INTRODUCTION TO SUPERSYMMETRY (PHYS 661) Instructor: Philip Argyres (pronounced “are–JEER–us”) Office: Room 334 Newman Lab., 255-4118, [email protected]. Course Outline I. Qualitative supersymmetry — 4 lectures, few indices a. Coleman-Mandula theorem b. Supersymmetric QM: vacua, superfields, & instantons II. Perturbative supersymmetry — 8 lectures, the basics a. Chiral multiplets b. Nonrenormalization theorems c. Vector multiplets III. Supersymmetric model building — 4 lectures, qualitative issues a. Supersymmetric standard model b. Soft supersymmetry breaking terms c. Messenger sectors IV. Non-perturbative supersymmetry — 9 lectures, mostly SU(n) SQCD a. Higgs vacua (& instantons) b. Coulomb vacua (& monopoles) c. Chiral theories V. Dynamical supersymmetry breaking — 2 lectures, if I get to it There is no text covering the contents of this course. Some useful references for its various parts are: 1. Qualitative supersymmetry: E. Witten, “Dynamical breaking of supersymmetry,” Nucl. Phys. B188 (1981) 513; S. Coleman, “The uses of instantons,” in The Whys of Subnuclear Physics (Plenum, 1979), and in Aspects of Symmetry (Cambridge, 1985). 2. Perturbative supersymmetry: J. Wess and J. Bagger, “Supersymmetry and supergrav- ity,” 2nd ed., Ch. I–VIII, XXII, App. A–C. Note: I will try to follow the notation and conventions of this book in the course. 3-5. Phenomenology and non-perturbative methods: Various reviews and papers which I’ll mention as we get closer to these topics. Prerequisites are a basic knowledge of QFT (gauge theories, path integrals, 1-loop RG) on the physics side, and an aquaintance with analysis on the complex plane (holomorphy, analytic continuation) as well as rudimentary group theory (SU(3), Lorentz group, spinors) on the math side. i This is a pass/fail course. There will be no tests or final. The grade will be based on participation and doing or discussing with me problems which will be given during the course of the lectures and will also be posted on the web page http://www.hepth.cornell.edu/~argyres/phys661/index.html which can also be found by folowing the appropriate links from the high energy theory home page. These problems are meant to be simple exercises, not lengthy or difficult research projects—so please keep solutions brief, and be sure to come talk to me if you are having trouble with them. My office hours are the hour after lectures and other times by appointment. ii I. Qualitative Supersymmetry 1. The Coleman-Mandula Theorem 1.1. Introduction This course is an introduction to 4-dimensional global N=1 supersymmetric field theory, so not, in particular, other dimensions, supergravity, or extended supersymmetry (except very briefly). I’ll introduce the basics of perturbative supersymmetry and apply them to a critical survey of models of weak-scale supersymmetry in the 1st half of the semester. The first two weeks will introduce supersymmetric quantum mechanics (QM) to try to separate the features special to supersymmetry from the complications of QFT in 3 + 1 dimensions. The 2nd half will be devoted to exploring the non-perturbative dynamics of supersymmetric FTs. Throughout the course I will introduce and use advanced QFT techniques (effective actions, RG flows, anomalies, instantons, ...) when needed, and I will try to emphasize qualitative explanations and symmetry-based techniques and “tricks.” Many of these techniques, though used frequently in QFT, are not usually taught in the standard QFT graduate courses. The aims of this course are two-fold. The first is to supply you with the wherewithall to evaluate the various claims/hopes for weak-scale supersymmetry. The second, and closer to my interests, is to use supersymmetric models as a window on QFT in general. From this point of view, supersymmetric FTs are just especially symmetric versions of ordinary field theories, and in many cases this extra symmetry has allowed us to solve exactly for some non-perturbative properties of these theories. This gives us another context (besides lattice gauge theory) in which to think concretely about non-perturbative QFT in 3 + 1 dimensions. Indeed, the hard part of this course will be the QFT, not the supersymmetry—someone who is thoroughly familiar with QFT should find the content of this course, though perhaps unfamiliar, relatively easy to understand. Finally, this course owes alot to Nathan Seiberg: not only is his work the main focus of the 2nd half of the course, but also much of this course is closely modeled on two series of lectures he gave at the IAS in the fall of 1994 and at Rutgers in the fall of 1995. 