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Quantum Field Theory*
Quantum Field Theory y Frank Wilczek Institute for Advanced Study, School of Natural Science, Olden Lane, Princeton, NJ 08540 I discuss the general principles underlying quantum eld theory, and attempt to identify its most profound consequences. The deep est of these consequences result from the in nite number of degrees of freedom invoked to implement lo cality.Imention a few of its most striking successes, b oth achieved and prosp ective. Possible limitation s of quantum eld theory are viewed in the light of its history. I. SURVEY Quantum eld theory is the framework in which the regnant theories of the electroweak and strong interactions, which together form the Standard Mo del, are formulated. Quantum electro dynamics (QED), b esides providing a com- plete foundation for atomic physics and chemistry, has supp orted calculations of physical quantities with unparalleled precision. The exp erimentally measured value of the magnetic dip ole moment of the muon, 11 (g 2) = 233 184 600 (1680) 10 ; (1) exp: for example, should b e compared with the theoretical prediction 11 (g 2) = 233 183 478 (308) 10 : (2) theor: In quantum chromo dynamics (QCD) we cannot, for the forseeable future, aspire to to comparable accuracy.Yet QCD provides di erent, and at least equally impressive, evidence for the validity of the basic principles of quantum eld theory. Indeed, b ecause in QCD the interactions are stronger, QCD manifests a wider variety of phenomena characteristic of quantum eld theory. These include esp ecially running of the e ective coupling with distance or energy scale and the phenomenon of con nement. -
Lattice Gauge Theory
Lattice Gauge Theory Hartmut Wittig Oxford University EPS High Energy Physics 99 Tamp ere, Finland 20 July 1999 Intro duction Lattice QCD provides a non-p erturb ati ve framework to compute relations between SM parameters and exp erimental quantities from rst principles Formulate QCD on a euclidean space-time lattice 3 with spacing a and volume L T 1 a gauge invariant UV Z S S 1 G F ][d ] e h i = Z [dU ][d ! allows for sto chastic evaluation of h i Ideally want to use lattice QCD as a phenomenologi ca l to ol, e.g. M f s Z 7! m + m 2 GeV m + u s K However, realistic simulation s of lattice QCD are hard... 1 1 Lattice artefacts cuto e ects lat cont p h i = h i + O a 1 a 1 4 GeV , a 0:2 0:05 fm ! need to extrap olate to continuum limit: a ! 0 ! cho ose b etter discretisati ons: avoid small p. 2 Inclusion of dynamical quark e ects: Z Y S [U ] lat G Z = [dU ] e det D + m f f lat detD + m =1: Quenched Approximation f ! neglect quark lo ops in the evaluation of h i. cheap exp ensive 2 Scale ambiguity in the quenched approximation: p Q [MeV ] 1 ;::: ; Q = f ;m ;m ; a [MeV ]= N aQ 3 Restrictions on quark masses: 1 a m L q ! cannot simulate u; d and b quarks directly ! need to control extrap olation s in m q 4 Chiral symmetry breaking: Nielsen & Ninomiya 1979: exact chiral symmetry cannot b e realised at non-zero lattice spacing ! imp ossibl e to separate chiral and continuum limits 3 Outline: I. -
Coupling Constant Unification in Extensions of Standard Model
Coupling Constant Unification in Extensions of Standard Model Ling-Fong Li, and Feng Wu Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 May 28, 2018 Abstract Unification of electromagnetic, weak, and strong coupling con- stants is studied in the extension of standard model with additional fermions and scalars. It is remarkable that this unification in the su- persymmetric extension of standard model yields a value of Weinberg angle which agrees very well with experiments. We discuss the other possibilities which can also give same result. One of the attractive features of the Grand Unified Theory is the con- arXiv:hep-ph/0304238v2 3 Jun 2003 vergence of the electromagnetic, weak and strong coupling constants at high energies and the prediction of the Weinberg angle[1],[3]. This lends a strong support to the supersymmetric extension of the Standard Model. This is because the Standard Model without the