DOCSLIB.ORG

Scattering Amplitudes LECTURE 1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Scattering Amplitudes LECTURE 1 Scattering Amplitudes LECTURE 1 Jaroslav Trnka Walter Burke Institute for Theoretical Physics, Caltech Center for Quantum Mathematics and Physics (QMAP), UC Davis Asian winter school, Okinawa, January 2016 The first instance of this phenomenon is extremely simple and trivial. Consider an analog of the “factorization channel” diagram (2.22), but connecting two black vertices. Because these vertices require that all the λ’s be parallel, it makes no physical di↵erence how they are connected. And so, on-shell diagrams related by, e (2.28) represent the same on-shell form. Thus, we can collapse and re-expand any chain of connected black vertices in anyway we like; the same is obviously true for white vertices. Because of this, for some purposes it may be useful to define composite black and white vertices with any number of legs. By grouping black and white vertices together in this way, on-shell diagrams can always be made bipartite—with (internal) edges only connecting white with black vertices. We will, however, preferentially draw trivalent diagrams because of the fundamental role played by the three-particle amplitudes. There is also a more interesting equivalence between on-shell diagrams that will play an important role in our story. We can see this already in the BCFW represen- tation of the four-particle amplitude given above, (2.20). The picture is obviously not cyclically invariant—as a rotation would exchange its black and white vertices. But the four-particle amplitude of course is cyclically invariant; and so there is another generator of equivalences among on-shell diagrams, the “square move”, [80]: (2.29) The merger and square moves can be used to show the physical equivalence of many seemingly di↵erentScattering on-shell diagrams. For amplitudes instance, the following two diagrams generate physically equivalent on-shell forms: 2 1 3 6 (2.30) 7 5 4 –16– Overview of lectures ✤ Review of scattering amplitudes Motivation On-shell amplitudes, kinematics Recursion relations, unitarity methods ✤ Geometric formulation Toy model: N=4 Super Yang Mills theory On-shell diagrams Amplituhedron Motivation Quantum Field Theory (QFT) ✤ Our theoretical framework to describe Nature ✤ Compatible with two principles Special relativity Quantum mechanics Perturbative QFT (Dirac, Heisenberg, Pauli; Feynman, Dyson, Schwinger) ✤ Fields, Lagrangian, Path integral iS(A, , ,J) = 1 F F µ⌫ + i D m A e 4 µ⌫ D D D L − 6 − Z ✤ Feynman diagrams: pictures of particle interactions Perturbative expansion: trees, loops Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron Theory: ge =2 1928 Experiment: ge 2 ⇠ Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron Theory: ge =2.00232 1947 Experiment: ge =2.0023 Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron 1957 Theory: ge =2.0023193 1972 Experiment: ge =2.00231931 Great success of QFT ✤ QFT has passed countless tests in last 70 years ✤ Example: Magnetic dipole moment of electron Theory: ge =2.0023193044 1990 Experiment: ge =2.00231930438 Dualities ✤ At strong coupling: perturbative expansion breaks ✤ Surprises: dual to weakly coupled theory Gauge-gauge dualities (Montonen-Olive 1977, Seiberg-Witten 1994) Gauge-gravity duality (Maldacena 1997) Incomplete picture ✤ Our picture of QFT is incomplete ✤ Also, tension with gravity and cosmology If there is a new way of thinking about QFT, it must be seen even at weak coupling ✤ Explicit evidence: scattering amplitudes Colliders at high energies ✤ Proton scattering at high energies LHC - gluonic factory ✤ Needed: amplitudes of gluons for higher multiplicities gg gg . g ! Early 80s ✤ Status of the art: gg ggg ! Brute force calculation 24 pages of result (k k )(✏ k )(✏ ✏ )(✏ ✏ ) 1 · 4 2 · 1 1 · 3 4 · 5 New collider ✤ 1983: Superconducting Super Collider approved ✤ Energy 40 TeV: many gluons! ✤ Demand for calculations, next on the list: gg gggg ! Parke-Taylor formula (Parke, Taylor 1985) ✤ Process gg gggg ! ✤ 220 Feynman diagrams, 100 pages of calculations ⇠ ✤ 1985: Paper with 14 pages of result Parke-Taylor formula ✤ Process gg gggg ! ✤ 220 Feynman diagrams, 100 pages of calculations ⇠ Parke-Taylor formula Parke-Taylor formula ✤ Within a year they realized Spinor-helicity variables 3 µ µ ˜ 12 p = σaa˙ λaλa˙ 6 = h i (1) (2) 23 34 45 56 61 12 = ✏ λa λ M h ih ih ih ih i h i ab b [12] = ✏ λ˜(1)λ˜(2) a˙ b˙ a˙ b˙ (Mangano, Parke, Xu 1987) Parke-Taylor formula m Fermi National Accelerator Laboratory FERMILAB-Pub-86/42-T March, 1986 ✤ Within a year they realized AN AMPLITUDE FOR n GLUON SCATTERING 12 3 = h i Mn 23 34 45 ... n1 h ih ih i h i STEPHEN 3. PARKE and T. R. TAYLOR Fermi National Accelerator Laboratory P.O. Box 500, Batavia, IL 60510. Abstract A non-trivial, squared helicity amplitude is given for the scattering of an arbitrary number of gluons to lowest order in the coupling constant and to leading order in the number of colors. *rated by Unlversitles Research Association Inc. under contract with the United States Department 01 Energy Birth of amplitudes ✤ New field in theoretical particle physics New methods and Uncovering new effective calculations structures in QFT “Road map” Explicit New structure New method calculation discovered which exploits it What are scattering amplitudes Scattering process ✤ Interaction of elementary particles ✤ Initial state i and final state f | i | i ✤ Scattering amplitude if = i f M h | i + + + ✤ Example: e e − e e − or e e − γγ etc. ! ! 2 ✤ Cross section: σ = d ⌦ probability |M| Z Scattering amplitude in QFT ✤ Scattering amplitude depends on types of particles and their momenta = F (p ,s ) Mif i i ✤ Theoretical framework: calculated in some QFT ✤ Specified by Lagrangian: interactions and couplings = ( ,g ) L L Oj k ✤ Example: QED = e Aµ Lint µ Perturbation theory ✤ Weakly coupled theory = + g + g2 + g3 + ... M M0 M1 M2 M3 ✤ Representation in terms of Feynman diagrams ✤ Perturbative expansion = loop expansion tree 1 loop 2 loop = + − + − + ... M M M M Divergencies ✤ Loop diagrams are generally UV divergent 4 1 d ` log ⇤ ⇠ (`2 + m2)[(` + p)2 + m2] ⇠ Z1 ✤ IR divergencies: physical effects, cancel in cross section ✤ Dimensional regularization: calculate integrals in 4+✏ 1 dimensions Divergencies ⇠ ✏k Renormalizable theories ✤ Absorb UV divergencies: counter terms Finite number of them: renormalizable theory