Supersymmetry Lecture notes for FYS5190/FYS9190 Paul Batzing and Are Raklev December 4, 2015 2 Contents 1 Introduction 7 2 Groups and algebras 9 2.1 Whatisagroup?.................................. 9 2.2 Representations................................. 12 2.3 Liegroups...................................... 13 2.4 Liealgebras..................................... 15 2.5 Exercises ...................................... 16 3 The Poincar´ealgebra and its extensions 19 3.1 TheLorentzGroup................................. 19 3.2 ThePoincar´egroup ............................... 20 3.3 The Casimir operators of the Poincar´egroup . ........ 20 3.4 The no-go theorem and graded Lie algebras . ...... 22 3.5 Weylspinors .................................... 23 3.6 The Casimir operators of the super-Poincar´ealgebra . ............ 25 3.7 Representations of the superalgebra . ....... 26 3.8 Exercises ...................................... 28 4 Superspace 31 4.1 Superspacecalculus .............................. 31 4.2 Superspacedefinition.. .. .. .. .. .. .. .. 33 4.3 Covariantderivatives. 35 4.4 Superfields ..................................... 35 4.4.1 Scalarsuperfields.............................. 36 4.4.2 Vectorsuperfields. .. .. .. .. .. .. .. .. 37 4.5 Supergauge ..................................... 38 4.6 Exercises ...................................... 39 5 Construction of a low-energy SUSY Lagrangian 41 5.1 Supersymmetry invariant Lagrangians . ....... 41 5.2 Albaniangaugetheories . 42 5.3 Non-Abelian gauge theories . 43 5.4 Supersymmetricfieldstrength. ..... 45 5.5 The (almost) complete supersymmetric Lagrangian . .......... 46 5.6 Spontaneoussupersymmetrybreaking . ...... 46 5.7 Supertrace...................................... 48 3 4 CONTENTS 5.8 Softbreaking .................................... 49 5.9 Thehierarchyproblem. .. .. .. .. .. .. .. .. 50 5.10 The non-renormalization theorem . ....... 51 5.11 Renormalisation group equations . ....... 52 5.12Vacuumenergy ................................... 54 5.13Excercises ..................................... 54 6 The Minimal Supersymmetric Standard Model (MSSM) 57 6.1 MSSMfieldcontent ................................ 57 6.2 Thekineticterms................................. 58 6.3 Gaugeterms .................................... 59 6.4 TheMSSMsuperpotential. 60 6.5 R-parity....................................... 61 6.6 SUSYbreakingterms ............................... 62 6.7 RadiativeEWSB .................................. 63 6.8 Higgsbosonproperties. 66 6.9 The gluinog ˜ .................................... 68 6.10 Neutralinos&Charginos. 69 6.11 Sleptons&Squarks............................... 71 6.12 Gaugecouplingunification. ..... 72 6.13Excercises ..................................... 74 7 Sparticle phenomenology 77 7.1 Modelsforsupersymmetrybreaking . ..... 77 7.1.1 Planck-scale Mediated Supersymmetry Breaking (PMSB)....... 78 7.1.2 Gauge Mediated Supersymmetry Breaking (GMSB) . 79 7.2 Supersymmetryathadroncolliders . ...... 80 7.3 Currentboundsonsparticlemasses. ...... 86 7.3.1 Squarksandgluinos ............................ 86 7.3.2 Sbottom................................... 87 7.3.3 Stop..................................... 87 7.3.4 Sleptons................................... 88 7.3.5 Charginosandneutralinos. 88 7.4 Supersymmetryatleptoncolliders . ...... 89 7.4.1 Current bounds at lepton colliders . 93 7.5 Precisionobservables. 94 7.5.1 Electroweak precision observables . ...... 94 7.5.2 (g 2) ................................... 95 − µ 7.5.3 b sγ .................................... 95 → 7.5.4 B µ+µ ................................. 96 s → − 7.6 Excercises...................................... 97 8 Supersymmetric dark matter 105 8.1 Evidencefordarkmatter(DM) . 105 8.2 WIMPmagic .................................... 107 8.3 Dark matter candidates in supersymmetry . ....... 108 8.3.1 Neutralino.................................. 108 CONTENTS 5 8.3.2 Sneutrinos.................................. 110 8.3.3 Gravitino .................................. 110 8.3.4 Others.................................... 110 8.4 Directdetection................................. 111 8.5 Indirectdetection............................... 111 8.6 Excercises...................................... 114 6 CONTENTS Chapter 1 Introduction The goal of these lecture notes is to introduce the basics of low-energy models of supersym- metry (SUSY) using the Minimal Supersymmetric Standard Model (MSSM) as our main example. The notes are based on lectures given at the University of Oslo in 2011 and 2013, and lectures at the NORDITA Winter School on Theoretical Particle Physics in 2012. The notes were originally made by Paul Batzing in 2011, but has since been embellished somewhat. Oslo August 2015 Are Raklev 7 8 CHAPTER 1. INTRODUCTION Chapter 2 Groups and algebras Rather than starting with the problems of the Standard Model, we will focus on the alge- braic origin of supersymmetry in the sense of an extension of the symmetries of Einsten’s Special Relativity (SR), which was the original motivation for work on what we today call supersymmetry. We first need to introduce some basic concepts used in physics for exploring symmetries, mainly groups and Lie algebras. 