Does Charged-Pion Decay Violate Conservation of Angular Momentum? Kirk T

Total Page：16

File Type：pdf, Size：1020Kb

Does Charged-Pion Decay Violate Conservation of Angular Momentum? Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (September 6, 2016, updated November 20, 2016) 1Problem + + Charged-pion decay, such as π → μ νµ, is considered in the Standard Model to involve the annihilation of the constituent quarks, ud¯,oftheπ+ into a virtual W + gauge boson, which + materializes as the ﬁnal state μ νµ. While the pion is spinless, the W -boson is considered to have spin 1, which appears to violate conservation of angular momentum. What’s going on here?1 2Solution A remarked by Higgs in his Nobel Lecture [3], “... in this model the Goldstone massless (spin- 0) mode became the longitudinal polarization of a massive spin-1 photon, just as Anderson had suggested.” That is, in the Higgs’ mechanism, the Sz =0stateofaW boson is more or less still a spin-0 “particle.” Likewise, Weinberg in his Nobel Lecture [4] stated that: “The missing Goldstone bosons appear instead as helicity zero states of the vector particles, which thereby acquire a mass.” A similar view was given in [5]. References [1] S. Ogawa, On the Universality of the Weak Interaction, Prog. Theor. Phys. 15, 487 (1956), http://physics.princeton.edu/~mcdonald/examples/EP/ogawa_ptp_15_487_56.pdf [2] Y. Tanikawa and S. Watanabe, Fermi Interaction Caused by Intermediary Chiral Boson, Phys. Rev. 113, 1344 (1959), http://physics.princeton.edu/~mcdonald/examples/EP/tanikawa_pr_113_1344_59.pdf [3] P.W. Higgs, Evading the Goldstone Boson,Rev.Mod.Phys.86, 851 (2014), http://physics.princeton.edu/~mcdonald/examples/EP/higgs_rmp_86_851_14.pdf [4] S. Weinberg, Conceptual Foundations of the Uniﬁed Theory of Weak and Electromag- netic Interactions,Rev.Mod.Phys.52, 515 (1980), http://physics.princeton.edu/~mcdonald/examples/EP/weinberg_rmp_52_515_80.pdf 1The possible need for an intermediate scalar boson was noted in [1], and pursued conceptually in, for example, [2]. 1 [5] N. Nakanishi, T*-Product and False Nonconservation of Angular Momentum in the Pion Decay, Mod. Phys. Lett. A 17, 89 (2002), http://physics.princeton.edu/~mcdonald/examples/EP/nakanishi_mpla_17_89_02.pdf 2.

Copy Link

Recommended publications

Basic Ideas of the Standard Model
BASIC IDEAS OF THE STANDARD MODEL V.I. ZAKHAROV Randall Laboratory of Physics University of Michigan, Ann Arbor, Michigan 48109, USA Abstract This is a series of four lectures on the Standard Model. The role of conserved currents is emphasized. Both degeneracy of states and the Goldstone mode are discussed as realization of current conservation. Remarks on strongly interact- ing Higgs fields are included. No originality is intended and no references are given for this reason. These lectures can serve only as a material supplemental to standard textbooks. PRELIMINARIES Standard Model (SM) of electroweak interactions describes, as is well known, a huge amount of exper- imental data. Comparison with experiment is not, however, the aspect of the SM which is emphasized in present lectures. Rather we treat SM as a field theory and concentrate on basic ideas underlying this field theory. Th Standard Model describes interactions of fields of various spins and includes spin-1, spin-1/2 and spin-0 particles. Spin-1 particles are observed as gauge bosons of electroweak interactions. Spin- 1/2 particles are represented by quarks and leptons. And spin-0, or Higgs particles have not yet been observed although, as we shall discuss later, we can say that scalar particles are in fact longitudinal components of the vector bosons. Interaction of the vector bosons can be consistently described only if it is highly symmetrical, or universal. Moreover, from experiment we know that the corresponding couplings are weak and can be treated perturbatively. Interaction of spin-1/2 and spin-0 particles are fixed by theory to a much lesser degree and bring in many parameters of the SM.
