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Lectures on Supersymmetry Part I: Basics Lectures on Supersymmetry Part I: Basics Yael Shadmi Technion Yael Shadmi (Technion) ESHEP2014 1 / 106 Motivation You have probably heard about the motivation for supersymmetry Through these lectures, this will (hopefully) become clear and more concrete But it's important to state at the outset: There is no experimental evidence for supersymmetry The amount of effort that has been invested in supersymmetry (theory and experiment) is thus somewhat surprising Yael Shadmi (Technion) ESHEP2014 2 / 106 [True: There is no experimental evidence for any underlying theory of electroweak symmetry breaking, which would give rise to the (fundamental scalar) Higgs mechanism as an effective description] [There is experimental evidence for BSM: dark matter, the baryon asymmetry|CP violation] Why so much effort on supersymmetry? Yael Shadmi (Technion) ESHEP2014 3 / 106 It is a very beautiful and exciting idea (I hope you'll see this in the lectures) And it's something completely different from anything we know in Nature Yael Shadmi (Technion) ESHEP2014 4 / 106 any BSM which the LHC discovers will be exciting: eg: composite Higgs, \technicolor": new strong dynamics: beautiful! (but we know QCD: asymptotic freedom, confinement, chiral symmetry breaking!) extra space dimensions |amazing! (but we have 3 already..) supersymmetry is conceptually new: it relates bosons and fermions Yael Shadmi (Technion) ESHEP2014 5 / 106 And so is the Higgs: the first spin-zero (seemingly) fundamental particle The herald of supersymmetry? The Higgs is the first scalar we see. Only scalars have quadratic divergences: supersymetry removes these divergences. We will see that in some sense: Supersymmetry makes a scalar behave like a fermion Given charged fermions: supersymmetry predicts spin-0 particles Yael Shadmi (Technion) ESHEP2014 6 / 106 Have we been wasting our time? Supersymmetry is NOT a specific model (certainly not mSUGRA, cMSSM, minimal gauge mediation..) There is a wide variety of supersymmetric extensions of the SM : Different superpartner spectra, different signatures In thinking about them: a whole toolbox: triggers, searches, analysis Even if it's not supersymmetry: may help discover something else Yael Shadmi (Technion) ESHEP2014 7 / 106 Supersymmetry supplies many concrete examples with: new scalars (same charges as SM fermions) new fermions (same charges as SM gauge bosons) [and for discovery: spin is a secondary consideration] potentially leading to missing energy displaced vertices (very) long lived particles disappearing tracks ··· Yael Shadmi (Technion) ESHEP2014 8 / 106 Plan Lecture I: Basics: Study a few simple toy models to understand the basic concepts and structure. These toy models=\modules" for building the supersymmetrized standard model. De-mystify supersymmetry: Why is it a space-time symmetry (extending translations, rotations..) Why does it remove UV divergences (fine tuning) Why do we care about it even though it's clearly broken? ··· Why is the gravitino relevant for LHC experiments? Yael Shadmi (Technion) ESHEP2014 9 / 106 Lecture II: The Minimal Supersymmetric Standard Model: Here we will put to use what we learn in I. Motivation (now that you can appreciate it..) The field content The interactions: NO FREEDOM (almost) The supersymmetry-breaking terms: freedom + determine experimental signatures. EWSB and the Higgs mass Spectra (the general structure of superpartner masses) Yael Shadmi (Technion) ESHEP2014 10 / 106 Lecture III: Superpartners in Action: LHC signatures General properties of spectra Decays Production Is it supersymmetry? (??) The three lectures will be somewhat independent. Yael Shadmi (Technion) ESHEP2014 11 / 106 Spacetime symmetry: Yael Shadmi (Technion) ESHEP2014 12 / 106 Spacetime symmetry: The symmetry we are most familiar with: Poincare: Translations x µ ! x µ + aµ: generator Pµ µ µ µ ν µν Lorentz transformations: x ! x + wν x generators: J (with w µν antisymmetric) (Throughout consider global, infinitesimal transformations) Yael Shadmi (Technion) ESHEP2014 13 / 106 contains: rotations: around axis k (with angle θk ): w ij = ijk θk eg for rotations around z: x 0 ! x 0 ; x 1 ! x 1 − θx 2 ; x 2 ! x 2 + θx 1 ; x 3 ! x 3 boosts: along axis k (with velocity βk ): −w 0k = w k0 = βk eg for boost along z: x 0 ! x 0 + βx 3 ;; x 1 ! x 1 ; x 2 ! x 2 ; x 3 ! x 3 + βx 0 Yael Shadmi (Technion) ESHEP2014 14 / 106 The algebra: [Pµ; Pν] = 0 [Pµ; Jρσ] = 0 (1) [Jµν; Jρσ] = i(g νρJµσ − g µρJνσ − g νσJµρ + g µσJνρ) Yael Shadmi (Technion) ESHEP2014 15 / 106 Let's \discover" all the above in a simple field theory: complex scalar field µ ∗ 2 2 L = @ φ @µφ − m jφj (2) Symmetry: transformation of the fields which leaves the Equations Of Motion (EOMs) invariant this is the case if action invariant, Lagrangian can