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Particle Physics 2 ๐ ๐๐ ๐ ๐ ๐ฒ+ ๐ ๐ฉ ๐พ± ๐ฑ/๐ ๐ ๐ ๐๐ ๐ ๐๐ ๐ธ ๐ฏ๐ ๐๐ ๐ Prof. Glenn Patrick U23525, Particle Physics, Year 3 University of Portsmouth, 2013 - 2014 1 Last Lecture Particle Physics 1 Course Outline Preliminaries - Assessment Preliminaries - Books Preliminaries - Course Material Particle Physics, Cosmology & Particle Astrophysics Natural Units Rationalised Heaviside-Lorentz EM Units Special Relativity and Lorentz Invariance Mandelstam Variables (s, t and u) Crossing Symmetry and s, t & u Channels Spin and Spin Statistics Theorem โ Fermions and Bosons Addition of Angular Momentum Non-Relativistic Quantum Mechanics (Schrödinger Equation) Relativistic Quantum Mechanics (Klein-Gordon Equation) Feynman-Stückelberg Interpretation of Negative Energy States 2 Todayโs Plan Particle Physics 2 Dirac Equation Dirac Interpretation of Negative States Stückelberg & Feynman Interpretations Discovery: Positron & e+e- Pair Production Decays โ Lifetimes Scattering โ Cross-sections Feynman Diagrams Quantum Electrodynamics (QED) Higher Order Diagrams LEP ๐ + ๐ โ โ ๐พ๐พ ๐พ example Running Coupling Constant (๐ผ) QED Renormalisation Electron/Positron Annihilation & precision LEP electroweak example 3 Correction - Assessment 40 hours of lectures across two teaching blocks plus 8 hours of tutorial classes. The main aim is to improve your understanding of fundamental physics. However, we cannot forget the small matter of your degreeโฆ. Not 3 hours as I said last week. 1 Final written examination (2 hours) โ 80% 2 Coursework questions and problems โ 20% ๏ ๏ ๏ ๏ Main thing is that you enjoy the course. We will try and focus on understanding the underlying concepts. Extra material/maths shown mainly to aid understanding. Guidance will be given over essential knowledge needed for exam. 4 Corrections - Timetable ๏ถ There is NO lecture this afternoon at 15:00 โ 17:00. The timetable was wrong. ๏ถ There are also 2 extra weeks added in March to the end of teaching block 2 (weeks 34 and 35). 5 Moodle Site Slides - pptx Slides - pdf 6 2013 Nobel Prize โ Particle Physics The Nobel Prize in Physics 2013 was awarded jointly to François Englert and Peter W. Higgs "for the theoretical discovery of a mechanism that contributes to our understanding of the origin of mass of subatomic particles, and which recently was confirmed through the discovery of the predicted fundamental particle, by the ATLAS and CMS experiments at CERN's Large Hadron Colliderโ. 7 Dirac Equation The Klein Gordon equation was believed to be sick, but today we now understand it was telling us something about antiparticles. To try to solve the problem of โve energy solutions, Dirac: ๏ถ Wanted an equation first order in ๐ ๐๐ก . ๏ถ Started by assuming a Hamiltonian, which is local and linear in p and of the form: H D ๏ฝ (๏ก ๏ p ๏ซ ๏ขm) ๏ฝ ๏ก1 P1 ๏ซ ๏ก 2 P2 ๏ซ ๏ก 3 P3 ๏ซ ๏ขm To also make the equation relativistically covariant, it has to be linear in ๐ป: ๏ถ๏ i ๏ฝ (๏ญi๏ก ๏ ๏ ๏ซ ๏ขm) ๏ถt Solutions of this equation were also required to be solutions of the Klein Gordon equation. Only true if: ๐ถ๐ ๐ = ๐, ๐ท๐ = ๐, ๐ถ๐ ๐ท + ๐ท๐ถ๐ = ๐, ๐ถ๐ ๐ถ๐ + ๐ถ๐ ๐ถ๐ = ๐ (๐ โ ๐) Coefficients do not commute so they cannot be numbers โ require 4 matrices. 