[Show full text]
Introduction to Supersymmetry
Introduction to Supersymmetry Pre-SUSY Summer School Corpus Christi, Texas May 15-18, 2019 Stephen P. Martin Northern Illinois University [email protected] 1 Topics: Why: Motivation for supersymmetry (SUSY) • What: SUSY Lagrangians, SUSY breaking and the Minimal • Supersymmetric Standard Model, superpartner decays Who: Sorry, not covered. • For some more details and a slightly better attempt at proper referencing: A supersymmetry primer, hep-ph/9709356, version 7, January 2016 • TASI 2011 lectures notes: two-component fermion notation and • supersymmetry, arXiv:1205.4076. If you find corrections, please do let me know! 2 Lecture 1: Motivation and Introduction to Supersymmetry Motivation: The Hierarchy Problem • Supermultiplets • Particle content of the Minimal Supersymmetric Standard Model • (MSSM) Need for “soft” breaking of supersymmetry • The Wess-Zumino Model • 3 People have cited many reasons why extensions of the Standard Model might involve supersymmetry (SUSY). Some of them are: A possible cold dark matter particle • A light Higgs boson, M = 125 GeV • h Unification of gauge couplings • Mathematical elegance, beauty • ⋆ “What does that even mean? No such thing!” – Some modern pundits ⋆ “We beg to differ.” – Einstein, Dirac, . However, for me, the single compelling reason is: The Hierarchy Problem • 4 An analogy: Coulomb self-energy correction to the electron’s mass A point-like electron would have an infinite classical electrostatic energy. Instead, suppose the electron is a solid sphere of uniform charge density and radius R. An undergraduate problem gives: 3e2 ∆ECoulomb = 20πǫ0R 2 Interpreting this as a correction ∆me = ∆ECoulomb/c to the electron mass: 15 0.86 10− meters m = m + (1 MeV/c2) × .
[Show full text]
Search for Non Standard Model Higgs Boson
Thirteenth Conference on the Intersections of Particle and Nuclear Physics May 29 – June 3, 2018 Palm Springs, CA Search for Non Standard Model Higgs Boson Ana Elena Dumitriu (IFIN-HH, CPPM), On behalf of the ATLAS Collaboration 5/18/18 1 Introduction and Motivation ● The discovery of the Higgs (125 GeV) by ATLAS and CMS collaborations → great success of particle physics and especially Standard Model (SM) ● However.... – MORE ROOM Hierarchy problem FOR HIGGS PHYSICS – Origin of dark matter, dark energy BEYOND SM – Baryon Asymmetry – Gravity – Higgs br to Beyond Standard Model (BSM) particles of BRBSM < 32% at 95% C.L. ● !!!!!!There is plenty of room for Higgs physics beyond the Standard Model. ● There are several theoretical models with an extended Higgs sector. – 2 Higgs Doublet Models (2HDM). Having two complex scalar SU(2) doublets, they are an effective extension of the SM. In total, 5 different Higgs bosons are predicted: two CP even and one CP odd electrically neutral Higgs bosons, denoted by h, H, and A, respectively, and two charged Higgs bosons, H±. The parameters used to describe a 2HDM are the Higgs boson masses and the ratios of their vacuum expectation values. ● type I, where one SU(2) doublet gives masses to all leptons and quarks and the other doublet essentially decouples from fermions ● type II, where one doublet gives mass to up-like quarks (and potentially neutrinos) and the down-type quarks and leptons receive mass from the other doublet. – Supersymmetry (SUSY), which brings along super-partners of the SM particles. The Minimal Supersymmetric Standard Model (MSSM), whose Higgs sector is equivalent to the one of a constrained 2HDM of type II and the next-to MSSM (NMSSM) are among the experimentally best tested models, because they provide good benchmarks for SUSY.
