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Clases de Master FisyMat, Desarrollos Actuales The of 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 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 and . The SUSY generators transform fermions into bosons and vice–versa, namely: FermionQ > > , Boson > > 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 ...

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 . 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 (nB = 2) Gravitational supermultiplet: • – 1 two–component Weyl –fermion with spin 3 (n = 2) ±2 F – 1 real spin–2 masseless (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 notede ˜L ande ˜R and called . One can write the fermion fields as e e ande ¯ = e† so that one has: ≡ L R two left-handed chiral supermultiplets for the : (e, e˜L), (¯e, e˜R∗ ). The same for all other and , 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 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 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 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 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 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 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). µ: a supersymmetric Higgs– parameter with a dimension of mass • (it is thus a really supersymmetric parameter, see discussion later....). Now, we have to introduce soft SUSY breaking: • To explicitely break SUSY without reintroducing the quadratic divergences (the so–called soft SUSY–breaking), we add by hand a collection of soft terms (of dimension 2 and 3): 1 = M ˜b˜b + M Σ3 w˜aw˜ + M Σ8 g˜ag˜ + h.c. Lgaugino 2 1 2 a=1 a 3 a=1 a 2 h 2 2 2 2 2 2 i 2 =Σ m Q˜†Q˜ + m L˜†L˜ + m u˜ + m d˜ + m l˜ Lsfermions i Q,i˜ i i L,i˜ i i u,i˜ | Ri| d,i˜ | Ri| ˜l,i| Ri| 2 2 = m H†H + m H†H + Bµ(H .H +h.c.) LHiggs 2 2 2 1 1 1 2 1 =Σ Au Y uu˜ H .Q˜ +Ad Y dd˜ H .Q˜ +Al Y l l˜ H .L˜ +h.c. Ltrilinear i,j ij ij Ri 2 j ij ij Ri 1 j ij ij Ri 1 j h i

9 This is a rather complicated and problematic potential indeed! It has too many parameters and thus it is not very predictive. • It leads generically to a very problematic phenomenology. • In the most general case (mixing and phases allowed): 105 free parameters! complex gaugino masses M , M , M : 6 • 1 2 3 3 3 hermitian mass matrices m ˜ : 45 • × F 3 3 complex trilinear coupling matrices A : 54 • × f 2 2 matrix for the bilinear B coupling : 4 • × Higgs masses squared, m2 , m2 : 2 • H1 H2 111–6 (due to constraints from symmetries and Higgs sector, see later)= 105. For “generic” sets of these parameters, it leads to very severe problems: large and dangerous/unobserved flavor changing neutral currents [FCNC]; • an unacceptable amount of additional CP–violation; • color and/or charge breaking minima that break SU(3) and U(1) symmetries; • it leads to an incorrect value of the Z boson mass (predictable), etc...... • We need more constraints on this MSSM!

10 6.3 The constrained MSSM’s A phenomenologically viable MSSM can be defined by making following assumptions: all soft SUSY–breaking parameters are real (no new source of CP violation); • the mass and trilinear couplings for sfermions are diagonal (no FCNC); • there is a 1st/2d sfermion generation universality (no problems with heavy quarks); • We define the phenomenological MSSM (pMSSM) with 22 free parameters: tan β: the ratio of the vevs of the two–Higgs doublet fields. m2 , m2 : the Higgs mass parameters squared. Hu Hd M1, M2, M3: the bino, wino and mass parameters. m , m , m ˜ , m˜, m : 1st/2d generation sfermion mass para. q˜ u˜R dR l e˜R m ˜, m˜ , m˜ , m˜, m : third generation sfermion mass para. Q tR bR L τ˜R At, Ab, Aτ : the third generation trilinear couplings. Au, Ad, Ae: the first/second generation trilinear couplings. 2 2 You can trade the masses m , m with the more ”physical” parameters µ and MA • Hu Hd (in fact: µ2 and Bµ can be determined from requirement of ESWB, as seen later).

