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?
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 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? Energy How much heat, light or ionization does a particle leave Topology Different interaction with different materials for different particles 4
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 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: 6
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 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 8
Second Quantization Fundamental interactions of matter 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 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 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: 10
Spin: Bosons and Fermions All particles carry a quantum of angular momentum Bosons Integer spin Fermions Half integer spin Symmetric wavefunctions Force carrying particles Antisymmetric wavefunctions Obey Pauli exclusion principle Matter particles (take up space!) Spin states: Projection of angular momentum Mz integer from -s to s 2s+1 spin states Spin 0 (scalar) 1 spin state, mz = 0 Spin ½ 2 spin states, mz = -1/2, 1/2 Spin 1 (vector) 3 spin states, mz = -1, 0, 1 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: 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? 13
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 computations Dimensional analysis is simpler (fewer units to keep track of) 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 15
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 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 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 H1 Then the true eigenvalues are The true eigenvalues and eigenfunctions can be expanded in a perturbation series 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 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 quantum number inside the ket Or you can use symbols to describe the state such as a neutrino or Schrödinger's cat 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 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 22
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 antimatter This is a symmetry of nature They can be pair-produced or annihilate with one another 23
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 24
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 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 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 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 interactions All we see is two electrons scatter 27
Scattering 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 cross section) Rules for scattering: The center of mass 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 28
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 29
Inelastic Scattering Incoming and Outgoing particles are different Center 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 ) 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 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 31
The 4 Fundamental Forces Gravitation Electricity and Magnetism Weak nuclear force 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 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 32