[Show full text]
JHEP05(2010)010 , 6 − Ing Springer and 10 May 4, 2010 9 : April 19, 2010 − : Tic Inflationary December 3, 2009 : Published Lar Potential, and No S
Published for SISSA by Springer Received: December 3, 2009 Accepted: April 19, 2010 Published: May 4, 2010 Light inflaton hunter’s guide JHEP05(2010)010 F. Bezrukova and D. Gorbunovb aMax-Planck-Institut für Kernphysik, P.O. Box 103980, 69029 Heidelberg, Germany bInstitute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary prospect 7a, Moscow 117312, Russia E-mail: [email protected], [email protected] Abstract: We study the phenomenology of a realistic version of the chaotic inflationary model, which can be fully and directly explored in particle physics experiments. The inflaton mixes with the Standard Model Higgs boson via the scalar potential, and no additional scales above the electroweak scale are present in the model. The inflaton-to- Higgs coupling is responsible for both reheating in the Early Universe and the inflaton production in particle collisions. We find the allowed range of the light inflaton mass, 270 MeV . mχ . 1.8 GeV, and discuss the ways to find the inflaton. The most promising are two-body kaon and B-meson decays with branching ratios of orders 10−9 and 10−6, respectively. The inflaton is unstable with the lifetime 10−9–10−10 s. The inflaton decays can be searched for in a beam-target experiment, where, depending on the inflaton mass, from several billions to several tenths of millions inflatons can be produced per year with modern high-intensity beams. Keywords: Cosmology of Theories beyond the SM, Rare Decays ArXiv ePrint: 0912.0390 Open Access doi:10.1007/JHEP05(2010)010 Contents 1 Introduction 1 2 The model 3 3 Inflaton decay palette 6 4 Inflaton from hadron decays 10 JHEP05(2010)010 5 Inflaton production in particle collisions 12 6 Limits from direct searches and predictions for forthcoming experiments 14 7 Conclusions 16 A The νMSM extension 16 1 Introduction In this paper we present an example of how (low energy) particle physics experiments can directly probe the inflaton sector (whose dynamics is important at high energies in the very Early Universe).
[Show full text]
Introduction to Supersymmetry
Introduction to Supersymmetry J.W. van Holten NIKHEF Amsterdam NL 1 Relativistic particles a. Energy and momentum The energy and momentum of relativistic particles are related by E2 = p2c2 + m2c4: (1) In covariant notation we define the momentum four-vector E E pµ = (p0; p) = ; p ; p = (p ; p) = − ; p : (2) c µ 0 c The energy-momentum relation (1) can be written as µ 2 2 p pµ + m c = 0; (3) where we have used the Einstein summation convention, which implies automatic summa- tion over repeated indices like µ. Particles can have different masses, spins and charges (electric, color, flavor, ...). The differences are reflected in the various types of fields used to describe the quantum states of the particles. To guarantee the correct energy-momentum relation (1), any free field Φ must satisfy the Klein-Gordon equation 2 − 2@µ@ + m2c2 Φ = ~ @2 − 2r2 + m2c2 Φ = 0: (4) ~ µ c2 t ~ Indeed, a plane wave Φ = φ(k) ei(k·x−!t); (5) satisfies the Klein-Gordon equation if E = ~!; p = ~k: (6) From now on we will use natural units in which ~ = c = 1. In these units we can write the plane-wave fields (5) as Φ = φ(p) eip·x = φ(p) ei(p·x−Et): (7) b. Spin Spin is the intrinsic angular momentum of particles. The word ìntrinsic' is to be inter- preted somewhat differently for massive and massless particles. For massive particles it is the angular momentum in the rest-frame of the particles, whilst for massless particles {for which no rest-frame exists{ it is the angular momentum w.r.t.
