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Notes on Seibergology EM-Like Duality SUSY and Applications

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Notes on Seibergology EM-Like Duality SUSY and Applications Notes on Seibergology EM-like duality SUSY and applications Flip Tanedo Laboratory of Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA E-mail: [email protected] This version: May 7, 2013 Abstract This is a set of unfinished LATEX'ed notes on Seiberg duality and related topics in phe- nomenology. It subsumes an older set of notes on metastable supersymmetry breaking. Corrections are welcome, just don't expect me to get around to it any time soon. Contents 1 Introduction 1 1.1 Pre-requisites ...................................... 1 1.2 A selected non-technical history ............................ 1 N = 1 Duality in SQCD 2 2 Nonperturbative SUSY QCD 2 2.1 Effective Actions .................................... 2 2.2 Gauge theory facts ................................... 3 2.3 Reminder: basic SUSY gauge theory facts ...................... 4 2.4 Moduli space ...................................... 6 2.5 The holomorphic gauge coupling ............................ 10 2.6 The NSVZ β-function .................................. 13 2.7 The Konishi Anomaly ................................. 17 2.8 Symmetries of SQCD .................................. 18 3 F < N: the ADS superpotential 19 3.1 Holomorphic scale as a spurion ............................ 19 3.2 The ADS Superpotential ................................ 20 3.3 ADS: F = N − 1, instantons .............................. 22 3.4 Deforming SQCD: Higgsing a squark ......................... 24 3.5 Deforming SQCD: mass perturbations ........................ 26 3.6 The coefficient of the ADS superpotential ....................... 27 3.7 Run, run, runaway ................................... 28 3.8 Gaugino condensation ................................. 31 3.9 Integrating in ...................................... 34 3.10 Relatied topics ..................................... 40 4 F = N and F = N + 1: Some special cases 40 4.1 't Hooft Anomaly Matching .............................. 40 4.2 Moduli Space ...................................... 41 4.3 N = F .......................................... 41 4.4 The ADS superpotential from nothing ........................ 45 4.5 F = N + 1: s-confinement ............................... 47 5 F > N + 1: Seiberg Duality 50 5.1 The Banks-Zaks Fixed point .............................. 50 5.2 Some facts about superconformal theories ...................... 53 3 5.3 2 N < F < 3N: the conformal window ........................ 55 5.4 Seiberg Duality ..................................... 55 5.5 Checks of Seiberg Duality ............................... 57 5.6 N + 1 < F < 3N=2: compositeness .......................... 60 5.7 RG motivation for Seiberg duality ........................... 60 5.8 The importance of the scale µ ............................. 62 5.9 Remarks on the meaning of the duality ........................ 63 6 The phase structure of SQCD 64 7 Siblings and Cousins of Seiberg Duality 64 8 Crouching Seiberg, Hidden Gauge Group 64 8.1 Mini review: nonlinear sigma model .......................... 65 8.2 Hidden gauge group ................................... 65 9 Relation to AdS/CFT 65 10 Some current directions of interest 66 11 Unsorted ISS and SUSY breaking notes 66 11.1 Questions originating from Yael's chiral ISS model .................. 66 2 N = 2 Duality 67 12 Seiberg-Witten 67 13 Gaiotto Dualities 68 Breaking SUSY 68 14 SUSY Breaking: history 68 14.1 SUSY breaking ..................................... 68 14.2 Dynamical supersymmetry breaking .......................... 68 14.3 What makes SUSY-breaking non-generic? ...................... 69 14.4 Metastable SUSY breaking ............................... 70 14.5 Problems with gaugino masses ............................. 71 15 Dynamical Supersymmetry Breaking 72 15.1 Motivation ........................................ 72 15.2 Toy example: SUSY QM ................................ 72 15.3 Basics .......................................... 73 15.4 The 3-2 Model ...................................... 74 15.5 The ITIY Model .................................... 75 15.6 Tools for Noncalculable Models ............................ 75 16 Gauge Mediated SUSY Breaking 75 16.1 SUSY Breaking refresher ................................ 75 16.2 Set up and features ................................... 77 16.3 EWSB and the µ-Bµ problem ............................. 81 16.4 Direct, semi-direct, extraordinary, and general .................... 84 17 The Nelson-Seiberg R-symmetry theorem 86 18 The Komargodski-Shih Gaugino Mass Theorem 88 18.1 Basics of SUSY breaking in Wess Zumino Models .................. 88 18.2 Tree-level SUSY and R-symmetry breaking ...................... 92 18.3 Application to Gaugino Masses and Model Building ................. 