Numerical Evaluation of Feynman Loop Integrals by Reduction to Tree Graphs
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Bhabha-Scattering Consider the Theory of Quantum Electrodynamics for Electrons, Positrons and Photons
Quantum Field Theory WS 18/19 18.01.2019 and the Standard Model Prof. Dr. Owe Philipsen Exercise sheet 11 of Particle Physics Exercise 1: Bhabha-scattering Consider the theory of Quantum Electrodynamics for electrons, positrons and photons, 1 1 L = ¯ iD= − m − F µνF = ¯ [iγµ (@ + ieA ) − m] − F µνF : (1) QED 4 µν µ µ 4 µν Electron (e−) - Positron (e+) scattering is referred to as Bhabha-scattering + − + 0 − 0 (2) e (p1; r1) + e (p2; r2) ! e (p1; s1) + e (p2; s2): i) Identify all the Feynman diagrams contributing at tree level (O(e2)) and write down the corresponding matrix elements in momentum space using the Feynman rules familiar from the lecture. ii) Compute the spin averaged squared absolut value of the amplitudes (keep in mind that there could be interference terms!) in the high energy limit and in the center of mass frame using Feynman gauge (ξ = 1). iii) Recall from sheet 9 that the dierential cross section of a 2 ! 2 particle process was given as dσ 1 = jM j2; (3) dΩ 64π2s fi in the case of equal masses for incoming and outgoing particles. Use your previous result to show that the dierential cross section in the case of Bhabha-scattering is dσ α2 1 + cos4(θ=2) 1 + cos2(θ) cos4(θ=2) = + − 2 ; (4) dΩ 8E2 sin4(θ=2) 2 sin2(θ=2) where e2 is the ne-structure constant, the scattering angle and E is the respective α = 4π θ energy of electron and positron in the center-of-mass system. Hint: Use Mandelstam variables. -
Pos(LL2018)010 -Right Asymmetry and the Relative § , Lidia Kalinovskaya, Ections
Electroweak radiative corrections to polarized Bhabha scattering PoS(LL2018)010 Andrej Arbuzov∗† Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141980 Russia E-mail: [email protected] Dmitri Bardin,‡ Yahor Dydyshka, Leonid Rumyantsev,§ Lidia Kalinovskaya, Renat Sadykov Dzhelepov Laboratory of Nuclear Problems, JINR, Dubna, 141980 Russia Serge Bondarenko Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141980 Russia In this report we present theoretical predictions for high-energy Bhabha scattering with taking into account complete one-loop electroweak radiative corrections. Longitudinal polarization of the initial beams is assumed. Numerical results for the left-right asymmetry and the relative correction to the distribution in the scattering angle are shown. The results are relevant for several future electron-positron collider projects. Loops and Legs in Quantum Field Theory (LL2018) 29 April 2018 - 04 May 2018 St. Goar, Germany ∗Speaker. †Also at the Dubna University, Dubna, 141980, Russia. ‡deceased §Also at the Institute of Physics, Southern Federal University, Rostov-on-Don, 344090 Russia c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ EW RC to polarized Bhabha scattering Andrej Arbuzov 1. Introduction The process of electron-positron scattering known as the Bhabha scattering [1] is one of the + key processes in particle physics. In particular it is used for the luminosity determination at e e− colliders. Within the Standard Model the scattering of unpolarized electron and positrons has been thoroughly studied for many years [2–12]. Corrections to this process with polarized initial particles were discussed in [13, 14]. -
Two-Loop Fermionic Corrections to Massive Bhabha Scattering