Infinite number: non-renormalizable theory ✤ Mostly only renormalizable theories are interesting ✤ Exceptions: effective field theories Example: Chiral perturbation theory - derivative expansion = + + + + ... L L2 L4 L6 L8 Different loop orders are mixed Analytic structure of amplitudes ✤ Tree-level: rational functions g2 2 Only poles ⇠ (p1 + p2) ✤ Loops: polylogarithms and more complicated log2(s/t) Branch cuts ⇠ Kinematics of massless particles Massless particles ✤ Parameters of elementary particles of spin S Spin s =( S, S) − On-shell (physical) particle Mass m p2 = m2 Momentum pµ ✤ Massless particle: m =0 p 2 =0 spin = helicity: only two extreme values h = S, S {− } Example: photon h =(+, ) − s =0 missing Spin functions ✤ At high energies particles are massless Fundamental laws reveal there ✤ Spin degrees of freedom: spin function s=0: Scalar - no degrees of freedom s=1/2: Fermion - spinor u s=1: Vector - polarization vector ✏µ s=2: Tensor - polarization tensor hµ⌫ Spin functions ✤ At high energies particles are massless Fundamental laws reveal there ✤ Spin degrees of freedom: spin function s=0: Scalar - no degrees of freedom s=1/2: Fermion - spinor u s=1: Vector - polarization vector ✏µ s=2: Tensor - polarization tensor hµ⌫ Polarization vectors ✤ Spin 1 particle is described by vector ✏µ 2 degrees of freedom 4 degrees of freedom ✤ Null condition: ✏ ✏ ⇤ =0 3 degrees of freedom left · ✤ We further impose: ✏ p =0 Identification · ✏ ✏ + ↵p Feynman diagrams depend on ↵ µ ⇠ µ µ gauge dependence Spinor helicity variables ✤ Standard SO(3,1) notation for momentum µ p =(p0,p1,p2,p3) pj R 2 ✤ We use SL(2,C) representation p + ip p + p p = σµ p = 0 1 2 3 ab ab µ p p p ip ✓ 2 − 3 0 − 1 ◆ 2 On-shell: p =det(pab)=0 Rank (pab)=1 Spinor helicity variables ✤ We can then write pab = λab ✤ SL(2,C): dotted notation pab˙ = λaλb˙ λ is complex conjugate of λ e ✤ Little groupe transformation λ tλ leaves momentum ! 1 p p λ λ unchanged ! ! t e e 3 degrees of freedom Spinor helicity variables ✤ Momentum invariant (p + p )2 =(p p ) 1 2 1 · 2 µ µ pµ = σµ λ λ p1 = σaa˙ λ1aλ1˙a 2 bb˙ 2b 2b˙ ✤ Plugging for momentae e µ (p p )=(σ σ ˙ )(λ λ )(λ λ ˙ ) 1 · 2 aa˙ µbb 1a 2b 1˙a 2b =(✏ λ λ )(✏ ˙ λ λ ˙ ) ✏ab ✏a˙ b˙ ab e1a e2b a˙ b 1˙a 2b Define: 12 ✏abλ1aλ2b [12] e✏ ˙ λe1˙aλ ˙ h i⌘ ⌘ a˙ b 2b e e Invariant products ✤ Momentum invariant s =(p + p )2 = ij [ij] ij i j h i Square brackets Angle brackets ij = ✏ λ λ [ij]=✏ λ λ h i ab ia jb a˙ b˙ ia˙ jb˙ ✤ Antisymmetry e e 21 = 12 [21] = [12] h i h i − ✤ More momenta (p + p + p )2 = 12 [12] + 23 [23] + 13 [13] 1 2 3 h i h i h i Invariant products ✤ Shouten identity 13 24 = 12 34 + 14 23 h ih i h ih i h ih i ✤ Mixed brackets 1 2+34] 12 [24] + 13 [34] h | | ⌘h i h i ✤ Momentum conservation n Non-trivial conditions: λiaλia˙ =0 i=1 Quadratic relation X e between components Polarization vectors ✤ Two polarization vectors λ ⌘ µ µ ⌘aλa˙ ✏µ = σµ a a˙ ✏+ = σaa˙ aa˙ ⌘ − [⌘ λ] Note that h ei e where are auxiliary spinors (✏+ ✏ )=1 ee · − ✤ Freedom in choice of ⌘ , ⌘ corresponds to ✏µ ✏µ + ↵ pµ ⇠ e ✤ Gauge redundancy of Feynman diagrams Scaling
Load more

Recommended publications
  • Glossary Physics (I-Introduction)
    1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay.
