jørgen beck hansen particle physics basic concepts particle physics
TRANSCRIPT
Niels Bohr Institute 2Jørgen Beck Hansen
Particle PhysicsBasic concepts
Setting the scale
Particle physicsis
Atto-physics
Niels Bohr Institute 3Jørgen Beck Hansen
Particle PhysicsBasic concepts
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).
Niels Bohr Institute 5Jørgen Beck Hansen
Particle PhysicsBasic concepts
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
Niels Bohr Institute 6Jørgen Beck Hansen
Particle PhysicsBasic concepts
Standard model NOT perfect:
• Origin of Mass?• Why 3 generations?
Interactons
Niels Bohr Institute 7Jørgen Beck Hansen
Particle PhysicsBasic concepts
Particle Physics and the Universe
Niels Bohr Institute 8Jørgen Beck Hansen
Particle PhysicsBasic concepts
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 constant and 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
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
Convert back to S.I. units by reintroducing ‘missing’ factors of ħ and cEXAMPLE: • 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
Niels Bohr Institute 9Jørgen Beck Hansen
Particle PhysicsBasic concepts
Particle Physics language: 4-vectorsParticles 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=a⋅ 0b0 – a1b1 – a2b2 – a3b3=a0b0 – a b⋅– Scalar product of momentum and space-time 4-vectors are thus:
x p=Et – x⋅ xpx – xypy – xzpz= Et – x p⋅Used in the Quantum Mechanical free particle wavefunction
– 4-momentum squared gives particle’s invariant massm2c2 ≡ p p⋅ = E2 ⁄ c2 – p2 or E2 = p2c2 + m2c4
Quick formulas
Niels Bohr Institute 10Jørgen Beck Hansen
Particle PhysicsBasic concepts
Relativistic Quantum mechanics – hueh?
• Take Schrödinger equation for free particle
The Klein-Gordon equation
Energy operatorMomentum 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 !!!!
Niels Bohr Institute 11Jørgen Beck Hansen
Particle PhysicsBasic concepts
• 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“
Niels Bohr Institute 12Jørgen Beck Hansen
Particle PhysicsBasic concepts
• 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 13Jørgen Beck Hansen
Particle PhysicsBasic concepts
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.
Niels Bohr Institute 14Jørgen Beck Hansen
Particle PhysicsBasic concepts
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 15Jørgen Beck Hansen
Particle PhysicsBasic concepts
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
Niels Bohr Institute 16Jørgen Beck Hansen
Particle PhysicsBasic concepts
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 20Jørgen Beck Hansen
Particle PhysicsBasic concepts
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 m2 = E2 + p2 - Virtual
Niels Bohr Institute 24Jørgen Beck Hansen
Particle PhysicsBasic concepts
Question: Derive 1/r dependency of electrical potential?
Niels Bohr Institute 25Jørgen Beck Hansen
Particle PhysicsBasic concepts
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 26Jørgen Beck Hansen
Particle PhysicsBasic concepts
• 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
• 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
αX characterizes strength of interaction at distances r ≤ R
Niels Bohr Institute 27Jørgen Beck Hansen
Particle PhysicsBasic concepts
• 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
• In the limit of very heavy MX, MX2c2 » q2, M(q) becomes a constant:
Propagator