VIEWS: 3 PAGES: 27 POSTED ON: 10/23/2012 Public Domain
Basic concepts Particle Physics Particle Physics Jørgen Beck Hansen Basic concepts Particle Physics Setting the scale Particle physics is Atto-physics Jørgen Beck Hansen Niels Bohr Institute 2 Basic concepts Particle Physics Basic concepts • Particle physics studies elementary “building blocks” of matter and interactions between them. • Matter consists of particles. – Matter is built of particles called “fermions”: those that have half-integer spin, e.g. 1/2 • Particles interact via forces. – Interaction = exchange of a force- carrying particle. • Force-carrying particles are called gauge bosons (integer spin). Jørgen Beck Hansen Niels Bohr Institute 3 Basic concepts Particle Physics Forces of nature Jørgen Beck Hansen Niels Bohr Institute 4 Basic concepts Particle Physics The Particle Physics Standard Model • Electromagnetic and weak forces can be described by a single theory -> the “Electroweak Theory” (EW) was developed in 1960s (Glashow, Weinberg, Salam). • Theory of strong interactions appeared in 1970s: “Quantum Chromodynamics” (QCD). • The “Standard Model” (SM) combines all the current knowledge. – Gravitation is VERY weak at particle scale, and it is not included in the SM. Moreover, quantum theory for gravitation does not exist yet. • Main postulates of SM: 1. Basic constituents of matter are quarks and leptons (spin 1/2) 2. They interact by exchanging gauge bosons (spin 1) 3. Quarks and leptons are subdivided into 3 generations Jørgen Beck Hansen Niels Bohr Institute 5 Basic concepts Particle Physics Interactons Standard model NOT perfect: • Origin of Mass? • Why 3 generations? Jørgen Beck Hansen Niels Bohr Institute 6 Basic concepts Particle Physics Particle Physics and the Universe Jørgen Beck Hansen Niels Bohr Institute 7 Basic concepts Particle Physics Tricks of the trade: UNITS and Dimensions • For everyday physics SI units are a natural choice • Not so good for particle physics: Mproton ~ 10-27 kg • Use a different basis - NATURAL UNITS Convert back to S.I. units by • Unit of energy : GeV = 109 eV = 1.602 x 10-10 J reintroducing ‘missing’ factors – 1 eV = Energy of e- passing a voltage of 1 V of ħ and c • Language of quantum mechanics and relativity, i.e. EXAMPLE: -2 – The reduced Planck constant and the speed of light: • Area = 1 GeV • ħ ≡ h/2 = 6.582 x 10-25 GeV s • [L]2 = [E]-2[ħ]n[c]m • c = 2.9979 x 108 m/s • [L]2 = [E]-2[E]n[T]n[L]m[T]-m – Conversion constant: ħc = 197.327 x 10-18 GeV m • Hence, n = 2 and m = 2 • Natural Units: GeV, ħ, c • Area = 1 GeV-2 x ħ2c2 • Units become Energy ► GeV Time ► (GeV/ħ)-1 Momentum ► GeV/c Length ► (GeV/ħc)-1 Mass ► GeV/c2 Area ► (GeV/ħc)-2 • For simplicity choose ħ=c=1 Jørgen Beck Hansen Niels Bohr Institute 8 Basic concepts Particle Physics Particle Physics language: 4-vectors Particles described by • Space-time 4-vector: x=(ct,x) where x is a normal 3-vector • Momentum 4-vector: p=(E/c,p) where p is particle momentum • 4-vector rules (recap) – a ± b = (a0 ± b0, a1 ± b1, a2 ± b2, a3 ± b3) – Scalar product (minus sign!) a⋅b=a0b0 – a1b1 – a2b2 – a3b3=a0b0 – a⋅b – Scalar product of momentum and space-time 4-vectors are thus: x⋅p=Et – xxpx – xypy – xzpz= Et – x⋅p Used in the Quantum Mechanical free particle wavefunction – 4-momentum squared gives particle’s invariant mass m2c2 ≡ p ⋅ p = E2 ⁄ c2 – p2 or E2 = p2c2 + m2c4 Quick formulas Jørgen Beck Hansen Niels Bohr Institute 9 Basic concepts Particle Physics Relativistic Quantum mechanics – hueh? The Klein-Gordon equation • Take Schrödinger equation for free particle and insert Momentum operator Energy operator • giving (ħ=c=1) • with plane wave solutions: • Problems: – 1st order in time derivative – 2nd order in space derivative NOT Lorentz invariant !!!! Jørgen Beck Hansen Niels Bohr Institute 10 Basic concepts Particle Physics • Take instead special relativity: E2 = p2 + m2 • and combine with energy and momentum operators to give the Klein-Gordon equation • Second order in both space and time - by construction Lorentz invariant • But second order is a problem! • Inserting a plane wave function for a free particles yields E2 = p2 + m2 that is E = ±√(p2 + m2) • Negative energy solutions? • Dirac equation: “ANTI-MATTER“ Jørgen Beck Hansen Niels Bohr Institute 11 Basic concepts Particle Physics • In 1928 Dirac constructed a first order form with the same solutions • where αi and β are 4 x 4 matrices and Ψ are four component wavefunctions: spinors Jørgen Beck