    [Show full text]
  • 12 Light Scattering AQ1
    12 Light Scattering AQ1 Lev T. Perelman CONTENTS 12.1 Introduction ......................................................................................................................... 321 12.2 Basic Principles of Light Scattering ....................................................................................323 12.3 Light Scattering Spectroscopy ............................................................................................325 12.4 Early Cancer Detection with Light Scattering Spectroscopy .............................................326 12.5 Confocal Light Absorption and Scattering Spectroscopic Microscopy ............................. 329 12.6 Light Scattering Spectroscopy of Single Nanoparticles ..................................................... 333 12.7 Conclusions ......................................................................................................................... 335 Acknowledgment ........................................................................................................................... 335 References ...................................................................................................................................... 335 12.1 INTRODUCTION Light scattering in biological tissues originates from the tissue inhomogeneities such as cellular organelles, extracellular matrix, blood vessels, etc. This often translates into unique angular, polari- zation, and spectroscopic features of scattered light emerging from tissue and therefore information about tissue
    [Show full text]
  • Solutions 7: Interacting Quantum Field Theory: Λφ4
    QFT PS7 Solutions: Interacting Quantum Field Theory: λφ4 (4/1/19) 1 Solutions 7: Interacting Quantum Field Theory: λφ4 Matrix Elements vs. Green Functions. Physical quantities in QFT are derived from matrix elements M which represent probability amplitudes. Since particles are characterised by having a certain three momentum and a mass, and hence a specified energy, we specify such physical calculations in terms of such \on-shell" values i.e. four-momenta where p2 = m2. For instance the notation in momentum space for a ! scattering in scalar Yukawa theory uses four-momenta labels p1 and p2 flowing into the diagram for initial states, with q1 and q2 flowing out of the diagram for the final states, all of which are on-shell. However most manipulations in QFT, and in particular in this problem sheet, work with Green func- tions not matrix elements. Green functions are defined for arbitrary values of four momenta including unphysical off-shell values where p2 6= m2. So much of the information encoded in a Green function has no obvious physical meaning. Of course to extract the corresponding physical matrix element from the Greens function we would have to apply such physical constraints in order to get the physics of scattering of real physical particles. Alternatively our Green function may be part of an analysis of a much bigger diagram so it represents contributions from virtual particles to some more complicated physical process. ∗1. The full propagator in λφ4 theory Consider a theory of a real scalar field φ 1 1 λ L = (@ φ)(@µφ) − m2φ2 − φ4 (1) 2 µ 2 4! (i) A theory with a gφ3=(3!) interaction term will not have an energy which is bounded below.
    [Show full text]
  • The Basic Interactions Between Photons and Charged Particles With
    Outline Chapter 6 The Basic Interactions between • Photon interactions Photons and Charged Particles – Photoelectric effect – Compton scattering with Matter – Pair productions Radiation Dosimetry I – Coherent scattering • Charged particle interactions – Stopping power and range Text: H.E Johns and J.R. Cunningham, The – Bremsstrahlung interaction th physics of radiology, 4 ed. – Bragg peak http://www.utoledo.edu/med/depts/radther Photon interactions Photoelectric effect • Collision between a photon and an • With energy deposition atom results in ejection of a bound – Photoelectric effect electron – Compton scattering • The photon disappears and is replaced by an electron ejected from the atom • No energy deposition in classical Thomson treatment with kinetic energy KE = hν − Eb – Pair production (above the threshold of 1.02 MeV) • Highest probability if the photon – Photo-nuclear interactions for higher energies energy is just above the binding energy (above 10 MeV) of the electron (absorption edge) • Additional energy may be deposited • Without energy deposition locally by Auger electrons and/or – Coherent scattering Photoelectric mass attenuation coefficients fluorescence photons of lead and soft tissue as a function of photon energy. K and L-absorption edges are shown for lead Thomson scattering Photoelectric effect (classical treatment) • Electron tends to be ejected • Elastic scattering of photon (EM wave) on free electron o at 90 for low energy • Electron is accelerated by EM wave and radiates a wave photons, and approaching • No
    [Show full text]
  • Optical Light Manipulation and Imaging Through Scattering Media
    Optical light manipulation and imaging through scattering media Thesis by Jian Xu In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2020 Defended September 2, 2020 ii © 2020 Jian Xu ORCID: 0000-0002-4743-2471 All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS Foremost, I would like to express my sincere thanks to my advisor Prof. Changhuei Yang, for his constant support and guidance during my PhD journey. He created a research environment with a high degree of freedom and encouraged me to explore fields with a lot of unknowns. From his ethics of research, I learned how to be a great scientist and how to do independent research. I also benefited greatly from his high standards and rigorous attitude towards work, with which I will keep in my future path. My committee members, Professor Andrei Faraon, Professor Yanbei Chen, and Professor Palghat P. Vaidyanathan, were extremely supportive. I am grateful for the collaboration and research support from Professor Andrei Faraon and his group, the helpful discussions in optics and statistics with Professor Yanbei Chen, and the great courses and solid knowledge of signal processing from Professor Palghat P. Vaidyanathan. I would also like to thank the members from our lab. Dr. Haojiang Zhou introduced me to the wavefront shaping field and taught me a lot of background knowledge when I was totally unfamiliar with the field. Dr. Haowen Ruan shared many innovative ideas and designs with me, from which I could understand the research field from a unique perspective.
    [Show full text]
  • 3 Scattering Theory
    3 Scattering theory In order to find the cross sections for reactions in terms of the interactions between the reacting nuclei, we have to solve the Schr¨odinger equation for the wave function of quantum mechanics. Scattering theory tells us how to find these wave functions for the positive (scattering) energies that are needed. We start with the simplest case of finite spherical real potentials between two interacting nuclei in section 3.1, and use a partial wave anal- ysis to derive expressions for the elastic scattering cross sections. We then progressively generalise the analysis to allow for long-ranged Coulomb po- tentials, and also complex-valued optical potentials. Section 3.2 presents the quantum mechanical methods to handle multiple kinds of reaction outcomes, each outcome being described by its own set of partial-wave channels, and section 3.3 then describes how multi-channel methods may be reformulated using integral expressions instead of sets of coupled differential equations. We end the chapter by showing in section 3.4 how the Pauli Principle re- quires us to describe sets identical particles, and by showing in section 3.5 how Maxwell’s equations for electromagnetic field may, in the one-photon approximation, be combined with the Schr¨odinger equation for the nucle- ons. Then we can describe photo-nuclear reactions such as photo-capture and disintegration in a uniform framework. 3.1 Elastic scattering from spherical potentials When the potential between two interacting nuclei does not depend on their relative orientiation, we say that this potential is spherical. In that case, the only reaction that can occur is elastic scattering, which we now proceed to calculate using the method of expansion in partial waves.
    [Show full text]
  • Lecture 7: Propagation, Dispersion and Scattering
    Satellite Remote Sensing SIO 135/SIO 236 Lecture 7: Propagation, Dispersion and Scattering Helen Amanda Fricker Energy-matter interactions • atmosphere • study region • detector Interactions under consideration • Interaction of EMR with the atmosphere scattering absorption • Interaction of EMR with matter Interaction of EMR with the atmosphere • EMR is attenuated by its passage through the atmosphere via scattering and absorption • Scattering -- differs from reflection in that the direction associated with scattering is unpredictable, whereas the direction of reflection is predictable. • Wavelength dependent • Decreases with increase in radiation wavelength • Three types: Rayleigh, Mie & non selective scattering Atmospheric Layers and Constituents JeJnesnesne n2 020505 Major subdivisions of the atmosphere and the types of molecules and aerosols found in each layer. Atmospheric Scattering Type of scattering is function of: 1) the wavelength of the incident radiant energy, and 2) the size of the gas molecule, dust particle, and/or water vapor droplet encountered. JJeennsseenn 22000055 Rayleigh scattering • Rayleigh scattering is molecular scattering and occurs when the diameter of the molecules and particles are many times smaller than the wavelength of the incident EMR • Primarily caused by air particles i.e. O2 and N2 molecules • All scattering is accomplished through absorption and re-emission of radiation by atoms or molecules in the manner described in the discussion on radiation from atomic structures. It is impossible to predict the direction in which a specific atom or molecule will emit a photon, hence scattering. • The energy required to excite an atom is associated with short- wavelength, high frequency radiation. The amount of scattering is inversely related to the fourth power of the radiation's wavelength (!-4).
