DOCSLIB.ORG

Theoretical Physics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Theoretical Physics 1090/33 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ON A PHASE STRUCTURE OF A THREE-DIMENSIONAL cp4 FIELD THEORY G.V. Efimov INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1990 MIRAMARE-TRIESTE T T r r IC/90/33 INTRODUCTION International Atomic Energy Agency In this paper we will investigate the phase structure of the quantum field models and United Nations Educational Scientific and Cultural Organization 1 (M) INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS - g (1.2) in a space-time of three dimensions R3. The Lagrangians (1.1 and 1.2) describe a one-component scalar field <p( x) and these Lagrangians are invariant under the transformation tp -* —<p. The mass 4 2 ON A PHASE STRUCTURE OF A THREE-DIMENSIONAL tp FIELD THEORY • m in (1.1) is positive, g is coupling constant, and tp0 in (1.2) is a real constant. If the coupling constant is small enough, the Lagrangian (I.I) describes the symmetric interaction, but the Lagrangian (1.2) describes the so-called situation of spontaneous symmetry breaking (SSB) (see, for example. Reft. 1 and 2, containing many earlier references). G.V. Efimov ** Our aim is to investigate the behaviour of the systems (1.1 and 1.2) in the strong cou- International Centre for Theoretical Physics, Trieste, Italy. pling limit, i.e., for 0 —> oo. We will investigate these Lagrangians by the method of canonical transformations. We have used this method for the investigation of the same models in R 2 [2]. By comparison with the Lagrangians in R2 we have the additional divergent diagrams ABSTRACT in R3. Nevertheless, the Lagrangians (1.1 and 1.2) remain to be super-renormalizable and we can write down all necessary counter-terms in the explicit form. We construct the Hamiltonians for Two models of a scalar field <p with the interaction Lagrangians gtp* andg(<p2 - f>l)z Lagrangians (1.1 and 1.2) and introduce operators with counter-terms. We formulate our problem 3 are considered in R . There are phase transitions in these models for a certain g = gc. In the as follows: first model gtp* the symmetry <p —» — <p keeps for any values of p. In the second model where the spontaneous breaking symmetry takes place for small g the symmetry <p —* — <p is restored for What representation of canonical commutative relations is suitable for different values of the dimensionless coupling constant g > gc. The effective perturbation coupling constant is small in both cases. (1.3) and what physical picture corresponds to this representation? 2 M1RAMARE - TRIESTE It turned out that the picture is quite different from the situation in R . Namely, the system (1.1) keeps the symmetry tp —> -tp (or all values of the coupling constant although a kind January 1990 of a phase transition takes place. The symmetry <p —» —ip is restored in the system (1.2) for \ > \c where \c is a critical coupling constant At last we want to stress especially that the effective perturbation coupling constant is small in both cases, i.e., the Lagrangians (I.I and 1.2) describe the systems with weak coupling for any values of g. In other words, the strong coupling regime in the sense that the perturbation expansion has a large coupling constant does not exist in the system s described by the Lagrangians To be submitted for publication. Permanent address: Joint Institute for Nuclear Research, Dubna, USSR. (1.1 and 1.2), T 3 to remove these divergences we should introduce in the interaction Hamiltonian (2.2) an operator 2. HAMILTONIAN v* IN R Rm containing counter-terms which cancel these divergences in perturbation calculations. This In this section we consider the theory of a scalar field <p in the space-time R ' which is operator Rm looks as described by the Lagrangian density 1 fi«= [ dS: \~Amp (x)*Cm *><*) (2.6) iv L2 C(x) = -<p(x)(O -T (2.1) where where gi and g4 are coupling constants. C« = 4! 93 04 Z , We denote coordinates by x = (10,3:1,12) - (zo,£) and momenta by fc • (fco,*i,to> = m Ea = j 4! 9\ Wm - j Vm + ~ 3! (2.7) We shall consider this theory in the Hamiltonian representation. The second-quantized These divergent diagrams can be computed, for example, in the Euclidean metrics (xo Hamiltonian for the Lagrangian (2.1) is written as iky) where the propagator of the field <p(x) is * 1 e"1" H (2.2) Here*2 - *2+Jt|+Jt|.(Jta) + x\ + s|,andfc » y^Jt2 + A22 + fcf We have (23) Here (2.9) Here an appropriate regularization should be introduced. (2.4) We can now say that the theory (2.2 and 2.3) is defined, i.e., the S-matrix The fields p( 1) and ir( z) are canonical variables and obey the canonical commutation relations S-Tcxpi-if* dtH?