93 19 The Landau Pole Problem 95 20 ISS: Metastable SUSY Breaking 95 20.1 Summary in words ................................... 95 20.2 Macroscopic Model I .................................. 96 20.3 Macroscopic Model II .................................. 106 20.4 Dynamical Realization: the ISS model ........................ 106 20.5 Immediate directions beyond the basic ISS model .................. 106 20.6 Extensions and related work .............................. 106 3 Appendix 107 A Notation and Conventions 107 A.1 Field labels ....................................... 107 A.2 Spacetime and spinors ................................. 107 A.3 Superfields and superspace ............................... 110 A.4 SUSY NLΣM ...................................... 110 A.5 2-component plane waves ................................ 113 A.6 OLD Notation and Conventions ............................ 113 B The Coleman-Weinberg Effective Potential 114 B.1 Quantum Mechanical Derivation ............................ 115 B.2 Diagrammatic Derivation ................................ 116 B.3 Functional Derivation .................................. 120 B.4 Integrating out fields .................................. 127 B.5 An illustrative example ................................. 127 B.6 Cutoff dependence ................................... 129 C Phases of Gauge Theories 130 D Review of Anomaly Cancellation 130 D.1 Overview and background ............................... 130 D.2 The Anomaly Coefficient ................................ 131 D.3 Cancellation of gauge anomalies ............................ 132 D.4 Comments on global anomalies ............................ 134 E Analytic Continuation into Superspace 134 E.1 Overview ......................................... 135 E.2 Preliminary results ................................... 136 E.3 Gaugino mass ...................................... 137 E.4 Analytic Continuation: Wess-Zumino soft terms ................... 139 E.5 Remarks ......................................... 141 4 1 Introduction These are personal notes originally based on the spring 2009 lectures by Csaba Cs´akiat Cornell University. They have been augmented with material following the author's interests in the field and draw heavily from some of the very excellent `Seibergology' review literature: • Intriligator and Seiberg's lectures on SUSY breaking [1] and electromagnetic duality [2] • Strassler's lecture notes on SUSY gauge theory [3] and the duality cascade [4]. • Terning's book on supersymmetry [5] and the accompanying slides which are available online. Additionally, there are several elements taken from various lectures, review articles, talks, current literature, and discussions with colleagues. I have tried to give credit and links to further literature where appropriate. To be clear, this document includes no new results and the pedagogical approach is an amalgamation from other sources. If unlisted, it is fair to assume that material and the style of presentation came from one of the aforementioned references. All I did was interpolate between several sources to weave a narrative which makes sense to me. Comments, constructive criticism, and corrections are especially welcome. 1.1 Pre-requisites The reader is expected to already be familiar with supersymmetric gauge theories at the level of an introductory graduate-level course, e.g. [6]. For more foundational reading on supersymmetric gauge theories there is now a plethora of review articles and recorded lectures available. As a rule of thumb any set of lectures after 1996 should review Seiberg duality while any lectures after 2006 should at least mention metastable vacua. Recent reviews which mention the metastable SUSY-breaking program include Dine's 2008 Cargese lectures [7] (see also his more recent review [8]) and Shirman's 2008 TASI lectures [9]. One may find further references in those papers. The textbooks by Terning [5] and Dine [10] also mention useful material in modest depth. The multimedia-inclined are encouraged to view Brian Wecht and Nathan Seiberg's lectures at the Isaac Newton Institute's 2007 \Gauge Fields and Strings" workshop1, Seiberg's lectures at PiTP 20102, or Csaba Cs´aki'slectures on supersymmetry3 Further references to appropriate pedagogical or otherwise important literature will be men- tioned as appropriate in this document. 1.2 A selected non-technical history [To do: This section to be written.] 1http://www.newton.ac.uk/programmes/SIS/sisw02p.html 2http://video.ias.edu/pitp-2010 3Available to the Cornell LEPP theory group. 1 1.2.1 Dynamical SUSY breaking 1.2.2 N = 1 electromagnetic duality and beyond 1.2.3 Metastable SUSY breaking N = 1 Duality in SQCD 2 Nonperturbative SUSY QCD The basic tools we will need come from supersymmetric QCD. We shall now review the key aspects of supersymmetric SU(N)
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