2.1 What is a group? Definition: The set G = g and operation form a group if and only if for { i} • g G : ∀ i ∈ i) g g G, (closure) i • j ∈ ii) (g g ) g = g (g g ), (associativity) i • j • k i • j • k iii) e G such that g e = e g = g , (identity element) ∃ ∈ i • • i i 1 1 1 iv) g− G such that g g− = g− g = e. (inverse) ∃ i ∈ i • i i • i A simple example of a group is G = Z with usual addition as the operation, e = 0 and 1 g− = g. Alternatively we can restrict the group to Zn, where the operation is addition − 1 1 modulo n. In this group, g− = n g and the unit element is again e = 0. Note that Z is i − i an infinite group, while Zn is finite, with order n (meaning n members). Both are abelian groups, meaning that the elements commute g g = g g . The simplest, non-trivial, of i • j j • i these groups is Z2 which has the members e = 0 and 1. The operation is defined by 0+ 0 = 0, 0+1=1and1+1=0. A somewhat more sophisticated example of a group can be found in a use for the Taylor 1Note that we use e as the identity in an abstract group, while I or 1 is used as the identity matrix in matrix representations of groups. 9 10 CHAPTER 2. GROUPS AND ALGEBRAS expansion2 1 2 f(x + a) = f(x)+ af ′(x)+ a f ′′(x)+ . 2 ∞ an dn = f(x) n! dxn n=0 X a d = e dx f(x). a d Notice the dubious black arts employed in the last equality. The operator Ta = e dx is called the translation operator (in this case in one dimension). Together with the (natural) 1 operation Ta Tb = Ta+b it forms the translational group T (1), where Ta− = T a. In n − • ~a ~ dimensions the group T (n) has the elements T~a = e ·∇. We next define some groups that are very important in physics. Definition: The unitary group U(n) is defined by the set of complex unitary n n matrices U, i.e. matrices such that U U =1 or U 1 = U . If we additionally × † − † require that det(U) = 1 the matrices form the special unitary group SU(n). The unitary group has has the neat property that for ~x, ~y Cn multiplication by a unitary ∀ ∈ matrix leaves scalar products unchanged: ~x′ ~y′ ~x′†~y′ = (U~x)†U~y · ≡ = ~x†U †U~y = ~x†~y = ~x ~y. · In a sense it doesn’t change the size of the vectors it acts on. We now extend our vocabulary for groups by defining the subgroup of a group G. Definition: A subset H G is a subgroup if and only if:a ⊂ i) h h H for h , h H, (closure) i • j ∈ ∀ i j ∈ 1 ii) h− H for h H. (inverse) i ∈ ∀ i ∈ a An alternative, equivalent, and more compact way of writing these two requirements is the −1 single requirement hi • hj ∈ H for ∀hi, hj ∈ H. This is often utilised in proofs. There is a very important type of subgroup called the normal subgroup. The importance will become clear in a moment. 2This is the first of many points where any real mathematician would start to cry loudly and leave the room. 2.1. WHAT IS A GROUP? 11 Definition: H is a proper subgroup if and only if H = G and H = e . A 6 a6 { } subgroup H is a normal (invariant) subgroup, if and only if for g G, ∀ ∈ 1 ghg− H for h H. ∈ ∀ ∈ A simple group G has no proper normal subgroup. A semi-simple group G has no abelian normal subgroup. a Another, pretty but slightly abusive, way of defining a normal group is to say that gHg−1 = H. Up to this point things hopefully seem pretty natural, if not exactly easy. We will now become slightly more cryptic by defining cosets. Definition: A left coset of a subgroup H G with respect to g G is the set ⊂ ∈ gh : h H , and a right coset of the subgroup is the set hg : h H . For { ∈ } { ∈ } normal subgroups H it can be shown that the left and right cosets coincide and form the coset groupa G/H. This has as its members the sets gh : h H for { ∈ } g G and the binary operation with gh g h (g g )h : h H . ∀ ∈ ∗ ∗ ′ ′ ∈ { • ′ ∈ } a Sometimes called the factor or quotient group. The coset group has a heuristic understanding as a division of groups, where the structure of the normal subgroup is removed from the larger group, hench the symbol. Finally, we need to introduce products of groups in order to discuss multiple symmetries. Definition: The direct product of groups G and H, G H, is defined as the × ordered pairs (g, h) where g G and h H, with component-wise operation (g , h ) ∈ ∈ i i • (g , h ) = (g g , h h ). G H is then a group and G and H can be shown to be j j i • j i • j × normal subgroups of G H. × Now, given both the definition of a coset group and the direct product of groups, we can see how to both make and remove products of groups. The normal subgroups can, again heuristically, be viewed as factors in groups. Because it has at least one guest star appearance in the text we also need the semi-direct product. Definition: The semi-direct product G ⋊ H, where G is a mapping G : H H, → is defined by the ordered pairs (g, h) where g G and h H, with component-wise ∈ ∈ operation (g , h ) (g , h ) = (g g , h g (h )). Here H is not a normal subgroup i i • j j i • j i • i j of G ⋊ H.