Quantum Chromodynamics and Quark Glucon Plasma Sea-A Abstraction and Attrition Model
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Arxiv:2007.10198V2 [Math.NA] 5 Nov 2020
ON THE VALIDITY OF COMPLEX LANGEVIN METHOD FOR PATH INTEGRAL COMPUTATIONS∗ ZHENNING CAIy , XIAOYU DONGz , AND YANG KUANGx Abstract. The complex Langevin (CL) method is a classical numerical strategy to alleviate the numerical sign problem in the computation of lattice field theories. Mathematically, it is a simple numerical tool to compute a wide class of high- dimensional and oscillatory integrals. However, it is often observed that the CL method converges but the limiting result is incorrect. The literature has several unclear or even conflicting statements, making the method look mysterious. By an in-depth analysis of a model problem, we reveal the mechanism of how the CL result turns biased as the parameter changes, and it is demonstrated that such a transition is difficult to capture. Our analysis also shows that the method works for any observables only if the probability density function generated by the CL process is localized. To generalize such observations to lattice field theories, we formulate the CL method on general groups using rigorous mathematical languages for the first time, and we demonstrate that such localized probability density function does not exist in the simulation of lattice field theories for general compact groups, which explains the unstable behavior of the CL method. Fortunately, we also find that the gauge cooling technique creates additional velocity that helps confine the samples, so that we can still see localized probability density functions in certain cases, as significantly broadens the application of the CL method. The limitations of gauge cooling are also discussed. In particular, we prove that gauge cooling has no effect for Abelian groups, and we provide an example showing that biased results still exist when gauge cooling is insufficient to confine the probability density function. -
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. -
Scaling Laws in Particle Physics and Astrophysics
SCALING LAWS IN PARTICLE PHYSICS AND ASTROPHYSICS RUDOLF MURADYAN Dedicated to the Golden Jubilee (1961-2011) of publication of the article by Geoffrey Chew and Steven Frautschi in Phys. Rev. Lett. 7, 394, 1961, where a celebrated scaling law J m2 has been conjectured for spin/mass dependence of hadrons. G. Chew S. Frautschi 1. INTRODUCTION: WHAT IS SCALING? Any polynomial power law f() x c xn , where constant c has a dimension dim f dimc (dimx )n exhibits the property of scaling or scale invariance. Usually n is called scaling exponent. The word scaling express the fact that function f is shape-invariant with respect to the dilatation transformation x x f ( x) c ( x)n n f() x and this transformation preserves the shape of function f . We say, following Leonhard Euler, that f is homogeneous of degree “n” if for any value of parameter f ( x) n f() x . Differentiating this relation with respect to and putting 1we obtain simple differential equation x f() x n f() x solution of which brings back to the polynomial power scaling law. There are tremendously many different scaling laws in Nature. The most important of them can be revealed by Google search of scaling site:nobelprize.org in the official site of Nobel Foundation, where nearly 100 results appears. Ten of them are shown below: 1. Jerome I. Friedman - Nobel Lecture 2. Daniel C. Tsui - Nobel Lecture 3. Gerardus 't Hooft - Nobel Lecture 4. Henry W. Kendall - Nobel Lecture 5. Pierre-Gilles de Gennes - Nobel Lecture 6. Jack Steinberger - Nobel Lecture 7. -
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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. -
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. -
Nobel Lecture by Toshihide Maskawa
WHAT DOES CP VIOLATION TELL US? Nobel Lecture, December 8, 2008 by Toshihide Maskawa Kyoto Sangyo University, Kyoto 603-8555, Japan. I would first like to thank the Royal Swedish Academy of Sciences and the Nobel Foundation for awarding me this honour, of which I had never even dreamt. I was born in 1940, the son of a furniture craftsman in the city of Nagoya, in Japan. My father wanted to change his job and was taking a correspondence course to become an electrical engineer, while he was a trainee furniture craftsman. However, he told me that he could not really understand sine and cosine, since he had not received a basic education. Eventually, though, he did manage a small furniture factory employing a few craftsmen and worked there himself. But this came to nothing because of the War, that reckless and miserable war which our country caused. After the War, he displayed in front of his house the door hinges, wood screws and other pieces of furniture, which remained at hand. They sold quite well. Getting a taste for selling, he became a merchant dealing with sugar as an ingredient for cakes. He still wanted to boast of his knowledge of electricity, but he could not find anyone suitable to explain it to. One day, though, he found a good tar- get: his son. In those poor days after the war, almost all the houses were without bath- rooms and so people went to the public bath. On his way to and from the public bath, he boasted of his knowledge: Why do three-phase current mo- tors rotate? Why don’t the solar and lunar eclipses occur every month? He explained proudly that it was because of the revolution planes of the earth around the sun, and of the moon around the earth, which are tilted at an angle of 5 degrees. -
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 . -
Wilson Loops in 3-Dimensional N=6 Supersymmetric Chern-Simons Theory and Their String Theory Duals Drukker, N; Plefka, J; Young, D