8 Dirac Equation In two dimensions, a natural set of matrices would be the Pauli spin matrices (from atomic physics for electron spin): ๏ฆ0 1๏ถ ๏ฆ0 ๏ญ i๏ถ ๏ฆ1 0 ๏ถ ๏ท๏ท ๏ณ 2 ๏ฝ ๏ง๏ง ๏ท๏ท ๏ณ 3 ๏ฝ ๏ง๏ง ๏ท๏ท ๏ณ 1 ๏ฝ ๏ง๏ง ๏จ1 0๏ธ ๏จi 0 ๏ธ ๏จ 0 ๏ญ 1๏ธ but there is no suitable fourth anti-commuting matrix for ฮฒ. Convenient choice is to instead use the Dirac-Pauli representation: ๏ฆ0 ๏ง ๏ง0 ๏ณ1 ๏ฝ ๏ง 0 ๏ง ๏ง1 ๏จ 0 0 1 0 0 1 0 0 ๏ฆ0 0 ๏ฆ0 0 0 ๏ญ i๏ถ 1๏ถ ๏ง ๏ง ๏ท ๏ท 0๏ท ๏ง0 0 i 0 ๏ท ๏ณ ๏ฝ ๏ง0 0 ๏ณ2 ๏ฝ ๏ง ๏ท 3 ๏ง1 0 ๏ท 0 ๏ญ i 0 0 0 ๏ง ๏ง ๏ท ๏ท ๏ง 1 ๏ญ1 ๏ง ๏ท ๏ท i 0 0 0 ๏จ 0๏ธ ๏จ ๏ธ 1 0๏ถ ๏ท 0 ๏ญ 1๏ท 0 0๏ท ๏ท 0 0 ๏ท๏ธ ๏ฆ1 0 0 0 ๏ถ ๏ง ๏ท ๏ง0 1 0 0 ๏ท ๏ข ๏ฝ๏ง 1 0 ๏ญ1 0 ๏ท ๏ง ๏ท ๏ง 1 ๏ญ 1 0 ๏ญ 1๏ท ๏จ ๏ธ 2 x 2 identity matrix These can be abbreviated as: (each element is a 2 x 2 matrix) ๏ฆ0 ๏ณ ๏ถ ๏ฆ1 0 ๏ถ ๏ท๏ท ๏ข ๏ฝ ๏ง๏ง ๏ท๏ท ๏ก ๏ฝ ๏ง๏ง ๏จ๏ณ 0 ๏ธ ๏จ 0 ๏ญ 1๏ธ Block matrix (2 x 2) 9 Dirac Equation Now that ๐ผ and ๐ฝ are matrices, the equation makes no sense unless the wave function ฮจ is itself a matrix with four rows and one column. This is the Dirac Spinor: (a four component wavefunction) ๏ถ๏ i ๏ฝ (๏ญi๏ก ๏ ๏ ๏ซ ๏ขm) ๏ถt ๏ฆ ๏1 (r , t ) ๏ถ ๏ง ๏ท ๏ง ๏2 (r , t ) ๏ท ๏ (r , t ) ๏ฝ ๏ง ๏3 (r , t ) ๏ท ๏ง ๏ท ๏ง ๏ (r , t ) ๏ท ๏จ 4 ๏ธ Plane wave solutions take the form: ๏ (r , t ) ๏ฝ u ( p )e๏i ( p.r ๏ญet ) ๏ where ๐ข(๐) is also a four-component spinor satisfying the eigenvalue equation: H p u ( p ) ๏บ (๏ก ๏ p ๏ซ ๏ขm)u ( p ) ๏ฝ Eu ( p ) There are four solutions: two with positive energy E = +Ep corresponding to the two possible spin states of a spin 1 2 particle and two corresponding negative energy solutions with E = - Ep. Electron spin is therefore included in a natural way (rather than the previous 10 ad-hoc attempts). All solutions also have positive probability density. Dirac Interpretation Dirac interpreted the negative energy solutions by: ๏ถ Postulating the existence of a โseaโ of negative energy states, which are almost always filled โ each with two electrons (spin โupโ and spin โdownโ): ๏ถ When an electron is added to the vacuum, it is confined to the positive energy region since all negative energy states are occupied (Pauli exclusion principle). ๏ถ When energy is supplied to promote a negative energy electron to a positive energy level, an electron-hole pair is created. The hole is seen as a charge +e and E > 0 state. A false start when Dirac identified negative energy electrons as protons. Eventually, persuaded by arguments of Weyl & Oppenheimer that +ve particle had to have the same mass as 11 the electron. Dirac Interpretation Photon with ๐ธ > 2๐๐ excites electron from โve energy state. ๐ธ Leaves hole in vacuum corresponding to a state with more energy (less negative energy) and a positive charge wrt the vacuum. Sea Problems with this picture. ๏ถ Bosons have no exclusion principle, so this picture does not work for them. ๏ถ It implies the Universe has infinite negative energy! Nonetheless, in 1931 Dirac postulated the existence of the positron as the electronโs antiparticle. 