The parameters Au, Ad, Ae are in general not relevant for phenomenology (they• enter only in ”light” flavor physics: (g 2) , edm, ....). − µ If you focus on a given particle sector (Higgs, gauginos, sfermions): only• few parameters to deal with and one can do model independent analyses.... a phenomenologically more viable model than the general MSSM. ⇒ 11 One can also use common soft–SUSY breaking terms in many concrete cases • (m = m = m ˜ ; m ˜, m˜ , m˜ ; A , A , A ; etc..) q˜ u˜R dR Q tR bR t b τ and one ends with an even more restrictive set of parameters, possibly < 10. much more predictive model than the general MSSM. ∼ ⇒ But almost all problems of the MSSM can be solved at once if soft SUSY–breaking parameters obey a set of universal boundary conditions at the scale MGUT. Underlying assumption: SUSY–breaking occurs in a hidden sector communicating with the visible sector only through gravitational interactions. some universal soft-SUSY breaking terms emerge if interactions are “flavor–blind”: ⇒ Besides g unification which fix the unification scale M 2 1016 GeV 1,2,3 GUT ∼ · Unification of gaugino, scalar masses and trilinear couplings at Q = MGUT Universal gaugino masses: M = M = M m 1 2 3 ≡ 1/2 Universal scalar masses: M ˜ = M˜ = MH m0 Qi Li i ≡ Universal trilinear couplings: Au = Ad = Al A δ ij ij ij ≡ 0 ij 2 Also: B and µ obtained from requiring EWSB and minimization of VHiggs µ2 = 1[tan2β(m2 tan β m2 cot β) M 2 ] 2 Hu − Hd − Z Bµ = 1 sin2β[m2 + m2 +2µ2] 2 Hu Hd Only 4.5 param: tan β , m1/2 , m0 , A0 , sign(µ)

12 All soft-SUSY breaking parameters at MS are then obtained through RGEs. 16 With MGUT 2 10 GeV and the SUSY scale defined as MSUSY √mt˜ mt˜ : ∼ · ∼ L R Evolution of sparticle masses 800

700 M3 ~ ~ mb ,Q 600 R L

m~t 500 R

m1 400 ___ 2 µ2 \/m 0 + mass (GeV) m 300 2 m M2 1/2 200 m~τ L m~τ R 100 m0 M1 0 2 4 6 10 17 10 10 10 10 10 Q (GeV) Radiative EWSB occurs since one gets M 2 <0 at the scale Q=v (from t/t˜ loops) H2 EWSB is more natural in the MSSM (µ2 < 0 from RGEs) than in the SM! ⇒

13 Another possibility for a very constrained MSSM is gauge mediated SUSY breaking, GMSB: soft breaking transmitted to MSSM fields via SM gauge interactions. The hidden sector for SUSY–breaking contains messengers fields, n /n /- • qˆ ˆl like pairs coupled to a gauge singlet chiral superfield Sˆ. The superpotential is W = λSˆqˆq¯ˆ + λSˆlˆl¯ˆ with fields Sˆ havings vevs. s and f • S soft SUSY-breaking are generated by (1 or 2) loop corrections at the scale M = λs • mes α (Mmes) Λ G M (M )= G Λ g( ) Σ N (m) G mes 4π Mmes m R m2(M )=2Λ2f( Λ ) Σ (αG(Mmes))2N G(m)CG(s) s mes Mmes m,G 4π R R A (M ) 0 (generated at the two–loop level). f mes ≃ with Λ = fs/s, G = U(1), SU(2), SU(3), m and s label messengers and scalars; f/g are one/two loop functions; N/C are Dynkin indices / Casimirs etc.... Thus, in the GMSB model there are six basic input parameters to start with

tanβ , sign(µ) , Mmes , Λ , nqˆ , nˆl and all soft-SUSY breaking terms are then obtained from the previous equations. NB: there is also the mass of the gravitino which is an independent/input parameter (and it playes a rather important role: it is very light, LSP, and make DM).

14 A third possibility for a very constrained MSSM is anomaly mediated SUSY breaking. In AMSB, SUSY breaking occurs also in a hidden sector (e.g. in an extra dimension) and is transmitted to the visible sector via, for instance, some super–Weyl anomalies. The gaugino masses, the scalar masses and the trilinear couplings are simply related to the scale dependence of the gauge and matter kinetic functions.