(t)\ (2.10) [pUKipCr')] = [ir(z),ir(5')]=0 is finite in each order of the perturbation expansion. [?<*),«<« ')]=i6(i~i'). (2.5) 3. CANONICAL TRANSFORMATION The integration in (2.3) is performed over a large "volume" V, here it is a square: Our aim is to realize what is this system (2.1) for different values of the coupling constants j3 and 04. In order to do it we shall use the method of canonical transformations. Let us perform and V = (2 L)2. All operators in (2.3) are taken in the form of normal products. the canonical transformation of our fields This theory contains ultraviolet divergences but it is super-renormalizable, i.e., it has only a finite number of the divergent Feynman diagrams. These diagrams are shown in Fig.l. In order <p(x) B (3.1) where the fields fl (z) and *( i) have the form (2.4) but their mass is equal to Hz= f dx M - m( 1 + f) . (32) where B is a constant field. The explicit form of this transformation (3.1) can be written but we will not S(m,M) =m2 -Af 4! do it here. 2 2 P(m, Af) = tn B + - 3AB) - A) + 4g4B) We now perform this canonical transformation (3.1) for the Hamiltonian (2.2 and 2.3) (3.8) and go over to the normal ordering in the new fields * (a;) and El ( x). We will use the well-known formulas B(m,M) ,Af) + S4(B*-6AB2 + 3A2)+ ^ ) (3.3) In Here: * : denotes the normal ordering of the fields <p(x) and <t> ( x) correspondingly, * is a constant 4! Kim, M) = - and A = A(m,M) = £>m(0) - D«(0) = . 3 Using the formulas (2.9) one can get The canonical transformation of the free Hamiltonian looks as 1 1 , M 1 l 64 ' = Ho1 + I dS: \^-=— + TO2B<I>(SC) + = ©2(i) + L(m, j (3.5) Jv I 2 2 —l(4ir)_3 where (3.9) - M2-m21 Let us put the coefficients of the operator : <t>2 : and * in Hi (3.6) equal to zero Using these formulas we are able to do the canonical transformation of the total Hamil- P(m,AO=0 . (310) tonian (2.2). One can obtain This requirement means that the total Hamiltonian cannot contain the linear term <t> at all and (3.6) can contain only the quadratic term : *2 : in the free Hamiltonian. In other words, the total Hamiltonian should have a "correct" form, i.e., the free Hamiltonian has a standard form and the Here H" is the free Hamiltonian of the fields <t> (x) and n (at) with the mass M. interaction Hamiltonian can contain field operators in degrees more than two. From a physical point of view, it means that our system consists of the scalar particles with the mass M and these noninteracting particles are described by the free Hamiltonian H". , G* =QA . (3.7) It should be noted that Eqs.(3.10) do not define the minimum of the energy E(m, M) in 2 The operator Jtu coincides with (2.6) where we should substitute (3.8) as it was in the <p* -theory in R 13]. It follows from that the counter-terms of the highest perturbation orders are introduced into the Hamiltonian (2.3) in order to remove all divergences m —* M, }]->(?}, jj —» GA from the S-matrix. We want to stress that it is the main reason so we cannot do the variational 5 i estimations for the Hamiltonian (2.2 and 2.3). At the same time the canonical transformations do Let us divide F into two subregions not introduce new divergences in the Hamiltonian. All lenns in the representation (3.6) are finite and the operator R contains all necessary counter-terms (see alto [4]). In other words, we can u where say that Eqs.(3.10) define the ground-state of our system. Introducing the dimensionless variables if if (3.16) ft . b^l^B (311) , x3 Then we shall consider that our system (2.1) is described by the representation (2.2 and 2.3) if (Xj,X«) £ F+ and by the representation (3.14) if e(X3,X«) e F_, So that the conditions (3.16) and using the formulas (3.9) one can get for Eq.(3.10) defines the phase structure of the system under consideration. 1 In addition we should remark that Eqs.(3.12) can have several solutions. Then we have -/ (l + |) + 3X4 (b - f) + 3X36 + 12X? In(l + /) = 0 to choose the solution which leads to the lowest vacuum energy. 6+ 2A (63 - 3/6) (3.12) 4 Thus our direct problem is to solve Eqs.(3.12) and to investigate the behaviour of the function e(X3)X4) (3.13).
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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[?].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • (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.
    [Show full text]
  • 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.