12 It was discovered 1 year later. Positron - First Antiparticle Discovered! Positron = the anti-electron 6 mm lead plate Discovered in 1932 by Carl Anderson + e photographing cosmic ray tracks in a cloud chamber. + e e- 13 ๐+ ๐โ Pair Production In 1933, Blackett & Occhialini observed pair production in a triggered cloud chamber and confirmed that Andersonโs particle was indeed Diracโs positron. ๐ธ + ๐๐๐๐๐๐๐ โ ๐+ + ๐โ + ๐๐๐๐๐๐๐ ๐โ ๐ธ ๐+ 14 Stückelberg Interpretation As early as 1941, Stückelberg cultivated the view that positrons may be understood as electrons running backward in time. E.C.G. Stueckelberg, Helvetica Physica Acta, 14 (1941), 588-594 15 Feynman Interpretation โThe problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the solutions of the Dirac equationโฆ..โ. July 1962 โฆIn this solution, the โnegative energy statesโ appear in a form which may be pictured (as by Stückelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approaching the potential and annihilating the electronโฆ. 16 Feynman Interpretation Waves can proceed backward in time Virtual pair production. Positron goes forward in time (4 to 3) to be annihilated and electron backwards (3 to 4) 17 One electron in the entire Universe? It was Feynmanโs PhD supervisor, John Wheeler, who suggested that there may be only a single electron in the universe, propagating through space and time! This obviously has a few problems! For example: โข You would expect equal number of electrons and positrons and yet we observe extremely few positrons. โข How do we account for the fact that electrons can be created/destroyed in weak interactions. โข This poor single electron would have had to traverse huge distances and be very ancient. World-line of single electron Nonetheless, Feynman kept the idea that positrons could simply be represented as electrons going from the future to the past18. Reminder: this is Applied Physics! This quantum theory is all very well, but what can we physically measure? Quantum Theorist Applied Physicist 19 Decays - Lifetimes In the case of particle decays, the most interesting physical quantity is the lifetime of the particle. ๏ถ Measured in the rest frame of the particle. We can define the decay rate, ฮ, as the probability per unit time the particle will decay. Similar to your Year 2 nuclear physics ๐๐ = โฮ๐๐๐ก Mean Lifetime (natural units) ฯ = 1 ฮ ๐ ๐ก = ๐(0)๐ โฮ๐ก โ ๐น๐๐ ๐ ๐๐๐๐๐๐ , ๐ข๐ ๐ ๐ = ฮ ๐ ๏ถ Most particles decay by several different routes. The total decay rate is then the sum of the individual decay rates: ๏ถ The lifetime of the particle is then: 1 ๐= ฮ๐ก๐๐ก ฮ๐ก๐๐ก = ฮ๐ ๐=1 ๏ถ The different final states are known as decay modes. ฮ๐ ๏ถ The branching ratio for iโth decay mode is: ๐ต๐๐๐๐โ๐๐๐ ๐ ๐๐ก๐๐ = ฮ๐ก๐๐ก 20 ๐ ๐ฒ ๐บ Meson Example Particle Data Group: http://pdg.lbl.gov/ 21 Scattering - Cross-Section na = no. of beam particles va = velocity of beam particles nb = no. target particles/area Incident flux F=nava dN = no. scattered particles in solid angle dฮฉ dฮฉ ๏ฝ sin ๏.d๏.d๏ช dN (d๏) ๏ฝ na va nb d๏ณ (d๏) ๏ฝ Fnb d๏ณ (d๏) ๏ฝ Ld๏ณ (d๏) Differential d๏ณ 1 dN cross-section d๏ ๏ฝ L d๏ Total cross-section Luminosity L = flux x no. targets (cm-2s-1) d๏ณ ๏ณ ๏ฝ๏ฒ d๏ d๏ Measured in barns. 