In terms of the gravitino mass m3/2, the β functions for the gauge ga and Yukawa Yi couplings and anomalous dimensions γi of chiral superfields, soft-SUSY breaking terms are: β βga Yi Ma = m , Ai = m ga 3/2 Yi 3/2 2 1 ∂γi ∂γi 2 m = (Σa βg +Σk βY )m i −4 ∂ga a ∂Yk k 3/2 RG invariant equations valid at any scale (which makes the model rather predictive). (the µ2 and Bµ terms are obtained as usual by requiring radiative EWSB). However, the picture is spoiled by tachyonic sleptons m2 < 0 in general! L˜ add a non anomalous contribution to soft masses c m2 to m2 to avodi that. ⇒ i 0 i In minimal AMSB with a universal m0, and coefficients ci = 1, the inputs are: m0 , m3/2 , tan β , sign(µ) and ci Makes AMSB also a rather predictive model.

15 6.4 The superparticle spectrum The basic Lagrangian, gives the currents, interactions and the corresponding states: but we need to turn the current eigenstates into the mass (physical) eigenstates. Terminology for the physical eigenstates in supersymmetric theories: The : the mixtures of the charged and the gauginos • W˜ ±, h˜± χ±,χ± 2/1 −→ 1 2 The : the mixtures of the neutral higgsinos and the gauginos • B,˜ W˜ 0, h˜0, h˜0, χ˜0, χ˜0, χ˜0, χ˜0 2 2 −→ 1 2 2 4 (for gluinos, there is no need of mixing, but include running and the hence the RC).. The sfermions: the mixed left- and right-handed sfermion states of same flavor • f˜ , f˜ f˜ , f˜ L R −→ 1 2 (mixing proportional to fermion mass: absent for sneutrinos and small for most sfermions). The Higgs bosons: two doublets of complex scalar fields H ,H 4 dofs: • 1 2 ≡ 3 degrees of freedom go to generate M +, M , M as usual; W W − Z ⇒ + 5 degrees of freedom are left in the spectrum: h,H,A,H ,H−. ⇒ To determine the masses and the mixing angles of all the mass (physical) eigenstates: one needs to find the relevant mass matrices and diagonalize them; better include RC?

16 The general mass matrix, in terms of M2, µ and tan β, is M √2M s = 2 W β , s sin β etc MC √2M c µ β ≡  W β  1 + if det C > 0 diagonalized by: U CV − U = , V = O M M → O− { σ if det < 0 3O+ MC (Pauli σ3 matrix to make the χ± masses positive and are rotation matrices) O± Simple analytical formulae for the masses mχ±1,2 and mixing angles in terms of M2, µ. For limiting cases, interpretation much simpler. µ M2, MW : 2 2 ≫ m M2 M µ− (M2 + µs2β) χ1± ≃ − W 2 2 m µ + M µ− ǫµ (M2s2β + µ) χ2± ≃| | W IN the limit µ : χ1± wino with m M2; χ2± higgsino with m = µ | | → ∞ χ1± ≃ χ2± | | In the opposite limit, M µ , M , the roles of χ±,χ± are simply reversed. 2 ≫| | Z 1 2 For neutralinos, the 4 4 mass matrix depends on parameters µ, M , tan β, M . × 2 1 ˜ ˜ ˜ 0 ˜ 0 In the ( iB, iW3, H1 , H2 ) basis, it is given by − − M1 0 MZsW cβ MZsW sβ 0 M2 −MZcW cβ MZcW sβ N = − M MZsW cβ MZcW cβ 0 µ  −M s s M c s µ −0  Z W β − Z W β − and can be diagonalized by a single real matrix Z. Again for µ M M : | | ≫ 1,2 ≫ Z 17 2 MZ 2 m 0 M1 2 (M1 + µs2β) s χ1 ≃ − µ W 2 MZ 2 m 0 M2 2 (M2 + µs2β) c χ2 ≃ − µ W 2 1 MZ 2 2 mχ0 µ + 2 ǫµ(1 s2β) µ M2sW M1cW 3/4 ≃| | 2 µ ∓ ± ∓ In the limit µ , χ0 is bino (M ), χ0 wino (M ) and χ0,χ0 higgsinos (µ). | | → ∞ 1 1 2 2 3 4  In the opposite limit, M , M , the higgsino/wino roles are again reversed. 1 2 → ∞ Finally, the gluino mass is identified with the gaugino mass parameter M3 at tree–level mg˜ = M3 In constrained models with boundary conditions at the high energy scale MU , the evolution of the gaugino masses given by RGEs 2 dMi gi Mi 33 2 = 2 bi , b1 = , b2 =1 ,b3 = 3 dlog(MU/Q ) − 16π 5 − where in bi all sparticles contribute to the evolution from Q to MU . RGEs related to 2 those of the gauge couplings αi = gi /(4π). With inputs at scale MZ and common value at M 2 1016 GeV, one has for gaugino mass parameters at the SUSY scale M : U ∼ × S M : M : M α : α : α 6:2:1 3 2 1 ∼ 3 2 1 ∼ With normalization factor 5 in α , we have the GUT relation M = 5 tan2 θ M 1M . 3 1 1 3 W 2 ≃ 2 2 µ M2 mχ0 m 2mχ0 M2, mχ0 mχ0 m µ. ≫ ⇒ 2 ∼ χ1± ∼ 1 ∼ 3 ∼ 4 ∼ χ2± ∼ µ M2 m 0 m m 0 µ, m 0 2m 0 m M2. ≪ ⇒ χ2 ∼ χ1± ∼ χ1 ∼ χ4 ∼ χ3 ∼ χ2± ∼