Hansen Niels Bohr Institute 12 Basic concepts Particle Physics Hmm – still negative energy solutions… • A hole created in the negative energy electron states by a γ with E ≥ mc2 corresponds to a positively charged, positive energy anti-particle • Every spin-1/2 particle must have an antiparticle with same mass and opposite charge • Today: E < 0 solutions represent negative energy particle states traveling backward in time. ➨ Interpreted as positive energy anti- particles, of opposite charge, traveling forward in time. • Anti-particles have the same mass and equal but opposite charge. Jørgen Beck Hansen Niels Bohr Institute 13 Basic concepts Particle Physics Particle physics’ first prediction ►DISCOVERY • In 1933, C.D.Andersson, Univ. of California (Berkeley): Observed with the Wilson cloud chamber 15 tracks in cosmic rays: Jørgen Beck Hansen Niels Bohr Institute 14 Basic concepts Particle Physics Feynman diagrams • In 1940s, R.Feynman developed a diagram technique for representing processes in particle physics. Electromagnetic vertex • Rules and requirements – Time runs from left to right Space – Arrow directed towards the right indicates a “Instantaneous” particle - otherwise antiparticle space-time moving – At every vertex, charge, momentum, and angular momentum are conserved (but not “At rest” energy) Time – Each group of particles has a separate style Jørgen Beck Hansen Niels Bohr Institute 15 Basic concepts Particle Physics Virtual processes • A process or particle is called virtual if E2 ≠ m2 + p2 • Such a violation can only be possible if ∆t x ∆E ≤ ħ • Forces are due to exchanged particles which are VIRTUAL • The more virtual (off- shell) a particle is - the shorter distance it can travel! Jørgen Beck Hansen Niels Bohr Institute 16 Basic concepts Particle Physics Jørgen Beck Hansen Niels Bohr Institute 17 Basic concepts Particle Physics Jørgen Beck Hansen Niels Bohr Institute 18 Basic concepts Particle Physics Jørgen Beck Hansen Niels Bohr Institute 19 Basic concepts Particle Physics A word on time ordering • The Feynman diagrams introduced in the book is based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) ►Results depend on the reference frame. • However, the sum of all time orderings is not frame dependent and provides the basis for modern day relativistic theory of Quantum Mechanics. • Energy and Momentum are conserved at interaction vertices • But the exchanged particle no longer has m2 = E2 + p2 - Virtual Space Virtual – space-like Real - On-shell Virtual -Time-like Time Jørgen Beck Hansen Niels Bohr Institute 20 Basic concepts Particle Physics Jørgen Beck Hansen Niels Bohr Institute 21 Basic concepts Particle Physics Jørgen Beck Hansen Niels Bohr Institute 22 Basic concepts Particle Physics Jørgen Beck Hansen Niels Bohr Institute 23 Basic concepts Particle Physics Question: Derive 1/r dependency of electrical potential? Jørgen Beck Hansen Niels Bohr Institute 24 Basic concepts Particle Physics Yukawa potential (1935) “The Fermi coupling constant” • Assuming that A is very heavy, the particle B can be seen as scattered by a static potential with A as source. The Klein-Gordon equation for the force mediating particle X [assume here that X is spin-0, but discussion is general] in the static case is: • The general solution is: • Here g is an integration constant. It is interpreted as coupling strength for particle X to particles A and B. Jørgen Beck Hansen Niels Bohr Institute 25 Basic concepts Particle Physics • Which reduces to the known electrostatic potential for MX = 0: • In Yukawa theory, g is analogous to the electric charge in QED, and the analogue of αem is αX characterizes strength of interaction at distances r ≤ R • An interesting case happens in the limit of very large MX, where the potential point-like. To determine the effective coupling for this case we will determine the Scattering Amplitude = Matrix-element Jørgen Beck Hansen Niels Bohr Institute 26 Basic concepts Particle Physics • Consider a particle being scattered by the potential thus receiving a momentum transfer q=qf – qi • Probability amplitude for particle to be scattered is • the Fourier-transform • Probability Amplitude = Matrix Element f(q) = M(q) and Scattering probability is proportional to |f|2 = |M|2. • Using polar coordinates, d3x = r2 sinθdθdrdφ, and assuming V(x) = V(r), the amplitude is Propagator • In the limit of very heavy MX, MX2c2 » q2, M(q) becomes a constant: Jørgen Beck Hansen Niels Bohr Institute 27