Limits of the Standard Model

Supersymmetry is one of the most plausible extensions of the Standard Model, since it is well motivated by the hierarchy problem, supported by measurements of the gauge coupling strengths, consistent with the suggestion from precision electroweak data that the Higgs boson may be relatively light, and provides a ready-made candidate for astrophysical cold dark matter. In the first lecture, constraints on supersymmetric models are reviewed, the problems of fine-tuning the electroweak scale and the dark matter density are discussed, and a number of benchmark scenarios are proposed. Then the prospects for discovering and measuring supersymmetry at the LHC, linear colliders and in non-accelerator experiments are presented. In the second lecture, the evidence for neutrino oscillations is recalled, and the parameter space of the seesaw model is explained. It is shown how these parameters may be explored in a supersymmetric model via the flavour-changing decays and electric dipole moments of charged leptons. It is shown that leptogenesis does not relate the baryon asymmetry of the Universe directly to CP violation in neutrino oscillations. Finally, possible CERN projects beyond the LHC are mentioned. Lectures given at the PSI Summer School, Zuoz, August 2002 1. Supersymmetry 1.1 Parameters and Problems of the Standard Model The Standard Model agrees with all confirmed experimental data from accelerators, but is theoretically very unsatisfactory [1]. 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 masses for all the neutrinos [2], as discussed in Lecture 2. The Big Issues in physics beyond the Standard Model are conveniently grouped into three categories [1]. These include the problem of Mass: what is the origin of particle masses, are they due to a Higgs boson, and, if so, why are the masses so small, Unification: is there a simple group framework for unifying all the particle interactions, a so-called Grand Unified Theory (GUT), and Flavour: why are there so many different types of quarks and leptons and why do their weak interactions mix in the peculiar way observed? 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 [3], 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. 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]. As discussed later, GUT predictions for the unification of gauge couplings work best if the effects of relatively light supersymmetric particles are included [5]. Finally, supersymmetry seems to be essential for the consistency of string theory [6], 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 [7], so this is the subject of Lecture 1, and it reappears in Lecture 2 in connection with the observability of charged-lepton flavour violation. 1.2 Why Supersymmetry? The main theoretical reason to expect supersymmetry at an accessible energy scale is provided by the hierarchy problem [4]: why is mW ≪ mP , or equivalently why is GF ∼ 1/m2W ≫ GN = 1/m2P ? Another equivalent question is why the Coulomb potential in an atom is so much greater than the Newton potential: e ≫ GNm = m2/m2P , where m is a typical particle mass? Your first thought might simply be to set mP ≫ mW by hand, and forget about the problem. Life is not so simple, because quantum corrections to mH and hence mW are quadratically divergent in the Standard Model: δm2H,W ≃ O( α π )Λ, (1) which is ≫ m2W if the cutoff Λ, which represents the scale where new physics beyond the Standard Model appears, is comparable to the GUT or Planck scale. For example, if the Standard Model were to hold unscathed all the way up the Planck mass mP ∼ 10 GeV, the radiative correction (1) would be 36 orders of magnitude greater than the physical values of m2H,W ! In principle, this is not a problem from the mathematical point of view of renormalization theory. All one has to do is postulate a tree-level value of m2H that is (very nearly) equal and opposite to the ‘correction’ (1), and the correct physical value may be obtained. However, this fine tuning strikes many physicists as rather unnatural: they would prefer a mechanism that keeps the ‘correction’ (1) comparable at most to the physical value [4]. This is possible in a supersymmetric theory, in which there are equal numbers of bosons and fermions with identical couplings. Since bosonic and fermionic loops have