18 The sfermion system is described by tan β, µ and 3 parameters for each species: m ˜ , m ˜ , A . fL fR f For 3d generation, a mixing mf has to be included. The sfermion mass matrices are: ∝ 2 2 2 mf + mLL mf Xf ˜ = 2 2 Mf mf Xf m + m  f RR  with the various entries given by 2 2 3L 2 2 mLL = m ˜ +(If Qf sW ) MZ c2β fL 2 2 2− 2 mRR = m ˜ + Qf sW MZ c2β fR 2I3L X = A µ(tan β)− f f f − They are diagonalized by 2 2 rotation matrices of angle θ , which turn the current × f eigenstates f˜L, f˜R into the mass eigenstates f˜1, f˜2. 2 2 1 2 2 2 2 2 2 2 m ˜ = mf + mLL + mRR (mLL mRR) +4mf Xf f1,2 2 ∓ − Note: mixing very strongh in the stop sector,q X = A µ cot β and generatesi mass • t t − splitting between t˜1, t˜2, leading to a light t˜1 state The mixing in the sbottom sector can also be strong for large X =A µ tan β. • b b− The same for the stau system andτ ˜ is in general the lightest sfermion at high tan β! • 1 In cMSSMs with universal m0 and m1/2 values at MGUT, the RGEs for the scalar masses are simple if the Yukawas are small (c(f˜) depend on I, Y, color):

19 2 2 m2 = m2 + 3 F (f)m2 , F = ci(f) 1 1 αU b log Q − ˜ 0 i=1 i 1/2 i b 4π i M2 fL,R i − − U 1   3 P 6 8 2  10 30 15 15 ˜ 3 ˜ 5 ˜ 3 ˜ L : , lR : 0 , Q : 2 , u˜R : 0 , dR : 0 2 ! 0 !  8  8 ! 8 ! 0 3 3 3 16 With the input values at Q = MZ, αU 0.041 and MU 2 10 GeV, one has 2 2 2 2 2≃ 2 2∼ × 2 2 mq˜ m0 +6m1/2 , m˜ m0 +0.52m1/2 , me˜ m0 +0.15m1/2 i ∼ ℓL ∼ R ∼ For the third generation squarks, the large Yukawa couplings have to be included. For instance, the approximate RGEs for top squarks [for small tan β values] are: 2 2 2 2 1 m˜ = m˜ m0 +6m Xt tL bL 1/2 3 2 2 ∼ 2 2 − 2 m˜ = m˜ m0 +6m1/2 Xt tR bL ∼ − 3 with the trilinear coupling given by the approcimate value X 1.3m2 +3m2 . t ∼ 0 1/2 In contrast to the first two generation sfermions, one has generically a sizable splitting between⇒ m2 , m2 at the weak scale, due to the running of the large top Yukawa coupling. t˜L t˜R Justifies the choice of different soft SUSY–breaking scalar masses and trilinear couplings for⇒ third generation and first/second generation sfermions [and for sleptons and squarks]. 2 Recal that already from mixing, m˜ lighter than all the other squarks • t1 Special status for the top squark.... ⇒

