The Higgs particles in the MSSM Abdelhak DJOUADI Theory Division, CERN, CH–1211 Gen`eve 23, Switzerland. Laboratoire de Physique Math´ematique et Th´eorique, UMR5825–CNRS, Universit´e de Montpellier II, F–34095 Montpellier Cedex 5, France. Abstract In these lectures, I review the Higgs sector of the Minimal Supersymmetric extension of the Standard Model (MSSM). In a first part, I discuss the link between ultraviolet divergences and symmetries and briefly introduce supersymmetry and the MSSM. In section 2, the two–doublet MSSM Higgs sector will be presented and the Higgs masses and interactions will be derived. In a third part I will discuss the decay modes of the various Higgs bosons, their production at LEP2 and the main experimental constraints on the Higgs spectrum. In section 4, I discuss the prospects for discovering the Higgs particles at the upgraded Tevatron, the LHC and at a high–energy e+e− linear collider, and summarize the possibility of studying their properties. Extended write–up of the Lectures given at the ”Ecole de GIF 2001”, LAPP Annecy (France), 10–14 September 2001 1 1. Divergences, symmetries and Supersymmetry In this section, we discuss the connexion between ultraviolet divergences and symmetries and present the main argument for introducing low energy Supersymmetry in the description of electroweak interactions: the cancellation of quadratic divergences to the Higgs boson mass. In a first part, we will summarize the electroweak symmetry breaking in a simple abelian model [for more details see the lectures of R. Cahn [1] at this school] and present the basic rules for calculating Feynman diagrams [2], which we will need later. In a second part, and after giving the rules for the calculation of loop diagrams, we discuss the radiative corrections to the electron and photon self–energies in this model, and then analyze the radiative corrections to the Higgs boson mass in the presence of only standard particles and when additional scalar fields are introduced. We finally present the basic features of Supersymmetry and the minimal supersymmetric extension of the Standard Model [3]. 1.1 Lagrangians and interactions In standard QED, the Lagrangian for a fermion of charge e and mass m: 1 L = − F F µν + iψγµ∂ ψ − mψψ + eψγµA ψ (1.1) QED 4 µν µ µ with Aµ and Fµν = ∂µAν − ∂νAµ, the electromagnetic field and tensor. The U(1) gauge invariance, i.e. the invariance of the Lagrangian under the local transformations 1 ψ(x) → eiα(x)ψ(x) ,A(x) → A (x) − ∂ α(x) (1.2) µ µ e µ µ implies that there is no AµA term in the Lagrangian so that the photon is massless. Let us now consider a complex scalar field coupled to itself and to an electromagnetic field Aµ 1 L = − F F µν + D φ∗Dµφ − V (φ) (1.3) scal 4 µν µ with Dµ the covariant derivative Dµ = ∂µ − ieAµ and with the scalar potential V (φ)=µ2φ∗φ + λ (φ∗φ)2 (1.4) The Lagrangian Lscal is again invariant under the local U(1) transformation: 1 φ(x) → eiα(x)φ(x) ,A(x) → A (x) − ∂ α(x) (1.5) µ µ e µ 2 For µ > 0, Lscal is simply the QED Lagrangian for a charged scalar particle of mass µ and with φ4 self–interactions. For µ2 < 0, the field φ(x) will acquire a vacuum expectation value and the minimum of the potential V will be at µ2 1/2 v φ = − ≡ √ (1.6) 2λ 2 2 Expanding the Lagrangian around the vacuum state φ,oneobtains 1 φ(x)=√ [v + φ1(x)+iφ2(x)] (1.7) 2 The Lagrangian becomes then (up to some interaction terms that we omit for simplicity): 1 L = − F F µν +(∂µ + ieAµ)φ∗(∂ − ieA )φ − µ2φ∗φ − λ(φ∗φ)2 4 µν µ µ 1 1 1 1 = − F F µν + (∂ φ )2 + (∂ φ )2 − v2λφ2 + e2v2A Aµ − evA ∂µφ (1.8) 4 µν 2 µ 1 2 µ 2 1 2 µ µ 2 Three remarks can then be made here: (i) There is a photon mass term in the Lagrangian, 1 2 µ − 2 2 MAAµA with MA = ev = eµ /λ. (ii) We still have a scalar particle φ1 with a mass 2 − 2 Mφ1 = 2µ . (iii) Apparently, we have a massless particle φ2, a would–be Goldstone boson. However, there is a problem. In the beginning, we had four degrees of freedom in the theory (two for the complex scalar field φ and two for the massless electromagnetic field Aµ) and now we have apparently five degrees of freedom (one for φ1,oneforφ2 and three for the massive photon Aµ). Therefore, there must be a field which is not physical at the end and indeed, in Lscal there is a bilinear term evAµ∂φ2, and we have to get rid of it. To do so, we notice that at first order, we have for the original field φ: 1 1 iφ2(x)/v φ = √ (v + φ1 + iφ2) ≡ √ (v + φ1)e (1.9) 2 2 By using the freedom of gauge transformations and by performing also the transformation, µ Aµ → Aµ − (1/ev)∂µφ2(x), the Aµ∂ φ2 term (and all φ2 terms) disappear in the Lagrangian. This is called the unitary gauge. The moral of the story is then, that the photon (two degrees of freedom) has absorbed the would–be Goldstone boson (one degree of freedom) and became massive (i.e. with three degrees of freedom): the longitudinal polarization is the Goldstone boson. The U(1) gauge symmetry is no more apparent and we say that it is spontaneously broken. This is the Higgs mechanism, with φ1 ≡ H being the Higgs field. Let us, in addition, couple this field to a fermion f (`alaYukawa): Lf = −λf . ψψφ (1.10) √ √ After SSB, i.e. φ → (v + H)/ 2, the fermion acquires a mass mf = λf v/ 2. Finally, we introduce two scalar fields φ1 and φ2 with a kinetic Lagrangian L | |2 | |2 − 2| |2 − 2| |2 kin = ∂µφ1 + ∂µφ2 m1 φ1 m2 φ2 (1.11) and with a part which describes the couplings to the scalar field φ after SSB (plus, eventually, terms in φ1φ2 that one can omit for simplicity) 2 2 2 2 2 LS = −λS|H| (|φ1| + |φ2| ) − 2vλSH(|φ1| + |φ2| ) (1.12) 3 Let us now summarize the basic rules for calculating Feynman diagrams, which we will need in the course of these lectures. • For each external (anti)–fermion with momentum p and spin s, we associate a spinor. The rules for the spinors and the propagators are summarized in the figure below. us(pi) us(pf ) pi pf p − i p +m −p¯ p¯i p −m = i p2−m2 f vs(−pi) vs(−pf ) The sum over the spins for the uu¯ and vv¯ combinations of the spinors gives − Σs us(p)¯us(p)=p + m,Σs vs(p)¯vs(p)=p m • For an external gauge boson Vµ with four–momentum q, one associates a polarization vector µ(q). The rules for the polarization vectors and for the internal propagators (in the unitary gauge) are shown below. 2 2 gµν −qµqµ/(k −MV ) ∗ µ(q) −i 2 2 (q) q −MV ν qµ ••qν q with the sum over the polarization given by ∗ − − 2 = ν µ = (gµν qµqµ/MV ) pol For the photon, one should discard the longitudinal components (qµqν) in the expressions µ above. Note that the transversality of the photon implies: µ · q =0. • For a scalar particle, one simply needs the propagator given by i/(q2 − M 2) q S • The Feynman rule for a given vertex is obtained by multiplying the term involving the interaction in the Lagrangian by a factor −i. One needs also to multiply by a factor n!where n is the number of identical particles in the vertex. In the case of the gauge boson couplings to fermions one has: − f ieγµ(vf af γ5) Vµ 3 2 3 Z : vf =(2I − 4ef s )/(4sW cW ) ,af =2I /(4sW cW ) f W √ f W : vf = af =1/(2 2sW ) f¯ γ : vf = ef ,af =0 4 The rules for Higgs bosons couplings to fermions and gauge bosons are given by f Vµ H H 2 · imf /v iMV /v gµν f¯ Vν while the trilinear and quadrilinear Higgs couplings to scalars are: S H S H ivλS iλS S∗ H S∗ • The basic relations for the treatment of the Dirac matrices are: µ {γµ ,γν} = γµγν + γνγµ =2gµν and p = pµγ (1.13) µ ν ρ σ 0 1 2 3 { } 2 γ5 =(i/4!) µνρσγ γ γ γ = iγ γ γ γ and γµ,γ5 =0,γ5 =1 Tr(1)=4, Tr(γµ)=0, Tr(γ5)=0 Tr(A1A2)=Tr(A2A1) , Tr(A1A2 ···AN )=Tr(A2 ···AN A1) For the contractions of the γ matrices, the rules are: µ µ ν µ γ γµ =2gµ + γµγ = gµ = 4 (1.14) µ µ γ γνγµ = γ (2gµν − γµγν)=2γν − 4γν)=−2γν µ ν ρ µν − ν µ ρ − ρ νρ − ν ρ − ν ρ ν ρ νρ γ γ γ γµ =(2g γ γ )(2gµ γµγ )=4g 2γ γ 2γ γ +4γ γ =4g while for the traces of γ matrices, one has Tr(γµγν)=Tr(2gµν − γµγν)=2gµν Tr(1) − γµγν ⇒ Tr(γµγν)=4gµν (1.15) Tr(γµ1 ···γµn )=Tr(γµ1 ···γµ2n+1 γ5γ5)=(−1) Tr(γµ1 ···γ5γµ2n+1 γ5) =(−1)2n+1 Tr(γ5γµ1 ···γµ2n+1 γ5)=+Tr(γ5γµ1 ···γµ2n+1 γ5)=0 µ µ ν σ µ1 µ2n+1 5 Tr(γ γ5)=Tr(γ γ γ γ5)=Tr(γ ···γ γ )=0 1 Tr(γµγνγ )= Tr(γαγ γµγν γ )=(1/4) Tr(γ γµγν γ γα) 5 4 α 5 α 5 µ ν α µ ν = −(1/4) Tr(γαγ γ γ γ5)=−Tr(γ γ γ5)=0 Using the same tricks as above, one can proof that the traces of 4 γ matrices are Tr(γµγν γργσ)=4(gµνgρσ − gµσgνρ + gµρgνσ) (1.16) µ ν ρ σ µνσρ Tr(γ γ γ γ γ5)=−4i 5 1.2 Loop calculations and the self–energy divergences 1.2.1 Rules for the calculation of loop integrals Let us first summarize the rules for the calculation of loop integrals, taking the simple example of a two point function with internal scalar particles and external Higgs bosons with momentum p and an HSS∗ coupling given by (ig).