opposite signs, the residual one-loop correction is of the form δm2H,W ≃ O( α π )(m2B −mF ), (2) which is <∼ m2H,W and hence naturally small if the supersymmetric partner bosons B and fermions F have similar masses: |mB −mF | <∼ 1 TeV. (3) This is the best motivation we have for finding supersymmetry at relatively low energies [4]. In addition to this first supersymmetric miracle of removing (2) the quadratic divergence (1), many logarithmic divergences are also absent in a supersymmetric theory [8], a property that also plays a rôle in the construction of supersymmetric GUTs [1]. Could any of the known particles in the Standard Model be paired up in supermultiplets? Unfortunately, none of the known fermions q, l can be paired with any of the ‘known’ bosons γ,WZ, g,H , because their internal quantum numbers do not match [9]. For example, quarks q sit in triplet representations of colour, whereas the known bosons are either singlets or octets of colour. Then again, leptons l have non-zero lepton number L = 1, whereas the known bosons have L = 0. Thus, the only possibility seems to be to introduce new supersymmetric partners (spartners) for all the known particles: quark → squark, lepton → slepton, photon → photino, Z → Zino, W → Wino, gluon → gluino, Higgs → Higgsino. The best that one can say for supersymmetry is that it economizes on principle, not on particles! 1.3 Hints of Supersymmetry There are some phenomenological hints that supersymmetry may, indeed, appear at the Tev scale. One is provided by the strengths of the different gauge interactions, as measured at LEP [5]. These may be run up to high energy scales using the renormalization-group equations, to see whether they unify as predicted in a GUT. The answer is no, if supersymmetry is not included in the calculations. In that case, GUTs would require sin θW = 0.214 ± 0.004, (4) whereas the experimental value of the effective neutral weak mixing parameter at the Z peak is sin θ = 0.23149 ± 0.00017 [10]. On the other hand, minimal supersymmetric GUTs predict sin θW ≃ 0.232, (5) where the error depends on the assumed sparticle masses, the preferred value being around 1 TeV [5], as suggested completely independently by the naturalness of the electroweak mass hierarchy. A second hint is the fact that precision electroweak data prefer a relatively light Higgs boson weighing less than about 200 GeV [10]. This is perfectly consistent with calculations in the minimal supersymmetric extension of the Standard Model (MSSM), in which the lightest Higgs boson weighs less than about 130 GeV [11]. A third hint is provided by the astrophysical necessity of cold dark matter. This could be provided by a neutral, weakly-interacting particle weighing less than about 1 TeV, such as the lightest supersymmetric particle (LSP) χ [12]. 1.4 Building Supersymmetric Models Any supersymmetric model is based on a Lagrangian that contains a supersymmetric part and a supersymmetry-breaking part [13, 7]: L = Lsusy + Lsusy×. (6) We concentrate here on the supersymmetric part Lsusy. The minimal supersymmetric extension of the Standard Model (MSSM) has the same gauge interactions as the Standard Model, and Yukawa interactions that are closely related. They are based on a superpotential W that is a cubic function of complex superfields corresponding to left-handed fermion fields. Conventional left-handed lepton and quark doublets are denoted L,Q, and right-handed fermions are introduced via their conjugate fields, which are left-handed, eR → Ec, uR → U c, dR → Dc. In terms of these, W = ΣL,EcλLLE H1 + ΣQ,UcλUQU H2 + ΣQ,DcλDQD H1 + μH1H2. (7) A few words of explanation are warranted. The first three terms in (7) yield masses for the charged leptons, charge-(+2/3) quarks and charge-(−1/3) quarks respectively. All of the Yukawa couplings λL,U,D are 3 × 3 matrices in flavour space, whose diagonalizations yield the mass eigenstates and CabibboKobayashi-Maskawa mixing angles for quarks. Note that two distinct Higgs doublets H1,2 have been introduced, for two important reasons. One reason is that the superpotential must be an analytic polynomial: it cannot contain both H and H, whereas the Standard Model uses both of these to give masses to all the quarks and leptons with just a single Higgs doublet. The other reason for introducing two Higgs doublets