CORE Metadata, citation and similar papers at core.ac.uk Provided by Queen Mary Research Online Wilson loops in 3-dimensional N=6 supersymmetric Chern-Simons Theory and their string theory duals Drukker, N; Plefka, J; Young, D For additional information about this publication click this link. http://qmro.qmul.ac.uk/jspui/handle/123456789/8211 Information about this research object was correct at the time of download; we occasionally make corrections to records, please therefore check the published record when citing. For more information contact [email protected] HU-EP-08/20 Wilson loops in 3-dimensional =6 supersymmetric Chern-Simons TheoryN and their string theory duals Nadav Drukker, Jan Plefka and Donovan Young Humboldt-Universit¨at zu Berlin, Institut f¨ur Physik, Newtonstraße 15, D-12489 Berlin, Germany drukker,plefka,[email protected] Abstract We study Wilson loops in the three-dimensional = 6 supersymmetric Chern-Simons N theory recently constructed by Aharony, Bergman, Jafferis and Maldacena, that is conjec- tured to be dual to type IIA string theory on AdS CP3. We construct loop operators in 4 × the Chern-Simons theory which preserve 1/6 of the supercharges and calculate their expec- tation value up to 2-loop order at weak coupling. The expectation value at strong coupling arXiv:0809.2787v4 [hep-th] 30 Oct 2008 is found by constructing the string theory duals of these operators. For low dimensional representations these are fundamental strings, for high dimensional representations these are D2-branes and D6-branes. In support of this identification we demonstrate that these string theory solutions match the symmetries, charges and the preserved supersymmetries of their Chern-Simons theory counterparts. -
Holographic Wilson Loops
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1675 Holographic Wilson Loops Quantum String Corrections DANIEL RICARDO MEDINA RINCON ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-513-0350-5 UPPSALA urn:nbn:se:uu:diva-349210 2018 Dissertation presented at Uppsala University to be publicly examined in Room FB53, AlbaNova Universitetscentrum, Roslagstullsbacken 21, Stockholm, Wednesday, 13 June 2018 at 10:30 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Gianluca Grignani (Università di Perugia). Abstract Medina Rincon, D. R. 2018. Holographic Wilson Loops. Quantum String Corrections. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1675. 67 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0350-5. The gauge-string duality has been one of the most exciting areas in theoretical physics as it connects strongly coupled field theories with weakly interacting strings. The present thesis concerns the study of Wilson loops in this correspondence. Wilson loops are observables arising in many physical situations like the propagation of particles in gauge fields, the problem of confinement, etc. In the gauge-string correspondence these observables have a known physical description at both sides, making them ideal probes for the duality. Remarkable progress from localization has lead to predictions at all orders in the coupling for certain Wilson loop configurations in supersymmetric field theories. Being the string theory weakly interacting, in principle we can use perturbation theory to calculate the corresponding quantities. However, current string calculations have only been successful at leading order and in rare cases, next to leading order. -
Hep-Th/9309067V1 10 Sep 1993 Etogto Saloe U Ue Hsi Setal Moder a Essentially Us Formalized Is Be This Can It If Tube
Strings, Loops, Knots and Gauge Fields John C. Baez Department of Mathematics University of California Riverside CA 92521 September 10, 1993 to appear in Knots and Quantum Gravity, ed. J. Baez, Oxford U. Press Abstract The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an approximation to QCD, in which the strings represent flux tubes of the gauge field. A heuristic path-integral approach indi- cates a duality between background-free string theories and generally covariant gauge theories, with the loop transform relating the two. We review progress towards making this duality rigorous in three examples: 2d Yang-Mills theory (which, while not generally covariant, has symmetry under all area-preserving transformations), 3d quantum gravity, and 4d quantum gravity. SU(N) Yang- Mills theory in 2 dimensions has been given a string-theoretic interpretation in the large-N limit by Gross, Taylor, Minahan and Polychronakos, but here we provide an exact string-theoretic interpretation of the theory on R × S1 for fi- nite N. The string-theoretic interpretation of quantum gravity in 3 dimensions gives rise to conjectures about integrals on the moduli space of flat connections, while in 4 dimensions there may be connections to the theory of 2-tangles. arXiv:hep-th/9309067v1 10 Sep 1993 1 Introduction The notion of a deep relationship between string theories and gauge theories is far from new. String theory first arose as a model of hadron interactions. Unfortunately this theory had a number of undesirable features; in particular, it predicted massless spin-2 particles. -
Wilson Loops in N = 4 SO(N) SYM and D-Branes in Ads5 × RP
PUPT-2618 Wilson loops in N = 4 SO(N) SYM 5 and D-Branes in AdS5 × RP Simone Giombi and Bendeguz Offertaler Department of Physics, Princeton University, Princeton, NJ 08544, USA Abstract We study the half-BPS circular Wilson loop in N = 4 super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of SO(N). We focus on the large N limit and present some simple quantitative tests of the duality with type IIB string theory in 5 AdS5 × RP . In particular, we show that the strong coupling limit of the expectation value of the Wilson loop in the spinor representation of the gauge group precisely matches the classical action of the dual string theory object, which is expected to be a D5-brane wrapping 4 5 a RP subspace of RP . We also briefly discuss the large N, large λ limits of the SO(N) arXiv:2006.10852v1 [hep-th] 18 Jun 2020 Wilson loop in the symmetric/antisymmetric representations and their D3/D5-brane duals. Finally, we use the D5-brane description to extract the leading strong coupling behavior of the \bremsstrahlung function" associated to a spinor probe charge, or equivalently the normalization of the two-point function of the displacement operator on the spinor Wilson loop, and obtain agreement with the localization prediction. June 22, 2020 Contents 1 Introduction1 2 Wilson loops in N = 4 SO(N) SYM4 2.1 Fundamental representation . .8 2.2 Large N saddle point of the SO(N) matrix model . -
Symmetries in Quantum Field Theory and Quantum Gravity
Symmetries in Quantum Field Theory and Quantum Gravity Daniel Harlowa and Hirosi Oogurib;c aCenter for Theoretical Physics Massachusetts Institute of Technology, Cambridge, MA 02139, USA bWalter Burke Institute for Theoretical Physics California Institute of Technology, Pasadena, CA 91125, USA cKavli Institute for the Physics and Mathematics of the Universe (WPI) University of Tokyo, Kashiwa, 277-8583, Japan E-mail: [email protected], [email protected] Abstract: In this paper we use the AdS/CFT correspondence to refine and then es- tablish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any \long-range" bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irre- ducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the \convex hull condition" of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary.