DOCSLIB.ORG

Supersymmetry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Supersymmetry Prepared for submission to JHEP Supersymmetry B. C. Allanacha aDepartment of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom E-mail: [email protected] Abstract: These are lecture notes for the BUSSTEPP Supersymmetry course, and are a snippet of my Part III 16 lecture course. You should already know some quantum field theory and group theory. You can watch videos of all of my Part III supersymmetry lectures on the web by following the link from http://users.hepforge.org/~allanach/teaching.html where the full lecture notes may also be found. I have a tendency to make trivial tran- scription errors on the board - please stop me if I make one. Plan Review spinors and the Lorentz algebra • Extend the Lorentz algebra to the global supersymmetry algebra • Build the states of the supermultiplets that are used in the minimal supersymmetric • standard model (MSSM) Introduce the MSSM. • A quick computer tutorial on using SOFTSUSY, a computer program that calculates the • MSSM particle spectrum and is usually the ‘first step’ in supersymmetry simulations etc. In particular, this unfortunately does not include the supersymmetric interactions: you need lectures about superspace and superfields for that. I welcome questions during lectures. Contents 1 Physical Motivation 1 2 Supersymmetry algebra and representations 2 2.1 Poincar´esymmetry and spinors 2 2.1.1 Properties of the Lorentz group 2 2.1.2 Representations and invariant tensors of SL(2, C) 3 2.1.3 Generators of SL(2, C) 5 2.1.4 Products of Weyl spinors 5 2.1.5 Dirac spinors 6 2.2 SUSY algebra 7 2.2.1 Graded algebra 7 2.3 Representations of the Poincar´egroup 9 2.4 = 1 supersymmetry representations 10 N 2.4.1 Massless supermultiplet 11 3 Introducing the minimal supersymmetric standard model (MSSM) 12 3.1 Particles 12 3.2 Supersymmetry breaking in the MSSM 13 3.3 Pros and Cons of the MSSM 15 4 Using SOFTSUSY 16 4.1 SOFTSUSY Tutorial 16 1 Physical Motivation Fundamental scalars (eg the Higgs) receive quantum corrections to their mass of order the heaviest mass scale in the theory (presumably M 1019 GeV), divided by a loop factor P l ∼ 16π2. The technical hierarchy problem is: how do we manage to end up with a 125 GeV ≈ Higgs boson in this case? Supersymmetry solves the hierarchy problem because the large quantum corrections cancel between fermions and bosons in the loops. Combined with the GUT idea, it also unifies the three gauge couplings at one single point at larger energies. A supersymmetric version of the Standard Model also provides the most studied candidate for dark matter. Moreover, it provides well defined QFTs in which issues of strong coupling can be better studied than in the non-supersymmetric models, and is an important ingredient in string theory. – 1 – 2 Supersymmetry algebra and representations 2.1 Poincar´esymmetry and spinors The Poincar´egroup corresponds to the basic symmetries of special relativity, it acts on space-time coordinates xµ as follows: xµ x′µ = Λµ xν + aµ 7→ ν Lorentz translation Lorentz transformations leave the metric tensor|{z}η = diag(1|{z}, 1, 1, 1) invariant: µν − − − ΛT η Λ = η They can be separated between those that are connected to the identity and this that are not (i.e. parity reversal Λ = diag(1, 1, 1, 1) and time reversal Λ = diag( 1, 1, 1, 1)). P − − − T − We will mostly discuss those Λ continuously connected to identity, i.e. the proper or- thochronous group1 SO(3, 1)↑. Generators for the Poincar´egroup are the hermitian M µν (rotations and Lorentz boosts) and P σ (translations) with algebra P µ , P ν = 0 h i M µν , P σ = i P µ ηνσ P ν ηµσ − h i M µν , M ρσ = i M µσ ηνρ + M νρ ηµσ M µρ ηνσ M νσ ηµρ − − h i A 4 dimensional matrix representation for the M µν is (M ρσ)µ = i ηµσ δρ ηρµ δσ . ν − ν − ν 2.1.1 Properties of the Lorentz group We now show that locally (i.e. in terms of the algebra), we have a correspondence • SO(3, 1) = SU(2) SU(2). ∼ × The generators of SO(3, 1) (Ji of rotations and Ki of Lorentz boosts) can be expressed as 1 J = ǫ M , K = M , i 2 ijk jk i 0i and the Lorentz algebra written in terms of J’s and K’s is [K , K ]= iǫ J , [J , K ]= iǫ K , [J , J ]= iǫ J . i j − ijk k i j ijk k i j ijk k Their2 linear combinations (which are neither hermitian nor anti hermitian) 1 1 A = J + iK , B = J iK i 2 i i i 2 i − i 1 0 These consist of the subgroup of transformations which have detΛ = +1 and Λ0 ≥ 1. 2 NB ǫ123 = +1. – 2 – satisfy SU(2) SU(2) commutation relations × Ai , Aj = iǫijk Ak , Bi , Bj = iǫijk Bk , Ai , Bj = 0 h i h i h i Under parity Pˆ, (x0 x0 and ~x ~x) we have 7→ 7→ − J J , K K = A B . i 7→ i i 7→ − i ⇒ i ↔ i We can interpret J~ = A~ + B~ as the physical spin. On the other hand, there is a homeomorphism (not an isomorphism) • C SO(3, 1) ∼= SL(2, ) . To see this, take a 4 vector X and a corresponding 2 2 - matrixx ˜, × x + x x ix X = x eµ = (x , x , x , x ) , x˜ = x σµ = 0 3 1 − 2 , µ 0 1 2 3 µ x + ix x x 1 2 0 − 3 ! where σµ is the 4 vector of Pauli matrices 1 0 0 1 0 i 1 0 σµ = , , − , . 0 1 1 0 i 0 0 1 ( ! ! ! − !) Transformations X ΛX under SO(3, 1) leaves the square 7→ X 2 = x2 x2 x2 x2 | | 0 − 1 − 2 − 3 invariant, whereas the action of SL(2, C) mappingx ˜ NxN˜ † with N SL(2, C) 7→ ∈ preserves the determinant detx ˜ = x2 x2 x2 x2 . 0 − 1 − 2 − 3 The map between SL(2, C) is 2-1, since N = 1 both correspond to Λ = 1, but ± SL(2, C) has the advantage of being simply connected, so SL(2, C) is the universal covering group. 2.1.2 Representations and invariant tensors of SL(2, C) The basic representations of SL(2, C) are: The fundamental representation • ′ β ψα = Nα ψβ , α, β = 1, 2 (2.1) The elements of this representation ψα are called left-handed Weyl spinors. The conjugate representation • ′ ∗ β˙ ˙ χ¯α˙ = Nα˙ χ¯β˙ , α,˙ β = 1, 2 Hereχ ¯β˙ are called right-handed Weyl spinors. – 3 – The contravariant representations are • ′α β −1 α ′α˙ β˙ ∗−1 α˙ ψ = ψ (N )β , χ¯ =χ ¯ (N )β˙ . The fundamental and conjugate representations are the basic representations of SL(2, C) and the Lorentz group, giving then the importance to spinors as the basic objects of special relativity, a fact that could be missed by not realising the connection between the Lorentz group and SL(2, C). We will see next that the contravariant representations are however not independent. Consider the different ways to raise and lower indices. The metric tensor ηµν = (η )−1 is invariant under SO(3, 1) and is used to raise/lower • µν indices. The analogy within SL(2, C) is • ˙ ǫαβ = ǫα˙ β = ǫ = ǫ , ǫ12 = +1,ǫ21 = 1. − αβ − α˙ β˙ − since ǫ′αβ = ǫρσ N α N β = ǫαβ det N = ǫαβ . ρ σ · That is why ǫ is used to raise and lower indices ˙ ˙ ψα = ǫαβψ , χ¯α˙ = ǫα˙ βχ¯ ψ = ǫ ψβ, χ¯ = ǫ χ¯β β β˙ ⇒ α αβ α˙ α˙ β˙ so contravariant representations are not independent from covariant ones. To handle mixed SO(3, 1)- and SL(2, C) indices, recall that the transformed compo- • nents xµ should look the same, whether we transform the vector X via SO(3, 1) or µ the matrixx ˜ = xµσ via SL(2, C) (x σµ) N β (x σν) N ∗ γ˙ = Λ ν x σµ , µ αα˙ 7→ α ν βγ˙ α˙ µ ν so the right transformation rule is µ β ν −1 µ ∗ γ˙ (σ )αα˙ = Nα (σ )βγ˙ (Λ ) ν Nα˙ . Similar relations hold for ˙ (¯σµ)αα˙ := ǫαβ ǫα˙ β (σµ) = (1, ~σ) . ββ˙ − Question: 1. Check the following identities: ˙ (¯σµ)αα˙ ǫαβǫα˙ βσµ = 1, σ1, σ2, σ3 ≡ ββ˙ − − − (σµ) (¯σ )γδ˙ = 2 δδ δγ˙ αβ˙ µ α β˙ µ ν ν µ β µν β (σ σ¯ + σ σ¯ )α = 2 η δα Trace σµσ¯ν = 2 ηµν – 4 – 2.1.3 Generators of SL(2, C) Let us define tensors σµν ,σ ¯µν as antisymmetrised products of σ matrices: i (σµν ) β := σµ σ¯ν σν σ¯µ β α 4 − α ˙ i (¯σµν ) β := σ¯µ σν σ¯ν σµα˙ α˙ 4 − β˙ which satisfy the Lorentz algebra σµν , σλρ = i ηµρ σνλ + ηνλ σµρ ηµλ σνρ ηνρ σµλ . − − h i They thus form a representation of the Lorentz algebra (the spinor representation). Under a finite Lorentz transformation with parameters ωµν, spinors transform as follows: i ψ exp ω σµν β ψ (left-handed) α 7→ −2 µν β α α˙ i ˙ χ¯α˙ exp ω σ¯µν χ¯β (right-handed) 7→ −2 µν β˙ Now consider the spins with respect to the SU(2)s spanned by the Ai and Bi: 1 1 i ψ : (A, B)= , 0 = J = σ , K = σ α 2 ⇒ i 2 i i −2 i 1 1 i χ¯α˙ : (A, B)= 0, = J = σ , K = + σ 2 ⇒ i 2 i i 2 i Let us just mention the identities3 1 σµν = ǫµνρσ σ 2i ρσ 1 σ¯µν = ǫµνρσ σ¯ , −2i ρσ known as self duality and anti self duality.
Load more

Recommended publications
  • Lecture 1 – Symmetries & Conservation
    LECTURE 1 – SYMMETRIES & CONSERVATION Contents • Symmetries & Transformations • Transformations in Quantum Mechanics • Generators • Symmetry in Quantum Mechanics • Conservations Laws in Classical Mechanics • Parity Messages • Symmetries give rise to conserved quantities . Symmetries & Conservation Laws Lecture 1, page1 Symmetry & Transformations Systems contain Symmetry if they are unchanged by a Transformation . This symmetry is often due to an absence of an absolute reference and corresponds to the concept of indistinguishability . It will turn out that symmetries are often associated with conserved quantities . Transformations may be: Active: Active • Move object • More physical Passive: • Change “description” Eg. Change Coordinate Frame • More mathematical Passive Symmetries & Conservation Laws Lecture 1, page2 We will consider two classes of Transformation: Space-time : • Translations in (x,t) } Poincaré Transformations • Rotations and Lorentz Boosts } • Parity in (x,t) (Reflections) Internal : associated with quantum numbers Translations: x → 'x = x − ∆ x t → 't = t − ∆ t Rotations (e.g. about z-axis): x → 'x = x cos θz + y sin θz & y → 'y = −x sin θz + y cos θz Lorentz (e.g. along x-axis): x → x' = γ(x − βt) & t → t' = γ(t − βx) Parity: x → x' = −x t → t' = −t For physical laws to be useful, they should exhibit a certain generality, especially under symmetry transformations. In particular, we should expect invariance of the laws to change of the status of the observer – all observers should have the same laws, even if the evaluation of measurables is different. Put differently, the laws of physics applied by different observers should lead to the same observations. It is this principle which led to the formulation of Special Relativity.