H1,2 is to cancel the triangle anomalies that destroy the renormalizability of a gauge theory. Ordinary Higgs boson doublets do not contribute to these anomalies, but the fermions in Higgs supermultiplets do, and pairs of doublets are required to cancel each others’ contributions. Once two Higgs supermultiplets have been introduced, there must in general be a bilinear term μH1H2 coupling them together. In general, the supersymmetric partners of the W and charged Higgs bosons H (the ‘charginos’ χ) mix, as do those of the γ, Z and H 1,2 (the ‘neutralinos’ χ 0 i ): see [1]. The lightest neutralino χ is a likely candidate to be the Lightest Supersymmetric Particle (LSP), and hence constitute the astrophysical cold dark matter [12]. Once the MSSM superpotential (7) has been specified, the effective potential is also fixed: V = Σi|F | + 1 2 Σa(D ) : F ∗ i ≡ ∂W ∂φi , D ≡ gaφi (T )jφ, (8) where the sums run over the different chiral fields i and the SU(3), SU(2) and U(1) gauge-group factors a. Thus, the quartic terms in the effective Higgs potential are completely fixed, which leads to the prediction that the lightest Higgs boson should weigh <∼ 130 GeV [11]. In addition to the supersymmetric part Lsusy of the lagrangian (6) above, there is also the superymmetry-breaking piece Lsusy×. The origin of this piece is unclear, and in these lectures we shall just assume a suitable phenomenological parameterization. In order not to undo the supersymmetric miracles mentioned above, the breaking of supersymmetry should be ‘soft’, in the sense that it does not reintroduce any unwanted quadratic or logarithmic divergences. The candidates for such soft superymmetry breaking are gaugino masses Ma for each of the gauge group factors a in the Standard Model, scalar massessquared m20 that should be regarded as matrices in the flavour index i of the matter supermultiplets, and trilinear scalar couplings Aijk corresponding to each of the Yukawa couplings λijk in the Standard Model. There are very many such soft superymmetry-breaking terms. Upper limits on flavour-changing neutral interactions suggest [14] that the scalar masses-squared m20 are (approximately) independent of generation for particles with the same quantum numbers, e.g., sleptons, and that the Aijk are related to the λijk by a universal constant of proportionality A. In these lectures, for definiteness, we assume universality at the input GUT scale for all the gaugino masses: Ma = m1/2, (9) and likewise for the scalar masses-squared and trilinear parameters: m20 = m 2 0δ i j , Aijk = Aλijk. (10) This is known as the constrained MSSM (CMSSM). The values of the soft supersymmetry-breaking parameters at observable energies ∼ 1 TeV are renormalized by calculable factors [15], in a similar manner to the gauge couplings and fermion masses. These renormalization factors are included in the subsequent discussions, and play a key rôle in Lecture 2. The physical value of μ is fixed up to a sign in the CMSSM, as is the pseudoscalar Higgs mass mA, by the electroweak vacuum conditions. 1.5 Constraints on the MSSM Important experimental constraints on the MSSM parameter space are provided by direct searches at LEP and the Tevatron collider, as compiled in the (m1/2,m0) planes for different values of tan β and the sign of μ in Fig. 1. One of these is the limit mχ± >∼ 103.5 GeV provided by chargino searches at LEP [16], where the fourth significant figure depends on other CMSSM parameters. LEP has also provided lower limits on slepton masses, of which the strongest is mẽ >∼ 99 GeV [17], again depending only sightly on the other CMSSM parameters, as long as mẽ − mχ >∼ 10 GeV. The most important constraints on the u, d, s, c, b squarks and gluinos are provided by the FNAL Tevatron collider: for equal masses mq̃ = mg̃ >∼ 300 GeV. In the case of the t̃, LEP provides the most stringent limit when mt̃ −mχ is small, and the Tevatron for larger mt̃ −mχ [16]. Another important constraint is provided by the LEP lower limit on the Higgs mass: mH > 114.4 GeV [19]. This holds in the Standard Model, for the lightest Higgs boson h in the general MSSM for tan β <∼ 8, and almost always in the CMSSM for all tan β, at least as long as CP is conserved 1. Since mh is sensitive to sparticle masses, particularly mt̃, via loop corrections: δm2h ∝ m4t m2W ln ( m t̃ m2t )