DESY 07-053 SFB/CPP-07-15 HEPT00LS 07-010 Two-Loop Fermionic Corrections to Massive Bhabha Scattering 2007 Stefano Actis", Mi dial Czakon 6,c, Janusz Gluzad, Tord Riemann 0 Apr 27 FZa^(me?iGZZee ZW.57,9# Zew^/ien, Germany /nr T/ieore^ac/ie F/iyazA; nnd Ag^rop/iyazA;, Gmneraz^ Wnrz6nry, Am #n6Zand, D-P707/ Wnrz 6nry, Germany [hep-ph] cInstitute of Nuclear Physics, NCSR “DEMOKR.ITOS”, _/,!)&/0 A^/ieng, Greece ^/ng^^n^e 0/ P/&ygzcg, Gnznerg^y 0/ <9%Zegm, Gnzwergy^ecta /, Fi,-/0007 j^a^omce, FoZand Abstract We evaluate the two-loop corrections to Bhabha scattering from fermion loops in the context of pure Quantum Electrodynamics. The differential cross section is expressed by a small number of Master Integrals with exact dependence on the fermion masses me, nif and the Mandelstam arXiv:0704.2400v2 invariants s,t,u. We determine the limit of fixed scattering angle and high energy, assuming the hierarchy of scales rip < mf < s,t,u. The numerical result is combined with the available non-fermionic contributions. As a by-product, we provide an independent check of the known electron-loop contributions. 1 Introduction Bhabha scattering is one of the processes at e+e- colliders with the highest experimental precision and represents an important monitoring process. A notable example is its expected role for the luminosity determination at the future International Linear Collider ILC by measuring small-angle Bhabha-scattering events at center-of-mass energies ranging from about 100 GeV (Giga-Z collider option) to several TeV. -
Lectures on Conformal Field Theory Arxiv:1511.04074V2 [Hep-Th] 19
Prepared for submission to JHEP Lectures on Conformal Field Theory Joshua D. Quallsa aDepartment of Physics, National Taiwan University, Taipei, Taiwan E-mail: [email protected] Abstract: These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students. arXiv:1511.04074v2 [hep-th] 19 May 2016 Contents 1 Lecture 1: Introduction and Motivation2 1.1 Introduction and outline2 1.2 Conformal invariance: What?5 1.3 Examples of classical conformal invariance7 1.4 Conformal invariance: Why?8 1.4.1 CFTs in critical phenomena8 1.4.2 Renormalization group 12 1.5 A preview for future courses 16 1.6 Conformal quantum mechanics 17 2 Lecture 2: CFT in d ≥ 3 22 2.1 Conformal transformations for d ≥ 3 22 2.2 Infinitesimal conformal transformations for d ≥ 3 24 2.3 Special conformal transformations and conformal algebra 26 2.4 Conformal group 28 2.5 Representations of the conformal group 29 2.6 Constraints of Conformal -
Infrared Divergences
2012 Matthew Schwartz III-6: Infrared divergences 1 Introduction We have shown that the 1, 2 and 3 point functions in QED are UV finite at one loop. We were able to introduce 4 counterterms ( δm , δ1 , δ2 , δ3) which canceled all the infinities. Now let us move on to four point functions, such as Ω T ψ( x ) ψ¯ ( x ) ψ( x ) ψ¯ ( x ) Ω (1) h | { 1 2 3 4 }| i This could represent, for example, Møller scattering ( e − e − e − e − ) or Bhabha scattering → ( e+ e − e+ e − ). We will take it to be e+ e − µ+ µ− for simplicity, since at tree-level this process only has→ an s-channel diagram. Looking→ at these 4-point functions at one-loop will help us understand how to combine previous loop calculations and counterterms into new observables, and will also illustrate a new feature: cancellation of infrared divergences. Although important results and calculational techniques are introduced in this lecture, it can be skipped without much loss of continuity with the rest of the text. Recall that in the on-shell subtraction scheme we found δ1 and δ2 depended on a fictitious photon mass m γ. This mass was introduced to make the loops finite and is an example of an infrared regulator. As we will see, the dependence on IR regulators, like m γ, drops out not in differences between the Green’s functions at different scales, as with UV regulators, but in the sum of different types of Green’s functions contributing to the same observable at the same scale. -
Introduction to Renormalization