    [Show full text]
  • The Homology Theory of the Closed Geodesic Problem
    J. DIFFERENTIAL GEOMETRY 11 (1976) 633-644 THE HOMOLOGY THEORY OF THE CLOSED GEODESIC PROBLEM MICHELINE VIGUE-POIRRIER & DENNIS SULLIVAN The problem—does every closed Riemannian manifold of dimension greater than one have infinitely many geometrically distinct periodic geodesies—has received much attention. An affirmative answer is easy to obtain if the funda- mental group of the manifold has infinitely many conjugacy classes, (up to powers). One merely needs to look at the closed curve representations with minimal length. In this paper we consider the opposite case of a finite fundamental group and show by using algebraic calculations and the work of Gromoll-Meyer [2] that the answer is again affirmative if the real cohomology ring of the manifold or any of its covers requires at least two generators. The calculations are based on a method of the second author sketched in [6], and he wishes to acknowledge the motivation provided by conversations with Gromoll in 1967 who pointed out the surprising fact that the available tech- niques of algebraic topology loop spaces, spectral sequences, and so forth seemed inadequate to handle the "rational problem" of calculating the Betti numbers of the space of all closed curves on a manifold. He also acknowledges the help given by John Morgan in understanding what goes on here. The Gromoll-Meyer theorem which uses nongeneric Morse theory asserts the following. Let Λ(M) denote the space of all maps of the circle S1 into M (not based). Then there are infinitely many geometrically distinct periodic geodesies in any metric on M // the Betti numbers of Λ(M) are unbounded.
    [Show full text]
  • 10 Group Theory and Standard Model
    Physics 129b Lecture 18 Caltech, 03/05/20 10 Group Theory and Standard Model Group theory played a big role in the development of the Standard model, which explains the origin of all fundamental particles we see in nature. In order to understand how that works, we need to learn about a new Lie group: SU(3). 10.1 SU(3) and more about Lie groups SU(3) is the group of special (det U = 1) unitary (UU y = I) matrices of dimension three. What are the generators of SU(3)? If we want three dimensional matrices X such that U = eiθX is unitary (eigenvalues of absolute value 1), then X need to be Hermitian (real eigenvalue). Moreover, if U has determinant 1, X has to be traceless. Therefore, the generators of SU(3) are the set of traceless Hermitian matrices of dimension 3. Let's count how many independent parameters we need to characterize this set of matrices (what is the dimension of the Lie algebra). 3 × 3 complex matrices contains 18 real parameters. If it is to be Hermitian, then the number of parameters reduces by a half to 9. If we further impose traceless-ness, then the number of parameter reduces to 8. Therefore, the generator of SU(3) forms an 8 dimensional vector space. We can choose a basis for this eight dimensional vector space as 00 1 01 00 −i 01 01 0 01 00 0 11 λ1 = @1 0 0A ; λ2 = @i 0 0A ; λ3 = @0 −1 0A ; λ4 = @0 0 0A (1) 0 0 0 0 0 0 0 0 0 1 0 0 00 0 −i1 00 0 01 00 0 0 1 01 0 0 1 1 λ5 = @0 0 0 A ; λ6 = @0 0 1A ; λ7 = @0 0 −iA ; λ8 = p @0 1 0 A (2) i 0 0 0 1 0 0 i 0 3 0 0 −2 They are called the Gell-Mann matrices.
    [Show full text]
  • Spin in Relativistic Quantum Theory∗
    Spin in relativistic quantum theory∗ W. N. Polyzou, Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242 W. Gl¨ockle, Institut f¨urtheoretische Physik II, Ruhr-Universit¨at Bochum, D-44780 Bochum, Germany H. Wita la M. Smoluchowski Institute of Physics, Jagiellonian University, PL-30059 Krak´ow,Poland today Abstract We discuss the role of spin in Poincar´einvariant formulations of quan- tum mechanics. 1 Introduction In this paper we discuss the role of spin in relativistic few-body models. The new feature with spin in relativistic quantum mechanics is that sequences of rotationless Lorentz transformations that map rest frames to rest frames can generate rotations. To get a well-defined spin observable one needs to define it in one preferred frame and then use specific Lorentz transformations to relate the spin in the preferred frame to any other frame. Different choices of the preferred frame and the specific Lorentz transformations lead to an infinite number of possible observables that have all of the properties of a spin. This paper provides a general discussion of the relation between the spin of a system and the spin of its elementary constituents in relativistic few-body systems. In section 2 we discuss the Poincar´egroup, which is the group relating inertial frames in special relativity. Unitary representations of the Poincar´e group preserve quantum probabilities in all inertial frames, and define the rel- ativistic dynamics. In section 3 we construct a large set of abstract operators out of the Poincar´egenerators, and determine their commutation relations and ∗During the preparation of this paper Walter Gl¨ockle passed away.