change by a total derivative µ L!L + α@µJ (3) α=(small) parameter of the transformation What's the symmetry of this theory? U(1): φ(x) ! eiαφ(x) (4) L is invariant this U(1) is an Internal Symmetry (NOT space-time) Yael Shadmi (Technion) ESHEP2014 16 / 106 Spacetime: Translations: x µ ! x µ + aµ (5) µ φ(x) ! φ(x − a) = φ(x) − a @µφ(x) (6) or µ δaφ(x) = a @µφ(x) (7) Lorentz transformations µ µ µν x ! x + w xν (8) µ µ µν φ(x ) ! φ(x − w xν) (9) so 1 δ φ(x) = w µνx @ φ(x) = w µν(x @ − x @ ) φ(x) (10) w µ ν 2 µ ν ν µ action is invariant Yael Shadmi (Technion) ESHEP2014 17 / 106 Algebra: 2 translations (with aµ, bµ) [δa; δb]φ ≡ δa(δbφ) − δb(δaφ) = 0 (11) 2 Lorentz: (with w µν, λρσ) νρ [δw µν ; δλρσ ]φ = iwµνλρσ · i fg (xµ@σ − xµ@σ) + permutationsg (12) Yael Shadmi (Technion) ESHEP2014 18 / 106 let's move now to a supersymmetric theory Yael Shadmi (Technion) ESHEP2014 19 / 106 A simple supersymmetric field theory Free theory with massive (Dirac) fermion of mass m 2 complex scalars φ+, φ− of mass m µ ∗ 2 2 µ ∗ 2 2 ¯ L = @ φ+ @µφ+−m jφ+j +@ φ− @µφ−−m jφ−j + (i=@−m) (13) [the labels +, − are just names, we'll see the reason for this choice soon] [This isn't the most minimal supersymmetric 4d field theory. \Half of it" is: a 2-component (Weyl) fermion plus one complex scalar. But Dirac spinors are more familiar so start with this.] Yael Shadmi (Technion) ESHEP2014 20 / 106 spacetime symmetry: translations, rotations, boosts: just as in our previous example f only difference: itself is a spinor, so transforms: (x) ! 0(x 0) (14) Yael Shadmi (Technion) ESHEP2014 21 / 106 actually, the L-handed and R-handed parts transform differently under Lorentz L (15) R with σi σi ! 0 = (1 − iθi − βi ) (16) L L 2 2 σi σi ! 0 = (1 − iθi + βi ) (17) R R 2 2 so it will be useful to write everything in terms of 2-component spinors Yael Shadmi (Technion) ESHEP2014 22 / 106 recall: we can write any R-handed spinor in terms of a L-handed one: ∗ R = −εχL (18) where 0 −1 " ≡ −iσ2 = (19) 1 0 Exercise: prove eq. (18) so we can write our Dirac spinor in terms of two L-handed spinors ∗ + and −: L = +, R = −" − + ! = ∗ (20) −" − g Yael Shadmi (Technion) ESHEP2014 23 / 106 let's write the Lagrangian in terms of these: µ ∗ y µ L = @ φ+ @µφ+ + +iσ¯ @µ + µ ∗ y µ + @ φ− @µφ− + −iσ¯ @µ − (21) 2 2 2 2 T − m jφ−j − m jφ+j − m( + " − + hc) T T exercise: Derive this. Show also that + " − = − " +, where ± are any 2-component spinors. Yael Shadmi (Technion) ESHEP2014 24 / 106 Can the spacetime symmetry be extended? Yes: there's more symmetry hiding in our theory: take a constant (anti-commuting) (L) 2-component spinor ξ p δ φ = 2 ξT " ξ + p + µ ∗ ∗ δξ + = 2 iσ εξ @µφ+ − mξφ− (22) and similarly for + ! − the symmetry transformations take a boson into a fermion and vice versa: THIS IS SUPERSYMMETRY! Yael Shadmi (Technion) ESHEP2014 25 / 106 exercise: Check that this is indeed a symmetry: 1. Show this first for m = 0. 2. Repeat for m 6= 0. Note that this only holds if the masses of the fermion and scalars are the same. Yael Shadmi (Technion) ESHEP2014 26 / 106 Note: For m = 0, the φ− − − and φ+ − χ+ parts decouple. each one is super-symmetric separately so: this theory is not the most minimal supersymmetric theory (half of it is) this is handy if we're to implement supersymmetry in the SM, because the SM is a chiral theory Yael Shadmi (Technion) ESHEP2014 27 / 106 is the symmetry we found indeed an extension of Poincare? it's surely a spacetime symmetry since it takes a fermion into a boson (the transformation parameters carry spinor indices) furthermore: consider the algebra: take the commutator of 2 new transformations with parameters ξ, η: µ µ y µ y µ [δξ; δη]φL = a @µφL with a = 2i ξ σ¯ η − η σ¯ ξ (23) a translation! Exercise: Check eq. (23). You will have to use the EOMs. Yael Shadmi (Technion) ESHEP2014 28 / 106 Our simple theory is supersymmetric. We have an extension of spacetime symmetry that involves anti-commuting generators. The supersymmetry transformations relate bosons and fermions. If the bosons and fermions had different masses: no supersymmetry. And let's count the physical dof's: on-shell we have fermions: 2 + 2 = 4 bosons: 2 + 2 = 4 (off shell: bosons same, but fermions: 2 × 4) Yael Shadmi (Technion) ESHEP2014 29 / 106 The vacuum energy Global symmetries ! Noether currents µ µ j with @µj = 0 so that there is a conserved charge: Z d Q = d 3x j 0(x) with Q = 0 (24) dt translations: conserved charge = Hamiltonian H so what we found above means that the anti-commutator of two supersymmetry transformations gives the Hamiltonian: fSUSY; SUSYg / H (25) Yael Shadmi (Technion) ESHEP2014 30 / 106 consider the vacuum expectation value of this: h0j fSUSY; SUSYg j0i / h0jHj0i (26) if SUSY unbroken: SUSYj0i = 0 (27) therefore h0jHj0i = 0 (28) The vacuum energy vanishes!! In a supersymmetric
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