    [Show full text]
  • Feynman Diagrams Particle and Nuclear Physics
    5. Feynman Diagrams Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 5. Feynman Diagrams 1 In this section... Introduction to Feynman diagrams. Anatomy of Feynman diagrams. Allowed vertices. General rules Dr. Tina Potter 5. Feynman Diagrams 2 Feynman Diagrams The results of calculations based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) depend on the reference frame. Richard Feynman 1965 Nobel Prize The sum of all time orderings is frame independent and provides the basis for our relativistic theory of Quantum Mechanics. A Feynman diagram represents the sum of all time orderings + = −−!time −−!time −−!time Dr. Tina Potter 5. Feynman Diagrams 3 Feynman Diagrams Each Feynman diagram represents a term in the perturbation theory expansion of the matrix element for an interaction. Normally, a full matrix element contains an infinite number of Feynman diagrams. Total amplitude Mfi = M1 + M2 + M3 + ::: 2 Total rateΓ fi = 2πjM1 + M2 + M3 + :::j ρ(E) Fermi's Golden Rule But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few. (Lowest order = fewest vertices possible) 2 4 g g g 6 p e2 1 Example: QED g = e = 4πα ∼ 0:30, α = 4π ∼ 137 Dr. Tina Potter 5. Feynman Diagrams 4 Feynman Diagrams Perturbation Theory Calculating Matrix Elements from Perturbation Theory from first principles is cumbersome { so we dont usually use it. Need to do time-ordered sums of (on mass shell) particles whose production and decay does not conserve energy and momentum. Feynman Diagrams Represent the maths of Perturbation Theory with Feynman Diagrams in a very simple way (to arbitrary order, if couplings are small enough).
    [Show full text]
  • 1. Physical Constants 1
    1. Physical constants 1 1. PHYSICAL CONSTANTS Table 1.1. Reviewed 2013 by P.J. Mohr (NIST). Mainly from the “CODATA Recommended Values of the Fundamental Physical Constants: 2010” by P.J. Mohr, B.N. Taylor, and D.B. Newell in Rev. Mod. Phys. 84, 1527 (2012). The last group of constants (beginning with the Fermi coupling constant) comes from the Particle Data Group. The figures in parentheses after the values give the 1-standard-deviation uncertainties in the last digits; the corresponding fractional uncertainties in parts per 109 (ppb) are given in the last column. This set of constants (aside from the last group) is recommended for international use by CODATA (the Committee on Data for Science and Technology). The full 2010 CODATA set of constants may be found at http://physics.nist.gov/constants. See also P.J. Mohr and D.B. Newell, “Resource Letter FC-1: The Physics of Fundamental Constants,” Am. J. Phys. 78, 338 (2010). Quantity Symbol, equation Value Uncertainty (ppb) speed of light in vacuum c 299 792 458 m s−1 exact∗ Planck constant h 6.626 069 57(29)×10−34 J s 44 Planck constant, reduced ~ ≡ h/2π 1.054 571 726(47)×10−34 J s 44 = 6.582 119 28(15)×10−22 MeVs 22 electron charge magnitude e 1.602 176 565(35)×10−19 C = 4.803 204 50(11)×10−10 esu 22,22 conversion constant ~c 197.326 9718(44) MeV fm 22 conversion constant (~c)2 0.389 379 338(17) GeV2 mbarn 44 2 −31 electron mass me 0.510 998 928(11) MeV/c = 9.109 382 91(40)×10 kg 22,44 2 −27 proton mass mp 938.272 046(21) MeV/c = 1.672 621 777(74)×10 kg 22,44 = 1.007 276 466 812(90) u = 1836.152 672 45(75) me 0.089, 0.41 2 deuteron mass md 1875.612 859(41) MeV/c 22 12 2 −27 unified atomic mass unit (u) (mass C atom)/12 = (1 g)/(NA mol) 931.494 061(21) MeV/c = 1.660 538 921(73)×10 kg 22,44 2 −12 −1 permittivity of free space ǫ0 =1/µ0c 8.854 187 817 .