Introduction to Renormalization A lecture given by Prof. Wayne Repko Notes: Michael Flossdorf April 4th 2007 Contents 1 Introduction 1 2 RenormalizationoftheQEDLagrangian 2 3 One Loop Correction to the Fermion Propagator 3 4 Dimensional Regularization 6 4.1 SuperficialDegreeofDivergence. 6 4.2 TheProcedureofDimensionalRegularization . .. 8 4.3 AUsefulGeneralIntegral. 9 4.4 Dirac Matrices in n Dimensions .................. 11 4.5 Dimensional Regularization of Σ2 ................. 11 5 Renormalization 12 5.1 Summary ofLastSection and OutlineoftheNextSteps . .. 12 5.2 MomentumSubtractionSchemes. 13 5.3 BacktoRenormalizationinQED. 14 5.4 Leading Order Expression of Σ2 for Small ǫ ............ 17 6 Outline of the Following Procedure 19 7 The Vertex Correction 19 8 Vacuum Polarization 22 CONTENTS 2 9 The Beta Function of QED 25 Appendix 27 References 30 1 INTRODUCTION 1 First Part of the Lecture 1 Introduction Starting with the Lagrangian of any Quantum Field Theory, we have seen in some of the previous lectures how to obtain the Feynman rules, which sufficiently de- scribe how to do pertubation theory in that particular theory. But beyond tree level, the naive calculation of diagrams involving loops will often yield infinity, since the integrals have to be performed over the whole momentum space. Renormal- ization Theory deals with the systematic isolation and removing of these infinities from physical observables. The first important insight is, that it is not the fields or the coupling constants which represent measurable quantities. Measured are cross sections, decay width, etc.. As long as we make sure that this observables are finite in the end and can be unambiguously derived from the Lagrangian, we are free to introduce new quantities, called renormalized quantities for every, so called, bare quantity. -
Two-Loop N F= 1 QED Bhabha Scattering: Soft Emission And
Freiburg-THEP 04/19 UCLA/04/TEP/60 hep-ph/0411321 Two-Loop NF = 1 QED Bhabha Scattering: Soft Emission and Numerical Evaluation of the Differential Cross-section a, a, b, R. Bonciani ∗ A. Ferroglia †, P. Mastrolia ‡, c, d, a, E. Remiddi §, and J. J. van der Bij ¶ a Fakult¨at f¨ur Mathematik und Physik, Albert-Ludwigs-Universit¨at Freiburg, D-79104 Freiburg, Germany b Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095-1547 c Theory Division, CERN, CH-1211 Geneva 23, Switzerland d Dipartimento di Fisica dell’Universit`adi Bologna, and INFN, Sezione di Bologna, I-40126 Bologna, Italy Abstract Recently, we evaluated the virtual cross-section for Bhabha scattering in 4 pure QED, up to corrections of order α (NF = 1). This calculation is valid for arbitrary values of the squared center of mass energy s and momentum transfer t; the electron and positron mass m was considered a finite, non van- arXiv:hep-ph/0411321 v2 11 May 2006 ishing quantity. In the present work, we supplement the previous calculation by considering the contribution of the soft photon emission diagrams to the 4 differential cross-section, up to and including terms of order α (NF = 1). Adding the contribution of the real corrections to the renormalized virtual ones, we obtain an UV and IR finite differential cross-section; we evaluate this quantity numerically for a significant set of values of the squared center of mass energy s. Key words: Feynman diagrams, Multi-loop calculations, Box diagrams, Bhabha scattering PACS: 11.15.Bt, 12.20.Ds ∗Email: [email protected] †Email: [email protected] ‡Email: [email protected] §Email: [email protected] ¶Email: [email protected] 1 Introduction The relevance of the Bhabha scattering process (e+e− e+e−) in the study of the → phenomenology of particle physics can hardly be overestimated. -
Collisions at Very High Energy
” . SLAC-PUB-4601 . April 1988 P-/E) - Theory of e+e- Collisions at Very High Energy MICHAEL E. PESKIN* - - - Stanford Linear Accelerator Center Stanford University, Stanford, California 94309 - Lectures presented at the SLAC Summer Institute on Particle Physics Stanford, California, August 10 - August 21, 1987 z-- -. - - * Work supported by the Department of Energy, contract DE-AC03-76SF00515. 