1 b = 10-24 cm-2 Event rate N ๏ฝ L๏ณ Cross-section quantifies rate of reaction. Depends on underlying physics.22 Feynman Diagrams โLike the silicon chip of more recent years, the Feynman diagram was bringing computation to the massesโ. Julian Schwinger 23 Feynman Diagrams โ The Problem ๐+ ๐โ โ ๐+ ๐โ Consider the calculation of the cross-section of one of the simplest QED processes. In the Centre of Mass frame: ๐โฒ = ๐, ๐โฒ = ๐ ๐ธ ๐ = ๐โฒ = ๐ = ๐โฒ = ๐ถ๐ 2 where ๐, ๐โฒ , ๐, ๐ โฒ are momenta ๐โ ๐ ๐ ๐โ ๐ ๐โฒ ๐+ ๐+ ๐โฒ โณ = QM amplitude for process. d๏ณ 1 2 Differential Cross - section ๏ฝ ๏๏ 2 2 d๏ 64๏ฐ ECM Bad News! Even for this simple process the exact expression is not known. Best that can be done is to obtain a formal expression for โณ as a perturbation series in the strength of the EM interaction & evaluate the first few terms. Feynman invented a beautiful way to organise, visualise and thereby calculate 24 the perturbation series. Quantum Field Theory - Basics โข In Quantum Field Theory, the scattering and decay of particles is described in terms of transition amplitudes. โข For a transition process |๐ โถ |๐ , the transition amplitude is written as ๐ ๐ ๐ , where S is an operator known as the Scattering Matrix or S-Matrix. โข Exact calculations of ๐ ๐ ๐ are not possible, but it is possible to use perturbation theory which allow approximate calculations of the transition amplitude. โข A bit like a binomial/Taylor series expansion: 1 โ 2 ๐ฅ 1 3 โ2 โ2 e.g. (1 + ๐ฅ) = 1 โ + ๐ฅ 2 + โฏ and then keeping the 2 2 first 2 terms in the expansion (for small values of x). โข Feynman diagrams are a technique to solve quantum field theory by calculating the amplitude for a state with specified 25 incoming and outgoing particles. Feynman Diagrams - Basics Annihilation Diagram External legs represent amplitudes of initial & final state particles. boson (wavy line) Final Initial vertex vertex State Internal lines (propagators) represent State amplitude of exchanged particle. time ๏ฎ Exchange Diagram fermion (solid line) CONVENTION fermion gluons anti-fermion H bosons Virtual Particle photons, W, Z bosons โข โข โข โข Time runs from left to right. Particles point in +ve time direction. Anti-particles point in โve time direction. Charge, energy, momentum, angular momentum, baryon no. & lepton no. 26 conserved at interaction vertices. Quark flavour for strong & EM interactions. Feynman Diagrams - Calculation ๏ถ Perturbation theory. Expand and keep the most important terms for calculations. ๏ถ Associate each vertex with the square root of the appropriate coupling constant, i.e. โ๐ผ. ๏ถ When the amplitude is squared to yield a cross-section there will be a factor ๐ผ ๐ , where n is the number of vertices (known as the โorderโ of the diagram). Second order Lowest order For QED: e2 ๏ก๏ฝ 4๏ฐ ๏ HL 1 137 Add the amplitudes from all possible diagrams to get the total amplitude, M, for a process ๏ transition probability. 