20 6.5 The Higgs sector of the MSSM 0 + H1 H2 In the MSSM, we need two Higgs doublet fields H1 = and H2 = 0 . H1− H2 The terms contributing to the scalar potential V come from three sources:  1 a 2 D terms, quartic S interactions: V = ( g S∗T S ) • D 2 a i a i i F terms of Superpotential: V = ∂W (z )/∂z 2 ∂W (φ )/∂φ 2 • F i | P iP i| → i | j i| 2 2 Soft terms: V = m H†H + m H†H + Bµ(H .H +h.c.) • soft 1 1 1 2P2 2 2 1 P Adding all terms the scalar potential involving the Higgs bosons is ⇒ 2 2 2 2 2 i j VH =m ¯ 1 H1 +m ¯ 2 H2 m¯ 3ǫij(H1H2 +h.c.) 2 | 2| | | − g2 + g1 2 2 2 1 2 2 + ( H H ) + g H∗H 8 | 1| −| 2| 2 2| 1 2| 2 2 2 2 2 2 2 with m1 = µ + m1, m2 = µ + m2, m3 = Bµ | | | 0| + 0 Development in terms of components H =(H ,H−),H =(H ,H ) • 1 1 1 2 2 2 2 1 2 + 2 2 0 2 2 2 + 0 0 VH = m1( H0 + H1 )+m2( H2 + H2− ) m3(H1 H2− H1 H2 +h.c.) 2 | 2| | | | | | | − 2 − g2 + g1 0 2 + 2 0 2 2 2 g2 + 0 0 2 + ( H + H H H− ) + H ∗H + H ∗H− 8 | 1 | | 1 | −| 2 | −| 2 | 2 | 1 1 2 2 | Now require that the minimum of VH breaks the symmetry GSM U(1)QED. min + + • So at V we have H =0 and at ∂V /∂H = 0 we have H→− =0; good for QED. h H h 1 i 1 h 2 i