supersymmetry, the extrapolation of 3 coupling constants from the values measured at low energies to unifi- cation scale do not intercept at a single point while in the supersymmetric extension, the presence of additional particles, produces the convergence of coupling constants elegantly[4], or equivalently the prediction of the Wein- berg angle agrees with the experimental measurement very well[5]. This has become one of the cornerstone for believing the supersymmetric Standard 1 Model and the experimental search for the supersymmetry will be one of the main focus in the next round of new accelerators. In this paper we will explore the general possibilities of getting coupling constants unification by adding extra particles to the Standard Model[2] to see how unique is the Supersymmetric Standard Model in this respect[?]. -
Critical Coupling for Dynamical Chiral-Symmetry Breaking with an Infrared Finite Gluon Propagator *
BR9838528 Instituto de Fisica Teorica IFT Universidade Estadual Paulista November/96 IFT-P.050/96 Critical coupling for dynamical chiral-symmetry breaking with an infrared finite gluon propagator * A. A. Natale and P. S. Rodrigues da Silva Instituto de Fisica Teorica Universidade Estadual Paulista Rua Pamplona 145 01405-900 - Sao Paulo, S.P. Brazil *To appear in Phys. Lett. B t 2 9-04 Critical Coupling for Dynamical Chiral-Symmetry Breaking with an Infrared Finite Gluon Propagator A. A. Natale l and P. S. Rodrigues da Silva 2 •r Instituto de Fisica Teorica, Universidade Estadual Paulista Rua Pamplona, 145, 01405-900, Sao Paulo, SP Brazil Abstract We compute the critical coupling constant for the dynamical chiral- symmetry breaking in a model of quantum chromodynamics, solving numer- ically the quark self-energy using infrared finite gluon propagators found as solutions of the Schwinger-Dyson equation for the gluon, and one gluon prop- agator determined in numerical lattice simulations. The gluon mass scale screens the force responsible for the chiral breaking, and the transition occurs only for a larger critical coupling constant than the one obtained with the perturbative propagator. The critical coupling shows a great sensibility to the gluon mass scale variation, as well as to the functional form of the gluon propagator. 'e-mail: [email protected] 2e-mail: [email protected] 1 Introduction The idea that quarks obtain effective masses as a result of a dynamical breakdown of chiral symmetry (DBCS) has received a great deal of attention in the last years [1, 2]. One of the most common methods used to study the quark mass generation is to look for solutions of the Schwinger-Dyson equation for the fermionic propagator. -
Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics Via the Dyson-Schwinger Equations
University of Adelaide School of Chemistry and Physics Doctor of Philosophy Regularization and Renormalization of Non-Perturbative Quantum Electrodynamics via the Dyson-Schwinger Equations by Tom Sizer Supervisors: Professor A. G. Williams and Dr A. Kızılers¨u March 2014 Contents 1 Introduction 1 1.1 Introduction................................... 1 1.2 Dyson-SchwingerEquations . .. .. 2 1.3 Renormalization................................. 4 1.4 Dynamical Chiral Symmetry Breaking . 5 1.5 ChapterOutline................................. 5 1.6 Notation..................................... 7 2 Canonical QED 9 2.1 Canonically Quantized QED . 9 2.2 FeynmanRules ................................. 12 2.3 Analysis of Divergences & Weinberg’s Theorem . 14 2.4 ElectronPropagatorandSelf-Energy . 17 2.5 PhotonPropagatorandPolarizationTensor . 18 2.6 ProperVertex.................................. 20 2.7 Ward-TakahashiIdentity . 21 2.8 Skeleton Expansion and Dyson-Schwinger Equations . 22 2.9 Renormalization................................. 25 2.10 RenormalizedPerturbationTheory . 27 2.11 Outline Proof of Renormalizability of QED . 28 3 Functional QED 31 3.1 FullGreen’sFunctions ............................. 31 3.2 GeneratingFunctionals............................. 33 3.3 AbstractDyson-SchwingerEquations . 34 3.4 Connected and One-Particle Irreducible Green’s Functions . 