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t/08/00061 Department of PHYSICS UNIVERSITY OF BERGEN Bergen, Norway Scientific/Technical Deport Ro. 110 Gauge Theories C. Jarlskog Department of Physics, University of Bergen Bergen, Norway Lectures given at "Advanced Summer Institute 197$ on Hev Phenomena in Lepton Hadron Physics", University of Karlsruhe - Germany, September k - 16, 1978. GAUGE THEORIES Cecilia Jarlskog Department of Physics, University of Bergen, Bergen, Korvay TABLE OF COMTETre 1. Introductory remarks 2. Ingredients of gauge theories 3. Symmetries and conservation lavs U. Local gauge invariance and quantum electrodynamics (QED) 5. Wenk interactions, quantum flavour dynamics (QFD) 6. Construction of the standard SU(2) x U(l) model 7. Unification of veak and electromagnetic interactions in the standard model ' 8. The weak neutral current couplings in the standard model 9. Spontaneous symmetry breaking (SSB) 10. Spontaneous breaking of a local symmetry 11. Spontaneous breaking scheme for the standard SU(2) x U(l) model 12. More quarks and leptons, masses and the Cabibbo angles. 13. The four quark and six quark models; CF - violation lit. Grand unification 15. Mixing angles in the Georgi-Glashov model 16. The Higgses of the SU(5) model 1?. The proton/neutron decay in SD(5) The metric and notations, in these lectures, are as in J.D. Bjorken and S.D. Drell, Eelativistic Quantum Fields, McGrav - Hill, 1965. s i. uRimwcnmr UMMB ID these lecture*, an introduction to the unified gauge the- ories, of weak and «l»ctromagn*tic interactions, ia given. Strong interactions are treated only briefly; they constitute the.thea* of Professor aachtaann'* lectures. The idea of unifying seemingly different force* of nature is not new. More than a century ago. Jams Clerk Maxwell (1831-1879) succeeded in unifying electricity and aegnetisa. Within the last decade most of the effort has gone into unifying weak force* with electroaagnetisa. Such a unification was suggested, by Schwinger, to be realized through a triplet of vector fields, whose universal coupling would generate both the weak and electromagnetic forces (J. Schwinger, Ann. ot Thy*. 2, 1*07 (1957)). Sala» and Ward (Ruovo Ciaento 11, 568 (1959)) suggested that the vector bosons should be introduced a la Yang and Mills. It is interesting to note that at that tine the muon-neutrino had not yet revealed itself; the known leptons e, u, \>e, were supposed to fora a triplet. The Schvinger model was extended by Glashow (Duel. Fhya. 22, 579 (196l)) who, in addition to the triplet of vector bosons, in­ troduced a new neutral vector boson. In the Clashow model, the two neutral vector bosons, denoted by Z3 and Z*, nix. The linear com­ binations Z3cos8 + Z'sin» and -Z'sinS + Z'cos6 were assumed to cor­ respond to the physical particles (photon and Z°, in present termi­ nology). Nowadays, 9 is usually referred to as the Weinberg angle. The model predicted neutral currents. Glasfaov prophesized: "For no choice of 9 is the interaction of neutral current small compared with weak interactions involving charged currents". t The model was constructed for leptons; we know that it is not easy to detect the leptonic neutral current, even when it is not small. In the Glashov model of 1961 the vector meson masses vere put in by hand and therefore the model is not renormalizable. The next major step was taken by Weinberg (Phys. Rev. Lett. 1£, 126k (1967)) who employed a spontaneous symmetry breaking mechanism to generate masses. He introduced four Higgs particles, whereof three disap­ pear by giving masses to the intermediate bosons and one (denoted by IIP) remains. The model relates the masses of the neutral and charged intermediate bosons, Itø = cos6 Mz. Weinberg guessed chat the model may be renormalizable. Salam and Ward had also been working on introducing masses via spontaneous symmetry breaking (see the talk by A. Salam in the pro­ ceedings of the 1968 Noble Symposium, Ed. N. Svartholm). The extension to hadrons, for four quarks, was given by Wein- bere (Phys. Rev. DJ, 1*12 (19T2)) «be utilised the so-called OOr- • mechanism (Olaahow, Iliopoulos and Maiaai, nys. «er. K, l8j In 1960's the unified model* and Yang-Mills type theories vere not particularly popular (however, t remember Veltmen giving • series of lectures in a Copenhagen summer school in the late 60's). Die unified theories beesas fashionable only after t'Hooft (Amsterdam International Conference 1971) demonstrated that the spontaneously broken gauge theories are renoraalisable. Two years after, in 1973, neutral currents vere discovered at CBU. Sine* then the unified theories have become mor* and more the standard framework for describing nature. nowadays, it is believed that, the weak, and electromagnetic interaction are described by a gauge theory based on the (non- abelian Lie) group 80(2) x Vil) and strong interactions follow a BU(3) in colour. One goes further and advocates a unification of all nongravitationol interactions, for example as in the SU(5) model of Ceorgi and Glashow (Fhys. Rev. Lett. JS, 1*38 (I97*)h Of course, ve have not yet seen any intermediate vector bosons {excepting the photon) nor any Higgs particles, which are the back bone of the modern unified theories; we have not had enough energy to produce any W or Z°, etc. The fact that the experiments are in an amazingly good agreement with the simplest gauge models makes us believe that ve are on the right track; with enough energy we shall be able to admire the glorious intermediate bosons. i> s. lmmaaxm or <uu» ramms i Gauge theories aisuaa that the Lagrangiaii formaliaa provide» a correct language in describing natura vhich is supposed to con­ sist of three classes of particlet: a) The fundaaental constituent! of aatter (leptons and quarks). b) Intermediate particles. c) Higgs particles. a) The fundamental constituents of aatter are feraions vitb spin }. There are tvo types of constituents: leptons and quarks (hadrons). The