motion: Example 1) Lagrangiaan 2) Euler-Lagrange vergelijking Equation of motion q: generalized coordinates pendulum

40. Lagrangian in Field Theory • Replace Lagrangian by a Lagrangian density in terms of field φ(x): • Least-action principle: • Euler-Lagrange equations: (classic) Niels Tuning (40)

41. Lagrangian  Equation of motion • spin-0 particles (Klein-Gordon) • spin-1/2 fermions (Dirac) • Photons Klein-Gordon equation Dirac equation Maxwell equations Niels Tuning (41)

42. QED

43. Break?

44. Local gauge symmetry Schrödinger equation (time-independent): Ψ’ stays solution of Schrödinger eq! Global phase: Local phase: Ψ’ solution of Schrödinger eq? No: • (How) can you keep the Schrödinger equation invariant ?

45. Phase Invariance • Quantum Mechanics: • Expectation value: • Observation is invariant under transformation: • But what happens to the Lagrangian density, ? • Depends also on derivative..., so: Niels Tuning (45)

46. Phase Invariance • Quantum Mechanics: • Expectation value: • Observation is invariant under transformation: • But what happens to the Lagrangian density, ? • Depends also on derivative..., so: • Replace by “gauge-covariant derivative”: Niels Tuning (46)

47. Phase Invariance • Quantum Mechanics: • Expectation value: • Observation is invariant under transformation: • But what happens to the Lagrangian density, ? • Depends also on derivative..., so: • Replace by “gauge-covariant derivative”: • And with : Niels Tuning (47)

48. Gauge Invariance We started globally: • Assumesymmetryψ→ψ’=ψeiα(x) • Keep Eqsvalid • Covariantderivative • Arbitrarygauge • Keep Eqsvalid • Needψ→ψ’=ψeiα Then we went local: Niels Tuning (48)

49. Gauge Invariance We started globally: • Assumesymmetryψ→ψ’=ψeiα(x) • Keep Eqsvalid • Covariantderivative • Arbitrarygauge • Keep Eqsvalid • Needψ→ψ’=ψeiα Then we went local: Why would you want to do that? Niels Tuning (49)

50. Quantum Electro Dynamics - QED • Let’s replace the derivative in the Dirac equation by the gauge-covariant derivative: • Gauge invariance leads to: gauge fields, and their interactions! Niels Tuning (50)

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