Quantum Mechanics Renormalization
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Penguin Decays of B Mesons
COLO–HEP–395 SLAC-PUB-7796 HEPSY 98-1 April 23, 1998 PENGUIN DECAYS OF B MESONS Karen Lingel 1 Stanford Linear Accelerator Center Stanford University, Stanford, CA 94309 and Tomasz Skwarnicki 2 Department of Physics, Syracuse University Syracuse, NY 13244 and James G. Smith 3 Department of Physics, University of Colorado arXiv:hep-ex/9804015v1 27 Apr 1998 Boulder, CO 80309 Submitted to Annual Reviews of Nuclear and Particle Science 1Work supported by Department of Energy contract DE-AC03-76SF00515 2Work supported by National Science Foundation contract PHY 9807034 3Work supported by Department of Energy contract DE-FG03-95ER40894 2 LINGEL & SKWARNICKI & SMITH PENGUIN DECAYS OF B MESONS Karen Lingel Stanford Linear Accelerator Center Tomasz Skwarnicki Department of Physics, Syracuse University James G. Smith Department of Physics, University of Colorado KEYWORDS: loop CP-violation charmless ABSTRACT: Penguin, or loop, decays of B mesons induce effective flavor-changing neutral currents, which are forbidden at tree level in the Standard Model. These decays give special insight into the CKM matrix and are sensitive to non-standard model effects. In this review, we give a historical and theoretical introduction to penguins and a description of the various types of penguin processes: electromagnetic, electroweak, and gluonic. We review the experimental searches for penguin decays, including the measurements of the electromagnetic penguins b → sγ ∗ ′ and B → K γ and gluonic penguins B → Kπ, B+ → ωK+ and B → η K, and their implications for the Standard Model and New Physics. We conclude by exploring the future prospects for penguin physics. CONTENTS INTRODUCTION .................................... 3 History of Penguins .................................... -
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. -
UNIVERSITETET I OSLO UNIVERSITY of OSLO FYSISK
A/08800Oé,1 UNIVERSITETET i OSLO UNIVERSITY OF OSLO The rols of penjruin-like interactions in weak processes') by 1 J.O. Eeg Department of Physics, University of OoloJ Box 1048 Blindern, 0316 Oslo 3, Norway Report 88-17 Received 1988-10-17 ISSN 0332-9971 FYSISK INSTITUTT DEPARTMENT OF PHYSICS OOP-- 8S-^V The role of penguin-like interactions in weak pro-esses*) by J.O. Eeg Department of Physics, University of Oslo, Box 1048 Blindern, 0316 Oslo 3, Norway Report 88-17 Received 1988-10-17 ISSN 0332-3571 ) Based on talk given at the Ringberg workshop on Hadronio Matrix Elements and Weak Decays, Ringberg Castle, April 18-22, 1988. Organized by A.J. Buras, J.M. Gerard BIK W. Huber, Max-Planck-Institut flir Physik und Astrophysik, MUnchen. The role of penguin-like interactions in weak processesprocesses . J.O. Eeg Dept. of Physics, University of Oslo, P.O.Box 1048 Blindern, N-0316 Oslo 3, Norway October 5, 1988 Abstract Basic properties of penguin diagrams are reviewed. Some model dependent calculations of low momentum penguin loop contributions are presented. CP- violating effects generated when penguin-like diagrams are inserted in higher loops are described. 1 Introduction The history of so-called penguin diagrams has. mainly been connected to the expla nation of the A/ = 1/2 rule and the CP-violating parameter e' in K —> -KIT decays. Many reviews discussing weak non-leptonic decays have been given1,2,3,4,6, and all the details will not be repeated here. As far as possible I will avoid to overlap with other speakers. -
Finite Quantum Gauge Theories
