Supersymmetry for Alp Hikers

These lectures provide a phenomenological introduction to supersymmetry, concentrating on the minimal supersymmetric extension of the Standard Model (MSSM). In the first lecture, motivations are provided for thinking that supersymmetry might appear at the TeV scale, including the naturalness of the mass hierarchy, gauge unification and the probable mass of the Higgs boson. In the second lecture, simple globally supersymmetric field theories are introduced, with the emphasis on features important for model-building. Supersymmetry breaking and local supersymmetry (supergravity) are introduced in the third lecture, and the structure of sparticle mass matrices and mixing are reviewed. Finally, the available experimental and cosmological constraints on MSSM parameters are discussed and combined in the fourth lecture, and the prospects for discovering supersymmetry in future experiments are previewed. 1. GETTING MOTIVATED 1.1 Defects of the Standard Model The Standard Model agrees with all confirmed experimental data from accelerators, but is theoretically very unsatisfactory. It does not explain the particle quantum numbers, such as the electric charge Q, weak isospin I , hypercharge Y and colour, and contains at least 19 arbitrary parameters. These include three independent gauge couplings and a possible CP-violating strong-interaction parameter, six quark and three charged-lepton masses, three generalized Cabibbo weak mixing angles and the CP-violating Kobayashi-Maskawa phase, as well as two independent masses for weak bosons. As if 19 parameters were insufficient to appall you, at least nine more parameters must be introduced to accommodate neutrino oscillations: three neutrino masses, three real mixing angles, and three CP-violating phases, of which one is in principle observable in neutrino-oscillation experiments and the other two in neutrinoless double-beta decay experiments. Even more parameters would be needed to generate neutrino masses in a credible way, associated with a heavy-neutrino sector and/or additional Higgs particles. Eventually, one would like to include gravity in a unified theory along with the other particle interactions, which involves introducing at least two more parameters, Newton’s constant GN = 1/mP : mP ∼ 1019 GeV that characterizes the strength of gravitational interactions, and the cosmological constant Λ or some time-varying form of vacuum energy as seems to be required by recent cosmological data. A complete theory of cosmology will presumably also need parameters to characterize the early inflation of the Universe and to generate its baryon asymmetry, which cannot be explained within the Standard Model. The Big Issues in physics beyond the Standard Model are conveniently grouped into three categories [1]. These include the problem of Unification: is there a simple group framework for unifying all the particle interactions, a so-called Grand Unified Theory (GUT), Flavour: why are there so many different types of quarks and leptons and why do their weak interactions mix in the peculiar way observed, and Mass: what is the origin of particle masses, are they due to a Higgs boson, why are the masses so small? Solutions to all these problems should eventually be incorporated in a Theory of Everything (TOE) that also includes gravity, reconciles it with quantum mechanics, explains the origin of space-time and why it has four dimensions, etc. String theory, perhaps in its current incarnation of M theory, is the best (only?) candidate we have for such a TOE [2], but we do not yet understand it well enough to make clear experimental predictions. Supersymmetry is thought to play a rôle in solving many of these problems beyond the Standard Model. As discussed later, GUT predictions for the unification of gauge couplings work best if the effects of relatively light supersymmetric particles are included [3]. Also, the hierarchy of mass scales in physics, and particularly the fact that mW ≪ mP , appears to require relatively light supersymmetric particles: M <∼ 1 TeV for its stabilization [4]. Finally, supersymmetry seems to be essential for the consistency of string theory [5], although this argument does not really restrict the mass scale at which supersymmetric particles should appear. Thus there are plenty of good reasons to study supersymmetry, and we return later to examine in more detail the motivations provided by unification and the mass hierarchy problem. 1.2 The Electroweak Vacuum Generating particle masses within the Standard Model requires breaking its gauge symmetry, and the only consistent way to do this is by breaking the symmetry of the electroweak vacuum: mW,Z 6= 0 ↔ < 0|XI,I3|0 > 6= 0 (1) where the symbols I, I3 denote the weak isospin quantum numbers of whatever object X has a non-zero vacuum expectation value. There are a couple of good reasons to think that X must have (predominantly) isospin I = 1/2. One is the ratio of the W and Z boson masses [6]: ρ ≡ m 2 W mZ cos 2 θW ≃ 1, (2) and the other reason is to provide non-zero fermion masses. Since left-handed fermions fL have I = 1/2, right-handed fermions fR have I = 0 and fermion mass terms couple them together: mf f̄LfR, we must break isospin symmetry by 1/2 a unit: mf 6= 0 ↔ < 0|X1/2,±1/2|0 > 6= 0. (3) The next question is, what is the nature of X? Is it elementary or composite? In the initial formulation of the Standard Model, it was assumed that X should be an elementary Higgs-Brout-Englert [7, 8] field H: < 0|H0|0 > 6= 0, which would have a physical excitation that manifested itself as a neutral scalar Higgs boson [7]. However, as discussed in more detail later, an elementary Higgs field has problems with quantum (loop) corrections. Those due to Standard Model particles are quadratically divergent, resulting in a large cutoff-dependent contribution to the physical masses of the Higgs boson, W,Z bosons and other particles: δmH ≃ O( α π )Λ, (4) where Λ represents the scale at which new physics appears. The sensitivity (4) disturbs theorists, and one of the suggestions to avoid it was to postulate replacing an elementary Higgs-Brout-Englert field H by a composite field such as a condensate of fermions: < 0|F̄ F |0 > 6= 0. This possibility was made more appealing by the fact that fermion condensates are well known in solid-state physics, where Cooper pairs of electrons are responsible for conventional superconductivity, and in strong-interaction physics, where quarks condense in the vacuum: < 0|q̄q|0 > 6= 0. In order to break the electroweak symmetry at a large enough scale, fermions with new interactions that become strong at a higher mass scale would be required. One suggestion was that the Yukawa interactions of the top quark might be strong enough to condense them: < 0|t̄t|0 > 6= 0 [9], but this would have required the top quark to weigh more than 200 GeV, in simple models. Alternatively, theorists proposed the existence of new fermions held together by completely new interactions that became strong at a scale ∼ 1 TeV, commonly called Technicolour models [10]. Specifically, the technicolour concept was to clone the QCD quark-antiquark condensate < 0|q̄LqR|0 >∼ ΛQCD : ΛQCD ∼ 1GeV, (5) on a much larger scale, postulating a condensate of new massive fermions < 0|Q̄LQR|0 >∼ ΛTC where ΛTC ∼ 1 TeV. Assigning the techniquarks to the same weak representations as conventional quarks, IL = 1/2, IR = 0, the technicondensate breaks electroweak symmetry in just the way required to reproduce the relation (2). Just as QCD with two massless flavours would have three massless pions and a massive scalar meson, this simplest version of technicolour would predict three massless technipions that are eaten by the W± and Z0 to provide their masses via the Higgs-Brout-Englert mechanism, leaving over a single physical scalar state weighing about 1 TeV, that would behave in some ways like a heavy Higgs boson. Unfortunately, this simple technicolour picture must be complicated, in order to cancel triangle anomalies and to give masses to fermions [11], so the minimal default option has become a model with a single technigeneration: