Beyond the Standard Model: supersymmetry I. Antoniadis1 and P. Tziveloglou2 CERN, Geneva, Switzerland Abstract In these lectures we cover the motivations, the problem of mass hierarchy, and the main proposals for physics beyond the Standard Model; supersymme- try, supersymmetry breaking, soft terms; Minimal Supersymmetric Standard Model; grand unification; and strings and extra dimensions. 1 Departure from the Standard Model 1.1 Why beyond the Standard Model? Almost all research in theoretical high energy physics of the last thirty years has been concentrated on the quest for the theory that will replace the Standard Model (SM) as the proper description of nature at the energies beyond the TeV scale. Thus it is important to know if this quest is justified or not, given that the SM is a very successful theory that has offered to us some of the most striking agreements between experimental data and theoretical predictions. We shall start by briefly examining the reasons for leaving behind such a successful SM. Actually there is a variety of such arguments, both theoretical and experimental, that lead to the undoubtable conclusion that the Standard Model should be only an effective theory of a more fundamental one. We briefly mention some of the most important arguments. Firstly, the Standard Model does not include gravity. It says absolutely nothing about one of the four fundamental forces of nature. Another is the popular ‘mass hierarchy’ problem. Within the SM, the mass of the higgs particle is extremely sensitive to any new physics at higher energies and its natural value is of order of the Planck mass, if the SM is valid up to that scale. This is several orders of magnitude higher than the electroweak scale implied by experiment. The way to cure this discrepancy within the SM requires an incredible fine tuning of parameters. Also, the SM does not explain why the charges of elementary particles are quantized. In addition the SM does not describe the dark matter or the dark energy of the universe. It does not explain the observed neutrino masses and oscillations and does not predict any gauge coupling unification, suggested by experiments. Here, we describe some of these arguments in more detail, starting from gravity. If we follow the intuition that the fundamental theory should describe within the same quantum framework all forces of nature, we need to extend the Standard Model in such a way that it consistently includes this force. However, this is particularly difficult for a variety of reasons. The most serious is that gravity cannot be quantized consistently as a field theory. Another one is that the new theory has to provide a natural explanation for the apparent weakness of gravity compared to the other three fundamental interactions. The gravitational force between two particles of mass m is 2 m 1/2 19 F = G ,G− = M =10 GeV . (1) grav N r2 N Planck If we compare this with the electric force e2 F = (2) el r2 1On leave from CPHT (UMR du CNRS 7644), Ecole Polytechnique, 91128 Palaiseau, France. 2Department of Physics, Cornell University, Ithaca, NY 14853, USA. 157 I. ANTONIADIS AND P. TZIVELOGLOU for, say the proton, we get 2 Fgrav GN mproton 40 = 2 10− . (3) Fel e So gravity is an extremely weak force at low energies (small masses, large distances) even compared to the weak force as shown in the table below. 16 Force Range Intensity of 2 protons Intensity at 10− cm Gravitation 10 38 10 30 ∞ − − Electromagnetic 10 2 10 2 ∞ − − 15 5 2 Weak (radioactivity β) 10− cm 10− 10− 12 1 Strong (nuclear forces) 10− cm 1 10− 33 The distance at which gravitation becomes comparable to the other interactions is 10− cm, the Planck length. The energy that corresponds to this length is of order the Planck mass and is 1015 × the LHC energy. Thus another great difficulty is that even if we manage to construct such a model,∼ its phenomenological verification will be extremely difficult. Next, we describe the mass hierarchy problem. The mass of the higgs particle is very sensitive to high energy physics. For example, the main one-loop radiative corrections from the virtual exchange of top quarks and the higgs particle itself are λ 3λ2 μ2 = μ2 + t Λ2 + (4) eff bare 8π2 − 8π2 ··· where λ is the higgs quartic coupling, λt is the top Yukawa coupling, and Λ is the ultraviolet (UV) cutoff, set by the scale where new physics appears. From the masses of all other particles and for reasons that we shall explain later, we know that the higgs boson mass should be at the weak scale. Requiring the validity of the Standard Model at energies Λ (100) GeV imposes an ‘unnatural’ order-by-order fine 2 O tuning between the bare mass parameter μbare in the Lagrangian and the radiative corrections. 