lecture 01 introduction to elementary particle...

Lecture 01

Introduction to Elementary Particle

Physics

Particle Astrophysics

Particle physics● Fundamental constituents of nature● Most basic building blocks● Describe all particles and

interactions● Shortest length scales available

● ~ 10-21 m

Astrophysics● Structure and evolution of the

universe● Composite objects at the largest size● Largest length scales

● ~1026 m

Particle Astrophysics● Combines the largest and smallest length scales● How do elementary particles and their interactions affect large

scale structure in the universe?● How can we use elementary particles as probes of cosmological

evolution?● What do astronomical observations tell us about fundamental

particles?

PHYS 2961 Lecture 01 3

Elementary Particles

What are the building blocks of nature?● Atoms● Subatomic particles: protons, neutrons, electrons● Sub-nucleonic particles: quarks● Force-carrying particles: photons, gluons, etc

● What is an elementary particle?Cannot be broken down into smaller constituents

● We cannot see “inside” it● No substructure● Point-like

● The study of elementary particles focuses on understanding what the fundamental particles are and how they interact

● New Physics is usually ascribed to new particles and/or new interactions


Detecting particles

We look for evidence of a particle interacting with a detector● Tracks

● Particle leaves a “trail” as it passes through material● Does it bend in B field? If so, which way?

● Energy● How much heat, light or ionization does a particle leave

● Topology● Different interaction with different materials for different particles


Describing Particles and Interactions

● Elementary particles are NOT classical● Point-like particles● Governed by quantum principles● We must describe EVERYTHING about a

physical system in quantum mechanical terms

A fundamental particle interacts with another fundamental particle by exchanging yet another fundamental particle

Or

● Composite particles (such as nuclei) can be described by their fundamental constituents

● The interactions can be described as a sum of the fundamental interactions

● This process can be coherent or incoherent


First Quantization

Schrodinger Equation:

H is total energy (KE + PE)

First quantization gives the relation:

Based on commutation relation:

From which we get the familiar form of the Schrodinger Equation:


What is First Quantization?

We treat the particles quantum mechanically, but the fields classically

Example: Hydrogen atom● Electron is treated quantum mechanically

● Follows uncertainty relation

● Wave function gives probability density for electron position

● Potential treated classically● Use Maxwell's equations

Result:Quantum description of electron

But NOT of the force (e.g. photon)

For particle physics we must go to the next step and quantize the field and interaction as well

→ Quantum Field Theory


Review of E&M

Recall the relation between the fields E, B, and their potentials, ϕ, A

Maxwell's equations still satisfiedAll of E&M can be summarized in 4 distinct quantities:ϕ and the 3 components of AWe can combine these 4 quantities in a 4-vectorA

μ, with μ = 0,1,2,3

A0 = ϕ, A

1 = A

x, A

2 = A

y, A

3 = A

z

All of E&M can be written in terms of Aμ


Second Quantization

Fundamental interactions of matter and fieldsTreat matter AND fields quantum mechanically

● Quantum Field Theory quantizes Aμ in a similar way to the

construction of the Schrodinger equation● The quanta of the field are particles● For A, the quanta are photons

● Full discussion beyond the scope of this class● See Advanced Quantum Mechanics by Sakurai

● With the Schrodinger equation, we had quantum particles (e.g. electrons) interacting with classical fields (e.g. electrostatic field)

● Now we have quantum fields● Electrons interact with photons


Forces and Interactions

In classical physics, and 1st quantization, a force is derived from a potential:

In QFT, this is replaced by the concept of interactions In QED, two charged particles interact by the exchange of photonsThe correct quantization method (e.g. A

μ) gives the correct classical limit

Forces are mediated by exchange particles (force carriers)

● Two electrons interact by exchanging a photon

● The photon carries momentum from one particle to the other

● Averaging over many interactions, F = dp/dt

● On average:


Spin: Bosons and Fermions

All particles carry a quantum of angular momentum

BosonsInteger spin

Symmetric wavefunctionsForce carrying particles

FermionsHalf integer spin

Antisymmetric wavefunctionsObey Pauli exclusion principleMatter particles (take up space!)

Spin 0 (scalar)1 spin state, m

z = 0

Spin ½2 spin states, m

z = -1/2, 1/2

Spin 1 (vector)3 spin states, m

z = -1, 0, 1

Spin states:Projection of angular momentumMz integer from -s to s2s+1 spin states


Units

In quantum physics, we frequently encounter Planck's constant:

In special relativity (and of course, E&M), we encounter the speed of light:

We can put them together for convenient, quick conversions:

Angular momenturm

Speed


Nothing magical about these Universal Constants

Consider the speed of light in different units

It has different numerical values, but light ALWAYS travels at the same speed!

Why does this conversion constant exist?

Because we measure time and distance in different units[space] = m, cm, miles, …[time] = s, h, years, …

Why don't we measure them in the same units so that c = 1 and is dimensionless?Same arguments apply for Planck's constant (ratio of energy to frequency, or time)Why don't we measure time and space in the same units as energy?