21 + Ignoring the fields H1 ,H2− to simplify, the relevant part of potential VH is then simply: V = m2 H0 2 + m2 H0 2 + m2(H0H0 +hc)+(g2 + g2)( H0 2 H0 2)2/8 H 1| 1 | 2| 2 | 3 1 2 2 1 | 1 | −| 2 | Some important remarks can be made on this scalar potential: 2 V = m2 H0 2 + m2 H0 2 + m2(H0H0 +hc)+ MZ ( H0 2 H0 2)2 H 1| 1 | 2| 2 | 3 1 2 4v2 | 1 | −| 2 | Quartic couplings fixed in terms of the gauge couplings, only 3 free parameters: •2 2 2 m1, m2, m3 (6 parameters and a phase in a general 2HDM). 2 2 m1,2 + µ real, only Bµ can be complex. But any phase in Bµ can be absorbed in phases• of H| ,H| V (MSSM) conserves CP. 1 2 ⇒ H If Bµ is zero, all other terms are positive and thus V = 0 only if H0 = H0 = 0. • H h 1 i h 2 i To have SSB (without CCB), we need m1,2,3 =0 Connection of gauge symmetry6 breaking and SUSY breaking!! ⇒ More precisely: in SM, symmetry breaking takes place with ad hoc choice µ2 < 0. In the MSSM, m2 > 0 at Q = M but t/t˜ in RGEs make m2 < 0 at Q = M : Hi GUT Hi Z radiative breaking of the electroweak symmetry (i.e. through radiative corrections). Symmetry breaking more natural and elegant than in SM. ⇒ To obtain the physical Higgs fields and their masses from potential VH, develop 0 + 0 H1 =(H1 ,H1−) and H2 =(H2 ,H2 ) into real (CP-even+charged Higgses) and imaginary (CP-odd Higgs+Goldstone bosons) parts and diagonalize the 2 2 mass matrices: 2 1 2 × ij = 2∂ VH/∂Hi∂Hj Re(H0 ) =v , Im(H0 ) =0, H =0 M |h 1,2 i 1,2 h 1,2 i h 1±,2i 22 To obtain masses M1, M2 and the mixing angle θ, two useful relations are: 2 2 2 2 2 2 Tr( )= M1 + M2 , Det( )= M1 M2 M 2 M sin2θ = M12 , cos2θ = M11−M22 √( )2+4 2 √( )2+4 2 M11−M22 M12 M11−M22 M12 First note that if you perform the first derivative of the scalar potential VH: 2 V =m2 H0 2+m2 H0 2+m2(H0H0+hc)+ MZ ( H0 2 H0 2)2 H 1| 1 | 2| 2 | 3 1 2 4v2 | 1 | −| 2 | you have, at the minimum, ∂VH/∂H1,2 = 0, leading to the two relations: m¯ 2 = m¯ 2 tan β 1M 2 cos(2β) , m¯ 2 = m¯ 2cotβ + 1M 2 cos(2β) 1 − 3 − 2 Z 2 − 3 2 Z The second derivative of VH gives you the relevant mass matrices: 2 2 2 2 2 2 m¯ 3 tan β + MZ cos β m¯ 3 MZ sin β cos β = − 2 2 2− 2 2 MR m¯ M sin β cos β m¯ cotβ + M sin β  3 Z − 3 Z  2 2 2 = m¯ 3 tan β m¯ 3 MI − m¯ 2 m¯ 2cotβ  3 − 3  2 For the CP–odd case, since Det I = 0, one eigenvalue is zero (the ) and the other one corresponds to theM CP-odd Higgs (A boson) with a mass: M 2 = m¯ 2(tan β + cotβ)= 2¯m2/ sin2β A − 3 − 3 The mixing angle θ is, in fact, just the angle β: 0 0 G cos β sin β Im(H1 ) = 0 A sin β cos β Im(H2 )    −   

23 In the case of the CP–even Higgs bosons, the determinant and trace give: Det 2 =M 2M 2 c2 M 2M 2 MR A Z 2β ≡ h H Tr 2 =M 2 + M 2 M 2 + M 2 MR A Z ≡ h H To obtain the CP–even Higgs masses, solve the second order equation: M 2(M 2 +M 2 M 2)=M 2M 2 c2 M 4 M 2(M 2 +M 2 )+M 2M 2 c2 =0 h A Z − h A Z 2β ⇒ h − h A Z A Z 2β The two solutions are then (h is the lightest CP-even ): M 2 = 1 M 2 + M 2 (M 2 + M 2 )2 4M 2M 2 cos2 2β h,H 2 A Z ∓ A Z − A Z h p π i The mixing angle α which rotates the fields is given by (and obeys 2 α 0) 2 2 2 −2 ≤ ≤ 2 12 (MA+MZ)sin2β MA+MZ tan2α = M = − 2 2 = tan2β 2 2 11 22 (M M )cos2β M M M −M Z− A A− Z In the case of the charged Higgs bosons, one obtains similarly to the A boson case: M 2 = M 2 + M 2 H± A W and the mixing angle θ is, also as for the A boson, simply the angle β. In then MSM, we have an important constraint on the lightest MSSM h boson mass: M min(M , M ) cos2β M h ≤ A Z ·| |≤ Z besides some other (also important) relations for the H, A and H± bosons: M > max(M , M ) and M > M H A Z H± W If we send MA to infinity, we will have for the Higgs masses and the angle α: M M cos2β , M M M , α π β h ∼ Z| | H ∼ H± ∼ A ∼ 2 −

