Physical Degrees of Freedom in Higgs Models
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Melting of the Higgs Vacuum: Conserved Numbers at High Temperature
PURD-TH-96-05 CERN-TH/96-167 hep-ph/9607386 July 1996 Melting of the Higgs Vacuum: Conserved Numbers at High Temperature S. Yu. Khlebnikov a,∗ and M. E. Shaposhnikov b,† a Department of Physics, Purdue University, West Lafayette, IN 47907, USA b Theory Division, CERN, CH-1211 Geneva 23, Switzerland Abstract We discuss the computation of the grand canonical partition sum describing hot mat- ter in systems with the Higgs mechanism in the presence of non-zero conserved global charges. We formulate a set of simple rules for that computation in the high-temperature approximation in the limit of small chemical potentials. As an illustration of the use of these rules, we calculate the leading term in the free energy of the standard model as a function of baryon number B. We show that this quantity depends continuously on the Higgs expectation value φ, with a crossover at φ ∼ T where Debye screening overtakes the Higgs mechanism—the Higgs vacuum “melts”. A number of confusions that exist in the literature regarding the B dependence of the free energy is clarified. PURD-TH-96-05 CERN-TH/96-167 July 1996 ∗[email protected] †[email protected] Screening of gauge fields by the Higgs mechanism is one of the central ideas both in modern particle physics and in condensed matter physics. It is often referred to as “spontaneous breaking” of a gauge symmetry, although, in the accurate sense of the word, gauge symmetries cannot be broken in the same way as global ones can. One might rather say that gauge charges are “hidden” by the Higgs mechanism. -
Perturbative Renormalization and BRST
Perturbative renormalization and BRST Michael D¨utsch Institut f¨ur Theoretische Physik Universit¨at Z¨urich CH-8057 Z¨urich, Switzerland [email protected] Klaus Fredenhagen II. Institut f¨ur Theoretische Physik Universit¨at Hamburg D-22761 Hamburg, Germany [email protected] 1 Main problems in the perturbative quantization of gauge theories Gauge theories are field theories in which the basic fields are not directly observable. Field configurations yielding the same observables are connected by a gauge transformation. In the classical theory the Cauchy problem is well posed for the observables, but in general not for the nonobservable gauge variant basic fields, due to the existence of time dependent gauge transformations. Attempts to quantize the gauge invariant objects directly have not yet been completely satisfactory. Instead one modifies the classical action by adding a gauge fixing term such that standard techniques of perturbative arXiv:hep-th/0411196v1 22 Nov 2004 quantization can be applied and such that the dynamics of the gauge in- variant classical fields is not changed. In perturbation theory this problem shows up already in the quantization of the free gauge fields (Sect. 3). In the final (interacting) theory the physical quantities should be independent on how the gauge fixing is done (’gauge independence’). Traditionally, the quantization of gauge theories is mostly analyzed in terms of path integrals (e.g. by Faddeev and Popov) where some parts of the arguments are only heuristic. In the original treatment of Becchi, Rouet and Stora (cf. also Tyutin) (which is called ’BRST-quantization’), a restriction to purely massive theories was necessary; the generalization to the massless case by Lowenstein’s method is cumbersome. -
Frstfo O Ztif
FRStfo o ztif * CONNISSARIAT A L'ENERGIE ATOMIQUE CENTRE D'ETUDES NUCLEAIRES DE SACLAY CEA-CONF -- 8070 Service de Documentation F9119! GIF SUR YVETTE CEDEX L2 \ PARTICLE PHYSICS AND GAUGE THEORIES MOREL, A. CEA CEN Socloy, IRF, SPh-T Communication présentée à : Court* on poxticlo phytic* Cargos* (Franc*) 15-28 Jul 1985 PARTICLE PHYSICS AND GAUGE THEORIES A. MOREL Service de Physique Théorique CEN SACLA Y 91191 Gif-sur-Yvette Cedex, France These notes are intended to help readers not familiar with parti cle physics in entering the domain of gauge field theory applied to the so-called standard model of strong and electroweak interactions. They are mainly based on previous notes written in common with A. Billoire. With re3pect to the latter ones, the introduction is considerably enlar ged in order to give non specialists a general overview of present days "elementary" particle physics. The Glashow-Salam-Weinberg model is then treated,with the details which its unquestioned successes deserve, most probably for a long time. Finally SU(5) is presented as a prototype of these developments of particle physics which aim at a unification of all forces. Although its intrinsic theoretical difficulties and the. non- observation of a sizable proton decay rate do not qualify this model as a realistic one, it has many of the properties expected from a "good" unified theory. In particular, it allows one to study interesting con nections between particle physics and cosmology. It is a pleasure to thank the organizing committee of the Cargèse school "Particules et Cosmologie", and especially J. Audouze, for the invitation to lecture on these subjects, and M.F. -
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. -
PDF) of Partons I and J
TASI 2004 Lecture Notes on Higgs Boson Physics∗ Laura Reina1 1Physics Department, Florida State University, 315 Keen Building, Tallahassee, FL 32306-4350, USA E-mail: [email protected] Abstract In these lectures I briefly review the Higgs mechanism of spontaneous symmetry breaking and focus on the most relevant aspects of the phenomenology of the Standard Model and of the Minimal Supersymmetric Standard Model Higgs bosons at both hadron (Tevatron, Large Hadron Collider) and lepton (International Linear Collider) colliders. Some emphasis is put on the perturbative calculation of both Higgs boson branching ratios and production cross sections, including the most important radiative corrections. arXiv:hep-ph/0512377v1 30 Dec 2005 ∗ These lectures are dedicated to Filippo, who listened to them before he was born and behaved really well while I was writing these proceedings. 1 Contents I. Introduction 3 II. Theoretical framework: the Higgs mechanism and its consequences. 4 A. A brief introduction to the Higgs mechanism 5 B. The Higgs sector of the Standard Model 10 C. Theoretical constraints on the Standard Model Higgs bosonmass 13 1. Unitarity 14 2. Triviality and vacuum stability 16 3. Indirect bounds from electroweak precision measurements 17 4. Fine-tuning 22 D. The Higgs sector of the Minimal Supersymmetric Standard Model 24 1. About Two Higgs Doublet Models 25 2. The MSSM Higgs sector: introduction 27 3. MSSM Higgs boson couplings to electroweak gauge bosons 30 4. MSSM Higgs boson couplings to fermions 33 III. Phenomenology of the Higgs Boson 34 A. Standard Model Higgs boson decay branching ratios 34 1. General properties of radiative corrections to Higgs decays 37 + 2. -
1 Hamiltonian Quantization and BRST -Survival Guide; Notes by Horatiu Nastase
1 Hamiltonian quantization and BRST -survival guide; notes by Horatiu Nastase 1.1 Dirac- first class and second class constraints, quan- tization Classical Hamiltonian Primary constraints: φm(p; q) = 0 (1) imposed from the start. The equations of motion on a quantity g(q; p) are g_ = [g; H]P:B: (2) Define φm ∼ 0 (weak equality), meaning use the constraint only at the end of the calculation, then for consistency φ_m ∼ 0, implying [φm; H]P:B: ∼ 0 (3) The l.h.s. will be however in general a linearly independent function (of φm). If it is, we can take its time derivative and repeat the process. In the end, we find a complete set of new constrains from the time evolution, called secondary constraints. Together, they form the set of constraints, fφjg; j = 1; J. A quantity R(q; p) is called first class if [R; φj]P:B: ∼ 0 (4) for all j=1, J. If not, it is called second class. Correspondingly, constraints are also first class and second class, independent of being primary or sec- ondary. To the Hamiltonian we can always add a term linear in the constraints, generating the total Hamiltonian HT = H + umφm (5) where um are functions of q and p. The secondary constraint equations are [φm; HT ] ∼ [φm; H] + un[φm; φn] ∼ 0 (6) where in the first line we used that [un; φm]φn ∼ 0. The general solution for um is um = Um + vaVam (7) with Um a particular solution and Vam a solution to Vm[φj; φm] = and va arbitrary functions of time only. -
Unitarity Constraints on Higgs Portals
SLAC-PUB-15822 Unitarity Constraints on Higgs Portals Devin G. E. Walker SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, U.S.A. Dark matter that was once in thermal equilibrium with the Standard Model is generally prohibited from obtaining all of its mass from the electroweak phase transition. This implies a new scale of physics and mediator particles to facilitate dark matter annihilation. In this work, we focus on dark matter that annihilates through a generic Higgs portal. We show how partial wave unitarity places an upper bound on the mass of the mediator (or dark) Higgs when its mass is increased to be the largest scale in the effective theory. For models where the dark matter annihilates