Basics of Quantum Chromodynamics (Two Lectures)
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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. -
Zero-Point Energy in Bag Models
Ooa/ea/oHsnif- ^*7 ( 4*0 0 1 5 4 3 - m . Zero-Point Energy in Bag Models by Kimball A. Milton* Department of Physics The Ohio State University Columbus, Ohio 43210 A b s t r a c t The zero-point (Casimir) energy of free vector (gluon) fields confined to a spherical cavity (bag) is computed. With a suitable renormalization the result for eight gluons is This result is substantially larger than that for a spher ical shell (where both interior and exterior modes are pre sent), and so affects Johnson's model of the QCD vacuum. It is also smaller than, and of opposite sign to the value used in bag model phenomenology, so it will have important implications there. * On leave ifrom Department of Physics, University of California, Los Angeles, CA 90024. I. Introduction Quantum chromodynamics (QCD) may we 1.1 be the appropriate theory of hadronic matter. However, the theory is not at all well understood. It may turn out that color confinement is roughly approximated by the phenomenologically suc cessful bag model [1,2]. In this model, the normal vacuum is a perfect color magnetic conductor, that iis, the color magnetic permeability n is infinite, while the vacuum.in the interior of the bag is characterized by n=l. This implies that the color electric and magnetic fields are confined to the interior of the bag, and that they satisfy the following boundary conditions on its surface S: n.&jg= 0, nxBlg= 0, (1) where n is normal to S. Now, even in an "empty" bag (i.e., one containing no quarks) there will be non-zero fields present because of quantum fluctua tions. -
18. Lattice Quantum Chromodynamics
18. Lattice QCD 1 18. Lattice Quantum Chromodynamics Updated September 2017 by S. Hashimoto (KEK), J. Laiho (Syracuse University) and S.R. Sharpe (University of Washington). Many physical processes considered in the Review of Particle Properties (RPP) involve hadrons. The properties of hadrons—which are composed of quarks and gluons—are governed primarily by Quantum Chromodynamics (QCD) (with small corrections from Quantum Electrodynamics [QED]). Theoretical calculations of these properties require non-perturbative methods, and Lattice Quantum Chromodynamics (LQCD) is a tool to carry out such calculations. It has been successfully applied to many properties of hadrons. Most important for the RPP are the calculation of electroweak form factors, which are needed to extract Cabbibo-Kobayashi-Maskawa (CKM) matrix elements when combined with the corresponding experimental measurements. LQCD has also been used to determine other fundamental parameters of the standard model, in particular the strong coupling constant and quark masses, as well as to predict hadronic contributions to the anomalous magnetic moment of the muon, gµ 2. − This review describes the theoretical foundations of LQCD and sketches the methods used to calculate the quantities relevant for the RPP. It also describes the various sources of error that must be controlled in a LQCD calculation. Results for hadronic quantities are given in the corresponding dedicated reviews. 18.1. Lattice regularization of QCD Gauge theories form the building blocks of the Standard Model. While the SU(2) and U(1) parts have weak couplings and can be studied accurately with perturbative methods, the SU(3) component—QCD—is only amenable to a perturbative treatment at high energies. -
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. -
Quantum Mechanics Quantum Chromodynamics (QCD)
Quantum Mechanics_quantum chromodynamics (QCD) In theoretical physics, quantum chromodynamics (QCD) is a theory ofstrong interactions, a fundamental forcedescribing the interactions between quarksand gluons which make up hadrons such as the proton, neutron and pion. QCD is a type of Quantum field theory called a non- abelian gauge theory with symmetry group SU(3). The QCD analog of electric charge is a property called 'color'. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of Particle physics. A huge body of experimental evidence for QCD has been gathered over the years. QCD enjoys two peculiar properties: Confinement, which means that the force between quarks does not diminish as they are separated. Because of this, when you do split the quark the energy is enough to create another quark thus creating another quark pair; they are forever bound into hadrons such as theproton and the neutron or the pion and kaon. Although analytically unproven, confinement is widely believed to be true because it explains the consistent failure of free quark searches, and it is easy to demonstrate in lattice QCD. Asymptotic freedom, which means that in very high-energy reactions, quarks and gluons interact very weakly creating a quark–gluon plasma. This prediction of QCD was first discovered in the early 1970s by David Politzer and by Frank Wilczek and David Gross. For this work they were awarded the 2004 Nobel Prize in Physics. There is no known phase-transition line separating these two properties; confinement is dominant in low-energy scales but, as energy increases, asymptotic freedom becomes dominant. -
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 . -
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. -
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. -
Quantum Chromodynamics 1 9