    [Show full text]
  • Random Perturbation to the Geodesic Equation
    The Annals of Probability 2016, Vol. 44, No. 1, 544–566 DOI: 10.1214/14-AOP981 c Institute of Mathematical Statistics, 2016 RANDOM PERTURBATION TO THE GEODESIC EQUATION1 By Xue-Mei Li University of Warwick We study random “perturbation” to the geodesic equation. The geodesic equation is identified with a canonical differential equation on the orthonormal frame bundle driven by a horizontal vector field of norm 1. We prove that the projections of the solutions to the perturbed equations, converge, after suitable rescaling, to a Brownian 8 motion scaled by n(n−1) where n is the dimension of the state space. Their horizontal lifts to the orthonormal frame bundle converge also, to a scaled horizontal Brownian motion. 1. Introduction. Let M be a complete smooth Riemannian manifold of dimension n and TxM its tangent space at x M. Let OM denote the space of orthonormal frames on M and π the projection∈ that takes an orthonormal n frame u : R TxM to the point x in M. Let Tuπ denote its differential at u. For e Rn→, let H (e) be the basic horizontal vector field on OM such that ∈ u Tuπ(Hu(e)) = u(e), that is, Hu(e) is the horizontal lift of the tangent vector n u(e) through u. If ei is an orthonormal basis of R , the second-order dif- { } n ferential operator ∆H = i=1 LH(ei)LH(ei) is the Horizontal Laplacian. Let wi, 1 i n be a family of real valued independent Brownian motions. { t ≤ ≤ } P The solution (ut,t<ζ), to the following semi-elliptic stochastic differen- n i tial equation (SDE), dut = i=1 Hut (ei) dwt, is a Markov process with 1 ◦ infinitesimal generator 2 ∆H and lifetime ζ.
    [Show full text]
  • Chapter 1 Group and Symmetry
    Chapter 1 Group and Symmetry 1.1 Introduction 1. A group (G) is a collection of elements that can ‘multiply’ and ‘di- vide’. The ‘multiplication’ ∗ is a binary operation that is associative but not necessarily commutative. Formally, the defining properties are: (a) if g1, g2 ∈ G, then g1 ∗ g2 ∈ G; (b) there is an identity e ∈ G so that g ∗ e = e ∗ g = g for every g ∈ G. The identity e is sometimes written as 1 or 1; (c) there is a unique inverse g−1 for every g ∈ G so that g ∗ g−1 = g−1 ∗ g = e. The multiplication rules of a group can be listed in a multiplication table, in which every group element occurs once and only once in every row and every column (prove this !) . For example, the following is the multiplication table of a group with four elements named Z4. e g1 g2 g3 e e g1 g2 g3 g1 g1 g2 g3 e g2 g2 g3 e g1 g3 g3 e g1 g2 Table 2.1 Multiplication table of Z4 3 4 CHAPTER 1. GROUP AND SYMMETRY 2. Two groups with identical multiplication tables are usually considered to be the same. We also say that these two groups are isomorphic. Group elements could be familiar mathematical objects, such as numbers, matrices, and differential operators. In that case group multiplication is usually the ordinary multiplication, but it could also be ordinary addition. In the latter case the inverse of g is simply −g, the identity e is simply 0, and the group multiplication is commutative.
    [Show full text]
  • Lecture Notes on Supersymmetry
    Lecture Notes on Supersymmetry Kevin Zhou [email protected] These notes cover supersymmetry, closely following the Part III and MMathPhys Supersymmetry courses as lectured in 2017/2018 and 2018/2019, respectively. The primary sources were: • Fernando Quevedo's Supersymmetry lecture notes. A short, clear introduction to supersym- metry covering the topics required to formulate the MSSM. Also see Ben Allanach's revised version, which places slightly more emphasis on MSSM phenomenology. • Cyril Closset's Supersymmetry lecture notes. A comprehensive set of supersymmetry lecture notes with more emphasis on theoretical applications. Contains coverage of higher SUSY, and spinors in various dimensions, the dynamics of 4D N = 1 gauge theories, and even a brief overview of supergravity. • Aitchison, Supersymmetry in Particle Physics. A friendly introductory book that covers the basics with a minimum of formalism; for instance, the Wess{Zumino model is introduced without using superfields. Also gives an extensive treatment of subtleties in two-component spinor notation. The last half covers the phenomenology of the MSSM. • Wess and Bagger, Supersymmetry and Supergravity. An incredibly terse book that serves as a useful reference. Most modern sources follow the conventions set here. Many pages consist entirely of equations, with no words. We will use the conventions of Quevedo's lecture notes. As such, the metric is mostly negative, and a few other signs are flipped with respect to Wess and Bagger's conventions. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Motivation.........................................3 1.2 The Poincare Group....................................6 1.3 Spinors in Four Dimensions................................9 1.4 Supersymmetric Quantum Mechanics..........................