    [Show full text]
  • 11. Light Scattering
    11. Light Scattering Coherent vs. incoherent scattering Radiation from an accelerated charge Larmor formula Why the sky is blue Rayleigh scattering Reflected and refracted beams from water droplets Rainbows Coherent vs. Incoherent light scattering Coherent light scattering: scattered wavelets have nonrandom relative phases in the direction of interest. Incoherent light scattering: scattered wavelets have random relative phases in the direction of interest. Example: Randomly spaced scatterers in a plane Incident Incident wave wave Forward scattering is coherent— Off-axis scattering is incoherent even if the scatterers are randomly when the scatterers are randomly arranged in the plane. arranged in the plane. Path lengths are equal. Path lengths are random. Coherent vs. Incoherent Scattering N Incoherent scattering: Total complex amplitude, Aincoh exp(j m ) (paying attention only to the phase m1 of the scattered wavelets) The irradiance: NNN2 2 IAincoh incohexp( j m ) exp( j m ) exp( j n ) mmn111 NN NN exp[jjN (mn )] exp[ ( mn )] mn11mn mn 11 mn m = n mn Coherent scattering: N 2 2 Total complex amplitude, A coh 1 N . Irradiance, I A . So: Icoh N m1 So incoherent scattering is weaker than coherent scattering, but not zero. Incoherent scattering: Reflection from a rough surface A rough surface scatters light into all directions with lots of different phases. As a result, what we see is light reflected from many different directions. We’ll see no glare, and also no reflections. Most of what you see around you is light that has been incoherently scattered. Coherent scattering: Reflection from a smooth surface A smooth surface scatters light all into the same direction, thereby preserving the phase of the incident wave.
    [Show full text]
  • Interactions of Photons with Matter
    22.55 “Principles of Radiation Interactions” Interactions of Photons with Matter • Photons are electromagnetic radiation with zero mass, zero charge, and a velocity that is always c, the speed of light. • Because they are electrically neutral, they do not steadily lose energy via coulombic interactions with atomic electrons, as do charged particles. • Photons travel some considerable distance before undergoing a more “catastrophic” interaction leading to partial or total transfer of the photon energy to electron energy. • These electrons will ultimately deposit their energy in the medium. • Photons are far more penetrating than charged particles of similar energy. Energy Loss Mechanisms • photoelectric effect • Compton scattering • pair production Interaction probability • linear attenuation coefficient, µ, The probability of an interaction per unit distance traveled Dimensions of inverse length (eg. cm-1). −µ x N = N0 e • The coefficient µ depends on photon energy and on the material being traversed. µ • mass attenuation coefficient, ρ The probability of an interaction per g cm-2 of material traversed. Units of cm2 g-1 ⎛ µ ⎞ −⎜ ⎟()ρ x ⎝ ρ ⎠ N = N0 e Radiation Interactions: photons Page 1 of 13 22.55 “Principles of Radiation Interactions” Mechanisms of Energy Loss: Photoelectric Effect • In the photoelectric absorption process, a photon undergoes an interaction with an absorber atom in which the photon completely disappears. • In its place, an energetic photoelectron is ejected from one of the bound shells of the atom. • For gamma rays of sufficient energy, the most probable origin of the photoelectron is the most tightly bound or K shell of the atom. • The photoelectron appears with an energy given by Ee- = hv – Eb (Eb represents the binding energy of the photoelectron in its original shell) Thus for gamma-ray energies of more than a few hundred keV, the photoelectron carries off the majority of the original photon energy.
    [Show full text]
  • 3. Interacting Fields
    3. Interacting Fields The free field theories that we’ve discussed so far are very special: we can determine their spectrum, but nothing interesting then happens. They have particle excitations, but these particles don’t interact with each other. Here we’ll start to examine more complicated theories that include interaction terms. These will take the form of higher order terms in the Lagrangian. We’ll start by asking what kind of small perturbations we can add to the theory. For example, consider the Lagrangian for a real scalar field, 1 1 λ = @ @µφ m2φ2 n φn (3.1) L 2 µ − 2 − n! n 3 X≥ The coefficients λn are called coupling constants. What restrictions do we have on λn to ensure that the additional terms are small perturbations? You might think that we need simply make “λ 1”. But this isn’t quite right. To see why this is the case, let’s n ⌧ do some dimensional analysis. Firstly, note that the action has dimensions of angular momentum or, equivalently, the same dimensions as ~.Sincewe’veset~ =1,using the convention described in the introduction, we have [S] = 0. With S = d4x ,and L [d4x]= 4, the Lagrangian density must therefore have − R [ ]=4 (3.2) L What does this mean for the Lagrangian (3.1)? Since [@µ]=1,wecanreado↵the mass dimensions of all the factors to find, [φ]=1 , [m]=1 , [λ ]=4 n (3.3) n − So now we see why we can’t simply say we need λ 1, because this statement only n ⌧ makes sense for dimensionless quantities.
