Supersymmetry and a Candidate for Dark MatterElizabeth LocknerDepartment of Physics, University of MarylandCollege Park, MD 20742April 20, 2007.The Standard Model has been a powerful tool in understanding the symmetries of nature andpredicting new phenomena. There are, however, some fundamental flaws in the model. Thetheory of supersymmetry has provided an extension to the Standard Model in an effort to resolvesome of the most troubling paradoxes. This paper provides an overviewof the most basicsupersymmetric models and outlines how these might provide a candidate for the mysteriousdark matter permeating our universe.1 Why a new theory?1.1 The Standard ModelOver the last 2500 years the idea has been devel-oped that there are indeed constituents of matterthat are fundamental ingredients to all that we seeand feel. The most commonly accepted theory ofthese fundamental particles and interactions todayis the Standard Model.According to the Standard Model there are threetypes of fundamental particles: leptons, quarks andforce carrying particles. Figure 1 shows the groupsof particles in the Standard Model. Not shown arethe anti-particle partners of each of the leptons andquarks. Strong evidence for each of these particleshas be en seen experimentally. The quarks and lep-tons are fermions, half-integer intrinsic spin, whilethe force carriers are bosons, integer intrinsic spin.These two different natures of the particles lead tovery different mathematical consequences with re-gard to relativistic quantum field theory. One ad-ditional particle is widely accepted as part of theStandard Model even though its existence has notyet been observed. This is the Higgs boson. Theexistence of the Higgs boson would provide a mech-anism to explain why the W ± and Z0have mass,while the photon and the gluon are required to bemassless.Figure 1: Simple diagram of the Standard Model1.2 The Hierarchy ProblemIn the framework of relativistic field theory one candevelop a Lagrangian for this system of interactingparticles. The mass of a particle is calculated us-ing perturbation theory giving a leading order massterm and then subsequent corrections. The first odercorrection (often referred to as “next to leading or-der” or NLO) is called a radiative correction. Theseinvolve Feynman diagrams with one loop. In orderto include the correction for all possibilities of the1one loop, an integral is taken over all momenta:Zd4k1k2This integral is divergent, so one must choose anupper limit to integrate to, Λ, at which point newphysics is expected to take over. The StandardModel does not include the force of gravity whichwould start to have effects of the same order as quan-tum mechanics at around the Planck scale, a mass ofaround 1019GeV. If not before, the Standard Modeltheory must be modified at this energy, so this is anupper bound for Λ. For fermions the integral for theradiative corrections to the mass scale is only loga-rithmically divergent as is shown in Eq.( 1) wheremfis the fermion mass and alpha is a coupling co-efficient. Integrating up to 1019GeV does not in-troduce significant corrections to the leading ordercalculation.δmf∝3α4πmfln Λ2m2f!<< mf(1)The Higgs boson, however, would be a scalar boson.This integral is quadratically divergent as is shownin Eq. (2), so when the upper bound of the integralis set at around the Planck scale the radiative cor-rections become much larger than the leading orderterms.δm2higgs∝α4πΛ2>> m2higgs(2)This instability implies that either the the-ory is not correct or that there is a lowermass/energy/momentum scale at which new physicsmust start to play a role.1.3 Unification of forcesThe relative strengths of each of the three forces in-cluded in the Standard Mo del have been measuredat various energies around the energy of our cur-rent universe, approximately 100 GeV and below. Inthis region the inverses of the coupling coefficientsseem to be linear functions of energy and seem toget closer together at increasing energy. Figure 2shows the extrapolation of the current measurementsto higher energies.The coupling strengths do approach one another,but they do not meet at the same point. This isunattractive to theoretical physicists. Grand Uni-fied Theories propose that there is an overarchingFigure 2: Dependence of force coupling strengths onenergy. U(1):α1, SU(2):α2(these together form theelectro-weak force), and SU(3):α3(strong force)symmetry that represents all of nature and may bebroken at low energies but ought to unify all theforces at high enough energies. The manner in whichthe coupling strength depends on energy changes ac-cording to what model is used for the physics at anygiven energy. Hence, if this linear trend is modi-fied by a new theory at higher energies the couplingcoefficients may actually unite at one point.1.4 How does supersymmetry help?The two dilemmas outlined above can be solvedrather elegantly with the introduction of an ex-tension to the Standard Model. Supersymmetryproposes additional terms in the Lagrangian thatwould add extra particles and interactions to thetheory. Namely, there would be a fermionic part-ner to the scalar boson Higgs particle. This helpssolve the problem of quadratic divergences as thefermionic and bosonic portions of the integral areadded with an opposite sign. Hence the fermionicquadratic divergences would cancel out the scalarbosonic quadratic divergences, leaving only the log-arithmic divergences for both. This re-stabilizes themass of the Higgs. Similarly, by adding in new par-ticles and interactions the coupling coefficients havea different dependence on the energy scale. Figure 3shows how a fine tuned supersymmetric model wouldpredict the coupling coefficients to meet at a single2point, and that the forces would be contained underone symmetry at higher energies.Figure 3: Dependence of force coupling strengths onenergy with supersymmetryThere are many different variations of supersym-metric models, each tuning the model in such a waythat the forces unify slightly differently. However,the opportunity to choose parameters in s uch a waythat the coupling coefficients both agree with cur-rent experimental evidence at our energy scale andconverge at higher energy scales is one reason super-symmetric models have become so widely supported.2 The Basics of supersymmetry2.1 The MathematicsTypical symmetries involve changing either an in-trinsic property of the field such as isospin or charge,or an external