24 This is the decoupling regime: all the Higgs bosons are heavy except for h.

The lightest h boson should be lighter than MZ and should have been seen at LEP2 (as we had √s 200 GeV masses M + M 180 GeV were accesible) LEP2 ∼ ⇒ h Z ∼ So what happened in this case? Maybe the MSSM was already ruled out at LEP2? No! This relation holds only at first order (tree–level) and there are strong couplings involved in the theory, in particular the htt and ht˜t˜ couplings of the top/stop quarks. The calculation of radiative corrections to M necessary. ⇒ h More generally, radiative corrections very important in the MSSM Higgs sector. A large activity for the calculation of radiative corrections in the last 30 years. The dominant corrections are due to top and stop quarks at one-loop level • (they can be calculated easily using the material given before for top/stop loops). 2 4 2 2 3g mt mt˜ ∆Mh = 2 2 log 2 2π MW mt It depends on m4 (quadratic) and log(m2/m2); thus large: M max M +40 GeV. t t˜ t h → Z This explains why the h boson has not been observed at LEP2; heavier than 90 GeV. The full one–loop corrections have been calculated: • – other important parameters such as µ, At and Ab appear at the subleading level. – the h boson mass is maximal (minimal) for a stop mixing parameter A 2M ˜(0). t ∼ Q 25 Approximate calculation for the dominant two–loop radiative corrections (in the effective •potential approach where momenta transfert small compared to mass of internal particles): – dominant QCD corrections large but are absorbed by making m pole m MS. t| → t| – the Yukawa corrections are rather small in the limit Mh = 0. Using full 1–loop and the 2–loop corrections in the effective potential approach: • – (α α ): including squark mixing and the gluino loops. O t S – (α2): including mixing and (α α ), (α α ). O t O b S O τ S

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5¼ 5¼ 6.6 Extensions of the MSSM There are several extensions of the MSSM which can be obtained when departing from the minimality assuptions (particle content, R-parity, couplings...) R-parity violating MSSM: To avoid fast , we do not need that both L and B numbers are conserved.

In the most general superpotential W , one can include ∆L=1 or ∆B=1 interactions: 1 ¯ W∆L=1 = 2λijkLiLje¯k + λijk′ LiQjdk + µi′LiHu 1 ¯ ¯ W∆B=1 = 2λijk′′ u¯idjdk Proton decay modes and experimental limits on /B and /L imply that λ′,” 1. ijk ≪ However, one introduces at least 45 new parameters in the general case... • there will be no stable LSP and thus no SUSY dark matter candidate... • But a rich phenomenology is possible (e.g. s channel sfermion production) • if it enters in phenomenology, it adresses the problem of small ν masses •

27 CP-violating MSSM One can allow for some CP–violating soft SUSY breaking parameters, in particular:

complex gaugino masses M1, M2, M3 (some phases but not all can be rotated away) and• higgsino mass parameter µ complex trilinear A couplings, in particular in the third generation sfermion scetor • f and hence At, Ab, Aτ . The MSSM Higgs sector stays CP–conserving at the tree–level (as seen before) but the complex parameters will enter at the one–loop level e.g. through complex µ and At. CP violation is needed for (direct) baryogenesis in the MSSM. • However, many new additional parameters will enter in the general case • Complicates the determination of the spectrum but leads to less fine-tunning. • Strongly constrained by data (n ..) and needs delicate (unatural?) cancelations. • edm No sign yet of any additionnal form of CP/ in B–factories etc... • One can also allow for flavor non–diagonal interactions, however: Parameters strongly constrained from FCNC, + B- physics... • Only adds complications/parameters (there is no theory motivation)... •

28 The NMSSM The µ problem: the parameter µ enters EWSB and the determination of MZ.

It must be of the order of other SUSY–breaking parameters such as mH1, mH2. But µ is a SUSY preserving parameter, comes from a superpotiential term W µHˆ Hˆ , ∝ 1 2 and, a priori, there is no reason for having µ M , M M .... ∝ Z SUSY ≪ GUT The solution is: µ is related to a vev of an additional field S with S = s h i The NMSSM: one introduces a gauge singlet superfield Sˆ into the superpotential 1 W = W + λHˆ Hˆ Sˆ + κSˆ MSSM 1 2 3 THe NMMSM spectrum is extended compared to the minimal one of the MSSM: 0 one additional state: χ1,...,5 • ⇒ + two additional Higgs particles H ,H ,H , A , A ,H ,H− • ⇒ 1 2 3 1 2 it is a less constrained and fine tuned model, with a richer phenomenology... ⇒ NMSSM MSSM Example: the upper bound on h boson mass is Mh = Mh + 20–40 GeV. LEP searches bounds are not valid as h can be heavier than 100 GeV.

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