    [Show full text]
  • Supersymmetric Grand Unification
    Supersymmetric Grand Unification Sander Mooij Master’s Thesis in Theoretical Physics Institute for Theoretical Physics University of Amsterdam Supervisor: Prof. Dr. Jan Smit August 18, 2008 Contents 1 The Standard Model 6 1.1 Introduction to the Standard Model . 6 1.1.1 Fundaments . 6 1.1.2 Contents of the Standard Model . 10 1.1.3 Writing down the SM Lagrangian . 12 1.2 Troubles and shortcomings of the Standard Model . 18 1.2.1 Reasons to go beyond the SM . 18 1.2.2 Righthanded neutrinos . 18 1.2.3 Renormalization and the hierarchy problem in the SM . 20 1.3 Running of Coupling Constants: a first clue for Grand Unification . 21 1.3.1 Renormalized Perturbation Theory . 22 1.3.2 Calculation of counterterms . 23 1.3.3 Callan-Symanzik equation and renormalization equation . 25 1.3.4 Running Coupling Constants . 26 2 Group Theoretical Backgrounds 31 2.1 Roots . 31 2.1.1 Simple roots . 32 2.1.2 Dynkin labels . 33 2.2 Weights and representations . 34 2.3 Applications . 35 2.3.1 Casimir operators . 35 2.3.2 Eigenstates . 36 2.3.3 Decomposition of tensor products . 36 2.3.4 Branching rules and projection matrices . 38 3 Grand Unification 40 3.1 Embedding the SM fields in SU(5) . 40 3.2 Writing down the SU(5) Lagrangian . 45 3.2.1 Fermion kinetic terms . 45 3.2.2 Fermion mass terms . 46 3.2.3 Gauge kinetic terms . 48 3.3 The Higgs mechanism in SU(5) . 48 3.4 Consequences of SU(5) unification .
    [Show full text]
  • Spontaneous Symmetry Breaking in Particle Physics: a Case of Cross Fertilization
    Spontaneous symmetry breaking in particle physics: a case of cross fertilization Yoichiro Nambu lecture presented by Giovanni Jona-Lasinio Nobel Lecture December 8, 2008 1 / 25 History repeats itself 1960 Midwest Conference in Theoretical Physics, Purdue University 2 / 25 Nambu’s background Y. Nambu, preliminary Notes for the Nobel Lecture I will begin by a short story about my background. I studied physics at the University of Tokyo. I was attracted to particle physics because of the three famous names, Nishina, Tomonaga and Yukawa, who were the founders of particle physics in Japan. But these people were at different institutions than mine. On the other hand, condensed matter physics was pretty good at Tokyo. I got into particle physics only when I came back to Tokyo after the war. In hindsight, though, I must say that my early exposure to condensed matter physics has been quite beneficial to me. 3 / 25 Spontaneous (dynamical) symmetry breaking Figure: Elastic rod compressed by a force of increasing strength 4 / 25 Other examples physical system broken symmetry ferromagnets rotational invariance crystals translational invariance superconductors local gauge invariance superfluid 4He global gauge invariance When spontaneous symmetry breaking takes place, the ground state of the system is degenerate 5 / 25 Superconductivity Autobiography in Broken Symmetry: Selected Papers of Y. Nambu, World Scientific One day before publication of the BCS paper, Bob Schrieffer, still a student, came to Chicago to give a seminar on the BCS theory in progress. I was very much disturbed by the fact that their wave function did not conserve electron number. It did not make sense.
    [Show full text]
  • Lectures on Effective Field Theory
    Lectures on effective field theory David B. Kaplan February 25, 2016 Abstract Lectures delivered at the ICTP-SAFIR, Sao Paulo, Brasil, February 22-26, 2016. Contents 1 Effective Quantum Mechanics3 1.1 What is an effective field theory?........................3 1.2 Scattering in 1D.................................4 1.2.1 Square well scattering in 1D.......................4 1.2.2 Relevant δ-function scattering in 1D..................5 1.3 Scattering in 3D.................................5 1.3.1 Square well scattering in 3D.......................7 1.3.2 Irrelevant δ-function scattering in 3D..................8 1.4 Scattering in 2D................................. 11 1.4.1 Square well scattering in 2D....................... 11 1.4.2 Marginal δ-function scattering in 2D & asymptotic freedom..... 12 1.5 Lessons learned..................................DRAFT 15 1.6 Problems for lecture I.............................. 16 2 EFT at tree level 17 2.1 Scaling in a relativistic EFT.......................... 17 2.1.1 Dimensional analysis: Fermi's theory of the weak interactions.... 19 2.1.2 Dimensional analysis: the blue sky................... 21 2.2 Accidental symmetry and BSM physics.................... 23 2.2.1 BSM physics: neutrino masses..................... 25 2.2.2 BSM physics: proton decay....................... 26 2.3 BSM physics: \partial compositeness"..................... 27 2.4 Problems for lecture II.............................. 29 1 3 EFT and radiative corrections 30 3.1 Matching..................................... 30 3.2 Relevant operators and naturalness....................... 34 3.3 Aside { a parable from TASI 1997....................... 35 3.4 Landau liquid versus BCS instability...................... 36 3.5 Problems for lecture III............................. 40 4 Chiral perturbation theory 41 4.1 Chiral symmetry in QCD............................ 41 4.2 Quantum numbers of the meson octet....................
    [Show full text]