35 3.5 Euclidean Field Theory . 39 3.6 QEDviaFunctionalIntegrals . 40 3.7 Regularization.................................. 42 3.7.1 Cutoff Regularization . 42 3.7.2 Pauli-Villars Regularization . 42 i 3.7.3 Lattice Regularization . 43 3.7.4 Dimensional Regularization . 44 3.8 RenormalizationoftheDSEs ......................... 45 3.9 RenormalizationGroup............................. 49 3.10BrokenScaleInvariance ............................ 53 4 The Choice of Vertex 55 4.1 Unrenormalized Quenched Formalism . 55 4.2 RainbowQED.................................. 57 4.2.1 Self-Energy Derivations . 58 4.2.2 Analytic Approximations . 60 4.2.3 Numerical Solutions . 62 4.3 Rainbow QED with a 4-Fermion Interaction . -
1.3 Running Coupling and Renormalization 27
1.3 Running coupling and renormalization 27 1.3 Running coupling and renormalization In our discussion so far we have bypassed the problem of renormalization entirely. The need for renormalization is related to the behavior of a theory at infinitely large energies or infinitesimally small distances. In practice it becomes visible in the perturbative expansion of Green functions. Take for example the tadpole diagram in '4 theory, d4k i ; (1.76) (2π)4 k2 m2 Z − which diverges for k . As we will see below, renormalizability means that the ! 1 coupling constant of the theory (or the coupling constants, if there are several of them) has zero or positive mass dimension: d 0. This can be intuitively understood as g ≥ follows: if M is the mass scale introduced by the coupling g, then each additional vertex in the perturbation series contributes a factor (M=Λ)dg , where Λ is the intrinsic energy scale of the theory and appears for dimensional reasons. If d 0, these diagrams will g ≥ be suppressed in the UV (Λ ). If it is negative, they will become more and more ! 1 relevant and we will find divergences with higher and higher orders.8 Renormalizability. The renormalizability of a quantum field theory can be deter- mined from dimensional arguments. Consider φp theory in d dimensions: 1 g S = ddx ' + m2 ' + 'p : (1.77) − 2 p! Z The action must be dimensionless, hence the Lagrangian has mass dimension d. From the kinetic term we read off the mass dimension of the field, namely (d 2)=2. The − mass dimension of 'p is thus p (d 2)=2, so that the dimension of the coupling constant − must be d = d + p pd=2. -
General Physics Motivations for Numerical Simulations of Quantum
LAUR-99-789 General Physics Motivations for Numerical Simulations of Quantum Field Theory R. Gupta 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Abstract In this introductory article a brief description of Quantum Field Theories (QFT) is presented with emphasis on the distinction between strongly and weakly coupled theories. A case is made for using numerical simulations to solve QCD, the regnant theory describing the interactions between quarks and gluons. I present an overview of what these calculations involve, why they are hard, and why they are tailor made for parallel computers. Finally, I try to communicate the excitement amongst the practitioners by giving examples of the quantities we will be able to calculate to within a few percent accuracy in the next five years. Key words: Quantum Field theory, lattice QCD, non-perturbative methods, parallel computers arXiv:hep-lat/9905027v1 20 May 1999 1 Introduction The complexity of physical problems increases with the number of degrees of freedom, and with the details of the interactions between them. This increase in the number of degrees of freedom introduces the notion of a very large range of length scales. A simple illustration is water in the oceans. The basic degrees of freedom are the water molecules ( 10−8 cm), and the largest scale is the earth’s diameter ( 104 km), i.e. the range∼ of scales span 1017 orders of magnitude. Waves in the∼ ocean cover the range from centimeters, to meters, to currents that persist for thousands of kilometers, and finally to tides that are global in extent. -
