Jørgen Beck Hansen Particle Physics Basic concepts Particle Physics.

Jørgen Beck Hansen

Particle PhysicsBasic concepts

Particle Physics

Niels Bohr Institute 2Jørgen Beck Hansen


Setting the scale

Particle physicsis

Atto-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).



Forces of nature



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



Standard model NOT perfect:

• Origin of Mass?• Why 3 generations?

Interactons



Particle Physics and the Universe



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



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



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 !!!!



• 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“



• 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



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.



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:



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



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!









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









Question: Derive 1/r dependency of electrical potential?



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.



• 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



• 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

Jørgen Beck Hansen Particle Physics Basic concepts Particle Physics.

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