Particle Physics

1. Particle Physics

2. Setting the scale Particle physics is Atto-physics Niels Bohr Institute

3. Basic concepts • Particle physics studies elementary “building blocks” of matterand interactionsbetween 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). Niels Bohr Institute

4. Forces of nature Niels Bohr Institute

5. 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: • Basic constituents of matter are quarksand leptons(spin 1/2) • They interact by exchanging gauge bosons (spin 1) • Quarks and leptons are subdivided into 3 generations Niels Bohr Institute

6. Standard model NOT perfect: • Origin of Mass? • Why 3 generations? Interactons Niels Bohr Institute

7. Particle Physics and the Universe Niels Bohr Institute

8. 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 • Unit of energy : GeV = 109 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 constantand the speed of light: • ħ ≡h/2 = 6.582 x 10-25 GeV s • c = 2.9979 x 108 m/s • Conversion constant: ħc = 197.327 x 10-18 GeV m • Natural Units: GeV, ħ, c • Units become • 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 ħ2c2 • For simplicity choose ħ = c = 1 Niels Bohr Institute

9. 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 Mechanicalfree particle wavefunction • 4-momentum squared gives particle’s invariant mass m2c2 ≡ p ⋅ p = E2 ⁄ c2 – p2 or E2 = p2c2 + m2c4 Quick formulas Niels Bohr Institute

10. 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 !!!! Niels Bohr Institute

11. 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 isE = ±√(p2 + m2) • Negative energy solutions? • Dirac equation: “ANTI-MATTER“ Niels Bohr Institute

12. 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 Niels Bohr Institute

13. Hmm – still negative energy solutions… • A hole created in the negative energy electron states by a γwith E ≥ mc2 corresponds to a positively charged, positive energyanti-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. Niels Bohr Institute

14. 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: Niels Bohr Institute

15. 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 • 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 Space “Instantaneous” space-time moving “At rest” Time Niels Bohr Institute

16. 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! Niels Bohr Institute

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20. 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 Niels Bohr Institute

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24. Question: Derive 1/r dependency of electrical potential? Niels Bohr Institute

25. 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. Niels Bohr Institute

26. 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 αemis αXcharacterizes 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 Niels Bohr Institute

27. 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: Niels Bohr Institute