9. Quantum chromodynamics 1 9. QUANTUM CHROMODYNAMICS Revised October 2013 by S. Bethke (Max-Planck-Institute of Physics, Munich), G. Dissertori (ETH Zurich), and G.P. Salam (CERN and LPTHE, Paris). 9.1. Basics Quantum Chromodynamics (QCD), the gauge field theory that describes the strong interactions of colored quarks and gluons, is the SU(3) component of the SU(3) SU(2) U(1) Standard Model of Particle Physics. × × The Lagrangian of QCD is given by 1 = ψ¯ (iγµ∂ δ g γµtC C m δ )ψ F A F A µν , (9.1) q,a µ ab s ab µ q ab q,b 4 µν L q − A − − X µ where repeated indices are summed over. The γ are the Dirac γ-matrices. The ψq,a are quark-field spinors for a quark of flavor q and mass mq, with a color-index a that runs from a =1 to Nc = 3, i.e. quarks come in three “colors.” Quarks are said to be in the fundamental representation of the SU(3) color group. C 2 The µ correspond to the gluon fields, with C running from 1 to Nc 1 = 8, i.e. there areA eight kinds of gluon. Gluons transform under the adjoint representation− of the C SU(3) color group. The tab correspond to eight 3 3 matrices and are the generators of the SU(3) group (cf. the section on “SU(3) isoscalar× factors and representation matrices” in this Review with tC λC /2). They encode the fact that a gluon’s interaction with ab ≡ ab a quark rotates the quark’s color in SU(3) space. -
Supersymmetry Min Raj Lamsal Department of Physics, Prithvi Narayan Campus, Pokhara Min [email protected]
Supersymmetry Min Raj Lamsal Department of Physics, Prithvi Narayan Campus, Pokhara [email protected] Abstract : This article deals with the introduction of supersymmetry as the latest and most emerging burning issue for the explanation of nature including elementary particles as well as the universe. Supersymmetry is a conjectured symmetry of space and time. It has been a very popular idea among theoretical physicists. It is nearly an article of faith among elementary-particle physicists that the four fundamental physical forces in nature ultimately derive from a single force. For years scientists have tried to construct a Grand Unified Theory showing this basic unity. Physicists have already unified the electron-magnetic and weak forces in an 'electroweak' theory, and recent work has focused on trying to include the strong force. Gravity is much harder to handle, but work continues on that, as well. In the world of everyday experience, the strengths of the forces are very different, leading physicists to conclude that their convergence could occur only at very high energies, such as those existing in the earliest moments of the universe, just after the Big Bang. Keywords: standard model, grand unified theories, theory of everything, superpartner, higgs boson, neutrino oscillation. 1. INTRODUCTION unifies the weak and electromagnetic forces. The What is the world made of? What are the most basic idea is that the mass difference between photons fundamental constituents of matter? We still do not having zero mass and the weak bosons makes the have anything that could be a final answer, but we electromagnetic and weak interactions behave quite have come a long way. -
The Minkowski Quantum Vacuum Does Not Gravitate
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by KITopen Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 946 (2019) 114694 www.elsevier.com/locate/nuclphysb The Minkowski quantum vacuum does not gravitate Viacheslav A. Emelyanov Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany Received 9 April 2019; received in revised form 24 June 2019; accepted 8 July 2019 Available online 12 July 2019 Editor: Stephan Stieberger Abstract We show that a non-zero renormalised value of the zero-point energy in λφ4-theory over Minkowski spacetime is in tension with the scalar-field equation at two-loop order in perturbation theory. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction Quantum fields give rise to an infinite vacuum energy density [1–4], which arises from quantum-field fluctuations taking place even in the absence of matter. Yet, assuming that semi- classical quantum field theory is reliable only up to the Planck-energy scale, the cut-off estimate yields zero-point-energy density which is finite, though, but in a notorious tension with astro- physical observations. In the presence of matter, however, there are quantum effects occurring in nature, which can- not be understood without quantum-field fluctuations. These are the spontaneous emission of a photon by excited atoms, the Lamb shift, the anomalous magnetic moment of the electron, and so forth [5]. This means quantum-field fluctuations do manifest themselves in nature and, hence, the zero-point energy poses a serious problem. -
9. Quantum Chromodynamics 1 9
9. Quantum chromodynamics 1 9. Quantum Chromodynamics Revised September 2017 by S. Bethke (Max-Planck-Institute of Physics, Munich), G. Dissertori (ETH Zurich), and G.P. Salam (CERN).1 This update retains the 2016 summary of αs values, as few new results were available at the deadline for this Review. Those and further new results will be included in the next update. 9.1. Basics Quantum Chromodynamics (QCD), the gauge field theory that describes the strong interactions of colored quarks and gluons, is the SU(3) component of the SU(3) SU(2) U(1) Standard Model of Particle Physics. × × The Lagrangian of QCD is given by 1 = ψ¯ (iγµ∂ δ g γµtC C m δ )ψ F A F A µν , (9.1) q,a µ ab s ab µ q ab q,b 4 µν L q − A − − X µ where repeated indices are summed over. The γ are the Dirac γ-matrices. The ψq,a are quark-field spinors for a quark of flavor q and mass mq, with a color-index a that runs from a =1 to Nc = 3, i.e. quarks come in three “colors.” Quarks are said to be in the fundamental representation of the SU(3) color group. C 2 The µ correspond to the gluon fields, with C running from 1 to Nc 1 = 8, i.e. there areA eight kinds of gluon. Gluons transform under the adjoint representation− of the C SU(3) color group. The tab correspond to eight 3 3 matrices and are the generators of the SU(3) group (cf.