2๏ฐ 2 Transition Rate ๏ฝ M ๏ด (phase space) ๏จ Contains the fundamental physics Fermiโs Golden Rule โJustโ kinematics 27 Phase Space Introduced by Willard Gibbs in 1901. Defines the space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. pp ๏ฎ ๏ฐ 0๏ฐ 0๏ฐ 0 s ๏ฝ 2050 MeV Example: 3 Body Decay Phase space illustrated by this 2-D plot of a three-body decay (a so-called โDalitz plotโ). The contour line shows the boundary of what is kinematically possible - i.e. the edge of phase space. Contour showing limit of kinematically available phase space. Crystal Barrel Experiment 28 Quantum Electrodynamics (QED) Early relativistic quantum theories had to be rethought after the discovery of the Lamb Shift in hydrogen. Sin-Itiro Tomonaga Julian Schwinger Detailed results disagreed with the Dirac Equation. Due to self-energy of the electron. Led to concept of renormalisation (Bethe). Willis Lamb, Robert Retherford, 1947 29 The Most Accurate Theory Anomalous magnetic moment of electron (g-2). Dirac theory predicts g=2 exactly, but this is modified by quantum loops and ae ๏ฝ ( g ๏ญ 2) 2 the difference is defined as โฆโฆโฆโฆ. ๐๐ ๐๐ฑ๐ฉ = ๐ ๐๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐. ๐๐ × ๐๐โ๐๐ ๐๐ ๐๐๐ = ๐ ๐๐๐ ๐๐๐ ๐๐๐. ๐๐ ๐ ๐ ๐ ๐๐ × ๐๐โ๐๐ ๐๐ ๐๐๐ โ ๐๐ ๐๐๐ = โ๐. ๐๐ ๐. ๐๐ × ๐๐โ๐๐ T. Aoyami et al, Tenth-Order QED Contribution to the Electron g-2 and an Improved Value of the Fine Structure Constant, ArXiv:1205.5368 [0.24ppb] [0.67ppb] Calculated from 12,672 Feynman diagrams! 30 QED Vertex Factor Coupling constant, ๐, specifies the strength between the 3 particles at each vertex. This is a measure of the probability of spin 1 2 fermion emitting or absorbing a photon. ๐โ ๐ถ ๐โ ๐ธ Coupling strength ๐๐๐ธ๐ท = ๐ Because ๐ผ = ๐ธ ๐2 , often we simply 4๐ use ๐ผ in place of ๐๐๐ธ๐ท Taking the 2 vertices, we get a total factor of ๐ผ ๐ผ = ๐ผ ๐ถ ๐โ ๐โ 31 Basic QED Processes and Vertices (8) Electron Bremsstrahlung ๏ญ ๏ญ ๐โ Photon Absorption ๏ญ ๐โ ๏ญ e ๏ฎ e ๏ซ๏ง e ๏ซ๏ง ๏ฎ e ๐ โ ๐โ Positron Bremsstrahlung e ๏ซ ๏ฎ e ๏ซ ๏ซ ๏ง ๐+ Photon Absorption ๐+ e ๏ซ ๏ซ ๏ง ๏ฎ e ๏ซ ๐โ ๐+ ๐+ + โ ๐ ๐ Annihilation e๏ซ ๏ซ e๏ญ ๏ฎ ๏ง ๐ Vacuum Production ๐โ โ Pair Production ๐ ๐+ ๐+ Vacuum Extinction e ๏ซ ๏ซ e ๏ญ ๏ซ ๏ง ๏ฎ vacuum vacuum ๏ฎ e ๏ซ ๏ซ e ๏ญ ๏ซ ๏ง time ๏ง ๏ฎ e๏ซ ๏ซ e๏ญ โ ๐+ ๐+ 32 Charged Leptons As well as electrons and positrons interacting with the EM field (i.e. the photon), QED also includes the interactions of the other charged leptons โ the muon (๐ ± ) and the tau lepton (๐ ± ). The Charged Leptons ๐ ๐ฑ = , ๐ช๐๐๐๐๐ = ±๐ ๐ ๐๐๐ ๐ ๐ = 0.511 ๐๐๐ ๐๐๐ ๐ ๐ = 105.66 ๐๐๐ ๐๐๐ ๐ ๐ = 1776.82 ๐๐๐ ๐ > 4.6 × 1026 ๐ฆ๐ ๐ = 2.2 × 10โ6 ๐ ๐ = 290.6 × 10โ15 ๐ All properties shared except for their very different masses and lifetimes. Lepton Universality Example Electromagnetic properties of muons and tau leptons Dirac magnetic moment are identical with those of electrons provided the ๐ ๐= S mass difference is taken into account. ๐โ Both the ๐ and ๐ decay by the weak interaction and we will return to this in a later lecture. where ๐โ is the mass of the lepton. 33 Also Charged Leptons and Quarks Same interaction strength for all charged leptons โ QED only cares about charge. ๐โ ๐โ ๐ถ ๐โ ๐โ ๐โ ๐ถ ๐ธ ๐โ ๐ถ ๐ธ ๐ธ Coupling less for quarks due to fractional charge. u ๐ ๐ถ ๐ ๐ธ u d ๐ ๐ถ ๐ ๐ธ d 34 QED Propagator Factor Propagator factor tells us about the contribution to the amplitude from an intermediate (or virtual) particle travelling through space and time. ๐โ ๐ถ Virtual Particles Do not have mass of a physical particle. q 2 ๏น m X2 ๏น E X2 ๏ญ p X2 Known as โoff โmass shellโ (e.g. not zero for photon) ๐โ Propagator 1 ๐2 ๐ธ ๐ถ ๐โ ๐โ โข Virtual photons have propagators proportional to 1 ๐ 2 , where ๐ is the four-momentum transfer between the vertices (or the four-momentum of the exchanged particle). โข Heavy bosons with mass ๐ have propagators 1 ๐ 2 โ ๐2 . 