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Jørgen Beck Hansen Particle Physics Basic concepts Particle Physics

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute2 Setting the scale Particle physics is Atto-physics

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute3 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).

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute4 Forces of nature

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute5 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

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute6 Standard model NOT perfect: Origin of Mass? Why 3 generations? Interactons

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute7 Particle Physics and the Universe

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute8 Tricks of the trade: UNITS and Dimensions For everyday physics SI units are a natural choice Not so good for particle physics: M proton ~ 10 -27 kg Use a different basis - NATURAL UNITS Unit of energy : GeV = 10 9 eV = 1.602 x 10 -10 J –1 eV = Energy of e - passing a voltage of 1 V Language of quantum mechanics and relativity, i.e. –The reduced Planck constant and the speed of light: ħ ≡ h/2 = 6.582 x 10 -25 GeV s c = 2.9979 x 10 8 m/s –Conversion constant: ħc = 197.327 x 10 -18 GeV m Natural Units: GeV, ħ, c Units become Energy ► GeVTime ► (GeV/ħ) -1 Momentum ► GeV/cLength ► (GeV/ħc) -1 Mass ► GeV/c 2 Area ► (GeV/ħc) -2 For simplicity choose ħ = c = 1 Convert back to S.I. units by reintroducing ‘missing’ factors of ħ and c EXAMPLE: Area = 1 GeV -2 [L] 2 = [E] -2 [ħ] n [c] m [L] 2 = [E] -2 [E] n [T] n [L] m [T] -m Hence, n = 2 and m = 2 Area = 1 GeV -2 x ħ 2 c 2

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute9 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 = (a 0 ± b 0, a 1 ± b 1, a 2 ± b 2, a 3 ± b 3 ) – Scalar product (minus sign!) a ⋅ b=a 0 b 0 – a 1 b 1 – a 2 b 2 – a 3 b 3 =a 0 b 0 – a ⋅ b – Scalar product of momentum and space-time 4-vectors are thus: x ⋅ p=Et – x x p x – x y p y – x z p z = Et – x ⋅ p Used in the Quantum Mechanical free particle wavefunction –4-momentum squared gives particle’s invariant mass m 2 c 2 ≡ p ⋅ p = E 2 ⁄ c 2 – p 2 or E 2 = p 2 c 2 + m 2 c 4 Quick formulas

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute10 Relativistic Quantum mechanics – hueh? Take Schrödinger equation for free particle The Klein-Gordon equation Energy operator Momentum operator and insert giving (ħ=c=1) with plane wave solutions: Problems: –1st order in time derivative –2nd order in space derivative NOT Lorentz invariant !!!!

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute11 Take instead special relativity: E 2 = p 2 + m 2 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 E 2 = p 2 + m 2 that isE = ±√(p 2 + m 2 ) Negative energy solutions? Dirac equation: “ANTI-MATTER“

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute12 In 1928 Dirac constructed a first order form with the same solutions where α i and β are 4 x 4 matrices and Ψ are four component wavefunctions: spinors

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute13 Hmm – still negative energy solutions… A hole created in the negative energy electron states by a γ with E ≥ mc 2 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.

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute14 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:

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute15 Feynman diagrams In 1940s, R.Feynman developed a diagram technique for representing processes in particle physics. Rules and requirements –Time runs from left to right –Arrow directed towards the right indicates a particle - otherwise antiparticle –At every vertex, charge, momentum, and angular momentum are conserved (but not energy) –Each group of particles has a separate style Time Space “At rest” “Instantaneous” space-time moving Electromagnetic vertex

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute16 Virtual processes A process or particle is called virtual if E 2 ≠ m 2 + p 2 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!

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute20 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. Time Space Virtual -Time-like Virtual – space-like Real - On-shell Energy and Momentum are conserved at interaction vertices But the exchanged particle no longer has m 2 = E 2 + p 2 - Virtual

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute24 Question: Derive 1/r dependency of electrical potential?

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute25 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.

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute26 Which reduces to the known electrostatic potential for M X = 0: In Yukawa theory, g is analogous to the electric charge in QED, and the analogue of α em is An interesting case happens in the limit of very large M X, where the potential point-like. To determine the effective coupling for this case we will determine the Scattering Amplitude = Matrix-element α X characterizes strength of interaction at distances r ≤ R

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Jørgen Beck Hansen Particle Physics Basic concepts Niels Bohr Institute27 Consider a particle being scattered by the potential thus receiving a momentum transfer q=q f – q i 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, d 3 x = r 2 sinθdθdrdφ, and assuming V(x) = V(r), the amplitude is In the limit of very heavy M X, M X 2 c 2 » q 2, M(q) becomes a constant: Propagator

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Feynman Diagrams Topic 7.3.

Feynman Diagrams Topic 7.3.

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