TASI Lectures on Scattering Amplitudes
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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. -
Beyond Feynman Diagrams Lecture 2
Beyond Feynman Diagrams Lecture 2 Lance Dixon Academic Training Lectures CERN April 24-26, 2013 Modern methods for trees 1. Color organization (briefly) 2. Spinor variables 3. Simple examples 4. Factorization properties 5. BCFW (on-shell) recursion relations Beyond Feynman Diagrams Lecture 2 April 25, 2013 2 How to organize gauge theory amplitudes • Avoid tangled algebra of color and Lorentz indices generated by Feynman rules structure constants • Take advantage of physical properties of amplitudes • Basic tools: - dual (trace-based) color decompositions - spinor helicity formalism Beyond Feynman Diagrams Lecture 2 April 25, 2013 3 Color Standard color factor for a QCD graph has lots of structure constants contracted in various orders; for example: Write every n-gluon tree graph color factor as a sum of traces of matrices T a in the fundamental (defining) representation of SU(Nc): + all non-cyclic permutations Use definition: + normalization: Beyond Feynman Diagrams Lecture 2 April 25, 2013 4 Double-line picture (’t Hooft) • In limit of large number of colors Nc, a gluon is always a combination of a color and a different anti-color. • Gluon tree amplitudes dressed by lines carrying color indices, 1,2,3,…,Nc. • Leads to color ordering of the external gluons. • Cross section, summed over colors of all external gluons = S |color-ordered amplitudes|2 • Can still use this picture at Nc=3. • Color-ordered amplitudes are still the building blocks. • Corrections to the color-summed cross section, can be handled 2 exactly, but are suppressed by 1/ Nc Beyond Feynman Diagrams Lecture 2 April 25, 2013 5 Trace-based (dual) color decomposition For n-gluon tree amplitudes, the color decomposition is momenta helicities color color-ordered subamplitude only depends on momenta. -
Maximally Helicity Violating Amplitudes
Maximally Helicity Violating amplitudes Andrew Lifson ID:22635416 Supervised by Assoc. Prof. Peter Skands School of Physics & Astronomy A thesis submitted in partial fulfilment for the degree of Bachelor of Science Advanced with Honours November 13, 2015 Abstract An important aspect in making quantum chromodynamics (QCD) predictions at the Large Hadron Collider (LHC) is the speed of the Monte Carlo (MC) event generators which are used to calculate them. MC event generators simulate a pure QCD collision at the LHC by first using fixed-order perturbation theory to calculate the hard (i.e. energetic) 2 ! n scatter, and then adding radiative corrections to this with a parton shower approximation. The 2 ! n scatter is traditionally calculated by summing Feynman diagrams, however the number of Feynman diagrams has a stronger than factorial growth with the number of final-state particles, hence this method quickly becomes infeasible. We can simplify this calculation by considering helicity amplitudes, in which each particle has its spin either aligned or anti-aligned with its direction of motion i.e. each particle has a specific helicity. In particular, it is well-known that the helicity amplitude for the maximally helicity violating (MHV) configuration is remarkably simple to calculate. In this thesis we describe the physics of the MHV amplitude, how we could use it within a MC event generator, and describe a program we wrote called VinciaMHV which calculates the MHV amplitude for the process qg ! q + ng for n = 1; 2; 3; 4. We tested the speed of VinciaMHV against MadGraph4 and found that VinciaMHV calculates the MHV amplitude significantly faster, especially for high particle multiplicities. -
MHV Amplitudes in N=4 SUSY Yang-Mills Theory and Quantum Geometry of the Momentum Space