35 Feynman Factors: Summary ๏ Each part of a Feynman diagram has factors associated with it. Multiply them all together to get matrix element ๐. ๏ Initial and final state particles use wavefunction currents: ๏ถ Spin-0 bosons are plane waves. ๏ถ Spin-1/2 fermions have Dirac spinors. ๏ถ Spin-1 bosons have polarisation vectors ๐๐ . ๏ Vertices have dimensionless coupling constants. ๏ถ In the electromagnetic case, ๐ถ = ๐. ๏ถ In the strong interaction, ๐ถ๐ = ๐๐ (more later). ๏ Vertices have propagators, ๐๐ , which is the momentum transferred by boson. ๏ถ Virtual photon propagator is ๐ ๐๐ ๏ถ Virtual W/Z boson propagator is ๐ ๐๐ โ ๐ด๐ ๐พ or ๐ ๐๐ โ ๐ด๐ ๐ ๏ถ Virtual fermion propagator is ๐ธ๐ ๐๐ + ๐)/(๐๐ โ ๐๐ 36 Feynman Rules for QED External Lines Spin ๐ ๐ Spin ๐ Incoming particle ๐ข(๐) Outgoing particle ๐ข(๐) Incoming antiparticle ๐ฃ(๐) Outgoing antiparticle ๐ฃ(๐) Incoming photon ๐ ๐ (๐) Outgoing photon ๐ ๐ (๐)โ Internal Lines (propagators) Spin ๐ Spin ๐ Photon ๐ Vertex Factors Spin ๐ ๐ Matrix Element Fermion Taken from Thomson, page 124 โ๐๐๐๐ ๐ ๐2 โ๐ ๐พ ๐ ๐๐ + ๐ ๐ 2 โ ๐2 Fermion (charge โ ๐ ) ๐ โ๐๐๐พ ๐ โ โณ๐ Product of all factors โ๐โณ = ๐=1 37 Example: ๐โ ๐โ โ ๐โ ๐โ scattering ๐โ ๐1 ๐ ๐โ ๐3 ๐ = ๐1 โ ๐3 ๐ธ ๐2 ๐โ ๐ ๐4 ๐โ ๐ข ๐3 ๐๐๐พ ๐ ๐ข ๐1 electron current โ๐๐๐๐ ๐2 propagator ๐ข ๐4 ๐๐๐พ ๐ ๐ข ๐2 tau current In lowest order, the amplitude by applying the Feynman rules to the above diagram is therefore: ๐ โ๐โณ = ๐ข ๐3 ๐๐๐พ ๐ข ๐1 โ๐๐๐๐ ๐ ๐ข ๐ ๐๐๐พ ๐ข ๐2 4 2 ๐ 38 Higher Order Diagrams Tree level Processes (or Born Diagrams) = diagrams that contain no loops Higher Order Corrections (or Radiative Corrections) = loop diagrams ๐ + Lowest order diagram ๐ โ ๐๐ โ ๐ถ ๐ธ ๐ถ Order = number of vertices in each diagram ๐+ Any diagram of order ๐ gives a contribution of ๐ผ ๐ ๐ถ ๐โ ๐โ Second order diagrams: ๐ โ ๐๐ โ ๐ถ๐ ๐+ ๐+๐+ ๐ถ ๐ถ ๐โ ๐ถ ๐ถ + ๐โ ๐ธ ๐โ Total amplitude: ๐๐๐ = ๐ถ๐ด๐ณ๐ถ + ๐ถ๐ +.. ๐ถ ๐ถ ๐ถ ๐ ๐ด๐ ๐+ ๐ถ ๐โ + โฆ Jargon โ Leading Order (LO) + Next-to-Leading-Order (NLO) 39 LEP ๐+ ๐โ โ ๐ธ๐ธ(๐ธ) Example Born level diagrams + Excellent agreement with QED Radiative corrections Different for each experiment due to phase space 40 Alpha โ Fine Structure Constant o Need to take care with the Fine Structure Constant, ๐ถ, and its role as a coupling constant measuring the strength of the EM interaction. o ๐ผ was introduced by A. Sommerfield in 1916 to explain the fine structure of the energy levels of the hydrogen atom. In particle physics, it is not really a constant. o This is because in QED an electron can emit virtual photons, which form virtual e+/e- pairs which โscreenโ the electron. ๐โ Vacuum polarisation loops. Correction: ๐โ ๏ก (0) ๏ก (Q ) ๏ฝ 1 ๏ญ ๏๏ก (Q 2 ) ๐โ 2 OPAL, CERN-EP/98-108 + ๐ ๐ โ ๐+ ๐+ Alpha not really a constant! 