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Home , Elementary particle

Lecture 01

Introduction to Elementary Particle Astrophysics

Particle physics Astrophysics ● Fundamental constituents of ● Structure and evolution of the ● Most basic building blocks ● Describe all and ● Composite objects at the largest size ● Largest length scales ● Shortest length scales available ● ~1026 m ● ~ 10-21 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? Elementary Particles

What are the building blocks of nature? ● ● Subatomic particles: , , ● Sub-nucleonic particles: ● Force-carrying particles: , , 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

PHYS 2961 Lecture 01 3 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? ● ● How much heat, light or ionization does a particle leave ● Topology ● Different with different materials for different particles

PHYS 2961 Lecture 01 4 Describing Particles and Interactions

● Elementary particles are NOT classical ● Point-like particles ● Governed by 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

PHYS 2961 Lecture 01 5 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:

PHYS 2961 Lecture 01 6 What is First Quantization?

We treat the particles quantum mechanically, but the fields classically

Example: Hydrogen ● is treated quantum mechanically ● Follows uncertainty relation

● gives probability density for electron position

● Potential treated classically ● Use Maxwell's equations

For we must go to the Result: next step and quantize the field and Quantum description of electron interaction as well But NOT of the force (e.g. ) →

PHYS 2961 Lecture 01 7 Review of E&M

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

Maxwell's equations still satisfied All of E&M can be summarized in 4 distinct quantities: ϕ and the 3 components of A We can combine these 4 quantities in a 4-vector

Aμ, with μ = 0,1,2,3

A0 = ϕ, A1 = Ax, A2 = Ay, A3 = Az

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

PHYS 2961 Lecture 01 8 Second Quantization

Fundamental interactions of and fields Treat 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 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

PHYS 2961 Lecture 01 9 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 photons

The correct quantization method (e.g. Aμ) 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:

PHYS 2961 Lecture 01 10 : and

All particles carry a quantum of

Bosons Fermions Integer spin Half integer spin

Symmetric wavefunctions Antisymmetric wavefunctions Force carrying particles Obey Pauli exclusion principle Matter particles (take up space!)

Spin 0 (scalar) 1 spin state, m = 0 Spin states: z Projection of angular momentum Spin ½ Mz integer from -s to s 2 spin states, m = -1/2, 1/2 2s+1 spin states z Spin 1 (vector)

3 spin states, mz = -1, 0, 1

PHYS 2961 Lecture 01 11 Units

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

Angular momenturm

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

We can put them together for convenient, quick conversions:

PHYS 2961 Lecture 01 12 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?

PHYS 2961 Lecture 01 13

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 computations Dimensional analysis is simpler (fewer units to keep track of)

PHYS 2961 Lecture 01 14 Warnings with Natural Units

Beware of reciprocal units

They work backwards with multipliers

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

There will be exceptions to using Natural Units Example: cross sections Units of area, should be eV-2 But we typically use cm2

PHYS 2961 Lecture 01 15 How Particles Interact

The : 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 (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 or opposite electric charges) or repulsion (like same charges) ● It can also be more exotic ● Change of particle type ● Creation of new particles and

PHYS 2961 Lecture 01 16 The Feynman Path Integral

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

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

PHYS 2961 Lecture 01 17 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

Then the true eigenvalues are

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

PHYS 2961 Lecture 01 18 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

PHYS 2961 Lecture 01 19 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 inside the ket

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

PHYS 2961 Lecture 01 20 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

PHYS 2961 Lecture 01 21 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 CONVERGE Other methods necessary

PHYS 2961 Lecture 01 22 Matter and

● 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 , which is equal and opposite

All fundamental fermions exist in pairs of matter and antimatter This is a symmetry of nature They can be pair-produced or annihilate with one another

PHYS 2961 Lecture 01 23 Feynman Diagrams

● 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

PHYS 2961 Lecture 01 24 Parts of a

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

PHYS 2961 Lecture 01 25 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

PHYS 2961 Lecture 01 26 Using Feynman Diagrams in a Perturbation Series

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

Zeroth order is described First order is described by one loop by “Tree Level” diagrams 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 interactions All we see is two electrons scatter

PHYS 2961 Lecture 01 27

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

Scattering is the collision of two particles Two incoming particles interact There is a probability for the interaction (characterized by the )

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

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

PHYS 2961 Lecture 01 28

Elastic scattering: Ingoing and Outgoing particles the same

Examples:

Electron electron scattering scattering

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

PHYS 2961 Lecture 01 29 Inelastic Scattering

Incoming and Outgoing particles are different Center of mass energy goes into new particles

Examples:

Neutrino scattering Electron

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

PHYS 2961 Lecture 01 30 Decays

Particles can decay into lighter particles Mass must always decrease In 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 and an electron (if the nuclear of the final state is lower)

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

PHYS 2961 Lecture 01 31 The 4 Fundamental Forces

Gravitation Electricity and Weak Strong 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 , and also explains mass generation ● Strong interactions describe the substructure of , as well as other exotic particles ● These combine to make up the of particle physics

PHYS 2961 Lecture 01 32