Outline Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes Towards the stringy interpretation of the loop MHV amplitudes. MHV amplitudes in N=4 SUSY Yang-Mills theory and quantum geometry of the momentum space Alexander Gorsky ITEP GGI, Florence - 2008 (to appear) Alexander Gorsky MHV amplitudes in N=4 SUSY Yang-Mills theory and quantum geometry of the momentum space Outline Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes Towards the stringy interpretation of the loop MHV amplitudes. Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes Towards the stringy interpretation of the loop MHV amplitudes. Alexander Gorsky MHV amplitudes in N=4 SUSY Yang-Mills theory and quantum geometry of the momentum space Outline Introduction and tree MHV amplitudes BDS conjecture and amplitude - Wilson loop correspondence c=1 string example and fermionic representation of amplitudes Quantization of the moduli space On the fermionic representation of the loop MHV amplitudes -
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 -
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. -
Scattering in N = 4 Super Yang–Mills and N = 8 Supergravity
Scattering in = 4 super Yang{Mills and = 8 supergravity N N Thesis by Enrico Herrmann In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2017 Defended May 15, 2017 ii c 2017 Enrico Herrmann ORCID: 0000-0002-3983-2993 All rights reserved iii Acknowledgments During my time as a graduate student at Caltech I had the pleasure to interact with a number of remarkable scientists. First and foremost I would like to thank my advisor, Mark Wise, for taking me on as a graduate student during my first year at Caltech. Even more so, I am thankful that he gave me the opportunity to follow my own research interests and for all his support along the way. Mark always had an open ear and good advice concerning practical matters like organizing travel funding or navigating the academic life in general. I am grateful that he also nominated me for a Dominic Orr Fellowship, which allowed me a one year break from all teaching duties. Along with Mark, I would like to thank my co-advisor, Jaroslav Trnka, from whom I learned an enormous amount about scattering amplitudes and physics in general. I very much appreciate all his invaluable support and encouragement throughout my graduate studies. Without him and his generous help, many of my conference visits and trips would not have been possible. In all the workshops he co-organized, Jaroslav enabled young scientists to present their research. In many respects, Jaroslav is my scientific role model. In addition to the mentorship I have received by Mark and Jaroslav, I have benefited significantly from the interaction with my collaborators Zvi Bern, Jacob Bourjaily, Sean Litsey, James Stankowicz, and Jaroslav Trnka. -
Scattering Amplitudes in Quantum Field Theory
Scattering Amplitudes in Quantum Field Theory European Physical Society Meeting Stockholm July 24, 2013 Zvi Bern, UCLA W q e º q g0 g g g g 1 Outline 1) Remarkable progress in scattering amplitudes. 2) Brief summary of new advances and ideas. 3) Example: applications to LHC physics. 4) Example: A duality between color and kinematics. 5) Example: UV surprises in supergravity theories. 2 Scattering amplitudes Scattering of elementary particles is fundamental to our ability to unravel microscopic laws of nature. Arrival of the Large Hadron Collider raises importance of collider physics and scattering amplitudes. Here we give some examples of advances of past few years in understanding and calculating scattering in quantum field theory. 3 Major Advance in Scattering Amplitudes “Impossible calculations” of scattering amplitudes in gauge and gravity theories now commonplace. A few highlights from past year: • Constructing large chunks of the scattering amplitudes of N = 4 super-Yang-Mills theory, towards a full construction. Alday, Arkani-Hamed, Basso, Bourjaily, Cachazo, Caron-Huot, Dixon, Duhr, Gehrmann, Golden, Goncharov, He, Henn, Heslop, Huber, Johansson, Kosower, Larsen, Lipstein, Lipatov, Maldacena, Mason, Pennington, Postnikov, Sikorowski, Sever, Spradlin, Trnka, Vergu, Vieira, Volovich and many others • New remarkable representations of gravity amplitudes inspired by twistor string theory. Adamo, Cheung, Hodges, Cachazo, Geyer, Mason, Skinner, etc • Advances in constructing string theory scattering amplitudes with large numbers of external legs. Broedel, Drummond, Green, Mafra, Schlotterer, Stieberger, Taylor, Ragousy, Terasoma, etc • Relations between gravity and gauge theory amplitudes. ZB, Bjerrum-Bohr, Carrasco, Davies, Dennen, O’Connell, Huang, Johansson ,Monteiro, Roiban , etc • NLO QCD multijet processes for LHC physics. -
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 -
Loop Amplitudes from MHV Diagrams - Part 1
Loop Amplitudes from MHV Diagrams - part 1 - Gabriele Travaglini Queen Mary, University of London Brandhuber, Spence, GT hep-th/0407214, hep-th/0510253 Bedford, Brandhuber, Spence, GT hep-th/0410280, hep-th/0412108 Brandhuber, McNamara, Spence, GT in progress 2 HP Workshop, Zürich, September 2006 Outline • Motivations and basic formalism • MHV diagrams (Cachazo, Svrcek,ˇ Witten) - Loop MHV diagrams (Brandhuber, Spence, GT) - Applications: One-loop MHV amplitudes in N=4, N=1 & non-supersymmetric Yang-Mills • Proof at one-loop: Andreas’s talk on Friday Motivations • Simplicity of scattering amplitudes unexplained by textbook Feynman diagrams - Parke-Taylor formula for Maximally Helicity Violating amplitude of gluons (helicities are a permutation of −−++ ....+) • New methods account for this simplicity, and allow for much more efficient calculations • LHC is coming ! Towards simplicity • Colour decomposition (Berends, Giele; Mangano, Parke, Xu; Mangano; Bern, Kosower) • Spinor helicity formalism (Berends, Kleiss, De Causmaecker, Gastmans, Wu; De Causmaecker, Gastmans, Troost, Wu; Kleiss, Stirling; Xu, Zhang, Chang; Gunion, Kunszt) Colour decomposition • Main idea: disentangle colour • At tree level, Yang-Mills interactions are planar tree a a A ( p ,ε ) = Tr(T σ1 T σn) A(σ(p ,ε ),...,σ(p ,ε )) { i i} ∑ ··· 1 1 n n σ Colour-ordered partial amplitude - Include only diagrams with fixed cyclic ordering of gluons - Analytic structure is simpler • At loop level: multi-trace contributions - subleading in 1/N Spinor helicity formalism • Consider a null vector p µ µ µ = ( ,! ) • Define p a a ˙ = p µ σ a a ˙ where σ 1 σ • If p 2 = 0 then det p = 0 ˜ ˜ Hence p aa˙ = λaλa˙ · λ (λ) positive (negative) helicity • spinors a b a˙ b˙ Inner products 12 : = ε ab λ λ , [12] := ε ˙λ˜ λ˜ • ! " 1 2 a˙b 1 2 Parke-Taylor formula 4 + + i j A (1 ...i− .. -
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. -
Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories
Unitarity Methods for Scattering Amplitudes in Perturbative Gauge Theories by MASSIMILIANO VINCON A report presented for the examination for the transfer of status to the degree of Doctor of Philosophy of the University of London Thesis Supervisor DR ANDREAS BRANDHUBER Centre for Research in String Theory Department of Physics Queen Mary University of London Mile End Road London E1 4NS United Kingdom The strong do what they want and the weak suffer as they must.1 1Thucydides. Declaration I hereby declare that the material presented in this thesis is a representation of my own personal work unless otherwise stated and is a result of collaborations with Andreas Brandhuber and Gabriele Travaglini. Massimiliano Vincon Abstract In this thesis, we discuss novel techniques used to compute perturbative scattering am- plitudes in gauge theories. In particular, we focus on the use of complex momenta and generalised unitarity as efficient and elegant tools, as opposed to standard textbook Feynman rules, to compute one-loop amplitudes in gauge theories. We consider scatter- ing amplitudes in non-supersymmetric and supersymmetric Yang-Mills theories (SYM). After introducing some of the required mathematical machinery, we give concrete exam- ples as to how generalised unitarity works in practice and we correctly reproduce the n- gluon one-loop MHV amplitude in non-supersymmetric Yang-Mills theory with a scalar running in the loop, thus confirming an earlier result obtained using MHV diagrams: no independent check of this result had been carried out yet. Non-supersymmetric scat- tering amplitudes are the most difficult to compute and are of great importance as they are part of background processes at the LHC.