Properties of Elementary Particles

and of Elementary s 01/11/2018 My Office Hours: Thursday 1:00-3:00 PM 212 Keen Building

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Consider the world at different scales... Cosmology - only gravity matters XXXXX Input: Mass distributions Human scale - primarily E & M (also gravity matters) Atomic scale - primarily E & M XXXXX Input: charge(s) / mass(es) of nuclei / electron, XXXXX Input: life times and decay modes scale - E & M, strong & weak interaction XXXXX Input: charge / mass of electron / nucleons physics - E & M, strong & weak interaction XXXXX Input: masses, particles, couplings Each model explains how, the smaller-scale theory gives why. Example: Newton s Law of Gravitation describes motion, but it is Relativity that explains the details.

1 Two spin- 1 2 families of fermions (leptons, quarks) 2 One spin-1 family of bosons + spin-0 Higgs Charge + 2 3 1 3-1 0

Forces: electromagnetic (γ), strong (g), weak (W ±, Z 0 ), gravitational (graviton) interaction - not part of SM Charge + 2 3 1 3-1 0

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The magnetic moment of a magnet is a quantity that determines the torque it will experience in an external magnetic field: τ = µ B Torque on a current-carrying loop: τ = I A B sin φ with I A = µ Remember: L = I ω = (mr 2 ) ( 2π t ) (for a point charge moving in a circle): I A = Q t πr 2 = Q t 2m e 2m e πr 2 = Q 2m e L XX Electron: µ = q/2m e L

& For spin: µ = q/2m e S (according to the classical theory).

s have properties: mass, electric charge, spin (permanent ang. momentum),... 1924: Experimental evidence of e spin (spin-magn. moment) 1924: Two orientations relative to external magnetic field. 1924: : S z = m S ( = h/2π) with (2m S +1) orientations Stern-Gerlach Experiment

& For spin: µ = q/2m e S (according to the classical theory). However: Stern-Gerlach experiment involves sending a beam of particles through an inhomogeneous magnetic field and observing their deflection. Results show that particles possess intrinsic angular momentum that is closely analogous to the angular momentum of a classically-spinning object, 1 But that takes only certain quantized values, 2 And is off by a proportional factor for the spin-magnetic moment: µ = g e 2m e S = g µb S with the Bohr magneton (defined in SI units) µ B = e 2m e.

I Existence of spin ang. momentum is inferred from experiments. is like a vector quantity, it has: 1 Definite magnitude, 2 Direction, spin orientation. has peculiar properties different from orbital ang. mom.: quantum numbers (QN) may take half-integer values. Direction of spin can be changed but a particle cannot be made to spin faster or slower. of charged particle associated with a magnetic dipole moment: µ = g S q 2m S with g s = spin g factor Classically, g S 1 only if mass & charge fill volumes with different radii.

II Existence of spin ang. momentum is inferred from experiments. is like a vector quantity, it has: 1 Definite magnitude, 2 Direction, spin orientation. For electrons: 1 S 2 = s (s + 1) 2 = 3/4 2 for spin S = 1/2. 2 S z = m s with m S = ±1/2 (for electrons in units of ). µ Sz = µ B = q 2m and µ S = 3 µ B XXXXXXXXXXXXXXXXxx (total spin magnetic moment)

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Bosons & Fermions I Fermions are particles with half-integer spin: m S = n 2, n = odd. Obey Fermi-Dirac statistics and the Pauli exclusion principle. (No two fermions in a system can have the same QNs.)

Bosons & Fermions II Bosons are particles with integer spin: m S = n. Obey Bose-Einstein statistics. (More than one boson in a system can have the same QNs.)

Quarks form particles called hadrons: 1 Meson: quark-antiquark pair (q q) π, η, etc. 2 Baryon: three-quark state (qqq) p, n,, etc.

Back to spin... Quarks form particles called hadrons: 1 Meson: quark-antiquark pair (q q) π, η, etc. 2 Baryon: three-quark state (qqq) p, n,, etc. Charged particles with spin have intrinsic magnetic moment: µ S = (q/m) S Dirac equation describes point-like spin- 1 2 particle (q, m): µ z = q 2m = g S µ B m S µ B (Bohr magneton for electron) In general: XX g S 2 for electron XX g S 3.83 for neutron XX g S +5.59 for proton

Dirac Equation Dirac equation yields four solutions for an electron: Two solutions describe the two spin states. But what are the other two solutions? Relativistic energy-momentum relation: E 2 = m 2 c 4 + p 2 c 2 and thus, E = ± m 2 c 4 + p 2 c 2 The other two energies correspond to negative energies!! What does negative energy mean? Equivalent to +E solutions for particles with opposite charge: XXX antiparticles Q Q