lepton family contains the observed ones w, e~, v , u~, v and T~ (and of course their antiparticles). The evidence for v , although indirect, is substantial. Leptons are integrally charged. Toe quark (hadron) family contains the "observed" quarks up (u), dovn (d), strange (s), charm (c) and beauty or bottom (b). The need for an additional quark, called truth or top (t) is ur­ gent. '*'-•» hope that Petra vill discover the truth. In the conventional approach, each quark exists in three var­ ieties (colours). For example, there are three up quarks, etc. b) In gauge theories, the existence of the constituents of matter, together with the Holy Principle of the local gauge invariance (discussed below), leads to the existence of intermediate vector (spin 1) bosons. Their job is to mediate interactions. The family of mediators contains Maxwell's photon (the only member directly seen so far) vhich mediates electromagnetism. We believe that there are further members, such as W*, Z which mediate weak inter­ actions at lov energies; gluons C., i»l, ..., 8 vho mediate strong interactions, etc. Mediators are integrally or fractionally charged depending on the theory. c) The Higgs particles are theoretically needed betes noires. Some day, perhaps we will learn to live without them, but so far their existence is indispensable. They are believed to be the origin of all masses except their own. We shall discuss their properties in section 10. 3. SYMMETRIES AND CONSERVATION LAWS The Holy Principle in gauge theories is the Principle of Local Gauge Invariance, as we shall discuss shortly. Before ve do so, 1st us remind ourselves, that th» concept of invarianc» (or covari- ance), which is vsry essential in physics, is osatly foraulated in Lagrangian languag». For «xaapl», th» principl» that th» lavs of physics ars independent of th» particular Lorsnts fraas chosen re­ quires th» Lsgrangian density to b» s scalar quantity; the equa­ tions of action aust b» covariant, »tc. [Exercise: State the physical principle» leading to the lavs of «nergy-aoaentua conservation, angular aoaentua conserva­ tion and parity conservation (if it had been true). Hov are these conservation lavs guaranteed in the Lagrangian formal­ ism?) Hot all conservation lavs have to do vith space and time. There are a rather large number of physically estsblished conserva­ tion lavs, vhich pertain to "internal" properties. Let us give a fev examples: a) Conservation of charge. b) Conservation of baryon number B, and lepton numbers Le. !• and LT. c) It is commonly believed that colour is an exact symmetry, and only the colour singlet states are observable. There are hovever theories in vhich the colour leaks out. Experimentally, there is, up to nov, no evidence against the conservation lavs a - c. Approximate conservation lavs have also been observed in nature, for example d) conservation of strange-, charm-, beauty-(ness) by strong and electromagnetic interactions. These are violated by weak interactions. e) Isospin symmetry in strong interactions. f) SU(3)-symmetry, again in strong interactions. We leave it to our readers to ponder on the physical prin­ ciples responsible for the symmetries above. We may ask whether the conservation lavs a) and bl are exact? Are all physical states, vhich go through our bubble chambers, calorimeters, etc, colour singlets? We shall return to some of these questions later on in these lectures. Let us conclude this section by simply noting that the experimentally established conservation lavs are easily incorpor­ ated as symmetries into our Lagrangians, even if ve do not under­ stand their origin. «. LOCAL GMJGB IIYARMM» ARD OjUAIHW UKIROIKIIAMICS (QH>) Vt «hall nov construct the siaplest «xaaple of a (auge theory ••ploying the principle of local gauge invarlanc*. Suppose that ve bad only a singl* Baseless femion f. then the fres Lagr angi an, describing f is given by t\ - i •f(x)Y|i £- »f(x), (1,-1 > Dote that i -r— is the four aoaentum operator in quantim Mechanics.
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  • Path Integral Approach to Residual Gauge Fixing

    Path Integral Approach to Residual Gauge Fixing

    Path Integral Approach to Residual Gauge Fixing Ashok Dasa, J. Frenkelb and Silvana Perezc a Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627-0171, USA b Instituto de F´ısica, Universidade de S˜ao Paulo, S˜ao Paulo, BRAZIL and c Departamento de F´ısica, Universidade Federal do Par´a, Bel´em, Par´a66075-110, BRAZIL In this paper we study the question of residual gauge fixing in the path integral approach for a general class of axial-type gauges including the light-cone gauge. We show that the two cases – axial-type gauges and the light-cone gauge – lead to very different structures for the explicit forms of the propagator. In the case of the axial-type gauges, fixing the residual symmetry determines the propagator of the theory completely. On the other hand, in the light-cone gauge there is still a prescription dependence even after fixing the residual gauge symmetry, which is related to the existence of an underlying global symmetry. PACS numbers: 11.15.-q, 11.10.Ef, 12.38.Lg I. INTRODUCTION Light-front field theories [1] have been studied vigorously in the past. Quantization on the light-front (as opposed to equal-time quantization) leads to a larger number of kinematical generators of the Poincar´ealgebra resulting in a trivial vacuum state of the theory [2]. Therefore, non-perturbative calculations can, in principle, be carried out in a simpler manner in such theories. More recently, it has been observed both in the conventional light-front frame as well as in a generalized light-front frame [3, 4, 5] that canonical quantization (within the Hamiltonian formalism) of a gauge theory in the light-cone gauge n A =0, n2 =0, (1) · where Aµ can be thought of as a matrix in the adjoint representation of the gauge group in the case of a non-Abelian theory, leads to a doubly transverse propagator [6, 7].