1. Introduction The past fifteen years of high-energy physics have seen the successful elucida- - tion of the strong, weak, and electromagnetic interactions and the explanation of all of these forces in terms of the gauge theories of the standard model. We are now beginning the last stage of this chapter in physics, the era of direct experi- - mentation on the weak vector bosons. Experiments at the CERN @ collider have _ isolated the W and 2 bosons and confirmed the standard model expectations for their masses. By the end of the decade, the new colliders SLC and LEP will have carried out precision measurements of the properties of the 2 boson, and we have good reason to hope that this will complete the experimental underpinning of the structure of the weak interactions. Of course, the fact that we have answered some important questions about the working of Nature does not mean that we have exhausted our questions. Far from r it! Every advance in fundamental physics brings with it new puzzles. And every advance sets deeper in relief those very mysterious issues, such as the origin of the mass of electron, which have puzzled generations of physicists and still seem out of - reach of our understanding. -
UNIVERSITY of CALIFORNIA Los Angeles Modern Applications Of
UNIVERSITY OF CALIFORNIA Los Angeles Modern Applications of Scattering Amplitudes and Topological Phases A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Julio Parra Martinez 2020 © Copyright by Julio Parra Martinez 2020 ABSTRACT OF THE DISSERTATION Modern Applications of Scattering Amplitudes and Topological Phases by Julio Parra Martinez Doctor of Philosophy in Physics University of California, Los Angeles, 2020 Professor Zvi Bern, Chair In this dissertation we discuss some novel applications of the methods of scattering ampli- tudes and topological phases. First, we describe on-shell tools to calculate anomalous dimen- sions in effective field theories with higer-dimension operators. Using such tools we prove and apply a new perturbative non-renormalization theorem, and we explore the structure of the two-loop anomalous dimension matrix of dimension-six operators in the Standard Model Ef- fective Theory (SMEFT). Second, we introduce new methods to calculate the classical limit of gravitational scattering amplitudes. Using these methods, in conjunction with eikonal techniques, we calculate the classical gravitational deflection angle of massive and massles particles in a variety of theories, which reveal graviton dominance beyond 't Hooft's. Finally, we point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible topological phases, or equivalently topological superconductors, on the worldsheet. We explain how the classification of fermionic topological phases, recently achieved by the condensed matter community, provides a complete and systematic classification of ten-dimensional superstrings and gives a new perspective on the K-theoretic classification of D-branes. -
On the Massive Two-Loop Corrections to Bhabha Scattering∗ ∗∗
Vol. 36 (2005) ACTA PHYSICA POLONICA B No 11 ON THE MASSIVE TWO-LOOP CORRECTIONS TO BHABHA SCATTERING∗ ∗∗ M. Czakona,b, J. Gluzab,c, T. Riemannc a Institut für Theoretische Physik und Astrophysik, Universität Würzburg Am Hubland, D-97074 Würzburg, Germany b Institute of Physics, University of Silesia Universytecka 4, 40007 Katowice, Poland c Deutsches Elektronen-Synchrotron DESY Platanenallee 6, D–15738 Zeuthen, Germany (Received November 2, 2005) We overview the general status of higher order corrections to Bhabha scattering and review recent progress in the determination of the two-loop virtual corrections. Quite recently, they were derived from combining a massless calculation and contributions with electron sub-loops. For a mas- sive calculation, the self-energy and vertex master integrals are known, while most of the two-loop boxes are not. We demonstrate with an ex- ample that a study of systems of differential equations, combined with Mellin-Barnes representations for single masters, might open a way for their systematic calculation. PACS numbers: 13.20.Ds, 13.66.De 1. Introduction Since Bhabha’s article [1] on the reaction − − e+e → e+e (1.1) numerous studies of it were