[Show full text]
The Higgs As a Pseudo-Goldstone Boson
Universidad Autónoma de Madrid Facultad de Ciencias Departamento de Física Teórica The Higgs as a pseudo-Goldstone boson Memoria de Tesis Doctoral realizada por Sara Saa Espina y presentada ante el Departamento de Física Teórica de la Universidad Autónoma de Madrid para la obtención del Título de Doctora en Física Teórica. Tesis Doctoral dirigida por M. Belén Gavela Legazpi, Catedrática del Departamento de Física Teórica de la Universidad Autónoma de Madrid. Madrid, 27 de junio de 2017 Contents Purpose and motivation 1 Objetivo y motivación 4 I Foundations 9 1 The Standard Model (SM) 11 1.1 Gauge fields and fermions 11 1.2 Renormalizability and unitarity 14 2 Symmetry and spontaneous breaking 17 2.1 Symmetry types and breakings 17 2.2 Spontaneous breaking of a global symmetry 18 2.2.1 Goldstone theorem 20 2.2.2 Spontaneous breaking of the chiral symmetry in QCD 21 2.3 Spontaneous breaking of a gauge symmetry 26 3 The Higgs of the SM 29 3.1 The EWSB mechanism 29 3.2 SM Higgs boson phenomenology at the LHC: production and decay 32 3.3 The Higgs boson from experiment 34 3.4 Triviality and Stability 38 4 Having a light Higgs 41 4.1 Naturalness and the “Hierarchy Problem” 41 4.2 Higgsless EWSB 44 4.3 The Higgs as a pseudo-GB 45 II Higgs Effective Field Theory (HEFT) 49 5 Effective Lagrangians 51 5.1 Generalities of EFTs 51 5.2 The Chiral Effective Lagrangian 52 5.3 Including a light Higgs 54 5.4 Linear vs.
[Show full text]
Arxiv:1801.05670V1 [Hep-Ph] 17 Jan 2018 Most Adequate (Mathematical) Language to Describe Nature
Quantum Field Theory and the Electroweak Standard Model* A.B. Arbuzov BLTP JINR, Dubna, Russia Abstract Lecture notes with a brief introduction to Quantum field theory and the Stan- dard Model are presented. The lectures were given at the 2017 European School of High-Energy Physics. The main features, the present status, and problems of the Standard Model are discussed. 1 Introduction The lecture course consists of four main parts. In the Introduction, we will discuss what is the Standard Model (SM) [1–3], its particle content, and the main principles of its construction. The second Section contains brief notes on Quantum Field Theory (QFT), where we remind the main objects and rules required further for construction of the SM. Sect. 3 describes some steps of the SM development. The Lagrangian of the model is derived and discussed. Phenomenology and high-precision tests of the model are overviewed in Sect. 4. The present status, problems, and prospects of the SM are summarized in Conclusions. Some simple exercises and questions are given for students in each Section. These lectures give only an overview of the subject while for details one should look in textbooks, e.g., [4–7], and modern scientific papers. 1.1 What is the Standard Model? Let us start with the definition of the main subject of the lecture course. It is the so-called Standard Model. This name is quite widely accepted and commonly used to define a certain theoretical model in high energy physics. This model is suited to describe properties and interactions of elementary particles.
[Show full text]
11. the Top Quark and the Higgs Mechanism Particle and Nuclear Physics
11. The Top Quark and the Higgs Mechanism Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 11. The Top Quark and the Higgs Mechanism 1 In this section... Focus on the most recent discoveries of fundamental particles The top quark { prediction & discovery The Higgs mechanism The Higgs discovery Dr. Tina Potter 11. The Top Quark and the Higgs Mechanism 2 Third Generation Quark Weak CC Decays 173 GeV t Cabibbo Allowed |Vtb|~1, log(mass) |Vcs|~|Vud|~0.975 Top quarks are special. Cabibbo Suppressed m(t) m(b)(> m(W )) |Vcd|~|Vus|~0.22 |V |~|V |~0.05 −25 cb ts τt ∼ 10 s ) decays b 4.8GeV before hadronisation Vtb ∼ 1 ) 1.3GeV c BR(t ! W + b)=100% s 95 MeV 2.3MeV u d 4.8 MeV Bottom quarks are also special. b quarks can only decay via the Cabbibo suppressed Wcb vertex. Vcb is very small { weak coupling! Interaction b-jet point ) τ(b) τ(u; c; d; s) b Jet initiated by b quarks look diﬀerent to other jets. b quarks travel further from interaction point before decaying. b-jet traces back to a secondary vertex {\ b-tagging". Dr. Tina Potter 11. The Top Quark and the Higgs Mechanism 3 The Top Quark The Standard Model predicted the existence of the top quark 2 +3e u c t 1 −3e d s b which is required to explain a number of observations. d µ− Example: Non-observation of the decay W − 0 + − 0 + − −9 K ! µ µ B(K ! µ µ ) < 10 u=c=t νµ The top quark cancels the contributions W + from the u and c quarks.