via fermion exchange, an upper bound is generated when unitarity breaks down around 8.5 TeV. Models where the dark matter annihilates via fermion and higgs boson exchange push the bound to 45.5 TeV. We also show that if dark matter obtains all of its mass from a new symmetry breaking scale that scale is also constrained. We improve these constraints by requiring perturbativity in the Higgs sector up to each unitarity bound. In this limit, the bounds on the dark symmetry breaking vev and the dark Higgs mass are now 2.4 and 3 TeV, respectively, when the dark matter annihilates via fermion exchange. When dark matter annihilates via fermion and higgs boson exchange, the bounds are now 12 and 14.2 TeV, respectively. Given the unitarity bounds, the available parameter space for Higgs portal dark matter annihilation is outlined. -
Fundamental Physical Constants: Looking from Different Angles
1 Fundamental Physical Constants: Looking from Different Angles Savely G. Karshenboim Abstract: We consider fundamental physical constants which are among a few of the most important pieces of information we have learned about Nature after its intensive centuries- long studies. We discuss their multifunctional role in modern physics including problems related to the art of measurement, natural and practical units, origin of the constants, their possible calculability and variability etc. PACS Nos.: 06.02.Jr, 06.02.Fn Resum´ e´ : Nous ... French version of abstract (supplied by CJP) arXiv:physics/0506173v2 [physics.atom-ph] 28 Jul 2005 [Traduit par la r´edaction] Received 2004. Accepted 2005. Savely G. Karshenboim. D. I. Mendeleev Institute for Metrology, St. Petersburg, 189620, Russia and Max- Planck-Institut f¨ur Quantenoptik, Garching, 85748, Germany; e-mail: [email protected] unknown 99: 1–46 (2005) 2005 NRC Canada 2 unknown Vol. 99, 2005 Contents 1 Introduction 3 2 Physical Constants, Units and Art of Measurement 4 3 Physical Constants and Precision Measurements 7 4 The International System of Units SI: Vacuum constant ǫ0, candela, kelvin, mole and other questions 10 4.1 ‘Unnecessary’units. .... .... ... .... .... .... ... ... ...... 10 4.2 ‘Human-related’units. ....... 11 4.3 Vacuum constant ǫ0 andGaussianunits ......................... 13 4.4 ‘Unnecessary’units,II . ....... 14 5 Physical Phenomena Governed by Fundamental Constants 15 5.1 Freeparticles ................................... .... 16 5.2 Simpleatomsandmolecules . ..... -
Modern Methods of Quantum Chromodynamics
Modern Methods of Quantum Chromodynamics Christian Schwinn Albert-Ludwigs-Universit¨atFreiburg, Physikalisches Institut D-79104 Freiburg, Germany Winter-Semester 2014/15 Draft: March 30, 2015 http://www.tep.physik.uni-freiburg.de/lectures/QCD-WS-14 2 Contents 1 Introduction 9 Hadrons and quarks . .9 QFT and QED . .9 QCD: theory of quarks and gluons . .9 QCD and LHC physics . 10 Multi-parton scattering amplitudes . 10 NLO calculations . 11 Remarks on the lecture . 11 I Parton Model and QCD 13 2 Quarks and colour 15 2.1 Hadrons and quarks . 15 Hadrons and the strong interactions . 15 Quark Model . 15 2.2 Parton Model . 16 Deep inelastic scattering . 16 Parton distribution functions . 18 2.3 Colour degree of freedom . 19 Postulate of colour quantum number . 19 Colour-SU(3).............................. 20 Confinement . 20 Evidence of colour: e+e− ! hadrons . 21 2.4 Towards QCD . 22 3 Basics of QFT and QED 25 3.1 Quantum numbers of relativistic particles . 25 3.1.1 Poincar´egroup . 26 3.1.2 Relativistic one-particle states . 27 3.2 Quantum fields . 32 3.2.1 Scalar fields . 32 3.2.2 Spinor fields . 32 3 4 CONTENTS Dirac spinors . 33 Massless spin one-half particles . 34 Spinor products . 35 Quantization . 35 3.2.3 Massless vector bosons . 35 Polarization vectors and gauge invariance . 36 3.3 QED . 37 3.4 Feynman rules . 39 3.4.1 S-matrix and Cross section . 39 S-matrix . 39 Poincar´einvariance of the S-matrix . 40 T -matrix and scattering amplitude . 41 Unitarity of the S-matrix . 41 Cross section . -
8. Quantum Field Theory on the Plane
