Non-Renormalization Theorems Without Supersymmetry
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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 -
PHY 2404S Lecture Notes
PHY 2404S Lecture Notes Michael Luke Winter, 2003 These notes are perpetually under construction. Please let me know of any typos or errors. Once again, large portions of these notes have been plagiarized from Sidney Coleman’s field theory lectures from Harvard, written up by Brian Hill. 1 Contents 1 Preliminaries 3 1.1 Counterterms and Divergences: A Simple Example . 3 1.2 CountertermsinScalarFieldTheory . 11 2 Reformulating Scattering Theory 18 2.1 Feynman Diagrams with External Lines off the Mass Shell . .. 18 2.1.1 AnswerOne: PartofaLargerDiagram . 19 2.1.2 Answer Two: The Fourier Transform of a Green Function 20 2.1.3 Answer Three: The VEV of a String of Heisenberg Fields 22 2.2 GreenFunctionsandFeynmanDiagrams. 23 2.3 TheLSZReductionFormula . 27 2.3.1 ProofoftheLSZReductionFormula . 29 3 Renormalizing Scalar Field Theory 36 3.1 The Two-Point Function: Wavefunction Renormalization .... 37 3.2 The Analytic Structure of G(2), and1PIGreenFunctions . 40 3.3 Calculation of Π(k2) to order g2 .................. 44 3.4 The definition of g .......................... 50 3.5 Unstable Particles and the Optical Theorem . 51 3.6 Renormalizability. 57 4 Renormalizability 60 4.1 DegreesofDivergence . 60 4.2 RenormalizationofQED. 65 4.2.1 TroubleswithVectorFields . 65 4.2.2 Counterterms......................... 66 2 1 Preliminaries 1.1 Counterterms and Divergences: A Simple Example In the last semester we considered a variety of field theories, and learned to calculate scattering amplitudes at leading order in perturbation theory. Unfor- tunately, although things were fine as far as we went, the whole basis of our scattering theory had a gaping hole in it. -
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− .. -
Reference International Centre for C), Theoretical Physics
IC/88/346 REFERENCE INTERNATIONAL CENTRE FOR C), THEORETICAL PHYSICS ZETA FUNCTION REGULARIZATION OF QUANTUM FIELD THEORY A.Y. Shiekh INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/88/346 §1 Introduction International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization In generalizing quantum mechanics to quantum field theory one is generally INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS confronted with a scheme plagued by infinities. If one is to make sense of such an ailment, one must find a way to deal with those infinities. In the usual interpretation this difficulty is viewed as stemming from not having yet accounted for the self- interactions which modify the masses and coupling constants to their observed values. A sensible strategy, in order to be useful, must result in only a finite ZETA FUNCTION REGULARIZATION number of physical constants being left, undetermined. Not all theories can be dealt OF QUANTUM FIELD THEORY * with in this way; which has the beneficial consequence of eliminating certain, otherwise prospective, theories of nature. This follows the line of reasoning that physics is more than descriptive, but predictive by virtue of being limited by the A.Y. Shiekh "" requirement of consistency. International Centra foe Theoretical Physics, Tri.fste, Ltaly, In order to proceed with such a theory, the infinities must be restrained (regulated) while the self-interactions are accounted for. Visiting a different space- time dimension is a popular way to regulate (dimensional regularization) as it explicitly maintains the theory's covariance, which may otherwise be difficult to hold on to during reckoning for the self-interactions (renormalization). -
Renormalization in Nonrelativistic Quantum Mechanics
Renormalization in Nonrelativistic Quantum Mechanics ∗ Sadhan K Adhikari†and Angsula Ghosh Instituto de F´ısica Te´orica, Universidade Estadual Paulista, 01405-900 S˜ao Paulo, S˜ao Paulo, Brasil July 30, 2018 Abstract The importance and usefulness of renormalization are emphasized in nonrelativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin exhibits ultravi- olet divergence. The use of renormalization techniques in these problems leads to finite converged results for both the exact and perturbative solutions. The renormalization procedure is carried out for the quantum two-body problem in different partial waves for a minimal potential possessing only the threshold behavior and no form factors. The renormalized perturbative and exact solutions for this problem are found to be consistent arXiv:hep-th/9706193v1 26 Jun 1997 with each other. The useful role of the renormalization group equations for this problem is also pointed out. PACS Numbers 03.65.-w, 03.65.Nk, 11.10.Gh, 11.10.Hi ∗To Appear in Journal of Physics A †John Simon Guggenheim Memorial Foundation Fellow. 1 1 Introduction The ultraviolet divergences in perturbative quantum field theory can be eliminated in many cases by renormalization to define physical observables, such as charge or mass, which are often termed the physical scale(s) of the problem [1, 2, 3, 4]. Ultraviolet divergences appear in exact as well as perturbative treatments of the nonrelativistic quantum mechanical two-body problem in momentum space interacting via two-body potentials with certain singular behavior at short distances [5, 6, 7, 8, 9, 10, 11] in two and three space dimensions. -
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. -
Thesis Presented for the Degree of Doctor of Philosophy
Durham E-Theses Hadronic productions of a Higgs boson in association with two jets at next-to-leading order WILLIAMS, CIARAN How to cite: WILLIAMS, CIARAN (2010) Hadronic productions of a Higgs boson in association with two jets at next-to-leading order , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/414/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Hadronic production of a Higgs boson in association with two jets at next-to-leading order Ciaran Williams A Thesis presented for the degree of Doctor of Philosophy Institute for Particle Physics Phenomenology Department of Physics University of Durham England August 2010 Dedicated to My Parents Hadronic production of a Higgs boson in association with two jets at next-to-leading order Ciaran Williams Submitted for the degree of Doctor of Philosophy August 2010 Abstract We present the calculation of hadronic production of a Higgs boson in association with two jets at next-to-leading order in perturbation theory. -