[Show full text]
New Heavy Bosons
Citation: M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019 update New Heavy Bosons (W ′, Z ′, leptoquarks, etc.), Searches for We list here various limits on charged and neutral heavy vector bosons (other than W ’s and Z’s), heavy scalar bosons (other than Higgs bosons), vector or scalar leptoquarks, and axigluons. The latest unpublished results are described in “W ′ Searches” and “Z ′ Searches” reviews. For recent searches on scalar bosons which could be identiﬁed as Higgs bosons, see the listings in the Higgs boson sec- tion. CONTENTS: Mass Limits for W ′ (Heavy Charged Vector Boson Other Than W ) in Hadron Collider Experiments WR (Right-Handed W Boson) Mass Limits Limit on WL-WR Mixing Angle ζ Mass Limits for Z (Heavy Neutral Vector Boson Other Than Z) ′′ Limits for Z − SM Limits for Z − LR Limits for Z − χ Limits for Z − ψ Limits for Z − η Limits for other Z − ′ Searches for Z with Lepton-Flavor-Violating decays − ′ Indirect Constraints on Kaluza-Klein Gauge Bosons Mass Limits for Leptoquarks from Pair Production Mass Limits for Leptoquarks from Single Production Indirect Limits for Leptoquarks Mass Limits for Diquarks Mass Limits for gA (axigluon) and Other Color-Octet Gauge Bosons Mass Limits for Color-Octet Scalar Bosons X 0 (Heavy Boson) Searches in Z Decays + Mass Limits for a Heavy Neutral Boson Coupling to e e− 0 + Search for X Resonance in e e− Collisions Search for X 0 Resonance in ep Collisions Search for X 0 Resonance in Two-Photon Process Search for X 0 Resonance in e+ e X 0 γ − → Search for X 0 Resonance in Z f f X 0 → Search for X 0 Resonance in W X 0 ﬁnal state Search for X 0 Resonance in Quarkonium Decays See the related review(s): W ′-Boson Searches HTTP://PDG.LBL.GOV Page1 Created: 8/2/2019 16:43 Citation: M.
[Show full text]
Quantum Field Theory I + II
Quantum Field Theory I + II Institute for Theoretical Physics, Heidelberg University Timo Weigand Literature This is a writeup of my Master programme course on Quantum Field Theory I (Chapters 1-6) and Quantum Field Theory II. The primary source for this course has been • Peskin, Schröder: An introduction to Quantum Field Theory, ABP 1995, • Itzykson, Zuber: Quantum Field Theory, Dover 1980, • Kugo: Eichtheorie, Springer 1997, which I urgently recommend for more details and for the many topics which time constraints have forced me to abbreviate or even to omit. Among the many other excellent textbooks on Quantum Field Theory I particularly recommend • Weinberg: Quantum Field Theory I + II, Cambridge 1995, • Srednicki: Quantum Field Theory, Cambridge 2007, • Banks: Modern Quantum Field Theory, Cambridge 2008 as further reading. All three of them oftentimes take an approach different to the one of this course. Excellent lecture notes available online include • A. Hebecker: Quantum Field Theory, • D. Tong: Quantum Field Theory. Special thanks to Robert Reischke1 for his fantastic work in typing these notes. 1For corrections and improvement suggestions please send a mail to [email protected]. 4 Contents 1 The free scalar field9 1.1 Why Quantum Field Theory?..............................9 1.2 Classical scalar field: Lagrangian formulation..................... 11 1.3 Noether’s Theorem................................... 14 1.4 Quantisation in the Schrödinger Picture........................ 17 1.5 Mode expansion..................................... 18 1.6 The Fock space..................................... 21 1.7 Some important technicalities.............................. 23 1.7.1 Normalisation.................................. 23 1.7.2 The identity................................... 24 1.7.3 Position-space representation.......................... 24 1.8 On the vacuum energy................................