    [Show full text]
  • Arxiv:1908.07505V1 [Gr-Qc] 20 Aug 2019 Ysrmne N Hbeo Tnl Nnt,T H Bulk
    MPP-2019-180 Gravitational memory in the bulk Henk Bart Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany [email protected] Abstract A method for detecting gravitational memory is proposed. It makes use of ingoing null geodesics instead of timelike geodesics in the original formulation by Christodoulou. It is argued that the method is applicable in the bulk of a space- time. In addition, it is shown that BMS symmetry generators in Newman-Unti gauge have an interpretation in terms of the memory effect. This generalises the connection between BMS supertranslations and gravitational memory, discovered by Strominger and Zhiboedov at null infinity, to the bulk. arXiv:1908.07505v1 [gr-qc] 20 Aug 2019 1 Introduction In General Relativity, there exists the gravitational memory effect, which was discov- ered by Zel’dovich and Polnarev [1], then studied by Braginsky and Thorne [2, 3] in the linearised theory, and at null infinity by Christodoulou in the nonlinear theory in [4]. It is a statement about how the relative distance between geodesics permanently changes after the passing of a burst of radiation. This effect is conceptually nontrivial in the following sense. Usually, one imagines that a ring of test particles subject to a gravitational plane wave oscillates in the + or × polarisation directions, and then returns to its initial state. The gravitational memory effect states that this is not true; the relative distance between test particles of the ring is permanently changed after the passing of the wave. The effect is often referred to as the Christodoulou memory effect, because Christoudoulou made the observation1 that gravitational backreaction in the linearised theory cannot be ignored.
    [Show full text]
  • Towards a Recursive Monte-Carlo Generator of Quark Jets with Spin
    INT Workshop, Feb. 24-28, 2014 Towards a Recursive Monte-Carlo generator of quark jets with spin Zouina Belghobsi* Laboratoire de Physique Théorique, Univ. de Jijel, Algeria Xavier ARTRU** Institut de Physique Nucléaire de Lyon, France *[email protected] **[email protected] Introduction • A jet model which takes into account the quark spin of freedom must start with quantum amplitudes rather than probabilities. • A « toy model » following this principle was built [X. Artru, DSPIN- 09] using Pauli spinors and inspired from the multiperipheral 3 model and the classical "string + P0 " mechanism • It generated Collins and Longitudinal jet handedness effects. • This model was too simplified: hadrons were not on mass-shell. • We present an improved model with mass-shell constraints. Outlines • Quark-Multiperipheral (Q-M) and String Fragmentation (SF) models. • SF model = particular type of Q-M model 3 • The semi-classical "string + P0 " mechanism : discussion and some predictions • Semi-quantization of the string fragmentation model with spin • ab initio splitting algorithm (for recursive Monte-Carlo code) • "renormalized" input • a non-recursive mechanisms : permuted string diagrams Two models of q+q hadronization Quark Multiperipheral (QM) model String Fragmentation (SF) model h h h …. h h h hN h3 2 1 N 3 2 1 q N q4 q3 q2 q0 q1 γ* String diagram Confinement built-in Feynman diagram with a loop Cascade of "quark decays" = bad model : confinement violated Quark momenta in the Quark Multiperipheral model. 1) time- and longitudinal components momentum h h h N h3 2 1 diagram pN q0 q N q4 q3 q2 qN q0 q1 p2 q2 q p1 1 γ∗ p0 q1 = internal momentum → no classical meaning.