Studies in Field Theories: Mhv Vertices, Twistor Space, Recursion Relations and Chiral Rings
STUDIES IN FIELD THEORIES: MHV VERTICES, TWISTOR SPACE, RECURSION RELATIONS AND CHIRAL RINGS Peter Svr·cek Advisor: Edward Witten A DISSERTATION SUBMITTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS April 2005 Copyright °c 2005 by Peter Svr·cek All rights reserved. ii Abstract In this thesis we study di®erent aspects of four dimensional ¯eld theories. In the ¯rst chapter we give introduction and overview of the thesis. In the second chapter we review the connection between perturbative Yang-Mills and twistor string the- ory. Inspired by this, we propose a new way of constructing Yang-Mills scattering amplitudes from Feynman graphs in which the vertices are o®-shell continuations of the tree level MHV amplitudes. The MHV diagrams lead to simple formulas for tree-level amplitudes. We then give a heuristic derivation of the diagrams from twistor string theory. In the third chapter, we explore the twistor structure of scattering amplitudes in theories for which a twistor string theory analogous to the one for N = 4 gauge theory has not yet been proposed. We study the di®erential equations of one-loop amplitudes of gluons in gauge theories with reduced supersymmetry and of tree level and one-loop amplitudes of gravitons in general relativity and supergravity. We ¯nd that the scattering amplitudes localize in twistor space on algebraic curves that are surprisingly similar to the N = 4 Yang-Mills case. In the next chapter we propose tree-level recursion relations for scattering am- plitudes of gravitons. -
Toward Construction of a Consistent Field Theory with Poincaré
Toward construction of a consistent field theory with Poincare´ covariance in terms of step-function-type basis functions showing confinement/deconfinement, mass-gap and Regge trajectory for non-pure/pure non-Abelian gauge fields ∗ Kimichika Fukushima † Advanced Reactor System Engineering Department, Toshiba Nuclear Engineering Service Corporation, 8, Shinsugita-cho, Isogo-ku, Yokohama 235-8523, Japan Hikaru Sato Emeritus, Department of Physics, Hyogo University of Education, Yashiro-cho, Kato-shi, Hyogo 673-1494, Japan This article is a review by the authors concerning the construction of a Poincar´ecovariant (owing to spacetime continuum) field-theoretic formalism in terms of step-function-type basis functions without ultraviolet divergences. This formalism analytically derives confinement/deconfinement,mass-gap and Regge trajectory for non-Abelian gauge fields, and gives solutions for self-interacting scalar fields. Fields propagate in spacetime continuum and fields with finite degrees of freedom toward continuum limit have no ultraviolet divergence. Basis functions defined in a parameter spacetime are mapped to real spacetime. The authors derive a new solution comprised of classical fields as a vacuum and quantum fluctuations, leading to the linear potential between the particle and antiparticle from the Wilson loop. The Polyakov line gives finite binding energies and reveals the deconfining property at high temperatures. The quantum action yields positive mass from the classical fields and quantum fluctuations produces the Coulomb potential. Pure Yang-Mills fields show the same mass-gap owing to the particle-antiparticle pair creation. The Dirac equation under linear potential is analytically solved in this formalism, reproducing the principal properties of Regge trajectoriesata quantum level. -
Unitarity and On-Shell Recursion Methods for Scattering Amplitudes
FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN PhD thesis Kasper Risager Larsen Unitarity and On-Shell Recursion Methods for Scattering Amplitudes Academic advisor: Poul Henrik Damgaard Submitted: 23/11/07 Unitarity and On-Shell Recursion Methods for Scattering Amplitudes by Kasper Risager Submitted for the degree of Ph.D. in Physics from the Niels Bohr Institute, Faculty of Science, University of Copenhagen. November 2007 Abstract This thesis describes some of the recent (and some less recent) developments in calculational techniques for scattering amplitudes in quantum field theory. The focus is on on-shell recursion relations in complex momenta and on the use of unitarity methods for loop calculations. In particular, on-shell recursion is related to the MHV rules for computing tree-level gauge amplitudes and used to extend the MHV rules to graviton scattering. Combinations of unitarity cut techniques and recursion is used to argue for the “No-Triangle Hypothesis” in N = 8 supergravity which is related to its UV behaviour. Finally, combinations of unitarity and recursion is used to demonstrate the full calculation of a one- loop amplitude involving a Higgs particle and four gluons in the limit of large top mass. Contents 1 Introduction 5 1.1ScatteringAmplitudesinTheoryandExperiment.......... 5 1.1.1 BackgroundProcessesattheLHC.............. 6 1.1.2 ApproachingQuantumGravity?............... 7 1.2ScatteringAmplitudesintheComplexPlane............ 7 1.2.1 The Unitary and Analytic S-Matrix............. 8 1.2.2 GeneralizingfromLoopstoTrees............... 10 1.2.3 FullAmplitudesfromTrees.................. 11 1.3AboutThisThesis........................... 12 1.3.1 Omissions............................ 12 1.3.2 Outline............................. 12 1.4ConclusionandOutlook........................ 13 1.4.1 ACompetitorintheCalculationRace............ 13 1.4.2 UnderstandingQuantumFieldTheories........... 14 2 Preliminaries 15 2.1ColourOrdering...........................