19 2 For example, for Λ (MPlanck) 10 GeV and a loop factor of the order of 10 , we get ∼O ∼ − μ2 10 2 1038 = 1036 (GeV)2. Thus, we need μ2 1036 (GeV)2 +104 (GeV)2.An 1-loop ∼ − × ± bare ∼∓ incredible adjustment is then required, at the level of 1 part per 1032: μ2 /μ2 = 1 10 32.Even bare 1-loop − ± − more, at the next and all higher orders, a new adjustment is required. The correction will be of the order of (100) GeV only beyond 17 loops: 10 2 N 1038 104 N 17 loops. If we want to avoid O − × ≤ ⇒ ≥ these unnatural adjustments we need 10 2Λ2 104 (GeV)2 Λ 1 TeV. The resolution of the mass − hierarchy problem should reside within the energy≤ reach of LHC.⇒ ≤ Another open question is the charge quantization: all observed colour singlet states have integer electric charges. The symmetry group of the SM is SU(3) SU(2) U(1) and the electric charge × L × Y given by Q = T3 + Y . The representations of the symmetry group of all fields are u2/3 q =(3, 2)1/6 q = d 1/3 − c u =(3¯, 1) 2/3 − c d =(3¯, 1)1/3 ν0 =(1, 2) 1/2 = − e 1 − c e =(1, 1)1 2 158 BEYOND THE STANDARD MODEL: SUPERSYMMETRY where the subscripts denote the hypercharges in the left column and the electric charges in the right one. However, the SM does not tell us anything about this choice of the hypercharges, which guarantees the observed electric charge quantization. Also, the experimental indication of gauge coupling unification is a feature of nature that asks for 2 an explanation. The renormalization group evolution of the three SM gauge couplings αi = gi /4π is given by dαi bi 2 1 1 bi Q = αi αi− (Q)=αi− (Q0) ln , (5) d ln Q −2π ⇒ − 2π Q0 where Q is the energy scale and bi are the one-loop beta-function coefficients. The extrapolation of the low energy experimental data at high energies under the ‘desert’ assumption indicates an approximate 15 16 unification of all couplings at energies of order MGUT 10 –10 GeV. By doing a more precise analysis however, using the current experimental precision, one finds that the Standard Model fails to predict such a unification. One more reason for going beyond the Standard Model is the existence of dark matter. Astro- nomical observations tell us that the ordinary baryonic matter is only a tiny fraction of the energy of the universe. There are observations that point towards the fact that a kind of non-luminous matter is out there and that it is actually much more abundant than baryonic matter, consisting of around 25% of the total energy density of the universe. A natural explanation is that dark matter consists of a new kind of particles that are stable, massive at the electroweak scale, and weakly interacting. We could mention many other important reasons that drive us beyond the SM, like neutrino masses, dark energy and so on, but we shall stop here. Even this brief and incomplete presentation shows that there are a lot of fundamental questions that the SM cannot address, so the search for a new fundamental theory is definitely justified. Modifying the SM, though, is not as easy as it might seem. If we try to identify the scale Λ of new physics by looking at what energies there are deviations from the SM predictions by observations, we shall be disappointed. With the exception of dark matter and the higgs boson mass, all experimental indications suggest that Λ is very high. For example, the ‘see-saw’ mechanism for the neutrino masses v2 gives a very large scale: m 10 2 eV M 1013 GeV, where v is the vacuum expectation ν ∼ M ∼ − ⇒ ∼ value (VEV) of the SM higgs. Similarly, the scale of gauge coupling unification is even higher: MGUT ∼ 1016 GeV. Even more, modifying the Standard Model in a way that solves the hierarchy problem is highly restricted by numerous experiments. Any new interactions could violate various Standard Model pre- dictions that have been experimentally tested with good accuracy, such as lepton and baryon number conservation, absence of flavour changing neutral currents (FCNC), CP violation, etc. For example, the- ories with baryon number violating operators like 1 μ c ¯ c 2 (¯qγ u ) γμd (6) ΛB can lead to proton decay: p =[uud] (e+[dd¯ ]) = e+π0,uue+d.¯ (7) → → The bound from the Superkamiokande experiment on the proton lifetime of τ 2.6 1033 years highly p ≥ × restricts the allowed scale: Λ 1015 GeV. B ≥ Similarly for the lepton number; in this case, the lowest dimensional operator that violates lepton number has dimension five: 1 2 (H∗) (8) ΛL 3 159 I.