1 137.035999074(44) 1 At Q2โmW2, ๐ผ~ 41 128 At Q2=0, ๐ผ= QED Renormalisation The strength of the coupling between a photon and an electron is determined by the coupling at the QED vertex, which until now we have assumed constant with value ๐. โข The value ๐ผ โ 1 137 is obtained from measurements of the static Coulomb potential in atomic physics. โข This is not the same as the strength of the coupling in Feynman diagrams, which can be written as ๐0 (and called the bare electron charge). โข The experimentally measured value of ๐ is the effective strength of the interaction which results from summing over all relevant higher order diagrams. โข There is an infinite set of higher-order corrections, including the ones belowโฆ Corrections to electron four vector current Correction to propagator (a) Lowest order (b) (c) (d) ๐ช ๐ 2 corrections to QED vertex (e) 42 QED Renormalisation There are no restrictions on the momentum, ๐, of the virtual particles and the self-energy terms include integrals of the form ๐๐ ๐, which is logarithmically divergent. Techniques to deal with this are beyond scope of this course, but basicallyโฆ.. 1. Use cut-off procedures in integrals. It turns out that this enables the calculations to be separated into two parts: a finite term and one which blows up. 2. Amazingly, all the divergent terms then appear as additions to the bare parameters such as mass (๐0 ), charge (๐0 ) and coupling constant (๐ผ0 ). i.e. ๐๐โ๐ฆ๐ ๐๐๐๐ = ๐0 + ๐ฟ๐ 3. The strategy is then to absorb the infinities into renormalisable masses and coupling constants. i.e. it means that we use the physical values as determined by experiment and NOT the values (๐0 , ๐0 , ๐ผ0 ) that appeared in the original Feynman rules that we wrote down. As already discussed, one consequence is that the coupling constant depends logarithmically on energy scale โนโนโนโนโนโนโนโน ๐ผ ๐2 ๐ผ(๐2 ) = ๐2 1 2 1 โ ๐ผ(๐ ) ln 2 3๐ ๐ 43 CLIC after LHC at CERN? CLIC= Compact Linear (e+e-) Collider 44 Electron Positron Annihilation Electron-positron colliders have been central to the development and understanding of the Standard Model. WHY? โข It is easier to accelerate protons to very high energies than leptons, but the detailed collision process of protons cannot be well controlled or selected. โข Electron positron colliders offer a well-defined initial state. โข The collision energy ๐ is known and it is tuneable (e.g. for scanning thresholds of particle production). โข Polarisation of electrons/positrons is possible. โข In proton collisions, the rate of unwanted collision processes is very high, whereas the point-like nature of leptons results in low backgrounds. โข Scattering of point-like particles can be calculated to very 45 high precision in theory. LEP Electroweak Measurements Example Cross-sections of electroweak Standard Model (SM) processes. Dots with error bars show the measurements, while curves show theoretical predictions based on SM. Precision results on fundamental properties of W boson & EW theory ๐๐ = 80.376 ± 0.033 ๐บ๐๐ ฮ๐ = 2.195 ± 0.083 ๐บ๐๐ ๐ต ๐ โ โ๐๐๐๐๐๐ = 67.41 ± 0.27% ๐ ๐ 1 = 0.984+0.018 โ0.020 ๐ ๐พ = 0.982 ± 0.042 ๐๐พ = โ0.022 ± 0.019 Electroweak measurements in electronpositron collisions at W-boson-pair energies at LEP, Physics Reports, 2013, in press 46 End CONTACT Professor Glenn Patrick email: [email protected] email: [email protected]