[Show full text]
Scalar Bosons and Supersymmetry
Le Boson H, SéminairePoincaréXIX (2014) 75 { 87 SéminairePoincaré Scalar Bosons and Supersymmetry Pierre FAYET Laboratoire de Physique Théoriquede l'Ecole´ Normale Supérieure, 24 rue Lhomond, 75231 Paris cedex 05, France∗ Abstract. The probable discovery of a spin-0 Brout-Englert-Higgs boson leads to further enquire about other fundamental scalars. Supersymmetric theories involve, in relation with the electroweak breaking, five such scalars at least, two charged and three neutral ones, usually denoted as H±; H; h and A. They also introduce spin-0 squarks and sleptons as the superpartners of quarks and leptons. Supersymmetric extensions of the standard model also lead to the possibility of gauge/BEH unification by providing spin-0 bosons as extra states for spin-1 gauge bosons within massive gauge multiplets. Depending on its properties the 125 GeV boson observed at CERN may then also be interpreted, up to a mixing angle induced by supersymmetry breaking, as the spin-0 partner of the Z under two supersymmetry transformations, i.e. as a Z that would be deprived of its spin. 1 The electroweak symmetry breaking Special relativity and quantum mechanics, operating within quantum field theory, led to the Standard Model of particles and interactions. It has met a long series of successes with the discoveries of weak neutral currents (1973), charmed particles (1974-76), gluons mediators of strong interactions (1979), W ± and Z's mediators of weak interactions (1983), and the sixth quark known as the top quark (1995). Weak, electromagnetic and strong interactions are all understood from the ex- ± 1 changes of spin-1 mediators, W 's and Z's, photons and gluons, between spin- 2 quarks and leptons, generically referred to as the constituents of matter.
[Show full text]
[Physics.Gen-Ph] 11 Jul 2020 I.Lpo Asinteraction Mass Lepton III
Particle generations in R0|18 dust gravity Robert N. C. Pfeifer∗ Dunedin, Otago, New Zealand (Dated: July 12, 2020) The R0|18 dust gravity model contains analogues to the particle spectrum and interactions of the Standard Model and gravity, but with only four tunable parameters. As the structure of this model is highly constrained, predictive relationships between its counterparts to the constants of the Standard Model may be obtained. In this paper, the model values for the masses of the tau, the W and Z bosons, and a Higgs-like scalar boson are calculated as functions of α, me, and mµ, with no free ﬁtting parameters. They are shown to be 1776.867(1) MeV/c2, 80.3786(3) GeV/c2, 91.1877(4) GeV/c2, and 125.16(1) GeV/c2 respectively, all within 0.5 σ or better of the corresponding observed values of 1776.86(12) MeV/c2, 80.379(12) GeV/c2, 91.1876(21) GeV/c2, and 125.10(14) GeV/c2. This result suggests the existence of a unifying relationship between lepton generations and the electroweak mass scale, which is proposed to arise from preon interactions mediated by the strong nuclear force. 4 CONTENTS 4. Second-order correction to Kℓ from the tau channel 24 I. Introduction 1 5. Dilaton corrections to ∆e(mei ) 25 4 6. Corrections to [Ke(θe)] from the muon II. Boson Mass Interactions 2 and electron channels 25 A. W mass 2 0 18 1. Boson loops 3 IV. Relationships from R | dust gravity 27 2. QL photon and scalar interactions 5 A.
[Show full text]