Finite quantum gauge theories Leonardo Modesto1,∗ Marco Piva2,† and Les law Rachwa l1‡ 1Center for Field Theory and Particle Physics and Department of Physics, Fudan University, 200433 Shanghai, China 2Dipartimento di Fisica “Enrico Fermi”, Universit`adi Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy (Dated: August 24, 2018) We explicitly compute the one-loop exact beta function for a nonlocal extension of the standard gauge theory, in particular Yang-Mills and QED. The theory, made of a weakly nonlocal kinetic term and a local potential of the gauge field, is unitary (ghost-free) and perturbatively super- renormalizable. Moreover, in the action we can always choose the potential (consisting of one “killer operator”) to make zero the beta function of the running gauge coupling constant. The outcome is a UV finite theory for any gauge interaction. Our calculations are done in D = 4, but the results can be generalized to even or odd spacetime dimensions. We compute the contribution to the beta function from two different killer operators by using two independent techniques, namely the Feynman diagrams and the Barvinsky-Vilkovisky traces. By making the theories finite we are able to solve also the Landau pole problems, in particular in QED. Without any potential the beta function of the one-loop super-renormalizable theory shows a universal Landau pole in the running coupling constant in the ultraviolet regime (UV), regardless of the specific higher-derivative structure. However, the dressed propagator shows neither the Landau pole in the UV, nor the singularities in the infrared regime (IR). We study a class of new actions of fundamental nature that infinities in the perturbative calculus appear only for gauge theories that are super-renormalizable or finite up to some finite loop order. -
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). -
Introduction to Flavour Physics
Introduction to flavour physics Y. Grossman Cornell University, Ithaca, NY 14853, USA Abstract In this set of lectures we cover the very basics of flavour physics. The lec- tures are aimed to be an entry point to the subject of flavour physics. A lot of problems are provided in the hope of making the manuscript a self-study guide. 1 Welcome statement My plan for these lectures is to introduce you to the very basics of flavour physics. After the lectures I hope you will have enough knowledge and, more importantly, enough curiosity, and you will go on and learn more about the subject. These are lecture notes and are not meant to be a review. In the lectures, I try to talk about the basic ideas, hoping to give a clear picture of the physics. Thus many details are omitted, implicit assumptions are made, and no references are given. Yet details are important: after you go over the current lecture notes once or twice, I hope you will feel the need for more. Then it will be the time to turn to the many reviews [1–10] and books [11, 12] on the subject. I try to include many homework problems for the reader to solve, much more than what I gave in the actual lectures. If you would like to learn the material, I think that the problems provided are the way to start. They force you to fully understand the issues and apply your knowledge to new situations. The problems are given at the end of each section. -
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. -
Direct CP Violation in B Meson Decays
Faculty of Sciences School of Physical Sciences Direct CP Violation in B Meson Decays Liam Christopher Hockley Supervisors: Professor Anthony Thomas Associate Professor Ross Young February 2020 Contents Frontmatter xi 1 Introduction 1 1.1 The Standard Model . 2 1.1.1 The Electroweak Force . 3 1.1.2 Quantum Chromodynamics . 4 1.2 Chirality . 6 2 CP Violation 9 2.1 Symmetries . 9 2.1.1 Parity . 9 2.1.2 Charge Conjugation . 12 2.2 C and P Violation . 13 2.2.1 Parity Violation in the Decays of 60Co Nuclei . 14 2.2.2 Charge Conjugation Violation in the Decays of Pions . 15 2.2.3 Time Reversal and CPT . 16 2.3 CP Violation . 17 2.3.1 CP Violation in Kaons . 17 2.3.2 CP Violation in B Mesons . 18 2.4 Matter-antimatter Asymmetry . 20 2.5 Belle Results . 21 3 Effective Hamiltonian 23 3.1 CKM Matrix . 23 3.2 Four Fermion Interactions . 29 3.3 Operator Product Expansion . 33 3.4 Tree and Penguin Operators . 34 3.4.1 Tree Operators . 34 3.4.2 Penguin Operators . 35 3.5 Effective Hamiltonian . 36 3.6 Wilson Coefficients . 37 4 CP Asymmetry 41 4.1 Formalism . 41 4.2 Choice of Diagrams . 44 4.3 Tree and Penguin Amplitudes . 48 5 Hadronic Matrix Elements 49 5.1 Naive Factorisation . 49 5.1.1 Fierz Identities . 50 5.2 Factorisation Approximation . 52 iii iv Contents 5.3 Form Factors . 54 5.3.1 Motivation . 54 5.3.2 Matrix Element Parameterisation . 55 5.4 Matrix Element Contraction . -