The QED Coupling Constant for an Electron to Emit Or Absorb a Photon Is Shown to Be the Square Root of the Fine Structure Constant Α Shlomo Barak
The QED Coupling Constant for an Electron to Emit or Absorb a Photon is Shown to be the Square Root of the Fine Structure Constant α Shlomo Barak To cite this version: Shlomo Barak. The QED Coupling Constant for an Electron to Emit or Absorb a Photon is Shown to be the Square Root of the Fine Structure Constant α. 2020. hal-02626064 HAL Id: hal-02626064 https://hal.archives-ouvertes.fr/hal-02626064 Preprint submitted on 26 May 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. V4 15/04/2020 The QED Coupling Constant for an Electron to Emit or Absorb a Photon is Shown to be the Square Root of the Fine Structure Constant α Shlomo Barak Taga Innovations 16 Beit Hillel St. Tel Aviv 67017 Israel Corresponding author: [email protected] Abstract The QED probability amplitude (coupling constant) for an electron to interact with its own field or to emit or absorb a photon has been experimentally determined to be -0.08542455. This result is very close to the square root of the Fine Structure Constant α. By showing theoretically that the coupling constant is indeed the square root of α we resolve what is, according to Feynman, one of the greatest damn mysteries of physics. -
Connecting the Quenched and Unquenched Worlds Via the Large Nc World
Physics Letters B 543 (2002) 183–188 www.elsevier.com/locate/npe Connecting the quenched and unquenched worlds via the large Nc world Jiunn-Wei Chen Department of Physics, University of Maryland, College Park, MD 20742, USA Received 16 July 2002; accepted 5 August 2002 Editor: H. Georgi Abstract In the large Nc (number of colors) limit, quenched QCD and QCD are identical. This implies that, in the effective field theory framework, some of the low energy constants in (Nc = 3) quenched QCD and QCD are the same up to higher-order corrections in the 1/Nc expansion. Thus the calculation of the non-leptonic kaon decays relevant for the I = 1/2 rule in the quenched approximation is expected to differ from the unquenched one by an O(1/Nc) correction. However, the calculation relevant to the CP-violation parameter / would have a relatively big higher-order correction due to the large cancellation in the leading order. Some important weak matrix elements are poorly known that even constraints with 100% errors are interesting. In those cases, quenched calculations will be very useful. 2002 Elsevier Science B.V. All rights reserved. Quantum chromodynamics (QCD) is the under- fermion determinant arising from the path integral to lying theory of strong interaction. At high energies be one. This approximation cuts down the comput- ( 1 GeV), the consequences of QCD can be studied ing time tremendously and by construction, the re- in systematic, perturbative expansions. Good agree- sults obtained are close to the unquenched ones when ment with experiments is found in the electron–hadron the quark masses are heavy ( ΛQCD ∼ 200 MeV) deep inelastic scattering and hadron–hadron inelastic so that the internal quark loops are suppressed. -
(Supersymmetric) Grand Unification
J. Reuter SUSY GUTs Uppsala, 15.05.2008 (Supersymmetric) Grand Unification Jürgen Reuter Albert-Ludwigs-Universität Freiburg Uppsala, 15. May 2008 J. Reuter SUSY GUTs Uppsala, 15.05.2008 Literature – General SUSY: M. Drees, R. Godbole, P. Roy, Sparticles, World Scientific, 2004 – S. Martin, SUSY Primer, arXiv:hep-ph/9709356 – H. Georgi, Lie Algebras in Particle Physics, Harvard University Press, 1992 – R. Slansky, Group Theory for Unified Model Building, Phys. Rep. 79 (1981), 1. – R. Mohapatra, Unification and Supersymmetry, Springer, 1986 – P. Langacker, Grand Unified Theories, Phys. Rep. 72 (1981), 185. – P. Nath, P. Fileviez Perez, Proton Stability..., arXiv:hep-ph/0601023. – U. Amaldi, W. de Boer, H. Fürstenau, Comparison of grand unified theories with electroweak and strong coupling constants measured at LEP, Phys. Lett. B260, (1991), 447. J. Reuter SUSY GUTs Uppsala, 15.05.2008 The Standard Model (SM) – Theorist’s View Renormalizable Quantum Field Theory (only with Higgs!) based on SU(3)c × SU(2)w × U(1)Y non-simple gauge group ν u h+ L = Q = uc dc `c [νc ] L ` L d R R R R h0 L L Interactions: I Gauge IA (covariant derivatives in kinetic terms): X a a ∂µ −→ Dµ = ∂µ + i gkVµ T k I Yukawa IA: u d e ˆ n ˜ Y QLHuuR + Y QLHddR + Y LLHdeR +Y LLHuνR I Scalar self-IA: (H†H)(H†H)2 J. Reuter SUSY GUTs Uppsala, 15.05.2008 The group-theoretical bottom line Things to remember: Representations of SU(N) i j 2 I fundamental reps. φi ∼ N, ψ ∼ N, adjoint reps. -