    [Show full text]
  • Electron Scattering Processes in Non-Monochromatic and Relativistically Intense Laser Fields
    atoms Review Electron Scattering Processes in Non-Monochromatic and Relativistically Intense Laser Fields Felipe Cajiao Vélez * , Jerzy Z. Kami ´nskiand Katarzyna Krajewska Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland; [email protected] (J.Z.K.); [email protected] (K.K.) * Correspondence: [email protected]; Tel.: +48-22-5532920 Received: 31 January 2019; Accepted: 1 March 2019; Published: 6 March 2019 Abstract: The theoretical analysis of four fundamental laser-assisted non-linear scattering processes are summarized in this review. Our attention is focused on Thomson, Compton, Møller and Mott scattering in the presence of intense electromagnetic radiation. Depending on the phenomena under considerations, we model the laser field as a single laser pulse of ultrashort duration (for Thomson and Compton scattering) or non-monochromatic trains of pulses (for Møller and Mott scattering). Keywords: laser-assisted quantum processes; non-linear scattering; ultrashort laser pulses 1. Introduction Scattering theory is the part of theoretical physics in which interactions of particles and waves are investigated at remote times and large distances, as compared with typical time and size scales of probed systems. For this reason, scattering theory is the most effective and, in many cases, the only method of analyzing such diverse systems as the micro-world or the whole universe. Not surprisingly, the investigation of scattering phenomena has been playing a central role in physics since the end of the nineteenth century, starting from the Rayleigh’s explanation of why the sky is blue, up to modern medical applications of computerized tomography, the very recent discovery of the Higgs particle and the detection of gravitational waves.
    [Show full text]
  • Introduction to Small-Angle X-Ray Scattering
    Introduction to Small-Angle X-ray Scattering Thomas M. Weiss Stanford University, SSRL/SLAC, BioSAXS beamline BL 4-2 BioSAXS Workshop, March 28-30, 2016 Sizes and Techniques Diffraction and Scattering Scattering of X-rays from a single electron I e Thomson formula for the scattered intensity from a single electron 2 I 0 2 1 cos( 2 ) 1 I e r0 I 0 2 r 2 2 e 15 r0 2.817 10 m I : Intensity of incoming X - rays 2 0 mc 2 : angle of observatio n Classical electron radius I e : Intensity of scattered X - rays The Thomson formula plays a central role for all scattering calculations involving absolute intensities. Typically calculated intensities of a given sample will be expressed in terms of the scattering of an isolated electron substituted for the sample. In small angle scattering the slight angle dependence (the so-called polarization factor) in the Thomson formula can be neglected. Interference of waves • waves have and amplitude and phase • interference leads to fringe pattern (e.g. water waves) • the fringe pattern contains the information on the position of the sources (i.e. structure) • in X-ray diffraction the intensities (not the amplitudes) of the fringes are measured “phase problem” constructive destructive Scattering from two (and more) electrons scattering vector q k k 0 two electrons 2 4 sin with k s and q q 2 F (q ) f exp( iq r ) f (1 exp( iq r )) e i e i 1 Note: … generalized to N electrons F(q) is the Fourier N Transform of the F (q ) f exp( iq r ) spatial distribution e i i 1 of the electrons … averaged over all orientations N sin( qr ) F (q ) f e i 1 qr using the continuous (radial) distribution ( r ) of the electron cloud in an atom sin( qr ) F (q ) f (q ) dr (r )r 2 with f (0 ) Z qr 0 atomic scattering factor Scattering from Molecules The scattering amplitude or form factor, F(q), of an isolated molecule with N atoms can be determined in an analogous manner: N F (q ) f (q ) exp( iq r ) i.e.
    [Show full text]