1.2. The Coleman-Mandula theorem, or, Why supersymmetry? Though originally introduced in early 70’s we still don’t know how or if supersymmetry plays a role in nature. Supersymmetry today is like non-Abelian gauge theories before the SM: “a fascinating mathematical structure, and a reasonable extension of current ideas, but plagued with phenomenological difficulties.” 1 Why, then, have a considerable number of people been working on this theory for the last 20 years? The answer lies in the Coleman-Mandula theorem, which singles-out supersymmetry as the “unique” extension of Poicar´einvariance in 3+1 or more dimensional QFT (under some important but reasonable assumptions). In the rest of this lecture, I follow the qualitative description of the Coleman-Mandula theorem given by E. Witten “Introduction to supersymmetry,” in Proc. Intern. School of Subnuclear Physics, Erice 1981, ed. A. Zichichi (Plenum Press, 1983) p. 305. A theory of 2 free bose fields has many conserved currents. x Exercise 1.1. Check that in a theory of 2 free bosons µ µ = ∂µφ1∂ φ1 + ∂µφ2∂ φ2, (1.1) L the following currents are conserved: Jµ = φ1∂µφ2 φ2∂µφ1, − Jµρ = ∂ρφ1∂µφ2 φ2∂µ∂ρφ1, (1.2) − Jµρσ = ∂ρ∂σφ1∂µφ2 φ2∂µ∂ρ∂σφ1. − There are interactions which when added to this theory still keep Jµ conserved. x Exercise 1.2. Check that Jµ is conserved if any interaction of the form 2 2 V = f(φ1 + φ2) is added to the 2 boson theory. However, there are no Lorentz-invariant interactions which can be added so that the others are conserved (nor can they be redefined by adding extra terms so that they will still be conserved). This follows from the Coleman-Mandula theorem Phys. Rev. D159 (1967) 1251: In a theory with non-trivial scattering in more than 1+1 dimensions, the only possible conserved quantities that transform as tensors under the Lorentz group are the usual energy-momentum Pµ, Lorentz transformations Mµν , and scalar quantum numbers Qi. (This has a conformal extension for massless particles.) The basic idea is that conservation of Pµ and Mµν leaves only the scattering angle un- known in (say) a 2-body collision. Additional “exotic” conservation laws would determine the scattering angle, leaving only a discrete set of possible angles. Since the scattering amplitude is an analytic function of angle (assumption # 1) it then vanishes for all angles. Concrete example: Suppose we have a conserved traceless symmetric tensor Qµν . By Lorentz invariance, its matrix element in a 1-particle state of momentum p and spin zero is 1 p Q p p p η p2. (1.3) h | µν | i ∝ µ ν − 4 µν Apply this to an elastic 2-body collision of identical particles with incoming momenta p1, p2, and outgoing momenta q1, q2, and assume that the matrix element of Q in the 2-particle 2 state p p is the sum of the matrix elements in the states p and p . This is true if Q | 1 2i | 1i | 2i is “not too non-local”—say, the integral of a local current (assumption # 2). x Exercise 1.3. Show that conservation of a symmetric, traceless charge Qµν together with energy momentum conservation implies µ ν µ ν µ ν µ ν p1 p1 + p2 p2 = q1 q1 + q2 q2 , (1.4) for an elastic scattering of two identical scalars with incoming momenta p1, p2, and outgoing momenta q1, q2. Show that this implies the scattering angle is zero. For the extension of this argument to non-identical particles, particles with spin, inelastic collision, see Coleman and Mandula’s paper. The Coleman-Mandula theorem does not mention spinor charges, though. So consider a free theory of a complex scalar and a free two component (Weyl) fermion = ∂ φ∂µφ + iψσµ∂ ψ. (1.5) L µ µ Again, an infinite number of conserved currents exist. x Exercise 1.4. In a theory of a free complex boson and a free 2-component fermion, show that the currents ρ Sµα =(∂ρφσ σµψ)α, (1.6) ρ Sµνα =(∂ρφσ σµ∂ν ψ)α, are conserved. (σ and σ obey the Dirac-like algebra σµσν + σν σµ ηµν . We will discuss 2-component spinors in detail in lecture 5.) ∝ Now, there are interactions, e.g. V = g(φψαψ + h.c.) + g2 φ 4, that can be added to this α | | free theory such that Sµα (with correction proportional to g) remains conserved. (We will return to this example in much more detail in later lectures.) However, Sµνα is never conserved in the presence of interactions. We can see this by applying the Coleman-Mandula theorem to the anticommutators of the fermionic conserved charges 3 Qα = d x S0α, Z (1.7) 3 Qνα = d x S0να. Z Indeed, consider the anticommutator Q , Q , which cannot vanish unless Q is iden- { να µβ} να tically zero, since the anticommutator of any operator with its hermitian adjoint is positive definite. Since Qνα has components of spin up to 3/2, the anticommutator has compo- nents of spin up to 3. Since the anticommutator is conserved if Qνα is, and since the Coleman-Mandula theorem does not permit conservation of an operator of spin 3 in an 3 interacting theory, Qνα cannot be conserved in an interacting theory.