performed, triggered by better and better ex- perimentation. Bhabha scattering is of interest for several reasons. It is a classical reaction with a clear experimental signature. ∗ Presented at the XXIX International Conference of Theoretical Physics “Matter to the Deepest”, Ustroń, Poland, September 8–14, 2005 ∗∗ Work supported in part by Sonderforschungsbereich/Transregio 9–03 of DFG “Com- putergestützte Theoretische Teilchenphysik”, by the Sofja Kovalevskaja Award of the Alexander von Humboldt Foundation sponsored by the German Federal Ministry of Education and Research, and by the Polish State Committee for Scientific Research (KBN) for the research project in 2004–2005. -
27073109.Pdf
KS00206460G R: FI DEOO9O5232O \; «DE009052320* Cover detail; De aardappeleters, Nuenen 1885 Vincent van Gogh Amsterdam, Van Gogh Museum (Vincent van Gogh Stichting) Cover design: Jaap Joziasse Cover photo: Loek van Beers Prepress: Hennie Spermon VOL <n 7 A luminosity measurement at LEP using the L3 detector Een wetenschappelijke proeve op het gebied van de Natuurwetenschappen. Proefschrift ter verkrijging van de graad van doctor aan de Katholieke Universiteit Nijmegen, volgens besluit van het College van Decanen in het openbaar te verdedigen op 25 Juni 1996, des namiddags te 15.00 uur precies, door Elisabeth Nikolaja Koffeman geboren op 3 november 1967 te Nuenen. Promotores: Prof. Dr P. Duinker Prof. Dr F. L. Linde Co-promotor: Dr M. H. M. Merk Manuscriptcommissic: Dr D. J. Schotanus Dr G. J. Bobbink The work described in this thesis is part of the research programme of the "Nationaal Instituut voor Kernfysica en Hoge-Encrgic fysica (NIKHEF)'. The author was financially supported by the 'Stichting voor Fundamenteel Onderzoek der Materie (FOM)'. ISBN: 90-9009549-7 Un savant dans son laboratoire n'est pas settlement un tech- nicien: c'est aussi un enfant place en face des phênomènes naturels qui l''empressionnent comme un conte defées. Nous ne devons pas laisser croire que tout progrès scientifique se reduit a des mécanismes, des machines, des engrenases, qui, d'ailleurs ont aussi leur beauté propre. Marie Curie-Sklodowska (1867-1934) Contents Introduction 1 1 Theory 3 1.1 Introduction 3 1.2 The Standard Model 3 1.2.1 Particles and fields -
1 Bhabha Scattering (1936)
1 Bhabha scattering (1936) Consider the scattering of an electron and a positron. We have already computed the annihilation of an electron/positron in finding the cross-section for producing muon pairs. It is also possible for the final particles to be an electron-positron pair. In addition, the initial particles may exchage a photon without annihilating. Since the particles are distinguish- able, we have only the t-channel diagram for this, without the crossed diagram. Let the in and out electrons have momenta p; p0, respectively and the initial and final positrons have momenta k;k0. Let the angle between the initial and final electrons be q. The general calculation is lengthy; here we restrict our attention to the ultrarelativistic case, where we can treat the m2 electron and positron as massless. This means we neglect terms of order E2 . For GeV or TeV colliders, this term is of order 10−8 or 10−14, so we incur no great error. We may drop m whenever it occurs additively. 1.1 The scattering matrix We have two Feynman diagrams, so the matrix element is the sum s t iM fi = iM fi + iM fi where s and t refer to the photon channel. The first, s-channel, diagram describes the annihilation-creation. The matrix element is the same as we use for the muon problem, except the particle masses are all equal, and here we want to call p;k the incoming momenta and p0;k0 the outgoing momenta. Therefore, instead of s 0 a −i qa qb b 0 iM = v¯ p (−ieg )u(p) h − (1 − x) u¯(k) −ieg v k fi q2 ab q2 we write s a −i qa qb 0 b 0 iM = v¯(k)(−ieg )u(p) h − (1 − x) u¯ p −ieg v k fi q2 ab q2 that is, we just need to interchange p0 $ k in our previous result, and we have q = p + k = p0 + k0.