Natural Units

Let's choose units of energy, electron volts, as our basis of measurement

Since c = 1 and is dimensionless

Since ћ = 1 and is dimensionless

Again, since c = 1 and is dimensionless

This greatly simplifies equations and computationsDimensional analysis is simpler (fewer units to keep track of)


Warnings with Natural Units

Beware of reciprocal units

They work backwards with multipliers

Converting a number in Natural Units to “Usable” unitsYou can Always convert back!Only requires dimensional analysis

There will be exceptions to using Natural UnitsExample: cross sectionsUnits of area, should be eV-2

But we typically use cm2


How Particles Interact

The fundamental interaction: Boson exchange

● In particle physics, the fermions that make up matter transmit force by interacting with one another

● This interaction is mediated by a boson exchange● One fermion (say an electron) emits a boson (say a

photon) which is absorbed by another fermion (say another electron

● The boson carries momentum and energy from one particle to the other

● The affect of this can be attraction (like gravity or opposite electric charges) or repulsion (like same charges)

● It can also be more exotic● Change of particle type● Creation of new particles and antiparticles


The Feynman Path Integral

Probability for photon to be emitted at point A and absorbed at point BSum up amplitude from all possible paths

Richard Feynman developed a method for computing interaction probabilitiesPath Integral (which adopted his name)


Perturbation Theory

Recall from Quantum Physics I:Assume you have a Hamiltonian with exact, known energy solutions:

But the true Hamiltonian has a perturbing term H1

Then the true eigenvalues are

The true eigenvalues and eigenfunctions can be expanded in a perturbation series


Bra-ket notation

Dirac introduced a shorthand notation for describing quantum states

Bra

Ket

Put the together to get a Braket

You can also use this for expectation values


More on bra-ket notation

You can operate directly on a ket

Or take expectation values of operators

You can use shorthand notation to describe the wavefunction in the bra and ket, and label any relevant quantum number inside the ket

Or you can use symbols to describe the state such as a neutrino or Schrödinger's cat


Calculating the Perturbation Series

What's important for us?

● A perturbing Hamiltonian can be expanded in a perturbation series

● The eigenvalues and eigenstates can be computed from expectation values of the perturbing Hamiltonian

● If the series for a system converges, we can describe that system by this series● Leading order● Next-to-leading order● Next-to-next-to leading order● etc

and so on


Perturbation Theory in Particle Physics

Can we use perturbation theory to describe fundamental particles and their interactions?

Sometimes

In many cases, the Hamiltonian can be described by a “free particle” term (H0) and

and “interaction” term (H1)

We describe interactions in leading order, next to leading order, and so on

This doesn't always work!

Low energy strong interactions DO NOT CONVERGEOther methods necessary


Matter and Antimatter

● Dirac developed a relativistic treatment of electrons● For the relativistic Hamiltonian for a free particle, start with

special relativity

● Dirac essentially took the square root of a QM version of this equation

● Since both the positive and negative square roots are solutions, there are both positive and negative energy solutions

● The negative energy solutions are interpreted as antiparticles that have all quantum numbers identical except electric charge, which is equal and opposite

All fundamental fermions exist in pairs of matter and antimatterThis is a symmetry of natureThey can be pair-produced or annihilate with one another


Feynman Diagrams

● Richard Feynman developed pictures to represent particle interactions● The “Feynman Rules” associate different mathematical factors for each part

of a diagram● By writing a diagram, you can directly read off the QFT factors to compute

interaction probabilities


Parts of a Feynman Diagram

Fermions are drawn as a solid line with an arrow● The arrow shows the flow of matter● Matter flows forward in time● Antimatter flows backward in time

Photons are drawn as a squiggly line

W/Z/Higgs bosons are drawn as a dashed line

Gluons are drawn as loopy line

Labels:● Bosons do not have arrows (neither matter nor antimatter)● Fermions typically have a label to identify the particle● Sometimes the bosons do too, when it is not obvious what it is

- - - - - - - - - -


Axes

● One axis represents time, and the other space● But unfortunately, there are two conventions● And diagrams seldom have the axes labeled

In this course, I will exclusively use time from left to right(same as the textbook)But keep in mind that when you look up a Feynman diagram you must know which axis is time


Using Feynman Diagrams in a Perturbation Series

Feynman showed that a perturbation series can be described by a series of Feynman diagramsOrder proportional to the number of loops

Zeroth order is described by “Tree Level” diagrams

First order is described by one loop diagrams

When two electrons scatter, is it a tree level, one loop, two loop process?

Answer: We don't know!

Remember the path integral formulation:Sum up ALL possible interactionsAll we see is two electrons scatter


Scattering

A large class of particle interactions fall under the class of scattering

Scattering is the collision of two particlesTwo incoming particles interactThere is a probability for the interaction (characterized by the cross section)

Rules for scattering:The center of mass energy can go into the final productsAs scattering energy increases, heavier final state particles are available

Scattering experiments:Particle accelerators can collide particles with each other or fixed targetsHigh energy particles (like in cosmic rays) can collide with other matter


Elastic Scattering

Elastic scattering: Ingoing and Outgoing particles the same

Examples:

Electron electron scattering Electron neutrino scattering

● Very analogous to classical elastic scattering● No kinetic energy is lost, it is transfers from one

particle to another


Inelastic Scattering

Incoming and Outgoing particles are differentCenter of mass energy goes into new particles

Examples:

Neutrino neutron scattering Electron positron annihilation

● Analogous to classical inelastic scattering● There is a transfer of kinetic and mass energy (KE is “created” or

“destroyed”)


Decays

Particles can decay into lighter particlesMass must always decreaseIn particle's rest frame, only mass energy available

Particles decay with a lifetime given by

Most common example:● Radioactive decay of nuclei● A neutron inside a nucleus can decay into a proton and an

electron (if the nuclear binding energy of the final state is lower)

Other examples:● Muons decaying to electrons and neutrinos● Exotic quark states (mesons) decaying into lighter mesons


The 4 Fundamental Forces

GravitationElectricity and MagnetismWeak nuclear forceStrong nuclear force

● Everything except gravity can be described by quantum field theory● E&M + Weak interactions are unified by the electroweak theory● This predicted the Higgs boson, and also explains mass generation● Strong interactions describe the substructure of nucleons, as well as other

exotic particles● These combine to make up the Standard Model of particle physics

lecture 01 introduction to elementary particle...

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