    [Show full text]
  • Gauge Generators, Transformations and Identities on a Noncommutative Space
    Gauge Generators, Transformations and Identities on a Noncommutative Space Rabin Banerjee∗ and Saurav Samanta† S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, India By abstracting a connection between gauge symmetry and gauge identity on a noncommutative space, we analyse star (deformed) gauge transformations with usual Leibniz rule as well as unde- formed gauge transformations with a twisted Leibniz rule. Ex- plicit structures of the gauge generators in either case are com- puted. It is shown that, in the former case, the relation mapping the generator with the gauge identity is a star deformation of the commutative space result. In the latter case, on the other hand, this relation gets twisted to yield the desired map. arXiv:hep-th/0608214v2 7 Mar 2007 1 Introduction Recent analysis[1, 2, 3, 4, 5, 6] of gauge transformations in noncommutative theories reveals that, in extending gauge symmetries to the noncommutative space-time, there are two distinct possibilities. Gauge transformations are either deformed such that the standard comultiplication (Leibniz) rule holds or one retains the unmodified gauge transformations as in the commutative case at the expense of altering the Leibniz rule. In the latter case the new ∗E-mail: [email protected] †E-mail: [email protected] 1 rule to compute the gauge variation of the star products of fields results from a twisted Hopf algebra of the universal enveloping algebra of the Lie algebra of the gauge group extended by translations. While both these approaches preserve gauge invariance of the action, there is an important distinction.
    [Show full text]
  • Introduction to Gauge Theories and the Standard Model*
    INTRODUCTION TO GAUGE THEORIES AND THE STANDARD MODEL* B. de Wit Institute for Theoretical Physics P.O.B. 80.006, 3508 TA Utrecht The Netherlands Contents The action Feynman rules Photons Annihilation of spinless particles by electromagnetic interaction Gauge theory of U(1) Current conservation Conserved charges Nonabelian gauge fields Gauge invariant Lagrangians for spin-0 and spin-g Helds 10. The gauge field Lagrangian 11. Spontaneously broken symmetry 12. The Brout—Englert-Higgs mechanism 13. Massive SU (2) gauge Helds 14. The prototype model for SU (2) ® U(1) electroweak interactions The purpose of these lectures is to give an introduction to gauge theories and the standard model. Much of this material has also been covered in previous Cem Academic Training courses. This time I intend to start from section 5 and develop the conceptual basis of gauge theories in order to enable the construction of generic models. Subsequently spontaneous symmetry breaking is discussed and its relevance is explained for the renormalizability of theories with massive vector fields. Then we discuss the derivation of the standard model and its most conspicuous features. When time permits we will address some of the more practical questions that arise in the evaluation of quantum corrections to particle scattering and decay reactions. That material is not covered by these notes. CERN Academic Training Programme — 23-27 October 1995 OCR Output 1. The action Field theories are usually defined in terms of a Lagrangian, or an action. The action, which has the dimension of Planck’s constant 7i, and the Lagrangian are well-known concepts in classical mechanics.
    [Show full text]
  • Characterization of SU(N)
    University of Rochester Group Theory for Physicists Professor Sarada Rajeev Characterization of SU(N) David Mayrhofer PHY 391 Independent Study Paper December 13th, 2019 1 Introduction At this point in the course, we have discussed SO(N) in detail. We have de- termined the Lie algebra associated with this group, various properties of the various reducible and irreducible representations, and dealt with the specific cases of SO(2) and SO(3). Now, we work to do the same for SU(N). We de- termine how to use tensors to create different representations for SU(N), what difficulties arise when moving from SO(N) to SU(N), and then delve into a few specific examples of useful representations. 2 Review of Orthogonal and Unitary Matrices 2.1 Orthogonal Matrices When initially working with orthogonal matrices, we defined a matrix O as orthogonal by the following relation OT O = 1 (1) This was done to ensure that the length of vectors would be preserved after a transformation. This can be seen by v ! v0 = Ov =) (v0)2 = (v0)T v0 = vT OT Ov = v2 (2) In this scenario, matrices then must transform as A ! A0 = OAOT , as then we will have (Av)2 ! (A0v0)2 = (OAOT Ov)2 = (OAOT Ov)T (OAOT Ov) (3) = vT OT OAT OT OAOT Ov = vT AT Av = (Av)2 Therefore, when moving to unitary matrices, we want to ensure similar condi- tions are met. 2.2 Unitary Matrices When working with quantum systems, we not longer can restrict ourselves to purely real numbers. Quite frequently, it is necessarily to extend the field we are with with to the complex numbers.
    [Show full text]