The Determination of the Strong Coupling Constant Arxiv
The Determination of the Strong Coupling Constant G¨unther Dissertori Institute for Particle Physics, ETH Zurich, Switzerland June 18, 2015 Abstract The strong coupling constant is one of the fundamental parame- ters of the standard model of particle physics. In this review I will briefly summarise the theoretical framework, within which the strong coupling constant is defined and how it is connected to measurable observables. Then I will give an historical overview of its experimen- tal determinations and discuss the current status and world average value. Among the many different techniques used to determine this coupling constant in the context of quantum chromodynamics, I will focus in particular on a number of measurements carried out at the Large Electron Positron Collider (LEP) and the Large Hadron Col- lider (LHC) at CERN. arXiv:1506.05407v1 [hep-ex] 17 Jun 2015 A contribution to: The Standard Theory up to the Higgs discovery - 60 years of CERN L. Maiani and G. Rolandi, eds. 1 1 Introduction The strong coupling constant, αs, is the only free parameter of the lagrangian of quantum chromodynamics (QCD), the theory of strong interactions, if we consider the quarkp masses as fixed. As such, this coupling constant, or equivalently gs = 4παs, is one of the three fundamental coupling constants of the standard model (SM) of particle physics. It is related to the SU(3)C colour part of the overall SU(3)C × SU(2)L × U(1)Y gauge symmetry of the SM. The other two constants g and g0 indicate the coupling strengths relevant for weak isospin and weak hypercharge, and can be rewritten in terms of 0 the Weinberg mixing angle tan θW = g =g and the fine-structure constant 2 α = e =(4π), where the electric charge is given by e = g sin θW. -
Basics of Quantum Chromodynamics (Two Lectures)
Basics of Quantum Chromodynamics (Two lectures) Faisal Akram 5th School on LHC Physics, 15-26 August 2016 NCP Islamabad 2015 Outlines • A brief introduction of QCD Classical QCD Lagrangian Quantization Green functions of QCD and SDE’s • Perturbative QCD Perturbative calculation of QCD Green functions Feynman Rules of QCD Renormalization Running of QCD coupling (Asymptotic freedom) • Non-Perturbative QCD Confinement QCD phase transition Dynamical breaking of chiral symmetry Elementary Particle Physics Today Elementary particles: Quarks Leptons Gauge Bosons Higgs Boson (can interact through (cannot interact through (mediate interactions) (impart mass to the elementary strong interaction) strong interaction) particles) 푢 푐 푡 Meson: 푠 = 0,1,2 … Quarks: Hadrons: 1 3 푑 푠 푏 Baryons: 푠 = , , … 2 2 푣푒 푣휇 푣휏 Meson: Baryon Leptons: 푒 휇 휏 Gauge Bosons: 훾, 푊±, 푍0, and 8 gluons 399 Mesons, 574 Baryons have been discovered Higgs Bosons: 퐻 (God/Mother particle) • Strong interaction (Quantum Chromodynamics) • Electro-weak interaction (Quantum electro-flavor dynamics) The Standard Model Explaining the properties of the Hadrons in terms of QCD’s fundamental degrees of freedom is the Problem laying at the forefront of Hadronic physics. How these elementary particles and the SM is discovered? 1. Scattering Experiments Cross sections, Decay Constants, 2. Decay Processes Masses, Spin , Couplings, Form factors, etc. 3. Study of bound states Models Particle Accelerators Models and detectors Experimental Particle Physics Theoretical Particle Particle Physicist Phenomenologist Physicist Measured values of Calculated values of physical observables ≈ physical observables It doesn’t matter how beautifull your theory is, It is more important to have beauty It doesn’t matter how smart you are.