Penguin-Like Diagrams from the Standard Model
Penguin-like Diagrams from the Standard Model Chia Swee Ping High Impact Research, University of Malaya, 50603 Kuala Lumpur, Malaysia Abstract. The Standard Model is highly successful in describing the interactions of leptons and quarks. There are, however, rare processes that involve higher order effects in electroweak interactions. One specific class of processes is the penguin-like diagram. Such class of diagrams involves the neutral change of quark flavours accompanied by the emission of a gluon (gluon penguin), a photon (photon penguin), a gluon and a photon (gluon-photon penguin), a Z- boson (Z penguin), or a Higgs-boson (Higgs penguin). Such diagrams do not arise at the tree level in the Standard Model. They are, however, induced by one-loop effects. In this paper, we present an exact calculation of the penguin diagram vertices in the ‘tHooft-Feynman gauge. Renormalization of the vertex is effected by a prescription by Chia and Chong which gives an expression for the counter term identical to that obtained by employing Ward-Takahashi identity. The on- shell vertex functions for the penguin diagram vertices are obtained. The various penguin diagram vertex functions are related to one another via Ward-Takahashi identity. From these, a set of relations is obtained connecting the vertex form factors of various penguin diagrams. Explicit expressions for the gluon-photon penguin vertex form factors are obtained, and their contributions to the flavor changing processes estimated. Keywords: Standard model, penguin diagrams, neutral change of quark flavours, rare processes. PACS: 11.30.Hv, 12.15.-y, 14.70.-e INTRODUCTION The Standard Model (SM) of elementary particles is based on the SU(3)×SU(2)×U(1) gauge group. -
Higgs Boson Mass and New Physics Arxiv:1205.2893V1
Higgs boson mass and new physics Fedor Bezrukov∗ Physics Department, University of Connecticut, Storrs, CT 06269-3046, USA RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Mikhail Yu. Kalmykovy Bernd A. Kniehlz II. Institut f¨urTheoretische Physik, Universit¨atHamburg, Luruper Chaussee 149, 22761, Hamburg, Germany Mikhail Shaposhnikovx Institut de Th´eoriedes Ph´enom`enesPhysiques, Ecole´ Polytechnique F´ed´eralede Lausanne, CH-1015 Lausanne, Switzerland May 15, 2012 Abstract We discuss the lower Higgs boson mass bounds which come from the absolute stability of the Standard Model (SM) vacuum and from the Higgs inflation, as well as the prediction of the Higgs boson mass coming from asymptotic safety of the SM. We account for the 3-loop renormalization group evolution of the couplings of the Standard Model and for a part of two-loop corrections that involve the QCD coupling αs to initial conditions for their running. This is one step above the current state of the art procedure (\one-loop matching{two-loop running"). This results in reduction of the theoretical uncertainties in the Higgs boson mass bounds and predictions, associated with the Standard Model physics, to 1−2 GeV. We find that with the account of existing experimental uncertainties in the mass of the top quark and αs (taken at 2σ level) the bound reads MH ≥ Mmin (equality corresponds to the asymptotic safety prediction), where Mmin = 129 ± 6 GeV. We argue that the discovery of the SM Higgs boson in this range would be in arXiv:1205.2893v1 [hep-ph] 13 May 2012 agreement with the hypothesis of the absence of new energy scales between the Fermi and Planck scales, whereas the coincidence of MH with Mmin would suggest that the electroweak scale is determined by Planck physics. -
