Clases de Master FisyMat, Desarrollos Actuales The Standard Model of particle physics and Beyond Date and time: 02/03 to 07/04/2021 15:30–17:00 Video/Sala CAJAL Organizer and Lecturer: Abdelhak Djouadi ([email protected]) You can find the pdfs of the lectures at: https://www.ugr.es/˜adjouadi/ 6. Supersymmetric theories 6.1 Basics of Supersymmetry 6.2 The minimal Supesymmetric Standard Model 6.3 The constrained MSSM’s 6.4 The superparticle spectrum 6.5 The Higgs sector of the MSSM 6.6 Beyond the MSSM 1 6.1 Basics of Supersymmetry Here, we give only basic facts needed later in the phenomenological discussion. For details on theoretical issues, see basic textbooks like Drees, Godbole, Roy. SUperSYmmetry: is a symmetry that relates scalars/vector bosons and fermions. The SUSY generators transform fermions into bosons and vice–versa, namely: FermionQ > Boson > , Boson > Fermion > Q| | Q| | must be an anti–commuting (and thus rather complicated) object. Q † is also a distinct symmetry generator: Q † Fermion > Boson > , † Boson > Fermion > Q | | Q | | Highly restricted [e.g., no go theorem] theories and in 4-dimension with chiral fermions: 1 , † carry spin– with left- and right- helicities and they should obey Q Q 2 .... The SUSY algebra: which schematically is given by µ , † = P , , =0 , †, † =0, {Qµ Q } µ {Q Q} a {Q Qa } [P , ]=0, [P , †]=0, [T , ]=0, [T , †]=0 Q Q Q Q P µ: is the generator of space–time transformations. T a are the generators of internal (gauge) symmetries. SUSY: unique extension of the Poincar´egroup of space–time symmetry to include ⇒ a four–dimensional Quantum Field Theory... 2 The single–particle states of the theory are in irreducible representations of the SUSY algebra defined above, and which are called supermultiplets. The fermions and bosons of the same supermultiplet are called superpartners. They must have the same mass and the same gauge quantum numbers. Three types of supermultiplets are needed to describe physics phenomena. Chiral (or “scalar”) supermultiplet (denoted by ζ with ζc = ζ and S): • – 1 two–component Weyl fermion with spin 1 (n = 2) ±2 F – 2 real spin–szero scalar = 1 complex scalar field (nB = 2) Gauge (or “vector”) supermultiplet (denoted by Aa and λ ): • µ A – 1 two–component Weyl gaugino–fermion with spin 1 (n = 2) ±2 F – 1 real spin–1 masseless gauge vector boson (nB = 2) Gravitational supermultiplet: • – 1 two–component Weyl gravitino–fermion with spin 3 (n = 2) ±2 F – 1 real spin–2 masseless graviton (nB = 2) eL Example: Ψ=(eR) with eL/R being 2–component Weyl left- and right fermions 1 Each spin-2 state has a complex spin–0 superpartner notede ˜L ande ˜R and called sfermion. One can write the fermion fields as e e ande ¯ = e† so that one has: ≡ L R two left-handed chiral supermultiplets for the electron: (e, e˜L), (¯e, e˜R∗ ). The same for all other leptons and quarks, except for the massless νL. 3 All the fields involved have the canonical kinetic energies with Lagrangians • µ µ 1 a µνa i =Σ (D S∗)(D S )+ iψ D γ ψ +Σ F F + λ Dλ Lkin i{ µ i i i µ i} a{−4 µν 2 a a} with D the usual covariant derivative. Note that the fields ψ(λ) have 4(2) components.] The interactions are specified by SUperSYmmetry and gauge invariance only: • a interactions scalar fermions gauginos = √2 i,a ga[Si∗T ψiLλa +h.c.] − − L −1 a 2 = ( g S∗T S ) Linteraction quartic scalar −2 aP i a i i All known interactions are determined by the gauge coupling constants g ,g and g • P P 1 2 3 (fundamental prediction of SUSY: the same coupling g in gauge and Yukawa interactions) At this stage, all this is a very simple and minimalistic theory/extension of the SM: • Everything in the theory is completely specified and ther is no adjustable parameter. The only freedom: choice of Superpotential W which should be SUSY and gauge invariant. It gives the scalar potential and the Yukawa interactions (among fermions-scalars). – W function of the superfields z only (and not of the conjugate fields z∗); ≡ i i – it should be an analytic function: there are no derivative interactions; – renormalizability: only terms of dimension 2 and 3 are present. ∂W 2 1 ∂2W W = ψ ψj +h.c. ⇒L − i | ∂zi | −2 ij iL∂zi∂zj To obtain all the interactionsP explicitly: makeP the evaluation of W at ∂W/∂z . i|zi=Si 4 The SUSY tree–level scalar potential has two distinct components Vtree = VF + VD. The F–terms come in W through derivatives with respect to all scalar fields Si: • i 2 i V = F F ∗ = W with W = ∂W/∂S . F i i i i | | i The D–terms correspond to contributions of the U(1), SU(2), SU(3) gauge groups: • P1 P 1 3 a 2 VD = 2 i DiDi∗ = 2 a=1 ( i gaSi∗T Si) . SUSY cannot be an exact symmetry of Nature since no scalars exist with the same mass as the known fermions (belongP to same superP multiplets)P SUSY must be broken. ⇒ Spontaneous SUSY breaking? This would mean that the Lagrangian is invariant under SUSY global transformations but the ground state 0> is not invariant: 0>= 0 and † 0>= 0. | Q| 6 Q | 6 µ However, recall that the Hamiltonian is related to the SUSY charges: , † P 0 {Q Q }∼ So that one has for the vacuum: < 0 H 0 > < 0 P 0 > < 0 † 0 >= E =0 | | ≡ | | ∝ |QQ | vac 6 In fact, the vacuum energy should always be positive: Evac > 0. 0 D 0 = 0 or D–term breaking: leads to CCB minima does not work! • h | | i 6 ⇒ 0 F 0 = 0 or F–term breaking: needs linear a Φ term in W requires SM singlet! • h | | i 6 i i ⇒ Solution: SUSY–breaking occurs in a hidden sector of particles which have no (or else very tiny) couplings to the visible sector of the theory. If the mediating interaction is flavor–blind, the SUSY-breaking terms are universal. Examples: gravity (mSUGRA), gauge (GMSB) and anomaly (AMSB) mediation... 5 There are many breaking schemes but none is fully satisfactory at the present moment: Explicit breaking by hand (but there are also several possibilities in this case). ⇒ We need SUSY breaking at low energies to solve the known SM problems: • – the quadratic divergences in the Higgs sector. – the unification of the coupling constants of SU(3) SU(2) U(1) . C × L × Y – the Dark Matter problem (the existence of a massive stable particle), etc. In the SUSY breaking process, we still need to preserve: gauge invariance, the renor- malizability,• and no quadratic divergence should appear (soft SUSY–breaking). “Low energy SUperSYmmetry” effective theory at low energy. ⇒ ≡ As discussed earlier, in SUSY theories, we have that: 1 – for each SM particle, there is a SUSY partner with has a spin 2 difference; – SUSY must be broken at a relatively low scale M = (1 TeV) SUSY O In order to solve the unification, hierachy and the dark matter problems of the SM. The MSSM is the most economic low energy SUSY extension of SM. 6 6.2 The minimal Supesymmetric Standard Model The MSSM is based on the following four basic assumptions: Minimal gauge group: GSM = SU(3)C SU(2)L U(1). • × × 1 The SM spin–1 gauge bosons [B, W1 3 and g1 8] and their spin– gaugino partners − − 2 [˜b, w˜1 3, g˜1 8] called binos, winos and gluinos are in vector superfields. − − Superfields SU(3)C SU(2)L U(1)Y Particle content Gˆa 8 1 0 Gµ, g˜ ˆ i µ W 1 3 0 Wi , ω˜i Bˆ 1 1 0 Bµ, ˜b Minimal particle content: • Superfield SU(3)C SU(2)L U(1)Y Particle content ˆ 1 ˜ Q 3 2 3 (uL,dL), (˜uL, dL) ˆ c 4 U 3 1 3 uR,u ˜R∗ ˆ c −2 ˜ D 3 1 3 dR, dR∗ Lˆ 1 2 1 (νL, eL), (˜νL, e˜L) ˆc − E 1 1 2 eR,e ˜R∗ Hˆ 1 2 1 (H , h˜ ) 1 − 1 1 Hˆ2 1 2 1(H2, h˜2) 7 – Three fermion generations [but as in the SM no νR...] and their spin–0 SUSY partners, the two sfermions f˜L, f˜R, combined in chiral supermultiplets. – No chiral anomalies ( Q 0) and fermion mass generation in a SUSY invariant way f f ≡ (no conjugate H∗ field for u–quarks) two chiral superfields with Y =+1 and Y = 1. P ⇒ − R–parity conservation: • To eliminate terms violating B and L quantum numbers (thus leading to proton decay): Introduce a discrete and multiplicative symmetry called R-parity or Rp defined as R = +1 for all ordinary SMparticles R =( 1)2s+3B+L p − ⇒ R = 1 for all the SUSY particles − The phenomenological consequences of Rp conservation are extremely important: – SUSY particles are always produced in pairs; – SUSY particles decay into an odd number of SUSY particles; – the lightest SUSY particle (LSP) is absolutely stable. At this stage, we have a globally supersymmetric Lagrangian in which: everything is specified by SUSY and gauge invariance; • there is no additional parameter compared to SM; • the only freedom is the choice of the Superpotential. • The most general Superpotential compatible with supersymmetry, gauge invariance, 8 renormalizability and R–parity conservation is given by: u i ˆ ˆj d ˆi ˆ ˆj l ˆi ˆ ˆj ˆ ˆ W = Yij uˆRH2.Q + Yij dRH1.Q + Yij lRH1.L + µH1.H2 i,j=gen X Y u,d,l denote the Yukawa couplings among the three generations of particles • ij (and which are simply a generalisation of the SM Yukawa interactions).