8. Quantum Field Theory on the Plane In this section, we step up a dimension. We will discuss quantum field theories in d =2+1dimensions.Liketheird =1+1dimensionalcounterparts,thesetheories have application in various condensed matter systems. However, they also give us further insight into the kinds of phases that can arise in quantum field theory. 8.1 Electromagnetism in Three Dimensions We start with Maxwell theory in d =2=1.ThegaugefieldisAµ,withµ =0, 1, 2. The corresponding field strength describes a single magnetic field B = F12,andtwo electric fields Ei = F0i.Weworkwiththeusualaction, 1 S = d3x F F µ⌫ + A jµ (8.1) Maxwell − 4e2 µ⌫ µ Z The gauge coupling has dimension [e2]=1.Thisisimportant.ItmeansthatU(1) gauge theories in d =2+1dimensionscoupledtomatterarestronglycoupledinthe infra-red. In this regard, these theories di↵er from electromagnetism in d =3+1. We can start by thinking classically. The Maxwell equations are 1 @ F µ⌫ = j⌫ e2 µ Suppose that we put a test charge Q at the origin. The Maxwell equations reduce to 2A = Qδ2(x) r 0 which has the solution Q r A = log +constant 0 2⇡ r ✓ 0 ◆ for some arbitrary r0. We learn that the potential energy V (r)betweentwocharges, Q and Q,separatedbyadistancer, increases logarithmically − Q2 r V (r)= log +constant (8.2) 2⇡ r ✓ 0 ◆ This is a form of confinement, but it’s an extremely mild form of confinement as the log function grows very slowly. For obvious reasons, it’s usually referred to as log confinement. –384– In the absence of matter, we can look for propagating degrees of freedom of the gauge field itself. -
Threshold Corrections in Grand Unified Theories
Threshold Corrections in Grand Unified Theories Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der Fakult¨at f¨ur Physik des Karlsruher Instituts f¨ur Technologie genehmigte DISSERTATION von Dipl.-Phys. Waldemar Martens aus Nowosibirsk Tag der m¨undlichen Pr¨ufung: 8. Juli 2011 Referent: Prof. Dr. M. Steinhauser Korreferent: Prof. Dr. U. Nierste Contents 1. Introduction 1 2. Supersymmetric Grand Unified Theories 5 2.1. The Standard Model and its Limitations . ....... 5 2.2. The Georgi-Glashow SU(5) Model . .... 6 2.3. Supersymmetry (SUSY) .............................. 9 2.4. SupersymmetricGrandUnification . ...... 10 2.4.1. MinimalSupersymmetricSU(5). 11 2.4.2. MissingDoubletModel . 12 2.5. RunningandDecoupling. 12 2.5.1. Renormalization Group Equations . 13 2.5.2. Decoupling of Heavy Particles . 14 2.5.3. One-Loop Decoupling Coefficients for Various GUT Models . 17 2.5.4. One-Loop Decoupling Coefficients for the Matching of the MSSM to the SM ...................................... 18 2.6. ProtonDecay .................................... 21 2.7. Schur’sLemma ................................... 22 3. Supersymmetric GUTs and Gauge Coupling Unification at Three Loops 25 3.1. RunningandDecoupling. 25 3.2. Predictions for MHc from Gauge Coupling Unification . 32 3.2.1. Dependence on the Decoupling Scales µSUSY and µGUT ........ 33 3.2.2. Dependence on the SUSY Spectrum ................... 36 3.2.3. Dependence on the Uncertainty on the Input Parameters ....... 36 3.2.4. Top-Down Approach and the Missing Doublet Model . ...... 39 3.2.5. Comparison with Proton Decay Constraints . ...... 43 3.3. Summary ...................................... 44 4. Field-Theoretical Framework for the Two-Loop Matching Calculation 45 4.1. TheLagrangian.................................. 45 4.1.1. -
Atomic Parity Violation, Polarized Electron Scattering and Neutrino-Nucleus Coherent Scattering
Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 959 (2020) 115158 www.elsevier.com/locate/nuclphysb New physics probes: Atomic parity violation, polarized electron scattering and neutrino-nucleus coherent scattering Giorgio Arcadi a,b, Manfred Lindner c, Jessica Martins d, Farinaldo S. Queiroz e,∗ a Dipartimento di Matematica e Fisica, Università di Roma 3, Via della Vasca Navale 84, 00146, Roma, Italy b INFN Sezione Roma 3, Italy c Max-Planck-Institut für Kernphysik (MPIK), Saupfercheckweg 1, 69117 Heidelberg, Germany d Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil e International Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, Natal-RN 59078-970, Brazil Received 7 April 2020; received in revised form 22 July 2020; accepted 20 August 2020 Available online 27 August 2020 Editor: Hong-Jian He Abstract Atomic Parity Violation (APV) is usually quantified in terms of the weak nuclear charge QW of a nucleus, which depends on the coupling strength between the atomic electrons and quarks. In this work, we review the importance of APV to probing new physics using effective field theory. Furthermore, we correlate our findings with the results from neutrino-nucleus coherent scattering. We revisit signs of parity violation in polarized electron scattering and show how precise measurements on the Weinberg’s angle give rise to competitive bounds on light mediators over a wide range of masses and interactions strengths. Our bounds are firstly derived in the context of simplified setups and then applied to several concrete models, namely Dark Z, Two Higgs Doublet Model-U(1)X and 3-3-1, considering both light and heavy mediator regimes.