Physics and Mathematics of Quantum Field Theory
Physics and Mathematics of Quantum Field Theory Jan Derezinski´ (University of Warsaw), Stefan Hollands (Leipzig University), Karl-Henning Rehren (Gottingen¨ University) July 29, 2018 – August 3, 2018 1 Overview of the Field The origins of quantum field theory (QFT) date back to the early days of quantum physics. Having developed quantum mechanics, describing non-relativistic systems of a finite number of degrees of freedom, physicists tried to apply similar ideas to relativistic classical field theories such as electrodynamics following early attempts already made by the founding fathers of quantum mechanics. Since quantum mechanics is usually “derived” from classical mechanics by a procedure called quantization, the first attempts tried to mimick this procedure for classical field theories. But it was clear almost from the beginning that this endeavor would not work without qualitatively new ideas, owing both to the new difficulties presented by relativistic kinematics, as well as to the fact that field theories, while possessing a Lagrangian/Hamiltonian formulation just as classical mechanics, effectively describe infinitely many degrees of freedom. In particular, QFT completely dissolved the classical distinction between “particles” (like the electron) and “fields” (like the electromagnetic field). Instead, the dichotomy in nomenclature acquired a very different meaning: Fields are the fundamental entities for all sorts of matter, while particles are manifestations when the fields arise in special states. Thus, from the outset, one is dealing with systems with an unlimited number of particles. Early successes of quantum field theory consolidated by the early 50’s, such as precise computations of the Lamb shifts of the Hydrogen and of the anomalous magnetic moment of the electron, were accompanied around the same time by the first attempts to formalize this theory in a more mathematical fashion. -
Department of Physics and Astronomy University of Heidelberg
Department of Physics and Astronomy University of Heidelberg Diploma thesis in Physics submitted by Kher Sham Lim born in Pulau Pinang, Malaysia 2011 This page is intentionally left blank Planck Scale Boundary Conditions And The Standard Model This diploma thesis has been carried out by Kher Sham Lim at the Max Planck Institute for Nuclear Physics under the supervision of Prof. Dr. Manfred Lindner This page is intentionally left blank Fakult¨atf¨urPhysik und Astronomie Ruprecht-Karls-Universit¨atHeidelberg Diplomarbeit Im Studiengang Physik vorgelegt von Kher Sham Lim geboren in Pulau Pinang, Malaysia 2011 This page is intentionally left blank Randbedingungen der Planck-Skala und das Standardmodell Die Diplomarbeit wurde von Kher Sham Lim ausgef¨uhrtam Max-Planck-Institut f¨urKernphysik unter der Betreuung von Herrn Prof. Dr. Manfred Lindner This page is intentionally left blank Randbedingungen der Planck-Skala und das Standardmodell Das Standardmodell (SM) der Elementarteilchenphysik k¨onnte eine effektive Quan- tenfeldtheorie (QFT) bis zur Planck-Skala sein. In dieser Arbeit wird diese Situ- ation angenommen. Wir untersuchen, ob die Physik der Planck-Skala Spuren in Form von Randbedingungen des SMs hinterlassen haben k¨onnte. Zuerst argumen- tieren wir, dass das SM-Higgs-Boson kein Hierarchieproblem haben k¨onnte, wenn die Physik der Planck-Skala aus einem neuen, nicht feld-theoretischen Konzept bestehen w¨urde. Der Higgs-Sektor wird bez¨uglich der theoretischer und experimenteller Ein- schr¨ankungenanalysiert. Die notwendigen mathematischen Methoden aus der QFT, wie z.B. die Renormierungsgruppe, werden eingef¨uhrtum damit das Laufen der Higgs- Kopplung von der Planck-Skala zur elektroschwachen Skala zu untersuchen.